Coupled heat and mass transfer in multicomponent systems: Solution of the Maxwell-Stefan equations

I ~ IN H E A T A N D M A S S TRANSFER 0094-4548/81/050405-12502.00/0 Vol. 8, pp. 405-416, 1981 ©Pergamon Press Ltd. Printed in the United States

COUPLED HEATAND MASSTRANSFERIN MULTICOMPONENT SYSTEMS: SOLUTION OF THE MAXWELL-STEFAN EQUATIONS Ross Taylor Department of Chemical Engineering Clarkson College of Technology Potsdam, New York 13676 (Cu',,t'L~nicatedby J.P. Hartnett and W.J. Minkowycz)

ABSTRACT The general problem of coupled heat and mass transfer in n component mixtures is formulated using a generalization of the Maxwell-Stefan equations. Approximate and rigorous methods for obtaining the mass and energy fluxes are developed under the assumptions of steady one dimensional transfer. Numerical examples demonstrate the often excellent agreement between the exact solution and the analytical solution of the linearized equations. Introduction Coupling effects between heat and mass transfer (i.e. the Soret and Dufour effects) have been known for a long time and the phenomenological theory is well understood [2].

There exist a number of published analyses of

coupled heat and mass transfer in mul ti component heterogeneous systems. The studies of Delancey and Chiang [3-5] and of Gal-Or and coworkers [6,7,16,17] are based either on the Fickian formulation or on the Onsager formulation for the constitutive equations.

The diagonalisation procedure of Toor [20] and

of Stewart and Prober [15] is used for uncoupling the equations. Solutions are then obtained in terms of the eigenvalues of the matrix of transport coefficients.

For gaseous mixtures these coefficients

can be estimated from

the kinetic theory of gases. The situation with regard to non-ideal liquid mixtures is less happy and experimental data are necessary to obtain the

405

406

R. Taylor

coefficients [8].

Vol. 8, No. 5

Indeed, for this reason, a number of the aforementioned

analyses have not been subjected to numerical tests and in some other cases the computations are performed with assumed values for the coefficients. I t has recently been appreciated that the Maxwell-Stefan equations provide a convenient way of describing multicomponent diffusion [ I l l and this formulation allows a method of predicting the transport coefficients from information on the constituent binary transport parameters. The generalization of the Maxwell-Stefan equations for coupled heat and mass transfer [9,12,14] may hold out the same promise for non-ideal multicomponent mixtures.

Thus, in

the absence of pressure gradients and external body forces the generalized Maxwell-Stefan (GMS) equations can be written as

di

=

xi d~i RT-~

n xiJ k - XkJi k=l c ~)ik

= z

xi

n z j=l

x.~ i.

J

J

d In T dz

i=l,2

"

.n

(1)

j~i q =-[x + ½cR i ,zj x i x.j 4)i j re#j] ~ dT- c R T

n-l i=IZ ~in l)in d.~

(2)

where the ~)ij are the GMS diffusion coefficients with Oji = ~)ij and where the mij(mji = -mij) are the multicomponent thermal diffusion factors for the i - j pair.

The clear advantage of using equations (1,2) which for ideal gases are

completely consistent with the kinetic theory of gases is that they allow the prediction of mul ti component transfer behavior from constituent 4)ij and ~ij (see [8,11 ]). The non-linearity of equations (I-2) makes an exact analytical solution impossible to obtain, even for the simplest of all models - one dimensional steady state diffusion and heat transfer.

The purpose of this communication

is to show that good results may be obtained by linearizing the Maxwell-Stefan equations and developing an approximate analytical solution. Theoretical Development In order to develop solutions of equations (5,6) i t is necessary f i r s t to relate the constituent chemical potential gradients to the mole fractions as [I0] xi

dui dz

n-l dxk = RT S rik d T k=l

i=l,2..n-I

where the rik are thermodynamic coefficients defined by

(3)

VOI. 8, NO. 5

SOLUTION OF THE ~ . T , - ~

xi @In Yi r i k = 6ik + x k ~ In xi

ECXIATIONS

407

i,k=l ,2..n-I

(4)

Equations (1,2) may now be w r i t t e n as n-l

dxk n xiJ k - XkJi r i k dz k=l k=l c ~)i k k~i

q = _ ~, dT

xi

n d In T z xj ~i j=l j dz

i = l 2 .n-I ' "

(5)

n-l n-l dxi - cRT ~ ~kn Dkn rki ~ i =l k=l

where ~' = ~ + ½ cR

n

(6)

2

i,j:l

x i x. ~. ~i" J ~j J

(7)

A solution of equations (5,6) is sought subject to the boundary conditions of a model of steady state one dimensional transfers z = O, x i = X i o , T = To;

z = a, x i = x i 6 , T = T~

(8)

For steady state mass and energy transfer in the absence of chemical reactions in a plane f i l m the molar and energy fluxes are constant.

