Synthesis of Optimal Chemical Reactor Networks with Simultaneous

Ind. Eng. Chem. Res. 1996, 35, 4523-4536

4523

Synthesis of Optimal Chemical Reactor Networks with Simultaneous Mass Integration Ajay Lakshmanan and Lorenz T. Biegler* Department of Chemical Engineering and Engineering Design Research Center, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213

The processes of reaction, mixing, and separation are crucial in any chemical process flowsheet. Process synthesis schemes have traditionally considered these processes sequentially in the flowsheet. In the last decade, the superior performance of reactive distillation and membrane reactor units has aroused considerable interest in the area of simultaneous synthesis of reaction, mixing, and separation along with the rest of the flowsheet. Recent incentives to develop new unit operations have resulted in synthesis schemes which focus on specialized cases of reactive distillation. In this paper we consider reaction, mixing, and separation by a mass separating agent. We integrate our reactor network synthesis algorithm, which is a mixed integer nonlinear programming formulation consistent with geometric attainable region concepts, with mass exchange network concepts. The streams in the process are characterized as rich and lean streams, and we develop a reaction-mixing-separation model which may be solved simultaneously with the rest of the process constraints in an attempt to mass integrate the process flowsheet. 1. Introduction Conventional chemical process synthesis schemes are based on units which perform unit operations such as reaction, separation, and heat or mass exchange separately. Various kinds of reactors are used in practice. These include plug flow reactors (PFRs), continuous stirred tank reactors (CSTRs), and fluidized and packed bed reactors. Separation processes include energy separating agent processes like crystallization, evaporation, and distillation and mass separating agent processes like absorption, desorption, and ion exchange. Although these schemes are simple, they neglect the synergy which could be achieved by integrating several distinct unit operations. The integration of synthesis schemes developed for various unit operations has received considerable attention in the past decade. The integration strategies developed may be classified into three categories: heuristic-based techniques, mixed integer nonlinear programming (MINLP) techniques for optimizing flowsheets, and techniques to synthesize the specialized case of reactive distillation units. Some examples of the heuristic-based approaches are the hierarchical design schemes proposed by Douglas (1985) and Glavic et al. (1988) to synthesize unit operations in the flowsheet. On the other hand, MINLP techniques for optimizing flowsheets with reactors and separators were proposed by many researchers. For example, Kokossis and Floudas (1990) extended their reaction synthesis algorithm to include reaction, separation, and recycle units; Balakrishna and Biegler (1993) developed a synthesis strategy for simultaneous reaction, separation, and energy integration. Both these approaches considered sharp split separations. MINLP techniques have also been used to synthesize mass exchange networks. Gupta and Manousiouthakis (1993) proposed an MINLP for estimating the minimum utility cost of mass exchange networks with variable supplies and targets of a single component; Papalexandri and Pistikopoulos * Author to whom correspondence should be addressed. E-mail: [email protected]. FAX: (412) 268-7139.

S0888-5885(96)00371-5 CCC: $12.00

(1994) proposed a multiperiod MINLP model for the synthesis of flexible heat and mass exchange networks based on an integrated superstructure representation of all heat and mass exchange alternatives. Finally, the superior performance of reactive distillation over conventional reaction and separation (Agreda et al. 1990) has triggered considerable research in this area. For example, Doherty and Buzad (1994) used residue curve maps to analyze equilibrium and kinetically controlled reactive distillation systems. Their work was limited to single reactions. Ciric and Gu (1994) developed a synthesis scheme for nonequilibrium reactive distillation processes using MINLP optimization. Other related work in the area of process integration includes improved strategies for simultaneous heat exchanger network and process synthesis (Duran and Grossmann, 1986), state space representations of a distillation network as a composite heat and mass exchange network (Bagajewicz and Manousiouthakis, 1992), synthesis of mass exchange networks with reactive mass separating agents and general nonlinear equilibrium relationships by discretizing the composition range into piecewise linear segments (Srinivas and El-Halwagi, 1994), and simultaneous synthesis of wasteminimizing and profit-maximizing process flowsheets through multiobjective optimization (Lakshmanan and Biegler, 1994). The strategies suggested by Manousiouthakis and co-workers (Bagajewicz and Manousiothakis, 1992 Gupta and Manousiothakis, 1993) and by Srinivas and El-Halwagi (1994) adapt the Duran and Grossmann (1986) strategy for heat exchanger network synthesis to the synthesis of mass exchange networks. For the sake of clarity of presentation of the research work in the rest of the paper, a short review of techniques for reactor network synthesis is given below; a more extensive review was done by Lakshmanan and Biegler (1996). The essence of reactor network synthesis is to find optimal reactor types, sizes, and arrangement which will optimize a specific objective function. Two significant mathematical programming strategies for synthesis of reactor networks are superstructure optimization and targeting. In superstructure optimi© 1996 American Chemical Society

4524 Ind. Eng. Chem. Res., Vol. 35, No. 12, 1996

zation, a fixed network of reactors is postulated, and an optimal subnetwork which maximizes the performance index is derived (Jackson, 1968; Chitra and Govind, 1985; Achenie and Biegler, 1988; Kokossis and Floudas, 1990). Recently, Smith and Pantelides (1995) proposed a synthesis technique for reaction and separation networks using detailed unit operation models. Complete connectivity among the units, both forward and recycle, was assumed in the superstructure. These approaches may be suboptimal since the solution obtained is only as rich as the initial superstructure chosen, and it is difficult to ensure that all possible networks are included in the initial superstructure. On the other hand, targeting for reactor network synthesis is based on the concept of the “attainable region” in concentration space (Horn, 1964). The attainable region is the convex hull of concentrations that can be achieved, starting from the feed point. Recently, Glasser et al. (1987) and Hildebrandt et al. (1990) developed a geometric targeting strategy to map the attainable region for the processes of reaction and mixing. PFR, CSTR, and differential sidestream reactor (DSR) trajectories and mixing lines were drawn to cover the attainable region. Although this is a rigorous and elegant method, the geometric techniques are difficult to apply beyond three dimensions. In addition, Feinberg and Hildebrandt (1995) developed rigorous mathematical proofs for various properties of the attainable region in higher dimensions. In the past decade, researchers have found that simultaneous reaction, mixing, and separation is an attractive alternative to conventional chemical processing schemes. This is attributed to the fact that equilibrium limitations on a chemical reaction may be overcome by simultaneous reaction and separation, where products are continuously removed from the reaction zone, thus enhancing overall conversion. Also, in the case of competing reactions, overall selectivity may be increased by continuously separating desired products from reactants. In addition, mixing with a sidestream allows for partial manipulation of concentration and temperature. Finally, such a unit may reduce the annualized cost of process equipment, since it reduces the number of units and provides for direct mass and heat integration. The superior performance of reactive distillation units (Agreda et al., 1990) and membrane reactors (Itoh, 1987) emphasizes a need for an efficient synthesis scheme for simultaneous reaction, mixing, and separation. This study proposes an MINLPbased approach that combines the reactor network synthesis algorithm (Lakshmanan and Biegler, 1996) with mass exchange network concepts. A simultaneous reaction-mixing-separation network is developed in an attempt to synthesize mass integrated process flowsheets. The next section reviews our optimization-based reactor network synthesis algorithm. In section 3, this algorithm is extended, and the reactor models are modified to consider simultaneous reaction and mass integration. The Duran and Grossmann (1986) strategy for synthesis of heat exchange networks is adapted to synthesize the mass exchange network targets. Finally, in section 4, we solve a few complex example problems and illustrate the feasibility of this approach. 2. Algorithm for Reactor Network Synthesis Motivated by the disadvantages of previous strategies for reactor network synthesis outlined in the introduction, Lakshmanan and Biegler (1996) proposed a com-

Figure 1. Reactor module.

