simulation of single- and multiproduct batch chemical plants for optimal design and operation

Computers and Chemical Engineering Vol. 7, No. 3, pp. 137-156, 1983 Printed in Great Britain.

0098-1354/83 $3.00 + .oo 0 1983 Pergamon Press Ltd.

REVIEW PAPER

SIMULATION OF SINGLE- AND MULTIPRODUCT BATCH CHEMICAL PLANTS FOR OPTIMAL DESIGN

AND OPERATION?

DAVID W. T. RIPPIN

Professor of Chemical Engineering, Swiss Federal Institute of Technology, E.T.H. Zentrum, CH-8092 Ziirich, Switzerland

(Received 8 July 1982; received for publication 9 December 1982)

Abstract-Reasons for the increasing interest in batch plants and their improved design and operation are discussed. The extensive literature on the optimal operation of individual batch equipment items is reviewed and summarized. Topics covered include simple reactors with various kinetic schemes, polymerization, reactors with decaying catalysts including deactivating enzymes, batch distillation and crystallization. The study of multi-item batch plants is a much more recent development. A personal perspective is given of methods currently available for the quantitative treatment of the design and operation of multiproduct and multipurpose batch plants. Interconnections between existing areas of competence are pointed out and some directions suggested for further developments.

INTRODUCTION

In the early days of chemical reaction engineering in the 1950s students might well have gained the impression that the ultimate mission of the chemical en- gineer was to transform old-fashioned batch processes into modern continuous ones.

With such a perspective it would be surprising to find that, today, thirty years later a significant proportion of the world’s chemical production by volume and a much larger proportion by value is still made in batch plants and it does not seem likely that this proportion will decline. There are indeed some products for which it is not possible, or at least it would be unreasonably demanding in time and resources, to develop reliable continuous processes. However, many more products which could be manufactured continuously are in fact made in batch plants on economic grounds. Batch production is usually carried out in relatively standardized types of equipment which can easily be adapted and if necessary reconfigured to produce many other different products. Thus, it is particularly suitable for low volume products such as pharmaceuticals or other fine chemicals for which the annual requirement can be manufactured in a few days or a few batches in tin existing plant. The flexibility of the production arrangements can also cope with the fluctuations or rapid changes in demand which are often characteristic of products of this type. Thus, the difficulties associated with sizing a dedicated plant, or building an unnecessarily small one, be it batch or continuous, are avoided.

Increasing interest in fine chemicals has been evident recently as part of the trend to products of high

tAn earlier version of this paper was presented at the CHEMCOMP Symposium in Antwerp, May 1982, or- ganized by Koninklijke Vlaams Ingenieursvereniging (Royal Society of Flemish Engineers) and appeared in the proceedings of that symposium.

added value with high technological content, particularly in developed countries with few indigenous raw materials. It is likely that most of these chemicals will continue to be produced by batch processes. One crit- icism of batch processes has been their lack of re- producibility and resulting fluctuations in product quality. This is being countered in many batch plants by the replacement of manual operation by computer controlled sequencing in which the criteria for a change of state are consistently applied and independent of the idiosyncracies of the operator.

In the fine chemicals industry emphasis has tradi- tionally been placed upon innovation and the ability to produce marketable quantities of novel products rapidly. However, in recent years increasing difficulties have been reported both in discovering new products and in obtaining official approval for their introduction, particularly in the pharmaceutical field. This increases competitive pressures on existing products, which will be further strengthened by the expiry of patent protection.

All these factors will call for increased efficiency in the design, planning and operation of batch chemical plants.

If quantitative assessments are to be made of the performance of multiproduct plants, or even of single product processes comprising many sequential stages, the use of computer assistance is almost mandatory, simply to handle the large volumes of data involved. Of course, hand calculations have been used in the past in assessing such plants but these assessments have usually been based on one characteristic or dom- inating product, with subsequent adjustments to incorporate the requirements of the other products.

However, the development of computer-aided methods for batch plants has lagged substantially behind the corresponding development for continuous plants. Some of the factors stimulating the development of computer-aided methods for con-

137

138 D. W. T. RIPPIN

tinuous plants in the late sixties and early seventies can be identified, for example:

l the demand patterns for the different products, which may be distributed over time.

0 External Demand: The construction of many large new continuous plants with competitive pressure on performance and requiring the efficient use of design manpower. l Clear Objective: Relatively clear definition of

the requirements of a design so that these could be quantitatively assessed.

0 Appropriate Means: Requirements for design calculations could be met reasonably well by currently available batch computing facilities.

It is proposed that the set of products and the sequence of tasks by which each is to be produced are both fixed, thus we are not considering the problem of process synthesis for batch plants. However, there are nevertheless so many different ways in which the components and their inter-relations can be specified or constrained that an enumeration of all possible combinations is out of the question.

0 Timeliness of Methodology: The theoretical problems involved were well matched to concurrent interests in other fields. Chemical engineers could make progress in computer flowsheeting and related problems by interacting with numerical analysts or computer scientists who were working in data manipulation, equation solving and optimization.

Some of the problems arising in the study of batch plants can be identified as follows, roughly in order of increasing size and complexity of the system considered:

-Understand and optimize the performance of tasks carried out in individual batch equipment items.

-Optimize the performance of a sequence of tasks in several equipment items to produce a single product.

In recent years increasing interest in the computer- aided design and operation of batch plants has been evident. For example sessions were devoted to the subject at the 12th European Symposium on Com- puter Applications in Chemical Engineering (Mon- traux, 1979) and at A.1.Ch.E. Meetings in Houston, Texas (April, 1981) and New Orleans (November, 1981).

--Choose equipment needed to meet specified requirements of one or more products.

-Arrange or rearrange equipment interconnections to best meet new or modified production requirements.

-Determine the manner and sequence in which products should be produced to meet most effectively a demand distributed over time.

Factors earlier favouring the development of CAD for continuous plants may in the eighties be in- creasingly evident for batch plants: current status of these factors may be reviewed briefly:

-Incorporate in planning procedures the effect of quantitative and qualitative changes and uncertainties.

l External demand-increasing interest in batch plants and competitive pressure for improvement in their performance have already been cited. l Clear objective-no single clear objective can

be formulated for work with batch plants. Many alternative objectives need to be formulated for different purposes and some such possibilities are discussed further below. l Appropriate means-diffuse, partly qualitative

or multiple objectives are not conveniently handled by batch computing. Further development and wider availability of interactive computing facilities will stimulate the study of interactive computing strategies to handle such problems more effectively.

Each of these problems is considered in turn in subsequent sections of this review. It will be seen that the level of knowledge is very diverse between the different problem areas. In some cases (e.g. the optimization of individual equipment items) conclusions can be drawn about the potential benefits of computer assistance, in other cases attention is drawn to a number of areas in which further work is needed before the benefits to be derived from a more quantitative approach can be assessed.

DESCRIBING THE BEHAVIOUR OF INDIVIDUAL EQUIPMENT ITEMS

The understanding and quantitative representation

l Timeliness of methodology-methods are now available for solving and optimizing quite large systems of, often highly constrained, equations such as arise in continuous flow-sheeting problems. In working with batch plants, many more of the alternatives which must be considered are discrete and hence call for integer programming or combinatorial methods. These are topics which are currently receiving consid- erable attention in other fields. Hence there should be opportunity for work in the batch area to benefit from these developments.

THE BATCH PROCESSING SYSTEM

In designing or planning the operations of a batch plant, the system components which must be considered are: l the process specification for each product, giv-

ing the necessary tasks and the sequence in which they must be carried out to proceed from raw materials to final product.

of batch operations is a problem in mathematical modelling which can be solved by the methods used for continuous processes. The requirements placed on the model of a batch operation will be somewhat more rigorous since it will have to describe the behaviour of the system over the whole range of conditions encountered in the batch process. These may vary very widely from the beginning to the end of the batch. For a continuous process on the other hand the model’s main purpose will be to make predictions at or in the neighbourhood of the steady conditions under which the plant will normally operate, thus much more attention can be concentrated on this restricted area. How- ever, the penalties for inaccurate predictions are usually higher in a continuous plant than in a batch plant. A continuous plant will usually be single product and the equipment items will be matched to one another at the nominal throughput, whereas a batch plant will normally be multiproduct and thus will be built with more inherent flexibility to accommodate the requirements of different products.

l the set of equipment items that can be made available to carry out the various tasks.

If the contents of the batch at any instant of time can be regarded as homogeneous, as, for example in an

Simulation of single- and multiproduct batch chemical plants for optimal design and operation 139

ideally mixed reactor, then the modelling can be expected to be relatively straightforward even if many characteristics of the batch change substantially with time. It will be necessary to model any reaction processes taking place uniformly throughout the batch and any transport processes taking place at the bound- aries. A representative state of the batch at any time is defined by point measurements or samples taken any- where within the batch. This measured state can be used immediately for mode1 development and related to the rate of progress of the reactions or other processes taking place. Of course, difficulties may still be encountered, for example with complex kinetics, particularly when intermediates are postulated which cannot be measured. However, the form in which the behaviour of a uniform batch should be represented is generally clear even if the details of the mode1 may require substantial effort to elucidate.

