A4 - SYNTHESIS OF ISOTHERMAL.pdf

Chcmicol Engineering Scirnee, Vol. 46, No. 516, pp 1361-1383, 1991. oab9 2509/91 s3.m + 0.00 Printed in Great Britain 0 1991 Pergamon Press plc

SYNTHESIS OF ISOTHERMAL REACTOR-SEPARATOR-RECYCLE SYSTEMS

ANTONIS C. KOKOSSIS and CHRISTODOULOS A. FLOUDAS+ Department of Chemical Engineering, Princeton University, Princeton, NJ 085444263, U.S.A.

(First received 12 March 1990; accepted in revised firm 31 July 1990)

Abstract-A systematic synthesis approach is presented for isothermal reactor-separator-recycle systems. The approach proposes a general superstructure of different reactors and separation tasks and features all the potential interconnections among the proposed units. The synthesis problem based upon the proposed superstructure results in a mixed integer nonlinear programming (MINLP) formulation in which the objective function involves both integer and continuous variables and is subject to a nonlinear set of constraints. A variety of objectives was selected for the synthesis problem such as the minimization of the total annual cost of the plant and the maximization of its profit, as well as objectives traditionally used for optimizing the performance of a reactor network such as the product yield and selectivity. Discussion of the results and comparison among the different solutions obtained provided the ground for conclusions related to the potential trade-offs and the performance of the isothermal chemical systems under consideration.

1. INTRODUCTiON

In most chemical processes reactors are sequenced by systems that separate the desired products out of their outlet reactor streams and recycle the unconverted reactants back to the reactor system. Despite the fact that process synthesis has been developed into a very active research area for the last two decades, very few systematic procedures have been proposed for the synthesis of reactor-separator-recycle systems. The proposed evolutionary approaches are always based upon a large number of heuristic rules to eliminate the wide variety of choices. Many of these heuristics are actually extensions of results obtained by separately studying the synthesis problem of reactor networks or separator systems and, therefore, the potential trade- OITS resulting from the coupling of the reactors with the separators have not been investigated.

The delay in the development of a general synthesis strategy that will set the basis for a rigorous and systematic search for the optimal reactor-separator-recycle configuration is mainly due to the diffi- culties arising from the large number of structural alternatives and the nonlinear design equations of the participating units. Instead of focusing on an optimal search procedure, the various proposed methods have restricted the synthesis problem into a limited search around a feasible operation point. Thus, although it is often possible to identify directions of potential im- provements, the finally obtained solution can never be claimed, structurally and/or operationally, to be even a local optimal point.

In the optimization of reactor-separator-recycle systems a very limited number of publications exist. Conti and Paterson (1985) proposed a heuristic evolutionary technique to solve the synthesis problem. First a hierarchy of heuristics is adopted that target to: (i) minimize process complexity, (ii) maximize process

Author to whom correspondence should be addressed.

yield, and (iii) select the distillation column which minimizes total heat load. According to the proposed hierarchy a base case design is quickly established where everything is specified. Incremental changes are then introduced to the system by successively relaxing the heuristics so that a low cost process topology is obtained at each stage. It is evident that since there is no unique way of relaxing any of the above heuristics, arbitrary decisions should often be made to provide directions of potential changes in the system. Per- forming a case study for the Van der Vusse reaction, the authors were able to compare their results with reported optimal solutions for the isolated reactor system. The comparison made clear that a design based upon the maximization of reactor yield is much inferior to the design based upon the integrated flow sheet.

Floquet el ul. (1985a, b) proposed a tree searching algorithm (a branch and bound method) in order to synthesize chemical processes involving reactor- separator-recycle systems interlinked with recycle streams. The reactor network of this approach is restricted to a single isothermal CSTR or PFR unit and the separation units are considered to be simple distillation columns. The conversion of reactants into products and the temperature of the reactor, as well as the reflux ratio of the distillation columns, were treated as parameters. Once the values of the parameters have been specified, the composition of the outlet stream of the reactor can be estimated and application of the tree searching algorithm on the alternative separation tasks provides the less costly distillation sequence. The problem is solved for sev- eral values of the parameters and conclusions are drawn for different regions of operation.

Pibouleau et al. (1988) provided a more flexible representation for the synthesis problem by replacing the single reactor unit by a cascade of CSTRs. They also introduced parameters for defining the recovery rates of intermediate components into the distillate,

CES *6-5/6-K 1361

1362 ANTONIS C. KOKOSSIS and CHRISTODOLJLOS A. FLOLJDAS

the split fractions of top and bottom components that are recycled toward the reactor sequence, as well as parameters for the split fractions of the reactor outlet streams. A benzene chlorination process was studied as an example problem for this synthesis approach. In this example, the number of CSTRs in the cascade was treated as a parameter that ranged from one up to a maximum of four reactors. By repeatedly solving the synthesis problem an optimum number of CSTRs was determined.

In this paper a general approach based upon mathematical programming techniques is proposed for the synthesis of reactor-separator-recycle systems. A superstructure is postulated with all the different alternatives for the reactor and separator network, as well as for their possible interconnections. Different separation tasks, different types of reactors, different reactor configurations and different feeding, recycling and bypassing strategies are considered. The synthesis problem is formulated as a mixed integer nonlinear programming problem (MINLP). The continuous variables include the stream flow rates and compositions of the reactors and separators while the integer variables describe the existence of the reactors and the distillation columns. The solution of the (MINLP) formulation will provide an optima1 configuration of the reactor-separator-recycle system.

2. PROBLEM STATEMENT

For a chemical process in which a reactor network with a reaction system of known kinetics is followed by a sequence of separation tasks that are required to extract the desired products and recycle the unconverted reactants, the synthesis problem consists of systematically determining the reactor-separator-recycle system that operates so that a given performance criterion (e.g. total cost or profit of the plant, yield or selectivity of desired products, conversion of reactants) is optimized. The solution of such a problem should provide information about:

(a) the reactor network (types and sizes of reactors, feeding strategy and interconnections among the reactors);

(b) the separator network (appropriate separation sequence and sizes of separators);

(c) the interconnection between the two networks via the allocations of the outlet streams from the reactors and the allocations of the recycles from the separators back to the reactors.

In the proposed synthesis problem the following assumptions are made.

(1) All separators are considered to be distillation columns while the available reactor units consist of continuous stirred tank reactors and plug flow reactors that are approximated by equal volume CSTRs (SCs). In the separation level, distillation was assumed to be the only method available,

although, as will become clear in the following sections. alternative separation methods can also be handled in a way similar to distillation.

(2) All distillation columns are simple (i.e. one feed and two products) and operate as sharp splits of the light and heavy key components. No dis- tribution of components is allowed in both the distillate and bottom products and configurations that include nonsharp separators, like those described by Aggarwal and Floudas (19901, are not considered.

(3) The thermodynamic state of the various streams in both the reactor and separation networks are supposed to be known. For the examples presented in this paper, the feed streams of all columns are saturated liquids at the pressure of the column. Distillate and bottom streams are also considered to be saturated liquids at their bubble points.

(4) The operating conditions of the various units (pressure, temperature, reflux ratio of the columns) are not considered as optimization variables and are fixed at nominal values. The heating and cooling requirements are directly provided by hot and cold utilities and, therefore, no heat integration takes place.

