chemical process design.pdf

8/14/2019 chemical process design.pdf

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Studies in Chemical Process Designand Synthesis

N. NlSH lDAY. A. LIU

Department o f Chemical EngineeringAuburn UniversityAuburn, Alabama 36830

andA. ICHIKAWA

Department of Systems ScienceTokyo Institute o f TechnologyMeguro-ku, Tokyo, Japan

II.O p t i m a l Sy n th es is of Dynamic P rocess Sys tems w i th Unce r ta in tyThe concept of the optimal synthesis of dynamic process systems withuncertain parameters is introduced. A structure parameter approach is used

to theoretically derive the necessary condition for the optimal performancesystem structure, and an effective algorithm for implementing the synthesismethod is presented. The results are applied to the optimal synthesis of areactor-separator system for the dynamic start-up of two reaction systems.

SCOPESignificant progress has been made in the past few yearson the synthesis of chemical process systems. The deter-

mination of the optimal type and design of processingunits, as well as their optimal interconnections within aprocess flowsheet, can now be done with some success bymany synthesis techniques. A good description of the use-ful techniques has been given by Hendry, Rudd, andSeader 1973), and a latest review of the state of the artof process synthesis can be found in a report by Mah1975).

One serious limitation with most recent studies of proc-ess synthesis is that they are mainly concerned with thesynthesis of steady state process systems. A wide varietyof important chemical process operations, such as thestart-up or shutdown of equipment, the batch or semi-batch operation of processing units, etc., however, fallinto the category of dynamic process systems. Furthermore,a large number of recent studies in chemical engineeringhave shown that certain types of processing units, such asthe parametric pumping and the cycling zone absorption,etc., can give better performance by periodic (dynamic)operations. Therefore, it is of practical interest in manysituations to consider the problems of dynamics and controlat the stage of process synthesis (Mah, 1975; Henley andMotard, 1972) rather than merely synthesizing the processby assuming a steady state operation and then compensat-ing the effect of dynamics by means of control after thesynthesis is completed. Unfortunately, except for the recentstudies by the authors (Nishida and Ichikawa, 1975;Nishida, Liu, and Ichikawa, 1975a), this important aspect

of including the dynamics and control considerations inthe process synthesis problem has been neglected in theliterature. Another important aspect of process synthesiswhich has not been given sufficient attention is the effectof uncertainty in process parameters. In the literature, onlythe optimal synthesis of steady state process systems withuncertainty has been studied recently (Nishida, Tazaki,and Ichikawa, 1974).

The objective of this work is to extend the recent re-sults on the optimal synthesis of dynamic process systems(Nishida and Ichikawa, 1975; Nishida, Liu, and Ichikawa,1975n) to include the effect of process parameter uncer-tainty. The problem considered may be stated briefly as:Given the process dynamics, how to synthesize the proc-ess flow sheet and to specify the process design and controlvariables subject to the uncertainty in process parametersso as to achieve an optimal dynamic operation of theprocess? The approach taken is to consider the synthesisproblem as a variable-structure dynamic optimizationproblem by using the so-called structure parameters. Thelatter are essentially the splitting factors of process streamsconnecting various processing units. A number of dynamicprocess synthesis problems with uncertainty are defined interms of structure parameters, design and control variables,and uncertain parameters. In particular, the concept ofthe minimax performance system structure is introduced.This structure will minimize the effect of the worst varia-tions in the uncertain parameters. Both theoretical develop-ments and computational aspects are examined, and illus-trative examples are also given.

CONCLUSIONS A N D SIGNIFICANCEThis work presents quantitatively several new conceptsin the optimal synthesis of dynamic process systems with uncertain parameters. Specifically, the structure parameterapproach has been extended to derive theoretically thenecessary conditions for several optimal synthesis problems

set of structure parameters as well as design and controlCorrespondence concerning this paper should be addressed to Y. A.Liu. N. Nishida is on leave from the Department of ;Management Sci- by using the calculus* n particular, the Optimaence, Science University of Tokyo, Shinjuku-ku, Tokyo, Japan.

AlChE Journal Vol. 22, No. 3 ) M a y , 1976 Page 539


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variables which will minimize the effect of the worstvariations in the uncertain parameters in the minimaxstructure is presented. An effective computational algo-rithm for the synthesis of the minimax structure based onthe gradient method (Nishida and Ichikawa, 1975;Nishida, Liu, and Ichikawa, 1975a) has also been de-veloped. Illustrative examples of the optimal synthesis ofa reactot-separator system for the dynamic start-up oftwo exothermic semibatch reactions are given. Based onboth the theoretical and computational results of this work,the following important observations may be stated:1. In applying the method for the optimal synthesis ofdynamic process systems with uncertainty as presented inthis work, it is not necessary to include all the processinterconnections in the variable-structure dynamic optimi-zation problem in conjunction with the structure param-eters. The use of engineering experience or physical in-sight in determining a restricted system structure for ini-tiating the optimization and the incorporation of an adap-tive procedure based on the necessary conditions for theoptimal structure for automatically modifying the resultingoptimal restricted structures as illustrated in this workwill both enhance the successful application of the methodfor solving high-dimensional dynamic process synthesis

problems. For example, an eighth-order dyn,imic processsynthesis problem has been solved requiring only moder-ate and reasonable computation time and computer stor-age.

2. Based on the computational results for the two il-lustrative examples presented, it is concluded that when-ever the uncertain parameters have a significant influenceon the dynamic behavior of the process to be synthesized,the minimax structure will be quite different from theoptimal structure with the nominal values of uncertainparameters. This is illustrated by the important effect ofthe uncertain inlet molar flow rate ratios of re-actants on the process dynamics of an exothermic auto-catalytic reaction system. The use of the minimax syn-thesis is recommended for such situations. On the otherhand, if the uncertain parameters do not affect the dy-namic behavior of the process significantly, the processstructures synthesized based on the nominal values ofthe uncertain parameters will be similar to those by theminimax method, and thre is no need to use the minimaxsynthesis. This is also illustrated in an example of the dy-namic start-up of an isothermal consecutive reaction system.

