FILE cop:

NBSIR 77-859

DO NOT REM?'

THE THERMODYNAMiC PROPERTIES OF

SLUSH HYDROGEN AND OXYGEN

Cryogenics Division

Institute for 3asic Standards

National Bureau of Standards

Boulder, Colorado 80302

Final Report

Nove Tiber 1977

Prepared for:

NASA-Johnson Soace Center

Hous:on. Texas 7705S

H.M. Roder

iMBSlR 77-859

THE THERMODYNAMIC PROPERTIES OF

SLUSH HYDROGEN AND OXYGEN

H.M. Roder

Cryogenics Division

Institute for Basic Standards

National Bureau of Standards

Boulder, Colorado 80302

Final Report

November 1977

Prepared for:

NASA-Johnson Space Center

Houston, Texas 77058

U.S. DEPARTMENT OF COMMERCE, Juanita M. Kreos. Secretary

Sidney Har-nan, Unaer Secretary

Jordan J. Barucn, Assistant Secretary -'or Science anc Tecnnoicgy

NATIONAL BUREAU OF STANDARDS, Ernest AmDle^ Acting Director
BACKGROUND

During the Gemini, Apollo, and Sky lab programs, cryogenic systems were

used as the source for fuel cell energy production. Throughout these

programs the stored cryogens became one of the main limiting consumables

involving vehicle lifetime. The critical nature of the cryogenic system

required very accurate premission analyses. Not only were quantity

calculations required but pressure range evaluation was also made to ensure

that mission operations could be safely performed.

A potential major improvement in Space Shuttle operations involves the

use of slush cryogens. Slush is the term used to describe a mixture of solid

and liquid phases of a cryogen. The advantage of slush over liquid cryogens

is a higher stored mass for the same occupied volume storage tank. Also,

the latent heat of the solid in the slush increases storage time before

venting occurs. This improved capacity and storage time make possible

increased space flight time and distance.

The Cryogenics Division of the National Bureau of Standards provided

the ?VT data and the equations of state which were used during the Apollo and

Skylab programs under the sponsorship of NASA. The basic data and computer

programs are extended in the present report to provide the support necessary

to an increased effort in slush research and mission analysis. This software

technology is now available for all future space programs and is expected to

have industrial application. ... Paralleling the basic data effort NASA has been a prime driver in

funding the development of slush cryogen technology. A slush hydrogen

program conducted from 1965 to 1970 at NBS investigated several production

methods, measured the pressure drop in flowing slush hydrogen, determined

maximum solid fraction and investigated several instruments to measure solid

fraction . An experimental program currently underway at NBS is expected to

demonstrate the safe and reliable production, storage and handling of slush

hydrogen and oxygen in ground support equipment at NASA launch facilities.

* See page 21, 22 of Nat. Bur. Stand. Technical Note 664 for references.

ii
CONTENTS

Page

1. INTRODUCTION 1

2. BASIC THERMODYNAMIC CONSIDERATIONS 2

2.1 General 2

2.2 The Quality, Q 4 3. THE SOLID-LIQUID REGION FOR HYDROGEN 5 3.1 The Reference Values 5 3.2 The Melting Pressure Equation 5 3.3 The Heats of Fusion at Elevated Pressures 6 3.4 Property Calculations 6 3.5 Thermodynamic Consistency Checks 7 4. THE SOLID-VAPOR REGION FOR HYDROGEN 7 4.1 The Reference Point 7 4.2 Vapor Pressure of the Solid 8 4.3 Density of the Solid 9 4.4 Heat Capacity of the Solid 9 4.5 Thermodynamic Consistency Checks 10

5. THE SOLID-LIQUID REGION FOR OXYGEN 11

5.1 The Reference Values 11

5.2 The Melting Pressure Equation 11

5.3 Property Estimation 12

6. THE SOLID-VAPOR REGION FOR OXYGEN 12

6.1 The Reference Point 13

6.2 Vapor Pressure of the Solid 13

6.3 Density of the Solid 13

6.4 Heat Capacity of the Solid 14

6.5 Thermodynamic Consistency Checks 14

7. THE COMPUTER PROGRAMS FOR SLUSH HYDROGEN AND SLUSH OXYGEN . . 15

7.1 Input 15

7.2 Output 15

7 . 3 Flow Charts 16

7.4 Program Listings, Hydrogen 16

7.5 Program Listings, Oxygen

.

16

7.6 Test Programs, Sample Output 16

717 Unit Conversions 18

8. ERROR ESTIMATES, FURTHER WORK . 18

9. SUMMARY 22

10. REFERENCES 2 3

iii
Page

APPENDIX A. PROGRAM LISTINGS HYDROGEN 26

APPENDIX B. PROGRAM LISTINGS OXYGEN 30

APPENDIX C. TEST PROGRAM HYDROGEN 3 3 APPENDIX D. TEST PROGRAM OXYGEN . 3 5 APPENDIX E. CONVERSION FACTORS, HYDROGEN 3 7 APPENDIX F. CONVERSION FACTORS, OXYGEN 37

SYMBOLS

P pressure

dP

dT derivative of vapor pressure or melting pressure

T temperature T' temperature shifted to a new scale

P density s

specific heat at constant pressure

V volume - 1/p • specific heat at constant volume

H enthalpy AH enthalpy difference

S entropy AS entropy difference

U internal energy cSAT specific heat of saturated solid or saturated liquid

Q quality

A, B, C , a , c , Po coefficients

equations

in vapor pressure or melting pressure

SUBSCRIPTS

liquid

vapor

solid

t

melt

liquid phase

vapor phase

solid phase

triple point

melting line

fusion

sublimation

vaporization

UNITS

The primary variables in the computer programs are Pressure in atmospheres

Density in moles/liter, and

Temperature in Kelvin.

Conversions to other SI units and units normally used in applied problems

are given in Appendices E and F.

iv
THE THERMODYNAMIC PROPERTIES OF SLUSH HYDROGEN AND OXYGEN

Hans M. Roder

Cryogenics Division

Institute for Basic Standards

National Bureau of Standards

Boulder, Colorado 80302

The thermodynamic properties of hydrogen and oxygen have been calculated

for temperatures both below and above the triple point. Values of the various

physical properties required for the computations are either taken from the

literature, or are extrapolated. If extrapolated, the extrapolations are based on the known behavior of other simple fluids. The results are presented in the form of computer programs which cover two regions of the

phase diagram, solid-vapor, and solid-liquid. Input to the programs is

temperature and quality. The properties returned include pressure, density,

enthalpy, entropy and internal energy. The present programs for slush

hydrogen and slush oxygen are an extension of and depend upon the equations

of state developed previously for these gases.

Key words: Computer programs; enthalpy; entropy; hydrogen; internal energy;

liquid; oxygen; PVT; quality; slush; solid; vapor.

1 . INTRODUCTION

Slush is the term used to designate a mixture of solid and liquid

phases of the gas under consideration. There are two major reasons for the

current interest in slush. First, since solid is more dense than liquid,

slush offers a way to carry a larger quantity of the cryogen in a container

of fixed volume, as compared to the liquid alone. Second, since the solid

melts, absorbing heat, slush offers a way to lengthen the time that the

cryogen can be stored in a fixed volume container before venting has to occur.We have included the solid-vapor region in this study because slush is

currently produced by a freeze-thaw process. This means the properties of the

vapor at temperatures just below the triple point are also of concern. Further,

the properties of this two phase region are of interest in another application,

subliming refrigerators, which are used to cool infrared detectors.

The properties of concern in this study are temperature, pressure,

density, enthalpy, entropy and internal energy. Internal energy was added

to the set because the principal applications at present involve containers

of fixed volume. The regions of the phase diagram covered in this report,

solid-vapor and solid-liquid, are both two phase regions. In analogy -o the

familiar quality used in liquid-vapor problems, we have included qualicy as

a parameter in this study.
The thermodynamic properties described in this report are consistent

with and should be viewed as an extension of the two equations of state

developed previously, the equation of state for hydrogen by Roder and

McCarty (1975) , and the equation of state for oxygen by Weber (1977a, 1977b)

The single phase regions of each gas for liquid, and fluid gas or vapor as well as the liquid-vapor two phase region are defined by these equations of

state. The present slush routines call on the equations of state to define

liquid properties along the melting line and vapor properties on the vapor

boundary of the solid-vapor region.

2. BASIC THERMODYNAMIC CONSIDERATIONS

2.1 General

Consider a pressure temperature phase diagram shown schematically in

figure 1. We recognize the melting line, the vapor pressure curve of the

liquid and the vapor pressure of the solid. These curves define the

boundaries of the various phases, solid, liquid, and vapor. They intersect

at the triple point. It is from a plot such as this that the triple point

takes its name. What this phase diagram does not show is the variation of

density. One unique aspect of any two phase regime liquid-vapor solid-vapor

or solid-liquid is that the pair P, T on a phase boundary encompasses a large range of densities. We could consider a plot in the three variables

pressure, density and temperature, however, consider instead the more

familiar T-S diagram shown schematically in figure 2. This diagram is also

a phase diagram. We recognize the single phase regions solid, liquid and

vapor. We can also see the various two phase regions where two phases are

in equilibrium. In a diagram of this kind the triple point gets stretched

into a line. The density variation can also be included. For example, the

density at point 1 in figure 2 corresponds to the density of the solid at the

triple point, point 2 to the liquid triple point density, and point 3 to the

vapor density at the triple point.

The regions already defined by the major equations of state are shown

shaded- We will, in this report, cover the two phase regimes solid-liquid

and solid-vapor. If we assume that the P-T relation of the boundary is

available, then to define a region we need only define the properties at the

boundaries. For the true slush region, solid-liquid, we would have to know

density, enthalpy, entropy and internal energy at both the liquid and solid

boundaries. The liquid side is already defined for us by the intersection

of the equation of state and the melting pressure curve. It remains to

define the other boundary in this case the solid side. The actual approach

depends on what pieces of experimental data are available to guide us, and

these are discussed in detail in sections that follow.

2
Melting line

LU

TEMPERATURE

Figure 1. Pressure vs. Temperature, schematic.

TEMPERATURE

;

,

Solid

-a

=3 Li

"o

00 >

CP.

quid^^--*-*s^

j/^Liquid

\Vapor

Vapor \

Solid & Vapor S;;;;;^; ENTROPY

igure 2. Temoerature vs. Entropy, schematic shaded

area is covered by the 3 2-term MBWR.

3
A fundamental relation governs the various two phase regimes. This

equation is called the Clausius-Clapeyron equation, and we will use it

extensively. Written out for each of the three two phase regimes the

equation reads as follows:

Solid-Liquid Solid-Vapor Liquid-Vapor

dP _ AH (Fusion) dP _ AH (Sublimation) dP _ AH (Vaporization)

dT T(V, ^ . .-,-V , . ,) dT T(V -V ,.,) dT T (V -V,. .,) liquid solid vapor solid vapor liquid

The equation relates a latent heat, the pressure-temperature derivative

at the appropriate phase boundary and the volume change corresponding to the

phase change in question. If the available data are redundant, then we are

in a position to make thennodynamic consistency checks. Quite often the list

of available data is minimal and the opportunity for consistency checks is

lost. When elements of data are missing we are forced to calculate the

unknown pieces of information from the Clausius-Clapeyron equation. For

example, the volume change on fusion is usually not measured experimentally.

