RATES OF POISONING IN FIXED-BED REACTORS J.
H . OLSON
Chemical Engineering Department, University of Delaware, Nemark, Del.
19711
The time-dependent activity of a fixed-bed reactor, which is being poisoned by coke formation or irreversible adsorption of a feed component, is evaluated. The activity remaining depends critically upon the interacting values of the Thiele modulus and the number of solid-phase diffusion transfer units.
and Gorring (1966) developed a model for the poisoning of a single catalyst particle in a catalytic reactor. 'The. purpose of this work is t o extend this model to three problenis which may occur in a fixed-bed reactor. These problenis may be described briefly as follows: A poison \vhicli is strongly adsorbed upon a spherical catalyst is supplied in the feedstock to a fixed-bed reactor. T h e poison accumulates upon the catalyst a t the entrance of the reactor to a greater extent than the catalyst toward the exit by a "jhell-progressive" mechanism. Thus the bed has developed "pore-mouth ' poisoning of a variable extent throughout the system; it is desired to compute the rate a t which the activity of the reactor decreases with time. This problem is important to the operation of a fixed-bed reactor, because it controls the time when the reactor must be regenerated. The model d.eveloped in this work applies both to poisons adsorbed from the feed stream (metal salts) and to poisons generated by the reactor (coke). The second problem is related to the first; coke is assumed to be formed as a by-product of the reaction system. I t is desired to find the total amount of coke irhich is deposited on the catalyst as a function of time. This datum is useful \\,hen {vriting a material balance around the reactor. I n addition, some reaction systems are limited entirely by the rate of formation of coke; it is, therefore, instructive to see how the formation of coke is affected by the physical parameters of the system. The rhird problem concerns the operation of a guard-bed adsorber. .4 guard-bed is a fixed-bed device which is used to remove a strong poison for an expensive catalyst by chemisorption upon a cheap catalyst; the guard-bed is similar in concept to an ion exchanger for water treatment. I t is desired to compute the leakage behavior of the guard-bed as a function of the physical pa.rameters of the system. The solution to these three problems is obtained from the numerical integration of .:he "ion exchange" equation by the method of characteristicj. I n dimensionless form, the ion exchange (i'ermeulen, 1958) equation is irritten as: ARBERRY
G A S FILM
C"shell-progressive"
\There CB is the dimensionless concentration of poison-producing material in the fluid phase; Q is the volumetric fraction of the solid phase saturated with poison. The shell-progressive mechanism requires that the poisoned portion of the catalyst be completely saturated. Z is the dimensionless distance through the bed: z/l ; and T is the dimensionless time parameter defined as
cg, BULK CONCENTRATION
Cs,
SURFACE CONCENTRATION
c,,
INTERIOR INTERFACE CONCENTRATION
DIMENSIONLESS PARTICLE RADIUS
Figure 1.
Shell-progressive poisoning model
The poisoned zone forms a sharp interface with the unpoisoned core; this interface moves toward the center of the spherical catalyst pellet
and dimensional variables are signified by lower case letters. The remaining parameters are defined in the nomenclature.) T h e rate equation for growth in the volume fraction absorbed in the solid phase is developed from the diffusion equation for spherical coordinates in which the accumulation term is ignored (Bischoff, 1963). Figure 1 shows a cross section of a spherical pellet in the reactor; the pellet is assumed to be poisoned from the surface to dimensionless spherical radius, R. T h e total mass flux of poison through the gas film, the flux through the spherical shell, and the rate of chemisorption of the poison at the interior radius, R, are all equal. Thus one finds that the rate of decrease of R is given as:
Using the series resistance concept one may eliminate C, and C1 from Equation 2. The dimensionless solid concentration and the dimensionless radius are related by the simple equation
Q=1-R3
(34
-dQ = 3 R2dR (3b) Upon introducing Equation 3b and dimensionless time into Equation 2 , one obtains
(Dimensionless variables are denoted by upper case letters VOL. 7
NO. 2
MAY 1968
185
where N , = (3 Dkl/rP2ui) is the number of solid diffusion transfer units in the bed; NBa = ( k Q r p / D kis) the Biot number for ) the Damkohler number for mass transfer; NDa= ( k p r p / D k is the poisoning reaction; and F = volume fraction vapor in bed. T h e parameter a>k is the effective diffusivity in the particle. Although it is assumed to reflect Knudsen diffusion, the parameter can originate from bulk diffusion without any change in this development. Equations 1 and 4 are the “ion exchange” equations which describe the transient buildup of poison in a reactor. T h e boundary conditions for the equations are assumed t o be:
E! a
I-
LL
0.I
0.01
C ( T , 0) = 1
(5b)