Figure 2. Differential sidestream reactor.

Figure 3. Discretized DSR element.

bination of superstructure and targeting techniques to synthesize optimal reactor networks. This approach incorporates attainable region properties derived from Feinberg and Hildebrandt (1995). Some of these properties are as follows: (i) the boundary of the attainable region in reaction and mixing consists of plug flow reactor trajectories and straight lines, (ii) recycle streams need not be considered to map the boundary, and (iii) CSTR, PFR, and DSR trajectories may be used to map the entire boundary of the attainable region. The reactor network synthesis technique is constructive. It considers multiple reactor paths at each stage by targeting the attainable region using reactor modules. A reactor module consists of a PFR and a CSTR in two dimensions (the space represented by two independent variables such as residence time, temperature, conversion, etc.) and a differential sidestream reactor (DSR) and a CSTR for higher dimensions. A separate PFR model need not be considered in higher dimensional problems, since the DSR without the sidestream is a PFR. A typical reactor module is shown in Figure 1, and illustrations of the DSR are shown in Figures 2 and 3. A binary variable is associated with each reactor path in a module. If the DSR is chosen in the ith reactor module, then Yid, the binary variable associated with it, is set equal to 1. Additional reactor modules are considered, and the superstructure is developed in a constructive manner. The algorithm is formulated as an MINLP. In order to keep the MINLP as compact as possible, several attainable region properties are incorporated into it. The steps involved in the constructive stagewise algorithm are as follows: (1) Solve an isothermal segregated flow model to obtain a lower bound on the solution. This formulation is often a linear program with a guaranteed global solution (Balakrishna and Biegler, 1993). (2) Initialize the first reactor module with the solution obtained from step 1 and optimize it with respect to a specific objective. The inlet conditions to this reactor module are the feed conditions. This yields an initial target to the attainable region. In addition, it does not

Ind. Eng. Chem. Res., Vol. 35, No. 12, 1996 4525

eliminate the other path; the binary variables associated with each path are set to 1 or 0. (3) Extend the reactor module with an additional reactor module. The feed to the second module is a convex combination of the fresh feed and the exit of the first module. If the extension improves the objective, then further extensions need to be considered, or else the optimal network is assumed to have been found. (4) Ensure that the feed to the ith reactor module extension may be the exit of any one or a combination of the exits of the previous (i - 1) modules or the initial feed. (5) Account for bypass streams by ensuring that the exit from the ith reactor module plus the bypasses from the exits of any one or a combination of the previous (i - 1) reactor modules forms the inlet to the (i + 1)th reactor module. The problem is formulated as in (P2.1):

max J(Xexit,τ)

(P2.1)

q,f,T

XCSTR(i) ) R(XCSTR(i),TCSTR(i))τc(i) + X0(i)

(2.1)

dXDSR(i)/dR ) R(X(R(i)),T(R(i))) + (q(R(i))Qside(i)/Q(R(i)))(Xside(i) - X(R(i))) (2.2) XDSR0(i) ) X0(i) XDSRexit(i) )

τ)

∫0t ∫0R max

∫0t

1)

∫0t

1)

∫0t

(2.3)

max

f(R(i))X(R(i)) dR(i)

(2.4)

max

f(R(i)) dR(i)

(2.5)

max

q(R(i)) dR(i)

(2.6)

(q(R(i)′)Qside(i)/Qexit(i) - f(R(i)′)) dR(i)′ dR(i) (2.7)

(i)

TDSRexit(i) )

∫0t

max

f(R(i))T(R(i)) dR(i)

(2.8)

i-1

Fif )

∑ Fki-1 k)0

(2.9)

i-1

FifXif )

∑ Fki-1Xk

(2.10)

k)0

Xif ) Xicin ) Xidin

(2.11)

Fif ) Fic + Fid

(2.12)

FifXi ) FicXic + FidXid

(2.13)

1 ) Yic + Yid

(2.14)

Nmod

FifXi ) Xexit ) XNmod,exit 0 e Fid e UYid

FijXi ∑ j)1

(2.15)

0 e Fic e UYic Yic ∈ {0,1}

Figure 4. Individual reactor module.

Yid ∈ {0,1}

Equations 2.1-2.8 constitute the reactor module at stage i made up of a CSTR (2.1) and a DSR (2.2-2.8). Equations 2.9 and 2.10 may be understood by studying Figures 4 and 5. In Figure 4, the feed conditions (Fif, Xif) to ith module may be the exit conditions of any one

Figure 5. Parallel structure.

or a combination of the previous (i - 1) modules. Hence, when the ith reactor extension is considered, bypasses from the exits of the previous (i - 2) modules and the feed are automatically considered. Similarly, parallel reactor structures up to the (i - 1)th module are also accounted for. This structure keeps the constructive MINLP algorithm as small as possible. In the ith reactor module, either the CSTR or the DSR may be chosen, and the exit conditions from the ith reactor module (Xi) are determined appropriately. This is represented by eqs 2.11-2.14. The exit stream from the ith reactor module forms the inlet stream to any one or a combination of reactor modules (i + 1) to Nmod (eq 2.15). Figure 5 depicts the parallel structure; here, the feed streams to the ith and (i + 1)th modules are identical, and the exits of these modules combine to form the inlet to the (i + 2)th module. Similarly, up to Nmod parallel structures may be realized, where Nmod is the number of reactor modules or stages in the optimization. The resulting compact MINLP structure is a constructive technique. Alternative reactor paths are considered by the use of reactor modules. The optimal reactor structure is determined at each stage, but other paths are not eliminated. Lakshmanan and Biegler (1996) discuss the advantages and disadvantages of this strategy. In addition, they suggest a good initialization procedure and remedies for possible local solutions, due to the bilinear eqs 2.10 and 2.13. In the next section, the reactor network synthesis algorithm is extended to consider mass integration. 3. Simultaneous Reaction, Mixing, and Separation Reaction, mixing, and separation have traditionally been considered by researchers as sequential processes in flowsheet synthesis. In some cases, simultaneous reaction and separation of components is more profitable than the sequential reaction followed by separation. Balakrishna and Biegler (1993) proposed a model for developing reaction-separation targets by considering sharp splits. The separated components were mixed with the reactor exit to achieve intermediate splits. In this section, we propose a general nonisothermal reactor model that allows for simultaneous reaction, mixing, and separation by mass exchange. The model shown


dXc(R)/dR ) R(Xc(R),T(R)) + (Qsideq(R)/Q(R))(Xcside C

∑γc(R)Xc(R)/F - γc(R)) c)1

Xc(R)) + Xc(R)(

Figure 6. Reaction-mixing-separation model.

