Adaptive Control of Chemical Engineering Processes

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ADAPTIVE CONTROL OF CHEMICAL ENGINEERING PROCESSES L. Rallager and S. Bay J'rgensen

,\I)St.t"Hct. :\ ~IJ)J) l:lelfbu"Ilr\\! controller ()IDmC) is proposed tor ttrultivariable, l'lbtlliucw" ('1100\1:(,,,1 cnj.!i r~l'in.Q; pt>Qc(;)SSOS which hV.\vh."lve t!istributedch..racter. 'lDflSC cOIlf;iH1;$ 01' H Linear' tinle nel'i.es 1l1Odo1. combined with a linear qua<:ir'at.ic optim1l1 conU'Ol $trnt;c~~,\' rnodiTied to permit on-line solution of the Hiccuti OC{Wltion. The moc.lcl sthLCtllt'O is selected based .upoo !! priori chcmic'-ll. cn,giIICcrin,1J; pr'QoC:C~fS (~\'rullUi(:s k/lowl(;.'dgc rod the model p.1.r'lIlICtcl"S are estimutecl usin,u i' f'CCllf'Siv(! l.'xtcndcc.1 least l:iCftUIl'C!:i method. ;\l!;\DSC is investigntedcxparimcntally on it l'1.ox<.':

Chcmh:nl cI1.ginccl'il\U .plunts contain 111 ,I.!;encr-<11 n numhcp of" (~101'.<.!;\' 1U1<1 mass CXChiU~'C proC'cSSC~ com/lill(''(\ wHit ctwlIlic:i\ 1 1'l.:';lctiol1 proC:'CA$(;S. Tn ei tilel' t).llC of pt"OC'Css at le(lst ~'() sh'Qalll1$ 01' c'~lQnent!'; (t)'C b1"Q\\,cllt into c:-l()$(l (dil'Cct 01' il'):lir'CCt) contact whet"'eltpOt1 the exchange Ot' 1't:'.K·tlOl'l QCCUI'S. rh:~ proce.sSOs Hl"'C .most often Clll'I'icc.l Qut in tubular units or equipment \dthl:11(lIiy sti.l.~, ilcnce ti1cpr-occs$(>$ tend to hn\'c diStI'ibnted ci1nrncter. In addition the pl'OCe&.:;Cs Hl'e bilir'le.u· (U1ti oftCl"l 1)()l1lin{~m' due' to the thel'1l1O
reilc,tion kinetic relationships.

In indus-

tt'i011 pl'aC'tiCt" $10\\' C'rulf\\->'Cs often OC'C'UP in in1.<:.'n1<11 Pl'oC'csschm"ilC'tcr'i,:t.ic',::< such 11S
deposition .00 hcnt 'cntal~'st

c;.,:clli.u,\\~' ,::1'r'aC'es

nnd

01' biolo,u;icn.1
111e control problems rOI' C'h~nicn1 pvoces$es atoe l'nainl).' to maintain Pl"Odllct quality \\'ithin spoC'ifhxl lilllits ;:1I1d if 1:hn1:; fnil!:> that') 1:0 maintain rwoduction itttt ;tllpossible.. In PI'ii1(:iplo these ('()lltrol pl"Oblems ure multi,'ill'iable dt~e to the J1tUlibel' of $tr'e~'IfIl$ Ol' cOtl1'X>nents iriv01 ycd. Trtldi tiorll.1.l1~· howover they I.);wc.often beensoh'cd ~\S multi looppt"'OblclUs. TIlis solution 11<,,\s been &''ttisfactot~, in the \ust mnjol'ity of' the cases. HOweVer ca TlI.lfIlber of' processes hm -e. I'e!l'1.'1ined difficult to contl"Ol tl1l.,15 requil'ing a multi \at'iable desiS!tl technique. \\'i th the inc1"&\Sed tendency

to\\,n.nls pl''OCe5S intensification ~' more c;on~ tro1 pl'Oblems m.1~· nm"Mta..,(!co'lsly be sol\-ed \\'Ili le l'ecogn:l;ldng thei 1" multi \
eel

e.'q)el'imenWl~' th \t mar~' chemical engineering processes oft:en can be satisfactoril~' controlled using .fllI,utiYariable techniques.

