Effective Diffusivities under Reaction Conditions (Isobutene in Fluorinated q-Alumina) N. S. Raghavan and L. K. Doraiswamy’ National Chemical Laboratory, Poona-8, lndia
The effective diffusivity and surface diffusivity of isobutene in a fluorinated 7-alumina catalyst (containing 2.0 wt YO F) were measured at the reaction temperatures from the analysis of moments of the chromatographic peaks. The activation energy for diffusion was found to be 1.55 kcal/g-mol. The contribution of surface diffusivity to the total effective diffusivity was found to be negligibly small and hence an attempt to compare the experimental effective diffusivities with the effective diffusivities predicted by three different physical models which do not take the surface diffusivity into account was made. The values predicted by Johnson and Stewart’s model with their predicted factor K = ’/3 agreed closely with the experimental values determined under reaction conditions.
Introduction The objective of this work was to evaluate the effective diffusivities of isobutene in a porous 7-alumina catalyst under conditions of reaction by a transient method. Earlier studies in diffusion were based on the steady-state method developed by Wicke and Kallenbagh (1941) which had a limited application to a single catalyst pellet having a narrow pore size distribution and continuous pore structure unlike the unsteady-state method which could take into consideration the existence of small and “dead-end” pores in addition to the continuous pore structure. Some of the unsteady-state methods (see, for example, Barrer (1949), Habgood (1958), and Gorring and de Rossett (1964)) proposed to evaluate the diffusivities, though found to be more suitable than the steady state method, had their use restricted to measuring diffusion of gases in molecular sieve solids. Van Deemter and co-workers (1956) proposed a theory correlating the height equivalent to a theoretical plate in a chromatographic column to the various transport processes taking place in the bed of solid particles and this theory was later used by Leffler (1966) and Eberly (1969) in evaluating the effective diffusivities of hydrocarbon gases in solid catalyst particles. Schneider and Smith (1968) developed a method based on the theory proposed by Kubin (1965a,b) and Kucera (1965) for relating the moments of the chromatographic curves to the rate constants associated with the various steps in the overall adsorption process and used this method for determining the adsorption equilibrium constants, rate constants, and effective diffusivities for various hydrocarbons on silica gel and obtained good agreement between the experimental and breakthrough curves. Galan and Smith (1975) also evaluated effective diffusivity and surface diffusivity of SO2 in silica gel at high temperature using the method of moments analysis. Dogu and Smith (1976) derived equations to evaluate the intraparticle diffusivities, adsorption equilibrium constants, and adsorption rate constants from the moments of the response peak from one end face of a catalyst pellet when a pulse of a diffusing component is passed over the other end face and illustrated the method with experimental data for helium (in nitrogen) and cyclopropane (in helium) pulses diffusing through alumina pellets. The advantage of this method is that axial dispersion is eliminated as the pulse response experiments are carried out with single catalyst pellets. Chromatographic methods of determining rate parameters in general have recently been reviewed by Furusawa et al. (1976).
The present work is based on the theory of moments analysis and is a continuation of an earlier investigation by the authors (Raghavan and Doraiswamy, 1977) concerning the adsorption equilibrium constants obtained for isobutene at reaction temperatures by two different methods, namely, kinetics of the reaction n-butene isobutene in the presence of fluorinated ?-alumina as catalyst and the analysis of the first absolute moment of the effluent concentration wave from a bed of catalyst particles. An attempt has also been made here to compare the effective diffusivity values obtained by this method using a packed bed of catalyst particles with the values predicted by some of the theoretical models which are of a purely physical nature. The Chromatographic Method Used. Analysis of Moments. The theory of moments analysis proposed by Kubin (1965a,b) and Kucera (1965) helped improving the theoretical treatment of gas-solid chromatography. Kubin (1965b) solved a set of differential equations which describe the movement of an adsorbable substance in a chromatographic column and obtained explicit expressions for the moments of chromatographic curves, c ( z , t ) by describing the pulse input by a square function c = c g at z =
0 for 0 < to^
c = Oat z = Ofor t
>
to^
(1)
where to^ is the injection time of the adsorbate gas of concentration co. Kubin defined the nth absolute moment b,’ of the function c ( z , t ) as
where m, =
La
tnc(z, t )dt
( n = 0,1, 2, 3, . . .)