in Figure 6 is based on the DSR model (Lakshmanan and Biegler, 1996) shown in Figure 2, except that separation occurs along the length of the reactor. The features of the model include (i) realistic separations of components by modeling the separation network as a mass exchange operation with a lean stream which facilitates nonsharp splits, (ii) sidestream mixing along the length of the reactor, to manipulate the temperature and composition, (iii) removal of components from the reaction-mixing system by the lean stream. The separation (γ(R)) and mixing profiles (q(R)) are modeled as a function of time along the length of the reactor. The cost of separation is a function of the lean stream flow rates, which are optimized by considering simultaneous mass integration and reaction. The sidestream which mixes with the reactor stream could be the feed stream to the reactor or any other process stream; sidestream addition alters the composition and temperature of the reactor stream according to the linear mixing rule. The reactor-mixer-separator model is incorporated with the constructive MINLP approach for reactor network synthesis to develop optimal reactionmixing-separation networks. 3.1. Reaction, Mixing, and Separation Model. The simultaneous reaction, mixing, and separation model is shown in Figure 6. The model has two distinct sections: the upper section is the reaction and mixing section, and the lower section is the mass exchange section. In Figure 6, Qin, Qside, Qout and Xin, Xside, Xout are the inlet, sidestream, and exit reactor volumetric flow rates and mass concentration vectors, respectively. Lm is the flow rate of the mth inert lean stream, ym in and ym out are the inlet and outlet composition vectors of the key components (kg key/kg inert) in the mth lean stream, entering and leaving the mass exchange section of the reaction mixing separation model. Here, a key component is defined as any component that may be separated in the mass exchange section. Define R to be a measure of time along the length of the reactor. Let mc(R) be the mass of components c in the reactor at R. Postulate a separation vector γ(R), and the separation function γc(R) (c ) 1, C; the number of components in the system) is a measure of the mass exchange taking place between the rich reactor streams and a stream lean in component c flowing through the mass exchange section. It is defined such that, between times R and R + δR, a mass fraction of species c equivalent to γc(R)δR leaves the reaction-mixing section and enters the mass exchange section of the model. The mass entering the mass exchange section is equal to the mass absorbed into the lean stream. In addition, a mixing function q(R) is defined at R; q(R)δR is the fraction of Qside entering the reaction section between R and R + δR. Constant density systems are considered for simplicity of the model. But, variable density systems may be considered by modifying the equations appropriately. A differential mass balance for species c on δR yields eq 3.4; the differential change in composition depends on the reaction rate, mixing, and separation that occur at R.

(3.4)

The first two terms of eq 3.4 are similar to the DSR equation (eq 2.2), and the third term accounts for separation. In fact, if all the elements of the vector γ(R) are identical (which implies that there is no relative separation between the components), then the third term vanishes, since

∑c γc(R)Xc(R) ) γc(R)∑c Xc(R) ) γc(R)F, i.e., ∑γc(R)Xc(R)/F - γc(R) ) 0 c

c ) 1, C

The differential change in volumetric flow rate over the section δR may be expressed as C

dQ(R)/dR ) Qsideq(R) -

∑γc(R)mcr(R)/F c)1

(3.5)

The exit mass flow rates from the reaction-mixing section (mcr exit), the mass exchange section (mcl exit), and the overall outlet flow (mc out) are given by eqs 3.6-3.8, and eq 3.9 represents the overall mass balance.

mcr exit ) (1 - γc(Rmax))mcr(Rmax) mcl exit )

∫0R

max

γc(R)mcr(R) dR

mc out ) mcr exit + mcl exit C

mc side +

(3.6) (3.7) (3.8)

C

∑mc(0) ) c)1 ∑mc out c)1

(3.9)

where the mass flow through the reaction-mixing section at R is defined as

mcr(R) ) Q(R)Xc(R)

(3.10)

The mixing function q(R) is normalized to 1, in eq 3.11, and eq 3.12 represents the mean residence time (Appendix A) of the components in the reaction-mixing section.

1)

∫0t

max

q(R) dR

(3.11)

∫0t ∫0R(q(R′)Qside -

τ ) (1/(Qin + Qside))

max

C

∑γc(R′)mcr(R)/F) dR′ dR c)1

(3.12)

Equations 3.4-3.12 represent the reaction-mixing section. To complete the description of the reactionmixing-separation model, a constraint which ensures thermodynamic feasibility of mass exchange and a mass balance constraint between the rich and lean streams must be considered. The thermodynamic feasibility constraint requires a finite characterization of the streams which exchange mass. Since the component separated may be a product of the reaction, streams at intermediate points along the reactor need to be included. Hence, the model is solved by discretizing the reaction-mixing-separation unit into finite elements, which yields a finite number of rich reactor streams and


(mc

out)

are represented as

mcr exit ) XcNe,end(1 - γcNe)QNe

(3.22)

Ne

Xci endγciQi ∑ i)1

mcl exit )

(3.23)

Figure 7. Discretized reaction, mixing, and separation model.

mc out ) mcl exit + mcr exit

lean streams to be included in the mass integration. Orthogonal collocation on finite elements may be used to discretize the reaction-mixing-separation model and make it easier to implement. The discretized DSR structure outlined in Figure 3 is adapted to suit this purpose. Figure 7 shows the discretized reactionmixing-separation model with separation at the end of each reacting element and mixing in front of the element. The γc(R) in the original model reduces to a mass split fraction at the end of each finite element i given by γc,i. The control profiles q and γ are assumed to be piecewise constant over each finite element. In the general nonisothermal case, temperature can also be a control profile. As a result of the discretization, eq 3.4 in the continuous model is represented by the set of eqs 3.13-3.16. Here, eq 3.13 represents the change in composition due to mixing and determines the inlet composition (Xci in) to the ith reacting element. Equation 3.14 represents the change due to reaction, and eq 3.15 extrapolates the compositions at the K collocation points to find the composition at the end of the ith element. Equation 3.16 determines the composition at the end of ith separator (Xci u).

Here, Ne is the number of finite elements in the reactor discretization. The normalization eq 3.11 is converted to a summation (due to the assumption that qi is piecewise constant) as shown in eq 3.25, and the mean residence time (τ) in the reaction-mixing section may be represented by eq 3.26. A simple proof of eq 3.26 is given in Appendix A.

Xci in ) φiXc side + (1 - φi)Xci-1 u

(3.13)

K

∑ XcikLk′(Rj) - R(Xcij,Tij)∆Ri ) 0

k)2

j ) 1, K;

Xcik ) Xci in ∀ k ) 1 (3.14) K

Xci end )

∑ XcikLk(Rend)

(3.15)

k)1

Xci uQi u ) mcri u

(3.16)

Similarly, eq 3.5, the differential change in volumetric flow rate, is split into eqs 3.17, which determines the volumetric flow rate in the ith element (Qi), and 3.18, which determines the volumetric flow rate after separation at the ith separator (Qi u).