A much more difficult situation is encountered when the contents of the batch are not homogeneous since measurements within the batch are then not generally representative. To describe the state of the system at any time some model must be available to characterize the non-homogeneities. These non-homogeneities may range from concentration or temperature gra- dients in an otherwise uniform phase through systems of fluids containing particles of fixed size to multi- phase systems such as immiscible liquids or crys- tallizations in which the size distributions of the different phases are changing as the batch progresses.

The problems of modelling such systems are clearly much more severe than for homogeneous systems. When only modest departures from uniformity are encountered, such as could, for example, be characterized as incomplete mixing in a single phase system, the methods used to describe such phenomena in continuous systems can often be adapted to the batch case. For example the concept of the residence time distribution is by definition not applicable to a batch operation in which no material flows in or out. How- ever, extensions of the residence time concept to the two environment models often used to characterize incomplete mixing can also be used to characterize incomple’te mixing in batch operations[l].t When large excursions from complete mixing occur even in otherwise homogeneous systems more physically based models, such as those incorporating concepts of turbulent diffusion[2] may be needed. At the other end of the scale of model complexity called upon to describe non-homogeneous systems are, for example, heterogeneous polymerization systems in which not only the kinetics, but also the interactions between the phases are very complex.

Developing and solving models for batch systems is not essentially different from the same type of work for continuous systems, except in so far as the variation of the system state with time during the batch may make more extreme demands on the model. Thus, no at- tempt will be made to catalogue the current state of the modelling of batch processes since this would quickly become a survey of the overall state of knowledge not restricted to the batch processing aspects, of the pro-

tGenera1 references are listed at the end of the paper. References to specific batch operations are preceded by a letter and may be found in Tables 4-7.

cesses and operations commonly encountered in batch plants.

It is assumed subsequently that a model is available which will represent the response of the system to changes in the design and operating variables. Of course this is by no means always the case, it is prob- ably the exception rather than the rule. For many batch systems design and operating conditions are determined by trial and error in laboratory or pilot scale experimentation. Nevertheless it is of interest to examine the potential benefits of optimal choice of design and operating variables for commonly occur- ring types of model. This gives an impression of the incentive for the pursuit of modelling studies with a view to optimization as an alternative to a programme of trial experiments.

OPTIMIZING THE PERFORMANCE OF INDIVIDUAL EQUIPMENT lTEMS

It is assumed that the amount of experimental effort necessary to establish by empirical means the best constunt conditions for a batch operation will not normally be unacceptably high. If this is the case the incentive for the pursuit of more detailed modelling and optimizing studies must be derived primarily from the benefits of imposing more sophisticated optimization policies such as changing the levels of the operating variables as the batch progresses.

Variational methods or Pontryagin’s Maximum Principle have been used for more than twenty years to determine the effect of optimal profiles of operating variables on batch performance and preliminary studies of the same problem were reported considerably earlier. The most popular studies have been of the effect of profiles of temperature and to a lesser extent of the addition rate of one reactant on the course of a batch reaction. Studies have also been made of other batch operations, such as the effect on batch distillation of imposing a profile in reflux rate.

To assess the benefits of an optimal profile, performance must be compared with a system on which the best constant conditions are imposed. However, the potential benefits of optimization are still highly dependent upon the objective function chosen to represent the system performance.

This effect can be illustrated by means of a simple example. For a reversible exothermic reaction it is well known that an optimal temperature profile exists in which the temperature is initially high to promote the speed of the forward reaction and is reduced during the batch to induce a favourable equilibrium composition near the end. For any specified operating time the maximum conversion using an optima1 temperature profile can be calculated. This calculation can be repeated for different operating times and the resulting curve in the conversion time plane bounds the region attainable by any manipulation of the temperature profile. This attainable region was first described by Horn[3] who showed that for any objective function that is a linear function of the conversion the point of optimum performance will always lie on the boundary of the attainable region. Curves for constant temperature operation can also be drawn in the conversion time plane and for any specified time the best constant operating temperature can be identified. Re- peating this calculation for different times will gener- ate a curve which forms an upper envelope to the

140 D. W. T. kPPIN

constant temperature trajectories and can be regarded as the boundary of the region attainable by isothermal operation. This second curve will, of course, lie below the boundary of the attainable region for all possible temperature profiles. For a wide class of objective functions the benefits of an optimal compared with a constant profile can be derived from these two bounding curves.

In many cases optimal conditions can be determined by simple geometrical constructions on these curves. Figure 1 shows the attainable regions for isothermal and optimal temperature profiles for a first order reversible exothermic reaction with the ratio of the activation energies for the reverse and forward reactions

E2 ,= 1.4.

As shown in Table 1 the advantage of the optimal profile over the best isothermal operation varies from 4 to 67% in these examples according to the objective function selected. This strong dependence of the benefits of optimization on the particular objective function chosen must be borne in mind when assessing results reported in the literature. These are almost always expressed in terms of maximum conversion at a given time or minimum time for a required conversion. With this information alone it is difficult or impossible to assess the incentive for optimization in terms of more realistic economically based objective functions.

Optimal profiles have been reported for batch reactors with many different types of kinetics. It is generally assumed that the reactor is homogeneous and the temperature or other control variables such as feed addition rate, can be freely chosen at any time, possibly subject to upper and/or lower constraints on its level. In this case the modelling requires only a representation of the kinetics of the reactions taking place. Unfortunately, for the bulk of the cases examined the quantity and character of the results reported leaves much to be desired. In a small number of cases results are presented in tabulated form for a comprehensive range of parameters allowing the potential benefits of optimization for other systems with the same kinetics to be assessed relatively easily. However, in the major- ity of cases results are presented for only a single or at best a very limited set of parameter values. A qualitative impression of the form, or possible forms, of the profile can be obtained and if significant benefits are demonstrated among the limited cases examined this can provide an incentive for closer examination in other cases. If no substantial benefits are demonstrated this does not exclude the possibility that they may be realisable in other regions of the parameter space for the same kinetics, although it will doubtless be discouraging to the potential user.

Unfortunately, there are a significant number of publications which give only the general form of an optimal profile and no quantitative results. Others, while evaluating the optimal performance, make no comparison with the performance under constant operating conditions. Such results may have a certain

Conversion 2

0.6

Fig. l(a)

Time t

I 1 1 . 1 2 3 4 5 6


@I

Conversion x

0.6

./’ /I’

,.‘b’. 6 .f/’

.‘/ .‘/

,G’ .;’

0.2

a A - 1 2 3 4 5 6

Conversion x

,o E “_d _ --AC’-

_ Y-A --

--- -c=K- - - _

06

(cl

--

Time t I I I 5 6

Fig. I. Reversible exothermic batch reaction comparison of performance for various objective functions. (a) Minimum time for fixed conversion and maximum conversion in given time: -, Isothermal operation; ------, Limit of performace attainable with isothermal operation; -.-.-.-.-.-, Operation with optimal temperature profile. (b) Minimum cost per unit of product. (c) Maximum profit per unit

time.

142 D. W. T. RIPPIN

Table 1. Comparison of performance of best isothermal reactor and reactor with optimal temperature profile

First order reversible exothermic reaction. Ratio of activation energies &/E, = 1.4.

Objective function

Constraint Best

isothermal performance

Optimal Relative advantage performance of optimal profile

Maximum Conversion

Given time t = 1.4 / 0.802 / 1 1.04 0.834

_...F t I I

Minimum time

Given conversion x = 0.85 2.83 I .72 I .67

Minimum cost Cost of Product per unit of Material cost CA operating c,t per batch x product time 1.04

Minimize F E Maximize e c,

g+t s construct tangent to boundary from x = 0, i = - 7.

I I

Maximum profit per batch in Product value Raw material Changeover time unlimited market Cd; cost c, between batches t, L__!L-

Maximize cp - ” C.4

~ E construct tangent to boundary from x = c, t = - t, t -I- 1, P

_...~

Fig. 1A

Fig. IA

Fig. IB

Fig. 1C

theoretical interest but their practical value is very limited.

Table 2 summarizes kinetic schemes for which the optimal temperature profile is known to be isothermal. In Table 3 the form of optimal non- isothermal profiles is given with an indication of the quality of the information available about the benefits of optimization.

The following qualitative guidelines may be given about the form of optimal tem~rature profiles but detailed calculations are needed to predict the potential benefits: l If the favoured reaction has the highest activa-

tion energy always operate at T,,,,,. l If the favoured reaction has the lowest activa-

tion energy and time is of no significance, operate at T,,,,,. If time is significant profile may be rising, falling or both. l If the favoured reaction has an intermediate

activation energy profile may be isothe~al, rising or falling.

When reactions of different order are taking place simultaneously there may be advantages in adding one of the reactants gradually during the course of the batch, If the favoured reaction has the highest order, all the reactants should be added at the beginning, but if the favoured reaction has the lowest order and the time available is iimited, an optimal feed addition rate profile can be found. In one reported case with two parallel reactions the final product concentration could be increased from 40% with the optimal amount of reactant added all at the beginning of the batch to 57% with the optimal feed

addition rate profile (Fl). In a further development it was shown that when profiles of both temperature and feed addition rate are allowed and the desired reaction has the lower activation energy the optimal policy will be either to add all the reactant at the beginning and impose a temperature profile or to maintain the maximum temperature and impose a profile on the feed addition rate. The choice depends on the relation of the respective activation energies and reaction orders. It is never optimal to have profiles in both temperature and feed addition rate simultaneously. In the above example the final product concentration could be further increased to 80% by allowing temperature variation (F2).