These

fluxes are made up of " d i f f u s i v e " and convective contributions as [ l ] Ni = Ji + x i N t '

i:l,2..n,

E = q +

n E Ni Hi i=l

(9)

where Nt is the total molar f l u x and the Hi are the constituent p a r t i a l molar enthal pies. Equations ( I - 9 ) form the basis of a f i l m model f o r coupled heat and mass transfer in multicomponent systems. below:

Two methods of solution are described

an exact numerical method and an approximate analytical solution

derived by l i n e a r i z i n g equation (5,6). Exact Numorical Sol u t i o n The complex n o n - l i n e a r solution

coupling in equations

must be o b t a i n e d by numerical methods.

formulation

is p a r t i c u l a r l y

temperature g r a d i e n t s

useful

( 5 , 6 ) moans t h a t an e x a c t The M a x w e l l - S t e f a n

i n t h i s c o n t e x t s i n c e the c o m p o s i t i o n and

can be r e l a t e d d i r e c t l y

to the i n v a r i a n t

molar and

energy f l u x e s as n-I dx k n x i Nk - x k Ni n d In T z rik dT = z - x i s xi ~.. - - i = l , 2 . . n - I k=l k=l c ~)i k j =l ij dz kti j#i

(I0)

408

R. Taylor

q

dT ~ dz

= -

RT n ~ 2 i,j=l

~ij(xi

Nj

Vol. 8, No. 5

xj Ni )

(II)

E x p l i c i t expressions for the composition gradients are most readily obtained by f i r s t

combining the n-I equations (I0) in n-I dimensional matrix form as

d(x) _ [ r ] - l [ @ ] ( x ) dn

+ [F]-I(~)

_ [F]-I(kT)

d In T dn

(12)

where n is a dimensionless distance defined by n = z/a.

[~] is a matrix of

rate factors with elements

@ii = ETin + k=IZ --kik;

@ij = - Ni

-

i~j=l,2..n-I

(13)

k~i The column matrix (~) has elements ~i = - Ni/kin

i=l,2..n-I

(14)

The k i j in equation (13,14) are binary type mass t r a n s f e r c o e f f i c i e n t s In addition (k T) is a column matrix of independent defined by k i j = c ~ i j / 6 . thermal d i f f u s i o n

ratios with elements

n

kTi = x i

z xj ~i j=l J

i=1,2, n

(15)

The temperature gradient can now be obtained from equation (6,9) dT

1

n

s dn - h (k=l

Nk Hk - E) - ~

i,jz

~ij(xi

where h is a heat t r a n s f e r c o e f f i c i e n t

(16)

Nj - xj Ni)

defined by h=x/a.

Equation (16) is then

inserted into equation (12) for the composition gradients. The evaluation of the fluxes consists e s s e n t i a l l y of i n t e g r a t i n g equations (12) and (16) across the f i l m s t a r t i n g from the known boundary condition at q = O.

Since 6, the f i l m thickness, is not usually known the

k i j and h must be estimated from hydrodynamic correlations detail

in [ I I ] .

as described in

Notice that equations (12) and (16) now contain no d i r e c t

reference to the f i l m thickness.

Thus a multidimensional

Newton-Raphson

procedure can be used to search for the set of fluxes that yields the known compositions and temperature at n = I . I t is unfortunate that considerable computation time may be required to obtain a single estimate of the fluxes.

This may be avoided by l i n e a r i z i n g

Vol. 8, No. 5

SOI/3TION OF THE MA~a~.T,-ST~AN .EQUATIONS

409

equations (5,6) and developing an analytical solution as described below. Approximate Analytical S o l u t i o n In order to apply to uncoupling procedure of Toor [20] and of Stewart and Prober [15] i t is necessary f i r s t to l i n e a r i z e equations (5,6) and then to combine them in suitable matrix form.

