NBS

PubliMons

NASA Reference Publication 1051

NBSIR 79-1611

A111D4 514247

Thermodynamic Properties

of Nitrogen Gas Derived From

Measurements of Sound Speed

Ben Younglove and R. D. McCarty

DECEMBER 1979

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Thermodynamic Properties

of Nitrogen Gas Derived From

Measurements of Sound Speed

Ben Younglove and R. D. McCarty

Tbermophysical Properties Division

National Engineering Laboratory

National Bureau of Standards

Boulder, Colorado

Prepared for NASA Langley Research Center

NASA National Aeronautics

and Space Administration

Scientific and Technical

Information Branch

1979

SUMMARY

A virial equation of state for nitrogen has been determined by use of

newly measured speed-of-sound data and existing pressure-density-temperature

data in a multiproperty fitting technique. The experimental data taken were

chosen to optimize the equation of state for a pressure range of 0 to 1 0 atm

and for a temperature range of 60 to 350 K. Comparisons are made for thermodynamic properties calculated both from the new equation and from existing

equations of state.

INTRODUCTION

Over the past several years, there has been renewed interest in gaseous

nitrogen in the pressure range of 0 to 1 0 atm and the temperature range of 60

to 350 K. This interest has grown out of the development of the cryogenic

wind-tunnel concept at the Langley Research Center (LaRC) of the National

Aeronautics and Space Administration (NASA) . This new wind-tunnel concept

uses nitrogen gas at cryogenic temperatures to obtain aerodynamic data at high

Reynolds numbers and at transonic Mach numbers. The need for an accurate equation of state became apparent as real-gas effects on flow simulation and datareduction requirements were studied. Although the well-known equations of

state (Jacobsen and Beattie-Bridgeman, refs. 1 and 2, respectively) were readily available, discussions between NASA and the National Bureau of Standards

(NBS) revealed that very little experimental data were available for gaseous

nitrogen at the time of their development to substantiate these equations of

state in the pressure range of interest. Because of the lack of experimental

data, it was believed that substantial uncertainty could exist in thermodynamic

quantities derived from either the Jacobsen or the Beattie-Bridgeman equation.

To eliminate this uncertainty and to provide an equation of state for gaseous

nitrogen optimized to the more limited pressure and temperature regions used

with cryogenic tunnels, NBS has taken sound-speed data (ref. 3) for the conditions of interest and has used a virial equation of state and a multiproperty

fitting technique (ref. 4) to arrive at an optimized equation of state.

The authors would like to thank Robert M. Hall and Jerry B. Adcock of

NASA-LaRC who contributed significantly to the contents of this report.

SYMBOLS

Symbols shown below in parentheses are those used in table I.

a (VEL) sound speed, m/sec

B (BO) second virial coefficient, liters/mol

( Bl ) first derivative of B
(B2) second derivative of B

c (CO) third virial coefficient, ( liters/mol)

(Cl) first derivative of C

(C2) second derivative of C

cp

(CP) specific heat at constant pressure, J/(mol*K)

c v(CV) specific heat at constant volume, J/(mol*K)

(DIF) percent difference between present values and reference 1 values

E (E) internal energy, J/mol

H (H) enthalpy, J/mol

P (PRES) pressure, atm (1 atm = 101.3 kPa)

R gas constant, 8.31 434 J/(mol*K)

S (S) entropy, J/(mol*K)

T ( TEMP) temperature, IPTS-68, K

V specific volume, liters/mol

P (DENS) density, mol/liter

MULTIPROPERTY FITTING

The use of sound-speed measurements for calculating thermodynamic properties is well established (refs. 5 to 7) . The sound speed relates to derivatives

of the primary thermodynamic quantities p, p, and T in a nonlinear fashion.

For example.

a

2 = ( 1 )

or, equivalently.

2
and

a

2

(3)

It is this nonlinearity which prevents the use of conventional linear leastsquares fitting techniques (ref. 4)

.

