Ind. Eng. Chem. Res. 1993,32, 1637-1644
1637
Hydrodesulfurization of Thiophene on Highly Deactivated Coal-Liquid Hydrotreatment Catalyst: Effect of Deposited Iron Sulfide A. P. Raje and D. B. Dadyburjor’ Department of Chemical Engineering, West Virginia University, Morgantown, West Virginia 26506-6101
The hydrodesulfurization of thiophene was studied over highly deactivated NiMo on alumina Catalysts, iron sulfide, and a combination of the two. Iron sulfide is one of the metals deposited along the periphery of the catalyst particle during deactivation. For the supported NiMo catalysts, the Arrhenius plots bend downward as temperature increases; i.e., the apparent activation energy decreases. This is as expected. However, at even higher temperatures, the plot for the highly deactivated NiMo catalyst bends upward again; i.e., the apparent activation energy increases. A two-step series reaction mechanism can be used to explain the results over the highly deactivated catalyst. Step 2 controls the reaction rate on NiMo catalysts, while on iron sulfide alone step 1 controls the rate. A model is presented describing diffusion and reaction within a catalyst pellet having a nonuniform activity distribution. The model predicts the Arrhenius plots obtained over deactivated catalysts, in particular, the increase in activation energy observed at high temperatures.
Introduction One of the main reactions involved in the hydrotreatment (HT) process is hydrodesulfurization (HDS). HDS converts the sulfur-containing heteroatom compound to hydrocarbons and hydrogen sulfide. Typically the catalysts used are a mixture of nickel or cobalt and molybdenum oxides supported on alumina. The catalysts need to be sufided to be in the active state. During the HT of coal-derived liquids, the catalysts undergo rapid deactivation. The deactivation is due to the deposition of coke and metals on the catalyst surface. These deposits are not caused by the heteroatom compounds taking part in the principal reactions of HT, but by other constituents present in the feed-coke from hydrocarbons and metals from organometallicconstituents and mineral matter. Tamm et al. (1981)determined that coke deposits uniformly over the catalyst surface whereas metals deposit preferentially around the periphery of the catalyst pellet. The major metal components in the deposits are iron, titanium, and calcium. Iron is typically present as iron sulfide on the catalyst surface. The existence of two types of active sites for the HDS of thiophene has been suggested before. However, the relative roles of the two sites are ambiguous. For example, Satterfield et al. (1975)studied the reactions of the HDS of thiophene and the hydrodenitrogenation (HDN) of pyridine occurring simultaneously on CoMo and NiMo catalysts. For thiophene HDS on these catalysts,two types of active sites were proposed, with different relative activities and different thiophene adsorption characteristics. However, each type of active site was said to be able to catalyze the conversion of thiophene to butene by itself without needing the presence of the other type. On the other hand, Lipsch and Schuit (1969)postulate that butadiene, an intermediate, is formed from thiophene on one type of active site of CoMo. The butadiene then desorbs from this site and is further hydrogenated to butene and/or butane on another type of active site. In this paper, we present evidence that the two steps in the mechanism need different types of active sites or functionalities. A catalyst active for thiophene HDS must possess both types of active sites in sufficient amounts. Iron sulfide and NiMo catalysts differ from each other in this respect. We also present the effect of the highly uneven distribution of the metal deposits on the overall activity of the deactivated catalyst pellet. If the metal
deposits are catalytically active, then their distribution around the periphery of the catalyst pellet leads to a nonuniform distribution of activity. The effect of this nonuniformity on the observed rate for thiophene HDS is primarily felt under reaction conditions where intraparticle diffusion exerts a strong influence on the overall rate of reaction. Simultaneous diffusion and reaction in nonuniform catalyst pellets has been analyzed by a number of investigators. Wheeler (1951)analyzed a pellet with an inactive poisoned shell and an active core to obtain rate constants relative to a uniformly active pellet. Subsequently, Shadman-Yazdiand Petersen (1972)and Corbett and Luss (1974)analyzed pellets having various continuous activity distribution functions (with respect to position) inside the pellet. These nonuniform pellets were characterized for activity maintenance with time for various mechanisms of fouling and poisoning of the pellets. Becker and Wei (1977)analyzed nonuniform pellets possessing sharp variations in activity: an active outer shell with an inactive core, vice versa, and a pellet having an active annular region with an inactive outer shell and core. These pellets were then rated for their deactivationcharacteristics for an impurity poison depending on the Thiele moduli for the poison and for the main reaction. Wheeler’s (1951) analysis yields the rate constant for a single pellet having an unchanging but nonuniform distribution of activity, while the other analyses concentrate on the variation of overall activity with time as the nonuniform pellet deactivates.