However, i t is seen from these

equations that the heat f l u x q is proportional to the temperature gradient whereas the d i f f u s i o n fluxes are dependent only on the gradient of the logarithm of temperature. Further, these fluxes are of widely d i f f e r e n t magnitudes and units ( t y p i c a l l y q = 103 J m-2s-I Ji : 10-3 kmol m-2s- I ) 9

The uncoupling procedure has meaning only i f the elements of the matrix of l i n e a r i z e d transport c o e f f i c i e n t s have i d e n t i c a l units and (preferably) s i m i l a r magnitude. This rules out simply assuming constant c o e f f i c i e n t s in equations (5,6) and combining them in a form that relates q and the Ji to dT/dz and the dxi/dz.

To overcome problems of a related kind Gal-Or et al.

non-dimensionalize the working equations.

Delancey and Chiang, on the other

hand, work with the constituent mass densities and s p e c i f i c enthalpy times density as independent variables.

As a r e s u l t the transport c o e f f i c i e n t

matrix does have consistent units even though the independent variables and fluxes do not. Inspection of equation (9) shows that E and q have the units of a molar f l u x m u l t i p l i e d by the units of enthalpy.

The d i f f i c u l t y

may

perhaps be overcome by defining "reduced" heat and energy fluxes by Jo = q/Href ; No = E/Href

(17)

(where Hre f is some (necessarily non-zero) reference enthalpy) then Jo and No have the units of a molar f l u x (hence the choice of symbols). The subscript zero is henceforth associated with thermal fluxes and c o e f f i c i e n t s .

The choice

of Hre f is a r b i t r a r y but should be i n v a r i a n t over the f i l m and be of s u f f i c i e n t magnitude to leave Jo and No with magnitudes s i m i l a r to those of the Ji and NiI f a reference state is chosen to be f l u i d at temperature Tre f the c o n s t i t u e n t molar enthalpies may be found from (assuming constant CPi) Hi = CPi(T - Tref)

(18)

A convenient choice of Hre f therefore is Hre f = CPm Tre f where CPm is a mean heat capacity of the mixture defined by

(19)

410

R. Taylor

Vol. 8, No. 5

n Nt CPm = ~ Ni Cpi i=l

(20)

I t is important to note that Hre f in equation (19) is a d e f i n i t i o n of convenience and does not affect the choice of reference state in any way. On observing that dT/dz = d(T - Tref)/dz a " c o n s t i t u t i v e equation" for the f l u x _To is r e a d i l y obtained by d i v i d i n g equation (6) by CPm Tre f to give nz ~, T L Tref) cRT CPm d~(z Tref ~ i , k = l

Jo -

dx i ~kn Dkn rki dz

(21)

Equation (21) may be l i n e a r i z e d by assuming the c o e f f i c i e n t s of the gradients are constant over the f i l m .

Clearly t h i s is not s t r i c t l y

true but i t is common,

in such cases, to evaluate the c o e f f i c i e n t s at the mean f i l m temperature and composition.

This suggests that a good choice of Tre f is the mean f i l m

temperature Tm = ½(To + T~).

I t is usual to choose To as the reference

temperature for the simpler problem where thermal coupling is neglected [ I I ] . However, with the present choice of Tre f the temperature dependence of equation (21) is considerably reduced (an additional argument in favor of this choice is given below). Turning to the diffusion fluxes equations (5) may be combined in n-l dimensional matrix form and inverted to give the independent Ji as (cf [lO] for the isothermal case) (j) = _ c[B]-l[r]

~z )

_ c[B]-I(kT) d dzln T

(22)

where the matrix [B] has elements Bii

= _x i_ Bin

+ nz -x-k k=l ~)ik k#i

Bij = _ x i

(4)i - Biln) ij

i~j=l , 2 . . n - I

(23)

Equations (21) and (22) are of the same form but include differing dependencies on the temperature gradient.

However, with Tref = Tm

d In T _ l dT l dT ~z Td-z ~ ~mmc[z is a good approximation even for quite high temperature gradients.

(24) Any other

choice of Tref (TO or T~ say) would weaken the approximation. Thus by defining column matrices of the n independent fluxes (N) and (J) with elements

Vol. 8, No. 5

No = E/Href;

SOI/3TION OF THE MAX~.T,-STEFAN E~.~TIONS

Ni = Ni;

3o = q/Href;

and a column m a t r i x o f n i n d e p e n d e n t t ° = (T - Tm)/Tm;

] i = Ji

fluid

i=l,2..n-I

properties

Ti = x i

411

(25)

(T) w i t h elements

(26)

i=l,2..n-I

then equations (17,20-22) may be combined in n dimensional matrix form as (j)

(27)