When a gas is bubbled continuously through a batch of liquid with which it reacts, the utiiization of the gas can be improved by supplying it at a rate which decreases as the reaction progresses. The system has been studied for a variety of objective functions similar to those of Table 1 and optimal profiles have been established. The benefits of the optimal profile over the best constant feed rate of gas were generally up to -5% but could be larger (F5).

Most of the studies of optimal profiles reported are for relatively simple kinetic schemes. However, some reports of studies of more complex systems have appeared.

Polymerization reactors Since the late nineteen sixties the benefits of ap-

plying optimal profiles to polymerization systems have been reported and some references are given to earlier work. The main features of publications in


Table 2. Kinetics for which optimal profile is isothermal

T=T_ Kl* A&P z E, ’ 4 Time arbitrarily long

K2 ‘P

A/

bW

4 ’ 4 Time arbitrarily long

K3 Any number of strictly parallel reactions among which the desired reaction has the highest activation energy.

K4 AltP5W

KS A#P:W

KlO A, + A,AP + W,

P+A$W,+ W,

4 ’ ~52

E, > E,, E3

E, ’ E2

Time arbitrarily long

Calculated optimal time



Calculated optimal temperature

2 p

K3 AA W,

%W *

K7 Al*P

2 w, /” \ ’ W2

T=T,, KS A&PAW 2

K6 ’ P A3

?W

E, between

E2 and E3

As long a time as possible

E, between

E, and E3 Calculated optimal time

E, < E2, Ej

E, < E2, E,



A: raw material, P: desired product, W: waste product. *References to kinetic scheme Kl ff see Table 4.

which quantitative results are given are summarized in Table 5. The most commonly studied system is styrene polymerization and the general objective is to reduce the time required to produce a polymer of specified characteristics. A rising temperature profile is generally required and a feed rate of initiator sufficient to maintain a constant rate of initiation when the temperature is changing (Pl 1). At constant temperature this condition is equivalent to a constant initiator concentration. In several cases use of the optimal profile is predicted to reduce operating time by more than 20% compared with isothermal operation and there is some experimental confirmation of the optimal performance (see Table 5).

The results for nylon 6 (P3, P7) refer to a continuous plug flow reactor but are included for comparison since the calculation procedure is identical to that of a batch reactor.

In some cases it is desired to maintain polymer properties as closely as possible. It is shown that a molecular weight distribution of minimum breadth is obtained by choosing the temperature so that the average length of dead chains being formed at any instant is constant (P5). In copolymerization a con-

stant ratio of the two components in the polymer chain is achieved by adjusting the temperature as the reaction progresses so that the ratio of the rates at which the two monomers are incorporated into the chain remains constant (PI, P13).

Reactors with decaying catalysts or deactivating enzymes

Conditions have been derived for optimal temperature profiles in batch reactors with deactivating catalyst. This work has subsequently been applied to biochemical systems catalysed by enzymes of which the activity is decaying. Szepe & Levenspiel (Cl) showed that for a single irreversible reaction and a catalyst decay rate independent of concentration a rising temperature profile is called for provided the activation energy of the decay is greater than that of the main reaction. (Otherwise the maximum temperature should be used.) When the rate expression of the main reaction is a product of a temperature dependent reaction rate constant, the catalyst concentration and an arbitrary function of the reactant concentration the increasing temperature should be chosen to maintain constant the product of the

144 D. W. T. RIPPIN

Table 3. Non-isothermal profiles

Falling

profile

o(l)

(K4)

(KS)

6%

Rising profile

62)

(K3)

(KS)

(Kg)

(Kg)

Mixed profile

rising and

falling parts

(KlO)

(Kll)

A+P

AltP:W

A+PltW

\ P

AP, I ‘W

AbAP

21 41

WI w2

A ‘P

< 28

A-5 W,

A+P:W,

*PAW,

A(W 2

I 3 A+I-+P

21 41

w w2

A, + A,AP + W,

P +A,-:W,+ W,

2A+P+ W,

P+AAW,+ W, 4

EI < E2

EI < ~52

E, i E2, Es or

E3 < E, < E,

If available time

is restricted

E, < 4, E, If available time is restricted

E, > E2 and E3 i E4 Very large E, < E2 improvements

possible

Mirror image of profile for A P P

E, between E2 and E,

If available time is restricted

E, > E, > E,

E3 < E, < E, Confirmed by experiment

E,<E,andE3>E4 Very large improvements possible

Al

E, < E2 A3

E, < E, < E3 < E4

Al

A2

Bl

A3

Bl

Al

Al

C

A3

B2

C

A = tabulated results, B = isolated results. C = no quantitative results, 1 = gains significant, 2 = modest, 3 = small.

reaction rate constant and the catalyst concentration. Similar, more complex conditions were derived for arbitrary power law kinetics (C2) and for other forms of kinetics (C3).

The biochemical systems studied (see Table 6) have included more complex reaction schemes, simultaneous optimization of pH and temperature profiles (C5) and a comparison of different objective functions (C7). Reported benefits of the profile over operation under constant conditions range up to 30%. A more complex problem can be formulated in which the same slowly decaying catalyst is used for a number of batches in sequence and different conditions are chosen for each batch to achieve some overall optimum. However, this case is not considered here.

BATCH DETILLATION

Reports on the use of optimal reflux profiles in batch distillation date from 1963, as indicated in Table 7(a). The systems studied are generally binary with constant relative volatility (with a value -2). Ternary systems are considered in (D6) and (DlO) and a multicomponent system using Wilson or Renon coefficients is referred to in (D14) but no details are given. A ten component mixture with constant relative volatilities is discussed in (D7) but for some of the calculations it is. treated as an effective binary. The number of ideal stages considered is generally small, up to 8, with 3 and 5 being common values. Exceptions are (D3) 12 stages, (D7) 30 stages, (D13) a large number of examples with five to thirty stages and (D14) which claims a capability for handling up

Simulation of single- and multiproduct batch chemical plants for optimal design and operation

Table 4. Optimal temperature profile in batch reactor. References to kinetic schemes

145

The appended list of references is arranged in chronological order to indicate the development of the subject. References which treat the kinetic schemes Kl to Kll identified in Table 2 and 3 are listed below:

Kl K2 K3 K4

K5

Tl

T2

T3

T4 T5

T6

T7 T8 T9

TlO

Tll T12

T13

T14 T15

T16

T17

Tl, T2, T5, T7, T8, T9*, TlO, T15, T16, T20, T28. T2, T3, T5, T7, T8, T9*, T18, T20, T30. T8, T29. T2, T3, T5, T8, T9*, TlO, Tll, T12, T13, T14, T16, T17, T18, T19, T20, T21, T24, T25, T27, T28. T9*, TIO, T16.

Denbigh, K. G.: Tmns. Faraday Sot. 40, 352-373 (1944). Bilous, 0. and Amundson, N. R.: Chem. Engng Sci. 5, 81-92 (1956). Bilous, 0. and Amundson, N. R.: Chem. Engng Sci. 5, 115-126 (1956). Denbigh, K. G.: Chem. Engng Sci. 8, 125-132 (1958). Horn, F. and Kuechler, L.: Chem. Ing. Tech. 31, l-11 (1959). Horn, F. and Troltenier, U.: Chem. Ig. Tech. 32, 383-393 (1960). Horn, F.: Chem. Engng Sci. 14, 77-89 (1961). Horn, F.: Z. J Elektrochemie 65, 209-222 (1961). Horn, F. and Troltenier, U.: Chem. Ig. Tech. 33, 413416 (1961). Siebenthal, C. D. and Aris, R.: Chem. Engng Sci. 19, 747-761 (1964). Lee, E. S.: A.Z.Ch.E.J. 10, 309-315 (1964). Coward, I. and Jackson, R.: Chem. Engng Sci. 20, 91 l-920 (1965). Douglas, J. M. and Denn, M. M.: Ind. Eng. Chem. 57, No. 11, 18-31 (1965). Denn, M. M. and Aris, R.: IEC Fund. 4,7-16 (1965). Ahn, Y. K., F&m, L. T. and Erickson, L. E.: A.I.Ch.E.J. 12, 534-540 (1966). Dyson, D. C., Horn, F. J. M., Jackson, R. and Schlesinger, C. B.: Can. J. Chem. Engng 45, 310-318 (1967). Fine, F. A. and Bankoff, S. G.: IEC Fund. 6,288-293 (1967).

K6 K7 KS K9 KlO Kll

T18

T19

T20

T21 T22

T23

T24

T25

T26

T27

T28

T29 T30 T31

T32

T33

TlO. T8. T22, T32. T4, T6*, T8. T23, T31, T33’. T26.

Millman, M. C. and Katz, S.: ZEC Proc. Des. Dev. 6, 447-451 (1967). Rothenberger, B. F. and Lapidus, L.: A.I.Ch.E.J. 13, 982-988 (1967). Evangelista, J. J. and Katz, S.: Znd. Engng Chem. 60, No. 3, 24-33 (1968). Lee, E. S.: A.I.Ch.E.J. 14, 977-979 (1968). Binns, D. T., Kantyka, T. A. and Welland, R. C.: Trans. Inst. Chem. Engng 47, 53-58 (1969). Newberaer, M. R. and Kadlec, R. H.: A.I.Ch.E.J. 17, 1381-1387 (1971). Umeda. T.. Shindo. A. and Ichikawa. A.: IEC Proc. Des. Dev. il, 102-iO7 (1972). Cormack, D. E. and Luus, R.: Can. J. Chem. Eng 50, 39&398 (I 972). Jaspan, R. K., Coull, J. and Andersen, T.S.: Ado. Ch;m. Ser. 109, 161-165 (1972). Crescitelli. S. and Nicoletti. B.: Chem. Enana Sci. 28. 463471 (i973).

