Optimum Design of Chemical Plants Witn Uncertain Parameters

7/28/2019 Optimum Design of Chemical Plants Witn Uncertain Parameters

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Eliezer, K. F.,M Bhinde, M . Houalla, D. H. Broderick, B. C.Gates, J. R. Katzer, and J. H. Olson, A Flow Microreactorfor Study of High-pressure Catalytic Hydroprocessing Reac-tions, I nd. Eng. Chem. Fundamentals, 16, 380 (1977).Gates, B. C., J. R. Katzer, and G. C. A. Schuit, Chemistry ofCatalytic Processes, Chapt. 5, McGraw-Hill, New York(1979).Himmelblau, D. M., C. R. Jones, and K. B. Bischoff, Deter-mination of Rate Constants for Complex Kinetics Models,Ind. Eng. Chem. F undamentals, 6, 539 (1967).Hoog, H., Analytic Hydrodesulfurization of Gas Oil, Rec.Trau. Chim.,69,1289 (1950).Houalla, M., D. H. Broderick, V. H. J. de Beer, B. C. Gates,and H. Kwart, Hydrodesulfurization of Dibenzothiopheneand Related Compounds Catalyzed by Sulfided CoO-MoO3/yAI 203: Effects of Reactant Structure on Reactivity, Pre-pri nts, ACS Div. Petrol. Chem., 22, 941 (1977).Kilanowski, D. R., Ph.D. thesis, Univ. Del., Newark (in prepa-ration).-, H. Teeuwen, V. H. J . de Beer, B. C. Gates, G. C. A.Schuit, and H. Kwart, Hydrodesulfurization of Thiophene,Benzothiophene, Dibenzothiophene, and Related CompoundsCatalyzed by Sulfided CoO-M003/-y-A1203: Low-PressureReactivity Study, 1. Catal. (1978).Kwart, H., G. C. A. Schuit, and B. C. Gates, to be published.Landa, S., and A. Mrnkova, On the Hydrogenolysis of Sulfur-Compounds in the Presence of Molybdenum Sulfide, (inGerman) CON Czech. C hem. Commun., 31, 2202 (1966).

Mitra, R. B., and B. D. Tilak, Heterocyclic Steroids: Part II -Synthesis of a Thiophene Analogue of 3-Desoxyestradiol,J. Sci . Ind. Res. (I ndi a), 15B, 573 (1956).Massoth, F. E., Characterization of Molybdena-Alumna Cata-lysts, paper presented at 5th North American Meeting ofthe Catalysis Society, Pittsburgh, Pa. (1977).Nag, N. K., A. V. Sapre, D. H. Broderick, and B. C. Gates,Hydrodesulfurization of Polycyclic Aromatics Catalyzed bySulfided CoO-Mo03/y-A1203: The Relative Reactivities,J . Cat& submitted for publication.Obolentsev,R. D., and A. V. Mashkina, Dibenzothiophene andOctahydrodibenzothiophene Hydrogenolysis Kinetics overAluminocobdt-Mo Catalyst, (in Russian) Doklady Akad.Nauk SSSR, 119, 1187 (1958); Kinetics of Hydrogenolysisof Dibenzothiophene on Al-Co-Mo Catalyst, Khim. Sera-organicheskikh Soedinenie Soderzh o Neftyakh ( n Russian)i Nefteprod Akad. Nauk SSSR, 2, 228 (1959).Rollmann, L. D., Catalytic Hydrogenation of Model Nitrogen,Sulfur, and Oxygen Compounds, J. Catal.,46, 243 (1977).Shih, S. S., J . R. Katzer, H. Kwart, and A. B. Stiles, QuinolineHydrodenitrogenation: Reaction Network and Kinetics, Pre-prints ACS Dio. Petrol. Chem., 22, 919 (1977).Urimoto, H., and N. Sakikawa, The Catalytic HydrogenolysisReactions of Dibenzothiophene, (in Japanese) SekiyuGakkaishi, 15, 926 ( 1972 .Manuscript receioed J anuary 23. 1978; rev is ion rece ived J une 5, andaccepted J u ly 18 , 1978 .