323

TI1C control designs applied generally have been bastlCi upon off line identif'ie<:i line.ar'izcc.i muthematical models. These design methods however have not been applie<:i signifiC£lJ)tly in pf'O(:ess. control practice; probably clue to the eff'ort required to develop a J1'atht)m.'ttieal model fron basic physics and. chemis,try 1lsingconservation principles. F'tn'thef'more it is often difficul t ~U"lcl costly to identif'y such a Il10del ~ p<''ll'tl)<" bec.1.use estimates of some parnmeteli'S m;.,:,' be difficult to obtain fromeXpet'iment.s, nrlCl partly because the model oCten will be non-linear in the parameters. Fin..'1.11Z" the pan."lf!leters in chemical. processes ~U"e usually time VID'yin.g, for instru1ce due toch.'1llges in interni''lJ. process ch.,rnctel"istic:s thus necessitating more Or' less re,p:ular updating of ~tel"S. COnsequanti:;." thc)'e areobviOUSoppot'tl.!niti.es for multh-iU"iable ndapth'e control design m;:".:hQcts. TI'le method p~"esented in this 1"'''lPOl' is basetl ltpOn an inclil~t orexpliCi t aWl"OaCh \dlertl $i.mple linear relationships are postulatec! be".."Ween measurements and contrel inputs. The i'elationShips are based upon !: pl'iori chemical en~p.neering processknO\dedge concemingblllk flow directions, eneI'g)'and mass transf'ers aad rough estimates of' the time del~"S invol\'00. Byestinlating the coefficient.sof' these relationships on. line it i.s possible .for the moclel to a.d2Ipt tQ sl~' ch
L. Hallager and S. Bay J~rgensen

324

regulator based on linear quadratic optimal control of processes with uncertain parameters. Borison (1975, 1979) extended the basic mini mum variance self tuning controller to the mul ti variable case. Keviczh.y ' and Hetthessy (1977) used a dead time transformation. Koivo (1980) extended Clarke and Ga'.vthrop's method to a \IDIO self tuning controller with an equal number of inputs illld outputs. Buchholt and Ki.imnel (198 ) investigated a one step control method on a two stage evaporator. Bayoumi et al . (1982) presented an algorithm for model order detel~ination and considered processes where the time delay is the same for all control s ignals. In chemical e ngineering processes it is often advantageous to use more measurements than controls. The algorithm presented herein is based upon an incremental multivariable time series model where the parameters are esti mated using rect~sive extended least squares. The controller design is based upon linear quadratic optimal control. S,\PEnDIE:\TAL rnOCESS ,\ chemic01 r'eactor is used as an example of a typical chemical ens;ineering: process . The pilot plant reactor cons idered consists of a tubular fixed bed of' catalyst particles , desi.<;>:ned to be essentiall,'.! adiabatic and with neglis;iblc hent capacity of the reactor wall. It is described in detedl by Hansen and Jors;ensen (1976). In this seri es of experillcnts Cl stream of hydrogcn containing 0.258 .75 mole ~;' oxygen i s fed to the packed catalY'st bed at a temperature of approximately 82 °C. T\:Ie steady' state total flow rate is 2 . 6 m",/cm /sec . The adi abatic t emperature loise during water formation i s 169 °c pr . nole ~ reacted oxygen. :2ual i tab ve Dynamical Description In the gas phase fixed bed catalytic reactor, investigated here, the heat capacity of the sas is negligible compared to the heat capa::ity of' the catalyst pellets , thus the thermal residence time is several hundred times great 21" than the gas residence time. Furthermore it is knOI\TI (Hansen, 1973) that the mass transport processes are sufficiently fast to nake quasi stationar~y descriptions of the nass- balances reasonable. Finally there is 3.lmost thermal equilibrium between the catalyst pellets and the gas and within the pel lets . A reasonable simplified mathematical nodel of the reactor therefore is a pseudo 'lomogeneous model consisting of one dynamic 'lon- linear partial differential equation for the temperature and one quasi stationary' differential equation for the oxygen balance . 30th including dispersive te~s.

quasi stationary functional of the inlet conditions and the temperature profile . b.