I I

Marroquin, G. and Luyben, W. L.: Chem. Engng Sci. 28, 993-1003 (1973). Skrzypek, J.: Inzyn. Chemiczna III, 3, 639-653 (1973). Skrzypek, J.: Int. Chem. Erzgrzg 14, 214-222 (1974). Burghardt, A. and Skrzypek, J.: Chem. Engng Sci. 29, 1311-1315 (1974). Gangiah, K. and Husain, A.: Can. J. Chem. Engng 52, 654-660 (1974). Hatipoglu, T.: Techn.-Chem. Labor, E.T.H. Zurich to be published (1982).

References to feed rate nrofiles Profie in aas sutmlv rate Fl

F2

F3

F4

Jackson, R. and Senior, M. G.: Chem. Engng Sci. 23, F5 Liebman, -D.: ETH Zurich, Dissertation No. 5657 971-980 (1968). (1975). Jackson, R., Obando, R. and Senior, M. G.: Chem. Engng Sci. 26, 853-865 (1971). Jackson, R., Obando, R. and Senior, M. G.: Adu. Chem. Series 109, 156159 (1972). Young, M. J.: Processing 27-33, March 1980.

*These references contain tabulated results for a range of values of the relevant parameters.

to 100 stages. (Dll) and (D12) refer to a packed column. The effect of hold-up is generally not treated although it is acknowledged to be important. (D2) attempts to account for it, (D6) discusses a case with hold-up in the condenser amounting to 10% of the reboiler and in (D7) hold-up equal to 5% of the boiler charge is distributed over the plates. (DIO) treats a three plate column with the hold-up on each plate equal to 5% of the boiler charge. (D14) claims to treat the effect of hold-up.

In (D8) and (D9) the effect is considered of a recycled waste cut taken between the desired products and returned to the still with the new charge for the next batch. (This procedure is also referred to as an industrial practice in (D2).) The possibility of simultaneous reaction is considered in (D14). The optimal policy is generally one of increasing reflux as the

batch progressed. In (D5) it is shown that for easy separations it may be optimal to have an initial period with zero reflux. The presence of hold-up can affect the reflux policy at the beginning and end of the batch. In (DlO), where hold-up is considered, an initial period of total reflux is called for, presumably to bring the liquid on the trays rapidly up to the equilibrium composition. At the end of the batch the reflux rises sharply and then falls to zero, presumably to remove the lighter component from the tray hold-up before this is dumped in the boiler as part of the bottom product.

In most cases the objective function considered is to minimize the time required to obtain a given amount of product of given composition. The advantage of the optimal profile over the policy of constant reflux rate was up to about 13% and the advantage

146 D. W. T. BI’PIN

Table 5. Optimal profiles applied to polymerization

Systems Objective studied function Constraints

Optimizing variable

Form of profile

Reported advantage over iso- Experim. therm. oper. results

1. Styrene P4, P5, P6, P8, P9, PlO, Pll

2. Methyl methacrylate P4, PS, PI2

3. Vinyl acetate PI2

4. Nylon 6 P3, PI

5. Copoly- merisation Pl, P13

1. Min. time P3, P4, P6, P8, P9, PlO, Pll, P12

2. Min. unreact- ed monomer P3, P7

3. Polymer characteristics PI, P2, PS, PI3

1. Conversion, no avge. chain length P4, PS,P9,PlO, PI l,P12.

2. Weight avge. in addition to constraints under 1) P6

3. Joint target region of required properties P8

4. Other properties P3. P7

1. Temperature Pl, P3, P4, PS, P6, P7, P9, PlO, P12,P13

2. Initiator addition rate P2, P4, Pll

3. Temp and initiator addition simultaneously P8, PI 1

generally increasing, but see (a)*, (b)*

generally to maintain constant initiator concentration

1. Time saving PS, P9, -10% PIO, PI 1, P4 P13

2. Time saving _ 20% or better P9, PlO, Pll, P12

3. Substantial . improvement in polymer properties Pl,PS,P13

(a)* PI, PS, P13: polymer properties achieved by adjusting temperature to maintain desired concentrations at every instant. Profile may rise or fall.

(b)* P3, P7: rising profile, possibly to upper limit followed by fall.

References to optimal profiles in uolvmerization PI

P2

P3

P4

P5

P6

PI

Ray, W. I% and* Gall, C. ‘E.,. Macromolecules 2, 425-428 (1969). Bandermann, F., Angew. Marko. Chem. 18, 137-157 (1971). Naudin Ten Gate, W. F. H., Proc. Intern. Cong. Use of Electronic Comp. in Chem. Engng, Paris, April 1913. Sacks, M. E., Lee, S. I. and Biesenberger, J. A., Chem. Engng Sci. 27, 2281-2289 (1972). Sacks, M. E., Lee, S. E. and Biesenberger, J. A., Chem. Engng Sci. 28, 241-257 (1973). Kwan, Y. D. and Evans, L. B., A.Z.Ch.E.J. 21, 1158-1164 (1975). Mochizuki, S. and Itoh, N., Chem. Engng Sci. 33, 1401-1403 (1978).

over a policy in which the reflux rate was constantly adjusted to maintain the top product composition constant was up to about 6% for binary problems without hold-up. In (D13) results are reported of 24 examples of binary separations with no hold up. In this case the objective function used is the profit per unit time. This is obtained from the value of the product of given specification obtained during the operating time minus the cost of the material charged to the batch, all divided by the total time per batch. (The latter is the operating time plus the changeover time between batches.) From this quantity is sub- tracted the operating cost per unit time which is

P8 Clough, D. E., Masterson, P. M. and Payne, S. R., Summer Computer Simulation Conference, pp. 279-285, Newport Beach, California, 1978.

P9 Chen, S. A. and Jeng, W. F., Chem. Engng Sci. 33, 735-743 (1978).

PI0 Chen, S. A. and Lin, K. F., Chem. Engng Sci. 35, 2325-2334 (1980).

PI 1 Chen, S. A. and Huang, N. W., Chem. Engng Sci. 36, 1295-1305 (1981).

PI2 Rao, B. B. and Mhaskar, R. D., Polymer 22, 1593-1597 (1981).

P13 Tirrell, M. and Gromley, K., Chem. Engng Sci. 36, 367-375 (1981).

regarded as being constant independently of whether the batch is operating or being changed (compare the last objective function of Table 1). When the optimum reflux profile and operating time are chosen to maximize the above profit criterion the amount of distillate obtained is up to 45% better than with either the constant reflux rate or the constant over- head composition policies. However, the improvement in profit resulting from the optimal profile is frequently as much as 20% and sometimes substantially higher. General criteria are derived from which predictions can be made of the benefit of using an optimal profile without the necessity of carrying out

Simulation of single- and multiproduct batch chemical plants for optimal design and operation

Table 6. Biochemical reactions with deactivating enzymes

147

Ref. Main reaction

Enzyme decay

Optimization Objective variables function

Benefit of optimal profile

C5 Hydrolysis of penicillin (P) with enzyme amidase (E)

C6 Isomerisation of Dglucose (G) to Dfructose (F) with glucose isomerase

(E)

C7 As C6 with additional enzyme deacti- vation by proteases (P)

C8 Hydrolysis of sucrose with invertase

Michalis EAE’ Menten P-A

21 13 p’ A’ decay reactions 2,3 and 4 are first order in the decaying component G+F E-E’ Both reactions decay 1st order in reactant and enyzme

first order

G+E$xeF E+E; +E E+P+E”+P

P+P’ 1 st order in reactants

Michaelis Menten S-+,9’

E+E’ decay 1st order

Temperature Fractional yield

PH of product minus cost of fraction of enzyme used up

Temperature Maximize final enzyme activity for given feed conversion and time

Temperature For fixed final final time conversion of F initial amount of enzyme

1. Minimum initial amount of enzyme

2. Maximum fraction enzyme unused

3. Minimum amount enzyme used

4. Minimum annual cost for fixed product requirement

Temperature Maximize final conversion for given final enzyme conversion and time.

-5%

-10%

Best conditions very dependent on objective function

2. up to _ 30%

3. up to -13%

4. No direct comparison

Analytical form given (some experimental results)

References on deactivating catalysts and enzymes

Catalysts Enzymes Cl Szepe, S. and Levenspiel, 0, Chem. Engng Sci. 23,

881-894 (1968). C5 Ho, L. Y. and Humphrey, A. F., Eiotech Bioeng 12,

291-311 (1970). C2 Pommersheim. J. M. and Chandra. K., A.I.Ch.E.J. 21, C6 Haas, W. R., Tavlarides, L. L. and ynek, W. J.,

A.I.Ch.E.J. 20, 707-712 (1974). C7 Lim, H. C. and Emigholz, K. F., Enzyme Engineering

(Edited by E. K. Pyle and H. H. Weetall), Vol. 3, pp. 101-l 14. Plenum Press (1978).

