NOMENCLATURE
A(0) = amplitude at z = 0, in bed A ( L ) = amplitude at z = L, in bed A ( z ) = amplitude, 0 Q z Q L, in bed A(z)
=
complex amplitude, 0 Q z Q L, in bed
A'(?) = amplitude, 0 C r < R, in pellet A'(?-) = complex amplitude, 0 < r Q R B = exponent, a s defined in text
concentration, gas phase, moles per cc. of gas Concentration, solid phase, moles per cc. of porous solid D molecular diffusivitiy, sq. cm. per second DL longitudinal diffusivity, sq. cm. per second D, = diffusivity in solid, sq. om. per second Ir' = adsorption coefficient L = bed length, cm. q = mean concentration in solid a t any time r = radial position in solid pellet R = pellet radius t = time U = actual mean linear velocity of gas in bed, em. per second w = angular frequency, radians per second x = mole fraction, in gas phase in bed, 0 < z C L ~ ( 0 ) = mole fraction, in gas phase, at z = 0 z ( L ) = mole fraction, in gas phase, at z L ZM = time-average mole fraction in gas phase Y I = diffusional admittance, magnitude of real part Y Z = diffusional admittance, magnitude of complex part z = longitudinal position in bed c cs
= = = =
3
p
=
e p
= fraction voids between particles in bed = gas density, moles per cc. = period of waves, seconds
r
6 x
(.)1'2
20,
= phase angle, radians
research grant. They wish also to thank J. Ben Rosen, Forrestal Research Center, Princeton University, for contributions in the development of theory. The Calco Chemical Division, American Cyanamid Co., through 0. C. Karkalits, performed surface area measurements on the experimental porous catalyst carriers which were contributed by the Refractories Division, the Norton Co. LITERATURE CITED
(1) Bernard, R. A., and Wilhelm, R. H., Chem. Eng. Progr.. 46, 233 (1950). (2) Deisler, P. F., Jr., Ph.D. dissertation, Department of Chemical Engineering, Princeton University, 1952. (3) Deisler, P. F., Jr., and Wilhelm, R. H., unpublished manuscript. (4) Kreezer, G. L., and Kreerer, E. H., Biol. Bull., 93, 197 (1947). (5) Kreezer, G. L., and Kreerer, E. H., J . Cellular Comp. Physiol., 30, 173 (1947). ( 6 ) Latinen, G. A., Ph.D. dissertation, Department of Chemical Engineering, Princeton University, 1951. (7) Rosen, J. B., J . Chem. Phys., 20, 387 (1952). (8) Roaen. J. B.. Dersonal communication to R. H. Wilhelm. Princeton University, Jan. 13, 1951. (9) Rosen, J. B., and Winsche. W. E.. J. Chem. Phys., 18, 1587 (1950). (10) Thiele, E. W., IND. ENQ.CHEM.,31, 916 (1939). (11) Wicke, E., and Brotz, W., Chem. Ing. Tech., 21, 219 (1949). (12) Wicke, E., and Voight, U., Angew. Chem., B19, 94 (1947). (13) Wilke, C. R., and Hougen, 0. A,, Trans. Am. Inst. Chem. Engrs., 41,445 (1945). '
RECEIVED for review January 3, 1953.
= labyrinth factor for porous solids
ACCEPTED April 7, 1953.
For supplemental information (Tables 111 through XI), order Document ACKNOWLEDGMENT
The authors are pleased to acknowledge the sponsorship of this work by the Shell Development Co., Emeryville, Calif., through a
3967 through the AD1 Auxiliary Publications Project, Photoduplication Service, Library of Congress, Washington 25, D. C. A copy may be secured by citing the document number and remitting $2.50 for photoprints or $1.75 f o r 35-mm. microfilm. Advance payment is required. Make checks or money orders payable to Chief, Photoduplication Service, Library of Congress.