The heat of fusion is normally measured only at the triple point, not at

elevated temperatures or pressures. If we wish to know the volume change on

fusion, say at the triple point, then we must calculate it from the

experimentally measured heat of fusion and the derivative of the melting

pressure equation.

2.2 The Quality, Q

The concept of quality is familiar to most engineers who have solved

liquid-vapor problems. The conventional quality is defined as the percent

(by mass) of the vapor phase present, or:

.,) for volume, and similarly liquid ^

liquid^

, . . , ) and liquid

liquid

Since it is easy to make an error we caution that the expression in teims of

density is

Q

= (V -V, . • .)/(V liquid vapor

-V

Q

= (H -H, . • -,)/ (H liquid vapor

-H

Q

= (S -S, . • -,)/(S liquid vapor

-S

Q

= (U -u, . • -,)/ (U liquid ' vapor

-u

Q

=

Pi • J • P liquid vapor

p ' vapor / ( P 1 • - P ) liquid vapor

From these equations it is apparent that Q = 0.0 on the liquid side, i.e.,

the denser side, and that Q = 1.0 on the vapor side, i.e., the less dense

side.

4
For the other two phase regimes solid-liquid and solid-vapor we will

use a defined quality Q defined in a manner quite analogous to that for

liquid-vapor above- Q will be 0.0 on the dense side, i.e., the solid, and

Q will be 1.0 on the less dense side. The equations read as follows:

Solid-Liquid Solid-Vapor

Q = (V-^'solid^/^^liquid-^solid^ Q = (^-^solid^/^^vapor-^solid^

Q = («-«S0lid)/(«liquid-«S0lid) Q = (H-H3^,.^)/(H^^p^^-H3^^.^)

Q = (S-S T.,)/(S, . .,-S ,.,) Q = (S-S T.,)/(S -S , . ,) ^ solid ' liquid solid solid ' vapor solid

Q = (U-U ..,)/(U, . .-,-U Q = (U-U ..,)/(U -U , . ,) solid^ liquid solid solid vapor solid

3. THE SOLID-LIQUID REGION FOR HYDROGEN

Prior work on the thermodynamic properties of this region include the

experimental work and the calculations by Dwyer, et al. ( 1965a, b) and by

Cook, et al. (1965) . In the present report the thermodynamic properties

along the liquid boundary of this region are defined from the intersection

of the 32-term modified Benedict-Webb-Rubin (MBWR) equation of state as presented by Roder and McCarty (1975) with the melting pressure equation

(Goodwin, 1962) . The elements of data available to describe the region are limited. In the sections that follow we describe the reference values chosen

for the property calculations, the melting pressure equation of the solid,

the heats of fusion at elevated pressures, the property calculations, and

thermodynamic consistency checks.

3.1 The Reference Values

Since the available set of data is limited we are forced to calculate

changes in properties for this region from the Clausius-Clapeyron equation.

To obtain values for the solid boundaries these changes are applied from

the liquid boundary. In effect, for a given temperature the values obtained

from the 32-term equation of state and the melting line are used as starting

point, or point of reference. At the triple point temperature the procedure

leads to values for the solid that are used as reference point for the

solid-vapor region (see section 4.1).

3.2 The iMelting Pressure Equation

The equation used is the one developed by Goodwin (1962), except that

we shift the temperature scale by 0.003 K to match the triple point

temperature and pressure of the liquid vapor pressure curve used with the

32-term MBWR. The shift insures that the melting pressure equation and the

liquid vapor pressure equation intersect at the triple point, 13.800 K and

0.0695 atm, which are the values chosen for the basic 32-term equation of

5
state. With the temperature shift Goodwin's equation becomes:

P - P

T^-^ = A e - ^ ^ BT'

where t' = T + 0.003 in kelvin, A = 30.3312, a = 5.693, B = 2/3, P^ = 0.0695

atm and = 13.803.

3.3 The Heats of Fusion at Elevated Pressures

These values were determined experimentally by Dwyer, et al. (1965b).

The authors represent the experimental results by a straight line in pressure.

We select the value of 117.277 J/mol (28.03 cal/mol, Woolley, et al., 1948)

for the heat of fusion at the triple point, partially to remain consistent

with the earlier thermodynamic calculations, and partially because this value

is quite close to the average of all experimental determinations (see table

9-3, Roder, et al., 1973). Shifting the Dwyer, et al. equation to pass

through the selected value the equation used for the heats of fusion

becomes

AHp . = (28.02693164 + 0.044149P) • 4.184 fusxon

where AH^ • is in J/mol and P in atm. The uncertainty in AH^; fusion ' ^ fusion

remains essentially as estimated by Dwyer, et al. (1965b), around 1%.

3.4 Property Calculations

To find the values of the various properties on the solid boundary we

calculate the volume and entropy changes from the Clausius-Clapeyron equation.

The melting pressure equation and the heats of fusion were given in the

previous sections. The relations are:

P = P,,elt(T^ .

dF ^^^melt^

dT dT

AH (fusion) V = V,. --.-AV^ . with AV^ . - solid liquid fusion fusion dP

dT

H T J ~ Ht . • — AHj- solid liquid fusion

, . -, = S, . . -, - AS^ . , with AS^ . = AH^ . /T. solid liquid fusion fusion fusion

6
The internal energy for the solid follows from the thermodynamic relation

U = K - PV.

3.5 Thermodynamic Consistency Checks

Since the available set of data is limited, the checks that can be made

are limited also. We can examine the change of volume on fusion as a function of temperature. We expect the volume change to decrease as the

temperature increases, it does. The values obtained from the heats of fusion

through the Clausius-Clapeyron equation are in excellent agreement with the

experimental volume changes published by Bartholome (1936) for normal

hydrogen as shown in table 1.

Table 1. Volume Changes on Fusion at Elevated Temperatures

T Bartholome This Report % 13.96 2.83 2.81 .58

16.43 2.31 2.26 2.15

18.24 2.03 2.02 .61

Considering nitrogen and argon we would expect that the entropy change

on fusion for hydrogen should decrease as the temperature increases.

Quantitative comparisons cannot be made; however, this expectation is at

least qualitatively fulfilled.

4. THE SOLID-VAPOR REGION FOR HYDROGEN

Prior work on the thermodynamic properties of this region include

calculations by Mullins, et al. (1961), the T-S chart by Sindt and Mann (1966)

and the survey by Roder, et al. (1973) . The thermodynamic properties along

the vapor boundary of this region are available from the intersection of the

32-term MBWR (Roder and McCarty, 1975) and the solid vapor pressure curve

defining the boundary of the region. The elements required to complete the

description of the region are the reference point for all properties, the

vapor pressure equation of the solid, the density of the solid, and the heat

capacity of the solid.

4.1 The Reference Point

we wish to integrate this region in a thermodynamically consistent way

with the basic 32-term equation of state. The simplest thing we can do is to

refer the properties of the solid at the triple point to those obtained from

the 32-term MBWR for the liquid at the triple point. Specifically, we require that at the triple point temperature:

7
vapor pressure of the solid = vapor pressure of the liquid

solid Ixquid fusion

solid liquia fusion

S = S — AS solid liquid fusion

Internal energy of the solid follows through the thermodynamic relation

U = H - PV. AHc . , the heat of fusion selected for the triple point is fusion ^ 117.277 J/mol (see section 3.3). AS^ , the entropy change on fusion is fusion '^-^ ^ defined thermodynamically as AH^ /T. AV_ . , the volume change on ^ fusion' fusion ^ fusion is calculated from the Clausius-Clapeyron equation and the melting

pressure derivative (see section 3.4). The result for the triple point with

= 13. 800 K is AV^ =2.86. cm'^/mol.

t fusion 4 Having established a reference point or benchmark we can look next at

the description of the remainder of the region.

4.2 Vapor Pressure of the Solid

One of the drawbacks of the earlier calculation by Mullins, et al. (1961)

is that the authors presented no explicit equation for the vapor pressure of

the solid, indeed they could not because one can either force exact

thermodynamic consistency or use an explicit vapor pressure equation, but not

both. Experimental measurements of the vapor pressure of solid hydrogen

exist to quite low temperatures (Woolley, et al., 1948; Borovik, et al . , 1960;

and Harrison, et al., 1962). However, we will use these values only as a check on the extrapolation of the vapor pressure curve to low temperatures.

The form of the vapor pressure equation is dictated by thermodynamics.

Integration of the Clausius-Clapeyron equation assuming the vapor phase to

be an ideal gas leads to the Kirchhof f -Rankine equation:

log P = A/T + B log T + C

We use this equation but subject it to the following constraints:

1. P^ = 0.0695 atm, T^ - 13.800 K. To insure that this equation

intersects the melting equation and the liquid vapor pressure equation

at the triple point which is defined by the liquid vapor pressure

equation used with the 32-term MBWR.

dP

2. ^ at the triple point temperature = 0.04566744 atm/K. This constraint

defines the derivative dP/dT of the solid vapor pressure curve at the

triple point temperature, and through the Clausius-Clapeyron equation

the heat of sublimation. The value of the derivative is chosen so that:

AH^ , , . ^. = AH^ + AH . ^.

. . sublimation fusion vaporization
where IH- is the value chosen in sections 3.3 and 4,1, and the rusion

IH is determined by the 32-term MBWR and its companion vaporizarion ^ liquid vapor pressure equation.

3. That the vapor pressure at 10 K be 0.00255940 atm. The last requirement

is the best compromise to insure thermodynamic consistency of the heat

of sublimation down to 10 K. The point is discussed in more detail in

the section on thermodynamic consistency below.

With 3 degrees of freedom and three constraints the coefficients of the

solid vapor pressure equation are uniquely determined. They are:

A = -90.77568949 B = 2.489830940 and C = 4.009857354

where P is in mm Hg and T is in K.

Intercomparison to the available experimental vapor pressures shows the

maximum deviation in pressure down to 10 K is less than 1%. The simple

equation cannot, however, represent the data down to low temperatures, say

3 or 4 K. In this temperature range the departure between predicted and

experimental vapor pressures increases to nearly 200%. Additional terms in

the vapor pressure equation improve the representation of the vapor pressures

but not the heats of sublimation calculated from the vapor pressure derivative.

4.3 Density of the Solid

From the survey by Roder , et al. (197 3) we select the solid volumes

published by Ahlers (1963) primarily because we expect these values to be in

accord with the heat capacity measurements selected in the next section. More

recent publications (Anderson and Swenson, 1974) prefer values which differ

by as much as 1.5% from the ones selected especially at the lower temperatures. Ahlers' results can be represented by a parabolic function from 6 to

13.3 K. We shift this function to pass through the designated value of the

solid density at the triple point. The resulting relation is:

rr^ = V = 22. 77963478 - 0. 031T + 0.005T^

^solid ^""^'-^

with V , . , in cm"/mol and T in K. We estimate that the values of the sclxa

solid should be in error by no more than 0.4% at the triple point, and that

the error could increase -o as ~uch as 1.5% at 4 K or lower.