PROSLEM FORMULATlONConsider a dynamic process system consisting of M-2)number of processing units, called subsystems, which con-vert raw material streams to desired product streams. Thedynamic behavior of each subsystem is described by (see

Figure la)yi t) = gi[yi(t), xi(i), 4,ui(t), pi, qi( t ) ly d o ) = yw i = 2,3,. . ,M - 1 (1)

The design and control variables di and q and uncertainparameters pi and qi, which are within their correspondingadmissible regions Di, Ui,Pi, and Qi are called the admis-sible design, control, and uncertain parameters, respec-tively. For simplicity, the emphasis in the following dis-cussions will be on the single-input and single-output sys-tem in which each processing subsystem as well as thewhole process system has only one input stream and oneoutput stream. Extensions to the multi-input and multi-output system such as separation units can be learnedfrom the similar formulation for the problem of optimalsynthesis of dynamic process systems without uncertainty(Nishida and Ichikawa, 1975; Nishida, Liu, and Ichikawa,1975~) ut are illustrated in the later example problems.It is also convenient to use an imaginary input subsystemand an imaginary output subsystem to represent the sys-tem input and the system output, respectively. The imagi-nary input subsystem serves to receive the raw materialstreams and to distribute them to other subsystems. Theimaginary output subsystem collects the output streams ofother subsystems and delivers them as the desired productstreams to the outside of the process system. No trans-formation of the state vectors is considered in the imagi-nary input and output subsystems, and hence

y i ( t ) = x i ( t ) ; i = l , M (2)The structure (flow sheet) of the process system is char-acterized by a set of structure parameters aij (i, = 1,2,. . .M defined by (see Figure lb)

Mxi(t) = aijyj(t), i = l , 2 , . . . (3)

The structure parameters ij are essentially the splittingfractions of process streams connecting tho individualprocessing units. They are scalar valued variables satisfy-ing the constraints

j=1

DESIGN CONTROLii T4Pi gj (t)

'ilUNCERTAIN PARA METERS

Fig. 1 a. Top : i l l u s t r a t i o n o f subsystem var iables. b. Bot tom: i l l us t ra -t i on of system structure and s t ruc tu re parameters .

Page 540 May, 1976 AlChE Journal Vol. 22, No. 3)


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Ma i j = l ( j = 1 , 2 , ...M )

1=1

As an illustration, in the reactor-separator system shownin Figure 2u, is the fraction of the feed stream fromthe imaginary input subsystem 1 which goes to the secondcontinuous stirred-tank reactor (CSTR), ubsystem 3. Forthe multi-input and multioutput subsystem such as theideal separator (SEP) shown in Figure 2u, superscriptsmay be used to represent different input or output styeams;for example, 2 ~ ~ + 1 ) 4s the fraction of the second outputstream from the separator, subsystem 4, which goes tothe imaginary output subsystem N + 1 as the desiredproduct stream. For convenience, the structure parametermatrix with elements aij (i, i = 1, 2, . . M ) s denotedby a, nd the vector [ a l ~ ,. . YNi] is written as a+The performance index of the process system to besynthesized, J, is measured by the sum of scalw per-formance indexes for all processing units or subsystems,Jt ( i = 1,2, . .M):

M

i=1M Ptt

i = lHere, the subsystem performance index J i is the time in-tegral of a given function p, which may represent the costof raw material, the sale of product, the operating costof the processing unit, etc. This form of subsystem per-formance index can be considered to be fairly general,since other forms can always be written in this form bydefining additional state variables and subsystems.

Based on the above description of the dynamic processsystem, the synthesis problem may be defined as: Findthe optimal admissible structure parameters ai j designsdi, and controls ui which satisfy the subsystem dynamics,(1 ) and (2 ) , to minimize the process performance index,(5) , subject to the uncertainty in process parameters. Inthis study, four types of performance indexes are usedwhich lead to Merent types of optimal synthesis prob-lems. These are described as follows.The first problem is to find the optimal values o struc-ture parameters, design, and control variables which willminimize the performance index, (5 ) , when uncertain

parameters are specified at their nominal values. The re-sulting system structure will be called the nominallyoptimal performance system structure or the nominallyoptimal structure. Mathematically, this is equivalent tothe minimization problem represented by

= Min J(v,wN)V

along with the constraints (1 ) to 4 ) , where v = (a ,d,u ) ~nd w = (p, q) T . The sohtion to this problem hasbeen presented in previous papers (Nishida and Ichikawa,1975; Nishida, Liu, and Ichikawa, 1 9 7 5 ~) .The next problem is to find the minimax performance

system structure or the minimax stmcture, which willcompensate the effect of the worst variation of uncertain

I I4

SEPt t3 q3 ,. : N + 1 ) 4 aq2 -

q224

UFig. 2 a. Top: in i t ia l ly restr icted system structure for example 1. A

Bottom: in i t ia l ly restr icted system structure f o r example 2.parameters by choosing the optimal set of structure pa-rameters, design, and control variables. This can be ex-amined mathematically by first studying the maximumincrease in the value of the performance index, (5),caused by the worst variation of uncertain parameters,when structure parameters, design, and control variablesare specified at their nominal values. This auxiliary prob-lem can be described by

J(vN,w)= MaxJ(x,y,uN,dN,uN,p,q)P.4= MaxJ(vN,w )W 7)

The system structure defined by (7 ) will be called thenominal performance system structure with worst uncer-tain parameters. Here, it is assumed that the upper and/orlower bounds of uncertain parameters can be measuredor estimated, and hencepiL is iu (8)q i L qi( t )4 qiu ( i = 1,2, . . 1 (9)( i = 1,2, . . m

