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Design and control of fixed-beds affected by catalyst deactivation

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Title:
Design and control of fixed-beds affected by catalyst deactivation
Creator:
Hong, Jan-Chen, 1955-
Publication Date:
Language:
English
Physical Description:
xi, 90 leaves : ill. ; 28 cm.

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Subjects / Keywords:
Bandwidth ( jstor )
Catalysts ( jstor )
Feedback control ( jstor )
Inlet temperature ( jstor )
Inlets ( jstor )
Kinetics ( jstor )
Oxidation ( jstor )
Reactor design ( jstor )
Temperature control ( jstor )
Temperature dependence ( jstor )
Catalyst poisoning ( lcsh )
Chemical Engineering thesis Ph. D
Dissertations, Academic -- Chemical Engineering -- UF
Genre:
bibliography ( marcgt )
non-fiction ( marcgt )

Notes

Thesis:
Thesis (Ph. D.)--University of Florida, 1984.
Bibliography:
Bibliography: leaves 88-89.
General Note:
Typescript.
General Note:
Vita.
General Note:
REPL*
Statement of Responsibility:
by Jan-Chen Hong.

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University of Florida
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University of Florida
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Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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DESIGN AND CONTROL OF FIXED-BEDS AFFECTED BY CATALYST DEACTIVATION







By

Jan-Chen Hong
























A DISSERTATION PRESENTED TO THE GRADUATE
COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL
FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

1984















ACKNOWLEDGMENTS

The author wishes to express his sincere appreciation to his research advisor, Dr. Hong H. Lee, for his guidance, patience, and constant encouragement throughout this work. Particular thanks are also due to Drs. G.B. Hoflund, C.C. Hsu, J.P. O'Connell and S. Svoronos for serving on the advisory committee.

The author also wishes to express his gratitude to his colleagues Laks Akella, Irfan Toor and Karen Klingman for many stimulating discussions, to staff members Tracy Lambert and Ron Baxley for their assistance in fabricating the experimental apparatus, and to Derbra Owete for typing the manuscript.

Sincere appreciation is extended to his parents, Mr. and Mrs. I-Mo Hong and to his wife, Shiao-Ing for their support and understanding throughout his graduate study.

















ii














TABLE OF CONTENTS

Page

ACKNOWLEDGMENTS ................................... ii

NOTATION .......................................... v

ABSTRACT .......................................... x

CHAPTER 1 INTRODUCTION ........................ 1

CHAPTER 2 THEORY OF ON-LINE ESTIMATION OF DE- 5
ACTIVATION AND CONTROL ..............

2-1 A Single Reaction ................... 5
2-2 Nature of Reactor Activity Factor.. 13
2-3 Multiple Reactions .................. 15

CHAPTER 3 SIMULATION AND EXPERIMENTAL VERIFICATION .............................. 20

3-1 Simulation of a Model Reaction System ................................. 20
3-2 ExDerimental Verification ........... 25
3-3 Concluding Remarks ................... 42

CHAPTER 4 OPTIMAL PIECEWISE CONTROL ........... 45

4-1 Problem Statement and Solution
Method .............................. 45
4-2 Optimal Piecewise Control Policies.. 49
4-3 Concluding Remarks .................. 57

CHAPTER 5 OPTIMAL CONTINUOUS CONTROL FOR COMPLEX REACTION ....................... 62

5-1 Problem Statement ................... 62
5-2 Necessary Condition ................. 64
5-3 Optimal Continuous Control Policies. 67
95-4 Concluding Remarks .................. 71

CHAPTER 6 OPTIMAL CONTROL AND DESIGN .......... 73

6-1 An Optimization Problem of Piecewise
Control and Design .................. 73



iii











PacTe

6-2 An Optimization Problem of Continuous
Control and Design ................... 76'
6-3 Concluding Remarks ................... 82

CHAPTER 7 CONCLUSIONS .......................... 84

REFERENCES ......................................... 88

BIOGRAPHICAL SKETCH ................................ 90












































iv















NOTATION

AiA preexponential factor, i=l, 2, q

a1 -AH2/PC

a2 -AH1 (I-AH2/AH1)/pCpb1

a3 (-AHI-bAH2)/PCp

a4 -AH2/PC

bilb 2 ratios of stoichiometric coefficients

C concentration of main reactant; integration
constant

C d desired outlet concentration of main reactant

C d modified Cd given by Eq. (2-16)

Cd initial value of Cd
0
C. inlet concentration of main reactant
in
C specific heat of reaction fluid

(Cp c specific heat of coolant

De effective diffusivity

E a, E activation energy for main reaction
i=l, 2, q

ED activation energy for deactivation reaction

F molar flow rate

F molar flow rate of reactant mixture at the
0 inlet

f apparent concentration dependence of rate of
main reaction; temperature and activity dependence of X in Eq. (4-2)



v











G. E */Rgi i = l,2,q,D

g concentration dependence of intrinsic rate
of reaction; temperature and activity dependence of rate of deactivation in Eq. (4-1) H reactor activity factor defined by Eq. (2-7)

c current value of H

H new value of H
n
(-AH) heat of reaction

h local activity factor defined by Eq. (2-5)

hc current value of h

h f h att =t f

h film heat transfer coefficient h new value of h
n
I integral defined by Eq. (2-20)

J performance index

K D rate constant for deactivation reaction

K1k 1kq K 2 k 2kq

K. equilibrium constant

K P k
q Ao I q
k intrinsic rate constant

kq adsorption equilibrium constant

k a apparent rate constant

ka preexponential factor for k a

k Prate constant for poisoning reaction evalus ated at pellet surface temperature




vi










m mass rate of reaction fluid; order of deactivation reaction mc mass rate of coolant

N concentration of poisoning species

n number of control steps in piecewise control

P total pressure

P A partial pressure for species A

Pi ~E ./E D' i,,:12,q
Q deactivation capacity of catalyst

q quantity defined by Eq. (2-21)

R 9gas constant R G global rate of main reaction

R t tube radius

r intrinsic rate

S selectivity

T temperature of reaction fluid

T transformed function of T

T ccoolant temperature T.i reactor inlet temperature

T T at z=O
ce c
t time

t f time at which catalyst is regenerated
or replaced in continuous control

t n time at which catalyst is regenerated
or replaced in piecewise control U overall heat transfer coefficient





vii










v superficial velocity

W weight of catalyst

X conversion at the outlet

AX bandwidth of conversion allowed before an
adjustment in T.i is made x conversion in the reactor

Y yield at the outlet

y yield in the reactor

Y O mole fraction of A at the outlet

YL mole fraction of A at the outlet

*1 mole fraction of C at the outlet
Z reactor length

z axial reactor coordinate

Greek Letters

a proportionality constant defined in Eq. (2-16)

3 j D


y fraction of catalyst deactivated

E: B bed porosity

11 cost incurred due to the reactor shut-down

p reaction fluid density

T holding time given in Eq. (2-1)

Subscripts

A species A; ethylene

B species B; ethylene oxide

C species C; carbon dioxide



viii











c coolant; current value; with constantconversion constraint D deactivated

d desired

e reactor outlet

i index for a step change in inlet temperature

in inlet

max the maximum value

min the minimum value

n new value

0 at t=O or z=O

q adsorption equilibrium constant

r at reference state

1 reaction path 1

2 reaction oath 2

Superscripts

optimal value

d
dt

normalized value















ix














Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy


DESIGN AND CONTROL OF FIXED-BEDS AFFECTED BY CATALYST DEACTIVATION

By

Jan-Chen Hong

August 1984

Chairman: Dr. H. H. Lee
Major Department: Chemical Engineering

On-line measurements of temperature and concentration

and the global rate at any reference state of catalyst can be used to estimate a measure of catalyst deactivation. A feedback control policy is obtained from this measure of catalyst deactivation for manipulating the reactor outlet conversion or yield in any desired manner. This feedback control enables one to obtain the desired conversion or yield without any knowledge of deactivation kinetics, and the results are applicable to any type of deactivation. The feedback control policy has been verified by experiment based

on ethylene oxidation reactions overasilver catalyst using a laboratory shell-and-tube reactor. Excellent agreements have been reached between the theory and the experiment.






x









The problem of finding the optimal piecewise control policy has been formulated as a multivariable optimization problem. The optimal policy is one of increasing conversion. The constant conversion policy corresponds to the limiting case where the number of piecewise steps approaches infinity. This optimal piecewise control is suitable for on-line implementation in conjunction with the on-line estimation of the extent of catalyst deactivation. An optimal continuous control policy for parallel and reversible reactions has also been solved using the technique of calculus of variations. It is shown that a direct substitution method is more efficient than the commonly used method of applying the Pontryacjin maximum.,principle.

The optimal control policies can easily be taken into consideration for reactor design. This combination of pro*cess design and control with due consideration of catalyst deactivation for both is shown to result in a substantial improvement of the reactor performance.



















xi














CHAPTER 1

INTRODUCTION

The usual practice of operating a fixed-bed with slow catalyst deactivation is to raise the reactor inlet temperature to compensate for the declining catalytic activity so as to maintain a desired conversion (Kovarik and Butt, 1982). The catalyst is regenerated when the required temperature becomes too high. The reactor is usually designed to give the desired conversion without any consideration for the catalyst deactivation. A summary of operational strategies for a reactor subject to catalyst deactivation is listed in Table 1-1.

A number of questions can be raised regarding these

usual practices: The first question is how we should control the operating conditions of a reactor designed in the usual way. This question has been the subject of many studies (Chou et al. 1967; Szepe and Levenspiel, 1968; Ogunye and Ray, 1971; Haas et al., 1974; Levenspiel and Sadana, 1978). The control policies resulting from these studies either lead to an open-loop control or require detailed knowledge of catalyst deactivation. Although considerable progress has been made in our understanding of catalyst deactivation, the deactivation is still the least understood of all facets involved in the quantitative description of a


1






2




Table 1-1 Summary of Operational Strategies for a Reactor
Subject to Catalyst Deactivation

1. Vary reactor temperature with time to maintain a constant

conversion with a constant reactor feed flow rate. A typical policy for large throughput (400 MM pounds per year) and

slow deactivation rates (months to years of catalyst life).

2. Vary throughput of the reactor feed while holding the reactor

temperature and conversion constant. A possible policy for

medium deactivation rates (weeks to months catalyst life)

and small to medium throughput (about 25 MM. pounds per year)

systems.

3. Allow the conversion to fall while holding the reactor feed

flow rate and reactor temperature constant. Similar applications as in Item 2.

4. Maintain the fresh feed rate and reactor temperature constant

and let the recycle flow increase. Similar application as

in Item 2.

5. Use a combination of reactors in parallel and the policies of

Items 1 or 3. Usually, with two reactors in parallel, one

will be off-line for catalyst regeneration while the other is operating. A typical policy for large throughput and medium to fast deactivation rates (days to months of catalyst life).

6. Continuous catalyst regeneration while maintaining constant

conversion, throughput, and reactor temperature. A typical

policy for large throughput, rapid deactivation systems

(hours-todays of catalyst life).

Reprinted from Cat. Rev..24, 499 (1982), by courtesy of Marcel Dekker, Inc.





3



fixed-bed. This uncertainty regarding the deactivation and the compounded effects of deactivation and diffusion in a fixed-bed make it quite unattractive to implement an openloop control policy. The second question, therefore, is whether a measure of catalyst deactivation can be estimated from process measurements and, if so, how a feedback control can be realized with the measurements made. In Chapter 2 a method of determining a measure of catalyst deactivation is presented and the result then utilized for a feedback control policy. This feedback control policy is demonstrated in Chapter 3 with a simulation of a model reaction system, and is put to the test with a laboratory fixed-bed where ethylene oxidation reactions take place over a silver catalyst.

It has been shown that constant conversion policy is optimal only for a single irreversible reaction affected by concentration-independent deactivation, when the reactor is controlled continuously (Chou et al. 1967; Lee and Crowe, 1970; Crowe, 1970). Thus, the third question regarding the usual practice of maintaining constant conversion is what the optimal policy will be if the reactor is piecewise controlled (for example, the temperature is adjusted only once a day, rather than continuously), or if the reaction is complex. The solutions to this question are presented in Chapters 4 and 5.






4



A much more important question than the three discussed above has to do with the inherent interrelationship between process controllability and process design. It is intuitively clear that the process design dictates the controllability, for the parameters involved in describing a process contain design parameters which in turn dictate the way the process can be controlled. It is clear therefore, that the best possible performance of the process can be attained when the process design and control are combined. For the reactor under consideration, this means that the best possible performance can be attained when the catalyst deactivation is taken into consideration not only for the reactor control but also for the reactor design. The question, therefore, is what the reactor size and the feedback control should be for the best performance of the reactor. In Chapter 6, the answer to this question is presented.
















CHAPTER 2

THEORY OF ON-LINE ESTIMATION OF DEACTIVATION AND CONTROL

An important problem in operating a fixed-bed with

catalyst deactivation is that of finding a means of estimating the extent of catalyst deactivation from on-line measurements because of the uncertainty regarding the deactivation. The problem of controlling the conversion in any desired manner using a feedback scheme is another. These two problems are solved theoretically in this chapter, starting with a single reaction and then proceeding to multiple reactions.

2-1 A Simple Reaction

Consider a shell-and-tube reactor in which a single

reaction takes place. Assuming negligible radial gradients and pseudo steady-state, one-dimensional balance equations can be written as

= -RG; T = Z(I-B)/v ;C C. (2-1)
dz 'T- B iz=o in


dT --AH (mCp) c dT T
dz p G mCp dz ; -Tz=o in



dTc 2Ur RZ
dzc 2U (mpc (T-T) T = T (2-3)
dz (mC) cc z=l in





5






6

where C and T are the concentration and temperature of the reactant, and T c is the countercurrent coolant temperature. Here m and m c are the reaction fluid and coolant mass rates, C p and (C p ) c are the specific heat capacities for the reaction fluid and the coolant, Z is the reactor length, z is the axial reactor coordinate normalized with respect to Z, v is the superficial velocity assumed constant in the radial direction, E B is the bed porosity, U is the overall heat transfer coefficient, and R t is the tube radius. The global rate based on pellet volume, R G' can be expressed as
R G = h(z;t)k a f(CK (2-4)

where k a is the apparent rate constant in the form of the Arrhenius relationship and f represents the apparent dependence of the rate on concentration and equilibrium constants K.. In the absence of diffusion and deactivation effects,
3
the apparent rate reduces to intrinsic rate. The activity factor h (Wheeler, 1955) is defined by



h(z;t) G D (2-5)
(R G)r

where the subscripts r and D denote reference-state and deactivated catalyst, respectively. According to the definition, h is unity for catalyst at the reference state. This activity factor decreases with time due to catalyst deactivation and thus deDends on time. It should be recognized in Eq. (2-5) that the definition of h is general in that h can







7


depend on the concentration C and thus is applicable to concentration-dependent deactivation. Since C is a function of z at a given time, h(C;t) is a subset of h(z;t). The pseudo steady-state assumption is based on the fact that the time scale of catalyst deactivation is many orders of magnitude larger than that for the reactor to reach a steady state.

For the purpose of deriving an expression for a measure of the extent of deactivation, we rewrite Eq. (2-1) with the aid of Eq. (2-4) as


dC -Th(z;t)k af(C,K) (2-6)


Here again, one can write h as h(C;t) in place of h(z;t) for dependent deactivation. This equation can be integrated from the reactor inlet to the outlet to give




IC.C TaTfCj) = { h(z;t)dz EH(t) in 0 (2-7)

where C e and C.i are the concentrations at the outlet and the inlet, respectively, at any given time. By definition, H is unity at reference state since the activity factor h is uniform at unity throughout the reactor. The quantity H is a measure of the extent of catalyst deactivation which represents the activity for the reactor and thus may be termed reactor activity factor." The value of H decreases as the outlet conversion decreases due to the catalyst deactivation for a given inlet temperature.






8


The temperature appearing in Eq. (2-7) needs to be related to concentration for the integration. Integrating Eq. (2-2) from the reactor inlet to z, we obtain



T .+[A](C. -C) -2~-(T T )(2-8)
in Pp~ in mC ce c


where T ceis the coolant temperature at z=O and T.i is the

reactor inlet temperature. Consider first an adiabatic reactor for which Eq. (2-8) reduces to



T =T. + [ IH (C. C) (2-9)
in ~PC in


Now that T is expressed in terms of C and readily measurable inlet conditions, the value of H can be calculated by simply carrying out the integration numerically with respect to C in Eq. (2-7) with the measured outlet concentration. For instance, the apparent rate constant k a(T) for an adiabatic reactor can be expressed as



k =k exp a g(2-10)
a a 0Ti + t[pcHJ (Cm C



where k ais the apparent preexponential factor and E ais the apparent activation energy. The equilibrium constants K can be expressed in a similar manner. For a shell-and-tube reactor to which Eq. (2-8) applies, the calculation of H requires measurement of coolant temperature along the reactor length. This measurement can be used in Eq. (2-3) for the






9


calculation of T as a function of z. With these T and Tc/ the corresponding change in C for a selected interval of Az can be obtained from Eq. (2-8) which in turn can be used in Eq. (2-7) for the calculation of H.

An immediate use of the reactor activity factor calculable from the temperature and concentration measurements is in the manipulation of the reactor inlet temperature for the purpose of maintaining the desired conversion. Suppose that the inlet temperature is adjusted intermittently, as in the usual operation of a reactor, to compensate for the declining activity. Consider an adiabatic reactor for clarity. If we let the subscript c denote the current quantities and n the new quantities resulting from a change in the reactor inlet temperature, Eq. (2-7) can be written twice to give


H = J1 h (z;t)dz = e dC (2-11)
c 0c fC inTk[aT;(T. i)c f(C,K i)


Hn = flhn(zt)dz f= kT(T f(,. (2-12)
n 0 nCin Tka[;Tin n jfCK


where C d is the desired concentration which will be attained by changing the current inlet temperature, (Tin)ci to the new inlet temperature, (T in n As we shall soon see in the next section, the activity factors h cand h depend mainly on the fraction of catalyst deactivated and slightly on the temperature difference between bulk fluid and pellet surface. If we neglect this change in the temperature difference for the time being, we can set h c(z;t) =h n(z;t) (2-13)






10




for the purpose of calculating the new inlet temperature that yields the desired outlet concentration. The justification here is that the fraction of catalyst deactivated changes negligibly for all points along the reactor while the inlet temperature is adjusted from the current value to a new value. Note that the justification is still valid for concentration dependent deactivation, i.e. h c (C(z);t)

- h n(C (z) ;t) Equations (2-1l) through (2-13) together with Eq. (2-10) can be used to obtain




C rd dC __ H

C.i Tk aexp a g f(C,K.
in a(Ti + -H(Cin C
in n inC C)


(2-14)

where H cis the known, current reactor activity factor calculated from Eq. (2-12) just prior to the change in the inlet temperature based on the current measured outlet concentration C eand inlet temperature (T. i) Equation(2-14) can now be solved for (T. i) by a numerical technique such as Newton's method or bisection method since the left hand side of the equation can be evaluated, given a value of (T. i)n. The value of (T. i) that satisfies Eq. (2-14) will yield the desired conversion and, therefore, the inlet temperature can be changed to (T.i ) nto maintain the conversion at the desired level.






11


The control policy of Eq. (2-14) is a piecewise algorithm for the manipulation of the inlet temperature. Therefore, it will maintain the conversion at the desired level only for a short period of time after a change in the inlet temperature and the conversion thereafter will gradually decrease with time until the inlet temperature is raised again. A bandwidth for the allowed decrease in conversion can be used to trigger the adjustment of the reactor inlet temperature. The control algorithm for a nonadiabatic reactor can be obtained in a similar manner. The equation corresponding to Eq. (2-14) is


f c ___________ d-d H (2-15)
C.i T a [;Tin )n JfCK


This equation, however, needs to be solved in conjunction with Eqs. (2-3) and (2-10) for (T. i) along with the measured T profile.

As indicated earlier, we assumed that the heat transfer resistance across the interface between bulk fluid and pellet surface is negligible in arriving at the control policies of Eqs. (2-14) and (2-15). If there exists a significant temperature difference between bulk fluid and pellet surface, Eqs. (2-13) through (2-15) no longer hold. The discrepancy between h cand h caused by the temperature difference can be compensated for by adjusting C d as follows:





12




C = Cd + a (C C ) i=l,2,... (2-16)
i ~i-i eCd =c d
0

where Cd. is the adjusted Cd when a step change in the inI
let temperature, which is triggered by the bandwidth constraint, is made at the ith step and a is a proportionality constant with a value between zero and unity. Thus, for a given decrease in the outlet concentration from the desired value, the piecewise feedback control algorithm can be stated as follows:
Cd.
d i dC
Cin Tk a [T; (Tin)ilf(C'K)= (Hci-i
in


i=l,2,... (2-17)


where (H )i_1 is calculated from


C
i1 dC
(Hc)i = C. k a [T;(T iC i=l,2,... (2-18)

in

(Hc) 0 = 1


with the measured outlet concentration and inlet temperature. Here, Cd. is given by Eq. (2-16). Each time the outlet
I
concentration reaches the allowed bandwidth, the new inlet







13



temperature (T. i) is calculated for the manipulation of the inlet temperature. All that is required for the feedback control is the global rate for catalyst at a reference state.

2-2 Nature of Reactor Activity Factor

Before we proceed to multiple reactions, let us examine the nature of the reactor activity factor. By definition, it is an integrated value of local activity factor (Eq. 2-7). The local activity factor, which is the activity factor at a point in the reactor, can be obtained by solving pellet conservation equations. Instead of solving the conservation equations, we utilize the global rate obtained by Lee and Butt (1982) for a reaction affected by uniform deactivation and diffusion to get a clear picture of the local activity factor. The global rate for this case is



(R)=- (1-Y) 3
RG)D L qL2-9
[2D ekI(C)] q 2-9

where

I- = {O g(ct)dct (2-20)




l.2E a (-AH) [2D ekI]
2h R2 (2-21)
m g







14


Here, y is the fraction of catalyst deactivated, g(C) the concentration dependence of the intrinsic rate of reaction, k the intrinsic rate constant, D e the effective diffusivity, L the characteristic pellet length, and h mthe film heat transfer coefficient. The local activity factor can be obtained from the definition of Eq. (2-5):


h=- (R G)D (R G)D =(l-y) (2-22)


The same results hold (Lee and Ruckenstein, 1983) for nonuniform deactivation if y is replaced with y which is y at the pellet surface, as long as y is less than, say, 0.5.

