Ind. Eng. Chem. Fundam. 1986, 2 5 , 476-482
476
Comparison of Matrix Approximations for Multicomponent Transfer Calculationst Thomas C. Young and Warren E. Stewart" Chemical Engineering Department, University of Wisconsin, Madison, Wisconsin 53706
Linearization and film models for multicomponent mass transfer predictions are summarized and compared with detailed boundary layer calculations. The linearization method is improved significantly by adjustment of the averaging rule for the reference state. Linearization is t h e most accurate of the approximation methods tested, both for steady and transient mass transfer operations.
Introduction Multicomponent mass transfer problems often arise in the design and simulation of chemical processes. Since the fundamental equations are coupled and nonlinear, these problems are normally done by approximate solution methods. The favored methods fall into two categories: (i) linearized solutions of the equations of change and (ii) exact solutions of one-dimensional models. This paper tests both approaches by comparisons with exact solutions for steady and transient multicomponent mass transfer problems. The effect of property-averaging methods on the accuracy of the approximations is also investigated. The linearized matrix theory of multicomponent mass transfer was developed independently by Stewart and Prober (1964) and by Toor (1964). In this theory the fluid properties are treated as constants, evaluated at a reference state, a,and a coupled system of linear partial differential equations is obtained. The resulting species continuity and flux equations take the following forms in molar units when homogeneous chemical reactions are absent.
a
c,-x
at
+ (c,v+-VX) = -(V*J*) J+ = -c,D,Tx
(2)
(Bl, ..., Bm)
J+ = P I J += (J,*,..., Jm*)T ii = P ' x = (XI1.
..., 2 , ) T
(3)
The resulting decoupled equations are formally identical with those of a set of m binary systems d dt
c,--X',
+ c,(v*.'TF,)
= -(C.J,*)
i = 1, ..., m
(6)
+ W ededicate this article to Olaf Andreas Hougen, our beloved teacher and inspirational leader. His wide-ranging interests included multicomponents diffusion, and he welcomed our efforts in this research area. 0196-4313/86/1025-0476$01.50/0
=
JAO*
...)(x.A.O- XA-1
k,'(aAB,
(8)
Rewriting this result in the notation of eq 4-7, we obtain the multicomponent expressions
jLo* = h,*(BL,
- fl,)
i = 1, ..., m
(9)
or, in matrix form
Jot = diag {kxm(fiL, ...)I( sio - si,)
(10)
Transformation back into the original variables gives the final multicomponent solution
Jo+= Pkx'P1(xO - xm)
(11)
in which k,' is the diagonal matrix of eq 10. The accuracy of the linearized solutions depends on the closeness of the physical properties to the chosen reference values. In the limit of vanishing driving forces, xo - x,, the linearization is exact. For larger changes of state, the property-averaging method becomes important. In this paper, all properties are evaluated a t a common reference state, given by either
x, = a,x,
+ (1
a,)xo
(12)
0, =
+ (1 - a,)wo
(13)
-
or
(4)
(5)
(7)
The boundary conditions on J* and x ar_e trans_formed analogously. If the resulting conditions on J1+, ..., J,+ and f,, ..., f,,, are manageable, the multicomponent problem may be solved by superimposing the solutions of the corresponding set of m binary problems. The binary solutions are often given in terms of mass transfer coefficients, k x * ,defined by
(1)
Equivalent equations may be written in mass- or volume-based units. Here x and J+ are m-dimensional vectors and D is an m X m matrix. The integer m is usually chosen as the number of independent fluxes, but with proper care in the construction of D, the full set of species 1, ..., n can be included. These equations may be decouplsd by introducing transformed fluxes and compositions, J,* and f[,for which the diffusivity matrix becomes diagonal
P'D,P = diag
Jz+= - - c a D l ~ i 1 i = 1, ..., m
a,w,
and the accuracy of the mass transfer predictions is studied as a function of the averaging parameter, a, or a d . An analytic solution to the one-dimensional multicomponent film problem was derived by Krishna and Standart (1976). The Stefan-Maxwell equations were integrated for steady molecular diffusion in a stagnant film of thickness 6. The vector, Jo+, of diffusion fluxes into t h e film was found to be J,+ = p-'@{exp(@)- $l(xO - x,)
(14)
with the following expressions for the elements of @ and cp: @ 1986 American Chemical Society
Ind. Eng. Chem. Fundam., Vol. 25, No. 4, 1986
\
kf i
i,j=
I,
n - 1 (15)
n- 1
(16)
CBij Kij =
T-
(17)
Here /3-l is a matrix of mass transfer coefficients evaluated for small molar fluxes N, and @{exp(@) - 11-lis a matric correction for the flows of the chemical species through the film. A linearized solution to the same film problem was derived by Stewart and Prober (1964). They calculated the vector of fluxes using eq 11, with
in which
and
