Ind. Eng. Chem. Res. 1993,32, 2159-2169
2159
Enhancement of Film Mass Transfer by Forced Convective Flow Perpendicular to a Heterogeneous Surface Zuping Lu, Douglas D. Frey,+and Alirio E. Rodrigues’ Laboratory of Separation and Reaction Engineering, School 4099 Porto Codex, Portugal
of
Engineering, University of Porto,
The enhancement of intraparticle mass transport by intraparticle forced convective flow has been recognized in numerous past studies when “large-pore” materials are used in the presence of large pressure drops. However, the enhancement of film mass transfer by forced convective flow perpendicular to the “heterogeneous” surface has not been previously studied. In this study, a “unit cell” model is used which describes the steady-state mass transfer in the concentration boundary layer for the case where forced convective flows parallel and perpendicular to a heterogeneous surface are present. The perpendicular flow field in the momentum boundary layer is determined using the Navier-Stokes and continuity equations. The mass balance equations are solved by the method of collocation on finite elements. The minimum enhancement of film mass transport to the heterogeneous surface, measured by the Sherwood number ( ( S h )= k/k*) due to the forced convective flow perpendicular to the surface, given by the Peclet number, Pe,, is assessed. When Pe, < 0.5, (Sh)jsnearly independent of Pe,, and when Pe, is larger than 5, ( S h ) = Pe,/2, i.e., k = v0/2, so that D = vo6/2. Mass transport from a fluid phase to a permeable heterogeneous surface, such as a polymeric membrane, adsorbent, catalyst, a support for mammalian cell culture, or a large-pore HPLC packing, can partly control the overall mass transfer process. In many cases, the rate of mass transfer to a heterogeneous surface can be significantly less than to a homogeneous surface. In general, the film mass transfer coefficient for a heterogeneous surface (k)can be defined as
klk* = f@) (1) where k* is the film mass transfer coefficient for a homogeneous surface and f@) is a correction function in which p refers to various parameters such as Reynolds and Schmidt numbers and others describing the system geometry (Herskowitz et al., 1979;Dudukovic and Mills, 1985; Juhasz and Deen, 1991). The limits off@) can be obtained by considering the mass transport to partially impermeable or nonadsorbable surfaces (Dudukovic and Mills, 1985). If the thickness of the concentration boundary layer is much smaller than the space between active sites (pores),then f@) = t, where t is the area fraction of active sites (i.e., the particle porosity) of a heterogeneous surface. If the thickness of the concentration boundary layer is much larger than the space between the active sites, then f@) = 1,which implies that the mass transport behavior of heterogeneous and homogeneous surfaces is similar. A single symmetry element or “unit cell” model was used to determinethe mass transfer rate to a heterogeneous surface for a stagnant fluid boundary layer by Keller and Stein (1967),Wakeham and Mason (19791,andDudukovic and Mills (1985). The effect of a nonuniform spatial distribution of active sites was also considered by Brunn (1984). Recently, the flow parallel to the surface in the concentration boundary layer was included in the “unit cell” model by Juhasz and Deen (1991). They concluded that when the local Peclet number Pe, is in the range
* To whom correspondence should be addressed, e-mail, arodriga fe.up.pt. t Yale University, Department of Chemical Engineering, New Haven, CT 06520-2159.
between 10and lo2,and for practical ratios of the thickness of concentration boundary layer to the space between active sites, mass transfer to a heterogeneous surface can be strongly influenced by convective transport between active sites. A brief review of this topic was also made by Juhasz and Deen (1991). For porous materials containing large pores (i.e., pore sizes above 1000 A) in the presence of large pressure gradients, relations describing the enhancement of the intraparticle diffusivity by convection were derived by Rodrigues et al. (1982) and can be expressed as follows:
where deis the “augmented” effective diffusivity and X = u,l/De is the intraparticle Peclet number relating the intraparticle convective velocity uo,for a particle with slab geometry with half-thickness 1 and intraparticle effective diffusivity D,. The effect of intraparticle convection on the performance of chromatographic processes can be described by the extended Van Deemter equation derived by Rodrigues et al. (1991): HETP = A + B/u+ Cf(X)u (3) where HETP denotes the height equivalentto a theoretical plate and u is the superficial velocity. The enhancement of film mass transfer by intraparticle forced convective flow perpendicular to the surface has been observed by Lu et al. (1992a). This effect has not been considered in previous studies of the systems where forced convection exists such as in membrane systems (Chaara and Noble, 1989; Brites and Pinho, 1991). The objectives of this study are, therefore, as follows: (i) Derive the velocity field in the momentum boundary layer and develop a mathematical model describing the steady-state film mass transport in the concentration boundary layer when forced convective flow parallel to a heterogeneous surface (Juhasz and Deen, 1991)and forced convective flow perpendicular to the surface through the active sites (pores) are taken into account. (ii) Solve the two-dimensional problem described in (i) by the numerical method of orthogonalcollocation on fiiite elements (Mills and Dudukovic, 1985; Lu et al., 1992a).