Optimum Design of Chemical Plantswith Uncertain ParametersA new strategy is proposed for the optimum design of chemical plantswhose uncertain parameters are expressed as bounded variables. The de-sign is such that the plant specifications will be met for any feasible valuesof the parameters while optimizing a weighted cost function which reflectsthe costs over the expected range of operation. The strategy is formulatedas a nonlinear programme, and an efficient method of solution is derivedfor constraints which are monotonic with the parameters, a case whicharises frequently in practice. The designs of a pipeline with a pump, ofa reactor-separator system, and of a heat exchanger network, all withuncertain technical parameters, illustrate the effectiveness of the strategyfor rational overdesign of a chemical plant.

1. E. GROSSMANNan d

R. W. H. SARGENTDepor tment of Chemical Engineeringond Chem ical TechnologyImperial Col legaLondon S.W.7, England

I n the design of chemical plants, it is very often thecase that some of the technical or commercial parametersare subject to significant uncertainty. I n order to over-come this difficulty, the procedure which is normallyused is to apply empirical overdesign factors to the sizesof the units, Other methods which predict more rationaloverdesign factors have been proposed, but their use inpractice has been very limited. The main reasons for thisis because the computational requirements of these meth-ods are expensive and/or because the objectives in thebasic strategy are not very realistic for design purposes.I n this paper, a strategy is proposed which attempts tocircumvent these difficulties. The objective of the pro-

0001-1541-78-1683-1021-$01.05. 0 The American Institute of Chem-ical Engineers, 1978.

posed strategy is to design chemical plants which areable to meet the specifications for a bounded range ofvalues of the parameters and which, at the same time, areoptimum with respect to a weighted cost function. Whenprobability distribution functions of the parameters areavailable, the expected value of the cost can be approx-imated by choosing appropriate weights. The mathemat-ical formulation of the strategy leads to a large nonlinearprogram, where the inequality constraints must be max-imized with respect to the uncertain parameters. Thepurpose of the paper is to derive a reasonable method ofsolution for the problem and to apply the strategy totypical design examples in chemical engineering, whereuncertainty is present in parameters such as pump effi-ciencies, friction factors, reaction rate constants, and heattransfer coefficients.

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CONCLUSIONS AND SIGNIFICANCEAn efficient method of solution for the nonlinear pro-

gram is derived for the case where the constraints aremonotonic with respect to the uncertain parameters, a

in Practice Owing to thesparsity of the constraints and to the monotonicity of thestate variables with the technical parameters. The flex-

ibility in the strategy lies in the fact that only bounds onthe parameters are required, but that when probabilitydistribution functions are available, the expected valueof the cost can be estimated. The results of the examplesshow that the objectives of the strategy are quite suitablefor design purposes and &at the computational require-ments are reasonable.

which arises

The optimum design of chemical plants operating atthe steady state can be represented by the nonlinearprogram, minC (d , z, e)

d .zs.t.h(d,z,.B) =o (1)

gw, z , e ) L 0Usually Equation (1) is solved for given nominalvalues of e = eN, and empirical overdesign factors areapplied to the optimum design variables to take intoaccount the uncertainty in the parameters. Clearly, thisprocedure is not very satisfactory, and different methodshave been proposed to evaluate the effect of the uncer-tainty in a more rational way.Some methods take a statistical approach and assumethat the probability distribution functions of the uncer-tain parameters are known (K ittrel and Watson, 1966;Wen and Chang, 1968; Weisman and Holzman, 1972;Watanabe et al., 1973; Lashmet and Szczepanski, 1974;Freeman and Caddy, 1975; Johns et al., 1976; Patersonet al., 1977). Other methods take a deterministic approach

(Takamatsu et al., 1973; Nishida et al., 1974; Dittmarand Hartmann, 1976; Kilikas, 1976) and assume thatthe uncertain parameters can be expressed as continuousand bounded variableseL 684p

where BL, P, are specified bounds, which can also be in-terpreted as confidence limits for some unspecified dis-tribution. In both approaches, the difference in the meth-ods lies mainly in formulating the basic objectives, be-cause the problem of optimum design with uncertainparameters is not well-defined.In this paper, a new strategy is proposed for the opti-mum design of chemical plants when the uncertainty inthe technical parameters is expressed as in (2). Its ob-jective is to design plants that are always able to meetthe specifications for any feasible values of the parametersand that, at the same time, are optimum with respect toa weighted cost function. This function can be used toestimate the expected value of the cost when probabilitydistribution functions for the parameters are available.

