Znd. Eng. Chem. Res. 1987,26, 2409-2413
2409
Solution of the Linearized Equations of Multicomponent Mass Transfer with Chemical Reaction and Convection for a Film Model Inci @entarlitand Amable Hortagsu* Department of Chemical Engineering, Bozazici University, Istanbul, Turkey
Solution of the coupled equations of multicomponent mass transfer with chemical reaction and convection for a film model is obtained in explicit matrix form. The previous attempts a t solving the equations of a film model were for solving the equations Nt d(x)/dz = C[D] d2(x)/dz2for the nonreacting mixtures or the equations [D] d2(x)/dz2 [k](x) = 0 for the reacting mixtures in which the convective term is neglected. I n this paper, the solution of the equations Nt d(x)/dz = C[D] d2(x)/dz2 C[k](x) is presented in its general form. The method developed also includes solutions for the cases when there is no reaction and when there is no convection. T h e solution obtained is used to calculate individual and total molar fluxes for gas-phase systems that involve homogeneous first-order reaction and higher order reaction with linearized kinetics a t various conditions.
+
+
Introduction and Review Multicomponent film models are of great importance in chemical engineering since they have been used to form the basis of design procedures for many separation processes. It was after Toor (1964) and Stewart and Prober's (1964) linearized theory that the multicomponent masstransfer equations have been represented by a set of coupled linear differential equations, and matrix techniques have been applied to obtain their solutions. The linearized continuity equations for an n + 1component system, in which Fick's law is used, for a film model in its general form are d(x) Nt - - C[D] dz dz2
-
d20+ (r)
Here, (x) is the mole fraction column matrix of dimension n X 1, [D] is the multicomponent diffusion coefficient square matrix of dimension n X n, (r) is the rate of reaction column matrix of dimension n X 1, Nt is the total molar flux,and C is the total molar concentration. By the generalization of Fick's law, the diffusive fluxes are
opposite to its own driving force, may not diffuse a t all even if it has a concentration gradient, or may diffuse in the absence of any driving force. Dual coupling also may cause unusual behavior depending on the relative magnitudes of the interaction (Toor, 1965). However, this diffusion-reaction coupling has not been treated extensively in the literature (Cussler, 1976). D,,i # j , of the [D] matrix in the linearized continuity equations represents the coupling effects of the multicomponent mixtures. These coefficients can be obtained experimentally or they can be derived from the simultaneous solution of the Maxwell-Stefan equations and the generalized Fick's law (Krishna and Standart, 1979). Cussler (1976) points out that the linearized multicomponent mass-transfer equations of a film model involving the reactive term can only be solved for the special case dz'
where the convective term is neglected. Solutions for eq 5 were obtained by Beek (1961), Wei (1962), Toor (1965), and Delancey and Chiang (1970). The continuity equations for a film model in the absence of reaction
The constituent molar fluxes (N) = (J) + Nt(x)
(3)
are composed of the diffusive and the convective terms, and the reactive fluxes for the first-order reactions are where [k] is the reaction rate constant matrix of dimension n X n. According to the linearized theory, the total concentration of the mixture and the matrix of multicomponent diffusion coefficients in these equations are assumed to be constant through the film. In multicomponent mass transfer with chemical reaction, dual coupling occurs: one is the reaction coupling depending on the local concentrations and the other is the diffusion coupling depending on the local concentration gradients (Toor, 1965). Possible consequences of diffusion couplings are that a species may transfer in a direction 'Present address: Department of Chemical Engineering,Yildiz University, Istanbul, Turkey. 0888-5885/87/2626-2409$01.50/0
were solved by Toor (1964) and Stewart and Prober (1964) through uncoupling and by Taylor (1982) without uncoupling, in terms of matrizants. The reason why the solution of the linearized equations including both reactive and convective terms for a film model has not yet been accomplished to date is that the [D] and [k] n x n matrices cannot be uncoupled simultaneously except for the commuting case which can hardly ever occur in reality. There is another approximate solution in the continuum approach that is based on the concept of the "effective diffusivity". One of the general effective diffusivity expressions is defined by Bird et al. (1960) and the other by Stewart (1954). In Bird's equation, the effective diffusivity is dependent on the molar transfer rates, and it can be calculated if the ratio of the molar fluxes are known or can be approximated. In Stewart's expression, for practical calculations, the relative mole fraction gradients may be replaced by the ratio of mole fraction differences. There are also simpler effective diffusivity formulas, one of which is used by Wilke (1950) which is a special case of Bird et 0 1987 American Chemical Society
2410 Ind. Eng. Chem. Res., Vol. 26, No. 12, 1987
al.'s equation. However, these simpler methods including the method of Burghardt and Krupiczka (1975) fail to account for diffusional interaction in multicomponent systems (Smith and Taylor, 1983). In addition to the continuum approach, there is also the molecular approach. In this second approach, the diffusion in multicomponent mixtures is accurately described by the Maxwell-Stefan equations. Exact analytical solutions of these equations for a unidirectional film model in the absence of a reaction were presented by Krishna and Standart (1976,1979). However, these solutions for the nonreacting case become inadequate when a reaction takes place throughout the film. The purpose of this paper is to present a method we developed for the solution of the linearized multicomponent mass-transfer equations with simultaneous chemical reaction and convection for a film model. In this method, a matrix variable which combines the mole fraction column matrix and its spatial derivatives is introduced. By this way, the second-order differential equations can be reduced into first-order differential equations. Then the model system solution is obtained in terms of the exponential functions of a 2n X 2n matrix.