Assumption (1) does not impose any restrictions on the potential of the proposed reactor network. Due to the fact that CSTRs and PFRs represent the two extreme types of reactor performance, superstructures of these units are capable of handling simple and complex types of reaction mechanisms since they can approximate reactors with various degrees of mixing and can provide for different feeding, recycling and bypassing strategies (Kokossis and Floudas, 1989, 1990). Furthermore, batch and semi-batch processes can also be studied from their space equivalent plug flow reactors.

Assumption (2) can be justified for a large number of chemical processes where product specifications require very pure products and high recoveries only should be considered. Although assumption (3) is a useful postulation that enables the effective formulation of the synthesis problem, assumption (4) should be considered as the most restrictive assumption of the approach. This is not only because noniso- thermally operated units that are excluded according to this assumption may appear more profitable choices for the synthesis problem, but also because the potential heat integration among the hot and cold streams of the reactor-separator-recycle system is always expected to reduce the energy requirements for the plant and result in a much less costly alternative solution. The nonisothermal operation and the heat integration issue will be addressed in a future publication.

3. THE SYNTHESIS APPROACH

Based upon the set of assumptions introduced in the previous section, it is intended to present a sys-

Synthesis of isothermal reactor-separator-recycle systems 1363

tematic strategy potent to handle the isothermal synthesis problem of reactor-separator-recycle systems. The basic parts that constitute the proposed approach consist of the following.

3.1. Derivation of cost models via simulation data The synthesis problem involves cost expressions of

the various units. Since the specifications of the feed streams of the columns are variables in this approach, a series of simulations is performed by slightly relaxing the split assumption in order to provide cost data for different feed flow rates and compositions. Sub- sequent regression analysis of these data determines the expressions for the cost of each column as a function of these variables.

3.2. Generation of the superstructure The basic idea behind the generation of the super-

structure is to include all different configurations for the reactor-separator-recycle system. In the reactor network, the superstructure should feature aItern- atives for reactors in series, reactors in parallel or series-parallel, reactors with multiple feeds and recycles, as well as bypasses around the reactor units. The superstructure should account for all the potential interconnections between the reactor network and the separator network, it should include all potential allocations of the recycle streams originated from the separators and leading to the reactor units, and, in the separation level, it should include all the alternative separation sequences.

3.3. Formulation of the synthesis problem Based upon the proposed superstructure, the syn-

thesis problem is formulated as a constrained optimization problem. Continuous, as well as integer, variables are introduced. The integer variables represent the existence of each particular reactor and separator unit; these variables are assigned a nonzero value if the associated unit exists in the solution and a zero value otherwise. The continuous variables consist of the flow rates and compositions of all streams, the volumes of the reactors, the operation time of the plant, and the costs associated with each unit. The objective function is generally a mixed integer nonlinear function which is subject to a nonlinear set of constraints defined by the mass balances around the various splitters and mixers of the superstructure, the mass balances around the reactors and separators, and the logical constraints.

3.4. The proposed solution algorithm The above mathematical formulation results in a

mixed integer nonlinear programming problem (MINLP), the solution of which is obtained by decomposition and iteration according to the generalized Benders decomposition algorithm (Geoffrion,

1972). According to the specified criterion (e.g. profit or venture cost, yield or selectivity of the product), the solution will provide information about an optimum structure of the reactor-separator-recycle system, as well as information about the flow rates and composition of the various streams and the sizes of the reactor and distillation units.

4. COST MODELS AND SIMULATION DATA

The formulation of the synthesis problem involves expressions for the capital and operating cost of the reactors and the distillation columns. Although for the reactor units such expressions are readily available, for the distillation columns additional effort is required. This is due to the fact that the feedfiow rate and composition of each column are considered as optimization variables in the synthesis problem. The cost of a column is clearly a function of these variables and, since the function itself is not known, it has to be extracted from a series of simulation data obtained for a range of these variables.

Guthries cost module (1969) summarizes the data necessary for estimating the cost of a column to be: (i) the shell size and material, (ii) the pressure of the column, (iii) the number, diameter and type of trays, and (iv) the heat duties of the reboiler and condenser. The tray sizing can be based on some reasonable values of the flooding factor. A value of 78% is used for columns with diameters greater than 1.2 m and 75% for columns with smaller diameters. Valve trays were used in all cases. The shell size and the material can generally be prespecified and the pressure of the columns has already been assumed to be known. With the economically optimal reflux ratio being 1.2 times the minimum (King, 1971), shortcut distillation calcu- latious were performed for a wide range of feed flow rates and compositions and provided the number of trays and the heat duties for the condenser and reboiler. Fixing the number of trays to the number calculated by the shortcut distillations, a second series of rigorous calculations provided additional information about the diameter of the trays. In this work, all the simulation data was obtained using the PROCESS flow sheeting system (Simulation Sciences, 1985).

Once the number of trays, the size of trays and the heat duties of the reboiler and the condenser have been calculated, the data necessary for estimating the cost of the columns have been obtained. Mathemat- ical expressions are next derived that provide the cost of a column as a function of its feed flow rate and composition. This is achieved by fitting various models to the simulation data and using regression analysis via SAS (Statistics Analysis Software). The capital cost Cosr:lp and the operating cost Cost:: of each column were expressed in terms of the compositions of the feed components of the column XFC,+_, (kth degree polynomial), as well as the feed Aow rate of the columns FC,,,. The cost expressions are in the general

1344 ANTONIS C. KOKOSSIS and CHRISTOD~ULOS A. FLOUDAS

form:

+a clo,.mXFCm..., + . . . + &mXFC& >

(11

+b hnX~Cm,w~ + . . + bL,m XFCk,,, >

G-9

where h, d,..,, &.,,, . - , 4,,col, bLOl. b,l col, . . . , bk m co, are constants available from the regression analysis for each column and Msd is the set of feed components of the column. The above expressions for the costs are nonlinear in both the feed flow rates and the compositions of the considered components. The cost dependence on the composition vector is expressed via a polynomial form that features positive and negative coefficients while the dependence on the flow rate is expressed in a bilinear form that includes the polynomial of the composition vector and the feed flow rate. The exact type of the nonlinearities, however, is dependent on the problem under consideration and the fitting expressions resulting from the regression analysis.

The capital cost Cmt,,,, Cap of the reactors is assumed to be a linear expression of the reactor volume in the form:

COStCaP - relf - Y,,.. + La, vi (3)

where Y,,,, and L, are respectively the fixed charge and variable charge investment cost of the reactor and V Tea= is the reactor volume. In all cases, the operating cost of the reactors is assumed to be negligible.

5. THE GENERATION OF THE SUPERSTRUCTURE

A reactor-separator-recycle system consists of two different networks: the reactor network and the separator network. The reactor network consists of the reactor units and the interconnecting streams associated with these units while the separator network consists of the different separation tasks that constitute the desired separation process. A reactor- separator-recycle superstructure should generally provide options for all the structural alternatives for the units (reactors, separators), as well as a complete stream network that will allow for all the different interconnections among them. At the reactor network level, the superstructure should feature all the possible arrangements of the reactors and should consider cases of different types of reactors with multiple feeds, recycles and bypasses, as in Kokossis and Floudas (1989, 1990), as well as different allocations for the recycled reactants that flow purified as bottom or distillate streams out of the separators. In the separation level, the superstructure should include all different separation sequences and all different interconnections among the separators and the reactors. It should

be noted that the specifics of a system may provide additional alternatives that cannot be included in a general purpose superstructure. These cases are not addressed by the present approach which intends to handle the synthesis problem of reactor-separator-recycle systems in a general form. Once the mechanism is known, however, the extension of the superstructure in order to accommodate special features is always possible.