Qi ) qiQside + Qi-1 u Qiu ) Qi[1 where

∑c Xci endγci/F]

φiQi ) qiQside

(3.17) (3.18) (3.19)

The mass flow rates at the end of the ith reacting element (mcri end) and the ith separator (mcri u) may be represented by eqs 3.20 and 3.21.

mcri end ) Xci endQi

(3.20)

mcri u ) Xci endQi[1 - γci]

(3.21)

Equations 3.6-3.8, which represent the mass flow rates leaving the reaction-mixing section (mcr exit), the mass exchange section (mcl exit), and the overall exit flow

(3.24)

Ne

qi ) 1 ∑ i)1 Ne

∑ i)1

(3.25)

C

Ri (

∑ γci-1 mcri-1 end/F - qiQside) +

c)1

RNeQNe ) τ(Q1 + Qside) (3.26) If there are sufficient elements, the discretized model represents the continuous case accurately. Moreover, in most practical situations, it is possible to maintain only a discrete separation profile. The only cases where a continuous separation profile occurs is reactive phase equilibria, and this may be approximated by using sufficiently small elements. Also, Balakrishna and Biegler (1993) have proved that, for reasonably small finite elements, the discretized separation target model is equivalent to the continuous model. In order to comprehend mass integration of rich and lean streams and formulate a model for the mass exchange section, it is essential to describe a mass exchange network (MEN) model. 3.2. Mass Exchanger Network Model. We consider the separator at the end of each reacting element to be a mass exchanger. Each reacting element yields a stream rich in component c, called a candidate “rich stream”, which flows through the separator at the end of that element. Hence, the Ne finite reactor elements yield Ne candidate rich streams (NR ) Ne), which may be mass integrated with m (m ) 1, NL) candidate lean streams (either process or external) available for mass exchange. The mass exchanger performs a mass transfer operation between the set of rich streams and the set of lean streams. Recently, researchers have developed strategies to target a MEN based on the heat exchanger network (HEN) targeting strategy of Linnhoff and Hindmarsh (1983). For this technique to be useful to develop a MEN, the concentration intervals or the inlet and outlet compositions of all the streams need to be known in advance (El-Halwagi and Manousiouthakis, 1990). However, while attempting to simultaneously synthesize the MEN along with the process, the supply compositions of the streams involved vary depending on the process conditions. Simultaneous MEN and process synthesis is made possible by eliminating the need for fixed concentration intervals. The Duran and Grossmann (1986) technique for developing targets for HEN synthesis with varying inlet and outlet temperatures is adapted to develop targets for MENs. Here, we propose a model for simultaneous MEN and reactor


Figure 8. Mass exchanger model.

network synthesis in an attempt to synthesize a mass integrated process flowsheet. A countercurrent mass exchanger is shown in Figure 8, xci in, xci out and ycm in, ycm out are the inlet (supply) and outlet (target) key component mass compositions (kg key component/kg inert components) of the ith rich and mth lean streams, respectively. Here, key components are components which may be separated in the mass exchanger, and inerts are components which may not be separated. The key component mass compositions are defined in terms of the mass concentrations (X) used in the analysis of the reaction-mixing section as

xci in ) Xckey i end/

∑

Xgi end

(3.27)

∑

Xgi u

(3.28)

g;g*ckey

xci out ) Xckey i u/

Figure 9. Pinch diagram.

intervals over which the equilibrium relationship is piecewise linear. In addition, only isothermal mass exchangers are considered. If the temperature varies inside the mass exchanger, then chemical potentials should be used instead of the mass compositions of components used in this analysis. Hence, the set of pinch point candidates for the rich (xp) and lean streams (yp) may be defined by

xp ) {xci in,σ(ycm in + )}

g;g*ckey

i.e., the inlet and outlet compositions to the ith mass exchanger are defined in terms of the composition at the end of the ith reacting element and the upstream composition at the end of the separator, respectively. In Figure 8, mri inert and Lm are the mass flow rates of the inert components in the ith rich stream and mth lean stream, respectively:

yp ) {ycm in,f(xci in) - } i ) 1, NR, m ) 1, NL

where σ(y) ) f-1(y) is the inverse of the equilibrium relationship. The mass lost by the rich streams must be equal to the mass gained by the lean streams: NR

∑ i)1

NL

mri inert(xci in - xci out) )

∑ Lm(ycm out - ycm in)

m)1

(3.30)

C

mri inert )

∑

Xgi endQi

(3.29)

g)1;g*ckey

The rich streams in this case are process streams. If the mass exchanger served the purpose of an enricher rather than a stripper, then the rich streams could be streams external to the process. The lean streams may be process streams or external streams which are available at a cost. The composition of the components in the rich and lean streams are limited by equilibrium relationships. In order that mass transfer occurs between the rich and lean streams, there should be a driving force for mass transfer, also called a minimum difference in composition () between component compositions in the rich and lean streams. Hence, for any specific rich stream composition x, the maximum thermodynamically achievable lean stream composition (y) is given by

y ) f(x) - where y ) f(x) is the composition of the lean stream in equilibrium with the rich stream composition x. This relationship enables us to locate the pinch point compositions, the points where the rich and lean stream compositions are at the limit of thermodynamic feasibility. In order to simplify the analysis, only those equilibrium relationships which define a one-to-one mapping between x and y are considered in this study. This assumption ensures that the set of pinch point candidates is completely defined by the inlet compositions of the streams involved. If nonlinear equilibrium relationships need to be considered, then the composition interval may be discretized into several small

In addition, the second law of thermodynamics should be satisfied for feasible mass exchange between the rich and lean streams (Figure 9). The composite rich stream should always be above the composite lean stream in a composition vs mass load diagram, which translates to

Zp(x,y) e 0

(3.31)

where

Zp(x,y) ) [mass lost by rich streams below the pinch point candidate] [mass gained by the lean stream below the pinch point candidate] ) mri inert[max(0,xp - xci out) - max(0,xp - xci in)] -

∑i

Lm[max(0,yp - ycm in) - max(0,yp - ycm out)] ∑ m i ) 1, NR, m ) 1, NL

A pinch is observed when Zp(x,y) ) 0. In the above analysis, it was necessary to define the variables in terms of key component compositions (kg key/kg inert) instead of mass fractions of components, since the mass flow rates of rich stream i and lean stream m vary as they pass through the mass exchanger. The inlet compositions of the lean streams are known (either given or determined by the process conditions), while the outlet compositions are bounded and are determined by eqs 3.30 and 3.31. Similarly, bounds on the rich stream compositions are available. The inlet compositions of the rich streams are determined by the reaction rate expressions and the mixing relationships, and the


outlet compositions are determined by eqs 3.30 and 3.31. Finally, the max operators in eq 3.31 are nondifferentiable; they are smoothed using the following approximation (Balakrishna and Biegler, 1993): 2

2 0.5

max(0,f) ≈ (f + η ) /2 + f/2

γ,T,φ

COmLm ∑ m

(P3.1) (3.13)

(3.21)

mcr exit ) XcNe,end(1 - γcNe)QNe

(3.22)

Ne

mcl exit )

Xci,endγciQi ∑ i)1

(3.23)

mc out ) mcl exit + mcr exit

(3.24)

Ne

qi ) 1 ∑ i)1 Ne

(3.25)

C

∑ ∑ γci-1mcri-1 end/F - qiQside) + RN i)1 c)1 Ri(

e

QN e )

τ(Q1 + Qside) (3.26)

∑

Xgi end

(3.27)

∑

Xgi u

(3.28)

∑ Xgi endQi g)1;g*ckey

(3.29)

xci in ) Xckey i end/

g;g*ckey

xci out ) Xckey i u/

g;g*ckey C

mri inert ) NR

∑ i)1

NL

mri inert(xci in - xci out) )

∑ Lm(ycm out - ycm in)

m)1

(3.30) Zp(x,y) e 0

(3.31)

where

X(0) ) Xo u ) Xo; Q(0) ) Qo u ) Qo Xcik ) Xci in ∀ k ) 1; 0 e φi e 1 ZP(xc,yc) )

∑i mri inert[max(0,xp - xci out) - max(0,xp - xci in)] Lm[max(0,yp - ycm in) - max(0,yp - ycm out)] ∑ m yp ) {ycm in,f(xci in) - }; xp ) {xci in,f-1 (ycm in + )}; ycm in ) ycm supply;

K

∑ XcikLk′(Rj) - R(Xcij,Tij)∆Ri ) 0

mcri u ) Xci endQi[1 - γci]

i ) 1, NR, m ) 1, NL

Xci in ) φiXc side + (1 - φi)Xci-1 u

k)2

(3.20)

