Efficient Optimization-Based Design of Membrane ... - ACS Publications

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Efficient Optimization-Based Design of Membrane-Assisted Distillation Processes Mirko Skiborowski,† Johannes Wessel,‡ and Wolfgang Marquardt*,§ AVT−Process System Engineering, RWTH Aachen University, 52064 Aachen, Germany S Supporting Information *

ABSTRACT: While simulation-based design has become a standard tool in process engineering, most commercial software lacks appropriate membrane process models. Beside the necessity of available membrane data the major obstacle for process design is the lack of a systematic design approach. The latter is especially true for hybrid processes, which despite their complexity provide high potential for process intensification and can facilitate considerable savings in energy, emissions, and capital investment. Thus, an efficient approach for the design of such hybrid processes is highly desirable. Based on a thorough review of the available methods for the design of membrane-assisted distillation processes, we identify current limitations and present a novel optimization-based method for conceptual process design. The formulation of a superstructure in combination with a model decomposition and a stepwise solution strategy facilitates a robust and efficient optimization-based design, which relies on rigorous thermodynamics and bridges the gap between shortcut calculations and a detailed equipment design. The method is demonstrated for three industrially relevant case studies.

1. INTRODUCTION In their article on the separation needs for the 21st century, Noble and Agrawal1 describe process intensification (PI) as a critical research need and identify hybrid separation processes as an important topic. Hybrid separation processes combine two or more unit operations, which contribute to a given separation task by means of different physical phenomena. They are integrated such that synergetic effects allow to overcome the limitations of the individual unit operations.2,3 Although hybrid separation processes are still not applied on a regular basis, the chemical industry is well aware of their potential.4 Membrane-assisted distillation processes constitute one possible type of a hybrid separation process, which offers significant potential for the separation of close-boiling or azeotropic mixtures. In contrast to distillation, membrane separation is not limited by vapor−liquid equilibrium (VLE) and can thus overcome azeotropes and distillation boundaries. High selectivity, low energy consumption, and a compact and modular design are further advantages of membrane separation.5 However, stand-alone membrane processes are usually not economically viable if a large permeate flow rate or high purities of both permeate and retentate streams are required.6 Membrane-assisted distillation combines the advantages of both separation principles, operates without an additional separation agent, and can result in reduced energy demand, emissions, and investment cost.7,8 In particular, combinations of distillation and pervaporation (PV) or vapor permeation (VP) have been subject to excessive research and are in operation in more than 100 industrial processes worldwide.5,9 The review articles of Lipnizki et al.10 and, more recently, of Suk and Matsura11 and of Gorak et al.3 provide a comprehensive discussion on process design options and applications. As most available and the best-performing industrial membranes are hydrophilic,9 the main area of © XXXX American Chemical Society

application for PV and VP is the dehydration of solvent mixtures. However, the development of organophilic membranes provides additional potential and new application areas in organic chemistry.6,12 A major obstacle for considering hybrid processes as design options in industrial practice is the complexity of the design task. Consequently, there is an urgent need for suitable design methods.4 Based on a analysis of the design task and a thorough review of available methods for the conceptual design, presented in section 2, we identify current limitations and present a novel optimization-based method for the conceptual design of membrane-assisted distillation processes. While the general approach to modeling and optimization of membraneassisted distillation processes can be applied for various membrane processes, the current article focuses on PV-assisted distillation, due to its practical importance and the proven potential. Section 3 presents the superstructures and model formulations, while initialization and solution strategies are described in section 4. The application of the suggested method is demonstrated by means of three industrially relevant case studies in section 5. The method not only allows for an efficient conceptual design but also bridges the gap between shortcut calculations and detailed equipment design. This feature is illustrated by a comparison of results obtained by the method introduced here with those of shortcut calculations and of optimization-based equipment design relying on detailed ratebased models. Finally, section 6 compiles conclusions and discusses some perspectives for future work. Received: June 20, 2014 Revised: August 15, 2014 Accepted: September 8, 2014

A

dx.doi.org/10.1021/ie502482b | Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Industrial & Engineering Chemistry Research

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Figure 1. Exemplary membrane-assisted hybrid process configurations for the separation of binary mixtures.

2. DESIGN OF MEMBRANE-ASSISTED DISTILLATION PROCESSES Design of membrane-assisted distillation processes starts with the identification of potential process variants and the selection of a suitable membrane. While first attempts have been made to computationally predict permeability and permselectivity to support membrane selection,13 a suitable membrane is generally identified based on database information and expert knowledge.14 Many process variants are possible even for the separation of binary mixtures.15 Skiborowksi et al.14 and Lutze and Gorak12 provide a short review on the generation of variants. Figure 1 exemplarily depicts three common configurations for the separation of a binary mixture by means of a membraneassisted hybrid process. Each of these configurations offers multiple design degrees of freedom (DDoF). For each distillation column, the number of equilibrium trays, the location of feed and side streams and the energy duties need to be determined. The membrane process can comprise of multiple stages, which have to be specified by membrane area, feed pressure, and temperature as well as permeate pressure. Additional DDoF may result for specific configurations. Since hybrid processes are strongly integrated, each design decision will affect the performance of the overall process. Therefore, all DDoF should be optimized simultaneously in order to take full advantage of both separation principles. Thus, not only the single units but also the hybrid process should be considered in its entirety early on in the design process. In order to support and guide process design for complex hybrid processes, Marquardt et al.16 proposed a synthesis framework that has recently been extended by Skiborowski et al.:14 The generation of process variants is followed by a first screening based on shortcut models in order to assess feasibility and effort of a separation. The most promising variants are subsequently optimized using more detailed models by means of mixed-integer nonlinear programming (MINLP). The superstructure model relies on mass and energy balances, rigorous physical property models, and either equilibriumcontrolled or kinetically limited mass transfer models. Finally, the best-performing flowsheet should be refined by optimizing a more complex rate-based engineering design model, which accounts for hydrodynamics, as well as mass and heat transfer resistances. The main idea of this synthesis framework14,16 is to systematically facilitate the consideration of a large variety of options and to reduce the experimental demand to a minimum by gradually increasing model complexity, while decreasing the number of process variants throughout the design process.

Various methods have been proposed for the design of membrane-assisted distillation processes. In order to motivate this work, a brief review of these methods is presented next, classifying them in the categories of the synthesis framework. 2.1. Shortcut Design. One of the first shortcut models for membrane-assisted hybrid process design is the minimum-area method presented by Moganti et al.17 and Stephan et al.18 This method is based on either Smoker’s equation or the McCabe− Thiele method for distillation and a simple rating model for membrane performance. Pettersen and Lien19 present a shortcut model for VP assuming constant average permeation flux. In contrast, the shortcut model by Pressly and Ng15 is based on a specified separation factor and membrane cut. It is implemented in a screening procedure to perform a break-even analysis of VP-assisted distillation processes. Fahmy et al.20 present a similar approach to calculate optimal separation factors and membrane cuts for the membrane units in a hybrid process. Recently, Ayotte-Sauve et al.21 presented a thermodynamically motivated shortcut approach, which is based on the power-of-separation concept introduced by Sorin et al.22 This optimization-based method operates on a superstructure and compares favorably with respect to the minimum-area approach. All these shortcut methods rely on assumptions that restrict their application to binary systems with (approximately) ideal vapor−liquid equilibrium (VLE) behavior. They also rely on rather simple models to predict membrane performance. However, the application of hybrid processes becomes most interesting in case of strongly nonideal mixtures. While various shortcut methods for nonideal distillation are available, only few shortcut methods for hybrid processes account for nonideal thermodynamics and utilize complex mass-transfer models to determine membrane performance accurately. To do so, Bausa and Marquardt23 proposed the combination of the pinch-based rectification body method (RBM)24 and a one-dimensional membrane model including a complex mass transfer model. Besides ideal cross-flow, the membrane shortcut assumes isothermal operation in case of a retentate or an infinite feed flow rate in case of a permeate product. 2.2. Conceptual Design Methods. Rautenbach et al.25 and Hömmerich and Rautenbach26 were the first to present a simulation-based approach for conceptual design of membraneassisted distillation processes. It is based on commercial simulation software and a user-defined routine for membrane performance calculation, which relies on rigorous thermodynamics and a local mass transport model. Sommer and Melin8 further extended this approach to account for multiple membrane modules and interstage heating. Several authors B