The minimax structure is then defined byJ(vo, w ) = Min Max J ( x , y, a, , u, p, q)

a d.u p.q= Min Max J(v,w) (10)v w

This minimax approach essentially considers the dynamicprocess synthesis problem with uncertainty as a differentialgame (Bryson and Ho, 1969; Ciletti and Starr, 1970) be-tween uncertain parameters and decision variables appear-i n g in the synthesis which consist of structure parameters,design, and control variables. While uncertain parame-ters attempt to maximize the performance index, the bestset of structure parameters, design, and control variablesin this worst-case synthesis is to be determined. In anearlier paper (Nishida, Tazaki, and Ichikawa, 1974), thisapproach was applied to the synthesis of steady state pro o

AlChE Journal Vol. 22, No. 3) May, 1976 Page 541


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ess systems with uncertainty, in which further discussionson the minimax synthesis technique were given. A moregeneral treatment of considering the minimax synthesistechnique as one of the several possible approaches forthe optimal synthesis of process systems with vector-valuedor multiple performance indexes can be found in Liu et al.(1976) . Here, it is sufficient to mention that although theminimax approach is usually a conservative one, the proc-ess system synthesized based on the minimax performanceindex gives the process designers an assurance that theactual performance of the process system will never beworse than the one calculated from (1 0) .The last performance index used in this study is J(uNYdN, uN, pN, qN) the nominal performance index. This re-quires all variables appearing in the performance index(5) to be specified at their nominal values. The corre-sponding system structure will be called the nominal per-formance system structure or the nominal structure.THEORETICAL DEVELOPMENTSNecessary Condi tions for Nom in al Performanc e SystemStructure with Worst Uncertain Parameters, J (vN , w* )In order to derive the necessary conditions by means ofvariational calculus, several assumptions must be made.The functions f i and gi are both twice continuously differ-entiable and satisfy the Lipschitz conditions. For eachsubsystem, the output steam state vector yi is uniquelydetermined when the input stream state vector xi, designvector di, and control vector ui are given. Also, ui is piece-wise continuous in the time interval [0, tr].

It is appropriate to introduce the Hamiltonians HiN andSiN asHiN = - i (xi, yi, c L ~ ,iN,pi, qi)+ hiTgi(xi,yi, diN,uiN,pi, qi) (11)

S i N = {UkTYiU.iN} i = 1,2, . ..)M , (12)Mk = l

where hi and pi are the adjoint vectors defined by

hiT = 0The admissible time-independent uncertain parameterspi i = 1, 2 , . . .,m are said to satisfy the weak mini-mum condition, if for given values of xi, yi, hi,p., iN, uiNyand qi, the time-averaged Hamiitonians

are either stationary with respect to pi or attain their mini-mum values when pi is on the boundary of the admissibleregion Pi. The time-dependent uncertain parameters qi ( t )( i = 1, 2, . . ., 2 are said to satisfy the minimum condi-tion if at any instant t s [ O , t f ] he function HiN attains itsminimum value for given values of xi, yi, hi, uiN, diN, andpi. Since the functionFiN is not an explicit function of u iandTiN is not an explicit function of pi, the minimizationof the time-averaged Hamiltonians or the weak minimiza-

tion of EiN and T N an be carried rmt sepcirately, theformer with respect to pi and the latter witl, respect tou i The following necessary conditions can tnen be ob-tained:If a system structure of a dynamic proce s system isthe nominal performance system structure wit ti worst un-certain parameters defined by the performance indexJ (vN, w* ), (7), the admissible time-independent uncer-tain parameters pi ( i = 1, 2, . ., m satisf:y the weakminimum condition, and the admissible time-dependentuncertain parameters qi(t) ( i = 1, 2, .. , 1 satisfy theminimum condition.Necessary Condi tions f o r Min im ax Performance SystemStructure, Ilvo,w*)

To determine the optimal set of structure parameters,design, and control variables which will mmimize theeffect of the worst variations in uncertain parameters cor-responding to the minimax structure, it is necessary todefine several new Hamiltonians Hi, ,xi,ndTi. Thesefunctions are essentially similar to the preceding ones, HiN,SiN,xN,nd sN.heir expressions can be written simplyby replacing the nominal values diN, uiN, and U.iN in11) and (12) and (1 4) and (1 5) by di, ui, and u iNrespectively. The admissible structure parameters and de-sign variables are said to satisfy the weak maximum con-dition if, for given values of xi, yi, Xi, and pi the time-averaged Hamiltonians xi and re, respectively, eitherstationary with respect to structure parameters and designvariables that lie in the interior of the admissi,> e regions,or attain their maximum values when the admissible struc-ture parameters and/or design variables lie on the bound-ary of the admissible regions. The admissible control vari-ables are said to satisfy the maximum condition if at anyinstant tc[O, tf] the Hamiltonian Hi attains it,; maximumvalue for given values of xi, yi, Xi, and +. By extendingthe analysis in the previous papers (Nishida and Ichikawa,1975; Nishida, Liu, and Ichikawa, 1975a, 1975b; Nishida,Tazaki, and Ichikawa, 1974), the following necessaryconditions can be obtained:If a system structure of dynamic process s .stem is theminimax structure, the optimal set of admissible structureparameters, design, and control variables and uncertainparameters uo, io, ui, pi*, and qi *( t )> espectively, is atthe saddle point of the Hamiltonians Eli and S i . In otherwords, the admissible structure parameters u0 nd designvariables dio satisfy the weak maximum conditions, andcontrol variables uio satisfy the maximum condition forgiven uncertain parameters, while the admissible time-independent uncertain parameters pi* satisfy the weakminimum condition, and the admissible timedependentuncertain parameters qi* ( t ) satisfy the minimum condi-tion for given structure parameters, design imd controlvariables. By denoting the weak maximizatioli of A withrespect to b under the constraints of B, C, . . aswmax {A; B, C, .. }, the weak maximization of withrespect to u.i can be represented byb