(3.32)

where η is a small number close to 0. A typical value of η appears to be about 4 orders of magnitude less than f. Figures 6-8 depict a countercurrent contacting pattern between the candidate rich reactor streams and the candidate lean streams. However, the discretized reactor-mixer-separator NLP model (eq P3.1) picks the optimal reactor-mixer-separator target, the pinch points, and the optimal lean stream flows irrespective of the contacting pattern. Formulation P3.1a further extends eq P3.1 and finds the best objective when only unidirectional flow of a lean stream (cocurrent or countercurrent contacting patterns) is considered through each of the individual separators in the reactionmixing-separation model. 3.3. Discretized Reaction-Mixing-Separation Model. The discretized reaction-mixing model equations (eqs 3.13-3.26) and the MEN equations (eqs 3.273.32) may be used to model the reactor-mixerseparator (Figure 7) as an NLP. The formulation is shown in P3.1; it allows the NR ()Ne, the number of finite reactor elements) candidate rich streams to exchange mass with the NL candidate lean streams irrespective of the flow pattern in the reactor-mixerseparator. The supply compositions of the lean streams are known in the case of an external lean stream or determined by the process conditions in the case of a process lean stream. As noted in the previous section, the outlet composition of the candidate lean stream is greater than (or equal to) the inlet composition. The lean stream flows are optimized by eqs 3.30 and 3.31. Please note that, in the preceding discussion, the streams are called candidate streams to allow for the possibility that they may not exchange mass. The solution of formulation P3.1 gives an optimal target for the reaction-mixing-separation model, the separation profile γci, the mixing profile φi, the temperature profile Ti, and the optimal lean stream flows by minimizing the utility costs of the mass separating agents (COmLm).

max J(mc out,τ) -

mcri end ) Xci endQi

j ) 1, K

ycm in e ycm out; xci out e xci in

(3.14) K

Xci end )

∑ XcikLk(Rend)

(3.15)

k)1

Xci uQi u ) mcri u

(3.16)

Qi ) qiQside + Qi-1 u

(3.17)

C

Qi u ) Qi[1 -

∑Xci endγci/F]

(3.18)

c)1

φiQi ) qiQside

(3.19)

The reaction-mixing-separation model (P3.1) may be incorporated within the MINLP approach for reactor network synthesis (P2.1; Lakshmanan and Biegler, 1996) to find optimal reactor networks. In this case, the reactor modules in the MINLP formulation consist of the reaction-mixing-separation unit, in place of the DSR, and a steady state model of a CSTR with separation (Appendix B), in place of an ordinary CSTR. Flow Pattern Selection: Simple Cases. To determine the flow pattern of the lean streams inside the mass exchange section of the reaction-mixing-separation model, a simple NLP model (P3.1a) with the flow rates of the lean streams fixed at the optimal values


ycmi in e ycm i e ycmi out, ycmi in e ycm i+1 e ycmi out ycm supply e ycm Ne+1 e ycm supply + ξm ycm supply e ycm 1 e ycm supply + (1 - ξm) ycmi ) mcm i/Lm; m ) 1, NL; i ) 1, Ne + 1; mcm 0 ) mcmNe+2 ) 0 Figure 10. Discretized reaction-mixing-separation model. Flow pattern selection (ξ ) 0).

0 e ξm e 1; ξm ) Ψm; Ψm{0,1}, Ψm ) 1 cocurrent; Ψm ) 0 countercurrent 4. Process Examples

Figure 11. Process flowsheet, example 1.

obtained from P3.1 may be solved. In formulation P3.1a, only those m candidate lean streams which were picked by (P3.1) are included. Each of the lean streams is discretized into Ne substreams which flow through the Ne distinct separators. Hence, the MEN involves NR ()Ne) rich streams and mNe lean streams. Here we consider only unidirectional flow through the Ne separators, either countercurrent or cocurrent with the rich stream. The inlet and outlet compositions and mass flow rates of these mNe substreams are determined by eqs 3.32-3.36. Binary variables Ψm are defined to pick the overall direction of flow for each of the m lean streams, and continuous variables ycm i and mcm i (c ) 1, C; m ) 1, NL; i ) 1, Ne + 1) are defined to specify inlet (ycmi in, mcmi in) and outlet (ycmi out, mcmi out) compositions and flow rates of the mNe substreams, as shown in Figure 10. In additon, a set of continuous variables ξm ()Ψm) is defined so that the binary variables appear only linearly in the MINLP formulation. The initial conditions are determined by the expressions shown below formulation P3.1a, depending on whether the overall flow is co- or countercurrent. The lean stream flows are fixed at the values obtained by solving (P3.1), and the constraints of formulation P3.1 are solved along with eqs 3.32-3.36. Of course, γci at the ith exchanger may be zero, in which case the lean stream flows through the exchanger without any change in its composition.

max J(mc out,τ) γ,T,φ

k1

A + B 98 C k2

C + B 98 P + E k3

P + C 98 G The reaction rate vector for the components A, B, C, P, E, and G respectively is given by

R(x) ) [-k1XAXB; -(k1XA + k2XC)XB; 2k1XAXB 2k2XBXC - k3XPXC; k2XBXC 0.5k3XPXC; 2k2XBXC; 1.5k3XPXC]

(P3.1a) where

(3.13 - 3.31) ycmi in ) ξmycm i + (1 - ξm)ycm i+1

(3.32)

ycmi out ) (1 - ξm)ycm i + ξmycm i+1

(3.33)

mcmi in ) ξmmcm i + (1 - ξm)mcm i+1

(3.34)

mcmi out ) (1 - ξm)mcm i + ξmmcm i+1

(3.35)

mcm i ) ξmmcm i-1 + ξmmcri-1 endγci-1 + (1 - ξm)mcm i+1 + (1 - ξm)mcri end γci (3.36) where

Mass integrated process flowsheets may be developed by incorporating formulations P3.1 and P3.1a for the reactor-mixer-separator into the MINLP algorithm P2.1. In P2.1, the DSR equations are substituted by the constraints of P3.1, and the steady state model of a CSTR (Appendix B) is used instead of a CSTR. Example 1 illustrates how mass integration could benefit a simple process flowsheet, and example 2 shows how P3.1 could be used to model a membrane reactor. Example 1: Reactor flowsheet integration with emphasis on mass integration of components in the reactor network. The reactor network synthesis algorithm is coupled with the simultaneous solution strategy for reactor flowsheet integration. In addition, mass integration of components in the reactor network with process and external lean streams may be considered. A modified form of the Williams-Otto flowsheet (Ray and Szekely, 1973) is shown in Figure 11. The raw materials A and B are fed to the reactor, where they react to form an intermediate C, desired product P, and byproduct E, which react further to produce waste product G:

k1 ) 5.9755 × 109 × exp[-457.2958/0.08206T] wt frac h-1 k2 ) 2.5962 × 1012 × exp[-571.6197/0.08206T] wt frac h-1 k3 ) 9.6283 × 1015 × exp[-762.1596/0.08206T] wt frac h-1, T ) 313 K The X’s denote the weight fractions of the components. The effluent from the reactor is fed to a mass exchanger, where the desired product P is preferentially absorbed into a lean stream. The inert components at the exit of


the mass exchanger are cooled in a heat exchanger, followed by a decanter, where the waste product G is separated from the other components. The waste G is then treated in a waste treatment plant. A fraction of the remaining components is purged and used as fuel, while the rest is recycled to the reactor. The desired product P, which is absorbed into the lean stream, is recovered in a distillation column. The bottoms product from the distillation column is a pure lean stream, also called a candidate process lean stream while mass integrating the flowsheet. In addition, a makeup lean stream is also available. Three cases are considered. Case 1: The simpler case of sequential reaction and mixing followed by separation in a mass exchanger. Case 2: Sequential reaction and mixing followed by separation in a mass exchanger and mass integration of the flowsheet. Case 3: Simultaneous reaction, mixing and separation and mass integration of the flowsheet. The objective function considered in the optimization is an annualized net profit which includes sales, cost of raw materials, waste treatment cost, sales and research expenditure, reactor utility cost, depreciation costs, annualized capital cost, and lean stream and column utility costs:

J ) (8400(0.5FP + 0.0068FD - 0.02FA - 0.06FB 0.1FG) - 0.124(8400)(0.5FP + 0.0068FD) 2.22FR - 0.1(6FRτ) - 0.33(6FRτ) - 8400CL where FA, FB are the mass flow rates of A, B. FP is the mass flow rate of pure P at the top of the distillation column. FD is the purge rate, and FR is the total flow of components within the reactor. The variable τ includes the residence time in the complete reactor network. The reactor cost is a function of residence time and is irrespective of the reactor type. This assumption is reasonable since the capital cost of the reactor is usually much smaller than the other costs. CL includes the cost of the makeup lean stream used and the utility cost of recovering any process lean streams and products (CL ) (5 × 10-3)Fext lean + utility cost of the column). The process lean stream recovered as the bottoms product of the distillation column is costed as a function of the utility costs of the column. The equilibrium relationship between the mass composition of P in the lean stream and the rich reactor stream is given by

yLP ) 0.4xRP The maximum mass composition of P in the lean stream is 0.2, and an unlimited supply of the external lean stream, at a cost of 5 × 10-3 $/kg of lean stream, is available. A supply of steam for the reboiler is available at a cost of 6.16 × 10-3 $/kg. The maximum allowable waste production is 300 kg/h of G, and the desired rate of production of P is 400 kg/h. Case 1: The simpler case of sequential reaction and mixing followed by separation in a mass exchanger. The flowsheet is as shown in Figure 11, where the reactor network synthesis algorithm (Lakshmanan and Biegler, 1996) was used to find the optimal reaction-mixing network and the mass exchanger was modeled using the mass exchanger network equations described in the previous section. A minimum composition difference () of 10-4 was used for the pinch analysis. The problem was modeled as an MINLP with 221 constraints, 225

continuous variables, and two binary variables (which choose the best reactor inside the reactor module). The model was solved in 0.52 CPU s on a HP-UX 9000/720. The nondifferentiable max expressions were smoothed using an η value of 10-4 in (3.32); no ill-conditioning of the equations was observed at this value of η. Only one rich stream (the exit stream from the reaction-mixing network) and one lean stream (the external lean stream) are involved in the MEN targeting. The streams are pinched at the inlet concentration of the lean stream to the mass exchanger. The maximum annualized profit was found to be negative, i.e., a loss of $2 011 880/yr was incurred to produce 400 kg/h of product: 612 kg/h of A and 1517 kg/h of B were consumed, 300 kg/h of waste G was produced, and 17 449 kg/h of the external lean stream was used in the mass exchanger. The optimal reactor network was found to be a PFR with a residence time of 0.10 h, and no side-stream mixing was observed. Case 2: Sequential reaction and mixing followed by separation in a mass exchanger and mass integration of the flowsheet. The same problem as in case 1 is solved with mass integration of other lean streams in the process with the rich reactor stream. In particular, the process lean stream (lean stream recovered at the bottom of the distillation column) is included in the MEN targeting. Hence, two lean streams are available for mass exchange with the rich stream from the reactor. The problem is modeled as a MINLP with 230 constraints, 234 continuous variables, and two binary variables and was solved in 0.55 CPU s on a HP-UX 9000-720 workstation. The streams are pinched at the process lean stream inlet composition, and the rich stream exchanges all its mass with the process lean stream. The maximum annualized profit was much better than that in case 1, but still negative, i.e., a loss of $546 126.91/yr was incurred: 400 kg/h of product P was produced, 612 kg/h of A and 1517 kg/h of B were consumed, and 17 449 kg/h of the process lean stream was used instead of the external lean stream. The flow rate of the process lean stream in this case is identical to the flow rate of the external lean stream in the previous case because complete separation of the desired product (P) from the lean stream is assumed in the distillation column, which means that the bottoms product is a pure lean stream. The higher profit is because the external lean stream is not used for mass exchange and the optimal reactor network is a PFR with a residence time of 0.1 h. Case 3: Simultaneous reaction, mixing, and separation and mass integration of the flowsheet. Simultaneous reaction, mixing, and separation are considered. The reactor modules in the reactor network synthesis algorithm consist of a reaction-mixing-separation unit, instead of a DSR, and a steady state model of a CSTR with separation instead of an ordinary CSTR. The reaction-mixing-separation unit is represented by the cross-flow discretization shown in Figure 7. Ne, the total number of reacting segments, is chosen to be 5. This discretization corresponds to five candidate rich reactor streams (R1-R5). Two candidate lean streams, L1 and L2, are available in the mass exchange section: L1 is the external makeup stream, and L2 is the process lean stream obtained as the bottoms product of the distillation column. Hence, five candidate rich streams and two candidate lean streams are included in the MEN targeting. The maximum allowed residence time (tmax) is constrained to remain below 0.1 h. The equilibrium


Figure 12. Pinch curve. Simultaneous reaction-mixing-separation, example 1.

Figure 14. Pinch curve. Simultaneous reaction-mixing-separation, cocurrent flow, example 1. Table 1. Comparison of Results in the Sequential and Simultaneous Cases

Figure 13. Optimal reaction-separation network, example 1.

relationships and the minimum composition difference were the same as those used in the sequential case. The model was incorporated into the reactor network synthesis algorithm. The MINLP optimization models were formulated within the GAMS modeling system, and DICOPT++ (Viswanathan and Grossmann, 1990) was used to solve the optimization problem. The results of the optimization (362 constraints, 368 continuous variables, and two binary variables, 1.11 CPU s on a HPUX 9000-720) were a maximum annualized profit of $21 065.30/yr at a production of desired product P of 400 kg/h. The external lean stream (L1) was not used in the mass exchanger; instead, the process lean stream recovered in the bottoms product of the distillation column (L2) was recycled to the mass exchange section of the reaction-mixing-separation unit. Here, 10 190 kg/h of L2 was used, and five pinches were observed at the inlet concentrations of four rich streams (R2-R5) and at the inlet concentration of lean stream L2. Figure 12 shows the composition vs mass exchange diagram. The composite rich stream almost touches the composite lean stream at the points where the streams are pinched. The mean residence time of the components inside the reactor network was found to be 0.091 h. To determine the flow pattern of the chosen lean stream (L2) through the reaction-mixing-separation model, formulation P3.1a was solved. The flow rate of the pure lean stream L2 was fixed at 10 190 kg/h (obtained by solving eq P3.1), and five rich streams and five lean streams (L2 is discretized into five substreams) are involved in the MEN targeting. A countercurrent contacting pattern is chosen, and the pinch diagram is the same as that in Figure 12, except that intermediate lean stream compositions (due to the discretization) need to be included. The pinch points and the mass exchanged between the rich and lean streams are identical to the solution obtained from (P3.1) above, i.e., 400 kg/h of desired product P is produced. The optimal reaction-separation network is shown in Figure 13. No sidestream mixing was observed. The fractions shown in Figure 13 represent the fraction of P in the rich

profit, $/yr feed A, kg/h feed B, kg/h waste G, kg/h external lean stream, kg/h process lean stream, kg/h product P, kg/h

sequential (case 1)

simultaneous (case 3)