A slow propagation of temperature changes. 1he thermal wave passes through the reactor i n 11- 30 minutes dependent on the flow rate.

The Control Probl em The control problem considered in this paper is control of the chemical loeactor (see Fig . 1). The plant has three inputs- temperature, composi tion "md flow rate of the entel'ing gas - in which disturbances can ent er and control actions be tm
DistLU"b
to maintain all or part of the temperature profile, to reduce the the~al load on the process.

b.

to lTk"intain the exit concentration as a measure of pl'OCluct quality.

11 . Set point tracking: c.

THEORY The problem of designing an ada.pti ve control ler for a chemical process i s divided into three parts. i. ii. iii.

~.

..'.. fast propagation of concentration and flow rate changes . The mass balance is a

\Iodel structure select ion Parameter estimation Control design

The selection of a model structure Id,ich is relevilllt for the given process is considered to be very importilllt in order to pose a \,ell condi t i oned estimation problem \\'1. th a reasonable lo\\' number of parameters. In separating the process identification (i illld ii) from the control design (iii) the separation theor y is used although it is not strictl~, valid. The certainty equivalence principl e is used as en ad hoc design principle in that the control design is hillldled deterministically, not accounting for the parameter cmcertainty' . ~lodel

The two important dynamical features of the reactor system are thus

to tro.ck chan.c;es in the set point smoothly with reasonable speed .

structure

A qualitative model structure ma~' be obtained from chemical engineering knO\dedge of quali tati ve process d~ namics. The ingredients in such a dynamic process description are knowl -

Adaptive Control of Chemical En gineer in g Processes edge 0[: i)

The sufficient number of conservation balances for momentum, mass and eners,'Y necessary to describe the process state with a for practical purposes reasonable accuracy .

ii)

The s ignificant capacities.

iii)

The bulk flow direct ions.

i v)

'{he presence of couplings among the balances.

chemical engineering processes t h e Ilynitmic Process Tntraacti ons ClTl10ng the 00LlIlces c:li1d the process inputs rmy be depicted clearly in a di
... , a B , i = 0, ... , b and C ' i = 1, ... ,c i i are assumed unknmm but constant or slmdy varying. The s i ze of the model is defined when n,m,a, b,c and d are :fixed and it is decided \\hich measurements and controls are to be used . If the model is not further structured it \\'ould be necessary to estimate n(n·(a+ l +c)+m(h+1)+1) paranteters (including constant terms). In the example process the nlIlTlber of parame ters is reduced using DYFID ( F i g . 1): a.

The distributed disturbances (concentration and flow rate) influences a ll measurements with a delay of 1 samp ling peri od (due to the sampling of the system).

b.

The thermal wave as obser'ved by a number' of' temperature measurements i s described essentially as a delily from the tempera ture input to the first meLlsurement and :from one measurement to the ne~t. This behilviour necessi t;ltes L1 ppope l' select ion of sLlmpl ing time in relation to model order .

(; Lven the process contrDl s Lmd measurements

it is possible to sel ect a reasonably low unl >C l' of JlLlT'L"unetel's in the 1Tn11 ti vseI"'VdbLe . \vhen a state of ; I h; d; mce \Vi th pret!omincmtl,V convective flow is meLls\u'ell ;1]] inputs which af:fect the parLiClli ;n" Ix)1;u1ce ;1r'C observable . lIsil"\.\:( the J)'I1'TIl it is thus possible to investigate wh ct hel"' essentiLll process inputs LiI'C quali t;ltive l y observLlbl e . [.'or the eXcUTlple system measurements of interrn] oXY,Q;en concentriJ.tion ;Ult! temperatupc cont; lin in:fornnti on from ;t"! I i.npttts 1\11ere<1S mcasUI"Cnlent of total flo\\l ]";tte e . .\.'; . Lit the reactor Ol,tlet only provi des infonlHLion ;1bout in] et :flow ritte. Tt is ;Issluned ti1Llt the input- output chta from the chClllic;ll r'eactOl' cm be described by a disc]'etc- time lllulti\';'lrLlble time-series model : III

