MODELING, CONTROL, AND OPTIMIZATION OF FIXED BED REACTORS by KISHOR G. GUDEKAR, B.S. A DISSERTATION IN CHEMICAL ENGINEERING Submitted to the Graduate Faculty of Texas Tech University in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY Approved December, 2002
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MODELING, CONTROL, AND OPTIMIZATION
OF FIXED BED REACTORS
by
KISHOR G. GUDEKAR, B.S.
A DISSERTATION
IN
CHEMICAL ENGINEERING
Submitted to the Graduate Faculty of Texas Tech University in
Partial Fulfillment of the Requirements for
the Degree of
DOCTOR OF PHILOSOPHY
Approved
December, 2002
ACKNOWLEDGEMENTS
I would like to express my sincere thanks to my advisor Dr. James B. Riggs for
his financial support, guidance, and patience throughout the project. I would like to
express my thanks to Dr. Karlene A. Hoo for her guidance in the project. I would also
like to thank Dr. Theodore F Wiesner, and Dr. Surya D. Liman for being a part of my
dissertation committee.
There are many people who have influenced my life. I am grateful to the
Kawathekar family (Rohit, Gouri, and Anuya) for their constant support, love and care.
Special thanks to Govindhakanan for his constant encouragement and motivation during
the times of frustration and disappointment and enlightening the views about life.
I am also grateful to the centaur2 group (Shriram, Shree, Parag, Alpesh, Namit,
Satish, Mukimd, Makrand, Kulin, and Dungar) for making my stay in Lubbock pleasant. I
am thankful to the rapchick group (Rahul, Sameer, Milind, Sachin, Simil, Puru, Doctor
Sunil, Vijay, Robin, Kirti, and Vinay) for making my stay memorable in Lubbock. I
cannot forget those late night parties and oxir regular visits to the recreation center.
I would like to thank my fellow graduate students Dale Slaback, Eric Vasbinder,
Danguang Zheng, and Tian for making my stay pleasant in the department. I wish to
express my thanks to Matthew Hetzel for his help with the computer problems.
Most importantly, this could not have been possible without constant support,
love, and encouragement from my parents, my brother and sister, and other family
members and friends back home.
TABLE OF CONTENTS
ACKNOWLEDGEMENTS n
ABSTRACT vi
LIST OF TABLES viii
LIST OF FIGURES ix
CHAPTER
1 INTRODUCTION 1
2 LITERATURE SURVEY 5
2.1 Modeling of Fixed Bed Reactor 5
2.2 Solution Procedure 9
2.3 Fixed Bed Reactor Control 10
2.4 Optimization 12
2.5 Multiplicity, Bifurcation Theory and Stability 12
3 MODEL DEVELOPMENT FOR A VINYL ACETATE REACTOR 18
3.1 Generalized Dynamic Model for a Fixed Bed Reactor 20
3.2 Steady State Vinyl Acetate Reactor Model 24
3.3 Orthogonal Collocation 30
3.4 Catalyst Deactivation Model 36
3.5 Nomenclature 38
111
OPTIMIZATION OF A VINYL ACETATE REACTOR 44
4.1 Model VaUdation 45
4.2 Offline Optimization Approach 54
4.3 Sensitivity Analysis 56
4.4 Onhne Optimization 58
4.5 Nomenclature 62
MODEL DEVELOPMENT FOR ETHYLENE OXIDE PROCESS 64
5.1 Process Description 64
5.2 Reaction Chemistry and Mechanism 67
5.3 Kinetics 68
5.4 Mathematical Modeling Assumptions 69
5.5 Mathematical Model of Ethylene Oxide Reactor 74
5.6 Orthogonal Collocation 77
5.7 Modeling Equations for Steam Generator 85
5.8 Modeling Equations for Gas-Gas Heat Exchanger 86
5.9 Modeling Equations for Separation System 88
5.10 Catalyst Deactivation Model 88
5.11 Nomenclature 89
OPTIMIZATION AND CONTROL OF ETHYLENE OXIDE PROCESS 94
6.1 Model Validation 94
6.2 Offline Optimization Approach 103
6.3 Contirol of Ethylene Oxide Reactor 106
IV
6.4 Nomenclature 110
7 BIFURCATION ANALYSIS OF ETHYLENE OXIDE PROCESS 112
7.1 Bifurcation Study of an Industrial Ethylene Oxide Process 112
7.2 Continuation Algorithm to Develop Bifurcation Diagram 113
NPSOL (1986) solver is used to find the empirical constants by minimizing the following
weighted objective function.
The model parameters are found by minimizing the following objective function.
mm Age
Obj = j ; day^O
4 Wf -Wt
/=! Wt
(\-W)
Ind,
rp Ind rp Pred f cin - cin \ 2 /
rp Ind rp Pred cout -' cour
L Ind
P„ Ind )'-
+ (•
rp Ind rp Pred
T:. Ind
rp Ind rp Pred \2 . (out •'•out \2
P„ Ind
By minimizing the above weighted objective function, we get the following
values of the model parameters:
a = 0.0125
y5= 0.0133
Y = 0.020.
6.1.2 Catalyst deactivation model validation
Figures 6.1 to 6.7 show the comparison between the model prediction and
industrial data over the entire life of the catalyst. It can be seen that the catalyst
deactivation model predicts the industrial data quite well.
Because of confidentiality reasons, we are not able to provide actual values of the
operating conditions (e.g., temperature and composition) in the figures and tables.
