transport coefficients of multicomponent gas mixtures

Physica A 159 (1989) 369-385 North-Holland, Amsterdam

TRANSPORT COEFFICIENTS OF MULTICOMPONENT GAS MIXTURES

KULDIP SINGH Department of Physics, G. G. N. Khalsa College, Ludhiana-141001, India

AK DHAM and S.C. GUPTA Department of Physics, Punjabi University, Patiala-147002, India

Received 11 July 1988 Revised manuscript received 4 April 1989

General expressions for the transport coefficients of multicomponent gas mixtures have been written in a form where the computations for a given N-component gas mixture and for any order can be done on the computer without feeding the explicit expressions ft r the matrix & elements. (General expressions, available in literature earlier, require separate computer programs for each order of calculation.) These expressions, not only save a lot of computational effort but also avoid cumbersome algebraic manipulations. Finally, a few results are given for transport coefficients of ternary gas mixtures.

1. Introduction

Recently there has been renewed interest to work out higher order approximations (within the framework of the Chapman-Enskog method) of various transport coefficients of multicomponent gas mixtures [l-5]. This is due to the fact that more and more accurate data have become available and one needs e~precsions to match the accuracy of the data [6-111. (Besides, this will enable us to have a better understanding of the intermolecular interactions.) One of the major efforts to work out higher orders in literature has been to write explicit expressions for square or curly brackets [12,13]. This was previously done for simple and binary gas mixtures and has been recently extended to thermal conductivity of multicomponent gas mixtures [l]. As is clear, the effort involved for writing the explicit expressions of the brackets is extensive and the major one. Based on the expressions, developing computer programs for each

0378-4371/ 89 / $03.50 0 Elsevier Science Publishers B .V. (North-Holland Physics Publishing Division)

370 Kuldip Singh et al. / Transport coefficients of multicomponent gas mixtures

order is an additional effort [1, 2, 16-20] (that is the reason why so many papers have appeared in literature [1-4, 15-20])*.

Recently we have reported an alternative formulation of the Chapman- Enskog method for binary and ternary gas mixtures [3-5, 15], where all the higher order computations of transport coefficients can be done on the computer without feeding explicit expressions of the brackets. As discussed in the text of the present work and earlier works [15] also, the expressions for the binary and ternary transport coefficients involve the elements of a matrix L (--Lii). The explicit expressions of Lii in terms of k-matrices (defined later) are fed to the computer for carrying out computation for binary and ternary gas mixtures. It is obvious from the structure of these expressions for Lij that for N- (>3) component mixtures the structures of Lij in terms of k~j will be much more complicated. Thus we conclude that the advantage gained in writing separate simple programs for binary and ternary gas mixtures is lost for N- (>3) component gas mixture.

The purpose of the present paper is to report a simple program where computations for a given order n and for a given N-component mixture can be done on the computer without having the necessity of modifying the program from nth order to the (n + 1)th order or from a N-component mixture to a (N + 1)-component mixture. This program is valid for all the potentials. Thus this approach is a generalization of all the previous works, i.e. the work on binary and ternary mixtures as reported by the authors [3, 4, 15] and that reported by Pavlov [2] and Assael et al. [1]. In section 2, we present the expressions for the multicomponent gas mixtures as needed for the present work. In section 3, we discuss the computational difficulties and algorithm for the computer program (detailed program will be published elsewhere), and finally (in section 4) some specific computations based on the present program are given. [The basic definitions and notations used in the present work are the same as in Kumar [23] and equations taken from that paper will be prefixed by K.]

2. General expressions for the transport coefficients

The Boltzmann equations for each component of an N-component gas mixture considering binary interaction with other components are given by [12-14]

* As is well known, the solution of the Boltzmann equation for ionized gases using the Chapman-Enskog approach shows a slower rate of convergence. Some attempts using similar or alternative formulations have been reported in literature [21,22].

Kuldip Singh et al. 1 Transport coeficients of multicomponent gas mixtures 371

Here, f;:fi = pi, where & is the distribution function for the ith component. On substituting the expansion of fi (eq. (K.32)) into (l), multiplying the resulting equation by +“‘(aiCi) and integrating them w.r.t. Ci, we have [3]

(i = 1,2,. . . , N) .

(2)

2.1. First approximation

To solve the above equations for Ft”” (i = 1,2, . . . , N) for the usual second approximation, multiply the right-hand side of these equations by (l/e) and expand Fi in powers of E. Comparing the powers of E on both sides of the resulting equations we get a hierarchy of equations for different approximations. The solution of the first approximation as in case of simple gas [23] is

The further approximations to F must satisfy the following relations:

00

c rF(oo) = 0

i0 9

r=l

0~ N

cc n a’Fw = i i im 0,

r=l i=l

cc n rF(lo) = 0 . i i0

r=l i=l

(3)

(4)

(5)

(6)

These equations are called the subsidiary conditions. They should be satisfied by any solution of F.