~"'( t) = /\ (q-l )~"'( t - 1) + Il(q -1 )u"'( t-eil + C(q

-I

)c(t)

The simplest opproocll is to select eqlli distont measurements, Cl sampli ng time equu.l to (or sI i,Q)It1y larger than) the d ela~' between two n e ighbouping measurements and set a=O, to describe convective propagation. 1\ more elaborate approClch is to select

u*(t) ( dim m) i s u(t)-u(t-1) where u(t) is a n input vector

In this wCly the domi n;mt thernnl d\ H1ITlics of the r'eactor a nd the influence o:f the inputs Cill1 be modelled I d th Cl reasonable munbep of par<1ITlCters.

TIle i' th state in equation (1) can be \\T'i tten : ~,( t)

-1

{x

T

T

T (t- 1) ,x (t-2), ...

,~

T

(t - a - l),

T

u (t-d), ... , u (t-d-b)) and

-1

) are rTlc'ltl'iX polynomials of order a , b and_ in the bac!",,'arc! shift operator q i.e.

), 13 (q

) ill1d C (q

1

where:

( dim n) is a zero nte~1S, white noise sequence

-1

+ e, (t)

1

is Lilt? de Lly fl'Cxn i nput to output ,'\ (q

Cl

shorter sampling time but use additional old measureme nt (i. e . a>O) , thus including a description of dispersive ef'fects . This descript i on adds :flexibi l i t\' in the selection of sampl i ng time and allo\\'s for liJ.rger flow pate disturb~U1ces.

ParLlmeter Estimat ion

~"'(t) (dim n) is ~(t) -~(t -1 ) \,11ere ~(t) is a meLlSlu'ement vector

e ( t)

325

r

,-\(q

-1

)

A

O

+ q

-1

'-\1 + ... + q

+q

-b

-a

Aa

~

The dimensions (n,m,a,b,c) and the delay d are assluned knm\i1l and t h e matrices ':\' i = 0,

(superscript i means the i' th roll' of the matrix) . The parameter' vector 0 , i s estimated using a recursi ve least square§ algor'i thm \\i. th vari able forgetting (For'tescue et al . 1981, Hallager 1983):

L. Hallage r and S. Bay

326

0 . (t+l)

0. + K.(t+1)£ . (t+1)

E. (t+1)

x.(t+1) - ~T(t+1)0.(t)

K (t+1) i

P.(t) • . (t)(5 . (t+1)+a.(t+1)) - 1

5. (t+1 )

.T ( t+ 1 )P . (t) •. (t+1)

~

~

~

H. (t+1) ~

a.(t+l) ~

~

~

~

~

~

~

~

~

~

~

~

J~rgensen

feature of chemical processes a method for offset el iminat i on is essential . Here integral ac tion of selected measurements is used . Consequently the identifi ed model i s reformulated to the following state space form :

~

~

2

1-5.(t+1)-E.(t+1) /V · ~

1

~

~

0

2

';

-2[H.(t+1)+( H.(t+1)+45 .( t+ 1 ))-] ~

~

( 1, ) y (t) =h* T (t-l), ... ,.';*T ( t - a - l),u* T t - ct-

~

... u

where : P. (t+1) i s the normalized covariance matrix ~ of the estimate error ( 0 . - 0. (t+1)) for a. = 1. l

~

~

K (t+1) i s the corresponding gain vector i Cl.

~

(t+1) i s the variable forgetting factor whi ch keeps a weighted s um of the sqLk.red a posteriori errors fixed at a target value ViO'

In this a lgorithm the forgetting factor stays wi thin reasonable limits except i f 5. (t+1) = 0 and £I (t+1) /Vo:,1 occurs s imultaneously , which is not likely. The algorithm is initiated wi th : 0 . (0) = 0 ~