(3)
and the second central moment ~2 as
(4) The first absolute moment PI’ represents the position of center of gravity of the’peak and the second central moment is a property of the width of the peak. Schneider and Smith (1968) proposed the following correlations for the moments
NCL Communication No. 2089. Ind. Eng. Chem., Process Des. Dev., Vol. 16, No. 4, 1977
519
v
=
10,35 c m / r c c
VOL ‘I. OF ISOBUTENE IN THE SAMPLE
Figure 1. Effect of isobutene concentration on first and second moments at 605.5 K. where
Table I. Physical Properties of Catalyst and Packed Column ProDertv Catalyst
and
(7) when K A = 0 and LOA = 0, and
where
in the above equations represents the total contribution of intraparticle diffusivity and the external mass transfer process to the second central moment. The experimental chromatographic curves of the effluent from the bed of adsorbent particles obtained a t different carrier gas velocities can be used to evaluate the first absolute and the second central moments as a function of velocity.
Surface area Diameter, d, Particle density, p p Pore volume, p v Pore radius, r Internal void fraction of particle, /3 Void fraction of packed column, N Internal cross-sectional area of column Length of packed column, L
Value Fluorinated 7-alumina [2% W/W Fl 141 m2/g 0.034 cm 1.41 g/cm3 0.32 cm3/g 42 X cm 0.485 0.38 0.1256 cm2 27.8 cm
61
Experimental Section One of the important conditions on which the theory behind the present work is based is the linearity of the adsorption isotherm. In the case of nonlinear isotherms, the moments of the chromatographic curves are known to depend on the concentration in the injected pulse. Therefore, in order to be sure that the data collected by us pertained to the linear region of the adsorption isotherm, we conducted a few experiments initially with a fixed carrier gas velocity and temperature but with varying concentrations of isobutene in the injected pulse. The outcome of these experiments with regard to both the first absolute and the second central moments is shown in Figure 1. As can be seen, the moments are independent of concentration up to 40 vol % of isobutene in the pulse. Thus a concentration of 25 vol% of isobutene was chosen for the actual runs. The second and equally important condition is that adsorption should be reversible. This condition was checked and found to be satisfied in our experiments as the area of the effluent peak from the column of adsorbent particles a t the 520
Ind. Eng. Chem., Process Des. Dev., Vol. 16, No. 4, 1977
lowest operating temperature (574.5 K) was found to be equal to that of the area of the peak obtained with an empty column. Chromatographic curves of isobutene on 7-alumina (containing 2 wt % F) a t temperatures ranging from 574.5 to 621 K and a t atmospheric pressure were obtained. The carrier gas velocity for this study ranged from 2 to 35 cm/s, and it was found that a t the lowest velocity the conversion of isobutene to n-butene was negligible a t all temperatures. Adsorbent. Fluorinated 7-alumina particles of -40 +60 B.S. mesh size were used as adsorbent after conditioning in the apparatus for 15 h a t the highest temperature of study with a continuous flow of dry nitrogen. The physical properties of the adsorbent are given in Table I. Gases. The “ultrahigh pure” grade nitrogen which was used as the carrier gas was dried by passing through a trap containing 5A molecular sieve and then through a tube containing copper bits heated to 300 “C in order to remove trace amounts of oxygen. Isobutene prepared by dehydration of tert- butyl alcohol over a cation-exchange resin was dried before use by passing through traps containing silica gel and fused calcium chloride. Mixtures of isobutene and nitrogen were obtained by adjusting the flow rates of the individual gases in capillary flow meters. Gas Chromatograph. An N.C.L. designed gas chromatograph provided with a separate fluidized bed heating bath was
I
0.5
I
I
1 .O
1
1
I
1.5
1
2.0
1
I
2.5
I 3.0
3.25
VY ,set
Figure 2. Chromatography of isobutene on v-alumina (dependenceof the reduced first absolute moment on z l u ) .
