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NATL INST OF STANDARDS & TECH R.I.C.

All 101909396

/NBS monograph QC100 .USSe \/l72;1983 C.2 NBS-PUB-C 1959

i/t NBS MONOGRAPH 172

U.S. DEPARTMENT OF COMMERCE / National Bureau of Standards

Liquefied Natural Gas

Densities: Summary of

Research Program at the

National Bureau of Standards

i

I
NATIONAL BUREAU OF STANDARDS

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Liquefied Natural Gas Densities:

Summary of Research Program at

the National Bureau of Standards

W. M. Haynes

R. D. McCarty

M. J. Hiza

Chemical Engineering Science Division National Engineering Laboratory

National Bureau of Standards

Boulder, CO 80303

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Annbler, Director

Issued October 1 983
Library of Congress Catalog Card Number: 83-600608

National Bureau of Standards Monograph 172

Natl. Bur. Stand. (U.S.), Monogr. 172, 241 pages (Oct. 1983) CODEN: NBSMA6

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1983

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Contents

Page

1. Introduction 2

2. Major Accompl ishments 4

2.1 Experimental Apparatus 4

2.2 Experimental Measurements 6

2.2.1 Pure Fluid Jata 6

2.2.2 Binary Mixture Jata 6

2.2.3 Mul ti component Mixture uata 7 2.3 Mathematical Models 8

3. LiMG Density Research at Uther Laboratories y

4. Acknowledgments 11

b. Keferences 13

6. Major Publications of LNG Density Project 19

6.1 Reference [5] - Magnetic Suspension Densimeter for

Measurements on Fluids of Cryogenic Interest 19

6.2 Reference [9] - Simplified Magnetic Suspension Densimeter

for Absolute Density Measurements 33

6.3 Reference [6] - Apparatus for Density and Dielectric

Constant Measurements to 35 MPa on Fluids of Cryogenic

Interest 37

6.4 Reference [13] - Measurements of the Orthobaric Liquid

Densities of Methane, Ethane, Propane, isobutane, and i\lormal

Butane 49

6.5 Reference [14] - Orthobaric Liquid Densities of Normal

Butane from 135 to 3U0 K as Determined with a Magnetic

Suspension Densimeter 59

6.6 Reference [16] - Orthobaric Liquid Densities and Excess

volumes for Binary Mixtures of low Molar-Mass Alkanes and

Nitrogen Between 105 and 140 K 65

6.7 Reference [17] - Orthobaric Liquid Densities ana Dielectric

Constants of (Methane + 2-Methyl propane) and (Methane +

("•J-Butane) at low Temperatures 89

ii 1
Page

6.8 Reference [18] - Liquid Mixture Excess Volumes and Total

Vapor Pressures Usiny a Magnetic Suspension Densimeter with

Compositions Determinea by Chromatographic Analysis:

Methane Plus Ethane 99

6.9 Reference [20] - Orthobaric Liquid Densities and Excess

Volumes for Mul ticomponent Mixtures of Low Molar-Mass

Alkanes and Nitrogen Between 105 and 125 K 107

6.10 Reference [21] - Measurements of Orthobaric-Liquid Densities

of Mul ticomponent Mixtures of LN6 Components (N2, CH^, C2^q,

C3H3, CH3CH(CH3)CH3, C4H1Q, CH3CH(CH3)C2H5 , and CgHj^g^

Between 110 and 130 K 117

6.11 Reference [34] - Four Mathematical Models for the Prediction

of LiMG Densities 127

6.12 Reference [35] - Mathematical Models for the Prediction of

Liquefied-i\latural-Gas Densities 205

6.13 Reference [25] - An Empirical Excess Volume Model for

Estimating Liquefied Natural Gas Densities 223

6.14 Reference [22] - Prediction of Liquefied-Watural-Gas (LNG)

Densities from New Experimental Dielectric Constant Data . 235
LIQUEFieu fJATUKAL GAS DENSITItS: SUMMARY OF RESEARCH

PROGRAjVI at the MATIOiNlAL BUREAU OF STAf^lUARDS

W. M. haynes, R. u. McCarty and M. J. Hiza

Chemical Engineering Science ui vision

National Engineering Laboratory

i\lational bureau of Standards

Boulder, Colorado 80303

This report summarizes the results of a project concerning the

densities of liquefied natural gas (LiMO) and its components. This

project, initiated in the Properties of Fluids Section of the

Cryogenics Division of the relational Bureau of standards in July 1972!, was carried out under the sponsorship of a consortiu.n of eighteen

energy companies .

The experimental part of this project has included the following

accomplishments: (a) development of a magnetic suspension densimeter

for absolute density measurements on liquids, including liquid mixtures

in equilibrium with their vapor, at temperatures from 90 to 300 K;

(b) orthobaric liquid density measurements on the major components of

LNG, which include nitrogen (95-12!0 l<), methane (105-160 K), ethane

(100-270 K), propane (100-288.7 K), isobutane (115-300 K), and normal

butane (135-300 K); (c) orthobaric liquid density measurements on approximately thirty-five binary mixtures of the above components for

all combinations except nitrogen + butane systems, primarily in the

temperature range of 105 to 130 K; ana (d) ortnobaric liquid density

measurements on twenty-seven multi component mixtures (105-120 K),

including several LNG-like mixtures with up to eight components. The

total uncertainty of a single density measurement is approximately 0.1

percent at low temperatures and decreases to approximately 0.06 percent

at room temperature. The estimated standard deviation of a single

density measurement is less than 0.02 percent.

Tne density data have been used to optirnize, test, and compare

several mathematical models as to their suitability for the calculation

of LNG densities for custody transfer. Hodels selected for optimization and testing included an extended corresponding states method, a hard sphere model, a cell model, and an empirical model due to Klosek

and McKinley. The ultimate goal of this project was to produce one or more mathematical models that could be used to predict the density of

any Li\lG mixture to within an uncertainty of 0.1 percent from an input

of pressure, temperature, and composition. After revisions based on the new experimental data from this project, each of the models

* British Gas Corp., Chicago Bridge and Iron Co., Columbia Gas Service Corp.,

Oistrigas Corp., Easco Gas LWG, Inc., El Paso Natural Gas, baz de France,

I'^arathon Oil Co., Mobil Oil Corp., Natural Gas Pipeline Co., Phillips Petroleum

Co., Shell International Gas, Ltd., Sonatrach, Southern California Gas Co.,

Tennessee Gas Pipeline Co., Texas Eastern Transmission Co., Tokyo Gas Co., Ltd.,

and Transcontinental Gas Pipe Line Corp.

1
investigated here satisfy this goal for typical LNG compositions. The

limitations and ranges of validity of the various models are discussed.

Also presented are techniques for predicting Li\iG densities from

dielectric constant measurements and from excess volume calculations.

The last section of this report consists of publications that

provide a complete and detailed account of the results of this

project.

Key words: binary mixtures; density; experimental data; liquefied

natural gas; magnetic suspension densimeter; mul ticomponent mixtures;

prediction methods; pure fluids.

1. Introduction

Uuring the past decade liquefied natural gas has become an increasingly

important commodity on the world energy market. This trend is expected to

continue into the foreseeable future. In the buying ana selling of LNG, the

basis of custody transfer is its heating value. The determination of the heating

value of LMG requires a knowledge of its density, which in turn is dependent upon

its composition, temperature, and pressure. Since, for example, an error of one percent in density can result in an inequity of between :i>luO,uuO and !>200,000

(at 1983 prices) per 125, OUU m-^ shipload of LNG, the accuracy to which the

density of the liquid can be determined is extremely important in transactions

involving LNG. A one percent error in density was not uncommon at the time this

project was initiated.

There are, at least, two means for determining the density of a large volume

of LNG. One isiethod is by direct field measurements using commercially available

densimeters. A second method consists of the use of a mathematical model or correlation to calculate a density based on direct measurements of the liquid

composition and temperature.

This report describes a project that has been concerned with the determination of the density by the second method. In actual transfer situations it is

likely that both methods will be used to complement each other. By either

method, the accuracy to which the density can be determined is limited by both

practical and state-of-the-art considerations. Tnis has resulted in setting a goal of 0.1 percent for the total uncertainty in the determination of the density

of L.MG.

2
In July 1972, a project was initiated at this laboratory with an ultimate

goal of providing one or more mathematical models that could be used to calculate

the density of any LNG mixture with a total uncertainty of less than 0.1 percent

based on a knowledge of the pressure, temperature, and composition of the liquid

mixture. The mathematical models would be developed and/or optimized using

density data for the major components of LnG and for binary mixtures of these

components. Density data for multi component mixtures of LNG components would be

used to evaluate and compare the performance of the mathematical models. Some of

the test mixtures would be selected to simulate commercial Li\iG compositions. LNG

consists primarily of methane (concentration level typically greater than 75

percent) with lesser amounts of ethane (up to i5 percent), propane (up to

b percent), butanes (up to 2 percent), nitrogen (up to 2 percent), and pentanes

plus heavier hydrocarbons (up to 0.5 percent).

The accuracy of a calculational technique based on experimental data can be no better than the input data from which the technique is developed. At the time

this project was initiated, comparisons of saturated (orthobaric) liquid density

data for the pure components of LNG exhibited differences as large as 0.5

percent, while each investigator was generally claiming inaccuracies of 0.1

percent or less. Furthermore, there were only a few sets of published data for

liquid mixtures containing LNG components, especially at low temperatures. These

data were generally limited in scope or had claimed inaccuracies larger than the

desired 0.1 percent. (The pure fluid and mixture data for LMb components from

other investigations are sumi.iarized in Section 3 of this report.) Thus, a major

task of the LNG density project at this laboratory has been to provide an accurate and internally consistent set of density data for the major components

of LiMG and for mixtures of these components. (These data could also serve as a basis for calibration or development of gauging and metering methods for Li^b and

for process design and operation of LNG facilities.)

This project was sponsored by a consortium of eighteen international energy

companies, five of which were from countries outside the United States. The

sponsors represented both buyers and sellers of Lf^iG. In carrying out this

research project, the National Bureau of Standards is serving in its traditional

role as an independent third party. All results of this project would be

published in the open literature and hopefully gain wide acceptance throughout

the Ll'^G industry, since qualified sponsor representatives from all parts of the

3
international LNG market would have closely scrutinized the progress of this

project throughout its entirety.

The major purpose of this report is to incorporate under a single cover a complete account of the LNG density project. The major accomplishments of the

project are summarized in Section while the detailed results are presented in

the papers that comprise Section 6. Most of these papers are reprints of publications, while a few are presented in their existing, prepublished form. During

the course of this research, several papers [1-4] have been published that

presented a summary or status report of the project. These papers are not

included in Section 6. The present report serves as the final report to the

sponsors of this project.

2. Major Accompl ishments

2.1 Experimental Apparatus

The experimental technique selected for performing the density measurements'

for this project had to satisfy relatively stringent criteria. It must be

capable of absolute density measurements of high accuracy and precision on a liquia, including mixtures, in equilibrium with vapor at cryogenic temperatures

and at pressures to approximately 4 MPa using an optical cell. A magnetic

suspension densimeter, based on an application of Archimedes' principle, was developed for this purpose.

Two apparatus [5,6], each incorporating a magnetic suspension technique in

the density determination, were constructed for the measurements on liquids,

including liquid mixtures, associated with this project. For both apparatus, the

total uncertainty of a single density measurement, which is taken as three times

the standard deviation plus the systematic error, is approximately 0.1 percent at

low temperatures and decreases to approximately U.06 percent at room temperature.

The estimated standard deviation of a single density measurement is less than

0.02 percent.

A brief description of the method for determining densities with either

apparatus is as follows. A small magnetic buoy is freely suspended in the fluid

whose density is to be determined. The density of the buoy is significantly

greater I by approximately an order of magnitude) than che density of the fluids

investigatea here. I'he force necessary to lift the buoy is supplied oy one or more air-core solenoids. The buoy is siiaintained at a stable (vertical) position

4
through the automatic regulation of an electronic servocircuit containing the

lift coil and either a differential transfoniier [5j or a differential

capacitor [7] that aetects the motion of tiie buoy. Ihe horizontal position of

the Duoy is maintained by the axially symmetrical, diverging fiela of the lift

coil

.

When the buoy is supported at the sanie position in vacuum and in the fluiu

of interest, the upward magnetic force on the buoy supplied by the air-core

solenoids gives a nieans for measuring the density of the fluid since tnis is the

force that must be added to the buoyant force to ualance the downward gravitational force. Thus, the density is deduced from measurements of the currents in

the support coils necessary to support the buoy in vacuum and in the fluia of

interest at the same temperature and at the same vertical position (determined

with a 1^5X microscope).

The mass and volume of the buoy must be determined from independent

measurements. The mass was determined using an analytical microbal ance. The

volume of the buoy at room temperature was determined by using distilled water as

a reference fluid of known density. Thermal expansion data [8] for barium

ferrite, tne material from which the buoy was fabricated, were obtained so that

the volume of the buoy at low temperatures could be calculated.

The first apparatus [5] could be used at temperatures between 90 and 30u K and at pressures to 5 MPa. After initial tests with a three-coil arrangement to

support the buoy, the densimeter was simplified considerably by reducing to a system employing only one coil [9J. The second apparatus LoJ, which employed

exactly the same technique for deteraiining density, was significantly different

from, and wiore versatile than, the first apparatus. Tne second apparatus con- tained a concentric cylinaer capacitor that was used to make dielectric constant

measurements on the same liquid samples for which density data were obtained.

The major reason for the construction of the second apparatus was a need in other

research projects (e.g., PVT and dielectric constant measurements on liquid

propane [10], isobutane [11], and normal butane [12]) for a higher pressure

capability (35 MPa) for the magnetic suspension densimeter. The extensiuon to

higher pressures was not needed in the LHG density project.

It should be noted that the second apparatus gave no improvement over the

first in the accuracy to wnich the density could De determined. The consistency

of the density data from both apparatus should be equivalent to that for either

5
apparatus alone. To ensure that this was the case, measurements on liquid

methane were used as a control throughout the project with both apparatus.

1.1 Experimental heasureinents

Pure Fluid Uata

The first experimental measurements for the Li\iG density project consisted of

the acquisition of orthobaric liquid density data for the major components of

lnG. Comprehensive results were obtained for nitrogen (95-12U K) [b], methane

(105-160 K) [5,13], ethane (100-270 K) [13], propane (100-288.7 K) [13],

isobutane (115-300 K) Li3], and normal butane (135-300 K) [13,14]. Oetailed com- parisons between the data from this project and independent results demonstrated

the need for an accurate and internally consistent set of data for the major com- ponents of LimO. Differences as large as 0.5 percent between che density data of

independent investigators were comiiion. At the time this project was initiated,

there were no published data for normal butane at low temperatures. (In the

course of evaluating the performance of the magnetic suspension densimeter,

density data were also obtained for saturated liquid argon (luu-12o K) and

ethylene (105-200 n) Lit>].)

2.2.2 Binary Mixture Data

Next, orthobaric liquid density measurements were carried out on thirty-five

binary mixtures l1d-18] containing the major components of limG. Most of the

binary mixture data was taken in the temperature range of 105 to 140 K. All

binary combinations of the six major components of Li^G were investigated in this

project, with the exception of the nitrogen + isobutane and nitrogen + normal

butane systems.

Prior to the LiMG density project, no low temperature, liquid density data

could be found in the literature for the following systems: nitrogen + ethane,

nitrogen + propane, ethane + isobutane, propane + isobutane, propane + normal

butane, and isobutane + normal butane, all of which have been investigated here.

All of the binary mixture measurements were carried out on liquid samples

condensed into the cell from gravimetrical ly prepared gas mixtures. This was considered to be the most accurate method to fix the compositions of tne liquid

mixtures. For all mixtures containing nitrogen and/or methane, total vapor

pressures have also been measured. For the methane-rich binary mixtures

containing either isobutane or normal butane, dielectric constant data [17] were

obtained simultaneously with the density results.

6
txtensive comparisons of the binary mixture data from the present study have

been made with available literature data. In general, differences were less than

0.1 percent, except that the density data here for the ethane + propane system

exhibited approximately an 0.8 percent discrepancy when compared with the data of

Shana'a and Canfield [19]. This was inconsistent with comparisons with the same authors for the methane + ethane and methane + propane systems as well as for

pure fluid results, where the agreement was typically better than 0.1 percent.

2.Z.3 Mul ti component Mixture Uata

In order to evaluate and test the mathematical models that have been

developed and optimized using the binary mixture and pure component data,

orthobaric liquid density measurements have been performed on twenty-seven

mul ticomponent mixtures [20,21] of Li^u components, primarily in the temperature

range of 110 to 120 K. The mul ticomponent mixtures investigated in this project

ranged from ternary mixtures (methane and binary combinations of nitrogen,

ethane, propane, and normal butane) to four- to eight-component methane-rich

(75-90 mole percent) mixtures containing up to 5 mole percent nitrogen, 15 mole

percent ethane, 7 mole percent propane, 5 mole percent butanes, and 0.44 mole

percent pentanes.

The compositions of some of the six- to eight-component mixtures were selected to simulate commercial LNG mixtures. The compositions of other

mul ticomponent mixtures were chosen to provide severe tests of the mathematical

models and to complement the use of binary mixture data in the optimization of

the models, txcept for three of the mul ticomponent mixtures, the compositions

were determined from the preparation of the gas mixtures by gravimetric means. For the other three mixtures, the compositions were determined by gas chromatographic analysis using a thermal conductivity detector. The chromatograph was calibrated with a gravimetrically prepared mixture. Vapor pressure data have

also been obtained for the mul ticomponent mixtures. For those mul ticomponent

mixtures investigated with the secona apparatus, dielectric constant

measurements iZt:} were performed simultaneously with the density measurements.

Although the pure fluid and binary mixture measurements from this project

have not included data for pentanes, five mul ticomponent mixtures containing

pentanes [21] have been investigated. It was thought that densities of these

systems could be predicted with the mathematical models using existing pure fluid

data for the pentanes from other sources [23,24] and predicting the interaction

7
parameters for binary mixtures containing pentanes from the behavior of the

binary mixtures studied in the LNb density project. (Expressions for representing the pentane data of Orrit and Laupretre [24] are reported by Hiza [25].)

High accuracy in these predictions is not required since LNG mixtures contain

relatively small fractions of pentanes.

2.3 Mathematical Models

AS mentioned earlier, one method of determining the density of LNG is to

predict that density with a mathematical model of the equation of state, given

the pressure, temperature, and composition of the Lt>i(a. The ultimate goal of this

study was to produce such a model (or models) that would be accurate to within

0.1 percent of the true density. The scope of this study was limited to

saturated liquids over a temperature range of 105 to 140 K with pressures to

2 MPa. The components of LisiG were assumed to be CH^, C2H5, C3H^, iC4Hj^Q,

nC4hj^Q, iC^Hj^^' "^5^12*

Four models were chosen to fit to the experimental data, tach of the

selected models represents a different approach to modeling the equation of state

of a fluid. These are an extended corresponding states model [26-27], a hard

sphere model [28], a cell model [^.9-31], and a graphical model due to Klosek and

McKinley [32]. During the course of optimizing the four models using the

experimental data, it became necessary to revise the functional form of the model

proposed by Klosek and McKinley. No revisions of the functional form of the

other three models were necessary, and only the adjustable parameters were changed to achieve the desired fit of the experimental aata. The final result

was that all four of the models originally chosen can be used to predict the

density of LNG to within U.l percent of the true density, given the temperature,

pressure, and composition of the Li^jb. uetails of the fitting procedures,

comparisons to experimental data, and computer program listings are given in

references [33-35].

Also during the course of this study, an excess volume model was developed

by Hiza [25]. This model also achieves the 0.1 percent criteria outlined above

under the same restrictions as the other models. Lastly, a model has been

developed by Haynes and ^lcCarty [22] that does not require either the temperature

or pressure as input information (all of the other models require a minimum of

temperature as input and most require both temperature and pressure). This model

8
requires an input of dielectric constant and composition of the li^iu and predicts

the density to within 0.15 percent of the experimental value.

All of the models have restrictions on pressure, temperature, and composition ranges, which must be defined for each model. The models are of widely

ranging computational complexity. The selection of a particular model is

dependent on the needs and objectives of the individual user. However, on the

oasis of the accuracy of the calculated density of a commercial linIu iiiixture, each

of the models appears to be equally acceptable.

One final comment about accuracy should be emphasized. No mathematical

model of the equation of state can be more accurate than the experimental data to

which it has been fit. Therefore, all of the accuracy claims are dependent upon

the accuracy of the experimental data from this study, which were used almost

exclusively to optimize and test the models.

3. LN6 Density Research at Other Laboratories

Due to the widespread interest in reliable data and the prediction of

properties of mul ti component liquefied natural gas mixtures, in particular

saturated liquid (buDble point) densities, a number of independent research

programs were conducted from whict> a significant amount of density results have

been published in the open literature. These include the studies at baz de

France (Morlet [oo]). Air Products and Cnemicals, Inc. (KloseK and

McKinley [32]), Institute of bas Technology (uonzalez, et al. iy/^^bj, Huebler,

et al . [39]), University of Kansas (Huang, et al . [40], Jensen and Kurata [41]),

University of Oklahoma (bhana'a and Canfield [19]), Shell Research, Ltd. (Boyle

and Reece [42], McClune [23]), university of Wyoming (i^iller, et al. [28,43 - 46]),

•k and Societe hiationale Elf Aquitane (Roche, et al . [47,48j, Orrit and Laupretre

[24,49], and urrit [50]). There is considerable variation in the precision and

accuracy of measurement and in the completeness of the data published from tnese

studies. Therefore, the contributions of these programs to the precise knowledge

of the density behavior of Lnu mixtures are also quite variable.

From comparisons with these independent results, it appears that the data

from the University of Wyoming are the most precise and are uniformly consistent

* Certain companies are identified in this paper for the purpose of clarity

only. Such identification does not imply any type of endorsement by the National

Bureau of Standards.

9
with the results of the NBS program. Direct comparisons of measurement precision

were also made between the NbS densimeter and the Wyoming apparatus. Measurements were made at the university of Wyoming on one binary mixture and four

mul ti component mixtures, prepared at NBS, with the apparatus calibrated with pure

metnane densities determined with the MBS densimeter [46]. Densities for the

binary mixture and three of the mul ticomponent mixtures were also measured with

the nBS densimeter. Cosnparisons were made through the extended corresponding

states model when measured temperatures were not exactly the same. The maximum

difference found was 0.10 percent for one mul ticomponent mixture point; the

remaining differences were less than 0.03 percent and random. Though the Wyoming

program was carefully planned, measurements for some of the important binary

mixtures (e.g., nitrogen + ethane, nitrogen + propane, ethane + propane, etc.) were not included, and the accuracy of the molar volume results (though not that

of derived excess volumes) depends directly on the accuracy of the density data

of tne low temperature liquid used to calibrate the apparatus.

The most extensive experimental measurements are those from the Elf Aquitane

program. Their precision of measurement does not appear zo oe quite as good as the measurements of tne University of Wyoming, and there is a systematic bias of

about -0.1 percent in a sample of mul ticomponent mixtures data [49] compared to

the extended corresponding states model optimized to the iMBi) results. Though not

all of the important binary mixtures were includea in their study, it is notable

that measurements were made on a nearly equimolar mixture of ethane + propane [bO]. txcess volumes derived from these data and their pure fluid data

between 105 and 140 K are in excellent agreement with those of the NBS program.

Most of their V*^ values are between -0.01 and -0.04 cm"^/mol compared to -0.03

to -O.Ob cm^/mol for a comparable mixture from the present study and about

-0.49 cm^/mol from the University of Oklahoma data at 108.15 K for a mixture

containing 58.5k: mole percent ethane.

With the exception of this large discrepancy in the University of uklahoma

data for the ethane + propane mixture, the remainaer of the data from the

University of Oklahoma, as well as those from Shell Research, Ltd., are of

comparable precision as those from the University of Wyoming and from £lf

Aquitane. From the Shell Research, Ltd. program, only the pure component density

data have been published. i>lone of the remaining programs have provided data

approaching the precision ana accuracy goals desired for custody transfer.

10
The proyrarns of Llf Aquitane and Shell kesearch, Ltd. provide the only low

temperature experimental densities for pure isopentane and normal pentane, and

these data sets are in good agreement, i.e., within about 0.1 percent. With the

small concentrations of pentanes normally encountered in Ll\lb mixtures, and the

fact that the densities from these investigations are generally within about 0.1

percent of the NBS values for the lower molar mass alkanes, it was felt that

additional measurements on the pure pentanes were unnecessary. In the NBS

program, the pentanes were included only in multi component mixtures, with

compositions much like Li\IG mixtures that would be encountered in practice. These

measurements were made to provide proof that the mathematical models developed

could properly account for the presence of pentanes.

Compared to the LUii research programs noted above, the NBS program is unique

in that a single set of data is provided of uniformly high precision and accuracy

for all of the pure components (i.e., nitrogen and the lower mass alkanes through

the butanes), all the possible binary combinations where liquid phase separation

does not occur, the important ternary combinations, and mul ticomponent mixtures

that contain the highest probable amounts of nitrogen, butanes, and pentanes.

The measurement method employed is a sophisticated state-of-the-art method for

which a detailed analysis of the measuresnent uncertainties have been made and

published, and statistical control of the experiments was assured throughout the

program by repeating measurements of the density of liquid methane with each

experimental run. The mathematical models optimized to the data are representative of existing theoretical and empirical models in wide use, from the most

simple to the most complex. The concurrent experimental and mathematical

modeling efforts also served to guarantee that sufficient data were obtained to

optimize and to identify the inherent limitations of each.

4. Acknowledgments

The contributions of the members of the LrJG Density Project Steering

Coiiiiiiittee are gratefully acknowledged. The Steering Committee, comprised of

representatives from each of the sponsoring companies, met quarterly from May 11,

1972 to February 15, 1979. The critique of research progress and numerous suggestions offered by the members were extremely valuable. Special thanks are

due i-lartin k. Cines of the Phillips Petroleum Company who served as Chairman of

the LNG Density Project steering Committee during that time, and who played a major role in obtaining sponsors for this project. Thanks are also due Louis A.

11
Sarkes of the American Gas Association for his general support of LNG properties

research and for his contriDutions to this project. The AGA coordinated the

funding from the sponsoring companies for this project. L. James Kemp of

i>outhern California Gas Company assumed Chairmanship of the Steering Committee

following the retirement of Martin R. Cines from Phillips Petroleum Company. The

authors woulo like to express their appreciation to Karen A. Bowie and

Lawrence 0. Liberman for their assistance in the preparation of the manuscript.

12
5. References

[I] riaynes, W. h.; hiza, M. J.; hcCarty, R. U. Densities of Lub for custody

transfer. Proceedings of Fifth International Conference on LiMb, vol. 2;

1977 August ii9-Septeniber 1; Uusseldorf, Germany. Chicago, 111.: Institute

of Gas Technology; i>ection III, Paper 11. 30 p.

IZ'j Haynes, W. iM.; McCarty, K. 0. The density of liquefied natural gas.

Proceeding of the Fifty-Eighth annual (GPA) Convention of the Gas

Processors Association; 1979 March 19-^11; Denver, Colo.: Gas Processors

Association, Tulsa, ukla. lu-ib.

[3 J Diller, u. t. L(\ib density determination, hydrocarbon Processing 56(4):

142-144; 1977 April.

[4] McCarty, R. u. LnG densities for custody transfer. Proceedings of the

Fifty-Sixth International School of Hydrocarbon measurements; Ib/Bl

April 14-16; Norman, Okla. 515-517.

[5j riaynes, W. H.; Hiza, H. J.; Frederick, V. Magnetic suspension

densimeter for measurements on fluids of cryogenic interest. Rev. Sci.

Instrum. 47(10): 1237-iii50; 1976 October.

[6] Haynes, W. M. Apparatus for density and dielectric constant measurements

to 35 MPa on fluids of cryogenic interest, d. Res. f^at. Bur. Stand.

(U.S.) b8(4): 241-252; 1983 July-August.

[7] Frederick, N. V.; Haynes, W. IM. Differential capacitance sensor as position detector for a magnetic suspension densimeter, kev. Sci. Instrum.

50(y): 1154-1155; 1979 September,

[bj Clark, A. F.; Haynes, w. M.; Deason, V. A.; Trapani, R. d.; Low

temperature thermal expansion of barium ferrite. Cryogenics ib(j):

267-270; 1976 May.

[9] Haynes, W. M. Simplified magnetic suspension cieiisimeter for absolute

density measurements. Rev. Sci. Instrum. 48(1): 39-41; 1977 January.

[10] haynes, W. M. Measurements of densities and dielectric constants of liquid

propane from 90 to 300 k at pressures to 35 MPa. J. Chem. Thennodynamics

15(5): 419-424; 1983 May.

[II] Haynes, W. M. Measurements of densities and dielectric constants of liquia

isobutane from 120 to 300 K at pressures to 35 MPa. J. Chem. Eng. Data, to

be published.

[12] riaynes, W. M. Measurements of densities and dielectric constants of liquid

normal butane from 140 to 300 K at pressures to 35 MPa. J. Chem.

Thermodynamics 15(9): 801-805; 1983 September.

13
[13] Haynes, A. M.; riiza, i4. J. Measurements of the orthobaric liquid densities

of methane, ethane, propane, isobutane, and normal Dutane. J. Chem.

Thermodynaiiiics 9(Z): i79-id7; 1977 February.

[14 j riaynes, w. H.; liiza, H. J. Orthobaric liquid densities of normal butane

from 13b to 3U0 K as detennined mth a magnetic suspension densimeter.

Advances in Cryogenic Engineering, Vol Zl, K. U. Timmerhaus and u. H.

Weitzel, ed. ^^ew YorK, f\iY: Plenum Press; 1976. 516-62!1.

[lb] riaynes, w. M. Measurements of the orthobaric liquid densities of argon

(100-120 K) and ethylene (105-200 K). Cryogenics 18(10): 621-623; 1978

October.

[16] iiiza, M. vJ.; riaynes, w. M,; Parrish, vL R. Orthobaric liquid densities and excess volumes for binary mixtures of low molar-mass alkanes and nitrogen

between 10b and 140 K. J. Chem. Thermodynamics 9(9): c573-896; 1977

September.

[17] Haynes, W. tA. Orthobaric liquid densities and dielectric constants of

(methane + 2-methyl propane) and (methane + n-butane) at low temperatures.

J. Chem. Thermodynainics 15(10): 903-911; 1983 October.

[18] Hiza, H. J.; Haynes, W. M. Liquid mixture excess volumes and total vapor

pressures using a magnetic suspension densimeter with compositions

determined by chromatographic analysis: methane plus ethane. Advances in

Cryogenic Engineering, Vol 23. K. D. Timmerhaus, ed. hiew York, wY: Plenum

Press; 1976. 594-601.

[19] Shana'a, M. Y.; Canfield, F. B. Liquid density and excess volume of light

hydrocarbon mixtures at -165°C. Trans. Faraday Soc. 64(549): 2281-2286;

1968 September.

[20] Hiza, M. J.; naynes, W. M. Orthobaric liquid densities and excess volumes

for mul ti component mixtures of low molar-mass alkanes and nitrogen between

105 and 125 K. J. Chem. Thermodynamics 12(1): 1-10; 1980 January.

[21] haynes, W. M. Measurements of orthobaric-1 iquid densities of

mul ticomponent mixtures of Li^b components (i\i2, CH^, ^2*^6* ^3*^8'

Cri3CH(CH3)CH3, C^H^q, CH3Ch(CH3)C2H5, and C^ri^^) between 110 and 130 K.

J. Chem. Thermodynamics 14(7): 603-612; 198*1 July.

[22] Haynes, W. ; McCarty, K. D. Prediction of liquefied-natural-gas (LNG)

densities from new experimental dielectric constant data. Cryogenics

23(8): 421-426; 1983 August.

14
[23] McClune, C. k. Measurement of the densities of liquefied hydrocarbons from

-100 to -ISO'^C (173 to 93 K). Cryogenics 16(5): 289-295; 1976 May.

[24] Orrit, J.; Laupretre, J. N. Density of liquefied natural gas constituents.

Advances in Cryogenic Engineering, Vol. 23. K. u. Timmerhaus, ed. New

York, NY; Plenum Press; 1978. 573-579.

[25] Hiza, M. J. An empirical excess volume model for estimating liquefied

natural gas densities. Fluid Phase Equilibria 2(2): 27-38; 1978 August.

[26] kowlinson, J. S.; watson, I. u. The prediction of the thermodynamic

properties of fluids and fluid mixtures - I. The principle of

corresponding states and its extensions. Chem. Eng. Sci . 24(10):

1565-1574; 1969 October.

[27] Mollerup, o.; Rowlinson, J. S. The prediction of the densities of

liquefied natural gas and of lower molecular weight hydrocarbons. Chem.

Eng. Sci. 29(b): 1373-1382; 1974 June.

l28] Rodosevich, J. B.; Miller, R. C. Calculation of Li^G excess volumes by a modified hard-sphere model. Advances in Cryogenic engineering, Vol. 19.

K. D. Timmerhaus, ed. New York, i\IY: Plenum Press; 1974. 339-345.

[29] Renon, ri.; Eckert, C. A.; Prausnitz, J. M. Molecular thermodynamics of

simple liquids. Ind. Eng. Chem. Fundam. 6(1): 52-5b; 1967 February.

[30] Eckert, C. A.; Renon, h.; Prausnitz, u. M. Molecular thermodynamics of

simple liquids. Ind. Eng. Chem. Fundam. 6(1): 58-67; 1967 February.

[31] Aloright, M. A. A model for the precise calculation of liquefied natural

gas densities. NbPA Tech. Pud. TP-3. 1973.

[32] Klosek, J.; McKinley, C. Densities of liquefied natural gas and of low

molecular weight hydrocarbons. White, J. W.; Neumann, A. E. S., ed.

Proceeding of the First International Conference on LNG; 1968 April 7-12;

Chicago, 111. Chicago, 111.: Institute of Gas Technology; Session 5,

Paper 22. 15 p.

[33] McCarty, R. D. A comparison of mathematical models for the prediction of

LivlG densities. iMat. Bur. Stand. (U.S.) NBSIR 77-867; 1977 October. 60 p.

[34] McCarty, k. u. Four mathematical models for tne prediction of Lng

densities. Nat. Bur. Stand. (u.S.) Tech. note i030; 1980 uecemoer. 76 p.

[35] McCarty, R. u. Mathematical models for the prediction of liquefiednatural-gas densities. J. Chem. Thermodynamics 14(9): S37-854; i98^

September.

15
[36] Morlet, J. The density and miscibility of liquefied hydrocarbon gases at

low temperatures. Rev. Inst. Francais Petrole 18: 127-143; 1963 January.

[37] laonzalez, M. H.; Subrainanian, T. K.; Kao, R. L.; Lee, A. L. Physical

properties of natural gases at cryogenic conditions. White, J.W.; Neumann,

A. £. S., ed. Proceedings of the First International Conference on LMG;

1968 April 7-12; Chicago, 111. Chicago, 111.: Institute of Gas Technology;

Session 5, Paper 21. 38 p.

[38] Gonzalez, H. H,; Lee, A. L. Qew and bubble points of simulated natural

gases. 0. Chem. Eng. ilata 13(2): 172-176; 1968 April.

[39] Huebler, J.; takin, B. E.; Lee, a. L. Physical properties of natural gas

at cryogenic conditions. Proceedings of the Second International

Conference on Li\ib, Vol. I; 197U October 19-23; Paris, France. Chicago,

111.: Institute of Gas Technology; Session 2, Paper 9. 10 p.

[40] Huang, E. T. S.; Swift, G. W.; Kurata, F. Viscosities and densities of

methane-propane mixtures at low temperatures and high pressures. AIChE J.- 13(5): 846-850; 1967 September.

[41] Jensen, R. H.; Kurata, F. Jensity of liquefied natural gas. J. Petrol.

Tech. 21: 683-691; 1959 June.

[42] Boyle, G. J.; Keece, D. bulk measurement of Li^lG. Proceedings of the Second

International Conference on LNG, Vol. I; 1970 October 19-23; Paris,

France. Chicago, 111.: Institute of Gas Technology; Session 2, Paper 5.

14 p. [43] Rodosevich, J. B.; Miller, R. C. Experimental liquid mixture densities for

testing and improving correlations for liquefied natural gas. AIChE J.

19(4): 729-735; 1973 july.

[44] Miller, R. C. Estimating the densities of natural gas mixtures. Chem. Eng.

81(23): 134-135; 1974 October.

[45] Pan, W. P.; hady, M. H.; Miller, R. C. Dielectric constants and

Clausius-Mossotti functions for simple liquid mixtures: systems containing

nitrogen, argon and light hydrocarbons.. AIChE J. 21(2): 283-289; 1975

March.

[46] Miller, R. C; Hiza, M. J. Experimental molar volumes for some Li\ib-rel ated

saturated liquid mixtures. Fluid Phase Equilibria 2(1): 49-57; i978

August.

16
[47] Rocne, Y.; Perret, J.; Sochi rol, l. uensi ty of L^lb and of its

constituents. Proceedings of the Second International Conference on LUG,

Vol. I; 1970 Uctooer 19-i;3; Paris, France. Chicago, 111.: Institute of Gas

Technology; Session 2, Paper b. 8 p. l48] koche, Y.; Perret, J.; Bochirol, L.; Brauns, P. Density of Li\lG and of its

constituents. White, J. W., ed. Proceedings of the Third International

Conference on LUG; 1972 September 24-28; Washington, D.C. Chicago, 111.:

Institute of Gas Technology; Session 11, Paper l6. li p. [49] Orrit, J.; Laupretre, J. M. A cal culational method for obtaining the

density of a liquefied natural gas. Advances in Cryogenic Engineering,

Vol. 23. K. u. Tiinmerhaus, ed. i^^ew York, ImY: Plenum Press; 197b. 56b-b79.

[bO] Orrit, J. t. Densities of liquefied natural gas components mixtures,

{submitted to Fluid Phase tquilibria) 1983 January.

17

6. Major Publications of LiMb Density Project

6.1 Reference [5]

Magnetic suspension densimeter for measurements on fluids

of cryogenic interest*

W. M. Haynes, M. J. Hiza, and N. V. Frederick

National Bureau of Standards, Institute for Basic Standards, Cryogenics Division, Boulder, Colorado 80302

(Received 24 June 1976) An apparatus incorporating a magnetic suspension technique has been

developed for density measurements on Hquids and liquid mixtures, particularly,

at saturation, at temperatures between 90 and 300 K and at pressures to 5 MPa

(approximately 50 atm). The feasibility of adapting this method, previously used

at room temperature, for low temperature use had been demonstrated in an

earlier study with a density measurement on saturated liquid nitrogen near its normal boiling point. The present apparatus, which is significantly improved,

and in most respects different from the earlier model, is described in detail. It includes a cryostat for continuous wide-range temperature control, a windowed

equilibrium cell particularly suited for studies of liquid mixtures, and a new

electronic servocircuit with a linear differential transformer for position control of the magnetic buoy. Extensive tests and density measurements have been

carried out to evaluate the performance of this apparatus. Densities of saturated

liquid nitrogen between 95 and 120 K and saturated liquid methane between 105 and 160 K are reported. The estimated standard deviation of a single density measurement is less than 0.02%. The total systematic error in the measurement

process from known sources is approximately 0.05%. The total uncertainty of a

single density measurement, which is taken as three times the standard deviation

plus the systematic error, is approximately 0.1%. Comprehensive comparisons of

the present results with previous experimental data are presented.

1. INTRODUCTION

With the advent of large-scale liquefaction and global

transport of natural gas, accurate equilibrium liquidphase densities have become important for custody

transfer. The density data for liquefied natural gas mixtures and their components have been generally characterized by the existence of large discrepancies (approximately 0.5%) between different sets of data for the pure components and by a limited amount of mixture

data. Hence, an apparatus has been developed to pro- vide accurate, consistent, and comprehensive density data for liquefied natural gas mixtures and their pre- dominant pure components at saturation conditions. These measurements will serve as a data base for test- ing and optimizing selected mathematical models (correlations) useful in predicting the densities of liquefied natural gas. In the present paper the apparatus and

experimental procedures are described in detail and

representative measurements for two components of

liquefied natural gas are presented.

A magnetic suspension densimeter, which is based on an application of Archimedes' principle, has been

selected for the present measurements primarily be- cause this instrument is suitable for an independent,

direct determination of the density of a saturated

liquid over wide ranges of temperature and pressure

within the precision and accuracy required for antici- pated technical applications. In an earlier study, ^ the

1237 Rev. Scl. Instrum., Vol. 47, No. 10, October 1976

feasibility of adapting the magnetic suspension tech- nique for density measurements at low temperature was

demonstrated with a measurement on saturated liquid

nitrogen near its normal boiling point. The present

apparatus (see Fig. 1 for a scaled assembly drawing),

which is a much improved version of the earlier model, incorporates a three-coil support system,^ a

cryostat for temperature control at any temperature

between 90 and 300 K, a windowed equilibrium cell suitable for projected studies on liquid mixtures at pressures to 5 MPa, and a new electronic servocircuit with a linear differential transformer that serves

as a position sensor for a barium ferrite magnetic buoy.

The present work has included an extensive evaluation of the use of a magnetic suspension densimeter

for absolute density measurements. (Here the term

"absolute density" is used to indicate that cryogenic

fluids were not used to calibrate the instrument.) Results of performance tests along with densities

for saturated liquid nitrogen between 95 and 120 K and

saturated liquid methane between 105 and 160 K are presented in this paper to demonstrate the low temperature performance of this new apparatus. The estimated standard deviation of a single density measurement is less than 0.02%. The total uncertainty of a single

measurement, which is taken as the sum of the syste- matic error and three times the standard deviation,

is approximately 0.1%.

In Sec. II the general principles of a magnetic

Copyright © 1976 American Institute of Physics 1237

19
suspension densimeter are outlined. Section III is devoted to a detailed description of the experimental

apparatus. A comprehensive discussion of the experi1238 Rev. Sci. Instrum., Vol. 47, No. 10, October 1976

mental procedures and measurements is given in Sec. IV. In Sec. V a summary of performance tests carried out in evaluating the instrument is presented. This secOenshneter 1238
tion also deals with the uncertainties in the density measurements. Finally, the experimental data for saturated liquid nitrogen and methane, along with

comparisons with independent measurements, are re- ported in Sec. VI.

II. PRINCIPLES OF OPERATION

A piece of magnetic material is magnetically sus- pended in a fluid at a stable equilibrium position

through the use of a closed-loop servosystem. When

the float (or buoy) is in free support, there are three forces acting on it. In the present system the magnetic float is more dense than the fluid in which it is suspended; thus, an upward magnetic force is added to the buoyancy force to balance the downward gravitational force. In the present work a three-coil (sur-core) system^ is used to supply the magnetic force. The three coils are aligned coaxially. Each coil is symmetric about an

axis in common with the cylindrical axis of the magnetic buoy. The three coils consist of a main coil, which supplies the major part of the field necessary

to lift the float, and a pair of gradient coils. The

horizontal position of the float, which is in the shape

of a right circular cylinder (length/diameter > 1) magnetized along its cylindrical axis, is maintained by

the axially symmetrical, diverging field of the main coil. The magnctx force on the ferromagnetic buoy is given

by

F' mag =M^ ' aZ

(1)

where M is the total magnetic moment of the buoy

and dH/dZ is the variation of the external field intensity along the common cylindrical axis of the coils. For the present system the float is a permanent

magnet for which the total magnetic moment (excluding

temperature dependence) can be represented by

M = .Wn + MiiH). (2) where Mn is the permanent moment, and M,{H) is the induced moment resulting from an applied field. At a fixed position along the cylindrical axis of an air- core solenoid. H = kl and

dH^ dZ

(3)

(4)

where / is the solenoid current and k and k' are constants dependent upon the number of turns and

dimensions of the coil. As mentioned earlier, the float is supported at a position relative to the main coil such that its horizontal

position is fixed. This results in the float being sup- ported at a distance below the main coil slightly larger than the inside radius of this coil. The float is supported approximately midway between the gradient

coils. These coils are connected in series such that their magnetic field intensity contributions cancel at the float position. (For a Helmholtz coil arrangement the magnetic fields would add at the midpoint.) Thus, the magnetic field intensity at the float position is due

solely to the main coil contribution. Similariy, the magnetic field gradients of the gradient coils add at the float position. For this three-coil arrangement and with a constant main coil current (/i,) (and subsequently the magnetic

moment of the float does not change), the density of the

fluid (p) is related to the gradient coil current (/c) by the following relation:

p = A + BIc (5) where A and B are constants to be determined by

calibration with fluids of known density. This relation- ship is valid whether the float is magnetically hard or

soft. Knowledge of the mass and volume of the float is not required to obtain relative density results using Eq.

(5). However, since the magnetic moment of the float

is temperature dependent, the constants A and B would

have to be determined at each temperature of interest. At low temperatures at least two fluids of known

density, one of which may be "vacuum," are needed

for calibrating the instrument for use in performing rela- tive density measurements. However, the densities of low temperature fluids are not known to sufficient accuracy to permit them to be used as calibration

fluids and, in fact, this had been one of the major

motivations for developing aii independent technique for density measurements on cryogenic fluids. In the previous work by Haynes and Stewart.' a new method was demonstrated for determining absolute

densities using the same three-coil arrangement, but which is dependent upon the magnetic properties of

the buoy material. If one uses a permanent magnetic

material for the float and its induced moment is zero

or. at least, very small compared to the permanent

moment, then the magnetic force on the float is given

by the relation.

(6) where C and D' are constants depending on the rela- tive position of the coils and float, the magnetic moment

of the float, and the dimensions and number of turns of the support coils. One should note that for Eq. (6) the main coil current (/.„) is not held constant. Now

the density of the fluid is given by the expression.

p = (m - C/.V - DIgW, (7) where m and V are the mass and volume of the float. The acceleration of gravity has been included in the constants C and D. (The mass and volume of the float must be determined by independent measurements.)

The validity of this equation can be tested experimentally. For measurements in a vacuum, Eq. (7) re- duces to the following relation:

1239 Rev. Sci. Instrum., Vol. 47, No. 10, Octotwr 1976 Denalcneter 1239

Zl
m = Cht + Die- (8) Table I. Parameters of support coils.

It should be emphasized that the critical assumption

in deriving Eq. (7) is that there is no induced moment

in the buoy resulting from changes in the magnetic

field intensity over the range of magnetic fields needed

for the density measurements. (For the present measurements this includes fields between 0.006 and

0.016 T.) The current in the gradient coils has been measured

as a function of the current in the main coil while

the float was suspended at a constant height in a given

fluid, usually vacuum, at constant temperature. The

currents were fitted by the method of least squares to either Eq. (6) or Eq. (8). The quality of this fit indicated the validity of the assumption concerning

the dependence of the magnetic moment upon the magnetic field. (Results will be presented in a later section.) For the present measurements the main coil current, and thus the magnetic field, could be varied over a range amounting to 25% of its maximum value. The minimum ratio of Istllc was determined by the

loss of horizontal stability of the float. From the above discussion it should be apparent

that relative density measurements using Eq. (5) Us, constant) can be carried out to complement the

results of absolute measurements using Eq. (7) with the same three-coil arrangement. In performing measure- ments using Eq. (7) for which the main coil current

is varied over a considerable range, it is experimentally difficult to fix or determine the position of

the main coil from vacuum to liquid measurements

within the desired precision. This problem does not

arise when making relative density measurements for which the main coil current is held constant. In

principle, both procedures should give identical density

ratios for the same fluids at constant temperature.

In practice, the relative density measurements can be

carried out to deduce the systematic error in the absolute density measurements that depends on the magnitude of the main coil current. These tests have

been performed in the present work and will be dis- cussed in a later section.

III. APPARATUS

A. General considerations and cryostat

An assembly drawing (approximately to scale) of the major components of the apparatus is shown in Fig. 1

.

The general dimensions and configuration of the cryostat

and equilibrium cell were determined by the dimensions

and configuration of the main coil and the two gradient

coils. (See Table I for coil dimensions.)

Criteria considered essential, and which have been

satisfied, in the design of the apparatus are as follows:

(a) all materials of construction in close proximity to the magnetic suspension assembly are nonmagnetic; (b) the entire assembly is rigidly supported so that the rela- tive position of the buoy, microscope, and support coils

1240 Rev. Sci. Inttnim., Vol. 47, No. 10, October 1976

Main coil Gradient coils

Length 6.0 cm 6.2 cm

Inside diameter 7.4 cm 25.4 cm

Outside diameter 20.24 cm 38.6 cm

Number of turns 5000 5000

can be maintained; (c) the magnetic buoy and the liquid

level are visible within a windowed cell capable of

withstanding working pressures up to 5 MPa; (d) the volume occupied by the liquid sample within the cell

is sufficiently large to render the unavoidable vapor volume of the access tubing, etc.. insignificant within

the accuracy goals of experiments involving liquid mixtures; (e) temperature gradients along the length of the working space of the cell are monitored and

reduced to a practical minimum, i.e., less than a total of 10 mK; (f ) the rate of refrigeration of the equilibrium

cell is adjustable and continuous; and (g) the temperature of the ceil can be controlled at any temperature

between 90 and 300 K. A concrete block structure provides a stable support for the cryostat, the coils, and the microscope.

The cryostat assembly is suspended from an aluminum

plate that spans the top of the concrete structure. The stainless steel central support tube of the cryostat is attached to this aluminum plate with a brass

collar (not shown in Fig. 1). The lower end of the

stainless steel tube is soldered to a copper tube to which a heat exchanger is soldered.

Refrigeration is provided by continuous transfer of

liquid nitrogen from the cryostat reservoir through the heat exchanger. Regulation of the refrigeration rate is accomplished by controlling the nitrogen vent rate to the atmosphere..

Refrigeration is supplied to the bottom of the equilibrium cell through the copper radiation shield and

to the top of the cell through two copper braided

straps. The radiation shield, which contains a slit aperture in a position corresponding to the cell window,

is attached to a copper plate at the top of the shield with Wood's metal to facilitate removal of the shield and alignment of the shield within the glass tail section. The glass tail is connected to the stainless steel vacuum

jacket through a stainless steel-to-glass transition joint.

B. Equilibrium cell The equilibrium cell, made of electrolytic tough

pitch copper, has an overall length of approximately

30 cm, including the copper-plated bellows at the bot- tom. The cell closure is a compression fitting with a

silver-plated, solid copper 0-ring. The closure plug, threaded backing ring, and compression screws are

stainless steel. The top of the cell is held in align- ment by a stainless steel tube, which is brazed to the closure plug and extends through the uppermost

cryostat plate. The upper end of the cell support tube

Densimeter 1240
is threaded through a nut and bearing assembly to al- low small adjustments in the vertical position of the cell and thus the vertical position of the magnetic buoy.

The bottom of the cell is thermally anchored to the

radiation shield by mechanical attachment of a flanged bellows section. A thin coat of commercially avail- able conducting grease is used to enhance thermal

contact between the bellows flange and the shield. Two thermal links of copper braid are also mechanically

attached to the top of the cell from the top plate of

the radiation shield using thin coats of conducting

grease between the contact surfaces. The outside diameter of the main part of the cell is 4.13 cm and that of the closure section is 7.60 cm. The internal working space is 1.99 cm in diameter

and approximately 20 cm long. More than half of the

internal volume, however, is occupied by the magnetic buoy- sensor coils assembly (~10 cm^) and the vapor bulb (~35 cm^). The vapor bulb is a slip fit in the top section of the cell and is attached directly to the closure plug. The internal free volume of the cell is approximately 20.5 cm^. Anticipating measurements on

mixtures, the volume (—0.3 cm^) in the annulus be- tween the vapor bulb and the cell wall has been made as small as possible, since this is normally part of the vapor volume.

The electrical lead-throughs for the sensor coils consist of three No. 32 coated copper wires encapsulated

with an epoxy adhesive in a copper capillary tube. This tube extends through the closure plug to the bot- tom of the vapor bulb. The epoxy adhesive, an

alumina-filled resin with elevated temperature curing

agent, was selected based on an earlier study^ of low temperature properties of epoxy adhesives. By

pressurizing with helium gas while the adhesive was

hot. the entire length of the capillary tube was filled with

adhesive to minimize the available vapor volume in the equilibrium cell. The cell window was designed to allow viewing the

liquid sample from the base of the vapor bulb down

to 2 cm from the bottom of the liquid space. The

viewing slit is 0.6 cm wide x 7.5 cm long. The window

consists of a piece of tempered Pyrex"* glass of 0.32-cm

thickness with semicircular ends and chamfered edges.

The seed between the glass and the cell is made with indium wire compressed in a racetrack groove machined in a flat surface on the cell. The glass is compressed against the indium seal with a hardened

beryllium copper plate secured by stainless steel screws. A rubber asbestos pad is placed between the beryl- lium copper plate and the glass to relieve thermal and

mechanical strains. The maximum working pressure of

the equilibrium cell is limited by the maximum working

pressure of the window.

C. Magnetic buoy

The buoy is a barium ferrite (BaFei20,9) magnet in the shape of a right circular cylinder magnetized

1241 R«v. Sci. Inttrum., Vol. 47, No. 10, October 1976

along its cylindrical axis. Its length and diameter are approximately 0.64 and 0.51 cm, respectively.

Barium ferrite is a magnetically hard, ceramic material with a density of approximately 5 x 10^ kg/m^. It

is somewhat porous and has a high electrical resistivity. The buoy used in the present work has been permanently magnetized in a saturation field of 1 T. After

initial cycling over maximum ranges of temperature

(90- 300 K) and magnetic field intensity (0-1.3 x 10^ A/m) the barium ferrite magnet exhibited no hysteresis over the ranges of currents and temperatures needed

in the density measurements. The magnet is operating

in a very small segment (5-13 x l(P A/m) of a broad

hysteresis loop. The slope of the B-vs-H curve is small and constant. The change in the magnetic moment of the

float with temperature varies from 0.2%/K at 300 K to

0. 1%/K at 100 K. as determined from the present meas- urements in vacuum. This variation in the magnetic

moment with temperature results in a random error in the density of approximately 0.002% at 100 K, increasing

to 0.004% at 300 K. This estimate is based on a

reproducibility in the temperature of 2 mK from vacuum

to liquid measurements.

Since the barium ferrite magnet is porous, it must

be coated with a material which does not allow fluid to penetrate into the buoy. Copper was found suitable

for this purpose. After a piece of barium ferrite was ground into the shape of a right circular cylinder

>^ith chamfered edges (0.25 mm deep), a conducting

layer of copper was uniformly plated onto the ceramic

magnet by chemical reduction. On top of this conducting layer, copper was electroplated to a thickness of approximately 0.25-0.50 mm: a much thicker coating

resulted at the edges. Then most of the copper was

removed with a diamond tool until the Copperplated barium ferrite magnet was in the shape of a right circular cylinder without chamfered edges. The final buoy had a minimum of 0.06 mm of copper over the

cylindrical surface and the end faces and at least 0.25 mm

on the edges where the magnet had been chamfered.

The edges were built up to a thicker coating since the plating strength would be weakest at these

locations. (For density measurements it is not necessary that the plating be symmetric about the center of mass of the buoy.) Gold was flashed (10"^ mm) over

the entire copper surface as a protective coating.

D. Electromagnetic support coiis Each of the coils is composed of two separate

coils of 2500 turns of epoxy-coated aluminum foil of approximately 0.025-mm thickness. The foil for the main coil has a width of 2.5 cm while that for the gradient coils is 3.0 cm. Selected coil parameters

are given in Table I. Each of the gradient coils is rigidly supported by three aluminum rods resting on the concrete structure. The

gradient coils have a separation distance of approximately 23 cm. No effort has been made to control the

Dontimator 1241

Z6
12V 12V 12V 12V

Y

J MAIN J COIL

1.35V KELVIN

Hg CELL VARLEY

(TEMP i DIVIDER

LOAD STABILIZED)

(CURRENT

CONTROL)

POWER SUPPLY

( 100 VOLTS)

-vw

DIFFERENTIAL

VOLTMETER

^THERMAL LAGGING ENCLOSURE

Fjg. 2. Current control circuit for main coil.

position of the gradient coils except through the control of the laboratory temperature. The dH/dZ of the gradient coils changes very slowly (2.5 x IO~''/cm) from a maximum at the float position. Since the change

in dHjdZ with position for the main coil is two orders of magnitude larger than that for the gradient coils, special care has been taken to minimize changes in the position of the main coil. First, the main coil is supported at its midpiane so that thermal expansion effects inside the coil are minimized.

A circular copper plate has been soldered to the brass tube on which the main coil is wound. A coil of aluminum

foil (2500 turns) is clamped tightly to each side of the copper plate with a thin layer of vacuum grease be- tween the aluminum foil and copper for maximum

thermal contact. The copper plate (and thus the coil)

is supported by quartz rods extending to the concrete

structure. Thus, any changes in the position of the midplane of the coil due to room temperature changes

would be negligible. Copper plates with cooling water have been placed on the outside faces of each section of the main coil such that the average temperature of the main coil is below room temperature

for maximum current (1 A). However, for absolute

density measurements, the temperature of the main coil changes from vacuum to liquid measurements (1.2 K

for nitrogen at 100 K) since a larger current is required

to support the float in vacuum than in the liquid for a given gradient coil current.

E. Support system electronics

The current in the main coil, which contributes the major portion of the magnetic force necessary to lift the buoy, is provided by the circuit of Fig. 2. The circuit

functions as follows: The main coil current flows through

a stable resistor Rjtd (01 i^' in the emitter of Qo (the current-controlling transistor). The voltage developed

across Rstd 'S compared with a reference voltage from

a Kelvin-Varley voltage divider by amplifier A,. The

output of A, drives the base of Qj and controls the current through Rstd and the main coil. Since the loop

gain of this circuit is exceedingly high (>I0*) and the thermal drifts are very low, the long-term stability of

the main coil current is of the order of 0.0019?^. This

current controller has a temperature coefficient of the order of 5 ppm/K and a sensitivity to power supply variations of less than 2 x 10~® A/V. A slow drift of

the coil current of 2.5 x 10~* A/'h has negligible ef- fect on the density measurements since the main coil current is measured each time that the gradient coil current is determined. (It should be noted that the precision and accuracy of the density results depend on

the short-term stability and not the absolute accuracy of

the standard resistor.) The gradient coils are included in a control circuit (Figs. 3 and 4) that senses and maintains the vertical position of the buoy. The requirements for stability of a magnetic support or levitation system are dis- cussed in detail in Refs. 5 and 6 and will not be repeated

here. Suffice it to say that the characteristic equation

for a magnetic levitation system has a root with a positive

real part, and for system stability the plot of the frequency response of the characteristic equation (the Nyquist plot) must encircle the origin once in the counterclockwise sense. This is accomplished by the duaJ-lead network composed of R,, R,, C,, and Cj in Fig. 4. A dual-lead network is required for this system

since minimization of the steady state offset or buoy

position change requires at least a first-order integrator

1242 R*v. Sci. Instrum., Vol. 47, No. 10, October 1976 0«nBlinet«r 1242
in the control loop. This integrator consists of amplifier

A, and R, and C,. The additional resistor R3 is re- quired to stabilize the combined integrator and dual-lead network. A block diagram of the electronics is presented

in Fig. 5 to complement the detailed circuit diagrams in Figs. 2-4. The circuit diagrams are presented in detail since this circuit is considerably different from any found in the literature. It should be noted that the

12V i20pfHC1550G ^-^^^ +12V

PHASE SHIFTER

TO THE CURRENT

CONTROL FOR THE GRADIENT COILS

Fig. 4. Position sensor electronics and compensation circuit.

COMPENSATION CIRCUIT

1243 Rev. Scl. Inttrum., Vol. 47, No. 10, Octob«r 1976 Densimeter 1243
current control amplifier for the main-coil constant- current source is the same as that for the gradient

coils. Perhaps the most interesting part of the support

system is the position sensor. In the past, position

sensing has been accomplished by either optical'-*

(ligjit beam-photomuitiplier tube arrangement) or

electronic' *"'^ (inductance pickup coil or differential transformer) means. The requirements of small size, low

power dissipation, high sensitivity, the ability to with- stand high pressure and low temperatures, and exceptional dimensional integrity resulted in the use of a linear

differential transformer (LDT) wound on a machinable

glass ceramic form. The LDT consists of a pair of coils equally spaced about a transverse rectangular opening

in the ceramic form inside which the buoy is suspended.

Each coil is wound with 34 turns of double silk- wrapped 50/44 Litz wire in circumferential grooves of 2.8-mm width and 6. 1-mm depth. The outside diameter

of the coils and the ceramic holder is 17.3 mm. The

complete transformer assembly has been potted with a

low-viscosity epoxy resin to prevent intrusion of the pressurized fluid into the windings.

The sensitivity of the LDT as measured at the phasesensitive detector (synchronous detector) is of the order of 1 V/mm. The noise voltage at the sensor amplifier

input is less than 10~* V/Hz"^. The operational

amplifiers in the compensation circuit have very low. off- set drifts and are thermally- lagged to a large copper

block in the amplifier chassis. Consequently, the vertical movement of the buoy due to operational amplifier

drifts and rf amplifier noise is less than 10"' mm.

The LDT is the major component of a ratio-trans- former-type bridge. Pan of the bridge is located at the top of the cryostat and is connected to the LDT through

long coaxial lines. The bridge is constructed so as to re- move proximity effects of the outer conductors of the coaxial lines. The neutral position of the buoy is controlled by a small variable capacitor between the center

lead and one side of the LDT. Conductance balance

is achieved with a resistor connected across the capacitor. Unbalance in the LDT due to the inductance

of the long leads is minimized by tuning the two halves of the LDT to resonance with matched capacitors

(as large as possible for'the 0.5-MHz exciting signal).

These capacitors are connected near the LDT outside

the sample cell but inside the vacuum space.

IV. EXPERIMENTAL MEASUREMENTS AND

PROCEDURES

A. Temperature and pressure

The techniques and instrumentation for the measure- ment and control of the temperature and pressure are standard. The primary temperature sensor, calibrated on the IPTS 1968, is a platinum resistance thermometer

secured in a well at the base of the equilibrium cell with Wood's metal. The calibration of the thermometer

Ihisiriii

CIrciits r-WJ—

3,

VtAifi Ctitrillii frtmi

«Mtl|l

Cntril ttniit Snrti

Hciliitir

ifl Friiiiicf Trusliriirlitiiriiir t ClBMKItlM

Circaits

tiltiii Ciitrillil Citfiit SMrti

Crciits

Fig. 5. Block diagram of a magnetic suspension densimeter:

G,, G; —gradient coils; M —mam coil; S,. Sj —sensing coils; B —

magnetic buoy.

has been checked against vapor pressure measurements

on liquid methane and nitrogen.

The uncertainty of the calibration is approximately

0.002 K. Due to the specifications of the potentiometric

measuring system, uncertainties in the temperature

amount to a maximum of O.OiO K at 1 10 K, increasing to 0.030 K at 300 K. The temperature of the sample

holder is controlled to better than 0.005 K, approximately the same as the reproducibility of the temperature measurements. A current of 1 mA for the thermometer is supplied

by an electronic constant current source. The voltage

drop across a 100 H calibrated standard resistor in series with the thermometer is monitored continuously with a

six-digit differential voltmeter. With this setup the uncertainty and repeatability of the current is approximately 0.002^. The voltmeter is checked periodically

against a microvolt potentiometer using a calibrated standard cell. The total uncertainty in the reported temperatures in Sec. VI is believed to be less than 0.03 K. (At room

temperature it would be less than 0.04 K.) The 0.03-K

total uncenainty consists of approximately 0.025-K. systematic error and 0.005-K random error. The syste- matic error was determined from the uncenainties in the calibration of the thermometer and in the potentio- metric measuring system.

The temperature of the cell is regulated by balancing

coarsely variable cooling with precisely controlled

electric heating. A control heater near the bottom of the

cell is connected to a dc power regulator, which is part of a measuring/regulation system that also includes a

six-dial microvolt potentiometer and a microvolt

amplifier. A second heater at the top of the cell is con- nected to a manual power supply. Independent heaters

at the top and bottom of the cell are used to minimize

temperature gradients along the length of the cell.

1244 Rev. Scl. Inttrum.. Vol. 47, No. 10, October 1976 Denaimeter 1244

2b
Temperature gradients are detected by monitoring the vapor pressure of the liquid in the vapor bulb. Pressures below 7 bars are measured with a 0- to 7-bar spiral quartz Bourdon gauge. This gauge has a

resolution of less than 3.5 x iO""* bar. It has been cali- brated against an air dead-weight gauge giving an uncertainty of approximately 0.1% over the full range. For

pressures above 7 bars, a double-revolution 0- to 20-bar Bourdon gauge is used. It has also been calibrated

against the air dead-weight gauge and can be read to the nearest 0.005 bar. Vapor pressure measurements have been used in the present work primarily as a means of monitoring

temperature gradients. They have also been used as a check against the calibration of the platinum resistance thermometer. In general, the vapor pressures for methane have been consistently lower than those of Prydz and Goodwin.'^ This pressure difference cor- responds to a temperature difference of 0.01-0.02 K. The discrepancies between the nitrogen vapor pressures of the present work and those of Wagner''' correspond

to an average temperature difference of less than 0.015 K. with no particular trend observed.

B. Position of buoy

The accuracy and precision of the density measure- ments depend on being able to suspend the float at the same position relative to the support coils in vacuum

and in the test liquid at the same temperature. A

125 X filar micrometer microscope is used to determine

the position of the float. It has a resolution of ap- proximately 5 X 10"" mm. The maximum error in the

position determination is 2 x 10~^ mm. which cor- responds to an error in density of less than 0.03%.

This error includes approximately equal contributions from the alignment of the microscope, so that the "apparent" ' position of the float is independent of the index of refraction of the fluid inside the sample ceil, and from the repositioning of the float for vacuum and

liquid measurements. The first contribution would be a systematic error while the second is random. The preci- sion of the density measurements depends primarily on

the reproducibility of the position of the buoy from vacuum to liquid measurements. As mentioned above, the microscope must be aligned

so that the "apparent" position of the float does net depend on the index of refraction of the fluid inside the sample cell. In viewing the float inside the cell one must

look through the glass tail of the cryostat and the sample holder window. The "apparent" position of the

float would be dependent on the refractive index of the

fluid inside the cell unless the light rays are perpendicular

to the interface between the test liquid and the inside surface of the sample holder window.

The procedure for adjusting or aligning the micro- scope is as follows. First, the float is positioned at rest on the ceramic holder at the same horizontal

position as when in suppon. Then a fiducial mark on the

1245 R«v. Sci. Instrum., Vol. 47. No. 10, Octobvr 1976

float is observed under two experimental conditions,

i.e., with the float first immersed in gas and then im- mersed in liquid. Both methane and nitrogen were used

to carry out this procedure. During this procedure

the microscope tilt is adjusted so that the "apparent"

position of the float does not depend on the refractive index of the fluid inside the cell within the resolution of the microscope. The temperature of the cell is controlled to better than 0.01 K during these observations. Care is taken to ascertain that the float is ob- served through the same parts of the windows as during

the density measurements. Necessarily a different part of the float is observed. This presents no special problem

since there are diamond tool marks the entire length of

the float that can be used as reference lines. This

procedure has been repeated frequently during the course of the reported measurements. It has been found

that a readjustment of the microscope tilt is necessary

only if the apparatus is perturbed by disassembly.

C. Volume of buoy

Absolute density measurements [Eq. (7)] with a magnetic suspension densimeter require a determination of

the volume of the magnetic buoy. In the past this has been accomplished either by using distilled water as a reference fluid of known density'^ or by making direct length and diameter measurements on a uniformly constructed float.' Here the volume has been determined

by the first method.

The volume (0.13485 cm' at 300 K) of the magnetic

buoy was determined within 0.02%. The distilled water was vacuum distilled to remove air. Any problems with bubbles inside the cell were rectified by pressurizing the water with helium gas. The volume of the buoy has been measured three times at 300 K and once at 290 K. These four measurements gave a standard deviation of 0.005% using thermal expansion data for barium ferrite to calculate the volume change from 290 to 300 K.

This resulted in a 99% confidence interval of rO.015%.

The systematic error in the volume determination de- pends on the uncertainty in the density of water. This should be less than 0.005%. '« The present paper repons density data for cryogenic

fluids. Thermal expansion data for barium ferrite were

used to calculate the volume of the float at iou temperatures. Recently the linear thermal expansion

of polycrystalline barium ferrite was measured at this laboratory at temperatures from 76 to 293 K with a quartz tube dilatometer." The volume of the float at 100 K is approximately 0.4% less than the room temperature value. Barium ferrite is somewhat anisotropic and its thermal expansion was measured both parallel and perpendicular

to the magnetization direction. An anisotropy of 15-20%

was observed. The estimated overall uncertainty of the thermal expansion measurements would correspond to an error of 6% in the adjustment to the volume.

This would produce an uncertainty of approximately

Oonalmotar 1245

<Ll
0.02% in the volume at 120 K. The uncertainty in the adjustment to the volume due to the contraction of

the copper plating is negligible.

The effect of pressure on the volume of the float

is negligible at the highest pressures encountered in the present work. The bulk modulus for barium ferrite should be less than that for copper, which is 1.35 X 10" Pa.'* This value would correspond to a volume

correction of approximately 0.001% at 2.5 MPa, the vapor pressure of nitrogen at 120 K.

D. Mass of buoy

The mass of the magnetic buoy was determined with

a 10-g capacity equal-arm microbaiance. A calibrated

class M weight set was used in the weighings. Pre- cautions were taken to assure that the copper plating

and gold protective coating provided an impervious

barrier to liquids under pressure. The buoy was rapidly

cycled between room temperature and 76 K. It was then immersed in liquid ethanol at 5-atm pressure and room

temperature for 4 h. The buoy was weighed twice after removal from the ethanol bath. The average of these weighings was 9 ftg higher than the mean of the initial four weighings. A significantly larger increase in the mass was expected if the plating on the float contained a

pinhole that allowed ethanol to penetrate into the porous

ceramic. Thus, at room temperature, the metallic coating appeared to be impervious after thermal cycling.

The reproducibility of liquid methane density measure- ments (see experimental results) after many cycles be- tween room temperature and 100- 140 K is further evidence of the resistance of the metallic coatings to damage from thermal cycling.

The final mass of the magnetic buoy (0.73706 g) was taken as the mean of all weighings, including the weighings after immersion in ethanol. corrected for air buoyancy. A 99% confidence interval for the six weighings, based on a standard deviation of 6 ^g. was ± 10

fig. The systematic error in the mass determination,

resulting primarily from the weight set cahbration, is believed to be less than 5 /ig.

V. EVALUATION OF PERFORMANCE OF

MAGNETIC SUSPENSION DENSIMETER

A. Performance tests The use of Eq. (7) for obtaining absolute densities has again been verified experimentally'. The gradient

coil current has been measured as a function of the main coil current over a range of approximately 20%

of the maximum main coil current. These measurements

were carried out while supporting the float in a given

fluid (including vacuum) at constant height at a fixed temperature. The various pairs of and Iq were

fitted by the method of least squares to Eq. (7). The

residual standard deviation, with the gradient coil current as the dependent variable, was less than 10"' A. Examination of the residuals showed that they were

1246 Rev. Sci. Instrum., Vol. 47, No. 10, October 1976

random and well approximated by a normal probability

distribution. These results demonstrated that within the precision of the current measurements the magnetic

moment of the barium ferrite float is independent of the magnetic field intensity over the range of fields (0.006- 0.016 T) considered in the present work.

Other coil configurations have been tried that gave re- sults that substantiated the observations given above.

First the bottom gradient coil was disconnected re- sulting in a two-coil system. The values of the constant (O in Eq. (7) for the two- and three-coil arrange- ments under otherwise identical experimental conditions were in agreement within the experimental uncertainty.

In other words, the magnetic moment of the float (contained in C) remains constant for significantly differ- ent magnetic field intensities. Next the pair of approximately identical (gradient)

coils were connected to the constant current source in a Helmholtz arrangement. The main coil was connected

into the servosystem. The float was first supported at a constant height at fixed temperature by the force supplied by only the main coil. The float was located

at the midpoint of the Helmholtz coils while the current in the Helmholtz coils was increased to 0.6 A. There was an insignificant change in the main coil current

(lO"* A) for this relatively enormous change in the magnetic field intensity (about an order of magnitude more than for the typical data point). The results of these

tests demonstrated that this technique could be used for absolute density measurements.

In carrying out density measurements with this tech- nique, the position of the main coil must remain fixed over a wide range of currents for the vacuum and liquid measurements. This means that the "constant" k' in Eq. (4) and the constants. C and D, in Eq. (7) must

be independent of the current in the coil. It was hoped

that the measures exercised in controHing the temperature and position of the main coil (see Sec. Ill D)

would result in such behavior. It was experimentally

impracticable to determine if k' was a constant by

monitoring the position of the coil. However, the effect of a change in the current in the main coil on the density measurements could be obtained in another manner. As shown in Sec. II the fluid density is linearly

proportional to the current in the gradient coils if the main coil current is held constant. Not only is the position of the main coil fixed, but also the magnetic field intensity at the midpoint of the gradient coils (float position) is constant. Relative measurements [Eq. (5)] carried out with vacuum, nitrogen, methane, and

propane at 110 K were not consistent with the density

ratios computed from absolute measurements [Eq. (7)] for the same fluids. These results showed that the "constants" {k'. C, and D) vary slightly with the main

coil current. This variation produces a density error

relative to the difference between main coil currents

for vacuum and liquid measurements (constant Ig) of approximately 1.4%/A. This corresponds to a density

adjustment of 0. l-0.2%/(g/cm^) to the results from Eq.

Densimeter 1246

Z8
Table 11. Estimated uncertainties in density measurements.

Percent error in density

Metliane Nitrogen Normal butane

Systematic errors 105 K 160 K 95 K 120 K 135 K 300 K

Mass ot tioat (oec. iv u) ±U.UU^ Volume of float at 300 K (Sec. IV C) ±0.02 ±0.02 ±0.02 ±0.02 ±0.02 ±0.02

Thermaiexpansioncoefficientofbariumferrite(Sec. IV C) ±0.022 ±0.017 ±0.023 ±0.021 ±0.020 ±0.001

Position of float (Sec. IV B) ±0.015 ±0.018 ±0.010 ±0.012 ±0.009 ±0.009

Position of main coil, determined from relative measurements. Eq. (5) (Sec. V A) -0.02 ±0.02 ±0.02 ±0.02 ±0.02 ±0.02 Temperature of sample fluid (Sec. IV A) ±0.010 ±0.021 ±0.023 ±0.076 ±0.004 ±0.008

Three times standard deviation ±0.045 ±0.054 ±0.030 ±0.036 ±0.027 ±0.027

Total uncenainty" ±0.09 ±0.10 ±0.07 ±0.12 ±0.06 ±0.06

" The total uncertainty was determined from the square root of the sum of the squares of the systematic errors added to three times the standard deviation.

relative measurements [Eq. (5)]. The density difference between the relative and absolute measurements, ex- pressed in terms of the difference in main coil currents between the vacuum and liquid points il, - If) for a constant gradient coil current, is independent of temperature. The temperature dependence of the magnetic

moment of the float must be included if the difference

is expressed in terms of density.

The relative measurements were carried out using the same experimental setup and parameters (mass, volume,

and position) as the absolute measurements. Thus the

total uncertainty in the adjustment determination should

only include the random error in the measurement

process. Thus an estimated uncertainty of ±0.02% in the adjustment is based on the comparisons with the refer- ence fluids since this density difference was larger than a 99% confidence interval from a fit of the differences between relative and absolute measurements.

B. Error analysis

Evaluation of the experimental parameters involved in the present work showed that the uncertainty in the density measurements depends primarily on the uncertainties in the determination of the volume of the

float, the relative position of the float and the main

coil, and the temperature of the sample fluid. The

precision of the density measurements is inversely

proportional to the difference between the density of the

fluid and the density of the float. This variation with

density cannot be observed for the measurements re- ported in this paper because of the density range covered. The precision also varies with temperature, corresponding to the change in the magnetic moment of the float with temperature. Table II summarizes the uncertainties in the density measurements for the nitrogen and methane results presented in this paper. Normal butane is also included

in this table, primarily to show the performance of the instrument at higher temperatures. Normal butane

densities, determined with this instrument at temperaDensimeter 1247

29

I

(7). Since all measurements are based on the volume de- termination using distilled water as a reference fluid of known density, the adjustment must be applied relative

to this point.

A further check on this density-dependent adjust- ment was accomplished by measuring the densities of two other fluids of known density at room temperature.

