diffusional extraction from hydrodynamically stagnant regions in porous media

The Chemical Engineering Journal, 11 (1976) 39-56 0 Elsevier Sequoia S.A., Lausanne. Printed in the Netherlands

Diffusional Extraction from Hydrodynamically Stagnant Regions in

Porous Media

R. J. WAKEMAN

Department of Chemical Engineering, University of Exeter, Exeter, Devon (Ct. Britain)

(Received 4 March 1975; in final form 19 September 1975)

Abstract

Partial removal of solute from a packed bed or porous medium is achieved by direct displacement during the initial passage of solvent. Some solute is retained by surface tension over the surfaces of the solid which constitutes the medium, at points of contact of neighbouring particles, and within the microstructure of porous particles. Solute in these regions is considered to be stagnant, particularly when the residual liquor saturation approaches its irreducible level, and removal of this solute becomes rate controlling during washing.

Rigorous formukztions for mass transfer by diffusion from stagnant zones in porous structures have been developed, based on the above physical models. The effects of altering process variables on the effluent solute concentration and fractional solute removal from the porous medium are analysed. Although reliable experimental data in this area are lacking the models compare favourably with available data.

Different theoretical models relate to different process operating conditions and the size of the particles forming the bed. Effects of particle shape are discussed and theoretical models are developed for two different shapes of particle. The role of gases entrapped within the medium upon resaturation by solvent after draining are also considered.

The importance of the work lies in many unit operations such as the recovety of flocculated materials, the removal of solutes from filter cakes, the separation of polymers produced by heterogeneous polymerization and the chemical extraction of useful minemls.

39

INTRODUCTION

Diffusional extraction from the porous structure of materials is important in many chemical processes such as the recovery of flocculated materials, the removal of solutes from cakes formed by filtration, the separation of polymers produced by heterogeneous polymerization and the chemical extraction of useful minerals. The media involved in these chemical processes are porous and complete solute removal from them in continuous operations is only possible over long periods of solvent application. Mass transfer to and from stagnant fluid around particle contact points has been studied by Fasoli and Mellil , Han and Bixler2, Levich et al. 3, and more recently by Wakeman and Rushton4.

Edeskuty and Amundson5 studied the effect of intraparticle diffusion in agitated static systems and Rosen6 solved the transient behaviour problem of a linear fixed bed system where the adsorption rate was determined by the combined effect of a liquid film and solid diffusion into spherical particles. Masamune and Smith7 treated similar problems on gas adsorption in beds of porous glass particles and discussed the effects of surface adsorption, pore diffusion and external diffusion on the overall adsorption process. A further relevant section of the literature is that devoted to solute movements in packed beds when axial mixing in the fluid phase is represented by a dispersion coefficient. The mathematical problem was solved by Lapidus and Amundsons and their solution was later used in washing studies by Shermana. Dispersion models, however, fail to describe the solute removal process when hold- up of solute occurs resulting in a highly asymmetric

40

distribution of solute in the solvent effluent. ‘Ihere are many other examples of similar problems in the literature, some of which are brought together by BuffhamlO and their similarities compared in the development of a unified time-delay model to describe the effects of intra- solid resistance and axial mixing on outlet response curves.

The present work is aimed at developing a model for solute mass transfer from regions in packed beds and porous media which are hydrodynamically stagnant. Stagnant zones can be induced by poor vessel design but this work is not intended to include these. Drainage of beds of coarse particles leaves bodies of liquid held at particle contact points. When the individual particles are porous, liquid is held within their internal microstructures. It is to these regions that the present work applies. In flow through beds of fine particles, essentially stagnant regions can also exist between neighbouring flow pores. Models for these processes have been suggested” *I*, but no analytical solutions to the mathematical problems were reached and mass transfer from each type of stagnant zone was considered separately. Simultaneous solute removal from the difficult zones is treated here by formulating differential equations from the time-dependent behaviour of a solute in the flow zone and in the dead zones.

R. J. WAKEMAN

(a) lb)

Fig. 1. Mass transfer from porous particle structures: (a) porous particle; (b) aggregated particles.