The pressure and the sound s_peed are computed from the density virial

expansion

The virial coefficients B and C are in turn represented by an assumed power

expansion in temperature as discussed later in the paper. Coefficients from

which B and C are computed are found by using multiproperty fitting of two

sets of data.

Multiproperty fitting is a technique which allows the use of different

kinds of thermodynamic data simultaneously in a least-squares estimation of the

adjustable parameters in an equation of state. In the present case this technique allows the simultaneous use of p-p-T data and sound-speed data to deter

mine a virial equation of state. The technique is especially appropriate for

this particular combination of data forms to determine the second virial coefficient B. An additional advantage is obtained in this case because the B i equation (4), which is the density virial expansion, and the B in

which is the pressure virial expansion, are interchangeable. Therefore, an expression for the sound speed may be derived from equation (5) which contains

only the variables p, a, and T (i.e., no density). These are the actual

measured variables. This expression is then combined with equation (4) by

multiproperty fitting to determine B in the density virial expansion. For

details of this process, see reference 4.

The sound speed was measured with a fixed-path-length cavity, or resonator

Electrostatic transducers were used for sound generation and detection. A

standing wave is produced by exciting the generating transducer with a variable

frequency. The frequency of the oscillator is adjusted for maximum received

signal, which occurs at the cavity resonance. At resonance there are an integral number N of half-wavelengths in the tube of length L,

p

= RTp (1 + Bp + Cp2

) (4)

pv = RT + Bp (5)

EXPERIMENTAL METHOD

3
L ( 6 ) NX

2

The sound speed is computed from

a = FX (7)

Generally, frequencies F were measured corresponding to resonances for

N = 6, 8, 10, 12, 14, or 16. Three to five resonance frequencies were mea- sured and the experimental sound speed was taken to be the average of the

velocities computed from the various frequencies.

The cavity wall is a tube of Invar 67 mm long and 1 0 mm in diameter. The

electrostatic transducers are at the ends of the cavity. The inside wall of

the tube was ground and polished to allow proper evaluation of the boundary

effects. The sound speed inside the tube is slower than it would be in an unbounded medium because of energy losses to the walls and ends via thermal

conduction and viscous drag. This difference in sound speed can be computed

accurately by using thermodynamic properties such as viscosity, thermal conductivity, heat capacity, and density of the gas medium (ref. 1). For the present

experimental method, the size of this correction was generally a few hundredths

of a percent in sound speed; however, it increased to about 0.5 percent in the

worst case, which occurred at low temperature and high pressure.

The active transducer surface was a thin, flexible plastic film (ref. 8)

,

coated on one side with aluminum of a thickness of approximately 50 nm (500 A).

The conducting side was placed on the ends of the tube and stretched tightly so

as to produce a flat surface. Immediately on the other side of the coated surface were placed the electrodes, which were copper discs. The signal strength

applied to the electrodes was 50 to 1 00 V at 1 to 30 kHz. The transducers produce sound of measurable intensity over this range of frequencies. Over the

frequency variation of the width of a single resonance the sound intensity is

essentially constant.

Temperatures were measured on the International Practical Temperature

Scale-68 (IPTS-68) with a platinum resistance thermometer calibrated by the

NBS Heat Section. Uncertainties in the measured temperature were estimated

to be 5 mK at the lower temperatures and approximately 50 mK at 350 K. Pressures were measured with a precision quartz spiral Bourdon tube, calibrated

at the National Engineering Laboratory with an air-dead-weight gage. Pressure

uncertainties were estimated to be less than 0.013 percent at 15 atm, increasing to 0.3 percent at 0.33 atm. The sample gas was ultrapure carrier type.

Analysis of the nitrogen gas by the supplier showed an oxygen concentration of

1 ppm and total hydrocarbon concentration of less than 0.2 ppm.