The densities of liquid samples of normal pentane

and ethanol were determined by the Mass and Volume

Section of the Mechanics Division of the National Bureau of Standards in Washington, DC, to an estimated

inaccuracy of less than 0.01%. Measurements on these

fluids at 300 K with the present apparatus using Eq.

(7), combined with the adjustment based on the relative density measurements at low temperature, gave

densities that agreed with the reference values to better than 0.02%. Thus, the densities presented in this paper

based on absolute measurements [Eq. (7)] have been

adjusted slightly to be consistent with the results of

Table III. Experimental results for saturated liquid nitrogen (molecular weight = 28.0134).

T lOCKpexp - ptalc) (K) (mol/n (mol/l)

95.000 25.6731 25.6755 -0.009

100,000 24.6400 24.6391 0.004

105.000 23.4998 23.4988 0.005

110.000 22.2079 22.2071 0.004

115.000 20.6719 20.6748 -0.014

120.000 18.6818 18.6856 -0.020

100.000 24.6361 24.6391 -0.012

105.000 23.4%3 23.4988 -0.01

1

110.000 22.2044 22.2071 -0.012

115.000 20.6787 20.6748 0.019

120.010 18.6800 18.6808 -0.004

95.075 25.6624 25.6605 0.007

100.075 24.6207 24.6229 -0.009

100.075 24.6279 24.6229 0.020

105.075 23.4841 23.4807 0.015

110.075 22.1840 22.1862 -0.010

115.075 20.6490 20.6492 -0.001

117.575 19.7356 19.7326 0.015

120.075 18.6525 18.6497 0.015

1247 Rev. Scl. Instrum., Vol. 47, No. 10, October 1976
tures between 135 and 300 K, will be presented in a forthcoming paper. The standard deviation given in the table for methane

at 105 K was based on twelve measurements. The

other standard deviations were computed relative to the density and temperature of this point. Standard deviations at the other temperatures and densities could not be estimated from statistical analysis since sufficient repetitive measurements were not taken for these points. Measurements on more dense fluids, such as argon and

krypton, have demonstrated the expected variation of

the precision with density. Measurements near room

temperature, especially those on distilled water, have

exhibited the expected improvement in the precision as the temperature is increased. The total uncertainty of a single density measure- ment was taken as the square root of the sum of the squares of the systematic errors plus an allowance of

three times the standard deviation for random error.

Vi. EXPERIMENTAL RESULTS

A. Nitrogen

The experimental saturated liquid densities for nitrogen are presented as a function of temperature

(IPTS 1968) in Table III. The explanation for some of

the nitrogen points not being at integral temperatures

involves the presence of a temperature gradient along

the length of the cell in some of the initial measure- ments with this apparatus. Each of the data points was

taken from a new charge of nitrogen. For each

datum point the vacuum measurements were carried out immediately before or after the liquid measurements. The same procedure was used for methane. The three data sets for nitrogen over the temperature range of 95- or 100-120 K were obtained at approximately

six-month intervals. Of the two data sets at integral

temperatures, one was taken with methane, and the other with nitrogen, in the vapor bulb. The third set was taken with no liquid in the vapor bulb. (For the 120.010-K point a slight temperature gradient was applied to remove bubbles from the sample liquid inside the

cell.) The nitrogen samples were taken from commercially

available, research grade gas. A purity specification of 99.99 mol% was given by the supplier. The gas has been analyzed chromatographically with a thermal conductivity detector and found to be within the specified

purity. The nitrogen gas was passed through a room

temperature molecular sieve trap, primarily for re- moval of water. The saturated liquid densities (p) of nitrogen have

been fitted as a function of temperature (7") by the method of least squares to an equation of the following form:

Table IV. Parameters of Eq. (9) for nitrogen.

a = 19.39217=' (0.11)" b3 = 23.32977 (4.8) Pr = 11.21 mol/l

b, = 26.01408 (2.5) 7", = 126.20 K a = 0.016%

t,„ = -39.497.S9 (6.5)

° These coefficients (a. 6,. fco, i>3) and those in Table VI were obtained from a least-squares program in which the experimental densities in g/cm' to five digits were convened to molar densities within the program.

" Standard errors of coefficients rn parentheses.

pc and Tr are selected values of the critical density

and temperature and a. b, are coefficients determined by

least squares. Equation (9) consists of a scaling law

modification'* to a generalized Guggenheim equation.^"

The parameters of Eq. (9) for nitrogen, along with the

residual standard deviation (cr) of the fit and the standard

errors of the coefficients, are given in Table IV. The

standard errors of the calculated densities in Table III are not presented since they are significantly smaller than the standard deviation. The main purpose of the above expression is to facilitate comparisons with in- dependent measurements. The choice of the critical point parameters is arbitrary in the sense that the critical point density can be varied by as much as 5% and the

critical point temperature by 0.2% without changing the

fit of the data within the estimated precision of the present measurements. For nitrogen the critical point values were obtained from the PVT data compilation

of Jacobscn^'.

In Fig. 6 independent experimental data available

in the literature are compared with the expression

representing the present nitrogen data over the temperature range of the present measurements. The present data are also shown. The experimental temperatures

0.6p

0.5 -

0.4 -

0 3 -

0.2 -

0.1 -

1 0 -

a.-0.1 -

-0.2 -

-0.3 -

-0.4 -

-0.5 -

-0.5 —

100 105 110 TEMPERATURE. K

J

\0.35 ^

I T \'+''-l"3 p- p, = a |1 - -j + I

^. (l -_ )

. (9)

7-c

Fig. 6. Deviation plot of experimental densities of saturated liquid nitrogen compared with values calculated from Eq. (9) using parameters from Table IV: O present data. 3 Brauns.^^ C Weber." A Rodosevich and Miller.^' • Goldman and Scrzise.^' Terry et al." O Streett and Staveley," Mathias ei at}*

1248 Rev. Sci. Instrum., Vol. 47, No. 10, October 1976 Densimeter 1248

3U
Table V. Results for saturated liquid methane (molecular weight = 16.04303).

Standard Temperature Mean density deviation Number of (K) (mol/l) (%) data points

105.000 26.9458 0.015 12 110.000 26.4985 0.019 11 115.000 26.0443 0.014 12 120.000 25.5721 0.016 17 125.000 25.0845 0.016 18 130.000 24.5775 0.017 16 135.000 24.0540 0.016 23

140.000 23.5067 0.018 5 145.000 22.9312 0.014 5 150.000 22.3218 2 160.000 20.9876 2

of other measurements have not been adjusted to the temperature scale of the present results. The data of Brauns," Weber,^^ and Rodosevich and

Miller.-'' each obtained with a technique different from

that used here, fall within 0.15% of the present results. The densities of Goldman and Scrase^^ and Terry er

al.-^ are consistently lower than the present results by

approximately 0.3-0.4%. This pattern is also followed

for methane by Terry et al.. and preliminary analysis of recent results for argon and krypton obtained with

the magnetic suspension densimeter indicate comparisons with Terry et al., for these fluids exhibit the same

behavior.

B. Methane

Methane has been used as a control fluid throughout the entirety of a project that has included

density measurements on six pure fluids and, at least, thirty mixtures of these components. As a result, well over one hundred data points have been accumulated

for saturated liquid methane, primarily in the temperature range 105- 150 K. Although these experimental

data are not presented in this paper, the results of an

analysis of the methane data are helpfiil in evaluating

the performance of the magnetic suspension densimeter

used in the present work. More than ten data points were obtained at each 5-K interval between 105 and

135 K. The mean of the experimental densities, along

with the standard deviation and the number of points

at each temperature, are given in Table V. (The

standard deviations are not given at 150 and 160 K since there are only two data points at each of these temperatures.) Each of the data points represents a new

methane scunple. The vapor bulb contained liquid

methane for all points.

Table VI. Parameters of Eq. (9) for methane.

a = 18.65812

6, = 6.712030

bi = -0.9472020

(0.035)"

(0.21)

(0.20)

Pc

190.555 K

10.16 mol/l 0.016%

The liquid samples were obtained from either of two

cylinders of different batches of commercial, research grade methane. The purity for each cylinder as specified

by the supplier is 99.99 mol% minimum. Analyses in this laboratory substantiated these specifications. As with

nitrogen the methane gas was passed through a room

temperature molecular sieve trap to remove moisture

and any heavy contaminants not detected by analysis.

The 123 experimental data points have been fitted by

the method of least squares to Eq. (9). Only three

coefficients were needed to fit the methane data within

the precision of the measurements. The fourth term in the equation was not statistically significant. The coefficients of Eq. (9), the standard errors of the coefficients, the residual standard deviation (cr) of the fit, and a selected critical density^** and temperature^" are given in Table VI for methane. Equation (9), with the

coefficients for methane, has been used to compare the present results with experimental data from independent

investigators. The methane data from the present work

are not shown on the deviation plot because of the large

number of points. All of the experimental densities of methane obtained in the present work will be pre- sented in tabular form in a future publication. For saturated liquid methane there have been considerably more measurements than for saturated liquid

nitrogen. All of the other measurements were obtained

with techniques different from that of the present work.

Some of the older, less precise data have been included

in Fig. 7 for the sake of completeness. It is satisfying

that the relatively recent data of Orrit and Olives,^'

McClune,^^ Goodwin and Prydz,^^ and Shana a and

l.2l1.0^

0.81-

0.6 lj 0.4^

0 2

s °

-0.2

-0.4 |-D

-0.6

6>

-0.8

-1.21-L

" Standard errors of coefficients in parentheses.

no 120 130 140 ISO 160 TEMPERATURE. K

Fig. 7. Deviation plot of experimental densities of saturated liquid methane compared with values calculated from Eq. (9) usmg

parameters from Table VI: Terry et al..-^ Q Orrit and Olives." A McClune.^ C Goodwin and Prydz." ® Shana a and Canfield.^ T Vennix,'* ® Gngor.^" V Davenport et al.." Klosek and

McKiniev.'* © Sinor and Kurata,^" A Jensen and Kurata." A Bloomer

and Parent,-" O Fuks et al.," • Keyes et al..*' Moran." O Van

Itterbeek ei al."

1249 Rev. Scl. Instrum., Vol. 47, No. 10, Octobsr 1976 Densimeter 1249

I
Canfield^'' agree with the present results within 0.1%.

However, as mentioned earlier, the densities of Terry

et al.^^ are systematically lower than the present results by 0.3-0.4%. The extrapolated densities of Vennix^^ at approximately 150 and 160 K are 0.4% larger than the present results although the discrepancy at 140 K is ap- proximately 0.01%.

C. Other fluids To demonstrate the performance of the magnetic

suspension densimeter at low temperature, this paper

presents only a representative cross section of the data

that have been taken in the present work. Data have

recently been obtained for other pure fluids (ethane, propane, isobutane. normal butane, argon, and krypton) over a density range extending to 2.3 g/cm^ (krypton)

and over a temperature range extending to 300 K (iso- butane and normal butane). These data will be published

in the near future. ACKNOWLEDGMENTS

The authors would like to express their appreciation

to the following individuals who have made significant contributions to the development of this apparatus.

M. R. Cines of Phillips Petroleum Co., Chairman of the LNG Density Project Steering Committee, was instru- mental in the development of support for this project.

W. G. Layne, D. L. Smith, and A. N DiSalvo as- sisted in the design and fabrication of the apparatus.

J. E. Cruz designed some of the electronics associated with the refrigeration and temperature control systems.

J. R. Whetstone determined the densities of liquid

samples of normal pentane and ethanol. which were used

as reference fluids. P. V. Tryon assisted with the

statistical analysis of the data.

* This work was carried out at the National Bureau of Standards under the sponsorship of British Gas Corp., Chicago Bridge and

Iron Co.. Columbia Gas Service Corp.. Distrigas Corp., Easco Gas LNG. Inc., El Paso Natural Gas. Gaz de France, Marathon Oil Co. . Mobil R and D Corp.. Natural Gas Pipeline Co..

Phillips Petroleum Co.. Shell International Gas, Ltd.. Sonatrach, Southern California Gas Co.. Tennessee Gas Pipeline. Texas

Eastern Transmission Co.. Tokyo Gas Co., Ltd.. and Transcontinental Gas Pipe Line Corp.. through a grant administered by the .\merican Gas Association. Inc.

' W. .M. Havnes and J. W. Stewart, Rev. Sci. Instrum. 42,

1 142 (1971).

- J. W. Beams. Rev. Sci. Instrum. 40, 167 (l%9).

R. M. McClintock and M. J. Hiza. Mod. Plast. 35, 172 (1958).

' The use of a trade name for clarity and completeness does not

signify an endorsement or recommendation by the National Bureau of Standards.

' J, B. Breazeale. Ph.D. thesis. University of Virginia (1955),

' H. S, Monon. Jr.. Ph.D. thesis. University of Virginia (1953).

' i. P. Senter. Rev. Sci. Instrum. 40. 334 (1%9).

« J. W. Beams, C. W. Hulburt, W. E, Lotz, Jr.. and R. M,

Montague. Jr.. Rev. Sci. instrum. 26. 1181 (1955),

' S, P. Almeida and T. H, Crouch, Rev, Sci. Instrum,

42. 1344 (1971).

'° M, G. Hodgins and J, W, Beams, Rev, Sci, Instrum,

42, 1455 (1971). " J. L, Hales, J, Phys, E 3, 855 (1970), " S, C, Greer, M, R, Moldover. and R. Hocken. Rev„ Sci, Instrum, 45, 1462 (1974). " R, Prydz and R, D, Goodwin. J, Chem, Thermodyn, 4, 127

(1972).

'• W, Wagner, Cryogenics 13, 470 (1973),

J, W. Beams and A. M, Clarke, Rev. Sci, Instrum, 33, 750 (l%2), M, Menache and G. Girard. Metrologia 9, 62 (1973). " A. F. Clark. W, M, Haynes, V, A. Deason. and R. J, Trapani. Cryogenics 16, 267 (1976), G. Simmons and H. Wang. Single Crystal Elastic Constants and Calculated Aggregate Properties: A Handbook (MIT

Press. Cambridge, MA, 1971), p, 181, " R. D. McCarty, Natl. Bur, Stand, (U,S,) Intern, Rept, 74-357

(1974), " Y. C. Hou and J. J. Martin. AIChE J. 5, 125 (1959). R. T. Jacobsen. Ph.D. thesis, Washington State University

(1972). ^ P. Brauns (private communication, 1973). " L. A. Weber. J. Chem. Thermodyn. 2, 839 (1970). " J. B. Rodosevich and R. C. Miller. AIChE J. 19, 729 (1973). " K, Goldman and N. G. Scrase, Physica 44, 555 (1%9). M. J. Terry, J. T. Lynch, M. Bunclark, K. R. Mancell, and L. A, K. Staveley, J, Chem, Thermodyn, 1, 413 (1969), W. B. Streelt and L. A, K, Staveley, Adv. Cryog, Eng, 13, 363 (1968), ^ E. Mathias. H, K. Onnes, and C, A, Crommelin, Commun.

Leiden 145C, 19 (1914). " J. D. Olson, J, Chem, Phys. S3, 474 (1975). R, D. Goodwin, Natl. Bur. Stand. (U.S.) Tech, Note No.

653 (1974), " J. Orrit and J. Olives, distributed at 4th International Conference

on Liquefied Natural Gas, Algeria ( 1974), ^^C. R. McClune, Cryogenics 16, 289 (1976), " R, D. Goodwin and R. Prvdz. J. Res. Natl. Bur. Std. (U.S.) 76A, 81 (1972), " M, Y. Shana a and F. B. Canfield, Trans. Faradav Soc, 64, 2281 (1968). A, J, Vennix, Ph.D. thesis. Rice University (1%5). A. F, Grigor, Ph.D. thesis, Pennsylvania State University (1966). A, J, Davenport, J. S. Rowlinson, and G. Saville, Trans, Faraday Soc. 62, 322 il%6).

J. Klosek and C. McKinley, Proceedings of the First International Conference on LNG, Chicago (1968), Paper 22, " J. E. Sinor and F. Kurata, J. Chem. Eng. Data 11, 1 ( 1%6),

R. H. Jensen and F. Kurata, J. Petrol. Technol. 21, 683 (1969),

^' O, T, Bloomer and J. D. Parent, Chem. Eng, Progr, Symp,

Ser. 49, 1 1 (1952).

"2 S. Fuks, J. C. Legros, and A. Bellemans. Phvsica 31, 606

(1%5). " F, G. Keyes, R, S. Tavlor. and L. B. Smith. J. Math. Phvs,

1, 21 1 ( 1922). " D. W. Moran. Ph.D. thesis. Imperial College, University of London (1959). " A. Van Itterbeek, O, Verbeke, and K. Staes, Physica 29, 742 (1%3),

r

1250 Rev. Sci. Instrum., Vol. 47, No. 10. October 1976 Densimeter 1250

6Z
6.2 Reference [9]

Simplified magnetic suspension densimeter for absolute

density measurements*

W. M. Haynes

National Bureau of Standards, Institute for Basic Standards, Boulder, Colorado 80302

(Received 24 September 1976) A magnetic suspension densimeter, incorporating three support coils, has been reduced to a system using only one coil. This simplifies considerably the design of the apparatus and

the procedures involved in the measurements. This instrument can be used for absolute

density measurements; i.e., it does not have to be calibrated with reference fluids of known

density.

During the past two decades the magnetic suspension

densimeter'"'^ has evolved into a versatile research instrument that has been used routinely for liquid density measurements to a precision of ten parts per million. (Reference 14 presents a detailed review of the principles, applications, and development of the magnetic

densimeter.) In the original design (Fig. 1) of the instru- ment a piece of soft ferromagnetic material is suspended

in a fluid of unknown density by the magnetic force produced by a single solenoid. The magnetic force on

the buoy is given by the relationdH

Hz

(1)

where M is the magnetic moment of the buoy, H is the

axial magnetic field intensity of the air core solenoid,

and Z is the distance along the vertical. The density

(p) of the fluid is related to the current (/) in the support

coil by

p = A + BP (2) where A and B are constants to be determined by cali- bration of the instrument with reference fluids of known

density. The above relation is based on the assumption

that the magnetic moment of the buoy is linearly proportional to the magnetic field intensity.

It is not unusual for the magnetic susceptibility of the magnetically soft material of the buoy to vary slowly

with H. Thus a relatively large number of reference

fluids are required to calibrate a one-coil instrument to maintain a given precision over its entire operating

range. The precision is inversely proportional to the

difference between the density of the fluid and the density of the buoy. To cover an extended density range

it is necessary to construct and calibrate buoys of dif- ferent densities. The calibration of the instrument must

be checked periodicEilly to test the buoy material for

hysteresis.

These procedural tasks were reduced considerably

with the development of a three-coil support system^

(Fig. 2) which allows variations mdHldZ while is held

constant. The three-coil arrangement consists of a main

coil, which supplies the major part of the force necessary

to support the buoy, and a pair of gradient coils con- nected in such a way that at the buoy position their magnetic fields cancel and their magnetic field gradients add. The magnetic moment of the buoy is held constant;

it depends only on the stability of the constant current source that supplies the main coil current. In the first applications of the three-coil artangement (with a magnetically soft buoy) the instrument still had to be cali- brated but now the density range for a given buoy was

increased significantly over that for a single-coil system.

Also the calibration was simplified since the density was

linearly proportional to the curtent in the gradient coils for a constant main coil current. The buoy may be

either magnetically soft or hard for the three-coil system

.

Recently, in the adaptation of the magnetic suspension

densimeter for use at low temperatures,"''^ the same

three-coil arrangement (Fig. 2) was employed to deter- mine the force on a magnetic material in a nonuniform

field; i.e., it was used to determine the variation of the magnetic susceptibility of the buoy material with H over

the range of magnetic fields needed for the density measurements. It was found that the magnetic

force on a barium ferrite buoy could be represented by

the expression

dH„ ^ dHc

dZ dZ

(3)

where the subscripts denote the main and gradient coils.

SUPPORT

COIL^

IHmicroscopeTj 4^

SENSOR

COILS

Fig. 1 . Schematic diagram of a magnetic suspension densimeter with one support coil.

39 Rev. Scl. Instrum., Vol. 48, No. 1, January 1977 Copyright © 1977 American Institute of Ptiyslcs 39

33
This result demonstrated that, over the range of magnetic fields considered (0.006-0.016 T), there was no in- duced moment in the barium ferrite buoy (a magnetically hard materieil with a permanent moment). Therefore, it . was possible to determine fluid densities without the use

of calibration fluids. The instrument constants [C, D in Eq. (3)] could be determined from measurements of the

gradient coil current as a function of the main coil current while the buoy was supported at a fixed position at constant temperature in vacuum. The mass and volume of

the buoy must be determined from independent measurements. Now that it had been determined with the three-coil arrangement (Fig. 2) that the magnetic moment of the barium ferrite buoy was independent of the magnetic

field strength over the range of magnetic fields needed

for the desired density measurements, the instrument

could be simplified considerably by reducing to a onecoil system (Fig. 1). The magnetic force on the buoy de- pended only on the magnetic field gradient (dH/dZ) and

not on the magnetic field intensity (H). Thus the single

(main) coil was connected into the servosystem and the gradient coils were eliminated.

It should be noted that a one-coil system can be used

for absolute density measurements as long as the varia- tion in the magnetic moment of the buoy with magnetic

field intensity is known. This information could be obtained by techniques other than that used in the present

work, e.g., a vibrating sample magnetometer. This is mentioned to emphasize that, with a knowledge of M(H)

of the magnetic buoy material, the initial apparatus

design could be simplified significantly with the use of

only one coil. To carry out density measurements with the one-coil

systerfl with a barium ferrite buoy, first the current necessary to support the buoy in vacuum ( /, ) at a given

position and temperature must be determined. Then the current necessary to support the buoy in a liquid (If) of unknown density at the same position and temperature is measured. The density of the fluid is related to

this current ratio by

(4)

where m and V are the mass and volume of the buoy.

It should be emphasized that no reference fluids are required for calibration of the one-coil system (or the

three-coil system) used in the present work. However,

the volume of the float may best be determined near room temperature using distilled water as a reference

fluid of known density and, in fact, this procedure was

used in the present work.'' There are several obvious advantages of the onecoil system over one using three coils; most are in- volved with the experimental design and procedures.

With elimination of the two gradient coils and the align- ment problems associated with them, the design and

construction of the apparatus are much simpler. The

constant current source is no longer needed. The

CONSTANT

CURRENT

SOURCE

MAIN 7 COIL

GRADIENT

COILS L iqMICROSCOPEl=l \ SENSOR

\ COILS

MAGNETIC BUOY

CONTROL

CIRCUIT

Fig. 2. Schematic diagram of a magnetic suspension densimeter with

three support coils.

vacuum points for a single-coil system require only a

single current measurement, while those for a three-coil system require a series of measurements of l^, and Iq while holding the buoy at a fixed position. Then the currents must be fitted by the method of least squares to

. Eq. (3) to obtain the instrument constants. Also the

error analysis is less complicated with a one-coil

s'ystem. There is one inherent disadvantage in the method for obtaining absolute densities with either the one-coil or

three-coil systems where the magnetic force depends

only on dH/dZ. The change in dH/dZ with position is significantly larger than the change in the product of H and dHjdZ with position. Thus a very precise deter- mination ( lO"'' mm) of the separation distance of the main

coil and buoy was required in the present work and has been described in Ref. 17. Experimental densities have been determined for saturated liquid argon and methane at temperatures be- tween 100 and 120 K with both the three-coil and one-coil systems. The results agreed within the imprecision of

the measurements, which was approximately 0.015%

for methane and 0.005% for argon. Thus the simpler

one-coil system is now utilized for all density measure- ments with the present instrument. The servosystem that is used with the single coil was

originally connected to the gradient coils in the threecoil system. No modifications to the servosystem as described in Ref. 17 were required for its use with the

single coil. The present series of density measurements in this laboratory with the magnetic suspension densimeter

represented a nontypical use of the instrument. In

the past the instrument has been used for highly precise

(usually less than 0.001%) measurements over a short density range at a single temperature. Here a relatively

modest precision (less than 0.02%) was the result of

40 Rev. Scl. Instnim., Vol. 48, No. 1, January 1977 Magnetic suspension densimeter 40

34
measurements over large density (0.3-2.3 g/cm^) and

temperature (95-300 K) ranges with a single buoy.

The total uncertainty of the present measurements was

approximately 0.1%.

In order to obtain the highest precision and accuracy

with the present instrument it was necessary to perform vacuum measurements immediately before or after the

liquid measurements. The possible use of the instrument

without repeating the vacuum measurements for each

datum point has been investigated. In other words the

precision and accuracy of the instrument has been

estimated, based on a single series of vacuum measure- ments as a function of temperature at the beginning

(and/or end) of a set of density measurements without

regard to the temperature and magnetic field cycling of

the buoy. The separation distance of the coil and the buoy must be maintained throughout the set of measurements.

In the present setup the position of the main coil was

fixed as described in Ref. 17. The position of the float was determined with a 125x microscope and the buoy

could be returned to a desired position either electronically or mechanically. Over a period of several months during the course of density measurements at low temperatures on several different fluids, the sample

cell was cycled between approximately 100 to 130 K

and 300 K more than twenty times. The standard deviation of the vacuum measurements at low temperatures was of the order of 0.05% with no systematic trend observed. This was a measure of the thermal hysteresis of the magnetic moment of the barium ferrite buoy over

rather large temperature cycles. This result demonstrated that densities with a total uncertainty of approximately 0.2% could be expected without repeating

vacuum measurements for each datum point. Thus, for density data in which the highest precision and accuracy

are not required, the use of a one-coil system and a barium ferrite buoy would result in a relatively simple

and efficient measurement process that requires few vacuum measurements.

* This work was carried out at the National Bureau of Standards under the sponsorship of British Gas Corp., Chicago Bridge and

Iron Co., Columbia Gas Service Corp., Distrigas Corp.. Easco Gas LNG, Inc.. El Paso Natural Gas, Gaz de France. Marathon

Oil Co., Mobil R and D Corp., Natural Gas Pipeline Co., Phillips Petroleum Co., Shell International Gas. Ltd., Sonatrach, Southern

California Gas Co., Tennessee Gas Pipeline, Texas Eastern Transmission Co.. Tokyo Gas Co., Ltd., and Transcontinental Gas Pipe Line Corp., through a grant administered by the American Gas Association, Inc.

' J. W. Beams. C. W. Hulburt, W. E. Lotz, Jr., and R. M. Montague,

Jr., Rev. Sci. Instrum. 26, 1181 (1955).

' J. W. Beams and A. M. Clarke, Rev. Sci. Instrum. 33. 750 (1962).

^ A. M. Clarke, D. W. Kupke. and J. W. Beams, J. Phys. Chem. 67. 929 (1%3).

* D. V. Ulrich. D. W. Kupke. and J. W. Beams, Proc. Nat. Acad.

Sci. U.S.A. 52, 349 (1964).

' J. W. Beams. Rev. Sci. Instrum. 40, 167 (1%9).

« J. P. Senter, Rev. Sci. Instrum. 40, 334 (1969).

' P. F. Fahev, D. W. Kupke, and J. W. Beams, Proc. Nat. Acad. Sci. U.S.A. 63, 548 (1969).

* A. M. Wims, D. Mclntvre, and F. Hynne, J. Chem. Phys. 50, 616 (1%9).

» R. Goodrich, D. F. Swinehart, M. J. Kellv, and F. J. Reithel, Anal. Biochem. 28, 25 (l%9).

J. L. Hales. J. Phys. E 3, 855 ( 1970). W. M. Haynes and J. W. Stewart, Rev. Sci. Instrum. 42, 1 142 (1971). " S. P. Almeida and T. H. Crouch, Rev. Sci. Instrum. 42, 1344 (1971). M.G. Hodginsand J. W. Beams, Rev. Sci. Instrum. 42. 1455(1971). " D. W. Kupke and J. W. Beams, Methods Enzymol. 26. 74 (1973). " S. C. Greer. M. R. Moldover, and R. Hocken, Rev. Sci. Instrum.

45, 1462 ( 1974).

'» S. C. Greer. T. E. Block, and C. M. Knobler, Phys. Rev. Lett. 34, 250 (1975). " W. M. Haynes, M. J. Hiza, and N. V. Frederick, Rev. Sci. Instrum.

47, 1237 (1976).

41 Rev. Sci. Instrum., Vol. 48, No. 1, January 1977 Magnetic suspension densimeter 41

6.3 Reference [b]

JOURNAL OF RESEARCH of the National Bureau of Standards

Vol. 88, No. 4, July-August 1983

Apparatus for Density and Dielectric Constant Measurements

to 35 MPa on Fluids of Cryogenic Interest

W. M. Haynes and N. V. Frederick

National Bureau of Standards, Boulder, CO 80303

Accepted: March 11, 1983

An apparatus has been develojjed for simultaneous measurements of fluid densities and dielectric constants

at temperatures from 70 to 320 K and at pressures to 35 MPa. A magnetic suspension technique, based on an

application of Archimedes' principle, is employed in the density determination, while a concentric cylinder

capacitor is used for obtaining the dielectric constant data. The apparatus can be used not only for determining densities and dielectric constants of compressed gases and liquids (including mixtures), but for saturated liquid and vapor properties as well. Also included is the capability for acquiring liquid-vap>or equilibrium data for mixtures. The total uncertainty of a single density measurement is estimated to be

approximately 0.1% for densities as low as 50 kg/m'; at lower densities, the uncertainty increases. The

imprecision of the density data is typically less than 0.02%. The total uncertainty in the dielectric constants is approximately 0.01%. Experimental data for a 0.85 CH4 + O.15 CjH^ mixture are given here to demonstrate

the performance of the apparatus.

Key words: Clausius-Mossotti function; compressed fluid; concentric cylinder capacitor; density; dielectric constant; excess volume; magnetic suspension densimeter; methane-ethane mixture; saturated liquid; vapor

pressure.

1. Introduction

A magnetic suspension densimeter [1,2]' was used in

a large-scale program to measure the orthobaric liquid

densities of the major components [1,3,4] of liquefied

natural gas (LNG) and mixtures [5-9] of these

components. This technique, based on an application

of Archimedes' principle, was selected for the LNG

density project for several reasons:

1) It is capable of absolute density measurements of

high accuracy and precision over wide ranges of

' Figures in brackets indicate literature references at the end of

this paper.

density, temperature, and pressure.

2) Calibration fluids are not required.

3) The technique can be used not only to measure

densities of compressed fluids, but also to measure

liquid and vapor densities along the coexistence

boundary.

The measurements for the LNG density project were concentrated in the temperature range from

100-140 K at pressures typically less than 0.2 MPa.

The apparatus was designed for a maximum pressure of 5 MPa. Near the end of the LNG density project,

the pressure range of the densimeter was expanded to

at least 35 MPa. The expansion was because of a need

to map the PVT surfaces of fluids with critical points

significantly above ambient temperature. (A gas expansion technique [10-13] used at this laboratory for PVT measurements on cryogenic fluids could not be

used for this application.) The expansion of the

densimeter's pressure capability resulted in a new

instrument, described here, significantly different from

and more versatile than the previous one [1] developed

specifically for the LNG density project.

241

.About the Authors, Paper: W, M. Ha\nes is with

NBS' Chemical Engineering Science Dixision. and

N. V, Frederick with its Electromagnetic Technology

Dixision. The work reported on was carried out at NBS under the sponsorship of the American Gas

.Association. Inc. and the National Aeronautics and

Space Administration.

37
Although many of the components of the new

densimeter changed, the technique used is the same. To detect the position of the magnetic buoy, a Hnear

differential capacitance sensor [12], compatible with

the higher pressure environment, has been developed

to replace an inductance sensor. Although the

properties of the buoy had been well characterized in the previous work [1], it was necessary to determine

the effect of pressure on its volume. A new support

coil for lifting the buoy and a new microscope lens combination for determining the position of the buoy

were also required in adapting the technique to higher

pressures. Since the dielectric constant of a fluid is closely

related to its density through the Clausius-Mossotti

function, a concentric cylinder capacitor was added

inside the sample cell to enable simultaneous

measurements of dielectric constant and density on the same fluid samples. Dielectric constant measurements

can serve as simple and reliable substitutes for density

measurements. The addition of the capacitor was

motivated to some extent by the fact that some

commercial densimeters being developed for custody

transfer applications in LNG transactions include

devices based on capacitance measurements.

The new apparatus incorporates a cryostat design

different from that employed with the previous

densimeter, but similar to ones used with other

instruments [10-13] in this laboratory. The cryostat is suitable for continuous temperature control between

70 and 320 K. A new high-pressure window design

developed for the equilibrium cell allows the position

of the buoy to be determined by optical means. With

the new cell and cryostat, it was not possible to change the position of the buoy by mechanical means;

this is now accomplished electronically.

The apparatus can also be used for liquid-vapor

equilibrium measurements on mixtures. Means have

been provided for mixing (recirculation of vapor

through liquid), sampling of vapor and liquid, and

subsequent composition analysis. Compared to the

previous densimeter [1], improvements have been

made in the characterization of the temperature,

pressure, and composition of the fluid samples. With

the previous apparatus [1], it was possible to observe

the liquid-vapor interface through a window that extended most of the length of the sample space. In adapting the magnetic suspension densimeter for pressures to 35 MPa over a wide temperature range, it was not practical to retain this feature. This feature,

although convenient, was not essential for any of the

measurements performed with the new apparatus.

The new apparatus was first used to complete

measurements for the LNG density project [7,9]. Extensive tests were made to ensure that the density

results obtained with the new apparatus were in agreement with those from the previous apparatus.

The consistency of the density data was one of the

most important considerations in the development of

mathematical models [15-18] for predictions of

mixture (LNG) densities. The apparatus was then used

to measure the densities and dielectric constants of

liquid propane [19-20], isobutane [19,21], and normal

butane [19,22] over temperature ranges from their

triple points to 300 K at pressures to 35 MPa.

2. Measurement Methods

2.1 Density

In the magnetic suspension densimeter used, a piece

of magnetic material (barium ferrite magnetic buoy in the shape of a right circular cylinder magnetized along

its cylindrical axis) is suspended freely by the force

produced from the axial magnetic field of a single air- core solenoid. The vertical motion of the magnetic

buoy is controlled by the automatic regulation of the

solenoid current with a closed-loop servocircuit that includes a differential capacitance sensor to detect the

position of the buoy. (The horizontal position of the

buoy is maintained by the axially symmetrical,

diverging field of the solenoid.) In the present system,

the magnetic buoy is more dense than the fluids in which it is suspended. Thus, an upward magnetic

force is added to the buoyant force to balance the

downward gravitational force. In earlier work [2] with a densimeter that utilized a three-solenoid arrangement to supply the magnetic

force, it was found that the magnetic moment of a barium ferrite buoy was independent of magnetic field intensity over the range of fields (0.006-0.016 T)

needed to support the buoy. Barium ferrite is a magnetically hard material with a permanent moment.

This meant that a one-coil system could be used to determine fluid densities without the need of

calibration fluids. To carry out density measurements

with a one-coil system and a barium ferrite buoy, first the current (7^,) necessary to support the buoy in vacuum at a given position (buoy-coil separation

distance) and temperature is measured. Then the current (If) necessary to support the buoy in a fluid of

unknown density at the same position and temperature

is determined. The density (p) of the fluid is related to these currents by the relation,

p=7<i-f). (1)

242
where m and V are the mass and volume of the buoy.

Measurement procedures for using this equation to determine fluid densities for the instrument developed

in the present work are discussed later. Also presented

is a detailed description of the magnetic suspension

system.

2.2 Dielectric Constant

A stable concentric cylinder capacitor was used for

dielectric constant measurements. First, the capacitance (Q with the fluid (of unknown dielectric

constant) between the cylindrical electrodes is measured. Then, at the same temperature, the vacuum

capacitance (CJ is determined. The dielectric constant

(f) is calculated from the relation,

€ = C/Q. (2)

3. Apparatus

3.1 Cryostat

The major features of the cryostat are shown to

scale in figure 1, an assembly drawing of the apparatus. This cryostat is similar to some used previously at

this laboratory [10-13]. Those modifications necessary

to adapt the cryostat for use with a magnetic suspension densimeter will be emphasized.

The cryostat was supported by a 1.3-cm-thick

aluminum plate suspended from concrete block

columns at a height 2.3 m above floor level. The

aluminum plate was reinforced with 10-cm-wide

stainless steel channel beams so that the position of the

cryostat was independent of the amount of liquid

nitrogen in the reservoir. Apparent changes in the

buoy-coil separation distance resulted if the position of

the cryostat was not maintained during the course of

measurements. The aluminum plate from which the

cryostat was suspended could be leveled and clamped

in place using four bolts in contact with the concrete

block columns.

The cryostat was fabricated from nonmagnetic

materials. The inner cylinders or cans (liquid nitrogen

reservoir, cold ring, guard ring, shield, cold wall) were copper; the outer vacuum jacket and access

tubes were primarily stainless steel; and the flanges,

support plates, and fittings were mostly brass. The

various cylindrical portions of the cryostat and the

central support (reflux) tube had to be aligned

(concentric and vertical) so the buoy could be

suspended symmetrically about the cylindrical axis of

the differential capacitance sensor. Many of the

dimensions of the cryostat comp>onents were

determined by the dimensions^ of the high-pressure cell and the support coil. The outside diameter of the

stainless steel vacuum jacket is 20.3 cm while the

outside diameter of the glass tail of the cryostat is 9.0 cm. There is 0.4-cm clearance on the diameter

between the cryostat tail and the support coil. The

Access Tubes for Capillaries. Coaxial Cables and Wires

LNj Fill

Vapor Pressure Bulb Copper

Sample Holder

Sample Fluid

Support Coil

Platinum Resistance Thermometer

Differential Capacitance Sensor Magnetic Buoy

Quartz Leg

Vacuum

Metallic Bellows

LN2 Reservoir

Reflux Tube ^ —Wood's Metal Joint Cold Ring

Guard Ring

Vacuum Jacket

Cold Wall

Shield

Capacitor

Cooling Coils

^ Vacuum Window

la

Microscope

High Pressure Window

Vapor Pressure Bulb

Glass Tail

Figure 1-Assembly drawing (approximately to scale) of an apparatus

for density and dielectric constant measurements to 35 MPa on

cryogenic fluids.

243

39
overall length of the cryostat is approximately 94 cm. The liquid nitrogen reservoir has a capacity of

approximately 7 x lO' cm' and needs refilling under

normal conditions about every 16 h. Metallic bellows were placed in the fill and' vent lines of the reservoir

to eliminate relative movement, as a function of

temperature, between the cell and the outer stainless

steel jacket of the cryostat. This step was necessary to maintain alignment of the capacitance sensor and buoy

located inside the cell with the support coil located

outside the cryostat, independent of the experimental

conditions. Five access tubes, which passed through

the liquid nitrogen reservoir, were available for introducing capillaries, coaxial cables, and wires into

the vacuum space and cell inside the cryostat.

The cryostat was comprised of one large vacuum

space; holes (3 cm diameter) were cut into the

electroformed copper cylinders attached to the liquid

nitrogen reservoir and the guard ring for visual

observation of the buoy. With this arrangement it was

not feasible to use exchange gas for fast-cooling the

cell. The exchange gas would have been in direct contact with the outer stainless steel can which

included a glass tail as its lower section. The glass tail was connected to the outer jacket through a stainless

steel-to-glass transition joint; to ensure integrity of this

seal, the joint should be kept at a temperature near

ambient.

Fast-cooling of the cell was accomplished using the

refiuxing action of nitrogen in the central support

tube. The maximum cooling rate was approximately

50 K/h. The reflux tube is evacuated during

measurements when the cell is controlled at a desired

temperature. For additional cooling, three flexible copper braided straps (not shown in fig. 1), each with

a cross sectional area of 5 mm^ per strap, have been

connected symmetrically from the top of the cold

shield to the cell. The cryostat was designed to facilitate ease in assembly and disassembly and to minimize alignment

problems of critical components. All demountable

joints had enough latitude to make alignment

reasonably straightforward. The outer vacuum jacket was attached to the top of the cryostat through a rubber o-ring seal using a split-ring assembly not shown in the drawing. The inner cans were attached

with either Wood's metal or screws. Once the joints were secured, the entire structure was rigid, and no

problems with alignment were encountered after numerous temperature cycles. A nut and bolt

(fiberglass) arrangement was installed between the

cold shield and the bottom of the cell to eliminate

pendulous vibrations of the cell.

3.2 Equilibrium Sample Cell

Figure 2 is a detailed cross sectional diagram of the

copper equilibrium cell. The overall dimensions of the

cell used here are roughly the same as for the cell used

with the earlier version [1] of the magnetic suspension

densimeter, except for an increase in the wall thickness

required for use at higher pressures. The new cell has an overall length of approximately 29.5 cm, primarily

determined by the position at which the buoy must be

suspended relative to the diverging field of the support

coil for horizontal stability of the buoy. The outside

diameter of the main part of the cell is 4.44 cm, while

Capillaries Reflux Tube

Coaxial Cables

Vapor Pressure Bulb

Copper 0-Ring

Outer Electrode

Inner Electrode

Capacitor

Grounded Guard

Copper Sample Holder

Support Tube

Platinum Resistance Thermometer

Magnetic Buoy

Differential Capacitance Sensor

Capillary

Sapphire Window

Indium Gasket

Vapor Pressure i

Figure 2-The equilibrium cell.

244

4U
the top closure section has a diameter of 8.10 cm. The

dimensions of the internal sample space are 1.90 cm

diameter by approximately 23 cm length. About one

third of the internal volume is occupied by the

concentric cylinder capacitor and the magnetic buoycapacitance sensor assembly. This results in an internal

free volume of approximately 43 cm\ as determined

by filling the cell with water. To facilitate easy access to the assemblies inside the

cell, a flange-gasketed seal was used to close the cell. The seal was effected by compressing a silver-plated,

solid copper O-ring between a stainless steel plug and

a smooth, flat ledge of the copper cell. This assembly was similar to that used earlier [1]. A thin-walled

stainless steel tube, used to support the cell, was

soldered into the cell lid (the stainless steel closure

plug). This support tube was also used for reflux gas as described earlier. For visual observation of the magnetic buoy

position, a new window assembly for pressures to 35

MPa has been developed. The assembly consists of a

sapphire disk (1.90 cm diameter by 0.63 cm thickness)

with chamfered edges, indium gasket, asbestos-rubber

backing ring to relieve thermal and mechanical strains,

and stainless steel plug and ring with four stainless

steel set screws to provide a pressure- and vacuumtight seal. It has a 0.63 cm diameter viewing area. An

effort was made to minimize the size of the high

pressure window assembly, especially in the horizontal

direction. This effort resulted from an attempt to minimize the inner diameter of the support coil so

current and power in the coil were not excessively

large.

A total of four capillaries has been introduced into

the sample space. Two capillaries are soldered into

collars in the cell lid; the other two are soldered into

the cell wall with their ends terminating at the bottom

of the sample space. This makes it possible to fill the

cell from either the top or bottom, or from both

simultaneously. Filling procedures can be critical when condensing liquid mixtures into the cell. Under

normal conditions, a pair of capillaries, one at the top

and one at the bottom, is available for recirculating

the vapor through the liquid for mixing; the other capillary at the top is for pressure measurements while the

free capillary at the bottom can be used for liquid

samphng. By having capillaries at both ends of the

cell, it is easy to remove low vapor pressure liquids

I from the cell. At approximately the same height at which the

|| magnetic buoy is suspended, a platinum resistance

1 thermometer has been soldered with indium into a

Ij

closed-end copper tube that has been soft-soldered into

I

-

a groove cut lengthwise in the cell wall. Vapor

pressure bulbs fabricated from copp>er (each with an

internal volume of approximately 2 cm^) have been

installed at the ends of the cell in close thermal

contact (soft-solder) with the outside surface. These

bulbs, along with differential thermocouples (chromelconstantan) secured at the same positions, are used to monitor temperature differences between the ends of

the cell. Four independent heaters of 110 to 160 Cl each for temperature control have been wound

bifilarly at different positions along the length of the

cell.A total of six coaxial cables enter through the lid of

the cell into the sample space in pressure-tight

assemblies. Three of the cables are for the capacitance

sensor, two for the cylindrical capacitor, and one is a

spare. Each coaxial cable consists of an outer stainless

steel sheath of 0.51 mm diameter insulated from a 0.13 mm diameter inner conductor with polytetrafluoroethylene. These cables have been found to be leak- tight along short lengths at gas pressures to 70 MPa at room temperature. Cables approximately 1 meter in length have been inserted into stainless steel capillaries

(1.07 mm outside diameter X 0.66 mm inside diameter)

that extend from inside the sample cell to a position

above the top of the cryostat where the coaxial cables are soldered into the capillaries. This means that continuous lengths of the coaxial cables extend from

inside the sample cell to a region in which the cables

are leak-tight. At low temperatures, the cables do not

seal because of the relatively large difference in the

thermal expansion coefficients of stainless steel and

polytetrafluoroethylene. The stainless steel capillaries are soldered into collars in the lid of the cell and at the

top of the cryostat.

3.3 Magnetic Suspension System

The magnetic suspension system, employed here in the density determination, uses some components from

the previous densimeter [1]. The magnetic buoy, with

its well-characterized properties deduced from the

earlier work, has been described in detail. The buoy is a barium ferrite magnet in the shape of a right circular

cylinder (0.51 cm diameter X 0.64 cm length) magnetized along its cylindrical axis. Barium ferrite is a magnetically hard, ceramic material with a density of ap- proximately 5x10^ kg/m^ Since barium ferrite is porous, the buoy was plated with copper to a thickness of approximately 0.06 mm. A thin (10"' mm)

protective coating of gold was flashed over the

copper. No problems were encountered with fluids penetrating the copper layer at pressures to 35 MPa.

245

41
Barium ferrite was selected as the buoy material since

its magnetic properties are consistent with absolute

density measurements [2]. Over the range of magnetic

fields needed to support the buoy, the magnetic mo- ment of the barium ferrite buoy had been found to be

independent of the magnetic field intensity.

The densimeter described here included a single

solenoid that supplied the force required to lift the

buoy. The simplification in the densimeter design in going from a three-coil to a one-coil system, which

resulted from a determination of the magnetic

properties of barium ferrite, has been discussed in detail in an earlier paper [2]. (Although no further use of a three-coil arrangement with gradient coils was

anticipated, the outside diameter of the new support

coil was made less than the inside diameter of the

gradient coils used in the previous work [1] in the

eventuality that a need for tests on new buoy materials

ensued.)

The new support coil is composed of two separate

coils of 2275 turns each of epoxy-coated aluminum foil of approximately 0.025 mm thickness and 2.5 cm

width. Each of the two coils, epoxied to a central

quartz tube at a separation distance of 1.3 cm, has an

inside diameter of 10.16 cm and an outside diameter of

roughly 22.2 cm. To be compatible with the new

higher-pressure cell, the inside diameter of the new

coil is somewhat larger than that used previously [1]. The quartz tube is attached to a fiberglass plate

supported by three quartz rods (2.54 cm diameter) that extend to another fiberglass plate resting on a part of

the concrete block structure, used for rigid support of

the entire apparatus. Water-cooled copper plates,

insulated with thin mylar sheets, have been placed in close thermal contact with all faces of the two coils. No problems have been encountered with this arrangement for mounting and cooling the coils for use at currents to 1.5 A, which corresponds to 270 W

heat dissipation in the coils. The servocircuit is essentially the same as that used

before [1] except for a new type of sensor for detecting the position of the buoy. Some problems had

been encountered with the earlier densimeter [1] with

pressurized fluid slowly penetrating into the coil windings of an inductance sensor, resulting in a slow

change in the position of the suspended buoy. Thus, a

differential capacitance sensor [13], a solid monolithic

structure, has been developed for position detection in the present work that entails measurements in fluids at pressures to 35 MPa. The sensitivity of the

capacitance sensor was at least equivalent to that of

the linear differential transformer used previously. By

balancing a bridge of which the sensor is a part, the

position of the buoy can be made insensitive to the

dielectric constant of the sample fluid. For a detailed

description of the capacitance sensor, see reference

[13].The previous densimeter [1] included mechanical means for fine adjustment of the position of the buoy

in going from vacuum-to-liquid measurements. The

new system does not incorporate this feature. Small

changes in the buoy position are accomplished

electronically in the new system by adjusting the

offset voltage of the integrator amplifier in the

servocircuit. A calibrated 0.5-fl standard resistor,

placed in series with the support coil, can handle the

relatively large currents needed to lift the buoy in the

present work. The measurement of the voltage drop

across this resistor, which is immersed in an oil bath, enters directly into the density determination.

A 125x filar micrometer microscope is used to determine the position of the buoy. A new lens combination, which includes an objective lens (38 mm)

with a larger working distance (less power) and a higher-power (14x) eyepiece, gives approximately the same magnification as with the earlier apparatus [1]. The microscope is rigidly mounted on an aluminum

support table, adjustable in three dimensions. The

entire assembly is secured firmly with bolts to the

massive concrete block structure.

3.4 Concentric Cylinder Capacitor

The capacitor for dielectric constant measurements was located in the top portion of the sample space, as seen in figure 2. The central support mandrel of the

capacitor was connected rigidly to the sample cell lid using a slotted brass tube, crimped at its ends, that clamped tightly over mating surfaces on the mandrel

and the lid. The same type of brass tube was used to connect the differential capacitance sensor to a nut at the bottom of the capacitor. The use of the brass tubes

for support of the components inside the cell resulted

in an extremely rigid assembly that provided means

for independent rotational orientation of each

component. The coaxial leads to the differential capacitance sensor located below the capacitor passed

through the brass tubes and a slot in the central

support mandrel of the capacitor.

The concentric cylinder capacitor design used in the

present work was based on that developed by

Younglove and Straty [23]. Two slightly different

capacitors are used with the present apparatus,

depending on whether the measurements are for mixtures or for pure fluids. For mixture measurements,

slots are cut into the cylinders parallel to the

246
cylindrical axes, similar to the ring and bar design of

Pan et al. [24]. This modification was made to minimize the chances for composition gradients in the

cell by allowing free passage of fluid mixtures between

the electrodes. The vacuum capacitance of the

capacitor with slotted cylinders was approximately 20

pF, while that for the one with solid cylinders was

about 33 pF.

The dimensions of the capacitors used here were

proportionally the same as those of Younglove and

Straty [23]; however, the overall size was significantly

smaller. The overall external dimensions of each

capacitor were 6.4 cm length X 1.77 cm diameter. For

the capacitor with slots, the widths of the five slots in each cylinder were 0.32 cm, while the slot lengths in the outer cylinder were 4. 1 cm and those in the inner

cylinder were 1.9 cm. The outside diameter of the

inner cylinder was 1.06 cm. The outer cylinder had a

thickness of 0.32 cm, while that of the inner cylinder was 0.24 cm. There was a separation distance of 0.38

mm between the cylindrical electrodes. Small pieces

of Kapton^ film (0.05 mm thickness) were used to

insulate the electrodes from the support assembly.

Both of the capacitors were fabricated from copper. A

thin protective coating of gold was flashed over the

surfaces.

4. Measurements

4.1 Temperature and Pressure The primary temperature sensor, calibrated on the

IPTS-68, is a platinum resistance thermometer, which

had been used with the previous densimeter [1]. The

uncertainty of the calibration is approximately 0.002

K. The potentiometric system for temperature

measurements gives uncertainties that range from

approximately 0.010 K at 100 K to 0.030 K at 300 K.

The temperature of the sample space is regulated

within a few mK, approximately the same as the

reproducibility of the temperature measurements. A

current of 1 mA for the thermometer, supplied by an

electronic constant current source, is determined to an

uncertainty of approximately 0.002%. The total uncertainty in the measured temperatures is estimated

to be less than 0.03 K.

' In order to describe wterials and experimental procedures

adequately, it is occasionally necessary to identify commercial

products by manufacturers' or trade names. In no instance does such

identification imply endorsement by the National Bureau of

Standards, nor does it imply that the particular product is necessarily the best available for that purpose.

Vapor pressure measurements on liquid nitrogen,

methane, ethane, and propane at temperatures from

100-290 K have been used to check the calibration of

the thermometer. Temperatures from vapor pressure measurements, using selected data from the literature,

generally agreed with measured temperatures

(platinum resistance thermometer) to better than 0.02 K. Further details on temperature measurements are presented in reference [1]. Temperature differences between the ends of the

cell are monitored with vapor pressure bulbs located

at the ends of the cell. The bulbs have been filled with

the fluids mentioned above, the selected fluid depending on the temperature range needed. With the

reflux tube evacuated, overall temperature differences were typically less than 0.01 K, or within the precision

of the vapor pressure measurements. Thus, a

differential thermocouple, with junctions placed near

the vapor pressure bulbs, was not needed as part of a control loop to regulate the temperature distribution

along the length of the cell. A third vapor pressure bulb was soldered in close

thermal contact with the guard ring. This vapor

pressure bulb was used to check a differential thermocouple between the guard ring and the cell. This thermocouple was part of a control circuit to maintain the temperature of the guard ring

approximately equal to that of the sample cell. The

guard ring was connected to a copper radiation shield

that surrounded the cell to provide an approximately

isothermal environment for the sample space. The techniques and instrumentation for control of

the temperature of the cell are standard. Four

independent heaters along the length of the cell are

available, if needed, to minimize temperature

gradients. Only the middle two heaters have been used

in the control circuit. It should also be noted that temperature gradients along the length of the cell can be monitored by observing the vapor pressure of the

liquid inside the cell as a function of the liquid level. Pressures of the fluid under test are usually

measured with a dual-range, precision oil dead-weight

gauge. Its sensitivity ranges from 2x10^ MPa at 3 MPa to 2X10"^ MPa at 35 MPa. The overall

uncertainty in pressure is approximately 0.01%,

increasing somewhat at lower pressure. With the high

range piston, this gauge cannot be used for pressures

less than 0.2 MPa; with the low range piston, the

lowest pressure is approximately 0.04 MPa.

A spiral quartz Bourdon-tube gauge with a range of

0-1.38 MPa is normally used for vapor pressure measurements. It has been calibrated against an air dead-weight gauge; maximum uncertainty in the

247

43
calibration was 70 Pa. The resolution of this gauge is better than 20 Pa. Both pressure gauges are characterized by relatively

small free volumes, approximately 0.5 cm' for each.

This is an important consideration when performing

mixture (phase equilibria) measurements with this apparatus, e.g., to minimize vapor space corrections.

4.2 Density

The accuracy and precision of density measure- ments with the magnetic suspension technique used

here depend on a knowledge of the mass and volume

of the buoy, along with the capability to determine the

position of the buoy with high resolution independent

of the medium in which the buoy is suspended. The

same barium ferrite buoy as used with the previous

densimeter was employed here; its properties have

been well characterized and are discussed elsewhere

[1]. The mass of the buoy is 0.73706 g and its volume

at 300 K is 0.13485 cm\ Thermal expansion data [25] for barium ferrite were previously obtained to calculate the volume of the buoy at low temperatures. The

change in the volume of the buoy for a temperature

change from 100-300 K is approximately 0.4%.

The earlier densimeter was used at relatively low

pressures compared to those used now. Based on the

properties of similar materials, it was estimated that the effect of pressure on the volume of the barium

ferrite buoy was negligible (<0.001%) for the highest

pressures (2.5 MPa) encountered with the previous

densimeter. Since the new apparatus is used at pressures up to 35 MPa, bulk modulus measurements

[26] have been carried out on a sample of barium

ferrite, the buoy material. Bulk modulus (Bj) data (in units of MPa) from 75-295 K are represented as a function of temperature {T in units of K) by the

expression,

Br= 6295^.01 629-

0.0206014

175.15 r-75

-1 )

(3)

where B2„= 1.3030 x 10' MPa, the bulk modulus at 295 K. Coefficients were determined from nonlinear

least squares. The correction to the volume of the

buoy at room temperature for a pressure of 35 MPa

amounts to 0.027%.

For density measurements, it is necessary to suspend

the buoy at the same position relative to the support

coil in vacuum and in the test fluid at the same temperature. Although the microscope lens combination (sec. 3.3) used here is different from that of the previous

densimeter, the present arrangement results in position

measurements of the same quality. The maximum

error in the position determination is 2 X 10'' mm,

which corresponds to an error in density of less than

0.03% for a density of 5 X 10^ kg/m\ or larger.

The position of the buoy is observed through a 0.63 cm thick sapphire window in the cell. It has been observed that the apparent position of the buoy changes

slowly with the pressure inside the cell; as the cell volume expands, the angle of the sapphire window

changes. The magnitude of this effect has been deter- mined as follows. First, the microscope is adjusted so

that the position of the buoy is independent of the

index of refraction of the fluid inside the cell. This

procedure is accomplished by observing the buoy

resting at a stationary position on the capacitance

sensor assembly as the cell is alternately filled with gas and liquid at a pressure slightly greater than 0.1 MPa;

e.g., methane at 120 K. The temperature of the cell is maintained during these tests. Then, the cell is filled with liquid at a pressure less than 0.1 MPa; either

propane, isobutane, or normal butane have proved

ideal for these tests at temperatures between 100 and

300 K. Then, the pressure inside the cell is increased

to greater than 35 MPa while observing the buoy at rest on the sensor assembly. The change in the index

of refraction for a 35-MPa change in pressure along an

isotherm for liquid propane, isobutane, or normal

butane is negligible compared to the change in refractive index in going from vapor-to-liquid for methane at 120 K. The change in position with pres- sure corresponds to a change in density with pressure of less than 5x 10"^ kg/(m'-MPa) for liquid methane at 120 K.

The currents in the support coil needed to suspend

the buoy in vacuum and in the fluid of interest (see eq

(1)) are determined by measuring the voltage drops

across a 0.5-fl standard resistor in series with the

support coil. The voltages were routinely measured to 5x10 * V with a high resolution differential-type

voltmeter, which corresponds to a change in density

of less than 0.01% for a density of 5 X 10^ kg/m\

4.3 Dielectric Constant

The dielectric constant is determined from a measurement of the ratio of the capacitance of the

concentric cylinder capacitor with fluid between the

electrodes to the capacitance under vacuum. The

capacitances are measured with a three-terminal ac

248

44
bridge operated at an oscillator frequency of 5 kHz.

Measurements to a resolution of 10"* can be obtained

with little difficulty with this bridge.

Both capacitors used in this work yielded equal

results within the precision of the measurements in

tests on liquid methane. Measurements on mixtures are

inherently more difficult and are generally characterized by greater scatter because of the difficulties associ- ated with obtaining homogeneous liquid mixtures in the sample space. Examples of dielectric constant measurements on pure fluids and mixtures with the apparatus described here are given in references [7,19-22,27].

Measurements of the vacuum capacitance are normally obtained just before or just after fluid measurements are performed. Since vacuum measurements

are required for each run in the density measuring

technique used here, there was no reason vacuum ca- pacitances should not be recorded at the same time.

Thus, there was no reason to acquire a calibration curve representing the vacuum capacitance as a function of temperature. Vacuum measurements can be

made to a precision of 10^ pF and are usually stable within 0.0002 pF when the sample cell is cycled

between low temperature and room temperature.

(Vacuum capacitances have been observed to be stable

to better than 10"^ pF for more than 20 temperature

cycles over a period of several months.) The total change in the vacuum capacitance from 300-100 K is approximately 0.3%. Based on tests by Younglove and

Straty [23] on a similar cylindrical capacitor, the effect of pressure on the capacitor was sufficiently small to

neglect. It is estimated that, based on the resolution of

the capacitance measurements and the stability of the

capacitor design, the total uncertainty in the dielectric constant measurements is approximately 0.01%.

5. Performance of Apparatus

5.1 Results and Discussion

Before completing measurements for the LNG

density project, it was necessary to ensure that the new densimeter yielded results consistent with those

from the previous densimeter. First, density data were

obtained for several components of LNG (e.g., methane, ethane, etc.); the data obtained with the new

apparatus agreed to better than 0.02% with the results

1 1,3,4] determined at the onset of the LNG density

project with the earlier version [1] of the magnetic

suspension densimeter. As mentioned earlier,

|| measurements on saturated liquid methane were used

as a check on the measurement process during the

I entirety of the LNG density project. Dielectric

constant data for liquid methane exhibited differences

of <0.01% when compared with the data of Straty

and Goodwin [28]. Next, data were obtained for a binary mixture of

methane and ethane, a system that had been

extensively investigated with the earlier densimeter.

The data for three mixtures of methane and ethane

had been used to optimize several mathematical

models [15-18] developed for prediction of LNG

densities. The most accurate and versatile of these

models was the extended corresponding states method

[15-17]. The new data for a methane + ethane mixture

would be compared with predictions from this model.

The mixture, for which data are reported here, had

also been used to cross-check results from this laborato- ry obtained with the magnetic suspension densimeter

with those from another laboratory that employed an

entirely different technique for determining density

[29]. The experimental density of Miller and Hiza [29] for this methane + ethane mixture at 110.08 K differed

by 0.01% from that calculated from the extended cor- responding states model [16]. The experimental orthobaric liquid densities, vapor

pressures, and dielectric constants of a 0.85147 CH4 +

0.14853 C2H6 mixture are presented as a function of

temperature in table 1. Excess volumes and values for the Clausius-Mossotti (CM) function, as well as values

for the excess function, are also given in table 1. The

excess volume (V^) is defined by the relation,

V^=V- J^x.V.ll+P.iPrP)]^ (4)

where V is the molar volume of the mixture at a given

temperature at saturation pressure p, V, is the molar

volume of component / at the same temperature at sat- uration pressure p, , x, is the mole fraction of component /, and /3, is the isothermal compressibility of component /. The CM function is defined by the

expression,

where p is the density and € is the dielectric constant. Then the excess Clausius-Mossotti function (CM^) for a liquid mixture is defined, analogous to V^, by the relation,

CM^ = CM-2;x,CM,, (6)

249

4b
Table 1. Orthobaric liquid densities (p) and dielectric constants (£) of 0.85147 CH4 + 0.14853 C^H,, mixture

(molecular weight = 18.1265 g mol ') as a function of temperature (D and pressure (P). PcsM- density calculated from extended corresponding states model; V^, excess volume; CM, Clausius-Mossotti function; CM^, excess Clausius-Mossotti function.

T

K

P

MPa

P __ CM CM^

mol-dm"' PCSM cm'-mol' cm^-mol"' cm'-mol'

115.00 0.116 25.3618 -0.059 -0.410 1.67297 7.2243 0.005

120.00 0.166 24.9911 -0.013 -0.493 1.66070 7.2220 0.000

125.00 0.232 24.5983 -0.045 -0.567 1.64828 7.2239 -0.001 130.00 0.316 24.1950 -0.036 -0.655 1.63561 7.2258 -0.003

where CM refers to the Clausius-Mossotti function of

the mixture at a given temperature at the saturation pressure of the mixture, and CM, is the ClausiusMossotti function of pure component / at the same

temperature and pressure as the mixture. Adjustments

of the pure component CM,'s to the saturation pressure of the mixture are sufficiently small to

neglect. In the calculation of and CM, , the pure component molar volumes of methane and ethane were calculated from equations in references [1,3] obtained from

fitting experimental orthobaric liquid densities deter- mined with the earlier version of the magnetic suspension densimeter used in the LNG density project.

Vapor pressures for methane and ethane were taken

from Goodwin [30] and from Goodwin at al. [31], re- spectively. Isothermal compressibilities for methane were taken from Rowlinson [32]; those for ethane

from Miller [33]. The dielectric constants of Straty and

Goodwin [28] were used for methane, while those of

Weber [34] were used for ethane.

Also presented in table 1 are comparisons between

the experimental densities (p„pt) from this work and

densities (Pcsm) calculated from the extended

corresponding states model [15-17]. The average

absolute deviation is 0.038%. This result, combined

with the pure fluid comparisons, demonstrates that data obtained with the new densimeter are consistent

with data taken with the previous instrument. The

total uncertainty of a single density measurement for these binary mixture data is estimated to be ±0.1%.

The present results are also consistent within

experimental error with the data point of Miller and

Hiza [29] for the same mixture.

The calculated excess Clausius-Mossotti values were

less than 0.07% of the mixture CM values for the

methane 4- ethane mixture data presented here. This

result was expected based on dielectric constant and

density measurements on other mixtures containing

nonpolar constituents [7,9,24]. It appears that the excess CM values show a slow decrease with

increasing temperature.

After the new apparatus had been thoroughly

tested, it was used to complete the measurements for the LNG density project [7,9]. Next, the apparatus was used to obtain data for propane [19-20], isobutane

[19,21], and normal butane [19,22] at pressures up to 35

MPa. The performance of the apparatus at high

pressures is demonstrated by these pure fluid data.

5.2 Error Analysis

Detailed discussions of the systematic and random,

errors involved in measurements with the magnetic

suspension densimeter used in the present work have

been presented elsewhere [1,3-9, 19-22]. The

uncertainty in the density measurements depends

primarily on the uncertainties in the determination of

the volume of the buoy, of the relative position of the

buoy and the support coil, and of the temperature of

the sample fluid. With the new apparatus, designed for higher pressures than the previous one, the effect of

pressure on the apparent position of the buoy,

resulting from slight movement of the cell window,

must now be included. Maximum uncertainty in the

density determination resulting from this effect is 0.02%.

The effect of pressure on the volume of the buoy,

which is extremely small at maximum design pressure of the cell, must also be considered. As discussed in section 3.3, the reduction in the buoy volume for a pressure of 35 MPa is 0.027%. The uncertainty

involved in this adjustment is negligible.

As discussed [1], the total uncertainty of a single

density measurement is taken as the square root of the sum of the squares of the systematic errors plus an allowance of three times the standard deviation for random error. The imprecision of measurement (or

standard deviation) is typically less than 0.02%. Both

the precision and accuracy of density measurements

with a magnetic suspension densimeter depend on the

difference between the density of the buoy and the

250

4b
density of the fluid. For the present arrangement,

where the density of the buoy is significantly larger

(typically an order of magnitude) than the density of

the fluids, the precision and total uncertainty change

slowly with fluid density. The total systematic error in the measurement process from known sources is approximately 0.05% at low temperatures, decreasing

to approximately 0.03% at room temperature. This

results in an estimated total uncertainty in the density

of approximately 0.1% at low temperatures and 0.06%

at room temperature. (Of course, the uncertainty in

the density determination also depends on the values

of the derivatives, (dp/dP)j- and (dp/dT)p, for the

particular region of the PVT surface for the fluid under investigation.)

The uncertainties in the dielectric constant determination have been discussed in detail (sec. 4.3 and refs. [7,9,19-22]). The total uncertainty in the dielectric constant measurement is estimated to be approximately 0.01%.

The estimates of the uncertainties in the density and

dielectric constant measurements can be tested to some degree by making comparisons with reliable data

from independent sources. Such comparisons have

been made for many fluids over wide ranges of experimental parameters (such as temperature, pressure, density, etc.) [7,9,19-22]. In general, these comparisons

have confirmed the estimates of the uncertainty levels. There has been little information presented in this paper concerning the problems associated with

mixture measurements compared to pure fluid measurements. The uncertainties involved in the

determination of the composition of mixtures have

been discussed in detail in previous papers [1,5-9].

The authors are grateful to the following individuals

who made significant contributions in the development of this apparatus. W. R. Bjorklund, A. N.

DiSalvo, W. G. Layne (deceased), and D. L. Smith

assisted in the fabrication of the apparatus. M. J. Hiza

participated in many fruitful discussions. H. M.

Ledbetter provided prepubhshed bulk modulus data

for barium ferrite.

6. References

[1] Haynes, W. M.; Hiza, M. J.; Frederick, N. V. Magnetic

suspyension densimeter for measurements on fluids of cryogenic interest. Rev. Sci. Instrum. 47(10); 1237-1250; 1976 October.

|2) Haynes, W. M. Simplified magnetic suspension densimeter for absolute density measurements. Rev. Sci. Instrum 48(1): 39^1; 1977 January.

[3] Haynes, W. M.; Hiza, M. J. Measurements of the orthobaric

liquid densities of methane, ethane, propane, isobutane, and normal butane, J. Chem. Thermodyn. 9(2): 179-187. 1977 February.

(4) Haynes, W. M.; Hiza, M. J. Orthobaric liquid densities of normal butane from 135 to 300 K as determined with a magnetic suspension densimeter. Advances in Cryogenic

Engineering, Vol. 21. Timmerhaus, K. D.; Weitzel, D. H.,

ed. New York, NY: Plenum Press; 1976 516-521.

[5] Hiza, M. J.; Haynes, W. M.; Parrish, W. R. Orthobaric liquid densities and excess volumes for binary mixtures of low

molar-mass alkanes and nitrogen between 105 and 140 K. J. Chem. Thermodyn. 9(9): 873-896; 1977 September.

[6] Hiza, M. J.; Haynes, W. M. Liquid mixture excess volumes and total vapor pressures using a magnetic suspension densimeter with compositions determined by

chromatographic analysis: methane plus ethane. Advances

in Cryogenic Engineering, Vol. 23. Timmerhaus, K. D., ed. New York, NY: Plenum Press; 1976; 594-601.

17] Haynes, W. M Orthobaric liquid densities and dielectric constants of (methane + isobutane) and (methane + normal

butane) at low temperatures. J. Chem. Thermodyn. to be

published.

[8] Hiza, M. J.; Haynes, W. M. Orthobaric liquid densities and

excess volumes for multicomponent mixtures of low molar- mass alkanes and nitrogen between 105 and 125 K. J. Chem. Thermodyn. 12(1): 1-10; 1980 January.

[9] Haynes, W. M. Measurements of orthobaric-liquid densities of multicompKDnent mixtures of LNG components (Nj, CH,.

CjHe, CjHg, CHjCH(CH3)CH3, C^H.o, CH3CH(CHj)C,H5,

and CjH.j) between 110 and 130 K. J. Chem. Thermodyn.

14(7): 603-612; 1982 July.

(10) Goodwin, R. D. Apparatus for determination of pressure- density-temperature relations and spyecific heats of hydrogen to 350 atmospheres at temperatures above 14 K.

J. Res. Natl. Bur. Stand. (U.S.) 65C(4): 231-243; 1961 October- December

.

[11] Prydz, R.; Straty, G. C. The thermodynamic properties of compressed gaseous and liquid fluorine. Natl. Bur. Stand.

(U.S.) Tech. Note 392; 1970 October; 182 p. (12] Straty, G. C; Prydz, R. Fluorine compatible apparatus for accurate PVT measurements. Rev. Sci. Instrum. 41(8): 1223-1227; 1970 August.

(13] Straty, G. C. (p.V,T) of compressed fluid ethene. J. Chem.

Thermodyn. 12(8): 709-716; 1980 August.

(14] Frederick, N. V.; Haynes, W. M. Differential capacitance

sensor as p>osition detector for a magnetic suspension

densimeter. Rev. Sci. Instrum. 50(9): 1154-1155; 1979 September.

(15] McCarty, R. D. A comparison of mathematical models for the prediction of LNG densities. Natl. Bur. Stand. (U.S.) NBSIR 77-867; 1977 October. 60 p.

251

47
1 16] McCarty, R. D. Four mathematical models for the prediction of LNG densities. Natl. Bur. Stand. (U.S.) Tech. Note

1030; 1980 December. 76 p.

1 17] McCarty, R. D. Mathematical models for the prediction of

liquefied naturai gas densities. J. Chem. Thermodyn. 14(9): 837-854; 1982 September.

|18j Hiza, M. J. An empirical excess volume model for estimating

liquefied natural gas densities. Fluid Phase Equilibria 2(2): 27-38; 1978 August.

1 19] Haynes, W. M.; Younglove, B. A. Dielectric constants of

saturated liquid propane, isobutane, and normal butane. Advances in Cryogenic Engineering, Vol. 27. Fast, R. W., ed. New York, NY: Plenum Press; 1982. 883-891.

120] Haynes, W. M. Measurements of densities and dielectric constants of liquid propane from 90 to 3(X) K at pressures

to 35 MPa. J. Chem. TTiermodyn. to be published.

[21] Haynes, W. M. Measurements of densities and dielectric constants of liquid isobutane from 120 to 300 K at pressures to 35 MPa. J. Chem. Eng. Data, to be published.

[22] Haynes, W. M. Measurements of densities and dielectric constants of liquid normal butane from 140 to 3(X) K at pressures to 35 MPa. J. Chem. Thermodyn. to be

published.

[23] Younglove, B. A.; Straty, G. C. A capacitor for accurate wide

range dielectric constant measurements on compressed

fluids. Rev. Sci. Instrum. 41(7): 1087-1089; 1970 July.

124] Pan, W. P.; Mady, M. H.; Miller, R. C. Dielectric constants and Clausius-Mossotti functions for simple liquid mixtures: systems containing nitrogen, argon and light hydrocarbons.

AlChE J. 21(2): 283-289; 1975 March.

125] Clark, A. F.; Haynes, W. M.; Deason, V. A.; Trapani, R. J. Low temperature thermal expansion of barium ferrite. Cryogenics 16(3): 267-270; 1976 May.

[26] Ledbetter, H. M. Fracture and Deformation Division, National Engineering Laboratory, National Bureau of

Standards, Boulder, CO 80303. private communication.

127] Haynes, W. M.; McCarty, R. D. Prediction of liquefied- natural-gas (LNG) densities from dielectric constant measurements. Cryogenics, to be published.

128] Straty, G. C; Goodwin, R. D. Dielectric constant and

polarizability of saturated and compressed fluid methane. Cryogenics 13(12); 712-715; 1973 December.

129) Miller, R. C; Hiza, M. J. Experimental molar volumes for some LNG-related saturated liquid mixtures. Fluid Phase

Equilibria 2(1): 49-57; 1978 August.

130] Goodwin, R. D. TTie thermophysical properties of methane, from 90 to 500 K at pressures to 700 bar. Natl. Bur. Stand.

(U.S.) Tech. Note 653; 1974 April; 274 p. 131] Goodwin, R. D.; Roder, H. M.; Straty, G. C. Thermophysical

properties of ethane, from 90 to 600 K at pressures to 700

bar. Natl. Bur. Stand. (U.S.) Tech. Note 684; 1976 August. 319 p. 132] Rowlinson, J. S. Liquid and Liquid Mixtures. 2nd Ed. London:

Butterworths; 1969. 51.

[33] Miller, R. C. Estimating the densities of natural gas mixtures. Chem. Eng. 81(23): 134-135; 1974 October.

134] Weber, L. A. Dielectric constant data and the derived

Clausius-Mossotti function for compressed gaseous and

liquid ethane. J. Chem. Phys. 65(1): 446-449; 1976 July.

252

4d
6.4 Reference [13]

M-700

J, Chem. Thermodyn. Vol 9, No. 2, 179-87 (Feb 1977)

Measurements of the orthobaric liquid

densities of methane, ethane, propane,

isobutane, and normal butane

W. M. HAYNES and M. J. HIZA

Cryogenics Division, National Bureau of Standards,

Institute for Basic Standards, Boulder, Colorado 80302, U.S.A.

{Received 9 August 1976)

The orthobaric liquid densities of the major components of natural gas have been deter- mined with a magnetic suspension densimeter. This paper reports results for methane

(105 to 160 K), ethane (100 to 270 K), propane (100 to 288 K), isobutane (115 to 300 K),

and normal butane (135 to 3(X) K). The imprecision of the measured densities is approximately 0.015 per cent ; the estimated overall uncertainty is 0.1 per cent at low temperatures

and decreases to 0.06 per cent at 300 K. A simple expression has been used to represent

the densities as a function of temperature. Comprehensive comparisons with the experi- mental results of other investigators are presented.

1. Introduction

Liquefied natural gas (lng) is expected to become an increasingly important commodity on the world energy market. The basis for sale of lng is its total heating value,

which requires a knowledge of both density and composition. A project was initiated

at this laboratory to provide orthobaric (saturated) liquid densities for the major

components of lng, and for mixtures of these components. The densities will be

used to develop a mathematical model or correlation that predicts the density of

LNG type mixtures with an inaccuracy of 0.1 per cent, given a knowledge of the composition and temperature of the liquid. In the development of an accurate mathematical model (correlation), it is important to have both an accurate and an

internally consistent set of density data. Before this project was started there were significant temperature ranges for which saturated (orthobaric) liquid density data did not exist for some of the major

components of lng. For nitrogen and methane there were discrepancies as large

as 0.5 per cent between different sets of data. Not only was it important to fill in

" This work was carried out at the National Bureau of Standards under the sponsorship of

British Gas Corp., Chicago Bridge and Iron Co., Columbia Gas Service Corp., Distrigas Corp., Easco Gas LNG, Inc., El Paso Natural Gas, Gaz de France, Marathon Oil Co., Mobil R&D Corp., Natural Gas Pipeline Co., Phillips Petroleum Co., Shell International Gas, Ltd., Sonatrach, Southern California Gas Co., Tennessee Gas Pipeline, Texas Eastern Transmission Co., Tokyo

Gas Co., Ltd., and Transcontinental Gas Pipe Line Corp., through a grant administered by the American Gas Association, Inc.

