Control Problems Associated With an Open Loop Unstable Chemical Reactor

CONTROL PROBLEMS ASSOCIATED WITH AN OPEN LOOP UNSTABLE CHEMICAL REACTOR C.While*

A batch reaction is carried out in a water cooled reactor on a small pilot scale plant. Control problems arose because of the inherent one loop instability of the reactor due to insufficient heat transfer area between the reactor contents and the cooling water in the jacket. This paper describes how a control system was designed which stabilised the reactor. DESCRHTION OF PROCESS simplified line diagram is shown in Fig 1. A gaseous or-ganic feed is introduced to the reactor via control valve CV,. There it isabsorbed by a liquid catalyst and polymerisation takes place. The product is more dense than the catalyst and concentrates at the bottom of the reactor. It is drained off at the end of a batch leaving the cat~st reaqy for reuse. A

The reaction is exothennic, cooling being provided by a circulatory flow of water to the reactor jacket. The water temperature :is controlled by adjusting control valves CV2 and CV) which add steam or water respectively to the circulating flow. The reactor contents Initially one p.)djle 1-'1 the gas phase. As rises and the second

are stirred by two paddles on a common shaft. is immersed in the catalyst and the other is the batch proceeds, the level in the re~ctor paddle comes into contact with the liquid..

DEFINITION OF PROBLEM Prior to this investigation the reactor had been operated w~th a controlled jacket inlet temperature and manual adjustments of the organic feed. Because of the inherent instability of the reactor this required the continuous attention of a technical grade operator who would watch the reactor temperature closely and adjust the organic feed every few minutes. A plot of reactor temperature and pressure during a typical batch is shown in Fig 2; as a batch takes five to six hours to produce it is not surprising that some batches were unsuccessful and losses of product occurred. A number of unsuccessful control schemes had been tried by t.he local instrument personnel, eg, reactor pressure controlled from organic feed flow with reactor temperature controlled from .ater/ steam addition and a cascade version with reactor temper&tu-~ cascaded onto the jacket inlet temperature. The objective of this st;.:dy was to design a control system which would stabilise the reactor and hence release the operator for other work and reduce the losses of projuct.

*ICI, Mond Div, Runcorn, Cheshire.

373

THE MATH.EHATICAL MODEL

The basic set of non linear equations which describe the process are listed in Appendix 1. These equations form the basis of a CSHP 370 rr:odel which was writ :. en so that the perfonnance of different control systems could be assessed. Initial compu t er simulations confirmed plant experienc e, ie, the reactor was open loop unstable and a control system was difficult to select and to tune. The equations were linearised and mar~pula ted into matrix form (see Fig 3). Note that this formulation has replaced the ste.:un and chilled. water inputs by a single input vari.3.ble (that is, a split range system in which steam is regarded as 'negative cold water'). The res~lt is a four output.t.. o input process. The eigenvalues of the state matrix A reveal the existence of two right hand plane poles (equivalent to time constants of 500 and 15800 second3) and once again confirming the inherent instability of the reactor. The ca use of the instability is demonstrated pictorially in Fig 4. Thi s shows the heat generated by the reaction Qri and the heat removed from the reactor Qro plotted against the reactor temperature T r at constant jacket temperature T j. At equilibrium Qro must equal Qri, hence three equilibrium states, A, B and C exist. The reaction demands ope cation at point B, otherwise undesirable byproducts are fonne:i. Any small disturbance at the steady state condi~ions of point B cause the reactor temperature to run away to either state A or C depending upon the direction of the original disturbance. Hence sta ble operation is possible at points A and C but not at point B. If the gradient of Qro can be made greater than the maximum gradient of Qri, then state B is a stable operating condition. Adjusting the j acket temperature causes point B to move during the Qri curve but does not affect stability. The gradient of Qro is mo st easily increased by increasing the he at transfer area between the reactor and the jacket contents. Calculations showed that a two fold increase in this area was necessary to ensure open loop stability. Note that the heat trr nsfer area increas es as th e batch proceeds due to the rise in level of the content s, this means U1at the reactor is more stable at the end of a batch than at the beginning. Operating experience confirms this. CONI'ROL SYSTEM DESIGN Several control systems were studied.; each one was defined. by a suitable choice of the feedback matrix C, where