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STRATEGIES FOR BOUNDED PARAMETERSOne way of reformulating the nonlinear program in(1) when treating the parameters as in (2) is by solvingthe following problem as suggested by Nishida et a1(1974):

min max C (d , z,e)d 0(3)

g ( d z ,O) 4 0OL 6 4p

With this formulation, the plant is actually optimized atthe value of the parameters that maximize the cost func-tion. Although the solution of the minimax problem pro-vides an upper bound for the cost, it does not ensurethat the resulting design will fulfil the constraintsg (d , z, e) 60 ( 4 )

for arbitrary values of e satisfying (2). An explicit wayof achieving this has been suggested by Kwak and Haug(1976) when C (d , z, 0 ) =C ( d ) by solving the problemminC(d)

s.t. maxgi(d,z, 0 ) 60 i =1,2,. .. ( 5 )eLeeeeoh(d,Z, e ) =o

When C = C ( d , z, e), Takamatsu et al. (1973) haveformulated a strategy in which Equation (1) is solvedfirst for 0 = ON , the nominal values. The cost functionand the constraints are then linearized at the solution,and a linear program is then solved for the value e =ei,which maximizes each component of ( 4 ) . This value Bis chosen at one of the corner points of (2) using engi-neering judgment, and, of course, the method is onlyvalid if such a valueei actually exists.FORMULATION OF A NEW STRATEGY

The strategies presented in the last section tend togive very conservative results for chemical plants, as aunique value of d must be chosen. The fact is that aftera plant has been built, the control variables can beadjusted in light of the actual conditions prevailing,and it may even be possible to modify the design. Thus,the vector d can be divided into two subsets of vari-ables: the fixed design variables u which define theinitially installed plant, and the control variables v whichare chosen during operation and can thereforc takeaccount of the actual values of the parameters 8. Forgiven u and 0, the control variables v will therefore bechosen tosolve the problem

min C(u,u, z, 8)vt zs.t.g(tl,U , 2, e) 0,

h(u,U , Z, e ) =0,thus defining optimum values ( 6 )

A Au =u (u,e), x =z ( u, l,C(U,0) =CCU, ~( u ,), X ( U ,e),eiA A

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In fact, since u may include variables defining designmodification or the portions of the plant actually in use,it may be necessary to add extra constraints to (1) toensure that the installed capacity of the plant is notexceeded or to guarantee that the modified plant re-mains within its operability limits. Hence the new dimen-sion t of the vector g of inequality constraint functionsin (6)will, in general, be greater thans.A first requirement ot the initial design strategy isto choose the design variables u so that no matter whatvalues of the parameters e actually occur, it is possibleto choose the control variables to meet the process ob-jectives; in other words, to choose u so that (6) hasfeasible solutions for all e satisfying (2). Within theselimitations, it is desirable to choose u so that the averagecost over the likely range of conditions is minimized,giving the formulationminE{C (u,e)}

11 (7). .tHere the expected value of the cost is formally definedin terms of the parameter joint probability densityf (8)

( 8 a )but it will usually be adequate to approximate this in-tegral by a finite weighted sum

j =lwhere the number of termsnand the weights uj , =1,2,. . . . n, should be chosen to fit the integral up to a cer-tain order (see Gaussian and Radau integration formulas).However, the density function f ( , e ) will rarely be knownsufficiently well to justify a high accuracy of fit, and itwill usually be sufficient to select a relatively small num-ber of points covering the likely range of the parametersand regard the uj as subjective probabilities for the setsBj, their choice being based simply on engineering judg-ment if no other information is to hand.If (8b) is used in (7),we are concerned with onlya finite number of parameter sets ej, i = I, 2, . .. n,and a further simplification arises if the maximizingparameter sets ei, i = 1, 2 .. . t for the constraints in(7) are taken as the &st t sets 8, with n t.We canthen combine (6), (7),and (8b) and reformulate theproblem as

nmin 2 ujC u ,u? zj, ej)u.uJ.zJ j =l

gi (u ,u , zt, ei) = max gi(u, ii, zi , ) 4 0,f =l ,2 ....... . (9)

If it isdesired to specify the ej in the weighted objectivefunction, for example to conform with a given integra-tion formula, the formulation in (9) may still be used,simply setting uj = 0 j = 1, 2, .. . . . t, with the re-maining ej, f =t+1,. .nspecified as required.I t is clear that at the solution of (9) we shall have

