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PRAMANA © Printed in India Vol. 43, No. 1,

_ _ journal of July 1994

physics pp. 81-89

Thermal diffusion factors of hydrogenic trace mixtures with helium by column calibration factor

G D A S G U P T A , A K D A T T A and S A C H A R Y Y A

Department of Physics, University College, Raiganj 733 134, India MS received 20 July 1993; revised I 1 March 1994

Abstract. Using column calibration factor (CCF) F~ for a given column, the temperature dependence of experimental thermal diffusion factors ~t r of hydrogenic trace mixtures in helium are accurately determined. This study, however, observes the inelastic collision effect in these trace mixtures when ct T by our CCF method are compared with those by the existing methods and theoretical ones respectively.

Keywords. Column calibration factor; thermal diffusion factor; thermal diffusion column;

inelastic and elastic collisions.

PACS No. 44"90

1. Introduction

In the existing column theory as developed by F u r r y and others I-1-3] the column geometry plays an important role in determining the exact value of thermal diffusion factor a T of a binary gas mixture. The column as such cannot yield the actual 0t T values both in trend and in magnitude with respect to temperature and composition as the binary molecular interactions are often called into play. T h e r m a l diffusion column is still far superior to any other a T measuring instruments as the equilibrium separation factor qe defined by

(Z~/Z~),op

qe--

(Zl/Zj)bo,

is very large even in the case of small mass difference between the c o m p o n e n t s of a mixture. Here, Zi and Zi are the mass fractions of the lighter and the heavier components respectively. Hence for a binary mixture of almost identical masses, shapes and sizes a calibrated T D column can safely be used to measure a reliable relative and small

~7 values. F o r this reason we have calibrated the given column of Slieker and de Vries [4] with known and reliable at T of H e - T 2 mixture to arrive at the column calibration factor (CCF) Fs from the relation:

In qmax = 0eTFs(rc, rh, L, T) (I)

where T is the m e a n temperature of Th and T c, T h and Tc being the hot and cold wall temperatures in K. r~ and r h are the radii of cold and hot wall of a column of geometrical length L. F~ is supposed to be an independent molecular model solely dependent on the column geometry at any mean t e m p e r a t u r e T i n K.

81

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A number of studies by Acharyya et al [ 5 - 7 ] and N a v a r r o et al [8] on F~ enabled us to study the temperature dependence of s T of D T and H T in helium only to explore the fact that the T D column is a reliable relative ~t T measuring instrument and to observe the inelastic collision effects in them. In this study, we estimate the experimental parameters a' and b' governing the very nature of variation of the available experimental In qe against pressure p of H e - D T and H e - H T gas mixtures [4], the hydrogenic components were never becoming larger than 5% in helium at three experimental temperatures.

The computed data of In qe of H e - D T and H e - H T against pressure in atmosphere are shown in figures 1 and 2 respectively to ensure that the least square fitted curves agree excellently with the experimental ones. F o r H e - H T an interesting feature is that unlike the usual behaviour, In qe becomes smaller with temperatures, not noticed earlier [7].

The hydrodynamical part of the column theory is excellently obeyed by H e - D T and for some selected experimental points of H e - H T as their p2/ln q~ against p4 were found out to be

p2/ln q~ = 0.7758 + 0.8161 p¢ at 338 K.

= 0.4938 + 0.7294 p4 at 378 K

= 0.3964 + 0"6024 p4 at 423 K and

p2/ln q~ = 9.0749 + 14-8368 p4 at 338 K

= 13.5999 + 64.6412 p4 at 3 7 8 K

= 14.6461 + 411.5226 p4 at 423 K respectively.

t~r C

1.00 0.8 0.6 0.4 0.2 0

~

H e - D T

L I I I

0.325 P 0-650 0.975 ( Pressure in arm.)

Figure 1. lnqe against pressure p in atmosphere for He-DT trace mixture, at T= 338, 378 and 423 K, 'O' experimental points.

82 Pramana - J . Phys., Vol. 43, No. 1, July 1994

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Thermal diffusion factors

T = 3 3 8 K 0 . 0 4 -

0 . 0 3

cD O "

- 0 . 0 2

o.o,

0 - 3 0 . 6 0 . 9

P

( Pressure in atm.)

Figure 2. Inqe against pressure p in atmosphere for H e - H T trace mixture at

= 338, 378 and 423 K, ' O ' experimental points.

