Optimal Membrane-Process Design (OMPD): A software product for optimal design of membrane gas separation processes

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Optimal Membrane-Process Design (OMPD): A Software Product for Optimal Design of Membrane Gas Separation Processes Yousef Mohammadi Dr , Takeshi Matsuura , Johannes C. Jansen Dr. , Elisa Esposito , Alessio Fuoco , ´ , Fausto Gallucci Prof. , Enrico Drioli , Ludovic F. Dumee Masoud Soroush Prof. PII: DOI: Reference:

S0098-1354(19)30945-7 https://doi.org/10.1016/j.compchemeng.2020.106724 CACE 106724

To appear in:

Computers and Chemical Engineering

Received date: Revised date: Accepted date:

7 September 2019 14 November 2019 8 January 2020

Please cite this article as: Yousef Mohammadi Dr , Takeshi Matsuura , Johannes C. Jansen Dr. , ´ , Elisa Esposito , Alessio Fuoco , Ludovic F. Dumee Fausto Gallucci Prof. , Enrico Drioli , Masoud Soroush Prof. , Optimal Membrane-Process Design (OMPD): A Software Product for Optimal Design of Membrane Gas Separation Processes, Computers and Chemical Engineering (2020), doi: https://doi.org/10.1016/j.compchemeng.2020.106724

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Highlights     

Software for optimal design of membrane gas-separation processes is presented. It generates several potential process design configurations. It finds optimal design specifications and operating conditions. It can handle any type or number of objective functions. It optimizes membrane units arranged in multi-step and/or multi-stage configurations.

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Optimal Membrane-Process Design (OMPD): A Software Product for Optimal Design of Membrane Gas Separation Processes Yousef Mohammadi1*, Takeshi Matsuura2, Johannes C. Jansen3*, Elisa Esposito3, Alessio Fuoco3, Ludovic F. Dumée4, Fausto Gallucci5*, Enrico Drioli3,6, and Masoud Soroush7*

1

Petrochemical Research and Technology Company (NPC-rt), National Petrochemical Company (NPC), P.O. Box 14358-84711, Tehran, Iran 2 Department of Chemical and Biological Engineering, University of Ottawa, Ottawa, Ontario K1N 6N5, Canada 3 Institute on Membrane Technology, National Research Council (CNR-ITM), Via Pietro Bucci 17/C, 87036 Rende (CS), Italy 4 Deakin University, Geelong, Institute for Frontier Materials, Waurn Ponds, Victoria 3216, Australia 5 Inorganic Membranes and Membrane Reactors, Eindhoven University of Technology, Dept. Chemical Engineering and Chemistry, PO Box 513, 5600 MB Eindhoven, The Netherlands 6 Hanyang University, WCU Energy Engineering Department, Room 917 9th Floor FTC Bldg., 17 Haengdang-dong, Seongdong-gu, Seoul 133-791, South Korea 7 Department of Chemical and Biological Engineering, Drexel University, Philadelphia, PA 19104, USA

November 13, 2019

REVISED VERSION

Submitted for Publication in Computers & Chemical Engineering Special Issue Honoring Professor Thomas F. Edgar

Keywords: Membrane Gas Separation; Process Design; Multi-objective Optimization; Software Product

* Corresponding authors: Dr. Yousef Mohammadi: [email protected] Dr. Johannes C. Jansen: [email protected] 2

Prof. Fausto Gallucci: [email protected] Prof. Masoud Soroush: [email protected] Abstract The optimal design of a membrane gas separation process requires minimizing several objective functions subject to nonlinear relationships among the optimizing variables. This article describes a novel software product, named Optimal Membrane-Process Design (OMPD), for the optimal design of membrane gas separation processes. The product generates several potential process design configurations and then searches the process design parameters and operating conditions spaces to arrive at optimal design specifications and operating conditions. It is able to consider every type and any number of operational, compositional, and economical objective functions in a computationally cost-effective manner. It calculates all Pareto optimal solutions in a single trial. It can optimize any number of membrane units arranged in multi-step and/or multi-stage configurations. It optimally places pairs of adjacent membrane units, either two-step or two-stage, while simultaneously considering several membrane types.