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Figure 2. Illustration of the superstructure for a distillation column with one side and two feed streams.

proposed similar approaches later using different simulation software.7,27−33 In order to circumvent a tedious simulation-based design, optimization-based approaches were proposed more recently. Eliceche et al.34 also use commercial software and a tailored membrane model but determine the best operating point of a given process structure using the optimization capabilities of the software. They further extended this approach to account for multiple membrane units and interstage heating by means of a sequential calculation strategy.35 Lelkes et al.36 and Szitkai et al.37 were among the first to propose the simultaneous optimization of process structure and operating point combining a distillation column superstructure38 and a PV membrane network.39 While the superstructure allows for the optimization of merely all DDoF, a simple Mergules model is used for equilibrium calculations and either a regression or an interpolation model is used to describe membrane performance. Kookos40 presented a similar approach but proposed a more elaborate membrane model. The distillation superstructure is based on mass balances, equilibrium calculations, summation constraints, and enthalpy balances (MESH model) but assumes ideal liquid and vapor phases as well as constant molar overflow (CMO). Recently, Naidu and Malik41 presented a more rigorous approach with several superstructure formulations and detailed thermodynamic property calculations for both distillation column and PV models. However, they do not offer any systematic strategy for the initialization and solution of this complex optimization problem. Caballero et al.42 presented a two-step approach, which corresponds to a combination of the second and third level of the process synthesis framework of Marquardt et al.,16 for grassroot and retrofit design of hybrid distillation−VP processes. They utilize a very simple shortcut model, comprising a combination of a modified Underwood equation and a split model for the membrane. The more rigorous superstructure is modeled in a process simulator and solved as a continuous nonlinear programming (NLP) problem by an external general purpose solver, relaxing all discrete variables. While all these approaches differ in assumptions and hence model formulation, they were only applied to binary mixtures. Although not all of these methods are limited to binary

mixtures, the model complexity increases considerably for multicomponent systems. Since hybrid processes are particularly promising for strongly nonideal and azeotropic mixtures, assumptions such as CMO or constant relative volatilities (CRV) should be avoided. An optimization-based approach for designing a process to separate a nonideal ternary mixture has only been reported by Brusis.43,44 2.3. Design by Means of Detailed Engineering Models. In order to determine a detailed process and equipment design, nonequilibrium (NEQ) models should be utilized for both distillation and membrane units, considering hydrodynamics, pressure drop, temperature, and concentration polarization, as well as additional mass and heat transfer resistances. Such models significantly contribute to the nonlinearity of the process model and further complicate simulation and optimization. Detailed engineering models of membrane separations have, for example, been described by Mariott and Sørensen45,46 and Soni et al.47 Their application to the design of a hybrid distillation−PV process has been presented by Barakat and Sørensen.48,49 They extended the superstructure of Kookos40 to facilitate the use of either distillation, PV, or a hybrid configuration and employed a genetic algorithm for optimization. A similar strategy was followed by Górak and co-workers: Detailed mass-transfer models were developed and implemented in Aspen Custom Modeler (ACM) for distillation and membrane units.50 Buchaly et al.,51 Roth et al.,52 and Koch et al.53 have built on these models and a modified differential evolution algorithm54,55 to optimize different variants of hybrid processes. While these publications demonstrate the feasibility of process optimization using detailed NEQ models for process and equipment design, they suffer from a high computational load (varying from several hours to days53) and require excessive experimental effort to determine mass transfer area and resistances.3 Thus, detailed NEQ models should only be applied if accurate correlations are available for the target equipment.56 Therefore, as envisioned by the process synthesis framework,14,16 the design space should be reduced as far as possible beforehand. Although the shortcut approach by Bausa and Marquardt23 can cover strongly nonideal and azeotropic VLE C



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vapor stream from one tray, or to selectively remove a single component, which is of special importance for initialization (cf. section 4). These balances are complemented by the summation constraints

behavior and relies on a reasonably accurate membrane model, it has not been validated by means of more rigorous optimization-based design, because an appropriate methodology considering rigorous thermodynamics, which is applicable to multicomponent mixtures, has not yet been presented in literature. Such a methodology is the main contribution of this paper. It builds on the ideas of Brusis43 and extends previously presented results.57 An efficient deterministic optimization is achieved by a decomposition of the process model and a sophisticated initialization and solution strategy.

nc

∑ xn,i = 1 nc

∑ yn,i = 1 and the equilibrium relations yn , i = K n , ixn , i ,

Kn,i =

(5)

γi(xn , i , Tn) ·Fp , i(Tn , p) ·φi0(Tn) ·pi0 (Tn) φi(Tn , p , yn , i ) ·p

,

i = 1, ..., nc (6)

with activity coefficients γi(xn,i,Tn), Poynting factors Fp,i(Tn,p), pure component fugacity factors φ0i (Tn), vapor pressures pi0(Tn), and fugacity coefficients for the vapor mixture φi(Tn,p,y) of component i. The exact formulation depends on the thermodynamic models selected to predict the individual quantities. In addition, the enthalpies hLn and hVn (cf. eq 2) need to be calculated as a function of composition, phase state, and temperature of the stream. The calculation of the distribution coefficients Kn,i and the solution of the equilibrium calculation is a complex task and the thermodynamic equations are the major source of nonlinearity in the model. They complicate the numerical solution and require a solid scaling. To reduce the inherent model complexity, the thermodynamic property calculations are transferred to an external function.59,60 This idea was first promoted by Brusis43 and later adopted by Kossack et al.63,64 Such model decomposition results in a drastic reduction of the number of iterations and a more stable solution.65 We utilize an extension of this approach,66 which not only transfers the thermodynamic property calculations but completely relocates the solution of the complex equilibrium calculations to the external function. This extended approach also facilitates a deterministic optimization for heterogeneous mixtures by means of a combination of a phase stability test and a model reformulation. The DDoF of a distillation column comprises the number of trays ntrays, the feed tray nfeed,tray and side stream tray nside,tray, the operating pressure p, as well as the reboiler and condenser heat duties QB and QC. In order to optimize the number of trays, the feed and side stream locations, the binary decision variables (bF1,n, bF2,n, bB,n, bR,n, and bS,n) need to be determined. However, not all of them are necessary. One of the stream locations can and should be fixed to reduce combinatorial complexity without neglecting possible design options. 3.2. Membrane Network Model. Membrane separation can be accomplished in either a single stage consisting of one or multiple parallel membrane modules, or in a multistage membrane network. The use of multiple membrane stages and interstage heat exchangers is typically used in PV to

+ F2z F2, ibF2, n − SzS, ibS, n + RDx R, ibR, n + RByB, i bB, n , (1)

