Mixed-Resistance Diffusion Kinetics in Fixed-Bed Adsorption under Constant Pattern Conditions Robert D. Fleck, Jr.,’ Donald J. Kirwan, and Kenneth R. Hall” Chemical Engineering Department, rniversity of Virginia, Charlottesville, V a . 22901
Numerical solutions of the equations describing adsorption (or ion exchange) in fixed-bed columns are presented for various combinations of mass-transfer resistances under constant pattern conditions. Breakthrough curves have been developed for pore, solid, and external diffusion acting separately and in combination for adsorption equilibrium described by the Langmuir or Freundlich isotherms.
A n a l y s i s or design of a fixed-bed column for adsorption (or ion exchange) involves predicting the coiiceiitration of t’he solute in the effluent stream as a function of time or total volume of effluent. The shape and time of appearaiice of this concentration history or “breakthrough curve” has a significant influence on the operation of the column. The brealrthrough curve is generally sigmoid, but its esact shape is a function of the stoichiometric capacity of the sorbent, the coluriin volume, the volumetric flow rate, the volumetric throughput, particle diameter, the equilibrium behavior of the system and the controlling mass transfer resistance(s). Earlier descriptioiis of breakthrough curves have been generally restricted t o approaches assuming either contant pattern or proportionate pattern coiiditions. The choice is dictated by the equilibrium behavior of the system. This study deals {Yith constant pattern, favorable equilibrium behavior. The existence of constant patterns has been demoiistrated experimentally and conditions for their appearance have been listed by Cooney arid Lightfoot (1965), Rosen (1954), and Lapidus and Kosen (1954). Cooper (1965) studied the transition from a step function to conetalit pattern behavior when passing from a “shallow” to a “deep” bed. Several specific studies have involved the constant pattern assumption. Esact solutions for solid diffusion controlling with irreversible equilibrium were obtained by Wicke (1939) and by Ijoyd, et al. (194ij. Gluekauf and Coates (1955) and Vermeuleri (1958) proposed empirical driviiig force relationships. Hall, et al. (1966), studied pore diffusion and solid diff usiori controlling cases whik Vermeuleri aiid Quilici (1970) related the Hall results to earlier 11-ork using driving force approximations. This study is an extension of the work by Hall, et al. Specifically, breakthrough curves will be generated via numerical techniques for several cases involving combined mass-transfer resistances uiider t h e influence of two different types of favorable equilibrium.
If the curve is conves downward, the equilibrium is termed favorable and constant pattern behavior is possible. An intermediate case is linear equilibrium and a limiting case is irreversible equilibrium. In these latter situations the curve is diagonal aiid horizontal, respectively. Several matheinatical relationships are available which describe this equilibrium, but this paper will employ only two commonly observed isotherms: Langmuir and Freundlich. If one assumes t h a t the adsorption sites have identical energies and that there is no interaction between adsorbed atoms, the Langmuir equation is
+
(1) q*IQ.w = K C / ( l KC) where q* is the solid-phase concentration of the solute in equilibrium with solute in the fluid phase, Q M is the ultimate sorptive capacity of the solid, K is a n eyuilibrium constant, and C is the fluid-phase solute coiiceiitration. Equation 1 mag be written relative to the feed stream as
y * j y o * = C(1
Present address, GettJ- Oil Co., Ilelaware CitL-, Del.
(2 1
where the subscript, 0, refers to the feed. Introducing a n equilibrium parameter, R = (1 KCo)-l, and dimensionless concentrations, S = C/Co and I- = y/qo*, eq 2 becomes
+
I?*
=
S/(R
+ (1 - R ) S )
(3)
The parameter R provides a quantitative description of the equilibrium regions: R = 0 for irreversible, R < 1 for favorable, R = 1 for linear and R > 1 for unfavorable. Freundlich, noting that many surfaces do not have sites of equal adsorption eiiergy, assumed a n exponential dist’ribution of sites as a function of the energy of adsorption. This resulted ill the equation q*/Q.M = kCw
(4)
where k aiid IY are system-depelident parameters. Proceeding as with the Langmuir isot,lierm, eq 4 may be put into the dimensionless form
y*
Equilibrium
Physical equilibrium is assumed to exist for solute between the fluid and adsorbed phases. The shape of a plot relating fluid concentration to solid (adsorbed) conceiitratiori provides a qualitative descriptioii of t h e adsorption process. If the curve is coiives upvard, the equilibrium is termed unfavorable ailti proportioilate pat,tern behavior is usually observed.
