I n d . Eng. C h e m . Res. 1990, 29, 1485-1492 Models. Fluid Phase Equilib. 1986, 30, 135. Scatchard, G.; Satkiewicz, F. G. (1964) Organic Hydroxy Compounds: Alcohol. In Vapor-Liquid Equilibrium Data Collection; Gmehling, J., Onken, U., Eds.; DECHEMA: Frankfurt, W. Germany, 1977; Vol. 1/2a, p 439. Schmidt, G. C. (1926) Organic Hydroxy Compounds: Alcohol. In Vapor-Liquid Equilibrium Data Collection; Gmehling, J., Onken, U., Eds.; DECHEMA: Frankfurt, W. Germany, 1977; Vol. 1/2a, p 225. Soave, Giorgio. Equilibrium Constants from a Modified RedlichKwong Equation of State. Chem. Eng. Sei. 1972, 27, 1197. Tobochnik, J.; Chapin, P. M. Monte Carlo Simulation of Hard Spheres near Random Closest Packing using Spherical Boundary Conditions. J. Chem. Phys. 1988, 88, 5824. Vargaftik, N. B. Handbook of physical properties of liquids and
1485
gases: pure Substances and Mixtures; Hemisphere: New York, 1975. Vimalchand, P.; Donohue, M. D. Comparison of Equations of State for Chain Molecules. J. Phys. Chem. 1989, 93, 4355. Wichterle, I.; Kobayashi, R. Vapor-Liquid Equilibrium of Methane-Propane System at Low Temperatures and High Pressures. J. Chem. Eng. Data 1972, 17,4. Zawidzki, V. J. (1900) Carboxylic Acids, Anhydrides, Esters. In Vapor-Liquid Equilibrium Data Collection; Gmehling, J., Onken, U., Grenzheuser, P., Eds.; DECHEMA: Frankfurt, W. Germany, 1982; Vol. 1/5, p 144.
Received for reuiew July 19, 1989 Reuised manuscript receiued December 28, 1989 Accepted March 8, 1990
Intraparticle Mass Transfer in the Liquid-Phase Hydration of Isobutene: Effects of Liquid Viscosity and Excess Product Enric Velo, Luis Puigjaner, and Francesc Recasens* Department of Chemical Engineering, E T S Enginyers Industrials de Barcelona, Uniuersitat PolitBcnica de Catalunya, Diagonal 647, 08028 Barcelona, Spain
The values for the effective diffusivity of isobutene (iB) in the liquid-filled pores of Amberlyst-15 (A-15) have been obtained from reaction measurements. The reaction conditions included four temperatures (303-333 K), various tert-butyl alcohol (TBA) concentrations (0.1-2.5 kmol/m3), and two sizes of catalyst particles (0.5-1 mm). Intraparticle diffusivity was found to increase with temperature and decrease with TBA concentration. Apparent values for the tortuosity factor change from 1.3 (at high temperature in pure water) to about 4.5 (at low temperature in TBA-rich solutions). The role of surface diffusion and the effect of the change in the solvent viscosity are discussed. Unlike gaseous-phase diffusion, the molecular diffusivity of a gas dissolved in a liquid is several orders of magnitude lower. Moreover, liquid-phase nonidealities coupled with adsorption phenomena occurring in liquid-filled pores may greatly affect intraparticle transport properties, and do so in an unpredictable manner. These two facts make the diffusional transport properties of the dissolved limiting reactant critical for obtaining high overall catalytic rates in multiphase chemical reactors. In general, various mechanisms contribute to the effective diffusivity observed in liquid pores, as summarized by Ramachandran and Chaudhari (1983). On one hand, restricted or hindered diffusion occurs whenever the pore diameter is similar in size to that of the diffusing species. Such is the case for fine-pore catalysts and zeolites (Satterfield et al., 1973; Satterfield and Katzer, 1971; Prasher et al., 1978; Savage and Javanmardian, 1989). On the other hand, if the catalyst is a strong adsorbent for the reactant, surface diffusion or migration (Satterfield, 1970; Smith, 1981) may contribute to the global transport. Surface diffusion is not fully understood. One typical indication of surface diffusion is that the effective diffusivity exhibits a negative temperature coefficient. That is, it is seen to decrease with increasing temperature. In fact, this constitutes an experimental verification of its presence. It has been shown by Komiyama and Smith (1974a,b) that the extent to which surface diffusion is operative depends both on the nature of the solvent for the same adsorbent-adsorbate system as well as on the nature of the adsorbent for a component adsorbing from a given solvent. The purpose of this paper is to explain the mechanisms of reactant diffusion for the direct hydration of isobutene (iB) to tert-butyl alcohol (TBA), whose stoichiometry is (H3C)&=CH2 + H20 + (H,C),COH (1) The reaction is catalyzed by Amberlyst-15 (A-15) par-