4,4 Heat Capac:ity of -che Solid

We require heat capacity values for the solid at saturarion . The values

selected were derived by Ahlers (1963) from his measurements of C^. in -he

3::.ngle phase solid. We orefsr these values to earlier measuremen-cs because

9
a closed loop third law check of entropy using these values is much more

consistent than checks using the earlier measurements (See Roder, et al.,

1965) . Since several conventional representations do not seem to apply to

the values for hydrogen, we settled on a simple interpolating polynomial

which returns Ahlers' values to within 0.3% from 14 K down to 5 K» Maximum

deviations down to 3 K are 1.65%, but the polynomial should not be used below

3 K. The power series used to represent C^^^ is:

C„-^ = -101. + 87.75 T - 26. 95833333 + 4.25 T"^ - 0.041666667

with Cg^^ in milli Joules/mole and T in K.

We use the heat capacity in the thermodynamic relations:

^« = ^l"" Sat = /^^ dT

T

to calculate the changes in enthalpy and entropy from the reference values

for the solid at the triple point.

4.5 Thermodynamic Consistency Checks

Once the vapor pressure of the solid is defined, we can perform a consistency check in the following way. We calculate the heat of sublimation

first by taking the difference between vapor and solid enthalpies at the

boundary, and second by calculating it from the Clausius-Clapeyron equation

using solid and vapor volumes and the derivative of the solid vapor pressure

curve. The enthalpy of the vapor is obtained from the 32-term equation of

state, we note that these enthalpies depend on the specific heat of the ideal

gas which is not defined below 10 K in the equation of state programs. The

enthalpy of the solid is obtained from the integration of the heat capacity

of the solid. The difference of these two values, the heat of sublimation,

calculated in this way is not particularly sensitive to the value of pressure

chosen. In the other calculation, however, the heat of sublimation depends

on the vapor volume obtained from the equation of state and on the derivative

of the solid vapor pressure curve and is quite sensitive to changes in

pressure. We select the value of the pressure at 10 K, the third constraint,

to minimize the departures between the heats of sublimation calculated by the

two different methods between 13.8 and 10 K. The intercomparison of the

heats of sublimation is shown in table 2; the departures do not exceed 0.1%.

10
Table 2. Intercomparison of the Heats of Sublimation of Hydrogen

T AH (sublimation) from AH (sublimation) from the Difference

K the boundaries J/mol Clausius-Clapeyron Eq. in percent

J/mol J/mol

10 961.234 960.300 0.10

11 979.262 979.169 0.01

12 996.088 996.584 -0.04

13 1011.351 1011.900 -0.05

13.8 1022.137 1022.137 defined to be the same

5. THE SOLID-LIQUID REGION FOR OXYGEN

With the exception of the melting curve and the heat of fusion at the

triple point prior work on the properties of this region is non-existent.

We are therefore forced to estimate all of the thermodynamic properties from

the Clausius-Clapeyron equation using other simple gases as model fluids.

The thermodynamic properties along the liquid boundary of this region are defined from the intersection of the 32-term MBWR equation of state and the

melting pressure equation presented by Weber (1977b). Weber's version of

the 32-term MBWR rather than an earlier version (Stewart, et al., 1972) is

used because it includes new values of PVT (Weber, 1977a) which go to high

pressures {"^ 800 bar). An additional advantage of Weber's formulation is

that the vapor pressure curve and the melting line intersect at the defined

triple point. In the sections that follow we describe the reference values

chosen for the property calculations, the melting pressure equation of the

solid, and the property estimation.

5.1 The Reference Values

Since the available set of data is limited, we are forced to calculate

changes in properties for this region from the Clausius-Clapeyron equation.

To obtain values for the solid boundaries these changes are applied from the

liquid boundary. In effect, for a given temperature the values obtained

from the 32-term equation of state and the melting line are used as a starting

point, or point of reference. At the triple point temperature the procedure

leads to values for the solid that are used as a reference point for the

solid-vapor region (see section 6.1). The enthalpy departure at the triple

point has been estimated by Weber (1977b) to be about 3 J/mcl.

5.2 The iMelting Pressure Equation

The equation used is the one developed by Weber (1970) as modified by

him (Weber, 1977a, b) to pass through the defined triple point. The equation

reads

11
with = 54.359 K, = 0.0014451 atm, P^ = 2637.33 atm, and c = 1.769.

5.3 Property Estimation

To estimate values of the various properties on the solid boundary of

this region we will use the Clausius-Clapeyron equation. To do this we must

make an educated guess as to how the heat of fusion, or the volume change on

fusion, or the entropy change on fusion depends on pressure (or temperature)

.

Considering the known behavior of hydrogen, argon, and nitrogen we postulate

that oxygen should obey the following qualitative criteria:

1. The heat of fusion should increase as either pressure or temperature is

increased,

2. The entropy of fusion should decrease as either pressure or temperature

is increased, and

3. The volume change on fusion should decrease as either pressure or temperature is increased.

Direct scaling of the volume change for oxygen from hydrogen, argon, or nitrogen (see page 53, Sindt, Ludtke and Roder , 1970) is not successful; one

of the criteria remains unfulfilled. Adopting a sort of an average between

the two extreme assumptions

1. AH^ . (T or P) = constant, and fusion ^ ' ' ^* '^^fusion ^"^ °^ ~ constant

allows us to fulfill all three criteria over a range of experimentally

plausible pressures, say up to 5000 psia. As for hydrogen we express the

heat of fusion of oxygen as a linear function of pressure

^^fusion " (106.3 + 0.01126 P) • 4.184

where ^^lf^sj_Qj^ iri J/mol and P is in atm. The value 106.3 cal/mol is

taken at the triple point from the paper by Giauque and Johnston (1929)

.

The uncertainty in the estimated heat of fusion should be no larger than the

variation estimated for it, i.e., a maximum of 6% at the highest pressures.

With an estimate for the heat of fusion in hand the property calculations

follow the relations given in section 3.4. Obviously, consistency checks

cannot be made for this region.

' 6. THE SOLID-VAPOR REGION FOR OXYGEN

Prior work on the thermodynamic properties of this region includes the

calculations by Mullins, et al . (1962) . The thermodynamic properties along

,:' 12
the vapor boundary of this region are available from the intersection of the

32-terin MBWR (Weber, 1977b) and the solid vapor pressure curve defining the

boundary of the region. The elements required to complete the description

of the region are the reference point for all properties, the vapor pressure

equation of the solid, the density of the solid, and the heat capacity of

the solid.

6.1 The Reference Point

The reference point for this region is defined in exactly the same way

as for hydrogen (see section 4.1). The heat of fusion at the triple point

was experimentally measured by Giauque and Johnston (1929) to be 106.3 cal/mol

(444.8 J/mol) . The triple point temperature, 54.359 K, and the melting

pressure equation are taken from Weber (1977b). The volume change on fusion

3 at the triple point becomes approximately 0.94 0g cm /mol.

6.2 Vapor Pressure of the Solid

The same general remarks apply to oxygen as they did to hydrogen. The

vapor pressure equation is of the Kirchhof f -Rankine type

log P = 1096^562485 _ 2.025578307 log T + 28.35976524

with T in kelvin and F in mm Hg. The constraints used are:

1. P^ = 0.0014451 atm, T^ = 54.359 K,

2. ||| = 0.000482427 atm/K, and

3. P = 0.000019200 atm at 44 K.

Intercomparisons with the available experimental vapor pressures of

Aoyama and Kanda (1936) down to the S-y solid transition show departures of

up to 30% in pressure.

6.3 Density of the Solid

Oxygen is an interesting solid because it occurs in three different

solid structures. We are interested here only in the so-called y-solid

which exists between the temperatures of 43.801 K (Kemp and Pickup, 1972) and

the triple point, 54.359 K. For this range of temperatures the thermal

expansion measurements of Manzhelii, et al . (1966) the absolute measurement

of Tolkachev and Manzhelii (1965) and the x-ray measurements of Barrett,

et al. (1967) are in reasonable agreement, i.e., within 0.6%. We represent

the experimental results with a parabolic function forced to pass through

the designated value at the triple point. The relation is

13
= V = 23.2808187 - 0.06772142868 T + 0.001339285715

p , . , solid solid

with V , . - in cm'^/mol and T in K. The solid densities should be in error soiia

by no more than 0.6% in the range of temperatures covered.

6.4 Heat Capacity of the Solid

The experimental measurements on the y-solid by Giauque and Johnston (1929)

were represented by Mullins, et al. (1962) with a simple polynomial which is

retained here

C^^„ = 16.908081 - 0.24181777 T + 0.0024809089 T^ SAT

with Cg^rp in cal/mol (4.184 J/cal) and T in kelvin. Departures of the

polynomial from the measured points are no larger than the stated error in

the measurements, i.e., about 0.5% in Cg^^f

6.5 Thermodynamic Consistency Checks

These checks are performed as described in section 4.5 for hydrogen,

except that here the temperature limitation is that of the B-y transition,

i.e., 43.801 K. Results are shown in table 3. Departures between the

different ways of calculating the heat of sublimation do not exceed 0.06%.

Table 3. Intercomparison of the Keats of Sublimation of Oxygen

AH , , . ^ . AH , , . , . sublimation sublimation

T, K from the boundaries from the C-C Eq. Difference

J/mol J/mol J/mol

44.000 8376.665 8376.109 .556

45.000 8359.501 8359.234 .267

46.000 8342.413 , 8342.344 .069

47.000 8325.380 8325.433 -.054

48.000 8308.379 - 8308.497 -.118

49.000 8291.388 8291.527 -.139

50.000 8274.383 3274.512 -.129

51.000 8257.341 8257.443 -.102

52. 000' 8240 . 238 . "

•, 8240.305 -. 067

53.000 8223.050 ' 8223.083 -.033

54 . 000 . . 8205. 751 8205.758 -. 007

54.359 ' 8199.509 8199.509 defined to be the same

14
7. THE COMPUTER PROGRAMS FOR SLUSH HYDROGEN AND SLUSH OXYGEN

The phase boundaries and representations of the different properties

along the phase boundaries described in the previous sections have been

combined into two computer programs. The programs are in the form of

subroutines

SLH2(TT, QQ, TP, PSL, DSL, HSL, SSL, USL)

SL02(TT, QQ, TP, PSL, DSL, HSL, SSL, USL)

where the variables in the parameter list are temperature, quality, triple

point key, pressure, density, enthalpy, entropy, and internal energy.

7 . 1 Input

The user must specify temperature and quality. He may specify the

triple point key depending on the temperature.

Temperature . The input temperature in kelvin should be between 10 and

22.60 K for hydrogen and between 44 and 58.2 K for oxygen. The programs

will return answers for temperatures less than the lower limit, but the

values returned become less and less reliable the lower the temperature. For

temperatures greater than the upper limit an error message is printed: "T is

too high, P will exceed 5000 psia."

Quality . A value for quality must be entered such that 0.0 <_ Q <_ 1.0.

A quality of 0.0 will cause the subroutine to return values on the denser

side, i.e., the values of the solid. A quality of 1.0 causes the subroutine

to return values on the less dense side, i.e., values of the vapor for the

solid-vapor region (temperatures below the triple point) , and values of the

liquid for the solid-liquid region (temperatures above the triple point) . If

the input Q is fractional the subroutine calculates the values of the

properties on both boundaries of the region and then performs the ratio

according to the equations given in section 2.2.

