Thermodynamic Properties of Air: Compressibility Factor, Z, and Related Functions, Study Guides, Projects, Research of Thermodynamics

Download Thermodynamic Properties of Air: Compressibility Factor, Z, and Related Functions and more Study Guides, Projects, Research Thermodynamics in PDF only on Docsity!

/ @ @ TABLES F

OF AI

R. L a 1

VKF

C. A. Neel, Scientific Computing Services ARO, Inc. Ol DEVEL 1 1 I I AIR FORCE SYSTE S COMMAND

US ~~AF

@

ASTIA (TISVV) ARLINGTON HALL STATION ARLINGTON 12, VIRGINIA

Department of Defense contractors must be established for ASTIA services, or have their need-to-know certified by the cogni- zant military agency of their project or contract.

'JJ

AEDC.TN.61.

ABSTRACT

Tables are presented for the thermodynamic properties of air from 90 to 1500 0 K using (1) temperature and pressure and (2) temperature and density as independent variables. The pressure tables range from 1 to 1200 atmospheres, whereas the density tables extend from 10- 7 to 400 relative atmospheres (or amagat units). The tabulated properties are the compres- sibility factor Z, the dimensionless functions E/RT, H/RT and SIR, together with pressure and density. The tables account for the so-called real gas effects, i. e., van der Waals 1 effects, at high pressures, but do not include the effects of the small amounts of dissociation at low pressures and high temperatures. The sources for the tables as well as the final tables are discussed.

i·'_J

CONTENTS

ABSTRACT ....

NOMENCLATURE.

INTRODUCTION

CONVERSION OF THE SOURCE DATA

ENTRIES AT CONSTANT DENSITY..

CHECKING AND INTERNAL CONSISTENCY

OF THE TABLES ............

NEGATIVE INTERNAL ENERGIES AND ENTHALPIES

REFERENCES

APPENDD{ ....................

TABLES

1. Reference^ Levels^ and^ Tabulated^ Forms^ of^ the Thermodynamic Properties of the Sources
2. Tables of Thermodynamic Properties of Air from 90 to 1500 0 K with Temperature and Pressure as Independent Variables
3. Tables of Thermodynamic Properties of Air from 90 to 1500 0 K with Temperature and Density as Independent Variables

I LLUSTRA TIONS

Figure

1. Locus of Points of Zero Internal Energy

AEDC·TN·61·

Page 3 7 9 10 11

12 13 17 19

2. Z as a Constant-Pressure Function of Temperature 22
3. Z as a Constant-Temperature Function of Pressure. 23
4. Internal Energy of Air at Constant Temperature... 24

" C>

AEDC.TN.61.

INTRODUCTION

The purpose of this work is to provide tables of the thermodynamic properties of air at temperatures below 15.9ooK which, together with the high temperature tables of Hilsenrath, Klein, and Woolley (Ref. 1), would cover the widest possible range of thermodynamic conditions in a uniform manner. The tables of Ref. 1 afford great convenience in that the dimensionless parameters Z, E/ RT, H/ RT, and S/ R and the pres- sure (in atm) are tabulated in terms of the absolute temperature (OK) and the common logarithm of relative density (amagats) as independent variables.

The tables of Ref. 1 are based mainly on theoretical calculations, and they extend from 1500 to 15, OOooK. Although they provide for the effects of dissociation and ionization, they have no regard for intermo- lecular forces and other van der Waals I effects. This lack is partially met by the tables of Gilmore (Ref. 2) which allow for gas imperfections, using second and third virial coefficients based on the Lennard-Jones potential. However, these latter tables provide entries only at 1000-deg intervals from 20000K upwards and only at densities from 10 to about 300 amagats.

The tables presented herein are based on those of Hilsenrath, Beckett, et al. (Ref. 3) and of Din (Ref. 4) whose entries are converted by machine-computed interpolation in compliance with the convenient for- mat of Refs. 1 and 2. The tables of Ref. 3 are constructed in terms of temperature and pressure as independent variables and with rather large intervals of pressure from 10- 2 to 102 atm. The tables of Ref. 4 extend from 90 to 450 oK, with pressures ranging from 1 to 1200 atm. However, the latter tables express thermal quantities in dimensional form, and they reckon entropy and internal energy from the state of the boiling liquid at one atm. On the other hand, in this country it is generally customary to use the internal energy of the ideal gas at absolute zero as the corre- sponding reference level. The earlier tables of Benedict and Hilsenrath (Ref. 5) for which the dimensionless form had not yet been adopted, also offer data not conveniently found elsewhere.