Table 11. Adsorption Coefficient Values for lsobutene from First Absolute Moment
Tema K
Adsorption coefficient for isobutene, Kin cm3/e
574.5 590.0 605.5 621.0
24.96 11.77 6.65 4.30
used for this study. The U-tube containing the catalyst particles was connected to the top flange of the heating bath and l/8-in. 0.d. stainless steel capillary tubing was used for connecting the U-tube to the carrier gas inlet and to the flame ionization detector. A six-port sampling valve permitted injection of a square function in concentration. A schematic diagram of the experimental assembly is shown elsewhere (Raghavan and Doraiswamy, 1977). A fixed quantity (0.5 cm3 in the present case) of isobutene-nitrogen mixture was injected into the U-tube through the gas sampling valve whenever the chromatographic curves were to be measured.
Results and Discussion Adsorption Equilibrium Constant from First Absolute Moment. Because of the direct proportionality between the concentration of gas and the deflection of the recorder connected to the F.I.D. a t the outlet of the column packed with the adsorbent particles, the moments of the chromatographic curves c ( z , t ) were determined using the observed deflection in place of concentration in eq 3 and 4. The integrals in the above equations were evaluated numerically using Simpson’s rule with the help of a digital computer. Equations 5,6, and 7 lead to
\
a
’ /
where &I’
= PI’ - F’l(inert)
I t can be seen from eq 10 that a plot of [&I’ - ( t 0 ~ / 2 ) ] / [ ( ( 1 - a ) / a ) p ]vs. z / u will yield a straight line passing through the origin. Such plots are shown in Figure 2 for the moments of the chromatographic curves obtained a t different temperatures for isobutene, and the adsorption coefficients calculated from the slopes of these plots are given in Table 11. In an earlier study (Raghavan and Doraiswamy, 1977), the adsorption coefficient values for isobutene on the same 7-alumina catalyst obtained from the analysis of the first-absolute moments of the chromatographic curves were compared with the values extracted from the kinetics of the reaction, n-butene 6 isobutene, and the two were found to agree to within about 12% in the temperature range of study. Evaluation of Effective Diffusivities and Axial Dispersion Coefficients from the Second Central Moments. The second central moments of the chromatographic curves, which depend on the width of the peaks as mentioned earlier, were obtained a t different carrier gas velocities for each temperature in the same manner as described for the first absolute moments. Modifying eq 8 and 9, we have
where
and
In the above equations, 6, and 6; represent the contributions to the second central moment by the external mass transfer process and the intraparticle diffusivity, respectively. The mass transfer coefficient Kf which generally depends on the velocity of the gas and the diameter of the particles has been shown to be independent of velocity a t low Reynolds Ind. Eng. Chem., Process Des. Dev., Vol. 16,No. 4,1977
521
Table 111. Diffusivities and Axial Dispersion Coefficients Ds [PpIPlKiB
Temp,
D, x 103,
K
cm2/s
574.5 590.0 605.5 621.0 652.0
a)k x
1.9930 2.0620 2.1400 2.2060 2.3480"
io3,
DsbplPIKi~
D, x 107,
Dk
Ea
X
lo2,
cm2/s
Tint
x 105
cm2/s
x 102
cm2/s
1.9540 2.0390 2.1245 2.1940 2.3480
3.127b 3.164b 3.176b 3.172b 3.19OC
3.9 2.3 1.5 1.2
5.374 6.721 8.790 9.599
1.9996 1.1280 0.7060 0.5470
4.72 5.49 6.80 7.86
-
-
-
-
a Calculated by extrapolating the Arrhenius plot of log De vs. 1/T. Calculated from the relationship De = DkP/Tint. from the relationship De = a)k = DkP/Tint.