Equations 5a and 5b denote a step input t o a bed initially free of poison. Hall et al. (1966) developed a solution for these equations by assuming that the adsorption band width in the column is constant. An obvious way t o avoid this assumption is direct integration of the equations by characteristics, the solution method used in this work. Before presenting these results, it is useful to develop the effectiveness factor for a bed of partially poisoned particles. If one assumes a first-order irreversible reaction occurs in the catalyst, then the local effectiveness factor (the effectiveness factor for a particle located a t a position in the bed where the radius of the poisoned core is R ) is given as: 3
where 7 = catalyst effectiveness factor; +b = (kTrp*/Dk)l/* is the spherical Thiele modulus (the Thiele modulus can be regarded as the ratio of the diffusion time, r p 2 / D Xto, the reaction time, k,-I; a large Thiele modulus implies that the reaction is faster than diffusive transport; k, = reaction rate for the catalytic system; NBo’ = Biot number for the catalytic system (one must distinguish NBo’from >VBo because the poisoning reactant and the catalytic reactant can have different effective Knudsen diffusivities). Shell-progressive poisoning can have a drastic effect upon the catalyst effectiveness; this result is particularly true when the Thiele modulus is large. T o find the over-all effect of the poisoning upon the activity of the bed, the linearity of the assumed kinetics permits simple averaging :
(7) where a ( T ) is the average effectiveness. T h e parameter of interest to the operator of a mixed-bed reactor is the total activity of the bed. T h e total activity is simply related to the fraction of the initial activity which remains after the poisoning process has started. This fractional activity ratio is simply
Numerical Analysis
The characteristic equations were integrated with a fourthorder Runge-Kutta-Gill subroutine programmed in Fortran I V . Copies of the program are available from the author. Results
Figure 2 shows the activity ratio of the bed as a function of the fractional time required to saturate the bed completely. The activity ratio is plotted for three values of the Thiele 186
l&EC FUNDAMENTALS
I
TIME ON STREAM
Figure 2.
Activity of b e d
The fraction o f the initial octivity of the b e d is plotted as a function of the time on stream. At time equal to unity, the reactor has been supplied with enough poison to saturate the b e d completely. The parameter N, describes the transport r a t e o f poison into the pellet while the Thiele parameter refers to the major chemical reaction
modulus and two values of the number of solid phase transfer units. The activity of the bed falls drastically for Ns = 1.0 as the ideal saturation time, T = 1.0, is approached. T h e effect is less drastic for the lower value of lV,; this result simply means that more poison has escaped from the bed as effluent. Curiously, the activity ratio for the iZr, = 1.0, $ = 100.0 curve is greater than the N , = 0.1, $ = 100.0 curve. This result may be interpreted as follo\vs: When the Thiele modulus is high, then the significant penetration of system reactant into the solid catalyst is very shallow. When the iV, parameter is large, then the axial solid concentration profile in the bed has narrow band ividth-i.e., there is a substantial portion of the bed totally unpoisoned for T < 0.4. Consequently the latter portion of the bed \rill have activity close to the initial value. O n the other hand, when the itr,parameter is small, the bed is poisoned more uniformly; this results in a rapid rate reduction for systems very sensitive to pore-mouth poisoning. Thus, the activity ratio for the high .Ys curve can be greater than the low N , curve for part of the poisoning cycle. T o generalize this analysis, there are two factors which affect the activity ratio. Low values of N , imply that the adsorption of poison occurs slowly; this generally tends to improve the productivity of the reactor. However, if the Thiele modulus is large, a uniformly poisoned bed is less effective than one poisoned with a sharp poison concentration profile; this result implies that a large value for ,V,is more nearly optimal. Thus, a quantitative calculation is needed to judge whether N , and should be increased or decreased in a particular system. (The effective D Lcan be altered in cracking catalysts by variation of the gelation procedure.) Figure 3, A and B, shows the position of the poisoned interface throughout the bed for three values of Ars. In accordance with the discussion above, the sharpness of the poisoned interface changes dramatically for the three values of N , used in this study. Further, the constant pattern or constant bandwidth assumption is not appropriate for most of the time of interest. A constant bandwidth pattern is established if the equilibrium is nearly irreversible and the system has been onstream long enough that the solid phaseat the entrance tothe bed is “nearly” saturated. (The word nearly is in quotes, because one must first define how closely one wishes to have “constant pattern” behavior.) Thus for itr, values of 0.1 and 0.333 constant
+
I 1.0
b
In II 0.8 0.6
-?