LnhnL + VnhnV = Ln − 1hnL− 1 + Vn + 1hnV+ 1 + Fh 1 F1bF1, n (2)

where and are the flow rate, composition vector, and enthalpy for the liquid and vapor phases on tray n. (F1, zF1, hF1), (F2, zF2, hF2), (S, zS, hS), (RD, xR, hLR), and (RB, yB, hVB) refer to the flow rate, composition vector, and enthalpy of the feed streams, the side stream, the reflux, and the boil-up stream. The binary variables bF1,n, bF2,n, bB,n, bR,n, and bS,n encode the decisions regarding feed, recycle, and side streams related to stage n. Consideration of an arbitrary composition zS,i and enthalpy hS for the side stream allows to withdraw either a liquid or a (Ln, xn, hLn )

i = 1, ..., nc

Kn,i is the distribution coefficient, which is a function of liquid and vapor phase composition xn and yn, equilibrium temperature Tn and pressure p. It derives from equal chemical potentials of the liquid and vapor phase, and reads as

Lnxn , i + Vnyn , i = Ln − 1xn − 1, i + Vn + 1yn + 1, i + Fz 1 F1, ibF1, n

+ F2h F2bF2, n − ShSbS, n + RDhRLbR, n + RBhBV bB, n

(4)

i=1

3. OPTIMIZATION MODEL The optimization model for a hybrid process constitutes an aggregation of superstructure models for each subprocess. The superstructure model for the distillation columns is based on the MESH equations, while the model of a membrane network comprises multiple membrane stages with interstage heating and possible interior recycles. Each membrane stage is modeled by one-dimensional, differential mass and enthalpy balances, incorporating a detailed and experimentally validated local mass transfer model. While all models rely on rigorous thermodynamics, no additional correlations for hydrodynamics or other mass transfer resistances are considered. The following subsections present the superstructure model formulations for the distillation column and the membrane network and describe their aggregation to hybrid process models. While the superstructure models are implemented in GAMS,58 equilibrium calculations and additional thermodynamic property calculations are performed by means of an external function.59,60 3.1. Distillation Column Model. Optimization of a single distillation column by means of a superstructure model was first introduced by Viswanathan and Grossmann38,61 for homogeneous ideal systems. Bauer and Stichlmair62 extended this approach to nonideal and azeotropic mixtures. Figure 2 shows a general superstructure for a distillation column with one side and two feed streams. It is based on the concept of a variable reflux column.63 The variable locations of feed and side stream trays as well as reflux and boil-up trays facilitate the decision on the discrete DDoF. The mass and enthalpy balances for an arbitrary tray n are given by

i = 1, ..., nc

(3)

i=1

(Vn, yn, hVn )

D



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Figure 3. Illustration of the general membrane network superstructure with variable permeate and retentate recycle and bypass streams for each membrane stage.

introduced into the economic model to annihilate the stage, if it is bypassed.

counteract the temperature drop caused by the evaporation of the permeating components. An elevated feed temperature usually results in an exponential increase in permeate flux, which, as a simple rule of thumb, is duplicated by a temperature elevation of 20 K.67 In the following, the superstructure for the membrane network is presented first, and the model for a single membrane stage is described in detail subsequently. While the superstructure and separation model can be applied in the same or a modified version to alternative membrane processes, especially the local mass-transfer model needs to be revised, if membrane processes other than PV or VP are considered. Refer to the book of Melin and Rautenbach67 for an elaborate description and appropriate model formulations for various membrane processes. 3.2.1. Superstructure. Figure 3 shows the superstructure of a multistage membrane network. A heat exchanger is used in front of each membrane stage to adjust the feed temperature, while another heat exchanger connected to the permeate effluent is used to generate the permeate-side vacuum pressure. Each heat exchanger is modeled similarly to the reboiler and condenser of the distillation column.63 While the preheater heat duty is determined from the temperature increase, the condenser cooling duty is computed assuming condensation of the vaporous permeate stream at permeate pressure. An additional pump can be utilized to increase the feed-side pressure, resulting in an elevated boiling point of the liquid feed, to approach the maximum operating temperature of the membrane material. Increasing feed temperature may also allow for operation at higher permeate-side pressure and the use of cooling water instead of more expensive cooling brine.44 An additional valve is used to reduce the pressure to the operating pressure of the distillation column or the outlet pressure. Santoso et al.68 showed that proper internal recycles in the membrane network can significantly improve efficiency and reduce cost. Thus, a possible recycle of either the permeate or the retentate is considered as an additional option. While it is possible to determine the number of membrane stages just by optimization of the membrane area, the introduction of additional effects limiting the driving force, such as concentration polarization, can result in numerical problems.69 In order to facilitate a robust optimization of the number of membrane stages, an optional bypass stream is integrated into the superstructure for each membrane stage. A stream directed to a membrane stage enters the preheater if bby,st = 0; otherwise, if bby,st = 1, the stream bypasses the membrane stage and is redirected to the following stage or to the outlet (cf. Figure 4). The bypass variables are also

Figure 4. Illustration of the bypass stream and structure of a single membrane stage st.

In order to avoid numerical problems due to bypassing a stage, membrane performance is always calculated from the potential feed stream of stage st, that is, L in,st = Fst

(7)

x in,st, i = xF ,st, i ,

i = 1, ..., nc

(8)

However, the stream to the next membrane stage st + 1 depends on the bypassing decision and is calculated from Fst + 1 = (1 − b by,st)Lout,st + b by,stFst x F,st + 1, i = (1 − b by,st)xout,st, i + b by,stx F,st, i ,

(9)

i = 1, ..., nc (10)

TF,st + 1 = (1 − b by,st)Tout,st + b by,stTF,st

(11)

Accordingly, the overall permeate stream and the membrane area also depend on the bypass variables. As every single membrane stage can be bypassed there are a lot of equivalent process structures. Additional constraints are introduced to avoid combinatorial multiplicity. In particular, b by,st ≤ b by,st + 1 ,

st = 1, ..., nst − 1

(12)

guarantees that all stages prior to a bypassed stage need to be bypassed, too. Computational experience from the investigated case studies showed that this formulation is superior to an alternative formulation demanding that all subsequent stages are bypassed. This can be attributed to the product specifications, which need to be satisfied for the last membrane stage in case of a retentate product. Bypassing this stage shifts the product specifications to the previous stage, which seems to complicate optimization. E



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rate Vout,st and compositions yout,st. The permeate pressure is limited by the condensation temperature TBout,st. Both, feed and permeate-side pressure are assumed constant for the membrane stage. As heat resistances through the membrane are usually negligible, the temperature on the retentate and permeate sides are assumed to be equal.71 The change of the retentate and permeate stream along the membrane length z is described by means of the differential mass and enthalpy balances

The DDoF of a membrane network comprise the number of stages nst, the feed side pressure pF, the permeate pressure pP, the use of a permeate or retentate recycle ξrecycle, the feed temperature for each stage TF,st and the membrane area for each stage Ast. The number of stages is determined by means of the bypass stream formulation. While the permeate pressure is usually fixed, the feed-side pressure is optimized to facilitate an increase in feed temperature, which also determines the preheater heat duties. 3.2.2. Separation Model. A membrane stage is usually comprised of multiple parallel membrane modules containing membrane sheets of fixed area. While it would be possible to model a membrane stage by means of a discrete number of parallel modules, the model complexity would be increased without any considerable effect on the optimal process structure.43 Thus, a fixed length lst and a variable width wst are assumed in each membrane stage st = 1,...,nst to manipulate the total membrane area Ast. Figure 5 presents a schematic representation of a membrane stage.

dLi = −Ji ·wst , i , ..., nc dz

(13)

dVi = Ji ·wst , i , ..., nc dz

(14)

J dh L = wst(hL − hV ) dz L

(15)

Li and Vi are the local molar flow rates of component i on the retentate and permeate sides, while Ji is the local flux of component i from the retentate to the permeate side. The molar enthalpy of the liquid retentate hL and the vapor enthalpy of the permeate flux hV are required in order to account for the temperature drop due to the evaporation of the permeate flux. They are calculated by means of external equations, similar to the distillation column model. Additional algebraic equations are required to relate the component flow rates to the overall flow rates and compositions: nc

L=

∑ Li , Li = xi·L , i , ..., nc i

(16)

nc

V=

Figure 5. Schematic of a membrane stage model.