49
180 W. M. HAYNES AND M. J. HIZA

gaps, but also to provide new independent measurements of sufficient accuracy to help resolve inconsistencies. In this paper orthobaric liquid densities for methane, ethane, propane, isobutane,

and normal butane are reported. Results for nitrogen were presented in an earlier paper.*" Major emphasis has been placed on the low temperature region of 105 to 140 K; however, measurements have been carried out to 160 K for methane, to 270 K for ethane, to 288.7 K for propane, and to 300 K for isobutane and normal

butane. The densities have been represented as a function of temperature with an

expression that is used to facilitate comparisons with other measurements.

The present measurements were carried out with a magnetic suspension densi- meter.'" In this method a magnetic buoy is freely suspended in the liquid of interest by the force generated from the axial oiagnetic fields of air-core solenoids. The

motion of the buoy is controlled by the automatic regulation of a servo-circuit. The magnetic force necessary to maintain the buoy at a given position is inversely

proportional to the buoyant force on the buoy. Thus, using Archimedes' principle,

along with measurements of the mass and volume of the buoy, the density of the

liquid is obtained.

2. Experimental

The experimental apparatus and its operation have been described in detail else- where.'" At low temp)eratures the experimental procedures for the measurements on

the hydrocarbons other than methane differed significantly from those for nitrogen

and methane. The density of a given fluid is determined from measurements of the magnetic force necessary to support a barium ferrite buoy in a vacuum and in the

fluid at the same position and temperature. For nitrogen and methane the vacuum

measurements were performed imipediately before or after the liquid measurements.

At low temperatures it has been found impossible to evacuate the sample cell within

a reasonable time aifter it has been filled with one of the heavier hydrocarbons. Most

of the liquid can be removed by pressurizing with helium gas; however, a liquid

film is left on the surfaces, including those of the buoy, inside the cell. The buoy

cannot be brought into support until the film is removed. Thus, for the heavy hydrocarbons at low temperatures, the vacuum points must be obtained before the liquid

measurements.

So that more than one point could be obtained in a given day for a heavy hydrocarbon at low temperatures, vacuum points were obtained at two temperatures

separated by 5 K before liquid was condensed into the cell. Then the liquid measure- ments were performed at each of these temperatures. Performance tests had demonstrated that the barium ferrite buoy does not exhibit any detectable hysteresis at low temperatures within the precision of the current measurements for a temperature

range of, at least, 20 K. After the vacuum and liquid measurements were performed for one of the heavy hydrocarbons at low temperatures the cell was warmed to a temperature above the normal boiling temperature of the test fluid for evacuation.

Methane was used as a control fluid during the heavy-hydrocarbon measurements

at low temperatures. Each day a new methane point was taken to insure that the

bO
LIQUID DENSITIES OF CH«, QHo, C3H,, i-CMio, AND w-C^Hjo 181 warm-up and cool-down of the apparatus did not affect the apparent position of the buoy from liquid-to-vacuum measurements. The position of the buoy was deter- mined with a high-powered microscope that had been adjusted initially so that the

jipparcnt position of the buoy did not >Jepend on the index of refraction of the fluid msidc the cell. It was found that the temperature cycling of the cell had no detectable

effect on the apparent buoy position.

Some of the results presented in this paper were taken with a one-coil system

instead of the three-coil arrangement described in the apparatus paper.*** The evolution to the use of only one coil is discussed in reference 2. All of the gases were of research grade quality. The minimum purities as specified

by the suppliers were 99.9 moles per cent for isobutane and normal butane and

99.99 moles per cent for methane, ethane, and propane. The gases were analyzed

chromatographically with a thermal-conductivity detector and found to be within

the specified purities except for isobutane. It was found that the isobutane contained

approximately 0. 1 5 per cent of normal butane. This relatively large amount of normalbutane impurity has a negligible effect (< 0.01 per cent) on the density results. The methane gas was passed through a room-temperature molecular-sieve trap

to remove moisture and any heavy contaminants not detected by analysis. The

other hydrocarbon gases were normally not passed through a molecular-sieve trap.

3. Results

The experimental orthobaric liquid densities of methane, ethane, propane, isobutane,

and normal butane are presented as a function of temperature (IPTS-68) in tables

1 to 5. The relatively large number of points for methane at any given temperature

resulted from the use of methane as a control fluid throughout the project. Although

the mean experimental densities of methane have been given in an earlier paper***

they are presented again here, along with other information (calculated densities,

etc.), so that the orthobaric liquid densities of all the low molecular-weight alkanes

TABLE I. Orthobaric liquid densities of methane, where F is the temperature (IPTS-68), /7„pt is the mean experimental density for n observations at a given temperature, />c»ic is the density

calculated from equation (1) and is the largest value of (pe^pt — Poaic)IPct\c

r/K /'«pt/mol dm"' n Pc.ic/moI dm"^ lOMn...

105.000 26.9458 12 26.9456 0.030

110.000 26.4985 11 26.5005 0.035

115.000 26.0443 12 26.0429 0.028 120.000 25.5721 17 25.5712 0.036 125.000 25.0845 18 25.0839 0.037 130.000 24.5775 16 24.5790 0.036

135.000 24.0540 23 24.0540 0.031 140.000 23.5067 5 23.5060 0.024

145.000 22.9312 5 22.9311 0.024

150.000 22.3218 2 22.3243 0.014

160.000 20.9876 2 20.9857 0.037

5i
182 W. M. HAYNES AND M. J. HIZA

TABLE 2. Orlhobaric liquid densities of ethane, whcrv T is the temperature (llTS-bS), is the experimental density, /i„.i„ is the density calculated from equation (I), and A is the value of (/'.«Pt - /'o»lo)//'c»lo

T /'•Kpt 10^.1 T

K mol dm"^ mol dm"^ K mol dm"^ mol dm " ^

100.000 21.3408 21.3388 0.009 170.000 18.6867 18.6869 -0.001

105.000 21.1585 21.1568 0.008 180.000 18.2793 18.2787 0.003 110.000 20.9746 20.9742 0.002 190.000 17.8612 17.8586 0.015 115.000 20.7927 20.7907 0.010 200.000 17.4289 17.4240 0.028 120.000 20.6022 20.6063 -0.020 210.000 16.9713 16.9720 -0.004

125.000 20.4186 20.4208 -0.011 220.000 16.4988 16.4989 -0.001

130:000 20.2317 20.2343 -0.013 230.000 15.9973 15.9994 -0.013

135.000 20.0461 20.0466 -0.002 240.000 15.4642 15.4670 -0.018

140.000 19.8566 19.8575 -0.005 250.000 14.8899 14.8919 -0.013

150.000 19.4751 19.4748 0.002 260.000 14.2610 14.2598 0.008 160.000 19.0857 19.0850 0.004 270.000 13.5493 13.5477 0.012

TABLE 3. Orthobaric liquid densities of propane, where T is the temperature (lPTS-68), pt^^x. is the experimental density, Pc&ic is the density calculated from equation (1), and A is the value of (/7sxpt — Pcalc)IPc&\c

T

K

PaxpC Poalo 10^A T

K

Pcalc lO^J mol dm"^ mol dm"^ mo! dm~^ mol dm"^

100.075 16.3065 16.3048 0.011 140.075 15.3751 15.3755 -0.002

105.075 16.1872 16.1885 -0.008 145.075 15.2588 15.2590 -0.002

110.075 16.0718 16.0723 -0.003 150.075 15.1400 15.1424 -0.016

115.075 15.9557 15.9562 -0.003 200.000 13.9560 13.9524 0.026 120.075 15.8411 15.8401 0.006 240.000 12.9271 12.9285 -0.010

125.075 15.7250 15.7241 0.006 270.000 12.0733 12.0742 -0.008

130.075 15.6085 15.6080 0.003 280.000 11.7622 11.7631 -0.008

135.075 15.4910 15.4918 -0.005 288.706 11.4790 11.4775 0.013

TABLE 4. Orthobaric liquid densities of isobutane, where T is the temperature (IPTS-68), pexpi is the experimental density, Pc«io is the density calculated from equation (1), and A is the value of (/>„pt — Pc»lc)//'calc T

K

Pnpt WA T

K

Poalc 10^J

mol dm"' mol dm"^ mol dm"' mol dm"'

115.075 12.7305 12.7313 -0.006 145.075 12.2353 12.2372 -0.015

120.075 12.6489 12.6491 -0.001 150.075 12.1534 12.1544 -0.008

125.075 12.5687 12.5669 0.015 228.000 10.8273 10.8263 0.009 130.075 12.4850 12.4846 0.003 288.706 9.6676 9.6687 -0.012

135.075 12.4015 12.4022 -0.005 290.000 9.6411 9.6417 -0.007

140.075 12.3215 12.3197 0.014 300.000 9.4300 9.4287 0.014
LIQUID DENSITIES OF CH4. CaHe, CaHe, /-QHio. AND n-QHio 183

TABLE 5. Orthobaric liquid densities of normal butane, where T is the temperature (IPTS-68),

/^xpt is the experimental density, />c»io is the density calculated from equation (1), and A is the value of (Pexpt — Poalo)//'o»lo T

K

10=J T

mol dm'^ mol dm"^ K mol dm"' mol dm"'

135.075 12.6517 12.6524 -0.005 165.075 12.1634 12.1659 -0.020

140.075 12.5706 12.5714 -0.006 170.075 12.0839 12.0846 -0.005

145.075 12.4920 12.4904 0.013 230.000 11.0911 11.0905 0.005

150.075 12.4089 12.4093 -0.003 288.706 10.0325 10.0324 0.001 155.075 12.3299 12.3283 0.014 290.000 10.0067 10.0073 -0.007

160.075 12.2484 12.2471 0.010 300.000 9.8103 9.8099 0.004

investigated in the present work are included in a single paper. All of the experimental

points for methane will be presented in a future report/^' Each methane point was

taken from a new filling of the cell. For the other hydrocarbons no more than two

points were taken from a single filling. The experimental densities p have been fitted as a function of temperature T to

the expression:

(p-p,)/mol dm-3 = a(l-T/TJ°-^' + ^ fe/l-T/r,)"+<'-^>/3', (1)

i= 1 which incorporates a scaling-law modification^*^ to a generalized Guggenheim

equation.*'* The coefficients o, f); determined by least squares, and selected values

of the critical teinperature 7^ and density for each fluid are given in table 6.*^"*^' Only three coefficients were needed to fit the results for methane, which covered a

relatively small temperature range compared with that for the other fluids. The residual standard deviations of the fit io equation (1) for each fluid are given

in table 6. These values substantiate the estimate of the imprecision of the density

measurements, which is approximately 0.015 per cent. The estimated inaccuracy in

TABLE 6. Parameters of equation (1),

(p - /)„)/moi dm"' = fl(i - r/re)"'" + i bi\ - 7/r„)<>

where Tc and Po are the critical temperature and density. The coeflRcients a, bi, fta, were obtained from a least-squares program in which the experimental mass densities to five digits were converted

to molar densities within the program. The standard deviations a and molar masses M are also given

Methane Ethane Propane Isobutane Norma) butane

a

h,b,TJK

Pc/mo] dm"'

10V<P>

M/g mol "

*

18.65812 6.712030 -0.9472020

190.555'"'

10.16<''> 0.016

16.04303

12.55205 13.43284 -19.00461

11.07716

305.33 <»' 6.86<»> 0.012 30.07012

8.684459

18.04086 -29.46261

16.43559

369.82 <«' 5.00"«' 0.011 44.09721

7.657535

8.145251 -13.10582

8.145894

408.13<">

3.80"°> 0.013 58.1243

7.286063

11.96308 -19.87592

11.60211 425.16<>»

392(11) 0.012

58.1243

53
184 W. M. HAYNES AND M. J. HIZA

the densities is O.I per cent at low temperatures and decreases to 0.06 per cent at 300 K. The total uncertainty in the reported temperatures is estimated to be less than

30 mK at 100 K and less than 40 mK at 300 K. These uncertainty limits in the temperature correspond to a maximum uncertainty of 0.02 per cent in the density

for the results reported in this paper. A detailed error analysis of the magnetic

suspension densimeter used in the present work has been given elsewhere.'"

Equation (1), along with the parameters given in table 6, has been used for comparisons of the present results with independent experimental data.'*' '^-25) Deviation

plots for ethane, propane, isobutane, and normal butane are presented in figures i

0.8

0.6

ciT 0.4

f 0.2

-0.2

-0.4

e

V e

1 r

e e

100 140 180 220 260 T/K

FIGURE 1. Deviation plot of experimental orthobaric liquid densities of ethane compared with

values calculated from equation (1) using parameters from table 6. O, present results; •, Shana'a and Canfield;"^' O, Chui and Canfield;'"' ©, Orrit and Olives;'"' A Rodosevich and Miller;'^^',

, McClune;'^*' , Klosek and McKinley;*^''' A, Douslin and Harrison;''*' 0, Maass and

Wright ;"»' V, Jensen and Kurata;'^''* T, Kahre.'">

through 4. The deviation plot for methane was presented in an earlier paper

thus, it is not included here. In comparing the results of other investigators with equation (1) some general

trends are observed. Below 140 K the densities of Shana'a and Canfield,"^' Chui

and Canfield,"^> Orrit and 01ives,'^*> Rodosevich and Miller," McClune,"''' and

Klosek and McKinley'*'^ are generally lower than the present results by 0.05 to

0.1 per cent. Exceptions to this trend are: for isobutane the densities of Rodosevich

and Miller"*' between 114 and 120 K are larger (maximum of 0.1 per cent) than

the present results and exhibit a significantly different temperature dependence;

and for ethane the change in density with temperature reported by Klosek and

McKinley"'' is appreciably larger than that observed in the present work.

54
0.4

0.3

0.2 -

^0.1

5 0

-0.1

-0.2

LIOUID DLNSirilS Ol- CH,. C.Ho. CM.,. /-C,H,„. AND //-t .H,., IS5

0.4V t -e-T" T

•• • V

0 0 n ^ o-o—o-®^

; * " o

0 a "a® A

Pn °

~0 JT

^0 0° DO

100 140 180 220 260 T/K

FIGURE 2. Deviation plot of experimental orthobaric liquid densities of propane compared

with values calculated from equation (1) using parameters from table 6. O, present results;

*, Siiwinski;'8' •, Shana'a and Canfield;''^' (D, Orrit and 01ives;<"* A, Rodosevich and Miller;""' n. McClune;"«' , KJosek and McKinley;'"' 0, Maass and Wright;"^' V, Jensen and Kurata;""' , Kahre;'"' Tomlinson;'"' ^, Seeman and Urban;'"' ®, NGAA;'"'; C, Van der Vet.'"'

0.3

J

0.2 -

5i: 0.1 -

o

-0.1 -

-O O-j,

® g 5^ d»a

• •

• • • •

e

c «

120 160 200 240 280 T/K

FIGURE 3. Deviation plot of experimental orthobaric liquid densities of isobutane compared

with values calculated from equation (1) using parameters from table 6. O, present results;

•, Sliwinski;'*' ®, Orrit and Olives;'"' A, Rodosevich and Miller;'"' , Mcaune;'"' T, Kahrc;"**' ®, NGAA;'"' €), Van der Vet."""

55
186 W. M. HAYNES AND M. J. HIZA

0.3 1 r

0.2

O.i -

0

-0.1

-0 2' I \ I I I \ i : \ \ i 120 160 200 240 280 r/K

' FIGURE 4. Deviation plot of experimental orthobaric liquid densities of normal butane compared

with values calculated from equation (1) using parameters from table 6. O, present results;

•, Sliwinski;'^' ®, Orrit and Olives;"*' , McClune;''«' Kahre;'^!' ®, NGAA;'"' C, Van

der Vet.<"'

Above 140 K Ihe data of Chui and Canfield/^^^ Orrit and Olives/'*^ and

McClune^'^' differ from the present results by less than 0.05 per cent. At higher

temperatures (above 280 K) the data of Sliwinski^^* for propane, isobutane, and

normal butane generally differ from the present results by less than 0.05 per cent. The orthobaric liquid densities reported by Douslin and Harrison*'^* for ethane at temperatures between 248 and 263 K were systematically larger than the present

results by 0.3 to 0.35 per cent. Some of the cider less precise (but frequently used)

data have been included on the deviation plots for the sake of completeness.

Ahhough Klosek and McKinley^'^' give densities for isobutane and normal

butane at temperatures between 105 and 133 K, these densities are not experimental,

and therefore, are not plotted on figures 3 and 4. A few comments on the reliability

of the omitted values are appropriate. Their densities were obtained from the Francis

equation.^^^^ For normal butane at temperatures below its triple-point temperature,

their results were systematically higher by less than 0.2 per cent than the densities obtained through extrapolation of the present results. ^'^^^ However, their isobutane

densities at temperatures between 116 and 133 K were 1.5 per cent higher than those

of the present work.

4. Summary

This research has provided accurate and self-consistent measurements of the orthobaric liquid densities of methane, ethane, propane, isobutane, and normal butane

at temperatures down to 100 K. Most of the measurements recently reported by

other workers differ from the present results by less than 0.1 per cent. In subsequent

papers, density measurements on liquefied mixtures of the major components of

liquefied natural gas will be reported.

66
LIQUID DENSITIES OF CH4, CzHe. Cjhi„, /-C,H,o, AND /;-C,H,„ 187 The authors would like to acknowledge R. D. McCarty and M. J. Brown for assistance

with the correlation and reduction of the data.

REFERENCES

1. Havncs, W. M.; Mi/a, M. J.; Frederick, N. V. Rev. Sci. Instnini. 1976, 47, 1237.

2. Hayncs, W. M. Rev. Sci. Instrtiin. In press.

3. Mcihaiie data arc available from authors upon request and will be published in a Nat. Bur. Stand. (U.S.) report.

4. McCarty, R. U. Nui. Bur. Stand. (U.S.), Internal Rept. 1974, 74-357.

5. Hou, Y. C; Martin, J. J. A.I.CIi J. 1959, 5, 125.

6. Goodwin. R. D. Nat. Bur. Siaiul. (U.S.), Tech. Note 653, 1974.

7. Olson, J. D. J. C/iem. Pliys. 1975, 63, 474.

8. Sliwinski, P. Z. Phys. Chem. (Frankfurt) 1969, 63, 263.

9. Das, T. R.; Eubank, P. T. Advances in Cryogenic Engineering Vol. 18. Timmerhaus, K. D.:

editor. Plenum Press: New York. 1973, p, 208.

10. Das, T. R.; Reed, Jr., C. O.; Ei.-bank, P. T. J. Chem. Eng. Data 1973, 18, 253.

11. Das, T. R.; Reed, Jr., C. O.; Eubank, P. T. J. Chem. Eng. Data 1973, 18, 244,

12. Shana'a, M. Y.; Canfield, F. B. Trans. Faraday Soc. 1968, 64, 2281.

13. Chui, C. H.; Canfield, F. B. Trans. Faraday Soc. 1971, 67, 2933.

14. Orrit, J.; Olives, J. distributed at 4th International Conference on Liquefied Natural Gas.

Algeria. 1974.

15. Rodosevich, J. B.; Miller, R. C. A.I.Ch..E.J. 1973, 19, 729.

16. McClune, C. R. Cryogenics 1976, 16, 289.

17. Klosek, J.; McKinky, C. Proc. 1st Int. Conf. on LNG, Paper 22, Chicago, 1968.

18. Douslin, D. R.; Harrison, R. H.J. Chem. Thermodynamics 1973, 5, 491.

19. Maass, O.; Wright, C. H. J. Am. Chem. Soc. 1921, 43, 1098,

20. Jensen, R. H.; Kurata, F. J. Petrol. Technol. 1969, 21, 683.

21. Kahre, L. C. J. Chem. Eng. Data 1973, 18, 267.

22. Tomlinson, J. R. Natural Gas Processors Assoc. Tech. Publ. TP-I, Tulsa, Oklahoma, 1971.

23. Seeman, F.-W.; Urban, M. Erddl und Kohle Erdgas Petrochem. 1963, 16, 117.

24. Tech. Conmt., Natl. Gas. Assoc. Am. Ind. Eng. Chem. 1942, 34, 1240.

25. Van der Vet, A. P. Congress Mondial du Petrol., Paris, Vol. II, 1937, p. 515.

26. Francis, A. W. Ind. Eng. Chem. 1957, 49. 1779.

27. Haynes, W. M.; Hiza, M. J. Advances in Cryogenic Engineering Vol. 21. Timmerhaus, K. D.;

Weitzel, D. H.; editors. Plenum Press: New York. 1976, p. 516.

57

6.5 Reference [14]

516-21^(1976)''^°^^"^'' Enpineerinp

. Vol 21. Plenum Press. New York.

ORTHOBARIC LIQUID DENSITIES OF

NORMAL BUTANE FROM 135 TO 300 K AS

DETERMINED WITH A MAGNETIC

SUSPENSION DENSIMETER*

W. M. Haynes and M. J. Hiza

Cryogenics Division, Institute for Basic Standards, Boulder, Colorado

INTRODUCTION

Of the principal constituents of natural gas, normal butane is the first in the

series of paraffin hydrocarbons that has a triple-point temperature significantly

higher than the normal boilingpoint (1 12 to 1 15 K) of methane-rich liquefied natural gas (LNG). Unlike isobutane, with a triple-point temperature (113.6 K) near the

normal boiling point of methane-rich LNG, normal butane freezes at the relatively

high temperature of 134.8 K. Thus any estimate of the contribution of n -butane

content to the molar volume (or molar density) of LNG requires a relatively long

extrapolation into the subcooled liquid region of n-butane. It is also known that r-butane is the first in the series of paraffin hydrocarbons exhibiting geometrical

isomerism, with nearly instantaneous equilibrium, which contributes to the temperature dependence of the molar density ['"^]. It follows that an analytical expression

that provides the most reliable means of extrapolating n-butane densities into the

subcooled liquid region cannot be based on correspondence alone, but must be based on extensive and accurate data above the triple-point temperature.

Other than references [^'^] that report results of saturated liquid density

measurements near room temperature, no references were found that report experimental orthobaric liquid densities of n-butane below 283 K. (The results given by

Klosek and McKinley at temperatures below the triple-point temperature are probably based on their own data at somewhat higher temperatures.) The purposes

of the present study were to obtain orthobaric liquid densities for n-butane that cover a relatively large temperature range at low temperatures, as well as some

measurements up to ambient temperature, and to provide an analytical representation of these data that is used for extrapolation into the subcooled liquid region to compare with other results [%

A relatively new technique for determining absolute densities of cryogenic fluids

['] was used in the present work. It is simple in principle and is based on a straightforward application of Archimedes' principle. Using a magnetic suspension

* This work w;is c;irrii'il out Ht the N.ition;il Bureau of Sl.indards under the sponsorship of the I.NCi

FX-nsity I'rojoit .Sli-«-iin>' ( omniiitcc, ihroiij'.h a I'.rant adniini'itercd by llic Anicricin (i ts Associ.ilinn. Inc 516

69
Orthobaric Liquid Densities of Normal Butane from 135 to 300 K 517

technique, a cylinder of magnetic material of known mass and volume is freely

suspended by air-core solenoids in a stable configuration in the fluid whose density is to be determined. At such a stable equilibrium position, the upward magnetic force is added to the buoyancy force to balance the gravitational force. Densities of the fluid can be determined directly from the measured currents in the coils necessary to support the magnetic "float" at the same position in vacuum and in the fluid at constant temperature. The vertical position of the float is controlled by the automatic

regulation of a servo-circuit, which includes a linear differential transformer for sensing inductively the position of the float. The horizontal position of the float is determined and maintained by the divergence of the magnetic field intensity.

EXPERIMENTAL

A complete description of the apparatus and a detailed discussion of the

performance characteristics are beyond the scope of this paper and will be published

elsewhere ['°]. Only those essentials of the experiment necessary for clarity are included here.

The feasibility of using a magnetic suspension densimeter for measurements on

cryogenic fluids had been demonstrated earlier with an absolute density measure- ment on liquid nitrogen at the normal boiling point The development of the

present apparatus, shown in Fig. 1, was based on experience with that apparatus and

incorporates a cryostat with continuous wide-range temperature control, a windowed equilibrium cell suitable for studies of the liquids and liquid mixtures of

interest here, a new servo-circuit for vertical position control of the magnetic buoy,

and a high-powered (125 x ) microscope for monitoring the position of the magnetic

buoy.

The magnetic buoy is a barium ferrite (BaFe]20i9) ceramic magnet in the shape

of a right circular cylinder, approximately 5 mm in diameter and 6 mm in length. It has been plated with a thickness of approximately 0.07 mm of copper over its entire

surface to prevent fluid from penetrating into the porous ceramic. Gold was flashed over the copper surface. The mass of the buoy was determined to better than 0.002%

with an analytical balance. The volume of the buoy was determined within 0.02% at room temperature by using distilled water as a reference fluid of known density. At

temperatures below ambient, the change in volume was calculated using recent thermal expansion data obtained for barium ferrite ["].

Evaluation of the experimental parameters involved in this method and extensive performance tests indicate that the uncertainty in the density measurements

depends primarily on the uncertainty in the determination of the position of the

buoy, relative to the main coil, from vacuum to liquid measurements. The position of

the float was determined to less than lO""* mm with a 125 x microscope. The position

of the main coil was maintained from vacuum to liquid measurements by supporting

the coil at its midplane with quartz rods and by controlling the temperature of the coil with water. The total uncertainty in the density measurements was estimated to be less than

0.1% at the lower temperatures; near room temperature the absolute error should

be less than 0.05%. The imprecision of the density measurements was less than

0.02% over the complete temperature range. A platinum resistance thermometer calibrated on the International Practical

Temperature Scale (1968) was used for the temperature measurements. The uncertainty of the calibration was approximately 0.002 K. Due to the specifications of the

bu
SIS W. W. Haj-nes and M. J. Hiza
Orthobuic Liquid Densities of Normal Butane from 135 to 300 K 519

potentiometric measuring system, uncertainties in the temperature amounted to a maximum of 0.0135 K at 135 K, increasing to 0.030 K at 300 K. The temperature

of the sample holder was controlled to better than 0.005 K, approximately the same

as the reproducibility of the temperature measurements.

The purity of the n-butane used in these experiments was determined by

chromatographic analysis with thermal conductivity detection. A 7.5-m liquid

partition column was used, capable of separating air (or nitrogen) from methane in small concentrations and the paraffinic from olefinic hydrocarbons. The only detectable impurity in this sample was air (or nitrogen) at approximately 25 ppm. The limits of detectability of the analyzer were approximately 20 ppm of air (or nitrogen) and

methane and approximately 10 ppm of ethane and the higher hydrocarbons.

The experin:ental densities for n -butane are given in Table I with the corre- sponding experimental temperatures. These data were fitted by the method of least squares to an equation of the form

where p is the density in moles/liter, is a characteristic density in moles/liter-

(approximately the critical density), T<. is a characteristic temperature in kelvin

(approximately the critical temperature), n is defined as l+5(/ — 1), and a, b, are

coefficients determined by least squares. The parameters of (1) also are given in Table I, as well as the standard deviation and the molecular weight.

The deviations of various experimental results [*'^] and several frequently used

literature values [^^"'•*] from the values calculated from (1) are plotted as a function

of temperature in Fig. 2. The accuracy with which equation (1), with parameters as

RESULTS

(1)

Table I. Densities of Saturated Liquid Norma] Butane and

Parameters of Equation (1)

moles/liter

Peak. moles/liter

100(p„p- Pcalc) T, K

12.6517

12.5706 12.4920 12.4039

12.3299

12.2484

12.1634 12.0839

11.0911 10.0067 9.8103

135.075

140.075

145.075 150.075

155.075

160.075

165.075 170.075 230.000

290.000

300.000

12.6524

12.5714

12.4904

12.4093

12.3283

12.2471 12.1659

12.0846

11.0905

10.0073 9.8099

-0.005 -0.006

0.013 -0.003

0.014 0.010 -0.020

-0.005

0.005 -0.006

0.005

Standard deviation 0.01 3% T^ = 425.0 K

= 3.876 moles/liter Molecular weight 58.12430

a - 7.4673 W332

- 1I.25V63703

hi - -18.03664709

/j-, - 11 22771351

6Z
520 \V. Nf. Haynes and IM. J. Hiza

• Jr

o It l llllIll|ltlN • il'IOlll —I r

^ mu

O flllCII

111 ];i ntmnm.

Fig. 2. Deviation plot of densities of

saturated liquid normul butane compared with values calculated from

equation (1 ); data from this work and

the Iiterature[*''-'^'''*]. (Note that

results from references 12 through 14

are not experimental data.)

Ill i?i KitriaiiK. I

Fig. 3. Deviation plot of

densities of liquid normal

butane of Klosek and

McKinley C] below the

triple-point temperature

compared with the extrapolated values calculated from

equation (1) for two cases:

(a) a least squares fit of the data from the present work

from 135 to 170 K using two coefficients and (b) a

least squares fit of the data from the present work from

135 to 300 K using four

coefficients. (Note that the

baseline represents two sets of extrapolated results obtained from the present data and that both curves

with designated points represent a single set of values from Klosek and

McKinley ['].)

given in Table I, can be extrapolated above 300 K has not been investigated, but

from the comparisons with the work of Sliwinski [^] and Kahre C] it appears that the

equation gives reasonably accurate values up to 370 K.

Two different methods of extrapolation to temperatures below 135 K are presented in Fig. 3. First, the present data from 135 to 170 K were fitted to equation

(1) using two coefficients (a = 7.318406322 and f>i = 3.479543073) and then

densities were calculated for temperatures below the triple point and compared with

the values of Klosek and McKinley Next, the present data from 135 to 300 K

were fitted, using equation (1) with four coefficients, and again comparisons were made with the values of Klosek and McKinley. The graph demonstrates the

significant differences obtained for the two extrapolation methods and also provides

comparisons with the only other published density values for normal butane at temperatures below 283 K. It should be noted that no effort has been expended in the present work to produce a "best" method for extrapolation into the subcooled

liquid region.

CONCLUSIONS

The results of this study provide the first set of experimental density data for the

orthobaric liquid phase of n-butane at temperatures between 135 and 300 K. The

b3
Orthobaric Liquid Densities of Normal Butane from 135 to 300 K 521

absolute error is estimated to be less than ±0.1%. The analytical expression selected

represents these data with a standard deviation less than the estimated imprecision of

measurement, i.e., ±0.02%. It is felt that the analytical expression given here

provides a reasonable means of extrapolation of n-butane densities into the subcooled region for liquefied natural gas applications.

ACKNOWLEDGMENTS

The authors wish to acknowledge the contributions of the LNG Density Project Steering Committee

and the financial support of the companies they represent. REFERENCES

1. R. A. Bonham and L. S. Bartell, J. Am. Chem. Soc. 81:3491 (1959).

2. R. A. Bonham, L. S. Bartell, and D. A. Kohl, J. Am. Chem. Soc. 81:4765 (1959).

3. A. L. Verma, W. F. Murphy, and H. J. Bernatein, /. Chem. Phys. 60:1540 (1974).

4. M. R. Cines, Phillips Petroleum Co., private communication (January 1974).

5. T. W. Schmidt, Phillips Petroleum Co., private communication (May 1974).

6. P. Sliwinski. Z. Phys. Oiem. (Frankfurt) 63:263 (1969).

7. L. C. Kahre, / Chem. Eng. Data 18:267 (1973).

8. J. Klosek and C. McKinley, in: Proceedings of First Intern. Conference on LNG, Chicago, Illinois (1968), Paper 22, Session 5. 9. W. M. Haynes and J. W. Stewart, Rev. Sci. Instr. 42:1142 (1971).

10. W. M. Haynes, N. V. Frederick, and M. J. Hiza, unpublished dat:i. 11. A. F. Clark, \V. M. Haynes, V. A. Deason, and R. J. Trapani, Cryogenics (to be published).

12. A. W. Francis, Ind. Eng. Chem. 49:1779 (1957).

13. Natural Gas Processors Supplies Association, Engineering Data Book, Ninth Edition, NGPA, Tulsa, Oklahoma (1972).

14. American Petroleum Institute, Project 44, Selected Values of Physical and Tliermodynamic Properties of Hydrocarbons and Related Compounds, Carnegie Press, Pittsburgh, Pennsylvania (1953). DISCUSSION

Question by R. C. Hendricks, NASA Lewis Research Center: Have you \ erified the results from your

primary platinum temperature sensor by comparison with a second platinum temperature sensor? If not, how did you assure yourself that the temperature values obtained in your study were accurate? Answer by author: No. Vapor pressures of pure methane were measured in the vapor bulb at the top

of the cell and in the equilibrium space to determine the reliability of temperature measurement and

control, as well as departures from isothermal conditions in the equilibrium cell.

04
6.6 Reference [16]

M-754

/. Chem. Thermodynamics 1977, 9, 873-S96

Orihobaric liquid densities and excess

volumes for binary mixtures of low molarmass alkanes and nitrogen between

105 and 140 K*

M. J. mZA, W. M. HAYNES, and W. R. PARRISH

Cryogenics Division, Institute for Basic Standards,

National Bureau of Standards, Boulder, Colorado 80302, U.S^.

{Received 21 February 1977)

A magnetic suspension densimeter has been used to determine ortbobaric liquid densities of gravimetrically prepared binary mixtures of the major components of liquefied natural gas (LNG) i.e. nitrogen, methane, ethane, propane, i-butane, and n-butane, generally

between 105 and 140 K. All binary combinations were included in this study, with the exception of nitrogen + i-butane and nitrogen + /r-butane. Uncertainties in the reported

liquid-mixture densities are discussed in detail. Comparisons are made between excess volumes computed from the present results and coirnarabie values from the literature.

It was found that the volumetric properties of binary liquid mixtures of the heavy hydrocarbons (those mixtures not containing nitrogen or methane) are closely approximated

by ideal mixing. Some observations are included on the use of excess volumes of the heavy

hydrocarbon systems to determine effective molar volumes of n-butane in liquid mixtures below its triple-point temperature. For mixtures containing nitrogen or methane, approxi- mate total vapor pressures are given.

1. Litroduction

Custody transfer of liquefied natural gas (LNG) on the world energy market has

imposed one of the most stringent requirements for accuracy in the prediction of

orthobaric (saturated) liquid mixture densities. Since heating value is directly related

to the density and composition of the liquid, it is desirable to be able to predict LNG

densities within a small known uncertainty, preferably within ±0.1 per cent, for any

condition encountered in commerce. The most promising theoretical methods for predicting the properties of fluid mixtures require that the pure-component characteristic parameters be combined to give the mixture characteristic parameters of the included binary pairs. It is well

known that small adjustments to these parameters can significantly affect the difference

" This woric was carried out at the National Bureau of Standards under the sponsorship of British Gas Corp., Oiicago Bridge and Iron Co., Columbia Gas Service Corp., Distrigas Corp., Easco Gas

LNG, Inc., El Paso Natural Gas, Gaz de France, Marathon Oil Co., MobQ Oil Corp., Natural Gas

Pipelkie Co., Phillips Petroleum Co., Shell International Gas, Ltd., Sonatrach, Southern California Gas Co., Tennessee Gas Pipeline Co., Texas Eastern Transmission Co., Tokyo Gas Co., Ltd., and

Transcontinental Gas Pipe Line Corp., through a grant administered by the American Gas

Association, Inc.
874 M. J. HIZA, W. M. HAYNES, AND W. R. PARRISH

between predicted and actual mixture properties. Though reasonable estimates of

deviations from the combining rules can be made in special cases/^^ the best values

for a given application and model must still be derived from accurate and consistent

experimental results at several temperatures for the binary mixtures and pure components of interest

For the development of a model to predict LNG densities reliably, for example, one needs suflBciently accurate densities of uniformly high precision for the pure

components and the possible binary combinations, as well as for selected multicomponent mixtures, to test and optimize the prediction method. To provide this data base, a comprehensive study was initiated in our laboratory to obtain orthobaric

liquid densities of the desired accuracy for the major components of LNG and for

their mixtures.

Previously, orthobaric liquid densities were reported for nitrogen^ and the low

molecular-weight alkanes—methane,^^'^^ ethane, propane, /-butane, and n-butane^''^ —as determined with a magnetic suspension densimeter.

The present study was conducted with the same apparatus to obtain orthobaric

liquid densities for gravimetrically prepared binary mixtures of these components,

generally between 105 and 140 K. All of the possible binary systems were studied,

with the exception of nitrogen -f- /-butane and nitrogen + «-butane. Based on the

phase equilibria for nitrogen + ethane^*^ and nitrogen + propane^'^ it was estimated

that the limits of miscibility for the nitrogen + butane systems would preclude the

possibility of obtaining experimental densities for these systems in the temperature

range of interest Where possible, total vapor pressures were also determined, but these are considered approximate since the main focus was to assure the reliability of the

density measurements.

Excess volimies, computed for each of the binary mixtures studied, show that

mixtures of the heavy hydrocarbons exhibit nearly ideal mixing. Effective liquidphase molar volumes for «-butane below its triple-point temperature were obtained

from the ethane 4- n-butane liquid-mixture densities by assuming that the small

excess volume remains constant in the temperature range studied.

Only a few sets of orthobaric liquid-mixture densities in the literature^*^ provide

data applicable to the LNG problem discussed above. These data are generally

Umited in either the temperatures or compositions covered, or have admitted in- accuracies significantly larger than the desired ±0.1 per cent. Comparisons are made

between excess volumes from the present study and comparable values taken from

the literature. Prior to the present study, there were no liquid densities in the temperature range of interest for nitrogen + ethane, nitrogen + propane, ethane -H i-butane,

propane + z-butane, propane + n-butane, or i-butane -I- w-butane.

2. Experimental

The magnetic suspension densimeter used in this study was discussed in detail else- where.^^* All of the results reported here were obtained with the one solenoid

arrangement^'^ With this arrangement a barium ferrite magnetic buoy of known

mass and volume is freely suspended, in vacuum and in the liqmd, by the force

66
UQUm DENSITIES FOR BINARY MIXTURES OF LNG COMPONENTS 875

generated from the axial magnetic field of a single air-core solenoid. The position

of the buoy is controlled by automatic regulation of the solenoid current with a

radio-frequency servo-circuit Since the magnetic force necessary to support the buoy

at a given position is dependent on the buoyant force, measurement of the solenoid

current needed to support the buoy in the liquid relative to vacuum at the same

position and temperature yields the liquid density directly.

The relation used to compute the density of the liquid with the one solenoid

system is

Pi-(mjn){l-(/i//r)}. (1)

where pi is the mass density of the Kquid, is the mass of the buoy, V^, is the volume

of the buoy, and /, and /, are the solenoid currents required to support the buoy at the same position in the liquid and vacuum, respectively.

The procedures used in this study were basically the same as those used in measuring

liquid densities of the pure fluids.^^*^*^^ Vacuum measiu:ements were made immediately before or after those for nitrogen 4- methane liquid-mixture points, and

each measurement was made with a separate fill. Since the temperature of the equilibrium cell had to be increased nearly to room temperature for effective removal

of the heavy hydrocarbons, it was necessary to make vacuum measurements before

introducing mixtures containing the heavy hydrocarbons. Measurements for these

mixtures generally were made in pairs at 5 K increments with a single fill. With few

exceptions a liquid-methane density measurement was made each day as a control on the measurement parameters. Liquid methane was used as the vapor-bulb fluid to provide the criteria for adjusting the control heat to the top and bottom of the

cell to minimize temperature gradients. The heater currents were adjusted so that the

methane vapor pressure was consistent with the temperature (IPTS-68) determined

with a platinum resistance thermometer mounted near the bottom of the cell. Vapor

bulb and equilibrium cell pressures were measured simultaneously with quartz

Bourdon-tube pressure gages (0 to 0.69 MPa) calibrated against an air dead-weight

gage. The estimated maximum uncertainty in calibration of these gages is about

±70 Pa. Since the imcertainty in the measured density ofa binary liquid mixture is inherently

larger than for the pure fluids due to the added uncertainty in composition, pre- cautions were taken to assure that the uncertainty in composition was minimized and

that the composition in the equilibrium cell was homogeneous.

Basically, there are three options available to fix the composition of a liquid

mixture to be studied experimentally. These are (1) liquid-phase sampling and analysis

relative to calibration mixtures, (2) introducing known quantities of each pure component and mixing within the experimental chamber, and (3) introducing the desired

mixture, prepared under carefully controlled conditions, into the equilibrixmi chamber

in the mixed state. Each method, or the various combinations, requires an exact accounting of the amount of each component at some point in the process. The

third method was considered to be the most desirable and potentially the most

accurate for this study.

The mixtures were prepared gravimetrically in thoroughly cleaned and dried metal

t)7
876 M. J. mZA, W. M. HAY>fES, AND W. R. PARRISH

cylinders, each with a free volume of about 3 dm'' and a tare mass of about 4 kg. In a few cases, cylinders with about 7 dm^ free volume and a tare mass of about

12 kg were used. AH gases were research grade, and were analyzed chromatographically before use. In addition, nitrogen and methane were passed through roomtemperature molecular-sieve adsorption columns to remove moisture and any heavy

contaminants not detected by analysis. The amount of each component added to a

cylinder was determined by difference weighings using a Class S weight set and a

precision equal-arm balance with a capacity of 25 kg. An identical cylinder filled with nitrogen gas at atmospheric pressure was used as ballast on the opposite pan.

The standard deviation of 10 repetitive weighings with the balance was determined

by the manufacturer to be 0.51 mg. The standard deviation of 1 1 repetitive weighings

as used in the preparation of mixtures in the laboratory was determined to be 0.67 mg.

Since the uncertainty in the Class S weights is ten times lower than the standard

deviation in the weighings, the uncertainty in the amount of substance of each component in the mixture prepared is dependent only on the random error in the weighingTABLE 1. Uncertainty 5;c in mole fraction of each component in the prepared mixtures ba^ed on

the total amount of substance n prepared

Component n = 1 raol rt = 5 mol

±0.00014 ±0.00003

CH« ±0.00025 ±0.00005

CaHg ±0.00013 ±0.00003

C3H9 ±0.00009 ±0.00002

C«Hio ±0.00007 ±0.00001

The estimated error in mole fraction of each component is given in table 1 for both

1 and 5 mol of mixture prepared. Since two weighings are required to determine the

amount of substance of each component added, the estimated errors are based on

6 times the standard deviation determined in our laboratory.

The composition, molar mass, and the total amount of substance prepared are given in table 2 for each of the binary mixtures included in this study. A comparison

of the uncertainties in component composition given in table 1 with the total amount

of substance of mixture prepared, given in table 2, is a direct indication of the uncertainty in composition of each mixture. Three of the mixtures given in table 2

(mixtures with x « 0.5 of methane -f- ethane, methane -i- propane, and ethane +

propane) were obtained from the U.S. Bureau of Mines in Amarillo, Texas. The

estimated uncertainties in component composition of these gravimetrically prepared

mixtures are roughly the same as those given in table 1.^'^ A schematic diagram of the apparatus as used to measure liquid mixture densities

is given in figure 1. This arrangement incorporates the same capabilities as the closed

loop vapor-recirculation apparatus used in liquid-vapor equilibrium measurements^'' except for vapor sampling. A window in the equilibrium cell allows visual

observation of the liquid sample from about 2 cm above the bottom of the cell cavity

66
LIQUID DENSITIES FOR BINARY MIXTURES OF LNG COMPONENTS 877

TABLE 2. Prepared binary mixtures : M denotes molar mass and n the total amount of substance prepared

Mixture Af/g mol

/i/mol

0.04752Na + 0^5248CH4 16.6119 6.30 0.30349Na + 0.6965 ICH* 19.6759 4.37 0.49242Na + O.SOTSSCH* 21.9375 6.00 0.05933Na + 0.94067CaH« 29.9481 5.42 0!02014Na + 0.97986C3Ha 43!7733 IJO

0.03794N2 + O.962O6C3H9 43.4870 3.73 0.06740Na + O.9326OC3H8 43.0132 123

O.35457CH4 + 0.64543CaH« 25.0%5 5.04 O.49325CH4 + 0.50675CaH, • 23.1513 25.56 O.68OO6CH4 + 0.31994CaH. 20.5309 4.87 0.29538CH« + 0.70462C3H8 35.8106 1.27 0.49637CH4 + OJO363C3H8 » 30.1720 2.19 O.7492OCH4 + O.25O8OC3H8 23.0790 3.54 0.85796CH4 + 0.14204C3Ha 20.0279 4.79 0.48687CH4 + OJI313/-C4H10 37.6362 1.63 0.58828CH4 + 0.41172n-C4Hio 33.3687 1.92 0.91674CH4 + 0.08326n-C4Hio 19.5467 3.85

0.50105CaH« + 0.49895C3Ha » 37.0689 2.23 0.67287CaH, + 0.32713C3H8 34.6588 3.04 0.68939CaH9 + 0.31 061 /-C:4Hio 38.7840 1.21 0.72436CaH« + 0.27564/-C4Hio 37.8030 1.78 0.65343CaH8 + 0.34657n-C4Hio 39.7929 0.83 0.6711 7CaH8 + 0.32883«-C4Hio 39.2952 0.96 O.4903OC3H8 + 0.50970/-C4Hio 51.2468 0.74 O.50326C3H8 + 0.49674j-C4Hio 51.0650 0.75 O.58692C3H8 + 0.41308/i-C4Hio 49.8915 0.53 O.6O65OC3H8 + O.3935O/1-C4H10 49.6169 0.54 O.6O949C3H8 + 0.39051/i-C4Hio 49.5749 0.58 O.47039/-C4H10 + 0.52961n-C4Hio 58.1243 1.03

• Obtained from the U.S. Bureau of Mines, Helium Operations, Amarillo, Texas.

up to the base of the vapor bulb. The vapor bulb, attached directly to the closure

plug, is a slip fit in the top section of the cell. The vapor leaves the top of cell through a capillary tube passing through the vapor bulb. The density equilibrium system was designed so that the vapor volume, excluding the free volume (65 cm^) in the

recirculation pump, is extremely small relative to the liquid volume. The vapor

volume includes approximately 0.42 cm^ in the cell, 0.43 cm^ in the access tubing

within the cryostat, and 3.3 cm^ in the pressure gage and tubing outside the cryostat

The volume occupied by the liquid is approximately 20.5 cm'. The vapor volumes

were calculated from known dimensions, the liquid volume was determined by filling

the cell with water, and the pump volume was determined by gas expansion. Without

the pump volume, the effect of the vapor volimies on composition and density of the

liquid is quite small and becomes important only at the higher mixture vapor pressures

(0.2 to 0.3 MPa).

60

69
878 M. J. HIZA, W. M. HAYNES, AND W. R. PARRISH

I Recirculation

Liquid

mixture

Magnetic

buoy

i J FIGURE 1. Schematic diagram of experimental apparatus.

The prepared gas mixtures were condensed directly and continuously into the

equilibrium cell through the capillary vapor inlet tube at the bottom of the cell until the last vapor bubble disappeared below the vapor bulb. This filling method

was used to provide continuous mixing of the liquid during the fill, analogous to vapor recirculation. For mixtures containing only the heavy hydrocarbons, the vapor

pressures were extremely low, and the amount of vapor space left in the cell was

imimportant. Nevertheless, the cell was filled with liquid as discussed above. There

were no visible bubbles when filling with these mixtures.

For methane + heavy hydrocarbon mixtures, comparison of the measured densities

before and after vapor recirculation provided a simple test of homogeneity of the

liquid mixture. For these mixtiires, the vapor phase is essentially pure methane at the temperatures included in this study. Ethane, the most volatile of the heavy components, has a vapor pressure of only 0.00383 MPa at 140 K.^^^^ Initially, the re- circulation pump was filled with pure methane gas at a pressure equivalent to the

vapor pressure of the mixture for the second experimental temperature, and isolated

from the system. The solenoid currents for a pure liquid methane point and two vacuum points, separated by 5 K, were taken first. The prepared binary gas mixture

to be studied was then condensed into the equilibrium cell in the manner discussed

above. Solenoid currents for the two liquid-mixture points were measured at tempera7U
LIQUID DENSITIES FOR BINARY MIXTURES OF LNG COMPONENTS 879

tures corresponding to those of the vacuum points, the lower-temperature point

generally being taken last Subsequently, the system was opened to the pump, and

vapor was recirculated through the liquid at one to three bubbles per second (probably

less than 1 cm^ min"^) for several minutes. A low recirculation rate was used to avoid entraiimient The pump was then turned off, and the solenoid current for the

liquid mixture point was remeasured. No change in density could be detected outside

the precision of the measurement for any of the reported binary-mixture points thus

tested. For mixtures containing heavy hydrocarbons only, the most direct means to determine that the liquid sample was homogeneous and of the same composition as

the prepared mixture was by liquid sampling and analysis. Ethane + propane was

selected for this test Liquid samples were withdrawn under constant hehum-gas

pressure (about 0.3 MPa) after the density measurements were completed. During

sample withdrawal, the solenoid current was monitored to detect any variations in density as a different part of the liquid sample surrounded the magnetic buoy. Due

to the extremely low vapor pressures, the compositions of these liquid mixtures are not subject to change by preferential vaporization during sampling. Each liquid

sample was analyzed chromatographically using the prepared gas mixture for cali- bration. For these liquid mixtures, no change in solenoid current could be detected

during sample withdrawal, and the composition of the liquid was found to be consistent with that of the prepared mixture within the precision of the gas analysis,

i.e. within a few hundredths of 1 mole per cent

Liquid samples of methane + ethane and methane + propane were also withdrawn and analyzed subsequent to the density measurements and recirculation of

the vapor. However, it was extremely difficult to obtain samples of consistent composition due to preferential vaporization. At the higher temperatures (^135K)

samples with compositions consistent with the prepared mixtxires could be obtained

by using the recirculation-pump volume as pressure ballast during withdrawal. At

lower temperatures it was necessary to pressurize the cell with helium gas to obtain

samples with compositions equivalent to those of the prepared mixtures. In general,

however, analyses of liquid samples of the methane + hydrocarbon mixtures were

mainly useful to determine that there were no gross discrepancies (greater than a few tenths of 1 mole per cent) in composition and that the liquid mixtures were not

inadvertently contaminated. No attempt was made to recirculate the vapor or to sample and analyze the liquid mixtures containing nitrogen.

3. Results and discussion

EXPERIMENTAL RESULTS

The experimental orthobaric liquid-mixture amount-of-substance densities are given

in table 3 as a function of temperature. The excess volumes F^, also given in table 3, were computed from the expression:

where is the molar volume, is the saturation pressure, x is mole fraction, and P

(2)

71
880 M. J. mZA, W. M. HAYNES, AND W. R. PARRISH

TABLE 3. Ortiiobaric liquid amount-of-subsiance densities p of binary mixtures of low molar-mass

alkanes and nitrogen. T, temperature (IPTS-68); ^ouc. P calculated from equation (4); p, pressure; V^, excess volume

Mixture

T

K

P 1 r\1 P'm, Pc*.lo 103 P

mol dm"' MPa cm' mol"*

0.04752N2 + 0.95248CH4 105.00 26.8476 -0.006 0.138 -0.144

110.00 26.4052 0.010 0.199 -0.258

115.00 25.9374 -0.002 0.263 -0.442

120.00 25.4522 -0.006 0.350 -0.889

125.00 24.9496 0.005 0.460

130.00 24.4210 0.007 0.583 135.00 23.8600 -0.017 0.730

140.00 23.2809 0.009 0.920 0.30349N2 + 0.69651CH4 100.00 26.8735 —0.010 0.345 -0.584

105.00 26.3393 0.023 0.466 -0.885

110.00 25.7686 -0.005 0.618 -1.353

115.00 25.1790 -0.017 0.801 -2.253

1 1A AA 24.5737 0.010 1.011 A CIA —4.510

0.49242N2 + 0.50758CH4 95.00 27.0801 0.006 0.330 -0.541

1 AA AA 100.00 26.4Jos 0.003 0.465 —0.777

105.00 25.8106 —0.012 0.637 — 1.154 110.00 25.1387 -0.016 0.844 -L803

115.00 24.4431 0.006 1.097 -3.056

120.00 23.7096 0.017 1.398 —6.110

125.00 22.S315 0.010 1.731 130.00 22.1005 —0.015 2.110 0.05933Na -I- 0.94067C2H8 105.00 21.4718 0.385 -0.423

110.00 21.2912 0.463 -0.590

115.00 21.0845 0.547 -0.804

120.00 20.8998 0.638 —1.385 0.02014Na + 0.97986C3Hs 110.00 16.2131 0.357 -0.208

115.00 16.0931 0.471 -0.289

0.037941^2 + O.yoZOoCaHs 105.00 16.4638 0.495 —0.310

110.00 16.3410 0.671 —0.392

0.06740N2 + O.9326OC3H8 100.00 16.8055 0.631 —0.427

105.00 16.7084 0.880 -0.628

0.35457CH4 + 0.64543CaH8 105.00 23.1032 0.002 0.0256 -0.382

110.00 22.8777 0.009 0.0392 -0.442

115.00 22.6478 0.007 0.0580 -0.505

120.00 22.4035 -0.050 0.0826 -0.554

125.00 22.1872 0.030 0.115 -0.673

130.00 21.9441 0.002 0.155 -0.798

0.49325CH4 + 0.50675CaH, 105.00 23.9619 0.031 0.0325 -0.525

110.00 23.6937 -0.004 0.0503 -0.569

115.00 23.4328 0.003 0.0749 -0.639

120.00 23.1559 -0.047 0.108 -0.697

125.00 218933 -0.024 0.150 -0.801

130.00 22.6290 0.007 0.205 -0.927

135.00 22.3581 0.024 0.272 -1.068

140.00 22.0765 0.009 0.355 -1.223

7Z
UQUID DENSITIES FOR BINARY MIXTURES OF LNG COMPONENTS 881

TABLE 3continued

iViiAiurc T P

L \J — ^ P

yz K mol dm"^ Pcala MPa cxn^ mol ^

0.68006CH« + 0.31994CaHa 105.00 25.1027 0.002 0.0416 -0.524

24 7802 0 004 0 Ofi45 —0 562

115.00 24.4612 0.003 0.0963 -0.621

120.00 24.1402 -0.001 0.138 -0.698

125.00 23.8212 0.002 0.193 -0.802

130.00 23.5007 -0.001 0.262 -0.933

O.29538CH4 + 0.70462C3Hi 105.00 18.5132 0.0270 —0.469

11000 18 3^24 0 0409 —0 524

0.49637CH4 + 0.50363C3H8 105.00 20.4909 0.005 0.0384 —0.727

110.00 20.3046 -0.002 0.0591 -0.814

115.00 20.1180 —0.006 0.0874 —0.915

120.00 19.9311 -0.007 0.125 -1.033

125.00 19.7471 0.014 0.173 — 1.179 iin on —0 004 0 2'?2 — 1 329

O.7492OCH4 + O^OSOCsHg 105.00 23.4768 0.005 0.0478 —0.700

110.00 23.2064 —0.009 0.0738 —0.783

115.00 22,9364 —0.004 0.110 —0.887

150 no 0 017 0 158 — 1 014

125.00 22.3818 -0.010 0.222 -1.141

130.00 22.1019 0.002 0.303 -1.310

O.85796CH4 + O.I42O4C3H8 105.00 —0.007 0.0516 -0.553

1 10 00 24 fi%'%\ 0 009 0 0801 —0.617

115.00 24.2941 0.006 0.120 -0.684

120.00 23.9491 0.002 0.173 -0.764

125.00 23.5941 -0.021 0.242 -0.855

130.00 23.2461 0.012 0.330 -0.992

0.48687CH4 + 0.51313i-C4Hio 110.00 17.3575 0.0629 —0.803

115.00 17.2076 0.0938 —0.884

120.00 17.0639 0.1361 —1.002

10.71v0 0 1852

0.58828CH4 + 0.4I172n-C4Hio 120.00 18.6495 0.1636 —1.312

125.00 18.4772 0.2281 —1.466 125.00 18.4853 0.2281 -1.489

130.00 18.3058 0.3183 -1.646

0.91674CH4 + 0.08326/i-C4Hio 105.00 25.1536 0.009 -0.599

110.00 24.7960 -0.021 -0.638

115.00 24.4512 0.010 -0.719

120.00 24.0910 -0.011 -0.797

125.00 23.7370 0.006 -0.916

130.00 23.3789 0.018 -1.063

135.00 23.0110 0.003 -1.234

140.00 22.6391 -0.015 -1.451

0.50105CaH, + 0.49895C3H, 105.00 18.3618 0.021 -0.040

110.00 18.2169 0.001 -0.036

115.00 18.0726 -0.012 -0.034

120.00 17.9282 -0.020 -0.033

125.00 17.7880 0.000 -0.047

/J
882 M. J. mZA, W. M. HAYNES, AND W. R. PARRISH

TABLE 3

continued

Mixture

T

K

P mol dm'^

10*" ——————

/>o*io MPa

yz

cm' mol"

1 in on 1 jU.UU 1 /.041i A A1 1 —U.Ull —U.U41 135.00 17.4988 0.009 -0.051

140.00 17.3526 0.013 -0.051

0.67287C2H8 + 0.3271 3C3H, 125.00 18.6192 -0.044

130.00 18.4646 -0.053 A nc-y 14U.UU —A ruze U.UOJ

A £.QQ1Q/^ ZJ -LA ^lAi^l .* LT 1 7 171 1^ 1 / .J / lo 1 A Al 7

1 AA 1 7 HAA 1 A A1 C +U.U1 J

O.72436C2H8 + 0.27564/-C4Hio 105.00 17.9779 +0.013

1 1 A AA 171 /.04U1 1 A AAO

1 AA —A Al A U.UIU

1 "JA AA

1 1 < AA 17 71 94 1 /.Zlo4 —U.Ujj 120.00 17.0824 -0.045

U.o/il/Cang -r U.jZo9j/i>C4riio 1 1 A An 17 </M7 A /Vie 115.00 17.3706 -0.000 -0.050

125.00 17.1031 0.009 -0.060

1 lA AA 1 if Qi^7X A A7n A Ai4'5 11^ AA 1/S 979< —A AA^ U.UUO A A>1< 140.00 16.6947 0.014 -0.054

lAC AA lUj.UU lA m9n —u.uuz

1 t A AA 110.00 14.ZI Jo A AA7 1 A A70 +U.U7CS 115.00 14.1219 0.011 +0.056

120.00 14.0257 0.001 +0.057 —A AA*^ O.UUZ _1_A AC/4 130.00 13.8342 -0.002 +0.052

A CAI'^X/^ XT _L A >4AiC7/1.' ^ Tjr 1 ^< AA 1"3 Q71 Q 1 A Ay17 +U.U4/

1 '3A AA 130.00 13.0737 1 A A</f +0.054

1 1 A AA 1/1 A9'}Q —U.Uj4

135.00 14.1748 +0.021

140.00 14.0786 +0.006

0,60650C3H8 + 0.39350/:-C4Hio i4o:oo 14.1343 +0.010

145.00 14.0333 +0.015

150.00 13.9346 +0.009

O.6O949C3H, + 0.3905I/»-C«Hio 115.00 14.6487 -0.017

. 120.00 14.5521 -0.036

0.47039/-C4Hio + O.5296I/J-C4H10 125.00 12.6943 +0.016

130.00 12.6133 +0.013

135.00 12.5271 +0.048

140.00 12.4447 +0.053

74
UQUID DENSITIES FOR BINARY MIXTURES OF LNG COMPONENTS 883

is the isothermal compressibility. Subscripts m and i refer to the mixture and to the

pure components, respectively. The pure-component molar volumes were calculated

from the fit of the experimental results/^* Only those for nitrogen, methane, and

ethane were adjusted to the mixture pressure where appropriate. Isothermal compressibilities for nitrogen and methane were taken from Rowlinson^"^ and those for

ethane were taken from Miller^^^^ assuming linear temperature dependence. Excess

volumes for a few representative methane + ethane points were also computed

using isothermal compressibilities derived from the recent methane data of

Goodwin.^**^ The excess volumes using Goodwin's values of P were less than 0.1 per cent different from the excess volumes computed using those of Rowlinson.

Rowlinson's values were used here only because excess volumes for methane and

nitrogen systems reported in the literature are based on his values. Thus, a direct

comparison of excess volumes can be made without assessing the contribution of

different compressibility values. The ethane compressibility values of Miller were

chosen for the same reason. Vapor pressures for nitrogen, methane, and ethane were

taken from Strobridge,^^^^ Goodwin,^^*^ and Goodwin, Roder, and Straty,^^^^

respectively.

Effective molar volumes for n-butane below its triple-point temperature (134.8 K)

were obtained from equation (2) by assuming that the small negative excess volume

of ethane -H «-butane was constant and equal to the average value of the excess volumes above the triple-point temperature of n-butane. These subcooled liquid

molar volumes were fitted as a linear function of temperature to obtain the equation

:

F(n-C4) = 65.63936 -H0.0992160(r/K). (3)

The molar volumes calculated from this expression at 135 and 140 K agree with the

experimental values^'^ within 0.01 per cent. At 105 K, the molar volume of n-butane

calculated from the equation given in reference 3 fitted to the experimental liquid

densities is 0.056 per cent larger than the value given by equation (3). Equation (3) was used to compute the excess volumes given in table 3 for all of the /i-butane

mixtures below 135 K. For i-butane, the liquid-phase equation given in reference 3 was used to obtain subcooled liquid molar volumes below the triple-point temperature

(113.6 K).

In examining the excess volumes for propane 4- n-butane and /-butane -f- n-butane

using n-butane molar volumes from equation (3), an interesting anomaly was noted.

The excess volumes for these systems, as shown in figure 2, appear to have a dis- continuity at about the triple-point temperature of n-butane. Molar volumes of

n-butane determined from these systems in the same manner as those determined

from ethane + n-butane are in good agreement with each other. However, the

molar volumes calculated from a linear fit of the values from propane + n-butane

and /-butane -I- n-butane are 0.14 and 0.09 per cent lower than those from equation (3) at 110 and 130 K, respectively, and the extrapolated volumes are about 0.075 per cent lower than the experimental liquid molar volumes at 135 and 140 K.

Though the differences noted are not very large, the imprecision of the density

measurements for these heavy hydrocarbons and their mixtures is small enough (see
884 M. J. HIZA, W. M. HAYNES, AND W. R. PARRISH

the ethane + w-butane and propane + /-butane mixtures in table 3) to suggest that

this behavior is real and would be worth investigating in more detail. This behavior

might also be important in the development of precise methods to predict densities

of mixtures containing n-butane.

The pressures listed in table 3 are considered only approximate mixture vapor

pressures, and not all of these were directly measured in the present experiment.

Pressures are given only for mixtures where sufficient measurements were made to allow either interpolation in temperature or composition for a given system or comparison with existing phase equilibria. For nitrogen + methane, several random

pressure measxirements were made which were consistent with those interpolated

from previous phase-equilibrium measurements in our laboratory. As a result,

all of the pressures listed for nitrogen + methane were obtained from graphical

0.10 r

'o

0.05 E

-0.05

;_rH +n-CH 0-04 per cent in K„ - _^ h , A Jo .04 per cent

110 120 130 140 r/K

150 160

FIGURE 2. Excess volumes of propane + n-butane and /-butane -f- /i-butane from the present

study. O, O.58692C3H9 4- 0.41308n-C4Hio; 0. O.6O65OC3H, -i- 0.39350/i-C*Hio: O, 0.60949 CjHa -i- 0.3905 In-CiHio; , O.47039/-C4H10 -f 0.52961/i-C4Hio.

interpolations of the phase-equilibrium measurements. Pressures for nitrogen +

ethane and nitrogen + propane were difficult to measure with any consistency since

they are very strong functions of composition. Most of these pressures were estimated

by comparison of the present measurements with the phase-equilibrium data of Lu

and his colleagues.^*' For methane -I- ethane, the pressure corresponding to the

second point from one filling of the equilibrium cell was generally consistent with

phase-equilibrium data,^^''^°^ while the first point was usually high by a few per

cent Pressures for these mixtures were smoothed to the pressures of the second point

by comparison with the data of Miller and Staveley^*'^ and of Miller, Kidnay, and

Hiza.^^'' For methane + propane, discrepancies in the pressures between points

taken with the same fill were not as apparent, and the values given in table 3 were

not smoothed. Comparisons between the ratios of measured to Raoult's law pressures from the present study with those from the measurements of Stoeckli and Staveley,^-^*

Cutler and Morrison,^^^^ and Calado, Garcia, and Staveley^^^' are shown in figure 3. Pressures given for methane + /-butane and methane + ;i-butane, though no phase

7b
UQUID DENSITIES FOR BINARY MIXTURES OF LNG COMPONENTS 885

1 I ! I r

0.8580CH4 + 0.1420CjHg

LOl—1 1 i 1 1 1 90 100 110 120 130 140 r/K

FIGURE 3. Comparison of ratios of actual to ideal total vapor pressures for methane + propane

at two compositions. O, present stxidy; g, Stoecidi and Staveley;'*" A, Cutler and Morrison , Calado, Garda, and Staveley.**"

equilibrium measiu^ments are available for comparison, would probably exhibit

departures similar to those shown in figure 3.

ANALYSIS OF ERRORS IN LIQUID-MDCTURE DENSITIES

Where density measurements were made at five or more temperatures, amount-ofsubstance densities p at constant composition were fitted as a function of temperature

to the expression previously used in representing orthobaric amount-of-substance

densities of the pure components

p-p. = a{i-(r/Tj}''-"+ i &.{i-(r/r,)}(^^<'-^)/'>. (4)

The least-squares coefficients a and Z>f, and the values of the critical temperature

and critical amount-of-substance density Pj for each mixture thus treated are given

in table 4. The fourth coefficient was not statistically significant in the fit of any

of these results. Since experimental critical parameters are rarely available for mixtxires at the desired composition, those used here were obtained from the correlation of

Chueh and Prausnitz.^^*^ The standard deviations from the fit of the experimental

densities to equation (4) are also given in table 4 as a percentage of p. The largest

standard deviation obtained was 0.029 per cent in p for (0.35457CH4 -I- O.64543C2H6).

The overall uncertainty in the measured densities of the binary mixtures is slightly

larger than for the pure fluids due to the added uncertainty in composition and is dependent on the type of mixture. Estimated uncertainties in the density measure- ments for three representative mixtures—nitrogen + methane, methane -f- ethane,

and ethane + n-butane—are given in table 5. The estimated errors due to mixture

preparation were determined from the uncertainty in mole fraction of each component in the mixture for total amount of substance of mixture prepared (tables 1 and 2). The errors due to vapor-volume correction were estimated by assuming a volume imcertainty equivalent to that portion of the vapor volume in the equilibrimn

cell at the experimental temperature. The sources of the other systematic errors arc

77
886 M. J. HIZA, W. M. HAYNES, AND W. R. PARRISH

TABLE 4. Parameters of equation (4) where To and are the critical temperature and critical amount-of-substance density of the mixture. The coefficients a, bu aud 63 were obtained from a

least-squares program in wliich the experimental mass densities to five digits were convened to amount-of-substance densities within the program; c is the standard deviation

Mixture a bi ToK

Pc 10= X

mol dm"' mol dm"' mol dm"^ mol dm"'

0.04752N2 -f- O.95248CH4 16.41033 18.47257 -11.57943 188.04 10.258 0.011 O.30349N2 -h 0.6965 ICH4 14.92272 20.19078 -10.85599 173.20 10.747 0.023 0.49242N2 + O.50758CH4 14.54530 20.60585 -10.07112 161.48 11.018 0.014 O.35457CH4 -1- O.64543C2HS 13.02060 6.275239 276.40 8.196 0.029 O.49325CH4 -1- 0.50675C2Ha 12.99333 7.219530 262.24 8.762 0.028 O.68OO6CH4 -i- O.31994C2H, 18.86369 -16.08578 19.91557 239.93 9.482 0.003 0.49637CH4 + 0.50363C3H, 9.168092 7.647523 320.33 7.372 0.009 O.7492OCH4 -f O.25O8OC3H8 6.367272 24.40468 -11.53932 271.15 9.163 0.013 0.85796CH4 + O.I4204C3H8 8.371599 23.49229 -10.69945 241.28 9.836 0.016 0.91674CH4 -f- 0.083 26/J-C4H10 8.975024 12.09284 238.92 11.044 0.015 0.50105CsHa + 0.49895CjH8 9.598348 5.440200 344.42 6.125 0.015

0.671 17CaH« + 0.32882n-C^Hio 8.334169 5.886011 359.88 6.082 0.013 0.49030C3Ha + O.S0970i-C^Hio 7.647038 4.065571 39163 4.472 0.007

TABLE 5. Estimated uncertainties in density measurements

m5p\/p

Source of error 0.49242Na 0.49325CH4 0.671 nC^H,

+O.50758CH4 -t-0.50675C2H9 -h0.32883/r-C4H;

Systematic errors llOK 130 K llOK 130 K llOK 130 K

Mass of float 0.002 0.002 0.002 0.002 0.002 0.002 Volume of float at 300 K 0.020 0.020 0.020 0.020 0.020 0.020 Thermal expansion coefficient of barium ferrite 0.022 0.020 0.022 0.020 0.022 0.020

Position of float 0.012 0.013 0.012 0.012 0.009 0.009

Position of main coil. determined from relative measurements 0.020 0.020 0.020 0.020 0.020 0.020 Temperature uncertainty of

0.030 K 0.016 0.023 0.007 0.007 0.005 0.005 Mixture preparation 0.001 0.002 0.001 0.001 0.002 0.002 Vapor volume correction 0.006 0.020 0.001 0.005 0.000 0.000

Total systematic error

• 0.042 0.048 0.039 0.038 0.037 0.036 Three times standard

deviation " 0.042 0.048 0.081 0.084 0.039 0.039

Total uncertainty " 0.08 0.10 0.12 0.12 0.08 0.08

" The total uncertainty was determined from the square root of the sum of the squares of the systematic errors added to three times the standard deviation from the fit to equation (4).

76
UQUID DENSITIES FOR BINARY MIXTURES OF LNG COMPONENTS 887

the same as those discussed earlier^^^ for pure-fluid measurements. The total uncertainty was taken as the square root of the sum of the squares of the systematic

errors plus three times the standard deviation for random error. The standard

deviations are those obtained from the fit of the results to equation (4) taken as a

percentage of the mixture amount-of-substance density at the specified temperature.

For the estimate of random errors for pure-fluid measurements given in reference 2,

the standard deviation of a single density measurement was computed relative to the

density and temperature of methane at 105 K for which repetitive measurements

had been obtained. This method of assessing random error could not be used here. Since the standard deviation for methane obtained from equation (4)^^^ was in good

agreement with that computed from repetitive measurements, standard deviations for

the binary mixtures obtained with equation (4) are considered a reasonable basis for

estimating the random error.

It is probable that the overall uncertainties in the densities for some of the mixtures

investigated in this study are larger than those shown in table 5. Uncertainties in the

results for nitrogen + ethane and nitrogen + propane could exceed the maximum

total uncertainty shown in table 5 by a few hundredths of a per cent Because of the

dew-point limitations on the pressure of the gas mixtures prepared and the high

liquid-mixture vapor pressures, it was difficult to maintain adequate driving force

during the fill to provide continuous condensation and mixing. Of these two systems,

results for nitrogen + propane are considered the least reliable. Also, during the course of this study, difficulties were encountered in attempting to obtain consistent

results for methane + /-butane where xiCH^) > 0.9. This difficulty could have been

due to a problem in the filling procedure or to a dew-point related problem in the

prepared gas mixture. Though this problem was not encountered in measurements

of the (0.91674CH4 -f- 0.08326«-C4Hio) densities, it is quite possible that the overall

uncertainties in these results are also larger by a few hundredths of a per cent than

the maximum given in table 5. Fitting the results for mixtiu-es to equation (4) does not allow a test of their consistency as a function of composition. Where densities are available for a given binary

system at several temperatures and at three or more compositions, preferably at mole

fractions of approximately 0.3, 0.5, and 0.7, excess volumes can be examined for

consistency using a temperature-dependent Redlich-Kister expansion. In this study,

compositions for methane -i- ethane and methane + propane were selected to allow

this treatment As shown in table 2, each of these systems also includes one mixture

obtained from a laboratory with many years of experience in preparing gravimetric

standards. Excess volumes for these systems were fitted to a Redlich-Kister expansion

of the form:

F^/cm^ mol-^ = XiX2[{ao + ai(T/K)+a2(T/K)^}

+ {bo + biiTIK) + b:,{T{KY} (2;c, - 1) -h {co + Ci(T/K) + c^(TlYif}Clx, - 1). (5)

where is the mole fraction of methane. The least-squares coefficients a„ and c,, and the standard deviation, in per cent of the average mixture molar volume, for

each system are given in table 6. The C| coefficients were not statistically significant

7y
888 M. J. mZA, W. M. HAYNES, AND W. R. PARRISH

TABLE 6. Parameters of equation (5) for methane -r ethane and for methane + propane; ,Vi is the methane mole fraction and a the standard deviation

methane + ethane methane + propane

Co -14.22J 84 -11.96725

0.2649993 0.236748

fla -0.001419196 -0.001429237

6o -18.87392 16.59465 0.3108472 -0.3212125

b2 -0.001349755 0.001398221

Co -52.93613

Ci 0.9887147 -0.004667488

l(yff<F-i> 0.030 0.040

for methane + ethane. The excess volumes calculated from equation (5) are compared in figure 4 with the experimental values at 110 K from the present study.

The standard deviation for methane + ethane obtained with equation (5) is in very good agreement with that obtained with equation (4) for the methane + ethane

mixture in table 5. However, the standard deviation for methane + propane obtained

with equation (5) is more than a factor of two higher than that obtained with equation

(4) for any of the individual mixtures in the set The higher value from equation (5)

is considered to be more indicative of the random error in the results for methane +

propane.

The standard deviation given in table 6 for methane + propane was the highest

value obtained in the analysis. With a total systematic error of about ±0.04 per cent added to three times this standard deviation (±0.12 per cent), the overall uncertainty

for methane + propane would be approximately ±0.1 6 per cent. With the possible

exceptions noted above (i.e. nitrogen + ethane, nitrogen + propane, and methane +

/i-butane), this is believed to be a suitable estimate of the maximum overall uncertainty

in the densities of binary liquid mixtures obtained in this study.

0 V 1 1 1 1 i

1I 1I 1 1

I I 0 0.2 0.4 0.6 0.8 1 .x(CH^

FIGURE 4. Excess volumes for methane + ethane and methane + propane from the present

study at 110 K. O, experimental; , calculated from fit to equation (5).

8u
UQUm DENSITIES FOR BINARY MIXTURES OF LNG COMPONENTS 889

COMPARISONS OF EXCESS VOLUMES

Equation (5) has also been used to compare the present results for methane + ethane

and methane + propane with those of other investigators at 108.15 K, a temperature

approximately common to all of their measurements. The excess volumes were taken

directly from the literature where reported, or otherwise were calculated from the

pure-fluid molar volumes in their data set Though some of the results were 0.1 to

0.3 K below 108.15 K, the differences in temperature do not affect the comparisons

of excess volumes appreciably. From equation (5), a change in temperature of 0. 1 K

results in a change in excess volume of 0.001 cm^ mol"* for methane + ethane and

about 0.002 cm^ mol"* for methane + propane.

In figure 5, excess volumes for methane + ethane calculated from equation (5) at 108.15 K are compared with those from the experimental data of Shana'a and

Canfield/^^> Rodosevich and Miller/^*** Pan, Mady, and Miller,^^"^' Klosek and

McKinley,^^'^ and Jensen and Kurata,^^'^ The data from references 26 and 27 are

at about 108 K and those from reference 28 are at about 107.9 K. The excess volumes

from Shana'a and Canfield and from Miller and his colleagues are in excellent agree- ment with the curve calculated from the present results. With the exception of one

point each from Shana'a and Canfield and from Miller and his colleagues, the

differences are equivalent to much less than 0.1 per cent in the mixture molar volume.

In contrast, the excess volumes from Klosek and McKinley and from Jensen and

Kurata are much less consistent with the present results. The differences from the

calculated curve are equivalent to 0.15 to 0.3 per cent in the mixture molar volume.

In figure 6, excess volumes for methane + propane calculated from equation (5)

81
890 M. J. mZA, W. M. HAYNES, AND W. R. PARRISH

I 1 —J I \ 1 0 0.2 0.4 0.6 0.8 1

FIGURE 6. Comparison of excess volumes for methane + propane at 108.15 K. , Calculated from fit of present results to equation (5); , Shana'a and Canfield;'*" A, Rodosevich and Miller-/^*' A, Pan, Mady, and MUler;'"> O, Jensen and Kurata.'29>

at 108.15 K are compared with those from the experimental data of Shana'a and

Canfield/^'^ Rodosevich and Miller/^^^ Pan, Mady, and Miller,^^''^ and Jensen and

Kurata.^^^^ For this system the excess volumes from Shana'a and Canfield and from

Miller and his colleagues are also in excellent agreement with the curve from the

present results with a maximum difference of about 0.06 per cent in the mixture

molar volume. The excess volume from Jensen and Kurata again is not consistent

with the present results. The difference is about 0.5 per cent in the average mixture

molar volume, in the opposite direction from that found for methane + ethane.