99
M + 4
o Q. E o o
M + 2
a:
4 M
• • A A
• J
A
•
Time
• Industrial data A Model Predicrtion
Figure 6.1 Comparison between industrial data and model prediction for ethylene reactor outlet composition
o M + 3
o Q. E o o g M + 2
M + 1
Time
• Industrial data A Model Prediction
Figure 6.2 Comparison between industrial data and model prediction for oxygen reactor outlet composition
100
M + 2.0r-
o I M + 1.6 o (J
g M+1.2
f M + 0.8
cr
M + 0.4
M
Time
• Industrial data A Model Predic:tion
Figure 6.3 Comparison between industrial data and model prediction for ethylene oxide reactor outlet composition
o I M + 12 o o 0)
"D x o ''^ M + 8
o
3 M + 4 o
^ ^
<D
M
Time
• Industrial data A Model Predic:tion
Figure 6.4 Comparison between industrial data and model prediction for carbon dioxide reactor outlet composition
101
TO+ 50
3
n (1)
u. b £ m ro O)
(I)
o
tor
<J
0)
ce
TO+ 40
TO + 30
TO + 20 1
i *
T0 + 10
£i
P *m -y • •
A
f^
•
TO
Time
• Industrial data A Model Predit:tion
Figure 6.5 Comparison between industrial data and model prediction for reactor outlet gas tempeature
TO+ 40
CO
I TO+ 30 E <u
§ TO+ 20 o
2 TO + 10 u ro 0)
a:
i ^ t^cM^
TO
S 6
A •
Time
• Industrial data A Model Prediction
Figure 6.6 Comparison between industrial data and model prediction for reactor outlet coolant temperature
102
TO+ 30
ro 0)
I TO+ 20
J5 o o u
= TO+ 10
^.
A „
^
TO
A A A A &^
Time
• Industrial data A Model Prediction
Figure 6.7 Comparison between industrial data and model prediction for reactor inlet coolant temperature
6.2 Offline Optimization Approach
In the ethylene oxide reactor, higher temperatures lead to excessive formation of
carbon dioxide and water which resuhs in loss of selectivity. Also, the lower
temperatures result in lower conversion and loss of productivity. There is a need for
conversion and selectivity balance which will be met by careful control of optimal shell
side inlet temperature profile. Since the catalyst deactivates over the period of time, shell
side inlet temperature is increased to compensate for the loss of activity. The manner in
which the shell side inlet temperature is changed affects the net profit of the process. This
motivates us to carry out an optimization study for this process. The optimization
problem involved determining the optimal shell side inlet temperature profile over the run
length of the operation by maximizing the net profit of the process.
103
Figure 4.8 illustrates the optimization procedure. The continuous temperature
profile over the length of the operation is expressed m terms of node values at specific
points in the time domain comiected together by smoothly varying mterpolatmg
polynomial. The optimizer uses these node values as decision variables. The intermediate
values for any particular temperature required by the simulator are provided by applying
cubic spline interpolation (Riggs, 1994).
The optimization was carried out for fixed catalyst life and fixed reactor inlet
conditions. The optimizer sought the four decision variables such that the profit fimction
(O) value over the entire run period was maximized.
^ = P,o-EO-P,-E-P^-0
The carbon dioxide separation cost is insignificant and not considered in the profit
function.
6.2.1 Offline optimization results
The optimizer (NPSOL, Gill 1986) calculates the optimum temperature profile,
which gives the profit improvement of 8.56% over base case.
From Figure 6.8, it can be seen that the optimum temperature for the initial period
is higher than the base case temperature. For the remaining period of time, the optimum
temperature is greater than or almost equal to the base case temperature profile.
104
TO+ 50
TO+ 45 (U
a TO+ 40 ro
I TO+ 35
*Z TO+ 30
Z TO+ 25 c g TO+ 20 O o TO+ 15 •o ro <u a:
TO+ 5
TO
Time
• Base case • Optimal A node values
Figure 6.8 Comparison between base case temperature profile and optimum temperature profile
6.2.2 Optimization for different production rates
Above offline optimization procedure is used to calculate profit for different
production rates. Three different production rates are considered: (a) 10% decrease over
the base case production rate, (b) 10% increase over the base case production rate, and (c)
20% increase over the base case production rate. Table 6.3 shows the percentage profit
improvement over the base case for different production rates
105
Table 6.3 Percentage profit improvement over the base case for different production rates
% Over the base case production rate % profit increase over the base case
10% increase
10% decrease
20% decrease
9.02
8.96
9.43
6.3 Control of Ethylene Oxide Reactor
As described earlier, the process studied consists of a feed effluent heat
exchanger, a muhitubular fixed bed reactor, a steam generator, and a separation system.
The exothermic heat of reaction from the reactor is removed by passing coolant on the
shell side of the reactor. A portion of the heated coolant is passed through a steam
generator to produce steam and the total coolant stream is recycled back to the shell side
of the reactor. Figure 6.9 shows a single-loop PI control system that uses the flow rate of
the coolant that is passed through the steam generator to control the inlet temperature of
the coolant to the reactor.
