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I n d . E n g . C h e m . R e s . 1989,
28,
1177-1184
1177
siderab le degree of leeway regarding conditions of fluc tu-
ating tempera ture and f low rate. In addition, the dual-
catalyst concept would be applicable to other reaction
systems in which competing reactions, e.g., partial and total
oxidation, result in the conversion versus temperature
curve going through a maximum.
Phillibert,
N.
G. An Investigation of Copper Mordenite Catalyst for
the Reduction of Nitric Oxide with Ammonia. M.S. Thesis, Th e
Univeristy of Massachusetts at Amh erst,
1985.
Pruce,
L.
M. Reducing NO, Emmissions at the B urner, in the Fur-
naces, and after Combustion. Power 1981, 125(1), 33-40.
Yamaguchi, M.; Matsushita, K,; Takami, K. R~~~~~ NO, from
H N O BTail Gas. Hydrocarbon P rocess. 1976,55(8), 101-106.
Registry No
NO, 10102-43-9;
NH,,
7664-41-7.
L i t e r a t u r e C i t e d
Nam,
I. S.
Exp erimen tal Studie s and Theore tical Modeling of Cat-
alyst Deactivation. Ph.D . Dissertation, The University of Mas-
sachuset ts at Amherst, 1983.
Received
f o r
review M a y
31, 1988
Revised manuscript received
M a r c h
29, 1989
Accepted
M a y
2, 1989
PROCESS ENGINEERING AND DESIGN
Temperature Control of Exothermic Batch Reactors Using Generic
Model Control
B a r r y J. C o t t
and
S a n d r o M a c c h ie t to *
Depar tment of Chemical Engineering and Chemical Tech nology, Imperial College of Science, Technology and
Medic ine , South Kens ington, London
SW7
2A Y, England
A
new model-based contro l ler for the in i t ia l hea t -up an d subsequ ent temper a ture ma in tenan ce of
exothermic batch reactors i s p resen ted . T he contro ller was developed by us ing th e Gener ic Model
Control framework of Lee and Sullivan, which provides a rigorous and effective way of incorporatin g
a nonlinear energy balance model of the reactor and the heat-exchange ap para tus into the controller .
It also a llows the use of the sam e contro l a lgor i thm for bo th heat -up a nd te mpe ra ture m ain tenance ,
thereby e l iminat ing the need to sw i tch between two sepa ra te contro l a lgor i thms as i s the case wi th
todays more comm only used s trategies .
A
determinis t ic on- line es t imator i s used t o de termine the
am ount a nd ra te of hea t re leased by th e react ion . This in format ion is , in tu rn , u t il ized t o de termine
th e ch an g e in j ack e t t emp er a tu r e s e tp o in t i n o r d e r to k eep th e r eac t io n t emp er a tu r e o n i t s d es i r ed
trajectory. T he performance of the new GMC-based controller is comp ared to tha t of the com monly
used dual-mo de controller. Simulation stu dies show th e new controller to be as good
as
t h e d u d - m o d e
controller for a nom inal case for which bo th controllers are well tun ed . However, th e new controller
i s s ho wn to b e m u ch m o r e r o b u s t wi th r e s p ec t t o ch ang es in p ro ces s p a r ame te r s an d to mo d e l
mis ma tch .
1. I n t r o d u c t i o n
Th e init ial heat-up from ambien t tempe rature and the
subsequ ent temperature control of exothermic batch re-
actors have always proved to be a difficult control problem
(Shinskey, 1979). Because the amoun t of heat released as
th e reaction mixture is hea ted up ca n become very large
very quickly, the reaction may become unstable and cause
the tem perature to run away if th e heat generated exceeds
the cooling capacity of the reactor. This runaway can
obviously cause great risk to plant personnel an d equip-
me nt an d can, even in the best case, result in a loss of th e
batch. Therefore, careful control
of
the rate of change of
the reactor temperature and minimization of the tem-
perature overshoot is required. On the other hand, from
a production p oint of view, the heat -up should be done as
quickly as possible in orde r to red uce the overall cycle time
of the reaction process. Therefo re, any control strategy
for heat-up must balance the needs of production with
those of safety and quality.
Traditionally, the problem has been approached through
the use of open -loop control theory t o establish, a priori,
0888-588518912628-1177$01.50/0
the optima l temp erature profiles and of standa rd feedback
control algorithms to achieve these profiles. Th e control
actions needed to bring the reactor contents to the desired
setpoint were obtained by solving an optimal control
problem with th e objective of m inimizing th e time to reach
the setpoint (Shinskey, 1979). The resulting strategies are
of the on-off or bang-bang typ e and con sist in applying
maximum heating until the reactor temperature is within
a specified number of degrees of the se tpo in t and then
switching to maximum cooling to bring th e rate of tem -
perature change to zero when the tem peratu re has reached
its final desired setpoint. At this point, stand ard feedba ck
controllers can be switched on and used to maintain the
temp erature. Th e most commonly used strategy of this
type in indus try is the dual-m ode controller of Shinske y
and Weinstein (1965), which uses a s tandard
PID
con-
troller for maintaining temperature.
