PORE- AND SOLID-DIFFUSION KINETICS IN FIXED-BED ADSORPTION UNDER CONSTANT- PATTE RN CON D IT10 N S KENNETH R. H A L L , ’ LEE C. EAGLETON,* ANDREAS ACRIVOS,3 A N D THEODORE VERMEULEN Department of Chemical Engineering, University of California, Berkeley, Calif.
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 b y 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.
c
oLuMN-performance studies for fixed-bed adsorbers and ion 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 ratecontrolling 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‘nfavorableequilibrium 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. T h e Langmuir equation can be written :
1
Present address, University of Oklahoma, Norman, Okla. Pennsylvania, Philadelphia, Pa. Present address, Stanford University, Stanford, Calif,
* Present address, University of 3
212
I&EC FUNDAMENTALS
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
y*
=
x
R
+ (1 - R)X
(4)
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
ADSORBED LAYER ON S U R F A C E
Figure 1 . particle
FLUID PHASE I N PORES
Diffusion paths within adsorbent
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 masstransfer mechanism and a given value of R < 1, a single theoretical curve for b:reakthrough, on the appropriate coordinates, represents all operating conditions. O n 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 constantpattern 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. Reaction a t the phase boundary D. Solid-phase diffusion I n 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 interface 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. I n 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
will be assumed a t 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 reduced to the equation-e.g., Equation 16-95a ( 7 7)-is simple statement
X = Y
(6)
Rosen (77): Lapidus and Rosen ( 7 2 ) ,and Cooney and Lightfoot (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 a n 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 considering the similarity of a fixed bed to a countercurrent moving bed operating a t 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. Assumptions. 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 a n exact model for rate behavior. Moreover, axial dispersion [treated by Acrivos (7) ] \vi11 be assumed to be too small to contribute to the shape of the breakthrough curves. 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 a n effective pore diffusivity [see ( 7 7 ) or (27) VOL. 5
NO. 2 M A Y 1 9 6 6
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 a t 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
=
rs
1
Yr2dr
where rs is the outer radius of the particle. with respect to time gives
Differentiation
which, upon combination with Equation 7, leads to
d Y --
3DPore(1- e )
dt
(d bX
(1 0)
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 constantpattern case, and is outlined here. Furthermore, as the addition 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 rz < 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
Y (= X)
= (1
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 fluidphase 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
~
ra3
or ri/ra
p0re.F +
- ri3
r63
=
T- I. I”
- x ) ’ /=~ t
(11)
Hence the problem becomes one of evaluating the behavior of r,. T h e derivative bY/bt in Equation 9 is zero for r , < r < ra, and in that range Equation 7 becomes:
the number of pore-diffusion mass-transfer units (or porediffusion NTU) is N p o r eE , ~ 15 Dpor,(l - e ) v / r , 2
(17)
and the external N T l j is N ~ F kjFa,v/F
(18)
Here e is the void fraction, and F the volumetric flow rate. Combination of Equations 6, 11, 13 (with r = r 8 ) , 14, 16, and 17 with Equation 10 now gives:
from which
r2(dX/br)L= const. With the limits X = X , a t r tion of Equation 13 gives:
=
ra and
=
g
X
=
(13)
0 a t r = ri, integra-
dX - 3Npore.~ Xs - 3Npore,~ Xs dT 15ra l/r, - l/r8 15 t-’ - 1
(19)
With Equations 6, 16, and 18 used to replace Y , t, and k , ~ , Equation 15 yields
X8
g =
I t now becomes necessary to introduce the external masstransfer 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 (77), 214
l&EC FUNDAMENTALS
When this expression is introduced into Equation 19, the final differential equation results:
R
R-l Figure 3. grid
&+I
Forward-difference calculation
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.
Lrn
(1
-
X)dT =
so'
T dX = 1
(23)
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:
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 . a ,gives Transformation of Equation 27, again for NJF the following expression explicit in X :
-
Equation 25 may be rearranged into several equivalent forms. T h e form most useful for graphical presentation of the solution is:
T
-1
=
(i + L, [
@(XI
N p o r e . ~NIF
+ N p o r e , ~ I N(Inf ~X + 1) N p o r e , ~ / N f+ ,~ 1
1
(29) 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
(27)
I n 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-
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 a t 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 =
x.
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:
15R
[R
ax
PX
2
ax
+ (1 - R)XI2bNT - am2 g z +
VOL. 5
NO. 2
M A Y 1966
(30) 215
For one boundary condition, Equation 10 may be put into dimensionless form
which must hold along with tions are :
(X),=1 = X . T h e other condi-
At @ = 0, dX/d@ = 0
(32)
At all @, for NT = 0, X = 0
(33)
for NT = co, X = 1
(34)
Equation 30 is amenable to numerical integration on a highspeed 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 :
Npore,dT-I) Figure 5. Breakthrough behavior for pore-diffusion kinetics Ordinate on probability scale
result. 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 ) . T h e 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 a t 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, constantpattern results do not apply within 0.01 on X below N = 50 to 80. Modifications of the calculation method were required a t 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 a t each NT value. A linear - ~ ( X O , Z ,NT extrapolation of the form XO,NT= x0.1,~~ xo.1,~~) was chosen, with cy = 1 ; in retrospect, cy = 0.5 would give faster convergence but would not affect the final
Table 1.
