1176
Ind. Eng. Chem. Res. 1992, 31, 1176-1182
BCS8451240). Also, the research on which this project is based was financed in part by the State of Delaware as authorized by the State Budget Act of Fiscal Year 1988. We would also like to acknowledge the helpful suggestions of Bob Lumpkin, Ken Robinson, and Al Van der Klay of Amoco and Bill Calkins and Rowena Torres-Ordonez of the University of Delaware. Also, we would like to thank Mary Jacintha and Cecil Dybowski of the University of Delaware for the solid-state CP/MAS 13CNMR analyses. Nomenclature A. = fraction of mineral matter in original coal, t = 0 A, = fraction of mineral matter at time t C = concentration of coal plus coal-derived material in the reactor (g/cm3) CDBE = concentration of dibenzyl ether (mol/cm3) C D B b = concentration of DBE at t = 0 (mol/cm3) Cs = concentration of biphenyl (mol/cm3) C, = concentration of biphenyl at t = 0 (mol/cm3) P = reactor pressure (psig, 900 psig = 6.3 MPa) Q = flow rate (cm3/s) tD = sample duration (8) t = contact or reaction time (8) tf = total run time = final sample time (8) T = reactor temperature ("C) V = volume of reactor (cm3) W , = initial charge of coal to reactor (9) w s = weight fraction of biphenyl in reactor wso = weight fraction of biphenyl in reactor at t = 0 W , = initial charge of biphenyl to reactor (g) W , = weight of solid collected in each sample (g) x = amount of concentrated coal-derivedmaterial needed for analytical workup (g) Z = sample size (cm3)
Greek Symbols p = T
density (g/cm3)
= mean residence time (9) = V/Q Registry No. DBE, 103-50-4; PhMe, 108-88-3; PhCHO, 100-
52-7.
Literature Cited Brunson, R. J. Kinetics of Donor-vehicleCoal Liquefaction in a Flow Reactor. Fuel 1979,58, 203. Cassidy, P. J.; Jackson, W. R.; Larkins, F. P.; Louey, M. B.; Rash, D.; Watkins, I. D. The Structure and Reactivity of Brown Coal 12. Timesampled Autoclave Studies: Reactor Design, Operation, and Characterization. Fuel 1989, 68, 32. Cronauer, D. C.; Jewell, D. M.; Shah, Y. T.; Modi, R. J. Mechanisms and Kinetics of Selected Hydrogen Transfer Reactions Typical of Coal Liquefaction. Ind. Eng. Chem. Fundam. 1979, 18, 153. Danckwerta, P. V. Continuous Flow Systems. Chem. Eng. Sci. 1953, 2, 1. Gibbins, J. R.; Kandiyoti, R. Development of a Flowing-Solvent Liquefaction Apparatus. Fuel Process. Technol. 1990, 24, 237. Korobkov, V. Yu.; Grigorieva, E. N.; Senko, 0. V.; Kalechitz, I. V. Kinetics and Mechanisms of Thermolysis of Dibenzyl Ether. Fuel Process. Technol. 1988, 19, 243. Laine, J.; Becerra, 0. A. Semi-continuous Flow Reactor Technique for Coal Liquefaction Studies. Fuel Process. Technol. 1985,11, 127. Maa, P. S.; Neavel, R. C.; Vernon, L. W. Tubing Bomb Coal Liquefaction Technique. Ind. Eng. Process Des. Dev. 1984, 23, 242. Shinn, J. H. From Coal to Single-stage and Two-stage Producta: A Reactive Model of Coal Structure. Fuel 1984,63, 1186. Simmons, M. B.; Klein, M. T. Free-Radical and Concerted Reaction Pathways in Dibenzyl Ether Thermolysis. Ind. Eng. Chem. Fundam. 1985,24, 55. Zwietering, Th. N. The Degree of Mixing in Continuous Flow Systems. Chem. Eng. Sci. 1959, 1 1 , 1.