7 . 0.4
0
m W
z
2 E
0.1
Y
0
2
V
0.4
d 0.2
0.01
0.I
I
TIME
Figure 4. I.o
i
Coke buildup
The total amount of coke adsorbed in the b e d is very nearly a linear function of time. As the parameter N, increases the bed tends toward piston-flow adsorption
0.8
J L W (o
0.6
3
0.4
-0
I
a
a
z
E- 0.2
P
2 K I-
0
0.2
0.4
06
OB
z
% z
1.0
Z , POSITION IN BED
u
Figure 3. Interfacial position of poisoned shell
+ J W + 3
A. The radial position of the poisoned shell (Figure 1 ) is shown as o function o f axial position (it dimensionless time, T = 0.4. The catalyst a t the entrance to the b e d is not completely poisoned for any value of the parameter N,. Therefore, the "constant bond width" assumption is not valid. E. Results displayed a r e similor to Figure 3 A except thot the time has advanced to T = 0.6. The camtant band width model is valid only for the system in which N, = 1.0
0
pattern behavior is not observed for dimensionless time, T , less than 1.0, because the solid phase is not saturated at the entrance to the bed. Constant pattern behavior is seen for N , = 1.0 and T > 0.4, since the solid phase meets the saturation condition [ Q ( O , 0.44) = 0.998, R(0, 0.44) = 0.01 1. T h e time period required t o establish saturation should be considered when the constant band width model is used (Olson, 1961). The data related to the second problem, the buildup of coke in a fixed-bed reactor, are displayed on Figure 4. T h e total coke in the bed is expressed as the fraction of total saturation and is found by integration of the coke concentration through the bed. T h e total coke is plotted as a function of dimensionless time for several values of the solid diffusion parameter. O n e hopefully might expect that power-law functionality would be observed, and further, the "one-half" parameter frequently associated with diffusion-limited systems would be found. Power-law behavior is observed during the early parts of the saturation process, but the apparent exponent is greater than one half (0.85). One may obtain a useful reduction in the coke level through a modest change in the Knudsen diffusivity for the system; in particular, changing the diffusivity from 0.333 t o 0.1 will increase the on-stream time by 50% for any total coke level below 30y0. Apparently, increasing the N , parameter from 0.533 t o 1.0 has little effect upon the time required to obtain a given low coke level.
0.1
0
0.01 0.01
0.1
I
TIME
Figure 5.
Guard-bed performance
?he outlet concentration of o guard-bed adrorber is a strong function of the poison diffusion parameter, N,. This graph is a leakage curve for the guard-bed adsorber
The break-through behavior of a guard-bed reactor is illustrated in Figure 5. T h e effectiveness of the unit is a strong function of the N , parameter. This conclusion is most easily demonstrated by an example. Suppose one is required t o reduce the poison content of a reactant stream to 0.01 of the inlet value, say from 200 p.p.m. to less than 2 p.p.m. Apparently, if N , is 0.1, the guard-bed never is effective. Marginal effectiveness is achieved by increasing N , t o 0.333, and a vastly improved bed is obtained when N , is 1.0. T h e N , parameter is increased linearly with the Knudsen diffusivity and the bed length, and the inverse square of particle diameter. The conclusion is obvious, however; the value of the N , parameter should always be greater than 1.0 for efficient utilization of the adsorptive-reactive capacity of the guard-bed. Conclusions
T h e transient shell-progressive poisoning of a fixed-bed reactor has a marked effect upon the effective activity of the reactor. For low values of the reaction Thiele modulus, the useful lifetime of the reactor is improved by decreasing the Knudsen diffusivity. O n the other hand, the converse is true for very high values of the Thiele reaction parameter. I t is therefore necessary t o compute the actual poisoning performance of a reactor to determine which way the Knudsen diffusivity or the reaction rate should be changed. VOL. 7
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187
The rate of coke buildup in a reactor is sensitive t o the Knudsen diffusivity when the parameter N , is in the range 0 t o 1, In particular, decreasing Na from '/a t o 1/10 will increase the on-stream time by 50% if the reactor is limited by coking. T h e performance of a guard-bed reactor is extremely dependent upon the value of N,. A threefold increase in N , can increase the guard-bed lifetime by 12; this increased improvement may yield better process design for systems in which a guard-bed appears.
