Thermal conductivity of the gas mixtures: Ar-D2, Kr-D2 and Ar-Kr-D2

Thermal conductivity of the gas mixtures: Ar-D2, Kr-D2 and Ar-Kr-D2

Gambhir, Saxena, Physica R. S. 32 2037-2043 S. C. 1966 THERMAL CONDUCTIVITY Ar-D,, Kr-D, OF THE GAS MIXTURES: AND Ar-Kr-D, by R. S. GAMBHIR*...

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Gambhir, Saxena,

Physica

R. S.

32

2037-2043

S. C.

1966

THERMAL

CONDUCTIVITY Ar-D,, Kr-D,

OF THE GAS MIXTURES: AND Ar-Kr-D,

by R. S. GAMBHIR*) Department

and S. C. SAXENA

of Physics, University

of Rajasthan,

Jaipur,

India

Synopsis Employing

the thick-wire

variant

been measured

for the gas mixtures

of composition

at temperatures

interpreted

on the theory

and Saxena. gases

Two

recently

conclusions

more

developed

emerge

of the hot-wire

cell, thermal

of Ar-D 2, Kr-Ds

conductivity

and Ar-Kr-Ds

35, 50, 70 and 90°C. These results are compared

of Hirschfelder, theories have

and its simplified

of thermal also

from this overall

been

conduction considered.

form as given

for mixtures Some

has

as a function and

by Mason

of polyatomic

interesting

and

useful

study.

1. Introdzcction. An apparatus for measuring the thermal conductivity of gases and gas mixtures has been employed successfullyl) 2) in this laboratory. It uses a thick-wire type hot-wire cell. Measurements on some pure gasesr) including deuterium, and binary and ternary combinationss) of He, Ar and Kr have already been reported in the temperature range 30 to 100°C. In this paper results on two binary systems and a ternary system are discussed, which include deuterium as a common gas in all systems. In particular, Ds-Ar, Ds-Kr and Ds-Ar-Kr systems are investigated at 35, 50, 70 and 90°C. These systems have never been studied earlier and further provide some interesting circumstances to infer about the adequacies of the theories of thermal conduction in polyatomic gases. For these reasons our choice fell on particularly these systems. The theory of the apparatus, its description, various corrections to be applied, procedure to be adopted in making a measurement, have been described at length in an earlier publicationr). The unit required to prepare mixtures is also explained by Gambhir and Saxenas). We therefore straightaway report the results. 2. Exfierimental results. A series of measurements of the average resistance of the cell wire, &, are taken for an appropriate value of current, I; as also the value for zero current at the bath temperature, Ro. This, *) Present address: Department

of Physics, Panjab University, -

2037 -

Chandigarh, India.

R. S. GAMBHIR

AND

S. C. SAXENA

along with the knowledge of the thermal conductivity of cell wire and cell dimensions, is enough to compute the conductivity of the gas. We do not report here the values of these different constants as well as of i& and I for the sake of brevity. Instead, we report the smoothened thermal conductivity, il, values at round temperatures which were obtained by the graphical plot of il data at each composition as a function of temperature. TABLE

I

Unlike interaction parameters for the exp-six potential Gas pair.

/

OL

1 dk (“W

1

dA”)

The direct experimental values mostly lie within a percent of the compromised curves. Only in four cases the departure was more and on the average it amounted to as much as 2.5 percent. The smooth round values are listed in table III for the two binary systems and in table IV for the ternary system. In these tables Xi, Xs, and Xa represent the mole fractions of the constituent gases and are in the descending order of their molecular weights. 3. Comparison with theory. By making certain approximations, Hirschf elders) gave a theory for thermal conduction in polyatomic gas mixtures. His expression for a multicomponent mixture of n. components is:

Here & is the true coefficient of thermal conductivity of pure component i, and 1: is the coefficient that would be obtained if all internal degrees of freedom were frozen. Similarly A:,, is the conductivity of the mixture with frozen internal degrees of freedom. Dz~ is the self-diffusion coefficient of the pure i-th component, and Dij the mutual diffusion coefficient of components i and j. Xi represents the molefraction of the i-th component. For ALi, the expression of Muckenfuss and Curtiss*) as modified by Mason and Saxenas) was used. For Ai, Ddi and Dfj expressions are given by Hirschfelder, Curtiss and Bird6). Theoretical calculations of A:, ;l~ix, TABLE

II

pa, values according to eq. (4) at 35°C Gas pair. Kr-Da Ar-Dz Kr-Ar

1

91%

I

9-1 2.318 1.922 1.431

THERMAL

CONDUCTIVITY

OF SOME

TABLE Comparison

of the experimental

values,

X,i,

III

and the theoretical

(Cal. s-1. deg-1.