2.2. Second approximation

Equating the power of E’ on both sides of eqs. (2) and substituting the first approximation (eq. (3))) i.e. taking one set of indices to be zero, we get the

following set of equations:

(i = 1,2,. . . , N) , j=l

(7)

372 Kuldip Singh et al. I Transport coefficients o f multicomponent gas mixtures

where

[k,],.,. = [ (~ , - IJ , , l~ , - ' , ,~,o) + (,~,-14,1,~,o, ,~,,') N nj ]

+ ~.= ~ (o,,,,14jl,~,,,', ~,o) ( i= 1 , 2 , . . . , N) j# i

(8)

and

[k,,],..,., = (~,,14,1~,0. ~,,,') ( i , j = 1, 2 , . . . , N ) . (9)

In eq. (8), the first two terms represent interactions between like molecules; eq. (9) and the last term of eq. (8) represent intermolecular interactions between unlike molecules. The explicit form of (vlDi[O) has been given in our earlier work [3] and that of J-matrices is given in appendix A. Each of these matrices is an infinite square matrix in p and v'. Since all the collision J-matrices are diagonal in l ~,r,d m, therefore all the k-matrices [23] are diagonal in I and m,

[ k , A . , . , -- ( k l ~ ) ~ . ~ . a , , . a m m . . (10)

Introducing a (N x N) K-matrix, each element of which is an infinite square matrix, we can write eqs. (7) as

~ = K 1 F , (11)

where

K __. / (k,,)..., (~,)~.~,

(k,2)...' "" "(k,N)..~' \

(12)

iF=

°:( -n,'P(;') \ - n 2'F~" ) ~,

- ~r ' ( , " ) l - - n N r N /

(,,ID, IO) ) (-ID. ~10) .

(,,IDol0)

(13)

(14)

Kuldip Singh et al. / Transport coefficients of multicomponent gas mixtures 373

Substituting the values of (vlD, lO), eq. (14) can be rewritten as

¢ = U~ll~/Ol~m0 + [U~)B[o - vr~ (c9~) log T)B[,latl + V~o6tES~ ) , (15)

where ~o, ~;t, go, gl, 1F and ~ are (N x 1) matrices while B, UCm tJ and K are (N x N) matrices. These matrices have been defined for the N = 3 case in appendix 1 of our earlier work [3].

Eq. (15) g i v e s , as sum of the scalar (~t0), vector (~l,) and tensor (812) parts and there is no coupling among the various terms of different tensor character. Therefore, eq. (15) gets separated into the following set of equations, i.e.

0 IF(or'°) (16) ,Co"°' = [ K ] . , . , ,

1 O~ ') = [KI liP'("") (17)

t--Jv, v' --m

2 1Ft"E) (18) O ~ z) = (K),,,,,, m '

l 1 (v'l) , ~ t ) = 0 = (K) , , , v, F m for 1 t> 3, ( 1 9 )

and

*(0 vO)-- U~l~lO~mO ,

,(~l) = (U~)B~o _ V,~o~)log T B ~ l ) 6 , 1 ,

O~"~ = ~ s ~ o ~ , ~ .

(20)

(21)

(22)

Using the Euler equations of motion [23], we get

U = 0 (23)

and

v~2= " all" B'I q

r/i g[i

where

"io ) n i m i 0 cl) log p - ,, PP / m i p "1

(24)

N ] Fi- ~ ~kFk •

k--1 (25)


In our work we need only two values of 1, i.e. 1--- 1 corresponds to the vector part while l = 2 corresponds to the tensor part. And for these two values, eq. (11) gets separated into the corresponding vector and tensor part equations•

Incorporating the subsidiary condition

N E _ I L-,(OI)

n~ui ~im = 0, (26) i = 1

the vector part of eq. (11) gets modified to

I 1 '

IIJ~, (~ ' )= [ r ] ' , F(~ ~ ') (27)

where

= - {~o,. log T)B'~t ] (28)

with

a t . _ .