Th e main problem with approaches of this type is tha t
the optimal switching criterion from heating to cooling,
usually based on the reactor temperature, is determined
a priori an d is therefore only valid for a specific range of
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Chemical Society
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Ind . Eng. Ch em. Re s., Vol. 28, No. 8, 1989
operating conditions. Because heat -up proceeds in an
open-loop manner and no feedback from the reactor is
used, there is no allowance for modeling errors or for
changes in process parameters. Th e net result is that the se
strategies lack robustness, and any deviation in the op-
erating conditions from those used to tun e the controller
may result in significantly poorer control performance.
Th e use of adap tive control algorithms would appear to
offer promising solutions to this problem, and there have
been several attempts in this direction. A recent paper
by Cluett et al. (1985) is typical of these atte mp ts. They
used a single adaptive control algorithm for both he at-up
and temperature maintenance but found that the algor-
ithm did n ot handle the s harp change from the heat-up
mode to the tempera ture m aintenance mode very well .
They s ta te tha t adapta t ion dur ing the heat -up mode
misleads the o peration of the adaptive system and find
th at , in practice, fully adaptive strategies give poor per-
formance. In the en d, they effectively revert back to a
dual-mode approach, where the PI D controller is s imply
replaced by a n adaptive controller just for the tem perature
maintenance p art of the profile. T herefore, the robustness
concerns of S hinskeys dual-m ode controllers also apply
to this controller.
A
more encouraging strategy was proposed by J uta n an d
Uppal (1984),who used a model-based app roach to esti-
mate the current amount of heat being released in the
reactor at any given moment in time.
This information
was used in a feedforward control structure designed to
counterbalance the effect of the h eat released.
In order
to com pensa te for modeling errors and for the lack of a
precise estima te of the he at released, they combined the
feedforward controller with a feedback controller.
Al-
though t his app roach overcomes many of the problems of
the open-loop strategies, the control performance reported
by the auth ors could be improved further. Th e reactor is
not smoothly delivered to the desired temperature, an d
there is the presence of significant overshoot in the reactor
tempe rature. These effects may be attrib uted to the lin-
earization necessary to implement the feedback control
action and to the manner in which the feedforward and
feedback effects are added.
This paper presents a new model-based controller design
for the heat-up and temperature maintenance of exo-
thermic batch reactors, which is derived from the Generic
Model Control (GMC) algorithm of Lee and Sullivan
(1988) and which uses the on-line heat-released estimation
concept of Jutan and Uppal (1984). GMC has several
advantages that make i t a good framework for developing
reactor controllers:
1. Th e process model appe ars directly in th e control
algorithm.
2 .
The process model does not need to be linearized
before use, allowing for the inherent nonlinearity of exo-
thermic batch reactor operation to be taken into account.
3. By design, GMC provides feedback control of the ra te
of change of the controlled variable. This suggests th at
the ra te of temperatu re change, which as mentioned above
is very important in heat-up operations, can be used di-
rectly as
a
control variable.
4. The relationship between feedforward and feedback
control is explicitly stated in the GMC algorithm.
5 .
Finally an d im portantly, the GMC framework per-
mits us to develop a control algorithm that can be used
for bo th heat -u p and tem pera ture main tenance and
therefore eliminates the need for a switching criterion
between different algorithm s; this should result in a much
more robust control strategy.
The paper wll begin by outlining the details of the GM C
controller design and the on-line method used for esti-
mating the current heat released. Th e designs
of
t h e
controller and th e heat-released estim ator are general in
nature and applicable in principle to the temperature
control of any exothermic and even endothermic batch
reaction systems.
A
specific reaction/rea ctor example is
presented to demonstrate the tuning and nominal per-
formance of the GMC controller. In order to provide a
comparison for the GMC controller, the design of a
dual-mode controller is then presented a nd implem ented
on the sam e reactor system. Finally, the performa nce of
the two s trategies is compared w ith respect to changes in
process conditions an d modeling errors, and th e robustness
of both controllers is evaluated.
2.
Generic Model Controller Design
2.1. Control Algorithm Formulation. Th e formula-
tion of a Generic Model Controller for temperature control
of exothermic batch reactors is quite straightforward.