(c) Solute in the stagnant zones is displaced by a diffusional mechanism into the flowing solvent.

(d) Once solute has diffused from the dead zones into the solvent it is carried through the porous medium by the solvent with plug flow in the main channels.

The differential equations are solved simultaneously with the appropriate boundary conditions using the Laplace

(e) At the start of the process the stagnant regions

transform technique. are totally filled with solute and air is not entrapped to form a further zone separating the dead and flow zones such that mass transfer is inhibited.

ANALYSIS

The mathematical model for this analysis is a packed bed of spherical particles which has been drained prior to washing. Each individual particle is porous and therefore retains solute after draining. In the theory it is of no consequence whether the flow zones are created by the dewatering driving force across the medium, or whether excess liquid in the bed is pushed out by the initial passage of solvent. The simplified physical model is shown in Fig. 1, and the following assumptions are made in deriving the differential equations.

(a) The time taken for the flow channels to be filled with solvent is short compared with the overall mass transfer period. (0) Cbl

(b) Residual solute exists only and uniformly around particle contact points and within internal particle Fig. 2. Stagnant zones between (a) angular and (b) spherical structures at the start of mass transfer. particles.

(f) The shape of the dead zone is illustrated by Fig. 2(a) and the porous particle is represented as spherical.

DIFFUSION IN POROUS MEDIA 41

Akthematical formulation The material balance on solute in the flow channel yields an equation of the form

(1)

In this equation a and a* are the areas available for mass transfer per unit depth of flow channel from the side pore and from the porous particle respectively, D is the molecular diffusivity and D* is an effective diffusivity from the porous particle.

Solute diffusing into the solvent from around the particle contact points is governed by the diffusion equation

W,Y, t) = D a%,Y, 1) at ax2

(2)

Similarly, for solute diffusing from the internal particle structure

Boun&y conditions In many industrial processes the solvent is contaminated by small quantities of solute impurities. Assuming this to be so, and noting assumption (a), the boundary conditions for eqn. (1) are

c(t, y) = c, at y = 0 for all t (4)

c(t,y)=c,att=Oforally (5)

The conditions for eqn. (2) are

at t = 0,

z(x, y, t) = co for ally

for t > 0,

az(x,Y, t) ax

= 0 at x = b for ally

z(x,y,t)=cCy,t)atx=Oforally

When there is no resistance to mass transfer the boundary conditions for eqn. (3) are

(6)

(7)

(8)

at t = 0,

@(r,y,t)=coforally, OQrGR (9)

42 R. J. WAKEMAN

for t > 0.

Why, t) ar

= 0 at r = 0 for ally

@(r,y,t)=c(y,t)atr=RforaUy

Solution in the Laplace variable Transformation of eqns. (l)-(3) yields, using the appropriate boundary conditions,

D d23x, y, s> dx2

=s Z(x,y,s)-co

(10)

(11)

(12)

(13)

(14)

Equations (13) and (14) are effectively second order ordinary differential equations. Taking the derivatives of their respective solutions and substituting the resulting equations into eqn. (12) yields a solution to the problem in terms of the Laplace variable:

E(y,s) -c;=(c, -c,) +- 1 1 -& (1 - eWyT@))

where

(15)

u6j

Solution A good approximation to the exponential function is given by the first three terms of its Maclaurin series. After substituting the-series expansion into eqn. (15), the inverse Laplace transforms may be found by expressing the hyperbolic functions in terms of negative exponentials and expanding as a series by the binomial theorem. Following this procedure and using tables of the relevant transform pairs, the inverted form of eqns. (15) and (16) is found to be

X1 - 2(X, -X2) exp +2 2 (-l)“(X, -X2(nt2)21exp - n=o

( (n +;rb’)] +

+2 $ {X3 -X,(n+2)21exp n=o

(-(n;‘:“‘]] +


(17)

for c > 0, where

xr = 1 +y/4ut (18)

(19)

X, = X1 + a*D*y/QR (20)

(21)

Equation (17) is the equation derived to describe simultaneous solute mass transfer from fluid retained around particle contact points and from that retained within porous particles or aggregates of particles. When the particles are large and non-porous a* = 0 and eqn. (17) reduces to the mass transfer equation previously developed4.