Steady-State Unidirectional Mass Transfer with Chemical Reaction For steady-state unidirectional mass transfer with first-order reaction, the linearized continuity equations simplify to
Multiplying each term by the inverse matrix [D]-' and dividing by C, eq 7 becomes
Here, (C) is the matrix of constants determined from the boundary conditions 2
=0
(XI
2
=L
(x) =
= (x,) (XL)
Equation 15 becomes
(Y)= 3(C)
(17)
by defining the matrix function 3 as
3= (18) The evaluation of the matrix of constants is facilitated if the matrices 3 and (C) are expressed in the partitioned form 9 = ["i" 9 1 7 3 z i k ) Szz(2)
(C) =
(6>
(19)
If the boundary conditions are applied, eq 16 becomes
Equations 20 and 21 also satisfy
From this equation, the matrix of constants can be determined. Then the solution of the differential eq 13 becomes
where
Nt [A] = -[D]-' [B] = [D]-'[k] C Equation 8 can also be written in simpler form as (X) -
[A](X)+ [B](x) = 0
(9)
The constituent molar flux for the ith component is N , = J , + x,N, i = 1,2,...,n (24) where
(10)
Choosing a new variable matrix (Y) defined by
n
i = 1,2,...,n
J , = -C ED,,dx,/dz J=1
(11)
where
(25)
The concentration gradients are found from eq 23 to be dx,/dz = Y k i = 1,2,...,n k = n+l, ...,272 (26) and the total flux from
Yi= nk
i = 1,2,...,n
Yi=i,
i = n + l ,...,2n
k = 1,2,...,n k = 1 , 2 ,...,n
(12)
J=1
for an n + 1 component system. In terms of the new variable matrix (Y), eq 10 can be written as a first-order differential matrix equation d(Y)/dz = [Al(Y)
(13)
where the partitioned matrix [A] is
Since N,, C, and [D] are treated as constants, the solution of eq 13 (qentarli, 1985) is (Y) = e[*l2(C)
n
Nt = C N ,
(15)
(27)
The determinacy condition is the diffusion through a stagnant gas, N,,, = 0. For the no convection case (equimolar countertransfer), Nt = 0 is taken as the determinacy condition. Even though the method is derived with first-order reaction, it can also be used for higher order reactions by using the linearized kinetic expression for the (r) matrix.
Computational Method Since eq 23 for each component is implicit in N,, the individual fluxes and the total flux require trial and error calculations for the case of transfer through a stagnant gas. The computation algorithm for the solution described is as follows.
Ind. Eng. Chem. Res., Vol. 26, No. 12, 1987 2411 Table I. Constituent Molar Flux Calculations for Carbon Dioxide (1)-Water Vapor (2)-Hydrogen (3) Ternary %stema
boundary conditions 1 2 1 2 1 2
xn
xL
0.4029 0.0000 0.3366 0.3302 0.5333 0.0133
O.OOO0 0.3666 0.3300 0.3364 0.0000 0.0000
constituent molar fluxes, g-mol s-* (cm2)-' x IO5 this work lit.* 4.8730 -4.6590 0.0657 -0.0521 16.0900 0.4422
4.8600 -4.6500 0.0663 -0.0528 16.0900 0.4340
"Diffusion distance = 1 mm, temperature = 40 "C, pressure = 150 mmHg. Wilke, 1950.