GMC requires a dynamic model of the process written in
stan dard state variable form. T he controller is formulated
by solving th e dyn amic process model for the derivative
of the co ntrolled variable, x and lett ing i t equal what is,
in effect, a proportional integral term operating on the
difference between the current value of
x
and its desired
value, xsP. Hence, the GMC control algorithm can be
written as
d x / d t =
K , x - x ) + K2
x P - X d t
1)
s
where
K ,
an d K z are tuning constants. For temperature
control of a batch reactor, a process model relating the
reactor temperature, TI, o the manipulated variable, the
jacket temperature, Tj, is required. Assuming th at the
amo unt of heat retained in the walls of the reactor is small
in comparison with the hea t transferred in the rest of the
system, an energy balance around the reactor contents
gives the required model:
d T I
+
uA(Tj
-
71)
(2)
where
W
is the weight of th e reactor conten ts, C, is the
mass heat capacity of the reactor contents, U is the
heat-transfer coefficient, A is the heat- transfer area, and
Q is the heat released by the reaction. W an d C p are
assumed to be constant at this point. Replacing TI for x
and T:P for xsP in eq 1,combining eq 1 and 2, and finally
solving for the manipulated variable, Tj, we obtain the
GMC controller:
_ -
d t WCP
WCP
Tj = T , + - K1 T:P - TI)
U A
Tjgives the jacket temperature trajectory required so
tha t the reactor temperature, I follows the desired tra -
jectory defined by the values of the G MC con stants,
K 1
and
K z .
As written, eq 3 gives the continuous form of the GMC
algorithm. In order to use GMC in a discrete system, the
integral must be evaluated numerically using the ap-
proximation
where
A t
is the sampling frequency of the controller.
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Ind. Eng. Chem. Res., Vol. 28, No.
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1989
1179
ermined for each new system. Ju ba and H ame r used this
approach w ith good success on their pilot p lant reactors.
In addition, they point out th at the heat-released e stimator
could easily be formulate d as a Kalm an filter in order to
improve the estim ates by making use of the s tructu re of
the noise model. Indeed, a recent paper by de Valligre and
Bonvin (1989) demonstrate the effectiveness of using
nonlinear Kalman fil ters in the estimation of the heat
released.
In this work, th e second approach, the determinis tic
on-line energy balance, was used because
it
is the most
general of the three a pproaches an d therefore is most ap-
propriate for the generality of the controller formulation.
We minimize the problems of unknown process parame ters
by choosing t o estima te Q / U A ather than Q itself. By
solving for Q/UA, he number of parameters needed to
be determined is minimized to the single group, WC,/ UA.
In addition, WC,/UA is the only parameter left in the
GMC co ntrol algorithm of eq 5 ,
so
effectively one param-
eter characterizes both th e estimator and t he controller.
To develop the estimator, eq 2 is solved for Q/UA to
give
Therefore, the discrete time version of eq 3 is
Ti@ ) TI(k)
Equat ion 5 gives not th e jacket temperature setpoint,
T;p@), ut th e actual jacket temperature, Tj(k),eeded a t
the next t ime interval to move the reactor temperature
toward its setpoint, T:P. If Tj(k)ere used directly as the
setpoint, then, because the dynamics of the jacket are not
accounted for in eq 5 , the resulting control would be
sluggish. Therefore, some form of dynamic compensation
of Tj(k ) ust be used. If the dynamics of the jacket are
assumed t o be first order (a reasonable assumption given
th e findings of Liptak (1986)), h en a difference equation
can be used
At(TjBP(k)
T.(k-l))
j
(6)
where
~
is the estimated tim e constant of the jacket. The
jacket temperature setpoint,
TrP,
can be obtained by sim-
ply rearranging eq 6. Therefo re, the following dynamic
compensator is obtained:
Tj k)= T, k-l + 1
1
Tj(Tj(k)T.ck-1
At
(7)
T he solution of eq 5 and 7 gives the actual setpo int value
for the jacket temperature controller to be used for the
next control interval.
2.2. On-Line Estimation of the Heat Released for
the
GMC
Controller. T he success of the GM C temper-
ature controller is largely dependent on our ability to
measure, estima te, or predict th e hea t released, Q , a t an y
given period in time. The re are three main techniq ues of
estimating Q on-line as discussed by Juba and Hamer
(1986): 1. direct use of detailed kinetic models; 2. det -
erminis tic on-line energy balances; and
3.
empirical
heat-released estimators.
For m ost reaction systems of industrial interest, the first
approach often proves not to be feasible because of th e lack
of good kinetic models. In a rapidly changing business
such as fine chemicals, there often is not enough time or
financial benefit in carrying our detailed kinetic studies
of th e reactions.
Deterministic on-line energy balances can also have
drawbacks. Th e largest problem is often the assumption
th at the h eat held in the reactor walls is small. If the heat
capacity of the reactor walls is not small, then a deter-
ministic energy balance requires th e solution of a system
of coupled differential equations with several unobservable
states such as the wall temperatures (Ju ba and H amer,
1986). Furth ermore, the number of process parameters
increases, and there may be difficulty in obtaining good
estimates for all of them.
