Ind. Eng. Chem. Res. 1993,32, 247-252
247
KINETICS, CATALYSIS, AND REACTION ENGINEERING Wrong-Way Behavior of Packed-Bed Reactors: Influence of an Undesired Consecutive Reaction Andrew 11% and Dan Luss* Department of Chemical Engineering, University of Houston, Houston, Texas 77204-4792
A sudden decrease in feed temperature or a sudden increase in the feed velocity may lead to a wrong-way behavior in a packed-bed reactor in which two consecutive, exothermic reactions occur. When the reactor has a unique steady state, the wrong-way behavior is associated with a downstream-moving temperature front that causes a transient temperature rise and a loss of yield of the desired product. Increasing the rate of the undesired reaction increases the transient temperature rise and the velocity of the temperature front. When the reactor has multiple steady states, the wrong-way behavior may lead to ignition via slowly upstream-moving temperature waves. This ignition leads to a permanent yield loss of the desired product and may require reactor shutdown followed by a new startup. The impact of the wrong-way behavior in the two-reaction system is more pronounced and more likely to be encountered in practice than in the single-reaction case. Introduction A unique dynamic feature of a packed-bed reactor is the wrong-way behavior, which causes the initial reactor response to be in the opposite direction of a change in the feed condition. For example, a sudden decrease in the feed temperature may lead to an initial temperature rise in the downstream section of the reactor. When the reactor may attain multiple steady states, this transient temperature rise may ignite the reactor and shift it permanently to a high-temperature steady state (Pinjala et al., 1988; Chen and Luss, 1989; Il'in and Luss, 1992). The first predictions of the wrong-way behavior were made by Matzos and Beskov (19651, Boreskov et al. (1965), and Crider and Foss (1966). It was observed experimentally by Hoiberg et al. (1971),Van Doesburg and DeJong (1976a,b),and Sharma and Hughes (1979a,b). A simple, conservative estimate of the wrong-way behavior was presented by Mehta et al. (1981). More-accurate predictions were presented by Pinjala et al. (1988), Chen and Luss (1989),and Il'in and L w (1992). The impact of the wrong-way behavior on the temperature excursion in an automobile convertor was studied by Oh and Cavendish (1982). The previous studies were all of a system in which a single chemical reaction occurs. However, in practice, many of the control and runaway problems caused by the wrong-way behavior are encountered in systems in which several reactions may occur simultaneously. Specifically, the wrong-way behavior may cause a reaction, which has a negligible rate under the original steady state, to occur at an appreciable rate during the transient period. This may lead to significant heat release, which, in addition to affecting the yield of the desired product, may damage the catalyst and/or the reator and even lead to runaway. Thus, it is of practical importance to gain a better understanding of the wrong-way behavior in multireaction systems. We report here a study of ita impact in a reactor in which two consecutive reactions A B C occur. This network represents a large number of important catalytic
--
reactions, such as partial hydrogenation, chlorination, oxidation, and many others. We examine in particular the impact of a sudden decrease in feed temperature and of a sudden increase in feed velocity. Mathematical Model The packed-bed reactor is described by a two-phase one-dimensional model that accounts for heat and mass transfer, dispersion, and convection. The model assumes that the concentration and temperature gradient within the pellets and along the radius of the reactor is small relative to the interfacial gradient (Carberry, 1975). We consider the case of two first-order consecutive reactions occurring in the bed. The dimensionless equations are
1 aY 1 a2Y aY + H(y -ys) + U(y-y,) = -- -
Le at
(l-')".= Le
Peh a z 2
at
where
and
0000-5005/93/2632-0247$04.00 JO 0 1993 American Chemical Society
az (3)
248 Ind. Eng. Chem. Res., Vol. 32, No. 2, 1993
0.2 0.18 -
A+B+C
0.16 0.14 0.12 0.1
The reference temperature T,is selected so that kl(Tr)= u/L. The corresponding boundary conditions are
0.08 0.061
z
'
0
z = o
'
'
'
0.2
'
'
0.4
'
'
I
'
0.8
0.6
1
Dimensionless axial position Figure 1. Transient temperature profiies following a sudden decrease in the dimensionlegs feed temperature in an adiabatic reactor from 0.075 to 0.07 for a case with and without undesired consecutive reaction. Peh = 250, yr = 0.08. For the consecutive reaction case, 8 2 = 0.04, 6 = 0.001.