*
Step 1. Calculate the interactive [D] matrix a t the mean compositions between the boundaries. Step 2. Estimate the total flux, Ne For the first iteration, take Nt = C c p cy=1 Dij(xio- xiL). For the subsequent iterations, ta& the Nt that is calculated by eq 27. Step 3. Calculate the partitioned matrix [A]. Then calculate function 3 by using its polynomial expansion (Bronson, 1970). Step 4. Calculate the concentration gradients at the boundary from eq 23. Step 5. Calculate Ji from eq 25, Ni from eq 24, Nt from eq 27. Step 6. Repeat from step 2 until Nt converges. The algorithm is used to calculate Ni for the transport with simultaneous convection and reaction. If the N;s for the case of no reaction are required, [k] matrix is equated to zero in the [A] matrix. For the no convection case (equimolar countertransfer), Nt is directly set equal to zero in the [A] matrix, and no iteration is required. By use of the proposed method of solution and the above algorithm, a single computer program is sufficient for the calculation of the various cases of mass transfer, i.e., transfer through a stagnant gas with or without reaction and equimolar counterdiffusion. For the trial and error calculations, rapid convergences were achieved by using the successive substitution method which is described above. Furthermore, there was no instability in all the examples tested throughout this work. Applications and Discussions To test the validity of the method of solution proposed here, calculations were first carried out for a system without chemical reaction. The results are shown in Table I. For the gas mixture C02-H20-H2, the calculations from
this work are in very good agreement with the results obtained by Wilke (1950) using the Maxwell-Stefan equations. Individual and total constituent fluxes for two reacting gas-phase systems, cyclopropane-propene-Ar-Ne and 12-H2-HI-Ar quaternary mixtures, were computed a t different conditions. In the cyclopropane (C)-propene (P)-Ar-Ne system, the first-order reaction (C P) takes place, and the rate expression is r, = kC,. For the 12H2-HI-Ar system, the reaction is second order and reversible (H2 + I2 e 2HI). Linearized kinetics was used as described by Delancey (1974) and Lee and Delancey (1974), and the linearized rate expression is r = kH2CH2+ ~I,CI,+ HIC CHI. Molar fluxes were first calculated for transfer through a stagnant gas by using the multicomponent diffusion coefficient matrix, [D]. To examine the diffusional interaction effects, the fluxes were recalculated by using the effective diffusivity formulas of Stewart (1954) and Burghardt and Krupiczka (1975). Enhancement factors (ratio of the mass-transfer rate with chemical reaction to the mass-transfer rate without reaction) were also determined for mass transfer with convection by using the [D] matrix. In addition, calculations for equimolar countertransfer (no convection) with the [D] matrix were carried out. As can be seen from Tables I1 and 111, for the cyclopropane-propene-Ar-Ne system, the molar fluxes calculated by using our method (method 1)and the effective diffusivity formula of Stewart (method 4) are in very good agreement. There is also good agreement in the respective fluxes for the 12-H2-HI-Ar system. However, the fluxes calculated from the effective diffusivity formula of Burghardt and Krupiczka (method 3) are smaller in Tables I1 and 111, always come out to be in the direction of the concentration gradients, and are generally higher in Table IV. It can be concluded that calculations based on methods 1 and 4 correctly account for diffusional interaction effects, whereas method 3 fails to account for this effect. The fact that lower values are predicted by method 3 for the cyclopropane-propene-Ar-Ne system show that the diffusional interaction effects are positive for this system. Similarly, higher values of fluxes from method 3, for the 12-H,-HI-Ar system, imply that the diffusional interaction effects are negative for this system. This type of behavior can also be recognized from the signs of the cross diffusion coefficients (Dij, i # j ) which are positive for the cyclopropane-propene-Ar-Ne system and mostly negative for the 12-H2-HI-Ar system. The effect of reaction as indicated by the enhancement factors are shown in the last columns of Tables 11-IV. In