and

The target value V~o may be chosen ~n~ t i ally based upon the des ~red memory length and the process noise level . With a reasonable va lue of V the value of S i s of minor importance . iO By extending •. (t) and 0. appropriately the coefficients ii'i the syst~m noise polynomi al matri x may be est imated using the a l gori t hm given above . The convergence properties of the result i ng extended least squares algorithm is however more restri cted than for the simple recursi ve wei ghted least squares (50derst ran et a l. 1978) . In general all parameter s are assL~ed completely unknown hence i t i s reasonable to ini ti ate a ll covari ance matrices at the same value and thus reduce computations considerably as it only wil l be necessary to calculate one estimator gain and one covariance matrix in each step (Borison , 1979 ) . In the present case however extensi ve use i s made of structural zeros , i. e . on the basis of a pri ori knowledge some parameters are set to zero . This may l ead to differences between the different Q. , K. and P . ' s as the correspondi ng terms 1re femoved~from Q. , and t hus may necessitate the updating on n~est imator gains and n covariance matri ces . However each estimator gain and covariance matrix will be of reduced dimension, compared to the full parameter case and therefore possibly require less computer memory and time.

"ET

(t~ - b) , z

T

(t- 1))

y( t) =z ( t - 1 )+E/"( t) and

1\*, B* Llre shown ~n Table 1 . of the contl'ol c l'i terion:

~1inimization

Jj\ =

/(:\)O~'(:\) :\-1

I

+

T

T

[y (I,J0Iy(k) + u"E (I.;)O,)u( k) ] k=O g i ves

u*(I<)

- L(k)y(l<)

where

clJld 5 (k) is calculclted by solving the rliccLlti equation: 5(k)=01 + I\*T ' 5 (k+1)' (A*-B*'L(I,J) subject to the condition :

5(~)

= 00 '

In the general self tuning case this solution is not strictly usable becLluse the system matrices A* and B* are not constant and their future vari at i on i s not known LlS the [JaI'ameters a r e estimated on- li ne . If:\ i s c hosen as 1, i.e . a onc- step criterion is employed, the solution i s Llppl icLlble . In the present case , however , there is known to be a the rmal time delay from inlet to outlet , I,hich usual ly i s 5- 10 times the sampling time . I t seems reasonable that the control criterion should at least cover this delay in orDer t o account for the pri mary convect ive effects of a ,Q;iven control input. Consequently the follol\i.ng strategy is Llttempted : " is set to infinity indicating that the stationary sol ution is searched for, the Riccati - ,~qua tion is iterated (the ' I\T'emg' I,ay) once at each sampling instant (starting from the last solution) and the nel\'est value of 5(k) i s .lsed in the calculation of the feedback gai ns . \Ihen the system parameters are constant the feedback gains converge to the stationary opti mal gains . Therefore, if ti1e parameters converge the feedback gains also I\i.ll converge to the corresponding optimal stationary values .

Control In order to eliminate drift \,hich is a comnon

The purpose of this strategy i s to approach t he true optimal solution as~mptot ically and

327

Adaptive Contr ol of Ch emica l Engi nee r ing Processes ye t only use modest amounts of computer time . It does not seem reasonable to invest a lot of computer time in finding the true stationary solution of the 8iccati - equation as long as the parameters , which are the basis of the calculations, are only varying, uncertain approxima tions . A consequence of using this strategy is a transient period during startup in which an unstable process may have to b e stabi lized by a s imple controller i f the fee(~ck -gains ilre far from optil1k,l .

The single tenns in A3 delay .

In model IV and V a sampling time of 3 and 2 min a nd two and three A and B matrices were used. The AO and BO were structured as a bove . 'f1"le following A and B matrices account ed fOl" the delay and dispersive effects:

A, = 1

EXPEIHMENTI\L SPECIFICATIONS The IVlul ti Input- Multi Output Self tuning Controller (MI~ IOSC) presented Gbove h <1s been implemented on G DEC-ll03 system \'Vhicll con trol s the fixed bed c hemical re<1c tor . The necessc\ry en,l.!;ineerin,g input encompasses spec ifications of model s tructure , mcl control purpose . The ration<11e for these specifications i s demonstn lted below for the example process . Model St ructures

& A4 accow,ts for the

00000

X 0 0

X 0 0 0 0

000 o 0 0 " o0 0 o 0 c!