A Auj = u (u , ej), z = z (u, e j ) , solving (6) for eachj = 1, 2, . . . n, and it is the inclusion of the constraintsinvolving maximization that guarantees the operabilityof the plant within the constraints over the entire possiblerange of the parameters (even if they are allocated azero weight, u =o, in the objective function).Equation (9) can be solved by standard nonlinearprogramming techniques, although in this general formit may well represent a formidable computing task, bothbecause of the large number of constraints if n is largeand because of the maximization subproblems in 0to be solved at each iteration of the variablesu,u , zj.Special Solutio n for Monotonic Constraints

Because of the heavy computing requirements for thegeneral problem, it is important to exploit any specialstructure in the equations for each particular designproblem.One special case, which results in substantial simplifica-tion, arises when the functions gi(u, u, z, 8) are mono-tonic in each & (with other variables fixed), for themaximization subproblems in (9) are then trivial. Infact, the solution of each subproblem must lie at a cornerpoint of the polytope defined by (2), as is clearly seenfrom the relevant Kuhn-Tucker conditions:

(10)for all k =1,2, .,. ..p, f=l,2, .....where i f agi/aek z 0, 88kL 6 8k 4 8ku, we must haveXk z 0 and hence either ek =ek L or fl k =ekU, dependingon the sign.Often, some or all of the equality constraints in (9)will be solved for the state variables zj for each choiceof u, u j , 0' (for example, using a flow sheeting package),rather than leaving these to be dealt with by the non-linear programming algorithm. In this case, these equalityconstraints take the form z =~ ( u ,, e), and the abovemonotonicity property must then hold for the functiong(u, 0, Q(u, 0, e), 6'). The monotonicity property, inboth of these forms, is surprisingly common, especiallyfor technical parameters such as heat and mass transfercoefficients, reaction rate constants, and efficiencies, whichderives from the fact that the constraints are sparse andthe performance relations tend to be monotonic in theseparameters. I t is therefore worth examining the problemin some detail.If the derivatives agi/aBI, have the same sign for allfeasible u, u, z, and e (which is usually the case), thecorner points ei, i =1,2, ... t can be determined onceand for all, rather than at each iteration. Further, be-cause of the sparsity, many constraints will be indepen-dent of particular parameters &, making it possible tochoose these arbitrarily in the ei and hence merge someof these sets to produce considerably fewer than t param-eter sets to guarantee feasibility over the full range ofparameter values. Finding p , the minimum number ofsets that contain all the active bounds, is a combinatorialproblem whose solution can be determined efficientlyusing the algorithm described in the Appendix. In fact,the points where the merging takes place need not onlybe the corner points in (2), and other points can beconsidered as well. A suitable choice, as shown in theAppendix, is to consider points whose components canbe any combination of lower, upper bounds, BiL, eiu,or additional points&, j =p +1, .. .. .n.In order to determine the active bounds, the sign ofthe gradients of the constraints must be determined.

November, 1978 Page 1023lChE J ournal (Vol. 24, No . 6)


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In practice, i f the relevant derivatives are not easilydetermined, it may be simpler to assume monotonicityand use finite diflerences based on the nominal valuesand one limiting value for each parameter. This requires( p + 1) process evaluations at any arbitrary value ofd if the state variables are obtained in the form z =@(d ,B ) . With the parameter values thus fixed, Equation(9) is reduced to solvingmin 2 ujC(u,oj, d, 6 ' )u.uI,.zJ j =l

j =l , 2 .....n ( 11)1.t. h ( ~ ,j, zj, ej ) =0)g ( u, + ~ j ,j) owhere n 5B, and #, i = 1, 2, . .. n are chosen so asto guarantee feasibility of the constraints and to weightthe objective function.I t is interesting to note that Equatioii (11) corres-ponds to a particular case of the nonlinear programformulated for the optimum design of multipurpose plantsby Grossmann and Sargent (1977), so that their ap-proach can be used to solve it.