-
Table 11. Constituent Molar Flux Calculations for Cyclopropane (1)-Propene ($)-Argon (3)-Neon (4) Quaternary System" constituent molar fluxes, g-mol s d (cm2)-' x lo6 boundary conditions comDonents Xn X I method le method 2d method 3O method 4f enhancement factorb 1 2 3 1 2 3 1 2 3
1 2 3
0.0500 0.2500 0.2500 0.1000 0.2000 0.1500 0.2000 0.1800 0.2600 0.1500 0.0500 0.2500
0.1000 0.1500 0.2000 0.0250 0.1500 0.2000 0.1850 0.1850 0.2550 0.1460 0.1000 0.1500
0.3474 4.2930 4.6180 1.7930 2.5280 0.2027 0.6616 0.3740 0.7486 1.1670 -0.2438 4.6060
-0.3164 2.4010 2.5280 1.5040 1.7340 -0.5843 0.3182 0.0482 0.2892 0.3487 -0.6511 3.4920
-0.3063 3.1180 2.8480 1.3800 1.2340 -1.4970 0.5263 -0.0766 0.3332 0.8802 -0.6314 4.4240
0.3183 4.2690 4.6030 1.7920 2.5280 0.1993 0.6619 0.3727 0.7482 1.1670 -0.2464 4.6040
1.0122 0.9990 1.0028 0.9980 1.0211 0.9646 1.0086 1.0432
s-l. For method 1. Method 1: with [D] matrix. " Diffusion distance = 0.2 cm, temperature = 823 K, pressure = 60 atm, k = 8.084 X dMethod 2: with [D] matrix, no convection. "Method 3: with Dieffof Burghardt and Krupiczka (1975). fMethod 4: with Dieffof Stewart (1954).
2412 Ind. Eng. Chem. Res., Vol. 26, No. 12, 1987 Table 111. Constituent Molar Flux Calculations for Cyclopropane (1)-Propene (2)-Argon (3)-Neon (4) Quaternary System" boundary conditions constituent molar fluxes. e-mol s-l (cm2)-' x io5 components XO XL method le method 2d method 3O method enhancement factorb 1 0.0500 0.1000 0.4125 -0.2888 -0.3401 0.3728 1.1246 2 0.2500 0.1500 4.5460 2.5170 3.4610 4.5180 0.9899 3 0.2500 0.2000 4.9350 2.7020 3.1620 4.9160 1 0.1000 0.0250 1.9710 1.6663 1.5320 1.9710 1.0314 2 0.2000 0.1500 2.6470 1.7980 1.3700 2.6450 0.9778 3 0.1500 0.2000 0.2167 -0.6244 -1.6620 0.2128 1 0.2000 0.1850 0.8484 0.4849 0.5843 0.8474 1.2253 2 0.1800 0.1850 0.2583 -0.0933 -0.0851 0.2472 0.6234 3 0.2600 0.2550 0.8001 0.3090 0.3699 0.7958 1 0.1500 0.1460 1.3520 0.4834 0.9771 1.3580 1.0930 2 0.0500 0.1000 -0.3649 -0.8066 -0.7009 -0.3657 1.4607 3 0.2500 0.1500 4.9230 3.7320 4.9110 4.9220 aDiffusion distance = 0.2 cm, temperature = 865 K, pressure = 70 atm, k = 0.0840 s+. bFor method 1. "Method 1: with [D] matrix. dMethod 2: with [D] matrix, no convection. eMethod 3: with BLeff of Burghardt and Krupiczka (1975). 'Method 4: with Breff of Stewart (1954).
Table IV. Constituent Molar Flux Calculations for Iodine (1)-Hydrogen (2)-Hydrogen Iodide (3)-Argon (4) Quaternary Systemn boundary conditions constituent molar fluxes, g-mol s-l (cm2)-' x io5 comDonents Xn XI method le method 2d method 3e method 4' enhancement factorb 0.1242 0.608 0.188 1 0.1455 0.681 0.627 1.1760 2 0.1200 0.1001 1.561 1.219 1.561 2.139 1.0714 1.031 0.124 1.237 3 0.3050 0.2498 1.126 0.8423 0.273 0.214 0.382 1 0.0500 0.0000 0.271 1.1143 3.084 2.940 3.034 2.935 1.0095 2 0.0800 0.0500 -1.159 -1.659 -1.039 1.0517 3 0.1800 0.2800 -0.499 0.310 1.151 0.763 1 0.1000 0.0250 0.730 1.1947 5.796 4.684 6.160 2 0.2000 0.1500 5.810 1.0253 -0.236 3 0.1500 0.2000 -1.318 0.556 -0.238 -10.7272 0.502 -0.210 1.334 0.520 1.2947 1 0.1500 0.1200 4.981 4.309 5.351 4.982 1.0291 2 0.1500 0.1000 0.1700 -0.114 -0.996 0.766 3 0.1500 -0.135 -0.8347 "Diffusion distance = 0.2 cm, temperature = 781 K, pressure = 60 atm, k, = 1.537 X lo3 cm3/(g-mol s), kb = 38.68 cm3/(gmol s). bFor method 1. Method 1: with [D] matrix. Method 2: with [D] matrix, no convection. e Method 3: with Qeffof Burghardt and Krupiczka (1975). 'Method 4: with Bo,,,of Stewart (1954).