Triple point key . For the special case when the input temperature is

exactly equal to the triple point temperature, 13,800 K for hydrogen and

54.359 K for oxygen, the user must specify through the triple point key, TP,

in which region he wants the subroutine to calculate the properties.

TP = 0.0 means that the program will return properties from the solid-liquid

region^ while TP = 1.0 insures that the program calculates the triple point

properties in the solid-vapor region.

7.2 Output

The output from the program is handled through the parameter list. For

a given input of T and Q (TP if appropriate) the output is unique, i.e.,

single-valued. The values returned are the slush pressure PSL in atm, the

15
slush density DSL in mol/L, the slush enthalpy HSL in J/mol , the slush

entropy SSL in J/mol-K, and the slush internal energy USL in J/mol.

7.3 Flow Charts

The slush routines are detailed in the flow chart of figure 3. The

various phase boundaries and the interiors of the solid-liquid and solid-vapor

regions are keyed according to the inset, which is just a repeat of the phase

diagram already shown in figure 2.

7.4 Program Listings, Hydrogen

A listing of the hydrogen program is given in Appendix A. The slush

hydrogen routine calls on the program deck of the parent 32-term MBWR (Roder

and McCarty, 197 5) . Those routines of the parent deck that had to be

modified to insure consistency around the triple point are also listed in

Appendix A. They are:

FUNCTION PRESSM(TT) , and

SUBROUTINE RHOl (PP , DD , TT)

.

New routines developed for the slush application and listed in Appendix A are:

FUNCTION DPDTMP(TT), and

FUNCTION FINDM(P,T,D)

.

The first is simply the temperature derivative of the melting pressure curve,

while the latter is an additional density finder. The FINDM routine differs

from the normal FINDD density finder in that it requires a reasonable trial

density in its calling parameters.

7,5 Program Listings, Oxygen

A listing of the oxygen program is given in Appendix B. The slush

oxygen routine calls on the program deck of the parent 32-term iMBWR (Weber,

1977b) . Routines not available with the parent deck, or those that had to

be modified are listed in Appendix B as well. They are:

FUNCTION FINDT(P,D)

SUBROUTINE T1(PP,DD,TA)

FUNCTION PMELT(PP)

FUNCTION PRESSM(TT) .V

FUNCTION DPDTMP(TT)

7.6 Test Programs, Sample Output

A sample program for hydrogen is shown in Appendix C ; in Appendix D for

oxygen. The output from the test program for hydrogen in the form of an abbreviated table of values with particular attention to the triple point is

given in table 4; the companion output for oxygen in table 5.

16
Subroutine SLH2 (TT,aQJP,PSL,OSL,HSL.SSL.USL)

Subroutine SLO2 (n.QQJP,PSL.DSL.HSL,SSL.USL)

Enter with temperature TT and quality QQ

QQ = 0 Properties on the denser side QQ = 1 Properties on the less dense side QQ Fractional —^

mixture properties

A - Solid below TP

B - Vapor below TP

C - Mixture below TP

D - Liquid above TP

E - Solid above TP

F - Mixture above TP

Special Case, Temp. TT = Triple Point

TP=0. for Solid-Liquid Properties

TP=1. for Solid-Vapor Properties

^ Calculate vapor properties below T.P ^ and mixture properties if appropriate

Figure 3. Flow Chart of the Subroutines SLH2

SLO,

17
The sample programs are intended to serve as test decks for running at

other installations. They include a call to every possible combination of T

and Q in the regions solid-liquid and solid-vapor. To round out the list of

examples the liquid-vapor region, which is obtained from the parent 32-term

MBWR, has also been illustrated.

The astute observer will note that the various properties in lines 4 and

7 of tables 4 and 5 and also lines 6 and 14 and lines 9 and 13 should be

pairwise identical, except of course for Q and TP. To the extent that these

lines are not identical thermodynamic inconsistencies remain. One of the

inconsistencies arises from the fact that we carry 10 digits in the

coefficients of the vapor pressure equations and the melting line. The

pressures returned from each of the different functions at the triple point

temperature are very nearly the same, however, they are not identical in

every bit of the computer word. Entering the 32-term equation of state with

slightly different values of the pressure results in slightly different

values of vapor densities returned. The disagreement in entropy between lines

6 and 14 arises from an inconsistency inherent in the equation of state. This

is shown simply by the fact that the AH between lines 13 and 14 of tables 4 and 5 divided by the triple point temperature is not equal to the AS betv/een

lines 13 and 14. Several other discrepancies are attributable to rounding

errors. ; 7.7 Unit Conversions

Common conversion factors for use with hydrogen are given in Appendix E,

while those for use with oxygen are shown in Appendix F.

8. ERROR ESTIMATES, FURTHER WORK

Whenever possible estimates of uncertainty were given in the sections

that described the fitting or representation of a particular piece of

experimental data. We summarize these uncertainties as maximum errors in

the different properties used in the computer programs in table 5.

18

oIo

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0^

c:

0)CP>1 o

CQ

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CM in

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CM

(O •1-4 ro PO

-H 0

in iH*

(•J CM ^

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ir^ CM in

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in in PO

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*> Q

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l-H t—i a: 3 0 -J 0 a Q. a Q.

«ar>

—

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(T>

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PO PO CD • • • J- J- m in in in

20
Table 6. Estimates of Maximum Uncertainties

Variable

temperature

pressure

density

Hydrogen Oxygen

0.01 K 0.01 K

1% at the triple point 1% at the triple point

increasing to 200% at very increasing to 31% at 44 K.

low temperatures. Negligible Negligible for pressures above

for pressures above 1 atm.

0.2% at temperatures above

T.P. 0.4% at the triple

point increasing to 1.5% at

4 K.

1 atm.

0.2% at temperatures above

T.P. 0.6% for temperatures

below the triple point and

above the 6-Y transition.

enthalpy,

internal

energy

entropy

1.2 J/mol for solid-liquid

1.5 J/mol for solid-vapor

0.08 J/mol-K for solidliquid

.146 J/mol-K for solid-vapor

3 J/mol at the triple point

increasing to 27 J/mol at the

highest pressures.

3 J/mol at the triple point

increasing to 6 J/mol at the

lowest temperature.

0.05 J/mol-K at the triple point

increasing to 0.46 J/mol-K at

the highest pressures.

0.05 J/mol-K at the triple point

increasing to 0.13 J/mol-K at

the lowest temperature.

Entropy errors are approximated by considering enthalpy errors divided

by the temperature and vice versa. For hydrogen the entropy departure for

the solid-vapor region is taken from the closed loop third law calculation

of Roder, et al. (1965) . For the solid-liquid region we use the experimental

estimate of 1% in measured heats of fusion published by Dwyer, et al. (196 5b)

For oxygen we use a value of 3 J/mol as error estimate in enthalpy (Weber,

1977b) and add to this either 6% of the heat of fusion to get the solid-liquid

departures, or the enthalpy equivalent of 0.5% of the values of C^^rp to get

the solid-vapor deviations. The rather large discrepancies described in an earlier version of this report (letter report, June 1977) have indeed been

resolved, i.e., reduced considerably, by using a PVT surface which is

consistent with the melting line and the triple point.

The work described here has led indirectly to a review of the molar

volume of solid oxygen (Roder, 1977). One conclusion of that review is that

the volume change on fusion at the triple point is uncertain by about 0.6%.

Further experimental work to accurately measure this volume change at

essentially zero pressure is recommended. Additional experimental work on oxygen may be required particularly if large scale use of slush oxygen and

hydrogen is likely. The sponsor should arrange to have the dielectric

21
constant of oxygen reviewed and measured for the solid since this property

will be required for quantity gaging. In section 5 we indicate that the

thermodynamic properties for the solid-liquid region for oxygen had to be

estimated. Some experimental measurements are required to define the

properties adequately. We suggest that measurements of the heat of fusion

or the volume change on fusion be conducted for oxygen at elevated pressures.

An alternative possibility is to measure heat capacity values for oxygen in

the solid state. Finally, a review of Prandtl and Grashoff numbers should

be made, to include values of C , C thermal conductivity and viscosity.

P V

Several of these properties are presently not defined for a slush regime, yet

they will be needed to define heat transfer characteristics.

9. SUMMARY

Computer programs to describe the thermodynamic properties of slush

hydrogen and slush oxygen have been developed. The range of temperature for

these programs is from several degrees below the triple point to several

degrees above the triple point. The upper temperature limit coincides with

2 a pressure limit of about 5000 psia (34 MN/m ) . In the case of oxygen the

experimental data are very limited and the properties calculated are in the

nature of a "best estimate." The equilibrium and thermodynamic properties

included are temperature, pressure, density, enthalpy, entropy, internal

energy, and quality. Input to the programs is temperature and quality, the

remaining variables are outputs.
10. REFERENCES

1. Abler s, G. , Some properties of solid hydrogen at small molar volumes,

Rept. No. UCRL-10757 (Sep 1963), Contr . No. W-7405-ENG-48 , (Ph.D.

Thesis)

2. Anderson, M. S., Swenson, C. A., Experimental compressions for normal

hydrogen and normal deuterium to 25 kbar at 4.2 K, Phys. Rev. BIO , 5184-91 (Dec 1974)

3. Aoyama, S. I. and Kanda, E., Vapour tensions of oxygen and nitrogen in

the solid state, Sci. Repts. Tohoku Univ. First Ser. Vol 23, No. 4,

107-15 (1935)

4. Barrett, C. S., Meyer, L., and Wasserman, J., Expansion coefficients

and transformation characteristics of solid oxygen, Phys. Rev. 163 , 851-4 (Nov 1967)

5. Bartholome, E., Zuer theinnischen und calorischen Zustandsgleichungen

der kondensierten Wasserstoff istopen. ***0n thermal and calorimetric

equations of state of condensed hydrogen isotopes, Z. Physik. Chem. B33 , 387-404 (1936).

6. Borovik, E. S., Grishin, S. F. and Grishina, E. Ya . , The vapor pressure

of nitrogen and hydrogen at low pressures, Soviet Phys. Tech. Phys. 5^, 506-11 (Nov 1960); English Translation from Zhur. Tekh. Fiz. 30.' 539-45 (May 1960)

7. Cook, G. A., Dwyer, R. F., Berwaldt, 0. E., et al.. Pressure-volumetemperature relations in solid parahydrogen , J. Chem. Phys. 43_, 1313-16

(Aug 1965)

8. Dwyer, R. F., Cook, G. A., Berwaldt, 0. E. and Nevins, H. E., Molar

voliame of solid parahydrogen along the melting line, J. Chem. Phys. 43 , 801-5 (Aug 1965a)

9. Dwyer, R. F., Cook, G. A., Shields, B. M. , et al.. Heat of fusion of

solid parahydrogen, J. Chem. Phys. 42_, 3809-12 (Jun 1965b) ,

10. Giauque, W. F. and Johnston, H. L. , The heat capacity of oxygen from

12 degrees K to its boiling point and its heat of vaporization. The

entropy from spectroscopic data, J. Am. Chem. Soc. 51_, 2300-21 (Aug

1929).

11. Goodwin, R. D., Melting pressure equation for the hydrogens. Cryogenics

2 , 353-55 (Dec 1962) .