Mf f k i = 1, OLCYki4 1 \ (16)k=l Jfor i = 1, 2, . . ., M . The optimal structure parameters asthe solution to (16 ) are of particular importance in the

synthesis of system structure and can be shown to be(Nishida and Ichikawa, 1975; Nishida, Liu, and Ichikawa,Page 542 M a y , 1976 AlChE J o u r n a l Vol. 22, No. 3)


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1975u, 19753)

In (17), (18), and (19)

is called the structure parameter function, and ki desig-nates a set of subsystem numbers k s for which the cor-responding M k i S are maximum; that iski = { k ; Mki = max Mli}1

(21)for i = 1,2, .. ,MThe condition (17) implies that if the output stream ofsubsystem i, yi, goes completely to subsystem j as an inputstream, that is, when aj i = 1, then the value of M k i is themaximum among all Mk;S for k = 1, 2, . .., M. The con-dition (18) suggests that if fractions of the output streamof subsystem i, yi, are fed to different subsystems k s, thatis, if 0 uki 1 and 2 CYki = 1, then all the subsys-tems k s which receive yi as input streams will have thesame maximum value of Mki.

Because of space limitations, the solution of other maxi-mization and minimization problems for finding the opti-mal set of design and control variables, as well as time-independent and time-dependent uncertain parameters inthe minimax structure, will not be described here. Inter-ested readers are referred to the similar analysis givenin the previous papers (Nishida, Liu, and Ichikawa, 1975u,197527; Nishida, Tazaki, and Ichikawa, 1974).

1M

k = l

COM PUTA TIONA L ASPECTSThe use of (17) to (19) for specifying the optimal

values of structure parameters requires the knowledge ofthe structure parameter function M ki . The latter dependson the values of adjoint vectors, which are solved fromthe adjoint equations (13). The adjoint equations, inturn, involve the optimal values of structure parametersto be determined. Fortunately, an iterative procedure suchas the gradient method has been shown to be very effec-tive in solving a similar boundary-value problem to ob-tain the optimal values of structure parameters, design,and control variables (Nishida and Ichikawa, 1975; Ni-shida, Liu, and Ichikawa, 1 9 7 5 ~ ) .n this study, in orderto enhance the applicability of the gradient method forsolving high-dimensional synthesis problems, this methodis applied primarily to optimize an initially restricted sys-tem structure. Specifically, a restricted system structureis the one in which some of the structure parameters arefixed at preassigned values, possibly at zero, and the restof the structure parameters, design, and control variables,and uncertain parameters are allowed to vary during theoptimization process. The optimal system structure ob-tained under this initially restricted condition will becalled the restricted optimal system structure. Based onthe preceding theoretical developments, the true optimalityof the restricted optimal system structure when no restric-tions on the structure parameters are included can be ex-amined by the magnitude of the structure parameter func-AlChE Journa l Vol. 22, No. 3)

tion. This aspect will be incorporated in the followingcomputational algorithm. For convenience, several nota-tions are first defined here:

Zi = {k; CYki is allowed to vary in the restricted structure}I i k; Yki is fixed at zero in the restricted structure}Ii = {k; 1 > CYk > 0 in the optimal restricted structure}--

where i = 1, 2, . ,M.A Min imax Structure Algorithm

Step 1. Assume an initially restricted system structureZi(j) ( i = 1, 2, . .., M, and initially j = 0), where thesuperscript ( stands for the jth iteration.

Step 2. Assume the initial estimates, (initiallyk = O , of the admissible structure parameters, where thesuperscript (k) denotes the kth trial.

Step 3. Solve the problem Min Max J defined by (10)with the fixed values of structure parameters, cuijck), (a)guessing v(~) initially r = 0) in the admissible region,where the subscript ( T ) stands for the rth trial within thecomputational routine of step 3; (b) determining theworst uncertain parameters w by solving the problemMax J with the fixed v(,); (c) returning to ( a ) and repeat-ing the procedure until JStep 4. f J vo, w ] = J [ c Y . . ( ~ - ~ )3 , 0s ~ 1 ,o tostep 6; otherwise go to step 5.Step 5. Compute the state and adjoint vectors andmodify the structure parameters for the (k+ l ) t hrial by

v w

W vo,w"] is obtained.

where e is a positive step size factor.

are such thatStep 6. If all of the computed structure parameters auc

then the optimal restricted minimax structure satisfies thenecessary condition for the optimal structure even when noinitial restrictions on the structure parameters are in-cluded, and the iterative optimization terminates. If anyof the structure parameters a l k is such that-

Mli > Mki; 1 E Zi'j), k E Zi ), i = 1, 2, . .. M (23)then 1 of ~ y l i ( j )should become a member of Z i ( 5 + 1 ) in thenext restricted structure, and all k's of &') should remainas members of Z t ( j + l ) . Also, all k s of Orki ) such that k E Zi(j)and k e Zi ( j ) are removed from Zi(j+l) .Step 7. Return to step 2 and repeat steps 2 to 7 until(22) is satisfied.