+ k’Co)/Cu(l + K C )
=
A-w
(5)
This isotherm is most useful for intermediate concentrations and is especially useful for describing adsorption froni solutions. Mechanism
The rate equation for the sorbent particle takes on different forms depending on the mechanism which controls the transfer Ind. Eng. Chem. Fundom., Vol. 12, No. 1, 1973
95
Table 1. Summary of Particle Equations and Boundary Conditions-Langmuir Particle equation
Case
Isotherm Boundary conditions
All
- ax - _ _ 2 bx - (R b(R
15R
Pore
(R
+ (1 - R)X)' dN,T
+-wx b(R2 Y = P/(R
Solid
+ external Solid + external Pore + solid
Same as pore
Pore
Pore
1
bY
+B) x
R
E)+
(i%
+
Same as pore
2 R ( 1 - R) 15(1 (R - l ) Y ) 3
A mathematical description of fixed-bed adsorption is based upon material balances on the adsorption column, a particle in the column, and the fluid layer surrounding a particle. ilssuming constant film resistance and diffusion coefficients, no longitudinal diffusion or radial concentration gradients, uniform bed porosity, uniform spherical particles, and linear flow of t h e bulk fluid, the balance equations are given by bV
=
bC
bg
at
at
- ~ - - p ~ -
P-
solid
General Equations for Fixed-Bed Adsorption
be
Same as solid
+
of solute from the fluid phase into the solid phase. Possible mechanisms are: (1) external mass transfer-mass transfer from the flowing fluid to the external surface of the sorbent through a fluid layer surrounding the particle; (2) pore diffusion-diffusion resulting from the concentration gradient which exists in the fluid-containing pores of the sorbent particle; (3) solid diffusion-diffusion in the adsorbed state as a result of condensed phase concentration gradients in the sorbent particle; (4) adsorption kinetics or phase boundary reaction. Since a n adsorption-kinetic resistance is rarely observed, local equilibrium will be assumed to exist between solid and fluid phases at each point in the pore. Any one mechanism, or combination of mechanisms, may be rate controlling for adsorption. Results are presented here for the cases in which the controlling combinations are external plus pore diffusion (in series), external plus solid diffusion (in series), pore plus solid diffusion (in parallel), and pore plus solid plus external diffusion (in parallel and series).
bed:F-=
+ (R - l ) Y = p - r3Y 5 b(R RY
Same as solid
+ solid + external
fluid: -
+ (1 - R)Y)
RY
1
-
+ (R - l ) Y R (1 + (R - 1)Y)2
bY
+
') %
solute per weight of "dry" solid), E is the void fraction of the bed, is the bulk density of t'he bed, v is the column volume, F is the fluid flow rate, t is the time, K , is the external mass transfer coefficient, A , is the particle surface per unit volume, x is the particle porosity, D, is t'he effective pore diffusivitg, and D , is the effective condensed phase diffusivity. General solutions will be obtained by making these equations dimensionless. I n addition t o bhe dimensionless concentrations defined earlier, the following variables will be introduced: (1) A = qa*pb/Co,the distribution coefficient; ( 2 ) iYp = 150, (1 - e ) v / r S 2 F ,the number of pore diffusion mass transfer units (T, is the radius of the particle and F is the volumetric flow rate) ; (3) S,= 15D,bv/rs2F, the number of solid diffusion mass transfer units; (4) iVF = KFA,vc/F, the number of external diffusion mass transfer units; ( 5 ) T = ( t - ( v ~ / F ) ) / ( h v / F )the , throughput parameter; and (6) (R = r/rs, a dimensionless radius in the particle. In addition to constant pattern behavior, it will be assumed that the solute contained in the pore fluid is negligible although solute transport through the pores is significant in some cases. Thus all solute held by the particle is in the condensed state (this convention is accounted for when fitting the experimental equilibrium data and is, therefore, errorfree). Equations 6,7, and 8 become, respectively
X
=
I'= 3
s,'
Y6i2d(R
(9)
~
where 6 is the mean fluid-phase concentration (mass of solute per volume of solution), 4 is the mean solid-phase (mass of 96 Ind. Eng. Chern. Fundam., Vol. 12, No. 1, 1973
Tables I and I1 present the specific forms of the equations and their boundary conditions which describe the cases of interest in this paper. I n these equations either X or Y has been eliminated using the assumption of local equilibrium and the appropriate isotherm. I n the tables, CY is N,,"F; 0 is iV,/X,, and y is N s / * V F .For all cases the initial and final conditions are: a t S T = 0, 8 = 0 ; at Y T = m , 2? = 1.