Steam Generator
r-><}^X
R E A C T O R
Heat Excli. Separation System
TC K-
Fiaure 6.9 Schematic of the reactor inlet coolant temperattire control system
106
The PI controller is tuned for a change of 1°C in the set point of the reactor inlet
coolant temperature. Figure 6.10 shows the response of the reactor inlet coolant
temperature. Coolant flow through the steam generator is manipulated to control the
coolant inlet temperature to the reactor. Figure 6.11 shows the response of the flow
through the steam generator. Coolant flow through the steam generator and bypass
coolant flow is mixed before sending it to the reactor. Since we want an increase in the
coolant inlet temperature to the reactor, flow through the steam generator decreases first
(this means an increase in bypass flow) to increase in the coolant inlet temperature to the
reactor.
TO+ 2.
TO+ 1.6
<u 3 ro » TO + 1.2 Q. E o o
Z TO+ 0.8 c j5 o o O
TO + 0.4
TO
Time
Figure 6.10 Response of controlled variable to 1°C increase in set point
107
TO+IOOOT
o TO+ 800 4-1
ro k. o c o u> E TO+ 600 ra 0)
In
o TO+ 400
0)
ro
5 TO+ 200
u.
TO
Time
Figure 6.11 Response of manipulated variable to 1*C increase in set point
The performance of the controller is checked against step in a disturbance (i.e.,
change in carbon dioxide inlet composition). A 0.5% mole increase in the carbon dioxide
reactor inlet composition is made. Figure 6.12 and Figure 6.13 show the response of the
reactor inlet coolant temperature and flow through the steam generator, respectively.
108
TO+ 0.12
£ 3
4-»
ro g. TO+ 0.08 E o
4->
c _ra o o o
.E TO + 0.04
0)
on
TO
Time
Figure 6.12 Response of controlled variable to 0.5% change in the disturbance
F + 160 T
o F + 140
Time
Figure 6.13 Response of manipulated variable to 0.5% change in the disturbance
109
6.4 Nomenclature
Age Age of the catalyst in days
a Activity of the catalyst (0 < a < 1)
C;" Industrial reactor outlet mole % of i"' component
C^""' Model Predicted reactor outlet mole % of /"• component
$ P^ Cost of Ethylene (—)
$ P^o Price of Ethylene Oxide ( —)
$ PQ Cost of Oxygen (—)
T - Outlet tube side temperature
T^J"'' Industrial reactor coolant inlet temperature
T^J'"' Model predicted reactor coolant inlet temperature
T '""^ Industrial reactor coolant outlet temperattire cout
Tcou^'^ - Model predicted reactor coolant outiet temperature
TJ"^ Industrial reactor gas inlet temperature
TJ'"' Model predicted reactor gas inlet temperature
T '"^ - Industrial reactor gas outlet temperature out
TJ''^ Model predicted reactor gas outlet temperature
t ^ - Catalyst life in years
110
W - weight, 0<W <l
Greek Letters
« , / ? , / Empirical constants
111
CHAPTER 7
BIFURCATION ANALYSIS OF ETHYLENE OXIDE PROCESS
7.1 Bifiircation Study of an Industrial Ethylene Oxide Process
For a long time, it has been recognized that the nonlinear behavior (i.e.,
input/output multiplicities) of chemical reactors might have an important effect on the
operation difficulty of such process (Seider, 1990). Bifiircation theory has been
recognized as a very useful tool to address the nonlinear pattern behavior of processing
systems subject to the variation of some parameters (Kuznestov, 1998).
In this work, the open-loop and closed-loop nonlinear bifurcation analysis of an
industrial ethylene oxide reactor is performed. Ethylene oxide is one of the most
important pefrochemical intermediates and one of the raw materials used for the
production of glycol, polyethylene glycol and glycol ethers. The production of the
ethylene oxide is a critical process because the reactor can generate eleven times as much
heat in a runaway condition as under normal operating conditions. Therefore, the safety
issues for an ethylene oxide reactor system are dominant as industry tries to operate them
in an economically advantageous marmer.
The aim of this work is to provide a first look into the operability problems faced
by ethylene oxide reactor and to perform open-loop and closed-loop bifurcation studies
using the benchmarked model of the ethylene oxide reactor system. The steady state
operability problem is addressed by using nonlinear bifurcation techniques. The ethylene
oxide commercial facility that we studied did not operate at higher operating
112
CHAPTER 7
BIFURCATION ANALYSIS OF ETHYLENE OXEDE PROCESS
7.1 Bifurcation Study of an Industrial Ethylene Oxide Process
For a long time, it has been recognized that the nonlinear behavior (i.e.,
input/output muhiplicities) of chemical reactors might have an important effect on the
operation difficulty of such process (Seider, 1990). Bifurcation theory has been
recognized as a very useful tool to address the nonlinear pattern behavior of processing
systems subject to the variation of some parameters (Kuznestov, 1998).
In this work, the open-loop and closed-loop nonlinear bifurcation analysis of an
industrial ethylene oxide reactor is performed. Ethylene oxide is one of the most
important petrochemical intermediates and one of the raw materials used for the
production of glycol, polyethylene glycol and glycol ethers. The production of the
ethylene oxide is a critical process because the reactor can generate eleven times as much
heat in a runaway condition as under normal operating conditions. Therefore, the safety
issues for an ethylene oxide reactor system are dominant as industry tries to operate them
in an economically advantageous maimer.
The aim of this work is to provide a first look into the operability problems faced
by ethylene oxide reactor and to perform open-loop and closed-loop bifurcation studies
using the benchmarked model of the ethylene oxide reactor system. The steady state
operability problem is addressed by using nonlinear bifurcation techniques. The ethylene
oxide commercial facility that we studied did not operate at higher operating
112
temperattires because of the associated risks of a reactor runaway. Therefore, stable
temperature control of the ethylene oxide reactor is important. An analysis of the stable
control region of the system is developed as a function of the operating temperature,
catalyst activity, and disturbance direction and magnitude. In the open literature there are
no published papers on the nonlinear bifurcation analysis of the ethylene oxide
manufacturing process.
