and solid-diffusion kinetics in fixed-bed adsorption under constant

PORE- AND SOLID-DIFFUSION KINETICS IN FIXED-BED ADSORPTION UNDER

CONSTANT- PATTE RN CON D IT10 NS K E N N E T H R . H A L L , ’ L E E C . E A G L E T O N , * A N D R E A S A C R I V O S , 3 A N D T H E O D O R E V E R M E U L E N

Department of Chemical Engineering, University of California, Berkeley, Cal i f .

In the favorable-equilibrium adsorption region, the constant-pattern form of the isothermal breakthrough curves is known only approximately for the solid-diffusion (or pore-surface-diffusion) mechanism, and is not soluble analytically for the pore-diffusion mechanism. Numerical solutions for both these cases, in a widely applicable dimensionless form, have been obtained for a range of Langmuir isotherms by stepwise computation on a digital computer. These numerical results merge smoothly into the respective analytic solutions for completely irreversible equilibrium; for this case, combinations of pore diffusion and external mass transfer are also analyzed.

oLuMN-performance studies for fixed-bed adsorbers and ion c exchangers involve particularly the concentration history or “breakthrough curve” of the column effluent-that is? the variation in effluent concentration with respect to time or volume of effluent. Bohart and Adams (3) and Wicke (22) first identified the existence of constant-pattern concentration histories, for irreversible adsorption, in the reaction-kinetics and solid-diffusion cases, respectively. Michaels (74) obtained the exact solution of the external mass transfer case for a range of favorable equilibria, while Glueckauf and Coates (7) and Vermeulen (27) have given approximate solutions for the solid-diffusion, constant-pattern case in the same range. In the present study, a more cxact solution to the solid-diffusion problem has been sought, along with a solution of the previously neglected problem in which pore diffusion is the rate- controlling mechanism.

Equilibrium

Theoretical considerations of breakthrough curves must deal with three main factors : equilibrium, mechanism, and stoichiometry. Equilibrium curves, or isotherms, relating solid-phase to fluid-phase concentration for adsorption of a single component from a binary fluid phase directly influence the behavior of the breakthrough curves. Four idealized types of equilibrium behavior can be recognized, each of which is approached by many experimental systems:

1. Irreversible equilibrium 2. Favorable equilibrium 3. Linear equilibrium 4. ‘C‘nfavorable equilibrium

Types 1 to 3 can be represented in specific instances by Langmuir-type behavior, and type 4 by an extension of this behavior-for example, to desorption in a system exhibiting Langmuir adsorption. The Langmuir equation can be written :

1 Present address, University of Oklahoma, Norman, Okla. * Present address, University of Pennsylvania, Philadelphia, Pa. 3 Present address, Stanford University, Stanford, Calif,

in which q is the solid-phase concentration of the solute being adsorbed, the asterisk (*) denotes an equilibrium value, Qrn is the ultimate sorptive capacity at high concentrations, K is the Langmuir equilibrium constant, and c is the fluid-phase concentration. For gas-phase systems, c may be replaced by partial pressure, p , with no change in the dimensionless treatment that will follo\v. The results of this paper apply only to cases where the equilibrium conforms to a Langmuir isotherm.

Equation 1 includes the description of the equilibrium between the feed concentration, GO, and the saturation level, qo*, which it may produce in the column. The relation for saturation conditions can be divided into the general relation, resulting in

Now a “constant separation factor” or “equilibrium parameter,” R, is defined as

(3)

Dimensionless concentrations, Y (= q/qo) and X (= c/co), are introduced, each bounded by the values 0 and 1. Equation 2 then becomes

x R + (1 - R)X (4) y* =

This may be solved explicitly for R, showing its similarity to the “relative volatility” in vapor-liquid equilibria :

(5)

(Asterisks denoting equilibrium could be applied to either Y or X.) The mean concentrations at any given cross section in a fixed-bed column will be designated Y and X, and Equations 4 and 5 apply to these values as well as to the point functions. R = 0 for the irreversible case, 0 < R < 1 for favorable equilibrium, R = 1 for the linear case, and R > 1 for unfavorable equilibrium.

Equilibrium considerations have a particularly interesting effect on the shapes of breakthrough curves. For values of R

212 I & E C F U N D A M E N T A L S

A D S O R B E D L A Y E R F L U I D P H A S E ON S U R F A C E IN PORES

Figure 1 . Diffusion paths within adsorbent particle

below 1, the curves tendl to attain a constant pattern and thus to become relatively “self-sharpening“ as they advance through an increasing length 01’ column. Thus, for a given mass- transfer mechanism and a given value of R < 1, a single theoretical curve for b:reakthrough, on the appropriate coordinates, represents all operating conditions. On the other hand, \vith R > 1, a proportionate pattern is set up which sho\vs no tendency to sharpen (become constant) but instead stretches out continually as the utilized column length increases. Intermediate behavior is encountered in the vicinity of R = 1. This study is concerned only with the constant- pattern case, for R < 1.

Mechanisms of Mass Transfer. There are four major rate-determining mechanisms for fluid-solid mass transfer:

A. Fluid-phase external mass transfer B. Fluid-phase pore diffusion C. D. Solid-phase diffusion

In the present study mechanisms B and D are of primary concern, the others being considered infinitely fast except in one case in which A is included. In other cases where external diffusion is slow enough to merit consideration, it would enter as a resistance in series .with B or D, but the resistance is not additive in a simple way.

The distinction between pore diffusion, mechanism B, and solid diffusion, mechanism D, is vital to the prediction of breakthrough behavior and will be examined in some detail. Adsorption is emphasizeid here, since the results are applicable mainly in this area. (Ion exchange is treated briefly in Appendix I .) As indic,ated in Figure 1, solute may enter a particle of adsorbent material directly from the exterior surface by nlovement in condensed form along the pore surfaces (identified as “solid” diffusion), or it may diffuse through the fluid phase held in the :pores and then be deposited in a stationary location on the pore surface. Whereas the concentration level will usually change abruptly a t the fluid-solid inter- face because of the “reaction” or phase change that occurs there, the pores as such exhibit a fluid-phase concentration that varies continuously from the exterior to the center of the particle. In pore diffusion: the controlling mass-transfer process occurs before the phase change, but in solid diffusion it occurs aftenvard. At different operating conditions, the same particle can show either solid- or pore-diffusion behavior, favoring the former if the pore-fluid concentration level is low and the latter if this level is high. Of the two mechanisms, the more rapid one will control, since they act in parallel. Since a reaction-kinetic resistance is rarely observed, local equilibrium

Reaction a t the phase boundary

will be assumed at each point within a pore, betlveen the fluid and the immediately adjacent solid.

