Synthesis and Optimization of Gas Permeation Membrane Networks

Ind. Eng. Chem. Res. 2004, 43, 4305-4322

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Synthesis and Optimization of Gas Permeation Membrane Networks Ramagopal V. S. Uppaluri,† Patrick Linke,*,‡ and Antonis C. Kokossis‡ Department of Process Integration, P.O. Box 88, University of Manchester Institute of Science and Technology (UMIST), Manchester M60 1QD U.K., and Process & Information Systems Engineering, School of Engineering, University of Surrey, Guildford, Surrey, GU2 7XH U.K.

We present a modeling and optimization framework for the design of gas permeation membrane networks. The modeling framework constitutes a generic superstructure of membrane units; recycle, feed, and product compressors; and vacuum pumps. Various flow patterns such as cross, countercurrent, and co-current flow can be embedded into the representation. Both conventional and novel membrane network configurations can be developed through structural optimization of the superstructure. The optimization is carried out using robust stochastic techniques in the form of simulated annealing for minimization of the total annualized network cost. Prominent industrial examples such as air separation and hydrogen recovery illustrate the potential of the design technology. 1. Introduction Membrane separation processes are becoming increasingly important in the chemical process industries. Different process applications adopt membrane processes such as gas permeation, pervaporation, reverse osmosis, ultrafiltration, microfiltration, dialysis, and electrodialysis for the recovery of desirable components from purge streams, concentration of solutions, clarification of process streams, and purification of waste streams. In addition, various hybrid processes also exist because of the coupling of membrane processes with other separation processes including adsorption, absorption, and evaporation. Most of the industrial applications utilizing membrane processes are confined to single-stage operation. However, strict environmental legislation and capital cost constraints demand the utilization of membrane networks in many applications where the partial or complete processing of retentate or permeate streams is subjected to multiple stages. Such a scenario is very much prevalent in membrane separation for gas purification and recovery. The design of the membrane separation problem has been confined to different subareas of research. These areas can be classified as the modeling of individual permeators, the analysis of the performance of simple membrane configuration, and the analysis of the economic performance of membrane processes through process synthesis and integration efforts. A number of publications address the modeling of membrane units. Among others, Weller and Steiner,1 Antonson et al.,2 Hogsett and Mazur,3 Boucif et al.,4 Rautenbach and Dahm,5 Giglia et al.,6 Krovvidi et al.,7 and Ruthven and Sircar8 have proposed simple and complex mathematical models for gas permeators to address binary mixtures. Shindo et al.,9 Li et al.,10 and Pettersen and Lien11 have presented multicomponent permeation models for various flow patterns. These * To whom correspondence should be addressed. Tel.: +44 (0)1483 689116. Fax: +44 (0)1483 686581. E-mail: p.linke@ surrey.ac.uk. † University of Manchester Institute of Science and Technology (UMIST). ‡ University of Surrey.

works have addressed the modeling aspects for membrane stages but have not provided a design basis for economic assessment. A number of works have addressed the analysis of the performance of a small number of known simple membrane network configurations. Stern et al.12 and Stookey et al.13 analyzed the performance of single-stage membrane permeators with recycle options. Mazur and Chan14 studied multistage systems for natural gas processing, and Kao15 investigated recycle strippers and enrichers. Pan and Habgood,16 Hwang and Thorman,17 Schulz et al.,18 and Kao et al.19 have studied membrane columns. Rautenbach and Dahm5 and Bhide and Stern20,21 have performed economic feasibility studies for various membrane network configurations with and without recycle streams. Pettersen and Lien22 have investigated various multistage recycle systems and hybrid systems. Agarwal and Xu23,24 and Agrawal25 provided broad guidelines based on process economics for two compressor cascades. Hinchliffe and Porter26 presented a simple cross-flow model for binary mixtures for quick economic assessment. All of these applications dealt with the comparative analysis of a few conventional membrane network configurations, and no systematic methods for the automated selection of highperformance network designs were presented. By adopting optimization technology, Lagunstov et al.27 developed a design approach to choose the optimal separation system that develops optimized design parameters for a single compressor recycle system. Lababidi et al.28 considered three different membrane network structures including a single-stage permeator, two stages in series, and a membrane column and applied local deterministic optimization methods to identify the optimal performances of the selected designs. Tessendorf et al.29 presented various aspects of modeling, simulation, design, and optimization of membranebased separation systems modeled using orthogonal collocation. The proposed model can handle multicomponent mixtures and considers the effects of pressure drop and energy balances. Qi and Henson30,31 proposed designs based on local optimization techniques in the form of nonlinear programming (NLP) and mixed inte

10.1021/ie030787c CCC: $27.50 © 2004 American Chemical Society Published on Web 06/23/2004

4306 Ind. Eng. Chem. Res., Vol. 43, No. 15, 2004

Figure 1. Schematics of a membrane stage.

ger nonlinear programming (MINLP) for the optimization of gas permeation membrane networks using spiral wound permeators. The approach is flexible enough to apply to multicomponent systems. The models proposed by Qi and Henson,30,31 however, can address design automation for a number of system parameters, such as the numbers of membrane units, components, and constraints imposed by product(s) requirements. Despite the significant progress, systematic approaches for the selection of optimal membrane networks from the set of all possible design configurations have not been proposed to date, and most design efforts resort to trialand-error-based selection among a small number of conventional network designs. Most recently, Marriott and Sorenson32,33 developed a general approach to modeling membrane modules considering rigorous mass, momentum, and energy balances. Their approach constitutes a feed-side flow model coupled with a permeate-side flow model and a local transport model for the membrane system. They employ the detailed models in superstructure formulations that they then optimized using genetic algorithms.33 Their application focuses on pervaporation systems, and because of the complex models employed, the optimizations have proved to be extremely computationally demanding. In contrast to previous efforts, this paper presents an optimization-based approach for the quick but systematic screening and scoping of conceptual membrane network design options. It capitalizes on previous developments in integrated reaction and separation process synthesis34 to develop a comprehensive gas permeation membrane network representation in the form of superstructures that capture all possible conventional and novel combinations of co-current, countercurrent, and cross-flow gas permeation membrane units. The superstructures are optimized using robust stochastic optimization techniques in the form of simulated annealing35 to extract those designs that exhibit the best economic performances. The proposed approach enables the design engineer to systematically and quickly determine the most economical membrane network structures. It overcomes major limitations of existing approaches, as it (i) screens among all possible structural and operational process alternatives that might exist for gas permeation networks, (ii) identifies the performance limits of the system with probabilistic performance guarantees, and (iii) is flexible enough to