23
12. Harrison, H. , Fite, W. L. and Guthrie, G. L. , Chemical reactions using

modulated free radical beams. The vapor pressure of solid hydrogen in

the temperature range from 4.7 degrees K to 11.1 degree K, Final Rept.

No. AFOSR-2357, Contr. No. AF 49 (638) -301, GA-2972, Pro j . No. 38 (Feb

1962).

13. Kemp, W. R. G. and Pickup, C. P., The transition temperature of solid

oxygen. Temperature, Its Measurement and Control in Science and Industry

Vol 4, Pt. 1, Proc. Fifth Symp. on Temperature, Washington, D.C., Jun

21-24, 1971. Instrument Society of America, Pittsburgh, Pa., pp 217-24

(1972).

-- \. 14. Manzhelii, V. G. , Tolkachev, A. M. and Voitovich, E. I., Thermal

expansion of crystalline nitrogen, oxygen, and methane, Phys. Status

Solidi 13_, 351-8 (1966) .

15. Mullins, J. C, Ziegler, W. T. and Kirk, B. S., The thermodynamic

properties of parahydrogen from 1 degree to 22 degrees K, Georgia Inst,

of Technol., Atlanta, Tech. Rept. No. 1, Contr. No. CST-7339, Pro j

.

No. A-593 (Nov 1961)

.

16. Mullins, J. C, Ziegler, W. T. and Kirk, B. S., The thermodynamic

properties of oxygen from 20 degrees to 100 degrees K, Georgia Inst,

of Technol., Atlanta, Tech. Rept. No. 2, Contr. No. CST-7339, Pro j

.

No. A-593 (Mar 1962) .

'

17. Roder, H. M. , The molar volume (density) of solid oxygen in equilibrium

with vapor, submitted to J. of Phys. and Chem. Ref. Data (1977).

18. Roder, H. M. , Childs, G. E., McCarty, R. D. and Angerhofer, P. E.

,

Survey of the properties of hydrogen isotopes below their critical

temperatures, Nat. Bur. Stand. (U.S.), Tech. Note 641 (Oct 1973).

19. Roder, H. M. , McCarty, R. D., A modified Benedict-Webb-Rubin equation of

state for parahydrogen-II , Nat. Bur. Stand. (U.S.), Internal Rept. No.

75-814 (Jun 1975) . . -,

20. Roder, H. M. , Weber, L. A. and Goodwin, R. D., Thermodynamic and related

properties of parahydrogen from the triple point to 100 degrees K at

pressures to 340 atmospheres, Nat. Bur. Stand. (U.S.), Monograph 94

(Aug 1965) .

21.- Sindt, C. F., Ludtke, P. R. , and Roder, H. M. , Slush and boiling

. methane characterization, National Bureau of Standards, Boulder, Colo.,

Unpublished report (Jul 19 70) .

22. Sindt, C. F. and Mann, D, B., Temperature-entropy diagram for parahydrogen

.triple-point region, Nat. Bur. Stand. (U.S.), Tech. Note 343 (Jun 1966).
23. Stewart, R. B., Jacobsen, R. T. and Myers, A. F., The thermodynamic

properties of oxygen and nitrogen, NASA-CR-128525 , Contr. No.

NAS9-12078 (Aug 1972)

24. Tolkachev, A. M. and Manzhelii, V. G. , Density of solidified gases,

Soviet Phys. Solid State 7_, 1711-3 (Jan 1966); Transl. Fiz. Tverd.

Tela 7, 2125-8 (Jul 1965).

25. Weber, L. A., P-V-T, thermodynamic and related properties of oxygen

2 from the triple point to 300 K at pressures to 33 Mn/m , J. Res. Nat.

Bur. Stand. (U.S.), A74_, 93-129 (Jan-Feb 1970).

26. Weber, L. A., Thermodynamic and related properties of oxygen from the

triple point to 300 K at pressures to 1000 bar, Nat. Bur. Stand. (U.S.),

Internal Report 77-865 (Oct 1977); also NASA Ref. Publ. No. RP-1011,

pages (Oct 1977). Available National Technical Information Service,

Springfield, VA, Order No. NASA RP-1011 (1977a).

27. Weber, L. A., A modified Benedict-Webb-Rubin equation of state for

gaseous and liquid oxygen, Nat. Bur. Stand. (U.S.), Internal Report

(to be published) (1977b)

28. Woolley, H. W. , Scott, R. B. and Brickwedde, F. G., Compilation of

thermal properties of hydrogen in its various isotopic and ortho-para

modifications, J. Res. Nat. Bur. Stand. (U.S.), 41, 379-475 (Nov 1948).

25
APPENDIX A. PROGRAM LISTINGS HYDROGEN

SU3Rl)UTINE SLH2(TT,UQ,TP,PSL,0SL,HSL,53L,JSL)

THERMOO YMmMIC PROPERTIES FOR SOLIO-VAPOR OR SOLID-LIQUID REGIONS

TEMPERATURE TT MUST BE BELOW 22.6 K QUALITY QQ, 0. IMPLIES PROPERTIES ON THE OESSER SIDE

1. IMPLIES PROPERTIES CN THE LESS DENSE SIDE

FkACTIONAL QQ BETWEEN J. AND 1. ?ERMISSA3LE

OTHER VALUES OF QQ INVALID

FOR THE SPECIAL CASE OF T=TRIPL£ POINT TP=0. MEANS SOLID-LIQUID

TP=1. MEANS SOLID-VAPOR

DATA (VPl=-90 .775 66 9^9)f{VP2 = 2.^69<330 9'+u),IIVP3='*.00 9a5735i4)

DATA tCPl = -i3l.)»(CP2 = d7.75),(CP3 = -26.95833333), (CPi*=k,25) , 1 (CP5=-0. 0Hlo66b6c67)

DATA (Vl = 22.7 79 63'+74) ,(V2 = -O.O3l),(V3 = J.O05)

DATA (TTP = 13. 800), tTUP = 22. 63), (H TP = -7 '3.206291) , ( ST P = l .49995 90 ) T=TT

Q=aQ

PSL = OSL=iHSL=SSL=JSL= u . IF ( T . LT .TUP ) GO TO 1 PRINT 2 FORMAT ( 5 X

,

3dHT TOO HIGH, P WILL EXCEED 5000 PSIA)

RETURN

IF(u.GE.U. .AND. U.LE.l.j) GO TO 5 PRINT 4 FORMAT ( 5 X , ItHG OUT OF RANGE)

RETURN

IF(T-TTP) 6,7,6

ENTER HERE FOR TRIPLE POINT

IF ( TP.EQ. J . j) GO TO a ENTER HERE FOR SOLIu-\^APOR PROPERTIES

P SL = E XP ( VP 1/T + VP2*AL0G (T ) +\/P3) /76a . 0P0T = PSL»C-VPi/T»*2«- VP2/T)

VSCL= V1*V2*T+V3*T**2

OSL=130G./VSOL

HSL = HTP-(CP1*(TTP-T)4-CP2/2.*(TTP»*2-T*»2)+C = 3/3.*(TTP»*3-T»*3)

1 <-CP4/4.*(:TTP»»'*-T»»u) + GP5/5.*(TTP»»5-T»*5) ) / lOOO.

SSL = ST3-(CP1*{AL0&(TTP)-ALGG(T))+CP2*(TTP-T)4-CP3/2.*(TTP**2-T**2)

1 <-GP4/3.*(TTP»*3-T»*3)+CP5/4.»(TTP**4-T*»i^))/1000 . USL = H5L-PSL/LJSL»i:i. 325

USOL=USL

IF (Q. EQ . G . L ) RETURN ,

- CONTINUE hcRt. FOR VAPOR PROPERTIES

OVAP=FINDM(FSL, T, C. u6)

VVaP=103u ./DVAP

DEL V=VVAP-VSOL . .

" 0£LM=CELV*T»GFDT* G. 1G1325

GELS=OELH/T

UVAP=HSL+DELH-PSL/(1GGO./(OELV+1033./DSL))»1Q1.325

DELU=UVAP-USOL

IF (Q. NE . 1 . ; ) GO TO 9 GSL = lQ03./(LiELV + li3GC./C5L)

HSL =HSL +u£Lrl

UGL = HSL-PSL/GSL»-iGl.325

SSL=5SL+DELS

RETURN

CONTINUE HERE WiTh FRACTIONAL Q 0 3L = 1'jGG./(Q*l)ELV + 1G:K./DSL)

HSL = u*OELH-t-HSL

SSL = Q*QclLS+SSL

2 6
RETURN

ENTER -^LRl FCR SOLlD-LliUlJ ^RO^ERTIES

PSL=?RE3SM ( T

dfgt=up:tmf (T

3SL=FINDM (F3L , T , 3 . ) H5L=ENThAL(PSL,3SL,T1

USL=HSL--SL/aSL*l01. 325

JLIG=USL

SSL=ENTkCP(3SL,T)

IF(G.E3.1.a) RETUkN

CONTINUE rMRE FCR SOLID PROPERTIES

JELH=(25.0^o9 3l5'+C.0'ilH9 = SL)'^•.18i^

DELV = OE^..-t/T/OPGT/0.a2!»2i79/H.16-^

CELS=OELH/T

OSL = ii:uu./{ljOO./CSL-G£LV)

USCL=iSL-OELH-PSL/DSL*lui. 325

DELU=ULIG-u3Cl

IFCQ.NE.b..) 3 0 TO 13

HSL = HSi. -GELH

S5l=SSL-CEl3

USL=USOL

RETURN

CONTINUE t^lRE. rilJ-i FRACTIONAL G

OSL = 130:./(.**DELV + 1CJC./:3L)

rtSL=rtSL-G*OELH

SSL=S5L-G*:EL5

USL=USL-G»GELU

RETURN

END

FUNCTION FR£SSM(TT)

CALCULATES '-.ElTING PRESSURE FRO'I InPjT TEMPERATURE

T=TT-t-j,JC3

PS=.Gc95+(T-13.3:3)*3:.33i2»EX= (-5.d93/T)+(T-13.3Q3)*2.0»T/3.0

PRE5S.M=?S

RETURN

END

SUSRO'JTINE RHul(PP,GD,TT)

1ST CUT AT RHO FROM F = A-t-3*T REALLt an iteration, BUT IT MAY EE SMA^L ENOUGH AND FdST ENOUGH

P IN ATM, T IN <, RHO IN MGLES/lITER

DIMENSION ncH0(i*3) ,A (i*3) ,5 (-3)

DATA((RHC(I),I=l,'.3)=l.,2.,3.,u.,5.,b.,7.,3.,9.,iO.,ll.,l2.,13.,lit

1., 15. ,16. ,17. ,13. ,19., 20. ,21. ,22. ,23. ,2*.. ,25. ,2b. ,27. ,23. ,29. ,3 3.,

25l.,32.,33.,3'*.,35.,3c.,37.,3c,,39.,'*0.,'+l.,'»2.,'+3.)