Table 11. Summary of Particle Equations and Boundary Conditions-Freundlich Particle equation
Case
Isotherm
Boundary conditions
All
Pore Solid
+ external Solid + external Pore + solid Pore
Pore
Same as solid Same as solid
+ solid + external
Same as pore
+ solid LANGMU I R
GI-h R. R*m Figure 1. Grid illustration of Crank-Nicholson method: 0 , known values; 0, assumed values; X, point about which derivatives are averaged
N I T -11
Figure 2. Breakthrough curves for pore, solid, or external diffusion-Langmuir isotherm FREUNDLICH
Solution of Equations
The equations of Tables I and I1 are not generally capable of analytical solution because of either noiilinearities or complex boundary conditions. Therefore, numerical solutions will be presented which merge smoothly with exact results and previously accepted solutions. Exceptions are the solutions for external diffusion acting alone-Michaels (1952) and Kirwan (1971) have given analytical solutions for the Langmuir and Freundlich isotherms. Langmuir: N F ( T - 1) = 1 -
Freundlich: N F ( T - 1) = 1
R In (1 - X) - In X 1-R
[
+ In
1
(12)
-
1. c,
e 6 X
2
-3
-2 2 *? +4 16 Y(T-1,
Figure 3. Breakthrough curves for pore, solid, or external diffusion-Freundlich isotherm
Figure 1 illustrates the grid network for the CrankYicholson method. All differences are averaged about the point X m , l ~ ~ + ~ The / , h . nonlinearities in the equations of this study were removed by the relation T h e implicit, six-point Crank-Sicholson method has been employed for all remaining computations. This finite difference scheme, discussed in detail by Von Rosenberg (19691, allows replacement of all derivatives with secoiid-order-correct finite difference approximations. For the cases under study, the boundary conditions have been treated separately in each case introducing the possibility of instabilities for these equations which are nonlinear.
+
(15) xCi,,VT+'/th = (XCR,NT+h x&,NT)/2 without leading t o instabilities for the conditions selected. Equation 15 does not allow straightforward solution of the equations but the following iterative technique proved feasible: (1) Input a n initial estimate of the concentration profile ( X R , ~ )(2) . Also estimate the concentration profile at the next time level ( x R , h ) . (3) Solve the required equations for X*m,h (refer to Tables I and I1 and Fleck (1970)). (4) ComInd. Eng. Chem. Fundam., Vol. 12, No. 1, 1973
97
Fj
-
w
0.2
-2
-4
-0.2
0
12
-4
N 11-11
Figure 6. Breakthrough curves for pore diffusion and external diffusion acting in series
YpOre
f T - I1
Figure 4. Breakthrough curves for pore diffusion control as a function of the equilibrium parameter-Freundlich isotherm
I
ment 1-2 orders of magnitude larger than that used in the explicit method employed by Hall, et al. (1966). Test runs were made using radial increments of 10 and 20 equally spaced divisions. Because the results agreed within O.l%, the larger increment was used throughout. An error tolerance of 10-8 mas accepted for terminating a n iteration. Because only relative values of NT are produced, the results must be plotted us. N ( T - 1) t o be comparable. This is possible by using the material balance relation as described by Hall, et al. (1966).
La
(1
- QdT
=
L1
TdX
=
1
(16)
Case 1: Pore, Solid, and External Diffusion Acting Separately. The results are presented in Figures 2 and 3
and illustrate limiting behavior. For Langmuir equilibrium pore and solid diffusion results agree exactly with those of Hall, et al. (1966). The abscissas of these and all other plots are based upon Ntotal as suggested b y Hiester, et al. (1963).