It is clear from Eq. (2-22) that the local activity factor depends on y and the quantity q which represents the heat transfer resistance across the bulk fluid-pellet surface interface. Since y changes very little while the inlet temperature is raised to a new value, the magnitude of q determines whether h ccan be set to h n Therefore, the proportionality constant a~ in Eq. (2-16) can be chosen in accordance with the magnitude of q: for q much smaller than unity, the value of at can be set to zero, its value increasing with increasing q. Although the conclusion is made from the result for independent deactivation, the same should be valid for dependent deactivation since the factor causing the discrepancy between h cand h nis still the same, i.e. the heat transfer resistance across the interface, which q represents.







15

2-3 Multiple Reactions

For simple parallel reactions, we can write

1
A -+ blB

2
A -b2C


where b1 and b2 are the ratios of stoichiometric coefficients. We restrict our attention here to an adiabatic reactor for brevity since a nonadiabatic reactor can be treated in the same manner as shown for a single reaction. Let 8 denote the ratio of local activity factors, i.e., =h 2(z;t)/h1 (z;t), the reactor conservation equations can be written as:

dCA = Thl(Z;t){k fl(CA' CB) + 3 ka f2(CACB)};
dz 1 a 1a 1 A Ba2 2A



CA = CA (2-23)



dCB = b1Th1(z;t)ka fl(CA' GB) C Bz = CB (2-24)

-z 1 Bl Iz-O Bin



T = Tin + a1(CAin CA) + a2(CB B i ) (2-25)
in in


where a1 = AH2 /pCp and a2 = AH1(I-AH2/AH1)pCpbI. Equation (2-25) results when the heat balance equations are combined with the mass balance equations. Here the global rates for







16


the first and second reactions are expressed as hk a 1fl1(C ACB) and hk a2f 2(C A C B), respectively. If the same active sites are responsible for both reactions, the local activities are the same, and is unity. otherwise, we assume that 3 is a function of time but constant along the reactor, i.e., =S(t).

The reactor activity factor can be obtained in the same manner as for a single reaction:



H1 h Tk e- B (2-26)

0 1 GCB inb 1T 1f1r(C ACB)


where CGB is the outlet concentration of species B. As Eq.
e
(2-25) indicates, the temperature can be expressed in terms of C Aand C Bfor the calculation of H but C Ahas to be related to GB to carry out the integration. Combining Eqs. (2-23) and (2-24), we have



dC A -1 kaf 2(C ACB)
dC B b 1k af( C CAi
1A A in Bi

(2-27)

which can be solved to obtain C A as a function of C BI C A= f(C B), by Fourth Order Runge Kutta Method or PredictorCorrector Method (Hornbeck, 1975). If is unity, the solution is straightforward. Otherwise, the concentrations






17

measured at the outlet, C A and CB. can be used as an additional boundary condition, i.e., CAI CB = CB = CA e for Eq. (2-27) to obtain the value of $.e

If species B is the desired species such that (C B)d is the desired outlet concentration, the control policy similar to Eq. (2-14) can be written as

(C B)d dC B

lC CB. b Tk ex:E a /Rg
in 1 a lo J (Tin)+ al1(CA i- C A)+a(C -C)H




fl1(C ACB) (2-28)

which follows from Eq. (2-24). Here again, H ic is the reactor activity factor for the reaction Path 1 calculated just prior to a change in the inlet temperature. Equation (2-28) can be solved for (T. i) with the aid of Ea. (2-27). It is noted that (T. i) should also be used in Ea. (2-27) in the calculation.

Consider now consecutive reactions, for which we have

1 2
A ) b 1B b 2C

With the same assumptions made for the parallel reactions, the conservation equations for an adiabatic reactor are



dC A -Th(z;t)k f (C C C
daz a a1 1 A'CB' Alzo A.i


(2-29)







18


dC -Th(z;t) {ak f2 (CA,CB) blk l(CA,CB)}
dz a22AC) a1flAC



C = CB (2-30)
z=0= Bin


T T in + a3(CA. CA)- a4(CB C Bin) (2-31)

in i


where a3 = (-AHI-bAH2)/pcp and a4=-AH2/PC If we follow the same procedures as for the parallel reactions, we have for the reactor activity factor:



H 1 hCAe dCA
hldZ = Tk(T)f1 (CA,CB) (2-32)

in

The relationship between CA and CB resulting from Eqs.

(2-29) and (2-30) is



dCB k f2 (CA" CB)
dCA bl+ka fl(CA,CB) ; CBCA = = CB (2-33)
-1 CA =A. in
Ain

The same procedures as for the parallel reactions can be

used to calculate H from Eq. (2-32)withtheaid of Eqs. (2-31) and (2-33).

The control policy for maintaining the desired outlet concentration of species B follows directly from Eq. (2-30) and the definition of H:







19


(CB)d dCB
HIC= T{ka2 [ (T in)n f2 (C A'CB)-blka [ (T in)n f (CA'C)}
in (2-34)


This can be solved for (T in)n with the aid of Eqs. (2-31) and (2-33). If, however, the control objective is to have maximum CB at the outlet, the choice of (CB)d in Eq. (2-34) must satisfy the necessary condition:


dCB = 0 (2-35)
dz z=l


Substituting Eq. (2-35) into Eq. (2-30) gives


k a(Te)f2[CA'(CB)d] = bK a (Te)f1[CAe (CB)d] (2-36)


which can be used to obtain (CB)d with the aid of Eqs. (2-31) and (2-33). The new inlet temperature, (Tin)n can then be obtained from Eq. (2-34).

The control policies of Eqs. (2-28) and (2-34) can be

modified using Eq. (2-16) to obtain the piecewise algorithms similar to Eqs. (2-17) and (2-18).
















CHAPTER 3

SIMULATION AND EXPERIMENTAL VERIFICATION

The theory in Chapter 2 is demonstrated here with a

simulation of a model reaction system. The theory is also put to the test with a laboratory shell-and-tube reactor in which ethylene oxidation reactions take place over a silver catalyst. The catalyst undergoes sintering under the reaction conditions.

3-1 Simulation of a Model Reaction System

For an illustration of the calculation of H and its

use for the reactor control, we consider the reaction system studied by Lee and Butt (1982a), which is summarized in Tables 3-1 and 3-2. For the reaction taking place in an adiabatic reactor, which is affected by diffusion and uniform, independent poisoning, the intrinsic rate of deactivation r p is given (Lee and Butt, 1982b) by


r p = k Pa (1-Y)N


where k Pa is the rate constant evaluated at the pellet sur,face temperature and N is the concentration of poisoning species. With the use of reactor point effectiveness (Lee and Butt, 1982a), the reactor behavior can be simulated in a straightforward manner. The simulation results are used here as the process output for the purpose of illustrating 20






21









Table 3-1 Equations for the Model Reaction System


dC TRGCz = Cn



dN --Tk N(l y);N = N
dz pa lz=0 in



dy 1 1kN(l -y); y
dt Q -P pa l~




weeRG 1 / 1.2 E (-All)


L2 De k(l-y) fJ g(C) dCJ 2h mR gT2




g(c) = C



T = Tin (-AH) (C Gin)


k = k 0exp(-E a/R 9T)


T s=T + (-AH) R GL/h


k pa=k Poexp (-E /R T )







22






Table 3-2 Parameters For The Model Reaction System


k = exp 12,000 + 14.6 (1/s); Ea = 23.76 kcal/mol





( = exp [3,600 + 3.86](cm3/mol)



C. = 1.81 x 10-5 mol/cm3
in



(-AH 7 3
S= 4 x 10 K cm /mol; -AH = 5.04 kcal/mol
pCP


-4 2
h = 3.245 x 104 cal/s.cm .K
m

-8 3
N. = 5 x-108 mol/cm
in

Q = 10-4 mol/cm3 cat. pellet



kp = exp 7000 + 3.27]; EP = 13.86 kcal/mol
p Tp

D = 10-3 cm2 /s
p

T = 1280K
max

X = 70% AX = 10%

T = 20s

T. = 691.8 K
in,o






23


the feedback control given by Eqs. (2-16) through (2-18). The control problem is how the inlet temperature should be manipulated whenever the outlet conversion decreases to a certain level from the desired conversion (bandwidth). Note that the intrinsic rate of deactivation r p is used here only to generate the process response. In the feedback control, r p is treated as an unknown and only the on-line measurements (simulated temperature and concentration) are used to manipulate the inlet temperature. The constraints are the maximum reactor temperature allowed and the final time at which the catalyst is regenerated as the time when the final bandwidth becomes one-tenth of the initial bandwidth. These constraints are also given in Table 3-2 along with the desired outlet concentration (conversion) and the reactor size. The initial inlet temperature for fresh catalyst is the one corresponding to the specified reactor size (T) and outlet conversion, which is 691.8K. Since the film heat transfer coefficient is quite small for the example problem and thus the value of q is relatively large, the proportionality factor a in Eq. (2-16) was set at

0.95.

The behavior of the model reactor resulting from the feedback control is shown in Figure 3-1. Initially, the inlet temperature is at its initial value and the outlet temperature, which is the maximum reactor temperature in










---- max



TT (YO
1200 800


(K)
750




ii
700 P







X
M% 65



551
o .3 .6 .9 1.2 1.5 1.8 2.1 2.4 2.7

txlO (sec) Figure 3-1. Temperature and Conversion Behavior of a Fixed-bed
Subject to Inlet Temperature Manipulation






25



this case, is well below the maximum temperature allowed. Due to the deactivation, however, both the conversion and the outlet temperature decrease with time and the conversion eventually reaches the lower bound given by the bandwidth, triggering an adjustment of the inlet temperature. The new inlet temperature (T in)l calculated by the algorithm of Eqs. (2-16) through (2-18) is seen to bring the conversion back to the desired level, the actual value being 69.96% as opposed to the desired value of 70%. A bisection method was used for the calculation of (T in) j* When the outlet temperature reaches the allowed maximum, which occurs at around t = 2xlO 4 s, the conversion can no longer be brought back to the desired level due to the temperature constraint. Consequently the bandwidth decreases as time increases and eventually the bandwidth becomes less than one-tenth of the original bandwidth, resulting in the reactor shut-down for catalyst regeneration according to the constraint imposed. In practice, however, the conversion may be allowed to decrease below 60% when the maximum temperature is reached. In the example, the average conversion is essentially maintained at the middle of the bandwidth, which is 65%. 3-2 Experimental Verification

We have shown the feasibility of control policies for the model reaction system. The theory has also been put to test using a shell-and-tube reactor. Experimental results obtained and comparisons with the theory are presented in this section.







26


3-2-1 Catalyst Preparation

The catalyst used in this study of ethylene oxidation was silver supported on fused alumina. The quarter-inchdiameter alumina obtained from Norton Chemical Company (Catalog No. SA-5202) was crushed into 8-9 mesh (0.200.24cm) with a rod miller, washed with distilled water, and then dried in the furnace at 373 Kovernight. These particles were porous with 0.7-1..3 m 2/g BET surface area and 1.9 g/cm bulk density. The alumina was impregnated with an aqueous solution of lactic acid and silver oxide at 368 K for 90 minutes. The solution contained silver oxide, 85% lactic acid and distilled water in the ratio of 1:2:2 by weight. After impregnation the excess liquid was removed and the particles were heated in a furnace for 16 hours at 643 K to decompose the silver lactate to metallic silver. The catalyst was further stabilized by oxygenating at 573 K for 100 hours. Since the color of these pellets was not uniform, only those with lighter color were picked for the experiment. The weight gain indicated that the catalyst contained 12.4% silver. The oxygen chemisorption at 423 K was 36 viZ/g.

3-2-2 Experimental Apparatus

The reactor used for ethylene oxidation reactions

was shell-and-tube type as shown in Fig. 3-2. The shelland-tube section was 60.1 cm long with outside diameter (o.d.) of 3.81 cm for the shell and 1.19 cm for the tube.









Silver Catalyst Pressure
Sh 11 Tube Pressure
Thermal Thermal Gauge
Thermal
Well A Coolant in Wel A
Thermocouples






Thermal Signals from Well B Te mocou les Well B le
Sample
Temperature Rotary Temperature Valve
Indicator Selector Controller/
Switch Circulatora
Gas
Chromatograph

I I

2Flow 4 Recorder
Controlle- 02 Integrator
e EtpCO2


Figure 3-2. Experimental Apparatus for Ethylene Oxidation Reaction






28


one thermal well (A, 0.794 cm o.d.) ran axially into the tube, another (B, 0.0318 cm o.d.) into the shell. Silver catalyst was packed into the annular section between the tube and thermal well A. The shell-and-tube consisted of three segments (from left to right in Fig. 3-2): preheating zone (20.3 cm), reaction zone (35.6 cm) and outlet zone (5.1 cm). Glycerin was used as coolant which circulated through the shell side using a Haake H1T22 temperature controller/circulator. A total of twenty temperatures were measured as shown in Fig. 3-2: eighteen in thermal well A and two in B.

All temperature points (thermocouples) were connected to a rotary selector switch which was in turn connected to a temperature indicator. The feed gases were introduced into the reactor at the desired mass flow rates and composition controlled by a multiple channel electronic mass flow controller. Two pressure gauges were installed, one at the inlet, the other at the outlet. The outlet composition was analyzed by a Tracor 550 gas chromatograph. The coolant flow rate was set high enough to maintain an essentially isothermal reaction zone. The reactor pressure was held constant at 1 atm.

3-2-3 Determination of Kinetics

Ethylene oxidation involves a triangle reaction network (Spath and Handel, 1974; Dettwiler et al., 1979)





29
1

C2H4 2 B4
(A) / (B)

/
/

2 3
CO2 + H2 0
(C)

The oxidation of ethylene oxide (reaction path 3) is significant only at temperatures above 473 K (Dettwiler et al., 1979; Akella, 1983). Since the experiments were conducted in the temperature range between 448 and 468 K, only epoxidation and combustion of ethylene (reaction paths

1 and 2) need to be considered. A material balance in an isothermal fixed-bed can be written as

dF A
d-- - h l' r + r2)W ; F A~ = = F A(3 1
d = 1 r1 2 Aj= Ao


dFB
dz 11 1 Bz= = FBo = 0 (3-2)


where F A and FB are the molar flow rates for ethylene and ethylene oxide, respectively, and W is the weight of the catalyst. Here r1 and r2 are the intrinsic rates for the reaction paths 1 and 2. The mechanism of ethylene oxidation is still controversial. Under the condition of constant oxygen partial pressure, however, the following LangmuiurHinshelwood rate expressions are reported by most investigators (Dettwiler et al., 1979):







30


r k lkPA (3-3)
1 = l+kqPA



r k 2 kqPA (mol/s.g-cat.) (3-4)
2 l+kqpA


where pA is the partial pressure of ethylene and kq is an adsorption equilibrium constant. Since the catalyst particles are small, there is no diffusion limitation and the intrinsic rates are essentially the same as the global rates.

Let x and y be the conversion and yield in the reactor, i.e. x = (FAo FA)/FAo and y = (FB FBo)/FAo* Then, Eq. (3-1) can be rendered dimensionless with the aid of Eqs. (3-3) and (3-4):



dx = W h1(k1+ k2) kqPYo(l-x) 0
dz Fo l+k qoyo(l-x)x z=0=


(3-5)
where F is the molar flow rate of reactant mixture and
o

is the mole fraction of ethylene, both at the inlet of the reactor. The ideal gas law has been applied to obtain Eq. (3-5). The reactor activity factor for the-reaction path 1, H1 can be obtained by rearranging Eq. (3-5) and integrating from the reactor inlet to the outlet:

= f k qp' 0 -Zn(l-X)
H1 hldz = PYx (3-6)

0 (k1+ k2)kqpW/F 0

where X is the conversion at the outlet.







31



Dividing Eq. (3-1) by (3-2) and nondimensionalization gives


dxr2 k 2
= 1 + = 1 + ; x = 0 (3-7)
dy 1 1 RI y=o


Since the right hand side of Eq. (3-7) is constant throughout the isothermal reactor, Eq. (3-7) can be integrated from the reactor inlet to the outlet to give

k
X = 1 + 2 2 (3-8)
Y 1

where Y is the yield at the outlet.

Equations (3-6) and (3-8), which are the integrated versionsof Eqs. (2-26) and (2-27), can be used for both the feedback control described in Section 2-3-3 and the determination of kinetics. Before proceeding to the control policies, consider the kinetics first. It is convenient to choose the activity of the catalyst at the time kinetic data are taken as the reference state activity. Then, both h1 and 8 are unity by definition, and Eqs. (3-6) and (3-8) can be rearranged as


[n T] =-kq [YoX]-+ kkqpWF (3-9)



and k2 = kl[ 1] (3-10)



It has been shown (Akella, 1983) that the selectivity, defined as S = Y/X, is fairly independent of ethylene







32

partial pressure but is dependent on the reaction temperature. This can also be seen from Ea. (3-8) where the left hand side is 1/S and the right hand side depends only on temperature. Thus, at any given temperature, X/Y is constant, and Eqs. (3-9) can be used to obtain k and k k q lq
by a linear regression of (Zn 1I) on (yAoX). The rate constant k2 can then be calculated from Eq. (3-10). By definition and the stoichiometric relationships, the conversion and yield at the reactor outlet can be related to the inlet and outlet gas compositions as


X = 1 l (3-11)
YAo


Y X -Y (3-12)
2YAo

where Y4o and YI are the mole fractions of ethylene at the inlet and outlet, respectively, y C is the mole fraction of carbon dioxide at the outlet.

We have shown that at a given temperature, the rate constants, k1 and k2, and the absorption equilibrium constant, kq, can be determined from the inlet and outlet gas compositions, Yo' yAI and yC, using Eqs. (3-9) through (3-12). The kinetic data for ethylene oxidation reactions are given in Table 3-3. The reactor was maintained isothermal by controlling the coolant flow rate at a high level. Due to the low ethylene concentration and the high coolant flow rate, the heat generated by the reactions was removed immediately by the coolant. The oxygen composition in the feed gas was maintain at 20% for all runs, with













Table 3-3 Kinetic Data for Ethylene Oxidation Reactions


RUN T(K) YoY(%) y'(%) y'(%) X(%) Y(%) S(%) S(%)

1 448 4.00 3.36 .487 16.04 9.95 62.0 62.4

2 448 6.00 5.28 .537 11.93 7.45 62.5

3 448 8.00 7.26 .551 9.23 5.79 62.7

4 458 4.00 3.10 .718 22.51 13.53 60.1 59.8

5 458 6.00 5.01 .788 16.55 9.98 60.3

6 458 8.00 6.94 .869 13.26 7.83 59.0

7 468 4.00 2.77 .971 30.79 18.65 60.6 59.7

8 468 6.00 4.60 1.12 23.31 14.00 60.1

9 468 8.00 6.52 1.23 18.48 10.81 58.5

F = 100 ml/min
0
P = 1 atm

W = 78.34 g







34



helium as the diluent. Given in Table 3-4 are the results of the linear regression of (Zn 1 X)o yO)frec

temperature. The values of (Zn 1 X)calculated from the regression model, (2n 1pXi) cal' are included in the table for comparison. The percent error in the table was calculated from


Ero n11X1x 100% (3-13)
Errr = (Zn 1
1XCal

The constants k 1and k were calculated from the slope and intercept of the regression line, while k 2 was calculated from Eq. (3-10). The values of these rate constants are given in Table 3-4 along with the coefficients of correlation. Note that the average selectivity, S, given in the last column of Table 3-3 was used to calculate k 1and k q

It was assumed that the temperature dependence of the rate and equilibrium constants are all of the Arrhennius form:

k A A. exp(- E.i/R 9T) i=l, 2, q (3-14)


The activation energies and pre-exponential factors were obtained from linear regressions of Zn k.i on l/T. The results are given in Table 3-5.

The experimental results show that the kinetics of ethylene oxidations reactions can be well represented in the form of Eqs. (3-3) and (3-4) with the following rate constants and equilibrium constant:












Table 3-4 Results of Linear Regression of n- YAoX


RUN T YX zn-_i- Pn11 Error k k k Corr.
IU xTX cal 1-9 -9 q
(K) (x10-3 (xl0-1 (xl0 (%) (xl0 (xl0 (xl0) Coeff.

1 448 6.417 1.749 1.766 -1.0 4.797 2.890 7.675 -0.9863

2 448 7.156 1.270 1.198 6.0

3 448 7.384 0.968 1.023 -5.3

4 458 9.005 2.551 2.526 1.0 6.631 4.453 7.089 -0.9960
uL
5 458 9.932 1.810 1.869 -3.2

6 458 10.61 1.423 1.388 2.5

7 468 12.32 3.680 3.697 -0.4 9.457 6.378 6.556 -0.9985

8 468 13.99 2.654 2.603 2.0

9 468 14.79 2.043 2.078 -1.7












Table 3-5 Results of Linear Regression of in k. on lI/T
1


i T l/T Pn k. (kn k.) Error A. E.i/R Corr.
1c 1 1 g
(x10 ) (xl0) (x10) (%) Coeff.

1 448 2.232 -1.196 -1.916 -0.0 3.618x10-2 7.098x103 -0.9991

458 2.183 -1.883 -1.882 0.1

468 2.137 -1.848 -1.849 -0.1
-l 3
2 448 2.232 -1.966 -1.967 -0.1 3.038x10- 8.279x103 -0.9980

458 2.183 -1.923 -1.920 0.2

468 2.137 -1.887 -1.888 -0.1
3
q 448 2.232 0.4340 0.4340 0.0 1.942 -1.647x10 -0.9998

458 2.183 0.4261 0.4260 0.0

468 2.137 0.4183 0.4183 -0.0


calculated from regression model







37


k= 3.618 x 10-2 exp(-7098/T) (mol/s.g-cat)


k2 = 3.038 x 10-1 exp(-8279/T) (mol/s.g-cat)


k = 1.942 exp (1641/T) (1/atm) (3-15)
q

0.04 atm < P Ao 0.08 atm


448 K < T < 468 K

The comparisons made in Table 3-6 between the experimental conversions and those predicted by the kinetic model show very good agreements, the maximum error being 1.9%. The same can be concluded for the yield and selectivity. It is seen from the table that the maximum errors are 2.3 and 3.4% respectively for the yield and selectivity.