Received for review June 10, 1991 Revised manuscript received December 3, 1991 Accepted December 31, 1991
A Graphical Method for Determining Pore and Surface Diffusivities in Adsorption Systems Peter G.G r a y and Duong D. Do* Department of Chemical Engineering, University of Queensland, S t . Lucia, Queensland 4072, Australia
The dynamics of adsorption in a single particle was modeled using two different mathematical formulations: two intrinsic diffusional resistances (macropore and surface diffusion) and one apparent diffusional resistance (pore diffusion). The relationship between the apparent and intrinsic diffusivities was determined and found to be a function of the adsorption isotherm nonlinearity. By combination of this relationship with an analytical solution of the pore (apparent) diffusion model, a method was developed whereby the macropore and surface (intrinsic) diffusivities could be determined directly from experimental adsorption uptake data without the need for any model fitting. This simple method was applied to experimentally measured adsorption dynamics of sulfur dioxide (Freundlich isotherm) and n-butane (Langmuir isotherm) on activated carbon, and was found to give macropore and surface diffusivities similar to that obtained using a full model fit. In many common gas- and liquid-phase adsorption systems the overall rate of the process is controlled by diffusional resistance. To make practical use of these adsorption processes, it is necessary to determine what diffusion mechanisms are acting and to quantify their respective diffusion coefficients. It is, therefore, advantageous to have a simple and quick method for determining these diffusivities from experimental adsorption dynamic
* Author to whom correspondence should be addressed.
data. The development of such a method is the objective of this work. The mechanisms acting over the macroscopic length scale of an adsorbent particle are macropore (molecular and/or Knudsen) diffusion and often surface (sorbedphase) diffusion. The macropore diffusion control model has been applied to many gas- and liquid-phase adsorption systems over a number of decades, where a standard formulation of this model is given by Ruthven (1984). As well as sorbate transport through the macropore void region
0888-5885/92/2631-ll76$03.00/00 1992 American Chemical Society
Ind. Eng. Chem. Res., Vol. 31, No. 4, 1992 1177 of a sorbent, it is possible, for sorbates that adsorb significantly, to have an additional flux in the sorbed or “surface” phase that is in parallel to the macropore diffusion mechanism (Yang, 1987). While mobility of molecules in the sorbed phase is smaller than that in the macropore void phase, their concentration is much higher, so that significant surface fluxes are possible. Given that most practically useful sorbents necessarily promote large sorbed-phase concentrations, it is often important to include a surface diffusion mechanism in theoretical models of sorption dynamics. Determining the contribution of each diffusion mechanism to experimentally measured adsorption dynamics can be a complex and time-consuming procedure. A standard method of separating and quantifying the effects of macropore and surface diffusion was used by Schneider and Smith (1968),where they studied the adsorption of ethane, propane, and n-butane on silica gel using the method of moments on chromatographic data. As the adsorption isotherms were linear, the extracted diffusion coefficient was a linear combination of the macropore and surface diffusivity. By measuring the adsorption dynamics at high temperature, where the surface flux was negligible and macropore diffusion dominated, the authors determined the particle tortuosity factor. At lower temperatures, where both macropore and surface fluxes were important, the macropore diffusivity was calculated from the tortuosity factor and hence the surface diffusivity obtained by subtraction. Even with the relatively simple system of Schneider and Smith (1968), where the linear isotherm allowed an analytical solution, the procedure for extracting the diffusivities was quite involved. For many practical adsorption systems where the adsorption isotherm is nonlinear, this procedure becomes even more complex, as the model must be solved numerically and fitted to the experimental data in an iterative manner. If a macropore diffusion control model is applied to an adsorption system where both macropore and sorbed-phase diffusion are significant, then the extracted diffusion coefficient is not the intrinsic macropore diffusivity, but rather it is an apparent diffusivity, where this apparent diffusivity is some combination of the intrinsic macropore and surface diffusivity. In this work we determine the relationship between the apparent and intrinsic diffusivities, which is influenced by the degree of nonlinearity of the adsorption isotherm. Using this relationship, we develop a method whereby apparent diffusivities extracted from experimental data (by fitting with a macropore model) can be used to determine the intrinsic macropore and surface diffusivities. The method is then further simplified to allow the intrinsic diffusivities to be directly extracted from single particle adsorption uptake data, without the need for any model fitting. The validity of the method is checked by applying it to experimentally measured adsorption dynamics.