2039

MIXTURES

cm-l)

thermal conductivity

for binary

mixtures

Xl

Kr-D2

1.000

2.29

35°C

0.822 0.446

4.89 12.1

(f

1.2)

12.6

(+4.1)

13.1

(+

8.3)

0.222

19.4

20.3

(+

20.8

(+

7.2)

0.084

26.0

27.2

(+4.6)

27.4

5.4)

0.000

32.4

32.7

(+

0.9)

(+ -

2.37 (-2.5) 4.98 (- 2.5)

5.17 (+

70°C

90°C

exp.

1.000

2.43 5.11

4.79

(-

2.0) 4.6)

1.2)

(+2.3)

13.6

(+

6.2)

0.222

20.2

21.0

(+4.0)

2j.5

(+

6.4)

0.084

27.0

28.1

(+4.1)

28.2

4.4)

0.000

33.4

33.8

(+

(+ _

1.2)

1.000

2.56

2.50 (-2.3)

0.822

5.39

5.22 (-3.2)

0.446

13.6

13.7

(_tO.7)

_ 5.43 (+ 14.1

(+

0.6) 3.7)

0.222

21.3

21.9

($2.8)

22.4

(+

5.2)

0.084

28.2

29.3

(+3.9)

29.4

4.3)

0.000

34.8

35.2

(+l.l)

(+ -

1.ooo

2.66 5.68

2.62

- 1.5)

5.45 14.3

- 4.0)

5.62

0.446 0.222

14.3 22.1

22.8

+3.2)

14.7 23.2

0.084

29.3

30.5

+4.1)

30.5

0.000

36.1

36.6

+1.4)

0.0)

1.000

4.38

4.33

-1.1)

0.796

7.46

7.33

-1.7)

0.437

14.1

0.242 0.101

1.1)

(+ (+

2.8) 5.0)

(+ _

4.1)

7.83 (+ 16.0

4.9)

(+ 13.4)

+5.6)

20.6

+3.0)

22.2

(+

7.8)

25.8

27.2

+5.4)

27.7

(+

7.4)

1.000

4.55

4.50 (-1.1)

7.78

7.59

(-

2.4)

0.437 0.242

14.9 21.3

15.5

(+4.0)

21.9

0.101

26.6

28.1

1.000 0.796

(-

14.9 21.2

0.796

4.78 8.21

8.10 (+

4.1)

(+2.8)

16.5 22.9

(+ 10.7) (+ 7.5)

(+5.6)

28.6

(+

4.72(-1.3) 7.95 (-3.2)

7.5)

8.48

(f

3.3)

16.1

(+1.9)

17.2

(+

8.9)

0.242

15.8 22.2

22.8

(+2.7)

23.8

(+

7.2)

0.101

27.9

29.3

( + 5.0)

29.8

(+

6.8)

0.437

1.ooo 0.796 0.437

brackets

_

13.1

Ar-D2

90°C

4.95

12.8

35°C

70°C

within

approx.

rig. 2.28 (-0.4)

0.822 0.446

0.822

50°C

5.01

4.93 (- 1.6) 8.29 (-4.1) 16.8 (+ 1.8)

0.242

8.64 16.5 23.1

23.8

(+3.0)

0.101

29.5

30.5

(-c3.4)

denote

1

-I

Gas pair.

50°C

Numbers values.