R .5 4
l/Ntotal =
3 .2 1
0 -10
0
5 VI
5
11
11-1)
Figure 5. Breakthrough curves for solid diffusion control as a function of the equilibrium parameter-Freundlich isotherm
pare X * a , h with X m , h and if t'he difference is larger than the prescribed tolerance, replaceXa,h by X*a,h and again solve the equations. ( 5 ) When the successive calculations agree within the t'olerance, proceed to the nest time step. (6) Continue this procedure until 2 3 0.99. I n every case rrhere t'he initial profile was varied, identical solutions were obtained. Tested assumptions were a uniform profile of 10-8 and a n exponential profile giving = 0.01 (the latter was found to require less computer time). This agreement and the fact that these numerical results merge smoothly with all known analytical results suggest validity for the results. Results and Discussion
A few general observations can be made before discussing specific results. Time increments, radial increments, and error tolerances had to be select'ed. The time increment had t o be small enough t o assure convergence of the solution and large enough to conserve computer time. I n general, the increments ranged from the order of 0.01 to 0.4 as the equilibrium parameter changed from 0.2 to 0.8. This represented a time iricre98
Ind. Eng. Chem. Fundom., Vol. 12, No. 1 ,
1973
l/NF
+ l/(Np + NE)
(17)
Although these cases were run for several values of the equilibrium parameter, only results for values of R or W equal to 0.2,0.6, and 0.8 are illustrated to reduce the number of figures to a manageable number. Tabular results are deposited with the ACS Xicrofilm Depository Service. W- equalling zero corresponds to published results for R equal to zero. Figures 4 and 5 present previously unpublished results showing pure pore diffusion and pure solid diffusion under equilibrium conditions represented by the Freundlich isotherm. The curves are less symmetric than the equivalent curves for the Langmuir isotherm as shown by Hall, et al. (1966), and reproduced in this work. Case 2 : Pore plus External Diffusion Acting in Series.
I n this case (and all further cases) the results are illustrated only for a value of the equilibrium parameter, R ( W ) ,equal to 0.2. Figure 6 presents the results and the trends indicated here are valid for all other values of the equilibrium parameter. As a check 011 accuracy, a run was made for a = 0,001. These results merged with the curve for pore diffusion acting alone. As might be espected, pore diffusion acting alone and external diffusion acting alone are represented by a = 0 and a = m , respectively, with finite values of a lying between these limits. Curves are presented for a = 0.5 to illustrate spacing. Case 3 : Solid plus External Diffusion Acting in Series. The same comments apply t o the results presented in Figure 7 as were discussed in case 2 . T h e solid diffusion curve was matched exactly by a solution with y = 0.001. Case 4: P o r e plus Solid Diffusion Acting in Parallel. Again, t'he earlier comments are valid for the results pre-
iv
.0.2
* 0.
= inlet solute concentration in fluid phase, moles/ft*
2
=
DS DP F h k
mean solute concentration in fluid phase, moles/fta
= particle diameter, ft = effective solid diffusivity, ft2/hr = pore diffusivity, ft2/hr
dP
=
K KF
volumetric flow rate, fta/hr
= dimensionless time increment = Freundlich equilibrium parameter =
Langmuir equilibrium constant
= external mass transfer coefficient, ft/hr
dimensionless radial increment number of external mass transfer units number of pore diffusion mass transfer units number of solid mass transfer units total number of mass transfer units = solute concentration in solid phase, moles/lb = equilibrium value of solute concentration in solid phase, moles/lb = mean concentration of solute in solid = ultimate sorptive capacity of particle phase = radial dimension of particle, ft = particle radius, ft = dimensionless radius = Langmuir equilibrium parameter = time, hr = throughput parameter, hr = column volume, ft3 = Freundlich equilibrium parameter = dimensionless solute concentration in fluid = dimensionless mean solute concentration in fluid = dimensionless solute concentration in solid = dimensionless mean solid phase concentration
= .
.
= = = =
.