Experimental Section Catalysts Used. The catalysts used were Shell 324M, NiMo supported on alumina. Some properties of the fresh, i.e., unused, catalysts are given in Table I. The catalysts used are cylindrical pellets of length 4 mm and diameter 0.8 mm. Pre-deactivated catalyst samples were obtained from a HT reactor for coal-derived liquids at Wilsonville’s Advanced Coal Liquefaction Test Facility in run 242.The coal used in this run was Illinois No.6. Catalyst samples were removed from the HT reactor at different times on stream, quantified as pounds of resid processed per pound of catalyst. Table I1 lists some properties of two of the catalyst samples used in this study. The samples used in the present study were removed from the reactor relatively late in the run and are, therefore, more deactivated and contain a large amount of metal deposits. In addition, the
o a a a - ~ a a ~ i ~ ~ ~ ~ ~0~ 1993 ~ - American i ~ ~ ~ Chemical ~ o ~ . ~Society o~o
1638 Ind. Eng. Chem. Res., Vol. 32, No. 8, 1993 Table I. Properties of Fresh Shell 324M Catalyst (Stiegel et al., 1985) composition 2.8 wt % Ni Mo 12.4 wt % surface area 180 m2/g 1.38 /cm3 pellet density 115 average pore diam
x
Table XI. Properties of Deactivated Shell 324M (NiMo/ Alumina) Catalysts (from Data in Stiegel et al., 1985) catalyst WIL 9 WIL 12
age (lb of resid/ lb of catalyst) 368 606
coke" (wt %)
9.8 10.7
metals content (wt %) Fe Ti Cab 0.6 0.6 0.2 1.11 0.8 0.19
a Coke estimated by sum of wt % C + H + N. Ca estimated by total metals % - (Fe % + Ti %).
T O VENT
RESERVOIR
I
$
1
PRE-HEATER
w
DRAIN
REACTOR
I
IG A S ~~
I-(
I
CHROMATOGRAPH SIX PORT VALVE
TT i
-@
CONDENSER
t
DRAIN
Figure 1. Continuous-flow microreactor unit.
sample WIL 12 was crushed to a size of less than 37 pm for additional experiments. Acid-washed iron sulfide, of approximately 95% purity, was also used as a catalyst, in the form of crushed particles. Experimental Procedure. A continuous-flow packedbed reactor was used to carry out the HDS of thiophene over the pre-deactivated catalysts. A flowsheet of this unit is shown in Figure 1. The reactor is a 3/8-in.-o.d. stainless steel tube heated by an electric furnace. The temperature in the reactor is monitored by three thermocouples placed at the inlet, outlet, and middle of the packed catalyst bed, respectively. At the inlet and outlet of the bed, the pressure is also measured. A solution of 5 wt % thiophene in m-xylene (inert solvent) is fed by means of a metering pump, mixed with a controlled flow stream of hydrogen, and then passed through the preheater to vaporize the liquid completely. The gas chromatograph has a flame ionization detector, and the column used for analysis of thiophene, butene and butane is a Super Q column from Alltech Associates, Inc. Surface areas of
catalyst samples were evaluated using a Pulse Chemisorb 2700 manufactured by Micromeritics Instrument Corporation. Approximately 14 g of the catalyst was placed in the reactor. The catalyst was pretreated by passing helium through the bed at 300 "C, 15 psig pressure, and a flow rate of 400 mL/hr for 4 h. Sulfiding of the catalyst was carried out at 300 "C and 15 psig pressure, in 380 mL/h of a mixture of 10% HzS in Ha for 4 h. Catalysts used include the following: (i) catalyst sample WIL 9; (ii) catalyst sample WIL 12; (iii)iron sulfide of average particle size 64 pm; (iv) catalyst sample WIL 12 crushed to size less than 37 km; (v) a mixture of 52 % crushed WIL 12 and 48% iron sulfide, both of size less than 37 pm. In addition, thiophene HDS was carried out with the reactor packed with0.5-mm glass beads. The void volume in the reactor was maintained approximately the same as the reactor void volume when filled with catalyst pellets. The HDS of thiophene was carried out starting at the lowest temperature possible (consistent with an experimentally-measurable amount of product butene) and subsequently at increasing temperatures. At each temperature, conversions were measured at two different flow rates of reactant thiophene, with at least three data points obtained at each flow rate. The same catalyst charge was used for activity measurements at all temperatures. Due to the wide range of temperatures at which activity measurements were carried out, 180-540 "C, different sets of thiophene flow rates were used at different temperatures. Liquid flow rates ranged from 5 to 200 mL/h. The flow rate of hydrogen was also changed, to maintain the mole fraction of thiophene in the reactant mixture at approximately 1% and that of hydrogen at 80%. At high temperatures and at high flow rates of thiophene, there was an appreciable pressure drop through the packed bed reactor. This was accounted for in the kinetic analysis as shown below.