7.2 Continuation Algorithm to Develop Bifiircation Diagram
Elementary catastrophe theory might be used in order to detect analytical
conditions under which input/output multiplicities could emerge. However, one of the
major problems related to the use of catastrophe theory is that it requires collapsing the
entire mathematical model into a single algebraic equation. The procedure is totally
impractical for large-scale models. Because of the complexity (higher dimensionahty) of
the ethylene oxide process model equations, a purely numerical procedure is used to
characterize the multiplicity behavior (bifurcation study) of the ethylene oxide process. A
numerical technique enabling us to obtain one branch of solutions (or more branches of
solutions mutually connected at branch points) is called the continuation technique
(Kubicek, 1983). The continuation algorithm can produce a continuous curve (consisting
of branches of solutions). For any other curve (branch of solutions), we need to obtain an
initial estimate of the solution in order to begin the continuation procedure. How can we
obtain all steady state solutions of tiie given set of equations? The task may be
particulariy demanding in situations where we have poor preliminary estimates of the
113
solution and chosen iteration method diverges. In such situations, we can use randomly
generated initial estimates of solution for every initial estimate of the solution: either the
iteration algorithm converges or diverges. The new converged solutions thus, obtained
are subsequently stored into a memory of solutions. If a sufficiently high number of
initial estimates are chosen, the probability of solving the problem is high. The number of
random initial estimates necessary can sometimes be rather high. In principle, this
technique enables us to start from a known solution and continuously compute solutions
along a chosen branch.
7.3 Stability of Steady State Solutions
Stationary state x of the differential equation is called locally stable if
l im | |x (0-x 11=0,
for x(0) chosen in a sufficiently small neighborhood of x , i.e., for x(0) such that
\\x(t)-x\\<5,
where (5 is a conveniently small number.
Consider a system of linear differential equations with constant coefficients
dx — = Ax dt
where x € i?". Let the only stationary solution be 3c = 0. If the eigenvalues of matrix A
are known, we can determine the stability of this unique stationary solution (Kubicek,
1983). If for every eigenvalue X-,
Re(l,.)<0 z = l,2,...,7z
114
the zero solution is stable and all frajectories approach it for r ^ oo . If on the contrary, at
least one eigenvalue has a positive real part, the solution is unstable since there are
trajectories that approach oo as r -^ oo .
7.4 Results and Discussions
7.4.1 Effect of manipulated variable
In Figures 7.1-7.3, a bifurcation diagram of the reactor coolant inlet temperature
using the coolant flow through the steam generator as the continuation parameter is
shown. The bifurcation diagram is obtained by solving the 428 nonlinear algebraic
equations for a given coolant flow through steam generator. First, a continuation
algorithm is used to generate the continuation (bifurcation) diagram. For this system of
equations to converge, it required more number of initial guesses. Therefore, the problem
is formulated in a different way to obtain the bifiircation diagram starting only with a
single initial guess. One more equation is added to the existing 428 nonlinear equations
for the reactor inlet coolant temperature. Instead of specifying the coolant flow through
the steam generator, reactor coolant inlet temperature (set point) is specified and 429
nonlinear algebraic equations are solved using MINPACK. This formulation of the
problem was found to be more efficient than the earlier one. Under nominal operating
conditions, the reactor displays output muhiplicities, i.e., for a given coolant flow through
the steam generator. There are two different reactor coolant inlet temperattires. The
nominal upper steady state temperature is unstable while the lower steady state is stable.
This type of bifiircation behavior is called saddle node bifurcation. In saddle node
115
bifurcation, there is only a single turning point and at this turning point the solution
from stable to unstable.
The numbers (i.e., temperature and composition) in the figures and tables are
scaled.
TO+ 30
5 TO+ 25
E TO+ 20
"5 TO+ 15 o
Z TO+ 10
•o ro
tc TO + 5
TO
,A
0.4 0.5 0.6 0.7 0.8
Fraction of flow through steam generator
0.9
- • stable • - - A- - - unstable
Figure 7.1 Bifiorcation diagram using the flow through steam generator as a continuation parameter
116
XO + 12 T
XO + 10
XO + 8
-A -
- A ' . . A-
c
XO + 6
XO + 4
XO + 2
XO
0.4 0.5 0.6 0.7 0.8
Fraction of flow through steam generator
0.9
- stable unstable
Figure 7.2 Bifurcation diagram using the flow through steam generator as a continuation parameter
iper
atur
e
)rte
n
Max
imum
rea
ctc
TO+ 50
TO+ 45
TO+ 40
TO+ 35
TO+ 30
TO+ 25
TO+ 20
TO+ 15
TO+ 10
TO+ 5
TO 0.4 0.5 0.6 0.7 0.8
Fraction of flow through steam generator
0.9
• stable unstable
Figure 7.3 Bifiircation diagram using the flow through steam generator as a parameter
117
7.4.2 Effect of disturbance
Here, we analyze the open-loop bifurcation behavior of the ethylene oxide reactor
process with respect to a process disturbance. Reactor inlet carbon dioxide mole fraction
is a major disturbance to the ethylene oxide reactor inlet coolant temperature control
system. Figures 7.4-7.6 show bifuration diagrams using reactor inlet carbon dioxide mole
fraction as the continuation parameter. The same procedure as described in section 7.4.1
is used to obtain the bifiircation diagram. Similar to the effect of manipulated variable,
the reactor displays output multiplicities (saddle node bifurcation). Thus, the above
bifurcation diagrams can be used to understand changes in stability on a given branch of
solutions.