It should be mentioned that the above computationalalgorithm may yield only a locally optimal solution to thesynthesis problem, depending upon the starting values ofthe structure parameters, design, and control variablesand uncertain parameters. Furthermore, the condition( 2 2 ) is generally only necessary and not sufficient, ex-cept for the linear separable dynamic systems. In orderto obtain a global optimum, the iterative optimization hasto be tried with different sets of starting values. However,the above algorithm has an advantage in that it is notnecessary to include all the possible interconnectionsamong the available processing units or subsystems in theinitially restricted system structure in step 1. Even ifsome of the necessary interconnections are omitted in theinitially restricted system structure, the adjustment pro-vided by step 6 in the above algorithm will automaticallymodify the initial structure properly. Consequently, this

c

May, 1976 Pose 543


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adaptive procedure can greatly enhance the successfuluse of engineering experience or physical insight in thedetermination of the initially restricted system structurefor the dynamic process system to be synthesized.ILLUSTRATIVE EXAMPLESExample 1

Consider a process synthesis problem in which the opti-mal interconnections of a maximum number of two avail-able continuous stirred-tank reactors (CSTR) and an idealseparator (SEP) as well as the optimal reactor volumes areto be determined for the start-up operation for a first-order reaction A B -+ C Here, A is the raw msterial,B the desired product, and C the undesired product. Ini-tially, either one or both of the two CSTRs will be filledwith an inert fluid D. At time t = 0, the start-up of thesystem is initiated by continuously supplying the reactantA. This i s continued for a certain finite.time period [0,tf]. The ideal separator removes completely products Band C from the rest of the reaction mixture, A and D .If it is assumed that the molar volume is the same forall the components and does not change with time duringthe reaction, the dynamic behavior of the two CSTRs canbe written in terms of the molar flow rates of the compo-nents as

K1 K 2

(24)14. -- XfC - iC) + K2yiB41biyiD =- %D - iD)

yis(0) = yisowhere i = 2 ,3 represents the subsystem number for thetwo CSTRs and s = A, B , C, D denotes the materialA, B, C, and D , respectively. x i is the input molar flow rateto the specified subsystem i and yi the output molar flowrate. The reaction rate constants K 1 and K2 are given by& ( T i ) = Kloexp(-EJRTi) and ( T i ) = K20exp( -E2/RTi), where R is the ideal gas constant, T i ( i =2 , 3 ) is the temperature of the CSTR with the subsystemnumber i, Klo and K 2 0 are both constants, and E l and Ezare the activation energy constants. The mean reactorholding time Bi in (24) is related to the reactor volume Vito be specified by

Bi = V i / ( a + X ~ B+ ~ i c XiD)p, i = 2 , 3 (25)The operating cost of a CSTR per unit time is

f i = uV~= aBip(x, + Xis + X ~ C+ x ~ D ) , i = 2 , 3(26)where a is the cost factor for the CSTR.Nominally, the reaction proceeds isothermally in theCSTRs. The temperature fluctuation of the CSTR fromthe nominal temperature is taken as the uncertain parame-ter variation, and hence

T i ( t ) =TiN + q i ( t ) t 0 10, tfl (27)The uncertain temperature variation q i ( t ) may cause theformation of the undesirable product C and the decreasein the rate of the formation of the desired product B.The dynamic lag of the ideal separator is negligiblewhen compared to those of the CSTRs, and hence the

subsystem equations for the separator areT O 0 0 0 1

y * k / O 0 1 00 1 %l o o o O Jf l 0 0 0 1l o o o i J

Here, y4I and ~4~ are, respectively, output streitm 1 whichcontains B and C and output stream 2 which contains Aand D. The cost of the separator per unit time is given by

(30)4 = b(X4A + X4B + X4C + X4D)Pwhere b is the cost factor of the separator.With reference to the initially restricted system struc-ture chosen for this example as shown in Figure 2a theimaginary input and output subsystems and their associ-ated objective functions per unit time are

y 1 = x1; f l = 0YN = XN; fN = - NTyN(31)

y N + l = XN+l;P N T = VNT;

f N + 1 = - N + l T y N + 1W N + l T = V N + l T

where v is the price vector associated with all the fourcomponents in the product stream. The specific processsynthesis problem is then to find the system structure pa-rameters as well as the reactor design volumes for theoptimal start-up of the given reaction system in the pres-ence of the uncertain reactor temperature variation so as tominimize the overall objective function

This represents an integral revenue of the desired productB minus the cost of operation and the penalty for theformation of the undesired product C or this time period.For illustrative purposes, the following numerical valuesare used throughout the example:

= 1.0, tf = 1.0, K~~= 0.5 x 104,K20 -= 0.8 x l o 9

E l = 8 400, E2 = 20 000, R = 2.0TiN= 525 K, i = 2,3

VN = (0.0, 3.0, -3.0, O . O T ,v N + 1 = (0.0, 0.0, 0.0, O . O ) T

X l ( t 4 0 ) = (0.0, 0.0, 0.0, l.O)T,y p = (0.0, 0.0,0.0, .O)T,

X l ( t >0) = (1.0, 0.0, 0.0, O.O)Ti = 2, 3 a = 1.0, b = 0.04

Figure 3a illustrates the nominally optimal structure de-fined by (6 ) and calculated by the gradient method fromthe initially restricted system structure shown in Figure2a. The resulting reactor design volumes and the value ofthe objective function (6) areV2 = 0.0946, Va = 0.0931, and

] aN,N,qN) = -0.01712Page 544 May, 1976 AlChE Journal Vol. 22. No. 3)


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st. 2 3 4CSTR STR EPV2 =0.0946 V3 -0.0931

II (0.8926) 1(0.1074)a) NOMINALLY OPTIMAL STRUCTUREFOR EXAMPLE I

V3 80.0877

(0.8672) I(0.1328)b )M IN -M A X STRUCTURE FOR EXAMPLE If i g . 3. Nominally opt imal and minimax structures for example 1.

where the design vector d represents the volumes of theCSTRs, that is, d = (V2,V3)T,and the time-dependentuncertain parameter vector q denotes the temperaturefluctuations of the CSTRs, namely, q = [ q 2 ( t ) , q 3 ( t ) l T .Notice that since the structure parameters ~ ~ 2 2 ,Y23, and a33in the initially restricted system structure shown in Figure2a are fixed as zero, the preceding theoretical develop-ments suggest that the structure parameter functionsshould be used within the computational algorithm tocheck the true optimality of the restricted optimal systemstructure represented by Figure 3a according to (22)and (23 ). The computed values of the structure parameterfunctions areMz l = 0.172, M a = 0.254, M B = 0.282, M M ~ 0.000M3l = 0.151, M32 = 0.267, M33 = 0.348, M M ~ 0.044M11 = 0.114, M a = 0.210, M a = 0.359,