Constant-Pattern Behavior. For linear and unfavorable isotherms, the shape of the breakthrough curves is nearly independent of the rate-determining mechanism. An approximate solution for either of these cases is given by the Thomas equation (78, 79). .4n exact solution, for the linear isotherm only, has been presented by Rosen (75, 7 6 ) ; this treatment applies to either pore diffusion or solid diffusion in series ivith a mass-transfer resistance outside the particle.

For isotherms appreciably more favorable than the linear case, in relatively “deep” beds, the constant-pattern treatment applies. This leads to results substantially simpler than those of Thomas and of Rosen. Simplification in the mathematical solutions is brought about because the material-balance relation, normally expressed as a well kno\vn partial differential equation-e.g., Equation 16-95a ( 7 7)-is reduced to the simple statement

X = Y (6)

Rosen (77): Lapidus and Rosen (72) , and Cooney and Light- foot (4) have shown that a constant pattern is given by the asymptotic solution for favorable equilibria in “deep” beds. Cooper (5) has examined the evolution of shallow-bed into deep-bed behavior. In the asymptotic treatment, Equation 6 occurs as an incidental result, unless there is axial diffusion.

I t is somewhat easier to demonstrate Equation 6 by starting with the constant pattern as an assumption. Thus Michaels (74) and Eagleton and Bliss (6) derived this equation by con- sidering the similarity of a fixed bed to a countercurrent moving bed operating at total reflux. ilrith the same assumption, Vermeulen (20) and Hiester, Vermeulen, and Klein ( 7 7 ) derived it from the differential material balance, Ivith the constant-pattern assumption being expressed as (bu/bt)x = constant, using the “chain rule’‘ involving partial derivatives of the three variables functionally related-for instance, X, t , and v-where t represents time qnd u , column volume.

In solving for pore-diffusion breakthrough curves, the approximation is made that the amount of solute held in the pores is negligible compared with the amount adsorbed by the solid phase-that is, although all of the solute is carried through the pores, all of it is held by the solid. The experimental adsorption isotherm customarily incorporates the same assumption, and the resulting consistency tends to minimize the error due to neglect of pore-phase behavior as a separate factor. Thus, the resulting treatment given here is only a necessary first step toward a total solution.

For both mechanisms, this treatment applies only where changes in temperature, during passage of the adsorption wave, can be neglected. When this assumption does not apply, accurate accounting for the shift in equilibrium may be more essential than having an exact model for rate behavior. More- over, axial dispersion [treated by Acrivos ( 7 ) ] \vi11 be assumed to be too small to contribute to the shape of the breakthrough curves.

Assumptions.

Pore Diffusion

If all the solute within a particle is considered to be held by the solid, the rate of diffusion into a uniformly porous, spherical particle is assumed to be given by Fick’s second law :

where D,,,, is an effective pore diffusivity [see ( 7 7 ) or (27)

VOL. 5 NO. 2 M A Y 1966 213

for the relationship between D,,,, and fluid-phase diffusivity ] ; e is the void fraction; r is the radius of any spherical shell within the particle; X is the local value of 6/60 at radius r , as opposed to the exterior bulk fluid concentration, X( = Y ) ; Y is the local value of q/qo*; and -2 is the distribution coefficient, qo*pa/co, with pa the bulk density of the bed.

The mean solid-concentration value, Y, averaged over the particle, is

Y = 1 Yr2 dr rs

where rs is the outer radius of the particle. with respect to time gives

Differentiation

which, upon combination with Equation 7, leads to

(1 0) d Y 3DPore(1 - e ) bX (d - - - dt

Solution for Irreversible Equilibrium (R = 0), Including Fluid-Phase Resistance. The analytic result for pore diffusion and R = 0 has been given by Acrivos and L’ermeulen (ZO), but the steps in the derivation were not reported. This solution provides a limiting condition for the general constant- pattern case, and is outlined here. Furthermore, as the addi- tion of a fluid-phase resistance does not increase the complexity of the derivation for R = 0, this more general case is included in the treatment.

When R is zero, Equation 7 is simplified by the phenomenon that the concentration wave penetrating into the particle fully saturates each spherical-shell element before advancing. If the saturation region has penetrated to a shell of radius r l , then dY/dt is zero for r z < r < rs, infinite a t r = r z , and zero for r < r , ; while Y is unity for r , < r < ra and zero for r < r,. Integration of Equation 8, over r , gives under these conditions

r63 - ri3

ra3 Y (= X) = ~

or

r i / r a = (1 - x) ’ /~ = t (11)

Hence the problem becomes one of evaluating the behavior of r,. The derivative bY/bt in Equation 9 is zero for r , < r < r a , and in that range Equation 7 becomes:

from which

r2(dX/br)L = const. = g (1 3)

With the limits X = X , a t r = ra and X = 0 at r = r i , integration of Equation 13 gives:

X8

g =

I t now becomes necessary to introduce the external mass- transfer rate equation and, for simplicity, three dimensionless parameters.

The external mass-transfer coefficient, k,, is treated in the usual way-for example, as Equation 16-39 of ( 7 7 ) ,

T - I .

I” p0re.F + I”

Figure 2. Breakthrough behavior for irreversible adsorption with pore-diffusion kinetics and external transport in series

Ordinate on probability scale

where a p represents the outer surface of solid particles per unit volume of packed bed and subscript F denotes the use of fluid- phase concentrations.

For use in column calculations, it is convenient to define three parameters to replace t , D,,,,, and k f r . The throughput parameter is

the number of pore-diffusion mass-transfer units (or pore- diffusion NTU) is

N p o r e , ~ E 15 Dpor,(l - e ) v / r , 2 (17)

and the external N T l j is

N ~ F kjFa,v/F (1 8)

Combination of Equations 6, 11, 13 (with r = r 8 ) , 14, 16, Here e is the void fraction, and F the volumetric flow rate.

and 17 with Equation 10 now gives:

(19) 3Npore,~ Xs - dX 3Npore.~ Xs

dT 15ra l /r , - l /r8 15 t - ’ - 1 -

With Equations 6, 16, and 18 used to replace Y , t , and k , ~ , Equation 15 yields

When this expression is introduced into Equation 19, the final differential equation results:

214 l & E C F U N D A M E N T A L S

R - l R & + I

Figure 3. Forward-difference calculation grid

Integration of Equation ;!I gives:

Here C is the constant of integration; this cannot be found in the usual way, since there is no value of T for which X is known. Instead, C is selected so that the solution satisfies the over-all material balance--namely, the difference between the solute in the fluid entering the column and the solute in the effluent after the pattern ?merges from the equilibrated column must equal the solute adsorbed by the column. This material balance requires that the following equation be satisfied.