: _ ~ [ B ] d(T) = - [ k ] d ( ~ ) dn dn

(N) : (]) + Nt(T ) = - [k] d(_~) + Nt(T )

(28)

where the matrix of transport coefficients, [B], is the partitioned matrix x' c Cpm [~] . . . . . . . . . . . [B] -l (k T)

,," R (~T)T[r] ,, CPm ,L. . . . . . . . . . . . . ! [B] -l [I']

;

[k] = ~-[~]

(29)

I

The only assumptions necessary for the development of equations (27,28) are those of constant average heat capacities leading to equation (18) and that the logarithmic temperature gradient may be approximated by equation (24). The former is common in engineering analysis; that the l a t t e r is also quite good is rapidly demonstrated by a few elementary calculations.

Equations (28)

may be linearized by assuming c[I~] remains constant over the film.

The

inverse of equation (28) gives a set of f i r s t order linear coupled differential equations with constant coefficients and are readily solved using the uncoupling procedures of Toor [20] and Stewart and Prober [15] although i t should be noted that equations (28) may be solved in an entirely different way which does not involve uncoupling. variables is [ l l , 1 8 ]

The solution, in terms of the original

(T - t o) = {exp[~]n - r l j } { e x p [ ~ ]

- rl~}-I

(T~ - to)

where [ ~ ] = N t [ k ] - I With the d e r i v a t i v e

(30) (31)

o f equation (30) the " d i f f u s i o n "

f l u x e s a t the o r i g i n

(n=0)

are obtained on combination w i t h equation (27) as

(JO) : [k][~]{exp[~] - rl ~}-I (t O _ T~)

(32)

The film invariant molar and energy fluxes follow from equations (9) on specification of an appropriate determinacy condition as described in detail

412

R. Taylor

by Krishna and Standart [ I I ] .

Vol. 8, No. 5

The computation of the fluxes requires an

i t e r a t i v e procedure unless Nt is specified.

These quantities are easily

calculated from a simple extension of the algorithm of Taylor [19] for coupled d i f f u s i o n in isothermal systems. The matrix [k] serves as an augmented matrix of low f l u x t r a n s f e r coefficients.

Since a is not known i t is usual, under the assumptions of the

l i n e a r i z e d theory [15], to calculate [k] as the same function of the complete matrix [~] from the correlations that are used to obtain the k i j from the binary d i f f u s i v i t i e s

~)ij"

Krishna and Standart [II ].

Further discussion on this point is provided by I t is worth noting that the present l i n e a r i z a t i o n

permits this to be done for the matrix augmented by the thermal coupling terms since the elements are properly reduced to a consistent set of units and physical s i g n i f i c a n c e ;

the element Z)II is equivalent to a thermal d i f f u s i v i t y

which occurs in heat transfer correlations in place of the d i f f u s i v i t y transfer c o r r e l a t i o n s .

The matrix function [~]{exp[~] - r l j } - I

correct [k] for the influence of f i n i t e

in mass

serves to

rates of mass transfer.

Equation (30) contains a number of previously published solutions as special cases.

For example, i f el2 = 0 then equation (30) for two component

mixtures is exact.

This r e s u l t again contains no errors for the heat flux

in a mixture with any number of constituents provided all the ~ i j are zero. The d i f f u s i o n fluxes remain coupled however and equation (31) reduces to equation (19) given by Taylor and Webb [23]. Comparison of Exact and ApI~roximate Solutions The two major assumptions in the analysis above (constant coefficients and equation (24) for the temperature gradient) are subject to verification by comparison with the exact numerical solution. To show that good results may be obtained from the linearized equations consider the transfer of carbon tetrachloride (1) through stagnant hydrogen (2).

This system is chosen since the molecules have widely different

properties and consequently the thermal diffusion factor ~12 will be as high as any that may be encountered (and a good deal higher than most).

Physical

and transport properties are calculated from the kinetic theory [9 ] or from the methods described by Reid et al. [13] as appropriate, and are assumed constant over the film in both the numerical and analytical solutions (this is done to test the v a l i d i t y of the linearization employed in the analysis). The boundary conditions, physical properties and computed estimates of Nl

VOI. 8, No. 5

SOLL'I~CN CF THE M ~ I ~ S T E F A N E Q U A T I O N S

413

and E (N2 is zero) are summarized in Table 1 for two examples with this system.