I t is usually desirable for the evaluation of the weightedcost function to include additional points beyond thosestrictly required to guarantee feasibility of the constraints.One obvious choice is to select the nomiiial value of theparameter eN , as this corresponds to the expected valueof 8. Another reason is that the performance of chemicalplants is usually specified at the nominal conditions ofoperation. When the probability distribution functionf ( 1 9 ) is available, the integral in (8) can be estimatedas follows.A grid is associated with the polytope in (2) by divid-ing each side [ 8 k L , &'I, k =1, 2, . . ..p into three equalintervals. This will then define 3p elements of volume,Vi, i = 1, 2 ,. . 3p. The p points that guarantee feas-ibility will be contained in different elements Vi, andadditional points to be considered should preferably alsobe located in different elements. If the distribution issymmetric, the volume in the center of the polytopewill contain the nominal values. However, when thedistribution is skewed, it is possible that a given elementVi contains the nominal values ON and one of the ppoints. For such a case, or when, in general, a givenelement Vi contains more than one point, the procedureis to assign only one nonzero weight per element, Theintegral in (8) is then estimated by using n of the 3 pelements Vi, n n, which contain at least one point,Bj , i =1, 2, . . . . . . . n. When n 'v 3p, the logical choicefor the weights in (8b) is

ri=bi / 2 bj11 ( 1 2 a )j = 1whereCi =J - . vf f(O)d# . . ..dop i =1,2, . . . . .n

However, when n


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TABLE .RESULTS OR THE PI PELI NE

Nominal design % overdesign D %overdesign589 547 812 437 - 0.720641 d6C01 g2 @30.8 0.1 0.1 667 662 1370140 68.6 0.757851 5.22/3 1/6 1/6 679 497 1 367000 68.3 0.762506 5.81/3 1/3 1/3 708 950 1 359960 67.4 0.773573 7.3

Minimax 885 972 1 327490 63.4 0.841103 16.7

factor f = a/Re.lB. Their nominal values are I )N = 0. 5and a N =0.04,and their bounds are0.3G qL 0.6, 0.026a L 0.06 (14)

The required power of the pump is given by5.6608566a 194.018 [ P -2130914T;i928 0.04D4.s4W = 71

(15)

W k W (16)

(17)

and must not exceed the installed power of the pumpA

The outlet pressureP( N/rn2) must satisfy the constraint1 013 529.28- -L 0

Clearly, for the pump aP0/aq >0, where Po is the outletpressure of the pump, for the pipeline aP/aP, >0, andaP/aa


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TABLE. RESULTSFOR THE REACTOR-SEPARATORYSTEM 17 13- V % overdesign-Nomna design 151.92 12.346603 8.312/ 3 1/6 1/6 160.71/3 1/3 1/3 165.7 13.8735 12.2

Minimax 185.60 15.5854 26.0c2 13.3962

I f we assume that the rate constants have an uncer-tainty of k20% with respect to their nominal values,they will be bounded as follows:0.326k B 50.480.084 kR 60. 12

0.0166 kx 0.0240.0086ky4 0. 012

The total cost of the system is givenbyc=Clv+C2 n- uk[a" F k(XAk +Xsk)k =l +BkFk ( z x k+X Y " ) ] ( 24)where the first term represents the cost of the reactorand the second term the cost of the recycle.Calculating numerically the gradient of (22), wefound that ( 22) is maximized at ~ B L , k ~ . , xu, kyuwhich is what we expected from the physics of theprocess. Here it is not obvious that the cost will be mono-tonic with the rate constants, keeping constraint ( 22)active. As a result of this, and in order to have an appro-priate weighting of c,apart from the nominal values, athird value for the parameters was taken at the oppositebounds of the ones that maximize (22), km, kRu, kxL,k Y L . The weight u1 was assigned to the nominal valuesand u2,us to the two other values. The problem is thento choose V, &, f l i t , k = 1, 2, 3 in order to minimize(24) subject to constraints ( 21) and ( 22) for the threevalues of the parameters described above.The problem was solved for two difFerent sets ofweights, for only the nominal parameters, and for theparameters at k n L , kRL , kxu, kyu, which most probablycorrespond to the solution of the minimax problem, al-though no attempt was made to check this, The resultsare shown in Table 3, where it can be seen that theoverdesign factors with the proposed strategy are muchsmaller than the one obtained with the minimax strategy.It is to be noted that constraint (22) was active in allthe cases.Heat E x c hanger N e t w o r k

The network to be optimized corresponds to the op-timal network 4SP1 in Lee et al. ( 1970) , where thedata and specifications can be found. The network isshown in Figure 3. The outlet temperatures T3, T6, Tlo,T lz , T15 were specified as inequalities, and the steam

Nomna design

1

I 1

14q 1 5Fig. 3. Hea t exchanger network.