accommodate for user preferences and problem specific modeling aspects. The next section presents the problem description and system representation. Section 3 provides a summary of the superstructure framework adopted for system design. Section 4 outlines a brief summary of the optimization methodology. Section 5 addresses the network synthesis methodology. Three industrially significant membrane network design problems are presented in section 6. 2. Problem Statement and Synthesis Representation The gas permeation membrane network synthesis problem considered in this work can be stated as follows: Given the feed and product specifications, the retentate and permeate pressures, and the economic information, determine the optimal membrane process network configuration and operation to carry out the separation to achieve the desired product specifications at minimum total annualized cost. In this work, simple membrane stage models have been used to formulate membrane superstructures from which the optimal process configurations are extracted. The representation accounts for various industrially prominent flow patterns such as co-current, countercurrent, and cross-flow that can occur in membrane modules. 2.1. Membrane Synthesis Modules. The representation of a membrane stage is adapted from the work by Mehta and Kokossis.36 They considered generic reactor units in multiphase systems to develop a superstructure for multiphase reactors, including both well-mixed compartments and plug-flow compartments. The membrane permeation stage is represented in this work as a pseudo-plug-flow compartment without side streams and reaction and with diffusional mass-transfer links between the retentate and permeate compartments. Membrane systems comprising high-pressure and low-pressure states are quantified by allowing two states in the superstructure (analogous to the phases in a two-phase plug-flow compartment representation). The development of the membrane permeator unit representation is illustrated in Figure 1. Membrane unit c consists of a retentate compartment and a permeate compartment operating at different states (high and low pressure). Hence, a membrane unit is represented by a

Ind. Eng. Chem. Res., Vol. 43, No. 15, 2004 4307

Figure 2. Schematic of membrane network superstructure (top) and sample designs captured by the superstructure (bottom left, conventional two-stage stripping cascade; bottom right, novel networks with partial product processing).

pair of compartments. By convention, the generic membrane units in state 1 represent the membrane compartment at high pressure (retentate pressure), and the generic units in state 2 represents the membrane compartment at low pressure (permeate pressure). The following assumptions are considered in proposing the model equations for a membrane permeator stage: (a) The component permeability is independent of pressure and concentration. (b) Concentration polarization is negligible on the membrane surface. (c) Pressure drops along the membrane can be neglected. The above assumptions are valid for gas permeation membranes, as substantiated in the literature.37 These equations govern the simulation of the membrane compartments. Convergence problems suggested by Shindo et al.9 are avoided in this work by the development of automated methods to solve balances across a countercurrent flow pattern. 2.2. Membrane Network Superstructure. The proposed approach is based on membrane network superstructure optimization. The superstructure development follows the process synthesis representations elaborated by Mehta and Kokossis36 and Linke and Kokossis.34 Mehta and Kokossis36 considered a multiphase reaction network superstructure framework with a complete stream network in each phase. Linke and Kokossis34 extended the superstructure framework to consider simultaneous and consecutive reaction sepa-

Figure 3. Summary of sets for superstructure representation.

ration systems as well as hybrid separation systems in the superstructure representation. These representations are adopted and modified to suit the membrane network synthesis problem. A schematic diagram of the membrane network superstructure consisting of two membrane stages is presented in Figure 2. A membrane stage is represented by two compartment models, one in each pressure state, that exchange mass via the diffusional link across the membrane (Figure 3). The superstructure features complete connectivity among the retentate compartments and from permeate to retentate compartments. The superstructure representation accounts for a num-


Figure 4. Functionality of binary splitters in the permeate compartment for different flow patterns (co-current, cross-, and countercurrent flow, respectively).

ber of stream compression options including vacuum pumps, permeate recycle compressors, and feed compressors. The superstructure generation also assumes one vacuum pump per membrane stage, which is required if a partial permeate recycle is allowed in the generic membrane units. To represent industrial membrane network configurations, the distribution of feed and retentate streams is restricted to the retentate side, and the distribution of permeate streams is restricted to the retentate side and products only. Recycle streams from the permeate side to the retentate side are realized through recycle compressors. The allocation of feed compressors and vacuum pumps can be adopted flexibly by incorporating subsequent cost functions. The stream network that connects the membrane compartments in the superstructure framework is visualized in Figure 2. Equations 1-7 in section 3 present the balance equations of the superstructure. The membrane network superstructure considers all possible structural design options that can be obtained from different combinations of membrane units. It not only includes the various conventional network designs that have been proposed to date but also captures novel combinations of membrane units that might offer improved performances over their conventional counterparts. The superstructures can be systematically screened

using optimization techniques to extract those network configurations that offer the best economic performances for a particular separation task. The various structural options considered are feed compression; interaction of feed streams with retentate streams and retentate product streams; interaction of permeate streams with retentate streams and permeate product streams; and various industrially significant flow patterns such as cocurrent, countercurrent, and cross-flow. The locations of the different compressors (feed and recycle) and the locations of the different stream mixers and splitters in the superstructure framework are illustrated in Figure 2. Mass-exchange links exist among the membrane compartments within a membrane unit to account for diffusional mass transfer across the membrane. The connectivity among the membrane unit compartments in different states reflects the different flow patterns (co-current, cross-, and countercurrent flow patterns) that might exist. Our representation adopts the concept of a shadow reactor compartment36 to the membrane units, as shown in Figure 4. With regard to the compression equipment, the total number of vacuum pumps employed in an optimized design depends on the values of the split fractions associated with the permeate compartment streams. Details regarding the allocation of vacuum pumps for various scenarios are presented in Appendix I.


Figure 5. Intercompartment variables for a membrane permeator stage.