C 0

0

E

¢-

0 . 2

0 . 1

He - DT ~ " / ~

5

r ~ ~ -

4

2

0 I I

3 0 0 3 5 0 T 4 0 0 4 5 0

( M e a n t e m p . in K)

Figure 3. Variation of ~T with T o f H e - D T trace mixture, 1. Our expt ~T from ln qmas and Fs; 2. Expt ~T(Maxwell case); 3. Expt ~T(Slieker case); 4. Theor e~x(elastic ) from Eq (7); 5. Theor ~x(inelastic) with Z o , = 300 from Eq (7); 6. Theor

~T(inelastic) with Z or calculated from Barua et al (1970) from Eq (8); 7. Theor

~T(inelastic) with Zro , calculated from Parkers [12] formula with adjustable

Z~o, = 7.08 [ 13].

In the absence of a n y reliable possibility to estimate the actual experimental 0t x of a mixture t h r o u g h the use o f m o l e c u l a r m o d e l we used the values of F~ already o b t a i n e d for the c o l u m n [7]:

Fs = - 66"52202 + 0-3502286 T - 4-1879 x 10 - 4 ~2.

Pramana - J . Phys., Vol. 43, No. 1, July 1994 83

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0 . 1 5

"8 0 . 1 0

1 5

0 . 0 1 5

- -- 0 . 0 1 0

H e - H T -"

= 6

}

~ 0 . 0 5 0 . 0 0 5 ~

P-

~'-. _...>.-~ - - . - ~

~.,,;~ ~ ~ ~

. . . .

0 f.--- -F' . . . . - c:-_ --~--_ _~_---/,~...~. _

3 0 0 3 5 0 ~. 4 0 0 4 5 0

( M e o n temp. in K )

Figure 4. Variation of rt' T with 7"of H e - H T trace mixture

1. Our expt ST from ln q=a, and Fs; 2. Expt nT(Maxwell); 3. Expt

~tT(Slieker); 4. Theor ~tT(elastic ) from Eq (7); 5. Theor ~tr(inelastic ) with Z or= 300 from Eq (8); 6. Theor ~r(inelastic) with Z o t calculated from Barua et al 1-12] from Eq (8); 7. Theor ~T(inelastic) with Zro t calculated from Parker [13] formula with adjustable Z ~ = 12.15. t o l

N o w

~T'S

of H e - D T and H e - H T were obtained from (1) and c o m p a r e d with those by the existing methods using column theory as well as the theoretical ~T'S based on elastic and inelastic [9] collisions in figures 3 and 4 respectively in order to reveal the existence of inelastic collisions in these mixtures.

2. Theoretical formulation to estimate experimental at T

Both ends being closed for the ideal column of length L, In qe of a gas mixture at any mean temperature T is given by

H L

In qe - (2)

Kc + K d

where H, K c and Ka are the functions of t r a n s p o r t coefficients of a gas mixture and proportional to p2, p4 and pO respectively, p being pressure in atmosphere.

In order to remove parasitic remixing effect, F u r r y and Jones [2] simply added a term Kp proportional to p4 to the d e n o m i n a t o r when (2) becomes

a' p 2

In q~ -- (3)

b ' - F p 4

84 Pramana- J. Phys., Vol. 43, No. 1, July 1994

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Thermal diffusion factors which is also written as

P2/In qe = b'/a' + (1/a')p 4 (4)

a' and b' are however related by

(nL/Kc)p 2 = a'(l + Kp/Kc) and (Kd/Kc)p* = b'(l + Kp/K~)

H, K c, Ka are the functions of the transport coefficient of a mixture and Kp is the remixing coefficient.

Again if c represents the intercept of the straight line of (4) we have

n = (K~/Lc) (5)

the exact expressions for H, K~ and Kd are given in our previous publications [5-7].

The estimation of the experimental s T through the existing formulations involved the shape factors taking account of the inherent asymmetry of the column geometry.

The mass density p, the viscosity coefficient r/ and the diffusion coefficient D were calculated from M T G L of Hirschfelder et al [10], the column shape factors and the force parameters required had already been reported earlier [7].

It is observed in figures 1-2 that as the pressure increases lnqe increases and becomes maximum when p = (b') 1/* for which (6 In qe/gp)= O.

We then have from (3)

In qm~x = (a'/2,,//ff) (6)

It is also observed in H e - H T , unlike H e - D T , that some experimental data of In qe are not in fit with the hydrodynamical part of the column theory as they have tendency to yield the negative intercept of p2/ln qe against p4 which is absurd unless inversion of ~r would take place. Hence we are bound to select some six or seven data from the reported graph to fix the values of In qm~x from the (6).