Though there are only a few methane + /i-butane points reported in the literature, a comparison can be made at 108.15 K in the methane-rich region between the excess volume interpolated from the present results for the (0.91674CH4 + O.O8326/J-C4H10)

mixture and excess volumes from the experimental data reported by Shana'a and

Canfield^^^' and by Miller.^^^' This comparison is given in figure 7. The excess volume

from Shana'a and Canfield was computed using their pure methane data and the

_n 0 1 I

1

^ ' 0.90 0.95 1 x(CH^

FIGURE 7. Comparison of excess volumes of methane + n-butane at 108.15 K. O, Interpolated

from the present study; , Shana'a and Canfield A. Rodosevich and Miller.'"'

82
UQUID DENSITIES FOR BINARY MIXTURES OF LNG COMPONENTS 891

molar volume of «-butane obtained from equation (3). The excess volumes of Miller are those reported in his paper. The «-butane molar volumes used by Miller are within 0.02 per cent of the values given by equation (3). The difference between the

excess volume interpolated from the present results and the curve drawn through the

excess volumes of Shana'a and Canfield and of Miller is about 0.14 per cent of the

mixture molar volume. Since the data from Shana'a and Canfield and from Miller

appear to be in as good agreement for methane + n-butane as for methane + ethane

and methane + propane, it is possible that at least part of the 0.14 per cent difference

is due to an unaccounted error in our results for the mixture, as suggested earlier in

this paper. Unfortunately, data are not available for comparison with the present

results for (0.58828CH4 + 0.41172n-C4Hio), the accuracy of which we have no

reason to question. Measurements for this mixture could not be made at the lower

temperatures since the n-butane content is the solid solubility limit at about 1 16 K.^^°^

-0.5 'I 5 1

1 2 -1.0 e

—

B

-1.5

1 -1 ._. 1 1 1 1 I I l_ 100 110 120 130 140 T/K

FIGURE 8. Comparison of excess volumes from the present study for methane + heavy hydrocarbon mixtures having x x 0.5, O, 0.49325CH4 + 0.50675C3H«; , 0.49637CH4 + 0.50363C9H«;

A, 0.48687CH« + 0.51313/-C«Hio; O. 0.58828CH* + 0.41172rt-C«Hio.

The only data for methane + /-butane in this temperature range are those of

Rodosevich and Miller^^^^ for two mixtures with mole fractions 0.9152 and 0.9462

of CH4. The excess volumes they report for these mixtures at 108 K are about half- way between their results for methane + propane and for methane + n-butane at the same methane mole fraction. The excess volumes for the mixture of methane -f- /-butane with x « 0.5 from the present study are equivalent to those obtained for

methane -I- propane at 110 K and the same composition. Excess volumes from the

present study are shown in figure 8 for mixtures with x « 0.5 of methane -i- ethane,

-f" propane, and + /-butane. Excess volumes are also included for (O.58828CH4 +

0.41172«-C4Hio) for comparison. It is clear that more extensive measurements of

high precision are needed to describe accurately the composition dependence of the

excess volumes for both methane + butane systems.

The small excess volumes given in table 3 for binary mixtures containing the

higher molar-mass alkanes—ethane, propane, /-butane, and /x-butane—show that at

83
892 M. J. HIZA, W. M. HAYNES, AND W. R. PARRISH

cryogenic temperatures the molar volumes for binary mixtures of these components

are closely approximated by the assumption of ideal mixing. The only other investi- gators who report data for binary mixtures of these alkanes at cryogenic temperatures

are Shana'a and Canfield/^^^ They report densities for several mixtures of ethane +

propane and for one mixture of ethane + /i-butane.

For ethane + propane, the excess volume reported by Shana'a and Canfieid at 108.15 K for the mixture (0.5852C2H6 + O.4148C3H8) is compared in figure 9 with

those determined from the present results at two similar compositions. If the excess volumes for ethane + propane exhibit an asymmetry similar to those for methane +

ethane (figure 5), the excess volumes for the two ethane + propane mixtures from

the present study should be nearly the same, with the excess volume at the higher

mole fraction of CjHg shghdy larger. Shana'a and Canfieid report a maximum

-0.1 h

I

-0.2 h

-0.3

-0.4

-0.5

1 0.1 per cent in

105 110 115 120 125 130 135 T/K

FIGURE 9. Comparison of excess volumes for ethane + propane. O, O, Present study at xiCilie) =- 0.50105 and 0.67287, respectively; , Shana'a and Canfieid'"' at .r(QH,) =- 0.5852.

excess volume for this system of — 0.485 cm^ mol~^ for a mixture containing

(O.7O36C2H5 + 0.2964C3Hg). The difference in excess volumes from the two sets of

measurements is surprisingly large, equivalent to approximately 0.8 per cent in the

mixture molar volume. Since the pure-fluid molar volumes for ethane and propane

from Shana'a and Canfieid are only higher than those determined with the present

apparatus^^^ by approximately 0.06 and 0.12 per cent, respectively, it is logical to suspect composition error or incomplete mixing as the source of this disagreement

To assess the efiect of composition uncertainty on the excess volume, it was assumed

that the mixture mass density from the present study was exact and that the actual

mole fraction of propane was 0.01 larger than the composition of the prepared

mixture with x » 0.5. This results in an error of — 0.062 cm^ mol"^ in the excess volume. If the excess volume reported by Shana'a and Canfieid were assumed to be

correct, it would be necessary for the actual mole fraction of ethane in the liquid in

the present study to be larger by 0.064 than that of the prepared mixture.

84
UQUID DENSITIES FOR BINARY MIXTURES OF LNG COMPONENTS 893

As discussed in the experimental section of this paper, chromatographic comparisons of the compositions of liquid samples for ethane + propane were in agree- ment with the composition of the prepared mixture within a few hundredths of 1 mole

per cent The fact that no change in density of the mixture could be detected during

sample withdrawal is considered evidence that the liquid mixture was homogeneous.

The fact that the two mixtures studied were prepared independently in two separate

laboratories is also confirmation that there was no significant error in the mixture

preparation.

Recent enthalpy of mixing measurements in binary mixtures of methane, ethane,

and propane reported by Miller and Stavele/"^ should also be referred to in regard

to the ethane + propane results discussed above. These investigators report that the

enthalpy of mixing at 112 K is about 129 Jmol"^ for methane + propane with

X = 0.5, about 69 J mol~^ for methane + ethane with x = 0.5, and probably less than 5 Jmol"^ for ethane + propane throughout the composition range. A comparison of ratios of the enthalpies of mixing with ratios of the excess volmnes for these

systems also tends to support the more ideal excess volumes obtained in the present

investigation.

Shana'a and Canfield include a discussion on the invalidity of the principle of

congruence based on a comparison of their excess volumes for ethane + propane

and methane + ethane with those for methane + propane as a function of the

equivalent carbon number. When the excess volumes from the present study are

used, methane -I- ethane shows the same departure as expected, but ethane 4- propane

is in excellent agreement with methane + propane. For a detailed discussion of the

method, the reader is referred to Shana'a and Canfield's paper.

In figure 10, the excess volume computed from the molar volume of the (O.SSSSCzHg

+ 0.1167«-C4Hio) mixture reported by Shana'a and Canfield is compared with the

excess volimies of this system computed from our results. The excess volume of

Shana'a and Canfield was computed using their density for pure ethane and the molar

0 I

1 1 1 1

i

1 1

l_o

-0.1

s

0.1 per cent in V„

-0.2' 110 115 120 125 130 135 140

FIGURE 10. Comparison of excess volumes for ethane + n-butane. O, -o-. Present study at xiCJcU) = 0,67117 and 0.65343, respectively; , Shana'a and Canfield""" at xiCiiU) - 0.8833.

61

bb
894 M. J. HIZA, W. M. HAYNES, AND W. R. PARRISH

volume of n-butane from equation (3). For this system the difference in excess volumes

is eqmvalent to about 0.25 per cent in the mixture molar volume. Since the molar

volumes for n-butane in the subcooled liquid region given by equation (3) were

obtained from the results for this system the lack of temperature dependence is an

artificial constraint For this system, if the actual mole fraction of n-butane in the

liquid mixture at 110 K was 0.01 larger than the composition of the mixtures prepared

for this study, an error of —0.119 cm^ mol~^ in the excess volume would result.

0 0.1 0.2 0.3 0.4 0.5 x(N2)

FIGURE 11. Comparison of excess volumes for nitrogen -i- methane. O, Present study; , Rodosevich and MiUer;"" A, Liu and Miller;"" V, Massengill and Millcr.<^='

Of the data available for liquid mixtures of nitrogen + methane,^*^ the excess volumes reported by Rodosevich and Miller, Liu and Miller,^^^^ and Massengill

and Milier^^^' were selected for comparison with excess volumes from the present

results. These comparisons are shown in figure 11. The excess volumes of Massengill

and Miller are from the data of Liu and Miller adjusted with additional nitrogen

measurements. All of the excess volumes of Miller and his colleagues appear to be

in agreement with excess volumes from the present results within 0. 1 per cent of the

mixture molar volumes. The height of the symbols is equivalent to about 0.1 per cent of the mixture molar volumes.

The fact that the nitrogen + methane excess volumes are in good agreement does

not mean that the mixture molar volumes are in agreement. It is worth noting that

the data reported by Liu and Miller and by Massengill and Miller were obtained with

bb
UQUID DENSITIES FOR BINARY MIXTURES OF LNG COMPONENTS 895 a calibration based on the molar volumes of saturated liquid argon reported by

Terry et alP^^ while all of the other data of Miller and his colleagues^^^* '^'^^ referred to here were obtained with a calibration based on molar volumes of saturated liquid

methane taken from Goodwin and Prydz.^^*^ From the discussion given earlier/^^

the nitrogen + methane molar volumes of Rodosevich and Miller should be higher

but consistent with the present results within about 0.1 per cent, while those of Liu

and Miller and of Massengill and Miller should be higher than the present results by about 0.4 per cent

4. Smninaiy

The measurements of the present study provide a set of orthobaric liquid densities

for binary mixtures of the major components of LNG which are generally consistent

with those for the pure components within the precision of the measurements. The

maximum random error in the densities of the binary mixtures is believed to be about

±0.12 per cent, roughly twice that for the pure fluid densities. The known systematic

errors in the densities for both the pure fluids and the binary mixtures were estimated

to be from 0.03 to 0.05 per cent As noted, the uncertainties in the densities for

methane + /j-butane with ^(CHJ > 0.9, and for nitrogen + ethane and nitrogen +

propane, could exceed these estimates.

The largest excess volume found was —6.11 cm^ mol"^ for nitrogen + methane

with X = 0.5 at 120 K. This is equivalent to 13 per cent of the ideal mixture molar

volume. In contrast, the present results show that molar volumes of mixtures of the

alkanes, excluding methane, are closely approximated by ideal mixing in the

temperature range studied. The most significant disagreement between the present

results and published data is the 0.8 per cent difference in density for ethane +

propane from that of Shana'a and Canfield.

Based on the excess volumes of the alkane mixtures, excluding methane, from the

present study, it was shown also that effective molar volumes of subcooled n-butane

in a binary liquid mixture may differ by a small but significant amount dependent on the other components present It is suggested that this behavior would be worth

investigating in more detail for n-butane and other fluids such as /j-pentane. There is also a definite need for additional measurements on the binary mixtures containing

both n-butane and /-butane with methane.

In subsequent papers, the results of measurements of orthobaric liquid densities

of multicomponent mixtures and the ability of theoretical models to predict the

measured densities will be reported.

The authors acknowledge the contributions of M. J. Brown, R. C. Miller, and R. D.

McCarty to the correlation of the results.

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3. Haynes, W. M.; Hiza, M. J. /. Chem, Thermodynamics 1977, 9, 179.

d7
896 M. J. HIZA, W. M. HAYNES, Aim W. R. PARRISH

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Weitzei, D. H.: editors. Plenum Press: New York. 1976, p. 509.

31. Liu, Y.-P.; Miller, R. C. /. Chem. Tfiermodynamics 1972, 4, 85. 32. Massengill, D, R. ; Miller, R. C. /. Chem. Tliermodynamics 1973, 5, 207,

33. Terry, M, J.; Lynch, J. F.; Bunclark, M.; ManseU, K. R.; Staveley, L. A. K./, Chenu Thermodynamics 1969, 1, 413.

34. Goodwin, R. D.; Prydz, R. /, Res. Nat. Bur. Stand. US. 1972, 76A, 81,

88
Reference [17j

0-37

J. Chem. Thermodynamics 1983, 15, 903-911

Orthobaric liquid densities and dielectric

constants of (methane + 2-methylpropane) and (methane + /i-butane)

at low temperatures^

W. M. HAYNES

Thermophysical Properties Division, National Engineering Laboratory,

National Bureau of Standards, Boulder, Colorado 80303, U.S.A.

( Received 12 January 1983)

Measurements of the orthobaric liquid densities and dielectric constants of methane-rich (methane + 2-methylpropane) and (methane + n-butane) have been obtained at temperatures between 110 and 140 K. Densities were determined with a magnetic-suspension densimeter, while a concentric-cylinder capacitor was used for simultaneous measurements of dielectric constant. These measurements were part of an experimental program that has provided a

consistent and comprehensive set of densities for the major components of liquefied natural gas (LNG) and their mixtures, which was used to develop mathematical models for the calculation or prediction of LNG densities. Along with the (methane -(- 2-methylpropane) and

(methane + n-butane) experimental densities and dielectric constants, are presented

experimental vapor pressures, as well as excess volumes, Clausius-Mossotti functions, and

excess Clausius-Mossotti functions, derived from the densities and dielectric constants. Comparisons are shown between the excess volumes of the present work and those from

independent measurement using an extended corresponding-states model that had been

optimized to the results from this work. The total uncertainty of a single density measurement

is approximately O.I per cent with a precision of a few parts in 10*. Dielectric constants are estimated to be accurate to approximately 0.05 per cent. A brief description of the apparatus, experimental method, and procedures is also presented.

1. Introduction

An experimental program*'"*' has been carried out at this laboratory to measure

with an uncertainty of less than 0.1 per cent the orthobaric liquid densities of the major components" of liquefied natural gas (LNG) and mixtures**- of these components. The pure fluid and binary mixture results have been incorporated into the development of several mathematical models*'"'^* that can be used to predict

within an uncertainty of less than 0.1 per cent the density of any LNG mixture at

" This work was carried out at the National Bureau of Standards under the sponsorship of the British Gas Corporation, Chicago Bridge and Iron Co., Columbia Gas Service Corp., Distrigas Corp., Easco Gas

LNG, Inc., El Paso Natural Gas, Gaz de France, Marathon Oil Co., Mobil Oil Corp., Natural Gas

Pipeline Co., Phillips Petroleum Co., Shell International Gas, Ltd., Sonatrach, Southern California Gas

Co., Tennessee Gas Pipeline Co., Texas Eastern Transmission Co., Tokyo Gas Co., Ltd., and

Transcontinental Gas Pipe Line Corp., through a grant administered by the American Gas Association,

Inc.

56

89
904 W. M. HAYNES

saturation from an input of the temperature, composition, and pressure of the liquid

mixture. Multicomponent-mixture results have been used to test the models.

Orthobaric liquid densities from this project have been previously reported for nitrogen,"' methane,*''-" ethane,*^' propane,*^* 2-methylpropane,''' and n-butane;*^-'*'

all binary combinations*** of those components except (nitrogen + 2-methylpropane)

and (nitrogen + n-butane); and selected multicomponent mixtures'^' ^' with up to

eight components that included 2-methylbutane and n-pentane in addition to those

mentioned above. In an earlier paper**' describing binary-mixture measurements,

the need for additional measurements on binary mixtures of methane with either 2-methylpropane or n-butane was cited. Experimental problems had been

encountered in the earlier study in obtaining consistent results for methane-rich

binary mixtures containing 2-methylpropane or n-butane. It was conjectured that these difficulties were associated with liquid methane condensing in the narrow

annulus around a vapor-pressure bulb that fitted tightly inside the top part of the

cell. Thus, the present work was carried out to resolve these inconsistencies and to provide values for these mixtures that would complement and extend the earlier

results. The earlier work provided results for {xCH4-F(l —xXCH3)2CHCH3} and

{xCH4-l-(l — x)C4H,o} with x « 0.5 while the present study includes results for methane-rich mixtures (x from 0.78 to 0.93). The density results from the present study were obtained using a magneticsuspension densimeter*'-'' that was part of a new apparatus,*'-^* which differed

significantly from that employed for the previous measurements.**' (The new

apparatus*^' was developed because a magnetic-suspension densimeter of higher- pressure capability, 35 MPa, was desired for other projects.) Although the new

apparatus exploited the same technique for density measurements as that used in the

earlier work and gave results that agreed, within experimental precision, with those

obtained with the previous apparatus, there were significant new and improved

features of the apparatus that were important to the present work. These will be

discussed in the next section. The new apparatus contained a capacitor that was used for obtaining dielectric constants along with the densities. Through the use of the Clausius-Mossotti

function, dielectric constant measurements can serve as a simple and reliable substitute for density measurements. In the custody transfer of LNG, densities are determined by direct measurements or indirectly using a calculational technique

based on measurements of composition and temperature. One technique for commercially available densimeters is predicated on a density determination

through the application of dielectric-constant measurements. Thus both the

dielectric constants and densities from this study can contribute to equitable custody

transfer of such a valuable commodity as LNG.

2. Experimental

The experimental method, apparatus, and procedures for density measurements

have been discussed in detail in other papers.*'"®' Thus, this section will focus on a

description of the technique and procedures for the dielectric-constant

90
DENSITIES OF {xCH^ + (l -x)(CH,)jCHCH3} AND {xCH^ + (l -x)QH,o} 905

measurements, along with a discussion of those features of the apparatus and

procedures that are specific to the present study.

The dielectric-constant measurements were performed with a concentric-cylinder

capacitor of a design similar to that of Younglove and Straty."^* There is one major

difTerence between their design and that used here. Instead of solid cylinders, the capacitor in the present work consisted of cylinders with slots cut parallel to the

cylindrical axes, similar to the ring-and-bar design of Pan et a/.'*** This modification was carried out for the same reason as that given by Pan et ai: to reduce as far as possible the chances for composition gradients in the cell. The copper capacitor was

located in the top portion of the cell, while the magnetic buoy and position sensor

for the magnetic-suspension densimeter were contained in the lower portion.

Assuming the absence of composition and temperature gradients, simultaneous

measurements of density and dielectric constant were made on the same liquid

samples.

The dielectric constant was determined from measurements of the ratio of the capacitance with liquid between the electrodes to the capacitance in vacuo. The

capacitance in vacuo was approximately 20 pF. The overall external dimensions of

the capacitor were 6.4 cm in length by 1.77 cm in diameter. The widths of the five slots in each cylinder were 0.32 cm, while the slot lengths in the outer cylinder were 4.1cm and those in the inner cylinder were 1.9 cm. Kaptonf film insulators of

0.05 mm thickness were used between the cylinders and the support assembly. The

entire capacitor assembly was fabricated from copper and subsequently plated with

gold. Capacitances were measured with a three-terminal a.c. bridge operated at an

oscillator frequency of 5 kHz.

Although the resolution of the measured capacitances was approximately

10"* pF, the overall uncertainty in the dielectric-constant measurements for the mixtures investigated here was approximately 0.05 per cent. The limiting factor was

the reproducibility of the measurements, which was based on measurements from

several fillings taken from the same cylinder of prepared mixture. The dielectric- constant determination was the most sensitive monitor inside the sample space to indicate the reproducibility of liquid compositions each time the cell was recharged.

In vacuo capacitances, measured to a precision of approximately 10~*pF, were

observed before each run and were stable to better than 10" pF over the duration

of this project.

The densities were determined using a one-solenoid magnetic-suspension

densimeter''* with a differential-capacitance sensor" used to detect the position of

the magnetic buoy. The total uncertainty" ** in the density determination was

estimated to be 0.12 per cent, which includes an allowance of three times the standard deviation for random error. As demonstrated in a following section, the precision of measurement was better than 3 x 10"*.

Procedures for preparing mixtures of LNG components gravimetrically have been

t In order to describe materials and experimental procedures adequately, it is occasionally necessary to

identify commercial products by manufacturer's or trade names. In no instance does such identification imply endorsement by the National Bureau of Standards, nor does it imply that the particular product is necessarily the best available for that purpose.

91
906 W. M. HAYNES

discussed previously.**' For the mixtures investigated here, at least 3.8 mol was

prepared for each composition. This corresponds to less than 0.0001 uncertainty in the mole fraction of CH4 and considerably less uncertainty for either

(CHajzCHCHa or C4H,o.

Vapor pressures were measured with a spiral quartz Bourdon-tube gauge with a range of 0 to 1.38 MPa that had been calibrated against an air dead-weight gauge. Maximum uncertainty in the calibration was 70 Pa. Vapor-pressure measurements

were generally reproducible to within 3 x lO"* MPa for different fillings of the cell with the same mixture.

Prior to the filling of the cell with each new mixture sample, a density

measurement was performed on liquid methane to serve as a check on the densitymeasurement process. The densities were obtained to ensure that the ahgnment of

the cryostat and cell windows had not changed as a result of temperature cycling of

the apparatus. The windows had been aligned, along with the microscope, so that the buoy-position determination was independent of the refractive index of the fluid inside the cell. The cell had to be warmed to room temperature to effect complete

removal of mixtures containing butanes. Measurements in vacuo needed for both

density and dielectric-constant determinations were obtained before each new fiUing

of the cell. A maximum of four points at 5 K increments was obtained for each

filling. The mixtures were condensed directly and continuously into the cell at a pressure

greater than the vapor pressure of methane either through two capillaries at the

bottom of the cell or simultaneously through capillaries at both the top and bottom

of the cell. Based on the consistency of the results, it was found that, within the

precision of the measurements, the density and dielectric-constant results were

independent of the filling routines used.

All of the measurements in this project were made on liquid mixtures in equilibrium with vapor. To minimize composition or density errors, it was of the utmost importance to reduce as much as possible the low-temperature vapor

volume in the sample space. The total internal volume of the sample cell was 43 cm-', while the low-temperature vapor volume was less than 0.2 cm-'. Assuming that the

vapor composition was pure methane for the mixtures studied here, the maximum

adjustment (decrease) to the measured densities because of differences in composition between the prepared mixtures and the mixtures condensed into the

cell was 0.045 per cent, which was at 140 K for the mixtures with x % 0.78. This

corresponded to a correction of 0.03 per cent in the dielectric constant.

3. Results and discussion

The experimental orthobaric liquid densities, vapor pressures, and dielectric constants for (methane -I- 2-methylpropane) and (methane -I- n-butane) are presented

as a function of temperature (IPTS-68) in table 1. Excess molar volumes and

Clausius-Mossotti functions, as well as excess Clausius-Mossotti functions, are also given in table 1. The excess molar volume is defined by the relation:

Kn' = K„-L^iK*{i+'Ci(pr-P)}. (1)

92
DENSITIES OF {xCH^ + (l -xKCHj^CHCHj} AND {xCH^ + (I -xjQH.o} 907

TABLE 1. Orthobaric liquid densities and dielectric constants of {xCH« + (l -xKCHjljCHCH,} and {xCHt + (l —xJC^Hio}. M, molar mass; T, temperature (IPTS-68); p pressure; p„ experimental amountof-substance density; p„p„ experimental density; p^,,^, density calculated from equation (4); V^, excess molar volume; e, dielectric constant; ^q^,, Clausius-Mossotti function; <^cm> excess Clausius-Mossolti

function

T

K

P

e <^CM 4>CM MPa mol dm '"

p„,t cm' mol'' cm' -mol"' cm' mol"'

0.92044CH4+0.07956(CHj)jCHCH3 (M = 19.3910gmor')

1 15.00 0.1254 24.3633 -0.016 -0.547 1.69665 7.7352 0.024 120.00 0.1805 24.0029 -0.006 -0.624 1.68372 7.7326 0.021 125.00 0.2521 23.6403 0.011 -0.726 1.67103 7.7322 0.020

130.00 0.3434 23.2752 0.037 -0.859 1.65838 7.7321 0.018

135.00 0.4567 22.8920 0.004 -1.001 1.64527 7.7326 0.017 22.5037 -0.030 -1.186 1.63196 7.7320 0.015

0.78329CH4 + 0.21671(CH3)jCHCHj (M = 25.1625 g-moP')

110.00 0.0782 21.9144 0.005 -0.837 1.81689 9.7662 0.050

115.00 0.1164 21.6652 0.001 -0.941 1.80487 9.7639 0.051

V. i o /

1

21.4136 -0.011 -1.065 1.79271 9.7605 0.050

125.00 0.2329 21.1668 0.006 -1.229 1.78094 9.7581 0.049

130.00 0.3170 20.9125 -0.008 -1.412 1.76906 9.7571 0.049

135.00 0.4208 20.6629 0.006 -1.648 1.75729 9.7543 0.047

140.00 0.5474 20.4082 0.001 -1.925 1.74568 9.7548 0.047

0.92788CH4 + 0.072 nQH.o (M = 19.0779 g - mol

190 1 ZVJ.VA/ on nU. 11 R^n OZU 24.2615 -0.006 -0.661 1.67289 7.5511 0.007

125.00 0.2547 23.8868 0.005 -0.756 1.66033 7.5524 0.005

130.00 0.3470 23.5047 0.000 -0.872 1.64764 7.5538 0.003

135.00 0.4616 23.1225 0.012 -1.031 1.63477 7.5528 -0.001

140.00 0.6023 22.7284 -0.011 -1.219 1.62190 7.5547 -0.002

0.92780CH* +0.07220C4H,o (M = 19.0813 g - mol

115.00 0.1270 24.6285 -0.586 1.68465 7.5445 0.002

120.00 0.1824 24.2783 -0.693 1.67243 7.5418 -0.003

125.00 0.2549 23.8999 -0.782 1.65994 7.5446 -0.003

0.77982CH4 + 0.220 18C4H,o (M = 25.3085 g- mol -')

120.00 0.1702 21.6066 0.003 - 1.288 1.78385 9.5877 0.006

125.00 0.2369 21.3549 -0.002 -1.444 1.77286 9.5925 0.007

130.00 0.3228 21.1020 -0.010 -1.633 1.76202 9.5989 0.009

135.00 0.4291 20.8555 0.016 -1.879 1.75113 9.6013 0.008 140.00 0.5576 20.5978 -0.007 -2.151 1.73990 9.6049 0.007

0.77762CH4 +0.22238C4H,o (M = 25.4011 gmol -')

115.00 0.1179 21.8054 -0.027 -1.135 1.79554 9.6122 0.005

120.00 0.1699 21.5601 -0.001 - 1.273 1.78485 9.6180 0.007

125.00 0.2372 21.3164 0.037 -1.444 1.77382 9.6193 0.004

130.00 0.3230 21.0605 0.021 -1.623 1.76272 9.6249 0.005

135.00 0.4291 20.801

1

-0.009 -1.836 1.75139 9.6291 0.005 140.00 0.5579 20.5448 -0.021 -2.107 1.74020 9.6328 0.004

93
908 W. M. HAYNES

where V„ is the molar volume of the mixture at a given temperature at saturation pressure p, V* is the molar volume of pure component i at the same temperature at saturation pressure pf, x, is the mole fraction of component i, and is the isothermal compressibility of component i. The Clausius-Mossotti function 0cm is defined by the expression:

<^cM = (£-l)/(£ + 2)p„, (2) where p„ is the amount-of-substance density and e is the dielectric constant. Then

the excess Clausius-Mossotti function </>cm for a liquid mixture is defined,

analogously to V^, by the relation:

0CM = 0CM-L^i<^CM.P (3) where </)cm refers to the Clausius-Mossotti function of the mixture at a given

temperature at saturation pressure and </»cm.b is the Clausius-Mossotti function of

pure component i at the same temperature and pressure as the mixture. Along an

isotherm, the Clausius-Mossotti function is a very slowly -varying function;

therefore, adjustments of saturated-liquid pure-component Clausius-Mossotti

values to the saturation pressure of the mixture are sufficiently small to neglect. In the calculation of and 0cm, i> the pure-component molar volumes of CH4,

(CH3)2CHCH3, and C4H10 were calculated from equations*^' used to fit the

experimental orthobaric liquid densities obtained with the first version of the

magnetic-suspension densimeter used in the LNG density project. For 2-methylpropane and n-butane, the equations were used for extrapolation below their triplepoint tempieratures. The molar volumes of CH4 were adjusted to the mixture

pressure using vapor pressures from Goodwin*'^' and isothermal compressibilities

from Rowlinson.'^^' The compressibility corrections for 2-methylpropane and n- butane were sufficiently small to disregard. The dielectric constants of the pure

components, used in the calculation of 0cm. i' ^^^e taken from Straty and

Goodwin'*^' for CH4 and from Haynes and Younglove<^^' for (CH3)2CHCH3 and

C4H10.

The binary-mixture results presented in this paf>er, along with the previous purefluid and binary-mixture results from the LNG density project, have been used to optimize several mathematical models'^" for the prediction of LNG densities. The

most accurate and versatile of the models is the extended corresponding-states

model,'^""' which will be utilized here to demonstrate the consistency of the results for (methane -I- 2-methylpropane) and (methane -f- n-butane) from the LNG density

project at this laboratory and to make comparisons with values for these mixtures

from other sources. In figure 1 the experimental results for {xCH4 + (l -xXCH3)2CHCH3} from the LNG density project and from Rodosevich and Miller*^°' are compared with values

calculated from the extended corresponding-states method. The differences are less than 0.1 per cent for all points, including the values for x % 0.5 obtained earlier**' in the LNG density project. Figure 2 shows comparisons of experimental results for {xCH4-|-(l — x)C4Hio} from the LNG density project, from Miller,*^" and from

Shana'a and Canfield,*^^* with values predicted from the extended corresponding94
DENSITIES OF {xCH^+(l -x){CH,)jCHCH,} AND {xCH^+d -x)C4»to} 909

1 1

a

1 i

_

^.

- Z » e e

O o

100 110 120 130 140 T/K

FIGURE 1. Comparison of experimental densities of {xCH4+(l -xXCHjjjCHCHj} from this work, from Hiza et al.,^** and from Rodosevich and Miller/^"" with values calculated from the extended

corresponding-states model."""* O, This work, x = 0.92044; A, this work, x = 0.78329; j^c, Hiza et al, X = 0.48687; V, Rodosevich and Miller, x = 0.9462; T, Rodosevich and Miller, x = 0.9152.

States model. In general, the agreement is good (within ±0.1 per cent) except for the

results for (0.92CH4 + 0.08C4Hioy*' obtained with the first version of the magneticsuspension densimeter used in the LNG density project. As reported earlier in this paper, experimental difficulties were encountered with that apparatus in performing

measurements on methane-rich mixtures containing heavy hydrocarbons. Therefore,

the results for (O.92CH4-I-O.O8C4H10) were not included in the development of the LNG mathematical models.***' The excess molar volume is plotted as a function of x in figure 3 for isotherms

at 115 K for {xCH4 + (l -xXCH3)2CHCH3} and {xCH4-H(l -x)C4Hio}. The

smoothed curves represent values calculated from the extended corresponding-states

model. The results from the LNG density project are shown on the figure. The

maximum excess molar volume for {xCH4-t-(l — x)C4Hio} at 115 K, — 1.24cm-'-mol~* at x = 0.65, was approximately 20 per cent larger than that observed for {xCH4-l-(l — xXCH3)2CHCH3} at the same temperature.

For those compositions with points at five or more temperatures, the experimental amount-of-substance densities p„ have been fitted as a function of

temperature T to the relation:

(Pn-Pn.c)/(niol-dm-^) = a(l-r/rj°-^^+ f - TO* (4) i= 1

T/K

FIGURE 2. Comparison of experimental densities of {xCH4-(-(l -x)C4H,o} from this work, from

Hiza et a/.,**' from Miller,'^" and from Shana'a and Canfield,'^^' with values calculated from the extended

corresponding-states model.*'"'" O, This work, x = 0.92788; •, this work, x = 0.92780; A, this work, X = 0.77982; A, this work, x = 0.77762; , Hiza et ai. x = 0.91674; , Hiza et ai. x = 0.58828; V.

Miller, x = 0.9506; T, Miller, x = 0.9206; 3^, Shana'a and Canfield, x = 0.8843.

9b
910 W. M. HAYNES

0 0.2 0.4 0.6 0.8 1 X FIGURE 3. Excess molar volumes of (a), {xCH^ + (l -xXCHjj^CHCHj} and (b), {xCH4 + (l — x)C4H,o} as a function of x. Smoothed curves represent values calculated from the extended

corresponding-states model.""'" Points from the LNG density project are shown: O, This work; ,

Hiza et al.'*>

where and p„ ,, are the critical temperature and the critical amount-of-substance

density; a and 6, are coefficients determined by least squares. (This equation had

been used previously*^' to represent the orthobaric liquid densities of the pure

components*'" of LNG and selected binary mixtures*** of these components.) For the

results presented here, only two coefficients were statistically significant in the fits. The critical temperatures and densities were obtained from Chueh and Prausnitz*^^'

using the pure-component critical-point values given in refei-ence 3. These fits were

relatively insensitive to the selected critical-point values. The results of the fits are given in table 2. The maximum standard deviation was 0.028 per cent in the density,

which agrees with that observed in the other binary mixture measurements reported

earlier**' for the LNG density project. Deviations of the experimental densities from

values calculated from equation (4) are given in table 1. The systematic errors involved in density measurements on mixtures with the magnetic-suspension technique used here have been discussed in detail elsewhere.*' *' The total uncertainty is taken as the square root of the sum of the squares of the systematic errors plus an allowance of three times the standard

TABLE 2. Parameters of equation (4) where and ^ are the critical temperature and critical amount-ofsubstance density of the mixture"

Mixture a

7;K

P-.c moldm"^

0.92044CH4 + 0.07956(CH3)2CHCH3 11.21295 10.64118 232.19 10.17 0.026 0.78329CH4 -l-0.21671(CH3)jCHCHj 6.265006 11.14928 285.80 9.77 0.007 0.92788CH4 + 0.072 1 2C4H , o 9.448745 12.23371 233.14 10.99 0.010 O.77982CH4 + 0.2201 8C4H , 0 4.436119 12.72402 297.59 10.31 0.012 0.77762CH4-i-0.22238C4H,o 4.238446 12.93083 298.37 10.29 0.028

"The coefficients (a, /»,) were obtained from a least-squares program in which the experimental mass

densities to five digits were converted to amount-of-substance densities within the program.

96
DENSITIES OF {xCH, +(1 -x)(CH,)jCHCH,} AND {xCH,+(I -xjQH.o} 91 1

deviation for random error. The justification for using the standard deviation from

equation (4) as the basis for estimating the random error was discussed in detail in references 3 and 4. Combining the maximum standard deviation of 0.028 per cent from the fits to equation (4) with the systematic errors from references 1 and 4, the

total uncertainty of a single density measurement for the mixtures studied here was

estimated to be ±0.12 per cent. As noted, computed values of the excess Clausius-Mossotti function, as defined in equations (2) and (3), are given in table 1. A distinct difference was observed in the behavior of (/)cm for the two mixtures investigated here. For {xCH4 + (l — x)C4Hio},

both nonpolar species, the calculated values were less than 0. 1 per cent of the mixture (pcM values. Within the precision of the measurements for {xCH4 + (l — x)C4Hio}, there was little, if any, evidence of a relation between the

<j!)cM values and the composition. There was some indication that the (/>cm values

decreased with temperature for the mixture with x = 0.93. For {xCH4 + (l — xXCH3)2CHCH3}, in which 2-methylpropane has a small dipole

moment, the (^cm values were more than 0.2 per cent of 0cm for x = 0.92 and as large as 0.5 per cent of 0cm for ^ = 0-78. The large 0cm values observed for binary

mixtures containing a species with a small dipole moment compared with mixtures

of nonpolar species exemplified the behavior observed by Pan et a/.*^*' in dielectric- constant measurements on mixtures of LNG components, including their results for {0.91CH4 + 0.09(CH3)2CHCH3} and {O.95CH4 + O.O5C4H10}.

The contribution of M. J. Hiza in the preparation of mixtures is gratefully

acknowledged.

REFERENCES

1. Haynes, W. M.; Hiza, M. J.; Frederick, N. V. Rev. Sci. Instrum. 1976, 47, 1237.

2. Haynes, W. M.; Hiza, M. J. Advances in Cryogenic Engineering, Vol. 21. Timmerhaus, K. D.; Weitzel, D. H.: editors. Plenum Press: New York. 1976, p. 516.

3. Haynes, W. M.; Hiza, M. J. J. Chem. Thermodynamics 1911, 9, 179.

4. Hiza, M. J.; Haynes, W. M.; Parrish, W. R. J. Chem. Thermodynamics 1977, 9, 873.

5. Haynes, W. M. Rev. Sci. Instrum. 1977, 48, 39,

6. Hiza, M. J.; Haynes, W. M. 7. Chem. Thermodynamics 1980, 12, 1.

7. Haynes, W. M. J. Chem. Thermodynamics 1982, 14, 603.

8. Haynes, W. M. J. Res. Nat. Bur. Stand. (U.S.). in press.

9. McCarty, R. D. Nat. Bur. Stand. (U.S.). Interagency Report NBSIR 77-867. 1977.

10. McCarty, R. D. Nat. Bur. Stand. (U.S.). Tech. Note 1030. 1980.

11. McCarty, R. D. J. Chem. Thermodynamics 1982, 14, 837.

12. Hiza, M. J. Fluid Pha.<ie Equilibria 1978, 2, 27.

13. Younglove, B. A.; Straty, G. C. Rev. Sci. Instrum. 1970, 41, 1087.

14. Pan, W. P.; Mady, M, H.; Miller, R. C. AIChEJ. 1975, 21, 283.

15. Frederick, N. V.; Haynes, W. M. Rev. Sci. Instrum. 1979, 50, 1 154.

16. Goodwin, R. D. Nat. Bur. Stand. (U.S.). Tech. Note 653. 1974.

17. Rowlinson, J. S. Liquids and Liquid Mixtures. 2nd edition. Butterworths: London, 1969.

18. Straty, G. C; Goodwin, R. D. Cryogenics 1973, 13, 712.

19. Haynes, W. M.; Younglove, B. A. Advances in Cryogenic Engineering Vol. 27. Fast, R. W.: editor. Plenum Press: New York. 1982, p. 883.

20. Rodosevich, J. B.; Miller, R. C. AIChEJ. 1973, 19, 729.

21. Miller, R. C. Chem. Eng. 1974, 81, 134.

22. Shana'a, M. Y.; Canfield. F. B. Trans. Faraday Soc. 1968, 64, 2281.

23. Chueh, P. L.; Prausnitz, J. M. AIChEJ. 1973, 19, 729,

97

Reference [lb]

ADVANCES IN CRYOGENIC ENGINEERING, Proc. Cryogenic Engineering

Conf. (Boulder, Colo., Aug 2-5, 1977) Vol 23, K. D. Timmerhaus,

Editor. Plenum Press, New York (1978)

M—6

LIQUID MIXTURE EXCESS VOLUMES

AND TOTAL VAPOR PRESSURES USING

A MAGNETIC SUSPENSION DENSIMETER WITH

COMPOSITIONS DETERMINED BY

CHROMATOGRAPHIC ANALYSIS: METHANE

PLUS ETHANE*

M. J. Hiza and W. M. Haynes

National Bureau of Standards, Boulder, Colorado

INTRODUCTION

The measurements discussed in this paper are part of a comprehensive

experimental program to obtain orthobaric liquid densities for the major

components of LNG and their mixtures using a magnetic suspension densimeter.

The present measurements on methane plus ethane were made to evaluate vapor

recirculation as a method of liquid mixing and the consistency of liquid-phase

compositions determined by analysis of liquid samples. The results demonstrate the

feasibility of using a magnetic suspension densimeter to obtain accurate isothermal

phase equilibria and liquid density data simultaneously, with compositions determined by chromatographic analysis.

The magnetic suspension densimeter, developed for absolute measurements of

cryogenic liquid and liquid mixture densities, f''^] is a sophisticated but straightforward application of Archimedes' principle. Owing to the experimental procedures

involved in the determination of temperature-dependent parameters, this instru- ment is particularly suited for efficient use in isothermal density measurements as a function of cither pressure or composition.

In measurements of orthobaric liquid densities reported elsewhere for the

major components of LNG l''^'*) and their binary mixtures ['], the potential for elHcient use of the densimeter at constant temperature was not utilized. In the

binary mixtures study ['] for example, measurements were made on gravimetrically

prepared mixtures as a function of temperature to minimize the uncertainty in composition of the liquid mixture. Vapor recirculation was used only as a test for nonhomogeneity of the liquid mixture. In that study, measurements were made on

* Work carried out at the National Bureau of Standards under the sponsorship of British Gas Corp.; Chicago Bridge and Iron Co.; Columbia Gas Service Corp.; Distrigas Corp.; Easco Gas LNG, Inc.; EI Paso Natural Gas; Ga^ dc France; Marathon Oil Co.; Mobil Oil Corp.; Natural Gas Pipeline Co.;

Phillips Petroleum Co.; Shell International Gas, Ltd.; Sonatrach; Southern California Gas Co.; Tennessee Gas Pipeline Co.; Texas Eastern Transmission Co.; Tokyo Gas Co., Ltd.; and Transcontinental Gas Pipe Line Corp., through a grant administered by the American Gas Association. Inc. 594

99
Liquid Mixture Excess Volumes and Total Vapor Pressures 595

methane plus ethane mixtures at about 35 and 68 mole % methane from 105 to 130 K and at about 49 mole % methane from 105 to 140 K. Times involved in the

steps of the procedure restricted measurements to two data points per day. A

vacuum point at each experimental temperature was taken prior to measurements

on the liquid mixture, since the equilibrium cell had to be warmed nearly to room

temperature for effective removal of the heavy component from surfaces within the

cell. Thermal hysteresis of the magnetic moment of the buoy, although very

small, dictates that, for the most precise density measurement [ ] large thermal

cycles (e.g., from 100 to 300 K) should be avoided between the vacuum and liquid

points. Since the position of the buoy (which must be the same within about

0.002 mm for the vacuum and liquid points) is determined with a 1 25 x microscope,

ambient temperature must remain constant during these measurements. If a series

of density measurements can be made (at least within a single day) without

changing the experimental temperature, only one vacuum point would be required

for the series, and the experimental time per data point would be reduced

significantly.

The results of isothermal measurements of liquid mixture densities and the

corresponding total vapor pressures are reported for the methane plus ethane

system at 125 and 135 K with liquid-phase compositions determined by chromatographic analysis. Excess volumes, computed for the mixtures studied, are compared

with those obtained from the densities of gravimetrically prepared mixtures and

with values calculated from a temperature-dependent Redlich-Kister equation

fitted to all the methane plus ethane data obtained with this apparatus. Excess

pressures, derived from the total vapor pressures, are compared with values from

the prepared mixtures and from phase equilibria data in the literature. In this discussion, total vapor pressure is used interchangeably with mixture vapor pres- sure and equilibrium pressure. EXPERIMENTAL

A schematic diagram of the apparatus as used in the present study is given in Fig. 1. It incorporates an equilibrium system patterned after the closed-loop vapor

recirculation apparatus used for liquid-vapKir equilibria measurements [^'^] and the

simplified one-coil magnetic suspension densimeter [^]. A window in the equilibrium cell provides visual observation of the magnetic buoy and the liquid sample.

With the simplified version of the densimeter, a barium ferrite magnetic buoy

of known density is freely suspended by the force generated from the axial field of a

single air core solenoid. Measurement of the solenoid current needed to support

the buoy in liquid relative to that in vacuum at the same position and temperature

yields the liquid density directly. The relationship used to compute the liquid

density is

P/=P/.[l-(/y/.)l (1)

where pi and are the mass densities of the liquid and the magnetic buoy, and // and are the solenoid currents required to support the buoy in the liquid and vacuum at constant temperature and at the same position relative to the support

coil. The procedures used in this study were basically the same as those followed in measuring the liquid densities of gravimetrically prepared binary mixtures [^]. Solenoid currents were measured first with pure liquid methane in the cell and then

100
596 M. J. Hiza and W. M. Maynes

L J

Fig. 1. Schematic of experimental apparatus.

with the cell evacuated at the temperature of interest. (A methane point was taken

each day as a control on the measurement process, and these were used in a

statistical analysis of the data.) Then the mixture to be studied was introduced into

the cell, and the desired density and pressure measurements were made.

For measurements on the prepared binary mixtures [*], an effort was made to reduce the vapor volume to a minimum. A 0 to 100 psi (0 to 0.69 MPa) quartz

Bourdon tube gauge with a free volume of approximately 0.5 cm^ was used for equilibrium pressure measurements. The recirculation pump, with a free volume of

approximately 65 cm', was isolated during the time the cell was being completely

filled and during the density measurements. For binary mixtures containing

methane plus ethane, the pump was filled with pure methane gas at the mixture

vapor pressure so that the liquid sample could be mixed by vapor recirculation,

without significant pressure change, while monitoring the density. This procedure was used as a test for nonhomogeneity of the liquid sample.

In contrast, for measurements on mixtures with compositions determined from

liquid sample analysis, a large vapor volume is desirable. A large vapor volume

minimizes pressure changes in the equilibrium cell during liquid sample withdrawal,

and thus reduces changes in the liquid composition due to preferential vaporiza101
Liquid Mixture Exceu Volumes and Total Vapor Pressures 597

tion. In the present study, equilibrium pressures were measured with a 0 to 100 psi

(0 to 0.69 MPa) quartz Bourdon tube gauge with an cstinnated free volume of

80 cm'. In addition, the recirculation pump was in communication with the equilibrium system at all times, and only enough liquid was added to the cell to raise the

liquid level at least 1 cm above the top of the magnetic buoy when in suspension,

which allowed about 6 cm' of vapor space in the cell. At 135 K, data were obtained at four compositions between 35 and 75 mole %

methane by starting with pure liquid ethane in the cell and adding incremental

amounts of pure liquid methane. At 125 K, data for an approximately cquimolar

mixture were obtained by adding pure liquid methane to the cell subsequent to measurements on a gravimetrically prepared mixture at 35 mole % methane. A

sufficient amount of the prepared mixture was removed through the liquid sample

line so that the liquid level, after adding methane, was a centimeter or so above the

suspended buoy. In each case the liquid sample was mixed by vapor recirculation.

The liquid sample line consists of a 25-cm length of 0.028-cm-ID tubing

exiting the bottom of the equilibrium cell and about 2 m of 0.048-cm-ID tubing

leading out of the cryostat to a small volume valve. The line to the chromatograph

consists of about 5 m of 0.159-cm-ID tubing. The sample receiver shown in Fig. 1 was sized to hold the gas equivalent of 1 cm' of liquid sample. This receiver

provides a sufficient volume of sample for several repetitive analyses and a method

of averaging composition variations which might occur during liquid sample withdrawal. If vapor samples had been analyzed in this study, these would have been

withdrawn directly from the recirculation pump. Additional vapor volume probably

would have been necessary to provide sufficient sample for purging and analysis.

Different liquid sampling and analysis procedures were used at the two

temperatures of this study. At 135 K, before a sample was taken for analysis, the

line from the densimeter to the chromatograph and the sample valve were evacuated and then purged with sample continuously for 6 min at a flow rate of approximately 120 stp cm'/min. For the first data point, with sample line initially containing nearly pure ethane, it was necessary to repeat the density and pressure

measurement and sampling procedure to flush the residual ethane from the lines. For these analyses, the sample receiver was not used, and the chromatograph, with

a 9-m liquid partition column, was calibrated with an equimolar methane plus

ethane standard. At 125 K, the same purging procedure was used, but the sample was then introduced into the sample receiver which had been maintained under vacuum. A recirculation pump ['] attached to the receiver was used to stir the

sample before analysis. Two analyses were made which were in agreement within

0.02 mole % of the composition of each component. In this case, the chromatograph with a 4-m silica gel column was calibrated with a 68 mole % methane plus

ethane standard.

Tht: imprecision in measurement of component concentration was determined

to be about ±0.0.'^ mole % with either separation column. The uncertainty in component concentration due only to errors in analyzer calibration is estimated to be about ±0.05 mole %. Considering all other known sources of systematic error

in the density measurements (' *'] foi methane plus ethane mixtures of the

compositions studied here, an uncertainty of ±0.05 mole % in component concentration combined with the other systematic errors would result in an estimated total systematic error of about ±0.05 % in the mixture density. This was determined

from the square root of the sum of the squares of the systematic errors. Similarly,

the estimated tot;il systematic error in the mixture vapor pressure measurement

102
598 M. J. Hiu and W. M. Haynn

due to errors in component concentration, gauge calibration, and temperature

uncertainty is approximately ±0.01 x lO' Pa. Owing to the small number of data points obtained, it is not possible to make

an estimate of the random error in these measurements. In attempting to establish

reliable liquid sampling procedures for these measurements, it was quite apparent

that obtaining representative samples is not easily accomplished for systems of this type, particularly at lower temperatures. Problems were encountered when there was inadequate vapor volume to prevent preferential vaporization, low sample

driving force, and insufficient purging of the sampl^ lines to the analyzer.

RESULTS

The methane plus ethane total vapor pressures, P„, and molar volumes, V„,

obtained in this study are given in Table I. Also given are the corresponding excess

values, and V^, determined from the difference between the actual mixture

value and the mole fraction average of the pure component values at the conditions

of the rnixture. The pure component properties used by Hiza et al. ['] were also used here.

The excess volumes in Table I and those obtained from measurements on the

gravi metrically prepared mixtures [ ] were fitted to a Redlich-Kister expansion of

the form = XxX2[{ao + a,T + a2T^)+(bo + biT + b2T^)i2x:-\) + ico+c,T + C2T^)(2x,-lf]

(2)

where jci and X2 are the mole fractions of methane and ethane, respectively. 7'he least-squares coefficients (a, and b,) are given in Table II. The c, coefficients were

not statistically significant in representing excess volumes for this system. The

standard deviation, given in Table II in percent of the average mixture molar

volume (0.035%), is slightly higher than the comparable value, 0.030%, obtained

in fitting excess volumes from the data on prepared mixtures only The

differences between excess volumes obtained in the present study and values

calculated with the coefficients from Table II, in percent of the molar volumes, are

also given in Table I.

Table I. Orthobaric Liquid Molar Volumes, Total Vapor Pressures, and Corresponding Excess Functions for Methane Plus Ethane with Compositions Determined by Chromatographic Analysis

[ Vfaic from equation (2) with coefficients given in Table II]

T,KX, ClU

X, QH, PaxlO'*' PaxlO"'* cni'/mole cm''/mole

"'^xlOO v„

135.00 0.3563 0.6437 2.043 + 0.280 46.056 -0.876 -0.020

0.4447 0.5553 2.505 +0.310 45.171 -1.026 -0.018

0.5934 0.4066 3.150 + 0.229 43.836 -1.126 +0.021

0.7415 0.2585 3.743 +0.099 42.690 -1.040 -0.066

125.00 0.5014 0.4986 1.515 +0.162 43.573 -O.S35 -0.032

* IxlO* Pa = 1 bar.

103
Liquid Mixture Excess Volumes and Total Vapor IVessures 599

Table II. Coefficients for Equation (2)

from a Least^Squares Fit of all Excess

Volumes for Methane Pius Etiiane

Obtained from Measurements with the

Magnetic Suspension Densimeter

«0 -14.89469

Ol 0.2770596

az -0.001473116

bo -37.98245

0.6427122 -0.002783724

0.035%

In Fig. 2, the excess volumes are compared with those, obtained from

measurements on prepared mixtures at 125 and 135 K and curves showing the

composition dependence of excess volumes calculated with the coefficients in Table

II. At 135 K, the two prepared mixture points shown at about 35 and 68 mole %

methane were obtained by extrapolation from a least-squares fit of the densities of

these mixtures as a function of temperature [']. All others are experimental data.

From the several sources of methane j)lus ethane liquid-vapor equilibria data

listed in a recent survey of the literature [ ], excess pressures were taken from the

measurements of Miller and Staveley ['] at 115.77 K and of Wichterle and Kobayashi at 130.37 K for comparison with excess pressures given here. This

comparison is given in Fig. 3. Miller and Staveley measured total vapor pressures of

prepared mixtures, while Wichterle and Kobayashi analyzed the liquid and vapor

METHANE HOLE FRACTION

0 0.2 0.4 O.e 0.8 1.0

Fij;. 2. Comparison of excess volumes for methane plus ethane from the present study with those from measuremrnts on prepared mixtures anil with curves calculated from cq. (2).

104
600 M. J. Hiza and W. M. H«ynw

O.JxlO

0.2x10

0.1x10

0

1 1

D

1 o

•

1 Preient Study Pr»p*rK) MUtur»»

CompoiltloM by C4i Antlysli - miler t St*vtl»y —

135 Kjr^ >v

Ulch(«rl» & Kob^yiihi. 1)0.3; K

/ y / y

1

125 IC>,^0

1

O

1 0 0.2 0.4 0.6 O.B 1.0

HETHANE MOLE FRACTION

Fig. 3. Comparisons of excess pressures for methane plus ethane from the present study with those from measurements on prepared mixtures and with values obtained from the hteraiure.

Compositions at the equilibrium pressure of the mixture. Two points from vaporpressure measurements at 125 K on prepared mixtures in the densimeter are also shown. There appears to be a discrepancy between tHe present data and those of

Wichterle and Kobayashi. To be consistent, excess pressures at 130.37 K should

fall between the values at 125 and 135 K, while those of Wichterle and Kobayashi

are predominantly higher than the present values at 135 K. However, a data point

of the latter authors at about 60 mole % methane, when plotted with these new

values and that of Miller and Staveley at the same composition, produces a smooth

curve of excess pressure as a function of temperature.

SUMMARY

The experimental results given here demonstrate the feasibility of efficiently

utilizing the magnetic suspension densimeter to obtain both isothermal phase

equilibria and liquid mixture density data, with compositions of the equilibrium

phases determined by chromatographic analysis. A more complete evaluation of

sampling procedures is needed to determine the best procedure to obtain

representative liquid mixture samples reliably. Although vapor-phase compositions were not analyzed in the present study, obtaining representative samples of the

equilibrium vapor would not present a difficult problem. Using excess volumes

from liquid density data on accurately prepared mixtures as a reference, measure- ment of the liquid mixture densities along with liquid-vapor equilibria measure- ments provides an excellent means of assessing the reliability of liquid-phase

compositions obtained in the phase equilibria measurements.

REFERENCES

1. \V. M. Hayncs. M. J. Hiza, and N. V. Frederick. Rev. Sri. hisir. 47:12M (1976).

2. W. M. Waynes. Rev Sci. Instr. 48:39 (1977)

3. W. M. Haynes anil M, J. Hiza, / Chern. Vifrmodynamics 9:179 (1977).

105
Liquid Mixture Excess Volumes and Total Vapor Pressures 601

4. W. M. Haynes and M. J. Hi/a, in: Adunnces in Cryogenic Engineering, Vol. 21, Plenum Press, New

York (1976), p. 516.

5. M. J. Hiza, W. M. Haynes. and W. R. Parriih, / Chem. Thermodynamics 9;873 (1977).

6. A. G. Duncan and M. J. Hiza, in: Advances in Cryogenic Engineering. Vol. 15, Plenum Press, New

York (1970), p. 42.

7. M. J. Hiza and A. G. Duncan. Rev. Sci. Instr. 40:513 (1969). , 8. A. J. Kidnay. M. J. Hiza, and R. C. Miller. Cryogenics 13:575 (1973),

9. R. C. Miller and L. A. K. Staveley, in: Advances in Cryogenic Engineering, Vol. 21, Plenum Press, New York (1976), p. 493.

10. I. Wichterle and R. Kobayashi, / Chem. Eng. Data 17:9 (1972).

DISCUSSION

Question by R. Kobayashi, Rice University: Are there any attempts to apply the maxnetic

densimeter to measure gas densities as well as liquids? Answer by author: The precision and accuracy of a magnetic suspension densimeter, which is a straightforward application of Archimedes' principle, depends on the difference between the density of

the float and the density of the fluid Thus, it is very difficult to make precise density measurements on a low-density gas. It is one of our long-range goals. (We have tried some gas measurements in selected

regions.) For the extremely low gas densities you need a low-density float as well as very precise measurements of the position of the float and of the force (current in electromagnetic coils) necessary to support the float.

106
6.9 Reference [2U]

A- 145

J. Chem. Thermodynamics 1980, 12, 1-10

Orthobaric liquid densities and excess

volumes for multicomponent mixtures of

low molar-mass alkanes and nitrogen

between 105 and 125

M. J. HIZA and W. M. HAYNES

Thermophysical Properties Division,

Natiorml Engineering Laboratory,

National Bureau of Standards,

Boulder, Colorado 80303, U.S.A.

{Received 15 March 1979)

A magnetic suspension densimeter has been used to determine orthobaric liquid densities of

gravimetrically prepared multicomponent mixtures containing the major components of

liquefied natural gas, i.e. nitrogen, methane, ethane, propane, isobutane, and normal butane, between 105 and 125 K. These results were obtained to provide a test of the capability of mathematical models to predict the densities of liquefied natural-gas mixtures. Combinations of

the subject components were chosen to provide the most severe test of the models and the

possibility of using the measured densities to optimize parameters of the models. Deviations are given between the experimental densities for each mixture and values predicted with an extended

corresponding-states model optimized to binary-mixture and pure-component orthobaric liquid densities obtained with the same apparatus. Uncertainties of the present results are discussed in relation to the experimental technique, the knowledge of the compositions of the liquid mixtures, and the comparisons between the experimental and predicted results. Approximate

total vapor pressures are also given for each mixture at the temperatures studied.

1. Introduction

The present study is part of an experimental program to obtain orthobaric liquid

densities for the major components of liquefied natural gas (lng) and their mixtures,

primarily in the temperature range from 105 to 140 K. The overall purpose of the program is to provide sufficiently accurate densities (±0.1 per cent) of uniformly high

precision ( + 0.02 per cent) for the pure components, the possible binary

combinations, and selected multicomponent mixtures, for optimization and testing of

prediction methods.

" This work was carried out at the National Bureau of Standards under the sp)onsorship of British Gas

Corp., Chicago Bridge and Iron Co., Columbia Gas Service Corp., Distrigas Corp., Easco Gas LNG, Inc., El Paso Natural Gas, Gaz de France, Marathon Oil Co., Mobil Oil Corp., Natural Gas Pipehne Co..

Phillips Petroleum Co., Shell International Gas Ltd., Sonatrach, Southern California Gas Co., Tennessee Gas Pipeline Co., Texas Eastern Transmission Co., Tokyo Gas Co. Ltd., and Transcontinental Gas Pipe Line Corp., through a grant administered by the American Gas Association, Inc. 0021-9614/80/010001 -I- 10 $01.00/0 © 1980 Academic Press Inc. (London) Ltd.

1

107
2 M. J. HIZA AND W. M. HAYNES

Previously, orthobaric liquid densities were reported for nitrogen,"* the low molar mass alkanes: methane," '^* ethane,'-^' propane,*^* isobutane,*^* and normal

butane and their binary mixtures,**' as determined with a magnetic suspension

densimeter.

In this paper, orthobaric liquid densities between 105 and 125 K, determined with

the same apparatus, are reported for 10 gravimetrically prepared multicomponent

mixtures containing methane. Total vapor pressures are also reported, but these are considered approximate since the main focus was to assure the reliability of the density measurements. As an indication of the consistency of the densities of the

multicomponent mixtures with those for the pure fluids and binary mixtures reported

earlier, deviations are given between the experimental densities for each

multicomponent mixture and values predicted with an extended corresponding states LNG density model with parameters determined from the binary mixture and pure- component results only. A brief discussion is also included on the consistency of other

multicomponent liquid densities in the literature with the results presented here.

2. Experimental

The magnetic suspension densimeter used in this study has been discussed in detail elsewhere.'^' ^' All of the results reported here were obtained with the one solenoid

arrangement'^' using the procedures followed in measuring the densities of the binary

liquid mixtures.*'*' The relation used to compute the density of the liquid with the one

solenoid arrangement is

P. = Pb{l-(V/v)}, (1)

where p, and are the mass densities of the liquid and the buoy, and /, and are the solenoid currents required to support the buoy at the same temperature and at the same position, relative to the solenoid, in the liquid and vacuum, respectively.

The multicomponent mixtures were prepared gravimetrically as gas mixtures in thoroughly cleaned and dried metal cylinders, each with a free volume of about

3.3 dm'', using research-grade gases which were analysed chromatographically for impurities before use. In addition, nitrogen and methane were passed through roomtemperature molecular-sieve adsorption columns to remove moisture and any heavy

hydrocarbon contaminants not detected by analysis. Estimated errors in component

mole fractions, based on six times the standard deviation of weighing,*"*' are given in table 1 for both 1 and 5 mol of mixture prepared.

Composition, molar mass, and the total amount prepared are given in table 2 for each of the multicomponent mixtures included in this study. Compositions given for mixtures containing only hydrocarbons are the same as the prepared compositions.

Compositions given for mixtures containing nitrogen have been adjusted slightly (by

as much as 0.0005 in the mole fraction of and CH4) to include all or most of the vapor-phase correction at each temperature studied. In making the composition

adjustments, the mole fraction of each component was rounded to the nearest 0.0001. These vapor-phase corrections to the compositions were made to reduce the final

1U8
LIQUID DENSITIES FOR MULTICOMPONENT lng MIXTURES 3 TABLE I. Uncertainty 5.v in mole fraction of each component in the prepared mixtures based on the total amount of substance n prepared.

Component 5an -- 1 mol n = 5 mol

±0.00014 ±0.00003 CH4 ±0.00025 ±0.00005 C^H, ±0.00013 ±0.00003 C3H8 ±0.00009 + 0.00002 C4H10 ±0.00007 ±0.00001

TABLE 2. Multicomponent mixtures studied : x, mole fraction ; M, molar mass ; «, the total amount of substance prepared

Mixture .xfN,) x(CHJ .v(CjH,) a{i-C4H,o) .x(n-C4H,o) M/g mol ' n/mol

A 0.34242 0.31372 0.34386 30.0903 1.78 B 0.80284 0.09902 0.09814 20.1852 4.93 C 0.85443 0.05042 0.04038 0.02577 0.02901 20.1885 1.89 D 0.3380 0.3414 0.3206 24.5861 4.66 E 0.1635 0.6704 0.1661 20.3301 4.66 F 0.0995 0.7977 0.1028 20.1181 4.67 G 0.0484 0.8526 0.0483 0.0507 18.7223 4.41 H 0.0490 0.8060 0.0468 0.0482 0.0500 20.7423 3.00

I 0.0554 0.7909 0.0560 0.0500 0.0477 20.9017 3.16

J 0.0425 0.8130 0.0475 0.0487 0.0241 0.0242 20.6168 3.09

adjustment of the mixture densities from a maximum of about 0.05 per cent to 0.01 per cent or less. The prepared gas mixtures were condensed directly and continuously into the windowed equilibrium cell through a capillary tube entering at the bottom until no

vapor space could be observed. This filling method provided continuous mixing of the

liquid while the gas mixture was being condensed into the cell. For one mixture

(mixture C in table 2) a comparison was made between the measured densities at 105 K obtained before and after the vapor phase was recirculated through the liquid

for several min at a bubble rate of 1 to 3 s"^ with an external vapor-recirculation

pump. No change in measured density was detected after recirculation. Subsequently,

liquid samples were withdrawn, and two comparisons were made chromatographically with the composition of the prepared mixture. During withdrawal of

liquid samples, the recirculation pump (with a free volume of about 65 cm^) was kept

open to the vapor space of the equilibrium system to reduce the amount of CH4

vaporized from the liquid in the cell. The liquid samples were, nevertheless, lean in CH4 by about 0.3 mole per cent, and mole fractions of heavy components in the

liquid were all higher than those in the prepared mixture. The mole fractions of CjH^,

and CjHg were higher by approximately 0.001, while the mole fractions of

(CHjIjCHCHj and C4H10 were higher by approximately 0.0006 and 0.0002,

respectively. Since it would be difficult, with this arrangement, to completely avoid

109
4 M. J. HIZA AND W. M. HAYNES

partial vaporization of methane from the Hquid samples, the main value of the

analysis was in the comparison of the mole fractions of the heavy components in the

liquid samples to those in the prepared mixtures on a methane-free basis. On this basis, the variations in the liquid-phase compositions observed for mixture C are not

considered unreasonable.

Attempts to study mixtures comparable to mixture C containing only one of the

butanes were unsuccessful. Differences between measured and predicted densities were three to four times larger than expected, even after several min of vapor

recirculation, and the mole fractions of the heavy components from analysis of liquid

samples were inconsistent with those in the prepared mixture. Analysis of a liquid

sample for one mixture containing mole fraction approximately 0.05 of

(CH3)2CHCH3, without C4H10, for example, gave mole fractions of CjH^ and C3H8

in the liquid approximately 0.002 higher than those in the prepared mixture, and a mole fraction of (CH3)2CHCH3 lower than that in the prepared mixture by about

0.0013. This inconsistency in composition could be interpreted as a non-uniform mole

fraction of (CH3)2CHCH3 in the liquid of up to 0.0033. Since the magnetic buoy is essentially a point sensor relative to the size of the cell, the direction of error in observed density depends on the composition of liquid actually surrounding the

buoy. From observations made during the course of this study and in the previous

study on binary mixtures,'*' it appeared that mixtures which were most likely to present problems in filling the equilibrium cell with a homogeneous liquid sample of

the same composition as the prepared gas mixture were those of hydrocarbons only

which contain the butanes and mole fraction 0.85 or more of CH4. Though mixture C

fits these criteria, we did not detect any abnormalities in the liquid mixture during

measurement, as noted above, which we could consider significant.

3. Results and discussion

EXPERIMENTAL RESULTS

The experimental orthobaric liquid-mixture amount-of-substance densities are given

in table 3 as a function of temperature. Pressures listed in table 3 are considered only

approximate mixture vapor pressures, and in a few cases, measurements were not made and values given were obtained by interpolation. Based on comparisons of

pressures measured for the binary mixtures with reported phase equilibria,*"** pressures reported here are thought to be either equal to or slightly higher than the

true multicomponent mixture vapor pressures. Also given in table 3 are the corresponding experimental excess volumes V^,

computed from

l^'=Kn-Kdea,. (2)

where

K,,,i = ^:XX{l+KrAPi-PJ}^ (3)

V is molar volume, p is saturation pressure, x is mole fraction, and k> is isothermal

compressibility. Subscripts m and i refer to the mixture and pure components.

110
LIQUID DENSITIES FOR MULTICOMPONENT lng MIXTURES 5

TABLE 3. Orthobaric amount-of-substance densities p„ of multicomponent mixtures of low molar-mass

alkanes and nitrogen : 7, temperature (IPTS-68); p, pressure ; K^, excess volume calculated from equations

(2) and (3); ^caic density calculated from the extended corresponding-states model as given in reference 18

Mixture T/K p/MPa pJmo\ dm ' V^lcm^ mol '

10'(p-Pc.ic)P^'

A 110.00 0.0427 20.5342 -0.573 -0.03

115.00 0.0637 20.3497 -0.646 -0.02

115.00 0.0637 20.3567 -0.663 -hO.Ol 120.00 0.0914 20.1786 -0.766 -h0.06 120.00 0.0914 20.1803 -0.770 -t-0.07 125.00 0.128 19.9858 — 0.848 -1-0.04 B 105.00 0.0477 24.9975 -0.533 -1-0.02 110.00 0.0747 24.6696 -0.587 -0.02

115.00 0.112 24.3515 -0.676 4-0.01 120.00 0.162 24.0335 — 0.790 -1-0.06 C 105.00 0.0515 24.8775 -0.596 -f-0.18 110.00 0.0818 24.5382 -0.638 -f-0.11 115.00 0.119 24.2083 -0.716 -hO.lO 120.00 0.170 23.8859 — 0.832 -1-0.14 D 105.00 0.719 24.9084 -2.108 -fO.07 110.00 0.949 24.5350 — 2.764 -1-0.04

1 1 J.UU 11 .ZJ7^71 Z*+.l*f —— 81 fi J .o 1 u — 0 06 E 105.00 0.369 25.8587 - 1 .069 -0.00

110.00 0.481 25.4598 -1.433 -0.01

115.00 0.618 25.0466 -2.052 -0.05

120.00 0.773 24.6359 -3.496 -0.04 F 105.00 0.282 25.3950 -0.845 -0.01

110,00 0.362 25.0153 -1.091 -0.04

115.00 0.455 24.6465 -1.542 -t-0.01 120.00 0.561 24.2667 -2.525 -1-0.04 G 105.00 0.158 25.8910 -0.513 -0.07

110.00 0.214 25.5081 -0.656 -0.07

115.00 0.283 25.1224 -0.900 -0.06

120.00 0.365 24.7283 -1.425 -0.04 H 105.00 0.204 24.7803 -0.726 -h0,03 110.00 0.255 24.4327 -0.870 -0,03

115.00 0.314 24.1039 -1.154 4-0.00 120.00 0.379 23.7707 -1.730 -1-0.05

I 105.00 0.193 24.8080 -0.786 -0.00

110.00 0.253 24.4664 -0.954 -0.06

J 105.00 0.183 24.8496 -0.717 + 0.04 110.00 0.252 24.5159 -0.870 -^0.03 115.00 0.338 24.1783 -1.109 -f-0,02 120.00 0.443 23.8577 -1.634 + 0.10

111
6 M. J. HIZA AND W. M. HAYNES

respectively. Pure-component molar volumes were calculated from the fit of the

experimental results."'^* Isothermal compressibilities for nitrogen, methane, and

ethane were calculated from the expressions

:

K7-(N2)/MPa"' = -0.01032 + 0.51368/(126.2- T/K), (4) Kr(CH4)/MPa-' = -0.001595 + 0.29488/(190.555- T/K), (5)

and

Kj-(C2H6)/MPa-' = {(r/250K) + 0.45} x 10-^ (6) where T is thermodynamic temperature. These expressions are considered adequate

representation of isothermal compressibilities of nitrogen and methane from

Rowlinson,*^' those of methane from Goodwin,'^' and those for ethane from Miller.*®* Isothermal-compressibility corrections were not made for molar volumes of propane,

isobutane, and normal butane. Vapor pressures for nitrogen, methane, and ethane were taken from Strobridge,'^* Goodwin,'^* and Goodwin, Roder, and Straty,'*°*

respectively.

UNCERTAINTIES IN MEASURED LIQUID-MIXTURE DENSITIES

The uncertainty in measured density of a liquid mixture is inherently larger than for a pure fluid due to the added uncertainty in composition. Previously, detailed

discussions were given on random and systematic errors associated with magnetic

suspension densimeter measurements for pure liquids'" and binary liquid mixtures.*** In each case the total uncertainty was taken as the square root of the sum of the squares of known systematic errors plus three times the standard deviation for random error. For measurements on pure liquid methane, the predominant component in liquefied natural gas, the total uncertainty in density was estimated to be about ±0.1

per cent, with approximately equal contributions from random and systematic

errors.*'* For measurements on (methane + propane) liquid mixtures, which typify the binary liquid mixtures studied, the total uncertainty was estimated to be about +0.16

per cent with approximately the same contribution from known systematic errors as

in pure liquid methane measurements.*'** The random error in the (methane +

propane) density measurements was estimated from the standard deviation obtained

by fitting the excess volumes, for the four different compositions studied, to a

Redlich-Kister expansion as a function of both composition and temperature. Fitting the results as a function of temperature at constant composition resulted in significantly smaller standard deviations, comparable to that for pure methane.

However, it was felt that the evaluation based on both temperature and composition

would give a more realistic estimate of the random error.

It is not possible to obtain a standard deviation for the multicomponent-mixture

results from which an estimate of random error can be made as a function of both

temperature and composition, nor are the results for any mixture sufficient to obtain

a meaningful standard deviation as a function of temperature only. However, there is

nz
LIQUID DENSITIES FOR MULTICOMPONENT lng MIXTURES 7 no reason to believe that the random error in density measurements for the multicomponent Hquid mixtures in the present study should be significantly larger

than that noted above for the (methane + propane) binary mixtures. Estimated

systematic errors are also about the same. The increased uncertainty in composition

of prepared mixtures due to the increased number of components present adds very

little to the total systematic error. There is a potential source of error, suggested by our experimental observations,

for mixtures containing high mole fractions of CH4 with higher molar-mass alkanes

including either or both butanes, but with no nitrogen. These are mixtures for which

the vapor phase is essentially pure methane. Thus, as the mole fraction of CH4 in the

liquid mixture is increased, the difference between the mixture vapor pressure and

that for pure methane becomes quite small, and the probability increases for condensing pure methane in the vapor space, particularly at the lowest temperatures,

while the prepared mixture is being condensed into the equilibrium cell. If methane

thus condensed were to accumulate in a location where it could not be readily

blended back into the liquid mixture, and if the mixture were homogeneous, analysis

of representative liquid samples would indicate the mixture to be lean in methane

similar to the effect of partial vaporization during liquid sampling. Consequently, this experimental test alone would not be sufficient to confirm a possible error in measured density due to such an occurrence.

In the top section of the equilibrium cell, the annulus volume (of approximately

0.3 cm^) between the vapor bulb and the cell wall"* is the only possible location

where methane might condense without being in good communication with the bulk

of the liquid mixture in the cell. If pure liquid methane (rather than methane vapor)

completely filled this annulus when mixture C, for example, was condensed into the

cell, the mole fraction of CH4 in the liquid mixture would be about 0.0024 lower and

the measured mass density would be about 0.2 per cent higher than for the prepared

mixture, with a variation in the density error of about 0.015 per cent between 105 and

120 K. For comparison, if methane were to condense in a 20 cm length of the vaporexit capillary with an internal diameter of 0.048 cm, the error introduced in the

densities for mixture C would be about 0.03 per cent, about the same as the imprecision of measurement. Due to vapor-phase enhancement of the mole fraction of CH4 in (nitrogen -I- methane) gas mixtures in equilibrium with the liquid,"' ''' the presence of nitrogen in the multicomponent liquid mixture, e.g. mixture J as against

mixture C, would reduce the probability of methane condensing out in the vapor

space.

DEVIATIONS FROM PREDICTED DENSITIES

Comparisons between experimental and predicted liquid densities for multicomponent mixtures of lng components primarily provide a test of the predictive (or

correlative) capability of a proposed lng density model. However, when the parameters of a model are optimized only to the pure-fluid and binary-mixtures

results obtained in the same experimental apparatus, these comparisons also can

provide information on the consistency of the results for the multicomponent

113
8 M. J. HIZA AND W. M. HAYNES

mixtures with those for the pure fluids and binary mixtures, within known Hmitations

imposed in the development of the model.

Of several models proposed for the prediction of orthobaric liquid densities of lng

mixtures,'*' the extended corresponding-states model'' ^' appears to be the most

general in application. McCarty'^*' has demonstrated that, when optimized to purefluid and binary-mixtures densities obtained with the magnetic suspension densimeter

of the present study, all of the pure-fluid and binary-mixtures results are correlated

generally within their estimated accuracy, and, in most cases, nearly within the precision of measurement. It was felt that comparisons of the values for multicomponent mixtures with values predicted from this model would be most likely to identify inconsistencies. Percent deviations of the experimental orthobaric liquid amount-of-substance

densities from values predicted with the extended corresponding-states model,

without further modification, are also given in table 3. The average deviation or bias

is -1-0.016 per cent, and the average absolute deviation is 0.05 per cent. The maximum

deviation is -1-0.18 per cent for the 105 K point for mixture C. Only the results for mixture C show a systematic bias of 0.1 per cent or larger. Excluding the results for mixture C results in an average deviation or bias of -1-0.003 per cent, an average

absolute deviation of 0.04 per cent, and a maximum deviation of -1-0.1 per cent.

It is also worth noting the significant results of comparisons made with values

predicted from two less complex models, which are more limited in their application.

These are both excess-volume models, the empirical model of Hiza'^'* and the modified hard-sphere model of Rodosevich and Miller,''^' with parameters optimized

to the same pure-fluid and binary-mixture values. Excluding results outside the

applicable range of each model, deviations of experimental from predicted densities are in substantial agreement with those of the extended corresponding-states model.

