Thermodynamic properties of a mixture of He and N2 at high pressures determined from speeds of sound

PHY$1CA ELSEVIER

Physica A 246 (1997) 45-52

Thermodynamic properties of a mixture of He and N2 at high pressures determined from speeds of sound Samirendra N. Biswas, Cornelis A. ten Seldam, Jan A. Schouten

*

Van der Waals-Zeeman Institute, University of Amsterdam, Valckenierstraat 6547, 1018 XE Amsterdam, The Netherlands

Received 24 April 1997

Abstract The equation of state and heat capacity of a mixture of He and N2 of equal composition have been determined at temperatures from 156.15 to 298.15 K and at pressures from 200 to 1000 MPa using a recently developed computational method. This is based on stepwise construction of p T isobars starting from an experimental reference pp isotherm and using the previously reported sound speed data of the mixture as a function of pressure and temperature. The derived p p T data have been fitted to a new empirical equation of state for the mixture with a r.m.s. deviation of 0.02%.

I. Introduction The equation o f state and thermodynamic properties o f several binary gas mixtures are o f considerable interest both from a theoretical and technological point of view. The theoretical interest is due to the fact that accurate thermodynamic data o f gas mixtures are very useful for testing the thermodynamic property prediction methods, such as, Monte Carlo and molecular dynamics computer simulation, using theories o f statistical mechanics. The technological interest is owing to the fact that thermodynamic data of binary mixtures at high pressures are often required in the industrial application of high pressure technology on these mixtures. This investigation deals with a particular binary mixture, namely, helium and nitrogen, o f nearly equal composition. This system is o f special interest because of the

* Corresponding author. 0378-4371/97/$17.00 Copyright @ 1997 Elsevier Science B.V. All rights reserved PIIS0378-4371(97)00338-5

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following reasons. Firstly, equation of state data of the pure components of this system, namely He and N2, have recently been determined at pressures up to 1000 MPa and at temperatures from 98.15 to 298.15 K [1,2] so that their potential energy functions can be evaluated for the application of the one-fluid model to this binary mixture. Secondly, the molecular shapes of both helium and nitrogen are simple and there is a relatively large difference in size and mass of their molecules. These would allow an easier approach to acquire knowledge of unlike interactions and to compare existing theories of fluid mixtures with experimental data. And finally, experimental equation of state data for this binary gas mixture over such a wide pressure and temperature range are not available in literature up till now. Recently, we have developed a new computational method for the determination of the equation of state and thermodynamic properties of a pure gas at high pressures from speed-of-sound data [1]. This is based on stepwise construction of density-temperature isobars starting from one experimental reference pressure--density isotherm and using speed-of-sound data for the gas as a function of pressure and temperature. The method has already been applied to the accurate determination of the equation of state and heat capacity of pure helium in the temperature range 98.15-298.15 K with pressures up to 1000 MPa [1]. This methodology is now applied to the determination of the equation of state and heat capacity of a He-N2 mixture for which the speed-of-sound data are now available over a sufficiently wide temperature and pressure range.

2. Outline of the computational method The basis of the method is the thermodynamic relation for the sound speed u:

1/u2 = ( OP/OP)T -- ( Tp2 /Cp )( OV/OT)2p,

(1)

in which Cp is the isobaric heat capacity of the gas and V - 1 / p is the molar volume. Accordingly, the input data necessary for the calculation are those of the sound speeds u (p, T), the reference pp isotherm at 298.15 K and the isobaric heat capacity Cp(T) of the mixture at a certain reference pressure. The speed-of-sound data of the mixture (He-N2:49.8-50.2 mole percent) in the temperature range 156.83-298.15 K with pressures from 200MPa to 1000MPa are obtained from those reported recently by Zhang and Schouten [3]. These u (p, T) data are extended to zero pressure by computing the sound speed u0 of the mixture at p = 0 using the ideal gas law equation

u0(r) =

~

(2)

in which R is the gas constant, M is the molecular weight and Cv is the isochoric heat capacity. The molecular weight of the binary mixture, Mini×, is calculated from the molecular weights of the components Mne, MN2, using the relation Mmix = XMHe "}- (1 -- X)MN2

(3)