(1 - X)dT = T dX = 1 (23) Lrn so' Solving Equation 22 for T, substituting into the second equality of Equation 23, and perfcfrming the indicated integration gives the constant of integration:

Elimination of C from Equation 22 with Equation 24 gives the solution for this case:

where

(26)

Values for @(X) are given in Appendix 11. These values were used to find the empirical equation:

~- @(X) = 2.39 - 3.59 dl - X ( 2 7 )

In the absence of external resistance ( N ~ F + m), @(X) = Npare,p(T - 1). The error in values of T calculated from Equation 27 for a given X is less than 1% in the range 0 < X 5 0.8 and less than 5:7, a t higher values of X ; the error is defined as the value of T from Equation 25 miniis the approxi-

Figure 4. kinetics

Breakthrough behavior for pore-diffusion

Linear coordinates

mate value from using Equation 27, divided by the total change in T from X = 0 to X = 1 .

Transformation of Equation 27, again for NJF - a, gives the following expression explicit in X :

Equation 25 may be rearranged into several equivalent The form most useful for graphical presentation of the forms.

solution is:

T - 1 = (i + L, [ @(XI + N p o r e , ~ I N f ~ (In X + 1) 1 N p o r e . ~ NIF Npore ,~ /Nf ,~ + 1

(29)

Equation 29 was used in preparing Figure 2. Solution for Favorable Equilibrium (R < 1) . METHOD.

As soon as R becomes finite, and Y and X at a given time are no longer simple functions of the radius, an analytic treatment appears to be impossible. The material-balance relation Y = X still holds, however, along with the boundary condition

Equation 7 can be made dimensionless by introducing Equations 16 and 17 in order to replace the D,,,, and time terms by the product N p , , , , ~ T (abbreviated as NT), and substituting @i (= Y / Y ~ ) for the radius. Moreover, combination with Equation 4 leaves only X as a dependent variable:

= x.

ax P X 2 ax [R + (1 - R)XI2bNT - am2 + g z (30)

15R

VOL. 5 NO. 2 M A Y 1966 215

For one boundary condition, Equation 10 may be put into dimensionless form

which must hold along with (X),=1 = X. tions are :

The other condi-

At @ = 0, dX/d@ = 0

At all @, for NT = 0, X = 0

(32)

(33)

for NT = co, X = 1 (34)

Equation 30 is amenable to numerical integration on a high- speed computer. The "parabolic" form of this equation suggested the use of the well-known four-point forward-difference integration method.

In this method of analysis the elements of Equation 30 become :

where X is the computational interval on @, and 1 is the interval on NT. Equations 30 to 38 are applied to a grid of @ and NT values to solve for the X value of each (R,NT) intersection; diagrammatically, in Figure 3, the three known values (x) are used to determine the unknown volume (0).

The choice of values for X and p is important in determining the speed, accuracy, and convergence of the computation. Trial runs with R = 0.5 indicated that X = 0.1 would give sufficient accuracy. With M , however, it was necessary to start with one value and to reduce it progressively until the solution converged-i.e., when successive values of X at a given HT agreed within 0.01, indicating an absolute accuracy on the order of 0.002. Table I lists the values of 1 required for convergence a t each value of R. At R = 0.6 to 0.8, constant- pattern results do not apply within 0.01 on X below N = 50 to 80.

Modifications of the calculation method were required at the boundaries @ = 0 and R = 1.0 and for the initial NT value. At m = 0 a singularity in the finite-difference approximation necessitated assuming the X value at each NT value. A linear extrapolation of the form XO,NT = x0.1,~~ - ~(XO,Z ,NT - xo.1,~~) was chosen, with cy = 1 ; in retrospect, cy = 0.5 would give faster convergence but would not affect the final

Npore,dT-I) Figure 5. Breakthrough behavior for pore-diffusion kinetics


result. At @i = 1.0, Equation 31 was applied to obtain XI.O,NT+~; subsequently the volume-average value Y (= X) was determined by numerical integration so as to check the boundary condition (X),=1 = X. This check always showed agreement to \tithin 0.010.

To set this program in motion, the integration was begun a t X = 0.01; for numerical convenience it \\as terminated at X = 0.99. At X = 0.01 (corresponding to the initial NT value) the following empirical concentration-distribution \vas assumed :

Y = YIexp [-?(I - R ) ] (39)

In this expression Yz is the value of Y in equilibrium with X, and y is a constant x.hich was evaluated to make Y equal to 0.01. Table I1 gives the values of Y z and y for each R value considered. The Fortran program for the computation is given in Appendix IV, on file with the American Documenta- tion Institute.

RESULTS. The pore-diffusion solution is presented graph- ically in Figures 4 to 7. This graphical representation has the advantage of easy interpolation ; tables of the computed values are included in Appendix 111. The computer calculation gives only relative, not absolute, values of NT; the positioning of each entire curve relative to the T scale is calculated by

Table 1. NT Convergence Interval, 1 (at y = 0.1) Pore Solid

R Diffusion Diffusion 0

0 . 2 0.0002 0.001 0 . 3 0.001 0.002

0 . 1 0.0001 0:001

0 . 4 0 .5 0 . 6 0 .7 0 .8

0.005 0 .01 0.01 0.01 0.01

0.005 0 .01 0.01 0.01 0 . 0 1

Table I I .

R 0 0 . 1 0 . 2 0 . 3 0 . 4 0 .5 0 . 6 0 . 7 0 .8

Constants in Assumed Initial Concentration Distribution

YI Y

0.0917 0.0481 0.0326 0.0246 0.0198 0.0166 0.0142 0.0125

27 90 12.25

7 . 5 0 5 . 0 4 3.47 2.39 1.58 0.946


Figure 6. Midpoint slope of breakthrough curves for pore- diffusion kinetics

Curve for R = 1 from Thomas

0.5 - 1.0 7

R = 0.2-0.4

/

Figure 7. Breakthrough behavior for pore-diffusion kinetics relative to unit midpoint slope

graphical integration, in which the area above the breakthrough curve is made equal to unity according to Equation 23.

Figures 4 and 5 have been plotted with an abscissa of Npore,p(T - l) , which provides complete generality (with respect to residence times and mass-transfer coefficients) for each curve at its respective R value. Figure 4 utilizes rec- tangular coordinates; it displays the S shape that is experi- mentally characteristic o f breakthrough curves, with greater symmetry about the midpoint than is found with either external mass transfer or solid-phase internal diffusion controlling. As R increases toward unity, the slopes diminish and the curves become more symmetrical. Figure 5 shows the results with a probability scale for the ordinate, which straightens and separates the curves.