ticles, a macroporous ion-exchange resin with sulfonic acid active groups. Since surface diffusion has been associated with this kind of catalyst (Gupta and Douglas, 1967; Komiyama and Smith, 1974a,b), structural information about macroporous ion-exchange resins is pertinent. A-15 is a styrene-divinylbenzenecopolymer in which the active groups have been introduced by sulfonation. The total number of active groups is 4.7 mequiv/g. The resin particles are nearly spherical beads composed of agglomerates of gel-type microspheres surrounded by a macroporous matrix (Pitochelli, 1975; Kun and Kunin, 1967). The estimated size of the microspheres is about 30 nm (Ihm et al., 1988). Fragmentary information (Ihm and Oh, 1984) suggests that only 5% of the active groups are on the outer surface of the microparticles, the remainder being within the gel structure. Pore-volume distribution studies, performed by mercury porosimetry on the dry catalyst, were presented in a previous paper (Leung et al., 1986) in terms of the cumulative pore volume vs diameter. The total porosity is 35% with an average diameter of 26 nm. About 75% of the pore volume is in pores smaller than 30 nm, presumably associated with the microparticle porosity. This particular pore structure can be relevant to the interpretation of intraparticle diffusion and reaction on A-15. We have reasoned (Leung et al., 1986) that because of the small size of the microparticles, with Thiele modulus of about the reaction is unlikely to be affected by diffusion within the microspheres. Thus, the measured values of the effective diffusivity in the absence of product correspond to the macropore volume. Recent studies by Ihm et al. (1988) postulate a two-phase, microporousmacroporous reaction diffusion model with different diffusivities in each region. Their results indicate that the effectiveness factor of the microparticles is near unity; hence, reaction is not diffusion-limited in the microparticle region.
0888-5885/90/ 2629-1485$02.50/0 0 1990 American Chemical Society
1486 Ind. Eng. Chem. Res., Vol. 29, No. 7 , 1990
The kinetic- and mass-transfer data for reaction 1 over A-15 are available from previous work (Leung et al., 1986; Vel0 et al., 1988) for the range of butene and alcohol concentrations found in industrial trickle-bed operation (Leung et al., 1987a; CAceres et al., 1988). For the above process, Gupta and Douglas (1967) found that surface diffusion contributed significantly to the overall diffusivity over Dowex-50, a resin similar in functionality to A-15, although of gel type. The data by Leung et al. (1986) indicate that, under irreversible conditions (without TBA), the measured diffusivity behaves normally with temperature, hence with no sign of surface diffusion. The conditions may be quite different in the presence of product. Vel0 et al. (1988) measured the hydration rates in the absence of external and internal mass-transfer gradients. The results show that TBA inhibits the rate more than is to be expected from the extent of the reverse reaction. This led us to introduce empirical adsorption constants in the denominators of the rate equations. As to the reactants, they do not seem to adsorb for the ranges of concentrations studied. This opens up the question as to whether the product may retard the global rate in some way relative to the effects of product on the diffusivity of the reactant. One obvious effect would be that the effective diffusivity of the reactant changed with the presence of a relatively large excess of product, a condition likely to prevail in catalyst pores in a multiphase reactor with a gas-limited reaction. Such an effect has been reported for the case of hydrocracking of oil residuum (Ternan and Packwood, 1986) where hydrogen diffusivity seems to be related to micelle formation following cracking. Our case would be similar, in that aqueous TBA is a strongly nonideal solution where aggregation-solvation depends on the alcohol concentration (Kenttamaa et al., 1959). For instance, Leung et al. (1987b) found that enthalpy and entropy changes for butene absorption changed dramatically with concentration, giving a 1000-fold increase in gas solubility. This suggests a variable degree of molecular clustering in the liquid solution. Therefore, the mass-transport properties of butene are expected to be significantly affected by changes in molecular aggregation. The purposes of our paper are as follows. First, we wanted to study the effect of TBA on the intraparticle diffusion rates under conditions of temperature and concentration of a multiphase chemical reactor such as trickle bed or slurry type. The study is pertinent because the contribution of the various diffusion mechanisms may change significantly with product accumulation due to conversion. Another objective was to extract appropriate correlations for the effective diffusivity as a function of temperature and composition. To this end, we propose a method to determine the transport rates of the reactant diffusing in liquid-filled pores of constant composition. This is based upon a solution of the diffusion equation for the case of excess product, with the effective diffusivity as a single unknown.