3

Graphical Abstract

Two main steps in OMPD.

4

1. Introduction Membrane gas separation processes are used increasingly to separate gases and generate high purity streams for industrial applications. Compared to other conventional separation technologies such as cryogenic distillation and swing adsorption, membrane technologies are easier to implement and scale up, require less manual handling and lower operational costs, and require lower footprints and energy levels [1-5]. For the past decades, membrane technologies have been used in hydrogen recovery [6], oxygen enrichment of air [7,8], CO 2 capture [9-11], air dehumidification [12,13], natural gas purification (sweetening, dehydration, removal of impurities), and helium recovery from natural gas [14-16], among others. Membrane separations have also been applied to upgrade biogas to pure methane [17, 18], to produce high purity nitrogen [19], recover olefins [20-23], and adjust syngas ratios [24]. The economic viability of a membrane-based gas separation process depends on the permeability/selectivity of the membrane material and the optimality of the design and operation of the membrane process. A membrane gas separation process should be able to compete with conventional gas separation processes in terms of capital and operating costs, as well as the purity of the gaseous products generated [25,26]. In order to design optimal membrane processes, a great number of variables should be considered and optimized. The axes of the design space include membrane types, permeabilities and selectivities of membranes, pressure ratios, flow rates, number and arrangement of membrane units, and the total membrane area of each unit [27,28]. An optimal membrane gas separation process design is usually a multiobjective optimization problem, leading to complex relationships. An important step in a membrane gas separation process design is the selection of an optimal process configuration; i.e., finding an optimal arrangement of membrane units and determining how the units should be connected to each other [29,30]. The most common and simplest arrangement in membrane gas separation is a single membrane unit configuration with no recirculation/recycle stream. In general, this configuration is far from optimality since membranes typically exhibit low selectivities, and higher separation efficiency can be achieved only through co-regeneration and circulation across multiple modules.

5

To generate a membrane process configuration consisting of multiple membrane units, one can consider multi-stage and/or multi-step configurations [31]. In the multi-stage configuration, the permeate of the first unit becomes the feed of the second unit, while in the multi-step configuration, the feed to the second unit is the retentate gas stream from the first unit. Multi-stage and/or multi-step configurations are usually designed with two, three, or four membrane units [25]. Other innovative arrangements similar to that of distillation columns have also been developed and put into practice [32-34]. These designs not only use recycle streams while compressing a fraction of either permeate or retentate streams, but also determine optimal locations for the feed streams inputs in modules layouts. The large dimension of the design space described above suggests a need for an efficient software product able to find optimal design specifications and operating conditions of membrane gas separation processes. Efforts have been made to design optimal membrane gas separation processes by applying classical single- and multi-objective optimization techniques [29,30,35-38]. Unlike single-objective optimization problems, multi-objective ones in principle have multiple solutions, known as Pareto optimal solutions. Classical multi-objective optimizers, such as weighted sum and ε-constraint, transform a multi-objective problem into a simplified singleobjective optimization problem with specific constraint(s). In addition, classical multi-objective optimizers mostly employ deterministic transition rules and scalarizing functions iteratively to solve vector optimizations and to detect a set of Pareto optimal solutions. Rarely, all Pareto optimal solutions can be found in a single trial when traditional optimizers are used. Hence, a traditional optimizer should be utilized several times, and each time the predefined adjustable parameters of the algorithm should be meticulously regulated to obtain a portion of the optimal Pareto front [39]. Unlike classical deterministic multi-objective optimization algorithms, artificial intelligence (AI)-based optimizers are capable of finding all Pareto optimal solutions efficiently in an evolutionary single simulation trial [40-44]. AI-based optimization techniques include swarm intelligence, simulated annealing, particle swarm optimization, and genetic algorithms. Among different Genetic Algorithms, Non-dominated Sorting Genetic Algorithm (NSGA-II), which is suitable for multi-objective optimization, is used in this study. This article describes a novel software product, named Optimal Membrane-Process Design (OMPD), for the optimal design of membrane gas separation processes. The product first 6

generates several potential process design configurations as an initial population, and then searches the process design parameters and operating conditions spaces to arrive at optimal design specifications and operating conditions. The software product does not have any limitations in terms of the type or the number of optimizing variables, the type or the number of objective functions, or the number of membrane units arranged in multi-step and/or multi-stage configurations. As a case study, the software product is used to design an optimal membrane process for the separation of nitrogen gas (N2) from methane (CH4).

