95
Ind. Eng. Chem. Fundam. 1082, 21, 95-97
Radial void fraction distribution and channeling in a fixed bed have been known to be functions of the particle-to-bed diameter ratio. For example, Benenate and Brosilow (1962) report that the high void fraction fluctuations near the wall extend as far as nine particle diameters into the catalyst bed. These wall effects are thus more important in small diameter beds than in large diameter beds. The influence of particle-to-bed diameter ratio on pulsing inception has not been understood (Sicardi et al., 1979). This communication proposes that pulsing in small or poorly packed beds should be more pronounced than in large or well packed beds.
Literature Cited
Charpentier, J. C.; Favier, M. AIChEJ. 1975, 27, 1213. Chou, T. S.; Worley, F. L., Jr.; Luss, D. Ind. Eng. Chem. Process Des. Dev. 1977, 76, 424. Chou, T. S.; Worley, F. L., Jr.; Luss, D. Ind. Eng. Chem. Fundam. 1979, 18. 279. Froment, (3. F.; Blschoff, K. 8. “Chemical Reactor Analysis and Design”; Wiiey: New York, 1979. Shah, Y. T. “Gas-LlquM-SoiM Reactor Design”; McGraw Hill: New York, 1979. Slcardi, S.; Gerhard, H.;Hoffman, H. Chem. Eng. J . 1979. 78, 173. Specchia, V.; Rossini, A.; Baldi, 0. Ing. Chim. Ital. 1974, 10, 171. Weekman, V. W., Jr.; Myers, J. E. A I C M J . 1964, 70, 951. Weekman, V. W., Jr.; Myers, J. E. AIChEJ. 1965, 7 1 , 13.
Exxon Research and Engineering Company Florham Park, New Jersey 07932
Ramesh Gupta* Robert M . Koros
Received for review January 29, 1981 Accepted October 26, 1981
Beimesch, W. E.; Kessler, D. P. A I C M J . 1971, 77, 1160. Benenate, R. F.; Brosilow. C. 8. AIChE J . 1962, 3 , 359.
Impact of Diffusional Limitations on Temperature-Time Profiles for Deactivating Catalysts The influence of external and internal diffusional limitations on temperature-time profiles, for constant-conversion operation of catalytic reactors, is Illustrated by means of a simple mathematical model. Alterations in the magnitudes of the apparent deactivation parameters, caused by the presence of such diffusional limitations, are examined.
In the commerical operation of catalytic reactors, the effects of catalyst inactivation are often countered by increasing the catalyst temperature with time on stream in order to achieve a constant conversion of reactant to products. The temperature-time relationship required for this mode of operation was derived by Chou et al. (1967) for a plug flow reactor with catalyst undergoing concentration-independent, first-order deactivation. More recently, the analysis of Chou et al. was extended to the general case of nth order deactivation by Krishnaswamy and Kittrell(1979), who also demonstrated the adequacy of the model to describe hydrocracking and reforming data. However, these analyses do not account for the effects of internal and external diffusional limitations which often exist in commercial operation of catalytic reactors. In hydrodesulfurization of residue, for example, increase in catalytic effectiveness i n c r m the rate of metal deposition which, in turn, accelerates the rate of decline in catalyst activity over time for the sulfur removal reaction (Shah and Paraskos, 1975). In another study, Krishnaswamy and Kittrell(1981a,b) examined the deactivation of diffusion-influenced main reactions caused by diffusion-free deactivation reactions. For the case of a fmt-order primary reaction with catalyst decay being described by a first-order, concentration-independent rate expression, they showed that the apparent rate of decay is onehalf the true rate under severe internal diffusion limitations, and it approaches zero when large resistances to external mass transfer exist. Their analysis provided the following expression for overall catalytic effectiveness =
[
coth 4‘- 1/4’ T T (coth 4’- 1/4’+ Bim/r$’ 3 6 Bim
)I
4 = 3h and 4’ = 3 h 6 For constant conversion of reactant, we have kiVM
= Constant = kiVOtlT,,
(l)
AA exp(-EdRT).rlot = AA exP(-EA/RTo)Vo where qo is the zero-time overall effectiveness factor. Rearranging, we obtain (3)
First-Order Deactivation The rate equation for concentration-independent deactivation can be written as
(4) Substituting eq 3 into eq 4 and rearranging, we obtain
(a) Negligible Internal and External Diffusion Limitations. The overall effectiveness factor of eq 1for C#J 0 and Bim becomes
-
-
Q)
=a
qo = 1.0
(6)
Integration of eq 5 between limits of unity and a gives
d-’
0196-4313/82/1021-0095$01.25/0
(2)
or
Vot
where the activity a is a function of time and R h = - ki/DA = Thiele modulus 3
KmR Bim = -= mass Biot number DA
(7) @ 1982 American Chemical Society
06
Ind. Eng. Chem. Fundam., Vol. 21, No. 1, 1982
Bim-1000.0
0 0
200
600
400
800
Time, h r s
Time, hrs.
Figure 2. Effect of external diffusion on temperature-time profiles.
Figure I. Effect of intemal diffusion on temperaturetime profiles.