1 ~ • I t

Ot 1

1 1 n 2 0[2 or2

1 1

~nN~N aN "'"

ot

(29)

1

(K) ' . =

1 1

( k , , ) - , (k ,2) ' , . - . (k ;~ ) - , 1 1 1

(k21)~, (k22),,~, .- . (kEN)~ ~,

1 1 1

~(k~,)~, ( k ~ 2 ) ~ , - . . (k~N)~,

(30)

1 1

where (ki;):'~., ( i=1. , 2 . . . . , N) are obtained from ( k i i ) , , . . ( i = !,"~,.. . , N ) by modifying the first row of each of these matrices such that

1

(k/j)0,, = ,~jS~,o ( ] = 1 , 2 , . . . , N). (31)

2.3. Explicit expressions for the transport coefficients

The solution of eq. (11) is

~ F = L ~ , (32)

KuMip Singh et al. I Transport coefficients o f multicomponent gas mixtures 375

where L is the left inverse of K. The various components of L, i.e. t (Lij)w,, are infinite matrices and like other matrices are diagonal in I and m,

! (Lq)~,,, =(LlT)),z,,Stt,Smm,. (33)

Now tensor and vector parts of eq. (32) can be written as

- - n l P ' V I, '0 '~

IF( m 1) ( [ . . t (1) , ~ / ~ _ / - ~ ( I ) log T ) B ' ~ I } . ' = a v

(34)

(35)

The coefficient of viscosity is related to the tensor part (l = 2) of IF as given below (using eqs. (K.54) and (K.72)):

N

kT ~ - '~'(°2)~ i=, ¢~i r i m ] - = - k T ( ~ F ~ " 2 ) ) = - V ~ T l m i x S ( m 2) , (36)

where ~ is the row matrix corresponding to the column matrix ~0- Substituting 1 (v'2) F m from eq. (34) in eq. (36), the coefficient of viscosity timex is

N

nmix .~. (~l.2~jo)= ~ (Lu)oo " (37) kT ~,j=l

Coefficients, of symmetric diffusion Dij and thermodiffusion D ~ are obtained by using eq. (35) and the expression for the average velocity (Ci) given by [13]

1 (o~) ~ (1) T3(1) - - F ire = - Dqd]m- D~-m log T Ot i

(f= 1,2 . . . . , N , ,

with

N %"

D i j = O j i , i~i([J-~a)Dij=(} and ~ P i r "=" " i = i P oi = 0 .

( I ) d j,, is given by eq. (25). The expressions for diffusion Dij and thermodiffusion DI r are

N

n ~] ___PJ (La)oo ] ' ) i l - - t • pn~a~ /=2 ttjaj

( i= 1 , 2 , . . . , N ) ,

I D#_ n (L#).;o + Dil Ft i Otitlj OlI

( i , i = l , 2 , . . . , N , i # j )

(38)

(39)

(40)


and

Dr___ 5 1 l ( L , ) 0 1 a~n~ j=~ aj

( i= 1 , 2 , . . . , N ) . (41)

The expression for the thermal flux vector for an N-component gas mixture is obtained from eqs. (K.58) and (K.59) and is given by

q~)= 5 kT ~, ni F~ ) - kT ~'. n, (11) (42) i=, a~ 2 i=1 a'-i. Fire "

The thermal conductivity is the coefficient of ( -0~)T) in the above expression for q~). Using eq. (35), we obtain the coefficient of thermal conductivity as

t I t

~mix ~ ~mix d- '~mix , (43)

where

"'m)L~ix 5 N t =~ k ~ (Zij) l l /otiot j (44) i,j=l

and

k7O7, i=1

(45)

with k r (i = 1, 2 , . . . , N) the thermal diffusion ratios defined by

N N

ED~jkr=Dr~ and EkT=0. (46) j = l i= l

F

L ij are the components of the matrix L' which is the inverse of the K'-matrix (eqs. (30), (31)). Indices (00), (01), (11) indicate the element of first row and first column, the element of first row and second column, and the element of second row and second column, respectively, for each of the matrices Lij or L,~ mentioned in eqs. (37), (39)-(41), (43).

In our earlier works [3, 4, 15] we have written explicit expressions for Lij separately for binary and ternary mixtures and then developed computer programs. But in the present work our major interest has been to develop a general computer program with which transport coefficients could be computed without any change in it, even when N (number of the components in mixture) or n (order of calculation) change.

Kuldip Singh et al. / Transport coefficients o f multicomponent gas mixtures 377

3. Computation of transport coefficients

Having obtained the general expressions, it is obvious that any computer program based on this set of expressions requires an explicit form of L 0 in terms of kq (similar to the procedure outlined in appendix B). Since this method is time consuming and quite laborious [3, 4], these expressions do not prove useful for computation for a N (>3) component mixture. Alternatively one could go over to Assael's approach [1] for multicomponent mixture. But this approach has also a limitation that one has to write computer programs for each order of calculation separately.