2. Mathematical Model Figure 1 schematically shows a membrane unit equipped with an A -selective membrane type and challenged with a feed composed of two components; i.e., Gas A and Gas B . The feed stream enters the membrane unit and gets split into two streams, the permeate stream (the one that passes through the membrane) and the retentate stream (the one that contains the rejected components). QF , QR , and QP are the total flow rates (cm3[STP] s-1) of the feed, retentate, and permeate streams, respectively, and x F , A , x R , A , and x P , A are the compositions (mole fractions) of A in the feed, retentate, and permeate streams, respectively.

Figure 1. Schematic representation of a typical membrane gas separation unit with a binary mixed gas (A and B) feed. The membrane is A selective.

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An overall mass balance and a mass balance on A for the membrane gas separation unit yield:

 QF  QR  QP   x F , A QF  x R , A QR  x P , A QP

(1)

Given the permeabilities of components A and B in a membrane (denoted by PA and

), which

are the properties of the membrane material in the membrane unit, the following relationships hold [45]: (

(2)

) (

)

(

)

(3)

where

where l is the membrane thickness (cm), Am is the membrane area (cm2),  A / B is the selectivity of A ,

and

are the logarithmic means of mole fractions of A and B in the feed

side: (

)

(

)

(4)

and p f and p p are the total pressures on the feed and permeate sides, respectively. The permeability is usually expressed in terms of Barrer (10 −10 cm3 [STP] cm cm-2 s-1 cm Hg-1). For the single membrane unit, the following inequalities hold:

 xR, A  xF , A   xP, A  xF , A   p f x A  p p x P , A

(5)

The first two inequalities in (5) state that the mole fraction of A in the permeate is greater than that of A in the feed stream, and that the mole fraction of A in the retentate stream is lower than 8

that in the feed stream. The last inequality in (5) reflects that the log-mean partial pressure of A in the feed side is always greater than the partial pressure of A in the permeate side. 3. Configuration Generation The optimal membrane gas separation process design requires an efficient configuration generator capable of generating all potential membrane gas separation process configurations. An optimizer then interacts with the configuration generator and finds an optimal configuration. The configuration generator should be able to enliven and dial in all potential configurations that satisfy membrane gas separation principles. The development of an appropriate and computationally optimal configuration generator is important in the proposed software product. The configuration generator should be equipped with several powerful and well-developed subroutines, based on the fundamental concepts of membrane gas separation processes. The optimal configuration generator stochastically allocates a membrane type to each membrane unit. It is aware of the main characteristics of different membranes of interest, such as permeabilities and selectivities. It should satisfy the following constraints: 1. There are three possibilities for both retentate and permeate streams leaving a membrane unit. In other words, the retentate or permeate streams leaving the i th membrane unit can be: i) forwarded to the next membrane unit (F), ii) sent back to the previous membrane unit (B), or iii) considered as a waste (W) or product (P) stream (see Figure 2). Hence, the directions of the retentate and permeate streams for each membrane unit are specified, while taking into account the items 2-4 below. 2. One outlet stream of each membrane unit must be an inlet stream of another membrane unit. In other words, either the retentate or the permeate stream leaving the i th membrane unit should be considered as the feed stream for the (i  1) th membrane unit. 3. The retentate and permeate streams of the i th membrane unit are not allowed to be simultaneously sent back to the (i  1) th membrane unit. So, if the generator, for instance, decides to forward the retentate stream of the i th membrane unit, the permeate stream of the same membrane unit can be either sent back or considered as waste or product gas. The first membrane unit may not have the backward option, and the last membrane unit may not have the forward option.