∑ Vi , Vi = yi ·V , i , ..., nc i

(17)

nc

Most conceptual design models as well as the shortcut model of Bausa et al.,23 assume ideal cross-flow. For a more detailed performance description, we assume cocurrent flow of retentate and permeate in a dead-end module, such that retentate and permeate composition are determined from integration along the flow direction. The differences between cocurrent and counter-current flow are in general assumed to be rather small.70 The membrane feed enters stage st with flow rate Lin,st, compositions xin,st, and temperature Tin,st on the retentate side, which operates at pressure pF,st. Mass transfer through the membrane occurs along the flow direction z with permeation flux J and compositions m. The permeated stream evaporates on the permeate side due to the vacuum pressure pP,st, which is sustained by condensation of the permeate stream with flow

J=

∑ Ji , Ji = mi ·J , i , ..., nc i

(18)

For the integration of the ordinary differential equations (ODEs) (eq 13−15) an efficient implicit Runge−Kutta72 method is used. The chosen Radau II collocation method72,73 attains an order of convergence of 2ncp − 1 for ncp stages (collocation points), assuming that the right-hand sides in eqs 13−15 are sufficiently smooth. In this study, we use a three-stage method (ncp = 3) and increase the accuracy of the integration further by subdividing the total membrane length into three finite elements (nfe = 3) of equal length.74 The discretization and the location of the collocation points are illustrated in Figure 6. The collocation method constructs a polynomial for which the differential

Figure 6. Illustration of the collocation points and finite elements for one membrane stage. F



Article V hout,st = hV (yst, n

equation is satisfied exactly at the initial and at each collocation point. The numerical integration is performed for each finite element fe = 1,...,nfe of each membrane stage st = 1,...,nst by means of the set of equations 0 Lst,fe,cp, i = Lst,fe, i +

dLst,fe,cp, i lst , AR dz n fe

i = 1, ..., nc

0 Vst,fe,cp, i = V st,fe, i +

dVst,fe,cp, i lst , AR dz n fe

i = 1, ..., nc

dhst,fe,cp lst AR dz n fe

(19)

(20)

(21)

Ji = Q i·DF,i

for all collocation points cp = 1,...,ncp simultaneously. The weighting matrix ⎛ 88 − 7 6 ⎜ 360 ⎜ ⎜ 296 + 169 6 AR = ⎜ ⎜ 1800 ⎜ ⎜ 16 − 6 ⎜ ⎝ 36

296 − 169 6 1800 88 + 7 6 360 16 + 6 36

results from the specific distribution of collocation points.73 Lst,fe,cp,i, Vst,fe,cp,i, and hLst,fe,cp represent column vectors for the retentate and the permeate flow rates, as well as the molar enthalpy of the retentate at the collocation points. The initial values for the flow rates and molar enthalpies (L0st,fe,i, V0st,fe,i, LL,0 st,fe) correspond either to the feed state of the membrane stage

0 V st,1, i = 0,

i = 1, ..., nc

i = 1, ..., nc

L ,0 L hst,1 = h in,st

(23) (24) (25)

or to the values at the end of the previous finite element for fe ≥ 2: 0 Lst,fe, i = Lst,fe − 1, ncp , i ,

i = 1, ..., nc

(26)

0 V st,fe, i = Vst,fe − 1, ncp , i ,

i = 1, ..., nc

(27)

L ,0 L hst,fe = hst,fe − 1, ncp

(28)

The resulting outlet streams of the retentate and permeate leaving membrane stage st, are represented by the last collocation point of the last finite element, that is, Lout,st = Lst, nfe , ncp

(29)

̀ xout,st, i = xst, nfe , ncp , i ,

i = 1, ..., nc

(30)

L hout,st = hst,L nfe , ncp = hL(xst, nfe , ncp , pF,st , Tst, nfe , ncp)

(31)

Vout,st = Vst, nfe , ncp

(32)

yout,st, i = yst, n

fe , ncp , i

,

i = 1, ..., nc

(34)

i = 1, ..., nc

(35)

which is based on experimentally validated permeances Qi. Though the driving force DFi is given by the difference in chemical potential,67 it is often simplified to the difference in fugacities, activities, or partial pressures, depending on the type of membrane process. Although constant permeances are often assumed for PV and VP, the model should cover additional nonideal effects in case the solvent, solute, and the membrane material interact strongly.12 To date, this is only possible by postulating a specific permeance model structure and identifying parameters from time-consuming experiments.12 For each case study investigated in this work, experimentally verified mass-transfer models are available (cf. Appendix A). Again, a model decomposition is performed. In particular, all thermodynamic property calculations required for the calculation of the permeate flux and the enthalpies as well as the necessary flash calculations are performed by means of an external function to solve the optimization problem robustly. 3.3. Process Superstructure Model. For each separation problem, a superstructure model consisting of the superstructures of the individual units and their possible interconnections is implemented in GAMS. The DDoF of the distillation column(s) and membrane network(s) are optimized together with the choice for the locations of their interconnections. Such simultaneous optimization is of special importance for the identification of an optimal design. The choice of side stream and recycle trays not only effects the feasibility and performance of the distillation column(s) but also the required membrane area(s).44 One external function handles all equilibrium, enthalpy, and other thermodynamic property calculations for the complete process. For process optimization, an objective function needs to be defined, to reflect both operating and capital cost in form of total annualized cost (TAC). Operating cost mainly depend on heat exchanger duties and membrane replacement. Capital cost estimation for distillation columns is based on sizing equations and cost correlations reported by Guthrie77 and adopted by Kraemer et al.,63 while the capital cost of the membrane process are simply correlated by the membrane area. All cost correlations and parameters are provided in the Supporting Information. For all case studies presented in section 5, a depreciation period of 10 years and an interest rate of 6% are assumed. The capital cost estimates are updated by means of the Marshall and Swift index.

−2 + 3 6 ⎞ ⎟ 225 ⎟ ⎟ −2 − 3 6 ⎟ ⎟ 225 ⎟ 1 ⎟ ⎟ ⎠ 9 (22)

0 Lst,1, i = L in,st x in,st, i ,

, pP ,st , Tst, nfe , ncp)

The collocation approach is exemplarily validated and compared to a finite difference method in Appendix B for the first case study discussed in section 5.1.1. 3.2.3. Local Mass-Transfer Model. The transport of a component through the membrane can either be modeled by means of the solution-diffusion (SD) or pore-flow (PF) model.67,75 However, the exact permeation mechanism for PV is not yet known76 and often both phenomena influence mass transfer through the membrane.12 Thus, in most cases, mass transfer is described by means of a semiempirical model for the local flux

L

L L ,0 hst,fe,cp = hst,fe +

fe , ncp

(33) G



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Figure 7. Solution strategy for the optimization of membrane assisted distillation processes.