C8 Ooshima, H. and Harano, Y., J. Chem. Engng Japan 13, 484-489 (1980).

1029-1032 (1975). C3 Sadana. A.. A.I.Ch.E.J. 25. 535-538 (1979). C4 Pommersheim, J. M., Tavlarides, L. L: and’Mukkavilli,

S., A.I.Ch.E.J. 26, 327-330 (1980).

the full calculation. In (D14) reductions in cycle time of as much as 4fk50°A due to the use of optimal profiles are reported.

When hold up is considered the benefits of the optimal profile appear to be somewhat greater than in the no-hold-up case (DlO) and the use of a recycled waste cut also gives a significant improvement in performance (DS).

In a time of increasing energy costs there is an incentive for further study of optimal strategies for batch distillation. A more systematic study of the effects of hold-up is needed and of the effects of recycled waste cuts or other operational

modifications. The study of systems with non-ideal equilibrium has not been attempted.

BATCH CRYSTALLIZATION

In crystallizers it is desired to grow crystals in minimum time by either cooling or evaporation. Attempts to cool or evaporate too rapidly result in the formation of large numbers of very small crystals, thus it is important to adhere to a constraint which limits the nucleation rate or the degree of supersaturation. In (Xl) a temperature profile is determined to give a constant nucleation rate, without explicit considera- tion of optimal control, and is compared theoretically

148 D. W. T. RIPPIN

Table 7. (a) Optimum profiles in batch distillation

D 1 Converse, A. 0. and Gross, G. D., IEC Fund. 2, 217-221 (1963). D2 Converse, A. 0. and Huber, C. I., IEC Fund. 4, 475477 (1965). D3 Price, P. C., I. Chem. Engrs Symp. Ser. 23, 96104 (1967). D4 Coward, I., Chem. EngngSci..22: 503-516.(1967). . ’ D5 Coward, I., Chem. Engng Sci. 22, 1881-1884 (1967). D6 Robinson, E. R., Chem.Engng &i. 24, 1661-i668 (1969). D7 Robinson, E. R., Chem. Engng Sci. 25, 921-928 (1970). D8 Mayur, D. W., May, R. A. and Jackson, R., Chem. Engng J. 1, 15-21 (1970). D9 Robinson, E. R., Chem. Engng J. 2, 135-136 (1971). DIO Mayur, D. W. and Jackson, R., Chem. Engng .I. 2, 150-163 (1971). Dll Keith, F. M. and Brunet, J., Can. J. Chem. Engng 49, 291-294 (1971). D12 Ramakrishnan, R., Can. J. Chem. Engng 49, 711 (1971). D13 Kerkhof, L. H. J. and Vissers, H. J. M., Chem. Engng Sci. 33, 961-970 (1978). D14 Egly, W., Ruby, V. and Seid, B., Comput. Chem. Engng 3, 169-174 (1979). D15 Murty, B. S. N., Gangiah, K. and Husain, A., Chem. Engng J. 19, 201-208 (1980).

Xl Mullin, J. W. and Nyvlt, J., Chem. Engng Sci. 26, 369-377 (1971). x2 Rees, N. W., Frew, 1. A., Battersham, R. J., Thornton, G. J., Chem. Engng J. 3, 301-303 (1972). x3 Frew, J. A., IEC Proc. Des. Dev. 12, 46&467 (1973). x4 Jones, A. G., Chem. Engng Sci. 29, 1075-1087 (1974). x5 Ajinkya, M. B. and Ray, W. H., Chem. Engng Commun. 1, 181-186 (1974). X6 Morari, M., Chem. Engng Common. 4, 167-171 (1980).

(b) Optimum batch crystallization

and experimentally withthe case of uncontrolled cool- Ing. The mean crystal size obtained in a given time could be increased by -40%.

A more complex example from the sugar industry is considered in (X2) and (X3). In an evaporative batch crystallizer two controls are used, feed rate and net evaporation and the system is subject to five constraints. Pontryagin’s principle is used to determine the optimal policy. This generally follows rather closely the constraint boundary limiting the rate of nucleation, thus lending support to the policy used in (Xl). A set of qualitative operating principles is derived, some of which are claimed to be novel in the sugar industry. No quantitative comparison of the optimal policy with other practices is given.

In (X4) Pontryagin’s principle is used to determine a temperature profile for cooling to produce maximum crystal size. A profile was found which was superior to the constant nucleation rate of (Xl) and the benefits of the optimum profile were even greater in an experimental test.

In (X5) Pontryagin’s principle is applied to the case studied in (X 1). The results are independent of those in (X4) and use a simpler model. The optimum profile derived is close to the constant nucleation rate of (Xl).

In (X6) it is shown that, for a simplified model, the optimum temperature policy maximizes the crystal growth rate at any instant subject to the constraints on temperature and the permitted degree of supersaturation.

OPTIMIZATION OF THE OPERATION OF A SEQUENCE OF EQUIPMENT ITEMS

A number of equipment items are to be arranged in series for the batch production of a single product. It is assumed that for each item the changeover time from one batch to the next is given and that the attainable performance for any input conditions to that item can be made available as a function of the

allowed operating time, for example as the curve re- presenting the boundary of the attainable region discussed earlier. This bounding performance curve can be regarded as monotonic and nondecreasing since if the performance were to deteriorate with further extension of the operating time the process could always be stopped at that point. The location of the boundary can be established by optimization procedures as already discussed or from purely empirical considerations of what is feasible for this equipment. The manner in which the curve is determined is of no significance for the subsequent discussion.

For each of the items a best operating time could be found in terms of an appropriate objective function, as already discussed. Suppose for simplicity that, for each item, the performance is expressed as the fraction of feed material converted to material suitab!e for the next stage and that an appropriate objective function is the amount of material converted per unit time. For each individual item the best operating time is located by the tangent to the performance curve from the time origin when the start of the operating curve is dis- placed from the origin along the time axis by a distance corresponding to the cleaning time for that item (Fig. 2). If the performance curve in terms of the new time origin isf(r), the objective function

,=f(t)

is maximized when

dP r?‘(t) -f(r) = O dt= t2

giving

f’(t) =-c&l t

Simulrition of single- and multiproduct batch chemical plants for optimal design and operation 149

e e c t, = cycle time. t,

(a) @) Fig. 2. Determination of joint cycle time for items in sequence. (a) With one item for each task. (b) With

two parallel items at stage g.

the tangency condition. The time t between successive batches, or the cycle time, made up of the sum of the changeover time and the operating time, resulting from this calculation will generally be different for each equipment item. For the moment we wish to consider a situation in which a common cycle time is adopted for all equipment items so that the batch leaving one item can be fed directly to the next. The selection of a common cycle time which takes account of the various performance curves of the different items can be made by an extension of the above procedure. We consider the case of two items with performance curves f and g but the treatment is readily extended to any number. With the simple assumption made above the overall yield is the product of the yields of the individual items. Thus, in this case

pA t

dP dt=

V’g +dfP -fg = ,, t=

giving

The solution can be demonstrated either by con- structing a tangent to the composite curvefg (Fig. 2) or by plotting the condition derived from the differentiation. It is apparent from this analysis that the best common cycle time for a sequence of units will always be longer than the longest of the best individual cycle times. Thus a cycle time optimization for an individual unit should not be undertaken without considering its potential effects on the performance of other units which will operate concurrently. The improvement in yield obtained by a small extension of operating time may not be justified for any single equipment item but may be significant when accumulated over several items.

If there are wide disparities in the time scales of the performance curves for the different items it may be appropriate to consider providing two (or more) of the slowly working items to which batches can be fed alternately from the more rapidly responding items. The cycle time of the slow items will then be twice (or in the general case n times) that of the more rapid ones. In the simple example introduced above the best cycle time can be found by multiplying the performance curves with appropriately adjusted time scales

150 D. W. T. RIPPIN

either construct tangent from origin to this curve or satisfy

dP Tt=

W’(tM2t) + 2gVtIf(t)) -f(tM2t) = o

t2

f’(t) I W(2t) 1 f(t) go=t-

The extension to more items in series, each with any specified multiplicity in parallel is obvious. The justification for including additional items in parallel must be made by a more detailed economic study in which the additional cost of these items is balanced against the improvement in performance achieved.

Up to this point the constraint has been imposed that all the items in the sequence must have either a common cycle time or a cycle time that is an integer multiple of the reference cycle time which is the inter- val between the production of successive batches. If this constraint is removed a cycle time can be chosen independently for each equipment item, but the capacities of the items must be such that each item can handle the average amount of material coming forward per unit time from the preceding item. For items operating in sequence a longer cycle time will imply a larger capacity and vice versa For units with arbitrary cycle times operating in sequence, intermediate storage will need to be provided between the items to accommodate the different cycles of delivery and re- moval of material and quantitative estimates have been made of this storage requirement[4]. It is more difficult to forecast if additional labour costs will be incurred by allowing equipment items to be operated with arbitrary rather than common cycle times. If optimal cycle times are to be chosen for all equipment items the objective function used must take account of the corresponding variations in equipment capacity and also of the cost of intermediate storage between items if this is likely to be significant.

OPTIMIZATION OF THE DESIGN OF A SINGLE PRODUCT PLANT

A process is characterized by the set of tasks which must be carried out to transform raw materials into the desired product. Each task is characterized by the requirement of processing capacity and the amount of a specified reference material in the feed to this task per unit of reference material produced by the task and also by the time required to complete the task. In the simplest case these requirements may be fixed by some prior specification, in other cases they may be dependent on design and operating variables of the task. A hierarchical structure can be developed for the models used to describe batch processing tasks.