TABLE.ACTIVEBOUNDSF THE SETOF CONSTRAINTS27)Constraints Activebounds for nonzero gradients

12345678910

UL2, UL 4U L 1W L l , uu2, U L 3UU l , U L 2 , U L 3 , U L 5UUl, uuzUU3, uu4UUluu2UUl, uu2, uu3U L I , U U 2 , uu3

temperature T s was controlled with a valve. This wasdone in order to introduce enough degrees of freedomin (11) as suggestedby Grossmann and Sargent ( 1977) .I t was also assumed that the latent heat of the steamwas givenbyA =- 19. 78211Ts2+15 535.673T s- 99 269. 2

( 25)where X is given in Joules per kilogram, and T s is thesteam temperature such that T sL 555. 3722"K.A +- 2Or ; / ,ncertainty in the value of all the heat trans-fer coefficients Ui , i = 1, 2, . .. 5, was considered. Theproblem is then to choose the areas Ai, i = 1, 2, . , 5,(m2) the steam temperatures Tsk, k = 1, 2, .. n andthe outlet temperatures of the cooling water TWOk,k =1,2, . . .n in order to minimize the total annual cost

5 I t-C = coAiB+ 2U ' ~ ( C ~ F S ~cwF wk) ( 26)where F sk, F w~, = 1, 2, . . .n, are the steam and cool-i =l k =l

TABLE. RESULTSFOR THE HEAT XCHANGERETWORKiz A1 A2 A3 A4 A59 959.7 24.659 70.446 40.637 3.602 1.735

81 a2 03 a4 a50.6 0.1 0.1 0.1 0.1 11 757.8 30.823 65.019 45.576 3.904 2.849% overdesign - 25 -7.7 12.2 8.4 64.2Page 1026 November, 1978 AlChE Journa l (Vol. 24, No. 6)


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ing water flow rates, and coyp, CS, cw arecost parameterswhose values are given in Lee et al. (1970). The con-straints are given by the heat balances and by the follow-ing inequalities:

k =1,2,. ,.n (27)

Ti', i= 3, 6, 10, 12, denote the specified outlet tem-peratures. Unlike Lee et al. (1970), the value of 6 wastaken as5/9" K .The signs of the gradients of the constraints in (27)were determined by using finite differences to determinethe bounds of the heat transfer coefficients which areactive when the constraints in (27) are maximized, andthey are shown in Table4.Merging all the active bounds by the procedure de-scribed in the Appendix, four minimum points werefound: = (uui, uuz, uus, uu4, U N ~ ) ,3 = ( U L I ,V U Z , U U S , U U 4 r UN5)y u4= ( U U h u L 2 , uL3, u L 4 , U L S ) ,Us = ( U L I ,UUZ,ULS,U U ~ ,N5).The weights UZ,US, ~ 4 ,u5,were assigned to these points and the weight u1 tothe nominal value of the parameters, U1 = ( U N ~ , N ~,U N ~ , N ~ , N ~ ) .he problem was solved for one set ofvalues for the weights, and also the nominal design wasdetermined. The results are shown in Table5.I t is interesting to note that the second area has anegative overdesign factor. This was because the eighthconstraint in (27) was active at the solution when Uz= Vuz. This means that if a larger area were used inexchanger 2, the temperature approach would be lessthan 6 for Uz =Vuz,as the heat flow rate of hot stream1 is smaller than the heat flow rate of cold stream 7.Finally, only the temperature of stream 3 was at itsspecified outlet temperature for the five values of theparameters, whereas the temperature of stream 6 wasalways above its specified outlet temperature.

CONCLUSIONSThe objectives of the strategy which has been pro-

posed seem to be quite appropriate for the design ofchemical plants with uncertain parameters. The sim-plified method of solution which has been derived formonotonic inequality constraints can be applied for avery large number of practical cases. The three numericalexamples which have been solved show that the pro-posed strategy provides a rational and economical wayof obtaining safe optimum designs when the specificationsor data are subject to uncertainty.

