branch and bound of integrated process designs

7/29/2019 Branch and Bound of Integrated Process Designs

1/11

A CAT

= concentration deviation entering controlled sec-= temperature deviation entering controlled sectiontion of the reactorof the reactor= void fraction of packed bed= frequency, radians/fluid residenre time

EP I =fluid density, g/cniaP P = solid density, g/cnia0Q = frequency, w /H PSUBSCRIPTS

'1 tionsS = denotes steady-state quantityliterature Cited

evaluation at positions of z n bed when used as= subsciipts. Otherwise values of z at these posi-

Ark, It., Ind. Eng. Chem58, 32 (1966).Biloiis, O., AInundson,N . R., A.Z.Ch.E. J . 2 , 117 (1956).Boreskov, G. K.,Slin'ko, hl. G., PureApp l .Chem10, 611 (1965).Crider, J . E.,FOSS, .S. , l.I.Ch.E. J . 12,514 (1966).

Crider, J .E.,FOSS,.S.,A.I.Ch.E. J . 14,77(1968).Douglas, J . M., Eagleton, L . C., IXD. NG.CHEM.FUNDAMEN-Hoiberg, J . A ., Ph.D . thesis, University of California, Berkeley,1969.Law, V . J ., Rodehorst, C. W., Appleby, A. E., von Rosenberg,D. U., paper 15c, 56th National Meeting of A .I.Ch.E., M ay1965, San Francisco, Calif.Pavnter, H.M. , flegelungs-technik: Moderne Theorien und ihreverwendbarkeit, Verlag R. Oldenbourg, Munchen, 1957;Report of Heidelberg M eeting, September 25-29, 1956.Poxell, B. E., SA J . 10,45 (1963).Simpkins, C. R., Ph.D . thesis, University of Delaware, 1966.Sinai, J .,Foss,A . S., A.I.Ch.E. ., tobepubli shed, 1970.Stangeland, B. E., Ph.D. thesis, University of California, Berke-Tinkler, J . D., Lamb, D. E., Chem Eng. Progr. Symp. Ser. 61,

RECEIVEDor review February 27, 1969ACCEPTEDAugust 7, 1969

TALS 1,116 (1962).

ley, 1967.No. 55, 155 (1965).

Work supported by fellowships and research grants from theNational Science Foundation and by a fellowship from the Stan-dard Oil Co. of California. Computational facilities were providedby the Computer Center, University of California at Berkeley.

Branch and BoundIntegrated Process

Synthesis ofDesignsK. F. Lee,' A . H. Masso,Z and D F. RuddChemcal Engineering Department, University of Wisconsin, Madison, TVis. 65706

The branch and bound techniques of problem solving can b e used to guide the design engineer dur ing theinvention of integrated process designs. When confronted with a design problem for which no method ofdesign i s known, the engineer gainfully can branch to simpler design problems which bound the originalproblem. Oft en, the opt imal solution to the original unsolvable design problem can b e inferred from thesolutions to bounding problems. These ideas ar e ill ustrated in the design of an energy exchange system.

Rm.m~.ux,Tittle theoretical guidance is currently avail-able during the synthesis of integrated process designs(Hwa, 1965; K esler and Parker, 1969; hlasso and R udd,1969; Rudd, 1968; Siirola et al., 1970). For the most part', theselection of process equipment and its integration into aprocess sheet are left to experience. T he engineer inventsprocesses wi th the full knowledge that more efficient process-ing schemes may have escaped him.I n this report we examine the branch and bound methodof problem solving (L awler and Wood, 1966) and demonstratehow these ideas can be used by the engineer as guidelinesduring the invention of process designs. Using branch andbound theory, methods have been devised for the systematicsynthesis of energy exchange systems. The systems generatedare known to be the best attainable acyclic designs.

Branch and Bound StrategyAn engineering design problem consists of a statement of atask to be performed, such as the maiiufact'ure of certain1 Present, address, Amoco Chemical Corp., Whiting, Ind. 463942 Present address, Shell Development Co., Emeryvil le, Calif.94608

products from crude feedstock; pertinent technical informa-tion, such as the performance characteristics of process equip-ment, limitations on sources of energy and raw materials,etc.; and pertinent economic information including an eco-nomic objective to be reached, such as maximum ventureworth.For most industrial design problems there are no formalprocedures which lead directly to the process design thataccomplishes the design objective. Should we be confrontedwith such an unsolvable or excessively difficult design prob-lem (problem A) we look for simpler design problems forwhich methods of design esist (problemsB).Suppose that one can invent a design problem B which ismore easily solved. This simpler problem might arise from aclever manipulation of the original design task, the technicalconstraints on the original problem, or the economics. Whatinformation can be gained about the solution to the originalunsolvable problem A by examination of the solution toalternate problems B?If problem B has been constructed to exhibit certainbounding properties, and if its solution sat'isfies the originalproblem in certain critical ways, considerable information48 l&EC FUNDAMENTALS VOL. 9 NO. 1 FEBRUARY 1970