-2 011 880 612 1517 300 17 449 0 400

21 065.30 599 1298 258 0 10 190 400

stream which enters the lean stream at the points indicated. Nonsharp separation of desired product P is realized by considering mass exchange between the rich and lean streams. The result obtained from formulation P3.1a may be verified by solving the reaction-mixing-separation model with cocurrent flow of the lean streams in the mass exchangers. Formulation P3.1a with Ψ ) 1 (cocurrent flow pattern) was solved. The process lean stream (L2) flow is again fixed at 10 190 kg/h. The profit decreases to $9508.72/yr, and the streams are pinched at three points, as shown in Figure 14. The total amount of mass exchanged in the cocurrent case is less than that in the countercurrent case, i.e., only 387 kg/h of P is produced with the same flow rate of the lean stream L2. This is the reason why formulation P3.1a automatically picks the countercurrent contacting pattern. A comparison of results between the sequential and simultaneous cases is shown in Table 1. Some striking features of the comparison are that a larger quantity of feed is used and more waste is produced in the sequential case to produce the same amount of desired product. Hence, simultaneous reaction and separation is better than sequential reaction and separation. This is because the desired product P is separated out as it is formed, and mass integration enables efficient utilization of the available lean streams. Example 2: Conversion of cyclohexane to benzene in membrane reactor using palladium. The reactionmixing-separation model formulation (eq P3.1) may be used to optimize the performance of a membrane reactor. In this study, we consider the reversible reaction of cyclohexane to benzene in a membrane reactor, using a palladium membrane, in the presence of cylindrical catalyst pellets (0.5 wt % Pt/Al2O3, 3.3 mm


duC/dL ) rcVr duB/dL ) -rcVr duH/dL ) -3rcVr - RH([PTruH/

∑ ui ) uC + uB + uH + uA,

∑ui]1/2 - [PTsvH/∑vi]1/2) ∑ vi ) vH + vA,

uA ) uA0, vA ) vA0, vA ) vB ) 0

Figure 15. Membrane reactor, example 2.

o.d., 3.8 mm high) (Itoh, 1987). The reaction occurs inside the membrane tube (17 mm i.d., 17.4 mm o.d.).

C6H12 T C6H6 + 3H2 Figure 15 shows the schematic diagram of the experimental setup. The feed to the reaction side (tube side) is a 19.7% saturated vapor of cyclohexane, with argon being the inert component of the feed stream. Also, argon is the purge gas (or lean stream) which picks up the hydrogen permeating to the shell side (separation side) through the palladium membrane. The reaction was carried out under isothermal (473 K) and isobaric (1 atm pressure) conditions. The objective is to maximize the conversion of cyclohexane to benzene. To simplify the problem, the dehydrogenation reaction on the inner surface of the palladium membrane was assumed to be negligible compared to that on the catalyst pellets. The permeation rate of hydrogen gas through the palladium membrane, QH, was assumed to obey the half-power pressure law; i.e., it is proportional to the difference between the roots of the hydrogen partial pressures in the reaction and separation side, pH, pH′:

QH ) RH([pH/P0]1/2 - [pH′/P0]1/2) where the permeation rate constant of hydrogen gas is

∏loDCo/(ln(ro/ri))

RH ) 2

P0 ) 1.013 × 105 Pa, Co ) 1280 mol m-3 at 473 K, D ) Fick’s diffusion constant ) 9.23 × 10-10 m2/s, experimental setup lo ) 140 mm, (ro/ri) ) 17.4/17.0, RH ) 4.47 × 105 mol/s The rate of consumption of cyclohexane, rc, is given by

rc ) -k(KPpc/pH3 - pB)/(1 + KBKPpc/pH3) where

k ) 0.221 exp(-4270/T) mol m-3 Pa-1 s-1 KB ) 2.03 × 10-10 exp(6270/T) Pa-1 KP ) 4.89 × 107.76 exp(3190/T) Pa3 The differential material balances over a dimensionless section dL of the reactor are given by

where ui and vi (i ) cyclohexane, benzene, hydrogen, and argon) are the flow rates of component i in mol/s in the reaction and separation sides, respectively. Vr is the gross volume of the reaction side. PTr and PTs are the total pressures in the reaction and separation sides. Itoh (1987) reports an optimal conversion of 99.7% when uC0 ) 2.90 × 10-7 mol/s and vA0 ) 11.8 × 10-5 mol/s. The equilibrium conversion in a PFR packed with the same catalyst is 18.7%. The value of KP shown above differs from the value published by Itoh (KP ) 4.89 × 1035 exp(3190/T) Pa3); the value shown here was derived from the published values of k, KB, and the equilibrium conversion of cyclohexane (the high value of KP published in Itoh’s paper will disregard the reverse reaction, if the published kinetics were used, because the second term in the equation for rc (pB) will be negligible compared to the first term when KP is large). The reaction-mixing-separation model was used to model the membrane reactor. The model was discretized into 10 elements. The reaction side streams were considered as rich streams (rich in hydrogen), and the streams on the shell side (separation side) were characterized as lean streams. The equilibrium relationship between hydrogen in the rich and lean streams was derived from the half-power pressure law (QH ) 0). A minimum composition difference of 10-6 was used in the MEN model. Ten rich streams and one lean stream are involved in the MEN. The inlet mass composition to the mass exchanger of the ith rich stream is determined by solving duHi/dL ) -3rciVr (i.e., the reaction rate is the same as in the membrane reactor, but permeation losses are not considered). The outlet mass composition of the ith rich stream is obtained by using the expression with permeation of components considered. Note that the mass compositions in the mass exchanger are defined as kg H2/kg inert. In addition, there is a change in density in the gas phase reaction. However, sidestream mixing is not considered while modeling a membrane reactor; hence, formulation P3.1 may be used without any modifications. The inlet composition of the lean stream is zero (pure argon stream). The optimization model is an NLP with 942 continuous variables and 906 constraints. The nondifferentiable max expressions were smoothed using an η value of 10-8. To verify the accuracy of the NLP model, the flow rates of cyclohexane in the reaction section and argon in the separation section were set to be uC0 ) 2.90 × 10-7 mol/s and vA0 ) 11.8 × 10-5 mol/s, RH ) 4.47 × 105 mol/s, and the conversion was found to be 99.7%, which is identical to that obtained by Itoh (1987). No pinches were observed in the MEN. Later, RH, uC0, and vA0 were allowed to vary and the same model was solved; it was found that a conversion of almost 100% may be achieved in a membrane reactor with only 15.06% of the RH in Itoh’s paper. The feed flow rates are, for cyclohexane in the reaction side, uC0 ) 2.90 × 10-7 mol/ s, and, for argon in the separation side, vA0 ) 13.808 ×


Figure 16. Pinch curve, example 2.