the examples studied, the enhancement factors are always greater than 1for the reactants and in most cases less than 1 for the products. Enhancement factors greater than 1 are obtained for the products when the direction of transfer of the species is in the opposite direction of the total molar flux. Negative enhancement factors in the last two examples in Table IV arise from the ratios of two molar fluxes which are in opposing directions. In these examples, calculations showed that the products exhibit reverse diffusion for the no reaction case. Dual diffusion-reaction coupling seems to cause a new direction in the mass transfer. As to be expected, the system 12-H2-HI-Ar exhibits reaction coupling effect since the reaction mechanism is second order and reversible with high reaction rate constants. On the contrary, the system cyclopropane-propene-Ar-Ne may not exhibit reaction coupling since the reaction is irreversible, first order with lower reaction rate. The molar fluxes calculated for transfer through a stagnant gas (method 1) and the fluxes for equimolar countertransfer (method 2) have large differences bath in magnitude and direction. Therefore, the assumption of equimolar countertransfer in multicomponent masstransfer design calculations must be taken with care specially for nondilute systems. In order to understand the dual diffusion-reaction coupling phenomena better, more calculations of reacting systems must be conducted. The method proposed here can be used for as many systems as possible as long as kinetic and diffusivity data are available. Of course, the
limitations of the model due to the assumptions of linearized diffusion and linearized kinetics should be taken into consideration. The systems chosen in this work involve constant molar total flux over the diffusion path. If a reactive system with varying total molar flux is chosen, mass-average velocity must be used since total mass flux is always constant as far as film models are concerned.
Acknowledgment Publication costs of this work were partly covered by a grant from BogaziGi University Research Foundation.
Nomenclature [A] = square matrix defined by eq 9 of dimension n X n [B] = square matrix defined by eq 9 of dimension n X n C- = total molar concentration (C) = column matrix of constants of dimension 2n X 1 [D]= multicomponent diffusivity matrix of dimension n x n aieff = effective diffusivity coefficient of species i 3 = partitioned matrix defined by eq 18 of dimension 2n x 2n [I] = square unity matrix of dimension n X n J, = diffusion flux of species i (J) = column matrix with elements J, of dimension n X 1 [k]= reaction rate constant matrix of dimension n X n k , = forward reaction rate constant k b = backward reaction rate constant L = film thickness N , = molar flux of species i
Ind. Eng. Chem. Res. 1987,26, 2413-2419
(N)= column matrix with elements N ; of dimension n x
fit'=total molar flux
1
ri = reaction rate of species i (r) = column matrix with elements ri of dimension n X 1 xi = mole fraction of species i (x) = column matrix with elements xi of dimension n X 1 (Y)= column matrix defined by eq 11 of dimension 2n X 1 z = distance across film [O] = square null matrix of dimension n X n Greek Symbols [A] = partitioned matrix defined by eq 14 of dimension 2n X 2n
C = summation over index Superscripts ( 2 ) = first derivative of (x)
(a) = second derivative of (x)
Subscripts
i, j , k = index indicating component number n = total number of components less one Matrix Notation
[I
= square matrix
( ) = column matrix [ I-' = inverse of square
matrix
2413
Literature Cited Beek, J. AZChE J. 1961,7(2), 337. Bird, R.B.; Stewart, W. E.; Lightfoot, E. N. Transport Phenomena; Wiley; New York, 1960;p 571. Bronson, R. Matrix Methods-An Introduction; Academic: New York, 1970. Burghardt, A.; Krupiczka, R. Znz. Chem. 1975,5,487,717. Cussler, E. L. Multicomponent Diffusion; Elsevier: Amsterdam, 1976;Chapter 9. Delancey, G. B. Chem. Eng. Sci. 1974,29,2315. Delancey, G.B.; Chiang, S. H. Ind. Eng. Chem. Fundam. 1970,9(3),