In their paper, Ju ba an d H amer use an empirically de-
veloped discrete-time transfer-function model of the re-
actor. The model was determined experimentally by sim-
ulating heat generation by the injection of steam into a
reactor full of water. The y then use time series analysis
to develop a transfer function relating th e reactor tem-
pera ture to the jacket in le t tempera ture and the heat
generation. T he model is then inverted to obtain an es-
timate of the heat released. Thi s method has the advan-
tage of accounting for all the dynam ics of the re actor, but
it has the disadvantage th at th e resulting model
is
specific
to the given reaction/reactor system and m ust be redet-
TjBp(k) =
T.(k-U
+ 1
1
Although the reactor temperatu re,
T,,
and the jacket
temperature, Tj , re available through direct measurem ent,
the derivative of the reactor tem perature must be esti-
mate d on-line from the d irect measu remen ts of
T,.
Th is
can often be difficult because num erical differen tiation is
very sensitive to measu reme nt errors. Th e performance
of the estimator can be dramatically improved by using
a high-order difference equation for calculating the de-
rivative and by using low-pass filters on the m easurem ents
used in the estimator to remove th e high-frequency noise.
In this work, we use a three-term difference equation
(Jennings, 1964) and exponential filters with time con-
sta nt s of
1
min on both the temperature measurements
and the es t imate
of
Q / UA. Thes e filters were only used
for
estimation; the measu red signals of
T,
and
Tj
were still
used directly in the GMC control equation.
The full description of the estimator then becomes
(9)
WC, dT1Jk)
UA
d t
Q/UA) (k ) TrJk) TjJk)
(12)
where the s uperscrip t f indicates the filtered value of T
Tj,
or QIUA.
T he estim ator described by eq 9-13 can be applied to
any reaction/reactor system by changing the parameters
W,
C,,
U , and
A
to reflect the system. Fur ther simplifi-
cations of the estim ator are possible. For example, it is
often possible, given the reactor dimensions an d the den-
sity of the reaction mixture, to develop a relationship
between Wand A.
So,
if th e jacket only surrounds the side
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1180 Ind. Eng. Chem. Res., Vol. 28, No. 8, 1989
Hot
Cold
TC - Temperature Connol
W
- Weight
Figure
1.
Batch reactor schematic diagram.
of a cylindrically shaped rea ctor, as shown in Figure
1,
he
relationship of
W / A
s given as
w
pirr2h
p r
(14)
A 2irrh
2
where p is the density of the reaction mixture, r is the
radius of the reactor, and
h
is the height of the reactor
mixture. Therefore, the expression for We,/
UQ
can be
replaced, for this special case, by,
- = - -
_ -
we, CpPr
(15)
U A 2U
T he determ ination of th e value of W e , / U A to be used
in the es timator and controller depends on w hether any
of th ese pa ram eters change significantly over the course
of th e reaction. If they change very little, W e , / U A may
often be dete rmin ed simply by performing an open-loop
step tes t for the jacket tem perature se tpoint with a cold
charge. At these low temp erature s, the reaction rat e is
practically zero, and eq
2
shows tha t, when there is no heat
released,
W C , / U A
is merely the time constant of the
system.
If
any of the values change significantly over the
course of the reaction, fur ther tests may have to be per-
formed to characterize these changes in
W e , / U A .
I t should be noted th at t he s tructure of the estimator
obtained in eq 9-13 is very similar to the stru ctu re of the
empirically derived estimator of Juba an d Hamer. The y
are both low-order difference equations and are based on
similar plant measurements. If, for some reason, the a p-
proach of Ju ba an d Ham er was preferred for a given ap-
plication, the empirical estimator would simply replace eq
- -
- -
9-13.
3
R eac to r S imu la t io n
T he reactor simulation used in this work is largely based
on a dynamic model developed for the Warren Springs
Laboratory (Pulley, 1986).
A
well-mixed, liquid-phase
reaction system
is
considered, in which two reactions are
modeled:
reaction 1
A + B - C
reaction
2
A + C - D
Component C is the desired product while D is an un-
wanted byproduct, an d the general operating objective is
to achieve a good conversion of C while minimizing the
production
of
D. Extens ive optimization of th e reactor
conditions was presented in the original reference.
Th e heat- and mass-transfer rates in the reactor are
assumed to be high enough so tha t the system is essentially
reaction rate limited. Therefore, the rate of production
of C and D is only depe nden t on the reactant concentra-
tions:
where
R 1
and R2 are the rates of production of C and D,
respectively, and
MA, MB ,
and
M c
are the number of moles
of components A, B, and C prese nt in the reactor a t any
given time. Th e rate constants, k l and k 2 ,are dependent
on the reaction tem peratu re through th e Arrhenius rela-
tion. Both reactions have a large hea t of reaction (AHl=
-41
840 kJ/kmol,
AH2 =
-25 105 kJ/ km ol), which makes
the overall reaction system strongly exothermic.
Heating a nd cooling of th e reactor contents is performed
through the use of a single-pass jacket system. Th e values
of the physical parameters of the reactor such as volume,
heat-transfer coefficients, and area were based on the
dimensions of the batch reactor presented by Luyben
(1973). Control of the jacket temperature is provided by
a tempera ture controller on the jacket inlet stream. Th e
heat exchangers needed to control this tem perature are not
modeled but are accounted for by basing the tim e constant
of th e jacket te mp eratu re response on typical figures given
by Lip tak (1986). Figure 1 presents a diagram of the
reactor system.