In order to design a plant the tasks comprising the process must be allocated to appropriate equipment items for which the size and operating conditions must then be determined.

The simplest allocation is to make a one-to-one assignment of each task to an appropriate equipment. This will generally be satisfactory if the time required for acceptable performance in the selected equipment is roughly the same for all tasks. A common’cycle time for the sequence of items can then be determined as already described. However, if there are significant differences between the time requirements of the

different tasks some modification of the allocation procedure will be called for.

Possible revised procedures may include: l merging of tasks; l parallel use of equipment items; l interrupting the chain of tasks; l breaking the chain of tasks. If a group of consecutive tasks all have time require-

ments that are appreciably shorter than the typical requirements of other tasks in the sequence it may be possible to merge the tasks by carrying them out in sequence in the same equipment item which will then have a cycle time comparable with the requirements of the other tasks. For example, in some processes it may be appropriate to carry out preheating, reaction and cooling consecutively in a single vessel if these operations have to be co-ordinated with other long tasks, but they may be carried out in separate vessels if their individual time requirements are comparable to those of other tasks in the sequence.

If the requirement of one task is significantly greater than that of the others then the cycle time of the chain of tasks can be reduced by installing two or more items in parallel for the long task. For this task successive batches are fed alternately to the parallel items. The choice of cycle time for such a system has already been discussed.

With the above two modifications of the allocation procedure, as in the original one-to-one allocation, a batch proceeds directly from one item to the next without significant waiting and the integrity of the batch is maintained, that is to say it is not mixed with material from any earlier or later batch during the processing sequence.

When batch integrity is not essential and the product of an item is sufficiently stable that it can be stored for a time equal to or greater than the batch cycle time before being transferred to the next item for further processing then the chain of tasks can be interrupted or divided into segments each with its own characteristic cycle time. A simple example of the interruption of the chain may be considered as an alternative to the installation of parallel equipment items for a task. For example two successive batches may be accumulated from the earlier tasks in the chain and processed to- gether in a correspondingly larger equipment item for twice the cycle time of the other items. The product of this operation is then used up in two successive batches. Some equipment cost is saved by this operation due to economy of scale since two equipment items in parallel are replaced by one of twice the capacity, but some additional inventory cost is incurred for the system with interrupted processing and the batch integrity is lost.

In a more general case when a chain of tasks is divided into segments with no special constraints on the ratios of the cycle times a sufficient inventory capacity must be available to provide buffering between the succeeding segments of the chain, as discussed above. The question of where and into how many segments the chain of tasks should be divided can be examined by an optimization procedure taking into account the equipment cost and the cost of inventory for the additional buffering capacity between the segments.

A chain of tasks is regarded as being broken rather than interrupted when the sequence of process tasks is


not continued in further items in the same plant. The intermediate product available at this point must, of course, be stable for an appropriate period of time. It could be processed further in another plant, or when sufficient supplies of the intermediate product had been accumulated this production could be stopped and the same plant items used, possibly with rear- rangement of their interconnections, for the further process tasks which must be carried out on the intermediate product. A plant in which two or more products are produced in sequential campaigns, only one product being produced at any time, is described as a multiproduct plant and will be discussed further below.

It has been implied in the previous discussion that the designer is free to choose the cycle times for the various batch items on the basis of information provided in a performance-time curve. However, there are many cases in which the cycle time for an item is defined and no variations are permitted from the conditions established at the laboratory or pilot scale. (The performance time curve for an item is effectively regarded as a step function.) In this case the cycle time for a chain of tasks will simply be the longest cycle time among them. Only a limited set of discrete alternatives will need to be considered for the use of parallel items or for the interruption of the chain of tasks.

THE MULTIPRODUCT PLANT

When several products are to be produced in the same plant the simplest arrangement is for the products to be produced in sequence so that at any time only one product is being manufactured in the plant. Such an arrangement will be referred to as a multiproduct plant. The tasks for one product will be allocated to a set of equipment items. The tasks for each of the second and subsequent products will be allocated within the same equipment set, supple- mented as necessary by additional items. The equipment must then be sized so that all the product requirements can be met.

As already discussed, a major problem in the design of single-product plants is the resolution of incompatibilities in the cycle time requirement of the different tasks. In a multiproduct plant in which the tasks for all the products have been allocated to appropriate equipment items, difficulties may also arise because of the different capacity requirements placed upon the same equipment item by the various products to be produced in it at different times.

To facilitate subsequent discussions the capacity requirements per unit of product of that task for product i which will be carried out in equipment item j is designated as the size factor S,. The size factor will usually be expressed in units of volume but other measures of capacity may, on occasion, be appropriate. Since the size factor is the capacity requirement per unit of final product all intermediate conversion factors must have been taken into account by mass balance or other appropriate calculations for the succeeding items.

In most of the subsequent discussion it will be assumed that the size factors as well as the item cycle times are constant. This assumption can usually be relaxed at the cost of some additional iterative calculation in the algorithms to be discussed.

The simplest design problem for a multi-product plant is to determine sizes of equipment items to meet the requirements of a set of products in a specified time when the allocation of all product tasks to equipment items is also specified in advance. Since the products are not produced simultaneously the same equipment item may be allocated to a task for each of the products.

If the cycle time for equipment itemj when used for product i is qj,, then the cycle time TLi for product i is the longest of these for the equipment items in use:

T, = max TV j

The size Bi of the batch which can be made of product i is limited by that equipment item for which the capacity 6. is smallest relative to the corresponding size factor S,:

B,=min 2 . i 0 r/

The time needed to produce an amount Qi of product i is

QJ’, Bi ’

thus to meet the required production of all products in a time H the constraint must be satisfied:

For a very simple problem with two products A and B and two equipment items the feasible region can be illustrated in the V,-V2 plane (Fig. 3). Along the line labelled “Product A” the ratio of the capacities of the two equipment items is equal to the ratio of their size factors for product A so that both items are fully utilized when product A is produced. Above this line V, is larger than necessary, hence this item is not fully utilized and below this line V2 is correspondingly not fully utilized when A is produced. The line labelled “Product B” plays a similar role for the second product. Corresponding contours of batch size for the two products are parallel to the coordinate axes and turn through 90” when they cross the corresponding product line. The equality form of the time constraint is satisfied at one point P on the “Product A” line and vertically above that point. Since increase of V, only serves to decrease the utilization of that item and does not increase the feasible batch sizes, this region is of no practical interest. A similar point Q can be identified on the product B line. In the region between the two product lines the batch size of product A is proportional to I’, and the batch size of product B is proportional to V,. Thus, the equality form of the time constraint is a section of a hyperbola joining the two points P and Q previously defined and choices of equipment size satisfying the equality form of the time constraint will lie along this curve. Feasible solutions on the bounding curve can be generated, for example, by assuming the capacities will be proportional to the size factors of one of the products when point P or Q will be identified. An

152 D. W. T. RIPPIN

not fully utilized

utiked for product A utilized for product B

(a)

Roduct

__.-.-.A . V, not fully utilized

for both products

V, - - - - -contours of ottoinable botch we for oroduct A

_ _ _ _ _ _ _ _ contours of attainable botch size for product B

Fig. 3. Multi-product equipment sizing. (a) Equipment utilization and batch sizes. (b) Feasible region in original variables. (c) Feasible region in transformed variables.

alternative procedure proposed by Flatz[5] will also lead to one of the extreme points P or Q. He postulates that one of the equipment items will be fully utilized for all products and its minimum necessary volume is determined. In our example if V, is chosen to be fully utilized this locates the vertical portion of the time constraint passing through P. When V2 is fixed the minimum size needed for P’, will be located at P. An alternative approach that has been used by Sparrow et a1.[6,7] to provide preliminary solutions in the MULTIBATCH program for sizing multiproduct plants is to postulate a hypothetical product with capacity requirements intermediate between those of the actual products. This was done by generating a size factor for each equipment item as a weighted sum of the size factors for the products actually using that item, the weighting factors being proportional to the production requirements of the corresponding products. The product line for the hypothetical product will lie between the extremes of those for the actual products and its intersection with the hyperbolic segment PQ will provide a solution which is a compromise between the different requirements.

An optimal solution can be selected from the feasible solutions if an appropriate objective function can be defined. The commonest form used has been the total equipment cost in the form

minimize 1 aj( Vj)Pj. I i

Contours of this objective function will locate a unique maximum on the boundary PQ.

In a larger number of dimensions the optimization can be carried out by a search routine capable of handling non-linear constraints. However, a variable transformation can be made which reduces the problem to one with only linear constraints, which should be more readily soluble, although the objective function is still non-linear. The reciprocals of the volumes

4j=$,

I

are chosen as the optimizing variables with the reciprocals of the batch sizes

s,=’ 4

as intermediate variables. The calculation of the product cycle times is un-

changed. The definition of the batch size can be rewritten as

1

0

sy &= “i”” y.

-&ii = max S&,.

This can be rewritten as a set of inequality conditions

hi 2 s&j for all i and j.

In the new variables the time constraint becomes

i

Thus, all the constraints are linear and the objective


function is

minimize 1 a,(4j)j>- so. I

The corresponding feasible region in the transformed variables is shown in Fig. 3.