The self-sharpening property of the constant-pattern region is illustrated by Figure 6, which relates the midpoint slopes

[ ( d X ( d T ) ~ . ~ l in the range 0 < R < 1 to the number of transfer units. Figure 7 shoivs the entire constant-pattern behavior in terms of a unit midpoint slope. This graph may be used in conjunction with Figure 6 for estimating breakthrough curves at intermediate R values; interpolation in the range 0.5 2 R 2 0 can be obtained through the empirical relation (derived from the numerical solutions) :

1

(dX/dT) o . 5 T - To.6 = { (2R)O.g5 [linear solution] +

( 1 - 2R)o.95 [irreversible solution]) (40)

which applies particularly for X < 0 . 5

Solid Diffusion

The rate of diffusion into a uniformly porous spherical particle for this mechanism, closely related to Equation 7, is

where D, is the particle diffusivity. Equation 9 may be applied again to yield an equation analogous to Equation 10

(5 2 ) z - - 30. (2) 7 = rg bt ra

(The boundary conditions for these equations are discussed below.) An analytic solution is available for these relations a t R = 0, but for 0 < R < 1.0 numerical techniques were again applied.

For the case of constant surface concentration and completely irreversible equilibrium, an analytic solution has been proposed by Wicke (22) and others, and Glueckauf and Coates (7) and Vermeulen (27) have proposed, respectively, linear- and quadratic- driving-force approximations. The resulting equations for constant-pattern behavior are:

EXACT. X =

Solution for Irreversible Equilibrium.

1 - - 6 " 1 - exp{--n2[+,N,F(T - 1) +0.64]) (43) r2 n = ~ n2

LIXEAR. X = 1 - exp[-$lN,F(T - 1) - 11 (44)

QUADRATIC. X =

{ 1 - exp[-#,N,p(T - 1) - 0.61])0.5 (45)

where $ is a numerical factor dependent upon R ( $ e = ?r2/15; I)~ and $, are discussed below); n is an integer (one of an infinite series) ; and Npp, the number of transfer units based on the solid-diffusion mechanism, is

u - 150,A u Npp % k,Fa, - = __ - F r2 E

where k,Fa, (= 15D,,i/r2) is the mass-transfer coefficient for solid diffusion.

Solution for Favorable Equilibrium (R < 1). The material-balance relation Y = X again holds in the constant- pattern region, along with the relation (X),=l = X. Intro- duction of N,pT and 6i renders the rate equation dimensionless

(47) azy 2 b y 1 5 ~ - - - + - -

bN,pT amz a bR

VOL. 5 NO. 2 M A Y 1 9 6 6 217

,ND ~ ( ' 7 - I )

Figure 8. kinetics

Breakthrough behavior for particle-diffusion

Lineur coordiiiutes

with the boundary conditioxs

At O i = 0, aY/ilcR = 0

For all a, at I\T,.FT = 0, Y = X = 0

at N,F'T = co, y = X = 1

(49)

(50)

(51)

The condition of constant surface Concentration does not hold, except for R = 0.

The sirriilarity between these equations and those for pore diffusion suggests a similar integration. A procedure was therefore employed which difrcred from the preceding treatment only in the way the boundary condition is applied at R = 1. In this case the condition ('Y)@=i = X was selected; Equation 48 was merely used for a check on the consistency of the calculation and showed agreement always within 0.005. Equations 35 to 38 were modified i n that all X ' s became Y's, and with this change the terrns of Equations 35 to 38 approximate Equation 47. The same iriitial concentration-distribution (Equation 39 and 'l'able 11) \vas assumed here as in the pore-diffusion case ; also, the finite-difference interval on CR was again A = 0.1, while the inrrrval on N,FT varied with R as indicated in Table I.

The Fortran program used i T again part of Appendix IV, on file wit!] the American L)oc~mientatiun Institute, and the nurrmical results are part of Appendix 111.

Figures 8 to 11 preserlt the solutions for this mechanism, i n a EOJXI exactly ;inalogou.r to Figurrs 4 to 7 . I t is noticeable

NpF (T- I )

Figure 9. Breakthrough behavior for particle-diffusion kinetics


that the solid-diffusion curves are more asymmetric, being steeper a t low X and less steep at high X than the pore-diffusion curves at corresponding R. This observation can be explained by analyzing the behavior of solid diffusion and pore diffusion (neglecting external mass transport) at R = 0. At low values of Y solid diffusion is the faster and more efficient of the two mechanisms, in that the solute is free to spread rapidly to all parts of the particle; while: for pore diffusion, the solute is laid down stepwise and immobilized in the particle. At high values of Y, solid diffusion becomes much less efficient, since the unfilled sites in the particle are widely distributed, but for pore diffusion (while it becomes less efficient than previously) the unfilled sites are still localized, thus allowing faster final saturation.

A simple test exists to determine which mechanism i s controlling for a given particle. Examination of Equations 18 and 46 indicates that pore diffusion is directly dependent upon the bulk fluid-phase concentration, lvhile solid diffusion (through A) is relatively independent of this factor. If a system is chosen such that the influent concentration, 6 0 , may be changed without greatly affecting the equilibrium, the response of the breakthrough curve upon varying G O is a good indication of the controlling mechanism.

Approximate Solutions for Favorable Equilibrium (R < 1). l h r e e common approximations are made to Equation 47: Glueckauf and Coates's linear-driving-force approximation (7), Vermeulen's quadratic-driving-force approximation (27), and the symmetrical reaction-kinetic treatment. Each of these methods involves a different approximation to the slope of the breakthrough curve, as follows:

QUADRATIC (53)


50 -

20 -

I 2 5 I O 20 50 100 200 500

IO0 I I

50 -

20 - Ln 0

5 -

0 . 5 k I 2

Npore, F

Figure 10. Midpoint slope of breakthrough curves for particle-diffusion kinetic!;

Curve for R = 1 from Thomas

0.6 0' I- Figure 1 1. Breakthrough behavior for particle-diffusion kinetics relative to unit midpoint slope

dY 241[X(1 - Y) - RY(1 - X)] - (54) - KINETIC - - -

These expressions may be improved by employing a multiplier, $, for NT, which is the ratio of the exact midpoint slope (determined from the numerical integration) to the midpoint slope calculated by the above expressions. Values of have been calculated and are presented in Figure 12. The curve of $1 for the linear and reaction-kinetic approximations is matched closely by

dNpFT 1 + R

For

I - < 1 . 0 " = R1.06 + 1.125(1 - R)1.06 -

the quadratic expressmion the correction is

1 $ ' R'.' + 1.688(1 - R)