Experimental Methods Reaction Runs. iB hydration runs were performed in a batch-recycle, liquid-full, packed-bed reactor operated differentially without resaturation as described below. An initially iB-saturated liquid was continually passed through the reactor and the fraction of converted iB measured as a function of time. Detailed descriptions of the apparatus, analytical procedures, catalyst manufacturer data and preconditioning, etc., are given elsewhere (Velo et al., 1988). A range of the measurements is given in Table I. Figure 1 schematically shows the reactor-reservoir setup. The
Table I. Range of the Measurements pressure atmospheric 303, 313,323, 333 temp, K Amberlyst-15 catalyst catalyst diam," mm 0.51, 1.07 9 x 10-3, 13 x 10-3 dry mass of catalyst, kg 2.2 x 10-6 liquid recycle flow, m3/s initial iB concn, kmol/m3 below lo-* 0.1, 0.5, 1, 2, 2.5 TBA concn, kmol/m3 3 x 10-3 total liquid vol, m3 a Effective wet sizes after dry sieving commercial Amberlyst-15 (Rohm and Haas, Barcelona).
I
1
1
i5 l
12
n
Figure 1. Apparatus: 1, reactor; 2, reservoir; 3, condenser; 4, thermostated bath; 5 and 6, pumps; 7, wet gas meter; 8, rotameter; 9 and 10, sampling septa; 11,valve; 12, bypass valve; 13, backpreasure valve.
packed-bed reactor, 1, was a stainless steel jacketed tube (1.21-cm i.d.), through which reactant was recycled from the reservoir, 2, with pump 5. Operating temperatures were controlled by flowing thermostated water from the bath, 4,with pump 6 through the jacket. Temperature readings were made with a 100-QPt resistor inserted into the bed. The reactant flow rate was measured with a rotameter, 8. The operating procedure for a reaction run was as follows. The reservoir was charged with an aqueous TBA solution with known concentration and heated to the reaction temperature with recycle through the reactor. After the reaction temperature was reached, the recycle flow was stopped and bubbling of iB a t constant pressure was started until saturation was achieved. The gas pressure at the reservoir overhead was held constant by use of the back pressure valve, 13. Once saturation was reached, the flow of gas was arrested and the absorber was isolated from the atmosphere by shutting all valves. This was the zero time for the experiment. At this moment, recirculation of the liquid from absorber to reactor was restarted and iB concentration vs time was followed by gas chromatography. High liquid flow rates as well as a small catalyst mass ensured differential-reactor conditions and iB conversions (per pass) below 10-15%. The high liquid flow rates used made external mass-transfer resistance negligible, according to the calculations made in our previous work (Velo et al., 1988). Viscometry. Kinematic viscosities of aqueous TBA were determined as a function of the same concentrations and temperatures of the reaction runs. A suspended-level, capillary Ubbelohde viscometer was used (Schott) in accordance with DIN 51562 standard. The temperatures of the thermostated bath were held constant to within f O . l
Ind. Eng. Chem. Res., Vol. 29, No. 7, 1990 1487 "C. Viscometer constants as a function of temperature were available. Kinetic energy corrections were unnecessary for the high flow times employed.