An alternative method of solution is given by the contour integral

c(t, y) = _!_ lim v+ih

2ni k-+00 I e%(s, y) ds (22)

Y - ih

which can be written as a real infinite integral:

u+iA A A

I esfF(s, t) ds = cur 5 eiTrC(u t in, y)i dr = iey7 eiTrc(v t ir, y) dr + I

eiTrZ\u + ir, y) dr (23) y - ih -A 0

where the variable of integration is s = Y t ir, Y being fixed. The inversion integral can therefore be written as

c(r, y) = F s O” {u(v, r, y) cos (7, t) - U(V, r, y) sin (7, 1)) dr (24)

0

F(u t i7, y) = U(Y, 7, y) + ~LJ(V, 7, y)

@{ e%(u + i7, y)} = u(y, 7, y) cos (7, t) - I@, 7, y) sin (7, r)

(25)

(26)

However, because of the form of F(s, y) it would be difficult to obtain W(e”‘c((v + ir, y)} and the first method of inversion described was therefore adopted.

An approximate solution would be possible by numerical inversion of the Laplace transformed equations, but this method suffers from the disadvantage that no analytical solution is formed. Comparisons between curves predicted from a numerical technique” and those from eqn. (17) are illustrated in Figs. 3 and 4, for two different sets of variables. The exit solute concentration curves for the overall mass transfer process are comparable over the initial time periods, but over longer periods the numerical and analytical curves deviate considerably with the analytical curves lying above the numerical. The variance between the two methods of solution is clearly shown by comparing mass transfer curves either from the particle contact points or from the porous particle structure.

Mrss transfer equation in dimensionless form The equations so far derived contain a number of parameters which are as yet undefined, although their physical

44 R. J. WAKEMAh

Vo4d Volumes or Solwant -

Fig. 3. Comparison between numerical and analytical results. Fig. 4. Comparison between numerical and analytical results.

significance is apparent; they are a, a *, b and R. For the shape of side pore under consideration the volume of side pores per unit volume of main pores is given by2,12

S -= ab v 1-s Q

(27)

The average wash liquor velocity based on the void cross sectional area for flow is

v = Q/Ae(l -S) (28)

Hence,

a = SAe/b (29)

In eqns. (27)-(29) S refers to the solute saturation of the packed bed due to residual liquid trapped between neighbouring particles only, and not to that solute held within the internal structure of the particles. By definition s < 1 .o.

Two different porosity terms must now be identified for the definition of a*-the particle porosity e*, and the porosity E of the bed external to the particles. The total bed porosity et is then given by

et=e+(l -E)E* (30)

DIFFUSION IN POROUS MEDIA

and the particle pore surface area per unit depth of main flow channel is

a* = 3( 1 - +*A/%

Using these relationships, eqn. (17) can now be written in a more convenient dimensionless form:

c--w _ S -- CO -CW t1nw - s)lf [

X5 -2(X5 -&)exp(-v)+

+ 2 f (-l)“{Xs - X&r + 2)2) exp 1

(n + 2)2r;2(1 - s) -

W II t

n=o

3( 1 - f)E*

+ e?j{sw(l -s))4 [XT-i(s)+ (1 +~~(l$~~)+2(X, -X,)exp (-n2(1~s’)+

+ 2 2 (X7 - Xa(n + 2)2} exp n=o (

(n + 2)27?2(1 -S) -

II t

W

3(1 - e)e*s 1

+e&+rW(l -s)>+ T&l -s) ( [ 1 - 2,zo (-1)” exp

1

(n + l)2E2(l -s) -

+

W 11

for W 2 0, where

xs = 1 t(1 -S)/4W

x _ g2(l - S)2 + 3(1 - e)e*

8- 2wa eW

45

(31)

(32)

(33)

(34)

(35)

(36)

and the dimensionless quantities are defined by

w=$;(l -s) (37)

(38)

(39)

The factor t is a function of the ratio of the characteristic residence time b2/D of solute in the stagnant fluid at the particle contact point to the residence time of solvent’in the main flow channel. Similarly, 17 is related to the characteristic solute residence time R2/D* in the porous particle or aggregate.