(55)

(56)

0 0.2 0.4 0 6 0.8 i.0 R

Figure 1 2. rate expressions

Numericul Quctors in solid-diffusiaii

Figure 13. Cornpurison of numerical and empirical solutions for particle-diffusion kinetics a t R = 0.3

Equations 52, 53, a i d 51 (including .i/j integrate, respectively, to

(523)

(1 + K j - ZKXI -- (1 - KjX? (1 + R) -- 2RXz - (1 - R)Xf

In ._ .-- +

' / Q l , 5 biQ. 2 M A Y 1966 219

for the run, often a better value can be had from an average of all runs or from separate eq iilibrium determinations. If the value of q* given by a material balance on a run disagrees markedly with the best value obtained from other sources, interpretation by the foregoing procedure may give invalid results.

In iiiterpreting the data for a two-rate-parameter case, one should first recall that it may not be possible to determine two unique values from a single run. For example, in “deep-bed” cases for R = 1 , both external mass transfer and particle diffusion (the latter approximated by a linear rate equation) give the well known Anzelius-Schumann-Furnas result (70, 78) ; the two rate parameters combine to a single apparent over-all coefficient, and there are an infinite number of pairs of the individual rate parameters that would give the same breakthrough curve. For shallow beds at R = 1 , the Rosen solution applies, but still the breakthrough curve is highly insensitive to N ~ / N ~ F .

For the two-rate case with R << 1 , inspection of Figure 2 shows that there is a definite change in shape of the computed breakthrough curves, which can be used to determine an approximate value for Npore,F/NIF, but a variation of less than 50% in this ratio would be difficult to detect when comparing experimental data with the solutions for this model. The recommended method of interpretation, based on the observation that all of the curves in Figure 2 pass through the two points X = 0.14 and X = 0.80, is as follows: Find the experimental values of (T - 1) a t these two X values; denote the difference between them by 6 . Then find l/Npore,F + l / N p = 6/1.75. A plot of X us. (T - l)/(l/N,,,,,p + l/N,F) can then be matched with Figure 2, for an estimate of Npore,F/N~F from which both N’s may be obtained.

If one has an acceptable value for NfF-e.g., from a general- ized correlation of mass-transfer coefficients-the two-parameter case has only one parameter unknown. As indicated by Equation 25, one can calculate T - 1 - (In X 4- l ) / N p for each data point, and a plot of +(X) against this quantity should give a straight line having the slope Npore,F. This last method of interpretation utilizes the fact that the solution including external mass-transfer resistance is the same as the solution without this resistance, except that the term (In X - 1) (l/N/F) is subtracted from T. This method of including the external mass-transfer resistance could not be used as an approximation for the Rosen solution at R = 1 , but it is be- lieved to be satisfactory at low values of R (< 0.3) for both the pore-diffusion and the solid-diffusion cases. Since no exact solutions exist, the error cannot be estimated.

This calculation is likely to follow one of the sequences given previously ( 7 7, 27). For the desired or proposed flow rate and column volume, the N value is calculated by Equation 17 or 46, with prior knowledge of the mass- transfer behavior. V8/stoie (= Au, or q o * p b u / c o ) can be determined from the column volume and the known concentration factors; and tstoic is Vstoic/F. For a known R, the breakthrough curve can be interpolated or calculated to give X as a unique function of N(T - l ) , NT, or T. With t = T X tStoio + (ue /F) , or V = T X Vstoio + U E , the abscissa of the pre- dicted breakthrough curve can be converted to practical time or fluid-volume units.

Nonisothermal Experiments. Constant-pattern behavior for solute breakthrough implies constant pattern for the thermal wave also. If an experimental breakthrough curve is available for the thermal wave, the equivalent value of R (relative to G O and q o * a t the feed temperature) can be determined for each X value on the breakthrough curve. The diffusivity and the linear flow velocity can be considered con-

DESIGN CALCULATION.

[In the relation in (20), Equation 34, corresponding to Equa- tion 53a, the denominator of the first term should read ( K 2 - l ) rather than ( K 2 + l ) . ] These relations were used \. I 1 a value for R of 0.3 in Figure 13, which illustrates their app 3 -h to the true numerical solution. The quadratic-driving o~ ;e approximation appears to yield the best fit, and should be wry suitable if an algebraic interpolation is desired.

Applications

Isothermal Experiments. INTERPRETATION OF DATA. The theoretical breakthrough curves presented above may serve to determine whether pore or solid diffusion has been controlling, or to provide numerical values of the respective mass-transfer coefficients for a given set of experimental breakthrough data. First the experimental isotherm must be fitted by the Langmuir relation, and a suitable R found. If the equilibrium has not been measured previously, it can often be deduced from breakthrough runs by the methods of Hiester et al . (9 ) .

Interpretation of data is easiest for the cases in which there is only one unknown rate parameter; these correspond to all the solutions discussed in this paper, except the case of two rate parameters, leading to Equation 25, which are considered later.

In all the simpler cases the experimental datz consist of a set of values of X us. t . For each value of X, knowing R, one can determine Np,,,,F(T - 1 ) or NpF(T - 1 ) from the appropriate plot (Figure 3 ,4 ,7 , or 8) and (T - 1) from t . One then plots Np,,,,F(T - 1 ) or N,F(T - 1 ) against (T - l ) , using linear coordinates. The points should fall on a straight line, indicating how well the data fit the model, and giving the diffusivity (Dpore or Do) from the slope Npore,F or NpF. In cases where the points show curvature, local slopes can be obtained to give local values of N ; when these are compared for different runs, an idea of why the model is defective may be obtained. For example, a concentration dependence of N might reveal the way in which the concentration was treated improperly in the model.

Alternatively, if R is known, the X value at T = 1 can be determined from Figure 4 or 8, and the experimental time or volume to reach this X value (less the filling time, ut/F, or void volume, UE) can be identified as tstolc or V,,,,, (where V is solution volume). The midpoint slope is measured on the experimental plot, as dX/dt or dX/dV, and multiplication by tatoio or V,,,,,, respectively, converts the midpoint slope to the dimensionless form in Figure 6 or 9. The numerical value can therefore be used in this figure, with the appropriate R, to provide an “experimental” value of N. D,,,, or D, can then be calculated by way of Equation 17 or 46; and the reason- ableness of the resulting value (and the extent of its constancy for different particle radii and different fluid-phase concentrations) can be used to determine the likelihood that either pore diffusion or solid diffusion is the rate-controlling mechanism for the experiment in question.