0888-5885/93/2632-2159$04.00/00 1993 American Chemical Society
2160 Ind. Eng. Chem. Res., Vol. 32, No. 9, 1993
inactive
active(pore)
inactive
0
0.2
0.4
0.6
0.8
0.6
0.8
1
5
&=OS, p=1 1
l-
I
C
0 .-c, 0
.-2
I
U inactive
active (pore)
inactive
E
Figure 1. Sketch of the “unit cell” (a) with a perpendicular flow entering the active site and (b) with a perpendicular flow leaving the active site. Only the uw component of the velocity due to perpendicular flow is shown.
(iii) Determine the effect on the film mass transfer coefficient of the forced convective flow through the permeable heterogeneous surface under various conditions.
Mathematical Model The two-dimensional mass transfer problem in the boundary layer at steady state considered here, as shown in Figure l a and lb, is similar to that considered by Dudukovic and Mills (1985) and Juhasz and Deen (1991). The heterogeneous surface contains active sites of width 2a in a “unit cell” of width 2b. A fluid boundary layer (the same thickness for the momentum and concentration boundary layers is assumed, see Juhasz and Deen, 1991) is adjacent to the surface. A constant bulk fluid solute concentration co is assumed. At the active sites, the solute concentration has a constant value cs. The bulk flow parallel to the surface is assumed to lead to a parabolic velocity profile in the boundary layer (Juhasz and Deen, 1991): u,, = i u 0 [ l - (1- y/6)2]
(4)
where uo is the average velocity in the concentration boundary layer and 6 is the thickness of the concentration boundary layer. When the forced convective flow perpendicular to the surface is included in the model, a number of complications arise. Consider a permeable particle with slab geometry in a fluid with a forced convectiveflow through the particle. Two concentration boundary layers are present adjacent to either side of the slab. In one concentration boundary layer the perpendicular flow is entering the heterogeneous surface (pore mouth) as shown in Figure la, and in the
q::I 0.4
0.2
00
0.2
&=OS,p=1
0.4
,
r
Figure 2. Stream lines in the boundary layer in absence of the bulk parallel flow (e = 0.5, fi = 1): (a, top) fluid entering the active site, (b,bottom) fluid leaving the active site.
other the perpendicular flow is leaving the surface (Figure lb). If the forced perpendicular convective flow has the same direction as the mass transport, it increases the transport rate; otherwise, it decreases the transport rate. The total effect of the forced perpendicular flowto a porous particle on the mass transfer is the sum of the effects on the two sides of the particle. Velocity Field in the Presence of Perpendicular Flow. The velocity field in the presence of the perpendicular flow in the boundary layer can be derived from Naviel-Stokes and continuity equations. The converging/ diverging nature of the perpendicular flow in the boundary layer will cause a second parallel flow uyxwhich will be directly added to the forced parallel flow given by eq 4. Since the perpendicular flow uyyis symmetric around the center of the unit cell, i.e., uyy(-x)= uyy(x),and the second parallel flow is an odd function relative to the x coordinate, i.e., uyx(-x)= -uy,(x), only half of the unit cell is considered in the followingdevelopment for the case of no bulk parallel flow. “Dead” flow regions where uyy= 0 and uyx= 0 are assumed as shown in Figure 2. The two-dimensional steady-state Navier-Stokes equation (Happel and Brenner, 1966; Churchill, 19881, can be written in dimensionless form, neglectingthe inertial term, as
where the function R, is the dimensionless vorticity. R,
Ind. Eng. Chem. Res., Vol. 32, No. 9,1993 2161 can be written in terms of the dimensionless stream function, q,as is given by The boundary conditions to eq 2 are f=O,
9=0
f = to, q = q,, 9 = f l , and d q = 0
.