4. SOLUTION STRATEGY Model decomposition and especially the relocation of the equilibrium calculations to the external function strongly reduces the number of nonlinear equations and variables that need to be handled by the solver in GAMS. However, still, a complex MINLP problem has to be solved, which requires careful initialization and a tailored solution strategy.78 In order to facilitate a robust optimization of the hybrid process configurations, we propose an incremental initialization and solution strategy: first, each distillation column and membrane network is initialized individually; then, the subprocesses are aggregated and finally optimized. The solution strategy is based on previous work for distillation63,66 and hybrid desalination69 processes. The main idea is to first determine a feasible solution utilizing the full process structure, that is, all column trays and membrane stages included in the superstructure are used. The optimal process is then determined by reducing the process structure and

adjusting the operating conditions. The solution strategy is illustrated in Figure 7. First, a suitable process superstructure is selected and initial estimates for the composition of connection streams between the subprocesses are determined. These estimates do not need to be exact design targets but may also be derived iteratively. In the first step of initializing the distillation columns (1), the tray compositions and temperature are determined either by means of a feed flash calculation or by linear approximation of the column profile. Afterward, a feasible design with a maximum number of equilibrium trays is calculated in two steps: first, only mass balances, equilibrium, and summation constraints (MES) are solved; then, energy balances (MESH) are added. All feed streams are kept constant with fixed binary variables. If a membrane network is connected to the distillation column via side and recylce streams, these streams are lumped during the initialization and a selective separation is assumed. This facilitates the operation of the column sections below and above the selective side stream in different distillation regions. H



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Figure 8. Superstructure of the hybrid PV−distillation process for the dehydration of ethanol.

5. CASE STUDIES Case studies of various complexity are presented to illustrate the potential of the suggested methodology. First, the dehydration of ethanol is investigated, considering two different hybrid process configurations with one and two distillation columns. Afterward two ternary separations in MTBE and acetone production are investigated. They involve a hybrid process with a single distillation column and a membrane network, which are connected by means of a side and a recycle stream. While the distillation column for MTBE separation operates in one distillation region, the operation of the distillation column in the hybrid process for acetone separation covers both distillation regions. For all case studies, feed and product streams are assumed to be at their boiling point, and a membrane length of 1 m is assumed. All thermodynamical property data is enclosed in the Supporting Information. The optimization-based designs are compared with those resulting from shortcut calculations23 and optimization, using detailed engineering design models.53 Each optimization was performed on a standard PC with an Intel Core i3-2100 processor with 3.1 GHz. The SNOPT solver of GAMS 22.7 was applied for the solution of the series of relaxed NLP problems. The computation times vary depending on the number of components and the size of the superstructure. However, none of the calculations, including initialization and intermediate steps, exceeded a computing time of 10 min. 5.1. Ethanol Dehydration Process. The dehydration of alcohols is of tremendous importance in the biochemical industry, with an estimated annual worldwide production of more than a 100 billion liters of bioethanol.80 As most of the operating PV units are used for ethanol and isopropanol dehydration,9 the design of hybrid processes for the separation of such binary azeotropic mixtures was subject to many publications.15,29,36,37,40,52 This problem is also used here as a first representative case study. 5.1.1. Hybrid Process with One Distillation Column. Bausa and Marquardt23 performed shortcut calculations for the dehydration of ethanol by means of a hybrid configuration, in which a distillation column is used to concentrate ethanol to a nearly azeotropic composition, and a PV separation is used as a polishing step to further purify the ethanol product stream. The calculations were performed for varying distillate purities in order to determine cost-optimal designs. For a fair comparison,

The initialization of the distillation column ends with a minimization of the reboiler duty, subject to purity specifications and additional process constraints. The initialization of membrane networks (2) is performed in a similar fashion. In a first step, permeate fluxes are calculated for all collocation points, based on fixed feed-side temperature and compositions, assuming ideal cross-flow on the permeate side. Afterward, the collocated differential mass and enthalpy balances on the feed-side are included. Adding collocation equations for the permeate-side extends the model to cocurrent flow. Subsequently, mass and enthalpy balances for the preheaters and permeate condensers and possible recycle streams are introduced. The initialization of the membrane network is completed with a minimization of the membrane area subject to feed-side temperatures, purity specifications, and additional process constraints. Prior to the optimization of the hybrid process, the subprocesses are aggregated and mass and enthalpy balances for each of the previously assumed fixed connection streams are introduced (3). Afterward, equations for equipment sizing and cost estimation are added and initialized. A first economic optimization determines the optimal process design using the full process structure. Finally, the hybrid process is optimized (4) for minimum TAC with all discrete and continuous DDoF subject to purity specifications and additional process constraints. As the process structure is fixed in steps 1, 2, and 3, only NLP problems have to be solved. The discrete decisions on the process structure are only considered in step 4. Instead of solving the resulting MINLP problem directly, it is solved as a series of successively relaxed MINLP (SR-MINLP) problems as suggested by Kraemer et al.63 in the context of distillation processes design. While a continuous distribution of feed, recycle, or side streams along the height of a distillation column79 presents an undesired but physically valid solution, this is not the case for the bypass stream formulation of the membrane network. The arithmetic mean temperature (cf. eq 11) represents only an approximation in case of partial bypassing but gives the same result as a thorough enthalpy balance in case of a discrete solution. The simplification is justified by the small error and the fact that the final result corresponds to a discrete solution. I



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In the optimal design, the ethanol stream is concentrated to 85.5 mol % by means of the distillation column, all six membrane stages are utilized and the feed pressure is elevated, such that the feed temperature of each membrane stage can be increased by the preheater to the maximum value of 100 °C. A comparison of the results from shortcut-based and optimization-based design for different sizes of the membrane network is given in Table 2.

they were replicated in this work with an adjusted cost model, resulting in a cost-optimal configuration at xD,ethanol = 0.852. In order to investigate the accuracy of the shortcut, an optimization-based design is performed for the superstructure shown in Figure 8. The top product of the distillation is purified by means of a membrane network. The permeate streams of all membrane stages are collected and reintroduced into the distillation column or mixed with the top product of the distillation column and recycled to the membrane network. The location of the reflux is fixed to the top tray of the distillation column, without restricting the possible design options (cf. section 3.1). The thermodynamic properties of the mixture and the performance of the membrane separation are quantified by the models used by Bausa and Marquardt.23 The nonideality of the liquid phase is described by means of the Wilson model. A complex semiempirical flux model (cf. Appendix A.1) represents the separation characteristics of the PVA/PAN membrane Mol1140 (GFT, Germany). The specification of the separation is given in Table 1. The distillation column is operated at atmospheric pressure, while

Table 2. Comparison of Optimization-Based Design Results with a Varied Number of Membrane Stages and Shortcut Results for Tmax = 100 °C and pP = 30 mbar 3 stages xD,ethanol (mol/mol) QB,column (MW) Amem,tot (m2) hybrid process dist. column investment heating cooling PV network investment memb. replacement heating cooling

Table 1. Specifications for Ethanol Dehydration feed retentate bottom

flow rate (mol/s)

EtOH (mol/mol)

150.76

0.4236 ≥0.9948

H2O (mol/mol) 0.5764 ≥0.999

the permeate pressure is assumed to be set to 30 mbar. The results for a superstructure with a distillation column with 100 equilibrium trays and a membrane network with six stages are depicted in Figure 9.