The experiments for the kinetic data lasted 9 hours. An additional run carried out right after the period with the same conditions as in Run No. 2 gave almost the same conversion and yield as shown in Fig. 3-3 at t = o and t = 9 hr. This is a good indication that no significant deactivation had occurred during the kinetic experiments. 3-2-4 Experimental Verification of the Control Policy

The shell-and-tube reactor at the end of the kinetic experiments was allowed to run continuously to verify the feedback control policy presented in Chapter 2. The feed rate was controlled at 100 ml/min with the ethylene, oxygen and helium concentrations of 6, 20 and 74%, respectively,











Table 3-6 Comparison Between Experimental Data and Kinetic model RUN X t X ca* Error y Yca Error S s cal Error

1 16.04 15.94 0.6 9.95 9.96 -0.0 62.0 62.4 -0.6

2 11.93 11.70 1.9 7.45 7.30 2.0 62.5 62.4 0.0

3 9.23 9.22 0.1 5.79 5.76 0.5 62.7 62.4 0.4

4 22.51 22.30 0.9 13.53 13.62 -0.6 60.1 61.1 -1.6

5 16.55 16.55 0.0 9.98 10.11 -1.2 60.3 61.1 -1.2

6 13.26 13.12 1.1 7.83 8.01 -2.3 59.0 61.1 -3.4 0

7 30.79 30.53 0.8 18.65 18.25 2.2 60.6 59.8 1.4

8 23.31 22.98 1.4 14.00 13.73 2.0 60.1 59.8 0.5

9 '18.48 18.34 0.8 10.81 10.96 -1.4 58.5 59.8 -2.1

tExperimental data


Kinetic model

Error = (Xix ca-1)-100%









20 80


-72


16 -64


56
or







8





4 466 K





0 100 200 300 400 500 600 700 800 900
t(hr) Figure 3-3. Conversion, Yield and Selectivity Behavior for Ethylene Oxidation Reaction






40

The temperature was kept constant at 448 K. The outlet gas compositions were analyzed daily and the corresponding conversion and yield were calculated. The control policy was that of maintaining the yield of ethylene oxide at 8% in a piecewise manner. The bandwidth was set in such a way that an adjustment was made whenever the yield decreased to 6.5%. The temperature adjustments and the corresponding reactor behavior over a period of 35 days are shown in Fig. 3-3. The first temperature adjustment occurred at t = 120 hr. at which time the yield dropped down to 6.5%. This triggered an adjustment of the temperature in accordance with the specification of the bandwidth;

Consider the temperature adjustment, in particular the procedures of calculating the new temperature which will bring the yield back to the desired level. The value of required to calculate H 1 from Eq. (3-6) is obtained directly from the experimental values of the conversion and yield at the current temperature. This in turn is used in Eq. (3-6) to calculate the current reactor activity factor for the reaction path 1, H lC* Rewriting Eq. (3-6) for the new temperature and using the relationship H lc = H In yields

k Py' X kn(l X
H q Ao d d
lc (k 1 + 6k 2 ) k q PW/F 0 (3-16)





41



Here, the subscript n denotes the new temperature and X d is the desired conversion. Since local activities change little during the short period of temperature adjustment, by definition, also changes little. This means that the same obtained from Eq. (3-8) can be used in Eq. (3-16) to calculate the new temperature. If the conversion X d is the one to be controlled, then Eq. (3-18) can be used to calculate T n since it is the only unknown in Eq. (3-18). In this case, however, the yield Y d is to be controlled. Hence, we need an additional relationship between X d and Y d* This relationship can be obtained by rewriting Eq. (3-8) for the new temperature



d + (k 2] (3-17)
Y d k 1
n

This relationship was used for the calculation of the new temperature in the experiment.

The experiments were conducted over a period of 884 hours. As shown in Fig. 3-3, the first adjustment was made at t = 120 hr. The temperature was raised to 454 K, which brought the vield to 7.95% for the desired yield of 8%. Two additional temperature adjustments were made at t = 336 hr.and t = 624 hr. The experimental values of the yield of ethylene oxide resulting from the temperature adjustments were both 7.92%. The behavior of conversion and selectivity are also shown in Fig. 3-3. It is interesting to observe that







42


the selectivity increases at 448 K, remains almost constant at 454 K and then decreases at 460 and 466 K. This behavior can be explained in terms of H 1 and H 2* As shown in Fig.

3-4 H1 nd H decrease with different speeds, resulting in the observed selectivity behavior. The results given in the figure show excellent agreements between the theory and the experiment.

The experimentally determined values of H 1and H were fitted to an nth order deactivation kinetics using a nonlinear regression. The results are


dH 1 kD e1 =508x 10 5exp (-8879/T)H 38


dH -8 6 191

=H k~ I2 =-1.936 x 10 -8exp (5707lT)H 6.1



The experimental values are plotted in Fig. 3-4 along with the calculated values, which are shown as solid curves. 3-3 Concluding Remarks

We have shown in Section 3-1 the feasibility of the

feedback control policies by simulating an adiabatic fixedbed where a single reaction affected by diffusion and poisoning takes place. The feedback control policies have also been put to test with a laboratory isothermal fixed-bed where multiple reactions affected by sintering take place.










1.2






H 1

0.8 A



H2




0.4


0 D3 Aexperimental data

0.2
regression model




0 100 200 300 400 500 600 700 800 900

t(hr) Figure 3-4. Activity Behavior for Ethylene Oxidation Reaction







44


The results in Fig. 3-1 and 3-3 show that one can indeed maintain the conversion or yield at the desired level with good accuracy. It is noted in this regard that the control policies can also be used to control the conversion or yield in any desired manner, since X d or Yd can be changed each time the temperature is raised.

The accuracy of the feedback control depends mostly on

the on-line measurements and Y) but slightly on the referencestate kinetics determined. For example, at t = 624 hr., (as shown in Fig. 3-3 when T is adjusted from 460 to 466 K), 5% error in the determination of yield results in 1.7K difference in the new temperature calculated, while 5% error in activation energy for reaction path 1 (E 1 ) leads to only 0.3 K difference. In calculating new temperature by use of Ea. (3-16), the on-line measurements of conversion and vield are used only on the left hand side of the equation, while the activation energy E 1 is used on both sides. Thus, any error in E 1 is likely to be cancelled out from both sides of the equation. This is a good indication that the feedback control is applicable even when some uncertainty exists in the kinetic model.















CHAPTER 4

OPTIMAL PIECEWISE CONTROL

The control policies discussed in the previous

chapters are all piecewise type, i.e. the control actions are taken at discrete intervals. The optimal control

problems for reactors affected by catalyst deactivation have been the subject of several studies (Chou et al., 1967; Szepe and Levenspiel, 1968; Crowe, 1970), and a constant conversion policy has been proved to be optimal for a single irreversible reaction with concentration-independent deactivation. These studies deal only with the continuous control, which is essentially the piecewise control with an infinite number of steps. However, the feasibility of extending the optimal continuous control policies to piecewise control schemes with a finite number of steps remains questionable. In this chapter, a solution method for the optimal piecewise control is developed and a comparison between continuous and piecewise control is made.

4-1 Problem Statement and Solution Method

Consider an isothermal fixed-bed where a single irreversible reaction affected by catalyst deactivation takes place.. The temperature is to be adjusted in steps from T 1to




45






46


Tn as shown in Fig. 4-1. The pseudo steady-state balance equations can be written as

dh =i
dt g(Tl, T2''..T nh) ; h t= 1 (4-1)



X = f(TI, T2,...T n, h) (4-2)


where h is the activity of the catalyst, X is the conversion at the outlet of the reactor, and g represents the dependence of deactivation rate on the temperatures (T) and activity (h). Equation (4-2) is obtained by integrating the mass balance equation from the inlet to the outlet of the reactor.

The optimization problem is to maximize the total

amount of feed converted over a fixed total reaction time, tn by choosing the best set of temperatures, [TI, T2,...,Tn]. That is

J = max J i = 1, 2,...n (4-3)
T.


with
n t.
J Xdt = Xdt (4-4)
J tn t ti_1


and subject to

Tm n< T. T
Tmn T max


t
At = ti ti_ constant --I!n i = 1, 2, ..n
n (4-5)











T
max


T
n





T Ti


Ti1


T2 T


Tmin




0 t 1 t 2 t i t n

Figure 4-1. Temperature Behavior for a Piecewise Control







48


Equation (4-1) can be solved, analytically or numerically, to give

h = h(T1, T21 ... T n, t) (4-6)

Integrating Eq. (4-4) with the aid of Eqs. (4-2) and (4-6) yields

J = J(T1,' T 20" Tn) (4-7)

Thus, the optimal piecewise temperature control

problem of Eqs. (4-3) through (4-5) is a multivariable optimization problem searching for n variables (TV, T 2.. T n), rather than a variational calculus problem searching for a continuous function (T(t)). The latter is often encountered in problems dealing with optimal continuous control. The stationary point of J can be obtained either by applying the necessary conditions



3T. = 0 1 = 1, 2,...n (4-8)

and then solving these n nonlinear equations numerically, or by direct numerical search (Beveridge and Schechter, 1970). It is worth noting that even for the simplest first order reaction and deactivation, the expression for J is still very complex and the partial derivatives in Eq. (4-8) are difficult to obtain analytically. Therefore, direct numerical search is a better method for obtaining the optimal temperature lc, T T T Th

optimal temperature policy leads directly to the optimal conversion policy through Eq. (4-2) with the aid of-Eq. (4-1).






49


4-2 Optimal Piecewise Control Policies

As an illustration, the optimal piecewise control

problem of Eqs. (4-1) through (4-5) was solved for a model reaction system of first order irreversible reaction with first order deactivation (Table 4-1). The multivariable optimization problem was solved by direct numerical search using the ZXMWD subroutine from International Mathematical and Statistical Libraries (IMSL). The ZXMWD subroutine is used for obtaining the global minimum or maximum for a multivariable function with constraints. The constrained maximization problem of

J = max J(TI, T2,...Tn) (4-9)
T 1 ...T n


with

Tmin< T.< T i1, 2,...n
-- 1 max

is transformed in ZXMWD subroutine to an equivalent, but unconstrained problem

Si2-l
J = max J[T min-+T )Sin T,...,T. +
n,...(max- mi m


(Tmax T min)Sin2Tn] (4-10)


where Ti is the transformed function of Tiand is now unconstrained. With this transformation, each possible global maximum, including any on the boundary, is transformed into a local maximum (Box, 1966).

Shown in Figure 4-2 are the optimal piecewise control policy for the model reaction. The optimal temperature






50







Table 4-1 Equations and Parameters for a Model Reaction
System for Optimal Piecewise Control



dh k h hlt=0= 1
dt D t


X = 1 exp(-Tk1h)



kD AD ex T~]



k1 A1 Tx(~i

t t
T(t) = Ti for (i-l)- t < i- nn i = 1,2,...n
1nn ; i =12 ..


Tmin _L Ti < Tmax i = 1,2,...n



AD = 5.00 x 106 hr- 1 = 5.00 x 106 S-1

GD = 1.25 x 104 K G = 1.00 x 104 K

t = 2000 hr T =20 S
n


T = 473 K
min
Tmax = 573 K in Figs. (4-2), (4-3) and (4-5)

= 673 K in Figs. (4-4), (4-6) and (4-7) n = 10 in Figs. (4-2) through (4-6)

= varies in Fig. (4-7)









1.0 630





.8 4-h 610





h .6 590

or .595 (--X T
orm595x= 573 K (K) Ln
x~~~a H

X .4 570





.2 550




0 .I 530
0 400 800 1200 1600 2000
t(hr)

Figure 4-2. Optimal Piecewise Control Policies for the Model Reaction,
T = 573 K







52

policy is to step the temperature up from 539.9 K until it reaches T max (573 K) at t = 1400 hr. and then to keep it at that level as shown in the figure. The corresponding conversion zigzags with increasing bandwidth until the temperature reaches its upper limit. Due to catalyst deactivation the conversion decreases in each time interval while the temperature remains constant. It is of interest to note that each temperature increase gives a conversion Peak which is slightly higher than the previous one, i.e.

X n (t i ) > X n (t i-l ) for T < T max

where n denotes the new conversion immediately following the temperature adjustment. In other words, the optimal piecewise policy is an increasing conversion policy, rather than a constant conversion policy which is the optimal for continuous control (Crowe, 1970). The optimal continuous control Policy for the same model reaction is shown in Fig. 4-3 for comparison. It is obtained by one-dimensional search of the optimal initial temperature with the aid of the constant conversion policy. It can be seen from the figure that the optimal continuous policy is a constant conversion policy followed by a constant temperature policy at T max* Comparison between Fig. 4-2 and 4-3 shows that the temperature in the continuous control starts 1.3 K lower than in the piecewise scheme and reaches the upper limit 42 hours later. It can also be seen that the conversion in the continuous control is lower than the original conversion in the uiecewise control, but stays in the zigzag zone of the piecewise scheme. The corresponding activity behavior is also plotted in Figs. 4-2 and 4-3.









1.0 630





.8 - 610





h .6 -590

T
or .579 (K) L

X max
.4 570





.2 T- -.550





0 1 I530
0 400 800 1200 1442 1600 2000
t(hr)
Figure 4-3. Optimal Continuous Control Policy for the Model Reaction,
T=573 K






54


Another major difference between the piecewise and the continuous control is that although the temperature in the optimal continuous control always ends at its upper limit for the model reaction studied here (Crowe, 1970), the temperature in the optimal piecewise control may end at a temperature below the upper limit. This is shown in Fig. 4-4 where the final temperature is 621.3 K, far below the upper limit, 673 K. However, the final temperature will approach 673 K as the number of control steps, n, increases.

A comparison of performance indices calculated from

Eq. (4-4) for seven different control policies is given in Table 4-2 with T 573 K and 673 K. The Performance inmax
dices calculated with the optimal piecewise temperature control Policy, T, are listed in Entry 1. In Entries 2 and 3, the temperature control policies are 1 K higher and lower, respectively, than T but still within the interval [T min' T max The control Policy T c in Entry 4 is obtained by solving the same optimization problem with an additional constraint of constant conversion. Note that by applying the constant-conversion constraint, the multivariable optimization problem of Eq. (4-9) is simplified to a single variable optimization problem

J max J (T 1) (4-11)
T 1

since T 2$' T 3 ... and T n are fixed when T 1 is specified. Here subscript c denotes the performance index obtained under the constant-conversion constraint. The performance








1.0 630

621.3 K
T max= 673 K
-h ma


--xh
.8 I 610





h .6 590

or .587 T
(K)
X !

.4 570


.2 T

.2 550

539.3 Ky-


0 I I I I I I 530
0 400 800 1200 1600 2000
t (hr)

Figure 4-4. Optimal Piecewise Control Policy for the Model Reaction, T = 673 K
max












Table 4-2 Comparison of Performance Indices for Different Control Policies



Entry Temperature Tmax 53KTmax=67K
Control Policy J. j


1 T = T 0.53512 100.00 0.55855 100.00
2 =T 105398 999705537 999
2 T = T + 1 0.53498 99.97 0.55837 99.97


4 T = T t 0.53503 99.98 0.55628 99.59
c
5 T = T*tt 0.49219 91.98 0.49219 88.12
t
6 T = T max 0.44007 82.24 0.04823 8.63

7 T = T mi 0.06268 11.71 0.06268 11.22



/1 = 1 / x 100%

with constant-conversion constraint


ttwith constant-temperature constraint







57

indices obtained from Eq. (4-11) are listed in Entry 4. In Entries 5 through 7, the temperature policies are all constant with respect to time. It can be seen from the table that T (the optimal temperature policy shown in Fig. 4-2) is indeed the optimal temperature control policy, since all the other policies result in smaller values of the performance index.

The single-variable optimization problem of Eq. (4-11) was solved numerically using Fibonacci search (Beveridge and Schechter, 1970). The results are shown in Fig. 4-5 for T max 573Kand in Fig. 4-6 for T max = 673 K. It can be seen from the figures that the conversion is brought back to the initial conversion each time the temperature is adjusted.

94-3 Concluding Remarks

A comparison between the optimal performance indices

obtained with and without constant-conversion constraint is shown in Fig. 4-7 as a function of the number of control actions (n). It is seen that as n approaches infinity, J and i c approach the same asymptotic value (0.56556 for the model reaction with T max = 673 K). This indicates that the continuous control (which corresponds to an infinite number of piecewise control actions) is superior to the piecewise control in terms of the performance index. On the other hand, j*> 99.5% when n>21 as shown in Fig. 4-7, implying that the piecewise control is almost as good as the continuous control if the number of control actions is large









1.0 630




.8 h 610


.605


.6\ -590

or 4--X T
(K) u,
T =573 K (K)
co
X max
.4 570



T J.
.2 550

540.-7 K _-0 1 a 530
0 400 800 1200 1600 2000
t (hr)

Figure 4-5. Optimal Piecewise Control Policy with Constant Conversion, T = 573 K
max









1.0 6330
1T = 673K

~max
-h 612.5 K I
B [ 1h
.8 -110


.617 -x

h .6 \590 or T
or (K)


X .4 -570

TF
-~ I-.2 r 550
---


I I I 530
0 400 800 1200 1600 2000
t(hr)

Figure 4-6. Optimal Piecewise Control Policy with Constant Conversion,
T =673 K
max






4UTPa4SUOD UOTSJ9AUOD-qUPqSUOD qnoq4TM PuP q4TM pauTp4qo S90TPUI GOUPUlaOja9d u amqaa uosTapdwoD -L-V eanbTa
u

OOT 08 09 ot, oz 0
1 1 1 1 1 1 1 1 S6

gssgs-o ulor w=ul r

co--U
P 96
%001 x



co=U
r
%OOT X -* L6
r



Xlew EL9
86



10

66





OOT






61


enough. In practice, the temperature adjustments are made intermittently because of the slow nature of catalyst deactivation in fixed-beds. From the fact that the curve for j* lies above the curve for it can be concluded
c
that the optimal piecewise control Policy is an increasing conversion policy as shown in Fig. 4-2 and 4-4. However, the constant-conversion policy can be used when number of control steps is large, say, 30.

The problem formulation and solution method presented in Section 4-1 is neither restricted to an isothermal reactor nor to a single irreversible reaction. For example, one can consider a reversible reaction taking place in an adiabatic fixed-bed. In this case, the reactor temperatures, Vs, in Eqs. (4-1) through (4-5) should be replaced by the temperatures at the inlet of the reactor, (T in) i S. Equation (4-2) might then become implicit in X because of the reversible reaction and the nonisothermal reactor. However, X can still be solved numerically. Thus, the problem remains a multivariable optimization problem which can be solved by application of an appropriate numerical method.














CHAPTER 5

OPTIMAL CONTINUOUS CONTROL FOR COMPLEX REACTION

In the experiment described in Section 3-2, we controlled the yield of ethylene oxide at a constant level in a piecewise manner, and verified the feasibility of the feedback control policy presented in Chapter 2.

However, the constant yield policy used is not an

optimal policy. It is the purpose of this chapter to obtain an optimal control policy for complex reactions which are parallel with the Langmiur-Hinshelwood rate expressions. We consider here only the optimal continuous control policy since we have shown in the previous chapter that one can use the optimal continuous control policy for piecewise control as long as the number of control steps is large enough. 5-1 Problem Statement

Consider a parallel reaction network with the LangmiurHinshelwood rate expressions for an isothermal fixed-bed such as the ethylene oxidation reaction described in Section 3-2-3. Assume the same active sites are responsible for both reactions, i.e., h1 = h2 = h and = 1. Equation (3-5) and (3-7)

can be rewritten as

dx Th(K1 + K2) (l-x) 0
dz 1 + K q(l-x) Xz=0 (5-1)

and
dx 1+ 2 2 x 0 (5-2)
dy K1 xy=o


62






63



where
T = WP/Fo= k k and Kq= P k 1 1 k~q 12= k2kq q Ao q

The rate of decay of the catalyst activity can be expressed as

dh Kh m h =1 (5-3)
dt D zI

where KD is the deactivation rate constant. Integrating Eqs. (5-1) and (5-2) from the inlet to the outlet of the reactor yields

K qX n(l-X) = Th(K1 + K2) (5-4)

K 1
Y X (5-5)
K1 +K2

where X and Y are the conversion of ethylene and the yield of ethylene oxide at the outlet of the reactor, respectively.

The optimization problem is to maximize the total amount of yield of ethylene oxide over a fixed total reaction time, tf, by choosing the optimal temperature-time policy. We assume that the temperature dependence of the rate constants and adsorption equilirbium constant are all of the Arrhennius form:

Ki = Ai exp(-Ei/R T) i = 1, 2, q, D (5-6)

Since K is a strictly monotonic function of temperature,
D
we can use KD as the control variable to replace T. The constants KIl, K2 and Kq can be expressed in terms of KD






64


P.
K. = K3 j= 1,2,q (5-7)
j jD
P.
where a = Aj/DQ and Pj = Ej/Eo


Thus, our optimization problem is


J = max Ydt =max Kr f K2 Xdt (5-8)
K DWt I0~ K D(t) J K1+K2


subject to two local restrictions, Eqs. (5-3) and (5-4). This is a fundamental problem in calculus of variations. The derivation of the necessary conditions for J to be maximum is presented in the next section. 5-2 Necessary Condition

Equation (5-3) indicates that

K = KD(h, h) (5-9)

where
dh
dt

Since KI, K2 and Kq are all functions of KD, Eqs. (5-4) and (5-5) give

Y = Y(KD, h) = Y(h, h) (5-10)

Thus, the optimization problem of Eq. (5-8) is an elementary problem in the calculus of variations, and the necessary condition for the optimization is an Euler-Lagrange equation (Beveridge and Schechter, 1970):


Y h = C (5-11)

where



and C is an integration constant.