S.N. Biswas et al. IPhysica A 246 (1997) 45 52

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Table 1 The calculated values of the sound velocity at zero pressure, u0(T), of a mixture of He (49.8 mol%) and N2 (50.2 mol%) at several temperatures T (K) u0 (m/s)

298.15 481.14

273.06 460.47

248.15 438.98

223.15 416.28

198.23 392.35

173.15 366.70

156.83 348.99

x being the mole fraction of helium. The ideal-gas heat capacity ratio (Cp/C~) for the mixture at p = 0 is obtained by making use of the arithmetic mixing rule (4)

Cp,m(T) = XCp,He 4- (1 -- x)Cp,N2(T)

and the ideal gas-law relation Cp,m(T) = C~,m(T) - R.

(5)

The ideal-gas heat capacity of helium at constant pressure, Cp,He, is taken to be 2.5R while the ideal-gas heat capacity of nitrogen at constant pressure, Cp,Nz(T) (in J/(mol K)), is calculated from the following polynomial equation given in Ref. [4]: Cp,Nz(T) = ml + m2T + m3T 2 + m4T 3 + m5 T4

(6)

with ml =

2.9109996 x 10,

m2 = -8.0820995

m4 = -3.6893228 x 10 -8,

x 10 - 4 ,

m3 = 8.6142037

x 10 - 6 ,

ms = 5.6750880 x 10 -11.

Finally, the sound speed u0(T) of the mixture at p = 0 is calculated at several temperatures in the range 156.83-298.15 K after introducing in Eq. (2) the values of the ideal-gas heat capacity ratio and molecular weight of the mixture as obtained by the above procedure. The calculated values of u0(T) for the mixture at several temperatures are given in Table 1. The values of u0(T) are combined with the experimental sound speed data at higher pressures reported by Zhang and Schouten [3] and the combined data are fitted to the following double polynomial equation by a least-squares analysis:

u(p, T) = Z

cij(1/T)i(p3/a)J"

(7)

ij

Here cij are the coefficients of the double polynomial whose numerical values are given in Table 2 (for u in m/s, p in MPa, T in K). In the above double polynomial, p3/4 is used instead of p in order to stretch the free interval between p = 0 and p=200 MPa which otherwise would be too small to accommodate the steep increase of u vs p for each T. Also 1/T is used instead of T itself since this modification gives a better fit. The fitted double polynomial can be used to calculate u at any desired p and T as required for solving the thermodynamic relation of Eq. (1). The calculation of u(T, p) is carried out in two stages. First, for a given T, seven T-dependent coefficients, a0

S.N. Biswas et al./ Physica A 246 (1997) 45-52

48 Table 2 Coefficients c/j o f Eq. (7):

u(p, T ) = ~J-]~ijciJ xi yi, with x=l/T; y=p3/4 (u in m/s; p in MPa; T in K) coo = 1.1867336443 x 103 Cl i =2.8275722816 × 103 c22= - 4 . 0 3 9 2 3 2 2 8 1 0 × 103 c43 = - 1.2741870937 × 105 c2o = 1.0518344750 × 108 co3 = - 2 . 4 9 9 8 3 0 2 2 3 7 × 10 - 3 co5= - 6.6656715494 × 10 - 8 co6=9.5630733011 × 10 -11

co1=1.3041573080 co2=1.7029615236× 10 -1 co4= 1.8115936281 x 10 - 5 clo = - 4 . 4 4 2 0 3 4 2 4 2 7 x 105 c15---3.6773212996 x 10 - 7 c33--4.1969755901 × 103 c34---8.5325825686; c3o = - 1.2539696356 × 101° c4o---5.8636584678 × 1011

through a6, are computed as polynomials in e=l/T: ao = coo + e[clo -1- e{c20 + e(c30 q- c40)}]; al = c01 + ec11;

a4 -- c04 q- e.2e.c34;

a2 = c02 + 2ec22; a5 = co5 + e'c15;

a3 = c03 + e.2e(c33 + ec43), a 6 = c06.

(8)

Next, for a given p at this T, u(T,p) is computed as a straightforward 6th degree polynomial, namely X--" a i (9) U= ~ iq, i

with q = (p)3/4.