GAS

the percentage

deviations

between

8.84 (+ 18.0 (+

2.3) 9.1)

24.7 30.9

6.9) 4.4)

(+ (+

calculated

and experimental

2040

R. S. GAMBHIR

AND

S. C. SAXENA

Dii and Dz~ are performed on the modified Buckingham exp-six potential7). The parameters for the pure gases used are those of Mason and Rice*) a) except for krypton for which parameters as determined from viscosity data by Bahethi and Saxenala) are preferred. The unlike interactions were then evaluated by employing the combination rules of Mason and Rice 8). These parameters are recorded in table I. Mason and Saxenall) have simplified f elders) by well defined approximations. that of the Wassiljewa form, viz.,

the Amix expression of HirschTheir final result is similar to

where

(3) Here Mg and Mj are the molecular weights of the constituents i and i respectively. The calculations of Amix according to the rigorous formula of Hirschfelder, eq. (l), are given in table III, column 4, for the binary systems; and in table IV, column 6 for the ternary mixtures. Values are also recorded in these very tables obtained on the basis of the approximate formula of Mason and Saxenall), eqns. (2) and (3). The numbers within brackets represent the percentage deviations of the calculated values from the experimental data. il-values for DZ were obtained according to the relation

(4 TABLE Comparison

Temp.

IV

of the experimental and the theoretical thermal conductivity ,,,ix (Cal. cm-l. s-1. deg-I), for the ternary system Kr-Ar-Ds

values,

“C

35

14.8

15.1

(+2.0)

15.7

(+6.1)

0.218

0.173

0.609

0.298

0.602

0.100

4.72

50

0.218 0.298

0.173 0.602

0.609 0.100

15.5 4.98

15.6 ($0.6) 5.01 (+os)

16.3 (+5.2) 5.23 (+5.0)

70

0.218

0.173

0.609

16.3

16.3

17.0

0.298

0.602

0.100

5.30

0.218 0.298

0.173 0.602

0.609 0.100

0.609 5.60

90

Numbers perimental

within results.

brackets

denote

the percentage

deviation

4.83 (t2.3)

(

0.0)

5.03 (+6.6)

($4.3)

5.25 (-0.9)

5.48 (+3.4)

16.9 ( 0.0) 5.49 (-2.0)

17.6 (+4.1) 5.72 (+2.1)

of the calculated

values

from

the ex-

THERMAL

where

CONDUCTIVITY

OF SOME

GAS MIXTURES

2041

R is the gas constant, M the molecular weight and 7 the coefficient

of viscosity. For obtaining r] of Ds, the existing viscosity values of Ha were smoothed by plotting as a function of temperature and these were then multiplied by 42. This procedure is reliable enough because q values of Ds so obtained agree well with the directly measured values of Kestin and Nagashimais) at 20 and 30°C. 10 values obtained entirely from theory differ on the average only by 0.8% from those obtained from 7 data. The experimental values of il for Kr and Ar were directly used. The prj values computed according to eq. (4) for all the three binary gas pairs at 35°C are recorded in table II. These very values were also used for jlmix calculations at 50, 70 and 90°C. This approach seeks justification from the fact that q~ifare very feebly dependent upon temperature. This is well establishedis) no doubt for mixtures of monatomic gases, but is also borne out by eq. (3). The only factor which makes it temperature dependent is @g/L;) and it is very feebly dependent on temperature. 4. Discussions. Saxena, Saksena, Gambhir and Gandhii”) have also given a theory for the thermal conduction in mixtures of polyatomic gases. These workers have improved the theory of Hirschfelders) by considering the translational and internal energy exchange. For all the systems considered in this paper the results of thermal conductivity obtained on this theoryi4) are identical with that of Hirschfeldera). The reason for this lies in the large values of the relaxation times characterising the internal-translational energy exchange for pure deuterium gas as well as between deuterium and krypton or argon molecules. In more technical language, it refers to large values for the quantities ZDa_Er and ZDa_Ar which makes the correction term negligibly small. Here 2 refers to the number of collisions required for translational-internal energy equilibration. We have consequently no separate numerical results for these systems to quote. Monchick, Pereira and Masonis) have recently given another expression for thermal conductivity of polyatomic gas mixtures. The final expression is quite tedious and involves several assumptions in its derivationi6). Numerical results do not appreciably differ from those obtained on the expression of Hirschfelders) for most of the common mixturesr5). We therefore do not employ this theory as Hirschfelder’s expression is reasonably adequate to interpret these systems. A close look of the listings of table III reveals some interesting conclusions. The rigorous theory a) 14) in its present form seems to be moderately adequate for the two binary systems in the entire composition range and at the temperatures investigated here. The average absolute deviation for all the mixtures is 3.3 percent. For Kr-Ds and Ar-Ds individually the corresponding numbers are 3.1 and 3.5, respectively. The experimental uncertainty