Theoretical Basis of the Method In order to determine the relationship between the apparent and intrinsic diffusivities, two different singleparticle models were derived and solved numerically. Two common isotherm types (Freundlich and Langmuir) were used to show the effect of isotherm nonlinearity. Pore plus Sorbed-Phase Diffusion Model. This model of adsorption within the sorbent particle allows for both macropore and sorbed-phase diffusion mechanisms; hence the diffusion parameters are the intrinsic diffusivities. The following assumptions were made in order to provide a mathematically tractible set of equations de-
scribing sorption in a porous sorbent particle. (i) The sorbent particle is viewed as a microporous solid penetrated throughout by a network of larger interconnected pores (macropores). The microporous solid phase adsorbs the bulk of the sorbate, while the macropores have negligible sorbate capacity. (ii)The carbon particle geometry is arbitrary (e.g., slab, cylinder, or sphere). (iii) The resistances controlling the sorption dynamics are due to macropore and surface diffusion processes (by surface diffusion we mean sorbed-phase diffusion through the microporous solid region, which is in parallel with the macropore diffusion mechanism). Micropore diffusion resistance is assumed to be negligible. (iv) The macropore and surface diffusion coefficients are independent of concentration. (v) Film diffusion resistance at the external surface of the particle is negligible. (vi) The particle is isothermal. (vii) The microporous solid region of the particle is in equilibrium with sorbate in the macropore via a nonlinear isotherm (Freundlich or Langmuir). Using these assumptions, the mass balance equation for the particle may be expressed as EM
ac
ac,
- + (1- CM) - = at at
symmetry exists at the particle center, and a t the outer surface r = R; CIR = Cb and C, is related to C by the Freundlich isotherm C, = KC1/” (Freundlich) C,SbC
c, = (Langmuir) 1 + bC The first term on the left in eq l a represents the adsorbate accumulation in the macropore void phase, and the second term is the accumulation in the microporous solid phase. The two terms on the right-hand side of eq l a describe the diffusional flux of sorbate through the macropore void and microporous solid phases, respectively. By defining the following nondimensional variables and parameters
7
= UlSPt -a
R2
(1 - eM)SSC,O
’
6= eMSPCO
(34
eqs 1 can be rewritten in nondimensional form as
x = 1; All =
Ab
where, for adsorption, the initial condition is
(4b)
1178 Ind. Eng. Chem. Res., Vol. 31, No. 4,1992 7
= 0:
A=A,=O
Ab=1
(44
0 extracted f values
- empirical equation
and where the nondimensional Freundlich and Langmuir isotherms are given by A, = A'/"' A, =
(1 1
(Freundlich)
+ X)A + XA
(54
(Langmuir)
The parameter 6 is the ratio of surface (sorbed phase) to macropore void phase diffusional fluxes, where for a system in which macropore diffusion totally controls the sorption dynamics 6 = 0, but for a surface diffusion controlled system 6 > 1. The parameter m describes the Freundlich isotherm nonlinearity, where the isotherm is linear for m = 1and nonlinear for m > 1. Lambda (A) is a measure of the Langmuir isotherm nonlinearity; the isotherm is linear, nonlinear, and rectangular for X N 0, X rrr 1,and X >> 1, respectively. Pore Diffusion Model. This model of adsorption within the sorbent particle assumes that the uptake dynamics are controlled only by a macropore diffusion mechanism. If this model is applied to an adsorption system where both macropore and sorbed-phase fluxes are significant, then the extracted diffusion parameter is an apparent diffusivity. The assumptions for this model are identical to the previous model with the exception of assumption iii, where we now assume that only macropore diffusion resistance controls the sorption dynamics. This model is equivalent to the previous model equations with 6 set to zero and with the nondirnensional time now based on the apparent diffusivity (BApp).Thus, the nondimensional time is given by 7
= b1a)Appt/R2
(6)
and the nondimensional pore diffusion equation becomes
Model eqs 4 and 7 were reduced to ordinary differential equations (ODE) using the orthogonal collocation technique (Villadsen and Michelsen, 1978) and then numerically integrated using an ODE solver. Using 11collocation points in the particle, typical computation times (depending on the isotherm nonlinearity and the value of 6) were 60-120 s for an Intel 20-MHz 80386180387 based computer. Equation Relating Apparent and Intrinsip Diffusivities. For a linear isotherm, the equation relating the apparent (aApp) and intrinsic (ap and as)diffusivities is quite simple and has been shown before (Schneider and Smith, 1968). In terms of our notation this equation is (1- %l)C,O 33s = Bp(1 + 6) (8) 3 A p p = 33P + CMCO
where the factor multiplying Bsis simply the slope of the linear isotherm. For the case of a nonlinear isotherm the relationship is more complex and cannot be derived analytically. To determine the form of the equation relating BApp, Bp,and as,the following procedure was used. (i) Simulated adsorption uptake curves were generated by the pore plus sorbed-phase diffusion model using various values of Bp and Ds at increasing degrees of isotherm nonlinearity (i.e., increasing m or A). (ii) Using the pore diffusion model, the value of BApp was varied until the model solution matched with the uptake curves generated in step i.