The proposed superstructure is best described by defining a basic set of mixers and splitters associated with the various units. These include the set of mixers MX:,, prior to each i,, CSTR, mixers MXSCi,, sk prior to each sk SC-unit of the i,, PFR, mix&s MXRC, associated with each recycled component rc, mixers MXSS,, prior to each separation sequence sq, splitters SL$ associated with each feed stream of mf component and splitters SLP in the outlet of each reactor i.

The superstructure of the reactor-separator- recycle system is then generated so that:

(1) streams originated from any of the SL$-, MXRC,, and SLp lead to any of the mixers MXiB and MXSC! z,=,=.sk; (2) streams originated from any of the SLput lead to any of the mixers MXSS,; (3) streams originated from each MXSS,, lead to each one of the mixers MXRC,,; (4) outiets of the MXF,,, the MXSCi,,.,,, and the MXSS, feed the CSTRs, the SCs and the leading separator of the sq separation sequence respectively; (5) splitters SLP are fed by the outlets of the reactors and mixers MXRC,, (apart from MXSS,,) by these streams of the separators that produce pure rc.

In order to illustrate examples of superstructures generated according to the above propositions, the examples of two different reaction mechanisms are considered. In the first example the reactor network consists of three CSTRs and the components to be separated are A, B and C with their relative volatility in this order. The consecutive reaction mechanism

kl kl n-E- C is assumed to take place with B denoting the desired product and A the fresh feed component. The potential separation tasks are A/EC, AS/C, B/C and A/B. The component which should be considered For recycling is A and flows as distillate from columns A/BC and A/B. The generated superstructure, shown in Fig. 1, features all the possible interconnections among the reactor units (streams 7, 8, 10, 11, 15 and 16), among the reactor and the separation units (streams, 5, 6, 12, 13, 17 and 18), and all the potential recycles from the separation network to the reactors (streams 25, 26 and 27).

The different configurations for the reactor- separator-recycle system can be obtained by eliminating the appropriate streams of the proposed superstructure. Thus, elimination of all but streams 1, 7, 11, 14, 18, 22, 24 and 27 results in the configuration

Synthesis of isothermal reactor-sepaiator-recycle systems 1365

1366 ANT~NIS C. KOKOSSIS and CHRISTODOULOS A. FLOUDAS

shown in Fig. 2(a) where the three CSTRs are connected in series, the separator system includes columns A/BC and B/C and a total recycle of A is fed into the first CSTR. Should all but streams 1, 2,3,4, 5, 9, 12, 14, 17, 21, 23 and 26 be eliminated from the superstructure, the configuration of Fig. 2(b) is obtained where the CSTRs are connected in parallel, the separator network consists of columns AB/C and A/B and the recycle stream from A/B feeds the second CSTR. A different configuration, illustrated in Fig. 2(c), is the case where streams 1, 2, 8, 9, 14, 17, 12, 21, 23, 25 and 27 only are activated. In the reactor network, the first and third CSTRs are connected in series and the second reactor is in parallel with the serial arrangement of the other two CSTRs. The separator system consists of columns AB/C and A/B, while the recycle of A is fed into the first and third CSTRs. As a final example, the reactor-separator-recycle configuration of Fig. 2(d) is shown. The configuration results by considering only streams 1, 7, 9, 12, 14, 17, 21,23, 25 and 27 from the superstructure of Fig. 1 and consists of reactors CSTR- 1 and CSTR-2 in series with CSTR-2 feeding column A/BC, the distillate of which feeds CSTR-3 and CSTR-1.

In case the reactor network consists of two CSTRs and one PFR and the assumed reaction mechanism is

of the form: As B 2 C the generated reactor- separator-recycle superstructure is shown in Fig. 3. As in the previous example, B is assumed to be the desired product while C is a byproduct, the recycling of which may or may not be desirable. Thus, the superstructure features the potential recycle streams 25, 26, 27, 32, 33 and 34 which lead toward CSTR-2, CSTR-1 and PFR-1. According to the proposed features for the plug flow reactor, each potential feed stream of the PFR (1,4, 10, 15,27 and 34) is split into a number of substreams equal to the number of the SCSTRs in the PFR representation. As a result, different feeding strategies can be obtained for the PFR unit as illustrated in the configurations shown in Fig. 4.

In Fig. 4(a), the reactor network consists of the CSTR-1 in series with the PFR, the inlet stream of which is unevenly distributed along the reactor.-The separator system consists of columns A/BC and B/C and A is the only recycled component. In Fig. 4(b), the reactor network is restricted to the plug flow reactor, columns AB/C and A/B constitute the separator system and components A and C are both recycled toward the PFR. A total recycle of A is fed in the front section of the PFR while a substream of the produced C is fed in the middle and ending section of the reactor. In the final example, shown in Fig. 4(c), the reactor network consists of all three reactors. The leading PFR unit is fed by fresh and recycled reactant A and is connected in series with a parallel arrangement of the two CSTRs. The separator system consists of columns A/BC and B/C. The recycled component A is feeding the PFR while the recycled component C is feeding the CSTRs.

4. VARIABLES AND PARAMETERS OF THE SYNTHESIS PROBLEM

The basic index sets used to define the variables and parameters of the synthesis problem include the set of components M = {ml, the set of reactors Z = (i}, the set of the columns L = (I} and the set of intermediates N = { nJ. The latter set consists of all groups of two or more components discharged in the same outlet stream of a separator.

A number of subsets of these sets are also introduced and include:

Subsets of M

Mfeed _ 1 -

Mbot = 1

Mdis _ I -

{m/me M is a recycled component} {mlmt M is a fresh feed component) {m/me M is a desired product] { mlm E M participating in rp E RP path kinetic expression} {mlmE M is a feed component of column IEL) {m/me M is a bottom component of column IEL) {rnlrn~ M is a distillate component of column I E L}

Subsets of Z ZcSrR = { ili~Z is a continuous stirred

ZPFR = {iii E Z is a pIug flow reactor}

Subsets of L J = { I[! E L is a leading column}

tank reactor}

L!F:d = {l/f EL with feed the intermediate n E IV) L.tz = {ZIIE L with bottom the intermediate n E N} I$: = (I//E L with distillate the intermediate n E N) L1.E = (ZIIEL produces recycled component

mEM) L;tm= {ZllsL produces desired mcMdp).

In addition to the above sets, there is also the set of the reaction paths RP = {rp} and the set SK = (sk} of the subunits (CSTRs) that represent each individual plug flow reactor.

The variables of the synthesis problem are defined over the above sets and subsets and consist of continuous and integer variables. The first set of continuous variables includes variables associated with the cost or the profit of the plant,

CO&: annualized cost of the plant Pro$r: annualized profit of the plant,

as well as variables associated with the cost of the various units and the operation time of the plant,

CustFap: capital cost of column 1 Co@? capital cost of reactor i CostFper: operating cost of column 1 8: total operation time of the plant.