Simulation work by Pulley (1986) indicates tha t an in-
itial charge that is equimolar in A and
B
produces the
greatest yield
of
C. Therefore, assuming the density of the
reaction mixture
is
th at of water and given the dimensions
of th e Luyben reactor, th e nominal charge to the reactor
was assumed to be 360 kg of
A
and 1200 kg of B. Fu r-
thermore, given some cost function, Pulley determined that
the op timal isothermal reaction tem perature typically fell
in the range 90.0-100.0
C,
so the final reaction temper-
ature was set to 95.0 C. Finally, the jacket temp erature
was assumed to be limited to the range 20.0-120.0 C due
to th e heat-exchanger capacities, and t he reaction m ixture
was assumed to be at 20.0 C at time 0.
Because measurement errors are always present when
working with real equipm ent, these were included in th e
simulation by adding noise to all temperature measure-
ments. In order to use an appro priate noise model, time
series analysis was used to de termine the noise models for
several tem peratu re indicators on the pilot plants a t Im-
perial College. A first-order moving average noise model
was found to
f i t
the majority of these temperature indi-
cators and was therefore used in this work.
A
full description of the reactor system and the values
of the parameters used is given in the Appendix.
4. C o mp ar i s o n
of
G M C w i t h T r a d i t i o n a l C o n tr o l
Strategies
4.1. Dual- Mo de Control. To provide a s tandard with
which
to
compare the performance of the GMC controller,
the commonly used dual-mode controller (DM controller)
was implemented. As mentioned in section
1,
dual-mode
control is an example of an open-loop hea t-up controller
followed by closed-loop feedback controller to maintain
temperature. It was originally developed by Shinskey a nd
Weinstein (1965) and fu rther discussed by Liptak (1986).
The DM controller consists of a sequence of control ac-
tions, each one carried o ut after t he reactor has reached
a certain condition. T he sequence of actions is as follows:
1. Full heating is applied (jacket temperature setpoint,
TjsP,set to its maximum value) until the reactor tempe r-
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Ind. Eng. Chem . Res., Vol. 28, No. 8, 1989
1181
I I I l 1 1 1 l 1 1 I l
Min
TjJ
, ,
T i
Jacket Temperature Setpoint Reactor Temperature
. - - - - - * .
Figure 2 Relationship of dual-mode controller constants.
Table I. Constants Used in Dual-Made C ontro l ler
~
E , = 4 0 C
TD-1 = 2.5 min
PL = 50.0 C
TD-2 = 2.0 min
K , = 26.25 C jacket/ C reactor
rI = 2.75 min
rD = 0.406 min
A t
=
0.2
min
ature,
T,,
is within
E,,,
degrees of th e desired reactor tem-
perature, T:P.
2.
Full cooling (jacket temperature setpoint set to its
minim um value) is then applied for TD-1 minutes.
3 Th e setpoint of the jacket tempera ture controller
is
then set
to
PL, t he preload temperature for TD-2 minutes.
4. A reactor temperature controller, typically a PID
type, is cascaded to th e jacket temperature controller and
its setpoint se t to T:p.
When properly tuned, the dual-mode controller is op-
timal (i.e., it brings the reactor contents to setpoint in
minim um tim e given th e constraints of the heat-transfer
system) as maximum heating is applied for as long as
possible and then full cooling is applied to bring the reactor
temperature to its new setpoint with no overshoot and
a
rate of change of zero. Figure
2
presents th e relationship
between the seven dual-mode control parameters. As-
suming tha t th e jacket temperature controller is considered
separately, there are a total of seven tuning constants to
be determined for the DM controller
Em,
L, TD-1 D-2,
and th e PID constants of th e reaction temperature con-
troller,
K,, 71,
and 7 ~ .
Th e DM controller was tuned in th e following manner.
First, the PID controller was tuned by performing an
open-loop s tep response tes t on t he jacket tempe rature
controller. Its setpoint
was
changed from 20 to 30 C when
f i e d with a normal charge, and a first-order-with-deadtime
model was fitte d to the response. Th e Cohen and Coon
tuning rules were then applied to yield th e values of K,,
T ~
n d TD. Second, the remaining four constants were
determined by running a series of simulations. After each
simulation run, th e performance of the DM controller was
analyzed, and t he values of t he pa rame ters were changed
using the rules outlined by Liptak (1986) in an attempt
to improve the controller's performance.