Heat and Mass Transport in a Fixed Catalyst Bed
during Regeneration J. J. V A N DEEMTER KONINKLIJKE/SHELL-LABORATORATORIUM. AMSTERDAM,
I
N MANY cases regeneration of a catalyst requires the removal of a coke deposit from the catalyst pellets by oxidation. The oxidation is effected by supplying a hot mixture of oxygen and inert gas (nitrogen or steam) to the catalyst bed. In a fked catr alyst bed in this way a more or less pronounced reaction zone is formed, which moves through the bed. I n the oxidation zone heat is produced, which gives rise to a temperature peak. In order to retain the catalyst activity, the temperature in the bed should not be allowed to rise above a fixed value determined by the nature of the catalyst. It is, therefore, desirable to be able to predict the temperature distribution during regeneration or, at least, to know the influence of the various factors determining the temperature profile. The most important factors are: the reaction rate, the oxygen concentration, the gas rate, and the heat transfer mechanism. As far as is known to the author, measurements of temperature profiles during fixed-bed regeneration have never been reported in the literature, nor is there a theoretical treatment of the problem. I n the present paper a very much simplified model is introduced which allows a complete mathematical solution of the differential equations of heat and matter. The assumptions are: June 1953
HOLLAND
1. During operation the rate of reaction is constant. 2 . The heat transfer from particle t o gas or reversely is so great that the temperatures of gas and solid can be assumed t o be equal. 3. There is no loss of heat in a lateral direction (adiabatic reactor). 4. I n the direction of gas flow the transport of heat and matter bv eddy diffusivitv can be neglected as compared with forced conviction. 5. The mean gas velocity is constant-i.e., density variations due to temperature gradients and chemical conversion are negligible.
Assumption 1 is a very bold one, as it neglects the influence of temperature, oxygen concentration, and coke concentration during operation. Assumptions 2 , 4, and 5 are frequent introduced in the theory of fixed-bed reactor design. Nevertheless, a reasonably qualitative picture may be expected, a t least when the temperature peak is sufficiently low and the rate of burning is more or less controlled by diffusion. According to Tu, Davis, and Hottel(3) who carried out experimentswith brush carbon pellets 25 mm. in diameter, diffusion has no influence on the over-all rate of conversion below 1000" K. From experiments with charcoal pellets of various sizes and shapes Khaikina
INDUSTRIAL AND ENGINEERING CHEMISTRY
1227
T h e temperature distribution and t h e transport o f mass d u r i n g regeneration (coke oxidat i o n ) of a fixed catalyst bed have been studied theoretically on a m u c h simplified model allowing a complete mathematical solution. T h e m a i n assumptions are: u n i f o r m and constant rate of reaction, negligible temperature difference between t h e gas and t h e solid material, adiabatic reactor (heat-insulating walls), and negligible longitudinal diffusion of heat and matter. Although these assumptions i m p l y a rather drastic simplification, t h e results indicate a t least qualitative correlations between t h e t i m e f o r regeneration, t h e height of t h e temperature peak, t h e velocity, and t h e depth of t h e oxidation zone on t h e o n e hand, and t h e process variables (oxygen concentration, coke concentration, and gas velocity) on t h e other hand. There i s a critical oxygen concentration of t h e incoming gas, above which t h e oxygen is n o t completely consumed in t h e bed. For sufficiently h i g h reaction rates t h e oxidation zone is very small. Then t h e rate of regeneration isdetermined either by t h e rate of heating of t h e bed (above t h e critical oxygen concentration) or by t h e rate of oxygen supply (below t h e critical oxygen concentration). For a very low reaction rate t h e oxidation zone comprises t h e whole bed. T h e rate of regeneration is t h e n m a i n l y determined by t h e rate of reaction. Increasing t h e gas velocity implies a lowering of t h e t e m perature peak, an increase of t h e rate of regeneration, and a larger oxidation zone.
