Ind, Eng. Chem. Fundam. 1981, 20, 399-403
single band of lower intensity at 2045 cm-’. Further evacuation causes no change in either band position or intensity. The bridged CO band remains unaffected throughout the evacuation procedure. The splitting of the linear CO band is in accordance with the results of Garland (1959), which he attributed to heterogeneity of the surface. Such spectral changes are currently being correlated with various catalyst treatments which in turn are related to catalytic activities. Acknowledgment This work was sponsored by the US.Department of Energy, Office of Basic Energy Sciences, under Contract No. E(11-1)-2579. Support for one of us (S.H.M.) was
399
provided by Center Industrial Sponsors. Literature Cited Blyholder, G., Neff, L. D. J. Phys. Chem. 1962, 66, 1484. DeKock, R. L. Inorg. Chem. 1971, 10, 1205. Eberly, P. E., Jr. J . phys. Chem. 1967, 71, 1717. Eischens, R. P. A m . Chem. Res. 1972, 5, 74. Elschens, R. P., Francis, S. A., Pllskln, W. A. J. Phys. Chem. 1956, 60, 194. Elschens, R. P., Pliskln, W. A. A&. &tal. 1958, 10, 1. Garland, C. W. J. Phys. Chem. 1959, 63, 1423. Hughes, T. R., Whlte, H. M. J. Phyus. Chem. 1967, 71. 2192. Kavtaradze, N. N. Int. Chem. Eng. 1966, 6 , 700. O’Nelll, C. E., Yates, D. J. C. Spectrochlm. Acta 1981, 17, 953. Vannice, M. A., Moon, S. H., Twu, C. C., Wang, S-Y. J . Phys. €1979, 12, 849.
Received for reuiew November 3, 1980 Accepted July 27, 1981
COMMUNICATIONS Simple Method for Approximating Activity Profiles in a Deactivating Adiabatic Reactor A method has been developed for predicting the activity profiles of adiabatic catalytic reactors when deactivation by fouling occurs. The method utilizes a “deactivation modulus” which may be evaluated from knowledge usually available from commercial plant data sheets, i.e., the reactor temperature profile and Its variation with time, and inlet and outlet reactant concentrations. The estimation may be made using only a desk calculator. Comparison with data from the literature shows statisfactory agreement and suggests that the method may be applicable to plant conditions.
native is to estimate the catalyst effectiveness factor from point values of the temperature and concentration in the reactor. The ratio of this effectiveness factor to the effectiveness factor immediately after start-up (i.e., with no catalyst deactivation present) should enable an estimate to be made of the reactor activity. This method does require an estimate of the concentration profile and activity to be made in addition to measurements of the temperature profile. A simpler approach uses the relative effectiveness factors obtained from a knowledge of only the reactor temperature profile and the inlet and outlet reactant concentration. The purpose of this paper is to show that the activity distribution of packed bed reactors may be well approximated by the latter procedure and to suggest ways of obtaining a suitable deactivation function which can be used for predictive purposes for a range of reactor deactivation conditions.
