Thermal conductivity of hydrogen-nitrogen and ...

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Thermal conductivity of hydrogen-nitrogen and hydrogen-carbon-dioxide gas mixtures To cite this article: P Mukhopadhyay et al 1967 Br. J. Appl. Phys. 18 1301

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BRIT. J. APPL. PHYS.,

1967, VOL. 18.

PRINTED IN GREAT BRITAIN

Thermal conductivity of hydrogen-nitrogen and hydrogen-carbon-dioxide gas mixtures P. MUKHOPADHYAY, A. DAS GUPTA and A. K. BARUA Indian Association for the Cultivation of Science, Calcutta, India MS. received 12th October 1966, in revised form 22nd February 1967 Abstract. The thermal conductivity of H2-N2 and H2-CO2 mixtures has been measured from - 15 to 200"c by using the thick-wire variant of the hot-wire method.

The experimental values have been compared with those calculated from the Hirschfelder-Eucken formula. Although fairly satisfactory agreement between theory and experiment is obtained it appears that for a precise interpretation of the thermal conductivity of polyatomic gas mixtures a better representation of the cross-relaxation phenomena is necessary. 1. Introduction It is now well recognized that accurate measurement of the thermal conductivity of gases may be of great value for a proper understanding of the interchange of energy between the internal and external degrees of freedom of polyatomic molecules. For pure polyatomic gases several attempts have been made to correlate the thermal conductivity with the rate of exchange of energy (e.g. Waelbroeck and Zuckerbrodt 1958, Srivastava and Barua 1960). When the exchange of energy between the different degrees of freedom is very fast, so that local chemical equilibrium is maintained (molecules in different quantum states being assumed to be different chemical species), the modified Eucken expression (Hirschfelder 1957a) for the thermal conductivity of a polyatomic gas is obtained. However, for most of the real gases, the condition of local chemical equilibrium is not satisfied and for a proper interpretation of the thermal conductivity it is essential to consider the effects of relaxation of the different types of internal energy. A significant advance in this respect has been made by Mason and Monchick (1962) who have reduced the formal kinetic theory of polyatomic gases (Wang Chang and Uhlenbeck 1951) to a tractable form by making a number of well-defined (though not necessarily very accurate) approximations. Mason and Monchick's treatment, which is particularly applicable to the case for which exchange is small but not negligible, has not yet been tested conclusively. For polyatomic gas mixtures Monchick et al. (1965) have made an attempt to extend the formulation of Wang Chang and Uhlenbeck (1951). However, non-availability of extensive and precise experimental data has till now prevented any real progress in the understanding of the thermal conductivity of polyatomic gas mixtures. With a view to obtaining a clearer idea of the mode of exchange of energy in a mixture we have measured the thermal conductivity of H?-N2 and Hz-CO2 mixtures over the temperature range - 15-2OO"c. At these temperatures the vibrational exchange may be neglected. The rotational-translational relaxation time for H, is widely different from those for N, and CO, so that the effect of impedance of heat transfer by unlike interaction will play a comparatively significant role. 2. Experimental For our measurements we utilized the thick-wire variant of the hot-wire method (Kannuluik and Martin 1933, 1934). The conductivity cell has already been described by several workers (Srivastava and Saxena 1957), however, in place of Perspex, a Teflon cap was used and sealing was done by Araldite natural powder. For the temperature range 80-2OO"c the apparatus was placed in an oil bath and for the range -15-20"c a low-temperature bath with a Freon compressor was used. The temperature control was in general to within 1301

1302

P. Mukhopadhyay, A. Das Gupta and A. K. Barua

-0.01 degc. During the whole experiment, several cells had to be used and in table 1 the constants of the cell which was used for the Hz-COz system (temperature range 80-2OO0c) are given as a typical example. The hydrogen and nitrogen were supplied by Indian Oxygen Ltd. and purity was 99-95%. Pure COz was obtained by heating BaC03. The gas mixtures were prepared in a suitably constructed gas mixing unit and the compositions were calculated by the law of partial pressures; some sample checks were made by a mass spectrometer (Associated Electrical Industries, MS-3 model). Table 1. Constants of the thermal conductivity cell at different temperatures

Temperature ("c) Thermal conductivity h of platinum wire (cal degc-1 cm-l sec-I) Temperature coefficient of resistance a: of platinum wire (degc-l) Resistance of cell wire (a) Cell constant (1 - C) Length 31 of cell wire (cm) Radius 1'1 of cell wire (cm) Internal diameter 2rz of tube (cm) Outer diameter 2Y3 of tube (cm)