10-5 mol/s. Figure 16 shows the composition vs mass exchange diagram for this case. Note that, even at the optimal solution, the driving force for mass transfer is very high. The results show that almost 100% conversion may be achieved in a membrane reactor which is only 0.1506 × 140 mm long or with a diffusion coefficient which is 0.1506D of the membrane used in the work done by Itoh et al. Here we see that a shorter membrane, or a membrane with a lower diffusion coefficient, is sufficient for this system, and virtually no further improvement is achieved with a larger membrane. Also, the feed flow rates are comparable to those published earlier, and the flow rate of argon in the separation side is slightly higher. Conclusions Simultaneous reaction and mass integration to develop optimal reaction, mixing, and separation networks is a relatively new concept. The model proposed considers nonsharp splits by defining the splits as a function of mass exchanged between the rich reactor streams and the lean streams in the mass exchange section. The examples demonstrate the feasibility and the advantages of using such a model. In addition, optimal reactor networks may be derived with fewer modules of the reactor network synthesis algorithm, since simultaneous reaction and mixing, combined with separation of certain species, will enhance the extent of the reaction. Hence, the model may enable the algorithm to achieve higher objective function values with fewer units, thus reducing the total capital costs involved. Also, it enables us to consider realistic separations from the system. Nomenclature COm ) cost of mth lean stream, $/kg f(R) ) residence time distribution in the PFR or DSR f(R(i)) ) f(R) in module i fij ) residence time distribution at reacting element i, collocation point j Fif ) inlet flow rate to the ith reactor module Fic, Fid ) flow rate through the CSTR and DSR in the ith reactor module

Fki-1 ) exit flow rate from module k, which is an inlet to module i - 1, k ) 0, i - 1 g ) extent of the reaction K ) maximum number of collocation points Lm ) flow rate of the inert component of the mth lean stream L ) Lagrange polynomial mcl exit ) mass of species c at the mass exchanger exit mc out ) total mass of species c leaving the reactionmixing-separation model mcr(R) ) mass of species c in the reaction-mixing section at R mcr exit ) mass of species c at the reactor-mixing section exit mcri end ) mass of species c in the reactor-mixing section at the end of reacting element i mcri u ) mass of species c in the reactor-mixing section at the end of separator i Ne ) number of finite elements in the reactor discretization Nmod ) number of reactor modules necessary to complete the reactor network representation q(R) ) fraction of Qside entering the DSR or reactionmixing section at R q(R(i)) ) q(R) in module i qij ) fraction of Qside entering the DSR at reacting element i, collocation point j Q(R) ) volumetric flow rate through the PFR, DSR, or reaction-mixing section at R Q(R(i)) ) Q(R) in reactor module i Qexit ) exit flow rate from the DSR (Qin + Qside) Qi ) volumetric flow rate in reacting element i Qi u ) volumetric flow rate at the exit of separator i Qin ) inlet flow rate to the DSR Qij ) volumetric flow rate through the reactor at element i, collocation point j Qside ) sidestream flow rate to the DSR R(X(R),T(R)) ) rate vector at R R(Xij,Tij) ) rate vector at element i, collocation point j t ) residence time tmax ) maximum residence time in the reactor network T(R) ) temperature inside the reactor at R TCSTR(i) ) temperature inside the CSTR in module i TDSR exit(i) ) temperature at exit of the DSR in module i Tij ) temperature inside the reactor at element i, collocation point j U ) the upper bound on the flow rate inside the reactor X(R(i)) ) concentration vector at point R inside the DSR in module i Xc(R) ) mass concentration of species c in the reactionmixing section at R XCSTR(i) ) concentration vector inside the CSTR in module i XDSR(i) ) concentration vector inside the DSR in module i Xi end ) concentration vector at the end of reacting element i Xi 0 ) concentration vector at the beginning of reacting element i Xic in ) concentration vector at the inlet to the CSTR in module i Xci in ) concentration of component c at the inlet to the ith reacting element Xci end ) concentration of component c at the end of the ith reacting element Xci u ) concentration of component c at the exit of the ith separator Xid in ) concentration vector at the inlet to the DSR in module i Xif ) concentration vector at the feed to module i X0(i) ) concentration vector at the reactor inlet in module i Xside ) concentration vector in the DSR sidestream

Ind. Eng. Chem. Res., Vol. 35, No. 12, 1996 4535 ycm in ) inlet composition (kg key/kg inert) of the mth lean stream ycm out ) outlet composition (kg key/kg inert) of the mth lean stream Yic ) binary variable denoting the existence/nonexistence of the CSTR in module i Yid ) binary variable denoting the existence/nonexistence of the DSR in module i Zp(xc,yc) ) thermodynamic feasibility constraint Greek Letters R ) time along the length of the reactor ) minimum difference in composition between rich and lean stream component compositions γc(R) ) mass fraction of c leaving the reaction-mixing section at R φi ) fraction of the inlet flow to the ith reacting element which was added by the sidestream η ) a small number close to 0 used in the hyperbolic approximation eq 3.32 ξm ) continuous variable which determines the flow pattern of lean stream m in the mass exchanger () Ψm) Ψm ) binary variable which determines the flow pattern of lean stream m in the mass exchanger τ ) mean residence time

Appendix A: Mean Residence Time Distribution in the Reaction-Mixing Section The mean residence time in the reaction-mixing section is defined as

τ)

∫0t

max

τ ) (1/(Qin + Qside))

∫0t ∫0R (q(R′)Qside max

∑γc(R′)mcr(R)/F) dR′ dR c)1 In the discretized model, from eqs 3.17 and 3.18

∑c Xci-1 endγci-1/F]

Hence,

∑i Qi∆Ri ) Q1(R1 - R0) + Q2(R2 -

R1) + ... + QNe(RNe - RNe-1) ) (Q1 - Q2)R1 + (Q2 - Q3)R2 + ... + QNeRNe since R0 ) 0 Ne

)

C

Ri (∑γci-1mcri-1 end/F - qiQside) + RN QN ∑ i)1 c)1 e

e

Appendix B: Steady State Analog to the Reaction, Mixing, and Separation Model (CSTR with Separation) Equations 3.4 and 3.5 represent the differential equations for the composition and volumetric flow rate. At steady state, dXc/dR and dQr/dR are equal to zero, which gives C

q(R)Qside )

∑γc(R)Xc(R)Q(R)/F

c)1 C

∑γc(R)Xc(R)/F) - Xc(R)γc(R) c)1

0 ) Rc(X,T) + Xc side(

QinγcXc out ) L(yc out - yc in) C

Qout ) Qin - (

∑γcXc out/F)Qin

c)1

QinXc in - QoutXc out ) L(yc out - yc in) mcl exit ) QinγcXc out mcr exit ) QoutXc out mc out ) mcl exit + mcr exit C

∑

c)1

C

mc(0) )

∑mc out

c)1

Literature Cited

C

τ(Q1 + Qside) )

C

∑γcXc out/F c)1

Rc(X,T)t ) Xc outγc - Xc in

where the subscripts in and out refer to the streams entering and leaving the homogeneous reactor.

Q(R) dR/(Qin + Qside)

Qi ) qiQside + Qi-1[1 -

Assume that the mixture is homogeneous, and define a split fraction γc such that γc(R) dR ) γc. Also, in the case of a CSTR, Xc(R) ) Xc out and Xc side ) Xc in. The differential equations reduce to simple algebraic expressions:

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Received for review June 28, 1996 Revised manuscript received September 18, 1996 Accepted September 18, 1996X IE960371H

X Abstract published in Advance ACS Abstracts, November 1, 1996.

Synthesis of Optimal Chemical Reactor Networks with Simultaneous

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