Two sets of tables are provided. * In the first set, temperature and pressure are adopted as independent variables. The second set are

*Grateful acknowledgments are due Mrs. Homer Stooksbury and Mrs. Wanda Little who participated in the final preparation and checking of these tables. Manuscript released by authors July 1961.

AEDC. TN·61.

is numerically equal to 1/273.16, the pressure being expressed in atmos- pheres, the temperature in oK, and the density, P / Po> in amagat units.

H/RT = (tabulated enthalpy in joules/mole x 0.12027)-443. T (3)

E/RT = (H/RT) - Z (^) (4)

Equation (4) is an adaptation of the definition of enthalpy (H = E+pV) where ZRT is substituted for pV and a division by RT is made.

SIR = (tabulated entropy in joules/mole - oK x 0.12027) + 10.374 (5)

The constants In Eqs. (3) and (5) are averages of numerous compar- isons of pairs of check points in Hilsenrath, Beckett, et al. (Ref. 3) and Din (Ref. 4). Variations of the constants from one pair of check points to another appeared only in the last significant figures.

p = tabulated pressure in atmospheres (^) (6)

The entries of Hilsenrath, Beckett, et al. (Ref. 3) already have the desired form with the exception of the enthalpy function which is converted by the relation

H/RT = x-ToT (7)

ENTRI ES AT CONSTANT DENSITY

The construction of Table 3 (constant density) naturally divides it- self into two parts, viz: the perfect and real gas domains. In the per- fect gas domain, extrapolations to low pressures utilize the facts that enthalpy is a function only of temperature and that isothermal changes of entropy depend only on pressure. The values of H/RT and S/ R at a pressure of 10- 2 atm given in Ref. 3 are used as points of departure. The extrapolations devolve from the following relations:

At given values of p/po and T

p = RT (p/p (^) o ) (8)

A EDC· TN.61.

H/RT H/RT at 10- 2 atm (9)

E/RT (H/RT) - 1 (10)

SIR '" SIR at 10- 2 atm - In (p/10- 2 )

where Eq. (11) is an adaptation of the classical expression for the entropy of a perfect gas.

( 11)

In the real gas domain, the entries of Table 3 are derived from those of Table 2, with the addition of data in the range of pressures from one to 10- 2 atm in Ref. 3. The isothermal loci of Z, E/RT, S / R and log p, plotted as functions of log P / Po' exhibit so little cur- vature in any given region that interpolations can safely be made with the ratio

log (p/po)m - log (p/p)n log (P/P)h - log (p/p)n

in which the subscript m denotes the desired value of p / Po' the sub- script n refers to the value of p / Po nearest to and less than (pi Po)m' and the subscript h identifies the value of p / Po nearest to and greater than (pi Po)m' This simple procedure is admissible because entries are available for close intervals of density. The real-gas interpola- tions were carried down (in density) as far as possible, and then ideal gas data were added to bring the density- entries down to 10- 7 amagats. No adjustments are necessary to smooth the transitions from one set of data to the other.

CHECKING AND INTERNAL CONSISTENCY OF THE TABLES

The checking of Table 2 consisted of two parts. The first was the comparison of numerous pairs of points in the converted tables and Hilsenrath, Beckett, et al. (Ref. 3). The agreement proved to be excel- lent; differences occurred only in the last significant figures. Then each thermodynamic property was plotted as an isothermal function of pres- sure. This provided a check on the smoothness of the data and occasion- ally revealed errors of conversion, and in a few instances, errors in the original data.

Checks for internal consistency were made in both sets of tables. The internal consistency was taken as the ability of the tabulated values

A E DC- TN·61-1 03

which the internal energy is zero (cf. Fig. 1), and this line rises very steeply as the temperature is raised to 1700K and higher.

Being a real gas effect, this phenomenon may be better understood with reference to the derivatives of the compressibility factor at low tem- peratures. Internal energy is a function of pressure and temperature so that E == f (p,T) (12)

Taking the total derivative,

dE == (~) dp + (aE) dT ap^ T aT p

( 13)

Expanding each term by formulas from Bridgman I s tables of thermo- dynamic formulas (Ref. 6),

and

Since for a real gas

( ~) ap T ==^ -^ T^ (~)aT p -^ p^ (~)ap T

(~~ )p

v == ZRTp

Eqs. (14) and (15) become:

( aE)ap (^) T ==^ -^ T^ [}L Z^ (~)aT 1> +^ R^ (az)ap (^) T]

Substituting Eqs. (17) and (18) into Eq. (13),

dE == - T [~ (~~~p + R (~!)T ] dp

( 14)

(15)

(16)

(18)

(19)

AEDC- TN-61-

For a perfect gas, Eq. (19) reduces to the ideal gas relationship:

dE = (C (^) p - R)dT = CvdT (20)

The partial derivatives of Z in Eq. (19) govern the behavior of the internal energy. Figure 2 shows Z as a constant-pressure function of temperature, whereas Fig. 3 gives Z as an isothermal function of pres- sure. From these it may be seen that the derivatives of Z may be positive, negative, or zero, depending upon the conditions of pressure and temperature. It is instructive to examine the numerical magnitude of these derivatives.

