Optimization of Multistage Olefin/Paraffin Membrane Separation Processes through Rigorous Modeling

In this work, we explore the capabilities of an NLP optimization model to determine the viability of facilitated transport membrane processes intended to replace traditional distillation currently employed for propane/propylene separation. An NLP optimization model for multistage membrane processes has been formulated, introducing the mathematical description of the facilitated transport mechanisms in the PVDFHFP/BMImBF4/AgBF4 membranes previously developed by our research group. For the first time, a simultaneous optimization of the process and the membrane material (i.e. carrier concentration) has been performed, thanks to the implementation of the governing equations for the fixed site and mobile carrier mechanisms. Once the model is solved in GAMS it returns the optimal membrane area, carrier loading and permeate pressure of each stage based on Net Present Value Cost (NPVC) minimization. Different process flowsheets were evaluated and the results show prominent reductions on NPVC for facilitated transport multistage processes when compared to distillation.


Introduction
Propane/propylene gaseous mixtures resulting from fluid catalytic cracking and steam cracking are commonly separated using high pressure or cryogenic distillation, which is associated to major energy and capital consumptions. 1 Through the last years, process intensification by means of membrane technology has emerged as a promising alternative to large, expensive and energy-intensive distillation units. 2 Many membrane materials have been reported for olefin/paraffin separation, including polymers, [3][4][5] and more complex materials, such as carbon molecular sieves (CMSs), [6][7][8][9] zeolitic imidazolate frameworks (ZIFs) [10][11][12][13][14] or facilitated transport membranes. 15,16 Among these, facilitated transport membranes can easily surpass the permeabilityselectivity trade-off of polymeric membranes thanks to the reversible reaction between the olefin and a carrier cation, typically silver, which is added to the membrane composition. 17,18 Facilitated transport membranes have been synthesized following different approaches, from supported liquid membranes (SLM) 19,20 to supported ionic liquid membranes (SILM) 21 that replace organic solvents with non-volatile room temperature ionic liquids (RTILs) 22 in order to avoid solvent losses through evaporation. 23 Recently, composite facilitated transport membranes prepared by solvent casting of a polyvinylidene fluoride-hexafluoropropylene (PVDF-HFP) polymeric solution containing the ionic liquid and the silver salt have been reported. 24 In these dense membranes, which feature a combination of fixed site and mobile carrier transport mechanisms, 25 selectivities up to 150 and propylene permeabilities higher than 1000 Barrer have been achieved that avoid the issues of supported liquid membranes.
Since a single-stage membrane process that produces polymer grade propylene and fuel grade propane simultaneously is only feasible for high concentrated feeds 26 , due to the purity-recovery trade-off inherent in membrane operation, the implementation of membrane technology to intensify the olefin/paraffin separation process can be carried out according to two different approaches. The first one involves the use of membrane modules along with new or existing distillation columns to create a hybrid process that reduces the required reflux ratio and its associated expenses. [27][28][29][30][31] The second approach achieves complete replacement of the distillation column with membrane technology by designing and optimizing appropriate multistage/multistep membrane processes. 32,33 In particular, several trade-offs should be balanced when designing a multistage membrane process based on facilitated transport membranes. Firstly, the total membrane area of each stage determines the flowrates and purities of the product streams in that stage. Thus, higher stage areas generate larger permeate flowrates at the expense of permeate purity. In addition, the transmembrane flux of the transported species in these membranes is strongly dependent on the carrier loading, as derived from the experimental analysis and the mathematical models. 21,25 However, high carrier concentrations imply high membrane cost per unit area, which could affect the process economics. Finally, higher transmembrane pressures increase the available driving force for the permeation but at the expense of higher recompression requirements.
Moreover, the whole process flowsheet can be optimized in order to obtain the optimal process configuration in terms of number of stages, mixers, splitters and compressors.
This type of optimization involves the design of superstructures that are solved using This article is protected by copyright. All rights reserved.
binary variables, which result in complex mixed integer nonlinear programming problems "MINLP". 34 In this regard, the works by Prof. Agrawal's group provide extensive insight on this design strategy. 35,36 However, MINLP problems are difficult to solve because they combine challenges of nonlinear and mixed integer programming, and require dedicated methods for its resolution. 37 Instead, while it is often possible to study all potential multistage configurations in one single superstructure, most studies dealing with superstructure optimization for membrane separation show that in many cases a two-stage configuration is optimal. [38][39][40] Therefore, in this work we focus on the optimization of the implicit trade-offs in two multistage facilitated transport processes, specifically, one conventional two-stage configuration and one two-stage configuration with a two-step second stage, commonly known as "two-and-one-half" stage process. 32 In this manner, the complex mixedinteger nonlinear formulations associated to superstructures can be replaced with a nonlinear programming problem. The membrane modules have been modelled as hollow fiber modules, which is the most adequate configuration for gas separation, featuring high packing densities and energy efficiency. [41][42][43][44] Furthermore, the mass balances in the membrane modules have been described as ordinary differential equations and have been solved using orthogonal collocation on finite elements. 45 Consequently, black box modeling approaches can be avoided, thus allowing the study of the transmembrane flux profiles along the fibers. These profiles will show how the optimization works in balancing the recovery-purity trade-off by varying the membrane total area of each stage. Additionally, a simultaneous optimization of the process and