Rxi = C B i / 6 for a planar film. Equation 20 requires modification for cylindrical or spherical annular films (Stewart, 1984), but eq 18 and 19 remain unchanged. Smith and Taylor (1983) compared eq 14 with several approximate solutions to problems of multicomponent diffusion in stagnant films. They found the StewartProber linearized film solution to be the best of the approximate methods considered, predicting fluxes that were generally within a few percent of the Krishna-Standart exact solutions. Mass transfer in process equipment seldom occurs through stagnant films. However, Krishna and Standart (1979) suggest that eq 14 be generalized to other flow situations by use of binary mass transfer coefficients as the elements ~~j ~ i= j kx(B;j,...) (21) with each diffusion coefficient Bij evaluated for a binary mixture of species i and j . This modification of the film model leads to difficulties. Equations 17 and 21 are seldom compatible, since binary mass transfer coefficients commonly vary as fractional powers of am.To reconcile eq 17 and 21, it then becomes necessary to replace the film thickness, 6, by a different value, 6 i j , for each pair of species in the mixture. The resulting low-flux mass transfer coefficient matrix, rl, with elements proportional to fractional powecs of Bij.,is formally different from the correct value, PkXP-l,given by the linearized solution. The Stewart-Prober film model handles other flow situations differently, using linearization to calculate the elements, kxi, on the right-hand side of eq 18: h,; = k x ( B j ,...I (22) T o reconcile eq 20 and 22 requires replacement of 6 with
477
a set of quantities, 6i, appropriate to the individual eigenvalues. Again, different expressions for 6 are obtained for annular films. This approach is fundamentally different from the Krishna-Standart method, since the film approximations cancel out in the limit of small fluxes, NLo. Thus, the Stewart-Prober film method is exact in the small-flux limit and differs from the detailed linearized method only in the form of the large-net-flux correction. In this paper, the detailed linearization approach and the above film models are compared with exact solutions for two realistic systems. The problems considered are (i) steady mass transfer from a flat plate in longitudinal laminar flow and (ii) transient mass transfer to or from a semiinfinite medium. The diffusion phenomena in these two systems are known from prior studies (Stewart, 1963; Stewart et al., 1970; Stewart and Sarensen, 1974) to be representative of a wide variety of mass transfer systems. Mass T r a n s f e r from a Flat P l a t e Mass transfer from a flat, plate in longitudinal laminar flow is asymptotically representative of a large class of fluid-solid mass transfer problems in the limit of large Schmidt numbers (Stewart, 1963; Stewart and Sarensen, 1974). It is also one of the few boundary layer systems for which exact multicomponent solutions are known. Stewart and Prober (1962) gave constant-property solutions for binary problems, and Prober (1961) calculated detailed variable-property solutions for a set of ternary problems. Stewart and Prober (1964) also calculated solutions based on linearization in a hybrid system of units. The massaverage velocity appears naturally in the equation of motion. However, in the problems they considered, the total molar concentration was constant, so the species continuity equations were formulated in molar units. This strategy required explicit use of two reference velocities: the mass average, v, and the molar average, v*. In the present work, new solutions are calculated by linearizing all equations in mass units. The problems are also reworked by the method of Stewart and Prober (1964) and by the film methods described above. Prober (1961) numerically integrated the boundary layer forms of the equations of motion and continuity for ternary mixtures of H,, N2, and C 0 2 a t 300 K and moderate pressures. The density was calculated from the ideal-gas formula. Viscosity and binary diffusion coefficients were computed from kinetic theory; the latter coefficients varied fivefold over the three pairs of species. An equimolar interfacial composition, xio = lI3,was specified, and interfacial fluxes were specified by position-independent parameters 2U-z
*(T)
i
K 10. = POU,
= 1, 2, 3
(23)
The boundary layer equations were integrated for 20 sets of Kio values to obtain the bulk compositions. These solutions are summarized in Table I. The binary constant-property solutions given by Stewart and Prober (1962) take the form
( u,z
-k,'- - II'(O,O,K,Sca) PaUm
SCa
ua
2
)'I2
(24)
Here
is a mass-flux parameter independent of position, z is the distance from the leading edge of the plate, and u&) is
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Ind. Eng. Chern. Fundarn.. Vol. 25, No. 4, 1986
Table I. Exact Solutions to Flat-Plate Mass Transfer Problems (from Prober (1961))a ..___.__I______ ____ ___
NL
co2
0.5239 0.2623 0.7953 0.6187 0.1546
0.1399
problem no.