DAT4((A(I),I=l,^3)=-.Z9599325 7i*l,-i.l335'+55l75,-2.i*35l37S102,-^.3:

ItrlCSg 31 7, -..; = 9t5 5393, -i.2559135Hj5, -12. 33c2b6955, -15. 733961531,-1

29. 57337; 659, -23. o3i:-* 1295, -23.036^58235, -32. 77255895 1 ,-37. 72663671

39, -•*2.:;«»£371352, -2--325>*l,-5'f.i9i, '+6 51^2, -60. 237558136, -66. 7267

•'3 >»77l, -73. 516513375, - .)i.6-f3765-*19, -33. j7990 = s9'*, -95. 751501525,-133

5 . 6 3 96 :; ^1,-1 11. 72 777 3*. 2 , -119. 799 7- 17 2 , -127. 786 579 , - 135. 9^-257 76 1 ,-1

6-»3.3333i332, -151. 265+9•:)D-^, -157. 996 3 8i+63 , -163. 7133357, -167. 27873589

7,-169.7l92<:3;3,-169.769939l-n-lo6.3;7233-*5,-16w.22 9 7o-f5'*,-l-*9.6;3l

3 3c3,-l3-.=;3w;21o,-116.1l756-f;i,-9-t.293ll7563,-67,52-f2-*7269,-3c.7*

923953 2-*, -1. 32 9557^977)

DATA((3(I),I=l,-.3)=. 3 35j 5 92335-9,. 1^ 575l-:3596,.2717Eo7:*97 9,.3 72 9 76

113 72:*,.ii.7952911-kc5,.5^:14l2766l5,.7j36-»366952,.33l2'*6-i>+lH2,.95923U-f

291 l-,l,,923iu2o55,1.23223677il, 1.3776-55569, 1 .52 9'* 993193, 1. 6533212

3266, 1.35-32-(-^9l3, 2. 33; 335-23, G.2l5i;C6l 3 9, 2. -'ll2o7-*7 3 5, 2. 6l95u3366

27
45 ,Z .3^0 7G't32l2, 3. J756'f5J8 76,3. 3 250 4a55'+ ,3. 5895 52500 8, 3 .de968g35i+l,

5^+. 165 85 93639, i*. '+73296 l9'ti+,<+. 820^02 20 69, 5. 181*0092761 , 5 . 5696'+6053 ,5 . 697725 2^+ 362, 6. it057l7553l, 6. 321 '2 ^22 39, 7. 2730 6 97 825, 7 , 7 1+ 046 1 65 3 U , 3 . 1

7 9 725755a2,d.6358'46a'+65,9.J'+73286 C7&,9.-»26 0953807, 9. 81 32535 0^+5 , 10 . 2 822281 36 3, 1J.03Q12934, 11. 05913823 , 11. 56530719'*)

10 P=PP

T = TT

IP = 1 PLO=0.

C CHECK INPUT DATA FOR RANGE

C INPUT RESTRICTIONS REMOVED

C IF{P.GT.35G.> GO TO 15

C IFiT.GT. 300.1) GO TO 15

C GO TO 13

C 15 PRINT 16

G 16 F0RMAT(2'+H INPUT GATA OUT OF RANGE)

C 0 0=u.Q

C RETURN

PTP=\/PN(l3.a)

IF ( P. GT . PTP ) GO TO 1

8

IFCT.GT .13.3) GO TO 16

oc=Q.caaoi

RETURN - C CHECK MELTING LINE

13 IF( T.GT , (+3 . 60 0) GO TO 23 . PS=='RESSM ( T ) IF(P-PS) 23,21,19

19 PRINT 22

22 F0RMAT(26m INPUT CONDITIONS IN 3CLI0)

21 30=39.3

RETURN

G PhASEFINOER •

. 23 IF( T.LT . 32. 938) GO TO 27

PFHASE=-5G . 6u 02 +1 .92 0 888*T

IF (P.GT.PPriASE) IP=15

GO TO 36

27 P\/AP = VFN(T)

IF(P-F\/AP) 36, 32,35

32 PRINT 33

33 FORMAT(53H INPUT PLACES YOU EXACTLY ON THE VAPOR PRESSURE CURVE)

3J=0SATL (T)

RETURN . . 3-: OEfJL = DSATL ( T) IF = QErJL

C START TA3LE LOOKUP HERE

3d CONT INUE

00 <+0 I = IP,h3

PCALC = A ( I) +d( I) *T

IF(P.LT.PCALC) GO TO kl

PLO=PCALC

"^J CON TI NUE

IF ( P . LT . PCALC) I = -»3 IF(P.LT.FCmlG) go TO i.1 C -PRINT ^7

C -7 FGRMAT(35h HIGH DENSITY, OUT OF RANGE FOR NOW)

0 L = H •+ , RETURN

Hi CCNTIfJUE

00 5Hj=i,ij

RHOF=J

RHOF = ,-^H0F/ IC . IF(I.EQ.l) GO TO 5u

AA = A (I-i 3 + (A I I

) -A ( I- 1) ) tRH OF

28
33=3(I-i) *(3(I)-B(I-1) )*RHGF

GO TO 52

33 AA=A(I)*RHGF

33=8(I)»kH0F

52 PCALC=AA-t-3o*T

IF(P.LT.PCALC) GO TO 55

54 ?LO=PGALC

55 F'^AC= (P-PLu ) / (PCALC-PLO) /lij

00 =1-1

uu =00 +KhOF-0. 1+FRAC

RtTURN

ENTRY Ti

FIRST GUESS FOR TEMPERATURE ITERATION OF FINOT

P = PP

0 = 00

00 60 1=1.^3

IF(0.LT.RH0(I) ) GO TO 61

c3 CONTINUE

I='*3

61 FPAC=0-RHO ( I-l)

IF(I.EQ.i) GO TO 63

AA=a(I-l)<-(A(I)-A(I-l)) FRAG

3B=B(I-1J*- (3{I)-a(I-l))*FRAC

GO TO c2

63 FRAC=0

AA=A(I)»FRAC

33=3 ( I) FRAG

62 TT=(P-AA)/33

END

FUNCTION OPaTMP(TT)

NEW FOR HYORGGEN* .BELTING PRESSURE DERlVAriy/E

OATA (TTP=l3.8u3),(Al=3Q.33l2),{AL=5.b^3)

T = TT + C. GO 3 A2=2./3.

EXPP=Ai»EXP (-AL/T

)

0PQTMF=(T-TTP)*(EXPP*AL/T**2+A2)+EXFP+A2*T

RETURN

ENO

FUNCTIUN FINO .'^(P,T,0)

C CHANGEO, TRIAL OENSITY VIA SUBROUTINE RHOl/Tl, Zk FES 1975

Z CHANGED TO Aui-lIT A TRIAL DENSITY FOR SLUSH USE 15 DEC 76

COM VON/D«TA/G,R, GAMMA, VPfuTP

DIMENSION G( 32) ,Vr ( 3)

TYPE DOUoLEGi R, GAMMA

TT = T IF ( a. NE . u . ) GO TO 2 CALL RhG1(P,j,TT)

: DO=D

DO IJ 1 = 1, 5j

C^LL PRESS ( PP , DO, IT)

.P 2 = P P IF(A35 { P-P2 ) -1 .E-7*c ) 20 , 20 ,

i C-LL DPDD(PP,OD,TT)

DP = PP

CjRR= (P2-P) /L.t- J = uC

IF(A3S (CORR) -1 .E-7» D ) 20 , 2u , lu

1^ QO=Ju-CGRR

2 : F I D M= -] J

ENO

29
APPENDIX B. PROGRAM LISTINGS OXYGEN

SUBROUTINE SLO 2 ( TT , QQ , TP , PSL , OSL HSL t S SL , US L

)

C THERMODYMAMIC PROPERTIES FOR SOLID-VAPOR OR SOLIO-LIQUIO REGIONS

C TEMPERATURE TT MUST 9E BELOW 58.2 K C QUALITY QQ, 0. IMPLIES PROPERTIES ON THE DENSER SIDE

C 1. IMPLIES PROPERTIES ON THE LESS DENSE SIDE

C FRACTIONAL QQ BETWEEN 0. AND 1. PERMISSA3LE

C OTHER VALUES OF QQ INVALID

C FOR THE SPECIAL CASE OF T=TRIPLE POINT TP=0. MEANS SOLID-LIQUID

C TP=1. MEANS SOLIO-VAPOR

DATA (VPl = -1096.562i»85) , (VP2=-2. 025578307) ,(VP3 = 28.3597652'*)

DATA ( CP1 = 16. 908081) , (CP2 = -0. 2 '18 1777) , ( CP3= C . 0 02i»8 09 0 89)

DATA (Vl = 23. 2803137) , (V2 = -0 . 067721'+2868) , (V3 = 0. 001339285715)

DATA (TTP = 5't. 359) , (TUP=5 8.20) , ( H TP = -66 25 , 9 7 72 6 ) , ( STP = 5 8 . 928i*7 8

)

T = TT

Q = QQ

PSL=OSL=HSL=SSL=USL=0

.

IF (T.LT . TUP) GO TO 1 PRINT 2

2 FORMAT (5X,35HT TOO HIGH, P WILL EXCEED 5000 PSIA)

RETURN

1 IF(Q,GE.O. .AND. Q.LE.i.O) GO TO 5 PRINT k

U F0RMAT(5X,14HQ OUT OF RANGE)

RETURN

5 IF(T-TTP) 6,7,8

C ENTER HERE FOR TRIPLE POINT

7 IF(TP.EQ.O.O) GO TO 8 C ENTER HERE FOR SOLIO-VAFCR PROPERTIES

6 PSL=EXP ( VP1/T+VP2*AL0G (T) +VP3) /760

.

DPOT=PSL» J-VP1/T»*2+VP2/T

)

VSCL = V1 + V2*T4-V3*T»*2

DSL=1COO./VSOL

HSL=HTP-(CPl*TTP+CP2*TTP*»2/2.+CP3*TTP*»3/3.-(CPl*T+CP2»T»*2/2.

1 +CP3»T»»3/3. ) ) 18^

SSL = STP- (CP1*AL0G (TTP) +CP 2 *TT P+CP 3 *TT P ** 2/2 . - ( CPl * A LOG ( T) CP2*T +

1 CP3*T»*2/2.) )*it.l8i+

USL=HSL-PSL/OSL'101 . 325

USOL=USL

IF(Q.EQ.G.G) RETURN

C ,: , CONTINUE HERE FOR VAPOR PROPERTIES

OVAP=FINDM« PSL,T ,0. 00 0 6)

VVAP=1C00 ./OVAP

OELV = V VAP-VSOL

DELH=OEL V*T* DP DT»0. 101325

OELS=OELH/T

UVA?=HSL+DELH-PSL/ (IQOC./ (DEL V^lOQO. /DSD) •101.325

DELU=UVAP-USOL

IF (Q.NE . 1. 0 ) GO TO 9 OSL=1QCO . /(OELVf lO'lQ ./OSL)

HSL=HSL+OELH

US L= HSL -PSL /OSL* 101. 32 5 SSL=SSL+OELS

RETURN

C CONTINUE HERE WITH FRACTIONAL Q

9 OSL=10C0./(Q»OELV+1000./OSL)

HSL=Q»0ELH4-HSL

SSL=G*CELS+5SL .