is the Thiele modulus based on the pseudo-first-order kinetic constant:
Diffusion-Reaction Theory with Excess Product Consider a liquid-full, batch recycle reactor packed with A-15 spheres with the liquid in the reservoir initially saturated with iB. A t zero time, recycle is started and iB begins to be consumed by the reaction. Our aim here is to derive an expression for the remaining iB as a function of time. Since the reactor is operated differentially, it can be further assumed that the rate is constant along the bed. Also, since iB concentrations are always 3 orders of magnitude less than CA (Table I), accumulation of TBA due to the reaction is negligible. Thus, CA and Cw will be constant in the bulk and pore fluid a t all times during a run. Under these conditions, iB will diffuse into a liquid of constant composition. The conservation equations of iB in the pore and bulk liquid, respectively, are
An exact, closed-form solution to eqs 2-8 is
kCw/(1
k1
+ KACA)
m
cL(o)/cLO
= CBe/CLO
+ n = l ((1-
4'Xe/un2) exp[(u,2 - 42)(o/a)I)/(l+ (3a/2)[~0th( 3 ~ , ) / 3 ~-,Coth2 ( 3 ~+~11) ) (12) where the first term accounts for the steady-state or equilibrium solution a t infinite time (pole s = 0). The series term accounts for the dynamic response of the system. The summation is carried out over the roots of the transcendental equation 3(un2- 4') + ( u [ ~ ucoth , ( 3 ~-~11) = 0 (13) where u, is related to the Laplace variable, s, by the relationship s = (u,2 - @)/a
(3) where De is the effective diffusivity in a multicomponent liquid mixture. The initial and boundary conditions for eqs 2 and 3, are CB=O for t = O , r > O (4) (5) aCB/ar = 0 for t > 0, r = 0
CB = CL for
t
> 0, r
=R
(6)
CL = CLo for t = 0 (7) In eq 2, an expression for the intrinsic rate of reaction is needed. The one given by Velo et al. (1988) will be used: (8) where the values for the kinetic and equilibrium constants are known functions of temperature. Equation 8 was developed from rate measurements exempt from masstransfer effects performed on A-15 of small size (124 pm). The above model involves the diffusivity as the only adjustable parameter, all others being known from independent experiments. These include the value of De in pure water from Leung et al. (1987a,b). Thus, performing runs a t various CA's and temperatures and fitting CL(t) with the solution from the model will provide knowledge of De as a function of composition and temperature relative to that in pure water. Method of Solution. With the above assumptions, eq 2-8 can be solved analytically (Velo, 1990). A solution is found by transformation in the Laplace domain, followed by identification of the poles of the transform, CL(s),by the method of Munro and Amundson (1950). Inversion of the transform is readily carried out by use of the Heaviside expansion theorem (Crank, 1975). Algebraic manipulations are made in terms of the dimensionless time: mk.
where
(11)
(14)
For given values of 9 and a,the roots of eq 13 are determined by a standard numerical algorithm. During numerical manipulation of the solution, the series in eq 12 can be truncated after a few terms, as it converges to a constant value to within the sixth decimal place. It is readily proved (Velo, 1990) that all the roots of eq 13 are real and negative in the s domain. This means that the model predicts a stable system, as is experimentally observed.
Results Reaction runs were performed in a differential batchrecycle reactor. Measurements comprised two catalyst sizes and four temperatures, with alcohol concentrations from 0.1 to 2.5 kmol/m3. The evolution of CL vs time allows curve fitting with the solution from the model in order to obtain the reactant diffusivities. Figures 2-4 show the effects of the above variables. Figure 2 shows the influence of the catalyst size upon the reaction rate for constant TBA concentration a t two temperature levels. The initial iB concentrations correspond to the different solubility of iB with temperature (Leung et al., 1987b). For a given temperature and TBA concentration, the rate is higher the smaller the size of the catalyst. Since liquid-to-particle mass-transfer resistance is negligible in these runs (according to the previously derived criteria of Vel0 et al. (1988)),the results of Figure 2 actually show the effects of intraparticle diffusion on the observed rates. The retarding effect of product concentration on the rate for constant temperature and particle size is seen in Figure 3. The initial concentration of iB increases with CA, due to the exponential increase in iB solubility in the presence of alcohol. Prior kinetic data in the absence of product from Leung et al. (1986) indicate that the kinetics is first order in iB concentration. It is seen instead that increasing CAdecreases the rate as judged from the initial slopes of the curves. This suggests an inhibitory effect of the product. This effect has been properly accounted for in eq 8. We show later that, in addition, the diffusivity of iB decreases substantially when alcohol is present. The influence of temperature is more complex. Figure 4 shows the results of four runs at different temperatures for a given particle size and constant TBA Concentration. First, the solubility of iB decreases with increasing temperature. However, an increase in reaction rate is observed in going from 303 to 313 K. Also, a higher deflection of
1488 Ind. Eng. Chem. Res., Vol. 29, No. 7, 1990
A 323
t
.