Keeping in view the limitation of our earlier approach [3, 4] and that of Assael [1] it is desirable to have expressions in an alternate form which should have the advantage of writing a simple and compact program for an N- component gas mixture for a given order calculation. Indeed this can be done as shown below:

It is known that the K-matrix is of (nN × nN) size for viscosity and diffusion while for thermal conductivity and thermal diffusion, it is of [(n + 1)N × (n + 1)N] size. Let the elements of the inverted K-matrix, i.e. L (=T), be denoted by Tij (i, j = 1, 2 , . . . , nN). These elements are related to L~j if proper partitioning of T = (T0) is done. Thus Lij will become submatrices of size (n × n) in T. The expressions of the transport coefficients for nth order calculation and for a N-component gas mixture in terms of T 0 are

N

nmix= E Tl+(i-l)n,l+(j-1)n, (48) kT ~,j=~

O i I --

N n ~ Pj T' j- 1),,}

IJnioti j=2 njotj { ~+(~-l)..l+( (i = 1 , 2 , . . . , N ) , (49)

?

Dij= n {T~+(i_~),,.t+(j_~),, }+D, , (i, j = 1 , 2 , . . , N , i C j ) , n i° t injaJ (50)

1 ,~N 1 D r = - " ~ " {Tl+~/_l)o,+t).2+(i_t~<,,+l~ } ( i = 1 . 2 . . . . . N)

Ivl i OL i j= 10gj (51)

and

, = 5 u Amix 2 k ~ 1 { T ; + ( i _ l ) ( n + l ) , 2 + ( j _ l ) ( n + l ) } ' (52)

i,j= 10LiO~j

where T' ( - L ' ) is the inverted K:-matrix.


3.1. Algorithm for the computer program

The transport coefficients can be computed from the general expressions (eqs. (48)-(52)) if the collision matrices (defined by eqs. (A.1)-(A.5)) are known. The latter are evaluated by finding the function X(NL, hi, n'll~, v~, )2) (eq. (A.2)) and the interaction integral Vt,,,, (eq. (K.86)) for unlike molecules. The various steps involved in the computation are:

(a) The function X ( N L , nl, n ' l l I v 1 v2) As is obvious, X( ) depends upon the indices N, L, n, l, n'. This set of

indices, consistent with the conditions on Talmi coefficients for which this function is nonvanishing, are evaluated. These conditions are

2 N + L + 2 n + l = 2v~ + 1~ , (53)

(L + l)~> 11 I> IL- II, (54)

n ' = ( v ~ - v2)+ n . (55)

Thus, in order to calculate the collision matrices one has to find this set of indices N, L, n, l, n ' for which X( ) is non-zero. To obtain this, we first calculate the upper and lower limits, for N, L, n and I. All these indices are positive integers, the lower limits being obviously zero. The upper limits can be established using eqs. (53)-(55). The range of variation of the various indices is

O~ L ~ ( 2 v l + ll)

0~<n<~ v t ,

0<~ l~<(2v 1 + l l ) . (56)

Next we are to calculate the sets of these indices consistent with eqs. (53)-(55). This is done in the following way:

(i) Fix N, L and n. Vary l between 0 to (2v~ 4- ll) and find the values of l satisfying eqs. (53) and (54). Note these sets of consistent values of N, L, n and I.

(ii) Fix N and n. Vary L between 0 to (2v~ 4- ll). For each value of L repeat step (i).

(iii) Fix N and vary n between 0 to 1,1. For each value of n go through step (ii).

(iv) Vary N between 0 to v~ and for each N repeat the above steps. (v) Alter obtaining all the possible sets of indices N, L, n and I for a given l~

and v~, we can find the values of n ' for given v~ and v: from eq. (55).

Kuldip Singh et al. / Transport coeflficients of multicomponent gas mixtures 379

All the factors in X( ) are defined by eq. (A.2) except for the Clebsch- Gordon coefficient (1~01201130) which is given elsewhere [36].

So, for specific values of N, L, n, I and n' the expressions for different X's can be written in terms of the mass ratios ai and aj.

(b) Calculation of the interaction integrals The interaction integral Vlnn,(ij) for specific values of l, n, and n' are

computed through the reduced collision integral /~e,s~, by a relation Oven elsewhere [23] (eq. (K.A.25); for ready reference see appendix A). (The computation of/~tt,s), has been extensively discussed in literature [24-26].) In order to save labour for computing/2 tt'~) * directly tabulated values can also be fed for computing t V n~,. These values are available for different potentials and temperature ranges, the references to which are available in literature [14,24,27-34].