--__H2
1
0.3361
-
H, 0.001 -0.002 0.0045
--0.009 0.028 4.028 - 0.070 -0.140 -0,002 0.002 0.003
0.3841 0.3930 0.3790 0.3740
0.3109
0.2930 0.28") 0.5744 0.2018
0,o 10 Cl.020 -0.020
__I___..___.
CO2
N2 --0.097 0.194 0.1955 -0.391 0.192 -0.192
0.196 -0.392
-0.100 0.200 -0.020
-0.480 -0.960
0.038 -0.038 -0.095 -0.190 - 0.022
-0.200
- 0.SOO
0.100 0.164 -0.164 -4.4 10
-1.000
-0.820
--1.000
0.0960
--0.050
0.1800
-0.100
0.110
(3.2355
-0.0012 0.00 12
0.120
0.0812
-0.120 -0.300
-0.0812 -0.203
0.003
n.ooK
0.600
0.100
-0.200 0.200
0.050
0.202 -0.202
0.200 -0.200 -0.500
0.200
-0.505
-0.200 -0.t500
-1.010
- 1.000
-0.406
.-
-0.200
0.020
0.022 0.055
0.3990 0.4670 il.5,too
total 0.100
0.200 -0.200 -0.500 1 .000
Table 11. RMS Error" (70) vs. Averaging Method for Flat-Plate Mass Transfer Problem -
-
__
linearization ____-____ methods hybrid method of
-.--_.
mdsb
_ I _ _
based
Stewart and Prober
1')
0.i
0.2
o.;i
film methods Stewart and Prober a, used a, used 4.78 4.78 5.02 6.06 .5.49 5 -16
0.8 0.9
6.03 6.66 7.35 8.10 8.91 9.75 10.6
i .O
11.6
0.4
0.5
_' -
0.6 I/.
6.03 6.71
7 1 0 of a vertical tube of infinite length. Components 2, ..., n of the gas mixture are noncondensable. At time t = 0, the gas is exposed to pure liquid species 1by removal of a barrier at y = 0, and mass transfer of species 1 occurs in the y direction. The interfacial gas-phase mole fraction xl0 is in equilibrium with the liquid, and the interface is maintained at position y = 0. The diffusion fluxes in the gas are described by the Stefan-Maxwell expression, with constant binary diffusivities Bij. The total concentration, c, in the gas is also considered constant. The molar total continuity equation gives
a
- -(EN,,) =0 aY i
into eq 18 and 19. The resulting expression for the dimensionless fluxes, Kio, of eq 23 is
[2
cir,]= P diag (g(Ai))P’(xo- xm)
- x . j Mj
for this system. Hence, cuy* is independent of y. The species continuity equations become
(35)
in which the left-hand expression denotes a column vector. Here
The Stefan-Maxwell equations for this spatially one-dimensional system are
a
Kjo
C-x = -A(x)J,* aY
SM, where
and hi is given in eq 26. Results from this model are also shown in Table 11. Approximate solutions to the problems in Table I were found by specifying the interfacial and bulk compositions and the total mass flux and then determining the individual fluxes. Property averaging was done for each method, using eq 12 or 13 to determine the reference composition. The interfacial derivatives, II’(O,O,K,A),were calculated by Lagrangian interpolation from the tables in Stewart and Prober (1962). Transformed variables log (n’),log (A), and -log (0.9 - K ) were used to enhance the accuracy of the interpolation. The results are given in Table I1 as functions of the averaging parameter, a, or a,. Linearization in mass units, eq 29-31, with a, = 0.40 gave the best approximation; the rms error norm of the calculated mass fluxes was only 0.55%. The linearization method of Stewart and Prober (1964) was