Conventional network configurations, as presented in the literature,37,38 can be extracted from the superstructure model. Apart from these known designs, there is a vast number of novel membrane network designs, such as networks with feed distribution, partial retentate and permeate processing, and different flow patterns, that are also embedded in the superstructure. Figure 2 shows a sample conventional design and a sample novel design that are captured by the superstructure representation. Network configurations such as membrane columns and membrane enrichers have not been embedded in the superstructure representation because of undesirable retentate stream expansion. The thermodynamic reasons for not considering such networks are evident in the literature.5,31 The pressure ratio across the membrane network is set as an optimization parameter in this work. Because preliminary design investigations for network design calculations assume a pressure ratio, this work is intended for comparative study. However, if necessary, the pressure ratio across the membrane network can be treated as an optimization variable in the synthesis framework. 3. Mathematical Formulation This section presents the mathematical model of the membrane network superstructure. The superstructure development follows the steps outlined by Linke and Kokossis.34 All sets, variables, and parameters used in the formulation are defined in the Nomenclature section. For reasons of clarity, the schematic of sets used for system representation are also shown in Figures 2

Table 1. Parameters for the Membrane Network Superstructure parameter 1 2 3 4 5

Xm,f d Perm Ps Nsc

6 7 8 9 10

Nc Amin Amax rmin XDs,pr,m

11

PDs,pr,m

12

RDs,m,pr

description mole fraction of component m ∈ M in the feed stream membrane thickness permeability of component m ∈ M on the membrane pressure of compartment c ∈ C in state s ∈ S number of subcompartments in the membrane compartment number of compartments in different states s ∈ S minumum area bound per permeator unit maximum area bound per permeator unit minumum split fraction of an active stream desired purity in product stream pr ∈ PRs in state s ∈ S for component m ∈ M desired product molar flow rate of component m ∈ M in state s ∈ S for product stream pr ∈ PRs desired recovery in state s ∈ S for component m ∈ M in product stream pr ∈ PRs

and 3. The intercompartment and intracompartment variables employed in the superstructure formulation are included in Figures 5 and 6, respectively. The definitions of the superstructure variables and sets follow as an extension of the modeling efforts of Linke and Kokossis.34 The defined variables are used to specify mass balance relations (in terms of equality constraints), as well as optimization bounds and product specifications (in terms of inequality constraints). Various parameters for the optimization problem are summarized in Table 1. These include parameters related to the membrane permeator stage, optimization bounds, and membrane network and product specification constraints. The mathematical model of the membrane network superstructure comprises the equations below.


Figure 6. Intracompartment variables for a membrane permeator stage.

model for gas permeation as37

Equality Constraints.

Balances around feed splitter SPF1

∑ θFC1,ic + θFP1,pr ) 1

pr ∈ PR1

(1)

ic∈C

pr ∈ PRs

(2)

c,ic∈C

Xs,c,sc,m )

Balances around product splitters SPs,pr, pr ∈ PRs, s ∈ S

∑ θPCs,pr,c + θPPs,pr ) 1 c,ic∈C

pr ∈ PR1

(3)

∑ O1,ic,mθCC1,c,ic +

ic∈C

OP1,pr,m θPC1,pr,c + OSC1,c,m

m ∈ M, pr ∈ PR1 (4)

Balances across product mixer MI1,pr, pr ∈ PR1 OP1,pr,m ) F1,mθFP1,pr +

∑ O1,ic,mθCP1,c,pr

m∈M

ic∈C

(5) Balances across state change mixer MSC1,c, c ∈ C OSC1,c,m )

∑ O2,ic,mθCC2,c,ic + OP2,pr,mθPC2,pr,c ic∈C

m ∈ M (6) Balances across product mixer MI2,pr, pr ∈ PR2 OP2,pr,m )

∑ O2,ic,mθCP2,c,pr

m∈M

(7)

ic∈C

Compartment balances for retentate compartment c ∈ C (in state s ) 1 ∈ S) sc ) 1, m ∈ M (8) OS1,c,1,m ) I1,c,m - GPc,1,m OS1,c,sc,m ) OS1,c,sc-1,m - GPc,sc,m sc ) 2, 3, ..., Nsc, m ∈ M (9) O1,c,m ) OS1,c,Nsc,m

OSs,c,sc,m

∑ OSs,c,sc,m

m∈M

Balances across mixer MI1,c, c ∈ C I1,c,m ) F1,mθFC1,c +

m∈M

Ac Perm (P1X1,c,sc,m - P2X2,c,sc,m) (11) Nc δ

where mole fraction Xs,c,sc,m is calculated using the expression

Balances around splitters SPs,c, c ∈ C, s ∈ S

∑ θCCs,c,ic + θCPs,c,pr ) 1

GPc,sc,m )

(10)

where GPc,sc,m is evaluated using the mass-transfer

s ∈ S, c ∈ C, sc ∈ SC, m ∈ M (12)

For the purposes of our work, i.e., to identify a few promising designs out of the vast number of design alternatives, the accuracy of the mass-transfer model is expected to be sufficiently suited. The model captures the trends observed in membrane systems and thus allows for the comparison of design options on a common basis. The balances for the retentate compartments are different for different flow patterns. For a co-current flow pattern, the balance equations are

OS2,c,1,m ) GPc,1,m

c ∈ C, m ∈ M

(13)

OS2,c,sc,m ) OS2,c,sc-1,m + GPc,sc,m c ∈ C; sc ) 2, 3, ..., Nsc, m ∈ M (14) O2,c,m ) OS2,c,Nsc,m

c ∈ C, m ∈ M

(15)

The balances for a countercurrent flow pattern are

OS2,c,sc,m ) OS2,c,sc+1,m + GPc,sc,m c ∈ C; sc ) 1, 2, 3, ..., Nsc - 1; m ∈ M (16) OS2,c,Nsc,m ) GPc,Nsc,m OS2,c,m ) OS2,c,1,m

c ∈ C, m ∈ M c ∈ C, m ∈ M

(17) (18)

The balances for a cross-flow pattern are

OS2,c,sc,m ) GPc,sc,m

c ∈ C, sc ∈ Nsc, m ∈ M (19)


O2c,m )

∑ OS2,c,sc,m

c ∈ C, sc ∈ SC, m ∈ M (20)

sc∈Nsc

The assignment of values to different binary variables (0 and 1) associated with vacuum pumps is given using the following logical expressions: For vacuum pump V2,pr, the expressions are

∀θCP2,c,pr ) 1.0

(21)

BV2,pr ) 0

∀rmin < θCP2,c,pr < 1.0

(22)

BV2,pr ) 0

∀rmin < θCC2,c,ic < 1.0

(23)

BV2,pr ) 1

OBJ ) Costmem + Costfc + Costrc + BV2,prCostV2,pr +

For vacuum pump V12,c, the expressions are

∀rmin < θCP1,c,pr < 1.0

(24)

BV12,c ) 0

∀θCP2,c,pr ) 1.0

(25)

BV12,c ) 0

∀θCP2,c,ic ) 1.0

(26)

BV12,c ) 1

vacuum pump costing equations being particularly complex. Together with the discrete choices that have to be made for the structural design options, membrane network optimization is a mixed integer nonlinear problem. Objective Function. The objective function is a complex function of different cost terms associated with the network cost. The variables and parameters associated with the objective function are defined in the Nomenclature section. Relations between the various economic parameters and network pressures are presented in Appendices II and III. The objective function is expressed as