N O TA TI O NAA(m2)=heat exchange areaA =[aij]=matrix of active boundsa(-) =coefficient in friction factor formulaBAlChE Journal (Vol. 24, No . 6)

=raw material (reactor example)

=matrix derived fromA by deleting rows

C = costfuhctionC, v , z =optimum values of C, u , z, respectivelyCIc={cl, - cp}=set of corner points of (2)C A ~mole/m3) =feed concentration (of A )c , ,, c , =cost coefficientsc1 ($/m3), c2($/mole) =cost coefficientsdd NdoD(m)= pipe diameterE {.} = expectation operator [with respect to distribu-f(e)f ( - ) = friction factorF , F R (mole/h) =actual and minimal production rateF Ao(mole/h) =feed flow rateF,, F,, =steam and cooling water flow ratesghZ ={1,2, - - t}= ndex setof constraints1, = running indexesz,!, r, s =row and column labels1={1,2, - - p } = ndex set of parametersK = index set of cornersIKI = number of elements inKlc, I =set element labelslcB, kn, kx, k y (h-1) =reaction rate constantsL , Q, R, S, T =index sets used in algorithm (Appendix)rn = dimension of vector dn = number of terms in weighted objective functionn' =number of volume elements Viodf = overdesign functionp = dimension of vector 0P (N/m2) =outlet pressurePo(N/m2) =pump delivery pressurer = dimensionof vector hR =desired productRe =Reynolds Numbers, tT iT, = steam temperatureTwo = cooling water outlet temperatureTi * = minimum temperature of streamiu = vector of fked design variablesU = heat transfer coefficient for exchangeru, w , x, y, SL =weights used in algorithm (Appendix)v = vector of control variablesV(m3) =reactor volumeVi = elements of volume in estimating the objectivefunctionW, W =Watt actual and installed power of pumpxA, xB, xR, xx , x y (mole/m3) = molar concentrations ofX , Y = undesirable by-productsz

A A h

= m vector of design and control variables= size determined in nominal design= size determined with strategytion f ( 0 )1= parameter joint probability density

= s vector or t vector of inequality constraint func-= r vector of equality constraint functionstions

= dimension of vector g= temperature of stream i

A

A, B, R, X , Y, respectively= r vector of state variables

Greek Lettersa(-) = recycle ratio of A andBp = costindexp( -) = recycle ratio of X andYS = minimum temperature approachT ( - ) =pump efficiencyeeN , BL , eu =nominal value, lower and upper bounds for0= p vector of parameters= Kuhn-Tucker multiplier, latent heat of steam=minimum number of points to guarantee feasibility= weights in objective functionajQ = equality constraint function

November , 1978 Page 1027


8/8

=empty set= integralof f ( 8 ) over Vi

LITERATURE CITEDDittmar, R., and K. Hartmann, Calculations of Optimal De-sign Margins for Compensation of Parameter Uncertainty,Chem. Eng. Sci., 31,563 (1976).Freeman, R. A., and J . L . Caddy, Quantitative Overdesign ofChemical Processes,AIChE I ., 21, 436 ( 1975).Garfinkel, R. S., and G. L . Nemhauser, Integer P rogramming,Wiley, New York (1972).Grossman, I . E., Problems in the Optimum Design of ChemicalPlant, Ph.D. thesis, Univ. London, England ( 19771.-, and R. W. H. Sargent, Optimum Design of Multi-purpose Chemical Plants, I nd. Eng. Chem., Yroceas DebignDevelop. to appear.J ohns,W . R., G. Marketos, and D. W. T. Rippin, The OptimalDesign of Chemical Plant to Meet Time-V arying Demandsin the Presence of Technical and Commercial U ncertainty,paper presented at the Design Congress, 76, at the Univer-sity of Aston in Birmingham, England (1976).K ilikas, A. C., Parametric Studies of L inear Process Systems,Ph.D. thesis, Univ. Cambridge, England ( 1976).K ittrel, J . R., and C. C. Watson, Dont Overdesign ProcessEquipment,Chem. E ng. Progr., 62, 79 ( 1966).Kwak, B. M., and E. J . Haug, Optimum Design in the Pres-ence of Uncertainty, 1. Opt. Theory Appl., 19, 527(1976).Lashmet, P. K., and S. Z. Szczepanski, Efficiency Uncertaintyand Distillation Column Overdesign Factors, 2nd. Eng.Chem. Process Design Develop., 13, 103 (1974).Lee, K. F., A. H. Masbo, and D. F. Rudd, Branch and BoundSynthesis of Integrated Process Designs, 2nd. Eng. Chem.Fundamentals, 9,48 (1970).Nichida, N., A. Ichickawa, and E. T azaki, Synthesis of Op-timal Process Systems with Uncertainty, 2nd. Eng. Chem.Process Design Deuelop., 13,209 ( 1974).Paterson, W. R., J . W. Ponton, and R. A. B. Donaldson, The