E -
333
1
" - I m
01
'
'
"
'
"
"
3600
0
1
"
7200
TIME, s Figure 4. Effect of temperature on the reaction rate (at constant C A = 1 kmol/m3, d, = 1 mm). 1 5 , , ,
T - 323 K
,
.
,
,
,
,
,
,
,,
Ca kmol / m 3
25 0
0
3600 TIME.5
7200
0
3600 TIME,
7200 S
Figure 5. Sensitivity of iB disappearance rate to values of effective diffusivity: (A) CA = 2.5 kmol/m3, (B)CA = 1.0 kmol/m3.
01 0
J
3
3600
7200
TIME, s
Figure 3. Retarding effect of alcohol concentration (at constant T = 323 K,d, = 1 mm).
the curves is observed a t higher temperatures, because chemical equilibrium is approached. This has recently been explained (Velo et al., 1988), as follows. While the first-order kinetic coefficient increases with temperature, the chemical equilibrium constant decreases, due to the exothermic nature of the reaction. I t is known from our previous studies that the effects of product concentration and temperature are interlinked. Thus, we have shown that the degree of inhibition by product increases with temperature. This is consistent with the inclusion of an inhibition constant, KA,in the denominator of the Langmuir-Hinshelwood type of rate expression, eq 8, with a positive temperature coefficient. The features presented so far are all embodied in the model equations for a batch-recycle reactor, eq 2-8. In particular, eq 8 incorporates the effects of product inhibition with temperature and the approach to equilibrium at a location within a catalyst pore. Prior rate data free from mass-transfer effects allow us to safely use eq 8 as an intrinsic rate equation. Particle size is accounted for
by use of a Thiele modulus, eq 10. This typically involves a rate constant, effective diffusivity, and pellet size. Evaluation of Reactant Diffusivities. Reactant diffusivities were calculated by time-domain curve fitting of the observed reactor concentration data, using a function minimization subrouting with linear constraints for the diffusivities (BCONF, IMSL, 1987). A least-squares objective function based on the experimental and calculated concentrations was defined for each reaction run. The parameter constraints were taken as 10 times the diffusivity in pure water (upper bound), down to 100th of it (lower bound) a t corresponding temperatures. With the thus determined diffusivities, the theoretical responses could be obtained. The continuous lines of Figures 2-4 were thus drawn with the optimized parameter, using the solution of the model, eqs 12 and 13. Figure 5 relates the sensitivity of the response curves to changes in the diffusivity for various settings of temperature and alcohol concentration. The continuous lines are for the fitted values of De,given by the correlation (eq 15). The dotted lines represent the calculated responses with the values of De in pure water measured by Leung et al. (1986). As seen, the sensitivity of the method is not very high, being best a t low temperature and low TBA concentration for intermediate to high reaction times. In general, the procedure provides fair estimates (*30%) of the diffusivity, with somewhat less accuracy a t low TBA levels. Compare, for instance, parts A and B of Figure 5.
Ind. Eng. Chem. Res., Vol. 29, No. 7, 1990 1489 Table 11. Intramrticle Diffusivities of Isobutene d,, mm 0.51 1.07 CA9 P 0.09 2.45 0.1 0.48 0.95 kmol/m3 IO'OD,, m2/s T,K
-~ ~
2.05
~~
303 313 323 333
3.7 5.1 8.0 11.0
1.7 4.5 5.1 7.3
0.7 1.2 1.7
1.8 4.0 5.1
2.4
5.8
2.6 3.3 6.2
2.7 3.5 5.2 9.1
1.1
1.5 2.8
@FromLeung et al. (1986) on 0.45- and 1.04-mm A-15 wet catalyst.