4 stages

key design values 0.869 0.860 9.784 8.714 6,978 5,181 TAC (M€) 5.388 4.764 1.970 1.764 0.240 0.223 1.563 1.392 0.167 0.149 3.418 3.000 0.996 0.756 0.872 0.648 0.102 1.447

0.109 1.487

6 stages

8 stages

shortcut

0.855 8.278 4,199

0.853 8.163 3,921

0.852 8.442 4,132

4.461 1.674 0.210 1.322 0.141 2.787 0.630 0.525

4.394 1.659 0.216 1.304 0.139 2.735 0.598 0.490

4.380 1.665 0.171 1.349 0.145 2.715 0.586 0.517

0.113 1.519

0.113 1.533

0.076 1.536

It is obvious that the approximation inherent to the shortcut calculation becomes very accurate for an increasing number of membrane stages. In particular, while the optimization-based design results for a hybrid process with 3 stages differs by about

Figure 9. Optimization-based results for the hybrid PV−distillation process for the dehydration of ethanol with one distillation column and six membrane stages. J



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Table 3. Results of Optimization-Based Design for Six-Stage Membrane Network with Different Permeate Pressure pP and Max Membrane Temperature Tmax Tmax = 110 °C xD,ethanol (mol/mol) QB,column (MW) Amem,tot (m2)

0.853 8.081 2,707

hybrid process dist. column investment heating cooling PV network investment memb. replacement heating cooling

4.043 1.635 0.206 1.291 0.138 2.408 0.422 0.338 0.126 1.522

Tmax = 90 °C key design values 0.862 8.995 9,512 TAC (M€) 5.988 1.809 0.218 1.437 0.153 4.179 1.365 1.189 0.094 1.531

pP = 21 mbar

pP = 50 mbar

0.853 8.159 3,270

0.860 8.826 7,702

4.167 1.642 0.200 1.303 0.139 2.525 0.500 0.409 0.112 1.504

5.611 1.785 0.225 1.410 0.150 3.826 1.118 0.963 0.118 1.628

Figure 10. Superstructure of the hybrid PV−distillation process for the dehydration of ethanol with two distillation columns.

of 35% in TAC, an elevation by 10 K results in savings of about 35% of membrane area and 10% of TAC. A better utilization of the cooling brine at a decreased permeate pressure facilitates savings of about 22% of membrane area and 7% of TAC, while an increased permeate pressure of 50 mbar results in a 85% increase in membrane area and 25% in TAC. These results emphasize the importance of an optimal utilization of the utilities. 5.1.2. Hybrid Process with Two Distillation Columns. As the driving force decreases significantly for high ethanol purities, a large amount of membrane area is required to reach very high purities. For example, in case of the design with six membrane stages (cf. Figure 9) about 40% of the overall membrane area is required to drive the purity from 98.8 mol % up to 99.5 mol % of ethanol in the last two membrane stages. To elucidate whether purification by means of a membrane network is the best choice, an alternative superstructure with two distillation columns (cf. Figure 10) is investigated. The final purification of the ethanol stream is now performed by means of a second distillation column, which, similar to pressure-swing distillation, operates at higher pressure. While Sommer and Melin8 performed a simulation-based design for a similar configuration for isopropanol/water separation, none of

+25% in TAC, +15% in energy, and +70% in membrane area from the shortcut-based design, the differences become negligible for a hybrid process with 8 stages. While the TAC of the distillation column is dominated by heating cost, the TAC of the PV network is dominated by cooling cost, due to the expensive cooling brine needed for permeate condensation. Although the latter does not directly correlate with the membrane area, the shortcut calculations remain extremely accurate for a large number of membrane stages. As each membrane stage operates at maximum feed temperature and the cost for cooling brine dominate the TAC for the PV network, further optimization-based design calculations were performed to investigate the effect of permeate pressure, pP, and the maximum membrane temperature, Tmax, on the cost optimal design. Table 3 lists the results of the calculations for a membrane network with six stages, with a varied Tmax of 90 and 110 °C and a varied pP of 50 mbar and 21 mbar. The latter was determined as a DDoF in the optimization, assuming a temperature increase of 10 K for the cooling brine and an additional minimum temperature difference of 10 K for each permeate condenser. While a reduction of the maximum temperature by 10 K results in a tremendous increase of 130% in membrane area and K



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Figure 11. Optimization-based results for the hybrid PV−distillation process with two distillation columns and three membrane stages.

The single-column hybrid process is economically superior to the other two alternatives. The TAC of the pressure-swing distillation can be reduced by more than 20% if three membrane stages are introduced. Optimization of the twocolumn hybrid process with six membrane stages yields a design with TAC of about 4.765 M€, where the second column only accounts for a contribution of 75 k€. The second column is therefore of minor importance and might even be eliminated if the algorithm would facilitate this. Thus, this result suggests that a hybrid process with only one distillation column is most favorable. 5.2. Separation Process for MTBE Production. The second case study deals with the separation of a ternary mixture of methyl tert-butyl ether (MTBE), n-butene, and methanol (MeOH) by means of an integrated hybrid process. This separation relates to the production of MTBE, which although mostly substituted by ethyl tert-butyl ether (ETBE) in Europe and the U.S. due to environmental concerns, is still produced on a scale of millions of tons per year.82 Bausa and Marquardt23 investigated this separation by means of their shortcut approach using an empirical model for the description of the separation performance of the plasmapolymerized membrane PERVAP 1137 (cf. Appendix A.2). Their results will again be compared to those of the optimization-based design. The specifications are given in Table 5. The superstructure of the process is depicted in Figure

the previous optimization-based approaches has considered a process configuration of similar complexity. The optimization was performed with a UNIQUAC model, which provides a better approximation of the VLE at higher pressures than the Wilson model with the given parameters.81 The results for a superstructure with 80 equilibrium trays for the first and 100 equilibrium trays for the second distillation column and a membrane network with three stages are illustrated in Figure 11. The reevaluated single-column hybrid process with six membrane stages, the two-column hybrid process with three membrane stages, as well as a common pressure-swing distillation process are compared in Table 4. Table 4. Comparison of Design Results for Different Process Configurations (UNIQUAC Model) 1 column, 6 stages xD,ethanol (mol/mol) QB,column (MW) Amem,tot (m2) hybrid process dist. column(s) investment heating cooling PV network investment memb. replacement heating cooling

2 columns, 3 stages

key design values 0.844 0.839 8.634 8.671 3968 1786 TAC (M€) 4.605 4.919 1.756 1.747 0.231 0.214 1.379 1.385 0.145 0.148 2.849 2.073 0.599 0.277 0.496 0.223 0.125 1.629

pressure swing

| 0.735 | 3.431

0.860 | 0.731 16.38 | 10.38

| | | |

6.322 3.081 0.334 2.317 0.130

1.099 0.147 0.942 0.010

| | | |

3.241 0.350 2.849 0.042

Table 5. Specifications for Butene, MTBE, and MeOH Separation23

feed distillate permeate bottom

0.107 1.465 L

flow rate (mol/s)

butene (mol/mol)

110

0.4547 ≥0.933

MTBE (mol/mol) 0.4547

MeOH (mol/mol) 0.0906 ≥0.990

≥0.999 dx.doi.org/10.1021/ie502482b | Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX


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Figure 12. Superstructure of a hybrid PV−distillation process for the separation of a ternary mixture with a membrane network connected to the distillation column by means of a side and a recycle stream.

Figure 13. Optimization-based process design result and comparison to shortcut results for the separation of butene, MTBE, and MeOH at the given specifications.

point and the temperature drop along the membrane is rather small due to the small permeate stream. However, due to the small contribution of the membrane unit to TAC it is possible to determine alternative process designs with multiple membrane stages but approximately the same membrane area and TAC. The results of the optimization-based design are compared to those obtained by the shortcut method of Bausa and Marquardt23 in Figure 13. For the shortcut calculations, the composition of the side stream (SSC) is selected manually according to a heuristic.23 It represents the composition with the maximum methanol content in the intersection of the

12: A membrane network is connected to a distillation column by means of a side stream and the retentate stream is recycled to the distillation column as a second feed stream. A superstructure with 100 equilibrium trays and four membrane stages is considered in the optimization. The distillation column is operated at an elevated pressure of 6 bar. Therefore, no further pressure increase is considered for the membrane network. The permeate pressure is set to 50 mbar. The optimal design utilizes the bypass stream option to exclude three of the four membrane stages and neither the preheater nor the possible retentate recycle are used. Both decisions make sense, as the feed stream is already at its boiling M



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Table 6. Results of Shortcut and Optimization-Based Design Calculations for the Separation Processes for MTBE Production Tmax = 100 °C

no Tmax pP = 50 mbar

shortcut xS,butene (mol/mol) xS,MTBE (mol/mol) xS,methanol (mol/mol) QB,column (MW) Amem,tot (m2)

0.000 0.803 0.197 2.310 279

hybrid dist. column investment heating cooling PV network investment memb. replacement cooling

1.487 0.528 0.129 0.369 0.031 0.959 0.045 0.035 0.880

key design values 0.000 0.771 0.229 2.099 228 TAC (M€) 1.396 0.477 0.115 0.335 0.027 0.919 0.037 0.029 0.853

pP = 50 mbar

pp = 60 mbar

0.112 0.696 0.192 2.106 380

0.112 0.696 0.192 2.113 380

1.554 0.460 0.097 0.336 0.027 1.094 0.059 0.047 0.987

1.290 0.461 0.096 0.338 0.027 0.829 0.059 0.047 0.723

Figure 14. Optimization-based process design result and comparison to shortcut results for the separation of butene, MTBE, and MeOH with temperature limit and pP = 60 mbar.