TO+ 30
Si TO+ 25
ZJ
"TO
0}
f TO+ 20 B "c ro o TO+ 15 o o - TO+ 10
t3 ro 0) CC TO + 5
TO 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14
Carbon Dioxide Reactor inlet mole fraction
0.15 0.16
- • — stable - • - A- - - unstable
Figure 7.4 Bifiircation diagram using the reactor inlet CO2 mole fraction as a continuation parameter
118
TO+ 50
TO+ 45
S. TO+ 40
% TO+ 35 Q.
i TO+ 30
% TO+ 25 ro (U ^ TO+ 20
I TO+ 15 X
i TO+ 10
TO+ 5
TO
.A'
0.07 0.08 0.09 0.1 0,11 0.12 0.13 0.14
Carbon Dioxide reactor inlet mole fraction
0.15 0,16
- stable - - • A- • - unstable
Figure 7.5 Bifurcation diagi"am using the reactor inlet CO2 mole fraction as a continuation parameter
XO + 7
XO + 6
g XO + 5 c: o '2 XO + 4 >
^ XO + 3 JJ
£ XO + 2 <u
XO + 1
XO
., ^•
0.07 0.08 0.09 0.1 0.11 0.12 0.13
carbon Dioxide reactor inlet mole fraction
0.14 0.15
• stable unstable
Figure 7.6 Bifurcation diagram using the reactor inlet CO2 mole fraction as a parameter
119
7.5 Runaway Boundary
In the case of a fixed bed reactor for an exothermic reaction, a temperature
maximum may be exhibited at some location along the reactor, which is generally
referred to as a "hot spot." The magnitude of this hot spot must be bounded within
specific limits, because it may seriously affect reactor safety and performance. The
magnitude of the hot spot depends on the system parameters, such as operating
conditions, physicochemical properties, and reaction kinetics. For specific values of the
system parameters, the hot spot may undergo large variations relative to small changes in
one or more of the operating conditions or system parameters. In this case, the reactor is
said to operate in a parametrically sensitive region. In practical applications, it is
desirable to avoid this operating region for safety of the process. This provides the
motivation to develop a runaway boundary for ethylene oxide reactor. In particular, we
will identify the runaway region for this reactor by applying the generalized runaway
criterion (Morbidelli and Varma, 1986b) using the maximum in the catalyst temperatiire,
e *, profile along the reactor as the objective. For this we need to define the objective
sensitivity, s(Oj;^), which is defined as
dd„' s(0, •,<!>) =
d(p
where represents the model-input parameter or operating condition such as coolant
flow through steam generator. But the more appropriate quantity m sensitivity analysis is
120
the normahzed objective sensitivity, S(0*;^). The normalized objective sensitivity of
the catalyst temperature maximum, 6 J, along the reactor length is given by
e, ' e d$ p " p r
In the present ethylene oxide process, the fixed bed reactor is too short for
developing a local temperature maximum (hot spot). Here, the axial temperature profile is
monotonically increasing, and so the maximum catalyst temperature value considered in
the sensitivity analysis is that at the reactor outlet. The critical conditions for reactor
runaway, according to the generalized sensitivity criterion, are then identified as the
situation in which the normahzed objective sensitivity, 8(0^ ; ^ ) , is maximized
(Morbidelli and Varma, 1986b).
As shown in Figure 7.7, the normalized sensitivity S(0* •,(!>) for the temperature
of the catalyst particle operating on the lower temperature branch (open-loop stable
operating point) decreases with a decrease in the flow through the steam generator and
becomes large at the bifurcation point. Thus, at the bifiircation point, the catalyst
temperature becomes very sensitive to the change in the coolant flow through the steam
generator. It is worth noting that when there exist multiple steady states for the reactor, its
runaway boundary is always coincident with the bifurcation point of the multiple steady
state regions.
Note that the sign of the normahzed sensitivity value has a particular meaning. A
positive (negative) value of the normalized sensitivity of the temperature maximum with
121
respect to coolant flow through the steam generator indicates that the temperature
maximum increases (decreases) as the magnitude of coolant flow increases. Thus, m this
case, the normalized sensitivity is negative, which indicates that the fransition from non-
runaway to runaway behavior occurs as the coolant flow through the steam generator is
decreased.
CO
73 CU N
0,2
0,1
0
-0.1
-0.2
-0.3
-0.4
-0.5
0.75
. • - •
H i _ • -It^
0.8 0.85 0.9 0.95
Fraction of Flow through steam generator
TO + 10
TO+ 8
TO+ 7
TO+ 6
TO+ 5
TO+ 4
TO+ 3
TO+ 2
TO + 1
TO
• Normalized sensitivity S - - - - - - - coolant inlet temperature
o o o
3 T3 CD
Figure 7.7 Temperature of the catalyst particle and its normalized sensitivity with respect to the flow through steam generator
Figure 7.8 shows the open-loop runaway region for the different catalyst activity.
Any arbitary size disturbance near the runaway boundary will make the reaction runaway
in the open-loop. The runaway region was obtained by calculating the bifurcation points
for different catalyst activity as shown in the Figure 7.9.