(10)

In the present computational procedure, the isobaric heat capacity data of the mixture are required only for the first selected isobar which corresponds to a pressure of 200 MPa. As the isobaric heat capacity data of the mixture at this pressure are not available in the literature, the required data have been evaluated from the previously reported Cp vs T data of pure He and N2 at 200 MPa by making use of the arithmetic mixing rule given in Eq. (4). The Cp vs T data for the mixture thus found have been fitted to a simple polynomial of the type 5

Cp = Z A j T J ,

(11)

j=0

A j being the coefficients whose values are given in Table 3. It should be pointed out

here that the arithmetic mixing rule given in Eq. (4) is strictly valid for ideal gases and not for gases at elevated pressures. However, it turns out that the derived EOS data of the mixture are not very sensitive to the Cp data of the mixture at 200 MPa used in the present calculation. Therefore, a large error in the calculated Cp data introduces only a small error in the final EOS data. The remaining terms necessary to solve the thermodynamic relation of Eq. (1) for (~3V/OT)p a r e the derivative (Sp/Op)r and the density p corresponding to the selected

S.N. Biswas et al. IPhysica A 246 (1997) 45 52

49

Table 3 Coefficients Aj of Eq. (11): Cp(T) = ~-]j A i Tj (Cp in Jmo1-1 K -1, T in K)

j

Aj ( J m o 1 - 1 K - j - I )

j

Aj, Jmo1-1 K - i - I )

0 1 2

--52.403563; 2.1003818; --2.0334528x10

3 4 5

9.5655115x 10 -5 - 2 . 1 9 8 1 7 0 4 x 10 7 1.9758679x10 10

2

Table 4 Values of fitted parameters of Eq. (12): p=A+Bp-l+Cp-2+Dp 'n (p in mol 1-1, p in MPa) A :- -5.5712017314 x 101 C :- 1.1124402533 x 104 m = 0.1330

B = -4.2336777435 x 102 D = 4.4379434008 x 101

isobars at the reference temperature, 298.15 K. These are calculated by using the following isothermal equation of state:

p = A + B p -1 + Cp -2 + D p -'n

(12)

whose parameters A, B, C, D and m are evaluated by fitting this equation to the experimental p (p) data of the same mixture reported by Zhang et al. [5] at T-298.15 K and p=200-1000 MPa. An additional data point at p=50 MPa, obtained from the previously reported EOS data of He+N2 mixtures by Pfenning et al. [4], has been included in this fit. The fitted values of the parameters A, B, C, D, m are given in Table 4. The stepwise construction of pT isobars starts from the reference temperature 298.15 K with the repeated use of the thermodynamic relation of Eq. (1). The computations are carried out from 200 to 1000 MPa with pressure steps of 10 MPa. The sound speeds at these pressures and at the reference temperature are easily calculated from Eqs. (7)(10) using the coefficients given in Table 2. Also, the density p and the first derivative (c3p/c3p)r at the selected pressures are easily obtained from Eq. (12) of which the coefficients are given in Table 4. Thus, the only data necessary to solve Eq. (1) for (~V/(~T)p are those of Cp as a function of pressure at the reference temperature. As the Cp(T) data for the mixture are only evaluated at p=200 MPa, an iteration procedure has been adopted to solve Eq. (1) for (OV/OT)p. This is described below. In the first iteration, approximate values of (~V/~T)p corresponding to the selected pressures are determined by solving Eq. (1) in which the value of Cp at 200 MPa is used for all higher pressures. In the second iteration for this isothermal series, the corrected values of Cp at all higher pressures are evaluated by using the thermodynamic relation p O

Cp = C~,,i - ] T ( ~ 2 V / O T 2 ) p d p pl

(13)

50

S.N. Biswas et al./Physica A 246 (i997) 45-52

in which Cp, 1 is the isobaric heat capacity for the first selected pressure i.e., P1=200 MPa. The required second derivative (O2V/8T2)p is evaluated by taking the difference of the computed values of (OV/OT)p of two successive isothermal series with a temperature interval A T:

( ~2 V/~T2)p = [(~V/OT)p,T - ( ~V/~3T)p,T-~T]( AT) -l,

(14)

in which A T = 1 K. Introducing the corrected value of Cp for each pressure step in the thermodynamic relation of Eq. (1), the corrected value of (OV/~3T)p at that pressure is determined. The values of (OV/c3T)p are easily converted to the corresponding values of (Op/OT)p. These are then used to calculate the densities corresponding to the selected pressures of the next isothermal series using the relation

p(T + A T) = p(T) + A T(c3p/aT)p,

(15)

p(T+A T) being the density for any particular pressure at temperature T+A T. Finally, the pp data obtained at T ÷ A T are fitted to the isothermal equation of state, Eq. (12), giving values of the parameters A, B, C, D, m pertaining to the temperature T-t-AT, which then provide the values of (OP/~p)r for this isothermal series. Thus, by repeating the whole process at each temperature with an interval of 1 K, the p T isobars of the mixture are constructed down to the lowest temperature 156.15 K.

3. Results and discussion

The calculated densities corresponding to several selected temperatures along 17 selected isobars are recorded in Table 5. The vertical columns for the densities together with that for the selected pressures also provide pp isotherms for the several selected temperatures. The number of significant figures for the recorded densities does not indicate the absolute accuracy, but extra digits are retained so that it is possible to compute the derivatives of the equation of state with greater precision. This is justified from the high precision of the least-squares fits of the derived pp data. During the construction of the p T isobars, the heat capacity at constant pressure, Cp, of the binary mixture is obtained as a function of temperature and pressure. The heat capacity at constant volume, Co, of the mixture is also calculated by using the thermodynamic relation

Cp/Co = u 2(~p/c3p)~.

(16)

The calculated values of Cp and Co at round values of the pressure with intervals of 50 MPa and at several selected temperatures are given in Tables 6 and 7. As shown, in the pressure and temperature range of the present computation, Co increases monotonically with pressure at all temperatures while Cp is practically independent of pressure at higher temperatures but slowly decreases with pressure at lower temperatures. A similar behaviour of the variation of Cp and Co with pressure and temperature was previously noted for the pure components, He and N2 (cf. Refs. [1,2]).

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Table 5 Calculated ppT data of a mixture of He (49.8 mol%) and N2 (50.2 mol%) along selected isobars T (K):

298.15

p(MPa)

p (T,p) (moll -1 )

200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950 1000

32.237 35.265 37.763 39.895 41.758 43.416 44.911 46.274 47.528 48.690 49.773 50.788 51.744 52.647 53.504 54.318 55.096

273.15

33.477 36.454 38.908 41.002 42.830 44.457 45.923 47.259 48.488 49.626 50.687 51.680 52.616 53.499 54.337 55.134 55.894

248.15

223.15

198.15

173.15

156.15

34.816 37.732 40.135 42.183 43.970 45.558 46.989 48.293 49.491 50.601 51.634 52.603 53.514 54.375 55.191 55.967 56.706

36.266 39.109 41.447 43.437 45.173 46.714 48.101 49.365 50.526 51.601 52.603 53.541 54.424 55.258 56.049 56.801 57.518

37.840 40.589 42.845 44.763 46.434 47.916 49.252 50.467 51.585 52.619 53.583 54.487 55.337 56.140 56.902 57.627 58.318

39.551 42.176 44.325 46.151 47.743 49.156 50.428 51.588 52.655 53.642 54.563 55.426 56.239 57.007 57.736 58.430 59.091

40.793 43.311 45.375 47.127 48.653 50.008 51.232 52.349 53.379

.... ------

Table 6 Isobaric hem capacity (Cp) of a mixture of He (49.8 mol%) and N 2 (50.2 mol%) ~ selected values of pressure and temperature T (K):

298.15

273.15

p (MPa)

Cp (T, P) (J mo1-1 K -1 )

200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950 1000

29.9 29.8 29.9 29.9 30.0 30.1 30.2 30.3 30.4 30.4 30.5 30.5 30.5 30.6 30.6 30.6 30.6

30.4 30.4 30.5 30.7 30.8 30.9 31.0 31.1 31.2 31.3 31.3 31.3 31.3 31.3 31.3 31.3 31.3

248.15

223.15

198.15

173.15

156.15

30.7 30.8 30.8 30.9 31.0 31.0 31.1 31.1 31.1 31.0 31.0 31.0 30.9 30.9 30.8 30.8 30.7