(6a)
.
.
cosh xfo-e
e@*
2(1- e)
poutKo,~o~ = sin 21-e -cash ;
‘lo-
1
=O
(6b)
In the above equations,the dimensionlessspace variables are defined as = x l b and q = y16, while the parameters e and @ are defined as alb and 6/a, respectively. 9 = 1 corresponds to the fluid leaving the active site, while 9 = -1 corresponds to the fluid entering the active site. The line separating the “dead” and flow regions is denoted by p(fo,qo)= 0; Le., if f 1 fo and q I qo, then 9 = constant. The velocity field is given by
r
The stream functions 9 from eq 10are plotted in Figure 2a and 2b, respectively. It can be seen that the stream lines are in agreement with physical intuition.
Development of the Mass Transfer Model Equation in Boundary Layer. The dimensionless steady-state mass transport equation in the boundary layer, when both parallel and perpendicular flows are present, is given by
(7)
and
with the boundary conditions 1a3
uyxIvo= - --
€0
The simplest expression for the vorticity which satisfies eq 5 is
n,
= *(4I2h
(9)
+
where denotes flow entering and - denotes flow leaving the active sites (Figure 1). The general solution of eqs 6 and 9 contains an infinite series with unknown coefficients al, and azn. Since we have only the boundary conditions given by eqs 6a and 6b, we used only the first two terms of the infinite series. A particular solution can then be obtained as
s i n % ‘ + 2f: sinh[ g(q- qO)] s i n $ ‘ ] (10) 2sb The velocities uyyand u,, can be obtained from eqs 7 and 8. The line function pb(f0,qo) which passes through the points P1(1,1)andPz(e,O)and which satisfies the conditions qO)]
is given by +
z 1-e
cash-( l - q o ) - 1 2(1- e) =O cash -- 1 2(1- e) (11)
Similarly, the line function pout(co,q0) which passes through the points Pl(1,l) and Pz(e,O)and which satisfies the conditions
In eq 13, f is the dimensionless concentration (c - c,)/(co - cs), and g ( q ) = u,,/uo is evaluated from eq 4 as g ( d = (312)[1- (1-
(14)
The functions h,(f,q) = uyxlvoand h,(f,q) = u,,/vo can be obtained from eqs 7 and 8. The Peclet numbers for parallel and perpendicular flow are Pe, = uo6/D
(15)
Pe, = XIBi, = vo6/D
(16)
where D is the molecular diffusivity, Bi, = k*l/D, is the Biot number for the “homogeneous”surface, and k* = D/6 is the film mass transfer coefficient for a homogeneous surface. The local film mass transfer coefficient is defined as
and the average Sherwood number is
2162 Ind. Eng. Chem. Res., Vol. 32, No. 9, 1993 1.o
0.8
cSh> 0.6
0.2
.dl
.1
10
1
where
100
P (19)
Using eq 18, (Sh) can be evaluated at q = 0 and q = 1 to yield (Sh)o and (Sh)l, respectively:
From previous work (Lu et al., 1992a), it is known that when mass transfer resistances are present both inside the particle and in the film and when intraparticle convection is important, the concentrations at both surfaces of particles are different, i.e., #,(in) # cb,(out). Only in the limiting case of no mass transfer resistance in the particle or in the film are they equal. If mass transport (adsorption) is from the fluid to the particles, i.e., co > c, it follows that c,(in) > c,(out), so dJ,(in) > $,(out). If mass transport is from the particle to the fluid (desorption), i.e., co < cg, it follows that c,(in) < c,(out) and again @,(in) > r$,(out) since they are both negative. However, &(in) and r$,(out) are usually very close in magnitude; i.e., the third term on the right side of eqs 20 and 21 is much smaller than the first two terms. This is the case for a porous particle in a fixed-bed adsorber or catalytic reactor, in which the concentration change across a particle is small. More generally, the assumption @,(in)= &(out) is reasonable in a film mass transfer controlled operation. Therefore, the minimum enhancement of the film mass tranefer by the forced perpendicular convectiveflow occurs when the concentrations at both sides of the particles are the same, in which case eqs 20 and 21 become
Figure 3. Sherwood number (Sh)as a function of j3 for different c (e = 0.2,0.5,0.8).Comparison between different types of boundary conditions at active sites in a stagnant film, i.e., Pe, = 0 and Pe, = 0 (-) constant concentration, (- - -) constant mass transfer flux.