2/11

can be gained. I n fact, one may even say that the solutionto a simpler problem is the optimal solution to the originalunsolvable problem.T he idea developed in this report is that one can approachextremely difficult design problems gainfully by branchingfrom the difficult problem to other simpler problems, andgenerate the solution to design problems which are unsolvablewhen attacked frontall y. T he branch and bound principlesoffer guidance in the generation of alternate problems andhelp i dentify the degree to ivhich the solutions of the alternateproblems are satisfactory solutions to the original designproblem.T he more easily solved alternate design problem B mustbe selected to bound the original problem A in the followingway. If OA (D) is the economic design objective to be maxi-mized in problemA by adjusting design D, D A s the sought-fordesign which maximizes that objective, and OB(D) is theeconomic objective for the alternate problem B, if Equation 1holds, problem B boundr problem A

and DAmust be feasible for problemB.T hat is, if the optimal solution to problemA were availableand applied to problem 13, that design would be feasible forproblem B (satisfy all the technical constraints but not neces-sarily at maximum objective function), and exhibit an equalor greater value for the economic objective. If so, problem Bis an upper bound for A . T he alternate problems to be usefulmust be so constructed.Now suppose we solve bounding problem B, the optimaldesign being DE. If this optimal solution satisfies the condi-tions of Equation 2, it is also the optimal solution to theoriginal unsolvable problem A .DE is feasible for problem A, and

Conditions 1and 2 are sufficient for DB to be the optimalsolution to design problemA .If the design engineer is clever and fortunate enough toinvent an alternate problem which is more easily solved andbounds the original problem, there is a means for detectingthe ways in which the solution to the alternate problem doesor does not satisfy the original problem. T he creative abilityof the engineer is then directed to the invention of certainkinds of alternate problems, and he is no longer withoutguidance when groping with an excessively difficult design.Generally it is necessary to branch and bound a number oftimes until one coilverges on the bounding problem desired.It is highly probable that convergence will not be achievedif the experience gained during the ini tially unsuccessfulbranching is not applied effectively to the invention of newbounding problems.For example, suppose that a bounding problem B had beeninvented by relaxing a constraint in the original problem onavailable steam supply, and that the design D B was con-structed. Suppose further (applying Condition 2) , design DBis not feasible when applied to problem A-for example,design DE may demand more steam than is really available.We have gained information that the steam constraint iscritical and perhaps in force in the yet unknown optimaldesign, DA , and we would be directed toward boundingproblems which include that constraint but perhaps aresimpler in other noncritical ways.

As another example, suppose DE is feasible for problem Abut the objective functions for the two design problems arenot identical when solved by design DB-that is,OB(DB)>OA(DE)

We might then attempt to modify the economics of problemBto generate a new bounding problem which does not bound Aso generously. I n this way the new bounding problem hasa better chance of being the agent by which designs aredeveloped to satisfy the conditionsof optimality.One might even change the design task completely toachieve a simpler bounding problem-for example, the designtask A might be the recovery of a light hydrocarbon from amixture of several heavier hydrocarbons. X useful boundingproblem B might be that of recovering the light hydrocarbonfrom a single heavier hydrocarbon, and the info] llation gainedby the solution to problem B may allon- one to circumventthe difficult problem of dealing wi th a large number of hydro-carbons during certain phases of the design.These simple principles are exploited to accomplish aformidable design task, the synthesis of an energy recoverysystem for an oil refinery. T he existing energy recovery systemdesign methods involve the use of empirical rules of thumbor the solution of extensive integer or heuristic prograil is.K either is completely satisfactory, the former offeiing noproof of optiniality and the latter being extremely difficult toimplement. For our purposes, the design of such energy re-covery sybtems is an unsolvable problem to be approachedby the branch and bound strategy.Synthesis o f Energy Exchange Systems

A major industrial design problem is the synthesis net-works of heat exchangers, heaters, and/or coolers to transferenergy between processing streams required for economicprocess operation. T he branch and bound methods are usedto invent the means for creating optimal energy exchangesystems. The results are significant in two ways. First', themethods are simpler to irnplement than the existing heatexchange network design procedures. Second, and most ini-portant, the design procedures form in themselves a conditionof optimality : T he first feasible design synthesized is knownto be the best possible solution to the problem within certainlimits on cyclicity of information flow.Problem Statement. T he design problem to be con-sidered has been defi ned by Alasso and Rudd (1969). T hereis a total of s liquid process streams, 71 of which are to beheated and the remaining n streams are to be cooled.Associated wi th the i th stream are its flow rate, toi, inputtemperature, t i i , output temperature, f i o , and heat capacity,c i aT he available auxiliary heat transfer media are saturatedsteam and coolii ig water.T he equipment includes heat exchangers of the shell andtube type operating as countercurrent single-pass units.T he heat transfer coefficients are known for all exchangers,heaters, and coolers. ll ini ni um allowable approach tempera-tures are specified for all equipment. T he cost of a heatexchanger is correlated to the required heat t'ransfer surfacearea by E =u A , ~here a and b are constants. Cooling watercosts $C, per pound and steam costs $ C, per pound areknown. T he total costs are computed on a yearly basis withfixed costs amortized linearly over years. Table I sum-marizes the basic data used in this study, and Table 11con-tains three example design situations used to illustrate theprocedures to be developed.