= 1:
The initial conditions corresponds to a steady state with a specified feed temperature yfl. At t = 0, the feed temperature is suddenly reduced to yn. The simulation in Figure 12 corresponds to a sudden change in feed velocity. The numerical method of the solution was described by Il'in and Luss (1992). Simulation Results and Discussion Extensive numerical calculations were carried out to determine the impact of the wrongway behavior in a system in which two consecutive, irreversible, exothermic, fmt-order reactions A B C occur, and in particular to compare it to that in a system in which only a single reaction takes place. Emphasis was on finding the magnitude of the transient temperature rise and the corresponding temporal change in the yield of the desired product B. In all the simulations reported here, the following parameter values were used: Le = 100, Le* = 66, Pe, = 4 Peh,MA = MB = H = 180, B1 = 0.04, and u = 1.5. The values of the remaining parameters are specified in each figure. The simulations show that the magnitude of the transient temperature rise following a sudden decrease in the feed temperature of the two-consecutive-reactionsystem exceeds that in a reactor in which only a single reaction A B occurs, even for a case in which the undesired consecutive reaction has a negligible impact on the steady state. Moreover, the wrong-way behavior can cause a large temporal decrease in the yield of the desired product. An illustration of this behavior is shown in Figure 1. When only the single reaction A B occurs (Figure 1,top), the maximal transient solid temperature rise following a sudden decrease in the feed temperature is ye = 0.133, exceeding by 46% the adiabatic temperature rise. On the other hand, in the presence of the undesired consecutive reaction (Figure 1, bottom), the maximal catalyst temperature rise (y, = 0.196) exceeds by 150% the initial steady-state temperature rise. Thia happens even though the steady-state temperature profiles of both cases are essentially identical.
1
0.8
0.6 4
K
0.4
0.2 0
--
-
1
0.8 0.6 m
X
0.4
0.2 0
0
0.2
0.4
1
0.8
0.6
Dimensionless axial position Figure 2. Transient concentration profiies of the reactant and desired product for the two cases shown in Figure l, bottom. 0.21
= = = k='=
' =",- - ---_
-
0.05 0.05
,
0.06
0.07
.
I
0.08
.
/
0.09
.
0.1
Feed temperature Figure 3. Dependence of the maximal catalyst transient temperature on the original (solid line) and the new feed temperature (dashed line) for a case with undesired consecutive reaction.
Ind. Eng. Chem. Res., Vol. 32, No. 2, 1993 249
1 temporal maximum, - - - 1 c
/
o.2
r--l
temporal maximum_ ' 0
W L
;a
z
0.12
-
~
n (D
initial steady state
-
5:
0.12
tinal steady state I
0.08
n
0.6
-
r3 v
-
.
,
final steady state
.
0.08
I
.
I
.
1
.
initial steady sta 0.8
,final
steady state
'
\ \
0.4L 0.2
x"
-
,
\
temporal minimum
OO
0.0005
6
\.
---
0.001
Figure 4. Dependence of the maximal catalyst transient (dashed line) and steady-state (solid line) temperature (top) and concentration of the desired product (bottom) on the ratio of the rate constante of the undesired-to-desiredreaction rate constant at the reference temperature for an adiabatic reactor.