Rssults Rate Constants. The only products observed for the HDS of thiophene were butene and a small amount of butane. Therefore, conversionswere calculated as the ratio of butenes and butane formed to total moles of thiophene, butene, and butane in the product stream. Carbon material balances were greater than 93 % . A Langmuir-Hinshelwood mechanism is usually considered for the HDS of thiophene (Satterfield and Roberts, 1968). At low concentrations of thiophene (and hence of product hydrogen sulfide), the adsorption terms can be neglected. Therefore, the rate equation simplifies to -rT = k'zpTpH (1) a second-order reaction. The additional assumption of a pseudo-first-order reaction is reasonable since the mole percentage of thiophene is approximately 1% in the reaction mixture while the hydrogen concentration is much larger. Then the hydrogen partial pressure can be assumed to be constant. This assumption simplifies the rate constant evaluation from conversion and space velocity data. Rate constants for thiophene HDS were calculated from conversion versus space velocity data at each temperature. Since there was an appreciable pressure drop in the packed bed reactor, especiallya t high temperatures and pressures, the pseudo-first-order rate constant was calculated as suggested by Fogler (1986). The total pressure, Pbt, at any point, 2, in the packed bed reactor is represented by Pbt = P,(1- (UZ)1/2 (2) where a is a constant dependent on the total pressure
Ind. Eng. Chem. Res., Vol. 32, No. 8,1993 1639 drop and length in the reactor. Using eq 2 in the conservation equation for a packed bed reactor gives the relationship for the observed pseudo-first-order rate constant, kom,
10
9 h
3 I
(3)
8
c)
$
7
M
where an average pressure is defined as
>
6
v
2 pin2+ plllpout + pou: Paw= (4) 3 pin + p o u t Several adjustments need to be made to the values of the pseudo-first-order rate constant. First, the rate constants evaluated in this study are per unit weight of catalyst. Catalysts with varying ages contain differing amounts of deposits. In order to compare evaluated rate constants for catalyst samples of different ages, the rate constants are corrected to refer to some constant, invariant weight of the catalyst. This weight is chosen as the equivalent weight of the fresh catalyst. The corrected rate constants are calculated using information in Table I1 and the following equation: (1- wt fraction coke wt fraction of metal compound) (5) Second, as the temperature and pressure varies, the concentration of hydrogen, CH,and the effectivediffusivity, DeK, in the catalyst pellet are changed. These parameters affect the pseudo-first-order rate constant in a number of ways. If the intrinsic kinetics control the rate of reaction, then the pseudo-first-order rate constant is proportional to CH, whereas the proportionality is to C d I 2 if internal (pore) diffusion exerts a strong influence on the reaction rate. Further, in the regime of strong pore diffusion influence, the dependence of the diffusion coefficient on total pressure varies with the diffusion mechanism-if Knudsen, then Deft. is independent of total pressure, if bulk, then Den is inversely proportional to the total Pressure. For the pseudo-first-order rate constants to be comparable, they must be adjusted so that they all refer to the same concentration of hydrogen and the same total pressure. The references values are
knew= k,/
P*wf = 40 psig
(64
and CHsef = 7.675 x lo4 mol/cm3 (6b) (The value in eq 6b corresponds to 0.8 mole fraction H2 at Pmfand Tref = 200 O C . ) The adjusted value of the rate constant depends upon whether intrinsic kinetics control or whether there is a strong pore diffusion resistance; if the latter, then whether diffusion is Knudsen or bulk. The three possibilities are given below:
Arrhenius Plots. Figures 2 and 3 show the Arrhenius plots obtained for the samples WIL 9 and WIL 12, respectively. Both plots contain three regions. At low temperatures, where the intrinsic kinetics can be expected
v
x
5
9
4 3 0012
001.1
0016
0018
0020
0022
1/T ( K ) Figure 2. Arrhenius plot for catalyst sample WIL 9.