VOL. 9 NO. 1 FEBRUARY 1970 I&EC FUNDAMENTALS 49


3/11

Table 1. Design DataSteam (saturated) pressure

Cooling water temperatureM aximum water outputM ini rnuni allowable ap-temperatureproachHeat exchangerSteam heaterWater cooler

efficientsHeat exchangerSteam heaterWater rooler

Over-all heat transfer co-

Equipnient down timeHeat exchanger cost pa-rametersCooling 17ater costSteam cost

962.5 p.s.i.a. for prob-450 p.s.i.a. for problems100F180F

lem 4SP15SP1 and 6SPl

20F25OF20F150 Btu/(hr)(sq ft) ( O F )200 Utu/(hr)(sq ft)("F)150 Btu/(hr)(sq ft)("F)380 hr/yr350, 0.65 X 10-5 $/lb1 X 10-3 $/lb

StreamNo.

1234

12345

123456

Table It. Example Design ProblemsFlow Input outpu t Heatrate Temp. Temp. Cap.

Problem 4SPl20,643 140 320 0 7027,778 320 200 0 6023,060 240 500 0 5025,000 480 280 0 8027,000 100 400 0 8042,000 480 250 0 7535 000 150 360 0 7036,000 400 . 0 7038,000 200 400 0 65

PROBLChl 5SPl

Problem 6SP120,000 100 430 0 8040,000 440 150 0 7036,000 180 350 0 9135,000 520 300 0 6831,000 200 400 0 8542.000 390 150 0 80

The synthesis problem is that of creating the network ofheat exchangers, coolers, and/or heaters required to performthe design tasks at a minimum cost. Further, it is assumedthat once a stream is committed to a heat exchanger, cooler,or heater as much energy as is technically feasible is ex-changed, and no stream splitting is considered.Boundi ng Problem. Even in relatively small industrialenergy exchange problems an enormously l arge number offeasible heat exchange networks can be synthesized;therefore the unguided comparison of feasible networks isnot an eff icient method of synthesis. T he source of thediffi culty is the large number of ways in which a streamand i ts residual can be used i n the exchange of energy.T hi s leads to severe combinatorial problems.A useful problem which bounds the original problem can becreated merely by relaxing one condition of network feasi-bility-namely, that a stream cannot be used in more than

one place at the same time. This condition confounds theengineer, since, because of it, i t is not possible to focus atten-tion on the processing of one stream at a time. For example,in processing stream 1one might consider heat exchanger withstream 2: but, perhaps stream 2 is best used in an exchangewith the residual of a previous exchange wit'h streams 3 and 4.If the need for considering these secondary int'eractions couldbe eliminated temporarily by allowing the simultaneousmultiple use of streams, the resulting processing problemwould be simplified enormously.Further, if we allow the simultaneous multiple use ofpri mary streams and heat exchange residuals, the cost ofprocessing the streams is less than in the original designproblem. Finally, if in solving this new and physically un-realistic problem, t'he simultaneous multiple use of processingstreams does not occur, the original problem is recovered.I t is evident then that relaxing the conditions on the simul-taneous multiple use of streams leads to a simpler problemwhich has valid bounding properties. However, proceduresmust be established to force the bounding problem away fromthe simultaneous multiple use of st'reams if the originaldesign problem isto be solved.Focus attention on the computat'ion of the bound on theprocessing cost. First, the costs of heat exchange bet'ween all

streams are computed and the residual unprocessed hot andcold streams determined. Then the costs of heat exchangebetjyeen the primary streams and residuals, and of exchangeamong residuals, are computed and the new residual hot andcold streams determined. Also, at each stage the costs aredetermined for processing the primary streams and theirresiduals by auxil iary heating or cooling. This kind of com-putation is performed until there are no residuals left toprocess. The result's are lists of ways in which each streamcan be processed and the cost for each method of processing.We adopt the convention that one half of the heat exchangecost between two streams is assessed to each of the tIvostreams.These computations are performed ignoring the fact t,hata stream niay find better use elsewhere, and for this reasonthe computations are simple and straight'forward, with noneof the conibinatorial problems which arise in the originalreal design problem.A t thi s point for each stream t'here exists a list of ways inwhich the stream can be processed by heat exchanges and/orauxiliary heating or cooling. A system can be synthesizedmerely by selecting one entry from each list, and the systemwill be feasible if the entries are selected so that no simul-taneous multiple uses of streams occur.However, these lists can be put to the more useful purposeof forming bounding problems which lead directly to theoptimal system. Suppose nom that the ways of processing agiven stream are ordered according to increasing cost ofprocessing. T he first entry in each list is then t'he lowest costmethod of processing each stream, disregarding the networkfeasibil ity requirements. The sum of these fi rst streani-processing costs will be at least as low as the lowest cost ofprocessing the streams feasibly. And, if by chance no simul-taneous multiple uses of bhe streams occur, this mode ofprocessing is feasible, and by Conditions 1and 2 the design isoptimal.If the sum of the fi rst entries from t'he ordered l ists does notform a feasible system, one moves up the lists to the nextlowest suni. If this new system is feasible, i t is optimal. Ifnot,, one moves to the third lowest sum. If this process iscontinued, the first feasible system found will be optimal.