slightly less accurate; the optimal a, of 0.50 produced a rms error of 0.93%. The film methods were significantly less accurate; the Stewart-Prober film method produced a rms error of 4.8%, and that of Krishna and Standart produced a rms error of 8.O%, even with optimal choices of the property-averaging constants. Unsteady-State Diffusion The binary problem of transient one-dimensional ordinary diffusion into a semiinfinite gas with constant c and ZIABwas solved by Arnold (1944); his solution is reviewed in section 19.1 of Bird et al. (1960). The small-flux limit of this solution was used by Higbie (1935) in his penetration theory of fluid-fluid mass transfer. The Arnold solution has been generalized to three-dimensional turbulent systems by Stewart et al. (1970), thus showing its
The initial and boundary conditions for the system are a t t < 0, for all y x = x, (42) a t t 1 0 ,y = 0 a t t 1 0, y = 0
x1 =
(43)
x10
i = 2, ..., n
Ni = 0
(44)
Let
Y = y/(4t)1/2
(45) (46)
Jy*
(4t)1/2 = -J,* C
(47)
Use of eq 45-41 reduces eq 39 and 40 to ordinary differential equations
dx
-= dY
-A (x)J y*
d dx -Jy* dY = (2Y - I?,)- dY
(49)
The boundary conditions are chosen as follows at Y = 0, x = x,
[ -,T2,,,:3
(50)
1 - x I ,o
at Y = o , J y * = rx
(51)
- X n - I ,o
thus reparametrizing the problem into initial-value form to shorten the computations. Exact solutions of eq 48-51
480
Ind. Eng. Chem. Fundam., Vol. 25, No. 4, 1986
Table 111. E x a c t Solutions t o One-Dimensional T r a n s i e n t Mass T r a n s f e r Problems
0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.010 0.040 0.100 0.200 0.333 0.t500 0.600
-0.800 000 -0.520 000 -0.360000 -0.240000 -0.160 000 -0.080 000 -0.040000 -0.020 000 -0.010 000 -0.002 000 0.004 076 0.016 456 0.041 96 0.086 98 0.152 96 0.249 0 0.317 2 0.3986 0.502 2 0.655 4
1 2 3 4 5 6 ? .
I
8 9 10 11 12 13 14 15 16 17 18 19 20
0.400 0.400 0.400 0.400 0.400 0.400 0.400 0.400 0.400 0.400 0.495 0.480 0.450 0.400 0.333 0.250 0.200 0.150 0.100 0.050
0.700 0.800 0.900
were computed by numerical integration, starting from Y = 0 and continuing until the logarithmic derivatives, d(ln x,)/d(ln Y),became negligible. These solutions are shown in Table 111. For comparison with the exact solutions, the linearization method was used to calculate the mass transfer rate of speces 1 for the same values of xo and x I m . The transformed compositions, f i (elements of P-lx),satisfy the uncoupled differential equations
= Y / ( m
(53)
c,, = rr/(Bf)1/2
(54)
Solutions to eq 52 are given by the analytic expression
-1, --
- erf
(T,
C,,/2) + erf (C,,/2) 1 + erf (CJ2)
0.06003 0.101 21 0.133 24 0.194 8 0.242 8 0.308 2 0.350 1 0.373 93 0.386 67 0.397 28 0.502 17 0.508 78 0.522 54 0.547 22 0.584 20 0.639 02 0.678 2 0.724 3 0.780 6 0.854 9
0.143 51 0.198 85 0.243 45 0.285 6 0.3188 0.356 8 0.377 7 0.388 70 0.394 31 0.398 86 0.497 83 0.491 22 0.477 46 0.452 78 0.415 80 0.360 98 0.321 8 0.275 7 0.219 4 0.145 1