OXOOO/and B, = 1 0 0 X o 0 o 0 0 X 0

i

Control Purpose 'f1"le pW'j)ose of control was selected as keeping or reaching a desired temperature profile. The wei,Wlting matrices were diL1gonal . The integr a l states were infini tc sumnations of TO 2 for model I and 11 and of TO ? ' TO . 6 and T : for models Ill - V. The state 'wcights l O were always 0 . 001, the welghts on the lntegral states were 0 . 25 except for model V . 2 where 0 . 1 \Vas llsed on all thl"ee integral states .

The five model st ructures shown in TGble 2 were investi~i1ted . In all mode l s the clel<1y
Thc input wei s~:ht s are shown in Table 2 . In the control examplcs g iven below no effort WGS put into tlmin,<:!; the control parameters . The experiments \\'cre Lmdertakcn without prior simulations .

The AO Illiltrix was structured as :

Ilw"ing identification independent Pseudo- 8andam Binary Sequences were applied simultaneoLlsly to each input. After a few samples solution of the 8iccati - equation was started. hl,cn the model parameters h ad become nearly constant the PRBS generation was stopped. ~ lodel 111 was identified in closed loop \\nile the other model s were identified i n open loop. :\ow the loop was closed , if not before, and thc controllep a llowed to br'ing the process to the clesi red stead~' state. ll1e parameters were stored a nd the experimental pr'Ogram was lmder'taken .

x

0 0 0 0 XXOOO

o

X X 0 0

( 2)

OOXXO OOOXX \,here X denotes ,1 parClJnetel' to be estimated . In addition to the deL1\' tenn from one measurement to the next (- ';,ubdiagonal elements), an i1uto l"egress ive tenn in each me<1surement i s included (- diL1gonal elements). This ex-pansion was employed to <1dd some flexibility to the mode l.

1ni tially all p<1rameters were set to zer'o , \" 0 to the values shown in Table 2 and B=l . 1

RESl'L TS

,~,\1)

DISCL;SSIO:\

ll1e 13 l1k,trix for model III \,'as structured as : Identificati on

x X X OXX X X OXX

o o

(3)

X X

In model I were only temperature control is employed, t h e two last columns \\'ere deleted from e,is matrix . In model 1 1 only the lTlI?asurements at the axial positions O. 2 anc~ l . 0 were used with a sampling time of 4 min. The four A- matrices were structured as :

Genepally the a priori specified model structure proved to be very reasonable at lo\\' oX\'"e n level as the models with the estimated p~~eters were able to explai n 8ifo or more of the observed vari ance . The first measurements were however not well modelled by models I - Ill. In an earlier version Hallager and Jorgensen (1981) appli ed an abso l ute val ue model i nstead of the i n cremental model applied here . They found that the first measurement was well modelled with a model s i mi lar in structure to model Ill , but wi th a

328

L. Hallager and S . Bay

second B- matrix. On- line tuning of ViO was performed in,sane instances in oruer to keep the forgett l ng factors from changing too often. In agreement \,'ith the earlier analysis , where a rn..'ygen level the model \,'as
J~rgensen

-v . 1 is obvi ous in fig. 3 ; the differences are mainly due to l ack of tuning of the control we i ghts. Servo behaviour The servo behaviour of \[DIOSC was investi gated through a gradual set-point change of the integral states, corresponding to @1 increase in the inlet oxygen concentration from 0 . 25 mole% - ca . 0.75 mole% over a time peri od of two hours . The controller stabi lized the reactor efficiently at the high oxygen level using the controlle r weights shown i n Table 2 and specified above .

CO~CLliSIO'-J

~ Iodel

structures containing model order, sampling time and structural zeros may be specified a priori based upon qualitative chemical engi neerin,g process kno\dedge. A number of this type of models with cLifferent number of measurement positions
REFERE\CES Bay'oumi, \1.]\1., K. Y. \Iong and \I.A . El - Bagoung (1981) . A self- tuning regulator fo r multi variable s\'stems. :\utomatica , 17, 575 . Borisson , L'. ~(1975). Self-tuning regulators - industrial application and multivariable theo~' . Report 7513, Dept . of ,-'lutomatic Control , LTH, Lund , S\\eclen . Bor:esson , t . (1979). Self- tuning regulators for a class of multivariable systems . Automatica, 15, 209- 15 . Buchholt, f . and ~I . KUrrrnel (198 1 ). A mul ti vari able self tuning regulator to control a double effect evaporator . Automatica ,