65


In order to use Eq. (5-11), hY has to be obtained first. Differentiating Eq. (5-7) w.r.t. h yields



3K. ~dK. 3K ~ p.K._ = (dK D iy 1i = 1, 2, q (5-12)
dK K D m



Multiplying h gives


3K* 1 1 -1 6 K
S1 KD ; = piKi (5-13)


Differentiating Eq. (5-4) w.r.t. h, multiplying & and using Eq. (5-13) yields

p K X + K (h X) + 1 (h X) = 3T(K + K2)h (5-14)
q q q Xh) +--- ( X P3T



where p3 = (p1K1 + p2K2)/(K1 + K2)



Rearranging Eq. (5-14) and using Eq. (5-4) for T(K + K2)h we have


x (P3- p )K X p3 n(l-X)
x = 1 (5-15)
K +
q l-X


Differentiating Eq. (5-5) w.r.t. A, multiplying h and using Eq. (5-13) we obtain






66


K K KI
1 1 3 1 AK
Y= Pl K1 X K + K1 + K2h X-p3K +K

(5-16)

Substituting Eq. (5-16) into (5-11) with the aid of Eq. (5-15) gives


K1 -1 fp )KX + P3kn(l-X)+(K + 1
K1 + K2 K + 1 (Pq- P3Kq 3q
q 1-X

(l-p1 + P3)X} = C (5-17)


Equation (5-17) is the necessary condition for J to be maximum. The integration constant C can be determined by a boundary condition:

h = hf at t = tf (5-18)


if hf is specified, or by a natural boundary condition (Denn, 1969):


Yi = 0 at t = tf (5-19)


if hf is unspecified.

The necessary condition can be simplified if the activation energies for both reactions are equal (E1 = E2), and the adsorption equilibrium constant K is temperature
q
independent (E = 0). In this case, Eq. (5-17) gives



A1 1 {PlAqX + pln(lX)+(A + 1x )X} = C

1 +2 (Aq + (5-20)







67


where A i is the preexponential factor. Since A 1 A 2 Aq, p, and C are all constants, Eq. (5-20) indicates that X = constant (5-21)

In general cases, however, the conversion is not constant. The optimal control policies have to be obtained by numerically solving the necessary condition, Eq. (5-17), along with the local constraints, Eqs. (5-3) and (5-4). 5-3 Optimal Continuous Control Policies

The predictor-corrector method (Hornbeck, 1975) initiated by fourth order Runge Kutta method was used to solve the ordinary differential Eq. (5-3) while Newton-Raphson iteration (Carnahan et al., 1969) was used for the nonlinear Eqs. (5-4) and (5-17). The kinetic data obtained in Section 3-2-3 and the same parameters used in the experiment were used for obtaining the optimal control policies. Table 5-1 lists these parameters along with the deactivation kinetics used. The boundary condition of Eq. (5-18) was used to obtain the integration constant in Eq. (5-17). This split boundary condition problem was solved by "shooting method" (Hornbeck, 1975). The results are shown in Figs. 5-1 and 5-2.

It can be seen from Fig. 5-1 that the optimal control policy is one of increasing conversion and yield. However, with different deactivation kinetics, the optimal policy becomes one of decreasing conversion and yield, as shown






68







Table 5-1 Parameters Used for the Optimal Continuous Control Policies

K = k k = 7.028 x 10-2 exp(-5451/T)(mol/s*g-cat.atm) K1 =ilk

-1
K2 = k2k = 5.900 x 10 exp(-6632/T)(mol/s*g-cat*atm)

-1
Kq = PAokq = 1.165 x 10 exp(1647/T) (-)


WP 6
T p = 1.134 x 10 (g-cat-atm-s/mol)
F
0
h = 0.4 (-)

t = 1000 (hr)

KD 4.0 x 102 exp(-6000/T) (1/hr) in Fig. 5-1
KD = 4.0 x 10 exp(-6000/T) (1/hr) in Fig. 5-1
3
4.0 x 10 exp(-7000/T) (i/hr) in Fig. 5-2

E1
pl =E = 0.9085 in Fig. 5-1
D Fig


= 0.7787 in Fig. 5-2

E2
p2 E- = 1.105 in Fig. 5-1
2 ED

= 0.9474 in Fig. 5-2

m= 1.0

E1/E2 = 0.822








.20



h/5 KD= 400 exp (-6000/T)
.16 Pl = 0.779T

h/5 P2 = 0.947 (K)



X .12 500


or
aN
Y .08 480




.04
.04 T -460





0 I I I I I I I 440
0 200 400 600 800 1000
t(hr)
Figure 5-1. Optimal Control Policies for a Parallel Reaction Network
with the Langmiur-HIiinshelwood Rate Expressions









.20


h/5 KD= 4000 exp (-7000/T)

.6 P1 = 0.909
1.10 T
T
h/5 P2 = (K)



.12 500

or


.08 do 480





.04 T- 460




0 440
0 200 400 600 800 1000
t (hr)
Figure 5-2. Optimal Control Policies for a Parallel Reaction Network with the Langmiur-Hinshelwood Rate Expressions






71


in Fig. 5-2. This decrease in conversion and yield is due to the large activation energy for deactivation which limits the increase in temperature.

The valuesof the performance index for constant-conversion and constant-yield policies have been calculated for comparison. The results indicate that the control policies shown in Figs. 5-1 and 5-2 are indeed the optimal Policies. However, the percent difference between the optimal policy and the constant-conversion policy is less than 0.1% in terms of the performance index. As mentioned earlier, the constant-conversion policy is the optimal policy when E 1 =E2 and E q= 0. For the model reaction studied, E 1 is close to E 2 (E 1/E 2 = 0.822) and E qis small (compared with E 1 and E 2). This explains why the constant-conversion policy is almost as good as the optimal policies shown in Figs. 5-1 and 5-2.

5-4 Concluding Remarks

We have derived the necessary condition for the optimal continuous control for a parallel reaction network with Langmiur-Hinshelwood rate expressions by direct substitution. The optimal control policies have been obtained by direct numerical integration of Eq. (5-3) with the aid of Eqs. (5-4) and (5-17). Although iterations are required for solving Eqs. (5-4) and (5-17), it takes no more than two iterations for each step in the t-direction because X and K D are smooth functions of t and the values in the previous step provide very good initial guesses for the current step. In fact, it







72


took less than 3 seconds CPU time of IBM 360/370 computer to generate all the data shown in Fig. 5-1, including 5 iterations for shooting the boundary condition, h = hf at t = tf. This efficiency can never be achieved if the Pontryagin maximum principle (Pontryagin et al., 1962) and the Hamiltonian are used (Ogunye and Ray, 1971).















CHAPTER 6

OPTIMAL CONTROL AND DESIGN

We address in this chapter the problem of how a fixedbed with catalyst deactivation should be designed to obtain the best possible performance when it is controlled optimally. In particular, we ask what the size of the reactor should be that maximizes a certain performance index when the temperature is also manipulated to maximize the performance index. This is a design practice in which process control is taken into consideration. In order to clearly demonstrate the interrelationshio between design and control and the advantage that can be gained by utilizing this interrelationship, we assume for the time being that the detailed knowledge of deactivation is available. 6-1 An Optimization Problem of Piecewise Control and Design

Consider an optimization problem of a single irreversible reaction taking place in an adiabatic reactor given by

J = max Jmax 1JfXdt
Tin ()Tin () f0 (6-1)


subject to the system equations of Eqs. (2-1) and (2-2) with (inC P)c = 0 for the adiabatic reactor under consideration




73







7 4,'


and subject to T. in (T. i)m with a fixed t f* The

problem is to maximize the performance index J, which is the conversion (X) integrated over the catalyst life t f per unit reactor volume and unit on-stream time, by properly choosing T and T.i (t). This two-parameter optimization problem can be solved one at a time by first searching for the maximum for a given T and then finding the value of T that yields the maximum. For a fixed T, it is a Bolza's problem (Bliss, 1961) and the extremum condition obtained by Chou et al. (1967) and Ogunye and Ray (1971) is to maintain the conversion at a constant level until the maximum allowed temperature is reached and then to remain at the maximum if only one rate constant is involved. For the deactivation problem under consideration the policy is not directly applicable since the rate constant cannot be separated out. However, the constant conversion policy is still a suboptimum which is often used in practical operation of chemical plants. Thus, we apply the constant conversion policy for the problem considered. Since the feedback control policy obtained earlier is for maintaining a constant conversion level, the problem is simply that of finding an initial value of T.i (or the desired conversion level) if we use the control policy, for T.i can be set at (T. i)m once (T. i)m is reached.







75

The optimization problem was solved for the reaction system in Tables 3-1 and 3-2 using the control algorithms of Eqs. (2-16) through (2-18) with the constraints of t f 10 5 sec. and (T in)max = 720 K. The typical behavior of T in W and the outlet conversion is shown in Fig. 6-1 for the bandwidth AX of 5% and the initial inlet temperature of 691 K (or conversion of 87.2%). In accordance with the constraint on t f and the extremum condition, T in stays at the maximum allowed temperature once it reaches the maximum and the conversion decreases with time, eventually reaching a conversion of 54.5% at t = tfl, at which time the catalyst is regenerated. As indicated earlier, the optimization problem for a chosen T is that of finding an initial conversion (or initial inlet temperature) that yields a maximum. The solutions obtained for various T are shown in Fig. 6-2 for a bandwidth of 0.05. A few observations can be made from the figure. The results show that for a given T, the extremum. condition does yield a maximum. Furthermore, the value of the performance index is much more sensitive to a choice of the initial inlet temperature (or the initial'conversion) at relatively lower values of the inlet temperature than at higher values. The locus of the maximum values of J for various T is given in Fig. 6-2 as the dotted line. As apparent from the figure, there exists a maximum at around T = 22 sec. The performance index normalized with respect to the index corresponding











720



710 T 2 e


(Tinax 2



690 90 80

x

M% 70



60



0 .2 .4 .6 .8 1.0

txlO- (sec) Figure 6-1. Inlet Temperature and Conversion Behavior for an Optimal
Control with Bandwidth of 0.05






77






T (sec) 28 26 24 22 20 18 16
3.6




3.4





3.2



T=

3.0 24 22 20 18





2.8





2.6




2.4
670 680 690 700 710 720

T ino (K) Figure 6-2. Performance Index as a Function of
Reactor Size and Initial Inlet
Temperature. The dotted line shows
the locus of J*.






78


to T 22 sec., when the reactor is operated optimally,

is shown in Fig. 6-3 as a function of the reactor size T. It is seen that the performance index [Eq. (6-1)] doubles (T of 8 sec. VS. T of 22 sec.) when the reactor is sized optimally. This result confirms the intuitive reasoning that the best possible reactor performance obtainable by process control is inherently dictated by process design. It also confirms the fact that a significant improvement can be made by combining process design with process control.

6-2 An Optimization Problem of Continuous Control and
Design

In this section, we give an example of design practice in which the optimal continuous control is taken into consideration. Here we consider an isothermal fixed-bed where a first order reaction subject I t to first order deactivation takes place. The system equations are the same as thosein Table 4-1, but the temperature is controlled continuously. The performance index is given by



J* = max j = max i [ ftfXdt
TT(t) Tt f 0

(6-2)

where p is the cost incurred due to the reactor shut-down for regenerating or replacing the spent catalyst. The constant conversion policy which is indeed the optimal















1.0






.9






T= 22 .8.7






.6





.5 -A

8 12 16 20 24 28 32 36

T (sec) Figure 6-3. Maximum Values of Performance Index for Various Reactor Sizes






80


control policy for this reaction system (Crowe, 1970) was applied to solve the optimization problem of Eq. (6-2) with the following rate constants and restrictions


k D = 1.0 x 10 4 exp(-8000/T) (1/hr)

k 1 = 1.0 x 10 6 exp(-10000/T) WS)

473 K < T < 673 K

t f = 2000 (hr)

p = 220 (hr)

The solutions obtained are plotted in Fig. 6-4. The behavior of the performance index is similar to that shown in Fig. 6-2, i.e. for a given reactor size (T), J depends on the initial temperature chosen (T 0 ) and does yield a maximum. The locus of the maximum value of J for various Of T is shown in Fig. 6-4 as the dotted line. Again, we can see from the figure there exists an optimal reactor size, i.e. T = 13 sec. Further observations of the figure indicate that a reactor of T = 15 second gives better performance than a reactor of T = 10 second does if both are controlled optimally (To = 595.9 K for T = 15 sec. and T 0 = 605.4Kfor T = 10 sec). However, the latter performs better than the former does if same initial temperatures are chosen, say, 620 K. This indicates that both the deactivation of catalyst and the corresponding process control should be considered during the design phase to choose an optimal reactor size.






81




T (sec) 25 20 15 10 5
2.4




2.2 T=10



2.0 15



20
1. 8



25

1.6

5



1.4




1.2
580 590 600 610 620 630

T 0 (K) Figure 6-4. Performance Index 'as a Function of
Reactor Size and Initial Temperature* The dotted line shows the locus of J






82

Thenormalized optimal performance index is plotted in Fig. 6-5 as a function of reactor size. It can be seen from the figure that the performance index can be improved significantly if the optimal size is chosen. 6-3 Concluding Remarks

We have so far assumed that detailed knowledge of catalyst deactivation is available. The same optimal control and design can still be carried out if some knowledge of deactivation is available, however uncertain it may be, since what is sought in such a case would be approximate values of the optimal size and inlet temperature and some directions as to what to choose. In the example considered in Section 6-1, for example, one would choose a relatively high initial conversion (or a high inlet temperature) rather than a low initial conversion in view of the sensitivity discussed earlier.














1.0





0.9


13
00
0.8





0.7





0.6 1
0 5 10 15 20 25 30 35 40

T (sec) Figure 6-5. Maximum Values of Performance Index for Various Reactor Sizes















CHAPTER 7

CONCLUS IONS

A measure of the extent of deactivation can be determined from the on-line measurements of inlet temperature and outlet concentration along with the intrinsic rate of a single reaction for an isothermal or an adiabatic fixedbed reactor. Additional measurements of coolant temperature are required for a nonadiabatic fixed-bed reactor. These results can be extended to multiple reactions in a rather straightforward manner. The measure Qf deactivation is determined in such a way that a piecewise feedback control scheme follows directly from it. This feedback control maintains the conversion within a band about a desired level of conversion. These results enable one to carry out feedback control of the reactor in any desired manner without any knowledge of catalyst deactivation. Further, the results are applicable to any type of deactivation; whether the deactivation is dependent or independent of reactant concentration and whether the deactivation kinetics are separable or nonseparable from main reaction kinetics.

The feedback control policy considered is a piecewise rather than continuous control. The piecewise control necessitated by the pseudo steady-state nature of deactivation





84







85


with respect to the reactor response time allows us to avoid one problem in any feedback control involving dead time due to sampling and another one associated with reactor transients. Inherent in the pseudo steady-state assumption is the fact that the rate of deactivation is much slower than the rate of reaction. In terms of time, the deactivation is of the order of months for the majority of reactions carried out in a fixed bed whereas the transients reach the steady state in a few residence-times which is of the order of seconds. Therefore, the problem associated with "wrong way" transients [e.g., Crider and Foss (1966)] can be avoided since by the time a control action is taken, the reactor reaches a new, stable steady state. Further, the effect of dead time, which is of the order of minutes, on deactivation can be neglected since the extent of deactivation changes little in minutes.

The feedback control policy has been applied to a

simulated adiabatic reactor where a single reaction affected by diffusion and poisoning take place, and to a laboratory isothermal reactor where ethylene oxidation reactions affected by sintering take place. The results show that one can indeed control the conversion or yield in any desired manner without any a priori knowledge of the deactivation kinetics, and that the aforementioned problems of wrong way transients and dead time do not occur during normal operation.






86

The problem of finding the optimal piecewise control policy can be formulated as a multivariable optimization problem. For a single irreversible reaction affected by independent deactivation, the optimal piecewise control policy is one of increasing conversion. However, the wellknown constant conversion policy used in the continuous control system can be applied to the piecewise control system with little loss in reactor performance if the number of piecewise control actions is large. The solution method developed is applicable to any reaction taking place in any reactor.

The necessary condition for optimal control in complex reaction systems (such as ethylene oxidation reactions) can be derived by direct substitution. The optimal control policy is either an increasing conversion policy or a decreasing conversion policy, depending upon the ratio of activation energy of the main reaction to that of the deactivation reaction. The direct substitution method is shown to be more efficient than the commonly used method that relies on the Pontryagin's minimum principle through a Hamiltonian formulation.

The problem of combining process control with process design, with due consideration of catalyst deactivation for both, can be solved in a straightforward manner using the feedback control scheme and the approach of reactor point effectiveness.







87


However, detailed knowledge of deactivation has to be known to solve this problem. It is shown that the choice of reactor size can substantially affect the reactor performance. The solution yields the optimal reactor size and an optimal way of manipulating the inlet temperature based on on-line measurements for those reactors for which detailed knowledge of deactivation is available. Even when there is some uncertainty about the knowledge, the procedure yields approximate ranges of the optimal values and also some directions as to what is to be chosen. These results enable one to fully account for catalyst deactivation for a substantial improvement in the reactor performance.






88






REFERENCES

Akella, L.M., "Reactor Design and Analysis for Exothermic Reactions and Characterization of Ethylene Oxidation Reactions," Ph.D. Dissertation, University of Florida (1983).

Beveridge, G., and R.S. Schechter, Optimization: Theory and Practice, McGraw-Hill, New York (1970).

Bliss, G.A., Lectures on the Calculus of Variations, The University of Chicago Press, Chicago (1961).

Box, M.J., "A Comparison of Several Current Optimization Methods, and the Use of Transformations in Constrained Problems," Computer Journal 9, 67 (1966).

Carnahan, B., H.A. Luther, and J.O. Wilkes, Applied Numerical Methods, Wiley, New York (1969).

Chou, A., W.H. Ray, and R. Aris, "Simple Control Policies for Reactors with Catalyst Decay," Trans. Instn. Chem. Engrs., 45, T153 (1967).

Crider, J.E., and A.S. Foss, "Computational Studies of Transients in Packed Tubular Chemical Reactors," AIChE J., 12, 514 (1966).

Crowe, C.M., "Optimization of Reactors with Catalyst Decay: I. Single Tubular Reactor with Uniform Temperature," Can. J. Chem. Eng., 48, 576 (1970).

Denn, M.M., Optimization by Variational Methods, McGraw-Hill, New York (1969).

Dettwiler, H.R., A. Baiker, and W. Richarz, "Kinetics of Ethylene Oxidation on a Supported Silver Catalyst," Helvetica Chimica Acta, 62, 1689 (1979).

Haas, W.R., L.L. Tavlarides, and W.J. Wnek, "Optimal Temperature Policy for Reversible Reactions with Deactivation: Applied to Enzyme Reactors," AIChE J., 20, 707 (1974).

Hornbeck, R.W., Numerical Methods, Quantum Publishers, New York (1975).






89

Kovarik, F.S., and J.B. Butt, "Reactor Optimization in the Presence of Catalyst Decay," Cat. Rev., 24, 449 (1982).

Lee, H.H., "Catalyst Sintering and Pellet Effectiveness," Chem. Eng. Sci., 36, 950 (1981).

Lee, H.H., and J.B. Butt, "Heterogeneous Catalytic Reactors Undergoing Chemical Deactivation, Part I: Deactivation Kinetics and Pellet Effectiveness," AIChE J., 28, 405 (1982a).

Lee, H.H., and J.B. Butt, "Heterogeneous Catalytic Reactors Undergoing Chemical Deactivation, Part II: Design and Analysis: Approach of Reactor Point Effectiveness," AIChE J., 28, 410 (1982b).

Lee, H.H., and E. Ruckenstein, "Catalyst Sintering and Reactor Design," Cat. Rev., 25, 475 (1983).

Lee, S.I., and C.M. Crowe, "Optimal Temperature Policies for Batch Reactors with Decaying Catalyst," Chem. Eng. Sci., 25, 743 (1970).

Levenspiel, 0., and A. Sadana, "The Optimal Temperature Policy for a Deactivating Packed-Bed Reactor," Chem. Enq. Sci., 33, 1393 (1978).

Ogunye, A.F., and W.H. Ray, "Optimal Control Policies for Tubular Reactors Experiencing Catalyst Decay," AIChE J., 17, 43 (1971).

Pommersheim, J.M., L.L. Tavlarides, and S. Mukkavilli, "Restrictions and Equivalence of Optimal Temperature Policies for Reactors with Decaying Catalysts," AIChE J., 26, 327 (1980).

Pontryagin, L.S., V.G. Boltyznskii, R.V. Gamkrelidze, and E.F. Mishchenko, The Mathematical Theory of Optimal Processes, Interscience Publishers, New York (1962).

Spath, H.T., and K.D. Handel, "Kinetics and Mechanism of the Oxidation of Ethylene over Silver Catalysts," Adv. Chem. Ser. Chem. React. Eng. II, 133, 395 (1974).

Szepe, S., and O. Levenspiel, "Optimal Temperature Policies for Reactors Subject to Catalyst Deactivation--I. Batch Reactor," Chem. Eng. Sci., 23, 881 (1968).

Wheeler, A., "Reaction Rates and Selectivity in Catalyst Pores," Adv. in Cat., 2, 249 (1955).