As already mentioned by Bausa and Marquardt,23 the temperature of the side stream exceeds the maximum temperature of 100 °C of the membrane material. They conclude to rather use a more expensive configuration with a side stream location above the feed tray. By introducing a temperature constraint for the side stream, the optimizationbased method can easily determine an optimal design that adheres to the maximum temperature. This configuration comes with a TAC of 1.554 M€ and is hence more favorable than a configuration with the side stream located above the feed tray, which comes with a TAC of 1.670 M€ according to the shortcut. The TAC can further be reduced by increasing the permeate pressure to pP = 60 mbar, rather than decreasing it as in the previous case study (cf. Table 6). Although this may sound counterintuitive, there is a simple explanation. While the increase in permeate pressure results in a decreasing driving force, the saturation temperature of the permeate and the

rectification bodies (RB) of the middle and the stripping section (cf. Figure 13). Although the optimal side stream composition (Sopt) differs from the heuristic selection, the shortcut provides a very good approximation of the optimal design. This can be attributed to the rather small amount of permeate and the inferior contribution of the membrane unit to TAC. Additional calculations confirmed that the approximation becomes worse if the methanol content in the feed is increased. Note that the membrane accomplishes the separation of methanol across the (simple) distillation boundary (SDB), while the distillation column operates only in the MTBE-rich distillation region. The optimization results are compared to the shortcut results with reevaluated cost in detail in Table 6. The optimizationbased design operates the process very close to the minimum membrane area Amin = 223 m2 and reboiler duty QB,min = 2.10 MW determined by the shortcut. N



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means of the NRTL and Redlich−Kwong model; a semiempirical flux model representing the separation characteristics of the Pervap 2201D membrane (Sulzer Chemtech, Germany) is used (cf. Appendix A.3). The configuration of the membrane-assisted hybrid process is similar to the previous case study. Thus, the superstructure in Figure 12 is also used here. However, in contrast to the previous case study, a much larger permeate stream needs to be separated by means of the membrane network, and the distillation column operates in two different distillation regions. The introduction of a selective side stream in the initialization (cf. Figure 7) is of special importance. At first, only the specifications for the distillate are enforced, while the amount of water in the side stream is increased incrementally. Finally, the specifications for the bottom product are added and the required heat duty is minimized. The distillation column is operated at atmospheric pressure, while the permeate pressure is 30 mbar. The results for a superstructure of the distillation column with 100 equilibrium trays and a membrane network with three stages are illustrated in Figure 15. In contrast to the previous case study, the composition profile indicates that the distillation column operates in both distillation regions. The side stream (Sopt) is located in the water-rich distillation region, while the retentate stream that is recycled to the distillation column (Ropt) is located in the IPA-rich distillation region. While the separation process models used in this work are not as detailed as the rate-based models of Koch et al.53 and the economic models are not the same, the resulting designs are very similar.53 Especially, the side stream and retentate recycle flow rates are in good agreement. While neglecting the transport resistances results in an underestimation of the required membrane area, the membrane costs have only a minor influence on TAC, as already stated by Koch et al.53 Thus, the results obtained by the simpler models suggested in this work can be used for initialization and a reduction of the

temperature elevation of the cooling brine are increased. Thus, the cooling brine demand, dominating TAC, can be reduced significantly. The optimal design for a process with pP = 60 mbar is illustrated in Figure 14. Due to the temperature constraint, the side stream contains 11.2 mol % of butene and less MeOH compared to the design without the temperature constraint. This results in a 70% increase in membrane area, which, however, causes only a minor cost increase due to the small permeate stream. The increase in permeate pressure only reduces the required amount of cooling brine, while the process design for pP = 50 mbar and pP = 60 mbar are virtually the same. 5.3. Separation Process for Acetone Production. The third case study deals with the separation of a ternary mixture of acetone, isopropyl alcohol (IPA), and water. The separation relates to the production of acetone by dehydration of IPA. The potential of the optimization-based design has already been demonstrated before,57 using a different membrane model. Here, the results of this work are compared to those presented by Koch et al.,53 which rely on detailed rate-based models for both unit operations. Therefore, the specifications of the separation task (cf. Table 7) and the models for the prediction Table 7. Specifications for Acetone, IPA, and Water Separation

feed distillate bottom

purity recovery purity recovery

flow rate (mol/s)

acetone (mol/mol)

20.53

0.438 ≥0.995 ≥0.995

IPA (mol/mol) 0.052

water (mol/mol) 0.510

≥0.995 ≥0.950

of thermodynamic properties of the system and the membrane performances are similar to those used by Koch et al.53 The nonideality of the liquid and vapor phase are described by

Figure 15. Result of optimization-based process design result for the separation of acetone, IPA, and water at pP = 30 mbar. O



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referring to the choice and utilization of utilities for example, has been demonstrated. While the results of previously reported shortcut-based designs were largely validated with the optimization-based design approach, limitations of the shortcut method could be pointed out as well. The higher level of detail of the novel optimization-based approach provides additional insight and a better understanding of the “best” process design. The results of the optimization-based method can also be used to reduce the problem space for further experimental investigations or for the subsequent refinement of the design with rate-based engineering design models. The presented approach can also be used in future work to determine research targets for material scientists by optimizing the permeance values or the maximum temperature for an arbitrary membrane, as advocated and demonstrated by means of a simulation-based approach by Ploegmakers et al.32 An extension of the current approach should include the choice of membrane material and utilities. As stated by Wynn et al.,9 the use of more expensive ceramic membranes can increase the operating temperature from 100 °C up to 200 °C. The right combination of different membranes and operating conditions for different stages may facilitate a separation without or at least with less expensive cooling brine. As these discrete decisions will further complicate the optimization problem, a hybrid optimization approach combining deterministic methods and a metaheuristic83 may be preferential.

design-space for the very involved optimization with detailed engineering design models. The optimization results using the method introduced in this work are presented in Table 8 for two permeate pressures and a Table 8. Results of Optimization-Based Design Calculations for the Acetone Production Separation Processes pP = 30 mbar

pP = 50 mbar

key design values

3 stages

1 stage

3 stages

recycle (−) Amem,tot (m2) TAC (M€) hybrid process dist. column investment heating cooling PV network investment memb. replacement heating cooling

0.312 220

0.800 305

0.612 221

5 stages 179

1.125 0.236 0.074 0.146 0.016 0.889 0.050 0.053 0.070 0.716

0.460 0.237 0.073 0.149 0.016 0.222 0.057 0.073 0.070 0.022

0.434 0.231 0.073 0.142 0.016 0.203 0.052 0.053 0.076 0.022

0.420 0.234 0.074 0.144 0.016 0.187 0.049 0.043 0.073 0.022

membrane network with one to five stages. For each design, a side stream of approximately 34.5 (mol/s) with 50.5 mol % of water and less than 0.1 mol % of acetone is withdrawn from the distillation column. While the heat duty dominates the TAC of the distillation column and depends heavily on the distillate purity specifications, the cooling duty for permeate condensation dominate the TAC of the PV network. An increase in permeate pressure facilitates the use of inexpensive cooling water instead of the expensive cooling brine required at 30 mbar. This leads to a significant reduction in TAC of about 60%. An additional utilization of more membrane stages does not only reduce the membrane area and TAC but also the retentate recycle flow rate. The size and structure of the distillation column and the energy duties are very similar to the design depicted in Figure 15. Again, these results are in good agreement with the findings of Koch et al.53 using a detailed engineering model.