Table IV. R M S E r r o r ( % ) i n T o t a l F l u x vs. Averaging Method f o r One-Dimensional T r a n s i e n t Mass T r a n s f e r Problem film methods linearization methods a
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
with VI
0.796 46 0.699 94 0.612 30 0.519 6 0.438 4 0.335 0 0.272 2 0.237 37 0.21902 0.203 86 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.400 0.400 0.400 0.400 0.400 0.400 0.400 0.400 0.400 0.400 0.495 0.480 0.450 0.400 0.333 0.260 0.200 0.150 0.100 0.050
molar units, a, used 3.89 3.19 2.56 2.04 1.69 1.60 1.81 2.22 2.74 3.31 3.91
mass units, a, used 14.4 12.6 10.9 9.28 7.70 6.23 4.94 4.03 3.80 4.42 b
and Prober, a, used 15.6 14.7 13.9 13.1 12.3 11.5 10.7 9.98 9.25 8.55 7.88
and Standart" 14.3 14.3 14.3 14.3 14.3 14.3 14.3 14.3 14.3 14.3 14.3
No variable properties appear in this method for these problems. Solution not converged.
-
(55) 1,-- 1 , o However, the terminal values of 1,are initially unknown (including Zfo, since P is to be calculated a t the reference state x, of eq 12). Evaluation of eq 7 at the interface using eq 55 yields an implicit expression for the mass transfer rate,
density p ( w ) by a constant and the function pu,(y,t) by p,uo(t). Taking D , = D,,, we get the approximate starting equation (j9)
for which a similar strategy of calculation can be used. The solution for the mass transfer rate is given implicitly by
in which
Rxl = &(I
+ erf (&,/T"*))
exp($,,'/T) =
(zii) - fia)EAV]" J
JIO* and the net fluxes, rameter, rx,by
4
&,, are related
(57)
to the total flux paand
T l 12
=
-c2
= ( T / 4 B L ) ~ / 2 r xi = 1,..., n
-
1
(58)
4 Li'i
The notation here is compatible with that of Bird et al. (1960) and Stewart et al. (1970). Results obtained by this method are compared with exact solutions in Table IV. A more severe test of the linearization is obtained by using mass units, thereby approximating the varying
T1/2
= --Cui
2
= ( ~ t / D ~ ) l / ~i u= ~1, ..., n
The quantities Cui are related to
-
1
r Xof eq 46 by
(62)
Ind. Eng. Chem. Fundam., Vol. 25, No. 4, 1986
481
Table V. Error ( % ) vs. Total Flux for One-Dimensional Transient Mass Transfer Problems film methods
linearization methods problem no. 1
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
rx -0.800 000 -0.520 000 -0.360 000 -0.240 000 -0.160 000 -0.080 000 -0.040 000 -0.020 000 -0.010000 -0.002 000 0.004 076 0.016 456 0.041 96 0.086 98 0.152 96 0.249 0 0.317 2 0.3986 0.502 2 0.655 4
norm average a
mass units, a, = 0.50 3.33 2.58 1.93 1.33 0.885 0.429 0.208 0.102 0.051 0.001 0.022 0.116 0.248 0.358 0.275 0.256 0.840 1.66 2.72 3.98
molar units, a, = 0.80 3.18 1.22 0.03 0.93 1.36 1.32 0.92 0.54 0.29 0.07 0.54 1.97 4.16 6.21 6.52 3.89 0.99 2.68 6.72 10.0
Stewart and Prober, a, = 1.00 18.9 13.1 9.28 6.12 3.95 1.84 0.87 0.42 0.20 0.03 0.05 0.22 0.57 1.21 2.23 4.00 5.54 7.75 11.3 18.1
Krishna and Standart” 19.2 13.3 9.17 5.46 2.62 0.59 2.36 3.29 3.77 4.17 6.11 6.53 7.40 8.98 11.41 15.2 18.1 21.8 26.9 35.4
1.60 1.07
3.80 2.68
7.88 5.28
14.3 11.1
No variable properties appear in this method for these problems.