50 I&EC FUNDAMENTALS VOL. 9 NO. 1 FEBRUARY 1970


4/11

Figure 1 . Cost matrix for problem 4SPIT. Termination of a stream- St ream in feas ib i l i t y* Technolog ica l in feas ib i l i t y0 M od i t l ed c ost

Solution to Problem 4SP1Four-stream problem 4SP1 is now worked out in detailto illustrate the method.Construction of Primary Cost M atrix. T he primary

streams are first segregated into cold and hot streams, andput in a matrix form as shown in Figure 1. Entries in thefi rst row and column are the names of the hot and coldstreams, respectively. Entries in the second row andcolumn are the costs of processing the streams by coolers(C) and heaters (H), respectively. T he next four entriesin the upper left-hand porti on of the matrix are one halfthe costs of the heat exchangers required to exchange heatbetween the two corresponding streams. I n exchange heatof one stream with another (matching the streams), as muchheat is eschanged as technically possible. When the match isphysically impossible because of certain technological l imita-tions-the inlet temperature of a hot stream is lower thauthat of a cold stream--we use * to denote such technologicalinfeasibility.A t this point we introduce the folloiving symbolic notation,which we call stream process statements, to represent theway by which astream is processed.i , H.j , C.( i , ) ,[ i , ] .T.

Stream i is processed to a _Heater.Stream j is processed to asool er.Cold stream residual of i after it matches wi th jHot stream residual o f j after i t matches with iT he stream has satisfied the task constraint and neednot be processed further (and thus Terminated).For example, statement (1,2), H means that the coldstream residual of 1,after i t matches with 2, is sent to a heater;

and statement [(3,4),2],T means that the hot stream resid-ual of 2 completely satisfies the task constraint after itmatches with the cold stream residual resulting from thematch of 3and 4.For a symbolic statement of processing one stream to bephysically reali zable, the process statement must satisfy thefollowing stream feasibility criterion:i l stream process statement is feasible when a streamnumber does not appear niore than once in that same statement.For example, process statement ((1,2),[3,2]), H is stream-infeasible, since 2 appears tn ce in the statement, whichnieans that 2 is simultaneously matched with 1 and 3. Weuse- o denote stream infeasibility.T he primary cost matrix can be expanded to includeresiduals from primary matchings. T his expansion ultimatelyleads to the formation of bounding problems. T he firstexpansion includes the matchings of the residuals wi th

primary streams and with residual streams. The heat ex-changer costs for the primary matchings are modified to takeon one half of the original costs. These modified costs (roundedto whole numbers) are assessed to each of the two matchingstream and used for computation of the cost of each processstatement.T he expansion of the cost matrix is a finite process whichterniinates when no niore residuals are left to be processed.Such will be the case when all the residuals are terminated-i.e., they satisfy the task constraints-and/or the processStatements contain all stream numbers-Le., all streams arebeing used. The final cost matrix for problem 4SP1 is shownin F igure 1.T he modified costs for matches involving residualscontain not only the immediate heat exchanger costs but alsoVOL. 9 NO. 1 FEBRUARY 1970 l&EC FUNDAMENTALS 5 1


5/11

Resid-ua l ofStream1

2

3

4

Table Ill. Pertinent Data for A l l PossibleResiduals for Probl em 4SP1Cost ofHeatExchanger(Non-modifled)

99053474047623756462093036699074 57409307451,1038241,0035171,0575341,1034768242371,0035645176201,057366

Temp., O FIn

287 4320230 5241 3320320289 1300320200278 5200269 1300460329 9460329 9430349 9353 2280298 1450387 1323 1332 7280340415 4

o u t320

200

500

280

the costs of heat exchanges which lead to these residuals.TableI11summarizes pertinent' data for each possible residual.Formation of Bounding Problem and OptimalityCriterion. A11 possible ways of processing any one streamcan now be listed from the final cost mat'rix (Table IV) .T he cost for each process statement' can be computedfrom the modified cost and/or the cost of heater or cooler.T he process statements for each stream are ordered inincreasing cost (T able IV ).F E A S I B I L I T YEST.A necessary and sufficient conditionthat a set of process statements physically satisfy all thetask constraints is that there must, be one and only one processstatement which is stream-feasible for each stream; and therecannot be a multiple use of any one stream. I n other words, ifan A , B match appears in one process statement, then A , Cor C,B matches cannot appear in any other statement.The formation of bounding problems and the optimali tycriterion can no^ be stated as follows. T he first boundingproblem with the lowest cost is formed by selecting thelowest entries from each list in T able IV . If this selected setsatisfies the network feasibi li ty test, the network is optimal.If not, we move to the next higher bounding problem. T hefirst bounding problem which satisfies the network feasibilitytest generates t,he optimal network.This procedure is illustrated in Table V. T he first boundingproblem does not satisfy the second part of the networkfeasibility test; stream 1 is in mult'iple use in three different

Table IV . l i s t of Process Statements for Each Streamfor Problem 4SP1Entry

10987654321

54321

7654321

121110987654321

Cost

32,38916,72614,7508,0606,2734,389806462366267

11,0667,7397,24455749537,55029,35725,28925,25911,0606,6926,556

21,54918,72515,14812,4448,6087,4006,4265,6242,627654514

7,975

places-to match stream 4 in entry 1 of stream 1, to matchstream 2 in entry 1of stream 2, and to match stream [3,4]in entry 1 of stream 4. The optimal network is shown schemat-ically in Figure 2. This optimal design was found afterexamining 84 design candidates, there being a total of (10)-(5)(7)(12) = 4200 dif ferent ways of processing the streams.Improvement and Refinement o f BasicBranch and Branch Method