Excluding the results for mixtures D and E and the 120 K point for mixture F, the maximum deviation from values predicted with the empirical excess-volume model is -1-0.14 per cent for the 105 K point for mixture C. Excluding the results for mixtures

containing nitrogen at the highest temperatures, the 120 K points for mixtures E

through J and the 115 K point for mixture D, the maximum deviation from values

predicted with the hard-sphere model is' -I- 0.1 8 per cent, also for the 105 K point for mixture C. In fact, deviations of the densities for mixture C from values predicted with

the hard-sphere model are the same as those given in table 3 for the extended

corresponding-states model.

Though the densities for mixture C are systematically larger than the predicted

values by 0.1 per cent or more, the deviations given in table 3 generally are random

and, for the most part, are significantly smaller than ±0.1 per cent.

CONSISTENCY OF OTHER MULTICOMPONENT lng MIXTURE DENSITIES

Orthobaric liquid densities for multicomponent mixtures of lng components are

scarce in the open literature, and only some of those available approach the accuracy

goal of the present study. Our primary source of references is a fluid-mixtures

bibliography"^' current to January 1975 and more recent references from a file

114
LIQUID DLNSn iliS FOR MULTI( 0MP0N|;NT i N(i MIX IUIU-S 9

maintained to update that bibliography. A summary of the results of comparisons of

literature data with values predicted with the extended corresponding-states model

are included here as an indication of the precision and consistency of available values

with those of the present study. The values are divided, for purposes of discussion,

into two groups: first, those with an apparent imprecision of more than ±0.3 per

cent, and second, those with an apparent imprecision of less than ±0.3 per cent. In the first group are the results reported by Morlet,*^^' Gonzalez and Lee,*^''

Klosek and McKinley,'*^* and Jensen and Kurata.*^^' Morlet'^°* reported results graphically for several ternary mixtures of (methane + ethane + propane) at 90.2 K.

These results appear reproducible within about ±1.0 per cent, and tabulated

densities for three compositions from a fit of them deviate from predicted values by — 0.5 to - 1.1 per cent. Gonzalez and Lee*^'* reported bubble-point densities for 10

methane-rich mixtures, simulating natural gas, down to 99.8 K. For the four

mixtures not containing helium or hexanes and higher hydrocarbons, comparisons

were made with predicted values at the lowest temperature reported. For three of the mixtures at 116.5 K, the densities deviate from predicted values by —0.60 per cent, + 0.25 per cent, and —26.15 per cent. The density for the fourth mixture at 144.3 K

deviates from the predicted value by —12.14 per cent. Klosek and McKinley**''*

reported densities for 13 multicomponent mixtures with three to seven components

from 94.37 to 125.04 K. Though their values deviate from predicted values by ± 1 per

cent, the average deviation or bias is only +0.04 per cent, the average absolute

deviation being 0.36 per cent. Since the values for each mixture generally deviate from

predicted values systematically, the variation may be due largely to an uncertainty in composition. Jensen and Kurata'^^' reported results for essentially three methanerich mixtures containing nitrogen, ethane, propane, and normal butane from 98.15 to 113.15 K. These values appear to have an imprecision of about ±0.5 per cent. From

comparisons made at 108.15 and 113.15 K, they deviate from predicted values by + 1.03 to +2.21 per cent. In the second group are the results reported by Shana'a and Canfield,*^^'

Rodosevich and Miller,*^*' Miller,*^' Orrit and Laupretre,*^*' and Miller and Hiza.*"*

Shana'a and Canfield'^^' reported results for two methane + ethane + propane

mixtures at 108.15 K. These deviate from predicted values by —0.09 per cent and

+ 0.15 per cent. Rodosevich and Miller*^"^' and Miller'^' reported results for three ternary mixtures and one quaternary mixture at 108.00 and 1 15.00 K. In addition,

results included in reference 26 for one ternary mixture obtained by Rodosevich at 108.00, 110.00, and 115.00 K were included here. From predicted values, the average

deviation or bias of these is —0.03 per cent, the average absolute deviation is 0.03 per

cent, and the maximum deviation is —0.06 per cent. Very recently, Orrit and

Laupretre*^^' reported a sample of their results for five methane-rich mixtures of four

to eight components containing nitrogen and hydrocarbons through pentane from

95.50 to 123.61 K. From predicted values, the average deviation or bias of these values is —0.10 per cent, the average absolute deviation is 0.10 per cent, and the maximum deviation is —0.175 per cent. The last investigation in the second group, that of Miller and Hiza,'^^' is not completely independent of the present study. Measurements were made between 105

115
10 M. J. mZA AND W. M. IIAYNi:s

and 1 15 K at the University of Wyoming with Miller's apparatus calibrated on pure- methane values obtained with the magnetic suspension densimeter of the present

study.' Mixtures studied were prepared in our laboratory. These included two

mixtures for which results are reported here (mixtures H and I) and two similar four- component mixtures without nitrogen. From predicted values, the average deviation or bias of these values is -1-0.04 per cent, the average absolute deviation is 0.04 per

cent, and the maximum deviation is -f- 0.1 02 per cent (for mixture I at 105 K). Though

there is a slight bias, these deviations are consistent with the accuracy of the

calibration.

Additional measurements are now in progress with a new design of the magnetic

suspension densimeter, significantly different than that used in the present study.

Measurements are being made on both binary and multicomponent mixtures for which the data are insufficient, and for multicomponent mixtures containing the pentanes. These new results will be reported in subsequent papers.

The authors acknowledge the contributions of M. J. Brown and R. D. McCarty

during the course of the present study.

REFERENCES

1. Haynes, W. M.; Hiza, M. J.; Frederick, N. V. Rev. Sci. Insrmm. 1976, 47, 1237.

2. Haynes. W. M.; Hiza. M. J. J. Chem. Thermodynamics 1977, 9, 179.

3. Haynes, W. M.; Hiza, M. J. In Advances in Cryogenic Engineering, Vol. 21. Timmerhaus, K. D.;

Weitzel, D. H.: editors. Plenum Press; New York. 1976. p. 516.

4. Hiza, M. J.; Haynes, W. M.; Parrish. W. R. J. Chem. Thermodynamics 1977, 9, 873.

5. Haynes, W. M. Rev. Sci. Instrum. 1977, 48, 39.

6. Rowlinson, J. S. Liquids and Liquid Mixtures. 2nd Ed. Butterworths : London. 1969.

7. Goodwin, R. D. Nat. Bur. Stand. U.S., Technical Note 653, 1974.

8. Miller. R. C. Chem. Eng. 1974, 81, 134.

9. Strobridge, T. R. Nat. Bur. Stand. U.S., Technical Note 129, 1962.

10. Goodwin, R. D.; Roder, H. M.; Straty, G. C. Nat. Bur. Stand. U.S., Technical Note 684, 1976.

11. Miller, R. C; Kidnay, A. J., Hiza, M. J, Am. Inst. Chem. Eng. J. 1973, 19, 145.

12. Kidnay, A, J.; Miller, R. C; Parrish, W. R.; Hiza, M. J. Cryogenics 1975, 15. 531.

13. Klosek, J.; McKinley, C. In Proceedings First International Conference on LNG. White, J. W.

;

Neumann, A. E. S.: editors. Institute of Gas Technology: Chicago, III. 1968, Paper 22, Session 5. 14. Albright, M. A. Nat. Gas Proc. Assn. Tech. Publ. TP-3, 1973.

15. Mollerup, J.; Rowlinson, J. S. Chem. Eng. Sci. 1973, 29, 1373.

16. Rodosevich, J. B.; Miller, R. C. In Advances in Cryogenic Engineering, Vol. 19. Timmerhaus, K. D.:

editor. Plenum Press: New York. 1974. p. 339.

17. Hiza, M. J. Fluid Phase Equilibria 1978, 2, 27.

18. McCarty, R. D. Nat. Bur. Stand. U.S.. Interagency Rept. NBSIR 77-867, 1977, 60 pp. See also: Haynes, W. M.; Hiza, M. J.; McCarty, R. D. In: Proceedings Fifth International Conference on LNG, Vol. 2. White. J. W.; McGrew, W.; Farmer, S.; Hansen, D.; Jacobshagen, H.: editors.

Institute of Gas Technology: Chicago, 111. 1977, Paper 11, Session III. 19. Hiza, M. J.; Kidnay. A. J.; Miller, R. C. Equilibrium Properties of Fluid Mixtures, A Bibliography of Data on Fluids of Cryogenic Interest. NSRDS Bibliographic Series. IFI Plenum: New York. 1975.

20. Morlet, J, Rev. Inst. Franc. Petrol. 1963, 18, 127.

21. Gonzalez, M. H.; Lee, A. L.J. Chem. Eng. Data 1968, 13, 172.

22. Jensen, R. H.; Kurata, F, J. Petrol. Technol. 1969, 21, 683.

23. Shana'a, M. Y.; Canfield, F, B. Trans. Faraday Soc. 1968, 64, 2281.

24. Rodosevich, J. B.; Miller, R. C. Am. Inst. Chem. Eng. J. 1973, 19, 729.

25. Orrit, J. E.; Laupretre. J. M. In Advances in Cryogenic Engineering, Vol. 23. Timmerhaus, K. D.:

editor. Plenum Press: New York. 1978, p. 566.

26. MUler, R. C; Hiza, M. J. Fluid Phase Equilibria 1978, 2, 49.

116
b.lu Reference [21]

A-279

J. Chem. Thermodynamics 1982, 14, 603-612

Measurements of orthobaric-liquid

densities of multicomponent

mixtures of lng components

(N2, CH4, C2H6, C3H8, CH3CH(CH3)CH3,

C4Hio» CH3CH(CH3)C2H5, and C5H^2)

between 110 and 130

W. M. HAYNES

Thermophysical Properties Division, National Engineering Laboratory,

National Bureau of Standards. Boulder, Colorado 80303, U.S.A.

{Received 14 September 1981)

A magnetic suspension densimeter has been used to measure the orthobaric-liquid densities of 1

7

multicomponent mixtures of the major components of liquefied natural gas (lng) at temperatures from 110 to 130 K. These mixtures ranged from a ternary mixture containmg

nitrogen, methane, and butane to 4-to-8-component methane-rich (74 to 90 moles per cent) mixtures containmg up to 5 moles per cent of nitrogen, 16 moles percent of ethane, 7 moles

per cent of propane, 5 moles per cent of the butanes, and 0.44 mole per cent of the pentanes. Some of the compositions were selected to simulate commercial lng mixtures. Results of vaporpressure measurements are also presented. The major purpose of this work was to obtain muliicomfKinent-mixture values that could be used to test mathematical models that have been developed for the prediction of lng densities. To demonstrate the consistency of the multicomponent-mixture values, comparisons are presented between experimental densities and

calculated values from an extended correspwnding-states method that was optimized to purefluid and binary-mixture results from the lng density project here. The total uncertainty of a

single density measurement is estimated to be approximately 0.1 percent, which includes an

allowance of three times the standard deviation for random error. The imprecision of measurement is a few parts in 10*.

1. Introduction

A project has been carried out at this laboratory to provide one or more

mathematical models that can be used to predict the density of a liquefied natural gas (lng) mixture at saturation within 0.1 per cent of its true value from an input of the

° This work was carried out at the National Bureau of Standards under the sponsorship of British Gas

Corp., Chicago Bridge and Iron Co., Columbia Gas Service Corp., Distrigas Corp., Easco Gas LNG, Inc., El Paso Natural Gas, Gaz de France, Marathon Oil Co., Mobil Oil Corp., Natural Gas Pipeline Co.,

Phillips Petroleum Co., Shell International Gas. Ltd., Sonatrach, Southern California Gas Co., Tennessee Gas Pijjeline Co., Texas Eastern Transmission Co., Tokyo Gas Co., Ltd., and Transcontinental Gas Pipw

Line Corp., through a grant administered by the American Gas Association, Inc.

37

117
604 W. M. HAYNES

temperature, composition, and pressure of the liquid mixture. Such a model would

serve as a basis for custody transfer of this valuable commodity. In order to develop

and test such models, an accurate and consistent set of densities was needed for the major components of lng and for mixtures of these components. Results from this project have been previously reported for (a) nitrogen,*' ' methane,''-^' ethane,'^*

propane,'^* isobutane.*^' and normal butane*^-^' (major components of lng), (b)

binary mixtures'* of these components, including all combinations except

(nitrogen + isobutane) and (nitrogen + normal butane), and (c) selected multicomponent mixtures*^' of these components with an emphasis on ternary

mixtures containing methane and binary combinations of nitrogen, ethane, and

propane. All of these results were obtained with a magnetic suspension densimeter,*" an instrument based on an application of Archimedes' principle. In an earlier paper giving results for multicomponent mixtures,*^' reference was made to experimental problems that resulted in inconsistencies in densities for methane-rich (80 moles percent or more) mixtures containing only hydrocarbons

with approximately 5 moles per cent of butanes. The resolution of these inconsistencies was one of the major objectives of the present series of measurements.

Also, more results were needed for 5- and 6-component LNO-like mixtures containing

less than 5 moles per cent of the butanes and/or less than 5 moles per cent of nitrogen,

and for 7- and 8-component LNO-like mixtures containing up to 0.44 mole per cent of

the pentanes to test and define the limits of the models for prediction of densities for compositions approximating those of commercial lng. These types of mixtures had

not been investigated in the first series of measurements. (In the mathematical

models,'^"^' results for pentanes were obtained from the literature""' from papers

that also included results for the other components of lng that agreed with results from the present project. The behavior of binary mixtures containing pentanes was

predicted by extrapolating the results of binary-mixture measurements on other lng

components.)

The results presented here were obtained with a new magnetic suspension

densi^neter*'" that was significantly different from, and more versatile than, the apparatus*'' that was used for most of the previous measurements of this project.

However, the new apparatus did employ the same technique for determining densities and gave results for pure fluids and binary mixtures that agreed, within the precision

of the measurements, with values obtained with the previous apparatus. [The new

apparatus was constructed because there was a need for a higher-pressure capability

(35 MPa) for the magnetic suspension densimeter in other research projects.]

New or improved features" " of the new apparatus that were pertinent to the study

of multicomponent mixtures here are as follows: (a) improved temperature control

and measurements, (b) improved handling of mixtures, especially the filling of the cell so as to obtain homogeneous liquid mixtures of known composition, (c) improved

vapor-pressure measurements, and (d) a capacitor for dielectric-constant

measurements. (The dielectric constants for the multicomponent mixtures have not been analyzed as yet due to a lack of values for some of the pure components.) The new apparatus was also used for a study of (methane -I- isobutane) and

(methane -h normal butane).***

118
DENSITIES OF lng MIXTURES 605

2. Experimental

The method of measurement,*' the apparatus," " and the experimental procedures

and measurements" '^ *'^ " '^' have been described in detail elsewhere. Only those

details that are different or essential for this papjer will be discussed here. A onesolenoid magnetic suspension system"^' was used to obtain the results reported here. Measurements on liquid methane were used as a control on the measurement

process during the course of the mixture measurements. A density measurement on

liquid methane was performed before each filling of the cell with a new mixture

sample. This procedure was followed to insure that the warm-up and cool-down

cycles of the apparatus did not affect the apparent position of the buoy from liquid-to- vacuum measurements. After mixture measurements were completed for each sample,

the cell was warmed to room temperature to facilitate complete removal of the heavy

hydrocarbons. A maximum of four points at 5 K increments was obtained for each

sample.

There were a total of four capillaries entering into the cell, two at the top and two at the bottom. A pair of these (one at top and one at bottom) was available for recirculation of the vapor through the liquid. The second capillary into the bottom of

the cell was available for liquid sampling. The second line into the top of the cell was

used solely for pressure measurements. This was the only capillary not used in the

filling process. The mixtures were condensed directly and continuously into the equilibrium cell either through the two capillaries at the bottom of the cell (analogous to vapor

recirculation) or through the recirculation and liquid-sampling capillaries at the top

and bottom simultaneously. An effort was made to determine if the homogeneity of

the liquid sample in the cell was dependent on the filling procedure. As indicated by

the consistency of the densities, it was found that the measured densities were

independent of the filling routines attempted.

All measurements were on saturated liquids. In the present work it was impossible,

and not essential, to observe the liquid- vapor interface through a small high-pressure

sapphire window positioned near the mid-point of the cell along its length. The cell was initially filled with each new sample until the mixture was in the compressed

liquid, or at a pressure a few tenths of a MPa above its vapor pressure. Then, small

amounts of liquid were withdrawn from the bottom of the cell until the saturation pressure was attained, as denoted by reproducibility of the pressure readings. It was

extremely important to have as small vapor volume as possible in the sample space. As the cell was warmed to the temperature of the next point, the pressure of the mixture was monitored as a function of temperature. This information could be used

to calculate the amount of vapor volume in the cell for the prior point and to confirm

that the liquid mixture had been at saturation. The vapor volume in the cell was

sufficiently small (0.2 cm^) so that the liquid went into the compressed state for a temperature change of less than 2 K.

It was thought that the most accurate method to determine the compositions of the

liquid mixtures was to prepare them by gravimetric means under carefully controlled

conditions and, subsequently, to introduce them in a mixed state into the equilibrium

119
606 W. M HAYNES

chamber. The preparation of mixtures of lng components, with the exception of

pentanes, has been described in previous papers.**-^' The pentane fractions were

obtained from bottles of commercially available research-grade liquids. The lowest

purities, as specified by the supplier from freezing-temp>erature determinations, were

99.99 and 99.93 moles per cent for isopentane and normal pentane, respectively. The

most probable impurity in normal pentane was isopentane while those for isopentane were neopentane and normal pentane.

With one exception (mixture C in table 1,3.1 mol), at least 4 mol was prepared for each mixture investigated here. This corresponds to less than 0.0001 uncertainty in mole fraction of methane in the prepared mixture and considerably less uncertainty

for each of the other components. Due to insufficient mixing in a base mixture, the

compositions of three of the mixtures (mixtures m, n, and p) were determined by

analyses of samples using a gas chromatograph with a thermal-conductivity

detector.*'^' The uncertainty in the compositions determined in this manner

corresponds to an uncertainty of roughly 0.03 per cent in the density. At least six samples, with subsequent analyses, were taken for each mixture. Half of these samples were taken after heating the cylinders, in which the gas mixtures were contained, for approximately 30 min with an infrared lamp. The compositions for the six samples

agreed within experimental uncertainty and did not exhibit any dependence on the temperature of the mixture being tested. The total liquid volume within the cell was approximately 43 cm^, as determined

by filling the cell with water. The total vapor volume of the sample space was

approximately 7.4 cm^ and can be divided into three parts as follows : (a) inside cell

0.2 cm-*, (b) access tubing within cryostat—1 cm^, and (c) access tubing and pressure gauge outside cryostat—6.2 cm^. The volumes of the tubing and pressure gauge were computed from known dimensions. The composition of the liquid in the cell should

be slightly different from the prepared mixture composition as a result of the vapor

volume. The magnitude of the adjustment to the measured densities to account for

this composition change, so that the experimental densities can be assigned to the prepared mixture composition, is discussed in the next section. For all measurements reported here, the cell was always isolated from a

recirculation pump that could be used to mix the liquid inside the cell. Based on the consistency of the densities for different filling procedures and new mixture samples

and from comparisons of the experimental densities with predicted values, it was not deemed necessary to recirculate the vapor through the liquid to mix the samples or to take liquid samples and analyze them. There were other means for gaining

information on the reproducibility of liquid compositions within the cell for different samples from the same prepared mixture and on the homogeneity of the liquid

sample, as discussed below.

Vapor pressures were measured for each of the mixtures investigated here using a

spiral quartz Bourdon-tube gauge (0 to 1.38 MPa) calibrated against an air deadweight gauge. The maximum uncertainty in the gauge calibration was estimated to be

approximately 70 Pa. Vapor-pressure measurements for different samples and filling

procedures generally agreed within 3 x lO"'^ MPa.

A capacitor composed of slotted concentric cylinders located in the top part of the

120
DENSITIES OF lng MIXTURES 607

cell was used for dielectric-constant measurements on the mixtures. The capacitor had

a vacuum capacitance of 19.8 pF. Readings could be made routinely to within

0.0001 pF. This device was the most sensitive monitor within the sample space to determine the reproducibility of liquid compositions for different filling procedures

and for different samples from the same mixture.

Although the cell was monitored for temperature gradients along its length using

vapor bulbs at the top and bottom of the cell in only a few runs, a differential thermocouple with junctions at the same locations as the vapor bulbs was used in all runs. The vapor-bulb measurements were used as a control on the thermocouple

measurements. The reflux tube was always evacuated while performing

measurements. At temperatures between 110 and 130 K, greatest power input into the cell to balance the cooling was 0.5 W. The guard ring was controlled at a temperature somewhat above the cell tempjerature. Under the above conditions, no

problems were encountered with temperature gradients along the length of the cell within the precision of the vapor-pressure measurements.

The vapor pressures and dielectric constants were obtained for different portions

(different heights) of the liquid sample from that of the density determination.

Assuming an isothermal sample space, information on composition (or density)

gradients along the length of the cell could be gained from the various measurements

at different positions within the sample space.

3. Results

Experimental orthobaric liquid densities and vapor pressures of mixtures of lng

components are presented in table I as a function of temperature (IPTS-68). The

compositions presented in this table were determined gravimetrically except those for mixtures m, n, and p. The compositions of these mixtures were determined by

analyses of the prepared mixtures using a gas chromatograph with a thermal

conductivity detector, as discussed in the preceding section. Small corrections were made to the measured densities to account for differences in compositions between the prepared mixtures and the liquid mixtures in the equilibrium cell, which resulted from some of the sample space being occupied with

vapor. The largest adjustment was less than 0.05 per cent, which was for mixture a at 125 K. All other corrections were less than 0.02 per cent. For all mixtures containing

hydrocarbons only, it was assumed that the vapor composition was 100 per cent methane. For those mixtures containing nitrogen, the vapor compositions were

estimated from partial-pressure calculations using available phase-equilibria data for (nitrogen -I- methane)."*' ' ^' Excess volumes defined by the relation

:

V^=V^-Yx,V,{\+K,(p-pJ\, (1)

i

are also given in table 1. In this expression V„ is the molar volume of mixture at a

given temperature at saturation pressure is the molar volume of pure

I'll
608 W. M HAYNES

TABLE 1. Orthobaric-liquid densities of multicomponent mixtures of lng components. M, Molar mass;

T. temperature (IPTS-68); p, pressure; eip.- experimental amount-of-substance density, /»j,p„ experimental density; V^, excess volume calculated from equation (1 ); density calculated from

the extended corresponding-states method as given in references 7 to 9

T _p_ P..C.P, 10^(P„p.-p„n.)

K MPa mol dm"^ cm' mol"' p„„

(a ) (0.0593 1 N 2 + 0.8907 ICH4 + O.O4998C4 H 1 0) (M = 18.8562 g mol"')

110.00 0.2400 25.3450 - 0.713 + 0.03 115.00 0.3145 24.9440 -0.983 +0.01

120.00 0.4082 24.5383 -1.595 +0.00

125.00 0.5196 24.1141 -0.04

(b) {0.86040CH4+0.04600C2Hf, + 0.04790C3H8+0.04570CH3CH(CHj)CH3|

(M = 19.9552 g mol"')

115.00 0.1186 24.2654 -0.631 -0.03

120.00 0.1710 23.9371 -0.741 +0.01

125.00 0.2387 23.5868 -0.842 -0.01

130.00 0.3248 23.2331 -0.970 -0.02

135.00 0.4320 22.8637 -1.110 -0.07

(c) {0.85378CH4+0.05178C,Hb+0.04703C3H8+0.04741CH3CH(CH3)CH3l

(M = 20.0838 g mol-')

115.00 0.1191 24.2100 - 0.640 - 0.03 120.00 0.1706 23.8779 -0.739 -0.02

125.00 0.2379 23.5324 -0.843 -0.04

130.00 0.3238 23.1834 -0.975 -0.05

(d) (0.85133CH4 + 0.05759C2H6 + 0.04808C3H8+0.04300C4H,o)

(M = 20.0092 g mor')

.115.00 0.1180 24.3243 -0.675 -0.01

120.00 0.1700 23.9965 -0.785 +0.03

125.00 0.2374 23.6586 -0.907 +0.05

130.00 0.3232 23.3108 -1.045 +0.05

135.00 0.4301 22.9634 -1.226 +0.09

(e) (0.84566CH4+0.07924C2H^+0.05060C3H8 + 0.02450C4H,o)

(M = 19.6051 gmol"')

115.00 0.1167 24.5569 -0.621 -0.01

120.00 0.1683 24.2126 -0.711 -0.01

125.00 0.2350 23.8698 -0.832 +0.03

130.00 0.3201 23.5204 -0.975 +0.07

(f) i0.04801N,+0.8094OCH4+0.04542CjH<, + 0.05050CjH8+0.04667CH3CH(CH3)CH3|

. (M = 20.6355 g- mor")

115.00 0.3005 24.1487 -1.129 - 0.04

120.00 0.3863 23.8075 -1.679 -0.03

125.00 0.4874 23.4518 -0.05

130.00 0.6125 23.0893 - 0.07
DENSITIES OF LNG MIXTURES 609

TABLE 1

continued

K MPa mol dm"^ cm^ mo!"' p^m

(g) (0.02628N J + 0.8 1 249CH4 + 0.08484C2 H<, + 0.0493 1 Cj Hg + O.O27O8C4 H , 0) (M = 20.0706 g mol -

'

)

115.00 0.2214 24.4562 -0.908 -0.02

120.00 0.2874 24.1119 -1.254 -0.01

125.00 0.3768 23.7507 -0.05

130.00 0.4793 23.3954 -0.03

(h) {0.85892CH4+0.11532C2H6 + 0.01341C3He+0.00530CH3CH(CH3)CH3+0.00705C4H,ol

(M = 18.5565 g mol"')

115.00 0.1185 25.0957 -0.480 -0.03

120.00 0.1706 24.7131 -0.534 -0.07

125.00 0.2372 24,3294 -0.613 -0.08

130.00 0.3225 23.9490 -0.730 -0.06

(i) {0.84558CH4 + 0.08153C2H, + 0.O4778C3He+0.01259CH3CH(CH3)CH3+0.01252QH,o:

(M = 19.5838 g mol"')

115.00 0.1166 24.5586 -0.620 +0.01

120.00 0.1680 24.2180 -0.716 +0.03

125.00 0.2348 23.8688 -0.826 +0.04

130.00 0.3188 23.5154 -0.963 +0.07

(j) ;0.00601Nj +0.9061 3CH4+0.06026C,Hfe+0.02154C3H8+0.00300CH3CH(CH3)CH3 + 0.00306C4H,o! (M = 17.8195 g- mol"')

115.00 0.1478 25.3834 -0.425 -0.01

120.00 0.2043 24.9894 -0.551 +0.02

125.00 0.2785 24.5702 -0.02

130.00 0.3722 24.1578 +0.02

(k) ;0.00973N2 +O.88225CH4 +O.O7259C2H, +O.O256IC3H8 +0.00490CH3CH(CH3)CH3 + 0.0049 2C4H,ol (M = 18.3094 g- mol-')

115.00 0.1639 25.2023 -0.556 -0.04

120.00 0.2247 24.8047 -0.698 -0.00

125.00 0.3022 24.4022 -0.03

(!) {0.01383Nj + 0.85934CH4+0.08477C:,H6+0.02980C3Hs+0.00519CHjCH(CH3)CH3 + 0.00707C4H,oi (M = 18.7496 g moi-')

115.00 0.1812 25.0384 -0.662 +0.06

120.00 0.2441 24.6661 -0.874 +0.07

125.00 0.3223 24.2880 +0.09

130.00 0.4222 23.8981 +0.10

(m)0.85341CH4 + 0.07898C2H,+0.04729C3H8+0.00854CH3CH(CH3)CH3 + 0.00992C4H,o + 0.00097CH3CH(CH3)CjH5+0.00089C5H,2: (M = 19.3587 g mol" ')

110 00 0.0787 25.0063 -0.517 -0.01

115.00 0.1172 24.6566 -0.581 -0.03

120.00 0.1686 24.3079 - 0.670 - 0.02

125.00 0.2351 23.9525 -0.777 -0.01

130.00 0.3210 23.5883 -0.901 0.00

i/:6
610 W. M. HAYNES

TABLE 1continued

K MPa mol dm"^ cm^ mol"' p„m

(n) !0.75442CH,+0.15401C2H6+0.06950CjH9+0.00978CH3CH(CH3)CH3+0.01057C4H,o + 0,0O089CH3CH(CH3)C,H5 + 0.0O083C5H,2i (Af = 21 .1060 g • mor '

)

110.00 0.0723 24.2529 - 0.633 - 0.05 115.00 0.1081 23.9619 - 0.740 + 0.01 120.00 0.1549 23.6535 -0.840 +0.01

125.00 0.2153 23.3351 -0.949 -0.01

(o) {0.00859N2+0.75713CH4 + O.13585C2H(,+O.O6742CjH8+O.01336CH3CH(CH3K:H3 + 0.01326C4H,o + 0.O0223CH3CH(CHj)C2H5+0.00216C5H,j; (M = 21.3094 g moj- ') 110.00 0.1155 24.1809 - 0.741 +0.03

115.00 0.1595 23.8731 -0.857 +0.02

120.00 0.2155 23.5709 -1.055 +0.05

125.00 0.2873 23.2644 +0.08

130.00 0.3744 22.9514 +0.11

(p) |0.00801N2+0.74275CH4 + 0.16505CjH6 + 0.06547C3H8+0.00843CH3CH(CH3)CH3 + 0.00893C4H,o + 0.00069CH3CH(CH3)C2H5+0.00067C5H,3; (Af = 21.0976 g mor')

110.00 0.1158 24.3141 -0.676 -0.05

115.00 0.1584 24.0160 -0.807 -0.02

120.00 0.2093 23.6937 -0.965 -0.06

125.00 0.2853 23.3804 -0.05

(q) 10.00599Nj +O.9OO68CH4 +0.06537C,Hfe +O.O22OOC3H8 +0.00291CH3CH(CH3)CH3 + 0.00284C4H,o + 0.00010CH3CH(CHj)C2H5 + 0.0001 IC5H, 2! (M = 17.9026 g mol"')

115.00 0.1456 25.3600 - 0.449 + 0.02

120.00 0.2024 24.9656 -0.571 +0.03

125.00 0.2762 24.5450 -0.03

130.00 0.3698 24.1289 -0.02

component i at the same temperature at saturation pressure p,, Xj is the mole fraction

of component i, and k, is the isothermal compressibility of component i. The pure-fluid molar volumes were obtained from expressions given in references 1,

2, and 16, while the compressibilities for nitrogen, methane, and ethane were obtained

from expressions presented in reference 6. Compressibility corrections for hydrocarbons beyond ethane were sufficiently small to neglect. For those mixtures

containing nitrogen, excess volumes were not computed for temperatures above

120 K. (The critical temperature of nitrogen is approximately 126.2 K.) Vapor

pressures for nitrogen, methane, and ethane were obtained from references 17 to 19, respectively.

To demonstrate the consistency of the results presented here, table 1 includes

comparisons between the experimental densities and the values calculated using an

extended corresponding-states method,'^"^' which was optimized to only the purefluid (except for pentanes) and binary-mixture values from the lng density project at

this laboratory. Of the mathematical models developed at this laboratory*^"'' for the prediction of lng densities, the extended corresponding-states method is probably the most accurate and certainly the most general since it handles the widest ranges of

124
DENSITIES OF lng MIXTURES 611

pressure, temperature, and composition. All other models'^"^' developed at this laboratory were characterized by limitations in the ranges of either temperature or composition covered for mixtures containing nitrogen and butanes. The maximum

deviation between the experimental and calculated densities using the extended

corresponding-states method was 0.11 per cent, while the average absolute deviation was 0.04 per cent. Comparisons of the present results with other mathematical models optimized to the pure-fluid and binary-mixture values from this project are presented in reference 8. Comparisons of multicomponent mixture values (lng components) from

other sources with those of the present study have been accomplished indirectly in earlier papers.'^' ^' The results from other work were compared with the calculated

values from the extended corresponding-states method that had been optimized to the pure fluid and binary-mixture values from this project.

Detailed descriptions of the random and systematic errors involved in density

measurements on liquids and liquid mixtures with a magnetic suspension densimeter

have been presented elsewhere. The total systematic error in the measurements on multicomponent mixtures should be little different from that for binary mixtures.

The added uncertainty in the composition of the prepared mixtures resulting from the

larger number of components is insignificant.

The random error in the multicomponent-mixture measurements was difficult to evaluate since, for most mixtures, there was not a sufficient amount prepared to make

a statistically significant number of repetitive measurements on different samples from

the same mixture. It is believed that the random error in these multicomponent

mixture measurements should not be significantly different from that for (methane -I- isobutane) and (methane + normal butane) measurements'^' (maximum

standard deviation of 0.024 percent) obtained with the same apparatus.

Based on the analysis of errors for density measurements on pure fluids and binary

mixtures with a magnetic suspension densimeter, it is believed that the maximum

total uncertainty in the densities for the multicomponent mixtures presented in this paper is approximately 0.12 per cent. The total uncertainty has been taken as the root mean square of the known systematic errors added to three times the standard

deviation for random error. This estimate was confirmed, to some extent, in the comparisons between the experimental results and the predicted values from the extended corresponding-states method. As reported earlier in this section, the maximum deviation was 0.11 per cent and the average absolute deviation was 0.04

per cent.

The contributions of M. J. Hiza in the preparation of mixtures, R. J. Richards in the analysis of mixtures, and R. D. McCarty and R. G. Sargent in the reduction and

analysis of results are gratefully acknowledged.

REFERENCES

1. Haynes, W. M.; Hiza, M. J.; Frederick, N. V. Rev. Sci. Insirum. 1976, 47, 1237.

2. Haynes, W. M.; Hiza, M. J. J. Chem. Thermodynamics 1977, 9, 79.

3. Haynes, W. M.; Hiza, M. J. Advances in Cryogenic Engineering Vol. 21. Timmerhaus, K. D.; WeiUel, D. H.: editors. Plenum Press: New York. 1976. p. 516.

125
612 W. M. HAYNES

4. Hiza, M. J.; Haynes, W. M,; Parrish, W. R. J. Chem. Thermodynamics 1977, 9, 873.

5. Haynes, W. M. To be published.

6. Hiza, M. J.; Haynes, W. M. J. Chem. Thermodynamics 1980, 12, 1. 7. McCany. R. D. Nail. Bur. Stand. (U.S.). Interagency Kept NBSIR 77-867. 1977.

8. McCarty, R. D. Natl. Bur. Stand. (U.S.). Tech. Note 1030. 1980.

9. McCarty, R. D. to be published.

10. Orrit, J. E.; Laupretre, J. M. Advances in Cryogenic Engineering Vol. 23. Timmerhaus, K. D. : editor. Plenum Press: New York. 1978, p. 573.

11. Haynes, W. M. To be published.

12. Haynes, W. M. Rev. Sci. Insirum. 1977, 48, 39.

13. Parrish, W R.; Arvidson, J. M.; LaBrecque, J. F. Natl. Bur. Stand. (U.S.). Interagency Rept NBSIR

78-887. 1978.

14. Parrish, W. R.; Hiza, M. J. Advarwes in Cryogenic Engineering Vol. 19. Timmerhaus, K. D.: editor. Plenum Press; New York. 1974, p. 300.

15. Kidnay, A. J.; Miller, R. C; Parrish, W. R ; Hiza, M. J. Cryogenics 1975, 15, 531.

16. Hiza, M. J. Fluid Phase Equilibria 1978, 2, 27.

17. Strobndge. T. R. Natl. Bur. Stand. (U.S.). Tech. Note 129. 1%2.

18. Goodwm, R. D. Natl. Bur. Stand. (U.S.). Tech. Note 653, 1974.

19. Goodwin, R. D.; Roder, H. M.; Straty, G. C. Natl. Bur. Stand. (U.S.). Tech. Note 684. 1976.

126
6.11 Reference [34]

Four Mathematical Models

for the Prediction

of LNG Densities

Robert D. McCarty

Thermophysical Properties Division

National Engineering Laboratory

National Bureau of Standards

Boulder. Colorado 80303

U.S. DEPARTMENT OF COMMERCE, Philip M. Klutznick, Secretary

Jordan J. Baruch, Assistant Secretary for Productivity, Technology and Innovation

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Issued December 1980

127
CONTENTS

Page

Abstract 1

1. INTRODUCTION 1

2. EXTENDED CORRESPONDING STATES 3

3. A HARD SPHERE METHOD 6

4. A REVISED KLOSEK AND MCKINLEY METHOD 11

5. THE CELL MODEL

.

13

6. USE OF THE METHODS 15

7. CONCLUSIONS 19

8. REFERENCES 20

Appendix A. Experimental Data 24

Appendix B. Computer Program and Equation Parameters for the

Extended Corresponding States Model 39

Appendix C. Computer Program and Equation Parameters for the

Hard Sphere Model 46

Appendix D. Computer Program and Parameters for the Revised

Klosek and McKinley Model 50

Appendix E. Computer Program for the Cell Model 54

Appendix F. Computer Programs 55

i i i

128
FOUR MATHEMATICAL MODELS FOR THE PREDICTION OF LNG DENSITIES*

Robert D. McCarty

Thermophysical Properties Division

National Engineering Laboratory

National Bureau of Standards

Boulder, Colorado 80303

Four mathematical models of the equation of state for LNG like

mixtures are presented. The four models include an extended

corresponding states model, a cell model, a hard sphere model and a revised Klosek and McKinley model. Each of the models has been

optimized to the same experimental data set which included data for

pure nitrogen, methane, ethane, propane, iso and normal butane, iso and

normal pentane and mixtures thereof. For LNG like mixtures (mixtures

of the orthobaric liquid state at temperatures of 120 K or less and

containing at least 60% methane, less than 4% nitrogen, less than 4%

each of iso and normal butane and less than 2% total of iso and normal

pentane), all of the models are estimated to predict densities to within

0.1% of the true value. The revised Klosek and McKinley model is valid

only for mixtures within the range of temperature and composition

specified above while the other three models are valid for a broader

range of pressure, temperature and composition. The experimental PVTx

data set used in the optimization together with comparisons are given

and listings of computer programs for each of the models are included.

Key words: Cell model; comparisons; computer programs; corresponding states;

equation of state; hard sphere; LNG; mixtures; PVTx data; revised Klosek and

McKinley.

1. INTRODUCTION

The purpose of this report is to present in final form the four mathematical

models which were optimized to the experimentally determined orthobaric liquid

This work was carried out at the National Bureau of Standards under the

sponsorship of British Gas Corp., Chicago Bridge and Iron Co., Columbia Gas

Service Corp., Distrigas Corp., Easco Gas LNG, Inc., El Paso Natural Gas, Gaz de

France, Marathon Oil Co., Mobil Oil Corp., Natural Gas Pipeline Co., Phillips

Petroleum Co., Shell International Gas, Ltd., Sonatrach, Southern California Gas

Co., Tennessee Gas Pipeline, Texas Eastern Transmission Co., Tokyo Gas Co., Ltd.,

and Transcontinental Gas Pipe Line Corp., through a grant administered by the

American Gas Association, Inc.

IZ9
PVTx data of Miller and Hiza [25], Haynes and Hiza [12], Haynes, et al . [11],

Hiza, et al, [14], Hiza and Haynes [15] and Haynes [9]. Interim results of the

project have been reported by Haynes, et al . [13] and by McCarty [23]. The

models reported in these two interim publications differ only slightly from those

presented here. A companion archival document, McCarty [24] with the same

results but in much less detail has been submitted to the Journal of Chemical

Thermodynamics. The intent of the documentation here is more in the vein of a

user's handbook. The above experimental data are for the liquid phase of

nitrogen, methane, ethane, propane, normal and isobutane and various mixtures

thereof.

The goal of the project (the project included the above referenced experimental work) was to produce one or more computer models which would predict the

density of LNG to within 0.1% of the true value from a knowledge of the

temperature, pressure and composition of the LNG. At the beginning of this study

LNG was defined as mixtures of the above components {N2, CH^, CgHg, C^Hg,

n and iC^H^q) and only the saturated liquid between 95 - 150 K was to be

considered. Near the end of the project n and iCgHj^g were added to the list of

allowable components but no experimental PVTx of pure n and iCgHj^2 binary

systems containing n and iCgH^^g were measured as part of the project. The

inclusion of these two components is based on data from Orrit, et al . [29] and

Orrit, et al . [30].

Four models were considered: the extended corresponding states model, a

hard sphere model, a cell model and a Revised Klosek and McKinley model. With

the exception of the Revised Klosek and McKinley model only pure fluid and binary

system data were used to optimize the models. In the case of the Revised Klosek

and McKinley model multi component PVTx data were used in the optimization

process.

2

130
Over a normal range of LNG composition and temperature all four of the

models predict densities which agree to within 0.1% of experiment. This is true

of all of the experimental PVTx measurements on LNG like mixtures made as part of

this project.

No equation or mathematical model based on experimental data can be more

accurate than the original data and therefore the extent to which the original

goal of 0.1% accuracy has been met depends entirely upon the accuracy of the

experimental data referenced above.

There is no reason to doubt the experimental data and therefore there is

every reason to believe that the goal of the project has been achieved.

2. EXTENDED CORRESPONDING STATES

The thermodynamic equations for the extended corresponding states method are

developed in a paper by Rowlinson and Watson [35] and only a very brief

description will be given here. Leach [20] developed transformation functions

for hydrocarbons which are called shape factor functions. Mollerup [27] and

Mollerup and Rowlinson [26] combined the earlier work with the equation of state

for methane by Goodwin [8] to produce a computer program to calculate the density

of LNG mixtures, which was further modified by Mollerup [28].

The computer program in Appendix F for the calculation of LNG densities

based on the extended corresponding states method is an extensive revision of the

Mollerup program. Earlier revisions were reported by McCarty [23] and Haynes,

Hiza and McCarty [13].

The extended corresponding states method is defined by the following

equations:

VP'^^ = yp^i.o^-i.o'^/^ii,o^

3

131
Si^P'T] = fii,o VP Ni,o/^ii.o> T/fii,o] tn(h,,_„) (2)

where Z is the compressibility factor, G is the Gibbs free energy. Pis pressure

and T is temperature. The subscript o denotes the reference fluid and the

subscript i denotes the fluid for which properties are to be obtained via the

equation of state for the reference fluid and the transformation functions

f . . and h.. „. The double subscript ii is introduced now to allow extension n ,0 n ,0

of the method to mixtures. The two defining eqs (1) and (2) are necessary since

there are two transformation functions. In this case the equation of state for

methane by McCarty [22] was chosen for the reference fluid. During the course of

the study it was necessary to modify the equation of state by McCarty [22] to

give a realistic vapor liquid phase boundary down to a temperature of 43 K. This

modification was necessary to acconmodate the very low reduced temperatures of

the heavier hydrocarbons and was accomplished without changing the performance of

the equation of state above the triple point of methane. The equation of state

is given in Appendix B.

The f. . „ and h.. ^ are defined as n ,0 11,0

and

Ni.o = (V?i,o/^S' *ii.o (Tp-V («)

where

^i.o = 1 - w^)[ni - n2 m T^_ + (n3 - r^^/^U^^ - n^)] (5)

and

4

132
The and are reduced temperature and volume, (i.e., =

^/"'^ii

^r.

~ ^^^ii^ ^^^^ fluid requires a unique which was estimated using pure fluid

experimental data. A single set of the n^'s are used for all fluids. The n^. 's

were estimated using all of the pure fluid experimental data from this study. The

Z^/Z^ is the ratio of the compressibility factors (Z^ = PqV^/RT^) at the critical

point. The parameters n . , n., w. and Z.. are given in Appendix B. All of

these parameters were estimated using the experimental PVT data set from this

laboratory and least squares estimation techniques.

The extension of the above to mixtures is now accomplished by the following

application of the following combining rules:

f .. = (f • . f • • )^^^ (9)

Nj,o - ^ij (2 ^i.o ^ 2 ^-0,0)

The £;^.j and the n^^ are binary interaction parameters determined by least

squares from the PVTx data for binary mixtures. These parameters are given in

Appendix B.

5

133
This method works quite well as may be seen in the comparisons in

Appendix A. It has indeed reproduced all of the present experimental data set to

within + 0.1% except for 14 out of a total of 285 experimental data points. Of

these 14 points, 11 are judged to have an uncertainty greater than 0.1%.

Figure 1 presents the deviations between the calculated and experimental

densities for these 14 points. Appendix A contains comparisons of calculated and

experimental densities for the entire data set. This is the best performance of

the four models presented here. No pressure, temperature or composition

restrictions have been placed on this model.

In the interim publications by McCarty [23] and Haynes, Hiza and

McCarty [13] some doubt about the accuracy of the calculated densities was

expressed because of the disagreement with a few binary and multi component

systems containing methane and butane. This disagreement has since been resolved

by additional measurements (Haynes [9], Haynes [10] and Miller and Hiza [25] on

some of the systems which agree with the predictions of the model but disagree

with the previous measurements. The net result of the new measurements is a very

slight change in binary interaction coefficients of the methane-butane and

nitrogen-butane system. These changes have no practical effect on LNG like

mixtures where the concentrations of N2 , iC^Hj^Q and nC^H^^Q are individually

less than 5%. In other words either the models presented here or those in the

interim publications may be used to predict the density of a LNG like mixture to

within 0.1% of the true density.

3. A HARD SPHERE ^ETH0D

The model of Rodosevich and Miller [33] is one of many modifications of the

Longuet-Higgins and Widom [21] model, and was chosen to be included in this study

6

134
as a representative example of the application of the hard sphere equation of

state concept to the correlation of PVTx data. The equation of state by

Rodosevich and Miller [33] is

pv . - 1

•«• y + y _ _a_

(11)

where the y = b/4V and a, b, and c are adjustable parameters, P is pressure, V is

specific volume, T is temperature and R is the gas constant. The equation is

applied to mixtures by assuming the one-fluid theory and applying the following

combining rules.

a„ = S S a. . x.x. (12) m ^. J. ij 1 J

b„ = 2 2 b.. x.x.

m ^ j ij 1 J

Cm = 23 Sc.. X.X. m ^ j ij 1 J

(13)

(14)

The mixing rules are:

1

1

T 3

2

^

(1 -J-ij) (15)

(^i^jj)

1/2 r b

2 ^ 1/2

11

b..b .

.

n JJ

(1 - (16)

c . . + c .

.

(17)

The parameters j- • and k.. are in this case the binary interaction

parameters. The a's, b's, c's, j^^'s and k..'s are given in Appendix B. The

excess volume is now calculated using the equation of state and

7

135
V,E (18)

where V and the V. are calculated via the eqs 11 through 17 and then

V,m

V.x. . V, (19)

where the is from eq (18) and the are from experimental data. The

values of in this case were calculated from the equations for the liquid

density of the pure fluids given in Appendix C.

The above equations are those of Rodosevich and Miller [33] and Rodosevich

[34] and only the J^^-'s and k^^-'s have been revised on the basis of the

'0 'J present new data set, and only binary systems data were used to estimate via

least squares the L^'s and k. .'s.

As the method is used here it is an excess volume method, and consequently

when the temperature of the mixture approaches the critical temperature of one of

the component fluids, the method fails. Since the critical temperature of

nitrogen is about 126 K, this method should not be used for mixtures containing

nitrogen at temperatures above 120 K. Eliminating the data points for mixtures

which contain nitrogen at temperatures above 120 K reduces the set from 285 to

251 PVTx points. Figure 2 is a percentage deviation plot containing all of the

data points from the set of 251 for which densities calculated by the hard sphere

method differ from the experimental density by more than 0.1%. Two things are

readily seen in comparing figs. 1 and 2; first, even though total number of

points has been reduced in the comparison set, the number of points for which

deviations exceed 0.1% in the hard sphere comparison, fig. 2, is far more than

for the extended corresponding states comparison, fig. 1. Second, the hard

sphere method becomes more uncertain for all mixtures, regardless of components

as the temperature exceeds 115 K.

8

13b
ifi 0)

c

•5c

o

a

CO

o

O

<

•Hc

(Ua

a

<

inm

CO

<

+ DO

< -

o

o

CO

o

04

w

3

I-

<

111 Q.

z

UJ

o o

o

o

I AllSNaa Nl NOIIVIABQ %

(U

4-1 CO « 4-1 iH CO 3 T3 y

iH m

CO H 00 u CM

C

CO ho

• <

iH 4J

CO (U X

4-1 e •H c

ID CO C

s a> (U •H u a

U CO

a> 4J <

cn X c

00c

c •H CO T) 4J

(U c c

$ o 1-1 4J Q. . o

CO a ^ re

CD u 4J

.-1 o CO

• u o

o

<u c <U

CO 4-1 x: c

4J (U 4J of u X

<ii 1—1

4-1 1—1 CO (U CO

0) x: u 4J CO to 00

•H

CO c 4-1 c •H <U o CO CO •H 3

4.1 3

CO CO O

•H 01 n > •H

4-1 t-l CO T) •H CO 4-1 CO a c

iH C B •H

cu o o < 13 CJ a

3

00

•H

9

137
xc0)::::::::::::::: a. Oi-ieMfn«*ONOmooo«3-iriv£>«vjvDPn

0)

B

1 1 '

t> <

\

-

> <

> + • • -

•1 •

ha X

1 . 1

o

o

CO

Hi

oc3

O ICM <

lU

0.

s

O H

o

o

0)

CMo

CMoI

Q (no c

•H

<—

1

c

V to

to w to •H c QD o o 4J o

fH (U

(d ID to o o

I.c •o CO

to )

1 (0 •H CO

Ui 1—

1

to P

to o.Q

c o o

o C4-( •H d) CD

Id x: 4J • 01 c

O. •HO o a CM

r-* c (U (0

0) 4J 60 o CO C s •HT5 O 3

Xi Id CD iH O U r* c

p •H

* W

o 4Ja c

c 0) CO

CO CO o

ac X (U 4J

AO Id -<

0) CO « to •H to c a> c >-i &0 to 3 3 a JJ n a. to c CO < u

o <u 9) •H c a •H s

to 1-1 •H to CO > c

0) V (0 Q to

AiiSNaa Ni NOiiviAaa %

3

60

•H

1X4

10

138
4. A REVISED KLOSEK AND MCKINLEY METHOD

The Klosek and McKinley method [18] is a totally empirical recipe for

calculating the density of a LNG-like mixture given the temperature and

composition. Pressure is not taken into account. However, this does not seem to

be a serious omission. The procedure proposed by Klosek and McKinley [18] is as

follows:

V„,, = E X,V, - kX,„^ (20)

where V„. is the volume of the mixture, X. and V. are the mole fraction and mix 11 t h

volume of the i component, X^.^ is the mole fraction of methane and k is a

correction factor obtained from a table or graph. The and k are obviously

temperature dependent and in addition k is dependent upon the molecular weight of

the mixture.

Using the present data set k was calculated for all of the expe.'imental data

points "where methane was present in the mixture and excluding all data points

where N2 was present in greater than S% concentration. Figure 3 shows a typical

isotherm for k, with N2 present (labeled k2) and without Ng present (labeled k^)

.

All of the isotherms available show similar behavior, i.e., all of the k's for

mixtures containing nitrogen (of about 5%) fall on one line and all of those for

mixtures without nitrogen fall on another. Since all of the mixtures with

nitrogen have about the same amount of nitrogen present (about 4.5%), the method

was modified by adding a term to take into account the nitrogen when it is

present. The equation becomes

\U - 2 ''i^

- * (^2 - "!> '<n/-°*25]

X,„^ (21)

where everything is the same as in eq (20) except that k, is read from one

11

139
140
curve and k2 is read from the other. Appendix D gives tables of values for

the V^, and ^2 which are spaced such that linear interpolation is adequate

in both variables (i.e., temperature or molecular weight). The k factors in

Appendix D have been obtained graphically from the mul ti component PVTx data of

Hiza and Haynes [15] and Miller and Hiza [25] as well as densities calculated

from the extended corresponding states method of section 2.

The limits of compositions of the revised Klosek and McKinley method are the

most severe of any of the methods given here. This method should not be used for

mixtures other than LNG like mixtures and for LNG like mixtures only when they

contain at least 60% methane, less than 4% nitrogen, less than 4% each of iC^H^Q

nC^H^Q and less than 2% total of ^^^^^12 "^5^12*

There are 40 experimental PVTx points from the original set of 285 which may

be considered LNG like and fall within the composition limits outlined above.

Figure 4 shows all of the deviations between calculated and experimental

densities in this 40 point comparison set which exceeds the 0.1% criterion. The

deviation trends for the revised Klosek and McKinley method (fig. 4) are very

similar to those of the hard sphere method (fig. 2) and in fact all of the

deviations in fig. 4 occur at temperatures at or above 115 K, therefore the

method can only be considered as accurate as the others for LNG like mixtures at

temperatures below 115 K.

5. THE CELL MODEL

The cell model considered here was originally proposed by Renon,

et al . [32]. In a paper by the same three authors which appeared simultaneously

(Eckert, et al . [7]), the cell model was applied to mixtures via Scott's [36]

two-fluid theory and a three parameter corresponding states theory. Albright

[2] further modified the method by modifying the mixing rules on the basis of a

13

141
UJ

fC

UJ

0.

Li

AllSNaa Nl NOIlVIAiQ %

c

o

o (0

fll w TO 4J

TO ce TO a

BO o

V

to

(J he /\

H

cn

CO

• I—

1

HIo rS CD s

1

<D p c w

(J c

<U

(D T3 vO

•>

t> c

jj CO (0

HI vg

OJ00o

1^

es

ta j: CO 4J

4J c o

> 1—

t

Ui c ac

0) o -a- o. u

« 0) B •H o

u 4J CJ 60

60

•H

4-1 (0 C iH c •H 3 o CO E

•H S o 4J iH .-4 CO CO i-H a:

01 CO > •H u 4J CO c CO •o •H •H

CO >«s c

c 4.J -a- H

V O

< CO V

0)s

60

14

142
proposal by Yuan [38] and by inserting a pressure dependence based on the

experimental liquid ethane data by Pope [31].

The optimization of this method was carried out by M. Albright [l] at

Phillips Petroleum Company in Bartlesville, Oklahoma and the details of this work

will be published elsewhere. The model is included here because it was optimized

to the same data set as the others and therefore the comparisons between

experimental and calculated densities given here in fig. 5 together with figs. 1,

2 and 4 provide a conrion basis of comparison with the other three methods. A

listing of the computer program is given in Appendix F.

The same data set as was used in the hard sphere method for comparison has

been used here, i.e., all of the data points for mixtures containing nitrogen at

temperatures 120 K and above have been taken out of the original 285 points

leaving a total of 251 data points.

As in the case of the other methods fig. 5 shows all of the points for which

the calculated and experimental densities differ by more than 0.1%.

6. USE OF THE METHODS

When the project started in 1972, the atomic weights of nitrogen, carbon and

hydrogen were taken from the 1961 carbon 12 scale, lUPAC [16]. During the course

of the investigation a revision, Atomic Weights of the Elements [3], to this

scale appeared. The revision changed slightly the atomic weights of carbon and

hydrogen, but since the changes were small (the maximum difference in any of the

densities used here is 0.003%), and because changing the atomic weights would not

change the relative results, the changes were not made. Therefore when using

the tables and programs in the appendices, the molecular weights given in the

tables and programs should be used to maintain consistency.

15

143
>-jz

lU

a

<

>

U

Q

O mixture 3

-0.2 -

+ 11

12 • 13

14

15

16

V 17

35

39

e 40 • 42

45 * 52

53

55

56

Appendix A

100 110 120 130

TEMPERATURE, K

140

Figure 5. Deviations greater than 0.1% between experimental and calculated

densities using the Cell model. The comparison set is all of the data

in Appendix A except those data points for mixtures containing N2 at

temperatures above and including 120 K. (251 data points)

16

144
The critical parameters used here are from: CH^, McCarty [22]; CgHg,

Sliwinski [37]; C3Hg, Das, et al . [4]; iC^H^Q, Das, et al . [5]; nC^H^Q, Das,

et al . [6]; ^0^^i2* Kudchadker, et al . [19]; and Jacobsen, et al . [17].

Errors in the input variables will of course, cause errors in the density

predicted by the models. In general, the error in density caused by an error in

the input varibles is a function of those input variables, and must be treated on

an individual basis. However, for LNG like mixtures certain general trends are

found. An error in the pressure must be at least 50% before it will have any

effect at all on the resulting density. An error in composition, unless it is of

the order of several percent, will cause the same relative error in density as it

will cause in the molecular weight of the mixture, i.e., if an error in

composition causes a 0.1% error in the resulting molecular weight, it will also

cause a 0.1% error in the predicted density.

The error in the calculated density due to an error in the input temperature

is a function of the composition and the temperature. Table 1 gives resulting

errors in density for a 1% error in temperature, for three hypothetical LNG like

mixtures

.

"In general the errors in density caused by an error in temperature are the

largest for mixtures containing a high concentration of the most volatile fluids,

CH^ and and correspondingly the errors decrease as the concentration of

the heavier hydrocarbons increases in the mixture. These errors are not a

function of which model is being used.

When using the extended corresponding states method, one should keep in mind

that twelve significant figures are required by the methane equation of state.

The hard sphere model also uses the methane equation from McCarty [22] and the

nitrogen equation of Jacobsen, et al . [17] to calculate compressibilities and

17

145
TABLE 1. Errors in Density Caused by an Error in the

Input Temperature of 1%.

Temoerature % Error in Density

*

K Mix A* Mix B

95 0.28 0.25

100 0.30 0.27

105 0.32 0.29

110 0.35 0.32

115 0.39 0.34

Mix C*

0.20

0.22

0.24

0.29

0.31

Mix A = 0.95 CH^, 0.05 N2

Mix B* = 0.9 CH4, 0.02 C^}^^, 0.02 C3Hg, 0.02 iC^H^Q, 0.02 nC^H^Q, 0.02

Mix C* = 0.6 CH^, 0.3 C2Hg, 0.02 CjHg, 0.02 iC^H^q, 0.02 nC^H^Q, 0.02

0.02 iC5H^2*

*Arbitrary LNG like compositions assumed for the purpose of illustrating the

effect of an error in the input temperature.

18

146
therefore requires twelve significant figures to insure the accuracy of the

calculated density. The other two models require only eight significant figures

to be carried along in the calculations.

7. CONCLUSIONS

On the basis of the performance of the four models given here and subject to

the composition and temperature restrictions already noted, it is estimated that

given the pressure, temperature and composition of LNG, any one of the four

models may be used to predict the density to within 0.1% of the true value. As

has already been mentioned (see section 1) the above accuracy statement is

dependent entirely upon the accuracy of the experimental data in Haynes, et al

.

[11], Haynes, et al . [13], Hiza, et al . [14], Haynes [9], Hiza and Haynes [15],

Miller and Hiza [25] and Haynes [10]. These data have been estimated by the

authors to be accurate to within 0.1% of the true value with a precision of a few

hundredths of a percent. The work on the models given here have provided no

basis for questioning the claims of the experimenters, in fact the ability of the

models to predict the densities of the multi component mixtures to within 0.1% of

the measured values tends to support the accuracy claims of the experimenters.

Interim results of this study were reported by Haynes, et al . [13] and

McCarty [23], both of which contain earlier versions of the mathematical models

given here. These earlier versions are only slightly different than the final

ones and for the purposes of calculating LNG densities either of the versions may

be used. The reader is, however, cautioned to read the limitations of each model

as defined in the earlier sections.

Computer programs for the four models are available at the Thermophysical

Properties Division of the National Bureau of Standards in Boulder, Colorado.

19

147
8. REFERENCES

[I] Albright, M. A., Private cormunicati on (1976).

[2] Albright, M. A., A Model for the Precise Calculation of Liquefied Natural

Gas Densities, NGPA Technical Publication TP-3 (1973).

[3] Atomic Weights of the Elements 1973, Pure and Applied Chemistry, Vol 37,

589-603 (1974).

[4] Das, T. R., Eubank, P. T., Advances in Cryogenic Engineering Vol 18,

Plenum Press, New York (1973), p. 208.

[5] Das, T. R., Eubank, P. T., Advances in Cryogenic Engineering Vol 18,

Plenum Press, New York (1973), p. 253.

C63 Das, T. R., Eubank, P. T., Advances in Cryogenic Engineering Vol 18,

Plenum Press, New York (1973), p. 244.

C73 Eckert, C. A., Renon, H. and Prausnitz, J. M., Molecular Thermodynamics of

Simple Liquids, lEC Fund 6_, 58 (1967).

[8] Goodwin, R. D., The Thermophysical Properties of Methane, from 90 to 500 K

at Pressures up to 700 Bars, Nat. Bur. Stand. (U.S.), Tech. Note 653

(1974)

.

C93 Haynes, W. M. , Orthobaric Liquid Densities of Methane + Isobutane and

Methane + Normal Butane Mixtures at Low Temperatures (to be published).

[10] Haynes, W. M., Measurements of Orthobaric Liquid Densities of

Mul ti component Mixtures of LNG Components (N2, CH^, ^2^6' ^3^8' "^^4^10'

nC^H^Q, iCgH^^' '^^5^12^ Between 110 and 130 K, to be published.

[II] Haynes, W. M., Hiza, M. J. and Frederick, N. V., A Magnetic Suspension

Densimeter for Measurements on Fluids of Cryogenic Interest, Review of

Scisntific Instruments 47, No. 10, 1237-50 (Oct 1976).

[12] Haynes, W. M. and Hiza, M. J., Measurements of the Orthobaric Liquid

Densities of Methane, Ethane, Propane, Isobutane, and Normal Butane, J.

Chem. Thermodyn., 9, 179 (1977).

148
[13] Haynes, W. M. , Hiza, M. J. and McCarty, R. D., Densities of LNG for

Custody Transfer, (Proc. Fifth International Conference of LNG,

Dusseldorf, Germany), J. W. White, W. McGrew, S. Farmer, D. Hansen, H.

Jacobshagen, editors; Institute of Gas Technology, Chicago, 111., Vol. 2,

Paper II, Section III (1977).

[14] Hiza, M. J., Haynes, W. M. and Parrish, W. R., Orthobaric Liquid Densities

and Excess Volumes for Binary Mixtures of Low Molecular Weight Alkanes and

Nitrogen Between 105 and 140 K, J. Chem. Thermodyn. 9_, 873 (1977).

[15] Hiza, M. J. and Haynes, W. M., Orthobaric Liquid Densities and Excess

Volumes for Multi component Mixtures of Low Molar Mass Alkanes and Nitrogen

Between 105 and 125 K, J. Chem. Thermodyn. 12, 1 (1980).

[16] International Union of Pure and Applied Chemistry, 1961 Carbon 12 Scale.

[17] Jacobsen, R. T., Stewart, R. B., McCarty, R. D. and Hanley, H. J. M.,

Thermophysical Properties of Nitrogen from the Fusion Line to 3500 R

(1944 K) for Pressures to 150,000 psia (10342 x 10^ N/m^), Nat. Bur.

Stand. (U.S.), Technical Note No. 648 (Dec 1973).

[18] Klosek, J. and McKinley, C, Densities of Liquefied Natural Gas and of Low

Molecular Weight Hydrocarbons, Proc. First Int. Conf. on LNG, IGT, Chicago

(1968).

[19] Kudchadker, A. P., Alani, G. H. and Zwolinski, B. J., The Critical

Constants of Organic Substances, Chem. Rev. 68, 659-735 (1968).

[20] Leach, J. W., Molecular Structure Correction for Applying the Theory of

Corresponding States to Non-Spherical Pure Fluids and Mixtures, Ph.D.

Thesis, Rice University (1967).

[21] Longuet-Higgins , H. C. and Widom, B., A Rigid Sphere Model for Melting of

Argon, Mol. Phys. 8, 549 (1964).

21

149
[22] McCarty, R. D., A Modified Benedict-Webb-Rubin Equation of State for

Methane Using Recent Experimental Data, Cryogenics 14_, No. 5, 276-80 (May

1974).

[23] McCarty, R. D., A Comparison of Mathematical Models for the Prediction of

LNG Densities, Nat. Bur. Stand. (U.S.), Interagency Report NBSIR 77-867

(Oct 1977).

[24] McCarty, R. D., Optimization and Comparisons of Mathematical Models for

the Prediction of Liquefied Natural Gas Densities, J. Chem. Thermodyn. (to

be published, 1980).

[25] Miller, R. C. and Hiza, M. J., Experimental Molar Volumes for Some LNG

Related Saturated Liquid Mixtures, Fluid Phase Equilibria 2^, 49-57

(1978).

[26] Mollerup, J. and Rowlinson, J. S., The Prediction of Densities of

Liquefied Natural Gas and of Lower Molecular Weight Hydrocarbons, Chem.

Engng. Sci. 29 (1973).

[27] Mollerup, J., Correlated and Predicted. Thennodynamic Properties of LNG and

Related Mixtures in the Normal and Critical Region, Adv. Cryog. Eng. 20,

172 (K. D. Timmerhaus, editor. Plenum Press, New York, 1975).

[28] Mollerup, J., Thermodynamic Properties of Natural Gas, Petroleum Gas, and

Related' Mixtures Enthalpy Predictions, Adv. Cryog. Eng. 23, 550 (K. D.

Timmerhaus, editor. Plenum Press, New York, 1978).

[29] Orrit, J. and Olives, J. F., Density of Liquefied Natural Gas and Its

Components, Distributed at the 4th International Conference on Liquefied

Natural Gas, Algeria (1974).

[30] Orrit, J. E. and Laupretre, T. M. , Density of Liquefied Natural Gas

Components, Adv. Cryog. Eng. 23, 573 (K. D. Tinmerhaus, editor. Plenum

Press, New York, 1978)

.

22

150
[31] Pope, G. A., Calculation of Argon, Methane and Ethane Virial Coefficients

at Low Reduced Temperature, Ph.D. Thesis, Rice University (1971).

[32] Renon, H., Eckert, C. A. and Prausnitz, J. M., Molecular Thermodynamics of

Simple Liquids, lEC Fund. 6, 52 (1967).

[33] Rodosevich, J. B. and Miller, R. C, Calculation of LNG Excess Volumes by

a Modified Hard-Sphere Model, Adv. Cryog. Eng. 19, 339 (1974).

[34] Rodosevich, J. B., Measurements and Prediction of Liquefied Natural Gas

Densities, M.Sc. Thesis, Univ. Wyoming (1973).

[35] Rowlinson, J. S. and Watson, I. D., The Prediction of the Thermodynamic

Properties of Fluids and Fluid Mixtures - I. The Principle of

Corresponding States and Its Extensions, Chem. Engng. Sci . 24, 1565

(1969).

[36] Scott, R. L., J. Chem. Phys. 25, 193 (1956).

[37] Sliwinski, P. Z., Die Lorentz-Lorenz Funktion Von Dampff ormigem und

Fluessiyem Athan, Propan and Butan, Phys. Chem. (Frankfurt) 63, 263

(1969).

[38] Yuan, I. C, Correlation and Prediction of Excess Properties for Selected

Binaries in the C^-Cj Hydrocarbon Boiling Range of Conformal Solution

Theory, D.Sc. Dissertation, Washington University (1971).

23

151
Appendix A. Experimental Data

The following is a list of all of the experimental PVTx mixtures data which

were measured during the course of this project. The data are from Miller and

Hiza [25]; Haynes and Hiza [12]; Haynes, et al. [11]; Hiza, et al . [14]; Hiza and

Haynes [15] and Haynes [9,10).

All of the data are for the ortnobaric liquid except for the data of Miller

and Hiza [25] which are for the single phase liquid phase, very close to the

orthobaric conditions. The units of the data are bars, moles per liter and

kelvin. The columns labeled RKM, HS, CELL and CS correspond to percentage

derivations between experimental and predicted densities by the Revised Klosek

and McKinley, hard spnere, cell and extended corresponding states models

respectively. The derivations are always calculated using

experimental-calcul ated densities.

24
MIXTURE NO 1 (Miller and Hiza [25])

0.85147 CH^ + 0.14853

Dr n T1 I'm RKM HS rci 1 l^tLL uo 077 11 n nfl IS 1 9filv 10 .IcOO .13 - .02 n9 oUt .Ulm

MT YTI IDP Wn 9 f U 1 » a NU ^ ^nl za

,

et al . ri4H

U .ooUUD 4^6

/lie .'HO oc 1 097 iUO .uu 9n Kino .14 .05 1 n .lU n7 • U/

1 1 n nn 9n CTnQ ^U .souy .07 - .01 n9 .Ut ni .Ul

.961 24.4612 115.00 20.5309 .05 - .04 -.03 -.03

1.380 24.1402 120.00 20.5309 .06 - .06 -.06 -.05

1 .3JU IOC nn 1^3 .UU 9n csno tU .DoUy .08 - .05 - n/i .U4 - nc .Ud

C. .O^U o-i cnn7 .3UU/ iOU .UU 9n c^no ^u .oouy .12 - .01 nn .UU - n9 .Uc

MT Y Tl IDT nl A 1 UKt DU 0 V H 1 z a

,

et al . ri4l)

U .470^3 Ln^ T U.DUD/D 4^6

OO QC1Q to . yoiy iUO .UU 09 1 CI 9 ^0 .1 DiO .14 .07 .11 n7 . U/ .OUo OO CQ07 iiU .UU 9T 1 CI .09 .03 n"? .Uo n"? .uo 7AQ ^'^ AT?!? lie; 113 .uunn 9"? 1 Rl "? ^0 . 19 10 .11 .03 .Ulm .uon'1

1 ORn i .UoU Co • 1 993 1icu9n nn .uu ^0.1310 9*? 1 c;i 7 .07 - .02 — .UO - n'^ • uo 1.500 22.8933 125.00 23.1513 .08 .00 -.05 .00 2.050 22.6290 130.00 23.1513 .14 .04 -.02 .03

C % 1 C\} tt .OOOi 109 .uu 9"^ CO .1910 IRI'^ .19 .06 .Ul . U9

99 n7fic; .U/ DO 1itudn .uunn 91to .1910 1 m .25 .05 . ndUH

M T V Tt 1 DC Ml A lUKt rtU 4 iniza. et al . [14])

U .0340/ l/n^ + U.D40HO 4^6

ci . iUoo 1 nc nn iuD .uu 9c nof^c _ .04 - .07 - .ut - n7 .u / 7Q7 99 Q777 cc ,ol 1

1

1 1 n nn iiU .UU 9c no^c t9 .uyot) - .Ui n/1 - nc . UD - nc .Uo .580 22.6478 115.00 25.0965 .03 - .02 -.06 -.03

.826 22.4035 120.00 25.0965 .00 - .06 -.11 -.07

1.146 22.1872 125.00 25.0965 .10 .04 -.03 .03 1.550 21.9441 130.00 25.0965 .10 .03 -.04 .02

25

153
MIXTURE NO 5 (Hiza, et al. [14])

0.85796 CH^ + 0.14204 ^3^8

P D T MW RKM HS CELL CS

.517 24.9622 105.00 20.0279 .01 .10 .10 .07 .817 24.6332 110.00 20.0279 - .02 .08 .07 .06

1 1 QQ lie: nn ^n n97Q .UH .Un n"? .UJ n"? 1.785 23.9492 120.00 20.0279 - .07 .00 .00 .00 2.415 23.5942 125.00 20.0279 - .09 - .06 -.04 -.05

3.290 23.2461 130.00 20.0279 - .11 - .06 -.01 -.04

MIXTURE NO 6 (Hiza, et al. [14])

0.74920 CH^ + 0.25080 ^3^8

.478 23 .4767 105.00 23.0790 .01 .02 .03 -.02

.738 23.2064 110.00 23.0790 - .03 .00 -.01 -.03

1 OQQ CC. . 7 JDH 110 •UU cO .U/ 7U " •Ulni .UUnn _- .Utn7 — m\jL 1.582 22.6665 120.00 23.0790 .02 .01 -.01 .01 2.216 22.3818 125.00 23.0790 - .03 - .04 -.04 -.01

3.029 22.1019 130.00 23.0790 .02 - .05 -.02 .01

MIXTURE NO 7 (Hiza, et al. [14])

0.49637 CH^ +. 0.50363 ^3^8

.384 20.4909 105.00 30.1720 - .05 .01 .04 -.03

.591 20.3046 110.00 30.1720 - .08 .01 .01 -.02

C\j .1 iOU 11 D .UU "in OU .

11 77n / ^ u .UD • Ul -— •VjC n? - n? 1.250 19.9311 120.00 30.1720 .00 .01 -.03 .00 1.730 19.7471 125.00 30.1720 .04 .03 -.01 .04 2.320 19.5546 130.00 30.1720 - .01 .01 -.03 .05

MIXTURE NO 8 (Hiza, et al. [14])

0.29538 CH^ + 0.70462 ^3^8

.271 18.5132 105.00 35.8106 - .19 - .08 -.05 -.12

.409 18.3624 110.00 35.8106 - .24 - .09 -.10 -.12

26

154
MIXTURE m 9 (Haynes [10])

0.92788 CH^ + 0.07212 "^4^10

pr n J no 1.820 24.2615 120.00 19.0779 .00 .04 -.04 -.04

2.547 23 .8868 125.00 19.0779 .00 .00 -.07 -.05

3.470 23.5047 130.00 19.0779 - .01 .03 -.09 -.07

4.616 23.1225 135.00 19.0779 .04 .03 -.06 -.05

6.023 22.7284 140.00 19.0779 .15 .04 -.03 -.03

MIXTURE NO 10 (Haynes [10])

0.92780 CH^ + 0.07220

H lU 1.270 24.6285 115.00 19.0813 .01 .08 .01 -.01

1.824 24.2783 120.00 19.0813 .07 .11 .03 .04 2.549 23.8999 125.00 19.0813 .06 .07 -.01 .01

MIXTURE NO 11 (Hiza, et al . [14]) '

0.91674 CH^ + 0.08326 "^4^10

.521 25.1536 105.00 19.5467 .21 .35 .33 .22 .810 24.7960 110.00 19.5467 .12 .26 .21 .15

1 216 24.4512 115.00 19.5467 .14 .24 .18 .15 1.753 24.0889 120.00 19.5467 .10 .19 .11 .11 2.472 23.7370 125.00 19.5467 .15 .18 .10 .13 3.374 23.3789 130.00 19.5467 .17 .19 .12 .16 4.509 23.0110 135.00 19.5467 .21 .19 .15 .18 5.887 22.6391 140.00 19.5467 .35 .20 .22 .23

MIXTURE NO 12 (Haynes [10])

0.77982 CH^ + 0.22018

1.702 21.6066 120.00 25.3805 .12 .14 .11 .06 2.369 21.3549 125.00 25.3805 .11 .08 .02 .04 3.228 21.1020 130.00 25.3805 .09 .00 -.07 .03 4.291 20.8555 135.00 25.3805 .12 .03 -.10 .07 5.576 20.5978 140.00 25.3805 .20 .13 -.15 .07

27

155
MIXTURE NO 13 (Haynes [10])

0.77762 CH^ + 0.22238

P D 1L T MW RKM HS CEL CS 1.179 21.8054 115.00 25.4011 .07 .15 .16 .02

I .oyy CL .3DUJ. lev .uu

ocCD .Hull Ami •U/ .U/ no 2.372 21.3164 125.00 25.4011 .10 .06 .01 .03 3.230 21 .0605 130.00 25.4011 .06 .03 -.10 .00 4.291 20.8011 135.00 25.4011 .02 .14 -.20 -.03

5.579 20.5448 140.00 25.4011 .09 .23 -.26 -.03

MIXTURE NO 14 (Hiza, et al. [14])

u

.

oooco "^4^10

3.183 18.3058 130.00 33.3687 - .32 .11 -.15 -.08

2.342 18.4853 125.00 33.3687 - .17 .00 .00 -.03

2.281 18.4772 125.00 33.3687 - .21 .05 -.04 -.08

1.636 18.6495 120.00 33.3687 - .18 .02 .07 -.06

MIXTURE NO 15 (Haynes [10])

0.92044 CH^ + 0.07956 ^^4^10

1.254 24.3633 115.00 19.3910 - .21 .05 .06 -.01

i .OUD 120.00 19.3910 - .24 - .uo 2.521 23.6403 125.00 19.3910 - .22 .01 .04 -.03

3.434 23.2752 130.00 19.3910 - .21 .03 .08 .00 4.567 22.8920 135.00 19.3910 - .22 .00 .09 -.02

5.950 22.5037 140.00 19.3910 - .14 .02 .14 -.02

MIXTURE NO 16 (Haynes [10])

0.78329 CH^ + 0.21671 i^4"l0

.782 21.9144 110.00 25.1625 - .33 .10 .15 .01 1.164 21.6652 115.00 25.1625 - .33 .05 .07 -.01

1.671 21.4136 120.00 25.1625 - .35 .01 .00 -.02

2.329 21.1668 125.00 25.1625 - .34 .00 -.03 .00 3.170 20.9125 130.00 25.1625 - .35 .04 -.07 .00 4.208 20.6629 135.00 25.1625 - .33 .06 -.06 .04 5.474 20.4082 140.00 25.1625 - .23 .09 -.04 .07

f

28

15b
MIXTURE NO 17 (Hiza, et al

.