2042

R. S. GAMBHIR

AND

S. C. SAXENA

associated with these measurements is about 2 percent. However, this good agreement between theory and experiment is due to the special nature of these molecules, the relaxation time for the translational-internal energy exchange being large. In an ambition to have a simpler expression for quick estimation of jlmix Mason and Saxenaii) tional approximations

introduced eqs. (2) and (3). In view of some addimade in deriving this expression over and above

that involved in eq. (l), it is clear that the success to be achieved is limited. This is confirmed by the records of table III when one finds that the average absolute deviation for all the mixtures is 5.6 percent. Thus, the over-all success of this formula is about the same as found by Mason and Saxenari) and other workers by interpretation of data on a large number of systems. The simplicity of this formula still seems to be a very strong point in its favour and it more than compensates for a bit inferior agreement found on its basis relative to the rigorous theory. Another very interesting aspect of the Mason and Saxena expression which needs mention is the use of low temperature vij values at higher temperatures. In the limited temperature range of present measurements this approach is well substantiated in conformity with more detailed investigations on monatomic gas mixturesl7) 1s) 19). This is a very useful and interesting result and needs more detailed investigations both theoretical and experimental. Comparison of experimental and calculated Amixdata presented in table IV for the ternary system Kr-Ar-Ds substantiates and confirms all the conclusions derived for binary systems. This is very encouraging, for only limited amount of information is available for multicomponent mixtures at the moment. The average absolute deviations of the calculated values according to the rigorous and approximate procedures from the experimental data are 1.0 and 4.6 percent, respectively. Acknowledgement. This work was supported through a contract from the council of Scientific and Industrial Research, New-Delhi, which also provided a research scholarship to one of us (R.S.G.). Received

18-4-66

REFERENCES 1) Gambhir,

R. S., Gandhi,

published. 2) Gambhir,

R. S. and Saxena,

3)

Hirschfelder,

4)

1957). Muckenfuss,

J. M. and Saxena, S. C., Mol. Phys.,

J. O., Sixth international C. and Curtiss,

S. C., Indian

Phys.,

to be

to be published.

Combustion

C. F., J. them.

J. pure and appl.

symposium,

phys. 29 (1958)

p. 351 (Reinhold 1273.

Pub. Corp.

THERMAL

5)

Mason,

6)

Hirschfelder, (John

CONDUCTIVITY

E. A. and Saxena,

S. C., J. them.

J. O., Curtiss,

Wiley

OF SOME

R. B., Molecular

Mason,

E. A., J. them.

8)

Mason,

E. A. and Rice,

W. E., J. them.

Phys. 22 (1954) 522.

9)

Mason,

E. A. and Rice,

W. E., J. them.

Phys. 22 (1954) 843.

12)

Kestin,

Phys. 22 (1954)

0. P. and Saxena, E. A. and Saxena, J. and Nagashima,

Theory

of Gases and Liquids

1964).

7)

10) Bahethi, 11) Mason,

2043

MIXTURES

Phys. 31 (1959) 511.

C. F. and Bird,

and Sons, Inc., New York,

GAS

169.

S. C., Indian

J. pure and appl. Phys. 3 (1964)

12.

S. C., Phys. Fluids 1 (1958) 361. A., Phys.

Fluids 7 (1964) 730.

Gambhir, R. S. and Saxena, S. C., Trans. Faraday Sot. 80 (1964) 38. S. C., Saksena, M. P., Gambhir, R. S. and Gandhi, J. M., Physica (1965) 333. 14) Saxena, L., Pereira, A. N. G., and Mason, E. A., J. them. Phys. 42 (1965) 3241. 15) Monchick, 13)

16) Math ur, S. and Sax e n a, S. C., Proc. Phys. Sot. (London), to be published. S. C. and Gambhir, R. S., Proc. Phys. Sot. (London) 81 (1963) 788, Indian 17) Saxena, and appl. Phys. 1 (1963) 318. 18) Sarsena, Gandhi,

19)

M. P. and Saxena, J. M. and Saxena,

S. C., Proc. Natl. Inst. Sci. (India) S. C., Indian

31 (1965) 26.

J. pure and appl. Phys. 3 (1965) 312.

J. pure