Having shown how the branch and bound method can beapplied to solve the simple four-stream problem, we are nowready to show how the basic method can be improved to ob-tain the optimal solution much more efficiently. B ut fi rst,we show why the basic method has to be improved.Drawback of Basic M ethod. I n the last part of the solu-tion to problem 4SP1, the bounding problem is formedby selecting one entry f rom each process statement l ist,and the optimality criterion states that the first systemwhich is feasible is optimal. F irst a "superlist" is formedwhich contains all possible combinations of dif ferent52 l&EC FUNDAMENTALS VOL. 9 NO. 1 FEBRUARY 1970


6/11

Table V. Formation of Bounding Problems and Opti mal Network for Problem 4SP1Bounding Entr ies an d Process StatementsProblem Stream 1 Stream 2 Stream 3 Stream 4 Net wo r k Ne t wo r kcost Feasibi l i ty

7,832 NO7,894 ?TO7,931 9 0

13,481 Yes

200" F

1 2 8 0 ' F

Figure2. Optimal network for problem 4SPI

entries from the four stream lists. This list is then re-arranged and ordered in increasing cost (the sum of fourentry costs). Then one simply goes up thi s ordered superlistto find the first feasible bounding problem. I n brief, thisprocedure involves forming all possible sums, sorting thesums, and testing network feasibility.There is no real great diff iculty in the first step. It is in thesecond step that the need for improvement arises.. It is clearfrom Table I V that the superlist array for four-stream problem4SP1 has a size of (10)(5)(7)(12)= 4200 elements. B ut as thenumber of streams increases, the problem becomes out ofhand. For example, we found in five-stream problem 5SP1that the superlist array has 3,397,680 elements! Even withoutmentioning the storage problem, i t is evident that the problembecomes overwhelming when the number of streams increases.Hence the basic method must be improved to handle problemswith larger number of streams.Branching Strategy. One way to reduce the size of theproblem is to replace the difficult design problem h by aset P = {1,2, . . . . } of problems that bound problem Ain the sense that they jointly satisfy the following general-ized bounding property :

OA(DA)5 OA(DA) (la)and D A must be feasible when applied to design problem kePwhere k is one of the problems in set P .Now suppose we obtain an optimal solution D k to eachproblem keP, and suppose

OK(DK)=maxOdDk)kr P

Figure 3.branching problemsReplacement of original design 4SPI by a set of

Then DK is an optimal solution to problem .4 if i t satisfiesthe condition:DK s a feasible solution to problem A, and

OK(DK)= OA(DK) (2a)Sow we go back to problem 4SP1 and show how theabove strategy can reduce the size of the problem. I n Figure3, the original difficult problem is represented by a node whichcontains all possible heat exchange networks. Branching offfrom this node, this problem is replaced by a set of five prob-lems:Problem 1. Setwork in which no match is made-Le., allProblem 2. All networks which contain at least the 1,2Problem 3. A411networks ryhich contain at least the 1,4Problem 4. All networks which contain at least the 3,2

streams are processed to heaters and/or coolersmatchmatchmatchProblem 5. All networks which contain at least the 3.4match

The above problems jointly satisfy the bounding propertyof Condition l a. R e proceed to find the optimal solutionsfor each problem in the set. Problem 1 s easy to solve; it is astraightforward calculation. Solutions to the last four prob-lems are similar and problem 3 is selected to show the details.The solution to problem 3 which includes all networks thatcontain the 1,4 match, can be referred back to the processstatement list in Table IV . All process statements that areincompatible (fail the network feasibility test) with the 1,4match are eliminated from the list. For example, entry 10forstream 1has to be eliminated because statement 1,H is in-compatible with the 1,4 match. So is entry 9, 8, and so forthVOL. 9 NO. 1 FEBRUARY 1970 I&EC FUNDAMENTALS 53


7/11

Table VI. Compatible List o f Process Statements f o rProblem 3 (o f Probl em 4SP1)Process

St ream Ent ry S tatement cos t1 1 (1,4),T 2672 2 2,c 11,0663 4 3,H 37,5503 (312) H 29 3572 ((3,2), 1,41),H 25,2891 (3,[1,41),H 25,2564 3 11.41.C 7,975

1 [3,21,c 7,739

6 4262,627Table VII. Comp atible L i s t o f Process Statements f o rProblem 3-3 (o f Problem 4SP1)

ProcessStream Ent ry S tatement cos t

1 1 (1 4) T 2672 1 2,c 11,0663 1 (3,[1,41),H 25,2561 [3,[11411,T 2,6274 2 [1,41,c 7,975

for stream 1. A fter all the incompatible statements areeliminated for all the streams, a new l ist of process statementsfor problem 3 is obtained (Table V I ). T he size of problem 3is (1)(2)(4)(3)= 24, which means that we have reduced thesize of problem by a factor of more than 170 (from 4200 to24). Problem 3 can now be solved by the basic method pre-sented above. But for the sake of illustration, suppose we arestill not satisfied with a problem with size 24 (itmay be toolarge to be solved by hand). One can continue the branchingstrategy and replace problem 3 by a setof problems as follows:Problem 3-1. Network in which no match is made after 1,4matchProblem 3-2. -411 networks which contain both 1,4 and 3,2matchesProblem 3-3. All networks which contain both 1,4 and3, 1,4]matches