Full Text
16
the first and second reactions are expressed as
hk f, (C.jC^) and Bhk f_(C,.,C_), respectively. If the same
a^ 1 A B 2 A B
active sites are responsible for both reactions, the local
activities are the same, and 6 is unity. Otherwise, we
assume that B is a function of time but constant along the
reactor, i.e., B=B(t).
The reactor activity factor can be obtained in the same
manner as for a single reaction:
H, =
h^dz
'B
'B,
dC
B
blTka1fl(CA'CB)
(2-26)
m
where Cg is the outlet concentration of species B. As Eq.
e
(2-25) indicates, the temperature can be expressed in
terms of C- and C_, for the calculation of H but C.. has to be
A B A
related to Cg to carry out the integration. Combining Eqs.
(2-23) and (2-24), we have
^A
dCB
1_
b.
1+6
ka2f2(CA'CB>
VILA'S*
; c,
= ca
CB = CB. in
in
(2-27)
which can be solved to obtain CR as a function of C_.,
CA= f(Cg), by Fourth Order Runge Kutta Method or Predictor-
Corrector Method (Hornbeck, 1975). If 6 is unity, the so
lution is straightforward. Otherwise, the concentrations


88
REFERENCES
Akella, L.M., "Reactor Design and Analysis for Exothermic
Reactions and Characterization of Ethylene Oxidation Reac
tions," Ph.D. Dissertation, University of Florida (1983).
Beveridge, G., and R.S. Schechter, Optimization; Theory and
Practice, McGraw-Hill, New York (1970).
Bliss, G.A., Lectures on the Calculus of Variations, The
University of Chicago Press, Chicago (1961).
Box, M.J., "A Comparison of Several Current Optimization
Methods, and the Use of Transformations in Constrained
Problems," Computer Journal 9_, 67 (1966).
Carnahan, B., H.A. Luther, and J.O. Wilkes, Applied Numerical
Methods, Wiley, New York (1969) .
Chou, A., W.H. Ray, and R. Aris, "Simple Control Policies
for Reactors with Catalyst Decay," Trans. Instn. Chem. Engrs.,
45, T153 (1967).
Crider, J.E., and A.S. Foss, "Computational Studies of Tran
sients in Packed Tubular Chemical Reactors," AIChE J., 12,
514 (1966).
Crowe, C.M., "Optimization of Reactors with Catalyst Decay:
I. Single Tubular Reactor with Uniform Temperature,"
Can. J. Chem. Eng., 4_8, 576 (1970).
Denn, M.M., Optimization by Variational Methods, McGraw-Hill,
New York (1969).
Dettwiler, H.R., A. Baiker, and W. Richarz, "Kinetics of
Ethylene Oxidation on a Supported Silver Catalyst," Helvetica
Chimica Acta, (^2, 1689 (1979) .
Haas, W.R., L.L. Tavlarides, and W.J. Wnek, "Optimal Tem
perature Policy for Reversible Reactions with Deactivation:
Applied to Enzyme Reactors," AIChE J. 2J3, 707 (1974).
Hornbeck, R.W., Numerical Methods, Quantum Publishers, New
York (1975).


CHAPTER 4
OPTIMAL PIECEWISE CONTROL
The control policies discussed in the previous
chapters are all piecewise type, i.e. the control actions
are taken at discrete intervals. The optimal control
problems for reactors affected by catalyst deactivation have
been the subject of several studies (Chou et al., 1967;
Szepe and Levenspiel, 1968; Crowe, 1970), and a constant
conversion policy has been proved to be optimal for a
single irreversible reaction with concentration-indepen-
dent deactivation. These studies deal only with the con
tinuous control, which is essentially the piecewise control
with an infinite number of steps. However, the feasibility
of extending the optimal continuous control policies to
piecewise control schemes with a finite number of steps
remains questionable. In this chapter, a solution
method for the optimal piecewise control is developed and
a comparison between continuous and piecewise control is
made.
§4-1 Problem Statement and Solution Method
Consider an isothermal fixed-bed where a single irre
versible reaction affected by catalyst deactivation takes
place. The temperature is to be adjusted in steps from T^ to
45


85
with respect to the reactor response time allows us to avoid
one problem in any feedback control involving dead time due
to sampling and another one associated with reactor tran
sients. Inherent in the pseudo steady-state assumption is
the fact that the rate of deactivation is much slower than
the rate of reaction. In terms of time, the deactivation is
of the order of months for the majority of reactions carried
out in a fixed bed whereas the transients reach the steady
state in a few residence-times which is of the order of
seconds. Therefore, the problem associated with "wrong way"
transients [e.g., Crider and Foss (1966)] can be avoided
since by the time a control action is taken, the reactor
reaches a new, stable steady state. Further, the effect of
dead time, which is of the order of minutes, on deactivation
can be neglected since the extent of deactivation changes
little in minutes.
The feedback control policy has been applied to a
simulated adiabatic reactor where a single reaction affected
by diffusion and poisoning take place, and to a laboratory
isothermal reactor where ethylene oxidation reactions affect
ed by sintering take place. The results show that one can
indeed control the conversion or yield in any desired manner
without any a priori knowledge of the deactivation kinetics,
and that the aforementioned problems of wrong way transients
and dead time do not occur during normal operation.


t (hr)
Figure 5-1. Optimal Control Policies for a Parallel Reaction Network
with the Langmiur-Hinshelwood Rate Expressions


86
The problem of finding the optimal piecewise con
trol policy can be formulated as a multivariable optimi
zation problem. For a single irreversible reaction affected
by independent deactivation, the optimal piecewise control
policy is one of increasing conversion. However, the well-
known constant conversion policy used in the continuous con
trol system can be applied to the piecewise control sys
tem with little loss in reactor performance if the number of
piecewise control actions is large. The solution method
developed is applicable to any reaction taking place in
any reactor.
The necessary condition for optimal control in complex
reaction systems (such as ethylene oxidation reactions)
can be derived by direct substitution. The optimal control
policy is either an increasing conversion policy or a de
creasing conversion policy, depending upon the ratio of
activation energy of the main reaction to that of the
deactivation reaction. The direct substitution method is
shown to be more efficient than the commonly used method
that relies on the Pontryagin's minimum principle through
a Hamiltonian formulation.
The problem of combining process control with process
design, with due consideration of catalyst deactivation
for both, can be solved in a straightforward manner using
the feedback control scheme and the approach of reactor point
effectiveness.


720
710
700
690
90
80
70
60
50
i 1 1 1 1 1 1 1 r
O'!
J I I i I J I I I
o .2 .4 .6 .8 1.0
txlO ^ (sec)
Figure 6-1. Inlet Temperature and Conversion Behavior for an Optimal
Control with Bandwidth of 0.05


64
P .
K. = a.Kn
3 3 D
where
j= 1/2,q
j Aj/V and Pj Ej/ED
(5-7)
Thus, our optimization problem is
J = max
KD(t)
Ydt
= max
KD(t)
K,
0
K1 + K2
Xdt
r
(5-8)
subject to two local restrictions, Eqs. (5-3) and (5-4).
This is a fundamental problem in calculus of variations.
k
The derivation of the necessary conditions for J to be
maximum is presented in the next section.
§5-2 Necessary Condition
Equation (5-3) indicates that
where
Kd = KD(h, h)
£ dh
h = dt
(5-9)
Since K^, K^ and are all functions of KD, Eqs. (5-4)
and (5-5) give
Y = Y(Kd, h) = Y(h, h)
(5-10)
Thus, the optimization problem of Eq. (5-8) is an elemen
tary problem in the calculus of variations, and the necessary
condition for the optimization is an Euler-Lagrange equa
tion (Beveridge and Schechter, 1970):
Y h Y; = C
h
where
Yh E
9 Y
*911
h, t
and C is an integration constant.
(5-11)


573 K
max
r~
(K)
570
550
cn
CO
1 1
1
1
1 1 1
l
0
400
800
1200
1600
2000
t (hr)
Figure 4-5.
Optimal Piecewise Control Policy
T
max
57 3 K
with Constant Conversion,


18
dC
B
dz
-Th(z;t) (6ka2f2(CA,CB) -b.k^lC^^)} ;
'B
= CB
z=0 in
(2-30)
T = T. + a-(C. C,)- a. C0 C
in 3 A. A 4 B B
in in
(2-31)
where = (-AH^-bAH2)/pC^ and a4=-AH2/pCp. If we follow
the same procedures as for the parallel reactions, we have
for the reactor activity factor:
H1 =
fC,
h^dz = -
dC,
Tkai(T)£i(CA.cE)
(2-32)
m
The relationship between C, and C., resulting from Eqs
(2-29) and (2-30) is
dC
dC
B
A
ka f2(CA,CB)
= ^^a/l^A'S
(2-33)
in
The same procedures as for the parallel reactions can be
used to calculate H f rom Eq.(2-32 ) with the aid of Eqs. (2-31)
and (2-33).
The control policy for maintaining the desired out
let concentration of species B follows directly from Eq.
(2-30) and the definition of H:


80
control policy for this reaction system (Crowe, 1970)
was applied to solve the optimization problem of Eq. (6-2)
with the following rate constants and restrictions
kD = 1.0 x 104 exp(-8000/T) (1/hr)
kx = 1.0 x 106 exp(-10000/T) (1/S)
473 K £ T £ 673 K
tf = 2000 (hr)
Vi = 220 (hr)
The solutions obtained are plotted in Fig. 6-4. The
behavior of the performance index is similar to that
shown in Fig. 6-2, i.e. for a given reactor size (x) J
depends on the initial temperature chosen (Tq) and does
yield a maximum. The locus of the maximum value of J for
various of t is shown in Fig. 6-4 as the dotted line.
Again, we can see from the figure there exists an optimal
reactor size, i.e. x = 13 sec. Further observations of the
figure indicate that a reactor of x = 15 second gives
better performance than a reactor of x = 10 second does if
both are controlled optimally (T = 595.9 K for x = 15
sec. and Tq = 605.4Kfor x = 10 sec). However, the latter
performs better than the former does if same initial tem
peratures are chosen, say, 620 K. This indicates that
both the deactivation of catalyst and the corresponding
process control should be considered during the design phase
to choose an optimal reactor size.


CHAPTER 7
CONCLUSIONS
A measure of the extent of deactivation can be deter
mined from the on-line measurements of inlet temperature
and outlet concentration along with the intrinsic rate of
a single reaction for an isothermal or an adiabatic fixed-
bed reactor. Additional measurements of coolant tempera
ture are required for a nonadiabatic fixed-bed reactor.
These results can be extended to multiple reactions in a
rather straightforward manner. The measure pf deactivation
is determined in such a way that a piecewise feedback con
trol scheme follows directly from it. This feedback con
trol maintains the conversion within a band about a desired
level of conversion. These results enable one to carry out
feedback control of the reactor in any desired manner with
out any knowledge of catalyst deactivation. Further, the
results are applicable to any type of deactivation; whether
the deactivation is dependent or independent of reactant
concentration and whether the deactivation kinetics are
separable or nonseparable from main reaction kinetics.
The feedback control policy considered is a piecewise
rather than continuous control. The piecewise control ne
cessitated by the pseudo steady-state nature of deactivation
84


29
1
(C)
The oxidation of ethylene oxide (reaction path 3) is
significant only at temperatures above 473 K (Dettwiler
et al., 1979; Akella, 1983). Since the experiments were
conducted in the temperature range between 448 and 468 K,
only epoxidation and combustion of ethylene (reaction paths
1 and 2) need to be considered. A material balance in an
isothermal fixed-bed can be written as
fA
dz
hl (ri + 8r2)W
(3-1)
dz
hiriw
FBO = 0 (3-2)
where FA and Fg are the molar flow rates for ethylene and
ethylene oxide, respectively, and W is the weight of the
catalyst. Here r^ and r2 are the intrinsic rates for the
reaction paths 1 and 2. The mechanism of ethylene oxidation
is still controversial. Under the condition of constant
oxygen partial pressure, however, the following Langmuiur-
Hinshelwood rate expressions are reported by most investiga
tors (Dettwiler et al., 1979):


10
for the purpose of calculating the new inlet temperature
that yields the desired outlet concentration. The justifi
cation here is that the fraction of catalyst deactivated
changes negligibly for all points along the reactor while
the inlet temperature is adjusted from the current value to
a new value. Note that the justification is still valid for
concentration dependent deactivation, i.e. hc(C(z);t)
= h (C(z);t). Equations (2-11) through (2-13) together with
Eq. (2-10) can be used to obtain
dC
-E /R
xk exp
cl
o
a
f (C,K
(T._)+
m n
f-AH]
pC

PJ
H
c
(2-14)
where Hc is the known, current reactor activity factor cal
culated from Eq. (2-12) just prior to the change in the in
let temperature based on the current measured outlet con
centration Ce and inlet temperature (T^r) Equation (2-14)
can now be solved for (T^n)n by a numerical technique such
as Newton's method or bisection method since the left hand
side of the equation can be evaluated, given a value of
n. The value of (T^n)n that satisfies Eq. (2-14) will
yield the desired conversion and, therefore, the inlet tem
perature can be changed to (T. ) to maintain the conversion
m n
at the desired level.


63
where
T = WP/F ,
o
Kn = k,k K = kk
"i q
2 q'
and K = p, k
q Ao q
The rate of decay of the catalyst activity can be expressed
as
dh
dt
V
m
h
z=o
1
(5-3)
where KQ is the deactivation rate constant. Integrating
Eqs. (5-1) and (5-2) from the inlet to the outlet of the
reactor yields
K X £n(1-X) = xh(K, + K_) (5-4)
q 12
Y
K,
Kl+ K2
X
(5-5)
where X and Y are the conversion of ethylene and the
yield of ethylene oxide at the outlet of the reactor, re
spectively.
The optimization problem is to maximize the total amount
of yield of ethylene oxide over a fixed total reaction
time, tj, by choosing the optimal temperature-time policy.
We assume that the temperature dependence of the rate con
stants and adsorption equilirbium constant are all of the
Arrhennius form:
K = Ai exp(-Ei/RgT) i = 1, 2, q, D (5-6)
Since is a strictly monotonic function of temperature,
we can use KD as the control variable to replace T. The
constants K., K0 and K can be exoressed in terms of K^:
1 2 q D


72
took less than 3 seconds CPU time of IBM 360/370 computer
to generate all the data shown in Fig. 5-1, including 5
iterations for shooting the boundary condition, h = hf at
t = tf. This efficiency can never be achieved if the
Pontryagin maximum principle (Pontryagin et al., 1962)
and the Hamiltonian are used (Ogunye and Ray, 1971).


17
measured at the outlet, Ch and C_, can be used as an
A o
e e
additional boundary condition, i.e., C
C = c = CA for
e
Eq. (2-27) to obtain the value of (3.
If species B is the desired species such that (Cg)^
is the desired outlet concentration, the control policy
similar to Eq. (2-14) can be written as
H1C
r (CBJd
'B.
in
b. xk exp
la. ^
lo
dC
B
-E /R
a g
(Tinn+ al(CA. CA)+a2
in
(C -C
v B B.
in
fl(CA,CB) (2-28)
which follows from Eq. (2-24). Here aqain, Hlc is the re
actor activity factor for the reaction path 1 calculated just
prior to a change in the inlet temperature. Equation (2-28)
can be solved for (Tin)n with the aid of Eq. (2-27). It is
noted that (Tj_n)n should also be used in Eq. (2-27) in the
calculation.
Consider now consecutive reactions, for which we have
1 2
A B b2C
With the same assumptions made for the parallel reactions,
the conservation equations for an adiabatic reactor are
dz
-Th(z;t)x(C^/Cg);
= c
z=o
A.
in
(2-29)


jx"*-0
m
6-2
Functi" £
in e index as a f Iniet
~ss?&S.SS* une
%**£%?* J*'
tk J-
Figure


The problem of finding the optimal piecewise control
policy has been formulated as a multivariable optimization
problem. The optimal policy is one of increasing conver
sion. The constant conversion policy corresponds to the
limiting case where the number of piecewise steps approaches
infinity. This optimal piecewise control is suitable for
on-line implementation in conjunction with the on-line es
timation of the extent of catalyst deactivation. An optimal
continuous control policy for parallel and reversible reac
tions has also been solved using the technique of calculus
of variations. It is shown that a direct substitution method
is more efficient than the commonly used method of applying the
Pontryagin maximum principle.
The optimal control policies can easily be taken into
consideration for reactor design. This combination of pro
cess design and control with due consideration of catalyst
deactivation for both is shown to result in a substantial im
provement of the reactor performance.
xi


74.
and subject to Tin i (Tin)max with a fixed tf The
problem is to maximize the performance index J, which is
the conversion (X) integrated over the catalyst life t^
per unit reactor volume and unit on-stream time, by proper
ly choosing x and T^n(t). This two-parameter optimization
problem can be solved one at a time by first searching
for the maximum for a given x and then finding the value of
x that yields the maximum. For a fixed x, it is a Bolza's
problem (Bliss, 1961) and the extremum condition obtained
by Chou et al. (1967) and Ogunye and Ray (1971) is to main
tain the conversion at a constant level until the maximum
allowed temperature is reached and then to remain at the
maximum if only one rate constant is involved. For the de
activation problem under consideration the policy is not
directly applicable since the rate constant cannot be se
parated out. However, the constant conversion policy is
still a suboptimum which is often used in practical opera
tion of chemical plants. Thus, we apply the constant
conversion policy for the problem considered. Since
the feedback control policy obtained earlier is for main
taining a constant conversion level, the problem is simply
that of finding an initial value of T^n (or the desired
conversion level) if we use the control policy, for T.
in
can be set at (T. ) once (T. ) is reached.
m max m max


CHAPTER 2
THEORY OF ON-LINE ESTIMATION OF DEACTIVATION AND CONTROL
An important problem in operating a fixed-bed with
catalyst deactivation is that of finding a means of esti
mating the extent of catalyst deactivation from on-line
measurements because of the uncertainty regarding the deac
tivation. The problem of controlling the conversion in any
desired manner using a feedback scheme is another. These
two problems are solved theoretically in this chapter,
starting with a single reaction and then proceeding to
multiple reactions.
§2-1 A Simple Reaction
Consider a shell-and-tube reactor in which a single
reaction takes place. Assuming negligible radial gradients
and pseudo steady-state, one-dimensional balance equations
can be written as
HI = ~tRg; t = Z(l-eB)/v
;C
= c.
z=o m
(2-1)
dT
dz
-AH
PC
tRg +
(mC )
mC
D
dT
dz
= T. (2-2)
z=o m
dT
c
dz
2U 7TRtZ
(mC )
P c
(T-T )
c
T
c
Z = 1
T
c.
m
(2-3)
5


52
policy is to step the temperature up from 539.9 K until it
reaches T (573 K) at t = 1400 hr. and then to keep it at
that level as shown in the figure. The corresponding con
version zigzags with increasing bandwidth until the tempera
ture reaches its upper limit. Due to catalyst deactivation
the conversion decreases in each time interval while the
temperature remains constant. It is of interest to note
that each temperature increase gives a conversion peak
which is slightly higher than the previous one, i.e.
X (t.) > X (t. .) for T < T
n i n l-l max
where n denotes the new conversion immediately following the
temperature adjustment. In other words, the optimal piece-
wise policy is an increasing conversion policy, rather than a
constant conversion policy which is the optimal for continuous
control (Crowe, 1970). The optimal continuous control policy
for the same model reaction is shown in Fig. 4-3 for compari
son. It is obtained by one-dimensional search of the optimal
initial temperature with the aid of the constant conversion
policy. It can be seen from the figure that the optimal con
tinuous policy is a constant conversion policy followed by a
constant temperature policy at Tmax Comparison between Fig.
4-2 and 4-3 shows that the temperature in the continuous con
trol starts 1.3 K lower than in the piecewise scheme and reaches
the upper limit 42 hours later. It can also be seen that the
conversion in the continuous control is lower than the original
conversion in the piecewise control, but stays in the zigzag
zone of the piecewise scheme. The corresponding activity be
havior is also plotted in Figs. 4-2 and 4-3.


CHAPTER 3
SIMULATION AND EXPERIMENTAL VERIFICATION
The theory in Chapter 2 is demonstrated here with a
simulation of a model reaction system. The theory is also
put to the test with a laboratory shell-and-tube reactor in
which ethylene oxidation reactions take place over a silver
catalyst. The catalyst undergoes sintering under the
reaction conditions.
§3-1 Simulation of a Model Reaction System
For an illustration of the calculation of H and its
use for the reactor control, we consider the reaction system
studied by Lee and Butt (1982a), which is summarized
in Tables 3-1 and 3-2. For the reaction taking place in an
adiabatic reactor, which is affected by diffusion and uni
form, independent poisoning, the intrinsic rate of deacti
vation r is given (Lee and Butt, 1982b) by
hr
r = k (l-y)N
P pa
where k is the rate constant evaluated at the pellet sur-
pa
face temperature and N is the concentration of poisoning
species. With the use of reactor point effectiveness (Lee
and Butt, 1982a), the reactor behavior can be simulated in
a straightforward manner. The simulation results are used
here as the process output for the purpose of illustrating
20


TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS
NOTATION V
ABSTRACT x
CHAPTER 1 INTRODUCTION 1
CHAPTER 2 THEORY OF ON-LINE ESTIMATION OF DE- 5
ACTIVATION AND CONTROL
§2-1 A Single Reaction 5
§2-2 Nature of Reactor Activity Factor.. 13
§2-3 Multiple Reactions 15
CHAPTER 3 SIMULATION AND EXPERIMENTAL VERIFI
CATION 20
§3-1 Simulation of a Model Reaction Sys
tem 20
§3-2 Experimental Verification 25
§3-3 Concluding Remarks...,, 42
CHAPTER 4 OPTIMAL PIECEWISE CONTROL 45
§4-1 Problem Statement and Solution
Method 45
§4-2 Optimal Piecewise Control Policies.. 49
§4-3 Concluding Remarks 57
CHAPTER 5 OPTIMAL CONTINUOUS CONTROL FOR COM
PLEX REACTION 62
§5-1 Problem Statement 6 2
§5-2 Necessary Condition 64
§5-3 Optimal Continuous Control Policies. 67
§5-4 Concluding Remarks 71
CHAPTER 6 OPTIMAL CONTROL AND DESIGN 73
§6-1 An Optimization Problem of Piecewise
Control and Design 73
iii


25
this case, is well below the maximum temperature allowed.
Due to the deactivation, however, both the conversion and
the outlet temperature decrease with time and the conversion
eventually reaches the lower bound given by the bandwidth,
triggering an adjustment of the inlet temperature. The new
inlet temperature (T^n)^ calculated by the algorithm of
Eqs. (2-16) through (2-18) is seen to bring the conversion
back to the desired level, the actual value being 69.96% as
opposed to the desired value of 70%. A bisection method
was used for the calculation of (T. ).. When the outlet
in 3
temperature reaches the allowed maximum, which occurs at
4
around t = 2x10 s, the conversion can no longer be brought
back to the desired level due to the temperature constraint.
Consequently the bandwidth decreases as time increases and
eventually the bandwidth becomes less than one-tenth of
the original bandwidth, resulting in the reactor shut-down
for catalyst regeneration according to the constraint im
posed. In practice, however, the conversion may be allowed
to decrease below 60% when the maximum temperature is reached.
In the example, the average conversion is essentially main
tained at the middle of the bandwidth, which is 65%.
§3-2 Experimental Verification
We have shown the feasibility of control policies for
the model reaction system. The theory has also been put
to test using a shell-and-tube reactor. Experimental results
obtained and comparisons with the theory are presented in this
section.