( 1 ) deduced that the rate of burning is proportional to the volume of the pellets rather than to the surface. This would imply a first-order reaction with respect to the coke concentration. FORMULATION OF DIFFERENTIAL EQUATIONS
+
Consider a slab z , z dz of unit area perpendicular to the direction of flow. Let v be the superficial gas velocity and c the oxygen concentration in moles per unit volume. The inflow of oxygen into the slab in a time dt is vcdt, the outflow is v(c dc)dt, and the loss of oxygen in the slab by oxidation is - U d z d t , if U is the oxygen reaction rate in moles per unit of volume per unit of time. The increase in oxygen concentration in the slab is made up of these three terms; hence, if E is the fractional void space in the bed:
The characteristics are, therefore, a set of m 2 straight lines parallel to the two planes z = u t / € and c = U t / € (or c = 0 if li = 0) in the c-z-t space. The integral surface of Equation 1 is formed by the characteristics going through the appropriate boundary curves. In the same way the characteristics of Equation 3 are found to be
E
dcdz = vcdt
- V(C
+ dc)dt - Udzdt
+
A e=pUt+B z = aut
+
where
and
or
ac
ac
at
a2
E - + v -
=
-U
I N I T I A L STAGE
The decrease of coke concentration per unit of time is equal to: bC’= at
-U
where c’ is expressed in equivalents of 1mole of oxygen The inflow of heat into the above-mentioned slab is equal to p,c,vTdt if p, is’the gas density, co is the gas heat capacity, and T is the temperature. The outflow of heat from the slab is poc,v(T d T ) d t . When H is the heat of reaction per mole of oxygen, the heat production in the slab is HUdzdt. On the other E P ~ C ~ ] hand, the amount of heat increase is equal to [( 1 - e)p,c, d T d z , where ps is the solid density and cs is the solid heat capacity. Balancing gives:
When no reaction occurs ( U = 0), Equation 3 describes the heat transfer between a porous solid and a fluid flowing through it for the limiting case of an infinitely high heat-transfer coefficient. ’VVith the boundary conditions
the ~olutionof Equation 3 simply reads:
+
+
(4)
as may easily be derived with the aid of the characteristics. (The initial temperature of the solid is chosen as the zero level.) Equation 10 is the limiting case of the solution for a finite heattransfer coefficient, which, as is well known, shovs a gradual decrease of the temperature of both the fluid and the solid in the region near z = cvvt (Figure 1). 4 n infinite heat-transfer coefficient, therefore, implies a velocity of heat transport by forced convection which is exactly equal t o LUU. For the case of heating by a gas the velocity, av, mill be very small compared with the gas velocity, v . Considering this result, the oxidation may assumed to be confined to the region 0 z aut. I n this region the solution of Equation 3 with U # 0 satisfying the boundary conditions of Equation 9 is found to be
(5)
The heat produced by reaction, therefore, is also transported with the velocity, QU. In the region z > aut again 2‘ = 0.
METHOD O F SOLUTION
For uniform and constant U Equations 1 and 3 are linear and of the first order. They can, therefore, be solved by the method of characteristics. The characteristics of Equation 1 are given by
T = To
Integration of Equation 4 yields z = Ut/€ c = Ut/€
1228
+f BA
<
avt, c'
= ,;c
- U ( t - z/ av )'i
J
= co
(16)
The results found so far are represented in Figure 3. T h e increase of oxygen concentration in the region aut aut, c
= co
- @Ut
(17)
CRITICAL OXYGEN CONCENTRATION
It is learned from Figure 3 or from Equations 14 and 16 that at a certain moment either the coke a t the entrance of the bed will be removed, or the oxygen will be wholly consumed in the oxida-
Figure 2. e-s-t Diagram initial stage
c
E
Next we will consider the material balance
(12) with the boundary conditions c = co z = 0, 1=O(z>O),c=O
In the region 0
< z < aut the integral surface is found by inter-
El
i
I I
II
Figure 4. Oxygen Concentration, Coke Concentration and Tern perat u re
Figure 5. Oxygen Concentration, Coke Concentration, and Temperature
t = to
t = to
tion zone. Equation 16 shows that the time for coke removal is equal to tl o( V t
Figure 3. Coke, Oxygen, and Temperature Distribution Initial stage
June 1953
=
c;/u
(18)
According to Equation 14 the oxygen is completely consumed when the Oxidation zone has reached a 20
= covju
INDUSTRIAL AND ENGINEERING CHEMISTRY
(19) 1229
which corresponds to a time to = Z O / f f v = CQ/au
(20)
When to > t i , the coke at the entrance will be removed before the oxygen is consumed. For t o < tl first the oxygen is consumed. It will be seen that the behavior of the regeneration process for these two cases is very different. The limiting case t o = tl corresponds to a critical oxygen concentration
c~crit= ace'
Equation 22 represents a straight line parallel to the line z = aut (Figure 7). Because the depth of the oxidation zone is wholly determined by the oxygen concentration, i t remains constant and equal to zo while moving. f
I
(21)
OXYGEN CONCENTRATION LOWER THAN CRITICAL
First the case co