The fluxes have been computed using the exact method and the

linearized theory for both m12 ~ 0 and ~12 = O. Note that i f ~12 : 0 the analytical solution is exact for binary mixtures. TABLE 1 Comparison of Exact and Approximate Film Models of Coupled Heat and Mass Transfer Boundary Conditions:

To = 345 K, T6 = 295 K

XlO = 0.25 (example I ) , Physical Prope__rties:

XIO = 0.75 (example 2),

XI6 = 0.1305

c = 3.810 x 10 -2 kmel m-2, ~12 = 3.73 x 10 -5 m2s-I

CPl = 8.591 x 104 , Cp2 = 2.894 x 104 J kmol -I K-I ' ~12 = 0.5011 (example I ) , ~12

:

0.3241 (example 2), x = 9.13 x 10 -2 (example I ) .

= 4.44 x 10 -2 (example 2) kg ms-3K" I . Fluxes

Example 1

Example2

Nl/~(kmolm-3s -I ) Linearized (m~O) Exact (mpO) Exact (~:0) E/~(kJ m-3s - I ) Linearized ( ~ 0 ) Exact (m/O) Exact (m=O)

0.9252 0.9250 0.8404 1.9325 x I0~ 1.9334 x 1~ 1.832 x I0 ~

7.208 7.204 7.086 1.8692 x 104 1 9068 x 104 1 6240 x lO4

The sole difference between examples 1 and 2 lies in the boundary composition Xlo.

Thus example l includes a r e l a t i v e l y low concentration

gradient and example 2 a much greater change in concentration over the film. The temperature gradient is the same in both examples.

Physical properties

are evaluated at the average conditions; consequently ~12 and ~, which are concentration dependent, d i f f e r . The agreement between the numerical and approximate analytical solutions for these cases is remarkably good, p a r t i c u l a r l y for example I .

The estimate

of E in example 2 is less accurate but d i f f e r s from the exact solution by no more than 2%!

I f ~12 is positive the heavier species tends to move toward

the cooler region.

Thus in this example the f l u x of species 1 is augmented

by : 10% by the moderate temperature gradient.

The energy flux is increased

414

R. Taylor

Vol. 8, No. 5

by the Dufour contribution to the heat f l u x and also through the increase in the convective term. Sample calculations have also been performed for a hypothetical ternary system.

This system was constructed by adding an imagined t h i r d species

which has molecular properties intermediate between those of the CCI 4 and H2 of the binary examples above.

I t has been found that e x c e l l e n t agreement

between the methods is c h a r a c t e r i s t i c o f - u n i d i r e c t i o n a l

transfers.

The

approximate solution leads to increased errors in cases where the fluxes have mixed signs.

The errors are usually more severe for the energy f l u x .

However,

the e r r o r in the estimates of constituent rates of t r a n s f e r are u n l i k e l y to be greater than 10% which is s u f f i c i e n t l y

accurate for most applications.

and Webb [18] have discussed in detail

the errors introduced by l i n e a r i z i n g

the Maxwell-Stefan equations for isothermal mass t r a n s f e r .

Taylor

Similar comments

apply here. Concl usions A f i l m model for coupled heat and mass t r a n s f e r in multicomponent systems has been described in this work.

One of the novel features of the

present study l i e s in the use of the generalized Maxwell-Stefan equations as c o n s t i t u t i v e equations rather than the more widely used generalized Fick's law.

An important advantage of the Maxwell-Stefan equations is that they

permit the prediction of multicomponent t r a n s f e r c o e f f i c i e n t s from information on the transport c o e f f i c i e n t s of the binary pairs that make up the mixture. An approximate analytical

solution for a film model has been developed by

l i n e a r i z i n g the c o n s t i t u t i v e equations~

A comparison with an exact numerical

solution has demonstrated the e f f i c a c y of the s l i g h t l y unusual l i n e a r i z a t i o n employed in this analysis. In view of the continuing i n t e r e s t in coupled heat and mass t r a n s f e r and since good results can be obtained from the l i n e a r i z e d equations there would appear to be a case for applying the Maxwell-Stefan formulation to more demanding physical situations such as surface catalyzed chemical reactions in boundary layer flows.