To estimate the uncertainty involved in the overall experimental procedure,

sound-speed measurements of helium gas were taken in the apparatus and were

4
found to be within 0.05 percent of known values. This value of 0.05-percent

uncertainty is taken to be the uncertainty in the present nitrogen measurements.

RESULTS

The density virial coefficients resulting from application of the multiproperty fit to sound-speed and p-p-T data are based on temperature expansions. The form of the second virial coefficient is expressed as

B BjT( 3 “j) /2

(j = 1 , 2, 3, . . ., 7) (8)

and the form of the third virial coefficient is expressed as

C CjT C

-

j) /2

(j = 1 , 2, 3, . . ., 5) (9)

The newly calculated numerical values of B and C are given in the following

table:

Virial

coefficient

Calculated numerical

values of B and C

B1 0. 2920642838 E- 02

b 2 -. 2334399080E+00

b 3 .742508651 8E+01

b4 -.11 64300428E+03

b 5

. 9253305729 E+03

b 6 -. 3351 637808E+04

b7

.1 992311 954E+04

Cl .21 89970095E-01

C2 -. 5949349694E-01

c 3 - . 2399796650E+02

C4

. 453424441 3E+03

c 5 -.2283997281 E+04

The values for the thermodynamic properties for nitrogen computed from

equations (8) and (9) for B and C are given in table I. Values are com- puted for isotherms starting at 60 K, increasing at 1 0-K increments to 110 K

and at 20-K increments from 110 to 350 K. The second and third density virial

coefficients with the first and second derivatives of each are shown at the top

of each page. For most isotherms the property values are computed at pressures

beginning at 0.2 atm and increasing to 10 atm in 0.2-atm increments. The values for two-phase conditions are indicated by the notation "2-PHASE BOUNDARY"

shown at the right side of the table. Values for the thermodynamic properties

are given beyond the two-phase boundary as an indication of the properties for

metastable conditions.

5
The tabular values of the thermodynamic properties are arranged for com- parison of the values computed from the virial coefficients with the existing

values from NBS Technical Note 648 (ref. 1). The first column for each case

is computed from the virial equation. The numbers of the different columns are presented as percent of deviation from TN 648 values in the graphs in figures 1 to 7. However, above approximately 150 K the differences are omitted from the

graphs because they generally are very close to the values of Technical Note 648

in that region.

The largest deviations (approximately 6 percent) were found for Cp and

cv for the values at saturation in figure 1 and for Cp for the 1 00-K isotherm in figure 2. Deviations of about 1 percent were found for internal energy

(fig. 3) and enthalpy (fig. 4) . Deviations of a few tenths of a percent were

found for entropy (fig. 5) , density (fig. 6) , and sound speed (fig. 7) . The

largest deviations in property values occurred for isotherms at 90, 100, and

110 K and at pressures near the two-phase boundary, for which the change in

sound speed with pressure is largest.

The sound-speed measurements (ref. 3) from which the virial coefficients

are computed were taken for isotherms from 80 to 350 K and at pressures from

0.33 to 1 5 atm. Figures 8, 9, and 10 show sound-speed deviations at 80, 150,

and 31 0 K, respectively, computed with the virial coefficient as the reference.

These three figures illustrate the accuracy of the three equations - virial,

Jacobsen, and Beattie-Br idgeman - for predicting the experimental sound speeds,

and that no one of these equations is clearly superior in terms of fitting the

sound speed for isotherms over the entire temperature range. However, as explained later, the virial equation is expected to be superior in predicting

specific heat capacities.

For figures 8 to 10, the largest deviations of predicted sound speeds are

at 150 K. Sound speeds calculated with the virial equation are 0.1 to 0.2 percent larger than those calculated from either the Beattie-Br idgeman or the

Jacobsen equation. For the 80-K isotherm the deviations are in the other

direction and become larger as the pressure increases, with the largest deviations occurring for the experimental data. The last three data points are believed to be incorrect, since they could not be represented by a virial

expansion for this isotherm taken separately. However, comparison of virial

surfaces computed with and without these points shows that the points do not

affect the derived surface to any degree of significance. At temperatures

larger than 1 50 K the agreement with the virial surface was generally as seen in figure 1 0.