Batch equipment items are often available in discrete sizes. The results of the continuous optimization referred to or any other feasible solution can be used as a starting point for a search procedure over the available discrete sizes. The possibility of using additional items in parallel to reduce the cycle time of an item can also be incorporated in this discrete search. The development of a branch and bound search procedure for this problem has been reported by Sparrow et al.[7].

Earlier work on the sizing of batch equipment items and associated semi-continuous items such as pumps for a single product plant was reported by Ketner[8] and Loonkar dc Robinson[9] and a geometric programming approach was used by Hellinckx & Rijckaert [ IO]. Subsequently Robinson & Loonkar [ 1 I] considered the multiproduct design problem.

More recently Grossmann & Sargent [12] have applied a general non-linear programming package to the sizing problem and used a special form of constraint to drive the relevant size variables to discrete values. Knopf, et al.[l3] and Wiede et aI.[14] have transformed the problem by geometric programming considerations and used generalized reduced gradient codes to solve the resulting continuous optimization problem. They explore various strategies for search- ing the tree of discrete possibilities.

EXTENSIONS TO THE MULTIPRODUCT PLANT PROBLEM

Production of products in parallel The procedure already described can also be used

with small modifications, when products are produced in parallel during some of the campaigns. Equipment items are once again assigned to all tasks for all products. If sufficient equipment items are available it may be found possible to produce two or more products simultaneously without any equipment items being used in common. If no product appears in more than one campaign, although some campaigns contain more than one product, the problem constraints can still be linearised. The time required to produce the specified amount of each product can be calculated and when two or more products are produced simultaneously the largest of these time requirements is the value to be used in the overall time constraint. Since the time requirement is linear in the reciprocal of the batch size the transformed constraint is linear as before.

If a product A is produced in more than one campaign the time requirement imposed on a campaign by that product is no longer explicitly defined. Instead it is necessary that the production capability summed over the campaigns in which that product participates is at least sufficient to meet its requirement QA. If the times assigned to campaigns i andj in which product A participates are Bi and 0, the production constraint takes the form:

The linearity of the constraints in the reciprocals of the batch sizes can no longer be preserved, so that a more general optimization routine would be required to determine appropriate equipment sizes.

The above discussion assumes that it is specified which products are to be produced simultaneously. It is also possible to leave this choice open to be determined by a design procedure. This enormously increases the magnitude of the problem since, for example, for an N product plant there are 2N - 1 ways in which products could be grouped for parallel production. Suhami & Mah[ 151 have treated this problem with the restriction that only a single item or group of items of each equipment type is available. This means that only those products can be produced simultaneously which have no equipment requirements in common and the number of alternatives which must be considered is enormously reduced. However, this number may still be large enough to make complete enumeration impractical. Suhami t Mah obtained a solution by randomly generating a limited number of product assignment patterns satisfying certain constraints and choosing the best of these.

The multi-plant problem An alternative to producing parallel groups of pro-

ducts in sequence is to divide the products into groups and produce individual products or groups of products each in its own independent plant. It may appear that the construction of several plants rather than one is bound to be more expensive in equipment, due to the loss in economy of scale, as several small items may have to be purchased for the individual plants rather than one large item. However, against this may be set the fact that products making highly incompatible demands on the capacity requirements of the equipment items no longer have to be produced in the same plant.

A more important factor which has not yet been considered is the differing inventory requirements of a set of plants compared with that of a single plant in which all the products are produced in sequence. The annual stock value of products produced from a batch plant can easily equal the cost of the plant itself and may sometimes be many times greater for example, where pharmaceutical or other high speciality products are concerned. In the single plant, sufficient material must be in stock at the end of a campaign of one product to meet the demand for that product during the production periods of all the other products until it is time for the next campaign of the original product. The total cost of holding this active stock depends upon the campaign lengths chosen but, in the pharmaceutical industry, for example, where up to a week’s production time may be lost at each product changeover in addition to losses of material, the campaign lengths cannot reasonably be less than a few weeks for each product. In contrast to this a single product plant needs no active stock. (Buffer stock to cover fluctuations or emergencies is a separate requirement for both types of plant.) Thus, the loss of economy of scale by manufacturing in several plants can easily be outweighed by savings in inventory cost. The independent plants producing the separate groups of products could be organised as strictly multiproduct plants with the products being produced sequentially one at a time or in any other way deemed appropriate.

CACE 7/3-B

154 D. W. T. RIPPIN

With N products the number of single or multiproduct plants which must be considered is again 2N - 1. If a rapid evaluation procedure is available for plant sizing, such as that of Sparrow et al. [7] it is feasible to cost such a set of plants for up to 9 or 10 products and with suitable assumptions about campaign lengths the associated inventory costs can also be included in this evaluation. If it is assumed that each product is manufactured in only one of the plants, the selection of which of the plants should be constructed to meet the production requirement at minimum total cost can be solved as a set partitioning problem[l6]. Such problems are familiar in air-crew scheduling and other allocation situations [ 171.

THE GENERAL DESIGN PROBLEM

In the methods discussed up to this point it has been assumed that all product tasks are assigned to equipment items, the only flexibility being provided by the opportunity to select two or more of a particular equipment item to operate in parallel in order to reduce the limiting cycle time of the sequence of tasks.

However, more choices than this are frequently open to the designer. He may allocate his tasks not to individual equipment items but to classes of equipment. Each class could contain several equipment items, not necessarily all of the same size. At any one time members of one class might be in use on different tasks of several different products. To some tasks a single item might be assigned, others might have two or more items operating in parallel and out-of-phase to reduce a cycle time and yet others might be operating in parallel and in phase to relieve a capacity bottle neck. At other times the distribution of the items in the class might be rearranged into quite different groupings. Furthermore, instead of allocating each product task to a specified class of equipment the designer may have the freedom to suggest alternative equipment classes in which a particular task might be carried out. It may not necessarily always be best to call for equipment of the cheapest class if more versatile but more expensive equipment will be on hand because of the requirements of another product.

A further degree of flexibility is provided by the possibility, already referred to briefly, of determining at what points long chains of tasks should be broken into subchains, resulting in intermediate products. These may then be produced sequentially or simultaneously.

The general design problem for a significant number of products is far too large to be treated compre- hensively at present. A partly heuristic approach has been proposed in which the problem is treated in a stage-wise fashion[l8, 191. First the products are assigned to groups with the intention of maximizing the use of items in common within each group. In the second stage the structure of each group is determined to minimize the requirements for numbers of equipment items and their interconnections. Only in the third and final stage are sizes determined for the equipment chosen for the various duties by the earlier parts of the procedure. The sizing method at this stage is similar to those discussed in earlier sections. The method is still under development but promising results have been obtained for quite large problems.

THE ORGANIZATION OF PRODUCTION IN

AN EXISTING PLANT

Single equipment items may again be treated as discussed earlier. However, the problems of making the best use of existing equipment in a multi-item plant are somewhat different from those of choosing the best equipment for a specified duty. Problem areas will be identified but not reviewed in detail.

The production capability of an existing plant can be assessed under various degrees of constraint, for example:

(1) A simulation problem. Determine the throughput of a plant for which all product tasks are assigned to equipment items in advance and the sequence in which batches of products are to be started is also given. It may also be necessary to specify rules for resolving conflicting demands for the same equipment item or other resource. Simulation studies of this nature can be made without difficulty. Reports of their use to answer “what if’ questions both with and without the occurrence of random fluctuations in the system or its environment have been appearing regu- larly in the literature for more than 20 yr[20,21].

(2) A sequencing problem. Determine the sequence in which a set of products should be produced to satisfy a specified sequence of demands in time. In a pure sequencing problem only one route through the plant will be specified for each product. In many cases it may be necessary only to find a feasible solution to the problem. In other cases a solution may be sought which is optimal with respect to some objective function, including factors such as time to fulfil all requirements, penalties for late delivery and con- versely inventory charges for products produced before they are required and changeover losses.

The simple problem of determining the sequence in which products should be produced to meet specified requirements in minimum time where no intermediate storage is available is often relevant to batch chemical production and has been the subject of renewed discussion recently[22,23].

More complex sequencing problems can arise when, for at least some products, more than one manufacturing route through the plant is available. Additional constraints may be imposed, for example, when waiting between certain tasks is not permitted due to product instability, simultaneous operation of certain tasks is restricted by the limited availability of a common utility such as steam, electricity or labour, or the priority sequence in a product chain needs to be respected by ensuring that intermediate products are manufactured in time to be available when required by a batch of a following product. A programme has been reported[24] for obtaining batch production sequences for systems of modest size, subject to such constraints by complete enumeration. An implicit enumeration with appropriate bounding procedures for problems with special structure has also been reported [25].

Recent reviews on scheduling may be cited in the operations research literature[26,27] and with particular reference to chemical plants [28].

(3) Task allocation and product sequencing. Deter- mine the routes through the plant as well as the sequence in which the products should be manufactured. Product routes are constructed on the basis of the availability of appropriate equipment and the

necessary interconnections. Several products may be produced simultaneously and the route followed by batches of the same product may differ from time to time. A plant of this type is generally described as a multi-purpose plant and has obvious affinities to the job shop found in the mechanical industries.

A strategy has been presented[29, 301 for identifying advantageous campaign structures for a given multi-purpose plant and assigning time to these campaigns to meet a product demand pattern in the most effective way. This procedure has been incorporated in the BATCHMAN program.