Results of this method are also shown in Table IV. Application of the Krishna-Standart film theory requires the substitution of the low-flux mass transfer coefficients for K ~ ~ : (64) The derivation of the film theory equations assumes the fluxes, N, to be independent of position; this is not true in the present system. Krishna and Standart (1979) suggest using No in the expressions for 9 and 8. Since only one component is transferred a t the interface, eq 14-17 may be rearranged to yield
For a ternary system, this gives x1,
=
x 3 , = x30 exp(
5(
a)13
(68)
Interestingly, the binary diffusion coefficient a)23does not appear in these results, and all physical properties that appear in this method are constants for the stated problem. The Stewart-Prober film model requires the substitution
Manipulation of eq 11 yields an implicit expression identical with eq 56, except that
Here is again given by eq 58. All calculations were performed for gaseous mixtures of isobutane, N 1 , and H2 a t 270 K and moderate pressure. Binary diffusivities were calculated by using kinetic theory and range from 0.08 to 0.64 cm2s-l. The molecular weights vary from 58.12 to 2.016, and the density varies from 6.80 X to 2.43 X g~ m - ~ For . the exact calculations, xo and rxwere fixed and x, was calculated. For the approximate calculations, x, and xlmwere specified from the exact solutions and the insolubility conditions of eq 47 were imposed; then the remaining compositions, xim,were calculated by iterating on the total mass-flux parameter, r x . Twenty isobutane condensation and evaporation problems were solved, covering a wide range of compositions and mass transfer rates. The exact solutions are given in Table 111. Approximate solutions were found for a wide range of averaging parameters in eq 12 and 13. The rms error in r xfor all the problems is reported in Table IV as a function of the averaging parameter. ‘The best approximation was found using linearization in molar units; a, = 0.50 produced the minimum rms error of 1.60% in rx. Linearization in mass units works fairly well, although the density varies widely; a, = 0.80 gave a minimum norm of 3.80% for this set of problems. Film models are inappropriate for this transient system, and the numerical results are correspondingly less accurate. The Stewart-Prober film method gives a minimum norm of 7.88%, and the Krishna-Standart film method gives a norm of 14.3%. The errors for each method in each problem are shown in Table V. The approximate methods become less accurate a t large concentration differences. For the linearized methods, this is caused by departures from the constant-property assumption. For the film methods, the corrections for net mass flux become less accurate a t large net fluxes. In the limit of small concentration differences, the linearization methods and the Stewart-Prober film method are exact, as expected. The Krishna-Standart
482
Ind Eng. Chem. Fundam, Vol 25, No. 4, 1986
number-mean molecular weight of mixture, p / c column vector of molar fluxes relative t o stationary coordinates, mol cm-2 s-l column vector of mass fluxes relative to stationary coordinates, g cm-* s-l modal matrix of A or D, appropriate to given units material flux ratio in eq 57 or 61 Schmidt number, u,lBABor w o / D l time, s mass average velocity, cm molar average velocity, cm s-l column vector of mole fractions similarity coordinate for transient diffusion problem, cm s-' spatial coordinates, cm net flux parameter in transient diffusion problem,
mzthod is inexact even in this simple limit.
Conclusions Both the linearized matrix method and the film methods utilize binary functions to solve multicomponent mass transfer problems. F3r boundary layer systems, the linearized matrix method predicts the species fluxes well, even when the physical property variations are rather large. This is demonstrated here by comparing constant-density linearizations with exact computations for boundary layers where the density changes by as much as a factor of 2.2. This method can directly handle complicated geometries and flows. including turbulence and other time-dependent phenomena. The Krishpa-Standart (1976) film method is an analytical solution for steady mass transfer in one-dimensional stagnant films, subject to the Stefan-Maxwell diffusion law with constant binary coefficients DL,. Use of this method should be restricted to such conditions. The linearized film solution given by Stewart and Prober (1964) is a useful compromise. The mass transfer coefficients are optimally predicted in the limit of small concentration differences. The film model is used only to correct for the net flux. This method is well tested for stagnant 5lms (Smith and Taylor, 1983) and is suggested for situations not covered by detailed theory on net flux corrections. For information on these corrections, see Bird et al. (19601, Prober and Stewart (19631, Stewart (1963, 1971. 1973), Stewart and Prober (1962). and Stewart et al. (197C)
Acknowledgment This work was supported by Grant No. DE-FGOS84ER13291 of the U S . Department of Energy, Division of Chemical Sciences (Office of Basic Energy Sciences), and by an Amoco Foundation fellowship award to Thomas C Young.