.1
1
10
100
1000
10000
100
1000
10000
PeX
1
10
PeX Figure 4. Comparison between constant concentration (full lines) and constant mass transfer flux (dotted lines) at active site without perpendicular forced convective flow Pe, = 0 (a) Sherwood number ( S h )asafunctionofPe,fordifferentc (t =0.2,0.5,0.8),(b)Sherwood number ( S h ) as a function of Pe, for different B (,9 = 0.1, 1, 10).
of the membrane are different; i.e., the third term on the right side of eqs 20 and 21 should be defined in these cases (see the example in the Appendix). In the remainder 01 this study, the Sh&wood number (Sh) will be calculated from eq 22. For the cases of the diffusion-convection cell (Cresswell, 1985) and of membrane separations (Bohrer, 1983; Chaara and Noble, 1989; Brites and Pinho, 19911, the effect of the perpendicular flow on film mass transfer should be determined simultaneously with the mass transfer processes occurring inside the cell or membrane to determine c,(in) and c,(out) individually since the bulk fluid compositions at both sides
Numerical Method Equation 16 constitutes a boundary value problem with discontinuous boundary conditions. These types of problems have been studied by Mills and Dudukovic (1980) and Mills et al. (1985), who found that the use of finite difference schemes to solve discontinuous boundary value problems often requires an extremely fine mesh size. They also found that these schemes are frequently inaccurate
Ind. Eng. Chem. Res., Vol. 32, No. 9,1993 2163
1 .o
0.8
f (5
f cs,rl>o.6 0.4
0.2 0.0
1 .o’o.o 1 .a
1 .o
0.8
0.8
rl
0.4
.
1
0.8
0.6
0.4 -’
0.2
0.2
0.0 -1.0
1
0.6 I
0.6
rl
1
1.07
-
0.01
-0.5
0.u
s
0.5
1.o
&=OS,p=1 ,Pe,=O, Pq=O
-1.0
I
1 0 1
1
-0.5
0.0
I
0.5
I
1.0
s
Pe,=100 ~ 4 . 5 p=1, , Pe,=O
Figure 5. Concentrationdistribution in the film without DerDendicdar forced convective flow Pe, = 0 (t = 0.5, f l = 1) in the stagnant boundary layer (a) Per = 0 and with a bulk parallel flow (b)Per =- lob.
and, in general, are much more inefficient than other techniques, such as weighted residual methods (least squares, Galerkin and collocation). The latter methods provide a systematic approach to the solution of discontinuous boundary value problems of any geometry. Davis (1984) compared the solutions of a boundary value PDE problem by using PDECOL (collocation, B-spline basis) and DISPL (Galerkin, B-spline basis) and found them to be in good agreement. The method of collocation on finite elements is useful for solving the boundary value problems (Finlayson, 1980; Leitao and Rodrigues, 1990; Sereno et al., 1991; Lu et al., 1992a). In this paper, the discontinuous boundary condition problem at steady state was solved by approaching the steady state from the transient solution; i.e., the term cu(r3flar) was added to the right side of eq 16, where 7 is dimensionless time and a = 0.01. The procedure used was as follows: Model equations were first reduced to a one-dimension problem in q by using collocation on finite elements in the f coordinate with 10 elements after first transforming to a new spatial coordinate defined by y = (I+ W2. Then, the resulting PDEs were solved by the PDECOL package using 10 elements in the q coordinate. The simulation results were tested in several cases by using more than 10 elements in the t and q directions. The details of the numerical method can be found elsewhere (Finlayson, 1980; Lu et al., 1992a). As a test of whether or not steady state was achieved, eq 18 was verified, i.e.
