temperature control of exothermic reaction

8/12/2019 Temperature control of Exothermic reaction

1/8

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

989 American

Chemical Society


2/8

1178

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.


3/8

Ind. Eng. Chem. Res., Vol. 28, No.

8,

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


4/8

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-


5/8

A

- 1 E

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


6/8

1182

Ind. Eng. Chem. Res., Vol. 28,

No. 8,

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


7/8

Ind. Eng. Chem. Res., Vol.

28, No. 8,

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.


8/8

1184

Ind. Eng. Chem. Res., Vol. 28,

No. 8,

1989

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.

Literature Cited

Cluett, W. R.; Shah,

S.

L.; Fisher, D. G. Adaptive C ontrol of a B atch

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de ValliBre, P.; Bonvin, D. Application of Estimation Tec hniques to

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Comput. Chem. Eng.

1989, 13, 11-22.

Jennings, W. First Course in Numerical M ethods; Th e M acmillian

Compa ny: New York, 1964.

Juba, M. R.; Ham er,

J.

W. Progress and Challenges in Batch Process

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Morari, M., M cAvoy, T., Eds.; El-

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Des .

Dev.

1984,23, 597-602.

Lee,

P.

L.; Sullivan, G. R. Generic Model Control. Comput. Chem.

Eng.

1988, 12, 573-598.

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Luyben, W. L. Process Modeling, Simulation and Control for

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Pulley,

R.

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G.

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4-7,

1965.

Received f o r review July 12, 1988

Accepted J a n u a r y 3, 1989

temperature control of exothermic reaction

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