In Fig. 2, the curve for p = 50 atm has the steepest slope. At 140 o^ K, (aZlaT)p has a value of O. 025/°K which is a maximum for p = 50 atm. This slope rapidly decreases and for temperatures greater

than 400 o^ K, where it is 8 x 10- 5 10 K, is almost zero and at very high

temperatures assumes a slight negative value. For pressures above 300 atm the derivatives are negative at all temperatures. Forp = 500atm and T = 2000K the partial derivative is equal to -1. 75 x 10- 3/°K.

In Fig. 3 the line for T = 130 0 K displays the steepest slopes. For

p = 30 to 35 atm (azi ap)T is - O. 077 I atm, whereas at p = 50 to 60 atm

a positive slope of 0.0041 atm is found; this positive slope is nearly constant for higher pressures. At 10000K the slope had decreased to

3.4 x 10- 4 /atm. As the temperature increases, the slopes of the curves

in Fig. 3 approach zero.

Examining the first term of Eq. (19) and assuming that the temper- ature is held constant, the effect of pressure on the internal energy may be seen. At a constant temperature, the internal energy will increase or decrease with an increase in pressure, depending upon the magni- tudes and signs of the partial derivatives of Z. If the portion of the term in brackets is positive, the internal energy will decrease with increasing pressure and,increase when the bracketed portion is negative.

At low pressures, (a zi aT)p is positive, and (azi a1p)T is negative.

When combined with the other terms withi.n the bracketed quantity, the result is a positive quantity because '--t ( ~i ) is numerically greater

than R (~!)T .p

At moderate pressures both derivatives become positive and will add, thereby causing the internal energy to decrease rapidly with an increase

in pressure. Between the low and moderate pressure regions (azi ap)T

decreases from its negative value to zero and then to increasing positive

AEDC.TN.61.

The tendency at all pressures for the sum of Z and T(aZ/ aT)p to approach unity as the temperature increases has the effect of reducing the temperature term of Eq. (19) to (C (^) p - R)dT. Cp is always positive and greater than R, making (C (^) p - R)dT a positive quantity. Thus an increase in temperature at constant pressure results in an increase in internal energy.

Combining the effects of pressure and temperature on the internal energy of the real gas it is found that a rise in temperature increases the internal energy of the gas, whereas an increase in pressure reduces the internal energy. At high pressures the depressing effect of pres- sure may become larger than the increase in internal energy caused by temperature alone. Under these conditions negative values of internal energy result. The necessary conditions for this effect may be seen in Fig. 1. All points to the left of the curve in Fig. 1 represent negative values of internal energy, whereas those to the right give positive internal energy. As pressure and temperature both decrease to zero the internal energy will always approach zero.· This is to be expected as the zero reference point for internal energy was chosen at OaK and zero pressure. As the temperature increases the pressure required to attain the state of zero internal energy becomes very large and ap- pears to increase exponentially.

At low temperatures the pressures necessary to obtain the negative internal energies are moderate, and negative values appear in Table 2. As the temperature increases this pressure becomes greater, and the negative values begin to disappear from the tables. At temperatures above 1700K the range of pressures in the tables is not great enough to generate negative values of internal energy.

REFERENCES

1. Hilsenrath, J., Klein, M., and Woolley, H. W. "Tables of Thermo- dynamic Properties of Air Including Dissociation and Ionization from 1500 0 K to 15, OOooK." National Bureau of Standards, AEDC-TR-59-20, December 1959.
2. Gilmore, F. R. "Additional Values for the Equilibrium Composition and Thermodynamic Properties of Air." Rand RM-2328, December 1959.
3. Hilsenrath, J., Beckett, C. W., et al. "Tables of Thermal Pro- perties of Gases." National Bureau of Standards Circular 564, November 1955.

AEDC-TN.61-I

4. Din, F. Thermodynamic Functions of Gases, Vols. 1 and 2. Butterworths Scientific Publications, London, 1956.
5. Benedict, W. S. and Hilsenrath, J. "Tables of Compressibility and Density of Air. 11 National Bureau of Standards Report 1192, December 1951.
6. Bridgman, P. W. A Condensed Collection of Thermodynamic Formulas. Harvard University Press, Cambridge, Mass. J 1925.