Another set of variables is related to the flow rates of the streams that constitute the proposed super-


c 9 f

-l!I!b! A Eel--h B L-J C

B II C (a) 4 J

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CSTR-2 CSTR - 3 B

(d)

Fig. 2. Reactor-separator-recycle configurations obtained from the superstructure of Fig. 1

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Synthesis of isothermal reactor-separa(or-recycle systems

s 0

1369

(b)

Fig. 4. Reactor-separator-recycle configurations obtained from the superstructure of Fig. 3.

structure and consist of the fresh feed stream flowrates,

FD,: total fresh feed stream of component m E M mJ FRi,m: fresh feed of reactant m E M w in reactor i FRSw>,: sk substream of FR,., (i E IPFR, sk E SK),

streams leading to or originated from the reactor units (CSTRs and SCs),

IN,: inlet stream of reactor i OUT,: outlet stream of reactor i TNSi,,k: inlet stream of sk SCSTR of reactor i (i.zlPFR, skESK) OUTSi,sk: outlet stream of sk SCSTR of reactor i (i E IPFR, sk E SK) BP+: bypass around sk SCSTR of reactor i (i E IPFR, sk E SK),

streams leading to or originated from the separator units (feed, bottom and distillate streams of the distillation columns),

FC,: total feed of column I B,: bottom product of column I D,: distillate product of column 1,

as well as streams that interconnect different units of the superstructure such as the streams among the reactors,

RR,.,: interconnecting stream from reactor i to reactor k; i, ke1 RRSi.r,,r: sk substream of RR,_, (itz I, k E IPFR, Sk E SK)

or streams connecting the reactor network to the separator system and vice versa,

MM,.,: reactor network recycle in front of column 1 merging with REC, (1 E J, m E M) RMi,I: stream from reactor i to leading column 1 E J MRSi,,.Sk 1 sk substream of MRi,,, (i E I PER, m E M 111, sk E SK) MRi,m: recycle stream of component rnE M to reactor i

1370 ANTONIS C. KOKOSSIS and CHRISTODOULOS A. FLOUDAS

REC,: total recycle of pure m (me M) a: payout time PG,: total purge stream of m (m~A4~). p: income tax rate

Figure 5 shows the reactor-separator-recycle superstructure (one PFR, one CSTR and a three-

SP,: sales price of component m PR,: minimum production of desired m E Mdp,

component separation process) along with the stream as well as the chemical process (reaction mechanism, variables that have been introduced above. A different reaction mixture) and the plug flow reactor repres- set of variables is associated with the reaction rates entation (maximum allowable SC units), and the volumes of the reaction units,

R,,.i: reaction rate of reaction rp in reactor i R&i,& reaction rate of reaction path rp in sk SCSTR of reactor c (iEIPFR) V,: volume of reactor i vSi.J* ; volume of sk SCSTR of reactor i (if IPFR, Sk E SK),

Y,~,,: stoichiometric coefficient of component m in reaction path rp N,,: number of SK elements k,: reaction constant for reaction path rp MVm: molar volume of m.

as well as the concentrations of the species that appear in the reaction rate expressions, 7. FORMULATION OF THE SYNTHESIS PROBLEM

CNi,,: concentration of component m in the output 7.1. Objective function(s): the difSerent cases

of reactor i The different objectives that have been considered

CNSi,sk,m: concentration of component m in in this approach include those related to the eco- nomics of the nlant. like the profit and the annualized

0 U TSi,,r. venture cost df the system, *and those related to the

A final set of continuous variables considers the efficient utilization of the reactants like the overail

molar fractions of all the streams of the superstructure yield and selectivity. The exDression for the annualized venture cost of the and consists of:

XFC,,,: molar fraction of component m in FC, XB,,,: molar fraction of component m in BI XD,,,: molar fraction of component m in D, XMKwsw: molar fraction of component m MRi,,r XIN,,,: molar fraction of component m in INi XOuTi,,k3,: molar fraction of component m OUT, XINSi,sk,nt: molar fraction of component m INS+ XOUTS,,,: molar fraction of component m OUTSi,,*.

plant is of the general form:

in

costann = - :.{z# CostfP + c COStfaP &I. 1 (4)

in where

in Costa = the annual venture cost of the plant Cosp = the total capital cost of reactor unit i

in Costpap = the total capital cost of column I Cosf;lper = the annual operating cost of column 1.

Apart from the continuous variables, the proposed formulation also includes integer variables that describe the structural alternatives of the reactor- separator-recycle system. The set of integer variables includes:

Zi: binary variables associated with reactor i Y,: binary variable associated with column 1

zsi,sli : binary variable associated with sk SCSTR of reactor i (i E IPPR).

The above binary variables represent the existence or not of each unit. Thus, if the reactor i (column r) participates in the superstructure, then Zi ( Yr) takes the value of 1. Otherwise it takes the value of 0. The introduction of the binary variables ZSiVslrr in addition to Zi, reflects the alternative of having activated only a subset of the available SCSTRs given by N,,.

Besides the continuous and integer variables of the problem, a set of parameters is introduced which is related to the costing and the specifications for the plant,

In Section 4 it was shown how to derive the capital and operating costs of the units. In the case where a particular unit does not participate in the solution the cost associated with the unit should be eliminated. Logical constraints that are introduced later in this section force the feed flow rates and the volumes of nonexisting reactors and coIumns to take zero values, thus eliminating the operating costs of the units and their variable charge investment costs. For the fixed charge investment costs also to be eliminated in the case where the unit does not exist, the expressions for the capital costs are written in the equivalent form:

Costpa~ = yiz, + 6, v,. (6) For the total profit of the plant the expression used

Synthesis of isothermal reactorseparator-recycle systems

A

1371

1372 ANTONIS C.Ko~ossrs and CHRISTODOULOS A. FLOUDA~

is:

Pro& = c c SP, 13. FC,,, . XFC,,, [E If,-; ?nehfLp

- ,s; /SP, 0. FD, - Cost (7)

where an upper bound is usually set for the total operation time 0 of the plant.

The efficient utilization of the reactants can be evaluated in terms of the overall yield defined as the molar rate of desired products over the molar rate of fresh feed reactants, the overall selectivity defined by the molar rate of desired products over the molar consumption rate of reactants and/or the overall conversion defined by the molar consumption rate of reactants over the molar fresh feed rate of reactants. Since detailed expressions are not generally possible unless the particular reaction mechanism is available, objective functions based upon these criteria are re- presented by the general symbol R.

7.2. Constraints of the synthesis problem The desired objective is subject to a set of con-

straints constituted by the mass balances and the logical constraints. The mass balances include:

(i) molar balances for the feed stream splitters, and PFR stream splitters

FD, - CFR~,~ = 0 mEMmf (8) i.zr

FL - c FRSis,k,, = 0 iE IPFR, mEMm/ (9) skcSK

RR,,, - c RRSi,;p,,t = 0 iE I, ipe ZPFR (10) sk&K

MRi,m - 1 MRSi,,,,k = 0 iEZPFR, rnE Mm (11) sk&K

(ii) molar balances for the mixers in the inlet of each CSTR unit

+ F%rq -ZNi.XZNi.,=Om~M,

i E ZCsTR, mf e Mm- (121

(iii) molar balances for the mixers prior to each SC unit

Bpip,sk - 1 XNSip,s* - I

+ OUTSip,sk- I _ XOuTSip,rk- 1.m

- INS+ XINSip,s~.rn = 0

(13) ipEIPFA, 1 < sk < N,,, mEM

INi, . XINips, + c RRSi,ip.sk . XOUT,., iEI

+ FRSip.ti,mf

+ m,&_~MRSip,m,.,k _ XMRm,.m

- BPip.sr+ I . XINSip.sk+ 1.m

- INSip,sk+ I . XINSip,sk+ 1.m = 0

ip E ZPFR, skESK, sk = 1, rnEM (14)