M 2 , ~ + i ) 4 0.000Consequently, the inequality Ma2 > M 2 2 indicates that arecycle of a portion of the product stream from theCSTR-2 into its inlet should not improve the system per-formance. From the results M43 > M23 and M4 3 > M33,the recycles of a portion of the product stream from theCSTR-3 into the inlets of the CSTR-2 and the CSTR-3,respectively, should also not give any better system per-formance. Hence, the nominally optimal structure shown inFigure 3a is truly optimal, even when no restrictions areincluded in the initially restricted system structure of Fig-ure 2a.By assuming that the reactor temperature fluctuationsq 2 ( t )and q 3 t ) are bounded according to (9), with

9iL= - .0K, qiu = 5.O0K, i = 2,3the computed nominal performance index with worst un-certain temperature fluctuations, (7 ) , isAlChE Journal Vol. 22, No. 3)

L I Iinfr0.0 0.2 0.4 0.6 0.8 1.0TIME tB

Fig. 4. Worst temperature var iat ions in the CSTRs in the nominal lyopt imal s tructure in example 1.

F - YIELD WITH THE Im

v )w5 s3 23a 2L3

--- YIELD WITHOUTI 1 U U . 1 r . y

I+. TEMPERATURE VARIATIONr r

tORST TEMPERTUREVARl ATlON

0.0 0.2 0.4 0.6 0.8 .oTIME, tFig. 5. Opt imal prof i les o f product molar f low rates in the nominal lyopt imal s tructures with and without the worst temperature var iat ions

i n the CSTRs in example 1.

Figure 4 shows the corresponding worst temperature fluc-tuations q 2 ( t ) and 9 3 ( t ) The bang-bang characteristicsof these temperature fluctuations suggests that the uncer-tain parameters in this worst-case synthesis tend to in-crease the Hamiltonians piNand ?TiN given by (14) nd(IS) o their maximum values by staying on the boundarThe intermediate switching of these uncertain temperaturefluctuations between qiL and qiu shown in Figure 4 is anindication of the relative importance of the integral reve-nue of the desired product B vs. the cost of operation andthe penalty for the formation of the undesired product Cwithin the start-up period [O t f ] . his is further illustratedin Figure 5, where the changes in the product molar flowrates with time with and without the uncertain tempera-ture fluctuations are compared. For example, the solidcurve in Figure 5 shows that the undesirable product C

May, 1976 Page 545

of the admissib e uncertain parameter regions giL or qiY


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is formed significantly during the period of 0.04 < t 0.465, the worsttemperature fluctuation of the CSTR-3 is seen to be atthe lower bound of 9 3 t ) ,and a decrease in the desiredproduct is observed. This is also consistent with the rela-tive importance of the effect of temperature change on theassumed reaction rate constants Kz and K3, in which thevalues of the activation energy and the preexponentialconstants are quite different.By using the nominally optimal structure of Figure 3aas the initially restricted system structure, a minimax sys-tem structure is synthesized by the preceding algorithm.This is shown as Figure 3b, with a computed performanceindex as

J a , do, q* ) = -0.1580The comparison of Figures 3a and 3b indicates that thevolumes of the CSTRs are reduced in the minimax struc-ture as compared with those of the nominally optimalstructure, but the stream split ratio ( ~ 2 4 ~hanges its valueonly slightly in both structures. Also, the values of thethree system performance functions J (aN N, qN) J (aN,dN, q'), and J ao,o, q* ) are not significantly different.This is mainly because of the fact that for the isothermalreaction considered, the dynamic behavior of the reactor-separator system is not strongly influenced by the fluc-tuations in the temperatures of the CSTRs. Consequently,for the present dynamic start-up problem, the optimalreactor-separator configuration and reactor design volumesbased on the nominal parameters can be used with a rea-sonable faith even in the presence of the parameter un-certainty. Generalizing the results of this example, it maybe suggested that whenever the uncertain parameters haveonly a moderate influence on the process dynamics, it issufficient to consider the use of nominally optimal systemstructure only, but not the minimax system structure. Theabove discussions, however, do illustrate the essential fea-tures of the proposed concept of the optimal synthesis ofdynamic process systems with uncertain parameters andan effective algorithm for providing the quantitative basisfor choosing the nominal and minimax system structures.Finally, it should be mentioned that the computation timefor using the gradient method in solving this examplehas been moderate and reasonable. For instance, a CPUtime of only about 10 min on an IBM 370/155 computeris needed to obtain the minimax system structure. Judg-ing from the inherent singular bang-bang characteristicsof the minimax solution as represented by (17) to (20),it may appear that other computational algorithms devel-oped for the minimax control problems as reviewed andpresented by Muralidharan and Ho (1975) may requireless computation time. The computational experience insolving the singular bang-bang control problem by theauthors, however, seems to suggest that a recently devel-oped algorithm based on the piecewise maximization ofthe Hamiltonian (Nishida, Liu, Lapidus, and Hiratsuka,1976u, 1976b) is most suitable for application to this dy-namic process synthesis problem.In the next example, the effect of uncertain fluctuationsin the reactor inlet molar flow rates of reactants on thedynamic start-up of an autocatalytic reaction system willbe presented. Here, the uncertain parameters stronglyaffect the process dynamics, which leads to some minimaxsystem structures that are quite different from the nomi-nally optimal system structures. The important problem ofapplying proper reactor temperature control to reduce theeffect of parameter uncertainty in this dynamic processsynthesis problem will also be discussed.