The reader may find other ways of interpreting the data, but the methods proposed avoid a trial selection of the rate parameters. Also, in the first method, the quality of the fit over the entire range of X is displayed.

In carrying out the procedures just outlined, it is assumed that the various parameters are known exactly, except the rate parameter. The greatest uncertainty often occurs with the equilibrium capacity, q*, which leads to an uncertainty in A. Although q* can be obtained from an over-all material balance


stant, with N T thus continually proportional to throughput volume or to elapsed time, unless the percentage change in absolute temperature is relatively large.

The breakthrough curve can then be broken into AX seg- ments, usually ranging between X = 0.01 and 0.99, with cuts a t X = 0.10, 0.50, and 0.90. In each segment, an average local value of R can be used, and the respective A(NT) value recorded. The successive increments can be added, plotted on a probability scale of X for smoothing, then transferred to a linear plot, on which graphical integration should be per- formed to determine or Vstoic. From this point on, the analysis is the same as fix constant R curves under isothermal conditions.

In regard to alternative approaches, Leavitt (73) has presented a general approach to the nonisothermal fixed-bed problem, while Acrivos (2) has outlined a suitable numerical- solution procedure that utilizes the method of characteristics.

Appendix I. Application to Ion Exchange Materials

The commercially available organic-based resins in current use, both cationic and anionic, generally do not contain internal pores of the type under discussion. The gel composition of these resins may be viewed as a stationary water phase, inter- laced at the molecular scale by polymer chains containing hydrophilic functional groups. The polymer structure in which it is imprisoned, and the nature of the functional groups “dissolved” in it, make the gel properties different from those of liquid outside the particle; and the only boundary between the free liquid and the gel or “solid” phase is a t the outer surface of the particle.

However, two important classes of ion exchangers do possess uniform internal pore structures; in such materials the pores have mean radii in the range of to 10-6 cm., and bring all parts of the solid structure into much closer contact with the liquid. Such materials are the inorganic greensands and zeolites, and the new ‘‘ macroreticular” organic resins.

Figure 1 illustrates the way by which pores subdivide the “solid” portion of a n ion exchanger particle. In this case the pore surfaces generally exhibit no special mobility, but the

entire solid may conduct the exchanging ionic species. Thus the two interwoven networks of pore phase and solid phase may provide alternative diffusion paths for the solute, and either path may be rate controlling in certain cases.

The above treatment for adsorption therefore applies equally well to two-component ion exchange in porous or macroreticular materials. In order to account for the prac- tically constant exchange capacity of the solid in the ion exchange case, a slightly different nomenclature is conven- tionally used. The dimensionless local concentrations become, for the liquid and the solid, respectively,

x = c/co Y = q / Q

where G and q are local concentrations of the ion of interest, Co is the total equivalent concentration of ions per unit volume of liquid, and Q is the total equivalent concentration per unit mass of solid. The separation factor then becomes

where x and y again denote mean free-liquid and solid concentrations (dimensionless) a t any given cross section of a fixed bed. The distribution parameter, A, is now Qpb/Co. The same material-balance relations apply as in adsorption, and again lead to the constant-pattern property y = x. The foregoing algebraic and numerical results for adsorption thus require only the substitution of x , x, y, y, and r for X, X, Y , Y, and R, in order to fit corresponding situations in ion exchange.

Appendix II. Pore-Diffusion Irreversible-Equilibrium Case Values of @ ( X )

X O W ) X N X ) 0 -1,205 0.01 -1.189 0.10 -1.033

0.60 +O ,098 0.70 0.422 0.80 0.815

0.20 -0,847 0 . 9 0 1.342 0.30 -0.641 0 .99 2.200 0.40 -0,425 1 .oo 2.493 0.50 -0.179

Appendix 111. Computer Results For Pore Diffusion

R = 0.2 R = 0.7 N(T - 7 ) X N(T - 7 ) X

-1.63 0.0111 -2.50 -1.13 0.1497 -2.00 -0.63 0.3243 -1.50 -0.13 0.4885 -1 .oo

0.37 0.6426 - 0 , 50 0.87 0.7641 0 1.37 0.8634 0.50 1.87 0.9335 1 .oo 2.37 0.9789 1.50 2.87 0.9974 2.00

2.50 3.00 3.50

0.0102 0,0441 0.1228 0.2411 0.3793 0.5190 0.6476 0.7586 0,8488 0.9174 0,9638 0,9882 0.9967

R = 0.3 R = 0.4 N(T - 7 ) :X N(T - 7 ) X N ( T - 7 ) X N(T - 7 ) X

-3.57 -3.07 -2.57 -2.07 -1.57 -1.07 -0.57 -0.07

0 .43 0.93 1 .43

0.01 01 0,0230 0.0500 0 ,0989 0 . ‘I 736 0.271 1 0 . ,3832 0 . ,4998 0.6124 0.’7146 0.8023

1.93 2.43 2.93 3.43 3.93

0.8732 0.9263 0.9616 0.9817 0.9918

-4.85 -4.35 -3.85 -3.35 -2.85 -2.35 -1.85 -1.35 -0.85 -0.35

0.15

0.0100 0.0171 0.0290 0.0485 0.0786 0.1225 0.1820 0.2566 0.3433 0.4375 0.5337

0.65 0.6269 1 . I 5 0.7127 1.65 0.7880 2.15 0.8506 2.65 0.9003 3.15 0.9366 3.65 0.9614 4.15 0.9773 4.65 0,9870 5.15 0.9927

(Continued on page 222)

VOL. 5 NO. 2 M A Y 1 9 6 6 221

Appendix 111 Continued

For Pore Diffusion R = 0.5 R = 0.6

N(T - 7 ) X N(T - 7 ) X N(T - 7 ) X N(T - 7 ) X -6.57 0.0100 -0.07 0.4905 -9 .00 0.0100 0 0.4979 -5.57 0.0209 0.43 0.5678 -8.00 0.0167 0.50 0.5578 -5.07 0.0301 0.93 0.6423 -7.00 0.0280 1 .oo 0.6162 -4.57 0.0431 1.43 0.7116 -6.00 0.0466 1.50 0.6719 -4.07 0.0612 1.93 0.7735 -5.00 0.0761 2.00 0.7236 -3.57 0.0858 2.43 0.8269 -4.00 0.1213 2.50 0,7706 -3.07 0.1182 2.93 0.8771 -3.00 0.1864 3.00 0.8122 -2.57 0.1596 3.43 0.9064 -2.50 0,2273 4.00 0.8788 -2.07 0.2106 3.93 0.9335 -2.00 0,2735 5.00 0.9251 -1.57 0.2709 4.43 0.9536 -1 .50 0.3247 6.00 0.9552 -1.07 0.3391 5.43 0.9784 -1 .oo 0,3799 7.00 0.9738 -0.57 0.4132 6.43 0,9904 -0.50 0.4381 9.00 0.9916