TEMPERATURE,^^^^
Calculated
(*K-I
Figure 4. Arrhenius plot of effective diffusivities. / 00 2
1
2
3
4
5
6
7
8
+, Figure 3. Chromatographyof isobutene on 7-alumina (dependence of the second central moment on 1 / u 2 ) .
numbers by Wakao and co-workers (1958).For packed beds a t very low Reynolds numbers, Wakao proposed the following correlation between the mass transfer coefficient, particle radius, and the molecular diffusivity
K f R = DAB
(15)
According to the above equation, the mass transfer coefficient a t low Reynolds number is inversely proportional to the particle radius and is independent of velocity. Since all our data with regard to the second central moment were collected a t very low Reynolds numbers, eq 15 could be considered as valid in obtaining the mass transfer coefficient Kf. The molecular diffusivities, DAB, were calculated a t all temperatures using the Lennard-Jones potentials. It can be seen that when the left side of eq 12 is plotted against l / u 2 , a straight line with an intercept of [6, bi] and a slope of (EA/CX will result. This is shown in Figure 3 which summarizes the chromatographic data for isobutene a t all the temperatures of study.
+
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Ind. Eng. Chem., Process Des. Dev., Vol. 16, No. 4, 1977
Since the values of the adsorption equilibrium constants and hence 60 were known from the analysis of first absolute moments and K f R could be obtained from the molecular diffusivity DAB,the calculation of 6, was made easier. From a knowledge of the values of the intercepts in Figure 3 and of 6,, the effective diffusivities at the different temperatures were calculated, and these are given in Table I11 as also the values of the axial dispersion coefficient obtained from the slopes. The effective intraparticle diffusion coefficient De was found to increase with increase in temperature and an Arrhenius plot of log D e vs. 1/T yielded a straight line as shown in Figure 4. The activation energy for diffusion, as calculated from this plot, is 1.55 kcal/g-mol. Separation of S u r f a c e Diffusivity from Overall Effective Diffusivity. For porous solids of narrow pore size distribution with a small pore radius and large surface area of about 100 m2/g and above, the existence of a diffusional process due to surface migration in addition to the pore volume diffusion has to be considered. The following correlation for the overall effective diffusivity in terms of Knudsen diffusivity and surface diffusivity can be written
Table IV. Comparison of Effective Diffusivities from Theoretical Models with Experimental Valuesa
Temp,
- K
Weisz and Schwartz model, D , x 103
Weisz and Schwartz model modified with Haynes factor, D, x 103
Johnson and Stewart model, D , x 103
Wakao and Smith model, D , x 103
Exptl chromatographic values, D, x 103
574.5 1.669 1.310 1.818 2.650 1.993 590.0 1.691 1.327 1.929 2.699 2.062 605.5 1.713 1.344 2.042 2.742 2.140 621.0 1.735 1.362 2.182 2.785 2.206 Note: (1) In the case of the Weisz and Schwartz model the mean pore dius was obtained from the relationship F = 3p,/s,. (2) The Knudsen diffusivity, Dk for the Johnson and Stewart model was obtainec from the relationship Dk = [2r/3][E]for transport within long cylindrical tubes (Wheeler, 1951).(3)The mean pore radius for the Knudsen diffusivity in the case of the Wakao and Smith model was evaluated from the relationship i: = ( J ” ’ r
dp,) /p, (Wakao and Smith, 1962).
where & and D , represent the effective Knudsen and surface diffusivity, respectively. The effective Knudsen diffusivity can be evaluated from the true Knudsen diffusivity, the internal void fraction of the catalyst particles, and the internal tortuosity factor from the equation
.fDk= - Dk