BV12,cCostV12,c + BV22,cCostV22,c (35) where

Costmem )

For vacuum pump V21,c, the expressions are

∀θCP2,c,ic ) 1.0

(27)

∀rmin < θCP2,c,ic < 1.0

(28)

∀θCP2,c,pr ) 1.0

(29)

BV21,ic ) 1 BV21,c ) 0

Costfc ) Cfcom I

BV21,c ) 0

∑

c∈C

Inequality Constraints. Inequality constraints for membrane network design are area bounds and product specifications. Lower and upper bounds for membrane area can be presented as

Ac g Amin Ac e Amin

c∈C c∈C

(30)

Cost

)

∑ OPs,pr,mθPPs,pr

Cvac I Cost

)

∑

c∈C

(

vac Cvac HP CAHP

ηvac

Cvac ∑ I c∈C s ∈ S, pr ∈ PRs, m ∈ M (33)

(

Cost

V21,c

)

∑

c∈C

vac Cvac HP CAHP

ηvac

(

Cvac HP

for product recovery, and

OPs,pr,mθPPs,pr g PDs,pr,m

ηvac

s ∈ S, pr ∈ PRs, m ∈ M (34)

for product flow rate. 4. Superstructure Optimization The mathematical model of the membrane network superstructure contains nonlinear terms. Bilinear terms arise from balances around mixers and splitters, and the balance equations describing the membrane units are highly nonlinear. The objective function contains nonlinear cost correlations for the compressors, with the

∑

c∈C

Cvac I

bvac

(39)

)

∑ O2,c,m m∈M

∑ O2,c,m +

m∈M

(38)

)

ηvac

Cvac HP

for product purity

)

bcom

∑ ∑ O2,c,m c∈Cm∈M

∑ O2,c,m +

m∈M

s ∈ S, pr ∈ PRs, m ∈ M (32)

OPs,pr,mθPPs,pr g RDs,pr,m F1,m

+

Cvac HP

V12,c

ηcom

∑ ∑ O2,c,m

ηvac

(31)

g XDs,pr,m

m∈M

(

∑ I1,c,m m∈M

Ccom HP

c∈Cm∈M

Product specification constraints can be expressed as

OPs,pr,mθPPs,pr

+

ηcom

vac Cvac HP CAHP

(37)

∑ I1,c,m

m∈M

Ccom ∑ I c∈C

V2,pr

(36)

bcom F1,m + Cfcom ∑ P ( ∑ F1,m) m∈M m∈M

com Ccom HP CAHP

Costrc )

CAmemAc ∑ c∈C

bvac

(40)

)

∑ O2,c,m

m∈M

ηvac

(41)

The objective function (eqs 35-41) is defined as the sum of the annualized costs of the membrane units, feed compressor, recycle compressors, and vacuum pumps (if vacuum application is considered on the permeate side). For cases without feed compression or vacuum applications or a combination of the two, these economic parameters associated with the feed compressor and vacuum pump can be set to zero to modify the cost function. For systems with permeate product compression, additional cost terms can be added to the objective

4312 Ind. Eng. Chem. Res., Vol. 43, No. 15, 2004 Table 2. Optimization Parameters for Membrane Network Superstructure Model Optimization Parameters parameter

value

initial annealing temperature final annealing temperature feed bounds split fraction bounds Nc Nsc

1010 10-4 [50 150]% of specified feed flow rate [0.05 1] 3 10 Move Probabilities

parameter

value

state1/state 2 compartment/stream permeate recycle/product mixer add/delete/modify compartment add/delete/modify stream cross-flow/countercurrent/co-current flow feed flow rate no. of points for splitting feed bounds

0.5/0.5 0.5/0.5 0.5/0.5 0.33/0.33/0.33 0.33/0.33/0.33 1.0/0.0/0.0 0.2 1000

function as a function of product stream flow rate and permeate product compressor economic parameters. The simulated annealing (SA) algorithm is used to optimize the membrane network superstructures. The degrees of freedom of the search include variables such as number of units, connections between generic units, and area optimization. Unlike deterministic approaches, stochastic optimization adopts probability-based random search. Stochastic algorithms perform global searches of the solution space and offer a number of advantages such as applicability to systems described by discontinuous functions, flexibility to handle problem-specific model adjustments, and generation of a number of closely fitted optimal solutions that can be passed to the designer for further analysis. A number of stochastic experiments are performed to generate a multitude of solutions that can be associated with a confidence of lying within the globally optimal domain. SA performs a single-point search and is particularly easy to implement. It has been reported to perform robustly for the type of systems under consideration here,34,36 although the efficiency of this approach was shown to be inferior to that of tabu searching.34 For the purposes of our work, i.e., to demonstrate the use of robust stochastic membrane network superstructures, the SA algorithm was chosen for its advantage of ease of implementation with simultaneous achievement of high levels of robustness. However, refined future implementations might adopt a more efficient stochastic search algorithm such as a tabu search to reduce the search times. The advantages of the stochastic search methods over deterministic local optimization methods have been discussed by Linke and Kokossis.35 Although stochastic search algorithms are powerful tools for the optimization of large-scale mixed integer nonlinear problems, they are said to be slower than their deterministic counterparts. Because deterministic global optimization methods are constantly being improved, it would be interesting to see applications of state-of-the-art deterministic global optimization algorithms to the type of problem presented in this work. Stochastic search algorithms deliver solutions that are optimal in a stochastic sense. In contrast to deterministic optimization methods, which converge to strict mathematical optima, the stochastic optimization methods follow a probability-driven search and converge a set of stochastic experiments to a stochastic optimum that is represented by a distribution of closely fitted

solutions. The stochastic optimization methods implemented in this work perform global searches, and the solutions obtained are independent of the initial states of the problem. The characteristics of the distribution of solutions obtained from a number of stochastic experiments under varying initial conditions can provide statistical guarantees of convergence. The solutions are expected to lie within the proximity of the globally optimal solution to the problem if they are identical to or very closely approach the performance of the best solution found among all stochastic experiments. Such a set of stochastically optimal solutions is characterized by a narrow distribution of the performances of the individual solutions found. The propagation of the SA algorithm is presented in Figure 7. The algorithm starts at a given Markov chain length and initial annealing temperature value. The existing state is changed to generate a new, temporary, feasible state by a stochastic move using perturbation probabilities. The objective function at the new state is calculated, which determines the acceptance or rejectance of the new, temporary, feasible state according to probabilistic acceptance/rejectance criteria. If the new state is rejected, the system reverts to the previous state. The Markov process is continued until the maximum number of allowed iterations is reached or the minimum number of new states are accepted by the process. Depending on the standard deviation of the accepted new states in the Markov loop, the annealing temperature is reduced according to a cooling schedule. After the temperature has been reduced, the Markov loop is repeated or stopped by applying the termination criteria. The acceptance criterion, cooling, schedule and termination criterion are implemented based on the available literature.35,39,40 The key algorithmic parameters are presented in Table 2. Move perturbations allow structural and operational changes in the superstructure model to be made. These moves for membrane networks can be classified broadly into two types, namely, structural and stream moves. Structural moves allow structural changes to occur by the addition or deletion of retentate and permeate compartments, modification of membrane areas between the area bounds, and selection of flow patterns. Stream moves provide variations in the existing stream network by the addition and deletion of streams, coupled with the modification of the split fractions associated with a pair of streams. Probabilities are allocated for