PROBLEMuSOLVE

OF PROBLEMI V!&BY A SE T OFPROB.EMS w w n

MUTUALLY BOUND PROBLEM

Figure 4.branch and bound methodAlgorithm for branching strategy of improved

There is no problem for match (1,4),2, since (1,4) is ter-minated asshown in Table VI .Referring to Table VI , the solution to problem 3-1 is simply(1,4),T ; 2,C; 3,H; [1,4] C which has a cost sum of 51,697.T o solve either problem 3-2 or 3-3, we again eliminate processstatements in T able V I that are incompatible with the com-mitted matches. The new list of process statements forproblem 3-3 is shown in Table V I I . T he sizeof problem 3-3is (1)(1) (1) (2) = 2, and we have now reduced the size of theproblem by a factor of 2100! Again the optimal solution toproblem 3-3 can be easily obtained by the basic method.The above procedure might be repeated until one is satis-fied wi th the size of the problem or no more replacement ispossible. The algorithm for thi s improved branch and boundmethod isshown in Figure4. Table VI11 shows how the originalproblem is reduced by the improved method. I n other words,

2nd 140

Table VIII. Comparis on of Size o f Problems f o r Different Levels o f Branching for Problem 4SP1Match Commit ted Match Commit ted

Level of Branching in 1st Level in 2nd Level Size ofBranching Prob lem Branching Branching Prob lem to Be Solved

No . of Problems

0 None . . . . . . 4200 11st 1 Xone . . . 1 52 1,2 . . . 45

4 3,2 . . . 1805 3,4 . . . 451-0 Sone . . . 12-1 1,2 None 12-2 3,4 42-3 (1,2),4 83- 1 1,4 Sone 13-2 3,2 43-3 3,[1,41 24-1 3,2 hone 14-2 1,44-4 1,[3,21 85-1 374 Koiie 15-2 1,25-3 1,[3,41 8

3 1,4 . . . 24

44-3 (3,2),4 8

4a Problems 2-2 and 5-2 are equivalent. Problems 3-2 and 4-2 areequivalent.

54 l&EC FUNDAMENTALS VOL. 9 NO. 1 FEBRUARY 1970


8/11

Table IX . Act ual Computation Procedure and Solution to Problem 4SP1 by Impr ovedand Refined Branch and Bound Metho dMatchCommittedOrder of Branching in 1st le ve lComputation P roblem Branching

1 1 Xone2 2-1 1123 3-1 1744 4-1 3125 5-1 3146 3-3 1, 47 3-2 1148 2-2 1,29 4- 2 3, 210 5 2 3, 4

11 2-3 1, 212 4-3 3,213 4- 4 31214 5-3 314a Optimal solution to original problem.

MatchCommittedin 2nd levelBranching. . .NoneNone

NoneKone3, 1, 413721141, 2(1,2),4(3,2),4

3J 4

1, [3, 211 , [ 3 J 4 1

Size of ProblemRefined Refinedprocedure procedurenot used used

1 11 11 11 11 12 24 14 44 04 08 18 18 18 1

Cost of OptimalSolution102, 55467,65456,858>56,858>56,85839, 216>39, 21613, 481.Same as 3- 2Same as 2-2>13, 481>13, 481>13, 481>13, 481

instead of solving the original problem of size 4200, we are tosolve 14 problems with size ranging from 1 to 8.Further Refinement to Enhance Effi ciency. A fter thebranching procedure is completed, the actual computa-tional labor can be reduced further by the following proce-dure.1.Solve a problem wi th the lowest size and obtain its opti-mal cost, Cp.2. Go to the next lowest size problem.3. If this problem is equivalent to one previously solved,i t has the same optimal solution as the previous one and neednot be solved. If not, go to the next step.4. For the next lowest size problem, if the lowest boundingproblem-].e., the lowest entries from each stream list-has acost higher than C P ,whether it is feasible or not, then theoptimal solution for this problem must have a cost higher than

C p, and thi s problem need not be solved (because i t cannotbe the optimal solution to the original problem). If not, go tothe nest step.5. L ook for any process statement IC which has the followiiigproperty :S

C k l 2 C P - c13 = 13 #zwhere Ckz s the cost of the kth entry process statement forstream i , and C,, is the cost of the fi rst (the lowest) entryprocess statement of stream3 other than i .Eliminate all process statements with such property for allstreams = 1, ,s. If there is no feasible solution to thisreduced problem (often this can be detected by simpleinspection), we can conclude that the optimal solution to thenonreduced problem must have a cost higher than CP ,and

the problem need not be investigated further. B ut if we findan optimal solution to this reduced problem (by the basicmethod), we compare its cost CR with C p . If CR >C p , wesimply go back to step 2. But if CR