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Figure 2. Schematic representation of the possibilities that exist for the retentate and permeate streams of a membrane unit.

4. If a retentate or a permeate stream is selected as a W or P stream, the stream will be either a waste gas (W) or a product gas (P), depending on the membrane type and the final purpose of the separation process. For instance, if the membrane is an A -selective type and the target separation to be achieved is an A -rich gas stream, then the permeate stream will be the product gas (P), and the waste gas (W) will be the retentate stream. On the other hand, if the membrane is a B -selective type and again achieving an A -rich gas stream is the purpose of separation, then the retentate and permeate streams will be the product gas (P) and waste gas (W) streams, respectively. 5. The mole fraction of each component in a backward stream must be exactly equal to the mole fraction of the same component in the feed stream with which it combines. 6. As the driving force is the cross-membrane pressure difference, and as all permeate streams have lower pressure than the feed, recompression is required for the permeate streams to be forwarded to the next membrane unit or sent back to the previous membrane unit. 10

7. The mole fraction of a component in the retentate or permeate stream must satisfy the conditions in Eq. (5). For instance, if the i th membrane unit selectively permeates a component A ; i.e., if it is an A -selective membrane type, the mole fraction of the component A in the retentate and permeate streams must be, respectively, lower and higher than the

mole fraction of the same component in the feed stream. Also, the partial pressure of component A on the feed side should be higher than that on the permeate side. In fact, initial guesses for the mole fractions of a component in the retentate and permeate streams should satisfy all constraints in Eqs. (3) and (5).

Having set the directions of the retentate and the permeate streams for all membrane units, the configuration generator determines the mole fractions of all retentate and permeate streams leaving the membrane units. To achieve this, it assigns an initial guess for the mole fraction of a component, for instance, in the retentate stream, and it calculates the mole fraction of the same component in the opposite stream (the permeate stream) by applying Eq. (3). It is worth mentioning that there are two possibilities in order to guess the mole fractions of a specific component in a retentate stream and its corresponding permeate stream. If the configuration generator first assigns a guess for the mole fraction of a component in the retentate stream, the mole fraction of the same component on the permeate side is determined through solving for using Eq. (3), for a given x R , A (a quadratic equation). On the other hand, if it first sets the mole fraction of a component in the permeate stream, the mole fraction of the same component in the corresponding retentate stream, x R , A , is calculated using Eq. (3), for a given x P , A (a nonlinear equation). Having regulated the directions of all retentate and permeate streams along with the mole fractions of different components in the streams leaving the membrane units, the configuration generator calculates the flow rates and the membrane area for each created membrane unit. The flow rates of all retentate and permeate streams are determined simultaneously by using the two mass balance equations of each membrane unit; i.e., Eq. (1). The configuration generator then solves the system of linear equations numerically using a built-in equation solver (Gaussian Elimination method). The configuration generator then calculates the membrane area for each membrane unit using Eq. (2).

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A configuration generator was written in the Pascal programming language (Lazarus 1.8.4 IDE) and compiled into 64-bit executable using FPC 3.0.4. A subroutine based on the socalled Mother-of-all Pseudo Random number Generators algorithm [46] was employed to produce the required random numbers for the configuration generation. The random number generation subroutine satisfied the tests of uniformity and serial correlation with high-resolution. The cycle length of the random number generator was 3 × 10 47. The process configuration generations were performed on a desktop computer with Intel Core i7−3770K (3.50 GHz), 32 GB of memory (2133 MHz), under Windows 7 Ultimate 64-bit operating system. The runtime for generating a process configuration composed of three membrane units was about 0.05 s.

4. Membrane Process Design Optimization The main objective of the membrane gas separation system design optimization is to minimize the cost of separation while achieving the predefined separation goals. From an economic standpoint, this often becomes a trade-off between the total membrane area and the recompression power needed for one or more recycle streams. On the other hand, the product quality, including both product purity and recovery, are the main preset technical goals for each membrane separation process design. Hence, different optimization problems can be defined. Examples are: 

Maximize the product purity: (

∑ ∑

)

(6)

where { 

Maximize product recovery: (

∑

)

(7)

where {

12



Minimize compression power: (∑

)

(8)

)

(9)

where { 

Minimize the total membrane area: in(∑ni

where n denotes the number of membrane units, STRi and STPi are the type of the retentate and i permeate streams of the i th membrane unit, respectively, and E recomp is the required

recompression power for the permeate stream of the i th membrane unit. Note that Equations (6) and (7) are based on the assumption that the membrane is a B -selective type and achieving an A -rich gas stream in the retentate is the purpose of the membrane gas separation process.