Inspection of the transient mole-fraction profiles of the reactant and desired product (Figure 2)indicates that the maximal transient concentration of the desired product at any time is rather close to that obtained during the steady state at this point. However, the transient temperature rise causes rapid consumption of the desired product (see t = 0.9 and 1.2 in Figure 2,bottom), so that its temporal exit concentration is much lower than at either the original (t = 0) or the new steady state (t = m). Figure 3 is a master diagram describing the maximal transient temperature rise as a function of the new and original steady-state temperature. The solid line in that figure describes the dependence of the maximal steadystate temperature on the feed temperature. A unique steady state exists for any feed temperature in this case. The dashed line denotes the dependence of the maximal temperature rise on the new feed temperature and the original feed temperature (denoted by the point at which the dotted and solid lines intersect). Aa found for the case of a single reaction (Pinjala et al., 1988; Chen and Luss, 1989),an appreciable transient temperature rise occurs only for states that originally had an intermediate or appreciable temperature rise, but not for those with a low temperature rise or with a steep temperature front in the upstream section of the reactor. For example, in this case, the appreciable temperature rise can occur for the original steady states b and c, but not a or d. As expected, the reaction B C always increases the transient temperature rise over that obtained when it does not occur. Using the derivation in the appendix of Pinjala et al. (1988),it can be shown that in this case the asymptotic maximal transient temperature rise for a large decrease in feed temperature is
-
Ay =
81 + 82 1-0
where 0 is velocity of the temperature front. The impact of the consecutive reaction depends on ita rate relative to that of the main reaction, ita heat of reaction, and the activation energy. Figure 4 describes the
o.2
Itempore1 minimum 0.25
0.5
4'4
- --
0.75
1
Figure 5. Dependence of the maximal catalyst transient (dashed line) and steady-state (solid line) temperature (top) and concentration of the desired product (bottom) in an adiabatic reactor on the ratio between the heata of the undesired to desired reaction.
dependence of the maximal transient temperature rise (top) and minimum in the yield of the desired product (bottom) on 6, the ratio between the rate constants of the undesired-to-desiredreaction rate constant at the reference temperature yr. For all the 6 values shown in the figure, the side reaction practically does not occur at the final steady state (the one with the new feed temperature) and has a rather small effect on the original steady state (maximal increase of 24% in adiabatic temperature rise and 23% reduction in yield of B at largest 6 value). For all 6 < 0.0003,the side reaction has a negligible effect also on the transient temperature rise. However, for 6 > O.OOO7, the side reaction has a very strong impact, and during the transient period it significantly increases the maximum temperature rise and completely consumes the desired product. Figure 5 shows the impact of the ratio of the heats of reaction of the undesired to desired reaction. For all the ratios shown in Figure 5,the side reaction does not affect the final steady state (the one corresponding to the new feed temperature) and has a small effect on the original steady state. Changhii this ratio between these two heats of reaction causes a monotonic increase in the maximal temperature rise and monotonic decrease in the yield of the desired product. When both are equal, the desired product is completely consumed during the transient period, while at all steady states the yield of B exceeds 0.7. The consecutive undesired reaction affects the wrong-way behavior in a cooled reactor in a way similar to that in an adiabatic packed-bed reactor; Le., it incream the transient temperature rise and decreases the yield of the desired product. A typical case is shown in Figure 6. Here, the maximal transient temperature (y, = 0.2011)exceeds by 2259 i the original steady-state temperature rise. When only the first reaction occurs, the maximal transient temperature rise Cr, = 0.1297)exceeds by only 44% the original steady-state temperature rise. Figure 7 describes the impact of the ratio between the heat of reaction of the undesired and that of the desired one. In this case, the undesired reaction does not affect either the initial or final steady state for all 0 < &/&
250 Ind. Eng. Chem. Res., Vol. 32,No. 2, 1993 0.21
0.19
1.4
.
0.16
1.1 h*
0
.
1
6
1
I
,
,
0.065
0.07
I
M
0.11
0.. .~ 061
'
'
'
'
'
'
'
'
'
0.04
0.055
0.05
0.06
0.075
Feed temperature
Figure 8. Dependence of the maximal transient temperature on the new feed temperature for the caee with undesired consecutive reaction (dashed line). Steady states are shown by solid line. Initial feed temperature 0.067 (ignition point). Peh = 50, U = 2, yw = 0.061, yr = 0.07, = 0.04, 8 0.005.