30.9 30.9 30.9 30.9 30.9 30.9 30.8 30.8 30.7 30.6 30.5 30.4 30.3 30.2 30.1 30.0 29.9

31.1 31.0 30.9 30.9 30.8 30.6 30.5 30.4 30.3 30.2 30.0 29.9 29.7 29.6 29.5 29.3 29.2

31.4 31.2 31.0 30.8 30.6 30.4 30.2 30.0 29.8 29.6 29.5 29.3 29.1 28.9 28.6 28.4 28.1

31.6 31.5 31.2 30.9 30.6 30.3 30.1 30.0 29.9 --

--

For the interpolation of the density data as well as for the calculation of thermodyn a m i c f u n c t i o n s , it is c o n s i d e r e d u s e f u l to c o n s t r u c t a n e q u a t i o n o f s t a t e f o r t h e m i x t u r e which represents the data well within their estimated errors. Therefore, the derived p p T data of the He-N2

m i x t u r e h a v e b e e n f i t t e d to t h e f o l l o w i n g e m p i r i c a l e q u a t i o n o f s t a t e

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Table 7 Isochoric heat capacity (Cv) of a mixture of He (49.8 mol%) and N2 (50.2 mol%) at selected values of pressure and temperature T (K):

298.15

p (MPa)

Cv (T, P) (J mol "1 K -1 )

200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950 1000

273.15

20.2 20.6 20.9 21.3 21.6 21.9 22.2 22.5 22.7 23.0 23.2 23.4 23.6 23.8 23.9 24.1 24.3

Table 8 Coefficients MPa)

20.6 21.1 21.5 21.9 22.3 22.7 23.0 23.3 23.6 23.8 24.0 24.3 24.5 24.7 24.8 25.0 25.1

248.15

223.15

198.15

173.15

156.15

20.9 21.4 21.8 22.2 22.6 22.9 23.2 23.5 23.7 23.9 24.2 24.3 24.5 24.7 24.8 24.9 25.1

21.1 21.6 22.0 22.4 22.7 23.0 23.3 23.6 23.8 24.0 24.2 24.3 24.5 24.6 24.7 24.8 24.9

21.3 21.8 22.3 22.6 23.0 23.3 23.5 23.8 24.0 24.1 24.3 24.4 24.5 24.6 24.7 24.8 24.8

21.8 22.3 22.7 23.1 23.4 23.7 23.9 24.1 24.3 24.4 24.5 24.6 24.7 24.7 24.7 24.7 24.7

22.3 22.9 23.3 23.7 23.9 24.2 24.4 24.6 24.7 ---------

Cij of Eq. (17): l n p = ~ i j Cijxi )d, with x=T; y=l.O/p °3°2 (p in moll -1 , T in K, p in

i

j

Cij

i

j

Cij

0 1 0

0 0 2

1

2

0 1 2 2

1 1 0 1

-1.0165037001E× 101 -1.4918761161 x 10 - 2 -2.8114747230×10 6 2.0051071589× 10 - 5

0

3

4.8994971564 1.8345168815x 10 -3 4.1919315107×101 --2.5545360880x 10 - 2 --6.4579279190× 101

using a least-quares procedure: In p = ~

Cijxiy j,

(17)

ij where

x = T , y = l . O / p °'3°2, a n d Cij a r e t h e c o e f f i c i e n t s w h o s e v a l u e s a r e g i v e n i n T a b l e 8.

T h e fit is f o u n d to b e e x t r e m e l y

good, the r.m.s, deviation being 0.02%.

References [1] [2] [3] [4] [5]

C.A. ten Seldam, S.N. Biswas, J. Chem. Phys. 96 (1992) 6163. S.N. Biswas, C.A. ten Seldam, J. Chem. Thermodyn. 23 (1991) 725. W. Zhang, J.A. Schouten, Fluid Phase Equilibria 79 (1992) 211. D.B. Pfenning, F.B. Canfield, R. Kobayashi, J. Chem. Eng. Data 10 (1965) 9. W. Zhang, S.N. Biswas, J.A. Schouten, Physica A 182 (1992) 353.