Single-Stage and Multistage Membrane Processes
In membrane process design the basic unit is the membrane stage, which can be defined as an operating unit, comprising one or more membrane modules, which performs a specific task different from any other membrane stages existing in the same process. A single-stage process, shown in Figure 1, is the simplest membrane process that can be designed, although some major limitations in membrane operations affect these processes. In particular, membrane material selectivity and the industrially reachable pressure ratio prevent satisfying high product purity and recovery simultaneously, and process engineers may have to sacrifice one of these specifications. Therefore, singlestage processes are often used for bulk concentration prior to further purification processes. 47 To overcome the limitations of single-stage membrane processes more stages could be interconnected, generating different multistage configurations. Staging a membrane process involves solving the trade-off between capital expenses (additional compressors and membrane modules) and product recovery. Although there are many multistage configurations, the two-stage and the so-called two-and-one-half stage processes, 48 are preferred over three stage or higher multistage processes. 32 In the two-stage process, shown in Figure 2, the recompressed permeate of the first stage is fed into a second stage for a further purification step. The retentate of the second stage is then recycled and mixed with the original feed. Another usual two-stage variation uses the retentate of This article is protected by copyright. All rights reserved.
the first stage as feed for the second stage. However, such configuration is commonly used when the retentate is the final product and the purity of the more permeant species is not a concern. 32 In comparison, the two-and-one-half process shown in Figure 3 uses a two-step separation for the second stage and the permeate stream of the second step is recycled to the first step feed. This configuration can achieve any desired concentration of the more permeable gas by controlling the relative size of the second stage modules. Moreover, to balance the tradeoffs and calculate the feed and permeate pressures and the membrane area of each stage, process optimization is required.