After five tuning runs, the response in Figure 3 was
obtained , while Table I gives the final values of th e tuning
constants used. I t can be seen that the DM controller
performs very well in this nominal case. Th e reactor
temperature is delivered to the desired setpoint with no
over- or undershoot and the transit ion between the
open-loop heat-up mode and the PID control mode
is
achieved without any disruption. Th e vigorous changes
in the jacket tem peratu re setpoint are caused by the high
gains used in the PID controllers and the noisy reactor
tem peratu re measurements. This could be reduced by
filtering the reactor temp erature before using it in th e PID
1
/I
2 8 1 ' l / I I I , I I I
0
10
28 38 48 59
b 8 18
88 98
180
118 128
T I M E (m in )
Figure
3.
Dual-mode controller response for nominal operation.
In tun ing the GMC controller, because overshoot was
undesirable,
was
set
to 10.0.
T he value of was obtained
by examining the tun ing cha rts given by Lee and Sullivan
and
recognizing tha t, w ith
f
=
10.0,
he controlled variable
should cross the setpoint at approximately
0.257.
There-
fore, to achieve a performance similar to the dual-mode
controller, was s e t t o 80.0 min.
Figure
5
presents the control performance of the GMC
controller for the nom inal case using these tuning co nstants
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1989
lZ81-1
0
18 2 8
3 8
4 8
5 8
b 8
78 8 8 9 8 188
8
128
TIME
( m i n l
Figure 5. Generic Model Con trollers response for nominal opera-
tion.
Table 11 Constants Used in GMC ontroller
= 10.0
r
= 0.5
m
p
=
1000
kg/m3
T~
= 1.0 min
At
=
0.2 min
7 = 80.0 min
C
= 1.8828 kJ / (kg
C)
U = 0.6807 kW/ ( m 2 C)
and the oth ers listed in Table 11. For this nominal case,
it
was assumed that the value of
W C , / U A
was known
precisely. Th is assumptio n will be relaxed in the next
section. I t can be seen th at GM C provides performance
similar to tha t of the D M con troller bu t with less drastic
changes in th e jacket temperatu re setpoint, especially as
the reactor temperature approaches the setpoint value.
Furtherm ore, the GMC controller provides all the control
actions from heat-up to temp erature maintenance without
having to change control algorithms. Th e only drawback
of the G MC app ears to be the existence of a small am oun t
of offset. Theo retical studies of the GM C controller by
Lee and Sullivan show tha t this offset will eventually be
eliminated and the desired tempera ture will eventually be
reached.
Again, the relatively vigorous movem ents of the jacke t
temperature setpoint after the setpoint has been reached
is caused by the use of noisy tem pera ture m easurements
directly in the GMC controller and could be reduced by
the introductio n of low-pass filters.
5. Robustness Evaluation
5.1.
Robustness Tests.
The previous section shows
th at both th e D M controller and G MC controller effec-
tively control the reactor temperature for the nominal
operatio n for which they were tune d. However, it is im-
portant to examine the robustness aspects of both con-
trollers with respect to chan ges in operating and process
parameters and with respect to model mismatch. This
becomes especially impo rtan t for exothermic batch reactor
as the reactor mu st always be op erated safely in spite of
these changes.
A full robustness analysis of th e Gen eric Mod el Control
formulation is beyond the scope of this paper. Lee and
Sullivan discuss the effects of simple linear m odel mis-
matc h in their original paper, bu t the extension of these
results to the n onlinear batch reactor system is not simple.
Therefore, we decided to investigate the robustness
properties of the two controllers through s imulation
studies .
Five
tests
were made in which th e two controllers, tuned
for the nom inal operation, were used to control an oper-
ation where some of the conditions have changed from
their nom inal value. Th e first simply involved changing
the overall amount of the charge from the nominal 1560
4 8
30
S e t p o
i n t
D u a l
Mode
2 8 1 I I , I
I
I ,
0 1 8 2 8 3 8 4 0 5 0 b 0 7 0
88
98 1 0 0 1 1 8 1 2 0
TIME
( m i n l
Figure
6.
Responses of controllers for weight change.
kg to
1300
kg. It represe nts a change in operating con-
ditions that could be caused by a deliberate change in
produc t dem and or an accidental failure of the charging
system. Th e second tes t involves the reduction of the
heat-transfer coefficient from its nom inal value to one 25%
less . This tes t s imulates a change in heat transfer tha t
could be expected due to fouling of the heat-transfer
surfaces. T he third tes ts the robustn ess of the controllers
in the face of change in the reaction che mistry. As stated
by Juba and Hamer (1986), the sensitivity of a given
control strategy to variations in reaction chemistry is of
great importance. In this case, the reaction rate of the first
reaction was increased to abou t
1.5
times th e original rate.
Thi s is also equivalent to th e presence of unmodeled re-
actions. Th e fourth case combined the las t two pertur-
bations in the operation, the decrease in the heat-transfer
coefficient and t he increa se in reaction rate . In each of
these four cases, the changes in opera ting or process pa-
rameters all push the reaction system closer to instability,
especially the fourth case, and therefore provide good tes ts
of controller robustness. T he fifth and final case involved
using th e same controllers
to
control an endothermic rather
than exo thermic reactor. Th is case represents
an
extreme
case of model m ism atch where the sign of the he at released
has actually been reversed.