R
=
NT Convergence Interval, 1 (at y 0.1) Pore Solid Diffusion Diffusion
0.8
216
@i=
1.0, Equation 31 was applied to obtain
was determined by numerical integration so as to check the boundary condition (X),=1 = X. This check always showed agreement to \tithin 0.010. T o set this program in motion, the integration was begun a t X = 0.01; for numerical convenience it \\as terminated a t 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) expression Yzis the value of Y in equilibrium with X ,
In this 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 Documentation Institute. RESULTS.The pore-diffusion solution is presented graphically 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 II.
R
Constants in Assumed Initial Concentration Distribution
YI
Y
0.0917 0.0481
27 90 12.25
0
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7
At
XI.O,NT+~; subsequently the volume-average value Y ( = X)
0.0001
0:001
0.0002 0.005
0.001 0.002 0.005
0.01
0.01
0.01
0.01 0.01
0.001
0.01 0.01
l&EC FUNDAMENTALS
0.01
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
0.0326 0.0246 0.0198
0.0166 0.0142 0.0125
7.50 5.04
3.47 2.39 1.58 0.946
[ ( d X ( d T ) ~ .in ~ l 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 a t 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) :
T
- To.6 =
1
{ (2R)O.g5[linear solution]
(dX/dT) o . 5
+
- 2R)o.95[irreversible solution])
(1
which applies particularly for X
(40)
< 0.5
Solid Diffusion
T h e rate of diffusion into a uniformly porous spherical particle for this mechanism, closely related to Equation 7, is Figure 6. Midpoint slope of breakthrough curves for porediffusion kinetics Curve for
R = 1
from Thomas
where D, is the particle diffusivity. Equation 9 may be applied again to yield an equation analogous to Equation 10
R=
z
-
0.2-0.4
/
X
1
7
(5
2)
= rg
-
=
6 r2 n
LIXEAR.
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 a n abscissa of Npore,p(T - l ) , which provides complete generality (with respect to residence times and mass-transfer coefficients) for each curve a t its respective R value. Figure 4 utilizes rectangular coordinates; it displays the S shape that is experimentally 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
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. Solution for Irreversible Equilibrium. 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 quadraticdriving-force approximations. T h e resulting equations for constant-pattern behavior are: EXACT.
Figure 7. Breakthrough behavior for pore-diffusion kinetics relative to unit midpoint slope
(2)
30.
bt
0.5- 1.0 7
" 1- exp{--n2[+,N,F(T =
n2 ~
X = 1
QUADRATIC. X
- exp[-$lN,F(T
- 1) +0.64]) - 1) - 11
(43) (44)
=
{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 a n integer (one of a n infinite series) ; and Npp, the number of transfer units based on the solid-diffusion mechanism, is Npp % k,Fa,
u - 150,A u -
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 constantpattern region, along with the relation (X),=l = X. Introduction of N,pT and 6i renders the rate equation dimensionless
1
5 bN,pT
~ azy - +- - - 2 b y
amz
VOL. 5
a bR
NO. 2 M A Y 1 9 6 6
(47)
217
NpF ( T - I ) Figure 9. kinetics
Breakthrough behavior for
particle-diffusion
Ordinate on probability scale
,ND ~ ( ' -7 I ) Figure 8. kinetics
Breakthrough behavior
for
particle-diffusion
Lineur coordiiiutes
with the boundary conditioxs
At Oi = 0, aY/ilcR = 0
For all
a,a t I\T,.FT=
(49)
0, Y = X = 0
a t N,F'T = co, y =
X
=
(50)
1
(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 a t R = 1. I n 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 218
I&EC FUNDAMENTALS
that the solid-diffusion curves are more asymmetric, being steeper a t low X and less steep a t high X than the pore-diffusion curves a t 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)
IO0
I
I
50 20 Ln
0
5-
0.2
0 0 . 5 k 2 I
5
IO
20
100 200
50
500
Npore, F
Figure 10. Midpoint slope of breakthrough curves for particle-diffusion kinetic!;
0.4
R
06
0.8
i.0
Figure 1 2. Numericul Quctors in solid-diffusiaii rate expressions
Curve for R = 1 from Thomas
I-
0.6 0'
Figure 1 1. Breakthrough behavior for particle-diffusion kinetics relative to unit midpoint slope
dY
241[X(1 KINETIC - - dNpFT
- Y) - RY(1 - X ) ] -
1 + R
(54)
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. T h e curve of $ 1for the linear and reaction-kinetic approximations is matched closely by I
"=
R1.06
+ 1.125(1 -
-