[14])

0.48687 CH. + 0.51313 ^^4^10

P D T MW RKM HS CELL CS .629 17 .3575 110.00 37.6362 - .62 .05 .11 .02 .938 17.2076 115.00 37.6362 - .70 .00 -.01 -.02

1.361 17.0639 120.00 37.6362 - .64 - .02 -.08 -.02

1.852 16.9156 125.00 37 .6362 - .69 - .07 -.17 -.04

MIXTURE NO 18 (Hiza, et al. [14])

0.95248 CH^ + 0.04752 ^2 1.380 26.8476 105.00 16.6119 - .05 - .04 -.04 -.02

1.990 26.4052 110.00 16.6119 .06 .00 .00 .04 2.634 25.9374 115.00 16.6119 .20 .01 .02 .06 3.500 25.4522 120.00 16.6119 .26 .03 .07 4.600 24 .9496 125 .00 16 .6119 .54 .04 .07 5.830 24.4210 130.00 16.6119 .69 -5.69 .03 .05 7.300 23.8600 135.00 16.6119 .77 -3.99 -.01 -.03

9.200 23.2809 140.00 16.6119 1.09 -2.69 -.02 -.07

MIXTURE NO 19 (Hiza, et al . [14])

0.69651 CH^ + 0.30349 N2 3.450 26.8735 100.00 19.6759 - .39 .06 .09 .00 4.661 26.3393 105.00 19.6759 - .56 .02 .07 .04 6.181 25.7686 110.00 19.6759 - .92 - .05 -.02 .00 8.010 25.1790 115.00 19.6759 -1.28 .13 -.08 -.03

10.112 24.5737 120.00 19.6759 -1.56 1.88 -.07 -.04

MIXTURE NO 20 (Hiza, et al . [14])

0.50758 CH^ + 0.49242 N2 3.303 27.0801 95.00 21.9375 -2.16 .11 .14 .06 4.468 26.4588 100.00 21.9375 -1.92 - .01 .05 .04 6.383 25 .8106 105.00 21.9375 -1.56 - .15 -.05 -.01

8.461 25.1387 110.00 21.9375 -1.29 - .17 -.09 -.03

10.740 24.4431 115.00 21.9375 1.12 .28 -.04 -.01

13.983 23.7096 120.00 21.9375 -1.79 3.41 .07 -.01

17.529 22.9315 125.00 21 .9375 -1.14 * .26 .00

21 .076 22.1005 130.00 21.9375 - .72 -2.84 .05 .10

* The hard sphere solution for the density of N2 failed.

29

157
0.67287 CgHg + 0.32713

P D 18.6192

18.4648

18.3059

18.1509

MIXTURE NO 21 (Hiza, et al. [14])

^3^8

T MW RKM HS CELL CS 125.00 34.6588 .00 .02 -.02 .00 130.00 34 .6588 .00 .02 .00 .01 135.00 34.6588 .00 .00 .00 .00 140.00 34.6588 .00 .01 .04 .02

0.50105

MIXTURE NO 22 (Hiza, et al. [14])

Hg + 0.49895 C3Hg

18.3618 105.00 37.0689 .02 .04 18.2169 110.00 37 .0689 .01 .00 18.0726 115.00 37 .0689 .00 -.03

17.9282 120.00 37.0689 - .01 -.05

17 .7880 125.00 37 .0689 .00 -.03

17.6412 130.00 37.0689 - .02 -.04

17.4988 135.00 37 .0689 - .01 -.01

17 .3526 140.00 37 .0689 - .03 .00

.02 .00

-.01

-.01

.00

-.02

-.01

-.01

MIXTURE NO 23 (Hiza, et al . [14])

0.67117 C2Hg + 0.32883 nC^H^Q

17.5047 110.00 39.2952 .00 .02 .00 17.3706 115.00 39.2952 .01 .00 .00 17.1031 125.00 39 .2952 .02 .00 .02 16.9626 130.00 , 39.2952 - .01 -.03 -.02

16.8285 135.00 39.2952 .00 -.01 -.01

16.6947 140.00 39.2952 .00 .03 .01

MIXTURE NO 24 (Hiza, et al . [14])

0.65343 C2Hg + 0.34657 nC^H^Q

17.2184 115.00 39.7929 .00 .00 .00 17.0824 120.00 39.7929 - .02 -.03 -.02

MIXTURE NO 25 (Hiza, et al . [14])

0.72436 CgHg + 0.27564 IC^H^q

17.9779 105.00 37 .8030 .01 .01 .01 17.8401 110.00 37.8030 .01 .00 .01 17.4235 125.00 37 .8030 .01 .00 .01 17.2825 130.00 37.8030 .00 .01 .01

30

158
MIXTURE NO 25 (Hiza, et al . [14])

0.68939 CgHg + 0.31061 iC^H^Q

p D T MW RKM HS CELL cs 17.3716 115.00 38.7840 .00 -.03 .00 17.2344 120.00 38.7840 _ .02 -.04 -.02

MIXTURE NO 27 (Hiza, et al

.

[14])

0.94067 C^H^ + 0.05933; N2 3.850 21.4718 105.00 29.9481 .02 -.01 -.03

4.630 21.2912 110.00 29.9481 .03 .02 .03 5.472 21.0845 115.00 29.9481 - .01 -.04 -.04

6.383 20.8998 120.00 29.9481 .72 .01 .00

MIXTURE NO 28 (Hiza, et al

.

[14])

0.60949 C3Hg + 0.39051

14.6487 115.00 49.5749 .01 -.02 -.01

14.5521 120.00 49.5749 .02 .00 .02

MIXTURE NO 29 (Hiza, et al

.

[14])

0.60650 C,Hq + 0.3935:

14.1343 140.00 49.6169 .00 -.03 .00 14.0333 145.00 49.6169 .00 .00 .00 13.9346 150.00 49.6169 .02 .05 .02

MIXTURE NO 30 (Hiza, et al

.

[14])

0.58692 C3Hg + 0.41308

14.6839 110.00 49.8915 .01 .04 .01 14.1748 135.00 49.8915 .03 -.06 -.03

14.0786 140.00 49.8915 .00 -.02 .00

MIXTURE NO 31 (Hiza, et al., [14])

0.49030 C,Ho + 0.50970 K.H,^

14.3080 105.00 51.2468 - .01 -.04 -.01

14.2136 110.00 51.2468 - .01 -.05 .00 14.1219 115.00 51.2468 - .02 -.03 .02 14.0257 120.00 51.2468 .01 -.04 .01 13.9300 125.00 51.2468 .00 -.04 .00 13.8342 130.00 51 .2468 - .01 -.04 -.01

31

159
MIXTURE NO 32 (Hiza, et al. [14])

0.50326 C,Ho + 0.49674 iC.H.n

3 8 4 10

P D T MW RKM HS CELL cs 13.9718 125.00 51.0650 .01 -.03 .01 13.8737 130.00 51.0650 - .01 -.04 -.01

MIXTURE NO 33 (Hiza, et al

.

[14])

0.97986 CjHg + 0.02014 ^2 3.567 16.2131 110.00 43.7733 .01 -.04 -.01

4.712 16.0931 115.00 43.7733 - .02 -.05 -.02

MIXTURE NO 34 (Hiza, et al. [14])

0.96206 CjHg + 0.03794

4.955 16.4638 105.00 43.4870 .02 -.03 .01 6.708 16.3410 110.00 43.4870 .00 -.04 .00

MIXTURE NO 35 (Hiza, et al. r 1 /I T \ L14J}

0.93260 C3Hg + 0.06740

8.795 16.7084 105.00 43.0132 .20 .11 .18 6.313 16.8055 100.00 43.0132 .04 -.07 .01

MIXTURE NO 36 (Hiza, et al. [14])

0.52961 nC^H^Q + 0.47039 iC^H^o

12.6943 125.00 58.1243 .01 -.03 -.01

12.6133 130.00 58.1243 .01 -.01 .01 12.5271 135.00 58.1243 .01 -.04 -.01

12.4447 140.00 58.1243 .00 -.02 .00

MIXTURE NO 37 (Hiza and Haynes [15])

0.80284 CH. + 0.09902 C^Hg + 0.09814 C3H0

.476 24.9975 105.00 20.1852 .02 .04 .06 .02

.747 24.6696 110.00 20.1852 - .03 .00 .00 -.02

1.119 24.3515 115.00 20.1852 .01 .02 .02 .01 1.616 24.0335 120.00 20.1852 .07 .07 .07 .06

32

IbU
MIXTURE NO 38 (Hiza and Haynes [15])

0.34242 CH4 + 0.31372 C^Hg . 0.34386 C3HJi P D T MW RKM HS CELL CS All 20.5342 110.00 30.0903 - .03 .02 .00 -.03

.636 20.3497 115.00 30.0903 - .01 .03 -.02 -.02

.638 20.3567 b W •www/ 115.00 . UJ • nfi UD m * U X .914 20.1786 120.00 30.0903 .12 .10 .04 .06 .914 20.1803 120.00 30.0903 .12 .11 .05 .07 1.280 19.9858 125.00 30.0903 .10 .08 .01 .04

MIXTURE NO 39 (Hiza and Haynes [15])

0.67040 CH^ + 0.16610 0.16350 N2 3.688 25.8587 105.00 20 .3301 .57 .01 .28 .00 4.813 25.4598 110.00 20.3301 .64 - .03 .27 -.01

6.181 25.0466 115.00 20.3301 .79 .02 .25 -.05

7.731 24.6359 120.00 20.3301 .70 .42 .31 -.04

MIXTURE NO 40 (Hiza and Haynes [15])

0.34140 CH. + 0.32060 ^2"6 0 33800 N 7.190 24.9084 105.00 24.5861 2.06 .01 .61 .07 9.494 24.5350 110.00 24.5861 2.55 . - .01 .64 .04 L2.372 24.1433 115.00 24.5861 3.72 .24 .65 -.06

MIXTURE NO 41 (Miller and Hiza [25])

0 .86786 CH. + 0.08181 C-H^ +

3 8 0.05033 iC^H'10 .466 24.9239 100.00 20.4561 - .17 .04 .11 -.03

.769 24.4200 108.00 20.4561 - .17 .01 .05 -.04

1.327 23.9676 115.00 20.4561 - .18 - .01 .02 -.05

MIXTURE NO 42 (Hiza and Haynes [15])

0.79770 CH^ + 0.10280 C3H3 + 0.09950 N^ 2.817 25.3950 105.00 20.1181 .17 .08 .24 -.01

3.620 25.0153 110.00 20.1181 .10 .02 .23 -.05

4.549 24.6465 115.00 20.1181 .08 .08 .33 .01 5.610 24.2667 120.00 20.1181 .02 * .44 .04

* The hard sphere solution for the density of N2 failed.

33

161
MIXTURE NO 43 (Haynes [9])

0.89071 CH, + 0.04998 "^4^10 + 0.05931

p D T HS CELL cs 2.400 25.3450 110.00 18.8562 - .23 .00 .01 .03 3.145 24.9440 115.00 18.8562 - .25 - .04 -.02 .01 4.082 24.5383 120.00 18.8552 - .38 * -.02 .00 5.196 24.3141 125.00 18.8562 - .32 * -.04 -.04

MIXTURE NO 44 (Miller and Hiza [25])

0.85317 CH^ + 0.050// 4^6 * 0.04855 C^Hg + 0.04751 S^O

1.038 24 .6056 110.04 20.1165 .02 .15 .10 .06 1.464 24 .2712 115.00 20.1165 .00 .11 .04 .02

MIXTURE NO 45 (Haynes [9])

0.85133 CH^ + 0.05759 4^6" 0.04808 C,ri„ +

3 8 0.04300

1 .180 24.3243 115.00 20.0092 - .04 .07 .00 -.01

1.700 Zd .yyob 120.00 20.0092 .00 .10 .02 .03 2.374 c6 .bboo 125.00 20.0092 .05 .11 .04 .05 3.232 23 .3108 130.00 20.0092 .04 .11 .05 .05 4.301 CL .9do4 135.00 20.0092 .08 .13 .11 .09

MIXTURE NO 46 (Haynes [9])

0.84556 CH^ + 0.07924 ^2^6 ^ 0.05060 CjHg + 0.02450 nC,H,„

4 10 1.167 24.5569 115.00 19.6051 .00 .04 .00 -.01

1.683 24.2126 120.00 19.6051 .02 .04 -.01 -.01

2.350 23.8698 125.00 19.6051 .08 .07 .03 .03 3.201 23.5204 130.00 19.6051 .12 .11 .09 .07

* MIXTURE NO 47 (Haynes [9])

0.86040 CH^ + 0.04600 '^2"6^ 0.04790 C3Hg + 0.04570 ^^4^10

1.186 24.2654 115.00 19.9552 - .14 .01 .04 -.03

1.710 23.9371 120.00 19.9552 - .10 .04 .07 .01 2.387 23.5860 125.00 19.9552 - .10 .01 .05 -.01

3.248 23.2331 130.00 19.9552 - .13 - .01 .06 -.02

4.320 22.8637 135.00 19.9552 - .18 - .07 .05 -.07

* The hard sphere solution for the density of N2 failed.

34

162
MIXTURE NO 48 (Haynes [9])

0.85378 CH^ + 0.05178 0.04703 CjHj

J + 0.04741

^^4"l0

P D T RKM MS CELL CS 1.190 24.2100 115.00 20.0838 - .14 .01 .04 -.03

1.706 23.8779 120.00 20.0838 - .14 .01 .04 -.02

2.379 23.5324 125.00 20.0838 - .13 - .02 .02 -.04

3.238 23.1834 130.00 20.0838 - .15 - .04 .03 -.05

MIXTURE NO 49 (Miller and Hiza [25])

0.85378 CH^ + 0.05178 0.04703 CjHj

J + 0.04741 ^^4^10

.972 24.5434 110.02 20.0838 - .14 .03 .06 -.01

1.429 24.2154 115.01 20.0838 - .12 .02 .05 -.01

MIXTURE NO 50 (Hiza and Haynes [15])

0.85260 CH^ + 0.04830 0.05070 CjHj

J + 0.04840 N^ 1.581 25.8910 105.00 18.7223 .01 - .04 .03 -.07

2.138 25.5081 110.00 18.7223 - .09 - .07 .03 -.07

2.827 25.1224 115.00 18.7223 - .05 - .06 .05 -.06

3.650 24.7283 120.00 18.7223 - .08 -* .08 -.06

MIXTURE NO 51 (;Haynes [9];) 0.85892 CH^ + 0.11532 ^2^6^ 0.01341 CjHj

J + 0.00705 nC^H^Q + 0.00530 iC^H^g

1.185 25.0957 115.00 18.5565 .05 - .02 -.01 -.03

1.706 24.7131 120.00 18.5565 .03 - .06 -.05 -.07

2.372 24.3294 125.00 18.5565 .02 - .07 -.05 -.09

3.225 23.9490 130.00 18.5565 .05 - .02 .01 -.06

MIXTURE NO 52 (Hiza and Haynes [15])

0.85442 CH^ + 0.05042 C2Hg. 0.04038 CjHj

J + 0.02901 S"l0 ^ 0.02577 iC^H^Q

.515 24.8775 105.00 20.1885 .09 .25 .25 .16 .818 24.5382 110.00 20.1885 .00 .17 .15 .09 1.190 24.2083 115.00 20.1885 .00 .15 .12 .08 1.695 23.8859 120.00 20.1885 .05 .19 .15 .13

MIXTURE NO 53 ( Haynes [9])

0.84558 CH^ + 0.08153 0.04778 CjHjj + 0.01252 nC^H^O + 0.01259 iC^H^Q

1.166 24.5586 115.00 19.5838 .01 .05 .04 .01 1.680 24.2180 120.00 19.5838 .04 .07 .05 .03 2.348 23.8688 125.00 19.5838 .08 .08 .08 .04 3.188 23.5154 130.00 19.5838 .10 .10 .12 .07

* The hard sphere solution for the density of N2 failed.

35

163
MIXTURE NO 54 (Haynes [9])

0.81249 CH^ + 0.08484 0.04931 C3Hg + 0.02708 nC^H^Q + 0.02628 N2

pr nu T1 nw KNn no rn 1 LtLL 2.214 24.4562 115.00 20.0706 .01 .01 .03 -.02

2.874 24.1119 120.00 20.0706 .00 * .04 -.01

3.768 23.7507 125.00 20.0706 .02 * .02 -.05

4.793 23.3954 130.00 20.0706 - .02 -3.67 .07 -.03

MIXTURE NO 55 (Hiza and Haynes [15])

0.79090 CH. + 0.05600 CpHg + 0.05000 CjHg + 0.04770 nC^H^Q + 0.05540 N2 1.933 24.8080 105.00 20.9017 .04 .10 .12 .05 24.4664 110.00 9n om 7 fi9 n9

MIXTURE NO 56 (Miller and Hiza [25])

0.79054 CH4 + 0.05597 ^2^6 " 0.04996 C3Hg + 0.04762 nC^H^Q + 0.05591 Ng

2.158 24.8354 105.03 20.9029 .15 .21 .23 .15

MIXTURE NO 57 (Haynes [9])

CH4 + 0.04542 0.05050 CjHg + 0.04667 iC^H^Q + 3.005 24.1487 115.00 20.6355 .02 -.02 -.03 -.04

3.863 23.8075 120.00 20.6355 - .14 * .00 -.03

4.874 23.4518 125.00 20.6355 - .12 * .01 -.05

6.125 23.0893 130.00 20.6355 - .18 -5.06 .05 -.08

MIXTURE NO 58 (Hiza and Haynes [15])

U .OUDUU CH^ + 0.04680 0.04820 C3Hg + 0.05000 iC^H^q + nu.utyuu nAQnn n 2.039 24.7803 105.00 20.7423 - .03 .08 .03 .01 2.550 24.4327 110.00 20.7423 - .09 - .02 -.06 -.06

3.135 24.1039 115.00 20.7423 .06 .00 -.02 -.02

3.790 23.7707 120.00 20.7423 - .10 * -.01 -.02

MIXTURE NO 59 (Miller and Hiza [25])

0.80545 CH4 + 0.04671 C2Hg. 0.04817 C3Hg + 0.04998 iC^H^Q + 0.04969 N2 2.273 24.7831 105.06 20.7476 .00 .10 .05 .03

* The hard sphere solution for the density of N2 failed.

r

36

164
MIXTURE NO 60 (Haynes [9])

0.90613 CH^ + 0.06026 0.02154 C3Hg + 0.00306 0.00300 iC^H^g

+ 0.00601 ^2

P D T MW RKM HS CELL CS 1.478 25.3834 115.00 17.8195 .03 - .01 .01 -.01

2.043 24.9894 120.00 17.8195 .08 .00 .05 .02 2.785 24.5702 125.00 17.8195 .04 .00 .03 -.02

3.722 24.1578 130.00 17.8195 .06 -1.08 .11 .02

MIXTURE NO 61 (Haynes [9])

0.88225 CH^ + 0.07259 0.02561 C3Hg + 0.00492 S^IO ^ 0.00490 iC^H^Q

+ 0.00973 N2 1.639 25.2023 115.00 18.3094 .09 .04 .07 .04 2.247 24.8047 120.00 18.3094 .06 .00 .04 .00 3.022 24.4022 125.00 18.3094 .04 .00 .03 .03

MIXTURE NO 62 (Haynes [9])

0.85934 CH^ + 0.08477 C^Hg. 0.02980 CjHg + 0.00707 nC^H^O + 0.00519 iC^H^Q

+ 0.01383 ^2 1.812 25 .0384 115.00 18.7496 .12 .06 .09 .06 2.441 24.6661 120.00 18.7496 .12 * .11 .07 3.223 24.2880 125.00 18.7496 .19 .00 .16 .09 4.222 23.8981 130.00 18.7496 .17 -2.12 .20 .10

MIXTURE NO 63 (Hiza and Haynes [15])

0.81300 CH^ + 0.04750 C^Hg. 0.04870 CjHj

J + 0.02420 0.02410 iC^H^g

+ 0.04250 ^2 1.834 24.8496 105.00 20.6168 .03 .12 .10 .05 2.807 24.5159 110.00 20.6168 .02 .08 .06 .03 3.384 24.1783 115.00 20.6168 .09 .06 .05 .03 6.039 23.8577 120.00 20.6168 .04 * .11 .08

MIXTURE NO 64 (Haynes [9])

0.85341 CH4 + 0.07898 C^Hg. 0.04729 CjHg + 0.00992 0.00854 iC^H^Q

+ 0.00089 "^5^12 + 0.00097 iC 5^12

.787 25.0063 110.00 19.3588 .01 .02 .02 -.01

1.172 24.6566 115.00 19.3588 .00 - .01 -.01 -.03

1.686 24.3079 120.00 19.3588 .02 .00 .00 -.02

2.351 23.9525 125.00 19.3588 .05 .01 .03 -.01

3.210 23.5883 130.00 19.3588 .06 .02 .06 .00

* The hard sphere solution for the density of N2 failed.

37

165
MIXTURE NO 65 (Haynes [9])

0.75442 CH^ + 0.15401 ^2^6 * 0.06950 CjHg + 0.01057 "10 0.00978 iC. + 0.00083 + 0.00089 iC,5"

P D T MW RKM HS CELL cs

.723 24.2529 110.00 21 .1060 - .02 .00 -.01 .04 1.081 23.9519 115.00 21.1060 .04 .04 .03 .01 1.549 23.6535 120.00 21.1060 .07 .04 .02 .01 2.153 23.3351 125.00 21.1060 .04 .01 .00 .01

MIXTURE NO 66 (Haynes [9])

0.75713 CH^ + 0.13585 0.06742 CgHg + 0.01326 "10 0.01336 iC, + 0.00859 \L + 0.00216 nCcH,„ + 0.00223 iCg ^12 1.155 24.1809 110.00 21.3094 .09 .07 .08 .04

1 .byb li .o/ol 115.00 21.3094 .12 .05 .05 .02 2.155 23.5709 120.00 21 .3094 .12 .00 .08 .05 2.873 23.2644 125.00 21.3094 .15 .00 .12 .08 3.744 22.9514 130.00 21.3094 .22 -1.42 .16 .11

MIXTURE NO 67 (Haynes [9])

0.74275 CH^ + 0.16505 C^Hg-H 0.06547 CjHg + 0.00893 nC^ "10 ^ 0.00843 iC^ ^10

+ 0.00801 + 0.,00067 nC5H^2 + 0.00069 iC^ "12

1 .158 24.3141 110.00 21.0976 .02 - .02 .00 .05 1.584 24.0160 115.00 21.0976 .09 .01 .02 .01 2.093 23.6937 120.00 21.0976 .03 .00 -.03 .06 2.853 23.3804 125.00 21.0976 .05 .00 -.01 .05

MIXTURE NO 68 (Haynes [9])

0.90068 CH^ + 0.06537 C,Hg + 0.02200 C3H0 + 0.00284 nC^ "10 0.00291 iC.

+ 0.00599 Np + 0.00011 nCcH,^ + 0.00010 H.

«

12 1.456 25.3600 115.00 17.9026 .05 .01 .03 .01 2.024 24.9656 120.00 17.9026 .08 .00 .06 .03 2.762 24.5450 125.00 17.9026 .03 .00 .03 .03 3.698 24.1289 130.00 17.9026 .02 -1.13 .07 .02

38

166
Appendix B. Computer Program and Equation Parameters for the

Extended Corresponding States Model

The program listings in Appendix F include the extended corresponding states

method described in section 2. To use the program in its present form one must

make one of the two possible calls to SUBROUTINE PDMIX{P ,D ,T.X) (lines LNG 1

through LNG 9). The two possible calls are:

CALL PDMIX(P,D,T,X)

or

CALL PMIX(P,D,T,X)

When the call to PDMIX(P,D,T,X) is made the input variables are: P (pressure in

bars); T (temperature in kelvin); and X which is a matrix of the mole fraction of

the components of the mixture in the following order:

x(i) = mole fraction of methane

X(2) = mole fraction of ethane

X(3) = mole fraction of propane

X(4) = mole fraction of normal butane

X(5) = mole fraction of isobutane

X(6) = mole fraction of nitrogen

X(7) = mole fraction of normal pentane

X(8) = mole fraction of isopentane

No other components are allowed and if one or more of the above components are

absent, a zero should be inserted in the appropriate matrix element. The program

then calculates a density and it is returned in the argument list as D (density

in moles/liter)

,

When a call to PMIX(P,D,T,X) is made all of the above are the same except

that the roles of P and D are interchanged, i.e., D is an input variable and P is

calculated by the program.

39

167
The range of the program is 90 to 150 kelvin for the saturated liquid phase

of any of the pure components of CH^ , ^2^^, ^3^8' "^4^10' ^^4^10' ^^2' "^4^12

or 'C^H^2 "lixture of those fluids. The program will extrapolate to

higher pressures (higher than saturation pressure) but the user is reminded that

such a calculation is an extrapolation and should be used with caution.

Other subprograms required:

SUBROUTINE MIX DATA, line LNG 95, Appendix F

SUBROUTINE DATA CH4, line LNG 162, Appendix F

FUNCTION FINDM, line LNG 217, Appendix F

FUNCTION SATL, line LNG 236, Appendix F

SUBROUTINE PROPS, line LNG 248, Appendix F

The equation of state from which Zq and Gq by eqs (1) and (2) may be

derived is:

40
P = pRT + p^(N,T + N„T^/^ + + N./T + N./T^) r "2' "3 "4'

+ p^in^l + + Ng/T + Ng/T^)

+ p^(N^qT + N^^ + N^2/'^) + P^(N^3)

+ P^lN^^/T + N^g/T^) + p^N^g/T)

+ P^i^^j/T + N^g/T^) + p^(N^g/T2)

+ P^(N2o/T^ + N2t/T^) exp(- yp^) (B-1 )

+ p^(N22/T^ + N23/T^) exp(- w^)

+ P^(N24/T^ + "25/"^^) ^""P^" '^P^^

+ P^N2g/T^ + N27/t'^) exp(- yp^)

+ P^^N28/T^ + N2g/T^) exp(- yp^)

+ p^^(N3q/T^ + N3^/T^ + ^32/1^ exp(- yp^)

The computer subroutine PROPS(PP,DD,TT) , lines "LNG 248 through LNG 371 in

Appendix F" are the FORTRAN statements for eq (B-1) and the derivative

(3P/3p)tThe parameters for eq (B-1) as applied to methane are given in table 2 and

in the FORTRAN SUBROUTINE DATA CH4, lines LNG 162 through LNG 216, Appendix F.

The parameters for eqs (5) and (6), (9) and (10) are given in table 3 and 4

and in the FORTRAN SUBROUTINE MIX DATA (IBASE), lines LNG 95 through LNG 161,

Appendix F

.

41

169
Table 2. Methane Coefficients Ni for Eq B-1

R

'10

•ll

'12

'13

0.08205616

0.0096

= -0.187027997685 x

10"

0.103387108009 x 10

N3 = -0.155387625619 x 10'

= 0.772311478564 x

10"^

N5 = -0.377103300895 x lo'

Ng = 0.846818843475 x

10"

= -0.496415884529

Ng = 0.869909352414 x 10^

N„ = -0.322821592493 x lo'

•0.395843026318 x 10

-4

0.266772318035 x 10

-1

•0.304010057839 x 10

0,191584507536 x

10"

= -0.195587933458 x 10

-3

'15 0.607479967879 x 10

'16

'17

'18

'19

'20

'21

'22

'23

74

•25

'26

'27

'28

'29

'30

'31

'32

-0.529609525984 x

10"

0.152264286004 x 10

-0.109952182842 x 10

0.191395549929 x 10

0.386470003746 x lo'

-0.157930582612 x lO'

0.195270144401 x lO'

0.165996081629 x lO'

0.603051146711

0.376485162808 x lO'

0.125593680622 x 10

-0.343570032513 x lO'

-0.540945094139 x

10'

0.185622284663 x 10

0.770786979245 x 10

-0.286868318650 x 10

0.372376961647 x 10

-4

-1

-3

-2

-2

-8

-5

-4

42

170
Table 3. Coefficients for Eqs 5 and 6.

"1 "4 = -4.1419Z

"2 = 0.919454 "5 = 0.444850

"3 = -4.01525

"6 = 0.356808 "8 = 0.893323

"7 = 1.02619

*

"9 = 0.761533

w (bar) (K)

3

(cm /mol) M No.

0.0109 45.956967 190.555 98.522 16.04303 1

0.110427 48.60314 305.5 146.2 30.07012 2

0.154837 42.445123 370. 200. 44.09721 3

0.176372 38.295398 425. 251.62 58.1243 4

0.150115 36.88998 408.1 263. 58.1243 5

0.0291791 33.542557 126.2 89 .827 28.0134 6

0.234320 33.812152 469.6 304. 72.15139 7

0.288886 31.988302 460.39 306. 72.15139 8

d

*Note: The large number of significant figures given for critical pressure

is necessary to reproduce the Z (Z = P V /RT ) in the least

squares fit of the data. P^'s have been converted to bar from

atmospheres.

43

171
• ot—

1

•oc

*o

orttoO"

UJ

i- • o ro

Hto

4—' c

+-> o c

14- M- c

(U0 > c0 0)

4-> to

(D tft J- S- O) (U

+-> c E

>>c 3

CO IZ

00

CM

«£) 0000000 CM t-H 00

CM 00 CTi cr> 0 0 0000 CD

r-. • • • • • • 0 0 0 0 0 0

0 CM (Ti 0 0 r«- <£> in CO CM 0 0 1—

1

0 CM 0 00 «a- CM CM CO 0 <—

1

V£) cr^ CO n cr> CM 1

I-H CM «3- 0 in CO ( ^ in 0 0 0 CM

• cr.

• • 00

• CO• 0 0 0 0 0

00 cn ^ <x> O CO

in CM o o

o ^ <a- .-H 1^

1—I a-\ o cn

ro CM

«3- ^

CM .-I 000r-l O

CM CTi CM CTi

CTi in

.-H O O O

ino >-<

00 cy> cr>

• 0 • • • • • 0 0 0 0 0 0 I—

1

00

CO (Ti 1—

1

00 CT^ 10 CO 0 CO cri ro

cn CTi 0 CTi

• • • • 0 0 .—1 0

0 ro

ro CM

I-H CM in ro CM in

00 100

CT>

• • • 0 0 0

00o

CO

00

cr> en

cr.

CM

CM ro ^ in 10 CO •I- .-H CM m 10 00

44

172
Table 5. Values for Checking Calculations Using Corresponding

States Equations.*

0.6975 CH^ + 0.156 C^Hg + 0.092 C3Hg + 0.029 nC^H^^ + 0.014 iC^H^Q

+ 0.0115

Temperature Density in moles/liter Pressure, bar

K

95 24.333 1 100 24.067 1.1

105 23.796 1.2

Included for check purposes only; these values are calculated from the

corresponding states model and are not experimental data.

45

173
Appendix C. Computer Program and Equation Parameters for the

Hard Sphere Model

The program listing that follows is for the hard sphere model described in

section 3. To use the program in its present form one must make the following

reference to the computer program:

DEN = RODEN(P,T,X)

where DEN is density in moles/liter, P is pressure in bars, T is temperature in

kelvin, and X is a matrix of the mole fractions of the components of the mixture

in the following order:

X(l) = mole fracti on of methane

X(2) = mole fraction of ethane

X(3) = mole fraction of propane

X(4) = mole fraction of normal butane

X(5) = mole fraction of isobutane

X(6) = mole fraction of nitrogen

X(7) = mole fraction of normal pentane

X(8) = mole fraction of isopentane

Note: the inclusion of the pentanes is due to Rodosevich and Miller [33] and no

optimization of parameters has been included in this work for mixtures with

pentane as a component.

The range of the program is 90 to 150 kelvin for the saturated liquid phase

of mixtures of CH^, C2Hg, C2Hg, nC^H^Q, iC^H^q, N2, nC5Hj^2 ^^5^12*

program will calculate densities of any of the pure components but they will be

from a different model (i.e., some from an equation of state (CH^ and N2) and

some from the equations for saturated liquid densities. Therefore in its present

form, extrapolation to higher pressures is possible but the reliability of the

results is questionable.

46

174
other subprograms required:

SUBROUTINE FM, lines LNG 435 to 529, Appendix F

SUBROUTINE ZERO, lines LNG 617 to 623, Appendix F

FUNCTION FIND VI. lines LNG 530 to 548, Appendix F

FUNCTION FIND Gl, lines LNG 549 to 560, Appendix F

FUNCTION EXCESS, lines LNG 561 to 616, Appendix F

FUNCTION FIND M, lines LNG 217 to 234, Appendix F

SUBROUTINE PROPS, lines LNG 248 to 371, Appendix F

SUBROUTINE DATA CH4, lines LNG 162 to 216, Appendix F

SUBROUTINE DATA N2, lines LNG 372 to 415, Appendix F

The parameters for eq (11), section 3 are given in table 6, and in lines

LNG 462 through LNG 466 in Appendix F.

The binary interaction parameters j.. and k.. in eqs (15) and (16),

section 3 are given in table 7 and in lines LNG 447 through LNG 461 in

Appendix F

.

47

175
Table 6» Coefficients for Eq 11.

Fluid S^. Fluid

No.

CH^ 2.755x10^ 3.676x10"^ 1.00 1

CgHg 7.773 X 10^ 4.158 x 10"^ 1.50 2

C3Hg 14.165x10^ 4.644x10"^ 1.67 3

nC^H^Q 22.733 x 10^ 5.051 x 10"^ 1.83 4

iC^H^Q 21.279 X 10^ 5.056 x 10"^ 1.79 5

Ng' 1.718 X 10^ 3.546 x 10"^ 1.03 6

nC5H^2 30.550 x 10^ 5.389 x 10"^ 1.91 7

iCgH^^ 42.946x10^ 5.706x10"^ 2.11 8

b. = (2)(3.14159)(6.025 x 10^"^)S^/3

48

17b
o

o

r-1 CVJ

I I o o

00o

—1 O CTi ID O <->

«3- rCM

O —I O O

—1 LO .—

1

o o o

Lo 1^ CO LO I—

49

177
Appendix D. Computer Program and Parameters for the Revised

Klosek and McKinley Model

The program listing and tables that follow are for the Revised Klosek and

McKinley model described in section 4. The method may be used in two ways.

First using the equation:

The V^, and k2 may be obtained from tables 8, 9 and 10 and the volume of

the mixture calculated. For example given the mixture of 0.8130 CH^ + 0.0475

C2Hg + 0.0487 C3Hg + 0.0242 nC^H^Q + 0.0241 iC^H^q + 0.0425 N2 and a temperature

of 105 kelvin.

The X.V. and X.W. are obtained from table 8.

l^X.V. = (.8130)(. 037113) + ( .0475) ( .047267) + (.0487)( .061766)

+ (.0242) (.076100) + ( .0241 )( .077538) + ( .0425) ( .042565)

= 0.0409453

Then the E X^W. = ( .8130) (16.04303) + ( .0475) (30.07012) + (.0487) (44. 09721)

+ (.0242) (58 .1243) + ( .0241) (58.1243) + (.0425)(28.0143)

= 20.6168 the molecular weight of the mixture

from table 9 k^ = .697 x 10"^

from table 10 k2 = .849 x

10""^

plugging all this into eq (D-1) gives

\u - -O^^BS

l/^mix - "mix - 24.842 moles/liter

This compares to the experimental value of 24.850 (Appendix A, mixture No. 63) to

within 0.03%.

50

178
The same result may be obtained by using the computer program in the

following way:

D = FMKM(T,X)

where T is temperature in kelvin and X is a matrix of mole fractions of the

components in the following order:

X(l) = mole fraction of methane

X(2) = mole fraction of ethane

X(3) = mole fraction of propane

X{4) = mole fraction of normal butane

X(5) = mole fraction of isobutane

X(6) = mole fraction of nitrogen

X(7) = mole fraction of normal pentane

X(8) = mol e fraction of isopentane

for the example: T = 105., X(l) = .8130, X(2) = .

X(4) = .0242, X(5) = .0241 and X(6) = .0425.

Other subprograms required:

FUNCTION VIDEAL, lines LNG 709 through 732, Appendix F

FUNCTION SAT, lines LNG 733 through 762, Appendix F

51

179
ro vo CO «—1

CO CO 00 CO VO evj »—( o O «—1

ro LO 00

«3- in «£> CJ^ o .—1

ro cr> I—1

ro ID cr> .-H ro ID a>

1—1

C\J oo

O ro Lf> r«. 0^ ro IT) 00

O CVJ vo cr> t—i ro ID ro

zn CT> 0^

O o o o O 1—1

«—1

«—1

CSJ CM CM ro ro ro .—1

00 00 CO 00 <T> cri cr> cri a> <Ti cyi LT) 1— O O o o o o o o O o O o o o o O o o o O O t—I

esj

ro cr> VO ro o 00 vo ro CM CM CM ro •sS- lO

O ro <—1

vo

00 o o »—t

CM ro «:J- LO VO 00 cr> o CM ro «d- VO 1^ cy>

.—1 ro ir> o CVJ ^ VO 00

O CM vo 00 o ro in CT> 1—1

ro ro

CTi cn Ch o o o o o .—1 t—1

f-H 1—1

I—1

CM CM CM CM CM ro ro 1—(

to 00 CO 00 00 CTi cri cr> cr> (T> CTi 0^ cr> Ol to oc o O o o o o o o o O o O o O o O o o O o o «—1

CMI

o

Q.

ro I—1 o •3- r» vo 00 CM ro t-H CM LO ir> IT) ro

00 LO LO 00 CM CM

O vo ro ro o 00 1

CTi 00 VO vo

LO o vo CM 00 LO ro r—( O

O CM vo 1—1

CO vo o LO CM ro

CM f-. 00 00 CTi cr> o CM CO ro LO vo o CM vo CTi •—1 z ro ro ro ro ro ^ lO LO LO LO lO VO VO o o o O o o o o o o o o o o o o o o o o o o

00CM

O. vo cn ro ro o r>x vo LO vo 00 o «3- CTi LO CM «—1 o >—t o 00 vo lO lO ^ ro ro ro ro ro LO VO 00

O) 1—1 o CM vo CO o CM vo 00 o CM <^ vo CO

O CM •ej- VO 00

O ro JZ zz vo vo vo vo vo r~. r~. 00 00 00 00 CO crv o^ <T> CTi cy> O 4-> r-x r~ r>x r~. 1

00 CM o o o O o O o o O O o o o o o o o O o O O t-H **- o

-a

=3cr

•a

0)

i-E

O)

O)

00

ro o o ..-

00

0) Si

vo

00LO C7> 2

CO .—1 LO a^ LO •—1 00 vo LO

O 0^ «—1

00 vo o o cn LO ro »—t o 00 vo LO LO CO ro ro CM CM CM

CO o CM vo 00 o 1—1

ro lO r-. cr> 1—1

ro lO CT> 1-H ro lO ro ^ LO LO LO LO LO vo vo vo vo vo vo r«» r>«. 00 00 00

r» p^ r>- r-- 1^ 1

CMU o O O o O o o o O o O o O O o o o o o o o .—1 c

00

bO

CM r~-. i—t LO LO ro CM CM VO cr> cr> vo <r ro ro (/) VO ro o f

LO CM o LO ro 1 LO ro CM

O 00 vo f—1

0)

"S- vo CO cr> 1 ro lO vo CO

O CM ro LO 1

f—1

ro ^ vo 00 o CM 3 o o o o «—1

I—1

I—1

CM CM CM CM CM CM ro ro ro ro ro «d- P^

vo vo vo vo VO vo vo vo vo VO VO vo VO VO vo vo vo vo vo vo vo <T> o o o o o o o o o O O o O O o o O O o O O O >CO

1—1 lO o ro ro LO 00 CM 00 LO vo 1—1 a> 1—1 4-> CO ro CTi o vo CM 00 1—1 1

1—1 00 LO CM o CO LO ro CM CM

m o CM ro LO r«-. 00 o 1—(

ro lO vo 00 o 1—1 ro LO r». CO o CM "3- 1—( c

vo vo vo VO vo vo 1

fx fx CO 00 00 00 CO CO CTl <y> o *o ^ "3- «d- "d- o o o O o o O o o o o O o o o o O o o O o Ooro

CTl 1—1 00 LO CM T—<lO lO CM VO vo

O lO o LO S- vo cr CM lO CTl ro 00 ro CTl VO ro 1-H o o 1-H CM LO CO ro o

vo CO o CM lO cri CM CTl CM LO 00 1-H ^ O ro vo o LO LO lO vo vo VO vo vo p~- f-«. r>^ CO 00 00 cn CTl o o o ro

ro ro ro ro ro ro ro ro ro ro ro ro ro CO CO ro ro ro «d- «d- «3- vo O o o o o o o O O o o o o o o o O O o o o O

VO

c=o

1-H 4J

tU

la 0)

V)

Z3 x: o C7> <u

OJ 0) o CM vo CO o CM ^ vo 00 o CM vo 00 o CM OS- vo 00 o CTl CT^ o o o o o 1-H 1-H 1-H 1—1 i-H CM CM CM CM CM ro "o

1— E

52

180
If) CSJ ir»if>oomoooocM <TtoocT>CM^ootOLr>cr>i

CM

CM

cn CM

IT)

«3- oo

ID O CM

LO

CSJ CSJ CM CO

CM

•a- CM CM>— iDOomtnooooLO i<—i«3-Lr)COLn'3-voo r^cocTiOi—iCM^ixJoo-—

I

«3- CM

00 O .-H CO

CO

CM o

CO

CM

CO

CTi

CM CM CO CO

CO

CSJ o o o LO ^ CO

v£) r- 00 oCM

LO

lOo

oCO oCO

CO

in LO o CM ,—I lO LO r«> CT> CO

CM

CO COo CO

LO COo00 o

t-H ,-( CM C^J CO CO

COo

CMCM m LO IT) o CT> VO lOLO LO r>. 00 oLO LOo

LOoCM

LO lf> O 00 ir> o CO ir> 00

COo

CM

CM

CTi LO CO

00

LOo

CMCM

LO

00 COCM

CO

LO o

t-H r-l 1-1 CM CM CO

o

*-> u

ID

CM

LO ocn

LO

LO lOCO

LO r-.

LOCO

00

LO LO O LO 1—

I

O CM oI

CO ocn

IT)

CM

o

u

CM oLO cn o

CMCM

CO

LO 00 CM oo

CM CO

co

*-> u

0)

ii-o-

(_?

(U

oCM

cn

00

o o m LO CO O CO ^ LO LO LO

LO O O LO o o CM .—I CM CO O CT>

r-^ 00 C3~l O CM CO

O O LO lO ^ 00 CM 1^ CO CO ^ ^

LOCO

LO LO

LO LO O O (Ti cn CM LO LO cn o

oCM

CM o o LO O CM CO o

CO

LO

CO o ooLO O O LO

cn o CM

LO 00

o

o

<u

i~o

o

oCM

cn

00

oin <n

LO LO

LO CM

<n

CO co

LO

CO CO cn

LO r>«. LO

cn

CO

CM

00

«3-

oCM oLO

i-< CM CM

CM

cn o

CMCM

00

CM

«5r CO

CM <n

LO

CM <—

•

p«« cn vo

CM

o

.—I CM omOLOOOOioLOO CMCoir)LococMir><nr}-o ^.-(.-H,—ir-tCMCMCMCO«:J- CM vo

CsJ CM

CO LO LO O

cn CO

LO

LOOOOOOOr-t,— LO r>« r>. 00 cn co

I

oooooooooLO r»» LO ^ IT) o o o CMo

CM CO CO LO

CO «3- LO r>» o o o o

O If) cn o LO O LO O If) O LO O t—I 1—I CM CM CO CO O If) cn cn o LO O If) O t—t r—l O lO O LO CM CJ CO CO

53

181
Appendix E. Computer Program for the Cell Model

The program listings for the cell model start at line LNG 763 and continue

on to the end of Appendix F. As is mentioned in section 4, no details of the

model are given here only the program listing. To use the program in Its present

form one must make the following reference to the computer program:

CALL ECKNON(P,D,T,X)

where P is input pressure in bars, D is the output density in moles/liter, T is

the input temperature in kelvins and X is a matrix of the mole fractions of the

components of the mixture in the following order:

x(i) = mole fraction of methane

X(2) = mole fraction of ethane

X(3) = mole fraction of propane

X(4) = mole fraction of normal butane

X{5) = mole fraction of isobutane

X(6) = mol e fraction of nitrogen

X(7) = mole fraction of normal pentane

X(8) = mole fraction of isopentane

54

182
Appendix F. Computer Programs

Listing of computer programs for all four models. See the sections on the

individual models for a list of subprograms needed for each model.

The programs are written in FORTRAN IV and are operational on a CDC 6600

computer.

55

183
SUBROUTINE PDMIX(P,D,T,X) LNG 1 C FOR A CALL TO PDMIX, P,T AND X ARE INPUT. P IS IN BAR.T IS IN LNG 2

C KELVIN AND D IS OUTPUT IN THE UNITS OF f^LES/LITER LNG 3

C FOR A CALL TO P MIX, D,T AND X ARE INPUT AND P IS OUTPUT, THE LNG 4

C UNITS ARE THE SAME LNG 5 C THE X MATRIX MUST CONTAIN THE MOL FRACTION OF THE ALLOWABLE FLUIDS LNG 6

C IN THE FOLLOWING ORDER, 1=C1,2=C2,3=C3,4=NC4,5=IC4,6=N2,7=NC5,8=IC5 LNG 7 C PLACE A ZERO IN THE ELEMENTS OF X WHERE THAT PARTICULAR GAS IS NOT LNG 8

C PRESENT LNG 9 DIMENSION ZATA(10,10),ATA(10,10),TC(10),VC(10),ZC(10),AC(10),W(10) LNG 10 1,PC(10) ,CF(9) LNG 11 DIMENSION THETA(10,10),TH(10,10),PHI(10,10),PH(10,10),F(10,10),FH LNG 12 1(10,10),H(10,10),HH(10,10),VR(10,10),TR{10,10),X(10) LNG 13 CONWON/DATA M/ZATA,ATA,TC,VC,W,TC0,VC0,AC0,ZC0,RR,R,0MEG0,AC,ZC,N LNG 14 1,PC,CF LNG 15 DATA(IE=0) LNG 16 D=0.0 $ PI=P/1. 01325 LNG 17 GO TO 4 LNG 18 ENTRY PMIX LNG 19 P=0.0 LNG 20

IP=1 LNG 21 GO TO 5 LNG 22

4 IP=0 LNG 23

5 CONTINUE LNG 24 IF{IE.GT.0)G0 TO 6 LNG 25 IE=1 LNG 26 CALL DATA CH4 LNG 27 IBASE=L=1 LNG 28 CALL MIX DATA(IBASE) LNG 29

6 CONTINUE LNG 30

DO 1 1=1, N LNG 31 F(I,I)=H(I,I)=1. LNG 32 THETA(I,I)=1. LNG 33

1 PHI{I,I)=1. LNG 34

DO 30 0=1,30 ^ LNG 35 HX=FXHX=0.0 LNG 36 DO 10 1=1, N LNG 37 FH{I,I)=F(I,I) LNG 38 HH{I,I)=H(I.I) LNG 39 IF(X(I).LT..0001)G0 T 0 10 LNG 40 F(I,I)=(TC(I)/TC(L))*THETA(I,I) LNG 41 H{I,n={VC{I)/VC(L))*PHI(I,I) LNG 42

10 CONTINUE LNG 43

DO 11 IA=1,N LNG 44 DO 11 IB=1,N LNG 45 IF(X(IA).LT..0001)G0 TO 11 LNG 46 IF(X(IB).LT..0001)GO TO 11 LNG 47 FAB=ZATA(IA,IB)*(F(IA,IA)* F(IB,IB))**.5 LNG 48

HAB=ATA(IA,IB)*( .5*H( I A, I A)**( 1 ./3 . )+.5*H( IB, IB)**( 1 ./3. ) )**3 LNG 49 HX=HX+X(IA)*X(IB)*HAB LNG 50 FXHX=FXHX+X(IA)*X( IB)*HAB*FAB LNG 51

11 CONTINUE LNG 52 FX=FXHX/HX LNG 53 PR0=PI*HX/FX LNG 54 TR0=T/FX LNG 55 DEN=D*HX LNG 56 IFdP.EQ.DGO TO 8 LNG 57 DD=SATL(TR0)*1000.+1. LNG 58

9 DEN=FIND M{PR0 ,TR0 ,DD) LNG 59 8 CONTINUE LNG 60 IF(DEN.LE.O.O) GO TO 33 LNG 61 VR0=1000./DEN LNG 62 DO 12 1=1, N LNG 63 56

184
IF(X( I) .LT. .OOODGO TO 12 LNG 64

VR( I , I )=VRO*PHI ( I

, I )/VCO LNG 65

TR( I ,1 )=TRO*THETA( I ,1 )/TCO LNG 66 IF{VR( I ,1 ) .GT.2. )VR( I,I)=2. LNG 67 IF(VR(I,I).LT..5)yR(I,I)=.5 LNG 68

TH( I,I)=THETA(I,I) LNG 69

THETA( I I) = l . + { AC( I)-OMEGO)*{CF( l)-CF(21*ALnR(TR( I T U+f rF(lUrF(4 70 1)/TR(I I))*{VR(I I)-CF(5))) 1 Kir; 711 X PH(I I)=PHI(I I) 72 PHKI I ) = ( 1 , + ( AC( I)-OMEGO)*( CF(6)*( VR( I n-rF(7U-rF(81*(VR( T \\ 1 NR 73 1-CF(9) )*ALOG( TR(I I) ) ) )*ZCO/ZC(

I

\ 1L. wr; liU 74

12 CONTINUE LNG 75 DO\J\J 13 1=1 A X J N11 LNG 76 IF(X( I) .LT. .OOODGO TO 13 1 NG 77 IFfABS ffFHfl I)-F(I I))/F(I lU GT OOnGD TO 30 1 Kir; 78 IFfABS ((HH(I D-Hd I))/H(I l\) GT 001 ^GO TO ^0 79 IFfABS ffTHfl D-THETAfl IlWTHETAfl l\\ GT 001 )Gn TO 30 80 IFfABS ffPHfl I)-PHIfI li)/PHIfI D) GT 001 ^GO TO ?0 1 wr; Rl 13 rONTINlIF l_ liU R?

GO TO 31 R3

30 CONTINUE 1 NG 84

33 PRINT 100 P DEN T LNG 85

100 FORMATf* ITTERATION FAILED AT* 3F10 ^) LNG 86 STOP LNG 87

31 n=nFN/HX 1 MR 88 THC=THETAf6 6) LNG 89

IFf IP EO 0)G0 TO 32 LNG 90 CALL PRESSf P,DEN,TRO) LNG 91 PI=1 .01325*P*FX/HX LNG 92

32 RETURN LNG 93 END LNG 94 SUBROUTINE MIX DATAflBASE) LNG 95 DIMENSION ZATAflO 10) ATAflO 10) TCflO) VCflO) ZCflO) ACflO) WflO) LNG 96

1 PCflO) CFf9) LNG 97 COMMON /DATA M/7ATA ATA TC VC W TfO VCO AfO ZCO RR R OMFGO AC 7C N LNG 98

1 ,PC ,CF LNG 99 nATAfPr=4R 3'ifi d? Qfi7R7 41 RQOOft 37 7Q4fi? 3fi 407'5ft 33 1Q3Q3) LNG 100 nATA(TC=1Q0 RtiR "^O^i "i ?7n 4?^ 4flfi 1 l?fi 2) LNG 1X\J01X nATA(W=lfi 04^0? "50 0701? 44 0Q7?1 Rft 1 ?43 "iR 1243 ?fi 0134) LNG 102 nATA(VC=Q8 R?? 1 4fi ? POO P^il fi? ?fi3 00 RQ fi?7) LNG 103 nATAfAr= mnp lin4?7 1'i4fl17 17fi'?7? l^lOll'i 0?Q17Q1) 1 MR 104

! NR 1 rm

1 NR 106

11 n?fi1Q ftQ??P? 7fi1^'^'^^ 11 07u / nATAf L/Mln\nlnvljt/1«'JU31H ATAf 1 9)=] OORI d ^ ( ATAf 1 ? ^ = 1 01 Q?? ^ ( ATAf 1 d 1=1 04.74^ \

/ j\Mlrt^ijO/~l«Ul-7c.t. / j\rMA\lj*T/"~l« UHt'+J / y

1lUOOR

1IvninVljJj ( ATAf 1 =— 1l. n^indft U jUtO ) ( ATAf 1 fi 1 = 1 ni noQ )

/ jV"l"V 1 jD / I •UlUU-? / ^

1 09

Af ATAf? 1 =1 nn^QQ ) fATAf? dl = 1 fllfilfil 110 PfATAf? f^l = 1 n7?fiQl fATAf? fil=1 n?1?7i fATAf^ 41 = 1 0017?! 1 NG 1 1

1

111 3fATAf3 \J\r\\r^\^y^l5)=1X»UXX"U/j\riIf^\JjVJ/ 01140) fATAf3 6)=1X*v<"TU\,^U/j\r^ir^V~j'^/ 04606) fATAf4 5)= aJJ/XX"/} 997114) LNG 112 4fATAf4 61 = 1 ?nPRf;) fATAfR fi) = l T?Rft9) LNG 1 1

3

1 1 O DATAfZATAfl ?)=1 ni1?7) f7ATAf1 "^1= QRRfiOR) fZATAfl 4)= 9R3130) 1 NG 1 1

4

lit

lf7ATA(l ^1= QRfiQ7R) fZATAfl fi)= 9'i?4'^0) (7ATAf? 3)= 999961) 1 NR 1 1

R

X XiJ 2fZATA(2 4)= 97???3) f7ATAf2 5)= 998886) fZATAf2 6)= 939622) LNG 116 3fZATAf3 4)= 98'5'i47) fZATAf3 5)=1 03099) fZATAf3 6) = . 912209) LNG 117 4fZATAf4 5)= 976416) fZATAf4 6)= 849200) fZATAfS 6)=. 857310) LNG 118 L=IBASE LNG 119 N=8 LNG 120 N1=N-1 LNG 121 PC(7)=33.37 LNG 122 PC(8)=31.57 LNG 123 TC(7)=469.6 LNG 124 TCf8)=460.39 LNG 125 VC(7)=304. LNG 126 57

ia5
f U V O / JUU •

1 NG 1?7

1 MG 1XlO?R UfRi=7? TiT^Q 1 NG 1?Q

Ml/ V ' / •tOHJtU 1 NG 1X<JU '^n

ML» Ar(ft^= \ O / vCOOOOU ?flRRRfi 1 MG 1 "^1 x*?x ATAfI 7^=ATAM RWl nfi 1 MG ATAf? 7^=ATA(? fl^=l 0? 1 MG xoo ataH 7^=ATAn f^)=^ m 1 NG XOH ATAfd 7i=ATA(fl Ri=l 1 NG X03 ATAK 7^=ATAK fl^ = 1 1 KIG XuU

ATA(fi 7i=ATA(fi R^=1 1LnuMG xo / ATAf7 R^=1 Lnu X JO 7ATAf1 7^=7ATAf1 R)= PR 1LnuKIG HQX07 7ATA(? 7i=7ATAf? Ri= QQ 1LnUKIG 1xtu40

7ATAn 7\=7ATA('^ R^= QQ 1LPIU KIG 141 XHX 7ATAM 7\s7ATAffl R>= QQ 1LnuKIG XHC 14?

7ATA(t; 7^=7ATA('i fli= QQ Lnu 1 KIG 14'^ XHO ZATA(6 7i=ZATA(6 8i= 9Q 1 KIG 144 7ATA(7 RWl

DO 3 J=l Ml LNG 146 J1=J+1 LNG 147 DO 3 K=J1 N LNG 148

ZATAfK J)=ZATA(J K) LNG 149 ATA(K J)=ATA{J K) LNG 150

DO 4 1=1 N LNG 151 ATAd I)=ZATA(I I)=l. LNG 152

PCO=PC(L) LNG 153 VCO=VC( WW vwvi»/L) LNG 154

TCO=TC(L) l^w IwV^/ LNG 155 OMEGOsArfL^ LNG 156

7ro=prn*vcn/Tcn/R £.ww ruw iww/ 1 WW/ rv I NG 157 X«J / DO 5 1=1 N LNG 158

zc( n=pc( I)*VC( I ) /TC( I ) /R LNG 159 X«^ 7 RFTIIRN IfiO X wv END LNG 161

^IIRRdllTTNF DATA rH4 Lnu 1 KIG Xuc

INTTIAI T7F<; THF FnilATION OF ^TATF mU^TANT^ TO MFTHAMF 1LnuKIG xo J DIMFN<;T0N RH?) VPfQ^ RTMI) 1LnuKIG 1X fi4 vH rnNWON/nATA/R R RAMMA VP DTP 1LnuKIG X03

niMFW^^TfUl u X I luno X n\xui AMO^ LnU 1 KIG xoo rnMMnN/<;ATr V/Ui ii iwn / on 1 w//an Lnu 1 KIG 167

1LnUKIR IDO RAMMA=- nOQfi 1LnUKIR r\\x/ A(11=1Q0 X » 555 ^ tj %j 1LnuKIG X f VJ

1LnUKIG 171 Af3^=1RX O • 40415fi47? 'TW*t X */W*T / L 1LnuKIG 17? X 1 c A(4)=7 34Q8921512 tL> KIG ivu 173 X 1 *J A(5)=-l 4313160833 1 NG 174 A(6)=A(7)=0 0 LNG 175 Xi ^ G(U\ 1)=- X/ .187027997685F-01 •XU/wkfJ^/wU^LwX LNG 176 X f w

G(u\ 2)= fe/ .103387108009E+01 •Xw^^w/XWwwJ^'wX LNG 177

G{ 3)=--155387625619E+02 LNG 178

G( 4)= 772311478564E+03 LNG 179

G( 5W- 377103300RQ5F+05 1LnuKIG 1xouRO

G( 6^= ft4fiR1Rfl4'^475F-n"? Lnu 1 NG 181 XOX

Gf 7)=_ 4Q641 5RR45?QF+nn 1LnuKIG XO^

G( 8)= 869909352414E+02 LNG 183

6( 9)=-.322821592493E+05 LNG 184

G( 10)=- .395843026318E-04 LNG 185 G{11)= .266772318035E-01 LNG 186

G( 12 ) =- .304010057839E+01 LNG 187 G{13)= .191584507536E-03 LNG 188 G{14)=-.195587933458E-03 LNG 189 58

186
G(15)= .607479967879E+01 1 NG 190

G ( 16 ) =- . 529609525984E-03 t 1X Q17 1. G(17)= .152264286004E-04 1 MR i.y 1Q?c 6(18)=-. 109952182842E-01 LNG 193

G(19)= .191395549929E-03 194 G{20)= .386470003746E+05 t X 7 <J G{21)=- .157930582612E+07 LNG 196 G(22)= .195270144401E+03 LNG 197 G(23)= .165996081629E+07 LNG 198 G(24)= .603051146711E+00 LNG 199

6(25)= .376485162808E+02 LNG 200 G{26)= .125593680622E-02 LNG 201

G( 27 )=- .343570032513E+02 LNG 202

G ( 28 ) =- . 540945094139E-05 LNG 203

G(29)= .185622284663E-02 LNG 204

G(30)= .770786979245E-08 LNG 205

G( 31 ) =- .286868318650E-05 LNG 206 G(32)= .372376961647E-04 LNG 207 VP(1)=4. 77748580 LNG 208 VP(2)=1. 76065363 LNG 209

VP(3)=-. 56788894 LNG 210 VP(4)=1. 32786231 LNG 211 VP(5)=1.5 LNG 212 VP(6)=.1159 LNG 213

VP(7)=90.68 LNG 214 VP(8i=190.555 LNG 215

END LNG 216 FUNCTION FIND M(P,T,D) LNG 217 SOLVES THE EQUATION OF STATE OF (ETHANE FOR DENSITY GIVEN P AND T LNG 218 DD=D LNG 219 TT=T LNG 220 DO 10 1=1 50 LNG 221 CALL PRESS (PP DD TT) LNG 222 P2=PP LNG 223

IF(ABS (P-P2)-l E-7*P)20 20 1 LNG 224 CALL DPDD(PP DD TT) LNG 225 DP=PP LNG 226 C0RR=(P2-P)/DP LNG 227 IFfABS (CORR)-l E-7*DD)20 20 10 LNG 228

DD=DD-CORR LNG 229 FIND M=0 LNG 230 RETURN LNG 231 FIND M=DD LNG 232 RETURN LNG 233 END LNG 234

FUNCTION SATL(T) LNG 235

CALCULATES THE SATURATED LIQUID DENSITY OF METHANE LNG 236 DIMENSION A(10) LNG 237

COMMON/SATC/A LNG 238 IF{T.GT.A(1))G0 TO 1 LNG 239

X=(l.-T/A(l)) LNG 240 SATL=A(2)+A(3)*X**(.35)+A(4)*X+A{5)*X**(4./3.)+A(6)*X**(5./3. LNG 241 )+A(7)*X**2 LNG 242 SATL=SATL/1000. LNG 243 RETURN LNG 244 SATL=1.E20 LNG 245 RETURN LNG 246 END LNG 247 SUBROUTINE PROPS(PP ,DD .TT) LNG 248 EQUATION OF STATE FOR METHANE AND NITROGEN LNG 249

DIMENSION X(33) LNG 250 DIMENSION B(33),G(32) LNG 251 EQUIVALENCE (B.X) LNG 252 59

18?
COftlON/DATA/G.R.GAMMA

DATA(ID=1)

DATA(IZ=1) 1 CONTINUE

D=DD

P=PP T=TT GM=GAMMA

D2=D*D

D3=D?*D

D4=D3*D

D5=D4*D

D6=D5*D

D7=D6*D

D8=D7*D

D9=D8*D

D10=D9*D

D11=D10*D

D12=D11*D

D13=D12*D

TS=SORT (T)

T2=T*T

T3=T2*T

T4=T3*T

T5=T4*T

F=EXP {GM*D2)

GO TO (100,200),K

ENTRY PRESS

K=l GO TO 1 100 CONTINUE

B( 1)=D2*T

B( 2)=D2*TS

B( 3^=D2

B{ 4)=D2/T

B( 5)=D2/T2

B( 6)=D3*T

B( 7)=D3

B( 8)=D3/T

B( 9)=D3/T2

B(10)=D4*T

B(11)=D4

B(12)=D4/T

B(13)=D5

B(14)=D6/T

B(15)=D6/T2

B(16)=D7/T

B(17)=D8/T

B(18)=D8/T2

B(19)=D9/T2

B(20)=D3*F/T2

B(21)=D3*F/T3

B(22)=D5*F/T2

B(23)=D5*F/T4

B(24)=D7*F/T2

B(25)=D7*F/T3

B(26)=D9*F/T2

B(27)=D9*F/T4

B(28)=D11*F/T2

B{29)=D11*F/T3

B(30)=D13*F/T2

B(31)=D13*F/T3

B(32)=D13*F/T4

LNG 253

LNG 254

LNG 255

LNG 256

LNG 257

LNG 258

LNG 259

LNG 260

LNG 261 LNG 262

LNG 263 LNG 264

LNG 265 LNG 266

LNG 267 LNG 268

LNG 269

LNG 270 LNG 271 LNG 272 LNG 273

LNG 274

LNG 275

LNG 276 LNG 277 LNG 278

LNG 279 LNG 280

LNG 281 LNG 282 LNG 283 LNG 284 LNG 285

LNG 286 LNG 287

LNG 288

LNG 289 LNG 290

LNG 291 LNG 292 LNG 293 LNG 294

LNG 295 LNG 296

LNG 297 LNG 298

LNG 299 LNG 300

LNG 301 LNG 302

LNG 303 LNG 304

LNG 305 LNG 306

LNG 307 LNG 308 LNG 309 LNG 310

LNG 311 LNG 312

LNG 313 LNG 314 LNG 315 60

188
102 P=0

DO 101 1=1,32

101 P=P+B(I)*G(I)

P=P+R*D*T

PP=P RETURN

ENTRY DPDD

K=2

GO TO 1 200 CONTINUE

F1=2.00*F*GM*D

F21=3.000*F*D2 +F1*D3

F22=5.000*F*D4 +F1*D5

F23=7.000*F*D6 +F1*D7

F24=9.000*F*D8 +F1*D9

F25=] 1 nn*c*ni n+n *m 1 F26=] 5 nn*r*rM 9+F1 *ni 7 D

1 -9-L . nn*n*T UU U 1 D0

9 -C • UU u 1

0

DD J -9— ^ • nn*n UU u

b

A4 =9"L. * nn*n/T UU U / 1 D c0 — c. .UU U/ 1

1

nD

c

0 -7-J.UUnn*n9*T Uc 1 D 1 D / -0 . UU

nD

Q0 - J . UU Ut:/ 1 B( n

y -0 . nn*n9 UU /t9 uc./ 1 <1 B 10 — H . UU uo 1 B 11 -4 . UU L) J B 12 =4.00*D3/T

B 13 =5.00*04

B 14 =6.00*D5/T

B 151 =6.00*D5/T2

B 16 =7.00*D6/T

B 17 =8.00*D7/T

B 18 =8.00*D7/T2

B 19 =9.00*D8/T2

B 20 =F21/T2

B 21 =F21/T3

B 221 =F22/T2

B 23 =F22/T4

B 24 =F23/T2

B 25 =F23/T3

B 26 =F24/T2

B 27' =F24/T4

B 28 =F25/T2

B 29 =F25/T3

B( 301 =F26/T2

B 3r =F26/T3

B( 321 =F26/T4

P==0 DO 201 1=1,32

201 P=P+B(I)*G(I)

P=P+R*T

PP=P RETURN

END SUBROUTINE DATA N2

C INITIALIZES THE EQUATION OF STATE CONSTANTS

DIMENSION G(32) ,VP(9) ,GI(11)

COMMON/DATA/G , R .GAMMA , VP , DTP R=8.20539E-2

GAMMA=-.0056

G( 1)= 0.136224769272827E-02

61

LNG 316 LNG 317 LNG 318

LNG 319 LNG 320

LNG 321 LNG 322 LNG 323 LNG 324 LNG 325

LNG 326 LNG 327 LNG 328 LNG 329 LNG 330 LNG 331 LNG 332

LNG 333

LNG 334

LNG 335 LNG 336 LNG 337 LNG 338

LNG 339 LNG 340 LNG 341 LNG 342 LNG 343

LNG 344

LNG 345 LNG 346 LNG 347 LNG 348 LNG 349

LNG 350

LNG 351 LNG 352 LNG 353

LNG 354

LNG 355 LNG 356

LNG 357 LNG 358 LNG 359

LNG 360

LNG 361 LNG 362 LNG 363 LNG 364

LNG 365

LNG 366 LNG 367 LNG 368 LNG 369 LNG 370 LNG 371 LNG 372 TO NITROGEN LNG 373

LNG 374 LNG 375

LNG 376 LNG 377

LNG 378

189
G( 2) = 0.107032469908591E 00 LNG 379

G( 3) = -0.243900721871413E 01 LNG 380

G( 4) = 0.341007449376470E 02 LNG 381

G( 5) = -0.422374309466167E 04 LNG 382

G( 6) = 0.105098600246494E-03 LNG 383

6( 7) = -0.112594826522081E-01 LNG 384

G( 8) = 0.142600789270907E-03 LNG 385

G( 9)= 0.184698501609007E 05 LNG 386

G(10)= 0.811140082588776E-07 LNG 387 G(ll)= 0.233011645038006E-02 LNG 388

G(12)= -0.507752586350986E 00 LNG 389

G{13)= 0.485027881931214E-04 LNG 390

G(14)= -0.113656764115364E-02 LNG 391 G(15)= -0.707430273540575E 00 LNG 392

G{16)= 0.751706548852680E-04 LNG 393

G(17)= -0.111614119537424E-05 LNG 394

G(18)= 0.368796562233495E-03 LNG 395

G(19)= -0.201317691347729E-05 LNG 396

G(20)= -0.169717444755949E 05 LNG 397 G(21)= -0.119719240044192E 06 LNG 398

G(22)= -0.975218272038281E 02 LNG 399 G(23)= 0.554639713151823E 05 LNG 400

G(24)= -0.179920450443470E 00 LNG 401 G{25)= -0.256582926077184E 01 LNG 402

G(26)= -0.413707715090789E-03 LNG 403

G(27)= -0.256245415300293E 00 LNG 404

G(28)= -0.124222373740063E-06 LNG 405

G(29)= 0.103556535840165E-04 LNG 406

G(30)= -0.538699166558303E-09 LNG 407 G(31)= -0.757415412839596E-08 LNG 408

G(32)= 0.585367172069521E-07 LNG 409

VP(1)=5. 1113192094 $ VP(2)=6.482667539E-1 LNG 410

VP(3)=-1. 5108730916E-1 $ VP(4)=7 .4028493342E-1 LNG 411 VP{5)=1.5 $ VP(6)=.123 $ VP(7)=63.15 $ VP(8)=126.26 LNG 412

VP(9)=0.0 LNG 413

DTP=31.0 LNG 414

- RETURN $ END LNG 415 FUNCTION VPN{TT) LNG 416

CALCULATES THE VAPOR PRESSURE OF BOTH METHANE AND NITROGEN LNG 417 DIMENSION G(32) ,VP(9) LNG 418

COMMON/DATA/G.R, GAMMA, VP .DTP LNG 419 T=TT LNG 420

X=(1.-VP(7)/T)/(1.-VP(7)/VP{8))

VPN=VP(6)*EXP(VP(1)*X+VP{2)*X*X+VP(3)*X**3+VP(9)*X**4+VP(4)*X*

LNG 421 LNG 422

1(1.-X)**VP(5)) LNG 423 RETURN LNG 424

END LNG 425

FUNCTION RODEN(P,T,X) LNG 426

THE HARD SPHERE MODEL. SEE SUBROUTINE FM FOR THE ARGUMENT LIST LNG 427 DIMENSION X(10) LNG 428

CALL FM(P,T,X,V,G) LNG 429

D=28. LNG 430 V=V+V IDEL(P,D,T,X) LNG 431 R0DEN=1000./V LNG 432 RETURN LNG 433 END LNG 434

SUBROUTINE FM(0,T,X ,V9 ,G9) LNG 435

PREDICTION OF EXCESS PROPERTIES WITH LHW POTENTIAL, COMP 1-METH LNG 436

ANE,COMP 2-ETHANE,C0MP 3-PROPANE .COMP 4 N-BUTANE ,COMP 5-1 -BUT LNG 437

ANE, COMP 6-NITR06EN. COMP 7-NORMAL PENTANE.COMP 8-ISOPENTANE LNG 438

ADV CRYO ENGR. VOL. 19 (1973)-REPR0GRAMED BY R. MCCARTY .2/22/74 LNG 439

ARGUMENTS ARE X-MOLE FRACTIONS, T-TEMPERATURE.0-PRESSURE,V9-EXCESS LNG 440

VOLUME, G9-EXCESS GIBBS ENERGY, FIRST THREE ARE INPUT, LAST TWO ARE LNG 441 62

190
OUTPUT, INPUT IS IN KELVINS AND BAR,

DIMENSION A{8),B(8) ,S(8) ,V(8) ,G(8) ,C(8,8) ,D(8,8) ,K(8,8)

,

1J(8,8),E(8),Y{8),0(8),X(10)

TYPE REAL J,K,N1

DATA(KEY=1)

DATA(J(1, 2)=-. 00388616), (J(l, 3)=-. 0120932) , (0(1,4)=-. 0231577)

,

A( J{ 1, 5)=- .0238349), (J (1,6)=-. 00997547), (J (1 ,7)=-. 0326), (J( 1,8) = B-. 0458), (J(2, 4)=-. 00400910) ,( J(2, 5)=-. 00812712) ,(J( 2, 6)=-. 0143976)

C, (J (2, 7)=- .003) ,(J(2,8)=-.004) ,(J(3,4)=+.0007615710),(J(3,5)=-.003

D83743),(J(3,6)=-.024014),(J(3,7)=0.0),(J(3,8)=0.0) ,(J(4,5)=.002221

E50),(J(4,7) = .0) ,(J(4,8) = .0),(J(5,6)=-.0576043),(J(4,6)=-.0576043),

F(J(5,7)=.0),(J(5,8)=0.0),(J(6,7)=-.04),(J(6,8)=-.05),(J(7,8)=.0)

G ,(J(2, 3)=-. 002162)

DATA(K( 1,2)=. 00298830), (K( 1,3)=. 0597378), (K( 1,4)=. 110893),

(

AK(1,5) = .100298) ,(K(1, 6) = . 0197290), (K( 1,7) = . 14), (K( 1,8) = . 1745),

B(K(2,4)=.0677703) ,(K(2,5) = .0346632) .(K(2,6) = .0529034),(K(2,7) = C.02) ,(K(2,8) = .03),(K(3,4)=.0249291),(K(3,5)=-.00838212) ,(K(4,5) = .0 D199213),(K(4, 6)=. 154365 ) ,(K(4,7) = .0) ,(K(4,8) = .0) ,(K(5,6) = .154365)

E, (K(5,7)=.0),(K(5,8)=.0),(K(6,7)=.15),(K(6,8)=.18),(K(7,8)=0.0)

F, (K(2.3) = .014527),(K(3,6) = .14719),(K(3,7)=0.0) ,(K(3,8)=0.0)

DATA(S=3.676E-8,4.158E-8,4.644E-8,5.051E-8,5.056E-8,3.546E-8,

15.389E-8,5.706E-8)

DATA(A=2.755E+5,7.773E+5,14.165E+5,22.733E+5,21.279E+5,1.718E+5,

130.550E+5,42.946E+5)

DATA( 0=1. ,1.5,1.67,1.83,1.79,1.03,1.91,2.11)

THESE ARE THE ACENTRICITY FACTORS (FOR MOLECULAR SHAPES, ETC.) *** DATA(Y=35.,45.,60.,75.,75. ,40. ,90. ,105.)

DATA(P1=3. 14159) , ( Nl=6 .025E+23) ,(R=8.3143 ) IF(KEY.EQ.0)G0 TO 1 KEY=0

DO 2 1=1,8

J(I,I)=0.0

K(I,I)=0.0

DO 2 M=I,8

J(M,I)=J(I,M)

2 K(M,I)=K(I,M)

1 CONTINUE

P=0*.l

IW=8

DO 10 1=1, IW

10 E(I)=0(I)

DO 15 1=1, IW

15 B(I)=(2./3.)*P1*N1*S(I)**3

DO 20 1=1, IW DO 20 M=1,IW

D(I,M) =(((B(I)**( 1./3. ) + B(M) ** ( 1./3.))/ 2. )*(1.-J{I,M) ))

1 **3

20 C( I 1( D( A2=0

E2=0

M ) ) * B(M)

, M ) = ( 1. - K( I, I, M ) **2 / ( B(I)

$ B2=0

$ V2=0

DO 25 1=1, IW V2=V2+X( I)*Y(I)

DO 25 M=1,IW

E2=E2+X(I)*X(M)*(E(I)+E(M))/2.

A2 = A2 + X(I) * X(M) * C( I, M ) 25 B2=B2+X(I)*X(M)*D(I,M)

V6=V2

DO 30 I = 1, IW A3 = A(I)

B3 = B(I)

E3 = Ed)

V2 = Yd)

( Ad)* A(M)

) ** ( 1./2.