a
X
0.4
0.2
0.15,
u 0.2
0
,
,
,
,
,
,
,
I
I
1
1
0.8
0.6
0.4
,
1
Dimensionless axial position Figure 6. Transient catalyst temperature (top) and desired-product-concentration (bottom) profiles following a sudden reduction in the dimensionless feed temperature in a cooled reactor from 0.081 to 0.075. Same parameters as in Figure 1, bottom, and yw = 0.075, U = 1, 8 2 = 0.06. 0.23 0.2
h"
timporal 'maximljq - - -
1
1
,
Y
,
,
I
,
I
,
I
,
I
,
0.8
I
I I I
X
0.11 0.081 0
___-_--initir final steady state '
'
0.5
'
'
1
'
'
1.5
'
'
2
Dimensional axial position
Figure 7. Dependence of the maximal catalyst transient (dashed line) and steady-state (solid line) temperature on the ratio between the heats of the undesired to desired reaction in a cooled reactor.
Figure 9. Dimensionless initial steady-state catalyst temperature and desired product concentration of the five cases shown in Figure
2. The wrong-way temperature rise is affected only for f12/f11 > 1.25. Note that for the adiabatic corresponding case (Figure 5), the impact was noticeable for much lower &/& values. The simulations of both the adiabatic and cooled reactor show that when a unique steady state exists, the velocity of the downstream-moving temperature front is always increased by the second reaction. However, the opposite trend was found when the reactor has multiple steady states. Computations by Eigenberger (1972)indicate that when a single chemical reaction occurs in a packed-bed reactor, three different stable steady states may be found for some operating conditions. This suggests that four stable steady states may exist for the two consecutive reactions. Thus, different types of bifurcation diagrams of max(y,) vs feed temperature are expected to exist in this case, depending on the shape of the steady-state bifurcation diagram. The solid line in Figure 8 describes one of these cases. Three stable steady states exist for 0.0565 < yf < 0.0585, and two stable steady states exist elsewhere in the multiplicity region of 0.052 < yf < 0.067. The profile of the temperature and product concentrationof the statea correaponding to the five points marked in Figure 8 are shown in Figure 9. We note that for the high-temperature states d and
e the product B is totally consumed within the reactor, while a relatively high yield of B is obtained for states b and c. Consider now a case in which the system is initially at state c (yf = 0.0671,shown in Figure 9. When the new feed temperature is higher than the extinction temperature Cyf = 0.0565),the wrong-way behavior generates rather slow backward movement of a temperature front, similar to the behavior found for the single reaction by Pinjala et al. (1988),Chen and Luss (19891,and Il'in and L w (1992). A sudden reduction of the feed temperature below 0.0665 and above 0.0585 (boundary of region with three stable states) causes an ignition of the reactor by a slow backward-moving temperature front (Figure 10,top). The velocity of this front is much slower than that of the forward-moving front for the unique steady-state case. The new ignited steady state has a sharp temperature-rise front a t the inlet to the bed. Note that in this case the new steady-state temperature exceeds the maximal transient temperature. The ignition is accompanied by permanent loss of yield of the desired product, and this may require a shutdown and new startup. When the feed temperature is reduced to a value between 0.052 (extinction temperature of multiplicity region)
8.
Ind. Eng. Chem. Res., Vol. 32, No. 2, 1993 261 '
0.15
Catalyst ' - - - 1Gas
1
0.13,
,
,
,
I
'
,
,
I
'
,
,
,
,
1
'
I1
0.13 h 0.11 c
h"0.09
0.07 0.05 0.15
0.07b' I
,
,
,
I
,
I
I
I
,
0.2
0.4
0.6
"
0.8
Dimensionless axial position Figure 12. Wrong-way behavior caused by a 40% increase in feed velocity. Parameter values same as in Figure 6.
0.13
0.11 h"
0.09 0.07
0.05h
'
I
0.2
I
'
I
'
0.4
0.6
'
J
'
I
0.8
1
Dimensionless axial position Figure 10. Transient temperature profiles following a sudden reduction of the dimeneionleea feed temperature from 0.067 for the same parameters as in Figure 8. 0.15
1
'
1
6 = 0.001
0.13 L
3 E
E
c)
R
Z
2
ij
m
&:
0.11
0.09 0.07 0.05 0.15
,
J
0.13
-
0.11
-
8
0.09
-
a
0.07 -
;.