Introduction Deactivation of commercial catalytic reactors is an important industrial problem. Catalyst fouling by deposition of coke is unique among the possible deactivation processes in that the deposition is always associated with the main reaction. Therefore, it cannot be eliminated by use of guard reactors or use of adsorbents as for impurity poisoning, or by decreasing the operating temperature as in deactivation by sintering. Previous work on the fouling of catalytic reactors has been mainly of a theoretical nature, but a number of attempts have been made to systematize experimental data. Relevant papers include those by Voorhies (1954), Froment and Bischoff (1961), and Sadana and Doraiswamy (1971). Nonisothermal conditions have been considered by Dumez and Froment (1976), Best and Wojciechowski (1976), and Kam and Hughes (1979a). All these studies demonstrate that the activity is not simply related to any other property of the reactor. In practice, although measurements can be made of some reactor operating variables such as temperature profdes, it is not possible to determine the reactor activity from these alone. There are currently no means available for measuring catalyst activity as a function of distance and thus there is the necessity for estimating these activity profiles. To do this, a number of possibilities exist. If infomation is available on the kinetics of the deactivation process, then this information may be incorporated with the kinetics of the main reaction to provide a predictive model for the process. Unfortunately, acquisition of deactivation rate data is time consuming and may not provide accurate information. This is so because of the slow rates of deactivation of many processes. An alter0196-4313/81/ 1020-0399$01.25/0
Theory and Mathematical Development If the deactivation kinetics are known, the activity profile can be predicted for both simple and complex kinetics under nonisothermal conditions (Kam and Hughes, 1979a). However, when the precise kinetics of deactivation are not known,use may be made of the effectiveness factor concept. If catalyst deactivation is reflected directly by a change in the effectiveness factor, then the use of a relative effectiveness factor, q/qo, defined as the ratio of catalyst effectiveness after a given time on stream to that at zero time, may give a measure of catalyst activity. To obtain relative effectiveness factors, both concentration and temperature profiles within the reactor need to be known. 0
1981 American Chemical Society
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Deactivation can occur by parallel fouling or by series fouling or by a combination of both mechanisms. The reactant is the coke precursor for parallel fouling whereas for series fouling the product is the coke precursor. The dimensionless rate equation for the combined parallel and series deactivation may be expressed as a s u m of these two separate deactivation mechanisms for fouling of active sites as
where 7 is the dimensionless process time and can be defined as k , C,l,,t (parallel fouling) or kfz ClblEIot(series fouling). The corresponding particle effectiveness factor is written as CY
f 16a-1f(a.B)Sd6
+
Figure 1. Typical temperature and concentration profiles for adiabatic fixed bed reactors for exothermic reaction.
In normal commercial operation of reactors, concentration profiles are not usually measured as this involves sampling points along the reactor length. Usually, just the inlet and outlet concentrations are known since these are readily determined. The measurement of temperature along the reactor, however, is comparatively simple and such measurements are frequently available. Concentration and temperature profiles are sometimes available from pilot plant data and typical plots of both quantities are given in Figure 1for an exothermic reaction, The concentration profile is seen to be a mirror image of the temperature profile. Thus, if inlet and outlet concentrations are available to fix the profile bounds, the shape of the concentration profile is determinable from the temperature profie. With both profiles available (albeit in approximate form), it is possible to calculate the point rates, and from these plus the temperature gradients the effectiveness factors may be estimated. Under pseudo-steady conditions and assuming plug flow the dimensionless mass and heat continuity equations for the reactor can be expressed as
The terms in the above expressions are defined in the Nomenclature section and have their usual meaning; D is the reaction modulus and is a form of Darnkohler number. For simplicitity, only irreversible reactions are considered but a generalized expression for the reaction term f(ao,80)for the main reaction A B + C may be written in Langmuir-Hinshelwood form as
-
Simpler intergral order kinetics could also, of course, be employed.
where
An alternative way of writing the effectiveness factor from eq 2 is
Since the thermicity factor, p, and the reaction modulus D can be evaluated easily, the effectiveness factor can be obtained if the temperature gradients, dtJo/d[ and the reaction function f(ao,6o) are known. In a previous publication (Kamand Hughes, 1979b) the relative effectivenessfactor, v / q o was used to approximate the catalyst activity. In the notation used here, this may be defined as 71 (Botad) IT=, rl'tO
= a(Bo,ao,S)lr=o
(7)
Use of this relation requires a knowledge of the average pellet activity S. This is generally not known in industrial practice and its determination would require a full mathematical model. For a specific process this would entail the use of a high-speed digital computer. If, however, the effectiveness factor ratio is written as in eq 6 the average pellet activity s does not appear in the relation. Designating the ratio from (6) as 9,we now have
The quantity 9 is designated the deactivation modulus and is a measure of the catalyst activity at any time and at any point within the reactor. Specificially, 9 is a function of the temperature gradient and reaction rate. Both of these are measurable quantities which are usually available from plant data. A further advantage is that 9 may be easily evaluated using a pocket calculator, since a fully simulated model is not required. For isothermal systems, 9 is obtained from eq 1 as
Although the deactivation modulus is derived from the mass or energy conservation equation by assuming a pseudo-steady state, it can be employed to approximate
Ind. Eng. Chem. Fundam., Vol. 20, No. 4, 1981 401
Table I. Values of Parameters Used in Simulations
t S os
250
1.5
10
5
20
6.25
16
t0.02
0-40
1.0
Y
.'*
1.2
J
I
0.5
n. 1
Yy=
40
" _-
I 0
I
0.5
s+
1.0
Figure 3. Comparison of deactivation modulus, 9,with simulated activity profiles for exothermic reaction. First-order fouling: (a) parallel fouling; (b) Series fouling; -, simulated activity; - - -, deactivation modulus.