46 R. J. WAKEMAN

The pendular ring theory for connate residual liquid held at the particle contact point of two touching spheres enables a relationship between b and the particle diameter d to be obtained. At S ~0.1, b/d 2 0.38 is predicted and this appears to fall in line with experimental data for coarse particles2y4. Hence, if D = D*, which is likely to be the case for coarse aggregated particles, .$ = 0.76~. However, in the case of porous particles or aggregates of fine particles it is more likely that D* <D, since D* is dependent on the internal pore structure of the particles.

RESULTS AND DISCUSSION

Final results can be plotted in a variety of ways. If solute recovery from the solvent is to be carried out, then a plot of the instantaneous concentration of solute in the solvent effluent against the void volumes of solvent used is of interest. From these data the fraction of solute removed from the porous medium can be calculated, as a percentage, using the following equation:

(W

where the value of S corresponds to the use of Wt void volumes of solvent (IV, = 0 at f = 0). The results are plotted in both ways in this study so that it can readily be seen how effective the mass transfer operation is under various process conditions.

Figures 5 and 6 illustrate the effect of altering the mass transfer parameters g and TV (when D = D*). If the solven flow rate is identical for each curve and the same depths of bed are being washed, then increasing the value of g (or 77) is tantamount to treating beds containing larger diameter particles. On the other hand, if beds of particles of various depths are being washed at different flow rates, then higher values of .$ (or q) are associated with shorter solvent residence times. Lengthening the solvent residence time in the medium serves to provide a more effective mass transfer; a similar effect is obtained when the same residence times are used to wash beds of smaller particles. These facts are most easily seen on a plot of F versus W (Fig. 6).

0 I 2 3 4 5 tl

Void Volumes d Soiwnt B

Fig. 5. Effect of particle size on mass transfer.


As the effective diffusivity is lowered, the time taken to remove any given quantity of solute from the bed is increased, as shown in Figs. 7 and 8. This situation occurs when, for example, ion exchange resins or flocculated beds of colloidal or near colloidal sized particles are washed, or in other circumstances where diffusion is from a network of extremely small tortuous passageways within the porous particle.

Figures 9-11 show that improved washing efficiency is obtainable with lower particle porosities and at higher bed porosities. Flocculated or aggregated particles generally exhibit higher overall bed porosities and permit greater solvent flow rates through the interparticle voids; hence improved washing can be expected in these cases (if the floes are sufficiently strong not to break down at normal washing pressures). Highly porous particles tend to contain a large quantity of solute within their structures but the diffusion area over the particle surface is larger. Figure 10 indicates that these greater volumes of solute lead to less efficient washing by any given quantity of solvent.

An intrinsic assumption in the derivation of eqn. (32) is that there is no interaction between mass transfer from the side pores and mass transfer from the porous structure. Although transfer from each region starts at the same time, the ratio of the mass diffusion rate from the side pores to that from the pores of the solid is high and some solute will diffuse from the porous particle into the fluid surrounding the particle contact points. Hence some interaction between the two zones would be expected, particularly after longer washing periods when the solute concentration in the side pores has decreased significantly.

An example of the quality of fit between the experimental data and eqn. (32) (with II* = 0) is shown in Fig. 12, for the washing of a centrifuged bed of non-porous glass beads. This model appears to describe the mass transfer process better in the case of larger particles4sr2.

I 2 3 4 5

Void Uhn~s of Solvent -

1

is

13

ti

!I

6

Fig. 6. Effect of particle size on fractional removal of solute.

Coarse particles Except for a normalization factor, eqns. (15) and (16) coincide with the characteristic distribution function of residence times in the porous medium. Introducing the factor u/y (which corresponds to normalization of the

48 R. J. WAKEMAN

Fig. 7. Effect of D/D* on mass transfer.

Fig. 8. Effect of D/D* on fractional removal of solute.

distribution to unity), the average pl and the central moments pk of the distribution function of the residence times are :


s-o. 6 - 0.36 s=5.0 y. 29421

S=O., c -0.36 5 =5.0 “1 =29.421

Fig. 9. Effect of particle porosity on mass transfer. Fig. 10. Effect of particle porosity on fractional removal of solute.