9/11

R EPL AC Ec ; BY cg

YESV

NO

400 F

SEL EC T TH E L OWESTCOST PROCESS STATE-M EN T S , F OR STR EAMLDOES Si CON-

WHICH NO PROCESSS T A T E M E N T H A SBEEN SEL EC TED

1 YES

C A L C U L A T E C A L C U L A T E

ARE THERE ANY

BEEN SEL EC TED

Figure 6. Algorithm for finding C* P

statements can be eliminated and the size of the problem canbe reduced further even without the application of theimproved and refined method. One way to find such Cp* is toexplore the lower portion of each list in Table IV . Supposewe focus our attention first on one stream, say stream 2.The lowest cost method of processing this stream is by thefirst entry-namely, [l ,2],T in Table I V. This statementdirects our attention now to stream 1.From the list for stream1,we select the lowest process statement that is not incompat-ible with 1,2 match, and we find statement ((1>2),4),T entry2). T hen we go to the list for stream 4 and select the loweststatement that is not incompatible with (1,2),4 match, andwe find statement [3,[(1,2),4]],C entry 6). Finally we go tostream 3 and select the lowest statement that is not incompati-ble with 3, [(1,2),4]match, and we find statement (3, [(1,2),4],-H (entry 3). We have now found the following feasible solu-tion, and its cost, Cp', is much closer to the true optimalsolution of 13,481.

StreamNo. Process Statement cost2 [I 121,T 4951 ((1,2) 4) T 3664 [3, J 411 7,4003 (3, [( 1,2) ,41) ,~ 11,060Cp' = 19,321

Generally, we can start with each of the four streams,follow through the procedure outlined above, and obtain acost Cp' for each feasible solution. The lowest feasible solu-tion costs will be taken as the closer upper bound, Cp*, forthe original problem. Figure 6 shows the algorithm for findingsuchCp*.For problem 4SP1, CP *was found to be 19,321.Once Cp* is found, we can eliminate all the process state-ments k in Table IV which have the following property:

j#By this procedure the original size (4200) of problem 4SPlis reduced to (7)(5)(3)(8) = 840, even without employing

150 F

323 F

196 FI 50' FI5OoF 360. F

400 F

Figure 7. Optimal network for problem 5SPI

*I400 F

4 1

150 Ft273 FI

t 150 F 1300' FFigure 8. Optimal network for problem 6SPI

the improved and refined procedure. This closer upper boundtechnique can also be applied to solve each branching prob-lem.Suboptimization. T he closer upper bound CP * providesanother good feature. Suppose for some reasons (timesaving, labor saving, or rough estimation) we will besatisfied with a solution that is within certain range, sayby0, of the optimal solution obtained so far. We mayeliminate any branching problem which has the followingproperty:

8 Ci j 5 0.95 CP *j =1

or eliminate any process statement k which has the followingproperty :

56 l&EC FUNDAMENTALS VOL. 9 NO, 1 FEBRUARY 1970


10/11

~ ~

Table X. Summary of Computational labor Involved in Solving Three Sample Problems

Problem4SP15SP16SPl

No. ofStreams

Construction of Cost M atrixSize of Final No. of H e a tCost Mat r i x Exchangers

4 17 X 175 53 X 526 225 X 2251550222

No. ofCoolers1338190

Improved and Refined Procedure Coupled with Closer Upper Bound TechniqueActua l Computat iona l Labor

S i z e of O r ig i na l S i z e of No. of Tota l No.Prob lem Problem (Approx . ) prob lems prob lems of problems CP*4SPl 4 x 103 1 10 12 19,3212 14 15sP1

6SPl3 x 1061 x 10

1 202 41 1252 14 28 116 3

24132

38,52752,419

No. ofHeaters1331129

O p t i m a lSolu t ion13,481

38,27835,108

Even though the optimal solution obtained by this proce-dure may not be the true optimal, the saving in the computa-tional labor might justify the use of this suboptimizationtechnique.Solution t o Problems 5SP1 and 6SP1

Problems 5SP1 and 6SP1 are solved by the improved andrefined branch and branch method coupled with the closerupper bound technique, and the solutions are shown in Figures7 and 8, respectively. Problem 5SP l has been solved by M assoand Rudd (1969) using the heuristic structuring method.T he optimal network obtained by the heuristic method is thesame as obtained by the branch and bound, even though theoptimal costs are slightly different because of the use of adifferent value for the equipment down time (380 instead of260 hours per year used by M asso and Rudd).Table X compares the computational labor involved insolving sample problems 4SP1, 5SP1, and 6SP1. T here is atremendous growth in the size of the original problem as thenumber of streams increases (lo3for four streams, lo6 forfive streams, and 10 for six streams). Nevertheless, the im-proved and refined procedure coupled with the closer upperbound technique enables us to reduce the overwhelmingproblem to problems with size ranging from 1to 16. A lthoughthis increases the number of problems to be solved (12 prob-lems for four streams, 24 problems for five streams, and 132problems for six streams), the rate of growth is much slowerthan the size of the original problem. Furthermore, most ofthese problems are of size 1, which is very easy to solve.Lee (1969) gives further detail s.Conclusionr