h
CI
c I
8
'1
c)
m
M
> v
4
v
c
0012
0014
0016
0018
0020
0022
1/T ( K ) Figure 3. Arrhenius plot for catalyst sample WIL 12.
to control the overall rate of reaction, the activation energy is approximately23 kcal/mol. This is the same activation energy observed on mildly-deactivated catalysts (Raje, 1992). At moderately high temperatures, the activation energies in Figures 2 and 3 are approximatelyhalf that of the activation energy in the corresponding low-temperature (kinetics-controlling)regimes. This, too, is similar to what was observed earlier for mildly deactivated catalysts. At these temperatures, diffusion is exerting a strong influence on the overall reaction rate. At the highest temperatures, though, the activation energy increases, approaching (again) the value observed for the kineticscontrolling region. This is in dramatic contrast to what was observed over mildly deactivated catalysts, for which the activation energy continues to decrease at high temperatures to an even lower value, approximately 1.1 kcal/mol. With the reactor fiied with glass beads, and having a void volume approximately the same as the reactor filled with catalyst pellets, there is very little conversion of thiophene, less than 59%,even at the highest temperatures. This indicates that the homogeneous or noncatalytic HDS of thiophene is not significant at the temperatures of this study. In each of the three temperature regions, experiments were conducted for sample WIL 12to test if the assumption of pseudo-fiist-order rate is valid. The square of the correlation coefficient for plots based on eq 3 is greater than 0.96 at reaction temperatures from 225 to 425 OC. This indicates that the first-order assumption is indeed valid, and that there is no change in the rate order at the highest temperatures of the Arrhenius plots. Experimentswith crushed catalyst samples were carried out to verify that the low-temperature region of the
1640 Ind. Eng. Chem. Res., Vol. 32, No. 8, 1993
The experiments with a mixture of crushed catalyst sample WIL 12 and iron sulfide give unexpected results. The expected rate constant for the mixture can be obtained by adding the observed rate constants for the two catalysts alone in proportion to their weight fractions, w:
.I
A
crushed WIL # 12
A
pellets WIL # 12
= kFeSWFeS + ‘WIL 12wWIL 12 (8) The rate This is given by the dotted line in Figure 5. constant experimentally observed for the mixture, however, is greater by a factor of about 4, indicating some sort of synergisticeffect exhibited by the two catalysts together. We discuss this below. kexp
0
:
-
v
T ! 0012
\
A
I 0014
0016
0016
0020
0022
1/T ( K )
Figure 4. Arrhenius plots for crushed sample WIL 12 compared to the Arrhenius plot for whole pellets of sample WIL 12. 10
ki
thiophene + I
-
A
WIL 12 (cruahed)
0
FeS
0
WIL 12
+ FeS
Y
Y
2 0.0012
Discussion Low-Temperature Reaction over Iron Sulfide and NiMo Catalysts. Raje (1992) has proposed a modified reaction mechanism for HDS of thiophene:
1
0.0016
0.0020
1/T (K-’)
Figure6. Arrheniua plots for crushed sample WIL 12 (size less than 37 pm), iron sulfide alone (size = 64 pm), and a mixture of 48% iron sulfide and 62% crushed WIL 12 (size of both less than 37 pm).
Arrhenius plot is indeed one in which the intrinsic kinetics control the overall rate of reaction. Figure 4 shows the Arrhenius plot for crushed sample WIL 12 (particle size less than 37 pm). For comparison, the Arrhenius plot for sample WIL 12 (pellet) is recast here. The rate constants for the two catalysts at the common (low) temperatures are comparable, and the activation energy observed is 22 kcal/mol for both catalysts at those temperatures. Therefore, the low-temperature data for the catalyst pellets are in the kinetics-controlling regime. The small increase in the intrinsic rate constant for the crushed catalyst over the pellets can be attributed to pore pluggingin the catalyst pellets. Pore plugging would deny to the reactants access to the active sites of the uncrushed pellets, but the accessibility of the active sites would be relatively unaffected for the crushed catalyst. Figure 5 compares the Arrhenius plots obtained for crushed iron sulfide (particle size approximately 64 pm), crushed catalyst sample WIL 12 (recast from Figure 41, and a mixture of crushed WIL 12 and iron sulfide (particle size of both are less than 37 pm). The rate constants in these plots are calculated from the rates per unit gram of the catalyst sample present in the reactor; note that the correction of the catalyst weight to a fresh-catalyst-weight basis for the NiMo samples has not be made. Figure 5 shows that the rate constant for pure iron sulfide is small, compared to the value for deactivated sample WIL 12. Also, the activation energy is about 5.5 kcal/mol, and is much smaller than the activation energy in the kinetics-controlling regime for samples WIL 9 and WIL 12, about 23 kcal/mol. The relative values of these lowtemperature parameters are discussed below.