N(T - 7 )

-10.98 -12.98

-8.98 -6.98 -5.98 -4.98 -3.98 -2.98 -1.98

-0.98 -0.48

-1.48

R = 0.7 X N(T - 7 ) X

0.0100 0.02 0.5019 0.0201 0.52 0,5451 0.0405 1.02 0,5877 0.0796 1.52 0,6292 0.1100 2.02 0.6690 0.1498 3.02 0.7419 0.2004 4.02 0,8040 0.2624 5.02 0,8547 0.3351 6.02 0.8944 0.3748 8.02 0.9466 0.4161 10.02 0.9741 0.4587 12.52 0.9899

R = 0.8 N(T - 7 ) X N(T - 7 ) X -12.66 0,0564 -0.16 0.4877 -11.16 0.0770 0.84 0.5429 -1n.16 0.0944 1.84 0.5970 .. .

-9.16 0.1152 2.84 0.6491 -8.16 0.1398 3.84 0.6979 -7.16 0.1687 4.84 0.7428 -6.16 0.2020 5.84 0.7832 -5.16 0.2400 6.84 0.8189 -4.16 0.2826 7.84 0.8500 -3.16 0.3294 8.84 0.8766 -2.16 0.3798 9.84 0.8991 -1.16 0.4330 11.84 0.9336

For Solid Diffusion R = 0.7 R = 0.2

N(T - 7 ) X N(T - 7 ) X N(T - 7 ) X N(T - 7 ) X -1.22 -1.12 -1.02 -0.92

-0.62 -0.42 -0.22 -0.02

-0.82

0.18 0.38 0.58

0.0100 0.78 0.7871 0.1366 1.28 0,8437 0.2300 1.78 0.8851 0.3032 2.28 0.9154 0.3633 2.78 0.9377 0 4579 3.28 0.9541 _ . - - 0.5336 5.78 0.9662 0.5945 4.28 0.9751 0.6454 4.78 0.9816 0,6888 5.28 0.9865 0.7262 5.78 0.9900 0.7587

R = 0.3 N(T - 7 ) X N(T - 7 ) X

-2.29 0.0100 0.71 0.7309 -2.09 0.0275 1.21 0.7912 -1 .a9 0.0637 1.71 0.8382 -1.69 0.1204 2.21 0.8746 -1.49 0.1893 2.71 0.9029 -1.29 0.2610 3.21 0,9248 -1.09 0,3299 3.71 0.9418 -0.89 0.3940 4.21 0.9550 -0.69 0.4523 5.21 0.9731 -0.49 0.5052 6.21 0.9839 -0.29 0.5530 7.14 0.9900

0.21 0.6532

-1.61 -1.41 -1.21 -1.01 -0.81 -0.61 -0.41 -0.21 -0.01

0.19 0.39 0.89

N(T - 7 ) -3.23 -2.73 -2.23 -1.73 -1.23 -0.73 -0.23

0.27 0.77 1.27 1.77 2.27

0.0100 1.39 0.8308 n .0798 1.89 0.8724 0.i922 2.39 0,9038 0,2961 2.89 0.9275 0.3837 3.39 0.9453 0,4577 3.89 0.9588 0.5209 4.39 0.9689 0.5753 4.89 0.9766 0.6228 5.39 0.9823 0.6643 5.89 0.9866 0.7010 6.39 0.9900 0.7753

R = 0.4 X N(T - 7 ) X

0.0100 0.0349 0.0988

2.77 0.8803 3.27 0.9045 3.77 0,9240

0,2060 4.27 0,9395 0.3304 4.77 0.9518 0,4484 5.27 0.9617 0.5511 5.77 0.9695 0.6371 6.27 0.9758 0,7079 6.77 0.9807 0,7656 7.27 0.9847

0.8500 8.20 0,9900 0.8123 7.77 0.9878

R = 0.5 R = 0.6 N(T - 7 ) X N(T - 7 ) X N(T - 7 ) X N(T - 7 ) X

-4.70 -4.20 -3.70 -3.20 -2.70 -2.20 -1.70 -1.20 -0.70 -0.20

0.30 0.80

0.0100 0.0204 0.0405 0,0759 0.1312 0.2056 0.2923 0.3827 0.4700 0.5501 0.6212 0.6830

1.30 ...

1.80 2.30

3.30 2.80

3.80 4.80 5.80 6.80 7.80

9.65 8.80

0.7360 0.7810 0.8187 0.8504 0,8767 0.8985 0.9315 0.9539 0,9690 0.9792 0.9860 0,9900

-6.90 -5.90 -4.90 -3.90 -3.40 -2.90 -2.40 -1.90 -1.40 -0 .90 -0.40

0.10

0.0100 0,0231 0.0518 0.1078 0.1491 0.1992 0.2567 0.3193 0.3843 0.4491 0.5117 0.5705

0.60 i . i o 2.10 3.10 4.10 5.10 6.10 7.10 8.10 9.10

10.10 11.90

0.6248 0.6739 0.7568 0.8210 0,8694 0.9052 C.9316 0,9507 0.9646 0.9746 0.9818 0.9900


Appendix 111 Continued For Solid Diffusion

N(T - 7 ) -10.70 -9.70 -8.70 -7.70 -6.70 -5.70 -4.70 -3.70 -2.70 -1.70 -0.70

0.30

R X

0 0100 0 0163 0 0267 0 0431 0 0685 0 1061 0 1583 0 2258 0 3061 1 3842 0 4838 0 5694

-__ = 0.7 N(T - 7 )

1 . 3 0 2.30 3.30 4.30 5.30 6.30 7.30 9.30

11.30 13.30 15.70

X 0,6471 0.7149 0,7722 0.8196 0.8581 0,8890

0,9479 0,9689 0.9815 0.9900

0.9135

N(T - 7 ) -18.30 -16.30 -14.30 -12.30 -10.30

-8.30 -6.30 -5.30 -4.30 -3.30 -2.30 -1.30

R X

0.0100 0.0170 0.0291 0.0491 0.0813 0.1307 0.2009 0.2440 0.2920 0.3439 0.3984 0.4542

= 0.8 N(T - 7 )

-0.30 0.70 1.70 2.70 4.70 6.70 8.70

10.70 12.70 16.70 20.70 23.30

X 0.5098 0.5640 0.6156 0.6638 0.7481 0.8153 0,8667 0.9050 0.9328 0.9669 0.9839 0.9900

Acknowledgment The authors express their gratitude to Alan S. Foss for many

helpful suggestions in regard to explanatory comments; to Burris B. Cunningham for a careful review of the manuscript; to Peter N. Tenney for assistance in programming the computer calculations; to Harry McViegh for trial calculations involving combined res,istances, made on an N.S.F. under- graduate research project a t the University of Pennsylvania; to J. C. Shorrock of Battelle Institute, Geneva, for helpful dis- cussions; and Edith P. Taylor for preparation of manuscript.