80.1 0.1703

120.1 0.1730

160.1 0.1746

200.1 0.1754

0.0028

0.0025

0.0023

0.0021

1.1005 0.9810

1,2160 0.9810

1.3330 0.9810

1.4522 0.9800

6.304 0 * 00506 0.301 0.500

The solution of the differential equation for heat flow along the heated wire can be written as (Kannuluik and Martin 1933, 1934)

where R is the resistance of the wire when a current of I A is flowing through it, Ro is the resistance of the cell wire at bath temperature for I = 0, h and Ku are the thermal conductivity of the wire and the gas respectively. The value of J was taken as 4.1855 J cal-1. If the cell is highly evacuated, the conduction through the gas can be replaced by the radiation loss hr, which is in general quite small. Under these conditions equation (1) may be written as

which may be used for the determination of A. The various corrections to be applied to Ku and the constants were determined by the methods which have been described earlier by several workers (Srivastava and Saxena 1957, Kannuluik and Carman 1952). In the actual experiment we have calibrated the apparatus by a standard gas to take into account any asymmetry in the cell through the factor 1 - C which is calculated from the relation K=K(1 - C)

(5)

K being the true conductivity and K' the value of Ku after corrections for radiation loss, wall effect, etc. have been applied. The cell was calibrated at each temperature with spectroscopically pure Ne (supplied by British Oxygen Co.) and the factor 1 - C at different temperatures was obtained by using the thermal conductivity data of Kannuluik and Carman (1952).

Thermal conductiuity of Hz-Nz and Hz-CO2 gas mixtures

1303

At every temperature three different currents were used and Ku values agreed to within 0.5% on average. Taking everything into consideration our data should be accurate to within 1 %. The results obtained for HpN? and Hz-CQz mixtures are shon-n in tables 2 and 3. For H2-N2 Gray and Wright (1961) have measured the thermal conductivity from 25 to 150°C by using a two-wire type of apparatus. We have compared by interpoiation our data with those of Gray and Wright (1962) at 9 9 . 1 " ~and the average deviation Table 2. Thermal conductivity of Hz-iVz mixtures at: different temperatures

Temperature ("c)

-14.9

0.1

P/, H?

x 10"

5.71 8.83 13.37 18.17 26.07 39.46

5.47 8.53 12.98 17.70 25.47 3838

0.0

5.80 9.08 13.55 18.58 26.99 40.41

5.96 9.21 13.87 19.00 27.24 41.21

5.80 8.87 13.37 18.32 26.34 40.41

6.13 9.37 14.29 20.11 28.13 42.06

6.29 9.72 14.65 20.05 28.74 43 e43

6.13 9.33 14.02 19.17 27.51 42.06

6.69 14.08 19.48 25.99 33.50 48.86

7.22 14.04 19.63 26.14 33.66 49.38

6.69 13.60 19.03 25.36 32.69 48.86

7.40 15.18 20.89 28.31 36.49 52.20

7.82 15.16 21.18 28.16 36.19 52.92

7.40 14.72 20.62 27.47 35.38 52.20

7.89 14.46 23.83 28.05 37.60 55.21

8.40 14.54 24.29 27.85 38.47 56.59

7.89 14.01 23.47 26.87 37.18 55.21

8.43 14.98 25.54 29.16 40.34 58.81

8.97 15.49 25.86 29.58 40.79 59.61

8.43 14.85 24.78 28.71 39.11 58.81

0.0

0.0

0.0 30.6 48.8 64.9 79.2

0.0 25.0 52.8 60.5 79.0 100.0

200.1

Kcalc

5.47 8.62 13.24 17.96 25.70 38.58

100.0

160.1

x 10'

0.0 19.1 40.1 57,5 77.8 100.0

30.6 48.8 64.9 79.2 100.0 120.1

Kcalc

(from (6) but using expt. values of KI and Kz) (cal degc-1 cm-1 sec-1)

19.1 40.1 57.5 77.8 100.0

80.1

x 10'

(from equation (6)) (cal degc-1 cm-1 sec-1)

19.1 40.1 57.5 77.8 100.0 20.1

Kespt

(cal degc-l cm-l sec1)

0.0 25.0 52.8 60.5 79.0

100.0

P. Mukhopadhyaj, A . Das Gupta and A . K. Barua

1304

Table 3. Thermal conductivity of HC2-02 mixtures at different temperatures

Temperature (‘c)

% H?