5.2. Weight Change. Figure 6 shows the resp onses for
both the DM controller and the G MC controller . I t can
be seen tha t the performance of the D M controller is de-
graded while that of the GMC controller has remained
essentially the same as that for the nominal case. Th e
reason for the D M controller degradation is tha t the value
of
E,,,
is specified for a full reactor. On a partly filled
reactor as is the case here, cooling does not have to be
applied as early since there is less thermal inertia.
Therefore, cooling is applied
as
f the reactor were full and
hence the undershoot of the reactor temperature. GMC ,
on the oth er han d, can account for changes in W directly
in the model and therefore does not have to be retuned
for each set of conditions. This is a great benefit whenever
batch sizes change frequently as a result of changing
product demands.
5.3. Heat-Transfer Coefficient Change. Figure
7
gives the responses of both controllers in response to a
changed heat-transfer coefficient, U. This change tests
the p erformance of the c ontrolle rs in light of a change in
unmeasured parameters. Therefore, in this test, the GM C
controller is used w ith its original esti ma te of
U.
Although
both controllers show a change in performance, the per-
formance of the DM controller has degraded mu ch further
than t ha t of the GMC controller . In this tes t , the value
of E , for the D M con troller is too small and full cooling
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1989
1183
I----HC
E
1 0 2
3 0
4
5 e
b a 7 8 0 9 0 1 0 8 i i e
1 2 0
T I ME ( m i n )
Figure 7. Responses of controllers for heat-transfer coefficient
change.
S e t p o
i n t
ual
Mode
. . . . . . . . . .
0 10 2 0 3 4 0 5 0 b B 7 8 0
9 0 100 1 1 0 1 2 0
T I ME ( m i n )
Figure 8.
Responses
of
controllers for reaction rate change.
is begun too late to prevent the reactor tem peratu re from
overshooting, because h eat can no longer be transferred
a t
as
high a rate
as
in the nominal case. Furthe rmo re, after
th e maximum reactor temperature has been reached (98.61
C for DM control , 96.66 C for GMC), the GM C controller
returns th e reactor back to setpoint in a m uch smoother
and quicker manner th an the DM controller. Th is situa-
tion represents a much more dangerous operation tha n the
previous one, because an overshoot in tem perat ure brings
the system much closer to instability.
5.4.
Reaction Rate Change.
Th e results of the third
tes t are given in Figure 8. Once again, it can be seen t ha t
th e DM controller's performance h as again deteriorated
by changing th e reaction rate. Th e maximum reactor
temp erature has r isen to 101.20 C. On the other hand,
the GMC controller's performance has changed very little
when compared with the nominal response. Th e im-
provement in performance is provided by the on-line
heat-released estimator
as
t can predict the speed at which
hea t is being released in t he reactor.
5.5. Heat-Transfer Coefficient and Reaction Rate
Change. Figure 9 shows the performance of the two
controllers for a case when th e reaction rate increases as
well as he heat-transfer coefficient decreases. This is the
most stren uous of the four
tests as
both changes force the
reactor system toward instability. From Figure 9, i t can
be seen th at the DM controller has not prevented a tem-
perature runaway in this case, whereas the performance
of the GMC controller is approximately the same as in
Figure
7 ,
where only the heat-transfer coefficient had
changed. Therefore, this case confirms the result tha t the
GMC controller
is
much more robust th an the DM con-
troller and, the refore, will provide not only bette r control
ise
4 e
el
2 0 ( i I I I
, ,
I
,
I
e 1 0
z e
3 8
4 8 5 8
b e 7 8
B B
9 e
l e e
i i ~
2 8
TIME h i n l
Figure 9 Responses of controllers for combined changes.
110
............................
.*---- *.
i m ]
SETPOINT
[ - - - -
GMC I
0
10
2 30 4 8 5 0 b 0 70 88 9 0
1 0 0 1 1 0 1 2 0
TIME
( m i n )
Figure
10.
Responses
of
controllers for endothermic reaction.
performance bu t also increase t he safety of operation.
5.6. Application to an Endothermic Reaction.
As
a final demonstration of the robustness of the new con-
troller, the reactor simulation was modified so t h a t t h e
reaction carried out was endothermic rathe r th an exo-
thermic, while still using the nom inal controllers. Th e new
heats of reaction used were +20 920 kJ /k m ol for the first
reaction and +16736 kJ /km ol for the second. Although,
as
expected, the dual-mode controller's performance suffers
greatly, as seen in Figure 10, the GMC controller's per-
formance has remained consistent. Th e overall response
of th e G MC controller is slightly slower when compared
to the nominal case, but this
is
largely due to t he fact th at
the jacket temperature setpoint is constrained a t 120 C,
and therefore the amount of heat transfer is limited. The
ability of the GMC controller to handle such extreme
model mismatch is du e to the ge nerality in its formulation.