67
where is the preexponential factor. Since A^, A^, Aq,
p^ and C are all constants, Eq. (5-20) indicates that
X = constant (5-21)
In general cases, however, the conversion is not constant.
The optimal control policies have to be obtained by numeri
cally solving the necessary condition, Eq. (5-17), along
with the local constraints, Eqs. (5-3) and (5-4).
§5-3 Optimal Continuous Control Policies
The predictor-corrector method (Hornbeck, 1975) initiated
by fourth order Runge Kutta method was used to solve the
ordinary differential Eq. (5-3) while Newton-Raphson itera
tion (Carnahan et al., 1969) was used for the nonlinear
Eqs. (5-4) and (5-17). The kinetic data obtained in Section
§3-2-3 and the same parameters used in the experiment were
used for obtaining the optimal control policies. Table 5-1
/
lists these parameters along with the deactivation kinetics
used. The boundary condition of Eq. (5-18) was used to
obtain the integration constant in Eq. (5-17). This split
boundary condition problem was solved by "shooting method"
(Hornbeck, 1975). The results are shown in Figs. 5-1 and
5-2.
It can be seen from Fig. 5-1 that the optimal control
policy is one of increasing conversion and yield. However,
with different deactivation kinetics, the optimal policy
becomes one of decreasing conversion and yield, as shown


22
Table 3-2 Parameters For The Model Reaction System
12,000
k = exp
+ 14.6
(1/s); E = 23.76 kcal/mol
3.
5 = exp
^§5 + 3.86
(cm /mol)
C. = 1.81 x 10 ^ mol/cm^
in
-AH
ipCpj
7 3
= 4 x 10 K cm /mol; -AH =5.04 kcal/mol
= 3.245 x 10 ^ cal/s.cm^.K
N^n = 5 x'10 ^ mol/cm^
-4 3
Q = 10 mol/cm cat. pellet
kP exp
7,000
+ 3.27
; E =13.86 kcal/mol
P
D =10 ^ cm^/s
P
T = 1280K
nicix
X = 70% AX = 10%
T = 20s
m ,o
691.8 K


CHAPTER 5
OPTIMAL CONTINUOUS CONTROL FOR COMPLEX REACTION
In the experiment described in Section §3-2, we con
trolled the yield of ethylene oxide at a constant level in
a piecewise manner, and verified the feasibility of the
feedback control policy presented in Chapter 2.
However, the constant yield policy used is not an
optimal policy. It is the purpose of this chapter to ob
tain an optimal control policy for complex reactions which
are parallel with the Langmiur-Hinshelwood rate expressions.
We consider here only the optimal continuous control policy
since we have shown in the previous chapter that one can
use the optimal continuous control policy for piecewise con
trol as long as the number of control steps is large enough.
§5-1 Problem Statement
Consider a parallel reaction network with the Langmiur-
Hinshelwood rate expressions for an isothermal fixed-bed such
as the ethylene oxidation reaction described in Section §3-2-3.
Assume the same active sites are responsible for both reac
tions, i.e., h^ = h^ = h and 3=1. Equation (3-5) and (3-7)
can be rewritten as
dx
dz
Th(Kx + K2)(1-x)
= 0
(5-1)
1 + K (1-x)
x
z=o
and
x
v=o
0
(5-2)
62


Table 3-3 Kinetic Data for Ethylene Oxidation Reactions
RUN
T (K)
ykow
Yl(%)
Yci(%)
X(%)
Y(%)
S(%)
S(%)
1
448
4.00
3.36
.487
16.04
9.95
62.0
62.4
2
448
6.00
5.28
.537
11.93
7.45
62.5
3
448
8.00
7.26
.551
9.23
5.79
62.7
4
458
4.00
3.10
.718
22.51
13.53
60.1
59.8
5
458
6.00
5.01
.788
16.55
9.98
60.3
6
458
8.00
6.94
. 869
13.26
7.83
59.0
7
468
4.00
2.77
.971
30.79
18.65
60.6
59.7
8
468
6.00
4.60
1.12
23.31
14.00
60.1
9
468
8.00
6.52
1.23
18.48
10.81
58.5
F = 100 ml/min
o
P =1 atm
W = 78.34 g


89
Kovarik, F.S., and J.B. Butt, "Reactor Optimization in the
Presence of Catalyst Decay," Cat. Rev., 2_4 449 (1982).
Lee, H.H., "Catalyst Sintering and Pellet Effectiveness,"
Chem. Eng. Sci., 36_, 950 (1981).
Lee, H.H., and J.B. Butt, "Heterogeneous Catalytic Reactors
Undergoing Chemical Deactivation, Part I: Deactivation
Kinetics and Pellet Effectiveness," AIChE J. 2J3 405 (1982a)
Lee, H.H., and J.B. Butt, "Heterogeneous Catalytic Reactors
Undergoing Chemical Deactivation, Part II: Design and
Analysis: Approach of Reactor Point Effectiveness," AIChE J.
28_, 410 (1982b) .
Lee, H.H., and E. Ruckenstein, "Catalyst Sintering and Re
actor Design," Cat. Rev., 2_5, 475 (1983).
Lee, S.I., and C.M. Crowe, "Optimal Temperature Policies
for Batch Reactors with Decaying Catalyst," Chem. Eng. Sci.,
25, 743 (1970) .
Levenspiel, 0., and A. Sadana, "The Optimal Temperature
Policy for a Deactivating Packed-Bed Reactor," Chem. Eng.
Sci., 33, 1393 (1978).
Ogunye, A.F., and W.H. Ray, "Optimal Control Policies for
Tubular Reactors Experiencing Catalyst Decay," AIChE J.,
17, 43 (1971).
Pommersheim, J.M., L.L. Tavlarides, and S. Mukkavilli,
"Restrictions and Equivalence of Optimal Temperature Poli
cies for Reactors with Decaying Catalysts," AIChE J., 26,
327 (1980).
Pontryagin, L.S., V.G. Boltyznskii, R.V. Gamkrelidze, and
E.F. Mishchenko, The Mathematical Theory of Optimal Processes
Interscience Publishers, New York (1962).
Spath, H.T., and K.D. Handel, "Kinetics and Mechanism of
the Oxidation of Ethylene over Silver Catalysts," Adv. Chem.
Ser. Chem. React. Eng. II, 133, 395 (1974).
Szepe, S., and 0. Levenspiel, "Optimal Temperature Policies
for Reactors Subject to Catalyst Deactivation--I. Batch
Reactor," Chem. Eng. Sci., 23_, 881 (1968).
Wheeler, A., "Reaction Rates and Selectivity in Catalyst
Pores," Adv. in Cat., 2, 249 (1955).


Silver Catalyst
Figure 3-2. Experimental Apparatus for Ethylene Oxidation Reaction


Table 4-2 Comparison of Performance Indices for Different Control Policies
Entry
Temperature
T
max
573 K
T
max
673 K
Control
Policy
J.
i
5§
i
J.
i
jf
l
1
*
T = T
0.53512
100.00
0.55855
100.00
2
*
T = T +
1
0.53498
99.97
0.55937
99.97
3
*
T = T -
1
0.53498
99.97
0.55837
99.97
4
*+
T = T
c
0.53503
99.98
0.55628
99.59
5
m m*+t
t
0.49219
91.98
0.49219
88.12
6
T = T
max
0.44007
82.24
0.04823
8.63
7
T = T
mm
0.06268
11.71
0.06268
11.22
J. =
1
J./J, x 100%
t
with constant-conversion constraint
+ + ,
with constant-temperature constraint


21
Table 3-1 Equations for the Model Reaction System
dC
dz
= tR,
= C.
z=0 in
dN
dz
xk N(1 y) ;
pa '
N
= N.
z=0 in
dy 1
dt Q rp
1
Q
k N (1
pa
y) ;
Y
t=0
0
where R., =
Vj
1/L
2Dek(l-Y)
g(c)dc
o
1.2 E (-AH)
a
2h R T2
m g
g (c) =
C
1+£C
T = T.
in pC
K PJ
-AH
(C C. )
in
k = k exp(-E /R T)
o ^ a g
T = T + (-AH) R_L/h
s G m
k = k exp (-E /R T )
pa po P g s


V
V
superficial velocity
w
weight of catalyst
X
conversion at the outlet
AX
bandwidth of conversion allowed before an
adjustment in T\n is made
X
conversion in the reactor
Y
yield at the outlet
y
yield in the reactor
Yko
mole fraction of A at the outlet
Yki
mole fraction of A at the outlet
yi
mole fraction of C at the outlet
z
reactor length
z
axial reactor coordinate
Greek Letters
a
proportionality constant defined in Eq. (2-16)
a .
3
Pi
Aj/AD -1 j =1 / 2 q
6
h2/h1
Y
fraction of catalyst deactivated
£B
bed porosity
y
cost incurred due to the reactor shut-down
p
reaction fluid density
T
holding time given in Eq. (2-1)
Subscripts
A
species A; ethylene
B
species B; ethylene oxide
C
species C; carbon dioxide
Vlll


8
The temperature appearing in Eq. (2-7) needs to be re
lated to concentration for the integration. Integrating
Eq. (2-2) from the reactor inlet to z, we obtain
T = T. +
in
-AH
PC
1 pJ
(C. -C) -
in
(mC )
P 9 (T T )
mC ce c
(2-8)
where T is the coolant temperature at z=0 and T. is the
ce ^ in
reactor inlet temperature. Consider first an adiabatic re
actor for which Eq. (2-8) reduces to
T
T. +
m
-AH
PC
PJ
(C
m
C)
(2-9)
Now that T is expressed in terms of C and readily measur
able inlet conditions, the value of H can be calculated by
simply carrying out the integration numerically with respect
to C in Eq. (2-7) with the measured outlet concentration.
For instance, the apparent rate constant k (T) for an adia-
3.
batic reactor can be expressed as
k
a
exp
'
-E /R
a g
T. +
l in
-AH
pc
rv
(C. C)
in J
(2-10)
where k is the apparent preexponential factor and E is the
o. a
apparent activation energy. The equilibrium constants K^
can be expressed in a similar manner. For a shell-and-tube
reactor to which Eq. (2-8) applies, the calculation of H re
quires measurement of coolant temperature along the reactor
length. This measurement can be used in Eq. (2-3) for the


31
Dividing Eq. (3-1) by (3-2) and nondimensionalization gives
p- = 1 + 3 = 1 + Sr^
dy r, k
= 0
(3-7)
y=o
Since the right hand side of Eq. (3-7) is constant through
out the isothermal reactor, Eq. (3-7) can be integrated from
the reactor inlet to the outlet to give
X
Y
1 + 8
(3-8)
where Y is the yield at the outlet.
Equations (3-6) and (3-8), which are the integrated
versionsof Eqs. (2-26) and (2-27), can be used for both
the feedback control described in Section §2-3-3 and the
determination of kinetics. Before proceeding to the control
policies, consider the kinetics first. It is convenient
to choose the activity of the catalyst at the time kinetic
data are taken as the reference state activity. Then,
both h^ and 8 are unity by definition, and Eqs. (3-6)
and (3-8) can be rearranged as
+
kiVw
(3-9)
and k^
(3-10)
It has been shown (Akella, 1983) that the selec
tivity, defined as S = Y/X, is fairly independent of ethylene


T
(K)
500
480
460
440
t (hr)
Figure 5-2. Optimal Control Policies for a Parallel Reaction Network with
the Langmiur-Hinshelwood Rate Expressions
o


78
to x = 22 sec., when the reactor is operated optimally,
is shown in Fig. 6-3 as a function of the reactor size x.
It is seen that the performance index [Eq. (6-1)] doubles
(x of 8 sec. vs. i of 22 sec.) when the reactor is sized
optimally. This result confirms the intuitive reasoning
that the best possible reactor performance obtainable
by process control is inherently dictated by process design.
It also confirms the fact that a significant improvement
can be made by combining process design with process con
trol .
§6-2 An Optimization Problem of Continuous Control and
Design
In this section, we give an example of design practice
in which the optimal continuous control is taken into
consideration. Here we consider an isothermal fixed-bed
where a first order reaction subject to first order deacti
vation takes place. The system equations are the same as
those in Table 4-1, but the temperature is controlled con
tinuously. The performance index is given by
*
J
max
T,T(t) J
max
T,T(t)
y
(6-2)
where u is the cost incurred due to the reactor shut-down
for regenerating or replacing the spent catalyst. The
constant conversion policy which is indeed the optimal


30
klVA
rl 1+k p
q^A
(3-3)
ro = (mol/s-g-cat.) (3-4)
2 1+kqPA
where is the partial pressure of ethylene and is
an adsorption equilibrium constant. Since the catalyst
particles are small, there is no diffusion limitation and
the intrinsic rates are essentially the same as the glo
bal rates.
Let x and y be the conversion and yield in the reactor,
i.e. x = (F. F,. )/F* and y = (F_ F ) /F, Then,
Ao A Ao 1 B Bo Ao
Eq. (3-1) can be rendered dimensionless with the aid of
Eqs. (3-3) and (3-4):
dx w w-Bkp yyko(1-x)
dz" vko 1+kqp4o(1-x)
x
z=0
0
(3-5)
where Fq is the molar flow rate of reactant mixture and
y^Q is the mole fraction of ethylene, both at the inlet of
the reactor. The ideal gas law has been applied to obtain
Eq. (3-5). The reactor activity factor for the.reaction path 1,
can be obtained by rearranging Eq. (3-5) and integrating
from the reactor inlet to the outlet:
H
1
1
h^dz
0
k py' X-£n (1-X)
(k.+8k_)k pW/F
1 q*- o
where X is the conversion at the outlet.
(3-6)


14
Here, y is the fraction of catalyst deactivated, g(C) the
concentration dependence of the intrinsic rate of reaction,
k the intrinsic rate constant, the effective diffusivity,
L the characteristic pellet length, and h^ the film heat
transfer coefficient. The local activity factor can be ob
tained from the definition of Eq. (2-5):
h =
(Vd
d
d
Y=1
= (1-Y)
1~q
1- (1-Y) 2q
(2-22)
The same results hold (Lee and Ruckenstein, 1983) for non-
uniform deactivation if y is replaced with y which is y
at the pellet surface, as long as y is less than, say, 0.5.
It is clear from Eq. (2-22) that the local activity factor
depends on y and the quantity q which represents the heat
transfer resistance across the bulk fluid-pellet surface in
terface. Since y changes very little while the inlet tem
perature is raised to a new value, the magnitude of q de
termines whether h can be set to h Therefore, the pro-
portionality constant a in Eq. (2-16) can be chosen in
accordance with the magnitude of q: for q much smaller
than unity, the value of a can be set to zero, its value in
creasing with increasing q. Although the conclusion is made
from the result for independent deactivation, the same should
be valid for dependent deactivation since the factor causing
the discrepancy between hc and hn is still the same, i.e.
the heat transfer resistance across the interface, which q
represents.


3
fixed-bed. This uncertainty regarding the deactivation and
the compounded effects of deactivation and diffusion in a
fixed-bed make it quite unattractive to implement an open-
loop control policy. The second question, therefore, is
whether a measure of catalyst deactivation can be estimated
from process measurements and, if so, how a feedback control
can be realized with the measurements made. In Chapter 2
a method of determining a measure of catalyst deactivation
is presented and the result then utilized for a feedback
control policy. This feedback control policy is demonstrated
in Chapter 3 with a simulation of a model reaction system,
and is put to the test with a laboratory fixed-bed where
ethylene oxidation reactions take place over a silver cata
lyst .
It has been shown that constant conversion policy is
optimal only for a single irreversible reaction affected ,
by concentration-independent deactivation, when the reactor
is controlled continuously (Chou et al. 1967; Lee and Crowe,
1970; Crowe, 1970). Thus, the third question regarding the
usual practice of maintaining constant conversion is what
the optimal policy will be if the reactor is piecewise con
trolled (for example, the temperature is adjusted only once
a day, rather than continuously), or if the reaction is
complex. The solutions to this question are presented in
Chapters 4 and 5.


20
80
0 1 1 1 1 1 I l I i I | I 1 I I I I
0 100 200 300 400 500 600 700 800 900
t (hr)
W
Figure 3-3.
Conversion, Yield and Selectivity Behavior for Ethylene Oxidation
Reaction


c
coolant; current value; with constant-
conversion constraint
D
deactivated
d
desired
e
reactor outlet
i
index for a step change in inlet temperature
in
inlet
max
the maximum value
min
the minimum value
n
new value
o
at t=0 or z=0
q
adsorption equilibrium constant
r
at reference state
1
reaction path 1
2
reaction oath 2
Superscripts
*
optimal value
d_
dt
-
normalized value
IX


15
§2-3 Multiple Reactions
For simple parallel reactions, we can write
1
A * b^B
2
A b2C
where and b2 are the ratios of stoichiometric coeffi
cients. We restrict our attention here to an adiabatic
reactor for brevity since a nonadiabatic reactor can be
treated in the same manner as shown for a single reaction.
Let 6 denote the ratio of local activity factors, i.e.,
B=h2(z;t)/h^(z;t), the reactor conservation equations
can be written as:
dCA
~ Th1(z;t){ka f1(CA, CB) + 6 ka f2(CA,CB)};
= C (2-23)
z=0 in
dCB
dz~~ ~ b1xh1(z;t)ka^ (CA, CB) ; Cg
= C (2-24)
z=0 in
T=Tin+al(CA. CA}
in
+ a2(CB CB. >
in
(2-25)
where = AH2/pCp and a2 = AH^(1-AH2/AH1)pCpb^. Equa
tion (2-25) results when the heat balance equations are combined
with the mass balance equations. Here the global rates for


DESIGN AND CONTROL OF FIXED-BEDS
AFFECTED BY CATALYST DEACTIVATION
By
Jan-Chen Hong
A DISSERTATION PRESENTED TO THE GRADUATE
COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL
FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1984


34
helium as the diluent. Given in Table 3-4 are the results
of the linear regression of (Jin on (y^QX) fr each
temperature. The values of (Jin y~y) calculated from the
regression model, (Jin are included in the table
1 x cal
for comparison. The percent error in the table
was calculated from
Error =
Jin-
1-X
- 1
(Jin
1-X'
Cal
x 100%
(3-13)
The constants and were calculated from the slope and
intercept of the regression line, while k^ was calculated
from Eq. (3-10). The values of these rate constants are
given in Table 3-4 along with the coefficients of correlation.
Note that the average selectivity, S, given in the last column
of Table 3-3 was used to calculate k, and k .
1 q
It was assumed that the temperature dependence of the
rate and equilibrium constants are all of the Arrhennius
form:
ki = Ai exp(- E./R T) i=l, 2, q (3-14)
The activation energies and pre-exponential factors were
obtained from linear regressions of Jin k^ on 1/T. The re
sults are given in Table 3-5.
The experimental results show that the kinetics of
ethylene oxidations reactions can be well represented in
the form of Eqs. (3-3) and (3-4) with the following rate
constants and equilibrium constant:


90
BIOGRAPHICAL SKETCH
The author was born on January 9, 1955, in Taipei,
Taiwan, Republic of China. He received his B.S. degree
in chemical engineering from National Taiwan University,
Taipei, Taiwan, in 1977. After serving as a Second Lieu
tenant in the Chinese ROTC for two years, he came to the
United States for graduate studies. The author received
his M.S. degree in chemical engineering from West Virginia
University, Morgantown, West Virginia, in 1981 and has
attended the Department of Chemical Engineering, University of
Florida, for Ph.D. study since then. His career interest
is teaching and research in catalytic reactor design and
/
in manufacturing semiconductor crystals.


on y' X
Ao
Table 3-4
Results of Linear Regression of £n
1
1-X
RUN T
(K)
1 448
2 448
3 448
4 458
5 458
6 458
7 468
8 468
y; x
1 Ao
(xlO-3)
tal-x
(xlO-1)
[^nl-x)cal
(xlO X)
Error
(%)
6.417
1.749
1.766
-1.0
7.156
1.270
1.198
6.0
7.384
0.968
1.023
-5.3
9.005
2.551
2.526
1.0
9.932
1.810
1.869
-3.2
10.61
1.423
1.388
2.5
12.32
3.680
V
3.697
-0.4
13.99
2.654
2.603
2.0
14.79
2.043
2.078
-1.7
(xlO
4.797
6.631
9.457
-g q
(xlO ) (xlO)
2.890 7.675
4.453 7.089
6.378 6.556
Corr.
Coeff.
-0.9863
-0.9960
-0.9985
u>
cn
9 468


65
In order to use Eq. (5-11) hY^ has to be obtained
first. Differentiating Eq. (5-7) w.r.t. h yields
9h
fdKil
f9Kl
ipiKil
1 1
ldKDJ
^ 3h'
i Kd J
,m
k h
i = 1, 2, q (5-12)
Multiplying h gives
3K.
3h
iPiKi]
-
l kd
, m
^ h J
= p.K.
i
(5-13)
Differentiating Eq. (5-4) w.r.t. h, multiplying h and
using Eq. (5-13) yields
pqKgX + Kg(h Xfi) + (h Xfi) = P3t(K1 + K2)h (5-14)
where p3 = (P1K1 + p2K2)/(K1 + K2)
Rearranging Eq. (5-14) and using Eq. (5-4) for x(K1 + K2)h
we have
h X; =
h
Kq + l^X
(5-15)
Differentiating Eq. (5-5) w.r.t. h, multiplying h and
using Eq. (5-13) we obtain


t (sec)
Maximum Values of Performance Index for Various
Reactor Sizes
Figure 6-3


87
However, detailed knowledge of deactivation has to be known
to solve this problem. It is shown that the choice of re
actor size can substantially affect the reactor performance.
The solution yields the optimal reactor size and an optimal
way of manipulating the inlet temperature based on on-line
measurements for those reactors for which detailed know
ledge of deactivation is available. Even when there is
some uncertainty about the knowledge, the procedure yields
approximate ranges of the optimal values and also some di
rections as to what is to be chosen. These results enable
one to fully account for catalyst deactivation for a substan
tial improvement in the reactor performance.