Having defined X( ) and Vtn,,(/j), we write collision matrices for unlike molecules from eqs. (A.1) and (A.2). For like molecules a i = a j and odd-/ terms do not contribute to the calculation of X( ). Thus Vt, o,(ij) is replaced by Vl,,,(ii) or VIn,,(]j) and J-matrices for like molecules may be written as

(~iv~lJi, l~,v2, c~,o)= ~lll2t~mtm2X( ) V t , , , ( i i ) . (57)

This matrix is symmetrical in the v-indices. With the knowledge of the collision matrices, we can now find the various transport coefficients. For a finite order (say nth) calculation of viscosity and diffusion coefficients, we truncate the infinite J-matrices to (n x n) size with v 1 and v 2 varying from 0 to (n - 1), and, for thermal conductivity and thermal diffusion ratios, these matrices are truncated to [(n + 1) x (n + 1)] size with vl and v 2 varying from 0 to n.

Using the J-matrices found above, the k-matrices can be obtained using eqs. (8) and (9). Then transport coefficients can be computed from eqs. (48)-(52).

4. Calculation and results

The present code has been tested for N = 3 , i.e. a Ne-Ar -Kr ternary mixture, though more complicated examples can also be discussed. The advantage of the present example is that one can compare the results of this work with something known. The known results are taken from our earlier works [3-5] where computations have been done only for LJ(12-7) and rigid sphere potentials using the expressions primarily developed for ternary mixtures. (One could take more familiar potentials for further work. It may be


emphasized that in the present work our interest is only to test the code.) Table I presents first order values T~?~ and higher order contribution ratios ¢(")

J r m i x

(=T(~x/T¢l)'mix, where - - m , x T('') denotes the contribution of first n orders to the transport coefficient Tmi~) of the various transport coefficients (i.e. viscosity Vlm~, thermal conductivity Am~ ~, symmetric diffusion Dq (i, j = 1, 2, 3) and thermal diffusion ratios k r (i = 1, 2, 3)) for equimolar composition of the Ne-Ar-Kr mixture at 300 K for the rigKl sphere and LJ(12-7) potentials. (The choice of the potentials is in continuation of our earlier works [3-5]. The force constants for the rigid sphere potential (RSP) have been taken from Chapman and Cowling [12] and those for LJ(12-7) from our earlier work [35].) The values agree exactly.

D(~) Having tested the code, we present some results about first order _q and higher order contribution ratios fo ° (") (t," I" = 1, 2, 3) (fo~,(n)_-- Di j(n)/D~j(~), where D~ ") is the contribution of first n orders to the diffusio'n coefficient Do) of symmetric diffusion coefficients (six in number) in table II for a Ne-Ar-Kr mixture at 300 K for the rigid sphere (RSP) and LJ(12-7) potentials. The results for asymmetric diffusion coefficients have already been presented in our recent research note [38]. The reasons for studying the symmetric diffusion coefficients are: (i) besides being used for reacting and polyatomie gases, they are consistent with Onsagar's reciprocity relations [13], (ii) thermal diffusion ratios k f of multicomponent gas mixtures are defined in terms of these coefficients (eq. (46)).

The computations of diffusion coefficients D~j for a Ne-Ar-Kr mixture are

Table I First order value --mix and higher order contribution ratio ¢(") for different transport coefficients

J Traix

[i.e. viscosity T/mix, thermal conductivity Ami ~, the symmetric diffusion D 0 (i, j = 1, 2, 3) and thermal diffusion ratios kf (i = 1, 2, 3)] of the Ne-Ar-Kr mixture (XNe = 0.30, XAr = 0.35, XK~ = 0.35) at 300K for the rigid sphere potential (RSP) and LJ(12-7) potential. [frmix(") = Tin,,/(.) .Tmix(l) , where Tmix(I) denotes the contribution of first n orders to Tm~.. The units of r/mix, Am,, and D,/ are IxP, ergcm -~ s -~ K -~ and cm 2 s -~ respectively, l

RSP

Frolaertv T (~) ft2) ¢ ( 3 ) f ( 4 ) r " " m~x ~ Tml x ~ Tm) x , Trot x

~/,,, 245.04 1.0146 1.0159 1.0161 A ..... 0.1757× 104 1.0332 1.0379 1.0396 Dl 1 0.8462 1.0406 1.0467 1.0467 D22 0.3739 1.0199 1.0218 1.0228 D 3 3 0.2299 1.0283 1.0319 1.0329 Di 2 -0.0681 1.0461 1.0536 1.0548 DI~ -0.1491 1.0394 1.0451 1.0466 D23 -0.1662 1.0176 1.0190 1.0200 k r - 8 . 3 9 9 x 1 0 -2 1.0472 1.1)586 1.0597 k~ -1.753 × 10-" 1.0204 1.0171 1.1)155 k r 10.1501 × 10 -2 1.0426 1.0514 1.0534