THE EFFEm OF UNCERTAINTIES ON THE DESIGN AND OPERATION OF BATCH PLANTS

Plants are subject to 0 uncertainties present in the design phase but

which are resolved after the plant has been built and brought into operation.

0 uncertainties of a random nature which recur throughout the life of the plant.

0 changes in the level or nature of the capability of the plant or the demands placed upon it by its environment.

Uncertainties at the design phase may include scale-up effects or what market demand will be realised. They can be allowed for by initially building a plant with sufficient flexibility or allowing the possibility of later modification or expansion if this proves to be justified. Quantitative treatments of this problem for a continuous plant have been presented[31,32] but work on the design of flexible batch plants has not been reported.

Random disturbances mav result in chances in product quality, throughput -of different equipment items or breakdown of these items. All of these can be dealt with to a limited extent by the provision of storage capacity at appropriate locations. The amount of storage which is necessary or economical will depend upon the length or autocorrelation of the disturbances. Equipment breakdown may be countered by reconfiguration of equipment or the provision of redundant items. Such strategies have not been systematically examined for multi-product batch plants. If a number of equipment items are arranged-in sequence and each is subject to independent random fluctuations about a common mean in the throughput of which it is capable, the provision of sufficient storage between the items can decouple these fluctuations so that the average throughput will be equal to the common mean value. If, however, there is no intermediate storage, the throughput at any time is limited by the item with the lowest current

capability. This means that the expected throughput of the whole chain is reduced and also the variance of the resulting overall performance is less than that of the individual components. If the expected performance of the sequence is to equal the desired throughput, overdesign of the order of one half to one standard deviation of the distribution of the original components may be called for to counter this concatenation effect [33].

Significant changes in the level or nature of demand or of plant capability are a recurring feature in fine chemicals production and the capability of deal- ing with such changes is often quoted as one of the major reasons why batch plants are preferred to continuous ones for such products. No systematic methods have been reported of assessing the re- silience or flexibility of a batch plant design. The flexibility to changes in the demand pattern of a given set of products can be assessed by considering the attainable region for the plant in the space of product requirements. For a given pure multi-product plant all the possible combinations of product requirements which can be manufactured lie within the attainable region bounded by a plane intersecting each of the product axes at the production capability when the plant is used for that product alone (Fig. 4). If the cycle time or size factor of a product on one of the equipment items is changed, this will influence the attainable region only if that item is correspondingly limiting in time or capacity for that product. The new attainable region can be constructed without difficulty for any range of such perturbations.

For a multi-purpose plant of the type treated by Mauderli, the attainable region for possible combinations of product requirements is constructed by the BATCHMAN program [29]. For a multi-purpose plant the attainable region generally extends above the plane intersecting the product axes at points corresponding to the single product capability of the plant. This reflects the fact that the greater freedom of the multi-purpose operation allows the assembly of product combinations for which simultaneous production is more efficient than any combination of sequential single product production. The effects of changes in process characteristics, reflected in cycle times and size factors, on the attainable region for a multi-purpose plant could be explored, but at the cost of considerably more computational effort than for a multi-product plant.

The qualitative effect of the introduction of one or more entirely new products has not been investigated. It could be conjectured that this effect will depend upon their degree of similarity to the products for which the plant was designed, but how that degree of


t n 0oundory of attahable

Requirement product L1

Fig. 4. Attainable region. (a) For multi-product plant. (b) For multi-purpose plant.

156 D. W. T. hPPIN

similarity can be quantitatively characterized is an 2. H. Belevi, J. R. Bourne & P. Rys, Chem. Engng. Sci open question. 36, 1649 (1981).

CONCLUSIONS 3. F. Horn, Z. fCr Elektrochem. 65, 209 (1961). 4. T. Takamatsu. I. Hashimoto & S. Hasebe’. Comout.

This review has given a personal impression of the current state of the study of batch plants and the use of quantitative methods for improved design and operation. The field is characterized by, on the one hand, a few strong points where problems have been clearly defined, methods developed and significant results obtained and, on the other, much more diffuse interconnecting regions where much work remains to be done, not least in identifying the true nature of the problems. It must be recognised that batch processing is used for such a wide diversity of products that the significance of different problem regions may well change from one application to another.

Chem. Engng 3, 185 (1979). _

W. Flatz, Chem. Engng, pp. 71-80 (25 Feb. 1980). R. E. Sparrow, G. J. Forder & D. W. T. Rippin, The Chem. Engr. No. 289, 520 (1974). R. E. Sparrow, G. J. Forder & D. W. T. Rippin, ZEC Proc. Des. Develop. 14, 197 (1975). !I. E. Ketner, Chem. Engng, pp. 121-124 (22 Aug. 1960). Y. R. Loonkar & J. D. Robinson, IEC Proc. Des. Develop. 9, 625 (1970). L. J. Hellinckx & M. J. Rijckaert, ZEC Proc. Des. Develop. 10, 422 (1971). J. D. Robinson & Y. R. Loonkar, Process Tech. Int. 17, 861 (1972). I. E. Grossmann & R. W. H. Sargent, ZEC Proc. Des.

- Develop. 18, 343 (1979).

A general procedure for classifying the structure of batch processing problems has been presented recently [34].

Areas in which methods are available: l Optimal operation of existing equipment items.

Methodology well established. Many results available but often not comprehensive and not well presented.

0 Design of truly multi-product plants at minimum equipment cost. Methods reasonably well established for discrete and continuous equipment sizes.

5. 6.

7.

8. 9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

l Equipment allocation and capacity planning for multi-purpose plant. Methods proposed but further evaluation required.

19.

l Short-term scheduling. Many general treatments available for other industries. Adaptation to specific requirements of batch chemicals reported but further developments needed.

Areas open to further study: l Coordination of individual equipment items to

give optimal operation of a multi-item plant. l Breakdown of a process task sequence into

subsidiary sequences for more efficient manufacture. l The role of inventory in design and operation. l Development of allocation and. design pro-

cedures for plants which are not purely multiproduct. l Improved scheduling procedures, more closely

adapted to the needs of batch chemical plants and capable of handling larger systems. l Treatment of effects of change and uncertainty.

20. 21.

22.

23.

24.

25.

26.

27.

28.

29.

30.

31.

32.

33. 34.

F. C. Knoof. M. R. Okos & G. V. Reklaitis. ZEC Proc. Des. Develop. 21, 79 (1982). W. Wiede, Jr., W. C. Yeh & G. V. Reklaitis, A.Z.Ch.E. Ann. Meeting, New Orleans, Louisiana, Nov. 1981. I. Suhami & R. S. H. Mah, IEC Proc. Des. Develop. 21, 94 (1982). M. Abellan, Diplomarbeit No. 3902, Techn. Chem. Labor, E.T.H. Ziirich, 1981. R. S. Garfinkel & C. L. Nemhauser, Integer Pro- gramming. Wiley, New York (1972). G. Gruhn, M. Grauer & G. Fichtner, Computerized control and operation of chemical plants. Proc. 14th Europ. Symp., pp. 45-51, Europ. Federation of Chem. Eng, Publication Series No. 17, Vienna, Sept. 1981. G. Fichtner, Porschungsbericht Techn. Hochschule “Carl Schorlemmer”. Leuna-Merseburg, G.D.R. (Aug. 1981). P. V. Youle, Comput. J. 3, 150 (1960). R. J. Miner, D. B. Westman & D. Cascio, Simulation 35, 125 (1980). I. Suhami & R. S. H. Mah, Comput. Chem. Engng 5, 83 (1981). D. W. T. Rippin & M. Hofmeister, A.Z.Ch.E. Ann. Meeting, New Orleans, Louisiana, Nov. 1981. U. M. Egli & D. W. T. Rippin, A.Z.Ch.E. Meeting, Houston, Texas, April 1981. H. Groelhn, Studienbericht No. 6, Inst. fiir Operations Research, E.T.H. Zurich (1977). B. G. Lagerweg, J. K. Lenstra & A. H. G. Rinnooy Kan, Ops Res. 26, 53 (1978). R. L. Graham. E. L. Lawler. J. L. Lenstra & A. H. G. Rinnooy Kan, Ann. Discrete Math. 5, 237 (1979). R. V. Reklaitis, A.2.Ch.E. Ann. Meeting, New Orleans, Louisiana, Nov. 198 1. A. Mauderli & D. W. T. Rippin, Comp. Chem. Engng 3, 199 (1979). A. Mauderli 8s D. W. T. Rippin, C.E.P., pp. 3745 (April 1980).

-_ __

GENERAL REFERENCES R. K. Malik & R. R. Hughes, Comp. Chem. Engng 3, 473 (1979).

References to specific batch operations are preceded by a letter and may be found in Tables 47.

1. M. Takao, T. Yamato, Y. Murakami & Y. Sato, J. Chem. Engng Japan 11, 481 (1978).

W. R. Johns, G. Marketos & D. W. T. Rippin, Trans. Institut. Chem. Enana 56. 249 (1978). D. I. Saletan & A.-VT Caselli, C.E.P: 59(5), 69 (1963). D. W. T. Rippin, Int. Symp. Proc. Systems En- gineering, Kyoto, Japan, Aug. 1982.

simulation of single- and multiproduct batch chemical plants for optimal design and operation

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