Example 2Consider a reversible autocatalytic reaction

K1A + B = B + BK 2

where the forward reaction is second order and the reversereaction is first order. The start-up operation is carried outin a similar manner as example 1, except that the availableprocessing units are only one CSTR and one SEP, andinitially the inert material C is present in the CSTR. Withreference to the initially restricted system structure shownin Figure 2b, the dynamic equations of the CS'L'R in termsof the molar flow rates of the components are

(33)1yzc =- Xzc - zc)ezwhere

is the mean holding time of the reactor, and the othersymbols in (33) have the same interpretations as those inSince the rate of reaction is significantly influenced byboth reactants and products, the fluctuations in the inletmolar flow rates of reactants are important uncertainparameter variations. These variations are assumed to be

expressed as

82 = VZ/(x2A f XZB + C ) P(24)

X l A ( t ) = X I A N ( t ) f 4 ( t )t e 10, trl (34)

X l B ( t ) = X I B N ( t ) - ( t )where xlaN(t) and X I B N ( t ) are the nominal values of theinlet molar flow rates of reactants A and B, respectively,and q t ) is the fluctuation in the flow rate. Furthermore,in the design of a reactor-separator system for the start-upof the temperature-sensitive exothermic, autocatalytic re-action system, great care is required in the proper controlof the system. Thus, the temperature T z t ) of the CSTRis to be manipulated in an optimal fashion so as to givethe best performance of the whole processing systemsubject to the constraint T z LL: T z t ) TZU. The objec-tive function per unit time of the CSTR is assumed to be

f z = aVz + cTZ2 (35)where a is the cost factor for the reactor volume and c isthe cost factor for the reactor heating and temperaturecontrol. The ideal separator removes only material B com-pletely from the rest of the reaction mixture A and C.Its subsystem equations and objective function per unittime are all similar to (28) to (30). The imaginary inputand output subsystems, their objective functions per unittime, and the overall objective function of the wholereactor-separator system are in the forms of (31) and(32). In addition, the following numerical values are usedin this example:

a = 1.0, b = 0.04, c = 5 x p = 1.0tf = 1.0, K~~ = 0.256 x 105, K , ~ 0.12 x 109,

E~ = 104 E~ = 2 x 104 T ~ L 4 0 0 0 ~ ~ U 6 0 0 0 ~R = 2.0

Paae 546 M a v . 1976 AlChE Journal Vol. 22, No. 3)


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TABLE . COMPUTEDALUESOF OBJECTIVEUNCTIONSIN EXAMPLE

Case 1 Case2 Case 3uN,qN) -0.13169 -0.27423 -0.41827uN,9 ) -0.08610 -0.20452 -0.32416I(a9 u9 9 ) - .08615 - .20538 - .32526J a9 UQ,SN - .13164 - .27356 - .41774

X l ( t 0) = (0.0, 0.0, 1 . O T~ l ( t 0)= (0.70, .30, O . O ) T

yio = 0.0,.0, .o)T, V N f 1 = (0.0, .0, . O ) TVN = (0.0, NB, 0.0)

where UNB is the sale cost factor of the desired product Bto be specified.Figures 6u-1 to 6 ~ - 3nd Table 1 show the compdtedresults of the nominally optimal structures and the corre-

sponding performance functions, I (aN, N,qN) for threedifferent values of the sale cost factor of the product B,V N B . The true optimality of these results obtained from theinitially restricted system structure has been verified byusing the conditions (22) and (23). Figure 6u-1 showsthat when the cost of separating the reaction mixtureis more expensive than the revenue for selling the productB, the output stream from the SEP which includes theunconverted reactant A and the inert material C does notreturn to the inlet of the CSTR. However, when the costsfor separating the reaction mixture and selling the prod-uct B are relatively the same, the nominally optimal struc-ture shown in Figure 6a-2 includes both recycles fromone output stream of the SEP to the inlet of the CSTRand from the output stream of the CSTR to its inlet.Finally, when the cost of product B is more expensivethan the previous two cases, the nominally optimal struc-ture does not include a recycle of the product stream ofthe CSTR to its inlet but has a bypass feed stream to theSEP, as shown in Figure 6a-3. This is because when theoperating cost of the SEP is relatively low comparedwith the cost of the product B, it will be economical tosend a certain amount of the fresh feed to the SEP andseparate the reactant A from the mixture. This then fol-lows by returning the unconverted reactant A to theCSTR rather than feeding all the fresh material to theCSTR. All of these will take advantage of the low separa-tion cost.In Table 1, the computed objective functions for thenominally optimal structure with the worst uncertainparameter variation, ] aN,N, q ) , show a considerableincrease from those without parameter uncertainty, j ( uN,dN, qN). In all cases studied, the computed worst varia-tions in the inlet molar flow rates are taken as the uppervalues of the bounded parameter; that is, q t ) = 0.05for t E [0, 11.

Figures 6b-1 to b-2 and Table 1 show the computedresults of the minimax system structures and the corre-sponding values of objective functions. The computedworst variations in the inlet molar flow rate are q (t) =0.05, t E [0, 11 for all three cases. The minimax structurefor case 1 indicates that the reactor recycle ratio a 2 2 in-crease slightly when compared with that of the nominallyoptimal structure. For case 2, the minimax structure issomewhat different from the nominally optimal structure,as can be seen from Figures 6a-2 and 6b-2. Although thenominally optimal structure includes a recycle streamfrom the SEP to the inlet of the CSTR, this recycle streamis not found in the minimax structure. For case 3, the

(a1NOMINALLYOPTIMAL STRUCTURE b)MIN-MAX STRUCTUREa-1): VNB- 1.0 b-1): VNB=l.o

L U0.1527) (0.1713)a-2):VNB= 1.5

A h0.1336 (0.8664)a -3):VNB=2.0 b-3):VN,.2.0

0.2280) 0.7720)Fig. 6. N o mi n a l l y o p t i ma l a n d m i n i ma x s t r u c tu r es f o r e x amp l e 2.