• USL = Q*0£LU4-USL

30
RETURN

C ENTER HERE FOR SOLIO-LIQUIO PROPERTIES

8 PSL=PRESSM( T)

0POT=0POTMP (T)

OSL = FINOM(PSL, T,i»0 .87)

HSL=ENTHALf PSL ,OSL,T)

USL = HSL-PSL/OSL*101. 32 5 ULIQ=USL

SSL=ENTROPCOSL,T)

IFfQ.EQ.l.O) RETURN

C CONTINUE HERE FOR SOLID PROPERTIES

OELH=( 10 6.3+0.01126*PSL)*«+.13'+

OEL\/ = OELH/T/0PDT/0. 0 2i»217g/i»,18t*

DELS=OELH/T

OSL=10 0 0./(10 0 0./DSL-OELV)

USOL=HSL-OELH-PSL/DSL»101.325

OELU=ULIQ-USOL

IFro.NE.O.O) GO TO 10

HSL=HSL-OELH

SSL=SSL-DELS

USL=USOL

RETURN

C CONTINUE HERE WITH FRACTIONAL Q

10 OSL= 100 0 ./(Q*DEL\/ + 100 0 ./DSL)

HSL = >-SL-Q*OELH

SSL=SSL-Q»DELS

USL=USL-Q*OELU

RETURN

ENO

FUNCTION FIND T{P»0)

C NEW FEB 1975

C COPIED FROM HYDROGEN FOR USE WITH OXYGEN

COHMON/OATA/G,R, GAMMA, VP » DTP

DIMENSION G(32)»VP(8)

TYPE DOUBLEGtR, GAMMA

pp = p D0=0

C USES A FIRST GUESS IN TEMPERATURE FROM Tl

CALL T1(PP,0D,TA)

TT = TA

DO 10 1=1,10

CALL PRESS(PP,OD,TT)

P2 = Po

IF(A8S (P-P2) -l.E-7*P) 20,20,1

1 CALL DPOT (PP.DO, TT)

DP = PP

CORR= {P2-P) /DP

IF(A8S (CORR3-1.E-5 520,20,10

13 TT=TT-CORR

20 FIND T=TT

RETURN

•END

SUBROUTINE Tl(Po,DC,TA)

C SPECIAL SETUP FO" TANK-P OXYGEN

TA=99.

RETURN

ENO

31
FUNCTION PMELKPP)

INVERSE MELTING PRESSURE FOR 02, INPUT P, ATM OUTPUT T,K

NEW WEBER REFERENCES

DATA (P0 = 26 37.33),(CC = 1.76 9)t<TTP = 5i+.359), ( P TP = 0 . 0 0 l^+^S 1)

BRA=(PP-PTP )/PO+1.0

TEMP=ALOG«BRA) /CC

PMELT=EXP (TEMP) *TTP

RETURN

END

FUNCTION PRESSM(TT)

MELTING PRESSURE FOR 02, INPUT T,K OUTPUT P,ATM

NEW WEBER REFERENCES

OAT 4 (P0 = 2637.33) , (CC = 1.76 9) , { TT PsS'* . 3 59 ) ,(PTP = 0.00l'ti*51)

PRESSM=PTP*PO» ( (TT/TTP) **CC-1.0)

RETURN

END

FUNCTION DPDTMP(TT)

DERIVATIVE OF MELTING CURVE FOR OXYGEN

NEW WEBER REFERENCES

DATA (PO = 26 37.33),(CC = 1.769),(TTP = 5^.359),(PTP = 0.001'+it51)

OPQTMP=PO»CC/TTP*»CC*TT** (CC-1.0)

RETURN

END
APPENDIX C. TEST PROGRAM HYDROGEN

PkCGRAM values ( input , OUTPUT)

C THIS PROGRAM TESTS THE SLUSH Hr3F.JG£N SUBROUTINE

CALL OATAPHZ

C C0M3lrjATIG,NS OF SOLID - VAPOR

PRINT 1 T = 12.

a=G

CALL SLH2 (T ,u» 0 . f PSL ,OSL , HSLf SSL ,USL)

PRINT 2,T,Q, PSL , DSL ,HSL ,SSL, USL

Q = J . 5 CALL SLH2(T,Q,a.»PSL,CSL,HSL,SSL,USL}

PRINT 3,T,U, PSLt DSL ,H3L ,SSL, USL

Q = l.

CALL SLH2(!T,a,a.tFSL»0SL,HSL,SSL,USL)

PRINT •,TtCi» PSL, DSL, HSL, SSL, USL

T=13. doa

TF = 1.

a=!j

CALL 5LH2(T,a,TP,PSL, DSL,HSL,SSL ,USL)

PRINT 5, T,vi, TP, PSL, USL, HSL, SSL, USL

Q = :.5

CALL SL H2 (T , a, TP, PSL , DSL , HSL, SSL ,USL}

PRINT b,T,U, TP, PSL, DSL, HSL, SSL, USL

Q=l.

CALL SLH2(T,a,TP,PSL,CSL,HSL,SSL,USL)

PRINT 7, T,u, TP, PSL, DSL, HSL, SSL, USL

C CUMBlrjATIONS OF SOLID - LIQUID

PRINT i T = 1 3 . c G J TF = u. Q=G . CALL -,Ln2 { T , J , TP, FSL , CSL , HSL , SSu ,USLJ

PRINT :i,T,Q,TP,PSL,DSL,HSL,SSL,USL

Q = J .5

CALL 3LH2(T,U,TP,FSL,CSl,H3L,SSL,USL)

PRINT1u,T,u,TF,PSL,DSL,HSl,SSL,USL

a=i.

CALL SL H2 CT , a, TP, ?:.L , DS L, HSL, SSL , USL)

PKlNTli,T,0,TP,FSL,CSL,HSL,5SL,USL

T = 15. a=c. CALL iLHZtT,a,u.,l-SL,DSL,HSL,S3L,U3L!;

'JRINTi2,T,U, PSL, CSL, HSL, SSL, USL

i = J . 5 CALL SlH2 CT,Q,j , ,FSi_ , CSL, -'SLiSSl ,USL3

ORI NT ii , T , : , ?S L , D SL , HS L , SSL , U SL

J = 1 • CALL SLH2!;T,G,j. ,^SL,GSL,HSl,33L,U3L)

P^I\T1'*,T,Q, PSL, DSL, H5u, SSL, USL

Z CC.13I;.- TI ji<5 oF LI'iUIQ - ;APOk FROi-^ T^E 32 TER^.

PRINT 15

T = l 3. -i: j

?SL = ;r-j(T)

33
3SL = FINl)M(PSL, T, Jtt, ) HSL=£NTHAL(PSL,3SL,T)

USL=HSL-F5L/0SL*1Q1. 32 5 SSL = ENTROP (OSL , T)

PRlNTlb,T,G, PSL,uSL,H5L,S5L,USL

Q=l.

a5L = FINLM(PSL,T,0.a6)

HSL=ENThAL (PSL » OSL , T ) n USL=HSL-PSL/DSL»lti. 325

SSL = eNTROP ( QSL , T)

PRrNTl7,T,C, FSL,u5L,HSL,3SL,USL

T=i5. aa

J

Q = j.

PSL = \/PN{T)

L)SL=FlNLiM<FSL,Tt38. ) HSL=ENTHAL(PSL,OSL,T)

USL=H5L-P5L/0SL*lLi. 3 25

SSL=ENTROP(DSL,T)

PRINTlbtTtu, PSL,u3L,HSL,SSL,USL

Q=l.

OSL=Fir.!UM (PSL ,1,0.^6)

HSL = ENTHAL (PSLi OSL , T ) U5L=HSL-PSL/QiL»i01. 325

SSL=£NTROP ( J5L , T)

PRINTi7,T,U» PSL, QSL

,

HSl ,SSL

7

3

i J

11

12

13

FOR.IAT

1 FORMAT

1 F 1 3 . o FORMAT

1 Fl 3.

S

FORMAT

1 Fl 3.

3

FORMAT

1 Fl 3.

d

FORMAT

i F13.

3

FORMAT

1 F13.d

FORMAT

1 * FORMAT

1 F13. 8 FORMAT

1 i FORMAT

1 F13.6

FORMAT

1 F13.2

FORMAT

1 F 1 3 . d FOR.IAT

1 F 1 3. 0 'FORMA T 1 FORMAT

1 Fl 3. 3 FORMAT

1 F 1 3. 9 E NL

///,52H1SGLIQ-VAP0R

P, ATM RmO, MOL/i

Ig X FlH

iux

Fl^i

1 J X Fl-f

1 u X Fli+

lOX

Fl-^

1 u X F 1^+ ///

^ » 1 U X F11+

1 b X Fl^

1 C X '-"iH

1 G X Fm

i u X Fl^

1 0 X FlH

// / ? , 1 j X Flw

lb X Fl^

F3.3,2<+H SOLID,

b, Fll. 3 , F IC . 3 , Fll ,

3

Fo.3,2-fH MIX, J IS

6, Fll. 3,F1C.3,F11.3

F3.3,2itH VAPOR,

6, Fll. 3, F IC . 3 , Fll . 3 Fd.3,2--H SOLID,

6, Fll. 3, FlC. 3, Fll,

3

Fj.i,2HH MIX, Q IS

o,Fll.3,Fli:i.3, Fll . 3 F6.3,2i+H VAPOR,

6, Fll. 3,Fl0.3,F11.3

ic» SOLID-LlCiUlO T,

T,1 RHu, MOL/L

F8.3,2'.H SOLID,

o, Fll. 3, Flu. 3, Fll.

3

F4.3,2-*H rilx, 1 IS

6, Fll. 3, Fi;. 3, Fll.

3

Fd.Z^i'^^ LIUUID,

6, Fll. 3, FIG. 3, Fll.

3

Fi.3,2'*H SOLID,

o,Fll.3,FlC.3,Fll.3

F8.3,2UH MIX, Q IS

c, Fll. 3, FlC. 3, Fll.

3

F'i.3,2'tH LIQUID,

o,Fll.3,Fi:.3,Fll.3

;2H LIOUIO-VAPOR T, TM RhC, MOL/L

Fi.J,2^H LIUUID,

o,F11.3,Fl3.3,F11.3

Fc.3,2'-H VAPOR,

c,Fli.3,rlL.3,F11.3

, USL

K H, J/MQL

, / ) FRACTIONAL

,/)

/^

,/ ) FRACTIONAL

/ )

,/)

K H, J/MOL

, /)

FRACTIONAL

, / )

, / )

, / ) FRACTIONAL

,/)

, / ) K FROM TH:

-\ f J/MOl

, / )

, / )

a TP

5, J/MOL-K U, J/MOL*,//)

,F3.1,5X

= , F3. 1 , 5X

, F3. 1 , 5X

, F3. 1 ,F5 .

= ,F3.1,F5.1

,F3. 1 ,F5 .1

S, J/MOL-K

, F3 . 1 , F5 . 1

= , F3 . 1 ,F5 . 1

, F3. 1 ,F5 .

,F3.1,5X

= ,F3.i,5x

, F 3 . 1 , 5 X

, 32 TERM EG.

S, J/MOL-K

, F 3 . 1 , 5 X

, F3 . 1 , 5x

a TP

, J/MOL*,//)

Q TP

, J/MOL',//)

34
APPENDIX D. TEST ?ROGR?uM OXYGEN

3RCGRAM VALUESIINPUT, OUTPUT)

THIS PROGRAM TESTS THE SLUSH OXYGEN SUBROUTINE

CALL 3ATA02

C0M3INATI0NS OF SOLID - VAPOR

PRINT 1 T=5a

Q=C.