Table 1 and the graphs of figures 1-2, revealed that In qm~x from (6) in terms of a' and b' are in good agreement with the graphically determined values earlier [7]. This establishes the fact that our choice of the In q, data with pressure particularly for the H e - H T mixture where the mass difference between the components is practically nil, is almost right.

3. Theoretical formulations to calculate aT

Theoretical ct r can, however, be estimated from q(J) v ,

1 S"~Zl- y~.~(6C* - 5) (7)

0~ T = - -

6[).,i3, [ X z +

where (6C* - 5) depends mainly on the temperature while the other factors involved in (7) are the complicated functions of composition, masses and thermal conductivities of gases and gas mixtures. The a T, calculated from (7), is presented in the 12th column of table l, and shown graphically in figures 3-4 for H e - D T and H e - H T respectively.

Pramana - J . Phys., Vol. 43, No. 1, July 1994 85

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O0 Table 1. Experimental and Theoretical ~tr values of binary gas mixtures with temperature. W I gT O System Exp_t a T with Hot Our wall Cold Mean In qm, x calibration Maxwell Slicker temp wall Temp a' in b' in computed Expt F, factor method shape shape Th in K T~ in K Tin K (atm) 2 (atm) 4 from Eq (5) Ref(6) Eq. (1) factor factor Theor. Ot T from Eq (6) elastic theor. method

Theoretical ~T from Eq (7) with Zr. 300 Barua Parker et al g~

He-HT He-DT 393 283 338 0"0674 0"6116 0"0431 4"001 0-0107 0-0014 0-0017 473 283 378 0-0155 0-2104 0-0169 6-026 0-0020 0-0028 0"0007 563 283 423 0-0024 0-3559 0-0064 6"691 0-0009 0-0010 0-0006 393 283 338 1-289 14)519 0-6284 44)01 0.1567 0-060 ff186 473 283 378 1"371 0-6670 0-8331 6"026 0.1382 0-054 0"187 563 283 423 1.660 0.6580 1"0232 6-691 0-1529 0-050 0-194 0-030 O- 121 0-055 0-002 0-030 0-124 0-037 0-009 0-030 0-129 0-005 0-005 0-105 0-100 0-158 0-158 0-104 0-105 0-188 0-154 0-103 0-109 0"226 0"159

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Thermal diffusion factors

The inelastic thermal diffusion factor au is given by according to Monchick et al [ 11 ] (6C u * - 5)/21j | 2j t / ~ . . . . 2it~ . . . . \ q_ _ 1

where the symbols have their usual meanings only the collision integral ratio Cu differs from C* u" In fact Cu is not symmetric with respect to the interchange of the indices i and j and is very sensitive to inelastic collision.

For a pure gas the exact values of 2~. t .... and 2 ~ ~,,,, is given by

tl rl /

{

t + ~Z,o, 2 Cint R + to in[

:)}-']

. (9) Here C t,..~ = 3R/2, h e constant value of translational heat capacity, Z o t is the rotational translational collision number for inelastic collision.

The nonspherical terms of (8) we used Hirchfelder-Euken expression [10] to calculate the thermal conductivity 2 ~ from i int

, __

n[Dii]l

Cin t

2~ i . , - (10)

1 + % / Z ~ ) ( O . / O u ) ~

Theoretical inelastic a.'s for H e - D T and H e - H T thus calculated from (8) with the help of (9) and (10) are shown in table 1 and also in figures 3-4 respectively for comparison with other experimental ~r values.

4. R e s u l t s a n d d i s c u s s i o n

The inherent asymmetry in the column geometry is however, taken into account by Maxwell, Slieker and Lennard-Jones dimensionless shape factors [7]. We calculated the experimental ar's of H e - H T and H e - D T trace mixtures at T = 338, 378 and 423 K respectively from (5) using those shape factors. Slieker's case does not involve any molecular model and it gives rather a rough estimation of the experimental aT while the former includes inverse fifth power potential. As the cold wall temperature Tc was held fixed experimental a T due to L - J case cannot b e applicable here. The ar thus obtained due to Maxwell and Slieker cases is presented in table 1 and shown graphically by the curves 2 and 3 respectively of figures 3 and 4.

The experimental aT from (1) as obtained in terms of In q,,ax of (6) and Fs is shown by curve 1 in figures 3 and 4. When they are compared with those due to Maxwell (curve 2) and Slieker (curve 3) it is found that so far as the trend is concerned the data due to Slieker agree better than those due to Maxwelrs shape factors. This is perhaps due to the fact that both Slieker and our method are free from any binary molecular model. As the mass difference between the components of a binary mixture decreases as in the case of H e - H T the agreement is more close.