ACCURACY OF THE CALCULATED VALUES OF THERMODYNAMIC PROPERTIES

The accuracy of calculated values of thermodynamic properties depends on

a number of different factors. Often an estimation of accuracy is made on the

basic agreement between the experimental value and the calculated value of the

property in question. An estimate made in this manner is an estimate of the

precision of the experimental data and almost always results in an assessment

of accuracy which is unrealistically small. Systematic errors in experimental

data are usually greater than the imprecision in those data.

6
A more realistic estimate of the accuracy of data is to compare those data

with similar data from other sources and then make an estimate of the agreement.

This method is probably conservative, but it is more realistic than equating

precision to accuracy. In this particular case a realistic estimate of the

accuracy of a property value may be determined by taking the maximum deviation

shown for a particular property illustrated in figures 1 to 7 . The one exception is the specific heat capacity, for which experience has shown that empirical

equations of state usually predict erroneous heat capacities near the saturated

vapor boundary (and other boundaries such as the saturated liquid and solid

boundaries) unless extreme care is taken to prevent the error (refs. 9 to 12).

This erroneous heat-capacity behavior is due to the high degree of flexibility

of most equations of state (the Jacobsen equation, for example, has 32 adjustable parameters) and the absence of- the data immediately beyond the saturation

boundary to control the behavior of the equation of state. In the case of the

saturated vapor boundary, additional strain is placed on the equation of state

because the equation must evaluate the region of densities between the vapor

and liquid phases in such a manner as to predict the same pressure given the same temperature but different densities (i.e., a van der Waals loop). This

problem is avoided by choosing the virial equation of state which has a minimum

of adjustable constants and by using only vapor-phase data (no van der Waals

loop required) with pressures of 15 atm or less.

Although multiproperty-fitting techniques used with p-p-T data together

with either sound-speed data or specific-heat-capacity data can reduce the error in specific-heat-capacity values as predicted by empirical equations of

state near the two-phase transition, neither of these latter two kinds of data

were available at the time the Jacobsen equation was formulated.

Based on the preceding arguments, the Cp and cv from the wide-range

equations of state near the saturated-vapor boundary are questionable. Therefore the maximum uncertainty of the virial Cp and cv is estimated to be

2 percent, which corresponds to the disagreement shown in figures 2 and 3 for temperatures for which saturated conditions are not a factor. The average

uncertainty in Cp and cv

is closer to 1 percent.

SUMMARY OF RESULTS

For many substances there is a lack of data in the gas region, especially near the two-phase boundary. As a consequence, the thermodynamic

properties of this region are generally found to be poorly characterized by

equations of state. To correct this situation for nitrogen, measurements of

the speed of sound in the gas phase have been combined with p-p-T data by

multiproperty-fitting techniques to produce a density virial expansion for

nitrogen. This new virial expansion should provide a more accurate representation of the thermodynamic properties (especially heat capacity) in this region.

The values of the virial coefficients found from this optimization are compared in detail to the thermodynamic-property values derived by Jacobsen.

The virial equation sound speeds are also compared to the new experimental

data and to those computed from the Beattie-Bridgeman equation and from the

Jacobsen equation for 80-, 1 50-, and 31 0-K isotherms.

7
Experimentalists should be encouraged to take more detailed data near two-phase boundaries, and especially in the gas region, since the behavior of

the thermodynamic surfaces near these boundaries is generally the most uncertain. However, in the case of the gas phase this may well be the most difficult region to take data.