L f - +
Figure 2. Comparison of relative effectiveness factors with simulated activity profiles for exothermic reaction: (a) parallel fouling; (b) series fouling; -, simulated activity; - - -,relative effectiveness factor.
transient fouling conditions. This is demonstrated in the second example in this paper. In the following sections, comparisons are made between the full simulated activity profiles and approximate activity given by the relative effectiveness factor (eq 7) and the deactivation modulus (eq 8) for various main reaction conditions and fouling by parallel and series mechanisms. Results and Discussion Values of parameters were chosen to avoid the occurrence of multiple steady states and are listed in Table I. These values are within the limits quoted by Aris (1975). Values of yf, the activation energy for the fouling reaction, were varied from 0 through 20 to 40. Comparison of Simulated Activity Profile with Relative Effectiveness Factor. As already described, the relative effectiveness factor is the value of 7 / q o obtained by use of eq 5 and 7. A comparison of these values is made with the full simulated activity profiles obtained by simultaneous solution of the material and energy balances for pellet and reactor, together with the general deactivation relation eq 4. Both simulated activities and relative effectiveness factors are plotted against dimensionless reactor length in Figure 2. From the results obtained only two cases are illustrated and these refer to exothermic reaction with /3 = 0.02 and yf = 40. Langmuir-Hinshelwood kinetics with n = 1are assumed for the main reaction while the fouling reaction is taken as first order. Figure 2a refers to parallel fouling for dimensionless times of 0.5 and 1.2 while the corresponding deactivation a t the same time for series fouling is shown in Figure 2b. For both fouling mechanisms it can be seen that while the relative effectiveness factor t / t opredicts the same shape of activity profile as the simulated activity, the 7/70values are lower. In Figure 2a, where parallel fouling occurs, the two curves differ appreciably, especially for longer process times, and the approximation of using v/v0 instead of the computed activity cannot be considered satisfactory.
Furthermore, the difference between the two curves increases appreciably toward the exit of the reactor. Figure 2b is for series fouling and better agreement is now ob~ ~ served between the simulated and t / profiles. Despite the lack of close coincidence between the simulated and v/70 curves, the latter do show the well known features of parallel and series fouling in fixed bed reactors. These are that parallel fouling gives increased deactivation at the reactor inlet, while series fouling gives worse deactivation near the reactor outlet. Comparison of Simulated Activity Profiles with Deactivation Modulus. In this section the simulated activities along the catalyst bed are compared to the deactivation modulus which is the effectiveness factor ratio determined from eq 8 using the temperature and concentration profiles in the reactor. The kinetics of the main reaction must be known for this method but the form and magnitude of the deactivation rate is not required. From the above information the deactivation modulus P may be evaluated. The determination of the deactivation modulus from eq 8 requires a knowledge of the temperature gradient along the reactor. When relatively steep gradients are present there is no problem, but when temperature gradients may be small, as at the reactor exit, then a least square correlation is necessary to avoid large errors in the determination of these gradients. Accordingly, this was incorporated into the present work. Figure 3 shows comparison plots for parallel fouling (Figure 3a) and series fouling (Figure 3b) for the set of conditions where the main reaction is Langmuir-Hins h e l w d with n = 1while the fouling reaction is first order in coking precursor. Exothermic conditions with p' = 0.02 and yf = 20 are assumed. For parallel fouling (Figure 3a) it can be seen that the deactivation modulus is in very good agreement with the simulated activity profile over the full range of the dimensionless reactor length, t. Additionally, this close correspondence is achieved even at the longer process time. These results are in contrast with those obtained using the relative effectiveness factor where there was no close correspondence between the two curves. The corresponding comparison for series fouling (Figure 3b) shows that while the values of the deactivation modulus do not differ too much from the simulated activity profiles, the shape of the deactivation modulus is slightly
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*
P
,
~~
L
: - 1
r
,.,
t
I
0
h
Y
, 3 -
L
I
__
- .--
___
0.32.