Discretized membrane model
Previous studies on membrane process optimization usually make use of simplified membrane models. These shortcut models are typically implemented within superstructures that are solved as mixed-integer optimization problems. 38 For processes where a more detailed description of the membrane separator is needed as in the case of hollow fibers modules, a perfect cross-flow model is usually employed. This model assumes plug-flow in the high pressure side of the fiber, usually the lumen side, and perfect mixed flow in the permeate side, i.e. shell side. 28 However, assuming that the feed flows through the lumen side may seem unrealistic in gas separation hollow fibers, whereby the active layer is often formed in the outer fiber surface. In addition, most membrane models assume fixed permeability/selectivity values for a given membrane material. In contrast, the rigorous facilitated transport model used in this work requires This article is protected by copyright. All rights reserved.
calculating the partial pressure profiles along the module. To address these issues, a fully discretized optimization model has been developed in this work.
The membrane model is depicted in Figure 4 and is based on the following assumptions: • The feed and permeate streams flow through the shell and lumen sides of the fibers respectively.
• The module operates isothermally and in steady state.
• The feed and permeate streams flow in co-current mode.
• Plug-flow is assumed at both sides of the membrane.
• The total feed and permeate pressures are operation constants.
• The only pressure drop in the membrane module is the transmembrane pressure.
The component molar flowrates are discretized according to the following mass balances: where j F and are the molar flowrate and the transmembrane flux of component , respectively, and dA is the fiber outer wall area differential element. The dimensionless fiber axial length is defined as: where L is the total fiber length. Rearranging terms, the mass balances (ODEs) and the boundary conditions can now be rewritten as: One of the main features of this work is that the membrane material optimization (i.e. carrier concentration) and the multistage process optimization are performed simultaneously. For this purpose, the rigorous facilitated transport model previously developed by this research group has been implemented in the optimization, 25 particularly for PVDF-HFP/BMImBF 4 /AgBF 4 membranes containing silver as carrier.
In this model, which was specifically developed for facilitated transport membranes that combine fixed and mobile carrier mechanisms, the permeability of the paraffin can be considered constant. Thus its transmembrane flux is described through the solutiondiffusion model: 49 where 3 8 is the paraffin permeability, is the active layer thickness and 3 8 and 3 8 are the feed and permeate partial pressure of the paraffin.
Conversely, the olefin flux, i.e. the preferentially transported species, is affected by the complexation reaction with the silver cations: The flux can be calculated as the sum of the contributions of three transport mechanisms: solution-diffusion, fixed-site carrier and mobile carrier: where is the complexation constant, 3 6 is the propylene solubility in the composite membrane, is the ionic liquid mass fraction in the membrane composition, is a fitting parameter for the fixed-site carrier mechanism and [ ] is the silver concentration in the membrane. The temperature dependencies have been This article is protected by copyright. All rights reserved.
omitted as isothermal operation has been assumed. The contribution of the Fickian diffusion to the olefin transmembrane flux (Eq.11) is negligible compared to the contribution of the facilitated transport mechanisms (Eq. 12-13) and, in consequence, it has not been included in the optimization. A detailed description of the model can be found in the original source. 25 Table 1 shows the values of the model parameters.
As commented before, the use of this specific model for facilitated transport avoids using a fixed permeability parameter and introduces carrier concentration as a decision variable. In this way, the membrane material and the multistage process can be optimized at the same time.
Finally, the ordinary differential equations (Eqs 4-7) are solved as algebraic equations after discretization through implicit Runge-Kutta collocation methods using 100 finite elements and 3 internal collocation points.