A. MASS-TRANSFER MODELS A.1. Ethanol Dehydration

The mass-transfer model for the PVA/PAN membrane Mol1140 (GFT, Germany) and the ethanol/water mixture Table 9. Parameters for PVA/PAN Membrane Mol114023 ṁ 0,i (kg/m ·h) ni Ei /9 (K) T0 (K) ci ai 2

6. CONCLUSIONS Although membrane process models have been available for some time, the design of membrane-assisted distillation processes is still limited due to a variety of complex configurations and the lack of a systematic design approach. In this work, we reviewed available design methods and presented an efficient and robust optimization-based approach to overcome current limitations and to bridge the gap between shortcut calculations and a detailed engineering design. Based on a problem decomposition, the separate solution of complex subproblems, and an incremental initialization and solution strategy, complex process configurations can be optimized robustly at low computational effort. A series of challenging case studies confirms the potential of the novel design method. The results show the versatility of the optimization-based approach. The exchange of the thermodynamic property model requires little effort, and the optimization of even strongly integrated process configurations with multiple unit operations can be performed efficiently. The identification of bottlenecks and possible improvements,

EtOH

H2O

0.0429 3.8393 6.064 363.15 0 0

5.253 0.0876 5.888 363.15 0.0011 0.0686

Table 10. Parameters for PERVAP 1137 Membrane from Bausa and Marquardt23

2i

butene

MTBE

MeOH

1.375 × 10−04

3.250 × 10−04

4.0 × 10−05

Table 11. Parameters for Pervap 2201D Membrane (Sulzer Chemtech, Germany)53 Q0,i Ai

acetone

IPA

water

0.0358 14.161

0.0358 14.161

106.39 8.5296

studied in the case study in section 5.1 was developed by Vier70 and utilized by Bausa and Marquardt23 for shortcut design. In a slightly rewritten form the permeate flux Ji = Q i·DF,i

i ϵ{EtOH, H 2O}

(36)

can be expressed by means of a permeance P



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Figure 16. Comparison between the results for permeation flux and permeate flow rate along the membrane by means of collocation (ncp = 3 and nfe = 3) and FD with 10 and 100 grid points.

Q i = ṁ 0, ifx , i ·fT , i ·fa , i and a driving force expression yp DFi = xiγi − i 0P pi

many) and the ternary MeOH/MTBE/butene mixture investigated in the case study in section 5.2 was proposed by Bausa and Marquardt23

(37)

⎛ p ⎞ JMeOH = 2MeOH⎜⎜γMeOHxMeOH + 0 P ⎟⎟ pMeOH ⎠ ⎝

(38)

with the auxiliary terms fx , i = exp(nix H2O)

fT , i

⎛E ⎛ 1 1 ⎞⎞ = exp⎜⎜ i ⎜ − ⎟⎟⎟ T ⎠⎠ ⎝ 9 ⎝ T0

fa , i

ai ⎡ ⎛ ⎛ yp ⎞2 ⎞⎤ ⎢ ⎜ yp ⎥ = ⎢ci i 0P − ⎜⎜ i 0P ⎟⎟ ⎟⎥ ⎜ ⎟ ⎝ pi ⎠ ⎠⎥⎦ ⎢⎣ ⎝ pi

0 (γMeOHpMeOH xMeOH − pP )

(42)

(39)

(40)

0 Jbutene = 2buteneγbutenepbutene xbuteneJMeOH

(43)

0 JMTBE = 2MTBEγMTBEpMTBE xMTBEJMeOH

(44)

The parameters were fitted to experimental data from Vier.70 Due to a lack of data for butene the parameter fit was performed for the binary systems MeOH/MTBE and MeOH/ pentane, whereas pentane is used as surrogate for butene, which they justify by the chemical similarity. The resulting parameters are listed in Table 10.

(41)

Thus, the difference in activity is utilized as driving force, while the permeance is represented by means of a complex and strongly nonlinear model, which is the product of a base permeance ṁ 0,i and three terms accounting for the influence of water composition f x,i, temperature f T,i, and activity fa,i. The activity coefficients γi, which depend on the retentate-side compositions x and temperature T, and the vapor pressure p0i , which depends on the temperature T, are determined from thermodynamic property models. Model parameters are taken from ref 23 and listed in Table 9.

A.3. Acetone/IPA/Water Separation

The mass-transfer model for the Pervap 2201D membrane (Sulzer Chemtech, Germany) and the acetone/IPA/water mixture studied in the case study in section 5.3 is based on the work of Koch et al.53 The permeate flux is described by means of a permeance and a driving force, Ji = Q i·DF,i

i ϵ{acetone, IPA, water}

(45)

which are given by

A.2. MeOH/MTBE/Butene Separation

The empirical mass-transfer model for the plasma-polymerized membrane PERVAP 1137 (Sulzer Chemtech GmbH, Ger-

Q i = Q 0, i·exp(Ai wwater) Q

(46)



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and the difference in fugacities DFi = xiγipi0 ϕi0 − yp ϕ i P i

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with the fugacity coefficient of saturated vapor ϕ and the fugacity coefficient of the vaporous permeate mixture ϕi. The two parameters for each component i are listed in Table 11. 0

B. VALIDATION OF COLLOCATION METHOD In order to demonstrate that the collocation method provides high accuracy, the results of the numerical integration are compared to a finite difference (FD) approach with equidistant discretization for the first and the last membrane stage of the 6stage network in Figure 9. The resulting values for permeate flux and permeate side flow rate are illustrated in Figure 16 for the collocation as well as the FD. For the first membrane stage, the permeate stream of the collocation approach differs by about 5% from the one computed by FD with 10 grid points and less than 1% if 100 grid points are used. The differences become smaller for the last membrane module, for which the permeate stream is also much smaller. However, the calculation of the local fluxes by means of the FD approach is very sensitive toward the initial estimate of the permeate composition. Even for a good initial guess, an oscillation may occur at the first discretization points (cf. Figure 16). The collocation approach provides a high level of accuracy, while the complex thermodynamic property calculations have to be computed only for nCPnFE collocation points per membrane stage.

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ASSOCIATED CONTENT

S Supporting Information *

Information on the economic models and thermodynamic property data for the three case studies. This material is available free of charge via the Internet at http://pubs.acs.org/.

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AUTHOR INFORMATION

Corresponding Author

*Email: [email protected]. Present Addresses †

Department of Biochemical and Chemical Engineering, Laboratory of Fluid Separations, TU Dortmund University, 44227 Dortmund, Germany. ‡ BASF SE, GTE/AB O925, 67056 Ludwigshafen, Germany. § Forschungszentrum Juelich GmbH, 52425 Juelich, Germany. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors acknowledge financial support of “Deutsche Forschungsgemeinschaft” for the project “Optimization-based framework for the synthesis of membrane-assisted hybrid processes” (MA 1188/32-1).

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REFERENCES

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