ems'?
generalized Schmidt number, defined in eq 26 composition functim in binary flat-plate problems film theory matrix defined in eq 16 film theory matrix defined in eq 15, mol cm2 s film thickness, cm similarity coordinate for transient diffusion problem; see eq 53 molar transfer coefficient at low fluxes, defined in eq 17, mol cm ' s viscosity, g cm s ' kinematic viscosity, cm2 s-' dimensionless net flux, defined in eq 19, 37, 58, or 62
density, g cm-' column vector of mass fractions 0,
Subscripts a reference state for physical property evaluation chemical species in a binary system A, B chemical species or transformed quantities in a i. j , k multicomponent system T molar units iL' mass units 0 interfacial state bulk state c)
Nomenclature a
A ('
D ? I D l
I J* J ).* j
averaging parameter in eq 12 or 13 coefficient matrix in Stefan-Maxwell diffusion law. s cm-' total molar concentration! mol cm-3 A .-l, Fickian diffusivity matrix, em2 s-' binary diffusion coefficient, cm' SS] eigenvalues, identical for D , and D,,cm2 s-' identity matrix column vector of molar fluxes relative to v*, mol cm-?s-1
column vector of transformed fluxes defined in eq lty, mol cni-? 5.' column vector of mass fluxes relative to v , g cn-? 5-1
k,
column vector of mass-flux parameiers defined in eq 2 3 molar transfer coefficient at small net flux. mol cm-?
k,'
molar transfer coefficient corrected for net flux, mol
KO
SK1
k,
cm-2 mass transfer coefficient at small net flux, g cm-2
kd'
mass transfer coefficient corrected for net flux. g
M,
molecular weight of species i
5-1 cm-2
s-.
Superscripts transformed quantity as in eq 3-5 corrected for net material flux T transposed array
Literature Cited Arnold, J. H. Trans. A m . I n s t . Chem. Eng. 1944, 4 0 , 361-378. Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. Transport Phenomena; Wiley: New York, 1960. Higbic, R. Trans. Am. Inst. Chem. Eng. 1935, 3 7 , 365-389. Krishna, R.; Standart, G. L. AIChE J . 1976, 22, 383-389. Krishna, R.; Standart, G. L. Chem. Eng. Commun. 1979, 3 , 201-275. Prober. R. Ph.D. Thesis, University of Wisconsin at Madison. 1961 Prober. R ; Stewart. W. E l n t J Heat Mass Transfer 1963. 6 , 221--229, 872. Smith, L. W.; Taylor. R. Ind. Eng. Chem. Fundam. 1983. 2 2 , 97-104 Stewart, W. E AIChE J . 1963, 9 , 526-535. Stewart, W. E. l n t . J Heat Mass Transfer 1971, 74, 1013-1031 Stewart, W. E. AIChE J . 1973, 19, 398-400; 1979, 25, 208 Stewart, W. E. Ind. Eng. Chem. Fundam. 1984, 2 3 , 268. Stewart, W. E.: Anjelo, J. B.; Lightfoot, E. N. AIChEJ. 1970. 1 6 , 771-786. Stewart, W E.; Prober, R . I n t . J . Heat Mass Transfer 1962. 5 . 1149-1163; 1963, 6 . 872. Stewart. W. E.; Prober R. Ind. Eng. Chem. Fundam 1964, 3 . 224-235. Stewart, W. E.; Sorensen, J. P. Chem. Eng. Sci. 1974, 29. 811-817. 833-837. Toor, H. L. AIChE J . 1964, 1 0 , 448-455, 460-465.
Reteiced f u r r e i i e i b June 16, 1986 Accepted July 29, 1986