0.4 0.2 1
3
7
5
9
Freundlich parameter, m Figure 1. Sorbed-phase flux factor, f , as a function of Freundlich isotherm nonlinearity (m).
0
extracted f values equation
- empirical
0
Cd
0.6
-
tj4
0
5
10
20
40
60
Langmuir parameter, A Figure 2. Sorbed-phase flux factor, f , as a function of Langmuir isotherm nonlinearity (A).
(iii) Steps i and ii gave data in the form of one value of BApp for each set of Bp, 6, and m or A. This information was then summarized in the form of an equation similar to eq 8. BApp + f6) (9) The factor f was found to be dependent on the isotherm nonlinearity, ranging from 1 for a linear isotherm ( m = 1 or X = 0) to an asymptotic value of 0.63 for a highly nonlinear isotherm ( m >> 1 or X > 10). Plots showing the dependence off on m and X are given in Figures 1 and 2, respectively. These curves were fitted to the following empirical equations. f = 0.63 + 0.37/(1 a(m - l)b); a = 1.48, b = 1.45 (Freundlich) (loa) f = 1.0 - aX/(l bX); a = 0.29, b = 0.78 (Langmuir) (lob)
+
+
The decrease off with increasing isotherm nonlinearity can be explained by examining eq 9. This equation shows that the contribution of sorbed-phase diffusion to the apparent diffusivity (aAp ) is due to the term fa, where this contribution is greatest for the linear isotherm case when f = 1. With increasing isotherm nonlinearity the sorbedphase concentration gradient becomes flatter; hence the flux through the sorbed phase decreases relative to the macropore void flux. However, the dynamic parameter 6, as defined in eq 3c, is based solely on equilibrium con, C and does not account for the effect centrations (Co and ) of isotherm nonlinearity on the sorbed-phase flux. Therefore, 6 is multiplied by the factor f to describe this nonlinearity effect.
Ind. Eng. Chem. Res., Vol. 31, No. 4, 1992 1179 The relationship between the apparent and intrinsic diffusivities (eqs 9 and 10) can be used to extract Bpand Bs from experimental data, without the need to use the more complex pore plus sorbed-phase diffusion model (eq 4a). It is simpler to fit the pore diffusion model (eq 7) to experimental adsorption uptake data to extract the apparent diffusivity (BApp) at various adsorbate concentrations (CJ. The data can be obtained from various adsorber configurations (e.g., gravimetric), providing the previously stated assumptions are valid and the same model is used to describe the particle (eq 7). By combination of the Freundlich or Langmuir isotherm with eq 9, the following equations are obtained.
--
-.-
0.0
0.2
(Freundlich)
JT
0 Sphere: numerical solution Cylinder: nurn. soh. V Slab: num. soh. Analytical solutions Modified analytical soh. (cyl.)
0.4
0.6
0.8
(non-dim.)
Figure 3. Comparison of analytical and numerical solutions of the pore diffusion model with Freundlich isotherm (rn = 2.17).
Rearranging eq 13b gives the expression for the apparent diffusivity Plotting BApp versus fCol/m-lin the Freundlich isotherm case, or SA, versus f / ( l bCo) in the Langmuir case, enables the intrinsic diffusivities (33, and as)to be determined. Approximate Analytical Solutions for the Apparent Diffusivity. For adsorption on a single particle, the method just outlined was further simplified by extracting BApp directly from the adsorption uptake curves, without the need for model fitting. This was done by using an analytical solution of the pore diffusion model to provide an explicit relation between BApp and the experimental uptake curve half-time (t0.5). Freundlich Isotherm. An analytical solution of the pore diffusion model, with Freundlich isotherm, was derived by Kozhushko et al. (19881, where their model formulation was equivalent to eq 7 with u1 1 or X > 0), otherwise the abcissas of Figures 4 and 5 (fcollm-land f / ( l + KO), respectively) will be invariant. (ii) The sorbedphase diffusivity (2)s) is assumed to be independent of concentration. If Ds increases significantly with increasing concentration, the plots of eqs l l a and l l b would be curved rather than a straight line (i.e., the slope of the line would increase with increasing Co); hence there would be errors in the extracted values of and as.