-4
-2
0
12
+4
N I T -11
Figure 7. Breakthrough curves for solid diffusion and extemal diffusion acting in series
1. 1
8
6
t
x
T
4
2,
lli -P
- .7
‘1 .
’7.
-d
-4
-2
3
+2
-c
5Y Y
N 11-11
Figure 8. Breakthrough curves for solid and pore diffusion acting in parallel
GREEKLETTERS =
a
sented in Figure 8. Here t h e curve for p = 0.001 merged with the pore diffusion curve as required. Case 5 : Pore plus Solid Diffusion Acting in Parallel with External Diffusion Acting in Series. Unfortunately, this case virtually defies graphical representation analogous t o Figures 6, 7, and 8. However, a qualitative description may suffice. Assuming t h a t initially pore and solid diffusion are acting in parallel and t h a t external diffusion is added in series, the result would be to shift the curve in the direction of the limiting case with external diffusion acting alone. Thus the curve describing case 5 will lie between t’he locus of points traced by the liriiiting resistances, Le., within the envelope bounded by the outer c,urvesfor each case in Figures 2 and 3. Conclusion
The Crank-Nicholson numerical technique proved to be a n escellent method of solving t,he nonlinear parabolic partial differential equations associated with fixed bed adsorption. The time improvement’ over the explicit method of Hall, et al. (1966), was very large. The solutions to the equations for combined resistances t o diffusion showed that the breakthrough curve always lies between the locus of limiting cases. The resultant graphs can be used as outlined by Hall, et al. (1966), to estimate magnitudes of diff ereiit diffusional resistances. Acknowledgments
One of the authors (R. D. F.) was supported b y a n KEDA Title IV Felloqship. The authors wish to thank T. R. Vermeulen for reading the manuscript and providing helpful comments. Nomenclature
A,
=
C
=
exteriial area of particles per unit volume of packed column, ft2/f t3 solute concentration in fluid phase, moles/ft3
P
NP/NF
= Ns/Np
Y
= Ns/NF = void fraction = porosity
E
X
of bed
Pb
= bulk density of bed, lb/ft3
A
=
distribution parrmeter; ratio of particle to fluid phase concentration
literature Cited
Boyd, G. E., Adamson, A. W., Myers, L. S., J. Amer. Chem. SOC. 69,2836 (1947).
Cooney, D. O., Lightfoot, E. N., IND.ENG.CHEM.,FUNDAM. 4, 233 (19651.
Cooper, R. S . , IND. ENG.CHEM.,FUNDAM. 4, 308 (1965). Fleck, R. D., Thesis, University of Virginia, Charlottesville, Va., 1970.
Gluekauf, E., Coates, J., Trans. Faraday SOC.51, 1540 (1955). Hall, K. R., Eagleton, L., Acrivos, A., Vermeulen, T., IND. ENG. CHEM.,FUNDAM. 5,212 (1966). Hiester N. K., Vermeulen, T., Klein, G., “Chemical Engineers Handbook,” 4th ed, Section 16, McGraw-Hill, New York, N. Y., 1963. Kirwan, D. J., personal communication, 1971. Lapidus, L., Rosen, J. B., Chem. Eng. Progr. Symp. Ser. 50, 97 (1 954).
Michaels, A. A., Ind. Eng. Chem. 44,1922 (1952). Rosen, J. B., Ind. Vermeulen, T., Aduan. Chem. Ens. Vermeulen, T., Quilii 179 (1970).
Von Rosenberg, D. U.,“Methods for the Numerical Solution of Partial Differential Equations,” American Elsevier Publishing Company, Inc., New York, N. Y., 1969. Wicke, E., Kolbid-2. 86, 298 (1939). RECEIVED for review February 4, 1972 ACCEPTED August 16, 1972 Tabular results of mixed-resistance diffusion kinetics in fixed bed adsorption will appear following these pages in the microfilm edition of this volume of the journal. Single copies may be obtained from the Business Operations Office, Books and Journals Division, American Chemical Society, 1155 Sixteenth St., N.W., Washington. D. C. 20036. Remit check or monev order for $17.00 For photocopy or $2.00 for microfiche, referring to code number FUND-73-95. Ind. Eng. Chern. Fundarn., Vol. 12, No. 1, 1973
99