2
i
I +
1 J
1 L
3
4
Figure 4. Deactivation modulus predicted from the concentration profiles given by Schlosser (1977)via eq 9 -, deactivation modulus (this work); - - -,concentration (from Schlosser).
different to the activity profile. This is especially apparent at the reactor exit where the deactivation modulus is convex upward for both process times, whereas the simulated activity profile is concave upward at this point. The apparent anomaly between the very good agreement obtained for parallel fouling and the less good agreement between the simulated and deactivation modulus profiles for series fouling is caused by the differing temperature gradients. At the reactor exit (6 = l), the temperature gradients for parallel fouling are greater than for series fouling and the latter becomes very small as the process time is increased. Because of the difficulties of determining gradients accurately when these are small, even with a least-squares correlation, the increased divergence for series fouling at larger process times is explainable. From the results obtained it is clear that the deactivation modulus gives a much closer correspondence to the simulated activity than the relative effectiveness factor does. The reason for this is probably due to the different way in which the approximating functions are evaluated. An examination of eq 5 and 7, from which the relative effectiveness factor is determined, shows that the activity term, S, appears in the expression. No dependence on S appears in eq 8 from which the deactivation modulus is evaluated. Therefore, the deactivation modulus should be less subject to error. For both comparisons, the shape of the simulated activity curve is followed, with parallel fouling showing the greatest deactivation at the reactor entrance and least at the reactor exit while converse behavior is shown for series fouling. Thus, although no assumptions are made regarding the mechanism of fouling in either instance, the approximation from the temperature profiles predicts the nature of the fouling process.
Examples of the Use of the Deactivation Modulus In order to demonstrate the method, two examples were chosen from the literature in which temperature and/or concentration profiles under deactivating conditions are quoted.
Figure 5. Comparison between deactivation modulus and simulated activity profiles given by Ervin and Luss (1970): (a) parallel fouling; (b) series fouling; -, Ervin-Luss profile; - - -,deactivation modulus.
In a recent review of catalyst deactivation, Schlosser (1977) modeled fouling for an isothermal plug flow reactor. The concentration profiles obtained for series fouling are represented in Figure 4 by dashed lines. No activity profiles corresponding to these reactant concentration profiles were given by Schlosser. In the preceding sections of this paper, we have shown that the deactivation modulus is usually a close approximation to the actual activity profile. Accordingly, values of the deactivation modulus were calculated from Schlosser's data using eq 9 and are plotted against reactor length in Figure 4. In this figure it should be noted that the time constant T* is now defined in the same way as by Schlosser, i.e.
All the profiles exhibit increased fouling toward the exit of the bed, which is a characteristic of series fouling. Thus, use of the deactivation modulus enables the activity profile in the reador to be approximated as a function of time and the results give qualitative agreement with the features expected of series fouling. A more rigorous test of the method would be expected in an analysis of nonisothermal data. In a transient analysis of a nonisothermal adiabatic reactor, E.rvin and Luss (1970) studied the effect of fouling on the reactor stability. Both simulated activity and temperature profiles were quoted in this paper. A comparison of the activity curves produced by Ervin and Luss for both parallel and series fouling with the profiles using the deactivation modulus is shown in Figure 5. For parallel fouling (Figure 5a) a comparison is made at process times corresponding to 450 and 1350 min. As can be seen, the deactivation modulus reflects very well the shape of the activity curve predicting a minimum at the identical axial position for both process times. Activities were reasonably well represented by the deactivation modulus, except at the minimum in the curves where the error was much greater. Better agreement was obtained at 1350 min even at the minimum value of the activity. However, at points where the activity was higher, toward the exit of the bed, better correspondence was obtained at both times and identical
Ind. Eng.