I
& = 0*005
1
20
10
30
-
C
f
0.05
--dj ,
0
- / - /-/
I wJ 1
0.2
1
0.4
1
1
0.6
,
1
00
0.8
1
Dimensionless axial position Figure 11. Transient temperature profiles following a sudden reduction of the dimensionless feed temperature from 0.067 to 0.051 for the same parameters as in Figure 8 except for 6.
and 0.0585,the system is again ignited by a backwardmoving front, but the new steady state is type d; i.e., the temperature peak is in the center section of the reactor. An example is shown in Figure 10, bottom. When the reactor is cooled below 0.052,a forwardmoving temperature front causes a large transient temperature rise. F'igure 11compares two such cases that have two different b values, i.e., different ratios of the rate constanta of the undesired to desired reaction at the reference temperature. It shows that an increase in this ratio, or the relative importance of the second reaction, decreases the velocity of the forward-moving reaction front. This behavior can be explained by the exietence of steady-state multiplicity for the b = 0.005,while a unique steady state exiata for 6 = 0.001. Existence of an ignited steady state either slows the thermal front or causes it to move back-
ward, depending on the new feed temperature. Wrong-way behavior in packed-bed reactors, and especially cooled ones, may be c a d by an inmease in the feed velocity (Il'in and Luss, 1992). Figure 12 illustratm a case in which a 40% increase in the feed velocity causes a transient temperature rise, which exceeds by 22% the maximal temperature rise of the original steady state. Clearly, the rational design of a control and operating procedure must take into account this surprising dynamic response.
Concluding %marks The examples show that a sudden decrease in the feed temperature to a packed-bed reactor in which two consecutive reactions occur may lead to a rather large traneient temperature rise and a corresponding sharp decline in the yield of the desired product. When a unique steady state exists for all feed temperatures, the wrong-way behavior causes a slow downstream-movingtemperature wave. The higher the rate of the undesired consecutive reaction, the larger the maximal temperature rise and the faster ita forward velocity. When multiple steady states exist for some feed temperatures, a sudden reduction of the feed temperature may lead to ignition via a slow backwardmoving temperature front. In this case,an increase in the rate of the undesired consecutive reaction decreases the velocity of the backward-moving temperature front. This ignition is associated with a permanent loss of yield of desired product, and may require a shutdown of the reactor followed by a new startup period. The simulations show that even a very low rate of an undesired consecutive reaction, which has a very small impact on the steady-state behavior, may have an important influence on the magnitude of the transient temperature rise and reduction in the yield of the desired product. In general, the impact of the wrong-way behavior in the two-reaction system is more pronounced and more likely to occur than in the single-reaction case. The simulations show that a wrong-way behavior may be caused by a sudden increase in the feed velocity. In a cooled reactor, the empty section of the tube on top of the bed usually tends to moderate any sudden changes in the temperature of the feed. Thus, wrongway behavior in a cooled reactor is much more likely to be caused by sudden changes in feed velocity than by changes in feed temperature. The understanding and ability to predict this interesting dynamic response of a packed-bed reactor is essential for a rational design of the control system. Acknowledgment
We wish to thank the National Science Foundation for partial support of the research.