70.5
=
alaAppt0.5/R2
= yf
(164
where the factor f is calculated using eq 10b and y = 0.0306, 0.0631, and 0.1967 for spherical, cylindrical, and slab geometry, respectively. Rearranging eq 16a, we obtain a)App
= "/fR2/a1t0.5
(16b)
Application of the Method System with Freundlich Isotherm. To test the validity of the method derived above, it was applied to data which had previously been analyzed using the fullpore plus sorbed-phase diffusion model (eq 4a). The adsorption dynamics of sulphur dioxide on large particles of a high surface area activated carbon were measured gravimetrically by Gray and Do (1990). The equilibrium capacity of the sorbent best fitted the Freundlich isotherm, with m = 2.17 and K = 1.02 at 298 K. The method was applied to adsorption uptake dynamics measured on a 0.2-cm-radius slab particle at 298 K with SO2concentrations of 0.5, 2, and 5 mol % . Using the simple expression derived above (eq 15b), the apparent diffusivity ( a A p p ) was calculated directly from the half-time (t0.5) of the adsorption uptake curves. The extracted values of aApp a t each sorbate concentration (Co) were then plotted versus fCgl/m-l,where f was calculated using eq loa. This plot is shown in Figure 4, where the regression line through the data points gave the intrinsic diffusivities (ap and 3s) according to eq l l a . The diffusion coefficients extracted by this method were 2)p = 0.056 cm2/s and 33s = 5.6 X lo+ cmz/s. By comparison, the values of the intrinsic diffusivities extracted using the full pore plus sorbed-phase model (Gray and Do, 1990) were Zip = 0.038 cm2/s and as= 1.0 X cm2/s. As can be seen, the simple analytical method gave values of the diffusivitiessimilar to those extracted using the more complicated method of numerical solution of the full model. Some discrepancy between the values extracted using the two different methods was not unexpected, as the simple method used only one point on each of the experimental uptake curves (Le., the time at 50% uptake),
Acknowledgment This project is supported by the Australian Research Council.
Ind. Eng. Chem. Res., Vol. 31, No. 4, 1992 1181
Nomenclature A = dimensionless macropore concentration = C/Co A, = dimensionlessmicroporous solid-phase concentration =
coordinate transformation (Kozhushko et al., 1988), we obtain
C,/Cr?
Ab = ratio of bulk concentration to initial concentration = cb/ cO
a = constant in empirical equation relating f and m or A, eq
10 b = constant in Langmuir isotherm (cm3/mol) or in eq 10 C = macropore concentration (mol/cm3macrovoid) Co = initial macropore concentration (mol/cm3) Cb = bulk-phase concentration (mol/cm3) C, = microporous solid-phase concentration (mol/cm3 microporous phase) CrO= sorbed concentration in equilibrium with Co (mol/cm3 microporous phase) C,, = maximum sorbed-phase concentration in Langmuir isotherm DDAB = molecular diffusivity (cm2/s) DApp= apparent diffusivity (cm2/s) De = effective diffusivity (cm2/s) DK = Knudsen diffusivity (cm2/s) D M = composite molecular diffusivity = (1/Dm + l/&)-' (cm2/s) XIDp= particle macropore diffusivity = ae/tM (cm2/s) Ds= sorbed-phase or surface diffusivity (cm2/s) f = factor showing decrease of sorbed flux with increasing m or A, eq 9 K = constant in Freundlich isotherm m = constant in exponent of Freundlich isotherm n = parameter in analytical solution of pore diffusion model (Freundlich) R = radius of particle (cm) r = particle radial coordinate s = particle shape factor (0 = slab, 1 = cylinder, 2 = sphere) t = real time coordinate = adsorption uptake curve half-time (time to 50% of fiial uptake) ( 8 ) Ul(r) = first stage uptake analytical solution of pore model (Freundlich) x = Dimensionless particle radial coordinate = r / R Greek Symbols a = coefficient in eqs 13d and 14c
B = coefficient in eqs 13d and 14c
y = coefficient in eq 16 6 = ratio of sorbed phase to macropore flux tM
= macropore porosity
A = Langmuir isotherm nonlinearity parameter u1 = nondimensional macropore capacity (usually