CALL SL02 (T ,Q, Q. ,PSL , DSL , HSL , SSL , USD

PRINT 2,T,Q, PSL»OSL,HSL,SSL,USL

Q=0.5

CALL SL02(T,Q,D.,PSL,[;SL,HSL,SSL,USL)

PRINT 3,T,Qt PSL,DSL,HSL,5SL,USL

Q=l.

CALL SL02(T,Q,Q.,PSL,0SL,HSL,SSL,USL}

PRINT i*,T,Q, PSL»OSL,hSL ,SSL,USL

To=l.

Q=C . CALL SL02(T,Q,TP,PSL,0SL,HSL,SSL,USL)

PRINT 5,T,Q,TD,PSL,CSLtHSL,SSL,USL

Q=0.=

CALL SL02f T»Q,TP,PSL»OSL,HSL,SSL,USL)

PRINT 6,T,Q,TP,PSL,DSL,HSL,SSL,USL

Q=l.

CALL SL02(T,Q,TP,PSL,DSL,HSL,SSL,USL)

PRINT 7,T,Q,TP,PSL,DSL,hSL,SSL,USL

COMBINATIONS OF SOLID - LIQUID

PRINT 8 T = 5i».359

TP=] , q=C.

CALL SL02fT,Q,TO,PSL,CSL,HSL,SSL,USL)

PRINT 9,T,Q,TP ,?SL»DSL,HSL ,3SL,USL

Q=u .5

CALL SL02<T,Q,TD,PSL,DSL,HSL,SSL,USL)

PRINT10,T,Q,TP,PSL,DSL,HSL,SSL,USL

0=1.

CALL SL02(T,0,T3,3SL,DSL ,HSL,SSL,USL)

°RINT11,T,Q,TP,PSL,DSL,HSL,SSL,JSL

T = 56.

Q=Q . CALL SLO2tT,O»0.,PSL,DSL,-^SL,SSL,USL5

P'INT12, T,3, 3SL,CSL,HSL,SSL,USL

0=0 . 5 CALL SL02{T,0,G.,3SL,OSL,-iSL,SSL»U3L)

PRINT13,T,0, 3SL , DSL ,-S;. ,SSL ,USL

Q = l.

CALL SL0 2(T,::,G. , - S L , 3 S L , --S L , S SL , USL

PRINTIU, T ,0, P3L, DSL,hSl ,33L,USL

C0H3INATI0NS CF LIQUID

PRINT 15

T=5«4,359

-'J^PZ^ "Rj-^ TME 32 TE = M

35
Q = 0.

PSL=VFNIT)

QSL = FINOM(PSL»T,i+l.)

HSL=ENTHflL(PSL,OSL, T)

USL=HSL-PSL/OSL* 10 1.325

SSL=rENTROPeOSL,T)

P1RINT16,T,Q, PSL,DSL,HSL,SSL

(3 = 1.

OSL=FINOM4PSL»T,0. 000 3)

HSL = ENTHAL8PSL ,OSL »T)

USL=HSL-PSL/OSL*10i. 32 5 SSL=ENTROPCOSL,T)

PRINT17,T,Q, PSL,OSL,HSLtSSL

T = 56.

Q=G.

PSL=VPN(T)

OSL = FINDM(PSL«T,i*l. ) HSL=ENTHALCPSL,OSL , T)

USL=HSL-PSL/OSL*101. 3 25

S3L=ENTROP60SL,T)

PRINT16»T,Q, PSL»OSL ,HSL,SSL

Q=l.

OSL=f INOMtPSL.T»0. 0003)

HSL=eNTHALCPSL»OSL,T)

USL=HSL-PSL/OSL*101. 3 25

SSL=ENTROP(IDSL,T»

PRINT17,T,Q, PSL,OSL,HSL,SSL

1 FORMAT (///, 52H1S0LID-VAP0R T,

1 P, ATM RHO, MOL/L

2 FORMAT C10X,F8. 3, 2i*H SOLID,

1 F13.8,Flif,6,F11.3,F10.3,F11.3

3 FORMAT ( 10X»F8. 3 ,2'+H MIX, Q IS

1 F13,8,Fli».9,F11.3,F10.3,F11.3

k FORMAT (13X,F8. 3, 2i»H VAPOR,

1 F13.8 ,Flt+.9,F11.3,F10.3,Fil,3

5 FORMAT (10X,F8. 3, 2'+H SOLID,

1 F13.8,Fli».6,Fll,3,F10.3,Fll,3

6 FORMAT (10X,F8. 3, 2'+H MIX, Q IS

1 Fl3.8,Fli+.9,Fll,3,F10.3,Fll,3

7 FORMAT( 1QX,F8. 3,2'+H VAPOR,

1 F13,8,Flt,,9,F11.3,FlG.3,F11.3

3 FORMAT (///, 52H SOLID-LIQUID T,

1 * P, ATM RHO, MOL/L

9 FORMAT(10X,F3.3,2'fH SOLID,

1 Fl3.8,Flt».6,F11.3,F10.3,Fll,3

13 FORMAT(10X,F8.3,2i+H MIX, Q IS

1 F13.8 ,Fltf.6,F11.3,F10.3,Fll,3

11 F0PMAT(1QX,F8.3,2'+H LIQUID,

1 Fl3.8,Fli+.6,Fll,3,F10.3,F11.3

12 FORMAT (10X,F3. 3, 2'+H SOLID,

1 Fl3.8,Fli».6,Fli,3,rl0.3 ,F11,3

13 FORMAT ( 10X,Fd . 3, 2i+H MIX, Q IS

1 F13.8,Fl<f,6,F11.3,Fl0.3,F11.3

i<4 •FORMAT(10X,F3.3,2!*H LIQUID,

1 F13.8,FH*.6,F11.3,F1C .3,Fli.3

15 FORMAT (///,52H LIQUID-VAPOR T,

1 P, ATM RHO, MOL/L

16 FORMAT{10X,F3.3,2«»H LIQUID,

1 F13.8,F1»+,6,F11.3,F10.3,F11,3

17 F0PMAT(1JX,F8.3,21+H VAPOR,

1 F13.8,Fli+.9,F11.3,Fl0.3,Fll,3

END

USL

,USL

,USL

,USL

K H, J/MOL S,

,/)

FRACTIONAL = , ,/)

,/)

t/)

FRACTIONAL = , ,/)

» ,/)

K H, J/MOL S,

» ,/)

FRACTIONAL = , ,/)

» ./)

» ,/J

FRACTIONAL = , ,/)

» ,/)

K FROM THE 32

H, J/MOL S,

,/)

/)

Q TP

J/MOL-K U, J/MOL»,//)

F3.1,5X

F3.1,5X

F3.1,5X

F3.1,F5.i

F3.1,F5.

F3.1,F5. 1

J/MOL-K

F3,1,F5.1

F3.1,F5.1

F3.1,F5.1

F3. 1, 5X

F3. 1, 5X

F3.1, 5X

TERM EQ.

J/MOL-K

F3.1,5X

F3.1,5X

Q TP

, J/MOL*,//)

Q TP

, J/MOL*,//)

36
APPENDIX E. CONVERSION FACTORS, HYDROGEN

Temperature

Pressure

Specific Volume

Internal Energy, EnthalpyEntropy, Specific Heat

Thermal Conductivity

Viscosity

Speed of Sound

Molecular Weight

Surface Tension

1.8 R = 1 K

14.695949 psia = 1 atir. = 1.01325 x 10^ N/m^

0.00794590 ft^/lb = 1 cm^/mol m

(1 cm^ = 0.001 liter = lO"^ m^)

0.213405 3TU/lb = 1 J/mol m

0.118558 3TU/lb R = 1 J/mol-K m

0.0578176 BTU/ft-hr-R = 1 mW/cm-K

0.067196897 lb /ft-s = 1 a/cm-s

m

3 .2808 /s = 1 m/s

12

2 . 01594 (on the C

0.5710147 X 10~^

= 12.000 scale)

lb./in = 1 dyn/cm

-5

:i dyn = n:

APPENDIX F. CON'VERSION FACTORS, OXYGEN

Temperature 1.3 R = 1 K

Pressure 14.695949- psia = 1 atm = 1. 01325 x 10^ N/m^

3 3 Specific Volume 0.0005005957 ft /lb = 1 cm /g nol

m

Internal Energy, Enthalpy 0.013444 6 3TU/lb^ = 1 J/g nol

Entropy, Specific Heat 0.0074692 3TU/lb R = 1 J/q mol-K

Thermal Conductivity 0.0573176 3TU/ft-hr-R = 1 mW/cm-K

Viscosity 0.067196897 lb /fr-s = 1 g/cm-s

Speed of Sound - 3.2808 ft/s = 1 m/s

Molecular Weight 31.99 8 8 Surface Tension 0 . 5710147 x 10 ^ lb -/in = 1 dyn/cm

(1 dyn = 10'^ N)

37
NBS-1 14A (REV. 7-73)

U .S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO.

NBSIR 77-859

2. Gov't Accession

No.

3. Recipient's Accession No.

4. TITLE AND SUBTITLE

THE THERMODYNAMIC PROPERTIES OF SLUSH HYDROGEN AND OXYGEN

5. Publication Date November 1977

6. Performing Organization Code 275.02

7. AUTHOR(S)

Hans M. Roder

8. Pert'orming Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Project/Task/Work Unit No 2750423

11. Contract/Grant No. P.O. #T6570C

12. Sponsoring Organization Name and Complete Address (Street, City, State, ZIP)

NASA-Johnson Space Center

Houston, Texas 77058

13. Type of Report & Period Covered

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 200-word or less (actual sutrtmary of most si^ificant information. If document includes a significant

bibliography or literature survey, mention it here.)

The thermodynamic properties of hydrogen and oxygen have been calculated for temperatures both below and above the triple point. Values of the various physical properties

required for the computations are either taken from the literature, or are extrapolateq.

If extrapolated, the extrapolations are based on the known behavior of other simple

fluids. The results are presented in the form of computer programs which cover two

regions of the phase diagram, solid-vapor, and solid-liquid. Input to the programs is-,

temperature and quality. The properties returned include pressure, density, enthalpy,

entropy and internal energy. The present programs for slush hydrogen and slush oxygen

are an extension of and depend upon the equations of state developed previously for

these gases.

T7. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name; separated by semicolons

)

Computer programs; enthalpy; entropy; hydrogen; internal energy; liquid; oxygen;

PVT; quality; slush; solid; vapor.

18. AVAILABILITY Unlimited

! For Official Distribution. Do Not Release to NTIS

rZ Order From Sup. of Doc, U.S. Government Printing Office

Washington, D.C. 20402, SD Cat. No. CH

1^ Order From National Technical Information Service (NTIS)

Springfield, Virginia 221=il

19. SECURITY CLASS

(THIS REPORT)

UNCLASSIFIED

20. SECURITY CLASS

(THIS PAGE)

UNCLASSIFIED

21. NO. OF PAGES

44

22. Price

$4.50

USCOMM-DC 29042-P7^'