u = -cx The elements of the C matrix are the c on t roller gains. Hence the closed loop process is described by

x = (A-BC)X

374

The characteristic equation of the matrix (A-OC) is a polJrnominal in A whose coefficients contain the controller gains. The rcots of the equation are the eigenvalues of the matrix. Applicb.tions of the Routh-Hurwitz teclmique to the coeffioients enabled stabili ty bounds to be defined. By these means it was shown that the reactor could be stabUised i f the reactor temperature was controllei from the chilled water/steam addition provided that the controller gain was gr..;ater than 2.2 and. that the organic feed flow was held constant. However, for process reasons, the preferred method of controlling the reaction is to hold both the reactor temperature and pressure constant. Hence the effect of a second control loop, reactor pressure to organic feed, was investi~ted. When the C matrix contai."'ls more than one nonzero elanent th9 above approach results in ve~ tedious algebra and consequentlY other design techniques were used. (Subsequent work obtainoo. similar stabUity bounds to those described belOW). At this stage in the study con tact was established vi th the staff at the control systems centre at UMIsr. The UMIsr techniques have been well described. in the literature 1,2 and permit the design of multivariable control systems using classical frequency domain techniques. This is achieved by selecting a precompensator matrix which makes the plant transfer function matrix diagonal dominant. Application of the teChnique to the equatio.'lS of F'l.g j sha.ted that the reactor could be stabilised by two proportional only controllers, one controlling the reactor temperature by adjusting the chilled water/ steam addition and the other controlling the reactor pressure by adjusting the organic feed. For stable operation the controller gains must be greater than 3. and O. respectively. The results of this work have been published elsewhere.:';,~ A third design method, matrix Ricatti, was tried. The Ricatti method designs a control system which minimises a quadratic function of the system states and control effort. It was felt that any penalty function which placed high penalties on deviations of the system states and low penalties on the control effort must result in a stable control system. The elanents of the R and H weighting matrices in the penalty function were chosen us ing judgement based on an acceptable level of variations (see Fig 5). The steady state solution of the Ricatti equations led to a control system which required measurements of all the system states. The concentration of the organic in the liquid phase is not easilY measured but dropping terms involving this measurement from the Ricetti solution does not appear to degrade its perfor.mance significantly. Taking into account transmitter and valve gains, the control scheme becomes W

=

W

= 4.54

w

10.3 (DVTr - MVTr) + 3.frl (DVPr - HVPr) + (DVTr - MVTr) + 0.598 (DVTj - MVTj)

375

1.4 (DVTj - t-l'rrj)

Where DVx

is the desired value of stat.e variable x

~Wx

is the mea sured value of state variable x

Several other R and H matrices were used, most led to SUl.ilar control schemes but on two occasions the Ricatti solution was Wlstable!

SIMULATION RESULTS As both designs were based on a linear model of the reactor, the simulation experiments on the non linear CSMP modal. were of great value in assessing their relative performance. Fig 6 shovs the effect of a positive increase in the temperature set point with the reactor controlled by the two simple loops. There is onJ..y a small amount of interaction between reactor temperature and pressure but the response exhibits an undesirable oscillatory tendency. Fig 7 sho'ois the performance of the Ricatti control system when subj ected to the same disturbance, note the improved d,ynamic response but the incre Dsed interaction. This is not surprising as the Ricatti method is not intended to produce a non.-intaracting control system. Further contact with UMIsr led to the following control system : W

Uw

::I

J.fr7 (DVPr - MVPr)

=

4.54 (DVfr - HVTr) + 0.598 (DVTj - MVTj)

This scheme is derived from the Ricatti solution by dropping further feedback terms. The performance of this control system is also shown in Fig 7. It has a similar dynamic response to the loptilDal l system but the interaction is much less. Similar results have been obtained for other disturbances. PLANT OPERATlllG EXPERIENCE