252 Ind. Eng. Chem. Res., Vol. 32, No. 2, 1993
Nomenclature C = concentration of reactant c = specific heat D = axial mass-dispersion coefficient E = activation energy H = dimensionlesa heat-transfer parameter, defined by eq 8 -AH = heat of reaction h = interfacial heattransfer coefficient h, = overall heat-transfer coefficient at wall k(TJ = reaction-rate constanta at T = T, k, = interfacial mass-transfer coefficient L = reactor length Le = Lewis number, defined by eq 8 Le* = modified Lewis number, defied by eq 8 M = dimensionless mass-transfer parameter, defined by eq 8 Pe = Peclet number, defined by eq 8 R, = universal gas constant r = reactor radius = particle radius = specific surface area ?= temperature T, = reference temperature, defined by k(TJ = u/L t = dimensionless time, defined by eq 8 t’ = time U = dimensionless heat-transfer parameter at wall, defined by eq 11 u = superfacial velocity XCy,) = dimensionless rate function defined by eq 7 x = dimensionless concentration, defined by eq 8 y = dimensionless temperature defined by eq 8 z = dimensionlessaxial-position coordinate, defied by eq 8 z’= axial position coordinate
3
Greek Letters
j3 = dimensionless heat of reaction, defined by eq 8 13 = rate-constants ratio defined by eq 8 c = void fraction of bed A = axial heat-dispersion coefficient p = density a = activation-energy ratio, defined by eq 8 Subscripts
A = original reagent B = desired product f = feed, fluid h = heat m = mass p = pellet, particle r = reference
s = solid w=wall 1 = desired main reaction 2 = undesired consecutive reaction
Literature Cited Boreskov, G. K.; Slinko, M. G. Modelling of Chemical Reactors. f i r e Appl. Chem. 1965,10, 611. Carberry, J. On the Relative Importance of External-Internal Temperature Gradients in Heterogeneous Catalysts. Znd. Eng. Chem. Fundam. 1975,14,129. Chen, Y.C.; Luss, D. Wrong-Way Behavior of Packed-Bed Reactors: Influence of Interphase Transport. AZChE J. 1989, 35, 1148. Crider, J. E.; Foes, A. S. Computational Studies of Transienta in Packed Tubular Chemical Reactors. AZChE J. 1966, 12, 514. Eigenberger, G. On the Dynamic Behavior of the Fixed-Bed Reactor in the Region of Multiple Steady States-1. The Influence of the Conduction in Two Phase Models. Chem. Eng. Sci. 1972, 27, 1909. Hoiberg, J. A.; Lyche, B. C.; Foss, A. S. Experimental Evaluation of Dynamic Models for a Fixed-Bed Catalytic Reactor. AZChE J. 1971,17, 1434. Il’in A. V.; Luss, D. Wrong-Way Behavior of Packed-Bed Reactors: Influence of Reactant Adsorption on Support. AZChE J. 1992,38, 1609. Matros, Y.S.; Beskov, V. S. Calculation of Contact Apparatus with Internal Heat-Transfer as an Object of Control. Khim. Prom. 1965, 5,357. Mehta, P. S.; Sams, W. N.; Luss, D. Wrong-Way Behavior of Packed-Bed Reactors: I. The Pseudo-Homogeneous Model. AICW J. 1981,27, 234. Oh, S. H.; Cavendieh, J. C. Transients of Monolithic Catalytic Convertors: Response to StepChangea in Feedstream Temperature as Related to Controlling Automobile Emissions. Znd. Eng. Chem. Process Des. Dev. 1982,21, 29. Pinjala, V.; Chen, Y. C.; Luss, D. Wrong-Way Behavior of PackedBed Reactors: 11. Impact of Thermal Diffusion. AZChE J. 1988, 34, 1663. Sharma, C. S.; Hughes, R.The Behavior of an Adiabatic Fixed-Bed Reactor for the Oxidation of Carbon Monoxide: I. General Parametric Studies. Chem. Eng. Sci. 1979a. 34, 613. Sharma, C. S.; Hughes, R. The Behavior of an Adiabatic Fixed-Bed Reactor for the Oxidation of Carbon Monoxide: 11. Effect of Perturbations. Chem. Eng. Sci. 1979b, 34,625. Van Doesburg, H.; DeJong, W. A. Transient Behavior of an Adiabatic Fixed-Bed Methanator: I. Experiments with Binary Feeds of CO or COz in Hydrogen. Chem. Eng. Sci. 1976a, 31,45. Van Doesburg, H.; DeJong, W. A. Transient Behavior of an Adiabatic Fixed-Bed Methanator: 11. Methanation of Mixtures of Carbon Monoxide and Carbon Dioxide. Chem. Eng. Sci. 1976b, 31, 53. Received for reoiew August 29, 1992 Accepted November 9, 1992