17 , 737 . fortescue, T . R., L . S . Kershenbaum ill1d B . E . Ydstie (1981). Impleme ntation of self- tuning regulators with variable forgetting fac -

Adapt i ve Cont r o l of Chemica l En gin ee ri ng Processes

329

tors . !\utom::,ticil , 17 , 83 1- 835 . lIa llage r , 1,. (1983) . \w ti variabl e self - bming control of a f ixed-bed chemi cal reac tor using st ructureci , linea r models . Ph . D. '1l1es i s , W1der prepar ation. H;:;lJu,c;er , I. . OJKl 5 .13. Jorgensen (1981) . Experimental invest i gation of s elf - tW1ing control of d ,\!;i..tS phase fixed - bed ciltal~'­ lic re~\ctor' Id th muJ tipl c inputs . Pape r no . 106 . 2 Clt 8 ' th IF . \C \\'orld Congr'ess , K\'oto , J; Ipan . lI;lllsen , K. (197 3 ). Simul at ion of the trilI1si ent behd\'iour of ii p i 1 ot plilI1t f ixed- b ed f'e"ctOl-' . Chem . En,U; . Sc i. , 28 , 7 23-734 . II; m sell, I, . ,met 5 .13. ,Jor,<.>;ensen( 1 9 76) . l)yn
.!2,

2:51 - ,1-1.

.\()

T

,\-'" =

u

,\

,\ I --

;1

- - - - 0

0 -

U

-

-

0

° °

- Cl - - - - - 0

0 - - - 0 T - - -

o -

()

()

-

-

F ()

T
\lode L

Cl

-

-

-

-

-I

n

T

4

5

\ lcas\ u"emc!1 t positions O.~ ; O . 4 ; O . G ; 0.8 ;

1. 0

Cl

l(T )

°° ° °

c-

1(T )

IT

4

TTI

4

5

~lS

I

T\

:,

"5

~s

I

llS

~lS

I

itS

\'.1

\ .2 Table 2 .

0 . ::' ;1. 0

as \'.1

}i

melt)

')

,)

I} ' m

i is the

\ I;lt! 'ic es COl' c-on L!'ol p !"Oi ) Lem . !ltunbc !' or intcg!'iti st .. les .

i'lL

1

()

-

mill

° ° °°

n* =

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r

c c c III III

c

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ContJ'ol , Input \';ei,<.cht

-10 , 5 , 20 , 25 , 25

1.00 2 . 00

1

-10 , 5 , 20 , 5,5

2 . 0,1.0 , 1.0

1

1

-10 , 10 , 20 ,10,10

2 . 0 , 1.0,1.0

2

1

Cl

2

\'w -10 , 20

3

c

J(T

b

20 , 5 , 10 , 5 ,5

4 . 0 , 2.0,2 . 0

20 ,20 , -10 , 20 , 20

4 . 0 , 2 . 0,2 . 0

l.i st of model structtu"'Cs iJl\'est i gated and the corresponding sampling times. ('.;ot e t h ilt the I1lUllber of IT1<,tI'ices used are a+l , b+ l and c)

L. Hallager and S. Bay

330

DYNI\ .' II C t'l-(()CESS

J~rgensen

INTRAACTTON DJAGHAM:

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,.

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s

Fig . 1.

Dynamic Process Intraaction Diagram (DYFID) Ior the fixed- bed reactor exampl e . The state variables are shown at the right hand side. The total set of temperature measurement positions considered in this paper are indica ted by crosses .

Fig. 2.

Reactor outlet temperature and temperature control for ~Iodels I and 11 in response to upstream temperature disturbance of +5 °c at time 0 min . Vert ical axis unit 1 °c , horizontal axis unit 5 min .

Fig . 3 .

Reactor outlet temperature and control s i gnals flow (F c ), Concentration (Cc) and Temperature (Tc) for ~Iodels 1II - V in response to upstream temperature disturbance of +5 °c at time 0 min . Units as Fi g. 2 plus 0.01 mole% oxygen for concentration and 0 .1 mg/cm 2 /sec Ior flolll rate .