An extension of the present analysis would provide

an i n t e r e s t i n g comparison with the work of Tambour and Gal-Or [16-17]. Nomenclature [B]

square matrix of inverted d i f f u s i o n c o e f f i c i e n t s with elements given by equations (23)(dimension n-I x n - l )

c

molar concentration

Vol. 8, No. 5

Cp Bik

[m]

SOI/II~C~ CF THE M~XI~T.T,-STEFAN EQUATIONS

415

heat capacity generalized Maxwell-Stefan diffusion coefficient for the i - j pair matrix of transport coefficients defined by equation (29) (dimension n x n_)

(BT)T E h 1 Ji

(J)

row matrix with elements Bi T = Sin Bin (dimension l x n-l) energy flux heat transfer coefficient defined by h=~/a partial molar enthalpy of species i molar diffusion f lux of species i column matrix of independent diffusive type fluxes defined by equation (25) (dimension n x l)

[k]

matrix of transfer coefficients defined by equation (29)

kij Ni

binary mass transfer coefficient defined by ~ii,, = c ~)ij/6 molar flux of species i

Nt

total molar flux

(N)

matrix of fluxes defined by equation (25) (dimension n x l ) heat flux gas constant

q R T xi z

temperature mol e fraction distance Greek Symbols

~ ij

thermal diffusion factor for the i - j pair in a multicomponent mixture

It]

matrix of thermodynamic coefficients defined by equation (4) (dimension n-l x n-l)

Yi

a c t i v i t y coefficient of species i film thickness

aik q

[~] [~] (T) (~)

Kronecker delta dimensionl ess di stance chemical potential of species i thermal conductivity of the mixture matrix defined by equation (13) (dimension n-l x n-l) square matrix defined by equation (31) (dimension n x n) column matrix of independent f l u i d properties defined by equation (26) (26) (dimension n x l) column matrix defined by equation (14) (dimension n-l x l)

416

R. Taylor

Vol. 8, No. 5

[] [ ]-i

square matrix of dimension n-I x n-I or n x n inverse of a square matrix

r

diagonal matrix with n or n-I non-zero elements

()

column matrix of dimension n or n-I

()T

row matrix of dimension n or n-I

0,5

pertaining to the position z = O, z = ~ (only on x i , T or ~; l a s t i f multiple subscripts)

0,1,2,3,

index denoting component number (0 refers to thermal fluxes and

i,j,k,n

coefficients)

m

quantity evaluated at the mean film composition and temperature

ref

reference References

I.

R.B. Bird, W.E. Stewart and E.N. L i g h t f o o t , Wiley, New York, 1960.

"Transport Phenomena,"

2.

S.R. de Groot, and P. Mazur, "Non Equilibrium Thermodynamics," North Holland, Amsterdam, 1962.

3.

G.B. Delancey and S.H. Chiang, A.I.Ch.E~J.,14, 665, 1968.

4.

G.B. Delancey and S~H. Chiang, Ind. Eng. Chem. Fundam., 9, 138, 344, 1970.

5.

G.B. Delancey, Chem. Engng. S c i . , 27, 555, 1972.

6.

B. Gal-Or and L. Padmanabhan, A . I . C h . E . J . ,

14, 709, 1968.

7.

B. Gal-Or, I n t . J. Heat Mass Transfer, I I , 551, 1968.

8.

K.E. Grew, "Thermal Diffusion" in Hanley, H.J.M. (Ed) "Transport Phenomena in Fluids," Marcell Dekker, New York, 1969.

9.

J.O. Hirschfelder, C.F. Curtiss and R.B. Bird, "Molecular Theory of Gases and L i q u i d s , " Wiley, New York, 1954.

I0.

R. Krishna, Chem. Engng. S c i . , 32, 659, 1977.

II.

R. Krishna and G.L. Standart, Chem. Engng. Commun., 3, 201, 1979.

12.

C. Muckenfuss, J. Chem. Phys., 59, 1747, 1973.

13.

R.C. Reid, J.M. Prausnitz and T.K. Sherwood, "The Properties of Gases and Liquids," 3rd Edition, McGraw-Hill, New York, 1977.

14.

G.L. Standart, R. Taylor and R. Krishna, Chem. Engng. Commun., 3, 277, 1979

!5.

W.E. Stewart and R. Prober, Ind. Engng. Chem. Fundam., 3, 224, 1964.

16.

Y. Tambour and B. Gal-Or, Physics of Fluids, 19, 219, 1976; 2_00, 880, 1977.

17.

Y. Tambour, Int. J. Heat Mass Transfer, 23, 321, 1980.

18.

R. Taylor and D.R. Webb, Chem. Engng. Commun. 7, 287, 1980.

19~

R. Taylor, Comput. Chem. Eng., In press, 1981.

20.

H.L. Toor, A . I . C h . E . J . ,

I0, 460, 1964.