Estimation of Uncertainty in Complex Networks Using aModular Process Simulator, paper presented at the 4thAnnual Research Meeting of the Institution of Chemical En-gineers in Swansea, England ( 1977).Sargent, R. W . H., and B. A . Murtagh, Projection Methods forNonlinear Programmhg, Math. Prog., 4, 245 (1973).Takamatsu, T., I. Hashimoto, and S. Shioya, On Design Mar-gin for Process Systems with Parameter Uncertainty,J. Chem. Eng. J apan,6,453 (1973).Watanabe, N., Y. Nishimura, and M. M atsubara, Optimal De-sign of Chemical Processes Involving Parameter Uncer-tainty, Chem. Eng. Sci., 28, 905 (1973).Weisman, J ., and A. G. Holzman, Optimal Process SystemDesign under Conditions of Risk, I nd. Eng. C hem. ProcessDesign Develop., 11, 386 (1972).Wen, C. Y ., and T. M. Chang, Optimal Design of SystemsInvolving Parameter Uncertainty, ibid., 7, 49 (1968).

APPENDIX: ALGORITHM FOR MERGING THE ACTIVEBOUNDS IN T HE M I N I M UM NUM BER OF POINTSin (9) can be represented by matrix A, A = [a,,], whereThe active bounds for the maximization of the constraints

0 if agl/ds, =01 f B J L is active, that isagl/dB, 0a., = (A 1

i c 2={1,2,. t} i c J ={1,2,,. p ]Any comer point in ( 2) will be represented by the ordered set

Ck = c1, c2, . ..cp}where C, =1or 2, i c J , and k c K , where K is the index setPage 1028 November, 1978

of all the comers. If 11.Step 2-define matrix B = [bij],by eliminating row r in Aif row s, s p r, exists in A, such that arj # a,j implies arj =0, i e J1 and a,j =a,j, J 2, where J l U J 2 =J, J1nP =$J .Define ZB, I B _ C I , the index set of rows in B .Step 3-( a ) set Q =+ T =L, where L = 1 ) is the indexset of the subset of corners CL = {C I , c2, . . . cp}such that cj+cj, i E SO, cj=1 or 2, i 1 E J . ( b) enerate one cornerCk ={ c1, c2 , . . . cp}where k T , cj = I or 2, e So, c j =ci , e SO, i E J . Include k in T. ( c ) determine Sk ={ilids, bij

c j for all i E J when bij # O}. If Sk # @, include k in Q. IfI Q I _L 1 go to (e). d ) calculate R =SkuSl, k, 1 8 Q, k +1.If R =SIC,exclude 1 from Q. If R =S1, excludek from Q. (e)if IT1 < I < ] ,go to (b) .

w i

i

1

Step & ( a ) calculate y =max (2, a}where a=I tQand 1if u S k - U S ++

(4.2)0otherwise

If y = IQI, the solution is ,U =y with the corners Ck, k E Q.(b) solve the set covering problemmin 2 q

kaQw 1= { keQ kflwQ (A3)

s.t. 2 uiqxq I i r ~ gq c Qxq =0,l 4 0where xq is associated with the existence of comer 4 in the setK and

1 f bij =cj in Cq for all c J when bij p 00otherwise

(A4)x q with the corners Ck, k E K

= { g$q ,=I}. Problem (A3) can be solved with an implicitenumeration algorithm as proposed in Garfinkel and Nem-hauser (1972).I t should be noted that the efficiency of the algorithm of thisAppendix stems from the following facts:1. / K - LI comer points are enumerated from the IKl pos-sible enumerations.2. Nomore than IQ corner points need to be stored.

{iq =Thesolution is then /I =

PER

3. The size of prob em (A3 ) is reduced effectively as it in-volves 101variables, IQI 4 IKI, and l Z~l constraints, IIBI 5 IZ .Since, in general, K is not unique for the same ,u, theweighted cost function in (11 will be affected by taking dif-ferent corner points. This difficulty can be circumvented tosome extent by defining for each corner Ck ={cl, c:! . . . cp},k c K, cj =3when aij =0 for all i E Zk, where 3 is associatedwith the nominal value of the parameters ON or, more generally,any of the additional points 6j = p + 1, . . . n. In this way,the multiplicity of the solutions can be considerably reduced.Manuscript received August 1, 1977; revision received M ay 30, andaccepted J une21, 1978.

AlChE J ournal (Vol. 24, No. 6)

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Optimum Design of Chemical Plants Witn Uncertain Parameters