KesptX

105

Kcalc

x 10’

Kca~cx 10’

(cal degc-l cm-1 sec-1)

(from equation (6)) (cal degc-1 cm-I sec-l)

(from (6) but using expt. values of KI and Kz) (cal degc-1 cm-lsec-1)

0.0 18.9 42.0 64.1 80.7 100.0

3.31 6.06 10.37 16.99 24.61 38.58

3.59 6.41 11.11 17.79 25.33 39.46

3.31 6.06 10.65 17.19 24.59 38.58

0.0 18.9 42.0 64.1 80.7 100.0

3.53 6.37 10.83 17.60 25.48 40.41

3.84 6.78 11.67 18.59 26.40 41.21

3.53 6.41 11.20 17.99 25.70 40.41

0.0 18.9 42.0 64.1 80.7 100.0

3.72 6.63 11.44 18.58 26.98 42.06

4.19 7.31 12.50 19.85 28.06 43.43

3.72 6.74 11.75 18.89 26.90 42.06

0.0

25.2 37.4 57.7 79.0 100.0

4.78 9.39 12.35 19.01 30.00 48.86

5.24 10.24 13.39 20 * 28 31.25 49.38

4.78 9.73 12.86 19.71 30-65 48.86

120.1

0.0 20.8 32.8 62.6 78.5 100.0

5.82 9.43 12.21 22.74 32.05 52.20

5.96 10.27 13.38 24.37 33.46 52.92

5.82 10.06 13.12 23.97 32.93 52.20

160.1

0.0 17.9 41.4 59.4 78.5 100.0

6.54 9.59 15.93 22.94 34.44 55.21

6.67 10.51 17.33 24.63 35.85 56.29

6.54 10.29 16.95 24.10 35.09 55.21

200.1

0.0 15.4 36.1 61.8 74.9 100.0

7.39 9.91 14.98 25.60 33.38 58.81

7.34 10.80 16.76 27.59 35.50 59.61

7.39 10.78 16.65 27.29 35.07 58.81

-14.9

0.1

20.1

80.1

between the two sets of data is only 1*7%,which is quite satisfactory. For the H2-COs mixture the only available thermal conductivity data are those of Ibbs and Hirst (1929) at 0”c. At this temperature the average deviation between our data and those of Ibbs and Mirst is 2.9 %, which is also satisfactory in view of the larger uncertainties in Ibbs and Hirst’s measurement.

Thermal conducticity of H2-Nz and HFCOZ gas mixtures

1305

3. Comparison with theory By assuming local chemical equilibrium the thermal conductivity of a binary polyatomic gas mixture may be written as (Hirschfelder 1957b)

where the subscripts 1 and 2 denote the components of the mixture. Kmix-mon is the thermal conductivity of the mixture when the gases are treated as monatomic and can be calculated by the Chapman-Enskog theory (Hirschfelder et al. 1954). DII, 0 2 2 and Dl2 are the self-diffusion and inter-diffusion coefficients and x's are the mole fractions of the components. However, for most gases local chemical equilibrium is not satisfied and there is relaxation of internal energy. More recently, Monchick et al. (1965) have derived a formula for the thermal conductivity of a polyatomic gas mixture on the basis of formal theory of Monchick et al. (1963) which is more or less an extension of the Wang Chang and Uhlenbeck (1951) theory for a polyatomic gas. However, in view of the uncertainties introduced in their formulae by the approximations, Monchick et al. (1965) have suggested that for polyatomic gas mixtures, at least for the present, it is best to use HirschfelderEucken formula, after correcting for the relaxation effects in the pure components (i.e. K1 and Kz in equation (6)), to interpret the thermal conductivity data. In the present paper our data have only been compared with the Hirschfelder-Eucken formula and in a subsequent paper we propose to look into the problem more deeply by considering the data for other systems as well. For all our calculations we have used the Lennard-Jones (12 : 6) potential. The force parameters taken were determined from viscosity data (Hirschfelder et al. 1954). Kmix-mon, Dii, 0 2 2 , 0 1 2 , Kl(mon) and &(mon) were calculated from the expressions obtained by the Chapman-Enskog theory. The unlike interactions were approximated by the usual combination rules. K1 and K2 were calculated from the expression obtained by Hirschfelder (1957a):

where Sf = pDCpf/Kmon and Cpf = $R. The Kmixvalues obtained from equation (6) are shown in column 4 of tables 2 and 3. In order to account for relaxation effects in polyatomk gases we have preferred to use the experimental values of K1 and Kz instead of the formula derived by Mason and Monchick (1962). The values of Kmix as calculated from equation (6) using experimental values of K1 and K? are given in column 5 of tables 2 and 3. 4. Discussion of results

For H2-Na mixtures, if we omit the pure components from our present considerations, the mean deviation between the experimental and the calculated value of the thermal conductivity (averaged over all compositions and temperatures) is 1.8 % assuming local chemical equilibrium and -2.2 % when the experimental values for the pure components are used. For H?-C02 mixtures the experimental conductivity values are 5-8 % lower than those calculated assuming local chemical equilibrium. The situation improves when the experimental values of the conductivities of the pure components are used to calculate the conductivities of the mixtures (column 4, table 3) and in this case the mean deviation between the experimental and the calculated values at a given temperature increases from 1 % to 5 % as the temperature increases from - 15 to 200°C. It should be pointed out that for both the mixtures (particularly for the H?-CQz system) the deviation between the experimental and the calculated values of the thermal conductivity is larger than the experimental uncertainty (-1 %). It should be noted that for the H?-N, system the experimental