Optimization details
The aim of this work is to optimize two multistage membrane processes: a) a two-stage and b) a two-and-one-half stage, intended to produce polymer grade propylene, and enriched propane, from a C 3 refinery stream minimizing the Net Present Value Costs (NPVC) of both configurations. The feed stream is considered to come from the partial condenser of a depropanizer column. In this manner a vapor stream would be available without further processing. The pressure, temperature and composition of the original feed, stream F in Figures 1-3, are fixed by common refinery specifications. 50 Table 2 This article is protected by copyright. All rights reserved.
displays the properties of the propylene/propane mixed stream, the target product purities and the process constants and constraints.
The decision variables of the model are: • The membrane area of each module.
• The carrier concentration of each membrane.
• The permeate pressure.
The mathematical standard form of the optimization problem can be described as: where v is the vector of decision variables, is the set of constraints (Eqs. [16][17] and ′ and " are the lower and upper limits of the decision variables, respectively. Finally, the objective function (NPVC) has been calculated as a combination of OPEX and CAPEX, correcting the operating expenses according to the time value of money: This article is protected by copyright. All rights reserved.
The parameters and constants used are shown in Table 3. A detailed description of the economic calculations can be found in a previous work. 51 Briefly, the compressor expenses have been calculated according to the Guthrie's Modular Method for Costing and Sizing 52 and the membrane cost is calculated using the market prices for its constituents and the optimized composition. To maintain a conservative estimation of the membrane cost, the lab-scale price of its constituents has been considered: where is the membrane cost per unit area, δ is the membrane thickness and is the molar mass of AgBF 4 . The membrane replacement cost has been introduced through a membrane replacement factor ( ), which is estimated as a percentage of the membrane capital cost.
The detailed equations used to assess the process economics are outlined in Appendix A. The optimization model renders a NLP problem solved in GAMS using as solver the multi-start heuristic algorithm OQNLP on a 3.40 GHz Intel® Core™ i7-3770 processor. The model statistics are detailed in Table 4. The model is solved in less than 0.1 seconds when CONOPT is used as local solver for OQNLP. The reported results are typically obtained in 500 to 1500 iterations depending on the complexity of each case.

Results and Discussion
This section presents and discusses the main results regarding the design and optimization of the proposed flowsheets. The feed stream to both flowsheets consists of This article is protected by copyright. All rights reserved.
an equimolar propane/propylene mixture and the objective is to obtain propylene and propane purities of 99.5 mol% and 95 mol% respectively. The Net Present Value Cost (NPVC) of both multistage processes will be compared with the NPVC of the reference distillation column. Heat integration strategies are not considered in the distillation base case, as they heavily rely on the specific layout of each olefin production unit.

Multistage membrane process optimization results
The optimized two stage flowsheet is displayed in Figure 5. The optimal design comprises two membrane stages showing considerable size differences. The first stage requires around 14000 m 2 of membrane area to generate a propylene depleted retentate stream, thus achieving the desired propane purity. However, the second stage area is almost ten times smaller and is intended to produce a high purity propylene permeate regardless of the retentate purity.
Regarding the optimal carrier loading, its value is allowed to vary between the This article is protected by copyright. All rights reserved.
loading of the second stage raises the membrane specific cost to 324 $/m 2 , the relative small size of this stage dampens the total membrane cost.
Finally, it should be noted that this multistage configuration generates a large reflux stream with high associated compression costs. Nonetheless, the optimal permeate pressure of both stages falls to 1 bar, promoting higher driving force in the modules at the expense of higher compression duty.
In order to gain better insight on the optimal stage design, the transmembrane flux profiles along the fiber modules have been calculated. Figure  propylene. The optimal solution involves continuing the operation past that point to be able to generate the optimal cut-off. Figure 7 shows the molar fractions of propane and propylene in the retentate and permeate streams. In contrast, Figure 6B shows the same profiles in stage M2, but in this case the propylene transmembrane flux remains almost constant due to the reduced membrane area required, which produces a polymer grade propylene stream as permeate but maintains a notable propylene concentration in the retentate.
This article is protected by copyright. All rights reserved. This is a good example of how optimization can solve the intrinsic tradeoff between productivity and purity, and displaces it towards high permeate volume production or high product purity depending on the specific task of each stage within the multistage process.
The "two-and-one-half stage" process optimization results are shown in Figure 8.  Table 5.
It should be noted that the optimal permeate pressure in both case studies and in all stages falls to 1 bar. Although the compression is a capital-intensive operation, the permeate pressure has a major influence on the objective function. A higher permeate pressure reduces the available driving force for the permeation, which involves larger This article is protected by copyright. All rights reserved. membrane areas and a higher recycle flowrate. In this manner, the reduction in the required recompression does not produce any savings, because the potential benefits are hindered by the large flowrate that needs to be recompressed and the CAPEX associated to extra membrane area.
A brief analysis of the transmembrane flux profiles in the "two-and-one-half" stage process reveals similar design criteria as in the first studied case. Figure 9A-C displays the propylene and propane transmembrane flux along the dimensionless axial length of the fibers module.
As commented before, the large membrane area required in the first stage is intended to achieve the desired propane purity in one single pass. As a result, most of the propylene permeates and its driving force is reduced until its transmembrane flux falls below the propane level, see Figure 9A. Notice that this is not detrimental when the product stream is the retentate, as is the case for stage M1. However, the subsequent stages, Figures 8B-C, are intended to produce a propylene-enriched permeate and consequently, the propylene transmembrane flux remains high along the fiber modules, thanks to the relatively small areas involved in these stages.

