cases the minimum lies between these two calculated values, showing that our assumptions were correct. Therefore the wall region in beds of cylinders is wider than in beds of spheres. Nomenclature
Dp D,
= particle diameter
x
= distance from wall, cylinder diameter = void fraction, dimensionless
e
d,, d, $
=
tube diameter
distances of first minimum from wall, cylinder diameters = Dt/Dp
=
literature Cited
Benenati, R. F., Brosilow, G. B., A.Z.Ch.E. J . 8, 359 (1962). Debbas, S., Rumpf, H., Chem. Eng. Sci. 21, 583 (1966). Haughey, D. P., Beveridge, G. S. G., Chem. Eng. Sci. 21, 905 (1966). Kimura, M., Nono, K., Kaneda, T., Chem. Eng. Japan 19, 397 (1955). Roblee, L. H. S., Baird, R. M., Tierney, J. W., A.Z.CI1.E. J . 4, 460 (1958). Thadani, M. C., Peebles, F. N., IND.ENG.CHEM.PROCESS DESIGN DEVELOP. 5 , 265 (1966). RECEIVED for review January 16, 1967 ACCEPTEDOctober 24, 1967
OPTIMUM OPERATION-REGENERATION CYCLES FOR FIXED-BED CATALYTIC CRACKING VERN W. WEEKMAN, J R . Applied Research @ Development Division, Mobil Research and Development Corp., Paulsboro, N . J . 08066
A method is presented for computing the optimal operation-regeneration cycle for a fixed-bed reactor such that the yield of desirable product is maximized. Where the lengths of the operation and regeneration cycle are of similar magnitudes, the optimum cycle times become very critical. A kinetic-mathematical model of catalytic cracking conversion, which describes experimental results well, is the key to the optimization procedure. It characterizes the cracking behavior of catalysts in terms of a reaction velocity constant and a catalyst activity decay velocity constant. The effect of these important parameters on the optimum operation-regeneration cycle timing is discussed. An economic optimum space velocity exists which, when coupled with the optimum operation-regeneration cycle, gives an over-all optimum design. The method is aimed specifically at fixed-bed catalytic cracking, but is applicable to other fixed-bed processes describable by similar reaction kinetics.
HE design of fixed-bed reactor systems in which the T c a t a l y s t activity decays during operation requires specification of the length of operation prior to catalyst regeneration or replacement. Current fixed-bed refinery processes, such as hydrocracking, desulfurization, and reforming, usually can be operated for many months before regeneration is required. Thus the regeneration or replacement time represents only a small fraction of the operation time. Smith and Dresser (1957) treated the catalyst replacement problem in reforming where temperature was raised to maintain yield. Regeneration time was not treated as a variable. Walton (1961) explored the economics of catalyst replacement for fixed-bed reactors with linear aging rates. The method was aimed primarily at processes, such as reforming and hydrocracking, where the regeneration time is negligible compared to the operation time. Kramers and Westerterp (1963) present a graphical method of determining optimum operation-regeneration cycle times from production rate data. The present report treats the case where the regeneration and operation times are of similar magnitude. A unique operation-regeneration cycle results in optimal efficiency for a single fixed-bed reactor or a train of fixed-bed reactors. The more rapid the decay of catalyst activity, the more critical becomes the operation-regeneration cycle timing. While the optimum cycle-timing problem will, of course, be influenced by interactions with other parts of the processing complex,
252
l&EC PROCESS DESIGN A N D DEVELOPMENT
such as heat exchange and distillation, the method of specifying the effect of cycle timing on reactor yield is a necessary part of any optimization study. I n specifying the unique operation-regeneration cycle for optimum reactor efficiency, a kinetic-mathematical model of the cracking process is employed (Weekman, 1968) that takes into account the decay in catalyst activity due to use. T h e optimum conditions are presented in terms of process variables, such as space velocity, catalyst-oil ratio, and reaction temperature, and the more fundamental parameters: intrinsic catalyst activity and decay velocity. Reactor Efficiency Criterion