(iv) moIar balances for the splitters in the outlet of each reactor

c RR,,, . XOUT,,, + c RM,,j. XOUTiv, ksl Id

- OUT,- XOUT,,, = 0 iEI, meM (15)

BPip.slr . XINS+,,, + OUTSip.slr . XOWTSip.sk,m

- OUT,; XOiJTip.m = 0 QxIPFR, skESK,

sk = NsK, me M (16)

(v) molar balances for the mixers prior to each leading column

c RM,,I. XOUT,,, - 1 MM,,, . XFC,,, rn.z&frc

- FC, . XFCI,, = 0 IEJ, mEM (17)

(vi) molar balances for recycle mixers of the superstructure

REG, - 1 MM,,,, . XFC,sm Id

-~MR,,,;XMR,=,=O mEM,mrEM (18)

REC, + PG, - c XFC,,, . FCI = 0 mr EMC ,ELiZ

(19)

(vii) summation of the volumes of the SC units

Vi - C VSi,sk = 0 ifz ZPFR skfSK

(20)

(viii) summation of the mole fractions of the streams

c XMR,,,, - 1 =O mrEMrC (21) m

C XIZVi_- 1=0 iEI (22) msA4

c XOUTiv, - 1 = 0 i~l (23) tn.&f

1 XZNSi,wa - 1 =O iEIPER, skeSK (24) n.


c Xi?,,,- 1 =o 1EL (27) mEM

c XL,., - l=O ZEL (28) ma&f

(ix) molar balances around each reactor unit

IN, . XINi,,, - 0 U Ti . X0 U T,,,

- vi . Crp.m - Rr,,i = 0 ieIcsrn, mEA (29) rl,

INSi,.pk . XINSi,mt - OUTsi. XOUTSi.,k_,

- VSi,,~Cv,p.m.RS,p,i,,* = 0 iEIPFR, mEM. VP

(30)

The reaction rates I&,,,, and RS,,,.,ti.i can be expressed as a general functlonf,, of the outlet stream concentrations times the reaction constant k as:

R,p_i - k,, . f,,(CNi,,) = 0 rp E RP, i E ICSTR,

rnEMP (31)

RS,p,i& - k,p S,p (CNSi.sk,m) = 0

rp E RP, i E IPPR, s~ESK, rn~M~ (32)

where the outlet concentrations are given in terms of molar fractions as:

CNi,, . (

c XOUTi,m. MV,,, - XOUT,,, = 0 me&f >

~EI, rn~M (33)

CNSi.s*,m (

C X0UTSi.sk.m . MVm rnEM !

- XOUTSi,sL.m = 0 iEiPFR, mEM, skESK (34)

(x) molar balances around each separator of the superstructure

FC, - B, - D, = 0 EEL (35)

D, - FC, c

c XFC,

= 0 IEL (36)

XDL, (

c XFC,,, nlE A#:~ >

- XFCI,, = 0

IEL, rnEMP6 (37)

X%n . (

c XFC,., - XFC,,, = 0 meMy >

IcL, mEMy (38)

(xi) molar balances for the intermediate components

c Bi. X6,, - c Di.XQ,, I6LE 1 E Ld. I.1

+ ,e$d FC, . XFC,,, = 0 n E N (39) r..

as well as the specifications for the desired products,

c XFCL, -FC, 2 PR, mEMdp rsp

(xii) logical constraints

(40)

The logical constraints are used in order to force a continuous variable of a nonactive unit of the superstructure to take the value of zero. They are of the general form:

var; - u . varpt 6 0

where VULVA denotes the continuous variable associated with unit k, vary denotes the integer variable that describes the existence of this particular unit and U is a large number. If the unit participates in the superstructure, varp takes the value of 1 and the above inequality yields:

varconL < U

while if the unit does not exist, varp becomes zero and the inequality yields:

rrar;O < 0

which, since the continuous variable is always positive, can be true only if uarrn is zero.

For the present formulation the following set of these constraints was introduced:

lNi - U.Zi

1374 ANTONIS C. KOKOSSIS and CHRWCODOULOS A. FLOUDAS

A final set of constraints imposes that all variables are non-negative (non-negativity constraints) or binary variables (integrality constraints) and is defined as

FD,, FRi,,, MRi,,, INi, OUT, 3 0

D,, XFC,.,, -X-B,,,, XD,.,, XMR,,,, 2 0

v,, I%.&, &,,ir R.%~,i,,~, CNi,,t, CJJSi,,r,, 3 0 (51)

zi, r,, ZSi,Sk E (0, 11. (521

The complete mathematical formulation of the MINLP problem is given in Appendix A.

8. THE PROPOSED SOLUTION ALGORITHM

The optimization problem that is formulated in the previous section consists of two types of variables: continuous and integer variables. The integer variables participate linearly in both the objective function and the constraints while the continuous variables participate linearly and nonlinearly. The nonlinearities include the polynomial expressions obtained from the simulation data, the expressions of the reaction rates and the bilinearities associated with the mass balances. Such constraints define a nonconvex feasible region and therefore there is no guarantee for the global optimum.

The proposed solution algorithm for the MINLP problem is based upon a decomposition of the ori- ginal problem and iteration according to the Gen- eralized Benders decomposition algorithm. The algorithm suggests the solution of a sequence of nonlinear programming (NLP) primal problems and mixed integer linear programming (MILP) master problems that are based upon the dual representation of the primal problem. The solution of each NLP problem provides an upper bound (lower bound) for a minimization (maximization) direction while the solution of each MILP provides a lower (upper) bound. The MILP solution also provides the integer variables to project on the next NLP problem and convergence of the algorithm is achieved whenever the MILP master problem fails to provide a new feasible integer combination.

The NLP subproblems of the solution algorithm are large scale nonlinear programming problems and initialization is possible by solving a so-called pseudo- primaI NLP subproblem. The pseudoprimal problem is the projection of the full primal problem in the subspace of the continuous variables that are associated only with the active units at the current iteration. Around a small region of the solution vector of this subproblem a relaxed problem is next solved that relaxes the equality constraints H(X) of the primal in the form - a d n(x) d 0: and minimizes the infeasibilities CL. The solution vector of the relaxed problem is used to initialize the primal problem. In the case where

an infeasible primal problem is found, the Lagrangian multipliers that are associated with the next MILP master problem are obtained by the solution of the relaxed problem.

The suggested solution procedure which is included in OASIS (Floudas, 1990) was automated in the computer program ORESS (Optimization of REactor Separator recycle Systems) that uses the high level modeling language GAMS (General Algebraic Modeling System) and the algorithmic development methodology APROS (Paules and Floudas, 1989). At each iteration the program calls the appropriate solvers for the various subproblems, updates relevant parameters and checks for stopping criteria. If the criteria are met it stops, otherwise it continues to the next iteration.

Due to the nonconvex type of nonlinearities in- volved in the mathematical formulation, infeasibilities are very likely to occur in the NLP subproblems in the case where improper initialization has been used or inappropriate bounds have been allowed for the continuous variables. In order to avoid unfortunate solution trials, ORESS dynamically sets and updates lower and upper bounds for all variables, generates a number of different starting points at each iteration and makes use of all of these points before it declares as infeasible a nonlinear programming (NLP) problem.