and the sum of the simulation values at different times in
several typical positions was required also to be less than 106, Usually the dimensionless times needed were between 0.2 and 0.3. The computing times were 30-40 s in an IBM RS/6000-530 computer for different conditions. Results and Discussions In the model equations just presented, only the version for a constant concentration boundary condition a t the active site was considered since the simulation results can easily be checked by eq 18. However, Juhasz and Deen (1991) employed a Galerkin finite element method with bilinear basis function and found that (Sh) was relatively insensitive to the type of boundary condition. In particular, they found that the difference in (Sh) calculated by using constant flux and constant concentration boundary conditions at the pore mouth was 8-9%. Similar results were obtained by Keller and Stein (1967). For comparison, we used the constant mass transfer fluxboundary condition in several calculations, in which case eqs 13c and 13d become
af/aq = -1,
= 0,
f=O,
q=1
Ie;
af/aq = 0,
(23) = 0,
Id
5 , (Sh) is almost linearly related to Pe,, Le., (Sh) = Pe,/2. This result can also be derived analytically from the model equations with some simplifications (see the-Appendix). (Sh) = Pe,/2 also implies that k = v,/2, i.e., D = v,6/2 at high Pey. The augmented intraparticle effectivediffusivity due to convection is given by I), = V,1/3 at high intraparticle Peclet numbers, as obtained by Rodrigues et al. (1992). The different coefficient for the two situations (1/2 and 113)is due to the different definition of mass transfer film in the two cases which leads to more efficient mass transport by the convective flow in the film than in the porous materials. The flow parallel to the surface minimizes the effect of the parameters j3 and e at small Pey. Among all parameters, it is Pe, that most affects (Sh) and explains why (Sh) can be higher than unity.
Conclusions The velocity field in the presence of both parallel and perpendicular flows in the boundary layer adjacent to a porous surface is derived from Navier-Stokes and continuity equations. An extension of the 'unit cell" model used to describe mass transfer in the concentration boundary layer considering the forced convective flows parallel and perpendicular to the heterogeneous surface was derived, and model equations were solved by the method of collocation on finite elements. The effect of the parameters Pe,, Pe,, e, and j3, where only Pe, and e are independent parameters, on the Sherwood number (Sh) was discussed. Sherwood number is insensitive to the type of boundary conditions, constant concentration, or constant mass transfer flux at the active site for the case of a stagnant film. It is sensitive to the type of boundary conditions for small e and j3 when convection is present. The Sherwood number (Sh) can be enhanced significantly by the forced convective parallel flow to a heterogeneous surface; however, it can never be larger than unity without the forced convective flow perpendicular to the surface. The minimum enhancement of Sherwood number by the forced perpendicular convective flow, i.e., perpendicular Peclet number (Pe,), was determined and was found not to depend on the perpendicular flow profiles. When Pe, < 0.5, (Sh) is nearly independent of Pe,. When Pe, > 5, (Sh) approaches Pe,/2, iLe.,I) = v,6/2. This result is equivalent to the result De = V01/3 found in the intraparticle diffusion/convection, in the convectioncontrolled region (Rodrigues et al., 1991). The Sherwood number can be larger than unity in the presence of the forced perpendicular flow; Le., the mass transfer coefficient to a heterogeneous surface with forced perpendicular flow to it can be higher than that to a homogeneous surface in the absence of perpendicular flow.