mentation consists of network initialization, network simulation, objective function evaluation, simulated annealing optimization, and stochastic moves for realization of various design changes. The various stages are coordinated as follows: Depending on the problem specific parameters associated with the membrane network, a simple initial structure is simulated, and the objective function value is calculated. Through the random application of a move, the system is brought to a new state (or structure) that is again simulated. The objective function is evaluated again for the new design and passed to the annealing optimizer to determine acceptance. If the new structure is accepted, the optimization procedure proceeds with the new state. If the structure is rejected, the optimization procedure continues from the previous state. The procedure continues until the termination criteria are satisfied. The initial membrane networks comprise one, two, or three membrane stages without a permeate recycle, as such simple initial networks allow an easy and rapid simulation to start the optimization procedures. Initial guess values for membrane area and compartment flow rates were obtained from a simple network simulation program for single and multiple cross-flow membrane stage cascades without any recycles. The nonlinear set of equations describing the membranenetworks provided as initial networks and generated after stochastic moves are solved using the NEQLU solver.41 The solver has proven robust in that only about 10% of the designs generated during the optimal search could not be simulated. The computational times required for the optimal searcher ranged from 2500 to 10 000 CPU s for Markov chain lengths from 20 to 50, respectively. The required product flow rates, recoveries, and product concentrations are ensured using a penalty function approach.42 The penalty function is calculated as a linear function of the deviation of the product specifications and is coupled to a penalty parameter. The penalty function is later added to the overall cost of the membrane network. As the optimization procedure continues, only those structures with the smallest penalties are considered and approved, yielding final solutions that do not violate the product specifications. 6. Illustrative Examples Figure 7. Flowchart for simulated annealing algorithm.

the various move types required for the optimization process. The SA algorithm allows uphill moves. Poor states are accepted with high probability initially, when the algorithm is operating at a high annealing temperature, thus facilitating start-up from poor networks. However, essentially only better states are accepted at lower temperatures as the search progresses and terminates. Because conventional networks are hidden in the superstructure, many prominent industrial configurations can be subjected to optimization if necessary. This can be realized by assigning perturbation probabilities (as presented in Table 2) so as to disallow stream and structure modification in the course of optimization. 5. Synthesis Methodology The overall synthesis methodology is outlined in Linke and Kokossis.34 An overview of the features of the implementation is provided in Figure 8. The imple-

Three industrially prominent applications of membrane networks are presented to illustrate the proposed approach. These examples include enriched oxygen production, hydrogen recovery from synthesis gas, and hydrogen recovery from refinery streams. All computations were performed on a Sun Enterprise HPC 4500 workstation. A superstructure of three membrane stages and 10 subcompartments is generated for each example. A number of optimal searches are started from different initial structures and random number generator seeds. 6.1. Air Separation Using Vacuum Pumps. 6.1.1. Problem Summary. The objective of this case study is to consider the optimal allocation of vacuum pumps for the production of oxygen-enriched air using vacuum on the permeate side. The membrane and process economics are obtained from Bhide and Stern20,21 for a poly(dimethyl siloxane) membrane. Product specifications include the desired product flow rate in terms of equivalent pure oxygen and product purity. Equivalent pure oxygen20,21 is expressed as


Figure 8. Implementation strategy for network optimization.

EPO2 ) OP2,pr,1θPP2,pr

XO2 - 0.209 0.791

(42)

XO2, the mole fraction of oxygen in the permeate product stream can be evaluated from the expression

XO2 )

OP2,pr,1θPP2,pr OP2,pr,1θPP2,pr

∑

m)1,2

m∈M (43)

Two subcases of network design were set up for two different oxygen purities (0.3 and 0.4 XO2) and 10 tons of EPO2. Air (20.9 mol % O2) is fed to the membrane system at a pressure of 1.07 bar. The optimization variables for this case study include the feed flow rate along with the network design variables. The network design parameters are summarized in Table 3. 6.1.2. Optimization Results. (a) 30% Oxygen and 10 tons of EPO2. A number of membrane network configurations that were obtained from the optimal searches for this case are presented in Figure 9. The best system configuration corresponds to a system with two stages in series with a total area of 369 m2 and a network cost of about $176,000. For their best design, Bhide and Stern20 reported a network cost of $230,000 for a total membrane area of about 380 m2. The higher membrane areas and thus network cost primarily resulted from the authors not considering feed optimi-

zation. The numerous network configurations obtained from optimization are either single-stage membrane systems or two-stage membrane permeators in series. The optimal feed flow rate obtained after optimization corresponds to a value of about 0.1915 kmol/s. The corresponding initial feed flow rate was specified at 0.2234 kmol/s. The optimization results suggest that the feed rate can be decreased to about 15% of the specified value to achieve the product specifications at low cost. A Markov chain length of about 30 appears to be suitable for obtaining solutions with low standard deviations (5%) (b) 40% and 10 tons of EPO2. Figure 10 shows a number of network configurations obtained for this case. The best network configuration corresponds to a membrane network cost of $290,000 featuring two membrane units, one with a high membrane area and the other with a low membrane area, and a permeate recycle stream. The increased design complexity over the previous case in terms of the allocation of vacuum pumps on the permeate side is evident from Figure 10. The optimal feed flow rate for this case was determined to be about 0.1094 kmol/s. The corresponding initial feed flow rate was specified at 0.13 kmol/s. The network design solutions indicate that the feed flow rate is a critical optimization variable contributing the membrane network cost. Higher feed flow rates result in increased recycle rates, which result in higher vacuum pump costs. The cost of compression significantly contributes to the total membrane network cost. From this


Figure 9. Optimal membrane networks for the production of 30% O2 permeate product.