The proposed approach is general and, under the assumptions imposed in Section 2, accounts for all possible structural and operational alternatives of the reactor-separator-recycle system. As an offset to the advantages of the approach, however, one should consider that the solution algorithm is restricted to provide a local solution. The solution, which evidently depends on the initial starting point, cannot be ciaimed to represent the global optimum, the search of which is not addressed by this paper. The global optimum search for nonlinear programming and mixed integer programming problems is itself an area of research and newly developed techniques that have provided promising results might also be proved ap- plicable for the reactor-separator-recycle synthesis problem as in the proposed formulation.

In the following section, the potential of the presented algorithm is illustrated in two different examples of chemical processes by considering a variety of specifications for the reactor network and the desired products, as well as by optimizing in terms of different objective functions.

9. EXAMPLES

9.1. Benzene chlorination The design of a benzene chlorination process is

considered as a first example. The chemical reactions of this liquid-phase process are given as:

kl C,H, + Cl, - C,H,Cl + HCl

LZ C,H,Cl + Cl, - C,H,Cl, + HCl.

Synthesis of isothermal reactor-separator-recycle systems 137.5

Further chlorination reactions can also take place but since they involve insignificant amounts of reactants they have been considered to be negligible. The kinetics of the process were studied by McMullin (1948), who showed that the chlorination of benzene (A), monochlorobenzene (B) and dichlorobenzene (C) is in all cases first-order and irreversible.

In the reaction level, pure A reacts to the desired product B, waste product C and hydrochloric acid. The kinetic constants are k, = 0.412 h- and k, = 0.055 h - . The hydrochloric acid produced is

eliminated at the reaction level output by a stripping operation whose cost is not taken into account. Although all the reactions are exothermic internal coil are used in the reactors to remove the evolved heat and therefore keep the temperature in the reactors constant.

In the separation level, unreacted A is separated and recycled toward the reactor network, valuable product B, of which the demand is assumed to be 50 kmol/h, and product C. The volatility ranking of these components is aA > ug > cc,. Thus, the possible separation tasks are: A/BC (column l), AB/C (column 2), B/C (column 3) and A/B (column 4).

Simulation results reported by Auzerais (198X) provided estimates for the capital and operating cost of all the sharp distillation columns of the chlorination process (purity specification: 99%) and are given along with the cost expressions for the reactors in Appendix B.

The synthesis problem was studied in four different cases. In the first three cases, the objective was to minimize the annualized total cost of a plant that produces a minimum of 50 kmol/h chlorobenzene by considering a variety of specifications for the reactor network. In the fourth case, different objective functions were used and conclusions were drawn by comparing the different solutions. The postulated superstructure always consisted of four CSTRs and four PFRs and each PFR unit has been approximated by a cascade of seven CSTRs (SCs). The large-scale optimization problem included 947 continuous variables, 36

51.35 kmolfir

integer variables and 1063 constraints. In all cases, a MIPS RC 2030 workstation computer was used.

Case 1: utilization of a least number of CSTRs. In this case a minimum of two CSTRs is required for the reactor network. The starting point of the algorithm was one PFR and four CSTRs along with columns 2 and 4 and the optimal reactor-separator-recycle system was found to consist of two CSTRs and 1 PFR and columns 1 and 3. In the optima1 solution, shown in Fig. 6, a fresh benzene stream (53.14 kmol/h) along with a total recycle stream of purified benzene (51.35 kmol/h) originating from the distillate of column 1, are feeding the first CSTR of the reactor network. The reactors are connected in a CSTR-PFR-CSTR series combination and feature volumes V,,,, = 3.427 m3, V,,, = 10.565 m3 and V ,-sTR = 3.482 m3 respectively. The production of chlorobenzene is the minimum required (50 kmol/h) and the total annual cost is $314,361. The reported solution was obtained in nine iterations with an average CPU time per iteration of 5.88 s per prima1 and 2.00 s per master problem.

Case 2: utilization of CSTRs and/or PFRs. No requirements were imposed for the reactor network in this case. Providing an initial guess of one PFR and columns 2 and 4, the minimum annual cost is found to be 5304,911 and the optimal structure consists of a single PFR and coIumns 1 and 3. Convergence of the solution algorithm required five iterations with an average consumption of 5.50 CPU seconds per primal and 1.78 seconds per master problem. In the optimal solution, shown in Fig. 7, both the volume requirements (V,,, = 17.478 m3) and the total recycle from the separation network (52.3 kmol/h) are greater than in case 1. Fresh feed is required at a rate of 53.147 kmol/h and no splitting for any of the feeding streams of the PFR was necessary.

Case 3: utilization of a least number of PFRs. A minimum number of three PFRs is required in this

II A 50 kmoLbr

[

B d C c - Fig. 6. Solution for the benzene chlorination process--case 1.

1376 ANTONIS C. KOKOSSIS and CHRISTODOULOS A.FLOUDAS

52.31 kmobhr

t

Fig. 7. Solution for the benzene chlorination processdase 2.

case. The initial guess for the reactor-separator- recycle system consists of four PFRs along with columns 2 and 4. The minimum annual cost is found to be $309,840 and the optimal structure, shown in Fig. 8, is an arrangement of three PFRs in series and columns 1 and 3. The first PFR features a recycle (10.10 kmol/h) and has volume Y - 5.562 m3 PFR-L - while the other two PFRs have volumes VPFR_a = 5.734 m3 and V PFR-3 = 5.873 m3 respectively. Al-

though a larger total cost is found for the plant, the total volume requirements, the fresh feed requirements (53.02 kmol/h) and the tota recycle from the separation network (47.98 kmol/h) are less than in the previous case. The solution algorithm converged in two iterations consuming 6.24 CPU seconds per primal and 1.98 CPU seconds per master problem.

Case 4: multiple objectives. The importance of ap- plying different performance criteria for the reactor- separator-recycle system is next studied by solving the synthesis problem using different objective functions. The different objectives include the profit of the plant, the annualized cost of the plant and the overall yield of the product. In order to facilitate comparisons among the solutions obtained, a minimum production of 50 kmol/h chlorobenzene is required for all but the last of the presented examples.

(a) Profit: providing an initial structure of one CSTR and one PFR with columns 2 and 4, the

maximum profit is found to be $1,224,038. The solution is obtained in two iterations and the average CPU time is 5.56 seconds per primal and 1.85 seconds per master problem. The optimal structure consists of three CSTRs and columns 1 and 3 is shown in Fig. 9. A fresh feed stream of 51.97 kmol/h is required and the recycled benzene stream is 159.43 kmol/h. Both the fresh feed and the recycle stream lead to the first CSTR. The CSTRs are connected in series and are of almost the same size: V,-STR_L = 4.88 m3, Vcsm_2 = 4.97 m3 and V,,,,_, = 5.04 m3. The annual cost

of this plant is $321,158, the yield 0.314 and the benzene conversion 0.326.

(b) Annualized cost: with the same initial structure as in the previous example, the minimum cost is found to be $293,324. The optimal structure, shown in Fig. 10, consists of two CSTRs with columns 1 and 3. The CSTRs are connected in series and their volumes are VCSI.R_I = 10.12 In3 and V,,,,_, = 10.47 m3. A larger feed stream (54.45 kmoI/h) and a much smaller recycle stream (54.23 kmol/h) are required than in the previous case. The profit associated with the plant is $1,166,420, the yield 0.460 and the benzene conversion 0.501. The solution algorithm converged in two iterations consuming 5.275 s per primal and 1.86 s per master problem.