Aa the necessary control equipnent lolas read..il.y available the simpl.e two loop system was installed on the plant. Reacto:t: tenperature and pressure recordings from a typical batch are shown in Fig 8. Note the small number of pressure setpoint changes in comparison with the original control method. The time at which the large exotherm begins coincides lolith the second paddle coming into contact with the liquid surface. This was confirmed by watching the current taken by the stirrer motor. The cause is unknown but is probably due to an effect, lolhich increases the gaseous absorption rate (frothing?). As the time at which the disturbance occurs can be predicted and its effect can be compensated for by manual intervention (ie by reducing gas pressure) no further investigation of this phenomena vas economically justified.

376

The plant J:lB.nagement regarded the sch3l1e as a success and did not considar installing the multivariable echane to be wurthwhile. A3 this is a pilot scale plant and future production scale plan cs will. be designed with su.f!,icient heat transfer area to be open loop stabl~, this is a reasonable decision.

CONCLUSIONS The UMIS! techniques provide an excellent method of designing control schenes fcr multivariable processes which result in a low degree of cross-coupling amongst the controlled variables. In general the solution will be a multivariable controller which would be most easily implemented as part of a computer control scheme, although in this case a scheme which used two proportional only controllers proved to be adequate. However, the results from any design technique which operates on a linearisation of a non linear process should be tested either by simulation on the full non linear model or against the results basad on further linearisations at different operating conditions. As a:l identical control scheme b2. sed on intuitive rather than theoretical design methods, was tried on the plant prior to this investigation, one might ask why closed loop stability was not achieved. The reasons are two-fold, firstly the system is open loop unstable, and secondly, to acnieve stability a high controller gain is required in the temperature loop, and the intuitive reaction is to decrease controller gains i f instability occurs.

ACKNO'ALEOC-EMENT S The author wishes to thank his colleagues at ICI for help and encouragement during the course of the study, the staff at the 'Control Systems Centre " UMIsr for their invaluable assistance and the directors of ICI, Mond Division for permiss ion to publish.

1

H H P..osenbrock 'Progress in the Design of Multivariable Control

2

Mac.1arlane A G J 'Multivari".ble Control SystElll Design Techniques A Guided Tour'. Prec lEE, Vol 117, May 70.

Systems ,. Control SysteilS Centre. Report No 115, UXIST, October 71.

Munro N and McMorron PD 'Multivariable Control System Studies Design Problem No 4', April 71, Control Systems Centre, UMI,';)"!'.

4

Munro N tDesign of Cont~ollers for an Open Loop Unstable Multivariable System using the Inverse Nyquist Array', Proe lEE, Vo1 119, No.9, September 72, p1377-1392.

377

APPENDIX 1

dASIC PlIYSICOCHEJ'ILICAL EctUATIONS k

First order rate constant

=

koe-E/RI'r Co * L Po

Equilibrium conc of organic in cataly s t

e

-To/Tr

Reaction rate

r

=

VckC

Absorption rate

n

=

KLa (CiI-C)

Liquid Phase Mass Balance

dC dt

=

(n-r)/Vc

Gas P.1Clse Mass Balance

dP dt

=

c(,

Belit generated in reactor

Qri

=

-re

11 ., - J• • ' '.l. l.J

Qro

=

UA(Tr-T j) + H(Tl~-Ta)

Qji

=

UA(Tr-T j) + (Wp+Ws)Cw.Ti

Qjo

•

(Wp+Ws) .CwoT j


=

(Qr~-Qro)/(p cVcCc+Cr)

dl' j/ dt

=

(Qji-Qjo)/ (~wV jCw-+Cj)

r o:novoo. from reactor

lle:;t, to jacket

·Heactor Heat

Balance

J ack ·; t Heat Balance

378

Tr HVg (W-n) ~H

APPENDIX 1 Page 2

LIsr OF VARIABLES Name Lt

A C

Cc Cw Cj

er Co~

E H k

ko

KL LH H n

Pr Po Qri Qro Qjo r

R t

Ta ·1' . . l

Tj T ')