I n a fixed-bed process requiring periodic regeneration, it is necessary to specify the length of time the reactor may be operated before regeneration. As catalyst decay will cause the product yield to decrease with time, it will be necessary to stop a t some point and regenerate or replace the catalyst. For very slowly decaying catalysts, such as in reforming or hydrocracking, the time necessary for regeneration has little or no effect on how long the operating (on-oil) part of the cycle will continue. Thus, the decision to regenerate after 6 months of operation is little affected by whether the regeneration can be accomplished in 2 or in 3 days. If, however, the regeneration and operation times are of similar magnitude, the decision as to the length of operation
time will be more strongly coupled to the required catalyst regeneration or replacement time. In general, it is possible to specify an optimum operation-regeneration cycle for fixedbed catalytic cracking. T o investigate optimum operation-regeneration cycles, it is necessary to develop a quantitative performance or efficiency criterion. The ratio of the total product obtained over any number of operation-regeneration cycles to that obtained if the reactor operated ideally at 1 0 0 ~ oconversion with no regeneration required proved to be a suitable criterion. O n this basis the reactor efficiency, E,, is:
E,
=
h7FutUctotal actual product F,t, total ideal product ~
(11
where the numerator represents the total amount of product obtained over N cycles of t, operation time and the denominator the total product possible if the fraction conversion were always 1, with no regeneration required. T h e conversion, - .5 u
z
YE .4 w U 0
6 .3 9 u .2
.I
Combining Equations 1, 2, and 3 gives the following expression for the reactor efficiency:
0 (4)
This expression allows one to investigate the existence of an optimum efficiency that depends on the operating time, the regeneration time, and the space velocity for a single isothermal
0
.I
.2
.3
4
.5
.6
OPERATE CYCLE TIME,HRS. Figure 1 . Reactor efficiencies for regeneration time independent of operation time Space velocity.
8.0 vol./(vol.)(hr.)
VOL. 7
NO. 2
APRIL
1968
253
creased, the size of the reactors required increases. Thus, as the space velocity is lowered to achieve a higher reactor efficiency, the capital cost of equipment increases. Clearly a n over-all economic optimum space velocity exists; greater operating revenue (from higher reactor efficiency) is traded off against the increased capital cost of achieving this efficiency. Figure 3 shows how the reactor efficiency changes as the intrinsic activity of the catalyst varies for a constant decay velocity. Thus catalysts of lower activity lead to lower reactor efficiencies, causing the optimum efficiency to occur at shorter operation times. A catalyst of infinite activity would exhibit an optimum efficiency at infinite operation time. Figure 4 reveals how the reactor efficiency varies with the decay velocity of the catalyst. For the same intrinsic activity, a more rapidly decaying catalyst will exhibit lower reactor efficiencies and the optimum efficiency will occur a t shorter operation times. Figure 5 presents the locus of optimum reactor efficiencies in terms of required operation and regeneration times for various values of the reaction and decay velocity constants. I t shows that rapidly decaying catalysts require a shorter operation time than less rapidly decaying catalysts for the same optimum regeneration time. Higher activity catalysts allow longer operation times for the same optimum regeneration time. Regeneration Time Dependent on Operation Time. For many kinds of catalytic processes the regeneration time may depend a t least partly on the length of the operation cycle. Thus, in catalytic cracking the amount of coke laid down on the catalyst is primarily dependent on the length of the operation part of the cycle. If consideration of heat balance or possible catalyst damage prevents varying the regeneration operation a t will, regeneration time becomes dependent on operation time.
As shown by Voorhies (1945), coke formation in catalytic cracking may be described as a negative order-Le., selfinhibiting-reaction as follows:
(5) where k, is the coke-forming reaction constant which is a function of both the type of charge stock and the type of catalyst. Carbon-burning kinetics for simple diffusion-free burning may be described as follows:
While carbon burning is usually first-order-Le., m = 1high concentrations of coke (more than a monolayer) can reduce the apparent order of the reaction. For constant oxygen concentration we can ask: for what values of m and n will an optimum operation-regeneration cycle exist? For the case of negligible initial coke on catalyst (C, = 0 when to = 0) Equation 5 can be solved to yield the Voorhies relationship, C, = bt,?