(41)

(42)

for a* = 0 and where Y=S/(l - s), i.e. for beds of coarse non-porous particles. The asymmetry coefficient 7 of the distribution function is

7 = I-&#= (W

Mass transfer from the regions under consideration is most likely to occur after the bed has been drained to a low level of saturation, i.e. S - 0.1. Some values for the asymmetry coefficient are Riven in Table 1 for various

50 R. J. WAKEMAN

Fig. 11. Effect of varying the bed porosity.

Fig. 12. Centrifugal washing of glass beads.

TABLE 1 The effects of S and E on the asymmetry coefficient

2 0.091 0.1 7.19 3 0.091 0.1 8.09 2 0.167 0.2 4.84 3 0.167 0.2 5.58

values of t and S, where it may be seen that non-symmetric curves will be obtained and the exit solute concentration versus time curve will have a long tail. It has already been pointed out3 that for turbulent liquid flows the tails of the distribution only disappear if the Reynolds number is of the order 104-105.

Fine particles The theory developed here applies to the continuous extraction of solute from a porous medium when solute is trapped in stagnant pockets of fluid. However, solute is not entirely held in the form of stagnant pockets but also as a film over the surface of the solids (or around the periphery of the flow channels through the porous medium . This film fluid is also effectively stagnant and a satisfactory model for this situation has already been developed’ 2

as a simple form of the problem studied by Goldstein14 in connection with fixed bed ion exchange problems.


Although the film mechanism probably also applies during the washing of beds of coarse particles, it is then only secondary to the principal mechanisms described so far. In mass transfer from beds of fine particles (smaller than about 100 pm) significant quantities of solute are retained in this manner after the initial passage of solvent and a film model accurately predicts washing performance, as is shown in Fig. 13 for the washing of a bed of 20 pm magnesium carbonate particles. The equation governing the solute concentration in the solvent effluent is r3

------1 -exp{-(n+w)}le(2(CLo)~}- i exp{-(n+6)}1,{2(52S)f}d6 c-cc, _

CO -GV 0 (45)

Further dataI indicate similarly that a film model gives better agreement with mass transfer performances over the initial stages of washing whereas the latter stages are better represented by the side pore model. However, further experimental work is required to determine precisely the factors controlling the regions of applicability of the models and the point where transfer from either the side pores or porous aggregates becomes the rate- controlling mechanism.

The mathematical solutions presented here seem reasonable for practical purposes. Any computations should be checked using eqn. (40) to ensure that values for F approach 100 asymptotically. An experimental study of the effects of particle size and particle size distribution on mass transfer from hydrodynamically stagnant regions in packed beds is in progress and should elucidate the relationships between the model variables and the operating parameters.

-Expor~m~ntal CWYP

= Equmm 145l,rL= 8.0

0 EqNeon 132) y= 41.8

5/q = O-76

E = 0.351

E’= 0.787 szo.2

Q = 4.34 x IO-’ m3k

L = 2.6 x lO-2 m d =20pm

0.01; ’ 0; n ’ ’ ’ ’ ’ n ’ * ’ s n ’ I.5 2 2.5 3

Vosd wluws of &en; ___c 3.5

Fig. 13. Solute removal from a magnesium carbonate bed.

52 R. J. WAKEMAN

Effect of particle shape The notion of the shape factor tar particles of irregular shape has usually been introduced empirically. Arisrs showed that, if the particle volume-to-surface ratio is taken as the characteristic dimension of an irregularly shaped particle, the steady state solutions of simultaneous diffusion and reaction are largelysindependent of particle shape. This characterization was also found to be inadequate for the transient problem.