42
the selectivity increases at 448 K, remains almost constant
at 454 K and then decreases at 460 and 466 K. This behavior
can be explained in terms of and As shown in Fig.
3-4, and H^ decrease with different speeds, resulting in
the observed selectivity behavior. The results given in
the figure show excellent agreements between the theory and
the experiment.
The experimentally determined values of and were
fitted to an nth order deactivation kinetics using a non
linear regression. The results are
dH. t __
^ = kD H^1 =-5.018 x 10 exp (-8879/T)* *'
dH
jr- = ~ kD H^2 = 1.936 x 10-8 exp (5707/T)H819
The experimental values are plotted in Fig. 3-4 along with
the calculated values, which are shown as solid curves.
§3-3 Concluding Remarks
We have shown in Section §3-1 the feasibility of the
feedback control policies by simulating an adiabatic fixed-
bed where a single reaction affected by diffusion and poi
soning takes place. The feedback control policies have also
been put to test with a laboratory isothermal fixed-bed
where multiple reactions affected by sintering take place.


G.
i
E./R i = 1,2,q,D
y
g
concentration dependence of intrinsic rate
of reaction; temperature and activity de
pendence of rate of deactivation in Eq. (4-
H
reactor activity factor defined by Eq. (2-7)
H
c
current value of H
H
n
new value of H
(-AH)
heat of reaction
h
local activity factor defined by Eq. (2-5)
h
c
current value of h
hf
h at t = tf
hm
film heat transfer coefficient
h
n
new value of h
I
integral defined by Eq. (2-20)
J
performance index
kd
rate constant for deactivation reaction
K1
k., k
i q
K2
kk
2 q
K.
D
equilibrium constant
K
q
P* k
Ao q
k
intrinsic rate constant
k
q
adsorption equilibrium constant
k
a
apparent rate constant
k
a
o
preexponential factor for k^
k
rate constant for poisoning reaction evalu
PS
ated at pellet surface temperature
vi


12
:d. Cd + a(Cd.
i l-l
- C
ei-l
1=1,2,
(2-16)
C
d
C
d
o
where is the adjusted when a step change in the in-
i
let temperature, which is triggered by the bandwidth con
straint, is made at the ith step and a is a proportionality
constant with a value between zero and unity. Thus, for
a given decrease in the outlet concentration from the de
sired value, the piecewise feedback control algorithm can
be stated as follows:
'd.
i
dC
Tk [T; (T. ) ]f (C,K.) (Hc) i-1
V-* CL HI J_
in J
i=l, 2
/
(2-17)
where (H ) .. is calculated from
c l-l
C
r e
(H ) -.
c l-l
i-1
dC
C.
in
ka[Ti(Tin,i-llf(C'K1)
i=l,2,... (2-18)
(H)
c o
1
with the measured outlet concentration and inlet tempera
ture. Here, is given by Eq. (2-16). Each time the outlet
i
concentration reaches the allowed bandwidth, the new inlet


9
calculation of T as a function of z. With these T and T ,
the corresponding change in C for a selected interval of Az
can be obtained from Eq. (2-8) which in turn can be used in
Eq. (2-7) for the calculation of H.
An immediate use of the reactor activity factor calcul
able from the temperature and concentration measurements is
in the manipulation of the reactor inlet temperature for the
purpose of maintaining the desired conversion. Suppose that
the inlet temperature is adjusted intermittently, as in the
usual operation of a reactor, to compensate for the declining
activity. Consider an adiabatic reactor for clarity. If we
let the subscript c denote the current quantities and n the
new quantities resulting from a change in the reactor inlet
temperature, Eq. (2-7) can be written twice to give
H =
c
H =
n
h (z;t)dz = -
c
r C
dC
C. ^atT;(Tin) ]f(C,K )
m J
h (z;t)dz = -
n
dC
C. ^a[T;(Tln)n]f(C,K )
in
(2-11)
(2-12)
where is the desired concentration which will be attained
by changing the current inlet temperature, (Tj_n)c, to the
new inlet temperature, (T^n) R. As we shall soon see in
the next section, the activity factors h and h depend main-
c n
ly on the fraction of catalyst deactivated and slightly on
the temperature difference between bulk fluid and pellet
surface. If we neglect this change in the temperature
difference for the time being, we can set
hc(z;t> = hn(z,t)
(2-13)


54
Another major difference between the piecewise and the
continuous control is that although the temperature in the
optimal continuous control always ends at its upper limit
for the model reaction studied here (Crowe, 1970) the tem
perature in the optimal piecewise control may end at a tem
perature below the upper limit. This is shown in Fig. 4-4
where the final temperature is 621.3 K, far below the upper
limit, 673 K. However, the final temperature will approach
673 K as the number of control steps, n, increases.
A comparison of performance indices calculated from
Eq. (4-4) for seven different control policies is given in
Table 4-2 with T = 573 K and 673 K. The oerformance in
max
dices calculated with the optimal piecewise temperature
*
control policy, T, are listed in Entry 1. In Entries 2 and 3,
the temperature control policies are 1 K higher and lower,

respectively, than T but still within the interval

^Tmin' Tmax^ T^e contrl policy Tc in Entry 4 is obtained
by solving the same optimization problem with an additional
constraint of constant conversion. Note that by applying
the constant-conversion constraint, the multivariable optimi
zation problem of Eq. (4-9) is simplified to a single vari
able optimization problem
J* = max J (T ) (4-11)
c m C _L
1
since T^,... and T^ are fixed when T^ is specified.
Here subscript c denotes the performance index obtained
under the constant-conversion constraint. The performance


0.6 1 | | | | | 1
0 5 10 15 20 25 30 35
t (sec)
Maximum Values of Performance Index for Various
Reactor Sizes
40
Figure 6-5.


75
The optimization problem was solved for the reaction
system in Tables 3-1 and 3-2 using the control algorithms
of Eqs. (2-16) through (2-18) with the constraints of t^ =
105 sec. and (T. ) = 720 K. The typical behavior of
in max 2^
T. (t) and the outlet conversion is shown in Fig. 6-1 for
in
the bandwidth AX of 5% and the initial inlet temperature
of 691 K (or conversion of 87.2%). In accordance with
the constraint on t_ and the extremum condition, T. stays
f in J
at the maximum allowed temperature once it reaches the maxi
mum and the conversion decreases with time, eventually
reaching a conversion of 54.5% at t = t^, at which time
the catalyst is regenerated. As indicated earlier, the
optimization problem for a chosen x is that of finding an
initial conversion (or initial inlet temperature) that yields
a maximum. The solutions obtained for various x are shown
in Fig. 6-2 for a bandwidth of 0.05. A few observations
can be made from the figure. The results show that for a
given x, the extremum condition does yield a maximum.
Furthermore, the value of the performance index is much
more sensitive to a choice of the initial inlet temperature
(or the initial conversion) at relatively lower values of
the inlet temperature than at higher values. The locus
of the maximum values of J for various x is given in
Fig. 6-2 as the dotted line. As apparent from the figure,
there exists a maximum at around x = 22 sec. The perfor
mance index normalized with respect to the index corresponding


11
The control policy of Eq. (2-14) is a piecewise algo
rithm for the manipulation of the inlet temperature. There
fore, it will maintain the conversion at the desired level
only for a short period of time after a change in the inlet
temperature and the conversion thereafter will gradually de
crease with time until the inlet temperature is raised again.
A bandwidth for the allowed decrease in conversion can be used
to trigger the adjustment of the reactor inlet temperature.
The control algorithm for a nonadiabatic reactor can be ob
tained in a similar manner. The equation corresponding to
Eq. (2-14) is
d dC = tt
n Tk [T;(T. ) ]f(C,K.) c
C. a in n 1
in J
(2-15)
This equation, however, needs to be solved in conjunction
with Eqs. (2-3) and (2-10) for (T. ) along with the measured
in n
Tc profile.
As indicated earlier, we assumed that the heat transfer
resistance across the interface between bulk fluid and pellet
surface is negligible in arriving at the control policies of
Eqs. (2-14) and (2-15). If there exists a significant tem
perature difference between bulk fluid and pellet surface,
Eqs. (2-13) through (2-15) no longer hold. The discrepancy
between hc and hn caused by the temperature difference can
be compensated for by adjusting Cd as follows:


mass rate of reaction fluid; order of de
activation reaction
mass rate of coolant
concentration of poisoning species
number of control steps in piecewise control
total pressure
partial pressure for species A
Ei/ED, i=l/2,q
deactivation capacity of catalyst
quantity defined by Eq. (2-21)
gas constant
global rate of main reaction
tube radius
intrinsic rate
selectivity
temperature of reaction fluid
transformed function of T
coolant temperature
reactor inlet temperature
T at z=0
c
time
time at which catalyst is regenerated
or replaced in continuous control
time at which catalyst is regenerated
or replaced in piecewise control
U
overall heat transfer coefficient


71
in Fig. 5-2. This decrease in conversion and yield is
due to the large activation energy for deactivation which
limits the increase in temperature.
The valuesof the performance index for constant-conver
sion and constant-yield policies have been calculated for
comparison. The results indicate that the control policies
shown in Figs. 5-1 and 5-2 are indeed the optimal policies.
However, the percent difference between the optimal policy
and the constant-conversion policy is less than 0.1% in
terms of the performance index. As mentioned earlier, the
constant-conversion policy is the optimal policy when E^=
and E = 0. For the model reaction studied, En is close to
q 1
E_ (E.. /E_ = 0.822) and E is small (compared with E. and
2 12 q 1
E^). This explains why the constant-conversion policy is
almost as good as the optimal policies shown in Figs. 5-1
and 5-2.
§5-4 Concluding Remarks
We have derived the necessary condition for the optimal
continuous control for a parallel reaction network with
Langmiur-Hinshelwood rate expressions by direct substitution.
The optimal control policies have been obtained by direct nu
merical integration of Eq. (5-3) with the aid of Eqs. (5-4)
and (5-17). Although iterations are required for solving
Eqs. (5-4) and (5-17), it takes no more than two iterations
for each step in the t-direction because X and are smooth
functions of t and the values in the previous step provide
very good initial guesses for the current step. In fact, it


I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
o A ^ 0
Spyros Svoronos
Assistant Professor of
Chemical Engineering
This dissertation was submitted to the Graduate Faculty of
the College of Engineering and to the Graduate Council, and
was accepted as partial fulfillment of the requirements for
the degree of Doctor of Philosophy.
/LMa/ ft.
Dean, College of Engineering
Dean for Graduate Studies
and Research


26
§3-2-1 Catalyst Preparation
The catalyst used in this study of ethylene oxidation
was silver supported on fused alumina. The quarter-inch-
diameter alumina obtained from Norton Chemical Company
(Catalog No. SA-5202) was crushed into 8-9 mesh (0.20
0.24cm) with a rod miller, washed with distilled water,
and then dried in the furnace at 373Kovernight. These
2
particles were porous with 0.7~1.3 m /g BET surface area
3
and 1.9 g/cm bulk density. The alumina was impregnated
with an aqueous solution of lactic acid and silver oxide
at 368 K for 90 minutes. The solution contained silver
oxide, 85% lactic acid and distilled water in the ratio of
1:2:2 by weight. After impregnation the excess liquid
was removed and the particles were heated in a furnace for
16 hours at 643 K to decompose the silver lactate to metal
lic silver. The catalyst was further stabilized by oxy
genating at 573 K for 100 hours. Since the color of these
pellets was not uniform, only those with lighter color
were picked for the experiment. The weight gain indicated that
the catalyst contained 12.4% silver. The oxygen chemisorp
tion at 423 K was 36 y£/g.
§3-2-2 Experimental Apparatus
The reactor used for ethylene oxidation reactions
was shell-and-tube type as shown in Fig. 3-2. The shell-
and-tube section was 60.1 cm long with outside diameter
(o.d.) of 3.81 cm for the shell and 1.19 cm for the tube.


40
The temperature was kept constant at 448 K. The
outlet gas compositions were analyzed daily and the corres
ponding conversion and yield were calculated. The control
policy was that of maintaining the yield of ethylene oxide
at 8% in a piecewise manner. The bandwidth was set in such
a way that an adjustment was made whenever the yield de
creased to 6.5%. The temperature adjustments and the
corresponding reactor behavior over a period of 35 days
are shown in Fig. 3-3. The first temperature adjustment
occurred at t = 120 hr. at which time the yield dropped
down to 6.5%. This triggered an adjustment of the tem
perature in accordance with the specification of the band
width .
Consider the temperature adjustment, in particular
the procedures of calculating the new temperature which
will bring the yield back to the desired level. The value
of 8 required to calculate from Eq. (3-6) is obtained
directly from the experimental values of the conversion and
yield at the current temperature. This in turn is used in
Eq. (3-6) to calculate the current reactor activity
factor for the reaction path 1, Hlc. Rewriting Eq. (3-6)
for the new temperature and using the relationship H^c= H^n
yields
H
(3-16)
lc
(k. + 6k) k PW/F
1 2 q o
n


ACKNOWLEDGMENTS
The author wishes to express his sincere appreciation
to his research advisor, Dr. Hong H. Lee, for his guidance,
patience,and constant encouragement throughout this work.
Particular thanks are also due to Drs. G.B. Hoflund,
C.C. Hsu, J.P. O'Connell and S. Svoronos for serving on the
advisory committee.
The author also wishes to express his gratitude to his
colleagues Laks Akella, Irfan Toor and Karen Klingman
for many stimulating discussions, to staff members Tracy
Lambert and Ron Baxley for their assistance in fabricating
the experimental apparatus, and to Derbra Owete for typing
the manuscript.
Sincere appreciation is extended to his parents, Mr.
and Mrs. I-Mo Hong and to his wife, Shiao-Ing for their
support and understanding throughout his graduate study.
11


538.6 K
O
1
Figure 4-3.
_l I I I I L_J 1 I 5 30
400 800 1200 1442 1600 2000
t (hr)
Optimal Continuous Control Policy
T = 573 K
max
for the Model Reaction,


50
Table 4-1 Equations and Parameters for a Model Reaction
System for Optimal Piecewise Control
dh
dt
V
1
X = 1 exp(-xk^h)
Ad exp
r
A^ exp
T(t)
T.
l
for
(i-1) < t <
n
i 1,2 . n
T <. T. < T
mm i max
; i = 1,2 ,. .. n
Ad = 5.00 x 106 hr
Gd = 1.25 x 104 K
t = 2000 hr
n
A1 = 5.00 x 106 S
G1 = 1.00 x 104 K
T = 20 S
T .
min
T
max
n
473 K
573
K in
Figs.
(4-2)
, (4-3)
and
(4-5)
673
K in
Figs.
(4-4)
, (4-6)
and
(4-7)
10
in
Figs.
(4-2)
through
(4-
6)
= varies in Fig.
(4-7)


630
610
t (hr)
Figure 4-2.
Optimal Piecewise Control Policies for the Model Reaction,
T
max
573 K
m


Figure 4 6. Optimal Piecewise Control Policy with Constant Conversion
T = 673 K
Ln


23
the feedback control given by Eqs. (2-16) through
(2-18). The control problem is how the inlet temperature
should be manipulated whenever the outlet conversion
decreases to a certain level from the desired conversion
(bandwidth). Note that the intrinsic rate of deactivation
r^ is used here only to generate the process response.
In the feedback control, r is treated as an unknown and
P
only the on-line measurements (simulated temperature and
concentration) are used to manipulate the inlet temperature.
The constraints are the maximum reactor temperature allowed
and the final time at which the catalyst is regenerated as
the time when the final bandwidth becomes one-tenth of the
initial bandwidth. These constraints are also given in Table
3-2 along with the desired outlet concentration (conversion)
and the reactor size. The initial inlet temperature for
fresh catalyst is the one corresponding to the specified
reactor size (x) and outlet conversion, which is 691.8K.
Since the film heat transfer coefficient is quite small for
the example problem and thus the value of q is relatively
large, the proportionality factor B in Eq. (2-16) was set at
0.95.
The behavior of the model reactor resulting from the
feedback control is shown in Figure 3-1. Initially, the
inlet temperature is at its initial value and the outlet
temperature, which is the maximum reactor temperature in


13
temperature (T^n)^ is calculated for the manipulation of
the inlet temperature. All that is required for the feed
back control is the global rate for catalyst at a reference
state.
§2-2 Nature of Reactor Activity Factor
Before we proceed to multiple reactions, let us examine
the nature of the reactor activity factor. By definition,
it is an integrated value of local activity factor (Eq. 2-7).
The local activity factor, which is the activity factor at
a point in the reactor, can be obtained by solving pellet
conservation equations. Instead of solving the conservation
equations, we utilize the global rate obtained by Lee and
Butt (1982) for a reaction affected by uniform deactivation
and diffusion to get a clear picture of the local activity
factor. The global rate for this case is
*RG*D
(l-V)*5
L
[2D kl(C)]h
where
I- =
rC
g(a)da
0
q
(2-19)
(2-20)
1.2E (-AH)[2D kl]
a e
2h R T
m g
q
(2-21)


44
The results in Fig. 3-1 and 3-3 show that one can indeed
maintain the conversion or yield at the desired level with
good accuracy. It is noted in this regard that the con
trol policies can also be used to control the conversion
or yield in any desired manner, since or can be
changed each time the temperature is raised.
The accuracy of the feedback control depends mostly on
the on-line measurements (X and Y) but slightly on the reference
state kinetics determined. For example, at t = 624 hr.,
(as shown in Fig. 3-3 when T is adjusted from 460 to 466 K),
5% error in the determination of yield results in 1.7K dif
ference in the new temperature calculated, while 5% error in
activation energy for reaction path 1 (E^) leads to only 0.3 K
difference. In calculating new temperature by use of Eq. (3-16)
the on-line measurements of conversion and yield are used only
on the left hand side of the equation, while the activation
energy is used on both sides. Thus, any error in E^ is
likely to be cancelled out from both sides of the equation.
This is a good indication that the feedback control is apDli-
cable even when some uncertainty exists in the kinetic model.


48
Equation (4-1) can be solved, analytically or nu
merically, to give
h = h(T1, T2,...Tn, t) (4-6)
Integrating Eq. (4-4) with the aid of Eqs. (4-2) and (4-6)
yields
J = J(Tr T2,...Tn) (4-7)
Thus, the optimal piecewise temperature control
problem of Eqs. (4-3) through (4-5) is a multivariable
optimization problem searching for n variables (T^, ...
Tn), rather than a variational calculus problem searching
for a continuous function (T(t)). The latter is often en
countered in problems dealing with optimal continuous con-

trol. The stationary point of J can be obtained either by
applying the necessary conditions
/ ^ T
-spf- = 0 i = 1, 2,...n (4-8)
i
and then solving these n nonlinear equations numerically,
or by direct numerical search (Beveridge and Schechter,
1970). It is worth noting that even for the simplest first
order reaction and deactivation, the expression for J is
still very complex and the partial derivatives in Eq.
(4-8) are difficult to obtain analytically. Therefore,
direct numerical search is a better method for obtaining
£ £ &
the optimal temperature policy, T^. T2...,T The
optimal temperature policy leads directly to the optimal conver
sion policy through Eq. (4-2) with the aid of.Eq. (4-1).