( 1. / 2. )

LNG 442

LNG 443

LNG 444 LNG 445 LNG 446 LNG 447

LNG 448

LNG 449

LNG 450

LNG 451

LNG 452

LNG 453

LNG 454 LNG 455

LNG 456

LNG 457

LNG 458

LNG 459 LNG 460

LNG 461 LNG 462 LNG 463

LNG 464 LNG 465

LNG 466 LNG 467 LNG 468

LNG 469

LNG 470

LNG 471 LNG 472 LNG 473

LNG 474

LNG 475

LNG 476

LNG 477

LNG 478

LNG 479

LNG 480

LNG 481 LNG 482 LNG 483

LNG 484

LNG 485 LNG 486

LNG 487

LNG 488 LNG 489

LNG 490

LNG 491 LNG 492

LNG 493

LNG 494

LNG 495

LNG 496 LNG 497

LNG 498

LNG 499 LNG 500 LNG 501 LNG 502 LNG 503

LNG 504

63

191
VI = FIND Vl( A3, B3, E3, R, V2, P, T)

Gl = FIND Gl( A3, B3, E3, R, V2, P, T)

Vd) = VI 30 Gd) = Gl

A3=A2

B3=B2

E3=E2

V2=V6

V1=FIND V1(A3,B3,E3,R,V2,P,T)

G1=FIND G1(A3,B3,E3,R,V2,P,T)

V7=V1

G7=G1

EXCEAS VOLUME AND GIBS ENERGY V9=0

G9=0

DO 35 1=1, IW V9=V9-X(I)*V{I)

35 G9=G9-X(I)*G(I)

W9 = -V9

H9 = -G9

V9=V9+V7

G9=G9+G7

VIWW=V(IWW)

RETURN

END FUNCTION FIND Vl( A3, B3, E3,R, V2, P, T ) SOLVES THE HARD SPHERE EQUATION OF STATE FOR VOLUME GIVEN P AND T

A2 IS THE CONSTANT A, B2 IS THE CONSTANT B, E3 IS THE ACENTRICITY

INDEX = 0

1 VI = V2

XI = B3 / ( 4. * VI ) F2 = (( 1. + XI + Xl**2) / { 1. - Xl)**3) * E3 -A3 / ( VI * R * T)

1- (P * VI) / { R*T)

F3 = A3 / (R*T*V1**2) - P/ (R*T)

2.* Xl**2)

)*M * VD)

1 E3 F3 = F3 - ((( XI + XI) + 3* ( 1. + XI + Xl**2)

1* XI) / (( 1. - XI V2 = VI - F2/ F3

IF( ABS( (V2 - VI) / V2) .LT. .00001 ) GO TO 2 INDEX = INDEX + 1 IF( INDEX .LT, 250 ) GO TO 1

? VI = V2 FIND VI = V2 RETURN

END FUNCTION FIND Gl( A3, B3, E3,R, V2, P, T ) CALCULATES THE GIBBS FREE ENERGY FOR THE HARD SPHERE EOS

VI = V2

XI = B3 / ( 4. * VI ) Gl =ALOG( 1. / ( 1. - XI )) +{3. *X1) / { 1. - XI) + ( 3. * Xl**2)

1 / (2. * ( 1. - XI )**2)

Gl = Gl - A3 / ( E3 * R * T*V1 ) + ( P * VI ) / ( E3 * R * T ) -1 .0

1 - ALOG( VI ) Gl = R * T * E3 * Gl FIND Gl = Gl RETURN

END FUNCTION EXCESS{P,DD,T,X)

CALCULATES THE EXCESS OR IDEAL' VOLUM DEPENDING ON THE ENTRY DIMENSION X(10),F(10)

KR=0

GO TO 1 ENTRY V IDEL KR=1

64

LNG 505 LNG 506 LNG 507 LNG 508

LNG 509 LNG 510

LNG 511 LNG 512

LNG 513 LNG 514

LNG 515 LNG 516

LNG 517 LNG 518

LNG 519

LNG 520

LNG 521 LNG 522

LNG 523 LNG 524

LNG 525 LNG 526 LNG 527 LNG 528

LNG 529

LNG 530

LNG 531 LNG 532 LNG 533

LNG 534

LNG 535 LNG 536 LNG 537 LNG 538 LNG 539 LNG 540

LNG 541 LNG 542 LNG 543 LNG 544

LNG 545 LNG 546 LNG 547 LNG 548

LNG 549 LNG 550

LNG 551 LNG 552

LNG 553 LNG 554

LNG 555 LNG 556 LNG 557 LNG 558 LNG 559 LNG 560

LNG 561 LNG 562 LNG 563 LNG 564

LNG 565 LNG 566 LNG 567
1 CONTINUE LNG 568 CALL ZERO(F) LNG 569

IF(X(1) .LE..000001)G0 TO 2 LNG 570 CALL DATA CH4 LNG 571 IF(T.GT.190.555)G0 TO 12 LNG 572 PM=VPN(T)+. 00001 LNG 573 DELP=P-PM LNG 574 D=SAT(T,1) LNG 575 CALL DPDDfOP.D.T) LNG 576 DELD=DELP/DP LNG 577 D=DELD+D LNG 578 F(1)=X(1)*1000./D LNG 579 GO TO 2 LNG 580

12 D=FIND M{P,T.DD) LNG 581 IF{D.LE. 0.0)0=1000. LNG 582 F(1)=X(1)*1000./D LNG 583

2 IF(X(2).LE. .OOOOODGO TO 3 LNG 584 F(2)=X(2)*1000./SAT{T,2) LNG 585

3 IF(X(3) .LE.. OOOOODGO TO 4 LNG 586 F(3)=X(3)*1000./SAT(T,3) LNG 587

4 IF(X(4) .LE. .OOOODGO TO 5 LNG 588 F(4)=X(4)*1000./SAT(T,4) LNG 589

5 IF(X(5) .LE. .OOOODGO TO 6 LNG 590 F{5)=X(5)*1000./SAT(T,5) LNG 591

6 IF(X{7) .LE.. OOOODGO TO 61 LNG 592 F(7)=X(7)*1000./SAT(T,7) LNG 593

61 IF(X(8) .LE.. OOOODGO TO 62 LNG 594 F(8)=X(8)*1000./SAT(T,8) LNG 595

62 IF(X{6) .LE.. OOOODGO TO 8 LNG 596 CALL DATA N2 LNG 597 IF(T.GT.126.6)G0 TO 7 LNG 598 PN=VPN(T) + . 000001 LNG 599 DELP=P-PN LNG 600 D=SAT(T,6) LNG 601 CALL DPDD(DP,D,T) LNG 602 F{6)=X{6)*1000./(D+DELP/DP) LNG 603

GO TO 8 LNG 60'! 7 D=FIND M{P,T,DD) LNG 605 IF{D.LE. 0.0)0=1000. LNG 606 F(6)=X(6)*1000./D LNG 607

8 V=1000./DD LNG 608 VS=0 LNG 609

DO 21 1=1,8 LNG 610

21 VS=VS+F(I) LNG 611

EXCESS=V-VS LNG 612 IF(KR.GT.O)EXCESS=VS LNG 613 CALL DATA CH4 LNG 614 RETURN LNG 615

END LNG 616 SUBROUTINE ZERO(X) LNG 617

C INITIALIZES THE COMPONENT MATRIX TO 0 LNG 618 DIMENSION X(10) LNG 619

DO 1 1=1,10 LNG 620

1 X(I)=0.0 LNG 621 RETURN LNG 622 END LNG 623

FUNCTION FMKM(T,X) LNG 624

C THE REVISED KLOSEK AND MCKINLEY METHOD, THE INPUT IS TEMPERATURE LNG 625

C AND THE COMPONENT MATRIX. TEMPERATURE IS IN KELVIN .OUTPUT IS LNG 626

C DENSITY IN MOLES PER LITER. THE ALLOWABLE COMPONENTS ARE C1,C2,C3 LNG 627

C NC4,IC4,N2,NC5,IC5 IN THAT ORDER. THIS METHOD SHOULD NOT BE USED LNG 628

C FOR MIXTURES WITH LESS THAN 60% METHANE, OR FOR MIXTURES CONTAININ LNG 629

C MORE THAN 4% NITROGEN OR MORE THAN ^% EACH OF NC4 OR IC4 OR MORE LNG 630 65

193
THAN 1% TOTAL OF NC5 AND IC5. DIMENSION TM(IOO) ,TN(100) ,X{10) ,0(8)

DATA( 0=16 .04303 ,30 .07012 ,44 .09721 ,58 .1243 ,58 .1243 ,28 .0134 ,72 .1513

19,72.15139)

DATA{(TM( I), 1=1. 10)=- .005,. 12,. 22,. 34,. 43,. 515,. 595,. 66,. 725,. 795)

DATA((TM( I), 1=11, 20)=-. 006,. 135,. 26,. 38,. 5,. 59,. 665,. 74,. 81,. 885)

DATA((TM( I), 1=21, 30)=- .007,. 15,. 3,. 425,. 575,. 675,. 755,. 83,. 91,. 99)

DATA{{TW( I) ,1=31, 40)=-. 007,. 165,. 34,. 475,. 635,. 735,. 84,. 92, 1.045,

A1.12)

DATA{(TM(I)

A,U245)

DATA{(TM{I)

A1.38)

DATA((TM(I)

A1.55)

DATA((TM(I)

11.64,1.75)

DATA{(TM(I)

Al. 86, 1.99)

DATA{(TM{I)

A2. 105, 2. 272)

DATA((TN{I)

DATA((TN(I)

DATA((TN(I)

DATA((TN{I)

DATA{(TN(I)

11.71)

DATA({TN(I)

DATA((TN(I)

12.45)

DATA((TN{I)

1,2.9)

DATA((TN(I)

A3. 32, 3. 52)

DATA((TN(I)

A3. 99, 4. 23)

IF(X(1).LT.

AW=0.0

DO 1 1=1,8

1 AW=AW+X(I)*Q{I)

VI=VIDEAL{T,X)

0=1

IF(.T.GE.95.)J=11

IF{T.GE.100.)J=21

IF(T.GE.105.)J=31

IF(T.GE.110.)J=41

IF{T.GE.115.)J=51

IF(T.GE.120.)J=61

IF(T.GE.125.)J=71

IF{T.GE.130.)J=81

JJ=J+9

W=15.

DO 5 1=0 ,JJ W=W+1.

IF(AW.GT.W)GO TO 5 GO TO 6

5 CONTINUE

1=00

6 DIF1=AW-W

0=1-1

FK=(TM( I )-TM{0 ) )*DIF1+TM( I

)

FK1=(TM( I+10)-TM(0+10) )*DIF1+TM( I+IO)

IT=(T+. 00001 )/5.

1=41 ,50 )=- .008 , .19 , .375 , .535 , .725 , .835 , .95 ,1 .055 ,1 . 155

1=51, 60)=-. 009,. 22,. 44,. 61,. 81,. 945, 1.065, 1.18, 1.28,

1=61, 70)=- .01,. 25,. 5,. 695,. 92, 1.055, 1.205, 1.33, 1.45,

1=71, 80)=-. 013,. 295,. 59,. 795, 1.035, 1.21, 1.385, 1.525,

1=81 ,90)=- .015,. 345,. 7,. 92, 1.2, 1.37, 1.555, 1.715,

1=91, 100)=-. 017,. 4,. 825, 1.06, 1.39, 1.59, 1.8, 1.95,

1=1, 10)=-. 004,. 1,. 22,. 35,. 5,. 6,. 69,. 78,. 86,. 95)

1 = 11, 20)=- .005,. 12,. 28,. 43,. 59,. 71,. 83,. 94, 1.05, 1.14)

1=21 ,30)=-. 007,. 16,. 34,. 49,. 64,. 79,. 94, 1.08, 1.17, 1.27)

1=31 ,40)=- .01 ,. 24,. 42,. 61,. 75,. 91, 1.05, 1.19, 1.33, 1.45)

1=41, 50)=- .015,. 32,. 59,. 77,. 92, 1.07, 1.22, 1.37, 1.52,

1=51, 60)=-. 024,. 41,. 72,. 95, 1.15, 1.22, 1.3, 1.45, 1.65, 2.)

1=61, 70)=- .032,. 6,. 91, 1.23, 1.43, 1.63, 1.85, 2. 08, 2. 3,

1=71, 80)=-. 043,. 71, 1.13, 1.48, 1.73, 1.98, 2. 23, 2. 48, 2. 75

1=81, 90)=- .058,. 95, 1.46, 1.92, 2. 2,2. 42, 2. 68, 3.,

1=91, 100)=-. 075, 1.3, 2. ,2. 4, 2. 6, 3., 3. 4, 3. 77,

00001) GO TO 20

LNG 631 LNG 632

LNG 633

LNG 634

LNG 635

LNG 636

LNG 637

LNG 638

LNG 639

LNG 640

LNG 641 LNG 642

LNG 643

LNG 644

LNG 645 LNG 646

LNG 647 LNG 648

LNG 649

LNG 650

LNG 651 LNG 652

LNG 653

LNG 654

LNG 655 LNG 656 LNG 657 LNG 658

LNG 659

LNG 660

LNG 661 LNG 662

LNG 663

LNG 664 LNG 665

LNG 666

LNG 667

LNG 668

LNG 669

LNG 670

LNG 671 LNG 672

LNG 673 LNG 674

LNG 675

LNG 676

LNG 677 LNG 678

LNG 679 LNG 680

LNG 681

LNG 682

LNG 683

LNG 684

LNG 685

LNG 686 LNG 687 LNG 688

LNG 689

LNG 690

LNG 691 LNG 692 LNG 693

194
DIF2=T-IT*5 LNG 694

IF(T.GE.135.)DIF2=T-130. LNG 695

IF(T.LT.90.)DIF2=T-90. LNG 696 FK=FK+{FKl-FK)*DIF2/5. LNG 697

IF(X(6) .LT..0001)G0 TO 17 LNG 698 FKN={TN(I)-TN(J))*DIF1+TN(I) LNG 699

FK1=(TN(I+10)-TN(J+10))*DIF1+TN(I+10) LNG 700 FKN=FKN+(FKl-FKN)*DIF2/5. LNG 701 FK=FK+{FKN-FK)*X(6)/.0425 LNG 702

17 FK=FK/1000. LNG 703

FMKM=1./(VI-FK*X(1)) LNG 704 RETURN LNG 705

20 FMKM=0.0 LNG 706 RETURN LNG 707 END LNG 708 FUNCTION V IDEAL(T,X) LNG 709

CALCULATES THE IDEAL VOLUME OF A MIXTURE FOR THE K AND M METHOD LNG 710 DIMENSION X(10) LNG 711 V=0 LNG 712 J=0 LNG 713

IF(X(6).GT. .0001. AND. T.GT. 115. )J=1 LNG 714

DO 10 1=1,8 LNG 715

IF(X( I) .LE. .OOOOODGO TO 10 LNG 716 IF(J.GT.O.AND.I.E0.6)GO TO 10 LNG 717 V=V+X( I)/SAT(T,I) LNG 718

10 CONTINUE LNG 719 IF(J.E0.1)V=V+X(6)/SATN2(T) LNG 720 VIDEAL=V LNG 721 RETURN LNG 722 END LNG 723

FUNCTION SATN2{T) LNG 724 CALCULATES A PSEUDO SATURATED LIQUID DENSITY FOR N2 ABOVE 115 K LNG 725

IF{T.LT.115.)G0 TO 1 LNG 726 DELT={T-115.) LNG 727 SATN2=SAT(115.,6)+DELT*(SAT(115.05,6)-SAT(114.95,6))/.l LNG 728 RETURN LNG 729

1 SATN2=SAT(T,6) LNG 730 RETURN LNG 731 END LNG 732 FUNCTION SAT(T,n LNG 733

CALCUALTES THE PURE FLUID DENSITIES FOR THE K AND M METHOD LNG 734 UNITS ARE DEG K AND MOLES/LITER LNG 735

DIMENSION A(7,8) LNG 736 DATA((A{I) ,1=1, 7)=190. 555, 10. 16, 18. 65812322, 6. 712030737, LNG 737

1-. 9472019702, 0.0,0.0) LNG 738

DATA((A(I) ,I=8,14)=305.33, LNG 739

1 6.86,12.55205121,13.43284373,-19.00461066,11.07715985,0.0) LNG 740 DATA((A( I) ,1=15, 21) =369. 82, 5., 8. 684458671, 18. 04085714, -29. 46261356 LNG 741 1,16.43559114,0.0) LNG 742 DATA({A(I) ,I=22,28)=425.16,3.92, LNG 743

1 7.286062567,11.96307859,-19.87591962, LNG 744 211.60211932,0.0) LNG 745 DATA((A(I),I=29,35)=408.13,3.8, LNG 746

1 7.657535400,8.145251283,-13.10582462, LNG 747 28.145894091,0.0) LNG 748

DATA{(A( I) ,1=36, 42) =126. 2, 11. 21, 19 .39216835, 26. 01408462, -39. 497587 LNG 749

191,23.32977312,0.0) LNG 750 DATA( (A( I) ,1=43, 49)=469. 6, 3. 285, -.0362004993, 59. 00202990, LNG 751 1-93.44193819,43.66780833,0.0) LNG 752 DATA({A{ I), 1=50, 56)=460. 39, 3. 271, 2. 946310456, 35. 50770979, LNG 753 1-57.41242993,28.15898339,0.0) LNG 754 IF{T.GT.A(1,I))G0 TO 1 LNG 755 X=(1.-T/A(1,I)) LNG 756 67

195
SAT=(A(2,I)+A(3,I)*X**(.35)+A(4,I)*X+A(5,I)*X**{4./3.)

l+A(6,I)*X**(5./3.))

RETURN

1 SAT=l.E+20

RETURN

END SUBROUTINE ECKNON(PIN,DOUT,TIN,Q)

MAIN: ECKERT-RENON DENSITY MODEL

THIS PROGRAM IS DESIGNED FOR THE CALCULATION OF LNG LIQUID DENSITIES ONLY AND WILL NOT

COMPUTE DENSITIES -ABOVE 160 K.

THIS PROGRAM IS CALIBRATED SOLELY AGAINST DATA

tCASURED BY UNITED STATES DEPARTMENT OF COtWERCE

NATIONAL BUREAU OF STANDARDS

CRYOGENICS LABORATORY

BOULDER, COLORADO

>>>>>> >>>>>>>>NOTICE<<<< <<<<<<< <<<<<<<<<<<<<<<<<

NO WARRANTY IS MADE OR IMPLIED AS TO THE ACCURACY

OF DENSITIES CALCULATED BY USE OF THIS MODEL.

USERS ARE AT THEIR OWN RISK IN THE USE OF THIS PROGRAM.

COMPONENT ID NUMBERS ARE

1 METHANE

2 ETHANE

3 PROPANE

4 N-BUTANE

5 I-BUTANE

6 NITROGEN

7 N-PENTANE

8 I-PENTANE

INPUT VARIABLES ARE PIN=PRESSURE IN BAR TIN=TEMPERATURE IN K 0=MATRIX OF MOLE FPRACTIONS OF COMPONENTS

OUTPUT VARIABLE IS DOUT=DENSITY IN MOLES/LITER

DIMENSION VH0LD(12) ,VPH0LD(12) ,0(8) TYPE INTEGER CNTl ,CNT2 .SWITCH TYPE REAL MWM,INCR,KIJ ,MW,MOL,LAM,LAMB

COmon /RUN/ID(12),X(12) ,NAM(2,12),C12(12,12),R(12) ,Z(12),S(12) ,VS 1(12) .TS(12) ,US(12) ,SG(12),EP(12),LAM(12),NC,MW(12),AIJ(12,12),PST(

212),TCT(12)

COMMON /DAT/TBI (12) ,VBI(12) ,TBIM(12) ,VBIM{12) ,C(12) ,SGIM{12) ,SIM(1

12),EPIM(12) ,PIM(12) ,SIN(12) ,VSIM( 12) ,USIM( 12) ,TSIM( 12) ,DVBIM( 12 ) ,P 2D(12) ,PV(12) ,RH0{12) ,VBIMP{12) ,TTM(12) ,MWM,CNT1 ,CNT2,T0LD(12) ,INCR

3(12) , SWITCH ,JPC,JPCS,JMIX, DENS, VEX, HEX, GEX,TMP,VMP, TP C0W0N/PAR/KIJ( 8, 8),AjI( 8, 8)

68

LNG 757 LNG 758

LNG 759 LNG 760

LNG 761 LNG 762

LNG 763

LNG 764

LNG 765 LNG 766 LNG 767 LNG 768 LNG 769 LNG 770

LNG 771 LNG 772 LNG 773 LNG 774

LNG 775

LNG 776 LNG 777 LNG 778

LNG 779 LNG 780

LNG 781 LNG 782 LNG 783 LNG 784 LNG 785 LNG 786

LNG 787 LNG 788

LNG 789

LNG 790

LNG 791 LNG 792

LNG 793

LNG 794

LNG 795

LNG 796 LNG 797 LNG 798 LNG 799

LNG 800

LNG 801 LNG 802

LNG 803

LNG 804

LNG 805

LNG 806

LNG 807 LNG 808

LNG 809 LNG 810

LNG 811 LNG 812 LNG 813

LNG 814

LNG 815

LNG 816

LNG 817

LNG 818

LNG 819

196
COMMON/UNITS/ITC,IPC,T,TMAX,DT,P,PMAX,DP ,PS LNG 820

SWITCH = 0 LNG 821 CALL ZEROl 1 NG THE NEXT VALUES ARE SET TO PREVENT UNDER/OVER FLOWS LNG 823

TMAX = 100.0 LNG 824 DT = 100 0 1 WG OCyJ PMAX = 100 0 1 NG

1 NG ft?7 DP = 100 0 oco

PS = 100 0 TALL INPlITfO) rnnF=n K^KJU I- \J ITC=0

IPC=0 1L_)VU NG fi13 P=PIN/1 01325 T=TIN LNG 835 IF(T GT (160 ) ) GO TO 19 LNG 836

CALI PZERO(CTMX) 1 NG 837 MWM = 0 1 NG 838

DO 9 I = 1 NC LNG 839 MWM = MWM + (X(I)*MW(I)) LNG 840

CONTINUE LNG 841

DO 11 I = 1,NC LNG 842

TBK I) = T / TS( n LNG 843 CALL VOLUME(TBI(I) VBI(I) JPUR NC) LNG 844

TBIM( I) = T / TSIM( I) LNG 845 CALL VOLUME( TBIM( I) VBIM(I) JMIX NC) LNG 846 rONTINUE LNG 847 VM = 0.0 LNG 848

no 1? T = 1 NC LNG 849 VM = VM + ( Xf I

^ *VBIM( n*VSIM( T ) 1 LNG 850

CONTINUE LNG 851 TMP = ( T * 1 8^ - (459 671 1 MG ft5? no 13 I = 1 wr LNG 853

vHOi n( T 1 = vRT n

)

1 NG

VPHOI n( 1) = VR TM(

T

) 1 MG O J J

1 NG R5fi VRINiPf vuiiir\i/\) = VRTf 1 u T 1 i. \ I 1

1 NG 857

TAI 1 PRF^( PP VR TMPf T ^ TRT f T 1 MPl O 30 VR I ( n = VRTMP (

T

) 1 NG 859

pp = (p * T^TMfTU / ll"sTM(Tl 1 MR VRTMP( I ) = VRTM( I

)

LNG 861 CALL PRES(PP VRIMPfll TBIM(Il tiC) LNG 862 VBIM( I ) = VBIMP( I

)

LNG 863

CONTINUE LNG 864

VMP = 0.0 LNG 865 DO 14 I = 1 NC LNG 866

VMP = VMP + (X( I )*VBIMP( I)*VSIM( I)

)

fill fill ' \ /\ \ X 1 vuiiir\i/ *«jxii\x// LNG 867

PD( I ) = VBIMP( I ) * VSIM( I

)

LNG 868

CONTINUE LNG 869

DENS=MWM/VMP LNG 870

D0UT=DENS*1000 . /MWM LNG 871 RETURN LNG 872

DOUT=0 0 LNG 873 RETURN LNG 874

END LNG 875

SUBROUTINE ZEROl LNG 876

COMMON /RUN/A(361) LNG 877

COMMON /DAT/B(224) LNG 878

DO 1 I = 1,361 LNG 879

A(I) = 0.0 LNG 880

DO 2 I = 1,224 LNG 881 B(I) = 0.0 LNG 882

69

197
RETURN LNG 883

END LNG 884

SUBROUTINE INPUT(Q) LNG 885

DIMENSION NNO( 8) ,SIGM( 8),EPSI( 8),LAMB( 8),NAME(2, 8),SN0{ 8) ,M0 LNG 886

1L( 8),CT{ 8),TCH( 8),VCH( 8),ECH( 8),AR( 8),AZ( 8), 0(8) LNG 887 TYPE INTEGER CNTl ,CNT2 .SWITCH LNG 888 TYPE REAL MWM,INCR,KIJ,MW,MOL, LAM, LAMB LNG 889

COMMON /RUN/ID(12),X(12),NAM{2,12),C12(12,12),R(12),Z(12),S(12),VS LNG 890

1(12) ,TS(12) ,US(12),SG(12),EP(12),LAM(12),NC,MW(12).AIJ(12,12),PST( LNG 891 212),TCT(12) LNG 892

COMMON /DAT/TBI(12) ,VBI(12),TBIM(12),VBIM(12),C(12),SGIM{12),SIM{1 LNG 893

12),EPIM(12) ,PIM(12) ,SIN{12) ,VSIM(12) ,USIM(12) ,TSIM(12) ,DVBIM(12) ,P LNG 894

2D(12) ,PV(12) ,RH0(12) ,VBIMP( 12 ) ,TTM( 12 ) .MWM.CNTl ,CNT2 ,T0LD( 12 ) , INCR LNG 895

3(12),SWITCH,JPC,JPCS,JMIX,DENS,VEX,HEX,GEX,TMP,VMP,TP LNG 896

COMMON /PAR/KIJ( 8, 8),AJI( 8, 8) LNG 897 DATA SIGM/ LNG 898

*0. 991000, 1.029000, 1.155000, 1.278000. 1.388752, 1.392995, LNG 899

*1. 47162, 1.47217/ LNG 900 DATA EPSI/ LNG 901

*0. 640000. 0.909000. 1.69800, 2. 237000, 2. 705262, 2. 545907, LNG 902

*3. 76195 .3.68336/ LNG 903 DATA LAMB/ LNG 904

*1. 053604. 0.986325. 1.227117, 1.408519. 1.473346. 1.461676, LNG 905

*1. 698790, 1.695012/ LNG 906 DATA MOL/ LNG 907 *28. 01600, 16. 04200, 30.06800. 44. 09400. 58. 12000. 58. 12000. LNG 908

*72. 146000, 72. 146000/ LNG 909 DATA SNO/ LNG 910

no.00000. 10. 00000, 10. 00000, 10.00000, 10. 00000. 10.00000. LNG 911 *10. 00000. 10. 00000/ LNG 912 DATA CT/ - LNG 913

126.0600, 190. 5600, 305. 4300,369.8200, 425. 1600, 408. 0300, LNG 914

469.65,460.39/ LNG 915 DATA TCH/ LNG 916

112.7699, 170. 9645, 256. 8311, 294. 7255, 340. 7867. 322. 6109. LNG 917 411.0100,403.3200/ LNG 918 DATA VCH/ LNG 919

26.01846,29.04010,40.88188,54.60301,68.72075.69.44272. LNG 920

84.90197.84.99830/ LNG 921 DATA ECH/ LNG 922 1393.000,1977.000,3695.000.4867.000,5886.652,5539.895, LNG 923 8186.000,8015.000/ LNG 924 DATA AR/ LNG 925 1 . 000000 , 1 .000000 , 1 .000000 , 1 .000000 , 1 .000000 , 1 .000000 , LNG 926

1.000000,1.000000/ LNG 927 DATA AZ/ LNG 928

10.00000,10.00000,10.00000,10.00000,10.00000,10.00000, LNG 929

10.00000,10.00000/ LNG 930

DO 4 1=2,6 LNG 931

4 X(I)=0(I-1) LNG 932

X(l)=0(6) LNG 933

X{7)=Q(7) LNG 934

X(8)=0(8) LNG 935 NC=0 LNG 936 DO 1 1=1,8 LNG 937 IF(X(I).LE.O.O)GO TO 1 LNG 938

NC=NC+1 LNG 939

X(NC)=X(I) ' LNG 940

ID(NC)=I LNG 941

1 CONTINUE LNG 942 DO 2 I = l.NC LNG 943

J = ID(I) LNG 944

R(I) = AR(J) LNG. 945 70

198
Z(I) = AZ(J) LNG 946 S(I) = SNO(J) LNG 947

TS(I)= TCH(J) LNG 948 US(I)= ECH{J) LNG 949

SG(I) = SIGM(J) LNG 950 TCT(I) = CT(J) LNG 951 EP(I) = EPSI(J) LNG 952 LAM(I) = LAMB(J) LNG 953

VS(I) = VCH(J) LNG 954 MW(I) = MOL(J) LNG 955

2 CONTINUE LNG 956

DO 3 I = l.NC LNG 957

DO 3 J = 1,NC LNG 958

M = ID(I) LNG 959

L = ID{J) LNG 960

C12(I,J) = KIJ(M.L) LNG 961 C12(J,I) = KIJ(L,M) LNG 962

AlJd.J) = WKM.L) LNG 963

AIJ(J,I) = AJKL.M) LNG 964

3 CONTINUE LNG 965 RETURN LNG 966 END LNG 967 END LNG 968 SUBROUTINE PZERO(TRT) LNG 969 TYPE INTEGER CNTl , CNT2 , SWITCH LNG 970 TYPE REAL MWM, INCR ,KI J ,MW ,MOL ,LAM,LAMB LNG 971 COf^ON /RUN/ID(12),X(12),NAM(2,12),C12(12,12),R(12),Z(12),S(12),VS LNG 972 1(12),TS(12),US{12),SG{12) ,EP(12) ,LAM(12) ,NC ,MW( 12 ) ,AIJ ( 12 ,12 ) ,PST( LNG 973

212) ,TCT{12)

COKNON /DAT/TBI ( 12 ; ,VBI( 12) ,TBIM( 12) ,VBIM( 12 ) ,C{ 12 ) ,SGIM( 12) ,SIM(1

12) ,EPIM(12) ,PIM(12) ,SIN(12) ,VSIM(12) ,USIM( 12 ) ,TSIM( 12 ) ,DVBIM( 12 ) ,P 2D(12) ,PV(12),RH0(12),VBIMP(12),TTM(12),MWM,CNT1,CNT2,T0LD(12),INCR

3(12) , SWITCH ,JPC,JPCS,JMIX, DENS, VEX, HEX, GEX,TMP,VMP, TP COMMON/PAR/KIJ( 8, 8),AJI( 8, 8)

COMMON /UNITS/ITC,IPC,T,TMAX,DT,P,PMAX,DP,PS

IF(NC.EO.l) TRT = T/TCT(1)

IF(NC.GT.l) CALL TCM(TRT)

IF(NC.GT.l) TRT = T/TRT

IF(NC.EQ.l) CALL PURE (TRT, TRT, R( 1 ) ,RXX) DO 1 I = 1,NC TCXX = T/TCT{I)

IF(NC.GT.l) CALL PURE( TCXX ,TRT,R( I ) ,RTR)

JPC = 0 IF{TRT.GT.( .87)) JPC = 2 IF(TRT.GT.(1.0)) JPC = 1 IF(NC.EO.l) RTR = RXX THERE ARE SEVERAL OTHER FORMS OF THESE EQUATIONS

REFER TO ORIGINAL ARTICLES FOR VALUES TO USE

S(I) = RTR * Z(I) - 2. * RTR + 2. SIM(I) = S(I)

SG(I) = (VS(I)/RTR ) **(l./3.)

EP(I) = US(I) / S(I)

C(I) = LAMd) * (2176. /((185.6)*(1. 98726)))

TTM(I) = TS(I)

TOLD(I) = TS(I)

INCR(I) = (20.0)

1 CONTINUE

JPUR = 0 CNTl = 1 CNT2 = 1 JMIX = 0

2 DEN = 0 71

LNG 974 LNG 975

LNG 976

LNG 977

LNG 978

LNG 979

LNG 980

LNG 981

LNG 982 LNG 983

LNG 984 LNG 985

LNG 986 LNG 987

LNG 988

LNG 989

LNG 990

LNG 991

LNG 992 LNG 993

LNG 994 LNG 995 LNG 996 LNG 997 LNG 998 LNG 999

LNGIOOO

LNGlOOl

LNG1002

LNG1003

LNG1004

LNG1005

LNG1006

LNG1007

LNG1008

199
DO 3 I = l.NC DEN = DEN + (X(I)

CONTINUE

SIM(I))

l.NC

(X(I)

l.NC

DO 4 I PIM(I) = CONTINUE

DO 6 I = SGIM(I) = 0.0

DO 6 J = l.NC M = I N = J IF(I.GT.J) M = IF(I.GT.J) N

=

* SIM(I)) /DEN

IF(I.EQ.J) GO TO 5 TRUDX = (AIJ(N,M)/(T/TTM(I)))

IF(TRUDX.LT. (-180.0)) GO TO 5 SGIM(I) = SGIMd) +{(PIM(J) *(((SG(I)**(l./3.) + SG(J)**(l./3.))

1/(2.0)) **3))* (AIJ(M,N) * EXP(AIJ(N,M)/(T/TTM(I)))))

GO TO 6

5 SGIM(I) = SGIM(I) +{PIM(J) *( ( (SG{ I )**(1 ./3. ) + SG(J)**(l./3.))

l/(2.0)))**3 ) 6 CONTINUE

DEN = 0.0

DO 7 I = l.NC

DEN = DEN + (PIMd)* SGIM(I)* SGIMd))

7 CONTINUE

DO 8 I = l.NC

SINd) =(S(I) * (SG1M(I)**(2.))) / DEN

8 CONTINUE

DO 9 I = l.NC

TEST = (1.0) - (SIMd) / SINd))

TEST = ABS (TEST)

IF(TEST.GT.(0.00001)) GO TO 10

9 CONTINUE

GO TO 12 10 DO 11 I = l.NC

SIMd) =(SIM(I) + SINd)) /(2.0)

11 CONTINUE

CNTl = CNTl + 1 IF(CNT1.GT.250) GO TO 12 GO TO 2 12 DO 14 I = l.NC

SIMd) = SINd)

EPIM(I) = 0.0

DO 14 J = l.NC

K = I L = J IF(J.LT.I) K = J IF(J.LT.I) L = I IFd.EO.J) GO TO 13 TRUDX = C12(L,K) / (T/TTM(I))

IF(TRUDX.LT. (-180.0)) GO TO 13 EPIMd) = EPIMd) +((PIM(J) * SORT{EP{I)*EP(J)))*

1(C12(K.L) * EXP{(C12(L.K) /(T/TTMd ) ) ) ) )

)

GO TO 14

13 EPIMd) = EPIMd) + (PIM(J) * SQRT(EP{I) * EP(J)))

14 CONTINUE

ASSUME NUMBER OF MOLS OF MIXTURE = 1.0 DO 15 I = l.NC

VSIMd) = Rd) * (SGIM(I)**(3.0))

USIMd) = SIMd) * EPIMd)

TSIM(I) - USIMd) / ((1.98726) * C(I))

72

LNG1009

LNGlOlO

LNGlOll

LNG1012

LNG1013

LNG1014

LNG1015

LNG1016

LNG1017

LNG1018

LNG1019

LNG1020

LNG1021

LNG1022

LNG1023

LNG1024

LNG1025

LNG1026

LNG1027

LNG1028

LNG1029

LNG1030

LNG1031

LNG1032

LNG1033

LNG1034

LNG1035

LNG1036

LNG1037

LNG1038

LNG1039

LNG1040

LNG1041

LNG1042

LNG1043

LNG1044

LNG1045

LNG1046

LNG1047

LNG1048

LNG1049

LNG1050

LNG1051

LNG1052

LNG1053

LNG1054

LNG1055

LNG1056

LNG1057

LNG1058

LNG1059

LNG1060

LNG1061

LNG1062

LNG1063

LNG1064

LNG1065

LN61066

LNG1067

LNG1068

LNG1069

LNG1070

LNG1071

200
15 CONTINUE

DO 16 I = 1,NC

TEST = (1.0) - (TSIM(I)/nM(I))

TEST = ABS(TEST)

IF{TEST.GT. (0.00001)) GO TO 17 16 CONTINUE

RETURN

17 DO 20 I = 1,NC

IF(TSIM(I) - TTM(I))18,20,19

18 TNEW = TTMd) - INCR(I)

IF(TNEW.EQ.TOLD(I)) INCR(I) = INCR(I)/ (2.0)

IF(TNEW.EQ.TOLD( I)) GO TO 18 TOLD(I) = TTMd)

TTM(I) = TNEW

GO TO 20

19 TNEW = TTM(I) + INCR(I)

IF(TNEW.E0.TOLD(I)) INCR(I) = IF(TNEW.E0.TOLD( I)) GO TO 19 TOLD(I) = HMd)

TTMd) = TNEW

20 CONTINUE . CNT2 = CNT2 + 1 IF(CNT2.GT.250) RETURN

GO TO 2 END SUBROUTINE PURE(TZ,T,Z,R)

DIMENSION A(15)

IF(TZ.LT.( .8653525)) R = Z IF(TZ.LT.( .8653525) ) GO TO 2 IF(T.LT.( .8653525)) R = Z IF(T.LT.( .8653525)) GO TO 2 IF{T.GT.(1.0)) TA = 1.0 IF(T.LE.(1.0)) TA = T DATA A/. 9184780, -.1530647,

INCR(I) / (2.0)

1090050 , .8073883 , 1 .441803

,

1-10.85944 ,-6 .041687 ,51 .26758 , .9062500,-108.9805 ,28 .88672,

2106.1406,-45.78125,-38.43750,20.56250/

R = 0.0

n = (TA - (.9267292)) / ( .7267517E-01

)

DO 1 K = 1,14

R = R + A(16-K)

R = R * TT

1 CONTINUE

R = R + Ad)

2 RETURN

END SUBROUTINE VOLUME(T,V ,J ,NC) TR = T IF(TR.GT.(1.00)) GO TO 4

V = 0.5

1 VT = d.O) + (0.1*TR*V**(4.0))

ERR = (1.0) - (V/VT)

TEST = ABS{ERR)

IF{TEST.LE.(0.00001)) GO TO 2 V = VT

GO TO 1 2 V = V**(3.0)

3 RETURN

4 V = (((({(10.06600 *TR)-24.79837)*TR)+23.260722)*TR)-6. 686880)

GO TO 3 END SUBROUTINE BETA(P,DV,T,V,KK)

73

LNG1072

LNG1073

LNG1074

LNG1075

LNG1076

LNG1077

LNG1078

LNG1079

LNG1080

LNG1081

LNG1082

LNG1083

LNG1084

LNG1085

LNG1086

LNG1087

LNG1088

LNG1089

LNG1090

LNG1091

LNG1092

LNG1093

LNG1094

LNG1095

LNG1096

LNG1097

LNG1098

LNG1099

LNGllOO

LNGllOl

LNG1102

LNG1103

LNG1104

LNG1105

LNG1106

LNG1107

LNG1108

LNG1109

LNGlllO

LNGllll

LNG1112

LNG1113

LNG1114

LNG1115

LNG1116

LNG1117

LNG1118

LNG1119

LNG1120

LNG1121

LNG1122

LNG1123

LNG1124

LNG1125

LNG1126

LNG1127

LNG1128

LNG1129

LNG1130

LNG1131

LNG1132

LNG1133

LN61134

ZOl
TYPE REAL AF,BF,CF,DF,EF,AG,BG,CG,DG,P,DV,T,V,F»F1,G,G1,DEN,EG

TYPE REAL C,B1,B2,B3

DIMENSION Cdl)

DATA AF/0.11566564E+02/.BF/-0.53510144E+01/,CF/-0.74598207E-01/,

lDF/0 .67068653E+00/ ,EF/-0 . 11939B87E+00/ , AG/0 .62721813E+01/,

2BG/0 .47698365E+00/ ,CG/-0 .16023080E+01/ ,DG/0 .49837746E+00/

,

3EG/-0.42639183E-01/

F « AF +(BF*V)+(CF*V*V)+ {DF*V*V*V)+(EF*V*V*V*V)

IF(F.LT.(0.0)) GO TO 1 Fl ' (BF+{CF*V*2.E+00)+{DF*V*V*3.E+00)+(EF*V*V*V*4.E+00))*

IT * EXP{F)

G ' AG+{BG*V)+{CG*V*V)+(DG*V*V*V)+(EG*V*V*V*V)

IF(G.LT.{0.00)) GO TO 1 Gl = (BG+(CG*V*2.E+00)+{DG*V*V*3.E+00+(EG*V*V*V*4.E+00))) * EXP(G

1)

Gl = -Gl DEN = Fl + Gl DV = (l.E+OO) / DEN IF(DV.GT.(0.0).0R.DV.LT.(-1.0E+00)) GO TO 1 GO TO 2 DATA C/- .118659822E+02 , - .5509099E+00 , .4172102E+01 .- .8996686E+00

,

l-.1500376E+01,-.1958324E+00,-.2788988E+01,.5195283E+00,

2.3734878E+01,.1361904E+00,.4671948E-01/

1 Bl = C(l) + C(2)*T + C(3)*V

B2 = C{4)*((V+C(5))**(2 ))*EXP((C(6))*({T+C(7))**(2 ))) B3 = C(ll) * (V+C(8))**(C(9))/(T**(C{10)))

DV = Bl + B2 + B3

DV = EXP(DV)

DV = - DV

2 RETURN

END SUBROUTINE PRES(A,B,C,NC)

TYPE REAL H,P,V,T,K1,K2,K3.K4,VT,PT

P = A

V = B

T = C

H = l.E-02

IF(H.GT.P) H = P ASSIGN 3 TO KK

1 CALL BETA{P,K1,T,V,NC)

Kl = H * Kl VT = V + ((0.5E+00) *K1) PT = P + ((0.5E+00)*H)

CALL BETA(PT,K2,T,VT,NC)

K2 »= K2 * H VT = V + ((0.5E+00) *K2) CALL BETA(PT,K3,T,VT,NC)

K3 = K3 * H PT = P + H VT » V + ( K3) CALL BETA(PT,K4,T,VT,NC)

K4 = K4 * H

V = V + (((Kl + (2.E+00*K2) + {2.E+00*K3) + K4) / {6.E+00)) ) P «= P - H IF(P.EQ.{0.0)) GO TO 4 IF(P.LT.(0.0)) GO TO 2 GO TO 1 2 GO TO KK , (3,4)

3 P « P + H

H « P ASSIGN 4 TO KK GO TO 1 4 A = P 74

LNG1135

LNG1136

LNG1137

LNG1138

LNG1139

LNG1140

LNG1141

LNG1142

LNG1143

LNG1144

LNG1145

LNG1146

LNG1147

LNG1148

LNG1149

LNG1150

LNG1151

LNG1152

LNG1153

LNG1154

LNG1155

LNG1156

LNG1157

LNG1158

LNG1159

LNG1160

LNG1161

LNG1162

LNG1163

LNG1164

LNG1165

LNG1166

LNG1167

LNG1168

LNG1169

LNG1170

LNG1171

LNG1172

LNG1173

LNG1174

LNG1175

LNG1176

LN61177

LNG1178

LNG1179

LNG1180

LNG1181

LNG1182

LNG1183

LNG1184

LNG1185

LNG1186

LNG1187

LNG1188

LNG1189

LNG1190

LNG1191

LNG1192

LNG1193

LNG1194

LNG1195

LNG1196

LNG1197

zoz
B = V

C = T RETURN

END SUBROUTINE TCM{TMC)

TYPE REAL NUM TYPE INTEGER CNTl ,CNT2, SWITCH

TYPE REAL MWM,INCR,KIJ,MW,MOL, LAM, LAMB DIMENSION VCI(6) ,TH(12)

COWION /RUN/ID(12),X(12),NAM(2,12),C12(12,12),R(12),Z(12),S(12),VS

1(12),TS{12) .US(12) ,SG{12),EP(12),LAM(12),NC,MW(12),AIJ(12,12),PST(

212),TCT(12)

COMMON /DAT/TBI(12) .VBI(12) ,TBIM(12) .VBIM( 12 ) ,C(12) ,SGIM( 12) .SIM(1

12),EPIM(12),PIM(12),SIN(12) ,VSIM(12) ,USIM(12) ,TSIM(12) ,DVBIM( 12) ,P 2D(12) ,PV(12) ,RH0(12),VBIMP(12),nM{12),MWM,CNTl,CNT2,T0LD(12),INCR

3(12), SWITCH .JPC.JPCS.JMIX, DENS, VEX, HEX.GEX.TMP.VMP, TP C0MM0N/PAR/KIJ( 8, 8),AJI( 8, 8)

COMMON/UNITS/ITC,IPC,T,TMAX,DT,P,PMAX,DP,PS

DATA VCI/ 1.44,1.59,2.27,3.18,4.03,4.21/

DO 1 I = l.NC

J = ID(I)

V = VCI(J)**(2./3.)

TH( I) = X( I) * V

L CONTINUE

nZ = 0.0

DO 2 I = l.NC

nZ = TTZ + THd)

I CONTINUE

DO 3 I = l.NC

TH(I) = TH(I) / nZ

5 CONTINUE

SUMl = 0.0

SUM2 =0.0

K = NC - 1 DO 5 I = 1,K

L = I + 1 DO 4 J = L,NC

TTZ = (TCT(I)-TCT(J))/{TCT(I)+TCT(J))

TTZ= ABS(TTZ)

T12 = (((((({TTZ*(-3.038))+(5.443))*TTZ)+(-1.343))*TTZ)

1 + (0.287))*TTZ) - (.0076)

T12 = T12*{TCT(I) + TCT(J)) * (0.9)

SUM2 = SUM2 + ((2.)*TH(I)*TH(J)*T12)

\ CONTINUE

SUMl = SUMl + (TH(I)*TCT(I)*(1.8))

5 CONTINUE

TMC = SUMl + SUM2 + (TH(NC)*TCT(NC)*{1.8))

V = 0.0

DO 6 I = 1,NC

J = IDd)

TH(I) = X(I) * VCI(J)

V = V + TH(I)

5 CONTINUE

DO 7 I = 1,NC

TH(I) = TH(I) / V

7 CONTINUE

TMM = 0.0 DO 9 I = l.NC

DO 8 J = l.NC

M = ID(I)

N = ID(J) NUM = (VCI{M)**(l./3.)) * (VCI(N)**(l./3.))

75

LNG1198

LNG1199

LNG1200

LNG1201

LNG1202

LNG1203

LNG1204

LNG1205

LNG1206

LNG1207

LNG1208

LNG1209

LNG1210

LNG1211

LNG1212

LNG1213

LNG1214

LNG1215

LNG1216

LNG1217

LNG1218

LNG1219

LNG1220

LNG1221

LNG1222

LNG1223

LNG1224

LNG1225

LNG1226

LNG1227

LNG1228

LNG1229

LNG1230

LNG1231

LNG1232

LNG1233

LNG1234

LNG1235

LNG1236

LNG1237

LNG1238

LNG1239

LNG1240

LNG1241

LNG1242

LNG1243

LNG1244

LNG1245

LNG1246

LNG1247

LNG1248

LNG1249

LNG1250

LNG1251

LNG1252

LNG1253

LNG1254

LNG1255

LNG1256

LNG1257

LNG1258

LNG1259

LNG1260

203
NUM = SORT(NUM)

DEN = {0.5)*((VCI(M)**(l./3.))+(VCI(N)**{l./3.)))

NUM = NUM / DEN NUM = NUM**(3.)

AKIJ = (1.0) - NUM

TCIJ = (1.0 - AKIJ) * S0RT(TCT(I)*TCT{J)*1.8*1.8)

TMM = Ttfl + TH(I)*TH(J)*TCIJ

8 CONTINUE

9 CONTINUE

TCMP = TMM + (10.0)

ICT = 0 TPO = TMM + (10.0)

10 TR = (T*1.8)/TCMP

NUM = (2901.01) -{(5738.92)*TR) +( (2849.85)*TR*TR)

1 + ((1.74127)/ (1.01 - TR)) NUM = NUM * (TR - (1.0))

IF(NUM.LT.(-180.)) DD = 0 IF(NUM.LT.(-180.)) GO TO 11 DD = EXP{NUM)

THE STATEMENT ABOVE MAY RESULT IN AN UNDERFLOW SENSE LIGHT

ON SOME OPERATING SYSTEMS WHEN THE NUMBER NUM IS A LARGE NEGATIVE NUMBER. THE LARGE NEGATIVE VALUE IS PROPER AND

THE CORRECT ANSWER FOR DD IS ZERO.

11 TCMP = TMM + ((TMC-TMM)*DD)

TEST = (1.0) - (TCMP/TPO)

TEST = ABS(TEST)

IF(TEST.LT.( .0001)) GO TO 12 ICT = ICT + 1

IF(ICT.GT.250) GO TPO = TCMP

GO TO 10

12 TMC = TCMP/(1.8)

RETURN

END SUBROUTINE BLOCK

TYPE REAL KIJ.AJI DIMENSION KIJ(8,8

COMMON /PAR/ KIJ, DATA KIJ/

*0. 000000,. 293E-07

*0. 001000, 1.088608

*0. 001000, 0.001000

*.188E-10, 0.001000

*.546E-06,.136E-05

*. 9690485, 0.000000

*1. 016231,. 9976706

1.000000,1.000000

*0. 995000. 1.000000

0.000000/

DATA AJI/

*0. 000000,. 161E-04

0.001000, 1.014967

0.001000,0.001000

.188E-10, 0.001000

.105E-04,.119E-05

.9987221,0.000000

1.005949,1.001225

1.000000,1.000000

0.995000,1.000000

0.000000/

END

TO 12

) ,AJI(8,81

AJI

,.911E-06.

,0.000000,

,1.098880

,0.001000

,0.001000

,.541E-06

,.9435714

,1.000000

,1.000000

,.0030498

,.188E-10

,1.078910

,.8578232

,0.001000

,0.001000

,0.000000

,1.000000

,1.000000

, .257E-06

,0.000000

,1.008061

,0.001000

,0.001000

,.119E-05

,.9968984

.1.000000

,1.000000

,.0003462

,.188E-10

,1.004105

,.9863463

,0.001000

,0.001000

,0.000000

.1.000000

.1.000000

.0.001000

,.463E-06

.0.000000

,1.146026

,0.995000

,0.001000

,0.001000

,1.000000

.1.000000

.0.001000

,.197E-06

,0.000000

,1.011399

,0.995000

,0.001000

,0.001000

,1.000000

,1.000000

,0.001000

,.462E-08

,.517E-06

,1.009559

,1.287762

,1.178993

,0.001000

,0.000000

,1.000000

,0.001000

,.184E-05

,.142E-06

,1.001862

,1.024894

..9566020

,0.001000

,0.000000

,1.000000

,0.001000,

..120E-03,

,.535E-06,

,0.000000,

..9868486,

,1.266327,

,0.950000.

,0.001000,

,1.000000,

,0.001000,

,.335E-06,

,.103E-04,

,0.000000,

,1.003289,

,1.025269,

,0.995000,

,0.001000,

,1.000000,

LNG1261

LNG1262

LNG1263

LNG1264

LNG1265

LNG1266

LNG1267

LNG1268

LNG1269

LNG1270

LNG1271

LNG1272

LNG1273

LNG1274

LNG1275

LNG1276

LNG1277

LNG1278

LNG1279

LNG1280

LNG1281

LNG1282

LNG1283

LNG1284

LNG1285

LNG1286

LNG1287

LNG1288

LNG1289

LNG1290

LNG1291

LNG1292

LNG1293

LNG1294

LNG1295

LNG1296

LNG1297

LNG1298

LNG1299

LNG1300

LNG1301

LNG1302

LNG1303

LNG1304

LNG1305

LNG1306

LNG1307

LNG1308

LNG1309

LNG1310

LNG1311

LNG1312

LNG1313

LNG1314

LNG1315

LNG1316

LNG1317

LNG1318

LNG1319

LNG1320

LNG1321

LNG1322

76

204
6.12 Reference [35]

A-246

J. Chem. Thermodynamics 1982, 14, 837-854

Mathematical models for the

prediction of liquefied-natural-gas

densities^

R. D. Mccarty

Thermophysical Properties Division, National Engineering Laboratory,

National Bureau of Standards, Boulder. Colorado 80303. U.S.A.

(Received 27 October 1980; in revised form 25 January 1982)

Three mathematical models of the equation of state for liquid mixtures simulating liquefied natural gas (LNG) are discussed and compared. The adjustable parameters for each model have

been optimized using the same set of experimental data, consisting of over 280 new (p, V. T. x) points taken at the National Bureau of Standards in Boulder. Colorado. It is estimated that each

of the models will predict LNG densities over its range of validity to within O.I per cent of the true values, given the pressure, temperature, and compyosition of the mixture. Deviation plots and a detailed performance evaluation are given for each model. The range of validity varies

slightly among the models but in general the range of the study included the saturated liquid from 90 to 135 K.

1. Introduction

A project was begun at the National Bureau of Standards in Boulder, Colorado in the summer of 1972, with an ultimate goal of obtaining one or more models of the equation of state of liquefied-natural-gas-like (LNG-like) mixtures which could be

used to predict the density of any LNG mixture with a total uncertainty of less than

0.1 per cent given the pressure, temperature, and composition of the liquid. LNG-like

mixtures for the purposes of this paper are defined as mixtures of N2, CH4, CjHft,

CjHg, «-C4Hio, and 1-C4H10 with mole fractions of N2, A1-C4H10, and i-C4H,o

restricted to 0.05 or less. The project was sponsored by a consortium of 18 international energy companies through a grant administered by the American Gas

Association, with the hope of providing a means to an equitable custody transfer of

large quantities of LNG. A series of papers reporting the results of experimental

measurements performed during this project have been previously published in this Journal and the present paper is a part of that series.*'"*' The purpose of this paper is to report an analysis of three models which are believed to have achieved the 0.1 per

cent total-uncertainty goal.

No equation of state can be more accurate than experimental data. Therefore,

experimental (p, V, T, x) values of comparable or greater accuracy and of adequate

scope were a necessary condition to the success of the project. At the time when this

° This work is a contribution of the National Bureau of Standards, not subject to copyright.

205
838 R. D. Mccarty

project was initiated a survey of the existing values revealed that such a set did not

exist, and a major experimental program was initiated. Over the next six years comprehensive (p, K T, x) results for the saturated liquid state of the major

components (Nj, CH4, CjH^, CjHg, j-C4Hio, M-C4H10) of LNG as well as their binary and multicomponent mixtures were obtained. These results are reported

elsewhere,""^' and will not be repeated here. However, a synopsis of the experimental

values is given in a later section.

2. Selection of models

A preliminary investigation of the existing equations of state of a mixture, revealed a multitude of possibilities from which to choose. Rather than relying on a single model

it was decided to select several which would represent the various categories of

models found in the literature, thus enhancing the possibility of achieving the 0.1 per

cent accuracy goal of the project.

The choices were made on the basis of past performance for other fluids, and

likelihood of success. An efibrt was also made to choose one model which would be

representative of a general category of approaches which could be identified as being

basically different from each other. On these bases three models were selected : the extended corresponding-states model*^' which, while it is complex, allows a great deal

of flexibility; a hard-sphere model*^* which has demonstrated capabilities; and the Klosek and McKinley*'°' model which is very simple in principle and totally

empirical. From time to time as the project continued other methods and models

were suggested by co-workers and the sponsoring companies but only two of these were investigated in any detail: (a) Watson's*' method, investigated to the point

where it became obvious that it was unsuitable without some major modifications;

(b) Hiza's*'^' new empirical method, found to work well for restricted ranges of

composition and temperature.

3. Data used

The experimental densities for saturated liquid of the pure components of LNG given

in references 1 and 7 have been used here both as data for optimization procedures

and as pure-fluid data for calculating the molar volumes of the components (see

equations 19 and 21). Table 1 presents these data in the form of coefficients to an

equation for the saturated liquid density and table 2 summarizes the data used from

references 1, 2, 3, and 7. References 5 and 6 contain 110 (p, V, T, x) points for a total of 27 different multicomponent mixtures which, with one exception, were used for comparison

purposes only. The data for the ternary system: (0.05931N2 -I- O.8907ICH4 +

0.04998«-C4Hjo) from reference 6 were used to obtain the N2-n-C4H,o binary

interaction parameters, which are also used for the N2-/-C4H10 binary interaction

parameters given in sections 4 and 5. No other (p, K T, x) values have been used in the development of the mathematical models in the following sections. The use of values from a single source was the intent from the beginning of the

206
MATHEMATICAL MODELS FOR THE PREDICTION OF LNG DENSITIES 839

TABLE 1. Saturated-liquid densities of the pure fluids

3 (p-p,)/(moldm-') = a(l-r/r,)°" -(- ^ b,{l-T/Ty

where and are the critical temfjerature and amount-of-substance density

CH4 C^H, C3H8 i-C4H,o

a^

TJK

pJ{mo\dm'^)

M/lgmoi" ')°

18.65812 6.712030 -G.9472020 0

190.555

10.16 16.04303

12.55205 13.43284 - 19.00461 11.07716

305.33

6.86 30.07012

8.684459

18.04086 -29.46261

16.43559

369.82

5.00 44.09721

7.657535

8.145251 - 13.10582 8.145894

408.13

3.80 58.1243

"-QHio '-C5H,/ n-CiHii'

a

byb2

b,TJK

^^/(mol dm"-')

M/(gmol"')''

7.286063

1 1 ,96308 - 19.87592

11.60211 425.16

3.92 58.1243

19.39217 26.01408 -39.49759

23.32977

126.20

11.21 28.0134

2.946310456

35.50770979 -57.41242943

28.15898339

460.39

3.271 72.15139

-0.0362004993

59.0020299 -93.44193819

43.66780833

469.6

3.285 72.15139

" TTie project under which the work reported here was done started in 1972. The molar masses M were

calculated using the values for C, H. and N accepted at that time. The values for H and C were changed

slightly by the 1976 Commission on Atomic Weights. For the sake of consistency we have continued to use

the previous values. The maximum error in density of any of the fluids resulting from using the newer

values is 0.003 per cent.

^ Towards the end of the project, f-pentane and ^-pentane were added to the list of possible components

in the LNG-like mixtures. No experimental (p. V. T) values for these two pure fluids were taken as part of this project; the constants in table 1 for the saturated amount-of-substance densities of these fluids were

derived from the results of McClune.'^*'

project, the purpose being to eliminate systematic differences which almost always

occur when values from more than one source are used. The development of the mathematical models was done simultaneously with the acquisition of the experimental values; this procedure allowed almost immediate consistency checks

between the experimental results for various different compositions of a binary

mixture as well as for comparisons with data from other sources.*'°'^^ '^~*^' These

comparisons verified, in many cases, the anticipated systematic differences in multisource data.

4. Extended corresponding states A theoretical development of the thermodynamic equations for the extended

corresponding states method may be found in a paper by Rowlinson and Watson''^'

and only those equations necessary for this particular application will be given here. Other important work on this method has been done by Leach'^' in developing

models for the shape factors and by MoUerup"^' and Mollerup and Rowlinson'^''

207
840 R. D. McCARTY

TABLE 2. Experimental measurements used, their temperature ranges T, to Tj, and the ranges of the mole

fraction x of the most volatile component in binary mixtures

Fluid or bmary mixture Reference /,/K X No. of points

/ 120 pure fluid 19

LI L H4 11 IUj 160 pure fluid 11

1 lUU 270 pure fluid 22 CjHs 1

1 Art 100 288 pure fluid 16

1 300 pure fluid 12

; LI I-L4M10 2 113 300 pure fluid 12 28 148 249 pure fluid 21 '-C3H,, 28 125 254 pure fluid 20

.XN2 + (1 -X)CH4 2 105 140 0.05 to 0.5 21 2 105 120 0.06 4 xN, + (l-x)C3Hg 2 105 115 0.02 to 0.07 6 xCH4 + (l-x)C,H, 2 105 140 0.35 to 0.68 20 xCH4 + (l -xjCjHg 2 105 130 0.3 to 0.85 20 xCH4 + (l-x)i-C4H,o 2, 3 110 140 0.48 to 0.92 17 xCH4 + (l -x)n-C4H,o 2. 3" 115 140 0.58 to 0.93 23 xC,H, + (l-x)C,H8 2 105 140 0.50 to 0.67 10 xC2H, + (l-x)/-C4H,o 2 105 130 0.69 to 0.72 6 xC,H^ + (l-x)^-C4H,o 2 110 140 0.65 to 0.67 8 xCjH« + (l-x)i-C4H,o 2 105 130 0.49 to 0.50 8 xCjH, + (l-x)n-C4H,o 2 110 150 0.58 to 0.61 8 xi-C4H,o + (l -x)n-C4H,o 2 125 140 0.47 4

" The results for (0.91674CH4 + 0.08326n-C4H,o) from reference 3 were not used because they appear to be inconsistent with later results'*' for nearly the same composition.

who combined the work of Leach'**' with an equation of state for methane by

Goodwin*^*" to produce a model for LNG-hke mixtures. The same general approach

as used by Mollerup'^^' is used here except that the shape-factor functions of Leach'^*

have been refitted to the pure fluid values experimentally obtained at this laboratory on this project**'^' and an equation of state by McCarty'^'' has been used for methane. Interim results of this study can be found in references 21 and 22. The

differences between the interim results mentioned above and the final results, given

here, are minor and the previously published equations may be used. More details on

the performance of the extended-corresponding-states method are given later in this section. The extended-corresponding-states method is defined by the equations:

z,(p,r) = Zo(p/i.,o/A.o.7-/y;,oK d)

G,(p, T) =/i,,oGo(p^,.o/yi-,.o^ T/yi,,o)-«^ln(Vo). (2) where Z is the compression factor, G is the Gibbs free energy, p is pressure, and T is thermodynamic temperature. The subscript 0 denotes a reference fluid (in this case methane) and the subscript ^ denotes the fluid for which properties are to be obtained

via the equation of state for the reference fluid and the transformation variables

0

and /i„ 0- The double subscript is introduced now to facilitate the extension later to mixtures. Two equations, (1) and (2), are necessary to define uniquely the

2Ub
MATHEMATICAL MODELS FOR THE PREDICTION OF LNG DENSITIES 841

transformation variables o and ''/..o- The equation of state for methane by

McCarty*^^' was constructed using a hypothetical saturation line for both liquid and

vapor down to 43 K. This construction is necessary to represent realistically the very low reduced temperatures of the heavier hydrocarbons and was accomplished

without changing the performance of the equation of state above the triple-point

temperature of methane. The equation of state for the reference fluid is

p = pRT + pHNiT+ N2T"^ + N,+NJT+ N^/T^) + p\N(,T + N,+Ns/T + Ng/T^)

+ ANioT+ N,,+N,,/T) + pHN,,) + p^(N^JT + N,JT') + p'(N,JT)

+ p'(N^,/T + N,s/T') + p^(N,^/T') + pHN,o/T' + N,JT')cxp{-yp')

+ pHN22/T'+N,,/T^)exp(-yp') + p'{N,JT' + N,s/T')exp(-yp')

+ p^{N2JT' + N2,/r')exp(-yp') + p''{N2s/T'+N2<,/T')exp(-yp') + p''{N,o/T'+N,JT'+N,2/T*)exp{-yp'l (3) where p is pressure, T is thermodynamic temperature, and p is amount-of-substance

density. The coefficients N, are given in table 3.

TABLE 3. Coefficients N-, of equation (4) for CH4''

R 0.831434X 10"' MPa dm' mol"' K"',

N, -0.189506118654 x 10"' dm^ mor^ K"',

-0.157446511658 X 10' dm*mor^ N4

-0.382099919632 X 10* dm^ mor' K\

N7 -0.502993394999 X 10"' dm' mol"'. -0.327098978594 X 10* dm'mor^K^ N.o

0.270307051249 x 10"' dm'^ mor*. N.2

N.3 0.194123002261 x 10"* dm'5 mor'. A/l4 Nl5 0.615529077453 x 10 dm'« mor* K\ N.6

N.7 0.154281787794 X 10"' dm'* mor« K, N.8

0.193931540966 X 10"* dm'" mor' K\ N20

N21 -0.160023162832 X 10" dm' mor^ K\ N22

^23 0.168195529711 x 10" dm'^ mor' K*. ^24

0.381473591215 x 10' dm" mo!"' K^ N26

N27 -0.348122335444 X 10' drn'^ mol"' K*, N28

0.188081779935 x 10"' dm" mor" K\ ^30

N3, -0.290669323872 X 10"" dm" mor" K\ N32

0.0096 dm^ mol"

0.104756987190 X 0.782544605655 x 0.858039193151 x 0.881435651333 X -0.401087946417

-0.308038191105 -0.198179473576

-0.536626852203 -0.111409049265

0.391590731296 X 0.197857473814 X 0.611041574405 x 0.127257796890 X -0.548112616636

0.780999906720 x 0.377310956389 X

10 dm" mor' K-'",

10' dm" mor' K,

10"* dm' mor^ K-',

10' dm" mol-^ K.

X 10 ' dm" mol-* KX 10 dm" mol * K,

X 10"* dm"* moI-" K,

X 10"* dm" mor'-K,

X 10"' dm'* mol-« K'.

10* dm« moj-' K',

10' dm" mol-' K',

10"' dm" mor' K',

10"^ dm'" moI-' K',

X 10"" dm" mol"" K'

10-' dm" mol-'^ K'. 10"* dm^ mor'^ K*.

° An existing wide-range equation of state for CH4 was used for the base fluid to facilitate the extension

of this work to a wider range of pressures and temperatures in the future. The large number of digits in the

coefficients given here is necessary to insure the validity of the calculated pressure in some areas of the

(p, T) surface.

The fa 0 and /j,i,o are then defined as

^..o = (^^/.o/l^o')0...o(77,K^K (5)

where the superscript " denotes the critical property and 7^' = 7/7^- and Vi = F/l^-- The 0„(7;\ V,') and 0,..o(T7, V^) are called shape factors, where

209
842 R D. McCARTY

and

(/),,o = (Zyzn[l+K-H'o);6,(l/'-b,)-M^1'-/'4)ln77l], (7)

where the Tl and V- are as before and w, is a fluid-dependent parameter.

The d/s and b-s are fluid-independent parameters and Zq/Z- is the ratio of the compression factors at the critical point. The a/s. i»/s, and Wi's were estimated by

least squares using the experimental data from (1) and (2) and simultaneous fitting

techniques. The parameters a^. bj, h^. and the critical properties are given in table 4.

TABLE 4. Coefficients for corresponding-states equations (6) and (7)

t;, = -0.109495. = 0.919454. = -4.01525. = -4.14192, = 0.444850

6, = 0..^56808. b. = 1.02619. = 0.893323. b^ = 0.761533

w

7; M

Fluid no. MPa K cm^ mol"' g mol

' '

0.029179 3.3542557 126.2 89.827 28.0134 6 CH4 0.0109 4.59570 190.555 98.522 16.04303 1 0.110427 4.860314 .^05.5 146.2 .30.07012 2 C,H« 0.154837 4.2445123 370 200 44.09721 3 0.176372 3.8295398 425 251.62 58.1243 4 '-QHio 0.150115 3.688998 408.1 263 58.1243 5 n-C.H,, 0.234320 3.3812152 469.6 304 72.15139 7 '-C5H,, 0.288886 3.1988302 460.39 306 72.15139 8

" The large number of significant figures given for critical pressure is necessary to reproduce the

(Zj = p^V^lRT^) in the least-squares tit.

Equations (6) and (7) were originally proposed by Leach'*'*' with h', as the acentric

factor, which allows the extension of the equations to any fluid with a minimum of

information about that fluid. Here, u', was treated as an adjustable parameter to provide for the greatest possible accuracy in the limited range of pressure and

temperature of interest. Yhe extension of the above to mixtures is accomplished through the equations:

/'U.0 = '/u(i''/i'o + i/'];:o)'. (9)

''x.o = Z,I,^.-^j''u.O' (10)

L.o^h.o = Z, Z, •^i-^j/v.o''.j.O' (11)

where .x, and are the mole fractions of the pure components.

The e,j and the r}^^ are binary interaction parameters determined by least squares from (p, V, T, x) values for binary mixtures as outlined in section 3. The 0 '^^'^ ''x.o are now the transformation variables to transform the equation of state of the reference fluid to an equation of state for the mixture specified by p, 7, and x. The ^,/s

and £,,'s for the corresponding-states method are given in table 5. Of the three models presented here, the extended-corresponding-states model is superior to the other two in most respects. It covers the widest range of composition.

210
MATHEMATICAL MODELS FOR THE PREDICTION OF LNG DENSITIES 843

TABLE 5. Binary interaction coefficients for corresponding states equations (8) and (9). Fluid numbers / are given in table 4

i.j \ 2 3 4 5 6 7 8

1 1 1.00514 1.01922 1.04243 1.05048 1.01009 1.06 1.06 2 1 1.00599 1.01616 1.02369 1.02127 1.02 1.02 3 1 1.00172 1.01140 1.04606 1.01 1,01 4 1 0.997114 1.20955 1,0 1.0 5

' 1 1,13889 1,0 1.0 6 1 1.0 1.0 7 1 1.0

8 1

i.j \ 1 3 4 5 6 7 8

1 1 1.01127 0.988608 0.983130 0.986978 0.953430 0.98 0,98 2 1 0.999961 0.972223 0.998886 0.939622 0.99 0.99

3 1 0.985547 1.03099 0.912209 0.99 0.99 4 1 0.976416 0.849200 0,99 0,99 5 1 0,857310 0.99 0.99 6 1 0,99 0.99

7 1 0.99

8 I

the widest range of pressure and temperature, and is probably the most accurate,

although the last is hard to substantiate. Of the approximately 285 experimental

[p. V. T. x) points reported in references 2 to 6, only 14 differ by more than 0.1 per

cent in density from the value predicted by this extended-corresponding-states model.

Of these 14 points 11 (see below) are judged to have an experimental uncertainty

greater than the 0.1 per cent. Figure 1 presents the deviations between calculated and

exf>erimental densities for these 14 points. Deviations between predicted and

experimental densities of the other 271 (p, K 7, .x) points are published in reference 23. No restrictions as to pressure, temperature, or composition (within the data from references 2 to 6) need be placed on the model. The other two models have

pressure-, temperature-, and composition-range restrictions. In preliminary accounts

(references 21 and 22), some reservations were expressed about the accuracy of the calculation methods because of the disagreement between experimental and

calculated densities for a few binary and multicomponent system which contained

methane and butanes as components. This disagreement has since been resolved by more experimental measurements which have been made'*-^' on mixtures of similar

composition and in some cases the same composition (more measurements on the same sample). As a result of the additional measurements'*-^' some of the unpublished multicomponent mixture results for systems containing methane and

butanes were discarded,'^' and some of the already published binary systems

containing (methane -I- n-butane) were judged to be uncertain by more than the 0.1 per cent in density. The published results'^ *' which are believed to be uncertain by more than 0.1 per cent are detailed in table 2. As a result of the additional values

211
844 R. D. Mccarty

+ +

8 + 9

+ V

+

0

8 A A

-0.2

100 110 120 r/K

130 140

FIGURE 1, Relative deviations exceeding 0.1 per cent between densities calculated by the extendedcorresponding-states method and all of the (p, K 7, x) mixture values taken in this project.'^'*' +, 0.91674CH4 -I- 0.8326w-C4H,o: , 0.06740N2 + 0.93260C,He; A. 0.0O859N2 -)- 0.75713CH4 +

O.lBSSSC^He -I- 0.06742CjH8 + 0.01326n-C4H,o + 0.01336i-C4H,o + 0.00216f7-C5H, ^ -I- 0.00223/- C'sH.j; O. 0.85442CH4 + 0.05042C2H6 O.O4038C3H8 -I- 0.02901n-C4H,o -I- 0.02577i-C4H,o ; V.

O.7492OCH4 -I- O.25O8OC3H8.

obtained in the experimental program, subsequent to the models published in references 21 and 22, very slight changes were made in the binary interaction

parameters: CH4-/-C4H,o, CH^-n-C^HiQ, N2-1-C4H10. and N^-n-C^Hio- These

changes have a negligible effect on density calculations for LNG-like mixtures, i.e. where the mole fractions of N2, i-C4H,o. and n-C4H,o are individually less than 0.05. Therefore, either the models presented here or those in references 21 and 22 may be

used to predict the density of an LNG mixture to within 0.1 per cent of its true value. The accuracy claim of 0.1 per cent in density must be qualified by specifying ranges

of pressure and temperature as well as components. For each of the three models the

qualifications will be different; in the case of the extended-corresponding-states

model the temperature range is 95 to 135 K for saturated liquid pressures. The

component restrictions are minimal, i.e. any combination of mole fractions of Nj,

CH4, CjH^, CjHg, «-C4H,o, and i-C4Hio, as long as liquid-liquid phase separation

does not occur. The total mole fraction of «-C5H,2 and /-C5H12 must not be more

than 0.02. The above restrictions are based on the performance of the model when

compared to the set of results from this laboratory Comparisons with other

sources have been made.'^^*'^^^^' With a few exceptions the comparisons have

revealed our values to be internally consistent to better than the 0.1 per cent accuracy claim given here. Rodosevich's values**^' are in agreement with ours to within 0.1 per cent in density but do not extend the range of pressure and temperature

upon which the extended-corresponding-states model is based. A comparison of

calculated and experimental densities from Rodosevich*'^' and Rodosevich and

Miller'^' is given in figure 2. Orrit and Laupretre'^"*' have published (p. V. T. x) values

for LNG-like mixtures which agree reasonably well with this model as shown in figure 3, but again contain no points which would expand the scope of the set used

here. Nunes et al.^^^^ have measured (p, V, T, x) for (methane -I- nitrogen) at low

temperatures in the compressed liquid to several tens of megapascals. These results

212
MATHEMATICAL MODELS FOR THE PREDICTION OF LNG DENSITIES 845

I

0.08

^ 0.04 -

-0.04 -

T/K

FIGURE 2. Typical relative deviations between densities calculated by the extended-correspwndingstates method and experimental densities from Rodosevich"*' and Rodosevich and Miller."' O. O.92O6CH4 + 0.0794«-C4H,o; A. 0.9262CH4 + 0.0738CjHg; . 0.9737CH4 + 0.0263C,Hs; V.

O.92O6CH4 + 0.0794n-C4H,o; . 0.9462CH4 + 0.0538i-C4H,o; 0.8409CH4 + O.IO86C2H6 +

0.0505N,; A. 0.8303CH4 + O.lOOlCjHe + O.O493C3H8 -(- 0.0203n-C4H,o.

0.2

8

-0.2

1 1 On ^

1 1 1 CQ On

0 0 A OA A ^ A

^ 0

1 1

0

1 1 I 100 110 120 T/K. FIGURE 3. Typical relative deviations between densities calculated by the extended-correspondingstates method and experimental densities from Orrit and Laupretre.*^*' V. O.OOQNj -I- O.95CH4 -1- O.026C2H6 + O.OlSCjHg; , O.OO9N2 + O.9OCH4 + O.053C2H6 + O.O28C3H8 -t- 0.012n-C4H,o; A,

O.OO9N2 + O.93CH4 + O.O26C2H6 -(- O.OI5C3H8 -I- 0.012n-C4H,o + 0.008i-C4H,o; O. 0.006Nj +

O.89CH4 -I- O.OSeC^H^, + O.O24C3H8 + 0.012n-C4H,o -I- 0.012i-C4H,o -I- O.OOSi-CsH.j; O. O.OIN^ -1- O.7OCH4 + O.I6C2H6 + O.O9C3H8 + 0.029«-C4H,o + 0.014i-C4H,o -H O.OOOSn-CsH,, + O.OOlIi- CjHij

agree very well with the extended-corresponding-states model with the exception of one isotherm (see figure 4) which again suggests some internal systematic differences. Finally Straty and Diller'"' have measured several hundred (p, F, 7, x) points for (methane -i- nitrogen) over a wide range of p, V, and T including compressed liquid

and gas. Figure 5 is a summary of the agreement of the corresponding-states model

discussed here with these results. Figure 5 demonstrates that the model is well behaved and reasonable when extrapolated beyond the range of the present data set. but that the extrapolation is limited by the range of p and T of the equation of state

213
846 R. D. Mccarty

0.4

S 0.2

:^ 0

-0.2

T= 110.09 K r= 116.3 K

200 300 p/MPa

FIGURE 4. Typical relative deviations between densities calculated by the extended-correspondingstates method and experimental densities from Nunes et a/.'^*' for (0.497CH4 + 0.503N2).

.(b)

/7~12.6 mol-dm

t 170 180 190 200 210 220 T/K

240 260 280 T/K. 300

-0.06

-0.08

-0.10

-0.12

-0.14

(c) .

'

. i>~25.9 moi'dm

110 120 T/K

130

FIGURE 5. Typical relative deviations between densities calculated by the extended-correspondingstates method and experimental densities from Straty and Diller'^"" for (0.501 15Nj -i- 0.49884CH4).

for the base fluid, i.e. 90 to 350 K with pressures to 35 MPa. No comparisons with or

calculations of the thermodynamic properties of mixtures have been made with any of

the models presented here and calculations of this sort are not recommended because

of the extreme emphasis placed on (p, K T, x) and the complete absence of

consideration of other thermodynamic properties during the development of the models. Table 6 contains amount-of-substance densities for purposes of checking the method.