kdK
-
butenes
ka
(9)
The overall rate of formation of butene is given by rg = c ~ / { ( l / h i+ ) (l/k&l (10) The two steps in this mechanism were shown to have different activation energies. On mildly deactivated NiMo catalysts, step 1 has an activation energy of 1.1 kcal/mol, while step 2 has an activation energy of 22 kcal/mol. Further, it was shown that step 2 is the rate-controlling step for these catalysts at low temperatures. As shown below, this series mechanism can account for the rate data obtained at low temperatures, i.e., in the kinetics-controlling regime, for the highly deactivated catalyst samples WIL 9 and 12, for pure iron sulfide, and for the catalyst mixture. As shown in Figure 5, iron sulfide has both a low rate of reaction and a low activation energy, as compared to NiMo catalysts. We propose that, on iron sulfide, the forward rate constant for step 1, kl, is low, as compared to k&. Therefore, the forwardstep 1 controls the intrinsic rate of reaction at all temperatures; i.e., the intrinsic rate constant for iron sulfide is kint,FeS = kl$eS (11) and the observed activation energy should correspond to the activation energy for step 1. This value is 5.5 kcall mol, as noted in the previous section. Therefore, the activation energy for kl for iron sulfide is comparable to that for NiMo catalysts. Further, kl for the iron sulfide is much smaller than kl for NiMo catalysts. This accounts for the low reaction rates obtained on crushed iron sulfide alone. The lower value of k l could be due to there being fewer active sites on iron sulfide relative to NiMo catalysts. On the other hand, on NiMo catalysts at the low temperatures, the second step, the reaction of the intermediate I, is the rate-controlling step (Raje, 1992). From eq 10, if kl is much greater than k&, then the intrinsic rate constant at low temperatures for NiMo is k,,NiMo = k2,NMoK (12) The activation energy observed at low temperatures is about 22 kcal/mol for sample WIL 12. The corresponding value for mildly deactivated NiMo catalyst is also 22 kcal/ mol (Raje, 1992), consistent with our explanation. A physical explanation for the different reaction rates is shown in Figure 6. The number of sites active for step 1 on iron sulfide is very small as compared to their number on NiMo catalysts. Therefore, the production of intermediate I on iron sulfide is low, and the formation of the final product butene is small. On the other hand, the
Ind. Eng. Chem. Res., Vol. 32, No. 8, 1993 1641 aite
0
Thiophene
site
I
Table 111. Reaction Rete Constants (cm8/(h.g of catalyst)) at Low Temperatures (Intrinsic) for NiMo and Iron Sulfide _____
-
Butene
I
k,
@
k-1
k,
rate constant kl3&
k2pd
h,mK
Arrheniue form value at 225 O C 1.65 X 109 exp(-5.55 X 109/RT) 6.27 6.06 X 1010 exp(-20.44 x 109/RT) 74.10 3.31 X 10ls expk26.30 X 10BIRT) 112.72
contains a small amount of iron sulfide (see Table 11). Also shown in Figure 5 are the overall observed rate constants obtained for a mixture of known amounts of WIL 12 and added iron sulfide. The low-temperaturedata from these two independent experiments give two simultaneous equations of the form of eq 14, and these can be solved to yield the unknown quantities k~,"d( and
Th
k2,Fe&.
Thiophene
h- I /
>
Butene Figure 6. Schematic diagram explaining thiophene HDS over iron sulfide, NiMo catalyst, and a mixture of the two.