The optimizer should be capable of handling both single- and multi-objective scenarios. In the case of a single-objective scenario, one of the objective function examples listed above can be selected, and in the case of multi-objective scenario two or more objective functions are considered simultaneously. OMPD solves multi-objective optimization problems using Non-dominated Sorting Genetic Algorithm II (NSGA-II) [47]. The implementation of NSGA-II is exactly the same as the single-objective versions of Genetic Algorithms except for the population sorting. The flowchart in Figure 3 shows how the optimization algorithm is implemented [48-52]. The input variables to be optimized are first encoded into a chromosome-like structure. Each chromosome as a potential solution for the optimization problem resembles a genotype as the codified version of its corresponding phenotype, e.g., a process configuration in the case of membrane gas separation process design. As described in the previous section, the main determining parameters for a membrane unit are the directions that the retentate and permeate streams choose after leaving the membrane unit, along with the composition of each stream. Hence, the codified membrane unit should convey adequate information on both the direction and composition of its main streams.

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Figure 3. Flowchart showing the different components of OMPD. 14

Figure 4 schematically represents the codification process of each membrane unit proposed in this work. Each membrane unit, say the ith membrane unit, can be codified into a substring composed of four closely interconnected genes, where the first two positions belong to the ith retentate stream and the next two genes contain the information of the corresponding permeate stream. More specifically, the first and the third genes of the aforementioned substring host the information on the directions of the ith retentate and permeate streams, respectively, and they can be F, B, or W/P. On the other hand, both the second and the fourth positions within the substring accept the composition of the permeating component in the ith retentate and permeate streams. Therefore, the defined substring as a genotype is capable of enlivening a membrane unit as its phenotype. In the case of complex process configurations composed of several interconnected consecutive membrane units, the substrings of all membrane units are connected to each other in the order given to form a chromosome codifying the whole process configuration (Figure 4).

Figure 4. Schematic representation of the codification process for each membrane unit. All generated membrane units are encoded into chromosome-like structures conveying the key information of corresponding membrane units.

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To shed more light on the proposed codification process described above, the two process configuration examples depicted in Figure 5 will be discussed in more detail. In the first process configuration, E1, the system is composed of two membrane units in series randomly placed by the configuration generator. For both membrane units, utilizing B -selective membrane type, the flow rates and membrane areas are calculated and reported. The first retentate stream is forwarded and is considered as the feed for the next membrane unit, while the retentate stream of the second membrane unit is considered as waste/product gas. In the second step, the permeate stream of the first membrane unit is considered as a waste/product gas and the permeate stream of the second membrane unit is sent back to the previous membrane unit. After recompression, the permeate stream of the second membrane unit is combined with the feed, which enters into the first membrane unit. In addition, the composition of the permeating component in all streams is proposed and regulated by the configuration generator. The information is compactly encoded into a chromosome that consists of eight genes applying the above-mentioned codification process (see Figure 4). In fact, this chromosome as a potential genotype can be properly decoded into the gas separation process design illustrated in Figure 5, E1, as its corresponding phenotype and vice versa.

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Figure 5. Examples of process configurations proposed by the configuration generator along with the corresponding genotypes. E1 represents a process configuration consisting of two membrane units all equipped with B -selective membrane type (permeances of gases A and B are assumed to be 20 and 50 GPU, respectively). E2 represents a process configuration consisting of three membrane units all equipped with A -selective membrane type (permeances of gases A and B are assumed to be 150 and 50 GPU, respectively).