(c) Overall yield (a): the maximum overall yield is found to be 0.5653 and the optimal structure, shown in Fig. I 1, consists of a single PFR and columns 1 and 3. The initial structure was one CSTR and columns 1

47 98 km&l-c 50 krnoLhr

A B

10.10 kmol/hr

Fig. 8. Solution for the benzene chlorination process--case 3.


r

L59.43 kmolfhr l 50kmol/hr

I3 E C Fig. 9. Solution for the benzene chlorination process-se 4(a).

-d 54.23 kmoyhr

54.45kmolhm j wm

V CSTB=10.12 In3 V csTR=10.47m3

Fig. 10. Solution for the benzene chlorination process-ase 4(b).

4 50 kmolfhr

B t

54.57kmoh V

PFR = 20.73~~~ * C

Fig. 11. Solution for the benzene chlorination process+ase 4(c).

and 3, and the solution algorithm converged in two iterations with an average consumption of 5.48 CPU seconds per primal and 1.85 seconds per master problem. In the final solution, the PFR has volume I,,,_, = 20.73 m3 and does not feature any side streams.

The required fresh feed stream is 54.57 kmol/h and much less recycle (33.87 kmol/h) is needed than in the previous cases. The annual cost of the plant is $785,509, the profit $667,109 and the benzene conversion 0.617.

(d) Overall yield (b): instead of a minimum production of chlorobenzene, a minimum consumption of 50 kmol/h of benzene is required. Starting from one CSTR and columns 1 and 3, the optimal reactor-

CES 46-5/6-L

separator-recycle system consisted, as in the previous case, of a single PFR unit with columns 1 and 3. However, in the final solution, shown in Fig 12, a PFR was at each upper bound (V,,,., = 30.0 m3), a much larger feed stream is required (78.96 kmol/h) and a larger recycle stream (49.00 kmol/h). Although a larger annual cost ($840,214) and less profit (5615,405) are found, the overall yield and the benzene conversion are, as in case (a), 0.5653 and 0.617 respectively. The solution algorithm converged in two iterations consuming 5.62 CPU seconds per primal and 1.75 seconds per master.

As might be expected, the annualized cost and the

1378 ANTONISC. KOKOSSIS and CHRISTODOULOS A. FLOUDAS

49.00 kmol/hr

t

78.96 kmob'hr

Fig. 12. Solution for the benzene chlorination process--case 4(d).

profit provide similar solutions that favour the CSTR structure. The overall yield, however, results in a different optimal structure, namely a PFR, an optimal operation that accounts for 54.5% of the maximum obtainable profit and an annual cost which is 267.8% greater than the minimum possible. Furthermore, by comparing the results from the last two examples, it becomes clear that taking the yield as the performance criterion of the reactor-separator-recycle system, no distinction can be made among solutions associated with favourable annual costs and/or profits. Since a synthesis problem based upon the maximization of the overall yield does not actually take into account the separation network but maximizes instead only the performance of the reactor system, these results should be considered indicative of the importance of the coupling between the reactor and separator network and should lead to the conclusion that unless the objective does not account for this coupling poor design results are obtained.

9.2. Production of ethylbenzene In this example the alkylation of benzene with

ethylene for the production of ethylbenzene is studied. The process is an intermediate stage of the production of styrene using the direct hydrogenation method and is carried out in the liquid phase. The following first order reversible reactions describe this proces:

C,H, +CH,=CH,, % C,H,CH,CH,

C,H,CH,CH, + CH, = CH 2 2 C,H,(CH,CH,),

Higher order alkylation products as well as other high boiling materials that are obtained as by- products of the process were not considered here. In the alkylators liquid benzene (A) reacts with a gaseous stream of pure ethylene to produce the desired ethylbenzene (B) and the coproduct diethylbenzene (C). In the separation level, B is obtained at a minimum rate of 10 kmol/h while both A and C have potential recycles to the reactor network.

The ranking volatility is aA > air > uc and, thus, the possible separation tasks are: A/BC (column l), AB/C (column 2) B/C (column 3) and A/B (column 4). For

each column separate simulations were necessary for different feed flow rates and compositions. Details for the simulations along with the expressions for the venture costs of the distillation columns and the capital cost of the reactors are given in Appendix B.

In the following examples, a superstructure of four CSTRs and four PFRs is postulated for the reactorseparator-recycle system. With a required minimum production of 10 kmol/h ethylbenzene, the synthesis problem is solved by considering two different cases for the reaction kinetic constants of the alkylation process. The objective is to minimize the annual cost of the plant and the large scale optimization problem consists of 1050 continuous variables, 36 integer variables and 1133 constraints. An MIPS RC 2030 workstation was used for the computational part of the solution algorithm.

Case 1. In this case the reaction constants used are: k, = k; = k, = k; = 0.4 h-i. Starting from a structure of one CSTR and columns 2 and 4, the solution algorithm converged in six iterations and consumed 6.12 CPU seconds per primal and 1.6 CPU seconds per master problem. The minimum annual cost of the plant is $79,272 and the optimal structure, shown in Fig. 13, consists of a single PFR and columns 1 and 3. The required volume for the PFR is V pFR = 3.68 m3 and the fresh feed benzene stream is 11.78 kmol/h. Although a total recycle of 36.28 kmol/h is found for the benzene, a relatively small portion (21.6%) of the produced diethylbenzene (2.27 kmol/h) is recycled.

Case 2. In this case the reaction constants used are: k, = k; = 0.4h- and k, = k, = 4.0 h- . With an initial structure of one PFR and columns 2 and 4, the minimum annual cost is found to be $157,253. The optimal structure, shown in Fig. 14, is a single PFR and columns 1 and 3. The fresh benzene stream is 16.84 kmol/h and the volume of the PFR is V,,, = 5.7X m3. In this case, a much larger benzene

recycle (223.57 kmol/h) is required and a larger portion (40.5%) of the produced diethylbenzene (11.5 1 kmol/h) is led toward the reactor network than in case 1. Such a result should be expected since the

1380

CNLc.,

costann costpap Cost-P c*st+ D, FL,,,

F&y,

FRSi,sk,,

FG I ICSTR

p-FR

IN,

INSi.sk

M iced

Mbt

MY

MM,,,

MRSi,,.sk

MK,m

MV,

MXRC,,

MXSS,,

iv N SK

ANTONIS C. KOKOSSIS and CHRJSTODOULOS A. FLOUDAS

concentration of component m~ M of OuTSi*sc annualized cost of the plant capital cost of column I E L capital cost of reactor i E Z operating cost of column 1 E L distillate product of column 1 EL total fresh feed stream of component mEM fresh feed of reactant m E MI component into reactor iE1 substream of FR+,, leading to subunit sk E SK of reactor i E I PFR total feed of column IE L set of the reactors set of the continuous stirred tank reactors set of the plug flow reactors inlet stream of reactor i E Z inlet stream into subunit skESK of reactor i e IPFR set of the leading columns reaction constant for reaction path rpERP set of the columns set of 1 EL columns with feed n E N intermediate set of I EL columns with bottom n E N intermediate set of I EL columns with distillate no N intermediate set of EEL columns producing recycled m E M} component set of Ic L columns producing recycled M E Mdp) component set of components set of recycled components set of fresh feed components set of desired products set of components participating in reaction path rp E RP set of components feeding column I E L set of components at the bottom of column IEL set of components at the distillate of column IE L reactor network recycle in front of column 1EL substream of MRi,, leading to subunit sk E SK of reactor i E lPFR recycle stream of component m E M to reactor i molar volume of rnE M mixers prior to reactor i E FSTR mixers prior to subunit sk E SK of reactor i E IPFR mixers associated with each recycled component rc E M mixers prior to each leading column sq i