Langley Research Center

National Aeronautics and Space Administration

Hampton, VA 23665

November 2, 1979

8
REFERENCES

1. Jaoobsen, R. T.; Stewart, R. B.; McCarty, R. D. ; and Hanley, H. J. M.

Thermophysical Properties of Nitrogen From the Fusion Line to 3500 R

(1 944 K) for Pressures to 1 50 000 psia (1 0 342 x 1 0

5 N/m^) . NBS Tech.

Note No. 648, U.S. Dep. Com., Dec. 1973.

2. Beattie, James A.; and Bridgeman, Oscar C.: A New Equation of State for

Fluids. II. Application to Helium, Neon, Argon, Hydrogen, Nitrogen,

Oxygen, Air and Methane. J. American Chem. Soc., vol. 50, no. 12,

Dec. 1 928, pp. 31 33-31 38.

3. Younglove, Ben A.; and McCarty, Robert D. : Velocity of Sound Measurements

for Nitrogen Gas at Temperatures From 80 to 350 K, and Pressures to

1.5 mPa. J. Chem. Thermo., vol. 1 2, 1 980 (to be published).

4. McCarty, R. D. : Determination of Thermodynamic Properties From the Experimental p-V-T Relationships. Experimental Thermodynamics. Volume 2 - Experimental Thermodynamics of Non-Reacting Fluids, Butterworth & Co.,

Ltd., c.l 975, pp. 501 -526.

5. Gammon, B. E.: The Velocity of Sound With Derived State Properties in

Helium at -175 to 150° C With Pressure to 150 atm. J. Chem. Phys.,

vol. 64, no. 6, Mar. 15, 1976, pp. 2556-2568.

6. Van Dael, W. ; and Van Itterbeek, A.: The Velocity of Sound in Dense Fluids.

Physics of High Pressures and the Condensed Phase, A. Van Itterbeek, ed.

,

John Wiley & Sons, Inc., 1 965, pp. 297-357.

7. Quinn, T. J.; Chandler, T. R. D. ; and Colclough, A. R. : Determination of

the Gas Constant by an Acoustical Method. Nature, vol. 250, no. 5463,

July 1 9, 1 974, p. 21 8

.

8. Mylar - Physical, Electrical, and Chemical Properties. Tech. Rep. TR-1

,

E. I. du Pont de Nemours & Co., Inc.

9. Roder, H. M. : The PVT Surface of Simple Liquids at Densities Near Melting.

NASA CR-1 55757, 1 977.

10. McCarty, R. D. : A Modified Benedict-Webb-Rubin Equation of State for

Methane Using Recent Experimental Data. Cryogenics, vol. 14, no. 5,

May 1 974, pp. 276-280.

11. Roder, H. M. ; and McCarty, R. D. : A Modified Benedict-Webb-Rubin Equation

of State of Parahydrogen-2. Rep. COM-7511132/8, Nat. Bur. Stand.,

June 1 975.

1 2. McCarty, R. D. : Thermodynamic Properties of Helium 4 From 2 to 1 500 K at

Pressures to 10^ Pa. J. Phys. Chem. Ref. Data, vol. 2, no. 4, 1973,

pp. 923-954.

9
TABLE

I.-

THERMODYNAMIC

PROPERTIES

FOR

NITROGEN

COMPUTED

FROM

VIRIAL

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for various thermodynamic properties X.

44
ro

CD

S-

=3CO

CO

CD

S- Q_

LBULA‘d

3 ^8l79Nl c d

3

[PULA ‘d

3)00 L

45

Figure

2.-

Percent

of

deviation

from

TN

648

values

of

specific

heat

at

constant

pressure

versus

pressure

at

isotherms

of

70

to

140

K.
IBLJLA , , RtQ N1 I P L wi L A . l .

. ]/ (»v^i

3

.l .

. 3) 00I

46

Figure

3.