TABLE 6. Values for checking calculations using corresponding-states equations. These are included for check purposes only; they are calculated from the corresponding-states model and are not experimental

values 0.6975CH^ -l-0.156CjHe O.O92C3H8 0.029n-C4H,o -h 0.014(-C4H,o -(-O.OllSNj

T/K /)/(mol dm ') p/MPa

95 24.333 0,1013250

100 24.067 0.1114575

105 23.796 0.1215900

214
MATHEMATICAL MODELS FOR THE PREDICTION OF LNG DENSITIES 847

5. Hard-sphere model

The hard-sphere model presented here is a refit of the model proposed by Rodosevich

and Miller'^' which is one of many modifications of the Longuet-Higgins and Widom

model,'^^* and was chosen to be included in this study as a representative example of

the application of the hard-sphere equation-of-state concept to the correlation of

(p, V,T,x) data. The equation of state from reference 9 is pV/RT=c(l+ y + y^)/{l-y)^-a/RTV, (12)

where a, b, and c are adjustable parameters and y = fe/4K The equation is applied to mixtures by assuming the one-fluid theory and applying the mixing rules:

flm = (13)

<^n, = I,I/oXiX^- (15)

The combining rules are

:

b,j=m!' + b]i')(\-j,^,\ (16)

a„=(fl„fl,,)''^(65/^.^>)"'(l-^.,K (17)

c.,= (c, + c^,)/2. (18)

The parameters j.^ and A:,j are now the binary interaction parameters. The as, b\. c's.

y,/s, and kijS are given in tables 7 and 8. The excess volume is calculated using the equation of state and

y^ = K-lM (19)

where and the V, are calculated via the equations (12) through (18) and then

K„ = I,^^x, + F^ (20)

where the is from equation (19) and the are experimental. The in this case were calculated from the equations for the liquid density of the pure fluids given in table 1. The above equations are the same as they originally appeared in Rodosevich and

Miller'^* and only the ji-s and /c./s have been revised on the basis of the (p, K T, x)

TABLE 7. Coefficients for the hard-sphere model of equation (12): h = 2nLSf/3 where L denotes Avogadro's constant and the Si's are the pure-fluid coordination numbers

Fluid fly(dm*mor^•MPa) Sy(dmmor c, Fluid no. 1

N, 1.718X 10' 3.546 X 10-' 1.03 6 CH, 2.755 X 10' 3.676 X 10"' 1 1 CjH, 7.773 X 10' 4.158 X 10"' 1.5 2 CjHb 14.165 X 10' 4.644 X 10"* 1.67 3 n-C^H.o 22.733 X 10' 5.051 X 10"' 1.83 4

<-C4H,o 21.279 X 10' 5.056 X 10"* 1.79 5

^15
848 R D. McCARTY

TABLE 8. Binary interaction parameters for the hard-sphere model of equations (16) and (17). See table 4

for identification of i and )

hi ij 1 2 3 4 5 6 7 8

I 0 -0.00388616 -0.0120932 -0.0231577 -0.0238349 - 0.00997547 -0.0326 0.0458 2 0 -0.002162 -0.00400910 -0.00812712 -0.0143976 -0.003 -0.004

3 0 0.000761571 -0.00383743 -0.024014 0 0

4 0 0.00222 ISO -0.0576043 0 0

5 0 -0.0576043 0 0

6 0 -0.04 -0.05

7 0 0

8 0

i-j 1 2 3 4 5 6 7 8

I 0 0.00298830 0.0597378 0.110893 0.100298 0.0197290 0.14 0.1745 2 0 0.014527 0.067703 0.0346632 0.0529034 0.02 0.03

3 0 0.024291 -0.00838212 0.14719 0 0

4 0 0.0199213 0.154365 0 0

5 0 0.154365 0 0

6 0 0.15 0.18

7 0

8

binary-mixture set as outlined in section 3. An attempt was made to optimize the a, b, and c parameters for each pure fluid on the basis of the new pure-fluid values;

however, the results were disappointing, and no perceptible improvement in the prediction of mixture prop»erties was achieved. The results did suggest, however, that the introduction of a temperature dependence to the a, b. and c parameters would

improve the performance of the equation of state for pure fluids. This method has one disadvantage not present in the extended-correspondingstates method which causes the method to be restricted to temperatures below 120 K

for mixtures which contain Nj. As the method is applied here, it is an excess-volume

method, and therefore when the temperature of the mixture approaches the critical temperature of one of the component fluids the method fails. Since the critical temperature of is about 126 K, calculations for mixtures containing N2 should not

be made with this method for temperatures above 120 K. This added restriction on

the method reduces the original set of 285 experimental (p, K 7, .x) points'^"''' to 251

.

Figure 6 shows all of the points within the new set of 251 for which calculated and

experimental densities differ by more than 0.1 per cent. Conclusions may be drawn

from the deviations shown in figure 6. First, even with a reduced total number of

points for comparison, the total number of deviations between calculated and

experimental densities for the hard-sphere model which exceed the 0.1 per cent

criterion, is far more than for the extended-corresponding-states model. Second, the method degrades for all mixtures regardless of components as the temperature

becomes greater than 115 K.

216
MATHEMATICAL MODELS FOR THE PREDICTION OF LNG DENSITIES 849

0.4

1 1 1 1

0

r, V „ - » 8

• •

1

1

1 1 100 110 120 T/K

130 140

FIGURE 6. Relative deviations exceeding 0.1 per cent between experimental densities and densities calculated by the hard-sphere method. The comparison set is about 250 (p, V, T. x) points from references 2 to 6. a. 0.92780CH4 + 0.07220n-C«H,o : V. 0.91674CH4 + 0.08326n-C*H,o ; C- 0.77982CH4 +

0.2201 8n-C4H,o; 0.77762CH4 + 0.22238n-C4H,o; O.58828CH4 + 0.41 172fl-C4H,o : O,

0.30349Nj + 0.69651CH4; •. 0.49242N2 + O.50758CH4; V, 0.06740N2 + 0.93260C3Hg; .

O.34242CH4 -I- O.31372C2H6 -I- 0.34386C3H„; 9. 0.33800N2 + 0.34140CH4 + 0.32060CjH6; ,

0.85317CH4 + 0.05077C2He -I- 0,04855C3H8 + 0.0475 ln-C4H,o; A. O.84566CH4 + 0.07924CjH6 +

O.O5O6OC3H8 + 0.02450n-C4H,o; A,0.85133CH4 + 0.05759C2H<, + O.O48O8C3H8 + 0.04300n-C4H,o

:

-l-,0.85442CH4 + O.O5042C2H6 + 0.04038C3He + 0.02901n-C4H,o + 0.02577i-C4H,o ; <>.0.05540N2

-I- O.7909OCH4 + O.O56OOC2H6 + O.O5OOOC3H8 + 0.04770n-C4H,o; »,0.04250N2 + O.8I3OOCH4 +

0.04750CjH6 + O.O487OC3H8 + 0.02420n-C4H,o + 0.02410i-C4H,o.

Figure 7 shows deviations greater than 0.1 per cent between experimental densities and those calculated by the hard-sphere method for all mixtures in the 251 -point

comparison set which are LNG-like, i.e. all multicomponent mixtures with 60 moles

per cent or more of CH4 but with 5 moles per cent or less of each of Nj, j-C4Hio, and

n-C^HiQ. The points above 120 K for those mixtures containing have been

FIGURE 7. Relative deviations exceeding 0.1 per cent between experimental densities and densities calculated by the hard-sphere method. The comparison set is about 60 (p. V. T, x) points for LNG-like

mixtures from references 4, 5, and 7. A. O.84566CH4 -1- 0.07924C2Ht -1- O.O5O6OC3H8 + 0.02450n- C4H,o; A. 0.85133CH4 -t- 0.05759C2Hfc + 0.04808CjH6 + 0.04300n-C4H,o ; -t-, 0.85442CH4 -I- 0.05042C2H<, -(- O.O4038C3H8 + 0.02901n-C4H,o + 0.02577i-C4H,o ; .0.04250Nj + O.SBOOCH* +

0.04750CjH6 -I- 0.04870CjH8 + 0.02420n-C4H,o + 0.02410i-C4H,o217
850 R. D. McCARTY

excluded. This newly-defined comparison set now contains 60 (p, K T, x) points.

Excluding the 5 points (see above) which have been judged to be experimentally

uncertain by more than the 0.1 per cent criterion as explained in the section on the

extended-corresponding-states model, we are still left with 4 points outside the 0.1 per

cent tolerance which did not appear on the extended-corresp)onding-states deviation

plot. All of these four deviations occur at temperatures greater than 120 K. Therefore

the restriction of 120 K as an upper limit in temperature must be placed on the method regardless of the comp>osition.

6. A revised Klosek and McKinley method

The Klosek and McKinley method" is a totally empirical method and was included

in this study because of its simplicity and its widespread use in the LNG industry.

Very eady in the study it became apparent that the original form of the method would

not work for mixtures in which was present, even at mole fractions of 0.04 or less. The original method was proposed as

Kn,x = I,^.^-MCHJ (21)

where F^j, is the volume of the mixture, x, and V, are the mole fraction and molar

volume of component / and the summation is taken over all components including

CH4, x(CH4) is the mole fraction of CH4, and A: is a correction factor obtained from a

TABLE 9. Molar volumes V, of saturated liquid of the pure components at temperatures T at intervals close enough to allow linear interpolation

CH4 C3H, n-C«Hio <-C,H„

T/K mor '

)

90 0.035441 0.046081 0.060461 0.074708 0.076084 0.037543 0.089173 0.089243 92 0.035649 0.046235 0.060632 0.074891 0.076274 0.038081 0.089379 0.089454 94 0.035861 0.046390 0.060804 0.075075 0.076466 0.038650 0.089586 0.089666 9^ 0.036077 0.046547 0.060977 0.075259 0.076659 0.039254 0.089793 0.089878 98 0.036298 0.046704 0.061151 0.075445 0.076853 0.039897 0.090000 0.090091 100 0.036524 0.046863 0.061325 0.075631 0.077047 0.040586 0.090208 0.090304

102 0.036755 0.047023 0.061501 0.075818 0.077243 0.041327 0.090416 0.090518

104 0.036992 0.047185 0.061677 0.076006 0.077440 0.042128 0.090624 0.090733

106 0.037234 0.047348 0.061855 0.076194 0.077637 0.043002 0.090833 0.090948

108 0.037481 0.047512 0.062033 0.076384 0.077836 0.043963 0.091042 0.091163

110 0.037735 0.047678 0.062212 0.076574 0.078035 0.045031 0.091252 0.091379

112 0.037995 0.047845 0.062392 0.076765 0.078236 0.046231 0.091462 0.091596

114 0.038262 0.048014 0.062574 0.076957 0.078438 0.047602 0.091673 0.091814

116 0.038536 0.048184 0.062756 0.077150 0.078640 0.049179 0.091884 0.092032

118 0.038817 0.048356 0.062939 0.077344 0.078844 0.050885 0.092095 0.092251 120 0.039106 0.048529 0.063124 0.077539 0.079049 0.052714 0.092307 0.092470

122 0.0394O4 0.048704 0.063309 0.077734 0.079255 0.054679 0.092520 0.092690

124 0.039710 0.048881 0.063496 0.077931 0.079462 0.056797 0.092733 0.092911

126 0.040025 0.049059 0.063684 0.078128 0.079671 0.059085 0.092947 0.093133

128 0.040350 0.049239 0.063873 0.078327 0.079880 0.061565 0.093161 0.093355 130 0.040685 0.049421 0.064063 0.078526 0.080091 0.064263 0.093376 0.093578

218
MATHEMATICAL MODELS FOR THE PREDICTION OF LNG DENSITIES 851

table or graph. The and k are temperature dependent and in addition k is dependent upon the mean molar mass of the mixture.

The revised equation is

K„,x = I.^iK-{'^i+(^2-'^iWN2)/0.04251x(CHJ, (22)

where quantities are the same as in equation (21) except that a second correction

factor has been added and x(N2) is the mole fraction of N2. Tables 9, 10, and 1 1 give

values for the V^^k^, and A. 2, which are spaced so that linear interpolation is adeqate in both variables {i.e. temperature and molar mass).

The pure-fluid volumes in table 9 are taken from the equations in section 3 with the exception of that of Nj above 115 K which is a linear extrapolation of the 115 K

point using the slope of the curve at that point. The k factors in tables 10 and 1 1 have

been obtained graphically from the multicomponent (p, F, 7, x) data of references 4

and 6 and from multicomponent densities calculated using the extendedcorresponding-states method.

TABLE 10. Correction factor for the Klosck and McKiniey method as a function of temperature T and

molar mass M

M/(gmol ') 16 17 18 19 20 21 22 23 24 25 7/K 10^fc,/(cm' mor' )

90 -0.005 0.120 0.220 0.340 0.430 0.515 0.595 0.660 0.725 0,795 95 -0.006 0.135 0.260 0.380 0.500 0.590 0.665 0.740 0.810 0,885 100 -0.007 0.150 0.300 0.425 0,575 0.675 0.755 0.830 0.910 0,990

105 -0.007 0,165 0.340 0.475 0.635 0.735 0.840 0.920 1.045 1,120 110 -0.008 0.190 0.375 0.535 0.725 0.835 0.950 1.055 1,155 1.245 115 -0.009 0.220 0.440 0.610 0.810 0.945 1.065 1.180 1,280 1.380 120 -0.010 0.250 0.500 0.695 0.920 1.055 1.205 1.330 1,450 1,550 125 -0.013 0.295 0.590 0.795 1.035 1.210 1.385 1.525 1.640 1.750 130 -0.015 0.345 0.700 0.920 1.200 1.370 1.555 1.715 1.860 1.990 135 -0.017 0.400 0.825 1.060 1.390 1.590 1.800 1.950 2.105 2.272

TABLE 11. Correction factor k. for the Klosek and McKiniey method as a function of temperature Tand

molar mass M

M/(g • mol

') 16 17 18 19 20 21 22 23 24 25 T/K lO^kjAcm^ • mol ~

)

90 -0.004 0.10 0.22 0.35 0.50 0.60 0.69 0.78 0.86 0.95 95 -0.005 0.12 0.28 0.43 0.59 0.71 0.83 0.94 1.05 1.14 100 -0.007 0.16 0.34 0.49 0.64 0.79 0.94 1.08 1.17 1.27 105 -0.01 0.24 0.42 0.61 0.75 0.91 1.05 1.19 1.33 1.45 110 -0.015 0.32 0.59 0.77 0.92 1.07 1.22 1.37 1.52 1.71 115 -0.024 0.41 0.72 0.95 1.15 1.22 1.30 1.45 1.65 2.00 120 -0.032 0.60 0.91 1.23 1.43 1.63 1.85 2.08 2.30 2.45 125 -0.043 0.71 1.13 1.48 1.73 1.98 2.23 2.48 2.75 2.90 130 -0.058 0.95 1.46 1.92 2.20 2.42 2.68 3.00 3.32 3.52 135 -0.075 1.30 2.00 2.40 2.60 3.00 3.40 3.77 3.99 4.23

219
852 R. D. Mccarty

-1 1 1

—

-J 1 I 110 120 130 140 r/K

FIGURE 8. Relative deviations exceeding 0.1 per cent between experimenul densities and densities calculated by the revised Klosek and McKinley method. The comparison set is about 60 (p, V, T. x) points

for LNG-like mixtures from references 4, 5, and 7. O. O.84566CH4 + 0.07924C2H6 + O.OSOeOCjHg +

0.02450n-QH,o; +, 0.85934CH4 + 0.08477CjH6 + 0.02980CjH8 + 0.00707«-C4H,o + 0.00519i- C4H,o; A,0.00859Nj + 0.75713CH4 + 0.13585CjHe, + O.O6742C3H8 + 0.01326n-C4H,o + 0.01336i- C4H,o + 0.00216n-C5H,j + 0.00223i-C5H,j.

% 0.2 -

I

The composition limits of the revised Klosek and McKinley method are the most

severe of any of the three methods given here. This method should not be used for LNG-like mixtures, unless they contain at least 60 moles per cent of CH4, less than 4

moles per cent of Nj, less than 4 moles per cent of each of /-C4H10 and n-C4H,o, and

less than 2 moles per cent in all of /-C5H,2 and n-CsHij.

There are 61 experimental (p, V, T, x) points from references 4 and 6 which fall within the composition limits outlined above. Figure 8 shows all of the deviations

between calculated and experimental densities in this 61-point comparison set which

exceed the 0.1 per cent criterion. Figure 8 exhibits the same sort of deviation trend as

is found in figure 7. In figure 8 all of the deviations are at 1 15 K and above; therefore, the claim of 0.1 per cent accuracy with this method must be restricted to temperatures

below 115 K.

7. Conclusions

On the basis of the performance of the three models and subject to composition and

temp)erature range restrictions as outlined earlier in this paper; it is estimated that,

given the pressure, temperature, and composition of LNG, any one of the three models presented here may be used to predict the density to within 0.1 per cent of the

true value. As was mentioned before, no model or method can be any better than the

experimental values used to develop it; therefore the above accuracy claim depends

strongly on the accuracy of the experimental values. The estimated uncertainty of the experimental values''"^' used here is 0.1 per cent or less, with a precision of a few

hundredths of a per cent. With the exceptions already detailed in sections 3 and 4, the

correlations presented here provide no basis for questioning the claims of the experimenters.

This entire study including the experimental work has provided valuable

experience in the problem of mathematically modelling the equation of state of a mixture, and the lessons learned therefrom are worth passing along to future workers

in the field. Having accurate and internally consistent experimental results has always

220
MATHEMATICAL MODELS FOR THE PREDICTION OF LNG DENSITIES 853

been an important factor in equation-of-state development for pure substances but in the case of the equation of state of mixtures it is absolutely essential. Without

internally consistent results the job is hopeless, especially if the accuracy of the equation of state is important.

The value of conducting the experimental and modelling program simultaneously

has also been clearly demonstrated, for without the parallel efforts in this project the experimental problems with some of the mixtures containing butane would never have been detected. Of the three methods presented here, the extended-corresponding-states model has

proved to be more accurate and more versatile than either of the other two. However,

over the very restricted range of composition and temperature of liquefied natural

gas, no one method seems to be more accurate than the other. Interim results of this study were reported by Haynes et al!^^^ and McCarty''"

both of which contain earlier versions of the mathematical models presented here. The earlier versions in references 21 and 22 are only slightly different from the present

ones, the main difference being in the CH4-C4H,o binary interaction parameters for the extended-corresponding-states method and the hard-sphere method ; however, for calculations of the density of LNG, the difference is inconsequential. The user of any

of these models is cautioned to read carefully the limitations associated with each

model given in the relevant individual section. Computer programs for the three models presented here are available from the Thermophysical Properties Division of the National Bureau of Standards in Boulder,

Colorado. A complete listing of these programs and more detailed instructions on how to use them is published elsewhere.'^^'

REFERENCES

1. Haynes. W. M.. Hiza. M, J. J. Chem. Thermodynamics 1977, 9, 179.

2. Hiza. M. J.; Haynes, W. M.; Parrish. W. R. J. Chem. Thermodynamics 1977, 9. 873.

3. Haynes. W. M. Orthobaric liquid densities of methane + isobulane and methane + normal butane

mixtures at low temperatures (to be published).

4. Hiza. M. J.; Haynes. W. M. J. Chem. Thermodynamics 1980. 12. 1.

5. Haynes. W. M. J. Chem. Thermodynamics 1982, 14. 603.

6. Miller. R. C; Hiza. M. J. Fluid Phase Equilibria 1978, 2. 49.

7. Haynes, W. M.; Hiza. M. J.; Frederick, N. V. Rew Sci. Instrum. 1976, 47. 1237.

8. Leach, J. W. Molecular structure correction for applying the theory of corresponding states to nonspherical pure fluids and mixtures. Ph.D. Thesis, Rice University. 1967.

9. Rodosevich, J. B.; Miller, R. C. Adv. Cryog. Eng. 1974, 19, 339.

10. Klosek, J.; McKinley, C. Proc. First Int. Conf. on LNG. Chicago. 1968, paper 22.

11. Watson, K. M. Ind. Eng. Chem. 1943, 35. 4. 399.

12. Hiza, M. J. Fluid Phase Equilibria 1978, 2, 27.

13. Shana'a. M. Y.; Canfield, F. B. Trans. Faraday Sac. 1968, 64. 2281.

14. Bloomer, O. T.; Parent, J. D. Chem Eng. Prog. Symp. Ser. 1952, 49, 11.

15. Orrit, J.; Olives, J. F. Distributed at 4th International Conference on Liquefied Natural Gas. Algeria. 1974.

16. Rodosevich, J. B. Measurements and prediction of liquefied natural gas densities. M,Sc. Thesis, University of Wyoming. 1973.

17. Rowlinson. J. S.; Watson. I. D. Chem. Eng Sci. 1969, 24, 1565.

18. Mollerup. J. Adv. Cryog. Eng. 1975, 20, 172.

19. Mollerup. J.; Rowlinson. J. S. Chem. Eng. Sci. 1974. 29. 1373.

20. Goodwin. R. D.; Prydz. R. J. Res. Sail Bur. Stand. (L .S.) Sect. A 1972. 76, 1-81.

221
854 R. D. Mccarty

21. McCany. R. D. Natl Bur. Stand. (U.S.). Interagency Report NBSIR 77-867. 1977.

22. Haynes. W. M.; Hiza. M. J.; McCarty. R. D. Proc. Fifth International Conference on LNG,

Diisseldorf Vol. 2. Institute of Gas Technology, Chicago, 111. 1977, Section III, Paper 11. 23. McCarty, R. D. Natl Bur. Stand. (.U.S.). Tech. Note 1030. 1980.

24. Ornt, J. E.; Laupretre, J. M. Adv. Cryog. Eng. 1978, 23, 573.

25. Nunes da Ponte. M.; Streett, W. B.; Staveley, L. A. K. J. Chem. Thermodvnamics 1978, 10, 151.

26. Longuet-Higgins. H. C; Widom. B. Mol. Phys. 1964, 8. 549.

27. Slraty, G. C: Diller, D. E. J. Chem. Thermodynamics 1980, 12, 937.

28. McClune. C. R. Cryogenics 1976, 16, 289.

222
6.13 Reference [25]

Fluid Phase Equilibria, 2 (1978) 27-38 27

© Elsevier Scientific Publishing Company, Amsterdam — Printed in The Netherlands

AN EMPIRICAL EXCESS VOLUME MODEL FOR ESTIMATING

LIQUEFIED NATURAL GAS DENSITIES

M.J. HIZA

Cryogenics Division, Institute for Basic Standards, National Bureau of Standards,

Boulder, Colorado 80302 (U.S.A.)

(Received August 22nd, 1977)

ABSTRACT

Hiza, M.J., 1978. An empirical excess volume model for estimating liquefied natural gas

densities. Fluid Phase Equilibria, 2: 27—38.

The mathematical model presented herein was developed to represent excess volumes at saturation for multicomponent liquid mixtures of nitrogen and the low molecular weight

alkanes between 105 and 120 K. Parameters of the model were determined from experimental

excess <volumes for binary liquid mixtures of nitrogen, methane, ethane, propane, isobutane,

and normal butane. Comparisons made with selected experimental excess volumes reported

in the literature for multicomponent liquid mixtures of the above components demonstrate

the predictive capability of the model in two simple forms. An extension of the model to include mixtures containing isopentane and normal pentane is also proposed. Pure component

molar volumes are given at 0.5 K intervals from 105 to 116 K to facilitate the use of the present model in estimating liquefied natural gas (LNG) densities.

BACKGROUND

Analysis of consistency and the prediction of orthobaric (saturated) liquid

molar volumes of multicomponent mixtures requires a reliable mechanism to

account for the effect of each component as a function of composition and

temperature. Of specific interest here are multicomponent liquid mixtures of

natural gas components; viz. nitrogen, methane, ethane, propane, isobutane,

and normal butane. A simple and reliable empirical model in this specific area

is most attractive because it offers an alternative to the more complex theoretical models for predicting orthobaric LNG (Liquefied Natural Gas) densities

for custody transfer.

At temperatures where all components present in the liquid mixture are subcritical, the logical subject of an empirical model is the liquid mixture excess volume. The development of such a model into a useable (predictive) form

requires a consistent set of density data for the pure components and for the

223
28

possible binary systems, each at several compositions, in the temperature range

of interest.

The empirical models of Klosek and McKinley (1968) and of Miller (1974)

are of this type. The first, using mixture molecular weight as the correlating

parameter, does not properly account for the presence of nitrogen in the mixture nor does it include adequate information on the temperature dependence

of excess volume. The second, basically a more sound approach, treats the

presence of nitrogen separately, uses an effective ethane mole fraction as the

correlating parameter for the hydrocarbon contribution, and includes significantly more information on temperature dependence, as well as pressure

effects. However, both models were based on relatively incomplete data sets and some of the features of each were, of necessity, based on educated guesses. For example, accurate orthobaric liquid density data for nitrogen + ethane,

nitrogen + propane, and binary mixtures of ethane, propane, isobutane, and

normal butane were not available at the time these earlier models were developed. In addition, both models were intended only for use with methane-rich

mixtures. Neither will correctly predict the excess volume at saturation of a simple equimolar liquid mixture of methane, ethane, and propane.

The empirical excess volume model proposed here was developed with due

consideration of the deficiencies in the models of Klosek and McKinley

(1968) and of Miller (1974). It is based on the self-consistent orthobaric liquid

densities for the pure components (Haynes et al., 1976; Haynes and Hiza,

1977) and for the binary mixtures (Hiza et al., 1977) obtained in the NBS LNG

density project. The excess volumes for the methane + butanes and nitrogen + methane reported by Miller (1974) and Rodosevich and Miller (1973) were

also used. The excess volumes reported by Miller (1974) and Rodosevich and

Miller (1973) are generally consistent with those from the NBS data within an imprecision equivalent to about 0.1 percent in the mixture molar volumes

(Hiza et al., 1977). All excess volumes reported in these references were calculated from:

= v„ -Ex,[y, + /3,v.(P,-Pn.)] (1)

where V is the molar volume, P is the saturation pressure, x is the mole frac- tion, and j3 is the isothermal compressibility. Subscripts m and i refer to the

mixture and to the pure components, respectively. In all cases the isothermal

compressibilities were taken from Rowlinson (1969) for nitrogen and methane,

and from Miller (1974) for ethane. Isothermal compressibility adjustments to

the molar volumes of propane, isobutane, and normal butane were not made

in deriving excess volumes from the NBS data. Neglecting isothermal compressibility adjustments to the molar volumes of propane and the high-molecular

weight alkanes has no significant effect on the results. FORMULATION OF THE EXCESS VOLUME MODEL

The present model is based on excess volume ratios relative to the welldefined methane + ethane and nitrogen + methane binary systems, treating

Z2A
29

the hydrocarbon + hydrocarbon and nitrogen + hydrocarbon contributions

separately. Eqns. 2, 3 and 4 describe this model in general terms.

Vf = Vic + Vgc (2)

where

12 and

X5

(3)

In these equations is excess molar volume and x is the component mole

fraction. The subscripts t, CC, and NC refer to total, hydrocarbon + hydrocarbon, and nitrogen + hydrocarbon, respectively. The subscript N refers to nitrogen and 1, 2, 3, 4, and 5 refer to methane, ethane, propane, isobutane

and normal butane, respectively. In eqn. (3), the excess volumes for methane +

ethane ( V12), etc. are taken at a pseudo mole fraction of methane, x\, equal

to the sum of the actual methane and nitrogen mole fractions, x^ + jc^. In eqn. (4), the excess volumes for nitrogen + methane (V^i), etc. are taken at the actual nitrogen mole fraction in the mixture, x^. Obviously, if no nitrogen

is present in the mixture Vf = Vcc eqn. (4) is not needed.

In this approach, it is assumed that contributions from heavy hydrocarbon + heavy hydrocarbon interactions are negligible. In the temperature range of

interest, this assumption is valid since experimental excess volumes for binary

mixtures of ethane, propane, isobutane, and normal butane are nearly zero (Hiza etal., 1977).

The values of 1^12 and V13 in cm^/mol in eqn. (3) are evaluated from a temperature dependent Redlich—Kister expansion of the following form:

Vl = XiXi[{ao + a^T + a^T^) + (bo + b^T + h^T^){2Xi - 1) + (Co + Cir+C2r2)(2x,-l)2] (5)

atx, = x\ = x-y + jcn and Xj = 1 — x\. T is the temperature in Kelvin. The coefficients of eqn. (5) obtained from the NBS data for the methane + ethane

and methane + propane systems are given in Table 1 . The coefficients for the

methane + propane system are the same as those presented earlier (Hiza et al., 1977); those for the methane + ethane system are slightly different from the

earlier values (Hiza et al., 1977) since a few points, obtained by vapor recircu225
30

TABLE 1 Coefficients of Eq. (5) for the methane + ethane and methane + propane systems

methane + ethane methane + propane

flo —14.89469 —11.96725

ai 0.2770596 0.236748

02 —0.001473116 —0.001429237

60 —37.98245 16.59465

61 0.6427122 —0.3212125

62 —0.002783724 0.001398221

Co 0 —52.93613

Ci 0 0.9887147

C2 0 —0.004667488

lation and composition analyses of liquid samples (Hiza and Haynes, 1978), were added to determine the coefficients given here. The applicable temperature ranges for these coefficients are 105 to 135 K for methane + ethane and

105 to 130 K for methane + propane. For convenience values of A = Qq + aiT + a2T^,B = 60 + &iT+ &2^^,and C= Co + c^T^ CzT^ are given at a few

specific temperatures in Tables 2 and 3 for methane + ethane and methane +

propane, respectively.

There are few data points available for the methane + isobutane and methane + normal butane systems. Thus, it is not possible to represent the excess volumes for these systems analytically. The ratios of excess volumes for the

methane + butane systems to the excess volume for the methane + ethane sys- tem were determined at several compositions in the methane rich region at approximately 108 K. For purposes of this correlation these ratios have been

assumed to be constant. These are

vfjvh = 2.0 (6)

and

F?5/VL = 2.5 (7)

where subscripts 4 and 5 refer to isobutane and normal butane, respectively.

Thus eqn. (3) reduces to

yEc(@.;) = vf,[i.(J-i),-j^

X4 ^ 1.5x5

) il-xl) (l-xl)j

(8)

With some loss in precision, the ratios of the excess volumes for the methane

+ propane system to the methane + ethane system above 30 mole percent

methane can be represented as a fimction of or J only. With this modification,

226
31

TABLE 2 Redlich—Kister coefficients at specific temperatures for the methane + ethane system

T A B

(K) (ao + aiT + a2T^) (60 + 6iT + 62^2)

105 —2.044536 —1.188226

108 —2.154678 —1.038889

108.15 —2.160881 —1.032738

110 —2.242838 —0.967168

115 —2.514795 —0.885297

120 —2.860408 —0.942612

TABLE 3 Redlich—Kister coefficients at specific temperatures for the methane + propane system

T A B C

(K) (ao + fliT + aaT^^) {bo ^ biT + baT^) (Co + CiT + C2r2)

105 -2.866048 —1.717276 —0.580142

108 -3.069086 -1.787450 —0.596522

108.15 -3.079914 —1.790298 —0.599547

110 -3.218738 -1.820251 —0.654118

115 -3.642889 -1.853315 —0.961468

120 —4.138503 -1.816468 -1.502193

eqn. (8) becomes

1 + (O.eOlbxl + 0.1664)

1.5Xe

^3

(9)

In examining the nitrogen + hydrocarbon data, it was noted that the excess volumes for the nitrogen + ethane and nitrogen + propane systems are essentially the same; and, at low nitrogen content, these and the excess volumes for

nitrogen + methane are almost linear functions of the nitrogen mole fraction

(up to about 10 mole percent for nitrogen in methane). Without the benefit

of experimental data, it was assumed that excess volumes for the nitrogen + butane systems are approximately the same as those for the nitrogen + propane system. Therefore, eqn. (4) was reduced to

0.81(1 -xl)-

1 +

where

2.8842

134.196

• 126.2- T

(10)

(11)

227
32

The applicable temperature range of eqn. (11) is 91 to 120 K. For temperatures

above 115 K, this equation should not be used for nitrogen concentrations in excess of 5 mole percent.

MODEL LIMITATIONS

In the applicable temperature range, i.e. 105 to 120 K, limitations of the

present empirical model are due primarily to insufficient or lack of data for

certain binary systems. The available data are inadequate to describe the temperature and composition dependence of excess volumes for either of the

methane + butane systems. Above 115 K, more data are needed for the nitrogen + methane system to adequately describe the composition dependence of

excess volumes for mixtures containing more than 5 mole percent nitrogen.

And finally there is a lack of any experimental information on properties of

the nitrogen + butane systems.

Therefore, betv^^een 105 and 115 K, this model should not be used for multicomponent LNG mixtures containing more than 10 mole percent of nitrogen,

isobutane, or normal butane. Between 115 and 120 K, the uncertainties of

predicted excess volumes are expected to become unacceptably large (>0.15%

of the maxture molar volume) for multicomponent LNG mixtures containing

more than 5 mole percent nitrogen. For mixtures containing only methane,

ethane, and propane, the model should be applicable from 30 to 100 mole

percent of methane from 105 to 120 K, and probably to somewhat higher

temperatures. It is believed that, within these limits of composition and temperature, the uncertainty in predicted excess volumes for multicomponent

LNG mixtures is equivalent to about ±0.15% of the mixture molar volume.

Best results are expected in the middle of the applicable temperature range,

i.e., 110—115 K, particularly if eqn. (9) is used instead of eqn. (8).

Some comparisons between experimental excess volumes for selected multicomponent mixtures of LNG components from the literature and predicted

results using both eqns. (8) and (9) are given in Table 4. In almost all cases, the

predicted excess volumes are less than the experimental values, and only for the

mixture containing 72 mole percent of methane with ethane and propane

(Shana'a and Canfield, 1968) does the difference exceed 0.15% of the mixture

molar volume. The average difference between predicted and experimental excess volumes is about —0.09% of the mixture molar volume. Comparisons are

also being made with excess volumes from experimental multicomponent mixture measurements in the NBS LNG density project, and these will be published

with the data in the near future.

ESTIMATING DENSITIES OF LNG, INCLUDING PENTANES

Use of the present model for estimating the orthobaric molar volume, and

thus the density, of a liquefied natural gas mixture at a desired temperature

and saturation pressure follows directly from eqn. (1). A consistent set of pure

228
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229
34

component molar volumes adjusted to the mixture vapor pressure at the same

temperature are needed to make this calculation.

The saturated liquid molar volumes of nitrogen and the alkanes through the

butanes are readily obtained from the expression

with the parameters given by Haynes et al. (1976) and Haynes and Hiza (1977).

In this expression p is the density in mol/1, T is the temperature in Kelvin, and

the subscript c denotes the critical point. For isopentane and normal pentane,

saturated liquid molar volumes between 105 and 120 K can be obtained by

extrapolation of the data of Orrit and Laupretre (1978) or of McClune (1976).

Densities from these investigators are generally within 0.1% of the NBS values

for the lower molecular weight alkanes. In his paper, McClune gives the constants

for a quadratic expression in temperature which he used to extrapolate the densities for isopentane and normal pentane down to 93 K. In the present paper,

the densities given by Orrit and Laupretre for isopentane (from 124.925 to

253.794 K) and for normal pentane (from 148.879 to 249.011 K) were fitted

to eqn. (12) with values of Pc and taken from Kudchadker et al. (1968).

These parameters of eqn. (12) are given in Table 5. Compared with molar

volumes between 120 and 105 K from McClune's extrapolation, molar volumes

obtained from the fit of the data of Orrit and Laupretre are larger for isopentane

by 0.14—0.15% and smaller for normal pentane by 0.09—0.29%. With the small

concentrations of pentanes normally encountered in LNG mixtures, these differ- ences are not significant.

Of common interest is the molar volume or density at a temperature where

the total vapor pressure of the liquid mixture is about atmospheric pressure.

For mixtures containing mole fractions of 0.85—0.97 methane, 0.03—0.09

ethane, and up to 0.016 nitrogen, 0.03 propane, 0.01 butanes, and 0.0023 pentanes, Giarratano and Collier (1977) report a calculated boiling point temperature range from about 106.5 K with a nitrogen mole fraction of 0.016 to about 113.2 K with no nitrogen. The pure component molar volumes given in Table 6 at 0.5 K increments from 105 to 116 K and at 120 K can be used

TABLE 5 Parameters of eqn. (12) for isopentane and normal pentane

(12)

isopentane normal pentane

Tc (K) Pc (mol/1)

a 2.946310

35.50771

-57.41243

28.15898

460.39

59.00203

—93.44194

43.66781

469.60

—0.03620055

3.271 3.285

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231
36

directly to calculate the ideal contribution to the mixture molar volume when

the mixture vapor pressure is about one bar. The molar volumes given in Table

5 for nitrogen were adjusted to one bar using isothermal compressibilities given

by the following expression

/3N2(bar-^) = -0.001032 + 0.051368/(126.2 - T) . (13)

The molar volumes given for all of the other fluids are the orthobaric (saturation) values. Values for isopentane and normal pentane were calculated from

eqn. (12) and the parameters given in Table 5. Though the isothermal compressibility correction has little effect on the methane molar volume at the

conditions considered here, one can determine the magnitude of the pressure

effect using the following expression to obtain the isothermal compressibility

for methane

:

/3cH4(bar-i) = -0.0001595 + 0.029488/(190.555 - T) (14)

Between 105 and 120 K, eqns. (13) and (14) represent isothermal compressibilities listed by Rowlinson (1969) within 4% for nitrogen and within 2%

for methane. At the same temperatures eqn. (14) gives isothermal compressibilities for methane that are smaller but within 3% of those taken from the

more recent data of Goodwin (1974).

There are no experimental data in the literature from which excess volume

ratios of methane + pentanes to methane + ethane can be derived. However,

for the present purpose estimates of these ratios were made by inspection of

the excess volume ratios of methane + propane, methane + isobutane, and

methane + normal butane to methane + ethane. Based on these estimates,

eqn. (9) can be revnritten to include the pentanes as follows:

V^ci@xl) = Vf2 1 + (0.6015x1 + 0.1664) -

1.5x5 2.2x6 3-1^7

n-xi) (1-xi) (i-xi)j (15)

In eqn. (15), subscripts 6 and 7 refer to isopentane and normal pentane,

respectively. For the nitrogen + hydrocarbon contribution, Vnc» to the total excess volume, Vf, of mixtures containing small amounts of pentanes, eqns.

(10) and (11) should be used without modification.

CONCLUSION

For liquid mixtures containing nitrogen and the low molecular weight

alkanes all at subcritical temperatures, the mathematical excess volume model

proposed here combined with the consistent set of experimental pure component densities now available, allow reasonable estimates of the orthobaric

liquid densities of mixtures containing these components within defined composition limits. For the special case involving some typical LNG compositions

232
37

over a limited temperature range (105 to 115 K), liquid mixture densities can

be predicted with the present method with an inaccuracy of ±0.15% or less. Additional LNG type mixtures data, consistent with the pure component and

binary mixtures data used here, would more clearly identify the composition

and uncertainty limits of the present model.

ACKNOWLEDGMENTS

The contribution of M.J. Brown to the reduction of the data is gratefully

acknowledged. Partial support of this work was provided by British Gas Corp.,

Chicago Bridge and Iron Co., Columbia Gas Service Corp., Distrigas Corp.,

Easco Gas LNG, Inc., El Paso Natural Gas, Gaz de France, Marathon Oil Co.,

Mobil Oil Corp., Natural Gas Pipeline Co., Phillips Petroleum Co., Shell Inter- national Gas, Ltd., Sonatrach, Southern California Gas Co., Tennessee Gas

Pipeline Co., Texas Eastern Transmission Co., Tokyo Gas Co., Ltd., and Transcontinental Gas Pipe Line Corp., through a grant administered by the American

Gas Association, Inc.

REFERENCES

Giarratano, P.J. and Collier, R.S., 1977. Evaluation of capacitance densitometry for LNG

mixtures with low nitrogen composition. Ind. Eng. Chem., Proc. Design and Development, 16: 330—336.

Goodwin, R.D., 1974. The thermophysical properties of methane, from 90 to 500 K at pressures to 700 bar. Nat. Bur. Stand. (U.S.), Tech. Note 653, 274 pp. Haynes, W.M. and Hiza, M.J., 1977. Measurements of the orthobaric liquid densities of

methane, ethane, propane, isobutane, and normal butane. J. Chem. Thermodynam.,

9: 179-187.

Haynes, W.M., Hiza, M.J. and Frederick, N.V., 1976. Magnetic suspension densimeter for measurements on fluids of cryogenic interest. Rev. Sci. Instrum., 47: 1237—1250.

Hiza, M.J. and Haynes, W.M., 1978. Liquid mixture excess volumes and total vapor pres- sures using a magnetic suspension densimeter with compositions determined by

chromatographic analysis: methane • ethane. In: K.D. Timmerhaus (Ed.), Advances in Cryogenic Engineering, Vol. 23, Plenum, New York, paper M-6 (in press).

Hiza, M.J., Haynes, W.M. and Parrish, W.R., 1977. Orthobaric liquid densities and excess volumes for binary mixtures of low molar-mass alkanes and nitrogen between 105 and

140 K. J. Chem. Thermodynam., 9: 873-896.

Klosek, J. and McKinley, C, 1968. Densities of liquefied natural gas and low molecular

weight hydrocarbons. In: J.W. White and A.E.S. Neumann (Editors), Proceedings of

First International Conference on LNG, Institute of Gas Technology, Chicago, 111., Paper 22, Session 5. Kudchadker, A.P., Alani, G.H. and Zwolinski, B.J., 1968. The critical constants of organic

substances. Chem. Rev., 68: 659—735.

McClune, C.R., 1976. Measurement of the densities of liquefied hydrocarbons from —100

to -180°C (173 to 93 K). Cryogenics, 16: 289-295.

Miller, R.C., 1974. Estimating the densities of natural gas mixtures. Chem. Eng., 81

(No. 23): 134-135.

Orrit, J. and Laupretre, J.M. 1978. Density of liquefied natural gas constituents. In: K.D. Timmerhaus (Ed.), Advances in Cryogenic Engineering, Vol. 23, Plenum, New York,

paper M-4 (in press), (see also: Orrit, J. and Olives. J.F., 1974. Density of liquefied

233
38

natural gas and its components. Unpublished report distributed at 4th International Conference on Liquefied Natural Gas, Algeria, 16 pp).

Rodosevich, J.B., 1973. Measurement and prediction of liquefied natural gas densities. M.S. Thesis, Univ. of Wyoming, Laramie, 83 pp. Rodosevich, J.B. and Miller, R.C., 1973. Experimental liquid mixture densities for testing

and improving correlations for liquefied natural gas. A.LCh.E.J., 19: 729—735.

Rowlinson, J.S., 1969. Liquids and liquid mixtures. 2nd edn., Butterworths, London, 371 pp. Shana'a, M.Y. and Canfield, F.B., 1968. Liquid density and excess volume of light hydrocarbon mixtures at —165°C. Trans. Faraday Soc, 64 (No. 549, Part 9): 2281—2286.

234
6.14 Reference [22]

A concentric cylinder capacitor hu been uaed to metture the orthobaric liquid

dielectric constants of multicomponent mixtures of the major components of liquefied natural gas ILNG} to an accuracy of approximately ± 0.05% at temperatures from 1 10 to 130 K. These mixtures ranged from a ternary mixture containing nitrogen, methane, and

normal butane to four to eight component methane rich (74 to 90 mol %l mixtures containing up to 5 mol % of nitrogen, 16 mol % of ethane, 7 mol % of propane, 5 mol % of

the butanes, and 0.44 mol % of the pentanes. Some of these mixtures were prepared to simulate commercial LNG compositions. Experimental densities previously reported for these mixtures have been combined vitith the mixture dielectric constant data to calculate

values of the Oausius-Mossotti (CM) function and the excess CM function. Pure component experimental CM functions for LNG components except for propane and isobutane have been combined with the mixture data in the development of a simple calculational technique for the prediction of LNG densities to an uncertainty of approximately ± 0. 15%

based on a knowledge of the composition and dielectric constant of the liquid mixtures.

In fitting the data, pseudo values of the CM function are derived for the slightly polar

components, propane and isobutane, while constraining the mixture excess CM function

to be zero.

Prediction of liquefied natural gas (LNG) densities

from new experimental dielectric constant data

W.M. Haynes and R.D. McCarty

Key words: liquafiad natural gas, dansity, dialaetrie oonttant

A large-scale project has been carried out at this laboratory to provide erne or more mathematical models to

predict the density of a liquefied natural gas (LNG) mixture

to within an uncertainty of 0.1%, given the temperature, composition and pressure of the liquid mixture. Such models

will serve as a basis for custody transfer. The mathematical models developed for this purpose have ranged from simple

empirical models'"* to those that are theoretically based

and, consequently, much more complex.

In order to develop and test such models, a comprehensive, accurate and consistent set of density data has been

obtained for the major components*-' of LNG and for mixtures'"*" of these components. Most of these data were

obtained with a magnetic suspension densimeter.'*''" Prior to the completion of the LNG density project, the apparatus was completely rebuilt because of the need to extend the

pressure capability of the apparatus.'^ During the reconstruction of the apparatus, it was decided to include a capacitor inside the sample space so that simultaneous measurements of dielectric constant and density could be performed on the same fluid samples. The new apparatus was subsequently used to complete the measurements for the LNG density project. The measurements with the new

apparatus included density and dielectric constant data

for binary mixtures of methane with either isobutane or normal butane'" and for multicomponent mixtxires' with up to eight components, some simulating commercial LNG

compositions. The experimental densities for the multicom- ponent mixtures were reported as a function of temperature and pressure in an earlier paper.' The dielectric constant measurements that were performed at the same time were

not published along with the density data since, at the time of the measurements and analysis of the density data, there were insufficient pure fluid dielectric constant data

for LNG components at low temperatures to analyse property the dielectric constant results. Thus, one of the major purposes of this paper is to present the dielectric constant data for multicomponent mixtiues of LNG components, for which the pressure density temperature

properties have been reported previously.

The dielectric constant of a fluid is closely related to

its density through the Qausius-Mossotti (CM) function.

In fact, dielectric constant measurements can serve as a simple and reliable substitute for density measurements.

In the custody transfer of LNG, densities are usually determined either by direct density measurements with a commercially available instrument or indirectly using a calculation based on measurements of composition and

temperature. One technique employed in commercial

densimeters is based on a density determination through

the appUcation of capacitance (or dielectric constant) measurements. The dielectric constant data for multicomponent

mixtures of LNG components have been combined with

the previously reported density data to develop a simple

calculational scheme for the prediction of LNG densities based on measurements of the dielectric constant and com

position of the liquid mixture. This procedure provides an

alternative approach to the complex models^"* that have been previously used for prediction of LNG densities. Additionally, the procedure presented here can be used as a

relatively simple check against any of the other LNG

density determination methods, whether by direct or

indirect mearis. It should be emphasized that, in most cases,

it is much easier to perform accurate and precise measurements of dielectric constant than of density.

The prediction of LNG and LPG (liquefied petroleum

gas) densities and heating values from dielectric constant 0011-2275/83/080421-06 $03.00® 1983 Burterworth & Co (Publishers) Ltd. CRYOGENICS .AUGUST 1983 421

235
measurements has been investigated by Miller and

colleagues.""" However, their work for LNG" was based

primarily on binary mixture data, along with two ternary mixtures, and did not include pentanes. The present study

probably includes the only experimental dielectric constant

data that have been obtained for multicomponent mixtures

simulating commercial LNG compositions.

Experimental

The experimental apparatus, procedures, and uncertainties have been described in detail elsewhere.''*"'" The

dielectric constant measurements were performed using a concentric cylinder capacitor with a vacuum capacitance of approximately 20 pF. The dielectric constant was

obtained from measurements of the ratio of the capacitance with liquid between the electrodes to the vacuum

capacitance. Capacitances were measured to a resolution of ICP pF, wtoch is equivalent to the stability of the vacuum

capacitance. Based on reproducibility of binary mixture

dielectric constant measurements'" with the same capacitor, the uncertainty in the dielectric constant data for the multicomponent mixtures investigated in the present work

should be no larger than 0.05%.

Density data' were obtained simultaneously with the

dielectric constant results for the mixtures investigated

here. The devices for measuring density and dielectric constant were contained in a single sample cell; therefore, measurements of these properties were made on the same

liquid samples. During the course of the mixture measurements, data were obtained tor liquid methane, prior to

filling the ceil with each new mixture sample, as a control on the measurement process for determining density. The

'dielectric constant data for liquid methane (usually at

1 20 K), obtained along with the density data, agreed to

better than 0.02% with the results of Straty and Goodwin.'*

Vacuum capacitances were measured for each new filling of the cell. From an analysis of the density measurement tech- nique, it has been shown that the total systematic error in the density deter.nination is approximately 0.05%.' Combined with the 0.05% uncertainty in the dielectric constant, it is estimated that the maximum uncertainty in the derived values of the Qausius-Mossotti function is 0.15%.

ResultsThe experimental orthobarlc Uquid dielectric constants of mixtures of LNG components are presented in Table 1 as a function of temperature (IPTS-68), pressure, and

density. As noted in a preceding section, the vapour pressure and density data that were obtamed simultaneously with

the dielectric constant results have been discussed in detail.' The compositions of the mixtures were determined gravi- metrically except those for mixtures m, n, and p. For these mixtures the compositions were determined by analyses of

the prepared mixtures using a gas chromatograph with a thermal conductivity detector. Small adjustments were made to the observed dielectric constants to account for differences in compositions between the prepared gas mixtures and the liquid mixtures condensed mto the equilibrium cell, which resulted from

vapour occupying a small part of the sample space. The

maximum correction was less than 0.03%, which was for mixture a at 1 25 K. Ail other corrections were of the order of 0.01% or less. For mixtures containing only hydrocar422

bons, the vapour composition was assumed to be pure methane. For mixtures with nitrogen as a constituent, the vapour composition was estimated from partial-pressure calculations using available phase-equilibrium data for nitrogen-methane mixtures."'" The experimental dielectric constants, e, have been combined with the experuiiental densities, p, to calculate values for the Qausius-Mcssotti function CM, from the

relation, TM • (rfl) ?

")

The excess Clausius-Mossotti function (CM^ ), analogous to

the definition of the excess volume of a liquid mixture, is given by the expression,

CM^ = CM -^xfiM,, (2)

i

where CM, as defined above, refers to the Clausius-Mossotti

fiinction of the liquid mixtw'e at a given temperature at saturation pressure, x^ is the mole fraction of component i, and CMj is the Qausius-Mossotti function of component i at the same temperature and pressure as the mixture. Along

an isotherm, the CM function is a very slowly varying

function; therefore, adjustments of the pure component

CMf$ to the saturation pressure of the mixture are sufficiently small to neglect. In the calculation of CM['s, the pure Quid orthobarlc Uquid densities were obtained from expressions given in references 1, 5, and 6. The dielectric- constants for nitrogen were taken from Ely and Straty" ; for methane from Straty and Goodwin'* ; for ethane from Weber^ ; and for propane,

isobutane, and normal butane from Haynes and Younglove.^' For the pentanes, no dielectric constant data could be found

in the literature for temperatures less than 223 K. Therefore, dielectric constant values for the pentanes at temperatures between 110 and 130 K have been estimated by a

linear extrapolation of higher temperature results. '*'"'" The values used for saturated liquid isopentane ranged from

2.138 at 1 10 K to 2.105 at 130 K, while for normal pentane

the values varied from 2.123 at 1 10 K to 2.092 at 130 K. These extrapolated values are estimated to be accurate to

better than ± 1%. An uncertainty of 1% in the dielectric constant for either of the pentanes corresponds to a change

of less than 0.0008 cm^ moF' in the mole fraction average of the CMi'% (2 XfCMi) for mixture o, which contains the

largest amount of the pentanes (approximately 0.2 mol% for each pentane) of the mixtures investigated here. A change

of 8 X 10"* cm^ mof' represents less than 0.01% of the mixture CM function. From the results presented in Table I . several observations concemmg trends in the behaviour of the excess CM

function can be noted. For the measurements of the present

study, the GVf^'s are generally small and positive and

decrease with increasing temperature. (For the same mixtures, the excess volumes were negative, increasing in magnitude (more negative) with increasing temperature.') The maximum observed excess CM function for the mixtures of

this study was approximately 0.25% of the mixture CM

fimction. Those mixtures that contam the largest fractions of the polar components, propane and isobutane, are characterized by the largest excess CM functions. This behaviour was expected based on dielecmc constant and density

CRYOGENICS . AUGUST 1983

236
Table 1. Onhobaric liquid dielectric oonstanti e and Clautiui-Motsotti functioni CM of multioomponent mixture of LNG

componenti

3 3 1^31 10'(P«<Pt-Pc*c> I0^{p„ot-P»

r, K MPa p„p„ mol dm"^ e CM, cm^ mol"^ CAr.cm^mol"'

a - 0.05931 Na + 0.89071 CH* + 0.04998 nC4H,o

(MW- 18.8562) 110.00 0.2400 25.3450 1.65978 7.1130 0.0068 - 0.095 - 0.058

115.00 0.3145 24.9440 1.64742 7.1160 0.0069 - 0.097 - 0.100

120.00 0.4082 24.5383 1.63497 7.1188 0.0071 -0.100 - 0.140

125.00 0.5196 24.1141 1.62203 7.1218 0.0069 -0.097 -0.181'

b - 0.86040 CH4 + 0.04600 CjHg + 0.04790 C3H8 + 0.04570 /C4H10

(MW' 19.9552) 115.00 0.1186 24.2654 1.70983 7.8852 0.0092 - 0.117 0.036 120.00 0.1710 23.9371 1.69759 7.8815 0.0043 - 0.055 0.082

125.00 0.2387 23.5868 1.68523 7.8832 0.0042 - 0.053 0.061^ 130.00 0.3248 23.2331 1.67269 7.8836 0.0026 - 0.033 0.056^ 135.00 0.4320 22.8637 1.65999 7.8870 0.0041 -0.052 0.013^

c - 0.85378 CH4 + 0.05178 CaHe + 0.04703 CaHg + 0.04741 /C4H10

(Af»V- 20.0838) 115.00 0.1191 24.2100 1.71285 7.9304 0.0115 - 0.145 0.005 120.00 0.1706 23.8779 1.70061 7.9288 0.0088 - 0.111 0.026

125.00 0.2379 23.5324 1.68823 7.9296 0.0078 -0.098 0.071^ 130.00 0.3238 23.1834 1.67570 7.9293 0.0055 - 0.069 0.01

9^

d - 0J5133 CH4 + 0.05759 CjHg + 0.04808 CgHe + 0.04300 nC4Hio

(AfkV- 20.0092) 115.00 0.1180 24.3243 1.70951 7.8632 0.0090 - 0.114 - 0.009

120.00 0.1700 23.9965 1.69758 7.8619 0.0052 - 0.066 0.008

125.00 0.2374 23.6586 1.68553 7.8621 0.0024 - 0.031 0.006^ 130.00 0.3232 23.3108 1.67330 7.8631 0.0003 - 0.004 - 0.007^ 135.00 0.4301 22.9634 1.66120 7.8645 - 0.0012 0.015 - 0.025'

e - 0.84566 CH4 + 0.07924 CjHe + 0.05060 CaHg + 0.02450 nC4H 10 (MW' 19.6051) 115.00 0.1167 24.5569 1.70224 7.7241 0.0028 -0.036 0.077 120.00 " 0.1683 24.2126 1.69016 7.7244 0.0008 - 0.010 0.073

125.00 0^350 23.8698 1^7791 7.7219 - 0.0024 0.031 0.106' 130.00 0.3201 23.5204 1.66557 7.7198 - 0.0098 0.127 0.132'

f - 0.04801 Nj + 0.80940 CH4 + 0.04542 CjHe + 0.05050 C3HB + 0.04667 /C4H10

{ArtV- 20.6355) 115.00 0.3005 24.1487 1.69935 7.8284 0.0195 - 0.249 - 0.090

120.00 0.3863 23.8075 1.68708 7.8273 0.0172 - 0.220 - 0.075 125.00 0.4874 23.4518 1.67464 7.8285 0.0165 - 0.211 -0.092'

130.00 0.6125 23.0893 1.66152 7.8248 - 0.043'

g - 0.02628 + 0.81249 CH4 + 0.08484 CjHg + 0.04931 C3H8 + 0.02708 /7C4H10

(AfW- 20.0706)

115.00 0.2214 24.4562 1.69916 7.7283 0.0150 - 0.194 - 0.083

120.00 0.2874 24.1119 1.68701 7.7278 0.0120 -0.155 -0.077

125.00 0.3768 23.7507 1.67487 7.7322 0.0134 - 0.173 - 0.133' 130.00 0.4793 23.3954 1.66264 7.7331 -0.145'

h - 035892 CH4 + 0.11532 CaH, + 0.01341 C3H8 + 0.00530 /C4H10 + 0.00705 /7C4H10

(MW- 18.5565)

115.00 0.1185 25.0957 1.68091 7.3711 0.0006 - 0.008 0.028 120.00 0.1706 24.7131 1.66863 7.3749 0.0021 -0.028 - 0.022 125.00 0.2372 24.3294 1.65622 7.3771 0.0012 - 0.016 - 0.052' 130.00 0.3225 23.9490 1.64364 7.3760 - 0.0030 0.041 -0.037'

CRYOGENICS . AUGUST 1983 423

237
Table 1. continued

7, K P, MPa p„pt,moldm^ e CAf, cm-* mol ^ CW.cm^mol ^ Paxpt Pexpt

i - 0.84558 CH4 + 0.081 53 C2H6 + 0.04778 CaHg + 0.01259 /C4H10 + 0.01252 /JC4H10

(MW= 19.5838) 115.00 0.1166 24.5586 1.70334 7.7334 0.0094 -0.122 -0.000

120.00 0.1680 24.2180 1.69122 7.7323 0.0062 -0.080 0.014

125.00 0.2348 23.8688 1.67899 7.7322 0.0035 -0.045 0.014^ 130.00 0.3188 23.5154 1.66656 7.7309 -0.0006 0.008 0.032^

j -0.00601 N2 + 0.90613 CH4 + 0.06026 CjHg +0.02154 CsHg + 0.00300 /•C4H10 + 0.00306 nC4Hio

(/WIV= 17.8195)

115.00 0.1478 25.3834 1.65987 7.1030 0.0057 -0.080 -0.025

120.00 0.2043 24.9894 1.64698 7.0991 -0.0005 0.007 0.031 125.00 0.2785 24.5702 1.63408 7.1013 -0.0011 0.016 -0.001^

130.00 0.3722 24.1578 1.62098 7.0989 0.033*

k - 0.00973 N2 + 0.88225 CH4 + 0.07259 C2H6 + 0.02561 CaHg + 0.00490 /C4Hio + 0.00492 nC4Hio

(MW= 18.3094) 115.00 0.1639 25.2023 1.67011 7.2448 0.0081 -0.112 0.155 120.00 0.2247 24.8047 1.65731 7.2456 0.0067 -0.093 0.144

125.00 0.3022 24.4022 1.64434 7.2455 0.0037 - 0.051 -0.055^

I - 0.01383 Nj + 0.85934 CH4 + 0.08477 CjHg + 0.02980 C3H8 + 0.00519 /C4H10 + 0.00707 /7C4H10

(MW= 18.7496) 115.00 0.1812 25.0384 1.67817 7.3638 0.0076 -0.103 -0.027

120.00 0.2441 24.6661 1.66553 7.3609 0.0024 -0.033 0.012

125.00 0.3223 24.2880 1.65268 7.3569 -0.0045 0.061 0.066^ 130.00 0.4222 23.8981 1.63968 7.3542 0.103^ m- 0.85341 CH4 + 0.07898 C2H6 + 0.04729 C3H8 + 0.00854 /C4H10 + 0.00992 /7C4H 10 + 0.00097 /CgHis + 0.00089 /7C5H 12 (MW= 19.3587) 110.00 0.0787 25.0063 1.71025 7.6552 0.0100 -0.131 0.013

115.00 0.1172 24.6566 1.69823 7.6572 0.0098 -0.128 -0.013

120.00 0.1686 24.3079 1.68611 7.6573 0.0078 -0.10? -0.014

125.00 0.2351 23.9525 1.67385 7.6576 0.0054 -0.071 -0.017^

130.00 0.3210 23.5883 1.66137 7.6578 0.0028 -0.037 -0.020^

n - 0.75442 CH4 + 0.15401 CjHe + 0.06950 C3H8 + 0.00978 /C4H10 + 0.01057 nC^H^Q

+ 0.00089 /C6H 12 +0.00083 ^CgH 12 (AfkV= 21.1060) 110.00 0.0723 24.2529 1.74897 8.2374 0.0193 -0.234 -0.053

115.00 0.1081 23.9619 1.73738 8.2338 0.0137 -0.166 -0.011

120.00 0.1549 23.6535 1.72575 8.2353 0.0131 -0.159 -0.028

125.00 0.215:^ 23.3351 1.71406 8.2390 0.0139 -0.169 -0.074^ o- 0.00859 Nj + 0.75713 CH4 + 0.13585 C2H6 +0.06742 C3H8 + 0.01336 /C4H 10 + 0.01 326 nC4Hio + 0.00223 /C5H ,3 + 0.00216nC5Hi2

(MW= 21.3094)

110.00 0.1155 24.1809 1.74764 8.2502 0.0136 -0.165 0.015

115.00 0.1595 23.8731 1.73600 8.2521 0.0135 -0.164 -0.008

120.00 0.2155 23.5709 1.72406 8.2486 0.0080 -0.097 0.033

125.00 0.2873 23.2644 1.71227 8.2473 0.0038 -0.046 0.049^ 130.00 0.3744 22.9514 1.70017 8.2447 0.082^

p - 0.00801 Nj + 0.74275 CH4 + 0.16505 CjHg + 0.06547 C3H8 + 0.00843 /C4H10

+ 0.00893 nC4H 10 + 0.00069 /C5H 12 + 0.00067 /JC5H12

(yVfkV= 21.0976) 110.00 0.1158 24.3141 1.74469 8.1790 , 0.0154 -0.188 -0.015

115.00 0.1584 24.0160 1.73307 8.1767 0.0109 -0.133 0.014

120.00 0.2093 23.6937 1.72140 8.1816 0.0137 -0.167 -0.046

125.00 0.2853 23.3804 1.70971 8.1826 0.0117 -0.143 -0.058'

424 CRYOGENICS . AUGUST 1983

238
Tabia 1. oontinuad

T, K P, MPa p„ot, mol dm-3 CW.cm^mol-^ C/lf^,cm='mo|-^ ^° (P^ofPc^) 10 (P«.ot-PFiT)

^•xpt P«j<pt

q - 0.00599 Nj + 0.90068 CH4 + 0.06537 C2H6 + 0.02200 C3H8 + 0.00291 /C4H10 + 0.00284 ^CaHio + O.OOOIO/C6H12 +0.00011 nCsHu

(MW= 17.9026) 115.00 0.1456 25.3600 1.66189 7.1274 0.0031 0.043 0.013

120.00 0.2024 24.9656 1.64901 7.1242 - 0.0024 0.034 0.058

125.00 0.2762 24.5450 1.63624 7.1286 -0.0008 0.012 . -0.004^

130.00 0.3698 24.1289 1.62319 7.1284 -o.oor

AflV, molecular weight; T. temparatura (IPTS-68); P. preuure;p,^, experimsntal deniitv; CM^, exceu aausiuj-MoMorti function calculated from eauation 2.p^^^^, density calculated from equation 3 using experimental CM{% from literature; Pfit. density calculated from aquation 3

using CM\ values for propane and isobutane determined from least squaras by fitting multicomponent mixture data. ^Data not used in fit to equation 3

measurements of binary mixtures of LNG components containing either propane or isobutane.'"-'^ For those mixtures

containing nitrogen, excess CM functions are not given for temperatures above 125 K since the critical temperature of

nitrogen is approximately 126.2 K, Combining (1) and (2) and assuming that the CM

function of the mixture is a mole fraction average of the pure component CM/'s, ie, CM^ = 0, then a density (Pcaic) of the mixture can be calculated from the relation,

Pcaic e + 2 xfiMi (?)

Differences between experimental and calculated densities from (3) are presented in the next to last column of Table 1. These differences are equivalent to the ratios of the excess CM functions to the mixture CM functions (CM^/CM). In other words, assuming CM^ = 0, it is possible to predict the

densities of the LNG-like mixtures investigated here within 0.25% from measurements of the dielectric constant, composition, and temperature of the liquid mixture. This would

not meet the 0.1% uncertainty level desired in LNG custody

transfer applications. However, by making minor modifications to the CMj's in (3), it is possible to reduce the density

uncertainty to 'near the desired 0.1% goal, as discussed in the next section.

Caiculationai technique

In the preceding section, it has been observed that the

densities calculated from (3), using available experimental

CM data for the pure components, can differ by as much as 0.25% from the experimental densities for the LNG-like

mixtures investigated here. In general, those mixtures that exhibit the largest differences contain significant amounts

of the slightly polar components, propane and isobutane. Subsequently, an effort was initiated to develop a simple mathematical procedure that could be used to predict LNG

densities to within an uncertainty of approximately 0.1%

from an input of dielectric constant and composition. This

caiculationai technique would incorporate new pseudo

values of the CM functions of propane and isobutane to account for the relatively large CM^ values observed for mixtures containing these constituents. The ranges of temperature and composition over which the caiculationai technique would be applicable

should be defined. The normal boiling point of LNG

mixtures is approximately the same as that of pure methane

(112 K); thus, the temperature range for the caiculationai scheme was selected to be 1 10 to 120 K. The compositions of the mixtures from the present study coyer ranges that encompass almost any that might be encountered in LNG

custody transfer applications. The extent to which the

caiculationai procedure can be used for compositions

outside the range of the present data is not known.

Since the CM function varies slowly as a function of temperature (or density) for a liquid along the saturation

curve, the CM value for each of the pure components of LNG is assumed to be constant frcm 1 10 to 1 20 K. The

CM values for the saturated liquid at 1 1 5 K are used for each of the nonpolar components. Then, new pseudo values of the CM function are determined for propane and isobu- tane (given in Table 2 along with the CMfs for the nonpolar

components) by a least squares fit of (3) using the mixture

data in Table 1 for temperatures between 110 and 120 K. These pseudo values for the CM function for propane and

isobutane are determined by constraining the CM function

for LNG mixtures to be equal to the mole fraction average of the pure component CM values, ie, for CM^ = 0. The new

pseudo values for the CM function determined from fitting the data are approximately I.l and 0.4% larger than experimental CM values^' for propane and isobutane, res- pectively. The pseudo values are substantially smaller (1 .5 to 2.0%) than the infinite dilution values deduced by Pan

et al." from binary mixture data for methane with either propane or isobutane.

In the last column of Table 1 , comparisons between

experimental densities (Pexpt) and values (pfit) calculated from (3) with CW,'s from Table 2 are presented. At temperatures from 110 to 120 K for the mixtures studied here,

all deviations between the experimental and calculated densities, using the pseudo values for the CM function for propane and isobutane, are less than 0.1% except for mixture

a at 120 K (0.14%) and mixture k at 115 K (0.15%) and at 120 K (0.14%). The deviation of greater than 0.1% for mixture a is likely the result of using a constant CM value

for nitrogen over a 10 K range for a mixture that contams a relatively significant amount (6%) of nitrogen. For temperatures between 1 10 and 120 K, the average deviation or bias between experimental and calculated densities is - 0.001%,

while the average absolute deviation is 0.044%. Above 1 20 K,

at temperatures for which the data were not used in the fit, the average absolute deviation was 0.058%. Maximum

CRYOGENICS . AUGUST 1983 425

^39
Table 2. ValuM of tha ClauHU»4Motsottj function for LNG componants to ba utad in (3) to caicuiata LNG

dansitias

CM- w'**!^ em^ Will mol"^ IIIUI

4.3984

Msthanfi 6.5490

Ethane 11.063

Propane (16.1612)

Isobutane (21.2305)

Normal butane 20.303

Isopentane 25.151 Normal pentane 24.868

Pwudo values for propane and Itobutana (In p«rwitt>atai I vwere datarminad from lean iquarvs analysii by fitting tfta multicomponant mixture dru of present rtudy. The CM functions for the other componanu are experimental nturatad liquid v«lua* at 11 5 K

deviation (0.18%) was for mixttue a at 125 K; this mixture

contains a relatively large amount of nitrogen, which has a

critical temperature of approximately 126.2 K.

Conclusion

Dielectric constant data are reported for multicomponent mixttues of LNG components at temperatures from

1 10 to 130 K. Density results obtained simultaneously with

the dielectric constant data for these mixtures had been

published previously. These density and dielectric constant

data have been used to develop a simple, alternative approach

for determining densities of LNG mixtures to within an

uncertainty of less than 0. 1 5% based on measurements of

the dielectric constant and composition of the Liquid mixture. Composition measurements are required in deter- mining LNG densities with any correlation or prediction method. It is a relatively simple and straightforward task

to perform accurate and precise dielectric constant measure- ments compared to direct density measurements. The

extent to which the calculation can be exuapolated beyond

the composition ranges of the mixtures investigated here is unknown. However, the compositions of the present study were selected to cover the ranges that would be encountered in most LNG custody transfer applications.

Authors

The authors are at the Chemical Engineering Science

Division, National Engineering Laboratory, National Bureau

of Standards, Boulder, Colorado 80303. Paper received 2 March 1983. The contributions of MJ. Hiza in the preparation of the mixtures is gratefully acknowledged. This work

was carried out at the National Bureau of Standards under

the sponsorship of British Gas Corp., Chicago Bridge and Iron

Co., Columbia Gas Service Corp., Distrigas Corp., Easco Gas LNG, Inc., El Paso Natural Gas, Gaz de France, Marathon Oil Co., Mobil Oil Corp., Natural Gas Pipeline Co., Riillips Petro- leum Co., Shell International Gas Ltd., Sonatrach, Southern

California Gas Co., Tennessee Gas Pipeline Co., Texan

Eastern Transmission Co., Tokyo Gas Co., Ltd., and Transcontinental Gas Pipe Line Corp., through a grand administered by the American Gas Association, Inc.

References

1 Kzz^HJ. Fluid Phase Equilibria 2 (\^l%)n

2 McCarty, RJ). Nat Bur Stand (US), Interagency Report NBSIR 77-867 (1977)

3 McCany, RJ). Nat Bui Stand (US), Technical Note 1030

(1980) 4 McCarty, RJ). / Oiem Thermodynamics 14 (1982) 837

5 Haynes, W.M., Hiza, MJ., Fredenck, N.V. Rev Sci Insrrum 47 (1976) 1237

6 Haynes, WJ*L, Hiza, MJ. / Chem Thermodynamics 9 (1977) 179

7 Hiza, MJ., Haynes, W.M., Paxriah, W.R. J Oiem Thermo- dynamics 9 (1977) 873

8 Hiza, MJ., Hayne*, W.M. / Oiem Thermodynamics 12

(1980) 1 9 Haynes, W.M. /Oiem Thermodynamics 14 (1982) 603

10 Haynes, W.M. Otthobaric liquid densities and dielectric constants of (methane isobutane) and (methane * normal

butane) at low temperatures, submitted to J Chem Thermodynamics

11 Haynes, W.M. /?«v Sci /njrrum 48 (1977) 39

12 Haynes, W.M. Apparatus for density and dielectric measure- ments to 35 MPa on fluids of ayogenic interest, submitted

10 J Res Nat Bur Stand (US)

13 Pan, WJ., Mady, MJL, MiUet, tLCAIOiE J 21 (1975) 283

14 Thompson, Ji., R.T., Miller, R.C Advances in Cryogenic

Engineering 25 (1980) 698

15 Uo, CC Miller, R.C Cryogenics 21(1981 ) 85

16 Straty, G.C, Goodwin, R.D. Cryogenics 13 (1973) 712

17 Patmh, W.R., Hiza, MJ. Advances in (Ilryogenic Engineering 19 (1974) 300

18 fCidnay, AJ., Miller, R.C., Paniih, W.R., Hiza, MJ.

Cryogenics 15 (1975) 531

19 Ely, J.F.. Stiaty, G.C / Chem Phys 61 (1973) 712

20 Weber, 1_A./ Oiem /^ys 65 (1976)446

21 Haynes, W.M., Vounglove, B«A. Advances in Cryogenic

Engineering 27 (1982) 883 22 Btaziet, D.W., Freeman. G.R. Can J Chem 47 (1969) 893

23 MopsilcFJ./Oiem/^yj 50 (1969 ) 2559

24 Moliton, A., Gerbier, J. Revue de Physique Appliquee

6 (1971) 273

25 Betinskii, B.A., Iknmov, Sh. Kh. Sovier Physics-Acousrtcs 18 (1973) 300

26 Dagg,I.R., Reesot,GX.Qjn//*>'j 52 (1974) 29

426 CRYOGENICS .AUGUST 1983

240
NBS-IMA (REV. 2-.ec) U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR

REPORT NO.

NBS Monogr. 172

2. Performing Organ. Report No. 3. Publication Date

October 1983

4. TITLE AND SUBTITLE

LIQUEFIED NATURAL GAS DENSITIES: SUMMARY OF RESEARCH PROGRAM AT THE NATIONAL BUREAU

OF STANDARDS

5. AUTHOR(S)

W. M. Haynes, R. D. McCarty and M. J. Hlza

6. PERFORMING ORGANIZATION (If joint or other tiian NBS. see /n struct/on sj

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract/Grant No.

8. Type of Report & Period Covered

Final

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. Stote, ZIP)

British Gas Corp.; Chicago Bridge and Iron Co.; Columbia Gas Service Corp.; Dlstrigas

Corp.; Easco Gas LNG, Inc.; El Paso Natural Gas; Gaz de France; Marathon Oil Co.; Mobil

Oil Corp.; Natural Gas Pipeline Co.; Phillips Petroleum Co.; Shell International Gas,

Ltd.; Sonatrach; Southern California Gas Co.; Tennessee Gas Pipeline Co.; Texas

Eastern Transmission Co.; Tokyo Gas Co., Ltd.; Transcontinental Gas Pipe Line Corp.

10. SUPPLEMENTARY NOTES

Library of Congress Catalog Card Number: 83-600608

Document describes a computer program; SF-185, FIPS Software Summary, is attached.

11. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a si gnifi cant This report summarizes tiie results of a project concerning the densities of liquefied natural gas (LNG)

and its components. This project, initiated in the Properties of Fluids Section of the Cryogenics Division

of the National Bureau of Standards in July 1972, was carried out under the sponsorship of a consortium of eighteen energy companies.

The experimental part of this project has included the following accomplishments: (a) development of a magnetic suspension densimeter for absolute density measurements on liquids, including liquid mixtures in equilibrium with their vapor, at temperatures from 90 to 300 K; ( b) orthobaric liquid density measurements

on the major components of LNG, which include nitrogen (95-120 K) , methane (105-160 K) , ethane (100-270 K)

,

propane (100-288.7 K) , isobutane (115-300 K) , and normal butane (135-300 K) ; (c) orthobaric liquid density

measurements on approximately thirty-five binary mixtures of the above components for all combinations

except nitrogen + butane systems, primarily in the temperature range of 105 to 130 K; and (d) orthobaric

liquid density measurements on twenty-seven mul t icomponent mixtures (105-120 K), including several LNG-like

mixtures with up to eight components. The total uncertainty of a single density measurement is approximately 0,1 percent at low temperatures and decreases to approximately 0,06 percent at rocm temperature. The

estimated standard deviation of a single density measurement is less than 0,02 percent.

The density data have been used to optimize, test, and compare several mathematical models as to their

suitability for the calculation of LNG densities for custody transfer. Models selected for optimization and testing included an extended corresponding states method, a hard sphere model, a cell model, and an empirical model due to Klosek and McKinley, The ultimate goal of this project was to produce one or more mathematical models that could be used to predict the density of any LNG mixture to within an uncertainty of 0,1 percent from an input of pressure, temperature, and composition. After revisions based on the new experimental data from this project, each of the models investigated here satisfy this goal for typical LNG compositions. The limitations and ranges of validity of the various models are discussed. Also presented

are techniques for predicting LNG densities from dielectric constant measurements and from excess volume

ca leu I at ions.

The last section of this report consists of publications that provide a complete and detailed account of the results of this project,

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolons)

Binary mixtures; density; experimental data; liquefied natural gas; magnetic

suspension densimeter; mult icomponent mixtures; prediction methods; pure fluids.

13. AVAILABILITY

[X] Unlimited

I I For Official Distribution. Do Not Release to NTIS

[X] Order From Superintendent of Documents, U.S. Government Printing Office, Washington. D.C. 20402.

Order From National Technical Information Service (NTIS), Springfield, VA. 22161

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PRINTED PAGES

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15. Price

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