P 4 40 * Q Qm r rz 7, r m R t tstoio

T

V V X

X

X

X Y

Y

Y

Nomenclature aP

CO

co DP = particle diffusivity D,,,, = pore diffusivity F = volumetric flow rate g k,, = mass-transfer coefficient for external fluid-phase

transport kpF = mass-transfer coefficient for solid (particle-phase)

diffusion K = Langmuir equilibrium constant N = number of transfer units N,, = number of transfer units, solid-diffusion controlling N,,,,, , = number of transfer units, pore-diffusion controlling

= partial pressure of solute = particle-phase concentration = saturation level of particle-phase concentration

= external particle area per unit packed volume

= concentration of solute in feed = total “equivalent” concentration of ions in liquid

C = concentration of solute in fluid phase

= a function of time, in Equation 14

= particle-phase exc6ange capacity = ultimate sorptive capacity of particle phase = radius of any spherical shell within the particle = radius of interior boundary = outer radius of particle = equilibrium parameter for ion exchange = dimensionless radius; r/r , = equilibrium parameter for adsorption = time elapsed from start of run = time elapsed before a stoichiometric amount of

solute has entered column = throughput parameter, or ratio of actual volume of

effluent to stoichiometric volume. This notation, adopted from Helfferich (8), corresponds to Z in ( 7 7 and i?7), e/z in (20), t / s in (9 and 70)

= column volume = solution volume = dimensionless local fluid-phase concentration in ion

exchange = dimensionless effluent concentration in ion exchange = dimensionless local fluid-phase concentration in

= dimensionless effluent concentration in adsorption = dimensionless local solid-phase concentration in ion

= dimensionless mean concentration of solid phase in

= dimensionless local solid-phase concentration in

adsorption

exchange

ion exchange

adsorption

YI = dimensionless solid-phase concentration at = 1,

Y = dimensionless mean solid-phase concentration in

Y = a constant, in Equation 39 6 = T interval between X = 0.14 and X = 0.80

= void fraction of bed x = finite-difference interval on A = distribution parameter, or ratio of particle- to fluid-

corresponds to De in (9-

in equilibrium with X

adsorption

E

phase concentration; 77, 20)

= finite-difference interval on NT = (1 - X)’i3, in Equation 26 and later = bulk density of bed

@(X) = function defined in Equation 26 ic = correction factor for midpoint slope in solid-diffu-

* (superscript) = value in equilibrium with accompanying

! P b

sion approximations

value of another variable

literature Cited (1) Acrivos, A., Chem. Eng. Sci. 13, l(1960). (2) Acrivos, A., Ind. Eng. Chem. 48, 703 (1956). (3) Bohart, G. S., Adams, E. Q., J. A m . Chem. Soc. 42, 523 (1920). ( 4 ) Cooney, D. O., Lightfoot, E. N., IND. ENC. CHEM. FUNDA-

( 5 ) Cooper, R. S., Ibid., 4, 308 (1965). (6) Eagleton, L. C., Bliss, H., Chem. Eng. Progr. 49, 543 (1953). (7) Glueckauf, E., Coates, J. I., J . Chem. Sod. 1947, p. 1315. (8) Helfferich, F. G., “Ion Exchange,” p. 472, McGraw-Hill,

New York, 1962. (9) Hiester, N. K., Radding, S. B., Nelson, R. L., Jr., Vermeulen,

T.J.1.Ch.E.J. 2,404(1956). (10) Hiester, N. K., Vermeulen, T., Chem. Eng. Progr. 48, 505

(1952). (1 1 ) Hiester, N. K., Vermeulen, T., Klein, G., “Perry’s Chemical

EnTineers’ Handbook,” 4th ed., Sec. 16, McGraw-Hill, New

MENTALS 4, 233 (1965).

YoFk, 1963.

No. 14, 97 (1954). (12) Lapidus, L., Rosen, J. B., Chem. Eng. Progr. Symp. Ser. 50,

(13) Leavitt, F. W., Chem. Eng. Progr. 58, No. 8 , 54 (1962). (14) Michaels, A. S., Znd. Eng. Chem. 44, 1922 (1952). (15) Rosen, J. B., Ibzd., 46,1590 (1954). (16) Rosen, J. B., J . Chem. Phys. 20, 387 (1952). (17) Rosen, J. B., Ph.D. thesis in chemical engineering, Columbia

(18) Thomas, H. C., Ann. N . Y. Acad. Sct. 49, 161 (1948). (19) Thomas, H. C., J . Am. Chem. SOL. 66, 1664 (1944). (20) Vermeulen, T., Aduan. Chem. Eng. 2, 148-208 (1958). (21) Vermeulen, T., Ind. Eng. Chem. 45, 1664 (1953). (22) Wicke, E., Kolloid-2. 167, 289 (1939).

University, 1952.

RECEIVED for review September 24, 1963 RESUBMITTED December 6, 1965

ACCEPTED December 6, 1965

30th Annual Chemical Engineering Symposium, University of Maryland, College Park, Md., November 1963. Work financed by Grant 08042 from the National Institute of General Medical Sciences, U. S. Public Health Service. Material supplementary to this article has been deposited as Document No. 8806 with the AD1 Auxiliary Publications Project, Photoduplication Service, Library of Congress, Washington 25, D. C. A copy may be secured by citing the document number and by remitting $1.25 for photoprint or $1.25 for 35-mm. microfilm. Advance payment is required. Make checks or money orders payable to Chief, Photoduplication Service, Library of Congress.

VOL. 5 N O . 2 M A Y 1 9 6 6 223

and solid-diffusion kinetics in fixed-bed adsorption under constant

Documents

Transcript of and solid-diffusion kinetics in fixed-bed adsorption under constant