(7)
where
1 Y=-
l + n
Solving Equation 6 at constant oxygen concentration with the boundary condition C, = bt,Y for t , = 0 gives:
K0=143.0hr:;
a=
.7
18.8 hr
.7
I .6
.6
G
.5
w' .5
>0 z
i 0
z W
g
W
.4
.4 LL LL W
LL LL W
U p
a
p .3
.3
0
0
a W a
a a
W
.2
.2
.I
.I
0 0
.I
.2 .3 .4 OPERATE CYCLE TIME, HRS
.5
.6
Figure 2. Reactor efficiencies at various space velocities for fixed regeneration time Regeneration time.
254
0.1 hr.
l&EC PROCESS DESIGN A N D DEVELOPMENT
0
0
Figure 3.
.I
.2 .3 4 .5 OPERATION CYCLE TIME, HRS.
.6
Effect of intrinsic activity on reactor efficiency Decay velocity, Space velocity.
01
= 15.0 hr.-I
8.0 vol./(vol.) (hr.)
Based On Eqns (5,6)
.6
cf=*ooo3~ Ib Cat
kb= 20hr-'
.5
w'
z2.4 w
0 LL LL
a
.3
0
La w
u .2
.I
0
0 V
0
.2
.I
.3
.4
.5
.6
OPERATION CYCLE TIME,HRS. Figure 4.
.5
.6
Figure 6. Reactor efficiency with regeneration time dependent on operation time
Effect of decay velocity on reactor efficiency
Voorhies coefficient, y = 0.5 First-order coke burning, m = 1
Reaction velocity, K O = 100 hr.-* Space velocity. 8.0 vol./(vol.)(hr.)
1.0
.I .2 .3 4 OPERATION CYCLE TI ME, H RS.
For the efficiency, E,, to approach zero as to 0, the exponent on to must be negative. Thus, if n > 0 and m 1 0, an optimum will exist in E, as t o is increased. This result shows that for zero or any positive-order coke burning a n optimum operation-regeneration cycle will exist, provided the Voorhies coefficient is between 0 and 1 (0 < y < 1). Figure 6 shows the reactor efficiency, E,, when the coke burning is first-order and the Voorhies coefficient y is 0.5. Because of the exponential burnoff, a final coke value of 0.03 weight % coke on catalyst was used in the computation. From Figure 6 we see that a t each space velocity a n optimum operation time results. As the conversion model given by Equation 2 is based on a second-order reaction system experiencing a first-order catalyst decay with time, the optimization technique will be valid for any fixed-bed systems exhibiting this behavior. -+
.9 .8
'
ui
.7
Er-
.6
z .5
0 -
r-4 w a .3
2' 0
.2 .I n
'VO
.I
.2
.3 4 .5 .6 .7 .8 .9 REGENERATION TIME, HRS.
113
Summary
Figure 5. Locus of operation-regeneration times for optimum reactor efficiency
Equation 8 describes the regeneration time required to burn the catalyst clean as a function of the operation time. We 0 as to + 0, a n optican observe from Equation 4 that if E, mum E, will be found as to is increased, since as to -+ 03 , E, + 0. Substituting Equation 8 into Equation 4 and taking the limit as to + 0 by 1'Hospital's rule gives: -+
For catalysts that temporarily decay during use, such as in catalytic cracking, it is possible to specify a unique operation-regeneration cycle that gives a n optimum amount of product from the reactor. A mathematical method for obtaining the optimal operation-regeneration cycle is presented in terms of the catalyst's reaction velocity and decay velocity constants. For systems that require frequent catalyst regeneration or replacement, the reactor efficiency is strongly affected by the choice of the operation-regeneration cycle timing. When the regeneration time is dependent on the operation time due to coke formation, it can be shown that for a wide range of coke-forming and coke-burning kinetics a n optimum operation-regeneration cycle will exist. V O L 7
NO. 2 A P R I L 1 9 6 8
255
Nomenclature
A = dimensionless reaction group, p,fk,/p,S = K,/S b = coking constant, Ib. of coke/(100 Ib. of cat.) (hr.)n = coke tvt. %, on catalyst C, Coz == oxygen concentration E, = reactor efficiency, Equation 1 f = void fraction F, = feed rate, lb./hr. k6 = burning reaction velocity constant k, = coking formation constant, (coke wt./cat. wt.). hr.-l k, = reaction velocity constant of cracking, hr.-I KO = combined reaction velocity constant, p8fkf/pl n = order of coke formation with time hi = number of cycles S = liquid hourly space velocity, v./(v.) (hr.) to = time of operation on oil, hr./cycle t, = time of regeneration, hr./cycle t m e & = mechanical switching time/cycle tt = (to t, tmech)‘V, total time, hr.