Ind. Eng. Chem. Res., Vol. 32, No. 9,1993 2167 De = effective intraparticle diffusivity, cm2/s b = "augmented" molecular diffusivity, cm2/s be= "augmented" effective diffusivity, cm2/s f = dimensionless concentration in the boundary layer k = film mass transfer coefficientto a heterogeneous surface, cm/s k* = film mass transfer coefficient to a homogeneous surface without perpendicular flow, cmjs K = 'apparent" film mass transfer coefficient, cm/s N, = mass transfer rate, mol/cm% Pe, = Peclet number parallel to the surface (= u,S/D) Pe, = Peclet number vertical to the surface (= v06/D= X/Bi,) Sh = Sherwood number (= k6/D = k/k*) uo = average velocity parallel to the surface in the boundary layer , cm/s u,, = velocity parallel to the surface in the boundary layer due to bulk parallel flow, cm/s uyx= velocity parallel to the surface in the boundary layer due to converging and diverging perpendicular flow, cm/s uyy= velocity perpendicular to the surface in the boundary layer, cm/s uo = average velocity perpendicular to the surface in the boundary layer, cmis r = space coordinate parallel to the surface, cm y = space coordinate perpendicular to the surface, cm Greek Symbols @ = ratio between film thickness and pore 6 = film thickness, cm e = the fraction of the active site (= alb)
radii (6/a)
6, = dimensionless concentration value stated by eq 26 y = dimensionless space coordinate q, qo = dimensionless space coordinate 1 = half thickness of slab geometry, cm X = intraparticle Peclet number (= u,l/D,) 4 = the dimensionless vorticity
t, to= dimensionless space coordinate
\k = the
dimensionless stream function
Appendix From the results reported above, at large Pe,, ( S h )does not depend on flow parallel to the surface and the geometry factors. Therefore, here a two-dimension boundary value problem without parallel flow and with a uniform perpendicular flow is considered. In this case, eq 13 can be written separately for the region in the film over the active site, i.e., { I e:
Boundary conditions are
Acknowledgment Financial support from FUNDACAO ORIENTE, JNICT, NATO CRG 890600, and EEC JOULE 0052 is gratefully acknowledged. Nomenclature a = half length of the active site or radii of pore, cm Bi, = Biot number (= k*l/D,) b = half length of a "unit cell", cm c = concentration in the boundary layer, mol/cmg c, = concentration in the bulk fluid, mol/cms cB = concentration at the active site, mol/cm3 D = molecular diffusivity, cm2/s
and for the region in the film over the impermeable surface, i.e., 1 1 t > e:
-+ (e@)2-e"f2 = 0 a2f2
h2
Boundary conditions are
8s"
2168 Ind. Eng. Chem. Res., Vol. 32, No. 9, 1993
and (A71
7=1, f p = l
z = L,
afl/at = af,/at
(A91
+ U,C
(~16) When the value Pe, = v,/k* is very high, we can obtain from above analyses
and at ( = E we have continuity between the fluid over the active sites and impermeable regions:
fl = f p and
K,(c - 0) = -o,(ac/az)
&,=k*[Pe,+Pe,
(A171
and
&,= k*[O-Pe, (0-c11= u,
The solutions can easily be obtained:
so, we can get the boundary condition in the Chaara and Noble (1989) problem: z = 0,
-o,(ac/az)
= UJC, - C )
(AN)
and =o (A20) These boundary conditions are just the Danckwerts boundary conditions, which means there is no film mass transfer resistance. From here it can be concluded that, in the presence of the large forced convective flow through the membrane, film mass transfer resistance is usually smaller than the mass transfer resistance inside the porous materials, since the enhancement of mass transport by the forced convective flow in the film is larger than that in the porous materials. z = L,
fort < { < 1 (All) where for forced convective flow leaving the active site
and for forced convective flow entering the active site rl, rp should be replaced by r{, rp‘, respectively
The relation between X,I and X p n should be verified by eq A9. The interesting mass transfer region is t S E. Therefore, putting the boundary conditions (eqs A3 and A4) in eq A10, we obtain
hnl 1 = Ccl[exp(rl) - exp(r,)l cos n=O 4
-o,(ac/az)
Literature Cited Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. Transport Phenomena; John Wiley & Sons: New York, 1960. Bohrer, M. P. Diffusional Boundary Layer Resistance for Membrane Transport. Znd. Eng. Chem. Fundam. 1983,22, 72-78. Brites, A. M.; Pinho, M. N. Maes Transfer in Ultrafiltration. J.Membr. Sci. 1991, 61, 49-63. Brunn, P. 0. Interaction Between Pores in Diffusion Through Membranes of Arbitrary Thickness, J.Membr. Sci. 1984,19,117136.