case study, it is evident that the feed flow rate is an important optimization variable that should be considered in design studies involving multiple membrane stages and recycle compressors. A Markov chain length of 40 consistently provided high-quality solutions with standard deviations over all optimal solutions obtained of less than 5%. 6.2. Hydrogen Recovery from Synthesis Gas. 6.2.1. Problem Summary. This case study targets membrane networks for the recovery of hydrogen from synthesis gas using polysulfone membranes. The membranes offer an excellent H2/CO selectivity of about 15. Because the feed is available at high pressure (22 bar), feed compression is not required in this case. Unlike the air separation problem, the product specifications are imposed in the form of the recovery and purity of hydrogen in the permeate product. The optimization of the feed flow rate is not considered in this case, as a fixed recovery is desired from the given feed stream. High-purity hydrogen (99% H2) at high recovery is required at a pressure of 10 bar, which is the pressure of the permeate stream. The objective function includes the costs of the membrane area and recycle compressors. The aim of this case study is to visualize the impact of various flow patterns on the total cost of membrane networks. The feed optimization is discarded by setting the probability associated with feed optimization moves to 0. Probabilities for the selection of various flow patterns are set to 1 and 0 to refer to the particular flow pattern of interest. The membrane and economic data for this case are presented in Table 4.

6.2.2. Optimization Results. The best network structures obtained for different flow patterns are reported in Figure 11. The countercurrent flow pattern inside the membrane units results in the lowest membrane network cost compared to networks with other flow patterns. It is interesting to observe that the performance variations between various flow patterns are fairly small, indicating that designs with co-current and cross-flow patterns might be viable options if they are seen to offer operational advantages for this particular problem. A possible explanation for the similar performances observed for the different flow patterns is the low pressure ratio across the membranes of 2.2. One should note that it is generally difficult to realize countercurrent flow patterns in large industrial membrane permeators. It is therefore important to consider permeate stream interactions with permeate compartments for industrial systems so that the network performance of various flow patterns can be improved to a higher degree. A Markov chain length of about 40 provides a set of quality solutions with a standard deviation of less than 5%. 6.3. Hydrogen Recovery from Refinery Streams. 6.3.1. Problem Summary. This case study addresses the separation of H2 from CH4 in refinery process streams. The aim of the study is to analyze the process economics associated with the upgrading of multicomponent purge streams to produce relatively pure hydrogen sources. The feed and membrane specifications are taken from the available literature.43 Because the permeability of the C4 species is not available, their

4316 Ind. Eng. Chem. Res., Vol. 43, No. 15, 2004 Table 3. Problem Data for the Production of Oxygen-Enriched Air Membrane and Network Specifications parameter

value

P1 P2 δ PerO2 PerN2 ∑F1,m 30% O2 40% O2 area bounds 30% O2 40% O2

1.07 bar 0.2675 bar 10-7 m 2.049 10-11 kmol/(m2‚s‚m‚bar) 9.509 10-12 kmol/(m2‚s‚m‚bar)

EPO2 XO2 capital cost labor cost

0.23 kmol/s of air (20.9 mol %) 0.13 kmol/s of air (20.9 mol %) [50 500] m2 [50 700] m2 product specifications 10 tons/day 0.3, 0.4 25% of installed compressor cost and membrane cost $15.792 Cost Data

parameter

value

Cmem A Cfcom I Cfcom P Ccom HP Ccom AHP Ccom I Cvac HP Cvac AHP Cvac I com η

23.4 $/m2 101 718 $/(kmol/s) 41 943 $/(kmol/s) 186.5 hp/(kmol/s) 294.5 $/(hp‚year) 3791.3 $/hp 5832.7 hp/(kmol/s) 294.5 $/(hp‚year) 32 500 $/hp 0.60 0.50 0.82 0.50 10 10

ηvac bcom bvac ELcom ELvac

Table 4. Problem Data for Hydrogen Recovery from Synthesis Gas Membrane and Network Specifications parameter

value

P1 P2 (Per/δ)CO (Per/δ)H2 ∑F1,m feed pressure area bounds

22.0 bar 10 bar 3.125 × 10-7 kmol/(m2‚s‚bar) 4.689 × 10-6 kmol/(m2‚s‚bar) 0.0225 kmol/s (75 mol %) 22 bar [200 2500] m2 product specifications H2 purity 99% H2 recovery 90% product pressure 10 bar Cost Data parameter Cmem A Ccom HP Ccom AHP Ccom I com η bcom ELcom

value 100.0 $/m2 3140.9 hp/(kmol/s) 320.8 $/(hp‚year) 3791.3 $/(hp) 0.60 0.82 3

permeability is assumed to be the same as that of the C3 species. The membrane, economic, and product data for this case are summarized in Table 5. Only crossflow is considered in the case study. In addition to the membrane area and recycle compression costs, the objective function considers the cost of product compres-

Table 5. Problem Data for Refinery Stream Upgrading parameter

value

P1 P2 (Per/δ)H2 (Per/δ)C1 (Per/δ)C2 (Per/δ)C3 ∑F1,m Xm,f H2 C1 C2 C3 feed pressure area bounds

12.0 bar 1.0 bar 9.7130 × 10-6 kmol/(m2‚s‚bar) 1.2393 × 10-7 kmol/(m2‚s‚bar) 2.1436 × 10-8 kmol/(m2‚s‚bar) 1.5491 × 10-8 kmol/(m2‚s‚bar) 0.044 052 kmol/s

H2 purity H2 recovery Cmem A Ccom HP Ccom AHP Ccom I ηcom bcom ELcom

35.54 mol % 25.74 mol % 19.35 mol % 22.7 mol % 12 bar [200 2500] m2 product specifications 95, 99% 95% 100.0 $/m2 13 109.8 hp/(kmol/s) 320.8 $/(hp‚year) 3791.3 $/hp 0.60 0.82 3

sors to elevate the permeate stream to the desired pressure (12 bar). The cost tradeoffs associated with a single-stage membrane system are presented in Figure 12. It can be seen that the recovery values are limited to the range of 0.40-0.93, with the product purity ranging from 0.80 to 0.97. The system failed to provide any product of purity higher than 97% hydrogen. In addition, the recovery value dropped sharply at higher purities of hydrogen. A linear decline in membrane area and network cost can be observed with an increase in permeate purity. Because a single-stage membrane system is limited by the hydrogen purity and recovery it can achieve, an alternative membrane network needs to be targeted that offers improved recovery and purity. The membrane network synthesis seeks designs that provide permeate product streams of 95% and of 99% purity. 6.3.2. Optimization Results. A number of optimal network configurations obtained from the optimal search are presented in parts a and b of Figure 13 for the cases of 95% and 99% hydrogen purity, respectively. The optimal network configurations feature permeate recycles from a stage operated at a low membrane area to a stage operated at a high area. In addition, the presence of two compressors can be observed for highpurity designs (99% H2). About 20% of network cost is attributed to the product compressors. Future research directions should target the development of membranes that could provide H2 on the retentate side and thereby reduce compression costs. 7. Conclusions This work adopts proven optimization techniques in the form of simulated annealing and a synthesis representation originating from reaction and separation process synthesis for the synthesis of gas permeation membrane networks. The process representation captures all possible combinations of membrane units and their flow patterns. The richness of the representation and the robust optimization method enables the identification of the most economical conventional and novel network configurations for the required product specifications. The superstructure model can be easily de-