+ +
GREEKLETTERS = decay velocity constant, hr. = cat./oil ratio, vol. of cat./vol. of total oil p = l/(n 1) (Voorhies coefficient) y P = time-average wt. fraction conversion X = dimensionless decay group, a/,%’ or at, = initial oil vapor density, lb./cu. ft. po = liquid charge density a t room temperature, lb./cu. ft. (Y
+
literature Cited
Kramers, H., Westerterp, K. R., “Elements of Chemical Reactor Design and Operation,” Academic Press, New York, 1963. Smith, R. B., Dresser, J., Petrol. Refining 36, 199 (1957). Voorhies, A., Jr., Znd. Eng. Chem. 37,318 (1945). Walton, P. R., Chem. Eng. Progr. 57, No. 8 , 42 (1961). Weekman, V. W., Jr., IND.ENG.CHEM.PROC.DESIGN DEVELOP. 7, 90 (1968). RECEIVED for review May 1, 1967 ACCEPTED October 9, 1967
DIFFUSION IN COMMERCIALLY M A N U F A C T U R E D PELLETED C A T A L Y S T S CHARLES N . SATTERFIELD AND P. JOHN CADLE Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Mass.
The diffusivity characteristics of 17 commercially manufactured pelleted catalysts and catalyst supports were measured by a steady-state gaseous counter-diffusion method. The results are analyzed in terms of a model which may b e visualized as assuming that the pores of various sizes are in parallel. A tortuosity factor i s invoked to account for deficiencies in the assumption that all the pores are oriented in the direction of diffusion and that they are of uniform radius. Except for two materials which apparently had been calcined at very high temperatures, all tortuosity factors fell between 3 and 7.
HE greatest degree of uncertainty in determining the effecT t i v e n e s s factor for a porous catalyst usually resides in estimating the effective diffusivity of reacting species in the porous structure. The problem is complex, especially for catalysts which possess a wide pore size distribution, the case for most pelleted or extruded catalysts, where the complications of the pore geometry are compounded by the fact that the mode of transport is frequently in the transition region between Knudsen and bulk diffusion. This problem has received considerable attention in recent years but in nearly all approaches a predictive model has been compared with diffusion measurements (sometimes combined with reaction studies) on large pellets prepared by pressing powdered material from one side only in unlubricated dies to different density levels in order to vary the pore structure. However, recent studies (Cadle and Satterfield, 1968a; Satterfield and Saraf, 1965) have shown that the porous structure in such laboratory pressed pellets can be highly nonuniform and this can strongly influence the diffusivity of gases through the porous solid. Hence the validation of models by such experimentation is unreliable. The commercial manufacture of pelleted catalysts differs from the above laboratory procedure in several important ways, so there has remained considerable uncertainty concerning the diffusivities to be expected in such materials. The most
256
l & E C PROCESS D E S I G N A N D D E V E L O P M E N T
important differences seem to be that most commercial catalysts are smaller than the laboratory pellets; binders and lubricants are usually employed in commercial catalyst mixes which reduces wall stresses during compaction ; double-ended pressing is used; and one of the final stages in the process is a calcination treatment which evolves gases and vapors throughout the mass. Parallel-Path Pore Model
*
The pore size distribution in most pelleted catalysts is such that at atmospheric pressure most gaseous diffusion occurs in the transition region where both Knudsen and bulk diffusion modes are of importance. These modes interact between themselves as well as with the pore structure and they are correlated by the transition region equation as presented, for example, by Scott and Dullien (1962), Evans, Watson, and Mason (1961), and Rothfield (1963). T h e important characteristics of this equation are that it degenerates into the Kuudsen equation when r/X is small (about 10) to the bulk diffusion equation. Thus it can be applied to any single pore size under the conditions of temperature and pressure normally encountered in practice, with the assurance that the valid volume diffusion equation is being used in each case.