number of active sites for step 1 on NiMo catalysts is sufficient for intermediate I to be produced in substantial quantities. Therefore, large quantities of the final product butene can be formed, and k& is the limiting term for NiMo catalysts. Low-Temperature Reaction Over Catalyst Mixtures. Figure 6 can also be used to explain the improvement in reaction rate obtained by using the combination of NiMo (sample WIL 12) and iron sulfide. For this combination, the NiMo catalyst, with a large number of active sites for step 1, ensures that enough intermediate I is formed. The intermediate can further get converted to product butene on either NiMo or iron sulfide, both of which have sites active for step 2. This is shown schematically in Figure 6. Since the rate-determining step over NiMo catalysts is step 2, the increase in rate observed for the catalyst mixture can be explained by assuming that iron sulfide enhances the rate constant for this step. As mentioned previously, the rate constant for forward step 1 over iron sulfide is small compared to that for NiMo. Therefore, iron sulfide has little effect on the rate constant for forward step 1 of the mixture. The reaction network over the mixture can be given as thiophene
=I
Nitdo/
butene
butene
Note that forward step 1 does not control the rate, as for pure FeS, since the presence of NiMo ensures sufficient active sites for the formation of I. Step 2 is (still) the rate-controlling step for the mixture. The observed rate constant on a weight basis for the mixture is given by &mix
= (k2,NiMoWNiMo + k2,FeSwFeS)K
(14)
K is the equilibrium constant for step 1 and is not dependent on the particular catalyst used. Determination of Intrinsic Rate Constants. Figure 5 shows the overall observed rate constants obtained for the crushed catalyst sample WIL 12. This sample actually
The rate constants obtained from these calculations are shown in temperature-dependent form in Table 111,along ~ t the h numerical values at a temperature of 225 O C . Since K is an equilibrium constant and does not depend on the catalyst used, the different values of the quantities kz,Fe& and k2,NiMoK mirror the differences between the rate constant for step 2 of the two catalysts. It can be seen that the rate constant for step 2 over iron sulfide is of comparable magnitude to the rate constant for step 2 over NiMo Catalysts at 225 O C . The forward rate constant for step 1 over iron sulfide (k1,FeS) can be determined from the overall observed rate constant for iron sulfide alone, shown in Figure 5. The rate constant for this third independent experiment is represented by eq 10, since step 2 is not the ratedetermining step over iron sulfide alone. Using the value of k 2 , ~ ~ &determined earlier, k l p e s can be calculated from equation (10). The temperature-dependent form of this rate constant and its value a t 225 "C are also provided in Table 111. The forward rate constant for step 1 over iron sulfide at 225 O C is an order of magnitude lower than the quantities k& for iron sulfide and for NiMo. The rate constant for step 1 over sample WIL 12, k l ~ ~ o , cannot be determined. We have assumed this step to be infinitely fast relative to the other step in the overall mechanism. The above discussion implies that iron sulfide in the presence of NiMo possesses appreciable catalytic activity for the HDS of thiophene. The other mineral deposits of Table I may have some catalytic effect as well, but it is not as significant. We have assumed that the other deposits have no catalytic effect. This assumption overpredicts values of kZ,NiMo, but the qualitative arguments would not change. The deactivated samples WIL 9 and 12 contain iron sulfide distributed mainly along the periphery of the catalyst particle. In light of the above discussion, the deactivated catalyst samples will then possess a nonuniform activity distribution. It is of interest to'see if this nonuniform distribution of activity can explain the rise in activation energy observed for these catalyst samples at high temperatures. Analysis for Nonuniform Activity Distribution. The analysis assumes an infinite flat plate geometry for the catalyst pellet. An idealized model of a deactivated catalyst pellet with significant metals deposition is shown schematically in Figure 7. The outer layer is of length aL and the inner layer is of length (1- a)L,where L represents half the width of the flat plate. The activity, activity distribution, and effective diffusivities for both inner and outer layers remain constant with time. The iron sulfide is deposited along the periphery of the catalyst particle, either at the pore mouths of the catalyst pellet or a t the outside of the pellet surface
1642 Ind. Eng. Chem. Res., Vol. 32, No. 8, 1993
in the form of microparticles. Thus, the average effective diffusivity of this outer layer, Dout,can be different from the effective diffusivity of the inner core,Dh, of the catalyst pellet. In the interior of the pellet, there is assumed to be no iron sulfide. The rate constant per unit volume, kin, then corresponds to that for the rate-limiting step 2 for NiMo alone. At low temperatures, the concentration gradient of the reactant, thiophene, is negligible throughout the pellet, and the observed rate constant per unit volume, k,, will correspond to the intrinsic rate constant kiat,vwithin the pellet,
k, = hint,,= (1- a)k, + akout
(15) At higher temperatures, diffusion exerts a strong influence on the overall reaction rate, and the interior of the pellet experiences a smaller reactant concentration than the outer portion of the pellet. As the temperature is increased, the reactant contacts less and less of the interior of the pellet, and the overall rate is thus affected more and more by the outside pellet layer. Eventually the overall rate is influenced exclusively by the reaction occurring in the outer layer of the pellet. The differential equations describing a single reaction occurring inside this pellet are, d2CA/dX2- (k,,t/Dout)CA = 0
0