times were predicted for the point in the reactor at which the reaction is completed. The reason for the poorer agreement between simulated activities and the deactivation modulus at the minimum values may be due to errors in extracting temperature gradients from published data. For series fouling, the corresponding profiles are shown in Figure 5b for process times of 900 and 1350 min. The deactivation modulus again follows the shape of the simulated activity profiles and the final activity is close to the simulated value especially for the longer process time. The deactivation modulus curves are somewhat conservative in predicting the sharp fall in activity at a slightly closer position to the reactor inlet. The above examples show that present method of employing the deactivation modulus to predict catalytic reactor deactivation due to fouling for both isothermal and nonisothermal conditions gives good agreement with results published in the literature. Furthermore, even though the method described is based on a pseudo-steady-state assumption, the comparison with transient results (such as those of Ervin and Luss) shows reasonable agreement. Conclusions A simple method of predicting the approximate activity profiles in adiabatic catalytic reactors has been developed which may be employed for various kinetics of the main and fouling reactions and with various parameter values. The method relies only on a knowledge of the variation of the temperature profile with time, the inlet and outlet concentrations, and the kinetics of the main reaction. No knowledge is required of the fouling mechanism or of the kinetics of fouling. All the required information can usually be obtained from commercial plant data and evaluation can be undertaken using a desk calculator. The method has been checked against a full simulation of the activity profile and also with two other examples available in the literature. The results obtained for exothermic conditions show that the deactivation modulus gives a good approximation to the original activity profile and predicts the type of fouling obtained in the particular system with which it is compared. Much better agreement is obtained using this deactivation modulus than when the relative effectiveness factor is employed to approximate the reactor activity. The method of the deactivation modulus should prove useful in predicting deactivation characteristics of real reactors under operating conditions provided the minimal data required are available. Nomenclature a, b = dimensionless concentration of reactant A and B, respectively AB* = coefficient of foulant-adsorption per unit volume of pellet CA, CB = concentration of reactant A and product B, respectively C,, = heat capacity De = effective diffusivity
Chem. Fundam., Vol.
20, No. 4, 1981 403
f ( ), g( ) = function of ( ) h = heat transfer coefficient h, h, = dimensionless heat of adsorption for reactant and product, respectively -AH = heat of reaction k = rate constant for main reaction It,, k, = rate constant for parallel and series fouling reaction, respectively kf2*= intrinsic reaction constant of fouling reaction per unit volume of pellet k, = mass transfer coefficient KA*, Kc* = dimensionless adsorption equilibrium constant of reactant A and product C , respectively n = order of the main reaction in the Langmuir-Hinshelwood kinetic expression Nu = modified Nusselt number, RhlK, R =_ catalyst pellet radius S, S = point and average catalyst activity, respectively Sh = modified Sherwood number, Rk,/D, t = process time T = temperature u = superficial linear flow rate Greek Symbols a = geometric factor p' = thermicity factor in (-AH/p,Cp>(C&=o y,yf = dimensionless activation energy parameter for the main and fouling reaction, respectively 6 = dimensionless catalyst radial coordinate c' = void fraction in packed bed 7 = effectiveness factor 0 = dimensionless temperature K, = effective thermal conductivity of particle 5 = dimensionless reactor coordinate pg = density of the gas stream T = dimensionless time, kflC%lt=otand kf$AOlt=ot for parallel and series fouling, respectively T* = time constant as used by Schlosser (1977), AB*/kf;Cdpo \k = deactivation modulus as defined in eq 8 and 9 s2 = reaction modulus ([k(l - ~ ' ) L / U ] ( C , ~ ~ =. ~ ) " - ~
Subscript 0 = refers to the bulk condition Literature Cited Aris, R. "The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts"; Voi. 1; Clarendon Press: Oxford, England, 1975. Best, D. A.; Wojciechoswki, B. W. Can. J. Chem. Eng. 1876, 54. 197. Dumez, F. J.; Froment, G. F. I d . Eng. Chem. Process Des. Dev. 1876. 75, 291. Ervin, M. A.; Luss, D. AIChE J . 1870, 16, 979. Froment, G. F.; Bischoff, K. B. Chem. Eng. Sci. 1961, 76, 189. Kam, E. K. T.; Hughes, R. Chem. €ng. J . 1879a, 78, 93. Kam, E. K. T.; Hughes, R. AIChE J . 1878b, 25, 359. Sadana, A.; Doralswamy, L. K. J. Catel. 1871, 23, 147. Schlosser, E. G. Int. Chem. Eng. 1877, 77, 41. Voorhies, A. Ind. Eng. Chem. 1954, 3 7 , 318.
Department of Chemical Engineering University of Salford Salford 11.15 4 WT United Kingdom
Andrea Brito-Alayon Ronald Hughes* Ezra K. T. Kam
Received for review April 2, 1980 Accepted June 23, 1981