OI
(A13a)
and
From eq 22, we get
Chaara, M.; Noble, R. D. Effect of ConvectiveFlow across a Film on Facilitated Transport. Sep. Sci. Technol. 1989,24(11), 893-903. Churchill, S. W. Viscous Flows: The Practical Use of Theory. Butterworth New York, 1988. Cresswell,D. Intraparticle Convection: Ita Measurement and Effect on Catalyst Activity and Selectivity. Appl. Cat. 1985,15,103-115. Davis, M. E. Numerical Methods and Modeling for Chemical Engineers. John Wiley & Sons: New York, 1984. Dudukovic, M. P.; Mills, P. L. A correction Factor for Mass Transfer Coefficients for Transport to Partially Impenetrable or Nonadsorbing Surface. AIChE J. 1985,31, 491-493. Finlayson, B. A. Nonlinear Analysis in Chemical Engineering; McGraw-Hill: New York, 1980. Happel, J.; Brenner, H. Low Reynolds Number Hydrodynamics; Prentice Halk New Jersey, 1965. Herskowitz, M.; Carbonell, R. G.; Smith, J. M. Effectiveness Factors and Mass Transfer in Trickle-Bed Reactors, AIChE J. 1979,25, 272-282.
If Pe, is very large, through the analogy between eqs A13 and A14, it can easily be obtained that the minimum Sherwood number ( S h ) = Pe,/2 when &(in) and &(out) are approaching. Now, let us look at the membrane separation problem treated by Chaara and Noble (1989). When the forced convective flow through the membrane is very high, the boundary conditions a t both sides of the membrane in their problem should be written as
Juhasz, N. M.; Deen, M. Effect of Local Peclet Number on M a Transfer to a Heterogeneous Surface. Znd. Eng. Chem. Res. 1991, 30, 556-562.
Keller, K. H.; Stein, T. R. A Two-dimension Analysis of Porous Membrane Transport. Math. Biosci. 1967,1, 421-437. LeitBo, A.; Rodrigues, A. E. Fixed-Bed Reactor for Gasoline Sweetening: Kinetics of Mercaptan Oxidation and Simulation of the Merox Reactor Unit. Chem. Eng. Sci. 1990,46,679-687. Lu, 2.P.; Loureiro, J. M.; LeVan, M. D.; Rodrigues, A. E. Dynamic of Pressurization and Blowdown of an Adiabatic Adsorption Bed IV Intraparticle DiffusionlConvection Models. Gas Sep. Purif. 1992a, 6(2), 89-100.
Ind. Eng. Chem. Res., Vol. 32, No. 9, 1993 2169 Lu,Z.P.;Loureiro, J. M.;LeVan,M. D.;Rodrigues,A.E. Intraparticle Convection Effect on Pressurization and Blowdown of Adsorbers. AZChE J. 19924 38,666867. Milb,P. L.; Dudukovic,M. P. Applicationof the Method of Weighted Residual to Mixed Boundary Value Problems: Dual-Series Relations. Chem. Eng. Sei. 1980,35, 1557-1570. Mills,P. L.; Lai,S. S.;Dudukovic, M. P. SolutionMethod for problems with discontinuous Boundary Conditions in Heat conduction and Diffusion with Reaction. Znd. Eng. Chem. Fundam. 1985,24,6471.
Noble, R. D.; Way, J. D.; Powers, L. A. Effect of External Mass Transfer Resistance on Facilitated Transport. Znd. Eng. Chem. Fundam. 1986,25, 460-452. Rodrigues, A. E.; Ahn, B.; Zoulalian, A. Intraparticle Forced Convection Effect in Catalyst Diffusion Meaaurementa and Reaction Design. AZChE J. 1982,28, 254-551. Rodrigues, A. E.; Lu, Z. P.; Loureiro, J. M. Residence Time Distribution of Inert and Linearly Adsorbed Speciesin Fixed-Bed
Containing "Large-Pore" Supports: Applications in Separation Engineering. Chem. Eng. Sei. 1991,46, 2765-2774. Rodrigues, A. E.; Lopes, J. C.; Lu, 2. P.; Loureiro, J. M.; Dias, M. M. Importance of Intraparticle Convection on the Performance of Chromatographic Praceeaes. 8% Intern. Symp. Prep. Chromat. "PREP-Ol", Arlington, VA; J. Chromatogr. 1992,590,93-100. Sereno, C.; Rodrigues,A. E.; Villadsen,J. The Moving Finite Element Method with Polynomial Approximation of any Degree. Comput. Chem. Eng. 1991,15,25-33. Wakeham, W. A.; Mason, E. A. Diffusion Through Multiperforate Laminae. Znd. Eng. Chem. Fundam. 1979,18,301-306.
Received for review February 1, 1993 Revised manuscript received April 15, 1993 Accepted May 11,1993