122
TO + 2.4
TO+ 2.
ro (1)
emp
c ro
coo
(U c
o ro C) C
10
10
TO
+ 1.6
+ 1.2
+ 0.8
TO + 0.4
TO
0.85 0.9
Activity of the catalyst
0,95
Figure 7.8 Boundary of the runaway region
TO+ 30
<u 3
ro n> F <i)
o
tin
c ro o o o o (J ni u> 01
TO+ 25
TO+ 20
TO+ 15
TO+ 10
TO+ 5
TO
I I X
X X X X ^ ?-X X X X
, X X .' X X • — • — ( X ,', X -•. X > X
X • X X X
X X
x" ^
X
B
• • E
• • • • • #
f -f—:
I \ 1
• • • • • • • X • X X X
a
• •
•
•
•
• •
•
•
•
0.1 0.2 0.3 0.4 0,5 0.6 0.7 0.8 0.9
Fraction of flow through steam generator
• a=1,0 a=0.95 a=0,9 X a=0.85 X a=0.8 • locus of bifurcation points
Figure 7.9 Locus of bifiircation points for different catalyst activity
123
7.6 Closed-Loop Nonlinear Bifurcation Analysis
The ethylene oxide reactor is always operated at higher coolant inlet temperature
(open-loop unstable operating point) because it gives higher ethylene conversion than the
lower coolant inlet temperature (open-loop stable operating point). Although the
operating point is open-loop unstable, a controller may be able to stabihze it.
7.6.1 Effect of disturbances
Inlet carbon dioxide composition (mole %) is the primary disturbance for the
reactor inlet coolant temperature control loop. The disturbance magnitude is changed in
the positive and negative direction for different values of the control loop dead time to
evaluate the performance of the PI controller. Figure 7.10 shows the runaway region for a
positive change in the carbon dioxide reactor inlet composition. It can be seen that for
dead times below 20 seconds the system goes unstable for almost a +10mole% change in
the carbon dioxide inlet composition. This is because the increased carbon dioxide partial
pressure (which is due to increased inlet carbon dioxide mole%) favors the partial
oxidation of ethylene and does not favor the complete oxidation of ethylene, which has
more heat of reaction.
124
o £
in o Q . E o o
CM
o o .c CD cn c
Q) > +
10,5
10
9.5
8,5
7,5
6,5
Stable
Runaway
12 14 16 18 20
Dead time (seconds)
22 24 26
Figure 7.10 Closed loop stability region for Carbon Dioxide disturbance change in the positive direction
Figure 7.11 shows the closed-loop runway region for a negative change in the
carbon dioxide reactor inlet composition. It can be seen that the system is very sensitive
to negative change in the reactor inlet carbon dioxide composition. This is because the
decrease in the partial pressure of the carbon dioxide in the reactor favors the complete
oxidation reaction, which has a high heat of reaction.
From Figures 7.10 and 7.11, we conclude that a decrease in the carbon dioxide
reactor inlet composition from nominal is a major disturbance as compared an increase in
the carbon dioxide reactor inlet composition. Only a decrease in the carbon dioxide
reactor inlet composition is studied.
125
10 15
Dead time (sec)
20 25
Figure 7.11 Closed-loop stability region for Carbon Dioxide disturbance change in the negative direction
7.6.2 Effect of detuning factor
Here we study how the controller aggressiveness and sluggishness affect the
runaway region. First, the controller is tuned for set point changes using 1/6' decay ratio
as the tuning criterion. Then the detuning factor is varied from 0.1 to 5.0, and each time
the disturbance magnitude is varied till the conttoUer goes unstable. Figure 7.12 shows
the effect of the detuning factor on the stability region for zero dead time in the control
loop. It can be seen that runaway boundary is sensitive to the detuning factor.
126
3.5
.2J o E ^ 3 o
o a. E 8 2,5
CN o o '3) ,E 2
1,5
o Oi
>
Unstable
Stable
2 3
Detuning factor ft
Figure 7.12 Effect of detuning factor on the runaway boundary
7.6.3 Effect of operating temperature
Here we study, how the disturbance affects the stability of the controller for
different operating temperatures (i.e., reactor inlet coolant temperature). For a fresh
catalyst (catalyst activity equal to one), the range of the operating temperature is
estimated by solving the steady state nonlinear algebraic equations. But as discussed
earlier, in industry, the reactor is operated above the bifiircation point; therefore, we also
considered the operating temperature range from the bifurcation point to the maximum
achievable reactor inlet coolant temperature. The following procedure is used to get the
runaway region. First, the PI controller is tuned for a given operating temperature set
point for 1/6" decay ratio as a tuning criterion. Then the disturbance magnitude is varied
127
until the controller goes unstable. From Figure 7.13, it can be seen that the system is very
sensitive to disturbances at higher operating temperatures.
3.5
2.5
2
o Q . E o o
CM o o ^ 1.5
.£ 1
0.5
Runaway
Stable
TO TO + 1 TO + 2 TO + 3 TO + 4 TO + 5 TO + 6 TO + 7 TO + 8
Operating temperature
Figure 7.13 Effect of operating temperature on the stability region
7.6.4 Effect of catalyst activity
Figure 7.14 shows the effect of operating temperature on the runaway boundary
for different catalyst activities. The same procedure as described for Figure 7.13 is
applied to get the runaway boundaries for different catalyst activities. It can be seen that
the sensitivity of the operating temperature to the change in the reactor inlet carbon
dioxide (mole%) decreases as the activity of the catalyst decreases.