1.0; I I I 1 1 1 I I

T I M E , tFig. 7. O p t i m a l p r o f i l es o f r e a c ta n t a n d p r o d u c t mo l a r f l o w ra tes i nthe min imax s t ruc tu re under d i f fe ren t cos t fac to rs fo r examp le 2.minimax structure is quite different from the nominallyoptimal structure as seen from Figures 6a-3 and 6b-3.Figure 7 illustrates the computed molar flow rates for com-ponents A, B, and C corresponding to the three casesshown in Figures 6b-1 to 6b-3. It is seen that with theincrease in the cost of the product B from case 1 to case3, the molar flow rate of y2B is controlled to increase alongthe entire start-up operation by adjusting the temperatureof the CSTR according to the optimal values shown inFigure 8.The preceding results for this example clearly showthat when the uncertain parameters in a given dynamicprocess synthesis problem have a significant effect onthe dynamic behavior of the process, this minimax struc-ture will be generally different from the nominally opti-mal structure with parameter uncertainty. In this case,the use of the minimax method is recommended, since itgives the process designers an assurance that the actual

M a y , 1976 Page 547lChE Journal Vol. 22, No. 3)


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50.0

I I I I I I I I 10.0 0.2 04 0.6 0.8 I oT I M E , I

Fig. 8. Opt imal reactor temperature control in the minimax st ructureunder di f ferent cost factors for example 2.

performance of the process system synthesized will neverbe worse than that of the minimax structure. Also, theresults shown in Figure 6 indicate that the optimal reactor-separator configurations for the controlled s tar t-up of thisautocatalytic reaction system are quite sensitive to thechanges in the cost of the produc t B . This suggests thata secondary performance index representing the sensi-tivity of the original performance index with respect tothe cost factor of the product B may be used to form aprocess synthesis problem with multiple performance mea-sures. Further discussions on this aspect can be found inLiu et al. (1976) .ACKNOWLEDGMENT

This work was supported in part by the National ScienceFoundation grant ENG 75-11293 and the Research Grant-in-Aid Program of Auburn University.

NO T AT IO NA = substance Aa = cost factor of reactorB = substance BbC = substanceCc = cost factorDd = design vectorE = activation energyfgJK = reaction rate constant&M = number of subsystems or structure parameterP = admissible region of time-independent uncertain

= time-dependent uncertain parameter= admissible region of time-dependent uncertainq t ) = time-dependent uncertain parameterR = ideal gas constantT = reactor temperaturetfUu = controlvectorV = volume of reactor

= cost factor of separator= admissible region of design vector or substance D

= performance function of subsystem= right-hand side of subsystem equation= performance index of process system= set of subsystems incident to y i

functionsparameter

parameter

= final time of process operation= admissible region of control vector

vwxyGreek Lettersa6 = step size factore1p = molarvolumeSuperscriptT = transposeN = nominalvalueU = uppervalueL = lower value* = worst valueo = optimalvalueSubscriptA, B, C , D = substances A, B, C, nd D(k) = kth triali , j , k . = number of subsystem

= decision variable vector in synthesis defined by= uncertain parameter vector defined b? (p, q)= input stream state vector= output stream sta te vector= structure parameters defined by (3) and 4 )= holding time of reactor= adjoint vector defined by (13)= adjoint vector defined by (13)

a, U ) T

LITERATURE CITEDBryson, A. E., Jr., and Y. C. Ho., Applied Optzmal Control,Chap. 9, Blaisdell, Waltham, Massachusetts ( 1969).Ciletti, M. D., and A. W. Starr, ed., DifferentiaI Games: Theoryand Applications, Proceeding of a Symposium of theAmerican Automatic Control Council, Amer. SOC. Mech.Eng., New York, ( 1970).Hendry, J. E., D. F. Rudd, and J. D. Seader, Synthesis in theDesign of Chemical Processes, AlChE J., 19, 1 [ 1973).Henley, E. J., and R. L. Motard, Conference Moderators andEditors, Proceedings of a National Science Founda tionConference on Research Needs and Priorities in ProcessDesign and Simulation, Univ. Houston, Tex., (Frb., 1972).Liu, Y. A., D. C. Williams, N. Nishida, and Leon Lapidus,Studies in Chemical Process Design and Synthesis, 1V.Optimal Synthesis of Process Systems with Vector-Valued orMultiple Performance Indexes, Submitted for publicationto AIChE J. (1976).Mah, R. S. H., Conference Director and Editor, SummaryRepo rt of a National Science Foundation C onference onInnovative Design Techniques for Energy Efiicient Process,Northwestern Univ., Evanston, 111.; NTIS PB 243651/AS(May, 1975).Muralidharan, R., and Y. C. Ho, A Piecewise-Closed FormAlgorithm for a Family of Minimax and Vector CriteriaProblems, l E E E Trans. Auto. Control, AC-20, 381 (1975).Nishida, N., and A. Ichikawa, Synthesis of Optimal DynamicProcess Systems by a Gradient Method, Ind. Eng. Chem.Process Design Develop., 14,236 ( 1975).Nishida, N., Y. A. Liu, and A. Ichikawa, Studies in ChemicalProcess Design and Synthesis I. Optimal Synthesis ofDynamic Process Systems, Paper No. 57d, AIChE NationalMeeting, Boston, available as microfiche from AIChE (Sept.,1975~) . , Studies in Chemical Process Design andSynthesis: 11.Optimal Synthesis of Dynamic Prc cess Systemswith Uncertainty, Paper No. 1G9d, AIChE Annual Aleeting,Los Angeles, available as microfiche from AIChE (Nov.,19753).Nishida, N., Y.A. Liu, Leon Lapidus, and S. Hiratsuka, AnEffective Computational Algorithm for Suboptimal Singularand/or Bang-Bang Control, I. Theoretical Developmentsand Applications to Linear Lumped Systems, AlChE J., 22,505 ( 1976)., An Effective Computational Algorithm for

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