T ~<, 11 '

rf

Vg V.j

VI

Description Mass transfer area Heat tranRf er " :- ,-~ ~__ Cone of o n;ol: ic :"':1 solution Speeifi~ heut of catalyst Specific heat of water Heat capacity ef jacket Heat capacity of reactor Equilibrium cone of organic in solution Base equilibrium Activation energy of organic Natural cooling coefficient First order rate constant Heaction rate frequency factor Hass transfer coefficient La tent heat of steam Molecular weight of organic Absorption rate of organic Reactor pressure Base pre ssure for equil cone Hel3.t gene rated in reactor He1j t removed from reactor Heat out of jacket Il.ellction rate Univer sal gas constant Time Ambient t emp Temp of water into j ;;. cket Temp of Hater out of j ecket Base temp for equil cone He.· ctor temp Temp of :oaturated steillll Chilled water temp Heat transfer coefficient Volume of catalyst. in r eactor Volume of g; s in reae tor Volume of water in j ;;cket Flow of water from pump Flow of stebJll to mix er Flow of chilled water to mixer Flow of TFE to reactor Capacity element cons tant Heat of reaction Density of c~talyst Density of water

379

380

.,

" I

\ --

-__. _ __"_____-L'_._____.____----..s.:---'____

.1-1_ _ _ _ __

~

J

4

5

F\~ ~<. ('f) f.l (\ '-'.tt\ w \~

C

i:) f\

\- t 0

(,Av-.\fo.\\
f

0(

d",-(/'Rd-

381

0 ~"""" ~ ""

r-.ee. ~

~e.M9~w(,

.

r

Tr I •

-4.464

Pr

382

fJ

10-4

10343 x 10-3

0.0

-4.4155

X

2. .. 077 x 10-4

10-3 . 0.0

1.350

X

1 rrJ

10-41

I Prl

10.0

0.0

I .,

15.675 x 10. 1

0.0

.,

v

+

5.096

•

•

X

• X

-1. 663

C

I

3 1.5!)3 x 10-

•

X

10-3

10-3

0.0

-7.m X

2.476

X

10.2

10·3

0.0

0.0

-1.104 x 10-3

I I

I Tjl I c

I

.l

F' ~

-1.J73 x 10-1

0.0

I lo.o +

:;

L1D_riMCl state Space lquaUQD8 for t.be R8actor

0.0

B

J l "

J

t

~~

jj 1'

~

Q

~

-1)

3C

r""\

f~ ~~ i~ ~

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11

&..

j

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of' ~; -\S

V'

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~

'--'

383

r ~i E C \ _ :~ i'>~l> L

2~~ '~~~:~

~3

: .....:,

~.

I

hccep t~ ble

Variable

• _. ~ . '

•

: " ':~ :'- ~~' -{

_ _ •. , - ' _

Level of

+ 2° C

Pr

+

r 1 0~

J~

3quu :-e

Va ri ~ t i on

fr

~_: :::::;

( X._

f

+ u

F~~ d.l ty

0 . 25

R -, ,)

1..

R3)

c

+ 2 kg/H 3

4-

R' I

10- 4

i{

Ww

+

384

+

-2 10

~/

k~

sec

6 x 10- 2 K[;I SGC

That is

'-'"+

36 x 10-4-

1

0

0

0

16

0

0

0

l~

0

0

0

I

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-, I

:i

F:iG 5

.3ELECTi UN O?

[' ~ :.:

A :=: =G~:~ -= _'j l ~

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0

I

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ft',l':" J.

_

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?2:; ~ :". L~ Y

r"l" ,; ;:(~ ~ .: I_J: ~

16 .

l

1• ==

1•

4 x l \F

11

r 4- x

II

1.

,"-,--

+ 2 °C

.

:i~J

Soefficients

11

Tj

W

!~ ! : )

:.J

4-

0.5 bars

-r. _ ,

'v .-...r\.

==

10) -,I I

104-

0 10

3

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--1

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