128
TO+ 5 TO+ 10 TO+15 TO + 20 TO + 25 TO + 30 TO + 35
Operating Temperature
-a=0.93 a=0.85 --X ••a=0.75 ^ ^ a = 0 . 6 5
Figure 7.14 Comparison between runaway boundary for different catalyst activity
7.6.5 How to detect runaway
The runaway situation can be detected from the gas temperature measurements
along the length of the reactor. Here, we have considered three different temperature
measurements at 60%, 80%, and 100% (i.e., reactor outlet) of the reactor length. The
slope is calculated between these temperature measurements. Figure 7.15 shows the
runaway boundary for different catalyst activity. J t is remarked that whenever the
temperature slope exceeds the runaway boundary, the reactor will become unstable.
129
Runaway
0.85 0.9
Catalyst activity
0.95
- • — Temp slope between measurement at reactor outlet and 80% of the reactor length
-A— Temp slope betw een measurement at 80% and 60% of the reactor length
Figure 7.15 Temperature slope for different catalyst activity
7.6.6 How to prevent runaway
The runaway situation can be avoided by setting the make up oxygen to zero
flow. One such simulated runaway situation is shown in Figure 7.16. In this figure, the
disturbance is infroduced after 5 seconds, which causes a runaway reaction (i.e., the
reactor temperature shoots up exponentially). The 25 C increase in temperature is caused
by the change in the heat transfer coefficients, which are modeled as a function of
temperature through thermal conductivity and viscosity calculations. The make-up
oxygen is reduced to zero after 1.4 min once runaway is detected, since the oxygen
partial pressure becomes low the reaction ceases and the temperature drops immediately.
After 1.5 mins, there is no oxygen in the reactor, thus no reaction. There is only heat
130
transfer between the coolant and the reaction medium. Thus, reducing the oxygen can
always prevent the runaway reactor.
TO+ 45
TO+ 40
& TO+ 35 "ro
g_ TO+ 30 E 0)
•;;; T O + 25 ro
I TO+ 20 "3 o o TO+15 "o ro
l2 TO+10
TO+ 5
TO
Oxygen set to zero
0.5 1 1.5
Time (min)
Figure 7.16 Response of the outlet temperature to eliminating oxygen in the feed when runaway observed
131
CHAPTER 8
CONTRIBUTION
Quina et al. (1999) presented the steady state analysis of the region of parametric
sensitivity and the range of operating conditions leading to the phenomenon of
temperature runaway for a fixed bed reactor (the selective oxidation of methanol to
formaldehyde), where the catalytic bed is partially diluted with inert packing. A complete
bifurcation analysis of a general steady state two-dimensional catalytic monolith reactor
model that accounted for temperature and concenfration gradients in both axial and radial
directions is studied by Balakotaiah et al. (2001). A single exothermic first-order reaction
was considered. Garcia et al. (2000) studied the steady state nonlinear bifurcation
behavior of a high impact commercial polystyrene continuous stirred tank reactor. Chang
(1984) presented an analysis of the various types of bifurcation that are caused by a
conventional, SISO PID controller on a general nonlinear system. Thus, most of the
studies are centered on the steady state bifurcation analysis of fixed bed catalytic reactors.
The steady state bifurcation analysis helps to understand the input/output multiplicity in
the reactor and the stable/unstable operating points with respect to certain
physicochemical paramefrs. For a reactor to operate in a rehable and safe marmer, not
only the steady state nonlinear bifiircation analysis but also the closed loop stability
analysis is important. Also the effect of important parameters e.g deadtime, disturbances
which affects the closed loop performance, can not be stiidied in steady state bifurcation
132
analysis. The effect of controller aggressiveness can be stiidied in a closed-loop stabihty
analysis by using a detuning factor.
The aim of the present work was to contribute to the open-loop and closed-loop
stability analysis in a heat integrated multitubular fixed be reactor used for ethylene oxide
production. In this regard, the bifurcation theory is used to study the stability issues.
The open-loop bifurcation study showed that under nominal operating conditions,
the ethylene oxide reactor system displays output multiplicities (saddle node bifiircation).
The nominal upper steady state is unstable while the lower steady state is stable. The
bifiircation plots were obtained by varying flow through steam generator (a manipulated
variable) and inlet carbon dioxide inlet composition (a disturbance to the reactor inlet
coolant temperature control loop). These results are particularly interesting because they
enable one to conclude that bifurcation analysis can be based on simple parameters such
as COj'"'"' (inlet carbon dioxide composition) and F^ (flow through steam generator),
therefore allowing their manipulation in order to avoid the risky operating conditions.
The analysis is based on a heterogeneous two-dimensional model of the reactor, which
predicts the temperature and composition in the radial and axial direction of the reactor.
Also the coolant temperature is not constant, but it varies in the axial direction on the
shell side of the reactor.
The inlet coolant temperature control of the ethylene oxide reactor is
economically important. This is because at higher temperature the selectivity of ethylene
oxide decreases and at lower temperatures the conversion of ethylene to ethylene oxide
decreases. So there is an optimal temperature profile at which both conversion and
133
selectivity is maintained by careful control of the reactor inlet coolant temperature. But at
the higher reactor coolant inlet temperature, the reactor is susceptible to runaway due to
high heat of reaction of complete oxidation of ethylene. Therefore, the closed-loop
stability analysis of this system is very important and will be useful in understanding the
safe operating regions. An analysis of the stable confrol region of the system is developed
as a function of operating temperature, catalyst activity, detuning factor, and disturbance
(reactor inlet carbon dioxide composition) direction and magnitude. The closed-loop
stability region was found to be sensitive to the negative change in inlet carbon dioxide
composition. The reactor system was also found to be more prone to instability at higher
operating temperatures and higher catalyst activity.