REC, RMi,,

RP RR,,,

RRsi~,r

RSrp,i,s:

SK

SL$ SLOU

SP:, u vi vsi,sk

XFG,,

X&t,, XDm,, XM%mr

XZNm,i

outlet stream out of reactor iEZ outlet stream out of subunit skE SK of reactor i E IPFR total purge stream of component WI EM minimum production of desired component m r Mdp annualized profit of the plant reaction rate of reaction rpE RF in reactor i4zl total recycle of pure m E M stream from reactor i E I towards leading column I E J set of the reaction paths interconnecting stream from reactor i E 1 to reactor k E I substream of RR,,* leading to subunit skESK reaction rate of reaction path rp E RP in subunit skE SK of reactor ieZPFR set of subunits (SCs) that represent each reactor i E ZPFR splitters of each FD,, splitters of each OUT, sale price of component WEE M large number volume of reactor ieZ volume of subunit sk ESK of reactor iEIPFR molar fraction of component rnE M in FC, molar fraction of component m EM in B, molar fraction of component m E M in D1 molar fraction of component me M in ML,, molar fraction of component rnE M in ZNi

xuu Ti,sk,m molar fraction of component rnE M in OUT,

XINSi,sk,nt molar fraction of component me M in

ZNSi,,, XOUTS,,, molar fraction of component me M in

OuTSi,sr, Y, binary variable associated with column

1EL

zi binary variable associated with reactor icl

ZSi.sk binary variable associated with subunit sk E SK of reactor i E IPFR

II

letters payout time income tax rate fixed charge cost of reactor i E 2 variable charge cost of i E I reactor total operation time of the plant fixed charge cost of column 1 EL stoichiometric coefficient of component m E M in reaction path rp E RP functional expression for the yield or selectivity


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Simulation Sciences Inc., 1985, PROCESS Simulation Pro- gram Input Manual. Fullerton, CA.

Sittig, M., 1965, Chemicals from ethylene. Chemical Process Monograph No. 8.

APPENDIX A. THE COMPLETE MINLP FORMULATION

Max Profit or min Cost or max a.

Subject to:

+ a;+,XFC:., >

= 0

+ bf, XFC&

>

= 0

CostfP - yizi + ai vi = 0

Cost= - 1 a z i

cosp + c Cost,P - p c cost;pe = 0 1.s 1 ZEl

Profit - x x SP;t.FC,.;XFC,,, IELdP rnEMdP 1. m

- me-&SP,. 0. FD, - Cost- = 0

FD,-_cFRi.,=O mcMmf iEl

c RR,,, XOUT,,, + 1 M&m, XM&,, ILE, ?hrsh#-=

+ FRi.,f - INi. XINi+, = 0 mgM, iEIs, mffM

BPip.sk- I XIN.%p.sr,- I + OuTSip.slr- I XOUTSip,st- 1.m

+ 1 RRSi,ip.,k X0 u T,.m + FRSi,wt.ml id

rt _Es)fRLr.slr. XMR,.., - BP,,.,, + 1 . XIN%.,, + 1.m - fN%,sh+ I XJW,.,I,+ ,.,.a = 0

ip 6 lpFR, skESK, sk = 1, mrM

FRi,, - c FRzY~,,~.,,, = 0 i E IPFR, m E Mm r&SK

RR,,+ - x RRS,,,,,., = 0 iE I, ipEIPFR Sk&K

MR,,, - x MRSi,,..r = 0 iEIPFR, mcM .TkSK

1382 ANTUNIS C. KOKOSSIS and CHRISTODOULOS A. FLOUDAS

c RR,,, . XOUT;,, + c RM,,, . XOUT(., !sEI 1d

- OUT{, XOUT;,, = 0 isl, meM

BPip,sk XINSip.,k + OU TSic,sk XUU TS,,s,k m

~ OUT,; XOUTi,,, = 0 ipsIPPR, skESK, sk = N,,

~RM,.I. XOUTi,, - 1 MM,,, XFC,,, i mIE.aP

- FC,-XFC,.,=O ieJ,m~M

c XFC,,,, . FG + z MM,., . XfG., IE.q I

lE~

REC, - c MM,.,, . XFC,,, le.4

- CMR~*,,- XMR,,, = 0 rnEM, mreM 1e,

REC, + PG, - x XFC,,; FC, = 0 mr~ Mrc f E r;;;

xXMR,V., - 1 =0 mrEM m

x XIN,,, - 1 = 0 if i me%4

c XOUT,,, - 1 = 0 ieI _M

c XINSi,,r,, ~ 1 = 0 ieiPFR, sk ESK nrEM

c XOUTSi,srs, ~ 1 = 0 iEiPFR, skeSK IncM

1 XFC,_,-- 1 =0 1sL nlEM

c X&., - l=O ICL REM

c XD,,, - l=O IEL WM

IN,. XINi,, - OUT,. XOUT+,,

- Vi .I ~.p,m R,,.i = 0 in IcsT, msM P

INSi,ti X~NS


Table 1. Parameters for example 1

Cost of utilities Steam Cold water

Sale and purchase prices Chlorine Benzene Monochlorobenzcne

Payout time Income tax rate

$21.67/10 kl yr $4.65./10 kl yr

$19.88/kmol $27.98/kmol $92.67/kmol

2.5 yr 0.52

In the fourth case, however, the capital costs of the reactors (CSTRs and PFRs) is type dependent and given by the following expressions:

cos@$ = 25,794.255. Z,,,, + 8178.003. V,,,, @IO) Costy$p = 3894.938-Z,,, + 49.332.715. VP,,. (B11)

The cost of cold and hot utilities, the prices of the reactants and products, and other parameters of the synthesis problem, are given in Table 1.

Example 2 For columns 1 and 2 the simulations were performed at

50, 100, 200 and 300 kmol/h and covered molar compositions 40.0-53.5% of A, 3-Z% of B and 4.5-24% of C. For column 3 the flow rates were 20, 50, 70 and 90 kmol/h and

the molar composition of B ranged from 60 to 79.5%. Finally, for column 4 the feed flow rates were 75, 85,90 and 95 kmol/h and the molar composition of A ranged from 45 to 64.5%. The purity specification of the simulations was set to 99%.

The annualized venture cost of each column was calculated from the simulation results and the values af the cost coefficients (payout time, income tax rate, unit price of utilities) used in the previous example. Within relative error less than lo%, regression analysis on the results provided the following expressions:

Vencosr, = 41,357.320. Y, + FC, (432.709 + 418.420. XFC,.,

- 152.456. XFC,.,)

Vencosty = 622,272.549, Yz - FC,

(562.283 + 633.467. XFC,.,

+ 389.655. XFC,,,)

Vencost\ = 54,966.389. Y, + FC,

(748.609. XFC,., - 588.673 XFC:,,)

Vencosty = 40,526.895. Yd + FC, (382.765 + 432.315 XFC,,,).

0312)

@I31

(B14)

0315)

The capital cost of the reactors was given by the expression:

Cust;Bpita = 12,760.43 - Zi + 14.059.78. Vi (B16) while the operating cost was assumed to be negligible.

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