The final values for De as a function of T and CA are given in Table 11. This shows that at a given temperature the diffusivity decreases with increasing TBA concentration for either set of measurements on the two catalyst sizes. The uncertain trend for the larger size is due in all probability to the inaccuracies explained above and to experimental error. For a given CAand size, the diffusivity clearly increases with temperature, a major result to be discussed later in connection with surface diffusion. Consistency of the results for a similar CA (compare, for example, columns 2 and 4 and 3 and 7 of Table 11) on the two particles, lends confidence to the method. It should be noted, however, that, for the lower alcohol concentrations, the hypothesis of our model, i.e., excess and constant alcohol concentration in a run, may not hold. Excluding the runs with small CA,the diffusivities can be well correlated with the following equation De = exp(-9.52 - 0.551CA- 3690/T) (15) CAup to 2.45 kmol/m3, CB below 0.01 kmol/m3, and T u p to 333 K. Figure 6 gives the temperature dependence of the diffusivity for the alcohol concentrations studied including that in pure water. The continuous lines represent the least-squares fits, eq 15. Within experimental error, a single positive activation energy, E = 31 kJ/mol, accounts for all data, regardless of CA.
Discussion It is clear that intraparticle diffusion is retarded by the presence of TBA. In the discussion that follows, it will be assumed that the effective diffusivity is related to molecular diffusivity and particle porosity as De = Dt/r, (16) with 7, stands for the apparent tortuosity factor of the macropore structure. In the absence of surface diffusion, this can be estimated by using pore diffusion models (Wheeler, 1951; Wakao and Smith, 1962) or else it can be taken simply as 7, = 4,as recommended by Satterfield (1970). Since the degree of swelling of A-15 beads in water and in TBA is similar (Rohm and Hass, 1972), and in view of our dilute solution ( 5 % ) , a change in the particle void fraction with CAcan be neglected. Assume further for a time that the tortuosity factor is also constant. With these two hypotheses, it is clear that De will behave in the same manner as D, the molecular diffusivity of iB at infinite dilution (see iB concentration in Table I). That is, De is just proportional to D. Suppose now that the molecular diffusivity is related to solvent viscosity by the StokesEinstein equation for flowing spheres; i.e., D p / T = constant (17) which is the starting basis of most modern correlations (Wilke, 1949). Therefore, for constant t and T,, to speak of the effective diffusivity and viscosity is in principle the
2.9
I
I
3.0
3.1
I
I
3.2
3.3
3.4
1 0 ~ 1 ~~ - 1 Figure 6. Arrhenius plots for the diffusivities of iB for various TBA concentrations. For regression lines, see text.
same, in view of eq 16 and 17. There is firm experimental evidence that the diffusion coefficient of hydrocarbons in water follows the relation of eq 17. A large number of examples are found (Reid et al., 1977) that show that the diffusivity of a given solute is about inversely proportional to solvent viscosity at the same temperature. On the other hand, Hayduk and Cheng (1971) found by examination of many nonaqueous systems that D was proportional to solvent viscosity raised to a power p , for a short range of temperatures, the power factor varying between -0.5 and -1.0 in most cases. For water-TBA mixtures, the value of p should be between -1.0 and that in pure TBA. By properly modifying the Leffler-Cullinan (1970) equation (to account for a mixed solvent) to make it consistent with the Hayduk-Cheng equation, we have (Dp)I-(P+l)x~ = (Dwpw)1-xWAILA-')~A (18) For a value of XA= 0.05 and taking the most unfavorable value for p = -0.5, one obtains Dp0.975= constant
(19)