LJ(12-7)

T(l)_.:. f ( 2 ) f G ) f!,,4) III1"~ J • mix J t mix J t mix

264.92 1.0025 1.0025 1.0025 0.1933× 10' 1.0076 1.0079 1.0079 1.0301 1.0103 1.0106 1.0106 0.4390 1.0023 1.0023 1.0023 0.2787 1.0052 1.0053 1.0053

-0.0754 1.0161 1.0167 1.0167 -0.1875 1.0091 1.0095 1.0095 -0.1962 1.0012 1.0011 1.0011 - 0 . 4 0 7 8 × 1 0 -~ 1.0137 1.0138 1.0138

0.1638× 10 -2 1.1690 1.1737 1.1732 0.3914 × 10 ~ 1.0072 1.0071 1.0071

Kuldip Singh et al. / Transport coefficients o f multicomponent gas mixtures 381

presented in table II for equimolar concentrations and for one component in trace while the other two are equal, at 300 K. Besides usual conclusions (i.e. first order values for RSP are lower and rate of convergence slower), we observe the following points:

(a) Only diagonal elements of D 0 show a systematic decrease with an increase of concentration of the ith component.

(b) The negative values of the off diagonal elements are consistent with the defining equations (eqs. (38)).

Table II

Symmetric diffusion coefficients D 0 (i, j = 1, 2, 3), the first order value D (~) and higher order 0

contribution ratio ct,) t= D ~"~ (~ s o o , --o /Do , n = 2, 3) for different compositions of the N e - A r - K r mixture

for the rigid sphere (RSP) and LI (12 -7 ) potentials

Diffusion RSP L J(12-7) (i) ¢(2) ¢(3) D(1) ¢(2) ¢(s) coefficient Mixture D o s % s % ,i so, i so,:

Dll I 0.8462 1.0406 1.0467 1.0301 1.0103 1.0106 II 46.8317 1.0470 1.0548 56.9334 1.0122 1.0126

III 0.5422 1.0430 1.0497 0.6796 1.0106 1.0110 IV 0.4873 1.0267 1.0291 0.5577 1.0069 1.0071

D22 I 0.3739 1.0199 1.0218 0.4390 1.0023 1.0023 II 0.2236 1.0258 1.0286 0.2593 1.0023 1.0023

I I I 33.0877 1.0234 1.0234 38.7845 I. 0027 1.0027 IV 0.1838 1.0234 1.0260 0.1508 1.0064 1.0066

D33 I 0.2299 1.0283 1.0319 0.2787 1.0052 1.0053 II 0.0505 1.0337 1.0384 0.1401 1.0055 1.0057

III 0.0315 1.0428 1.0494 0.1296 1.0106 1.0109 I V 21.2227 1.0047 1.0049 52.0875 1.0006 1.0006

Di2 I -0 .0681 1.0461 1.0536 -0 .0754 1.0161 1.0167 II - 0.0399 1.0470 1.0548 - 0.0435 1.0197 1.0207

III -0 .0509 1.0472 1.0556 - 0.0521 1.0218 1.0227 IV -0 .2235 1.0273 1.0304 -0 .2743 1.0070 1.0072

DI~ I -0 .1491 1.0394 1.0451 -0 .1855 1.0091 1.0094 II - 0.0939 1.0470 1.0548 - 0.1175 1.0108 1.0112

III -0 .1290 1.0430 1.0496 -0 .1618 1.0106 1.0110 - 0 3 8 7 0 !_0!87 ! o908 -0 .4887 !.0043 ! O_f),~

D, 3 I - 0.1662 1,0176 1.0190 - 0.1962 1.0012 1.0011 II - 0.1054 1.0257 1.0280 - 0.1848 1.0023 ! .0023

Il l -0 .1455 1,0213 1.0232 -0 .1724 1.0014 1.0013 IV - 0.4203 0.9980 0.9973 - 0.5095 0.9987 0.9986

Mixture XN~ XAr XKr

I 0.300 0.350 0.350 II 0.005 0.500 0.495

III 0.495 0.005 0.500 IV 0.500 0.495 0.005


(e) Only for RSP, the ratio of higher order to first order (f~o"l) satisfy the relation

f (n) : ¢(n) Dii J D q

as long as i is the lightest component and is in trace.