The theoretical ~r based on elastic collision theory, (7), as shown by curve 4 in Pramana - J. Phys., Voi. 43, No. 1, July 1994 8 7

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figures 3-4 appears to be temperature independent. Unlike H e - H T , H e - D T however show slightly lower value at higher temperature. The inelastic ct T as calculated from (8) with Zro t = 300, show its positive temperature dependence as represented by curve 5 in figures 3-4.

When ~u were calculated with the available rotational translational collision number of Barua et al [12] an interesting feature is that the curve 6 of figures 3 - 4 coincide with tiT's of our C C F method. This fact prompted us to adjust Zro ~ from Parker's formula [13]. Using Z o t = 2.78, 2.91 and 3.05 for H T and Zro , = 4.78, 5.0l and 5-23 for D T at 338, 378 and 423 K respectively inelastic ctu's are then estimated for both H e - D T and H e - H T trace mixtures and were shown by curve 7 in figures 3 - 4 respectively for comparison with other CtT'S.

With Z o, determined by us inelastic theoretical CtT'S curve 7 (15th column of table 1) so far as the magnitude and trend are concerned in the case of H e - D T , support our 0tT'S curve 1 (9th column of table l) and onl.y in trend with ~r's due to Slieker ( l l t h column of table 1). In the case of H e - H T these theoretical ctT's almost coincide with our efT's, but in trend with the experimental ctr's due to Maxwell.

All these comparison of CtT'S SO far obtained thus reveal that inelastic collisions play an important role in such trace mixtures. Again the variation of In qm,x against T f o r H e - H T is given by

In qma, = 0"89879 -- 4"2097 X 10 -3 T + 4"9647 x 10 -6

"~2

showing that at T-~ 429 K, In q=,, m a y b e zero as shown in figure 5. The isobaric H e - H T mixture may yield an interesting p h e n o m e n o n of inversion of both in qma, and ct x with respect to temperature like isobaric system N 2 - C O as studied in our recent publication of Saha et al 114]. The system H e - H T deserves a detailed study of measurements of in qe against pressure for its different composition and temperatures.

1 . 0 0

E 0 . 5 0

H e - D T

_ H e

H e - H T

0 ~ -1"- . . . ~ I

3 0 0 3 5 0 4 0 0 4 5 0

( M e a n t e m p . in K )

Figure 5. Variation of In qma~ with temperature Tin K for He-DT He-T z and H e - H T mixtures.

88 Pramana - J. Phys., Vol. 43, No. I, July 1994

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Thermal diffusion factors

We are, therefore, now in a position to conclude that C C F is an accurate ~x determining factor of isotopic, nonisotopic and isobaric ctx's of binary gas mixtures.

The functional relationship of F~ with re, rh, L and T s h o u l d be studied both from the theoretical and experimental viewpoints.

References

[1] W H Furry, R C Jones and L Onsager, Phys. Rev. 55, 1083 (1939) [2] W H Furry and R C Jones, Phys. Rev. 60, 459 (1946)

[3] R C Jones and W H Furry, Rev. Mod. Phys. 18, 151 (1946) [4] C J G Slieker and A E deVries, J. Chim. Phys. 60, 172 (1967)

[5] S Acharyya, A K Das, P Acharyya, S K Datta and I L Saha, J. Phys. Soc. Jpn. 50, 1603 (1981)

[6] S Acharyya, I L Saha, P Acharyya, A K Das and S K Datta, J. Phys. Soc. Jpn. 51, 1469 (1982)

[7] S Acharyya, I L Saha, A K Datta and A K Chatterjee, J. Phys. Soc. Jpn. 56, t05 (1987) [8] J L Navarro, J A Madariage and J M Saviron, J. Phys. Soc. Jpn. 52, 478 (1983) [9] T K Chatterjee and S Acharyya, J. Phys. B7, 2277 (1974)

[10] J O Hirschfelder, C F Curtiss and R B Bird, Molecular theory of oases and liquids, (John Wiley, New York, 1964)

[11] L Monchick, S I Sandier and E A Mason, J. Chem. Phys. 49, 1178 (1968)

[12] A K Barua, A Manna, P Mukhopadhyay and A Dasgupta, J. Phys. B3, 619 (1970) [13] J G Parker, Phys. Fluids 2, 449 (1959)

[ 14] I L Saha, A K Datta, A K Chatterjee and S Acharyya, J. Phys. Soc. Jpn. 56, 2381 (1987)

Pramana - J. Phys., Vol. 43, No. 1, July 1994 89

References

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