6. Conclusions
A
model-based control strategy using th e Generic M odel
Control algorithm was developed and applied to the
heat-up and subsequent temperature control in an exo-
thermic batch reactor. GMC provides a method in which
nonlinear feedforward a nd feedback effects can be com -
bined properly. In the nominal case, th e resulting con-
troller has been shown to provide similar performance to
a well-tuned dual-mode controller. However, the new
controller is much more robus t with respect to changes in
measurable and unmeasured process parameters.
Fur-
thermore, because th e GM C controller works directly on
the ra te of the change of the jacket temp erature, the ad-
ditional protective rate of change constraint control stra-
tegies such as those described by Lipta k (1986) are un-
necessary.
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Ind. Eng. Chem. Res., Vol. 28,
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Acknowledgment
k , = exp(kll -
k I 2 / ( T ,
+ 273.15))
B.J.C. thanks the Association of Commonwealth Univ-
ersities for financial support in the form of a Common-
wealth Scholarship.
Nomenclature
C p = mass heat capacity of reactor contents, kJ /(k g C)
C,,$
= molar heat capacity of component i , kJ/(kmol
C)
AH
=
heat of reaction for reaction i kJ/kmol
At = sampling frequency of GMC controller, s
E , = approach tem perature difference for dual-mode con-
h = height of reactor, m
Kl
=
GMC controller constant
1
K 2 =
GMC controller constant 2
K , =
dual-mode controller PID gain, C jacket/ C reactor
k , = rate co nstant for reaction
i ,
kmol-' s-l
k,
=
rate constant 1 for reaction i
ki2
= rate constant 2 for reaction
i
Mi = number of moles of component i , kmol
MW, = molecular weight of component i , kg/kmol
P L = preload tem peratu re of dual-m ode controller, C
Q
= heat released in reactor, kW
p
= density of reactor co ntents, kg/m3
r
=
radius of reactor, m
R , =
reaction rate of reaction
i ,
kmol/s
T = temperature, C
T =
first-order time constant (s) or GMC tuning constant
t = time, s
TD = dual-mode controller PI D derivative time, s
TD-1 = length of time full cooling is applied in dual-mode
TD-2
=
length of time preload is applied in dual-mode con-
r =
dual-mode controller PID integral time,
s
U
= heat-tran sfer coefficient of reactor, kW / (m2 C)
V = volume, m3
W = reactor weight, kg
x
= controlled variable
S u b s c r i p t s
1
= reaction
1 (A B - )
2 = reaction
2
(A
+
C
- )
A =
component
A
B = component B
C = component C
D = component D
f
= filter
j =jacke t
r = reactor
S u p e r s c r i p t s
= actual value before addition of measurement noise
( k ) = at the kth time interval
sp = setpoint
Appendix: Batch Reactor Model
troller, C
controller, s
troller, s
= GMC tuning constant
Equations:
dn/i,/dt
= -R1- R2
k 2 =
exp(k21- k Z 2 / ( T r 273.15))
W =
MWAMA + MWBMB
+
MWcMc
+
M W f l D
Mr =MA
+ MB
+ M c +
MD
cpr
=
( c p ~ ~ AcpBMB+
Cp$C
+ Cp&D)/Mr
v =w/p
A
=
2 V / r
Qj
= U A (
Tj' - Tr'
)
Q,
= -AHlR1-
AH2R2
dT,' Qr + Q j
d t MrCpI
- -
--
dTJ'
FjpjCpJCTj P Tj') - Qj
- -
-
d t VjpjCpJ
T, =
T,
- 0 . 8 6 6 ~ ( ~ -
where
dk)
s normally distributed with
oa = 0.1
C
T.
= T.'
+
a(k )
-
0.866a(k-')
J J
where dk)
s
normally distr ibu ted with ua
= 0.1
C.
Physical Properties and Process Data. MWA = 30
kg/kmol, MWB
=
100 kg/kmol , MWc = 130 kg/kmol,
M W D
=
160 kg/kmol,
C,, =
75.31 kJ/(km ol
C),
Cp,
=
167.36 kJ/(k mo l C) ,
C =
217.57 kJ/(k mo l C) , C, =
334.73 kJ/(k mo l C) , k F
= 20.9057 k12 =
10000,
kzf =
38.9057,
k22
= 17000, AHl = -41840 kJ/kmol, AH2 =
-25
105
kJ/kmol , p
= 1000
kg/m3, r
= 0.5
m ,
U
= 0.6807
k W / ( m 2 C),
pj
=
1000
kg/m3, CpJ
=
1.8828 kJ /(k g C) ,
Fj = 0.0058 kg/s n d Vj = 0.6912 m3.
Initial Conditions at =
0
MAo 1 2 kmol, MBo= 1 2
kmol, Mco= 0 kmol, MDo= 0 kmol, T,O= 20 C, a nd TO
=
20
C.
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Received f o r review July 12, 1988
Accepted J a n u a r y 3, 1989