Figure 10. Optimal membrane network configurations for 40% O2 permeate product.

veloped and adjusted for networks with diverse features such as those with or without feed compressors, vacuum pumps, and product compressors. The approach is successfully applied to a number of illustrative examples. It is able to identify novel designs with significantly improved performances (cost reductions in excess of 20%) over previous designs reported in the literature.

Once a set of promising designs is developed using the proposed approach, each of these designs needs to be investigated further using detailed membrane unit models such as those presented by Marriott and Sorensen.32 The approach described here can be used to significantly reduce the number of design alternatives to be assessed using detailed models and can thus


Figure 11. Optimal network configurations for synthesis gas enrichment.

significantly reduce the times involved in designing membrane systems. Notation Sets C ) {c} ) set of membrane compartments

F1 {f )1} ) fresh feed stream in state s ) 1 ∈ S to represent feed on the retentate side FCOM ) feed compressor located before the feed splitter SPF1 M ) {m} ) number of components in the gas stream PRs ) {pr ) 1} ) product stream in state s ∈ S


Figure 12. Single-stage membrane system tradeoff.

RCOMc ) one recycle compressor for each compartment c ∈C S ) {s ) 1,2} ) pair of states for representing both highpressure and low-pressure states SC ) {sc} ) set of subcompartments Parameters Amax ) maximum area bound for the permeator unit Amin ) minumum area bound for the permeator unit factor ) plant stream factor m&s ) Marshall and Swift index Nc ) number of compartments in different states s ∈ S Nsc number of subcompartments in the membrane compartment PDs,pr,m ) desired product molar flow rate of component m ∈ M in state s ∈ S for product stream pr ∈ PRs Perm ) permeability of component m ∈ M on the membrane Ps ) pressure of compartment c ∈ C in state s ∈ S RDs,m,pr ) desired recovery in state s ∈ S for component m ∈ M in product stream pr ∈ PRs rmin ) minumum split fraction of an active stream T ) temperature of the gas stream, K UC ) utility cost, $/hp‚year XDs,pr,m ) desired purity in product stream pr ∈ PRs in state s ∈ S for component m ∈ M Xm,f ) mole fraction of component m ∈ M in the feed stream d ) membrane thickness γg ) ratio of specific heats of gaseous mixture entering compressor Mixers and Splitters BS2,sc,c ) binary splitter associated with subcompartment sc ∈ SC in compartment c ∈ C in state s ) 2 ∈ S MI1,c ) mixer at the inlet of every compartment c ∈ C in state s ) 1 ∈ S MIs,pr ) mixers on product stream pr ∈ PRs in state s ∈ S MSC1,c ) mixers to mix all streams leaving state 2 and undergoing state change to enter compartment c ∈ C in state s ) 1 ∈ S SPF1 ) splitter on the feed stream f ∈ F1 SPs,c ) splitter at the outlet of each compartment c ∈ C in state s ∈ S SPs,pr ) splitter on product stream pr∈ PRs in state s ∈ S Variables Ac ) area of membrane permeator unit bcom ) exponent for power cost function for recycle and feed compressors BV12,c ) binary variable for vacuum pump V12,c allocated on each permeate compartment whose permeate stream undergoes partial recycle

BV2,pr ) binary variable for vacuum pump V2,pr allocated on the product stream pr ∈ PR2 BV21,ic ) binary variable for vacuum pump V21,ic allocated for all permeate streams undergoing complete recycle to compartment ic ∈ C in state s )1 ∈ S. bvac ) exponent for power cost function for vacuum pumps Ccom AHP ) annual cost per unit horsepower (hp) consumed for the recycle compressor, $/ (hp‚year) Cvac AHP ) annual cost per horsepower (hp) consumed for the vacuum pump, $/(hp‚year) Cmem ) annualized cost per unit membrane surface area A (including manufacturing, installation, and replacement costs), ($/m2)/year Ccom HP ) horsepower constant per unit flow rate for recycle compressor, hp/(kmol/s) Cvac HP horsepower constant per unit flow rate for vacuum pump, hp/(kmol/s) capital cost per unit horsepower (hp) for recycle Ccom I compressor, $/(hp) Cfcom capital cost per unit molar flow rate for feed comI pressor, [$/(kmol/s)]/year ) capital cost per unit horsepower (hp) for vacuum Cvac I pump, $/hp Costfc ) annualized feed compressor cost, $ Costmem ) annualized membrane area cost, $ Costrc ) annualized recycle compressor cost, $ CostV12,c ) annualized cost of vacuum pump (one per compartment) allocated V12,c on each permeate compartment stream, $ CostV2,pr ) annualized cost of single vacuum pump V2,pr on the product stream pr ∈ PR2, $ CostV21,ic ) annualized cost of single vacuum pump V21,ic allocated for all permeate compartment streams that undergo complete recycle to compartment ic ∈ C in state s )1 ∈ S, $ Cfcom ) power cost per unit molar flow rate for feed P compressor, [$/(kmol/s)]/year ELcom ) equipment life for compressors, year ELvac ) equipment life for vacuum pumps, year F1,m ) molar feed flow rate of component m ∈ M to the membrane network in state s ) 1 ∈ S (retentate side) GPc,sc,m ) molar flow rate of component m ∈ M arising due to gas permeation from subcompartment sc ∈ SC in compartment c ∈ C in state s ) 1 ∈ S (retentate side) to subcompartment sc ∈ SC in compartment c ∈ C in state s ) 2 ∈ S (permeate side). I1,c,m ) input molar flow rate of component m ∈ M to compartment c ∈ C in state s ) 1 ∈ S OBJ ) total annualized cost, $ OPs,pr,m ) molar product flow rate of component m ∈ M from product mixer MIs,pr pr ∈ PRs in state s ∈ S. Os,c,m ) output molar flow rate of component m ∈ M from compartment c ∈ C in state s ∈ S. OSC1,c,m ) molar flow rate of component m ∈ M from state change mixer MSC1,c, c ∈ C OSs,c,sc,m ) output molar flow rate of component m ∈ M from subcompartment sc ∈ SC in compartment c ∈ C in states s∈S Xs,c,sc,m ) mole fraction of component m, m ∈ M, in subcompartment sc ∈ SC in compartment c ∈ C in state s∈S ηcom ) compressor efficiency for recycle compressor ηvac ) efficiency for vacuum pump θCC1,c,ic ) fraction of stream from splitter SP1,c toward the mixer MI1,ic where ic ∈ C in state s )1 ∈ S θCC2,c,ic ) fraction of stream from splitter SP2,c, c ∈ C, in state s ) 2 ∈ S toward the state change mixer MSC1,ic, ic ∈ C, in state s )1 ∈ S.