By rearranging eq 3 we obtain the activity vs. temperature relationship for this case
ii( T1 -
u = .XP[
EA
91
ferent conversion levels or space velocities for the two cases). (c) Presence of Internal and External Diffusion Resistances. For the specific case when (6'2 15.0 and Bim is finite, eq 1 reduces to llot =
Combining eq 7 and 8 gives the diffusion-free temperature-time relationship of Chou et al. (1967)
t
E
-EAeEd/RTo( 1-exp Ed
)])
[Ed(
-
1
1
R
T
To
Ad
(9)
h
-
(b) Severe Internal and Negligible External Diffusion Limitations. For Bim 00 and (6'2 15.0 eq 1can be approximated as = t/;;/h
and = l/h
The severely internal diffusion-limited activity-time, activity-temperature, and temperaturetime relationships are, respectively
(8a and
t = -= A "IRTo{ 1-exp Ed
Ad
[- - - 'd(
R
1
T
To
)]I
3h;im a
+ Bim
+
1
)
and 3h BimBim
Combining eq Ib with eq 3 and 5, we obtain an expression for the fouling rate
FR =
170
$(
ll0 = l(
The initial slope of the temperaturetime curve is termed the fouling rate and is given by
llot
1200
800
400
1000
=
dt
T,
[
I
Bim -T02e-Ed/RTo 2(3h Bim) AdR EA
+
(lob)
Equation 10b indicates how the fouling rate, and hence the initial slope of the temperature-time curve, is altered by the combined influence of external and internal mass transport resistances. This alteration in the temperature-time profile is shown in Figure 2 for three values of the mass Biot number (with common To). General n th-Order Deactivation The development of the diffusion-limited time-temperature profile for first-order deactivation can be easily extended to the case of a general nth order deactivation process. Equation 4 can be rewritten as da -A d eXp(-Ed/RT)Un (11) dt An analogous development yields the following expressions for the diffusion-free and severely internal diffusion limited systems
(ga)
It is interesting to note that the activation energy for deactivation is not altered by the presence of severe internal diffusion limitations. However, the apparent preexponential factor is one-half the true value. The corresponding expression for fouling rate then becomes
and
Hence, the temperature-time profile will have an initial slope which is one-half that for the diffusion-free system. This is illustrated by the plots of Figure 1, where Toeis assumed to be the same for (6'= 0 and (6'115 (i.e., dif-
respectively. It is immediately apparent that for nth order deactivation, the presence of severe internal diffusion limitations
97
Ind. Eng. Chem. Fundam. 1982, 21, 97
alters both the apparent deactivation activation energy and preexponential factor. It is expected that external diffusion resistances will further modify the magnitudes of both deactivation parameters. Since the activation energy for deactivation is unchanged by increasing diffusion resistance for a first-order deactivation process, this suggests a means of distinguishing between first and higher orders of deactivation. Note, however, that the expressions for fouling rate obtained from eq 12 and 13 are independent of n and identical with eq 10 and loa, respectively. Nomenclature a = fractional catalyst activity AA = preexponential factor for primary reaction, h-' Ad = preexponential factor for deactivation, h-' Bim = mass Biot number = (K,.R/D,) DA = effective diffusivity of reactant mthin spherical particle, cm2/h E A = activation energy for primary reaction, cal/g-mol E d = activation energy for primary reaction, cal/g-mol FR = fouling rate, O F f h h = Thiele modulus, ( R / 3 ) (ki/DA)'I2 ki = intrinsic reactant ( R / 3 ) constant, h-l k d = deactivation rate constant, h-I K , = mass transfer coefficient, cm/s
R = radius of spherical particle, cm t = process time, h T = catalyst temperature O R To= initial catalyst temperature, O R Greek Letters
vot = time-dependent overall effectiveness factor time-dependent overall effectiveness factor 4 = dimensionless group = (3h) 4' = dimensionless group = 3h.a1/2 qo =
Literature Cited Chou, A.; Ray, W. H.; Arls, R. Trans. Inst. Chem. Eng. 1987, 45. T153. Krishnaswamy, S.; Kimell, J. R. AIChE J . 1981a, 27, 120. Krlshnaswamy, S.; Klttrell, J. R. AIChEJ. WOlb, 2 7 , 125. Krlshnaswamy, S.; Klttrell, J. R. Ind. Eng. Chem. Process Des. Dev. 1979. 78, 399. Shah, Y. T.; Paraskos, J. A. Ind. Eng. Chem. Process Des. Dev. 1975, 14,
366. Gulf Research 8 Development Co.,Harmarville. PA.
Department of Chemical Engineering University of Massachusetts Amherst, Massachusetts 01003
S. Krishnaswamy* James R. Kittrell
Received for review February 5, 1981 Accepted September 8,1981
CORRESPONDENCE Comments on "A Simple Device for the Determination of Sphericity Factor" Sir: The short paper by Subramanian and Arunachalam
(IdEng. . Chem. Fundam. 1980 19,436) contains an error. The Ergun equation in the form used by these authors is not valid for vertical flow of a dense fluid, i.e. a liquid, without a correction for the gravitational potential of the fluid. For a fluid flowing downward, the correct expression is (using the nomenclature in the paper, and neglecting the inertial term)
_-hp = 150(1 - e)'pV L
4s2D,2gce3
- gp gc
If this equation is used in lieu of eq 2 in the paper, then the following expression is found
"=
[
150L(1 - t)'By
D,2t3g
]
11'
Note that for the special case of Hl = L, eq 5 reduces, correctly, to -AP = 0, and when this value is inserted into eq 1 we find V = 0, which is not correct since a granular bed just filled with a liquid will still drain, implying V # 0. On the other hand, the form of Ergun's equation valid for a dense fluid predich a definite value for V when AP = 0. Faculty of Engineering Science J. M. Beeckmans The University of Western Ontario London, Ontario, Canada N6A 5B9
Response to Comments on "A Simple Device for the Determination of Sphericity Factor" Sir: Professor Beeckmans is correct in his observations. The term L in the Pressure drop equation and hence the fador L/Hl in eq 4 should be deleted. However, the values of L/H1 in the present investigation were very low. The
0198-4313/82/1021-0097$01.25/0
results and the conclusions are not affected. P. Subramanian Vr. Arunachalam*
Department of Chemical Engineering Regional Engineering College Tiruchirapalli, 620015,India
0 1982 American Chemical Society