Appendix A

The collision J-matrix is given by [23]

(~,~,,IJl~,~,~, oljO) = 6lll2~mlm 2 Z X ( N L , nl, n'll, vt t,2)Vt..,(ij) , N L n l n '

where

2

X(NL, nl, n'll, ~,~ z,z) = ( - 1) "' +"~+"+"' Nm'N"tN"'t N~ltlN,..q

olj o.2(ILll ) × -~ ,~ ,

with

(A.1)

(A.2)

P = 2 N + L , p = 2 n + l and p ' = 2 n ' + l

and

o.(l,lz13)=(i),,+,2_t3J(21~ + 1)(212 + 1) " 47r(2l 3 + 1) (1101201130),

where (1~0120li30) is the Clebsch-Gordan coefficient [36]. Applying the trans- , , v t u t a U U l i p ~ u I J c ~ t y a t l i au l ix t ; O C l l l t ; l C l l t b , LIIC o m c r m r e e collision matrices c a n

be written as

(aiv, lJl~io, % ~ ) - 6,,,.6m,,, z E N L n l n '

)v',,,,(ij), Oli

N L n l n ' - X ( / )V,,,,,(ij) , (A.4)

" N L n l n '

Of/. - P - p '

- x ( ) v ' , , , , . ( q ) , ( A . 5 )

Kuldip Singh et al. I Transport coefficients of muiticomponent gas mixtures 383

where X ( ) =- X ( N L , nl, n '[l I v 11,2). Between other components, the structure of J-matrices is similar to that given above.

The interaction integral t V,,,,,(ij) is related to the reduced collision integral g](t.s), by the following relation:

=

tr2(2~r) ' 'z E ( p + l ) ! ( ( 2 1 - 4 r + 1)--(--1)/-2r) p,, ( p + ½)! l - 2 r + 1

x alrB(nl, n ' l , p)~-~(l-2r,p) , , (A.6)

where

, (2 t - 2r)! a r = ( - 1)r 2tr!( l _ r)!(l - 2r)! (A.7)

The range of variation of p and r is

l < ~ p < ~ ( l + n + n ' ) ,

0 ~< r ~< 1/2 for 1 even,

O<~r<~(l - 1)/2 for 1 odd ,

and B(nl , n ' l , p) are the Moshinsky brackets and tabulated by Brody and Moshinsky [37].

A p p e n d i x B

For a binary gas mixture, the K-matrix is

k,, (B 1) k12~ k21 k.,-,,. } '

where all the symbols are defined in eqs. (8) and (9). q'he inverse of this matrix (i.e. L) has the form

( Ll' L'2 ) L2 ! L22 . (B.2)

In order to compute transport coeffid'mts (eqs. (37), (39)-(41), (43) for N = 2) expressions of L# are fed to the computer. The explicit form for one of the elements (e.g. L~2 ) can be written as (for more details see ref. [15])

384 Kuldip Singh et al. / Transport coefficients of muiticomponent gas mixtures

L,2 = - {[(ktz)-tk,,-- (k:2)-'kztl-'}(k,,.)-'. (B.3)

The structure of expressions for other Lq will be of similar type [15]. For a ternary gas mixture, the K-matrix is

ktt kt2 k13~ k21 k22 k23] • k31 k32 k33]

(B.4)

The invers L-matrix is of (3 x 3) size and has the form

L11 LI2 LI3 L21 L22 L23 / . L3I L32 L33]

( B . 5 )

Explicit expressions for one of the elements (e.g. L12 ) can be written as (for details see ref. [3])

L12 = _ { [ ( k , a ) - t k , 2 - ( k 2 3 ) - ' k E 2 ] - ' [ ( k , a ) - ' k , , - ( k 2 3 ) - ' k z , ]

- [ ( k 2 3 ) - ' k 2 2 - ( k 3 3 ) - ' k 3 2 1 - t [ ( k 2 3 ) - ' k z I - - (k33)-Ik31]} -t

× { [ ( k t 3 ) - ' k , 2 - ( k z 3 ) - l k 2 2 1 - ' ( k z 3 ) - '

+ [ ( k 2 3 ) - ' k 2 2 - ( k 3 3 ) - ' k 3 2 1 - ' ( k 2 3 ) - ' } " (B.6)

The structure of the expressions for other Lq is of similar type [3]. For N = 4 and N = 5, the K-matrix will be of (4 x 4) and (5 x 5) sizes and

there will be 16 and 25 k-matrices in them. The corresponding L-matrices will of sizes (4 x 4) and (5 x 5), respectively. To compute the transport coefficients we need 16 and 25 explicity expressions of L# for 4 and 5 component gas mixtures, respectively, which can be very much time consuming and complicated.

We are grateful to the referee for useful comments.

References

[1] M.J. Assacl, W.A. Wakeham and J. Kestin, Int. J. Thermophys. 1 (1980) 7. [2] A.V. Pavlov, Soy. Phys. Tech. Phys. 26 (1981) 89.