+

1306

P. Mukhopadhyay, A . Das Gupta and A . K. Barua

conductivity values of the mixture are higher than the calculated values, whereas the reverse is true in the case of Hz-CO2 mixtures. One source of error which may influence the calculated values of the thermal conductivity is the uncertainty of the combination rules used to obtain the inter-diffusion coefficients D E . For the H F N ~system the difference between the experimental and the calculated values of D1z is quite small (this has been confirmed further by D I Zvalues obtained from viscosity data of mixtures by Weissman and Mason (1962)) and the error in the Dle values used is not likely to influence the thermal conductivity values significantly. For the He-COz system experimental 0 1 2 values are about 1 % higher than the calculated values. However, sample calculations made at 20.1 O c show a change of less than 0.1 % in the calculated thermal conductivity values. The combination rules are also used to calcuIate another quantity B12:#,which is a ratio of the collision integrals and has been tabulated as a function of the reduced temperature (Hirschfelder et al. 1954). BIZ*is required in the calculation of Kmix-mon by the ChapmanEnskog theory. However, since BIZ* changes very slowly with temperature, any error in the combination rules will not affect it significantly. Consequently, we may conclude that the discrepancy between the experimental and calculated values of the thermal conductivities of the mixtures is due to the failure of the Hirschfelder-Eucken theory. It should be pointed out that one major source of complications in the heat conductivity of polyatomic gases is the relaxation effects in the energy tracsfer between the translational and internal energy of the molecules. The relaxation effects in the pure components are taken into account approximately by utilizing experimental conductivity values in the Hirschfelder-Eucken formula (equation (6)). However, an important factor which cannot be considered satisfactorily at present is the cross-relaxation effects between the components cf the mixture. The same point has been stressed by Gray and Wright (1962) from an analysis of their thermal conductivity data on Hz-Ne mixtures. Monchick et al. (1965) have extended the treatment of Wang Chang and Uhlenbeck (1951) of the heat conductivity of polyatomic gases to mixtures. The results obtained by them are, however, not really encouraging. Another factor which may affect the heat conductivity of the He-COe system is the comparatively large quadrupole moment of CO2 which is likely to affect the inelastic collisions. We shall consider all these problems more deeply in a subsequent paper by including data for other systems as well. Acknowledgments The authors are grateful to Professor B. N. Srivastava for his kind interest in the work. References GRAY,P., 2nd WRIGHT,P. G., 1961, Proc. Roy. Soc. A, 263, i61-88. -1962, Proc. Roy. Soc. A, 267, 408-16. HIRSCHFELDER, J. O., 1957a, J. Chem. Phys., 26, 274-81. - 1957b, Proc. 6th Symp. (Int.) on Combustion, Yale University, 1957 (NewYork: Reinhold),

pp. 351-66. HIRSCHFELDER, J. O., CURTISS,C. F., and BIRD,R. B., 1954, Molecular Theory of Gases andliqitids (New York: John Wiley). IBBS,T. L., and HIRST,A. A., 1929, Proc. Roy. Soc. A, 123, 134-42. KA~NJLUIK, W. G., and CARMAS,E. H., 1952, Proc. Phys. Soc. B, 65, 701-9. KASNULUIK, W. G., and MARTIN,L. A., 1933, Proc. Roy. Soc. A, 141, 144-58. -1934, Proc. Roy. Soc. A, 144, 496-513. ~ I A S O NE. , A., and MOSCHICK,L., 1962, J. Chem. Phys., 36, 1622-39. MONCHICK, L., PEREIRA, A. N. G., and MASON, E. A., 1965, J. Chenz. Phys., 42, 3241-56. MVNCHICK, L., Y m , K. S., and MASON,E. A., 1963, J. Chem. Phys., 38, 1283-7. SRIVASTAVA, B. N., and BARUA,A. K., 1960, J. Chem. Phys., 32, 427-35. SRIVASTAVA, B. N., and SAXENA, S. C., 1957, Proc. Phys. Soc. B, 70, 369-78. WAELBROECK, F. G., and ZUCKERBRODT, P., 1958, J. Chem. Phys., 28, 524-6. WAXGCHASG, C. S . , and UHLENBECR, G. E., 1951, University of Michigan Engineei.iiig Research Report, No. CM-681. WEISWAS,S., and MASOX,E. A., 1962, J. Chem. Phys., 37, 1289-1300.