In conclusion, it seems reasonable to use the relation given by the Stokes-Einstein equation to correlate diffusivities for our range of alcohol concentrations. Figure 7 shows the values for De/T and D t / T plotted vs measured viscosities at the same temperature. The values for De are those from the reaction runs (Table 11). The values for the molecular diffusivities used in Figure 7 are those measured experimentally at 303 K in water by Gehlawat and Sharma (1968) and corrected for temperature and viscosity according to eq 17. As seen, the experimental data points correlate very well with a slope of about -1.86, far different from the slope of the dotted line (-1.0) suggested by eq 17. It is now important to note that the horizontal distance between the two lines gives an idea of the values for the tortuosity factor and how it changes with composition. Assuming that 7, is a geometrical parameter associated with pore structure alone, we can write Dt/T 7, = (20) De/T Equation 20, together with the values plotted in Figure 7, allows calculation of the approximate values for the apparent tortuosity factor as a function of temperature and composition. The results are given in Table 111, which
1490 Ind. Eng. Chem. Res., Vol. 29, No. 7 , 1990 -1
pore-volume distribution studies of Leung (1987a,b) indicate that a notable percentage of the volume of A-15 occurs in the micropore region. Despite its sulfonic acid functionality, A-15 has a styrene-divinylbenzene matrix, which is hydrophobic in nature, hence, with a large affinity for the butene-also a hydrophobic molecule. From these considerations, the conjecture of adsorption of iB, and hence surface diffusion, is reasonable. The phenomenon of surface migration is not well understood as yet (Satterfield, 1970; Komiyama and Smith, 1974a,b). It is generally associated with a negative temperature coefficient of the effective diffusivity, provided that reactant adsorption is an exothermic process (Smith, 19811, as is usually the case. In eq 22, the change in the group KBD,/D with temperature can be expressed as
1
10
--
c-
r
Y
C,
h
05
u)
0
u
-12
10
0
. b-
0 "
-1 3 10
I
10-4
I
I
I
1 I 4 I I
I
I
lo3
/
1
/
1
1
1
10-2
V I S C O S I T Y , kg / m s
Figure 7. Correlation of D,/T and Dc/T with solvent viscosity. The results include the values in pure water. Table 111. Apparent Tortuosity Factors of Amberlyst-15 CA, kmol/m3 a t 303 K a t 313 K a t 323 K a t 333 K 00 2.0 1.9 1.5 1.3 0.48 2.5 2.4 1.7 0.93 2.0 2.1 1.7 1.3 2.05 3.4 3.4 2.5 2.45 4.5 3.6 3.8 3.5
"From the data by Leung et al. (1986).
includes the values in pure water. Apparent tortuosities are seen to decrease with temperature and to increase with TBA concentration very markedly, except at high CAwhere the tortuosities remain somewhat more constant with T. The Role of Surface Diffusion. Various mechanisms may contribute to intraparticle transport in liquid-filled pores. Besides ordinary or bulk diffusion, surface migration and restricted diffusion may occur. Although restricted diffusion may lead to an apparent variable tortuosity with product accumulation (Savage and Javanmardian, 1989), it is unlikely to be operative in our case. Generally, the size of reactant and product molecules is required to be similar to that of the pores. Thus, we shall focus our discussion on surface diffusion. When this is present, the effective diffusivity can be expressed as a combination of pore-volume diffusion and the surface migration due to a gradient in adsorbed concentration (Smith, 1981; Ramachandran and Chaudhari, 1983). If reactant adsorption is linear, with an adsorption equilibrium constant, KB, and if adsorption equilibrium is assumed, the observed diffusivity can be written in terms of the surface and bulk diffusivities as De = (Dt + KBDs)/7 (21) where T is a true tortuosity factor, assumed to be common for both diffusion mechanisms (Komiyama and Smith, 1974b). From eqs 16 and 17, it follows that 7 . = 7/[1 + K B D , / ( D ~ ) ] (22) so that the variation in (Table 111) can be discussed in terms of the group KBD,/Dt. Let us see now that there are sufficient hints to make surface migration of iB a likely mechanism of transport. Gupta and Douglas (1967) determined De for iB on Dowex-50, a gel-type, sulfonic resin catalyst similar to A-15, but of a microporous type. They found that surface diffusion explained the results of a decreasing D, with T. The
Usually the heat of adsorption is a large quantity for an activated process, so it is usually expected to be larger than the activation energy of surface diffusion. Hence, one expects -AH > E,. Furthermore, if -AH - E, >> Ed, the observed effect will be a decrease in De with increasing temperature. In our case, the opposite is found for De,and hence, the apparent tortuosity factor decreases. This would mean that in eq 23 -AH - E,