Kttldip Single et al. I Transport caeficients of m~lticc~mponertt gas mistuws 3x5

(31 Kuldip Singh, A.K. Dham and S.C. Gupta, J. Phys. B: At. Mol. Ph,s. 1fi (1983) 245. [4] Kuldip Singh, A.K. Dham and S.C. Gupta, J. Phys. B: At. Mol. Phis. 16 (1983) 2613. [5] Kuldip Singh, Ph.D. Thesis, Punjabi University, Patiala. 1985. [6] A.A. Clifford, R. Fleeter, J. Kestin and W.A. Wakeham, Physica A 98 (1979) 467. (71 J.M. Hellemans, J. Kestin and S.T. Ro, Physica 71 (1974) 1. [8] J. Kestin, H.E. Khalifa and W.A. Wakeham, J. Chem. Phys. 65 (1976) 5186. [9] J. Kestin, H.E. Khalifa and W.A. Wakeham, J. Chem. Phys. 67 (1977) 4254.

[lo] J. Kestin and S.T. Ro, Ber Bunsenges Phys. Chem. 80 (1976) 619. [ll] S. Mathur, P.K. Tondon and S.C. Saxena, Mol. Phys. 12 (1967) 569. (121 S. Chapman and T.G. Cowling, Mathematical Theory of Non-Uniform Gases (Cambridge

Univ. Press, Cambridge, 1970). (131 J.H. Ferziger and H.G. Kaper, Mathematical Theory of Transport Processes in Gases

(North-Holland, Amsterdam, 1973). (141 J.O. Hirschfelder, C.F. Curtiss and R.B. Bird, Molecular Theory of Liquids and Gases

(Wiley, New York, 1964). [lS] A.K. Dham and S.C. Gupta, J. Phys. 8: At. Mol. Phys. 8 (1975) 2898. [16] M.J. Assael, M. Dix, A. Lucas and W.A. Wakeham, J. Chem. Sot. Faraday Trans. I 77

(1981) 439. [17] E.A. Mason, J. Chem. Phys. 27 (1957) 754. [18] S.C. Saxena and R.K. Joshi, Physica 29 (1963) 870. [19] R.K. Joshi, Phys. Lett. 15 (1965) 32. [20] R.S. Devoto, Phys. Fluids 9 (1966) 1230. [21] R.E. Robson, J. Phys. B: At. Mol. Phys. 5 (1972) 837. [22] A.J.M. Garret, J. Plasma Phys. 27 (1982) 136. [23] K. Kumar, Aust. J. Phys. 20 (1967) 205. (241 F.J. Smith and R.J. Munn, J. Chem. Phys. 41 (1964) 3560. (251 H. O’Hara and F.J. Smith, Comput. Phys. Commun. 2 (1971) -17. [26] G.C. Maitland, M. Rigby, E.E. Smith and W.A. Wakeham. Intermolecular Forces (Clarcn-

don Press, Oxford 1981). [27] Y. Singh and Dass A. Gupta, J. Chem. Phys. -52 (1970) 3055. [2S] M. Klein, H.J.M. Hanley and F.J. Smith and P. Holland, U.S. Ikpartmcnt of Commcrcc.

Report No. NSRDS-NB7-47 (1974). [29] P.D. Neufeld and R.A. Aziz, Microfiche Publication, New York. Report NAPS-1987 (1973). [30] J.A. Barker, W. Fock and F. Smith, Phys. Fluids 7 (1964) 897. [31] S.K. Kim and J. Ross, J. Chem. Phys. 46 (1967) 818. (321 E.A. Mason, J. Chem. Phys. 22 (1954) 169. [33] SE. Love11 and J.O. Hirschfelder. University of Wisconsin, Theoretical Chem. Rcpt. ARL

62-421 ( 1962). (341 R.M. Sevast’yanov and N.A. Zykov, Teplofiz, Vysok. Temp 9 (1971) 46. [35] A.K. Dham and S.C. Gupta. Ind. J. Pure & Appl. Phys. 1.7 (1979) 427. [36] U. Fano and G. Racah, Irreducible Tensorlal Sets (Academic Press. New York. 1959). [37] T.A. Brody and M. Moshinsky. Tables of Transformation Brackets (Universidad dc Mexico.

hA0V&A 1 om\ . ..V~f~.F. 11 VW,.

[3S] Kuldip Singh. A.K. Dham and S.C Gupta. Ind. J. PUW R AppI. h\.s. 26 ( 19X‘S) 675. _

transport coefficients of multicomponent gas mixtures

Documents