344. Krishna, R.; Standart, G. L. AZChE J. 1976,22,383. Krishna, R.;Standart, G. L. Chem. Eng. Commun. 1979, 3, 201. Lee, S. T.; Delancey, G. B. Chem. Eng. Sci. 1974,29,2325. gentarli, I. Master Thesis, Boiaziqi University a t Istanbul, 1985. Smith, L. W.; Taylor, R. Znd. Eng. Chem. Fundam. 1983,22(1),97. Stewart, W. E. NACA Technical Note 3208, 1954. Stewart, W. E.;Prober, R. Znd. Eng. Chem. Fundam. 1964,3(3),224. Taylor, R.Ind. Eng. Chem. Fundam. 1982,21(4),407. Toor, H.L. AZChE J. 1964,10(4),460. Toor, H.L. Chem. Eng. Sci. 1965,20,941. Wei, J. J. Catal. 1962,1, 526. Wilke, C. R. Chem. Eng. Prog. 1950,46(2),95.
Received for review September 2, 1986 Accepted August 3, 1987
On the Liquid Flow Distribution in Trickle-Bed Reactors R. 0. Fox Department of Chemical Engineering, Kansas State University, Munhattan, Kansas 66506
The previous analytical expression for the liquid flow distribution in trickle beds found by maximizing t h e so-called configurational entropy is reviewed, and a novel derivation is offered, more closely modeling data found in the literature. The novel distribution is a n analytical expression describing the liquid flow distribution recently found by employing a Monte Carlo simulation of the very large lattice model (VLLM), thus reducing this computationally time-consuming procedure t o a simple equation. The resultant distribution is a function of the total flow rate through the trickle bed, the bed permeability as expressed by the number of open channels through the bed, and the minimum liquid flow rate through a given channel. 1. Introduction The objective of this paper is to review the current theory describing the liquid flow distribution in trickle-bed reactors based on an entropy maximization procedure (Crine and L'Homme, 1984;Crine and Marchot, 1983) and to offer a novel theoretical derivation of the same distribution. The latter better describes the recent experimental findings of Ahtchi-Aliand Pedenen (1986). These authors developed the very large lattice model (VLLM) and conducted Monte Carlo simulations to study the resultant liquid flow distribution. The novel distribution presented in this work reduces these time-consuming Monte Carlo simulations to a simple analytical expression. The liquid flow distribution in a trickle-bed reactor determines the degree of irrigation of the packing and, in turn, can have a considerable effect on the overall performance. To date, most design procedures for these reactors rely on empirical models which are difficult to apply with any certainty to novel operating conditions. It is therefore of some interest for the advancement of the design of trickle-bed reactors to develop a priori theoretical models which relate the behavior of the system a t the level of a single packing element to the overall performance. The theory proposed by Crine and Marchot (1981a,b) based on percolation theory is an attempt to model the 0888-5885/87/2626-2413$01.50/0
trickle bed starting on the level of a single particle; these results are then averaged over the entire trickle bed by using the liquid flow distribution (also see Crine and L'Homme (1984)). They proceed by calculating local transport properties which are functions of the local liquid flow rate. These local properties are averaged over the entire trickle bed by using the liquid flow distribution derived by maximizing the "configurational entropy" of the system. The liquid flow distribution is thus obviously an important quantity in this theory since it determines the "average" behavior of the trickle bed. In the second section of this paper, the derivation presented by Crine and Marchot (1983) of the liquid flow distribution is critically reviewed. The resultant distribution is in the form of a classical exponential distribution, a decreasing function of the flow rate. Recently, Ahtchi-Ali and Pedersen (1986) conducted an experimental investigation of the liquid flow distribution and found that the exponential form did not correspond to their findings. The experimental distributions exhibited a relative maximum which cannot be modeled by using a liquid flow distribution that is a strictly decreasing function of the flow rate. It is possible, however, to derive an a priori liquid flow distribution exhibiting a relative maximum; this is the topic of the third section. In the final section, the two 0 1987 American Chemical Society