T he branch and bound method for the synthesis of heatexchange networks has two attractive features. First, all thecomputation procedures are simple and straightforward ;

all work was carried out by hand calculation. Second, anoptimality criterion can be stated, whereas previous methodsof heat exchange network design using empirical rules ofthumb or the heuristic approach offer no proof of optimality.T he branch and bound method illustrates two points.First, a difficult or seemingly unsolvable design problem canbe solved by replacing it with a bounding problem which ismuch easier to solve; and in designing heat exchange networkthis bounding problem is created merely by relaxing onecondition of network feasibi lity, that a stream cannot be usedin more than one place at the same time. Second, the size ofan overwhelming problem can be reduced by the branchingstrategy.Finally, none of the designs generated here exhibit cyclicinformation flow. The methods used to generate the initialcost matrix involve the matching of primary streams toprimary streams, primary streams to residuals of primarymatchings, and so forth. Every stream statement necessaril ycontains an initial primary-primary match or no match.Designs exist with no primary-primary matches, and theseexhibit the cyclic information flow missing in this paper.Such cyclic designs are very few in number in design problemsexamined. A lthough no cyclic designs were less costly thanthe opti mal acycli c designs generated; we have no proof thatthis holds in general.Nomenclaturea, b =heat exchanger cost parametersc = heat capacityC =cost of cooler (plus cost of water)C =cooling water costC, = steam costC k l = cost of kth entry process statement for streamC P =cost of optimal solution obtained upto that pointCR =cost of optimal solution for a branching problemCP =cost of a feasible solutionCP* =closer upper bound

VOL. 9 NO. 1 FEBRUARY 1970 l&EC FUNDAMENTALS 57


11/11

D = system design lit erature CitedEHmn0 = economic objective functionp, =steam pressurest = stream temperatureUw = flow rate7aT- = stream infeasibility* = technological infeasibility

= cost of heat exchanger=cost of heater (plus cost of steam)= number of hot streams to be cooled= number of cold streams to be heated= total number of process streams=over-all heat transfer coefficient=minimum allowable approach temperature difference= yearly equipment down time= termination of a stream

Hwa, C. S., M athematical Formulation and Optimization ofExchanger Networks Using Separable Programming, A . I.Ch. E./I. Chem. E. oint meeting, London, 1965.Kesler, M. G., Parker, R. O., Chem Ens. Progr. Symp. Sei-. 65 ,No. 92 (1969).Lawler, E. L., Wood, D. E., Operations Res. 11, No. 4, 699-719(1966).Lee,K.F.,Ph.D. thesis, University of Wisconsin, 1969.Masso, A. H., Rudd, D. F.,A. . Ch.E. . 15 , No. 1,lO-17 (1969).Rudd, D. F.,A . I . Ch.E. . 14, NO.2,343-9 (1968).Siirola, J . J .,Powers, G. J .,Rudd, D. F.,A . . Ch.E. J ., in press,RECEIVEDor review April 11, 1969ACCEPTEDovember 28, 19691970.

Work financed in part by the National Science Foundation.

Near-Optimal Control by Trajectory ApproximationTubular Reactors with Axial Dispersion

LeRoy 1. Lynn, Ell iot S. Parkin, and Raymond 1. ZahradnikDepartment of Chemcal Engineering, Carnegie-Mellon University, Pittsburgh, Pa. 16215

A new method for obtaining near-opt imal control policies is based on the use of weighted residual techniquesto solve approx imately the state and adjoint dif ferent ial equations which result from application of Pontrya-gins maximum principle to the optimal control problem. The technique, trajectory approximation , is app li edto the determination of optimal temperature profiles for a tubular reactor with axial dispersion, and theresults are compared to numerical solutions obtained by quasi linearization. The results ind icate that themethod is computationally sound. In fact, i t could le ad to the realizati on of on-line near-op tim al control formany chemical processes.

A K EW method for computing near-optimal control policiescould lead to real- time near-optimal control of chemicalprocesses. I l lustrative results are presented for near-optimaltemperature profiles in a tubular reactor with axial disper-sion.T he mathematical formulation of optimal control policies isrelatively advanced as a result of the pioneering iyork ofPontryagin et al. (1962). Considerable extensions and de-velopments to the optimal control problem may be attributedto hthans and Falb (1966), Fan (1966), Leitmann (1967),and L apidus and Luus (1967), to mention but a few. How-ever, the computational procedures proposed to implementthis theory have, in general, been complex iterative schemes,requiring sophisticated methods of numerical analysis. Amethod for obtaining near-optimal control policies is presentedhere, that could considerably simplify and shorten the com-putational effort involved in determining optimal controlpolicies for complicated nonlinear chemical processes. I tis referred to as trajectory approximation.T he method discussed is especially suitable for the deter-mination of near-optimal control policies for processes in the

chemical industry-for example, the computation of opti maltemperature profiles for tubular reactors is of continuinginterest. Solutions for various chemical kinetics in a tubularplug-flow reactor have been obtained by B ilous and Amund-son (1956) and Siebenthal and Aris (1964). Lee (1964) hasdiscussed the use of a gradient technique for determiningoptimal temperature profiles, and Flynn and L apidus (1969)applied nonlinear programming to the same problem.Zahradnik and Parkin (1969) first applied the trajectoryapproximation algorithm to the determination of near-optimal temperature profiles for a tubular reactor withoutaxial dispersion. They demonstrated that the two-pointboundary value problem to be solved in the maximum prin-ciple formulation of the control problem can be reduced to asystem of algebraic equations in a few variables, capable ofbeing solved very rapidly on a digi tal computer. This methodis an equally effective way of computing near-optimal tem-perature profiles for a tubular reactor with axial dispersion.Results are obtained for the concentration profiles which areindistinguishable f rom the exact solutions over a wide rangeof axial Ieclet numbers.

5 8 I&EC FUNDAMENTALS VOL. 9 NO. 1 FEBRUARY 1970

branch and bound of integrated process designs

Documents