CHAPTER 6
OPTIMAL CONTROL AND DESIGN
We address in this chapter the problem of how a fixed-
bed with catalyst deactivation should be designed to
obtain the best possible performance when it is controlled
optimally. In particular, we ask what the size of the
reactor should be that maximizes a certain performance index
when the temperature is also manipulated to maximize the per
formance index. This is a design practice in which process
control is taken into consideration. In order to clearly de
monstrate the interrelationship between design and control
and the advantage that can be gained by utilizing this
interrelationship, we assume for the time being that the
detailed knowledge of deactivation is available.
§6-1 An Optimization Problem of Piecewise Control and Design
Consider an optimization problem of a single irre
versible reaction taking place in an adiabatic reactor
given by
*
J =
max
T, T.n(t>
J =
max
T, T. (t) Ttf
m
Xdt
(6-1)
subject to the system equations of Eqs. (2-1) and (2-2)
with (mC ) = 0 for the adiabatic reactor under consideration
P C
73


37
= 3.618 x 10 2 exp(-7098/T) (mol/s.g-cat)
k2 = 3.038 x 10 1 exp(-8279/T) (mol/s.g-cat)
k = 1.942 exp (1641/T) (1/atm) (3-15)
0.04 atm <. P, <. 0.0 8 atm
Ac
448 K £ T £ 468 K
The comparisons made in Table 3-6 between the experi
mental conversions and those predicted by the kinetic
model show very good agreements, the maximum error being
1.9%. The same can be concluded for the yield and selec
tivity. It is seen from the table that the maximum errors
are 2.3 and 3.4% respectively for the yield and selectivity.
The experiments for the kinetic data lasted 9 hours.
An additional run carried out right after the period with
the same conditions as in Run No. 2 gave almost the same
conversion and yield as shown in Fig. 3-3 at t = o and t = 9
hr. This is a good indication that no significant deacti
vation had occurred during the kinetic experiments.
§3-2-4 Experimental Verification of the Control Policy
The shell-and-tube reactor at the end of the kinetic
experiments was allowed to run continuously to verify the
feedback control policy presented in Chapter 2. The feed
rate was controlled at 100 ml/min with the ethylene,
oxygen and helium concentrations of 6, 20 and 74%, respectively,


28
One thermal well (A, 0.794 cm o.d.) ran axially into the tube,
another (B, 0.0318 cm o.d.) into the shell. Silver catalyst
was packed into the annular section between the tube and
thermal well A. The shell-and-tube consisted of three
segments (from left to right in Fig. 3-2): preheating zone
(20.3 cm), reaction zone (35.6 cm) and outlet zone (5.1 cm).
Glycerin was used as coolant which circulated through the
shell side using a Haake HT22 temperature controller/circu-
lator. A total of twenty temperatures were measured as
shown in Fig. 3-2: eighteen in thermal well A and two in
B.
All temperature points (thermocouples) were connected
to a rotary selector switch which was in turn connected to
a temperature indicator. The feed gases were introduced into
the reactor at the desired mass flow rates and composition
controlled by a multiple channel electronic mass flow con
troller. Two pressure gauges were installed, one at the
inlet, the other at the outlet. The outlet composition was
analyzed by a Tracor 550 gas chromatograph. The coolant
flow rate was set high enough to maintain an essentially
isothermal reaction zone. The reactor pressure was held
constant at 1 atm.
§3-2-3 Determination of Kinetics
Ethylene oxidation involves a triangle reaction net
work (Spath and Handel, 1974; Dettwiler et al., 1979)


19
Hic=-
(CB)d
'B.
in
dC
B
(2-34)
This can be solved for (T. ) with the aid of Eqs. (2-31)
in n M
and (2-33). If, however, the control objective is to have
maximum Cg at the outlet, the choice of (Cg)^ in Eq. (2-34)
must satisfy the necessary condition:
dC
B
dz
= 0
z = l
(2-35)
Substituting Eq. (2-35) into Eq. (2-30) gives
ka ,d] = 6Va1(Te)fltCA'(CB)dI
2 e 1 e
(2-36)
which can be used to obtain (C_.) with the aid of Eqs.
B d
(2-31) and (2-33). The new inlet temperature, (T^n)R can
then be obtained from Eq. (2-34).
The control policies of Eqs. (2-28) and (2-34) can be
modified using Eq. (2-16) to obtain the piecewise algorithms
similar to Eqs. (2-17) and (2-18).


CHAPTER 1
INTRODUCTION
The usual practice of operating a fixed-bed with slow
catalyst deactivation is to raise the reactor inlet tem
perature to compensate for the declining catalytic activi
ty so as to maintain a desired conversion (Kovarik and Butt,
1982). The catalyst is regenerated when the required tem
perature becomes too high. The reactor is usually designed
to give the desired conversion without any consideration
for the catalyst deactivation. A summary of operational
strategies for a reactor subject to catalyst deactivation
is listed in Table 1-1.
A number of questions can be raised regarding these
usual practices: The first question is how we should con
trol the operating conditions of a reactor designed in
the usual way. This question has been the subject of many
studies (Chou et al. 1967; Szepe and Levenspiel, 1968; Ogunye
and Ray, 1971; Haas et al., 1974; Levenspiel and Sadana,
1978). The control policies resulting from these studies
either lead to an open-loop control or require detailed
knowledge of catalyst deactivation. Although considerable
progress has been made in our understanding of catalyst
deactivation, the deactivation is still the least understood
of all facets involved in the quantitative description of a
1


2
Table 1-1 Summary of Operational Strategies for a Reactor
Subject to Catalyst Deactivation
1. Vary reactor temperature with time to maintain a constant
conversion with a constant reactor feed flow rate. A typi
cal policy for large throughput (400 MM pounds per year) and
slow deactivation rates (months to years of catalyst life).
2. Vary throughput of the reactor feed while holding the reactor
temperature and conversion constant. A possible policy for
medium deactivation rates (weeks to months catalyst life)
and small to medium throughput (about 25 MM pounds per year)
systems.
3. Allow the conversion to fall while holding the reactor feed
flow rate and reactor temperature constant. Similar appli
cations as in Item 2.
4. Maintain the fresh feed rate and reactor temperature constant
and let the recycle flow increase. Similar application as
in Item 2.
5. Use a combination of reactors in parallel and the policies of
Items 1 or 3. Usually, with two reactors in parallel, one
will be off-line for catalyst regeneration while the other is
operating. A typical policy for large throughput and medium
to fast deactivation rates (days to months of catalyst life).
6. Continuous catalyst regeneration while maintaining constant
conversion, throughput, and reactor temperature. A typical
policy for large throughput, rapid deactivation systems
(hours to days of catalyst life) .
Reprinted from Cat. Rev. .2_4 499 (1982) by courtesy of Marcel
Dekker, Inc.


32
partial pressure but is dependent on the reaction tem
perature. This can also be seen from Eq. (3-8) where the
left hand side is 1/S and the right hand side depends only
on temperature. Thus, at any given temperature, X/Y is
constant, and Eqs. (3-9) can be used to obtain k and knk
q 1 q
by a linear regression of (£n j-_^) on (y'Aox) The rate
constant k2 can then be calculated from Eq. (3-10). By de
finition and the stoichiometric relationships, the conversion
and yield at the reactor outlet can be related to the inlet
and outlet gas compositions as
X = 1 -
Y = X -
y'.
A1
y;
Ao
Yr
Cl
2y'7
(3-11)
(3-12)
Ao
where y'_ and y' are the mole fractions of ethvlene at the
Ao A1
inlet and outlet, respectively, y^ is the mole fraction of
carbon dioxide at the outlet.
We have shown that at a given temperature, the rate con
stants, k^ and k^, and the absorption equilibrium constant, k^,
can be determined from the inlet and outlet gas compositions,
y^o' Y'a1 an<^ Yci' using Eqs. (3-9) through (3-12). The kinetic
data for ethylene oxidation reactions are given in Table 3-3.
The reactor was maintained isothermal by controlling the coolant
flow rate at a high level. Due to the low ethylene concentra
tion and the high coolant flow rate, the heat generated by the
reactions was removed immediately by the coolant. The oxygen com
position in the feed gas was maintain at 20% for all runs, with



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PAGE 102

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PAGE 103

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O 20 40 60 80
n
Figure 4-7. Comparison Between Performance Indices Obtained With and Without
Constant-Conversion Constraint
100


Table 3-5
Results
of Linear Regression
c
4-1
0
k. on 1/T
i T
1/T
(xlO-3)
In k.
i
(xlO)
(&n k )^ ,
i cal
(xlO)
Error
(%)
i
A.
l
E. /R
i g
Corr.
Coeff.
1 448
2.232
-1.196
-1.916
o
o
i
3.618xl02
7.098xl03
-0.9991
458
2.183
-1.883
-1.882
0.1
468
2.137
-1.848
-1.849
-0.1
2 448
2.232
-1.966
-1.967
-0.1
3.038xl0_1
8.27 9x10 3
-0.9980
458
2.183
-1.923
-1.920
0.2
468
2.137
-1.887
-1.888
1
O

q 448
2.232
0.4340
0.4340
0.0
1.942
-1.647xl03
-0.9998
458
2.183
0.4261
0.4260
o

o
468
2.137
0.4183
0.4183
1
o

o
f
calculated from regression
model


4
A much more important question than the three discussed
above has to do with the inherent interrelationship be
tween process controllability and process design. It is
intuitively clear that the process design dictates the
controllability, for the parameters involved in describing
a process contain design parameters which in turn dictate
the way the process can be controlled. It is clear
therefore, that the best possible performance of the process
can be attained when the process design and control are
combined. For the reactor under consideration, this means
that the best possible performance can be attained when the
catalyst deactivation is taken into consideration not only
for the reactor control but also for the reactor design.
The question, therefore, is what the reactor size and the
feedback control should be for the best performance of the
reactor. In Chapter 6, the answer to this question is pre
sented .


49
§4-2 Optimal Piecewise Control Policies
As an illustration, the optimal piecewise control
problem of Eqs. (4-1) through (4-5) was solved for a model
reaction system of first order irreversible reaction with
first order deactivation (Table 4-1). The multivariable
optimization problem was solved by direct numerical search
using the ZXMWD subroutine from International Mathematical
and Statistical Libraries (IMSL). The ZXMWD subroutine
is used for obtaining the global minimum or maximum for a
multivariable function with constraints. The constrained
maximization problem of
J* = max J(T T ,...T ) (4-9)
T . T X Z n
1 n
with
T. < T. < T i = 1, 2, n
mm i max
is transformed in ZXMWD subroutine to an equivalent, but
unconstrained problem
J* = max J[T +(T T )Sin2T,,...,T +
= = mm max mm 1 mm
1 n
(T T )Sin2T ]
max mm n
(4-10)
where T. is the transformed function of T-and is now uncon-
l i
strained. With this transformation, each possible global
maximum, including any on the boundary, is transformed into a
local maximum (Box, 1966).
Shown in Figure 4-2 are the optimal piecewise control
policy for the model reaction. The optimal temperature


1.2
H
1
or
3
4^
U)
Figure 3-4. Activity Behavior for Ethylene Oxidation Reaction


66
* Yh
P1
K
1
Substituting Eq.
Eq. (5-15) gives
+ K2X + Kl + K2h Xh P3 K1 + K2
(5-16) into (5-11) with the aid of
X
(5-16)
K,
K1 + K2 K +
S { q 1-X
(1-Pl + P3)X f = C (5-17)

Equation (5-17) is the necessary condition for J to be
maximum. The integration constant C can be determined by
a boundary condition:
h = hf at t = tf (5-18)
if h^ is specified, or by a natural boundary condition
(Denn, 1969):
= 0 at t = tf (5-19)
if hf is unspecified.
The necessary condition can be simplified if the acti
vation energies for both reactions are equal (E^ = E2),
and the adsorption equilibrium constant K is temperature
9
independent (E =0). In this case, Eq. (5-17) gives
**4
(A1 + A2J (Aq + 1-X'
IT \-piAqx + P^nd-xj + iA + jzx):
x> = c
(5-20)


41
Here, the subscript n denotes the new temperature and
is the desired conversion. Since local activities change
little during the short period of temperature adjustment,
3, by definition, also changes little. This means that
the same 3 obtained from Eq. (3-8) can be used in Eq.
(3-16) to calculate the new temperature. If the conversion
is the one to be controlled, then Eq. (3-18) can be
used to calculate T since it is the only unknown in Eq.
(3-18). In this case, however, the yield is to be con
trolled. Hence, we need an additional relationship between
X^ and Y^. This relationship can be obtained by rewriting
Eq. (3-8) for the new temperature
X,
= 1 + e
d
n
(3-17)
This relationship was used for the calculation of the new
temperature in the experiment.
The experiments were conducted over a period of 884
hours. As shown in Fig. 3-3, the first adjustment was
made at t = 120 hr. The temperature was raised to 454 K,
which brought the yield to 7.95% for the desired yield of 8%.
Two additional temperature adjustments were made at t = 336
hr. and t = 624 hr. The experimental values of the yield of
ethylene oxide resulting from the temperature adjustements were
both 7.92%. The behavior of conversion and selectivity are
also shown in Fig. 3-3. It is interesting to observe that


Figure 4-1. Temperature Behavior for a Piecewise Control


Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
DESIGN AND CONTROL OF FIXED-BEDS
AFFECTED BY CATALYST DEACTIVATION
By
Jan-Chen Hong
August 1984
Chairman: Dr. H. H. Lee
Major Department: Chemical Engineering
On-line measurements of temperature and concentration
and the global rate at any reference state of catalyst can be
used to estimate a measure of catalyst deactivation. A feed
back control policy is obtained from this measure of cata
lyst deactivation for manipulating the reactor outlet con
version or yield in any desired manner. This feedback con
trol enables one to obtain the desired conversion or yield
without any knowledge of deactivation kinetics, and the re
sults are applicable to any type of deactivation. The feed
back control policy has been verified by experiment based
on ethylene oxidation reactions over a silver catalyst using
a laboratory shell-and-tube reactor. Excellent agreements
have been reached between the theory and the experiment.
x


81
T (sec)
VK)
Figure 6-4. Performance Index as a Function of
Reactor Size and Initial Temperature*
The dotted line shows the locus of J .


Table 3-6 Comparison Between Experimental Data and Kinetic Model
RUN
X+
x i *
cal
Error ^
Y
Ycal
Error
S
S n
cal
Error
1
16.04
15.94
0.6
9.95
9.96
-0.0
62.0
62.4
-0.6
2
11.93
11.70
1.9
7.45
7.30
2.0
62.5
62.4
0.0
3
9.23
9.22
0.1
5.79
5.76
0.5
62.7
62.4
0.4
4
22.51
22.30
0.9
13.53
13.62
-0.6
60.1
61.1
-1.6
5
16.55
16.55
0.0
9.98
10.11
-1.2
60.3
61.1
-1.2
6
13.26
13.12
1.1
7.83
8.01
-2.3
59.0
61.1
-3.4
7
30.79
30.53
0.8
18.65
18.25
2.2
60.6
59.8
1.4
8
23.31
22.98
1.4
14.00
13.73
2.0
60.1
59.8
0.5
9
18.48
18.34
0.8
10.81
10.96
-1.4
58.5
59.8
-2.1
Experimental data
*
Kinetic model
§Error = (X/Xcal-1)100%


61
enough. In practice, the temperature adjustments are made
intermittently because of the slow nature of catalyst de
activation in fixed-beds. From the fact that the curve
~ ~ *
for J lies above the curve for J it can be concluded
c
that the optimal piecewise control policy is an increasing
conversion policy as shown in Fig. 4-2 and 4-4. However,
the constant-conversion policy can be used when number of
control steps is large, say, 30.
The problem formulation and solution method presented
in Section 4-1 is neither restricted to an isothermal
reactor nor to a single irreversible reaction. For example,
one can consider a reversible reaction taking place in an
adiabatic fixed-bed. In this case, the reactor tempera
tures, TVs, in Eqs. (4-1) through (4-5) should be replaced
by the temperatures at the inlet of the reactor, (T. ).'s.
Equation (4-2) might then become implicit in X because of
the reversible reaction and the nonisothermal reactor.
However, X can still be solved numerically. Thus, the
problem remains a multivariable optimization problem which
can be solved by application of an appropriate numerical
method.


Page
§6-2 An Optimization Problem of Continuous
Control and Design 78
§6-3 Concluding Remarks 82
CHAPTER 7 CONCLUSIONS 84
REFERENCES 88
BIOGRAPHICAL SKETCH 90
IV


I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
Hong H. Lee, Chairman
Associate Professor of
Chemical Engineering
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
Gar B. Hoflund
Associate Professor of
Chemical Engineering
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
(L/i,,. (Xj ~Hu
Chen-Chi Hsu
Professor of Engineering
Sciences
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
J S
John P. O'Connell
Professor of Chemical
Engineering


NOTATION
(C )
P c
f
preexponential factor, i=l, 2, q
-AH9/pC
z P
-AHX(1-AH2/AH1)/P0pb1
(-AH,-bAH)/pC
1 2 p
-AH~/pC
2 p
ratios of stoichiometric coefficients
concentration of main reactant; integration
constant
desired outlet concentration of main reactant
modified given by Eq. (2-16)
initial value of C,
d
inlet concentration of main reactant
specific heat of reaction fluid
specific heat of coolant
effective diffusivity
activation energy for main reaction
i=l, 2, q
activation energy for deactivation reaction
molar flow rate
molar flow rate of reactant mixture at the
inlet
apparent concentration dependence of rate of
main reaction; temperature and activity de
pendence of X in Eq. (4-2)
v


57
indices obtained from Eq. (4-11) are listed in Entry 4. In
Entries 5 through 7, the temperature policies are all
constant with respect to time. It can be seen from the table
k
that T (the optimal temperature policy shown in Fig. 4-2)
is indeed the optimal temperature control policy, since all
the other policies result in smaller values of the performance
index.
The single-variable optimization problem of Eq. (4-11)
was solved numerically using Fibonacci search (Beveridge
and Schechter, 1970). The results are shown in Fig. 4-5
for T = 573Kand in Fig. 4-6 for T = 673 K. It can be
max ^ max
seen from the figures that the conversion is brought back
to the initial conversion each time the temperature is ad
justed .
§4-3 Concluding Remarks
A comparison between the optimal performance indices
obtained with and without constant-conversion constraint is
shown in Fig. 4-7 as a function of the number of control ac-
. *
tions (n). It is seen that as n approaches infinity, J and
k
approach the same asymptotic value (0.56556 for the model
reaction with T x = 673 K). This indicates that the con
tinuous control (which corresponds to an infinite number of
piecewise control actions) is superior to the piecewise
control in terms of the performance index. On the other
hand, J > 99.5% when n>21 as shown in Fig. 4-7, implying
that the piecewise control is almost as good as the con
tinuous control if the number of control actions is large


900
850
800
n
:)
750
700
650
75
65
55
1300
T
e
(K)
2.7
Lgure 3-1.
Temperature and Conversion Behavior of a Fixed-bed
Subject to Inlet Temperature Manipulation


82
Thenormalized optimal performance index is plotted in
Fig. 6-5 as a function of reactor size. It can be seen
from the figure that the performance index can be improved
significantly if the optimal size is chosen.
§6-3 Concluding Remarks
We have so far assumed that detailed knowledge of ca
talyst deactivation is available. The same optimal con
trol and design can still be carried out if some knowledge
of deactivation is available, however uncertain it may be,
since what is sought in such a case would be approximate
values of the optimal size and inlet temperature and some
directions as to what to choose. In the example considered
in Section §6-1, for example, one would choose a relative
ly high initial conversion (or a high inlet temperature)
rather than a low initial conversion in view of the sen
sitivity discussed earlier.


46
as shown in Fig. 4-1. The pseudo steady-state balance
equations can be written as
dh
dt
g(T1,
Tn'h)
1
(4-1)
X = f(T1, T2,...Tr h)
(4-2)
where h is the activity of the catalyst, X is the conver
sion at the outlet of the reactor, and g represents the
dependence of deactivation rate on the temperatures (T)
and activity (h). Equation (4-2) is obtained by integrating
the mass balance equation from the inlet to the outlet of the
reactor.
The optimization problem is to maximize the total
amount of feed converted over a fixed total reaction time, t ,
by choosing the best set of temperatures, [T^, T2,...,T ].
That is '
k
J = max J i = 1, 2,...n (4-3)
T.
i
with
J =
n
n
n
t.
i
Xdt =
t i=l
n
Xdt
'i-1
and subject to
T min i max
(4-4)
t
At = t. t. = constant = i = 1, 2,
i i-i n '
n
(4-5)


68
Table 5-1 Parameters Used for the Optimal Continuous Con-
trol Policies
K. = k.k = 7.028 x 10 2 exp(-5451/T)(mol/sg-catatm)
J- x q
K2 = ^2^c = ^'^OO x 10 1 exp(-6632/T)(mol/sg-catatm)
K = p k = 1.165 x 10"1 exp(1647/T) (-)
CJ AO CJ
WP 6
T = = 1.134 x 10 (g-cat-atm*s/mol)
Fo
hf = 0.4 (-)
tf = 1000 (hr)
Kd = 4.0 x 102 exp(-6000/T) (1/hr) in Fig. 5-1
= 4.0 x 102 exp(-7000/T) (1/hr) in Fig. 5-2
E1
Pi = = 0.9085 in Fig. 5-1
1
= 0.7787
P2
jr = i-ios
ed
= 0.9474
in Fig. 5-2
in Fig. 5-1
in Fig. 5-2
m = 1.0
e1/e
2
0.822


7
depend on the concentration C and thus is applicable to con
centration-dependent deactivation. Since C is a function of
z at a given time, h(C;t) is a subset of h(z;t). The
pseudo steady-state assumption is based on the fact that
the time scale of catalyst deactivation is many orders of
magnitude larger than that for the reactor to reach a steady
state.
For the purpose of deriving an expression for a measure
of the extent of deactivation, we rewrite Eq. (2-1) with
the aid of Eq. (2-4) as
^ = -xh(z;t)k f(C,K.) (2-6)
dz a j
Here again, one can write h as h(C;t) in place of h(z;t)
for dependent deactivation. This equation can be integrated
from the reactor inlet to the outlet to give
C
r e
dC
Tk (T) f (C,K .)
JC. a 3
m
-I
h (z; t) dz = H (t)
0
(2-7)
where C and C. are the concentrations at the outlet and
e m
the inlet, respectively, at any given time. By definition,
H is unity at reference state since the activity factor
h is uniform at unity throughout the reactor. The quantity
H is a measure of the extent of catalyst deactivation which
represents the activity for the reactor and thus may be termed
"reactor activity factor." The value of H decreases as the
outlet conversion decreases due to the catalyst deactivation
for a given inlet temperature.


1.0
630
h
or
X
t (hr)
ui
U1
Figure 4-4.
Optimal Piecewise Control Policy
T
max
673 K
for the Model Reaction,


6
where C and T are the concentration and temperature of the
reactant, and is the countercurrent coolant temperature.
Here m and m are the reaction fluid and coolant mass rates,
c
C and (C ) are the specific heat capacities for the re-
P P <-
action fluid and the coolant, Z is the reactor length, z
is the axial reactor coordinate normalized with respect to
Z, v is the superficial velocity assumed constant in the
radial direction, is the bed porosity, U is the overall
D
heat transfer coefficient, and is the tube radius.
The global rate based on pellet volume, can be expressed
as
R_ = h(z;t)k f(C,K.) (2-4)
Lj a
where k is the apparent rate constant in the form of the
a
Arrhenius relationship and f represents the apparent depen
dence of the rate on concentration and equilibrium constants
Kj. In the absence of diffusion and deactivation effects,
the apparent rate reduces to intrinsic rate. The activity
factor h (Wheeler, 1955) is defined by
h (z ; t) =
(RG*D
(Vr
(2-5)
where the subscripts r and D denote reference-state and de
activated catalyst, respectively. According to the defini
tion, h is unity for catalyst at the reference state. This
activity factor decreases with time due to catalyst deacti
vation and thus depends on time. It should be recognized
in Eq. (2-5) that the definition of h is general in that h can