Figure 13. Optimal membrane network configurations for the production of 95% (top two designs) and 99% (bottom two designs) pure hydrogen permeate product.

θCPs,c,pr ) fraction of stream from splitter SPs,c c ∈ C in state s ∈ S toward the product mixer MIs,pr, pr ∈ PRs, s ∈ S. θFC1,c ) fraction of feed from feed splitter SPF1 toward compartment c ∈ C in state s )1 ∈ S. θFP1,pr ) fraction of feed from feed splitter SPF1 toward product mixer MI1,pr, pr ∈ PRs θFP1,pr ) fraction of feed from feed splitter SPF1 toward product mixer MI1,pr, pr ∈ PRs θPC1,pr,c ) fraction of stream from product splitter SP1,pr, pr ∈ PRs, in state s ) 1 ∈ S toward compartment mixer MI1,c c ∈ C in state s ) 1 ∈ S.

θPC2,pr,c ) fraction of stream from product splitter SP2,pr, pr ∈ PRs, in state s ) 2 ∈ S toward state change mixer MSC1,c, c ∈ C, in state s ) 1 ∈ S. θPPs,pr ) fraction of stream from splitter SPs,pr entering product streams pr ∈ PRs, s ∈ S

Appendix I. Vacuum Pump Allocation The logic behind the allocation of vacuum pumps is developed with the basic information that a single vacuum pump is usually connected to all permeate compartment streams undergoing no recycle and enter-


ing the permeate product stream. When a permeate stream undergoes a partial recycle, a single vacuum pump has to be connected to the permeate compartment. If the recycle is not performed in this way, then two vacuum pumps have to be connected, one corresponding to vacuum application on the permeate product stream and the other to the stream undergoing partial recycle. Hence, three different scenarios can be identified for the allocation of vacuum pumps: (a) Permeate Streams Undergoing No Recycle. This case is very common in systems without any state change recycle streams. In such a scenario, all permeate streams undergoing no recycle can be connected together, and a single vacuum pump can be allocated for these streams. The vacuum pump defined for this purpose is V2,pr. (b) Permeate Streams Undergoing Partial Recycle. In this case, as proposed in the theory, a single vacuum pump, V12,c, is allocated for all permeate compartments whose permeate stream undergoes partial recycle. (c) Permeate Streams Undergoing Complete Recycle. In this case, a vacuum pump can be assigned for all streams undergoing complete recycle and entering the state change mixer MSC1,c. As presented in the Appendix III, the cost expressions for vacuum pumps can be expressed as a function of the feed flow rate to the vacuum pump. Hence, without introducing any further mixers and splitters, binary variables could be enough to model the cost scenarios for all three different types of vacuum pumps, namely, product stream pump, compartment pump on the permeate side, and compartment pump on the retentate side. Subsequent cost expressions for three different scenarios along with logical expressions are presented in section 3.

UC ) 0.0373 $/(hp‚year)() 0.05 $/kWh)

(47)

factor ) (days operated/year) x 24 h

(48)

The costing equations can be expressed by the following set of equations com

fixed cost

)

Ccom I

(

)

bcom

com Ccom HP ng

ηcom

operating cost ) Ccom HP

(49)

com Ccom AHP ng

(50)

ηcom

is the molar flow rate of the gas through where ncom g the compressor. Appendix III. Cost of Vacuum Pumps The following equations are applicable for costing of a vacuum pump. Costing equations are taken from Bhide and Stern20 and Douglas.44 Power parameter Cvac HP is calculated using the equation

Cvac HP )

[( )

γg 1.013 8314 T 745.3 γg - 1 P2

]

(γg-1)/γg

(51)

where P2 < 1.0 bar. The temperature of the gas stream is taken to be the temperature at which the membrane system is operated. The installed compressor cost parameter Cvac is calI culated as20

) Cvac I

1 32 500 ELvac 10

(

)

(52)

Appendix II. Cost of Compressors The following equations have been developed from the literature44 to address the cost of recycle compressors. Similar equations apply for feed and product compressors. Power parameter Ccom HP is evaluated using the expression

Ccom HP )

[( )

γg P2 8314 T 745.3 γg - 1 P1

(γg-1)/γg

]

-1

(44)

where P2 g 1.0 bar. For vacuum applications, the inlet pressure is taken to be 1 bar by gas expansion. The value of γg is taken as 1.4. The temperature of the gas stream is taken to be the temperature at which the membrane system is operated. The compressor cost parameter Ccom is calculated I using the following equation

Ccom I

1 m&s ) (517.5)(3.11) ELcom 280

(

)

(45)

The operating cost parameter is calculated by multiplying the utility cost by a factor for yearly operation as

Ccom AHP ) (UC)(factor) where

The operating cost parameter is calculated by multiplying the utility cost by a factor for yearly operation as

Cvac AHP ) (UC)(factor)

(53)

UC ) 0.0373 $/(hp‚year)() 0.05 $/kWh)

(54)

factor ) (days operated/year) x 24 h

(55)

where

The costing equations can be expressed by the following set of equations vac

fixed cost

)

Cvac I

(

)

vac Cvac HP ng

operating costvac ) Cvac HP

ηvac

(

bvac

(56)

)

vac Cvac AHP ng

ηvac

(57)

is the molar flow rate of the gas through where ncom g the vacuum pump. Literature Cited

(46)

(1) Weller, S.; Steiner, W. A. Engineering aspects of separation of gases: Fractional permeation through membranes. Chem. Eng. Prog. 1950, 46, 585.

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Received for review October 24, 2003 Revised manuscript received May 12, 2004 Accepted May 17, 2004 IE030787C

Synthesis and Optimization of Gas Permeation Membrane Networks

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