NPVC comparison
The NPVC of both membrane processes and the base case distillation are itemized in Table 6. The results reveal the prominent potential of facilitated transport membrane processes to replace the traditional distillation and to sharply decrease the investment This article is protected by copyright. All rights reserved.
expenses when implementing the two stage membrane process. On the other hand, the introduction of an additional separation step in the "two-and-one-half" stage flowsheet produces minor savings compared to the two stages process and, therefore, a less complex process may be preferred. It is worth noting that the main difference between the base case distillation and the membrane processes is not in the capital expenses but in the operating costs, which is consistent with the use of process steam in the distillation reboiler. As expected, process intensification through membrane technology plays here a major role in energy saving.
The cost data included in Table 6 for the distillation column correspond to the base case reported in our previous paper, 29 that is, for a distillation column without heat integration. However, OPEX could be overestimated since the distillation columns can be heat integrated. Calculations reported in the literature show that heat-pump based heat-integrated columns (also known as vapor recompression design) are capable of achieving up to 75% energy savings. 14,26,53 Those same studies note that such designs are not routinely employed due to operational and control challenges. Instead, heatintegration can be implemented through other parts of the plant generating extra amount of low-grade heat, such as, quench water. 14 A detailed heat-integration study for distillation columns is beyond the scope of this study, since it would require analyzing the complete process flowsheet where the C3 splitter is included. However, our estimates show that even for a 50% heat-integrated column (i.e., with 50% of the reboiler duty provided by heat-integration), our proposed separation process with This article is protected by copyright. All rights reserved.
membrane modules can achieve savings in energy requirement over heat-integrated distillation.

Comparison for an Extreme Purity Specification
Although propylene purities higher than 99.5% are not demanded by the polypropylene industry, it is instructive to consider a final comparison between the two-stage and twoand-half stage optimization models. For a propylene permeate specification of 99.9% we observe that the two stage process becomes infeasible and has no solution. In contrast, the two-and-half stage model is able to obtain any permeate concentration, and consequently satisfies this specification. As shown in Table 7, the optimum for this process is achieved at a much higher cost, with more than double the NPVC.

Conclusions
The complete replacement of propane/propylene distillation processes by membrane technology strongly relies on adequate membrane processes and materials design. In this work, the simultaneous optimization of multistage processes and facilitated transport membrane composition reveals potential Net Present Value Cost reductions of around 45% compared to the standalone distillation. Additionally, the optimization results showed minor differences between the two studied multistage processes. Thus, the simpler two-stage layout may be more adequate in a real scenario. Furthermore, full discretization of the model reveals how the recovery-purity trade-off and the position of This article is protected by copyright. All rights reserved. This article is protected by copyright. All rights reserved.          (C) of the "two-and-one-half" stage process. This article is protected by copyright. All rights reserved. Membrane thickness (µm) 20 Silver loading b (mol l -1 ) 0-6 a 1 Barrer = 3.348x10 −16 mol 3 m m −2 Pa −1 s −1 .
b Silver loading delimited according to the experimentally studied concentration range. 24,25  This article is protected by copyright. All rights reserved. This article is protected by copyright. All rights reserved. This article is protected by copyright. All rights reserved. This article is protected by copyright. All rights reserved.