Based on this study, the ethylene oxide reactor can be operated at higher
temprature which will improve the profitability of the system without substantially
increasing the risk of a reactor runaway. It is also shown that shutting off oxygen feed to
the reactor can always prevent the runaway reactor. This sttidy represents the first open-
loop and closed-loop bifiircation sttidy of an industtial reactor system.
134
CHAPTER 9
DISCUSSION, CONCLUSIONS, AND RECOMMENDATIONS
The primary objectives of the research work were as follows:
1. For the ethylene oxide process:
a. develop a detailed mathematical model for the process,
b. benchmark the developed model against industrial data,
c. study offline optimization and control of the process based on economic objective
function,
d. study open-loop and closed-loop bifurcation of the process;
2. For the vinyl acetate process:
a. develop a detailed mathematical model for the reactor,
b. benchmark the developed model against industrial data,
c. study offline and online optimization of the reactor based on economic objective
function.
In both cases, the developed mathematical model was derived from a generalized
model for a multitubular gas phase solid catalyzed reaction based on certain assumptions.
For a vinyl acetate reactor, a steady state two-dimensional homogeneous model was
developed. For an ethylene oxide reactor, a two-dimensional dynamic heterogeneous
model was developed. Radial derivatives are approximated by orthogonal collocation and
axial derivatives were approximated by orthogonal collocation on finite elements. Most
of the assumptions were justified by satisfying existing criterion and some assumptions
135
were by actual calculations. For example, in the case of ethylene oxide reactor, the
effectiveness factors for both the reactions are calculated by simulating temperature and
concenfration profiles inside a catalyst. From the results, it was shown that the
intraparticle resistance could be neglected since the effectiveness factor is closed to unity
for both the reaction systems.
A base case operating condition was chosen from the industrial data at which the
catalyst was assumed to be fresh (activity =1). The developed model was benchmarked
against the industrial data. The model parameters were estimated through regression
analysis by minimizing the weighted error between the model predicted output values
(e.g., reactor outlet temperature and composition) and the industrial data for the base case
operating point.
Since the catalyst loses activity due to sintering and impurities in the feed, it was
required to take care of this effect in the model through catalyst deactivation. Based upon
the catalyst deactivation models available in the literature, a suitable model was selected.
The catalyst deactivation model was benchmarked against the industrial data over the
period of operation to represent the real process. In the case of deactivation model, the
model parameters were estimated through regression analysis by minimizing the total
error between the model predicted output values (e.g., reactor outlet temperatures and
compositions) and industrial data over the length of the operation. A comparison between
the industrial data and model prediction showed that the model predicted the industrial
data quite well for both the processes.
136
Since the catalyst deactivates over the period of operation, operating temperature
is increased to compensate for the loss of activity. But the increased operating
temperature can affect the selectivity of the desired product. Therefore, offline
optimization is done for both the vinyl acetate and ethylene oxide process, using a steady
state process model to find an optimal operating temperature profile which maximizes the
profit of the process.
To see the effect of the model parameter's uncertainty, sensitivity analysis is
carried out by perturbing the model parameter values by 10% and re-running the
optimization algorithm for offline optimization. The sensitivity analysis for vinyl acetate
process optimization showed that the results were sensitive to the uncertainty in the
model parameters.
For the vinyl acetate reactor, online optimization is done by updating the model
parameters online. Online optimization showed more profit improvement as compared to
offline optimization.
Nonlinear bifiircation analysis of ethylene oxide reactor has been carried out to
study the safe operating regimes. Under nominal operating conditions, the ethylene oxide
reactor system displays output multiplicities (saddle node bifiircation). The nominal
upper steady state is unstable, while the lower steady state is stable. The closed-loop
stability of the reactor was found to be sensitive to the negative change in inlet carbon
dioxide composition. The reactor system was also found to be more prone to instability at
higher operating temperatures and higher activity catalyst. Based on this sttidy, the
reactor can be operated at higher temperature which will improve the profitability of the
137
system without substantially increasing the risk of a reactor runaway. It is also shown that
shutting off the oxygen feed to the reactor can always prevent the runaway reactor.
The following recommendations are made for possible future study.
In case of the ethylene oxide process, the carbon dioxide inlet compositon
disturbance comes from the separation system; therefore, it can be measured before it can
affect the reactor inlet coolant temperature control system. Thus, a feedforward control
system will be more useful to take care of reactor inlet carbon dioxide composition
disturbance.
A confroUer with a nonliear control law can be designed for a reactor inlet coolant
temperature control system (in case of ethylene oxide process) in the vicinity of the
closed-loop runaway boundary. This will allow to operate the reactor at higher operating
temperatures especially when the catalyst activity decreases.
In case of the ethylene oxide production, a moderator (e.g., chlorine compound) is
added in controlled amounts which improves the selectivity to ethylene oxide while
inhibiting the total oxidation reaction (ethylene to carbon dioxide and water). The catalyst
activity depends on the amount of moderator added to the reactant mixture. Therefore,
the effect of the moderator on the actalyst activity can be considered in the catalyst
deactivation model.
138
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