Percent

of

deviation

from

TN

648

values

of

internal

energy

versus

pressure

at

some

isotherms

from

70

to

130

K.
100 K CO

kO

+-> 03

O)

13CO

CO

cu

s- D_

C\J

a;

ClawwO

Cl04wDW

Cl<u>

04 I l (0 -C

4-> C • O w

Idow

0)Di—I

to>

00Z

Ehg

Cl

4-1 co

4J <U

rO

g •iH O > cn

d) T3 4J

(0

4JC(1) o

Cl

0) 04

aiCl3

cn

•pH

LB

M • A H/( 8fr 9l I I e L

J

L

A • • H ) 00 1

47

isotherms

from

70

to

130
o o I I

I 9 U L A ,. Q I? Q M 1 I P LU L A . L .

. S/ (»V^N±

S _L .

. S)0QT

48

Figure

5.-

Percent

of

deviation

from

TN

648

values

of

entropy

versus

pressure

at

isotherms

of

70

to

110

K.
100 K

49

isotherms

from

70

to

210
170 K

-M

03

a>

s3300

00

aj

s_ QOI

[PULA , ,8^9N1 IP l J LA„ . l .

. e/(°‘/:?,Ni -L eL • • e ) ooi

50

Figure

7.-

Percent

of

deviation

from

TN

648

values

of

sound

speed

versus

pressure

at

some

isotherms

from

70

to

170

K.
Beattie-Bri

dgeman

equation

51

Figure

8.-

Percent

of

deviation

from

the

virial

equation

of

sound

speed

at

80

for

Beattie-Bridgeman

equation,

Jacobsen

equation,

and

experimental

data.
Beattie-Bridgeman

equation

52

Figure

9.-

Percent

of

deviation

from

the

virial

equation

of

sound

speed

at

150

for

Beattie-Bridgeman

equation,

Jacobsen

equation,

and

experimental

data.
0.03|—

Beattie-Bri

dgeman

equation

[B LJ la b/

( b-L^^ a b)ooI

53

Figure

10.-

Percent

of

deviation

from

the

virial

equation

of

sound

speed

at

310

K

for

Beattie-Bri

dgeman

equation,

Jacobsen

equation,

and

experimental

data.
1. Report No. NASA RP-1051

2. Government Accession No. 3. Recipient's Catalog No.

4. Title and Subtitle THERMODYNAMIC PROPERTIES OF NITROGEN GAS

DERIVED FROM MEASUREMENTS OF SOUND SPEED

5. Report Date December 1979

6. Performing Organization Code

7. Author(s) Ben Younglove and R. D. McCarty

8. Performing Organization Report No NBSIR 79-1611

L-13308

9. Performing Organization Name and Address National Bureau of Standards

Department of Commerce

Washington, DC 20234

10. Work Unit No. 505-31-53-01

11. Contract or Grant No.

12. Sponsoring Agency Name and Address National Aeronautics and Space Administration

Langley Research Center

Hampton, VA 23665

13. Type of Report and Period Covered Reference Publication

14 Sponsoring Agency Code

15. Supplementary Notes This work was done with an interagency agreement between MBS and NASA and funded

with NASA purchase orders L-55450A and L-69159A.

16. Abstract

A virial equation of state for nitrogen has been determined by use of newly mea- sured speed-of-sound data and existing pressure-density-temperature data in a multiproperty-fitting technique. The experimental data taken were chosen to

optimize the equation of state for a pressure range of 0 to 10 atm and for a temperature range of 60 to 350 K. Comparisons are made for thermodynamic properties calculated both from the new equation and from existing equations of

state

.

17. Key Words (Suggested by Author(s)

)

Equation of state

Thermodynamic properties

Speed of sound

p-p-T data

18. Distribution Statement Unclassified Unlimited

Subject Category 28

19. Security Classif. (of this report) 20. Security Classif. (of this page) 21. No. of Pages 22. Price' Unclassified Unclassified 53 $5.25

For sale by the National Technical Information Service, Springfield, Virginia 22161 NASA-Langley , 1979

National Aeronautics and THIRD-CLASS BULK RATE

Space Administration

Washington, D.C.

20546

Official Business Penalty for Private Use, $300

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