SUBSCRIPTS = bulk fluid s = surface I = interior interface of poisoned shell
B
Appendix: Evaluation of Dimensionless Groups
NBo. The effect of external gas mass transfer upon the system is reflected through the Biot number. This parameter can be estimated from the following dimensionless equation :
Acknowledgment
T h e author extends his sincere appreciation to Chung Lim for her assistance in programming this problem and t o the Houdry Process and Chemical Co., Philadelphia, Pa., for permission t o publish the work.
For the range of N , parameters used in this work, NBahas only a small effect upon unit performance. N n ~ ; I t is useful t o relate the Damkohler number and the
Nomenclature
Thiele modulus for the reaction which produces the poisoning. T h e poisoning reaction rate constant is written as
DIMENSIONAL VARIABLES b = stoichiometric coefficient, moles (solid phase)/moles (fluid phase) c = fluid phase concentration, mole/cc. cE0 = bulk concentration a t bed inlet, moles (fluid phase)/ cc. 9 B = gas phase diffusivity, :q. cm./sec. D~ = effective Knudsen diffusivity in solid phase, sq. cm./ sec. k , = fluid-phase mass-transfer coefficient, cm./sec. k i p = intrinsic rate constant of poisoning reaction, cm./sec. k, = effective surface reaction rate constant for poisoning reaction, cm./sec. k , = effective solid-phase reaction rate constant, sec.-l 1 = bed length, cm. 1, = thickness of poisoning reaction zone q = average concentration in solid phase, moles (solid phase)/cc. qw = concentration in solid phase at saturation, moles/cc. r = radius parameter, cm. rp = radius of particle, cm. s = surface area, sq. cm./g. t = time, sec. vi = interstitial velocity, cm./sec. z = bed length variable, cm. u = kinematic viscosity, sq. cm./sec. p, = density of pellet, g. cc. DIMENSIONLESS VARIABLES A = activity ratio of the bed CB = ( c E / c B 0 ) , bulk concentration in the fluid phase F = volume fraction of bed in fluid phase N B= ~ ( k o r p / D k ) Biot , number for mass transfer ND, = (kprp/ak), Damkohler number NR., = (2 uirBF/u), Reynolds number N , = (3 Dkl/r,2ui), solid diffusion number NB, = ( ~ B / u ) ,Schmidt number Q = ( q / q m ) , concentration in solid phase R = r / r p , radius parameter T = [ ( u ~ r)/l]{Fbc,"/[(I - F ) q m ] ) time , parameter 2 = z / l , length parameter q = catalyst effectiveness factor = mean catalyst effectiveness factor for bed = ( k , r , 2 / ~ k ) 1 Thiele /2, modulus
-
$
188
l&EC FUNDAMENTALS
where k , = poisoning rate constant, cm./sec.; s = surface area/mass catalyst, sq. cm./g.; p p = particle density of catalyst, g./cc.; kip = intrinsic rate constant of the poisoning reaction, cm./sec.; and 1, = length of the poisoning reaction zone. Equation A2 is developed to acknowledge that if the intrinsic rate constant, kip, is very large, then the poisoning If one reaction occurs in the very narrow zone defined by I,. compares the Damkohler number and the Thiele modulus, the following simple relation is found
where
I t has been argued (Carberry and Gorring, 1966) that $ p is of the order 200 for a shell-progressive mechanism; if one assumes that the ratio ( I P / r p ) should be of the order (0.7 X then the Damkohler number is approximately 20. T h e value of ND, does not affect the results of this analysis in a significant way. literature Cited
Bischoff, K. B., Chem. Eng. Sci. 18, 711 (1963). Carberry, J. J., Gorring, R. L., J . Catalysis 5 , 529 (1966). Hall, K. R.. Eagleton, L. C.. Acrivos, A.. Vermeulen,. T.,. IND. ENG.CHEM.FUNDAMENTALS 5,212 (1966): Olson, J. H., D. Eng. thesis, Yale University, New Haven, Conn., 1961. Vermeulen, T., Advan. Chem. Eng. 2, 148 (1958). RECEIVED for review August 30, 1967 ACCEPTED December 26, 1967