The effects of particle shape on mass transfer from porous particles cannot be deduced readily but for non- porous particles mass transfer from differently shaped bodies of fluid around particle contact points can be compared. A simple mass transfer equation can be derived from eqns. (1) and (2) when &/at = 0 as’

-(2n-l&u-1)

(46)

This equation applies to the parallel-sided pore shown in Fig. 2(a) (as does eqn. (32)), a shape of pore usually associated with angular particles 2. This solution is compared with the full solution given by eqn. (32) in Table 2, where it is seen that numerical values from the two solutions converge after longer periods of washing. However, the fluid body at the contact point of two adjacent spheres is better depicted in Fig. 2(b) and the solute material balance for this shape is written in degenerated bipolar coordinates as

where (Y = d2r/(r2 t z2) and fl= d2z/(r2 + z2) are the degenerated bipolar coordinates; r and z are cylindrical coordinates. This equation is integrated over the range 0 < cr < 00, --flu < /I < &, with the boundary conditions

TABLE 2 Comparison between eqns. (32)and (46) for mass transfer from non-porous angular particles

W cc - +.r)/(co - c,)

~=lo,S=o.l,e*=O E = 22.4, S = 0.1, E* = 0

eqn. (46) eqn. (3.2) eqn. (46) eqn. (32)

1 0.0123 0.00729 0.00549 0.00326 2 0.00515 0.00468 0.0023 0.00209 3 0.00395 0.00369 0.00177 0.00165 4 0.00333 0.00314 0.00149 0.0014 5 0.00294 0.00278 0.00131 0.00124 L 0.00265 0.00252 0.00119 0.00113 I 0.00244 0.00232 0.00109 0.00104 8 0.00227 0.00216 0.00102 0.000967 9 0.00213 0.00203 0.000955 0.000909

10 0.00202 0.00192 0.000904 0.00086

(48)


The method of solving this equation with eqn. (1) is presented in the Appendix, whence the solution is

where the h, are the roots ofJe@) = 0. Mass transfer from the stagnant fluid at the contact points of angular and spherical particles is compared in

Table 3, where it can be seen that, for the same value of the mass transfer parameter, beds of spherical particles are more easily washed.

All the side pore models described have implicitly assumed a uniform distribution of monosized pores through- out the depth of the porous medium. For the side channel lengths to have a distribution density would be more realistic. Also, a bed is more likely to contain a random distribution of a mixture of parallel-sided and wedge- shaped side pores. It is recognized that diffusion across a stagnant boundary is unlikely to be the sole means of solute removal from these regions. There is also the possibility of some mixing occurring at the interface due to buoyancy effects and differences in kinematic viscosities of the two fluids.

Effect of entrapped gases When the porous medium is being filled with solvent after drainage has taken place, gas may be trapped and hence form a third fluid separating the solvent from the residual solutes. The mass transfer time is then dependent on the physicochemical properties of the entrapped gas. Such a situation can, in principle, be analysed by a

TABLE 3 Mass transfer from fluid at the contact points of angular and spherical particles

W

- 1 2 3 4 5 6

s7 9

10

cc - cwmo - cw)

[ = 7.071, s = 0.1

ev. (46/, eqn. (491, angular spherical

0.01 74 0.0312 0.00727 0.0123 0.0055 8 0.00883 0.00471 0.00703 0.00415 0.00586 0.00375 0.00501 0.00345 0.00436 0.0032 0.00382 0.003 0.00337 0.00282 0.00299

E = 11.18,S= 0.1 [=11.18,S=0.2

eqn. (46). ev. /49/, eqn. (46), eqn. (49/, angu Iar spherical angular spherical

-- 0.011 0.0198 0.0246 0.0441 0.0046 0.00834 0.0103 0.0187 0.00353 0.00618 0.00794 0.0139 0.00298 0.00506 0.00669 0.0113 0.00263 0.00434 0.0059 0.00973 q.00237 0.00382 0.00534 0.00858 0.00218 0.00343 0.00491 0.00771 0.00203 0.00312 0.00457 0.00701 0.00191 0.00287 0.00429 0.00644 0.00181 0.00265 0.00406 0.00596

method similar to the foregoing when only a portion of the side pore is filled with solute and the remainder with gas. The solute diffusion time through the gas phase must be taken into consideration.

This problem is important, for example, in extraction from partially weathered ores which have been subjected to partial leaching by natural causes. In these cases the pore space within the particles may be filled with gases which are only slightly soluble in the solvent. Diffusional transfer cannot then occur until such time as the gas in the pores has been absorbed. An approach to this problem has been suggested16 by considering forced convection mass transfer of gas into a laminar film of liquid and assuming the validity of Henry’s law, to estimate the time delay before diffusional transfer can take place.

54 R.J.WAKEMAN

CONCLUSIONS

A formulation is developed describing the rate of mass transfer of solute from hydrodynamically stagnant regions in packed beds and porous media. The problem is stated by writing mass balance equations for each of the stagnant regions considered and for solvent flow through the medium; solving these yields an analytical solution for the solute concentration variation with time.

Computed results describing the effects of the various parameters on the mass transfer process are also presented Different ways of plotting the final results are discussed and compared. Plots of the percentage fraction of solute removed from the porous medium against void volumes of solvent appear to be the most useful in practice.

Several factors are thought to affect the diffusional extraction process; the effects of particle size and shape have been analysed and the delay in extraction caused by the trapping of slightly soluble gases in the pores is discussed. It is also recognized that the pores will have a size distribution, which will alter the shape of the final washing curve.

The significance of this paper lies in providing a rigorous formulation for diffusion extraction from the various types of stagnant regions in packed beds and filter cakes that occur in important operations in many chemical processes.

NOMENCLATURE

a a* A b CO9 Y) CW CO

d D D* F 10

Jo JI e

L s S t lJ W X,Y z(x, Y, t)

area of the side channel in the walls of the flow channel, [Lz L-l ] pore area of the porous aggregate (or particle) in the walls of the flow channel, [Lz L-l ] surface area of the porous medium, [L2] side channel length, [L] solute concentration i’n the wash solvent, [M L-s] solute concentration in the input wash solvent, [M Le3] initial solute concentration in the retained liquor, [M L-s] particle diameter, [L] effective diffusivity of solute from the side pore, [L2 T-l] effective diffusivity of solute from the porous particle, [L2 T-l ] fractional removal of solute, defined by eqn. (40) modified Bessel function of the first kind of zero order Bessel function of the first kind of zero order Bessel function of the first kind of first order solvent flow rate, [L3 T-l] radial coordinate, [L] radius of aggregate (or particle), [L] Laplace transform parameter saturation volume of retained liquor per unit volume of bed voids external to the porous aggregates time, [T] solvent velocity, [L T-l ] voids volumes of solvent, defined by eqn. (37) orthogonal coordinates, [L] solute concentration of fluid in the side pores, [M Ld3]

Greek symbols Y asymmetry coefficient f volume of voids external to the aggregates per unit volume of porous medium e* volume of voids in the aggregate per unit volume of aggregate

et total void volume of the medium per unit volume of medium 77 mass transfer parameter, defined by eqn. (39)

DIFFUSION IN POROUS MEDIA

pk

Z, y, t)

central moments of distribution mass transfer parameter, defined by eqn. (38) solute concentration of fluid within the porous aggregate (or particle), [M Lm3] dimensionless pseudo-time dimensionless distance

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APPENDIX

The solution to eqn. (47) is found by making the simplification p < o, which is reasonable at small values of S. Equation (47) can then be written

a2z(r, 0 : 1 W, O_ 1 az(r, 0 ar2 r ar D at (Al)

which has the Laplace transform

where

n=F(r,s)-l/s

Using the transforms of eqn. (48) the general solution to this equation is

Inversion of eqn. (A4) yields

z(r, t) - 1 = (co - 1)

1 1 - 2 2 J”@nr’b) exp - ‘$

n=l WIG”) ( 11

Hence

W, t) -Ir=b=(c~-l)~ 2 exp(-y) ar

n=1

55

(A3)

(A4)

(A5)

(Ah)

56 R. J. WAKEMAN

Substituting this into eqn. (1) with a* = 0, ac(t, r)/ay = 0, and integrating gives the solution

(A7)

The volume of side pores per unit volume of flow pores can be expressed in terms of the residual saturation at the start of this diffusion washing:

abv 2s -=- Q 1-S

Combination of eqns. (A7) and (A8) yields eqn. (49).

(A8)

diffusional extraction from hydrodynamically stagnant regions in porous media

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