The second process configuration, E2, is more complex. It is composed of three membrane units equipped with A -selective membrane type in series. E2 is encoded into a chromosome consisting of twelve genes (four genes for each membrane unit). In this typical example, the first and the second permeate streams are forwarded to be considered as the feed for 17

the second and the third membrane units, respectively, while the W/P choice is made for the last permeate stream. Moreover, the first and the second retentate streams are considered as waste/product gas streams, whereas the last retentate stream is sent back to be combined with the first permeate stream and feed the second membrane unit. It is clear that the genes transferring the streams’ composition can accept real values between zero and 100% (i.e., the mole percent of the permeating component in the gas streams), while those hosting information on the streams’ direction are allowed to explore a more limited search space; i.e., F, B, or W/P. Having defined the chromosome structure and its genes, an initial population of potential solutions (i.e., a number of chromosomes) is generated in a stochastic manner. The fitness of each chromosome; i.e., the cost of separation and/or the product quality which can be achieved by that chromosome/solution, is separately determined recalling the configuration generator. The competency of each chromosome is evaluated and quantified according to the predefined target(s); i.e., the performance indices (6)-(9). For instance, if the final aim of membrane gas separation process is to achieve appropriate levels of both recovery and purity of the target component in the product gas stream(s), then the optimizer merely focuses on precise calculation of the flow rate(s) and composition(s) of the target component in the product stream(s). The calculated fitness value(s) for each chromosome is separately compared with the preset corresponding target value(s) to determine the degree of goodness of each chromosome. Having evaluated the fitness of each chromosome, the optimizer evolves the potential solutions towards the global optimum, applying selection, mating, crossover, and mutation operators as intelligent genetic manipulators. It should be noted that the recombination and manipulation of the chromosomes as encoded membrane gas separation process designs will more probably result in unacceptable process configurations. In fact, the outputs of crossover and mutation steps (i.e., new process configurations) may not be capable of directly satisfying all the aforementioned constraints for a logical membrane gas separation process design. Therefore, another tool was designed and developed to refine and regulate the chromosomes leaving the crossover and mutation processes. This tool, a ‘configuration modifier’, obeys exactly the same protocols comprehensively described in the previous sections, and attempts to correct the new configurations with minimum manipulations. In fact, the configuration modifier intelligently supervises the genetic operators in 18

reconstructing the nominated members of the current population and evolving them into newcomers with mathematically logical process configurations. The heuristic optimization process is repeated by recalling both:  The configuration modifier to correct the members of the new population (i.e., new encoded process designs); and  The configuration generator to precisely calculate both the flow rates and membrane areas for membrane units encoded by the newcomers. It stops whenever one or more evolved solution(s) satisfy the predefined target(s). The main adjustable parameters applied in the developed multi-objective heuristic optimizer are listed in Table 1.

Table 1. Parameter values used for the heuristic optimization. Optimization Parameter Initial population size Sorting mechanism Selection mechanism Mating mechanism Crossover mechanism Crossover rate Mutation mechanism Mutation rate Number of iterations

Value 10,000 Non-dominated sorting Elitism Roulette wheel Double-point 50 % Uniform single-gene 15 % 1,000

Considering the algorithm in Figure 3, an intelligent optimizer based on NSGA-II was written in the Pascal programming language (Lazarus 1.8.4 IDE) and compiled into 64-bit executable using FPC 3.0.4. The developed optimizer was able to communicate/recall both the configuration generator and modifier in a swift manner. The Mersenne Twister pseudorandom number generator was used to produce the required random numbers [53]. The membrane gas separation process design optimization was performed on a desktop computer with Intel Core i73770K (3.50 GHz), 32 GB of memory (2133 MHz), under Windows 7 Ultimate 64-bit operating system. The runtime was approximately 4.6 and 6.2 hours for process optimizations that consist of two and three membrane units, respectively. For each case, 10,000,000 process configurations were generated, heuristically manipulated, and modified at 1,000 epochs; i.e., 10,000 chromosomes were managed for each iteration. 19

5. A Case Study 5.1. Problem Statement To evaluate the performance of the software product, it was used to develop an optimal membrane process design for the separation of nitrogen (N2) from methane (CH4). Process design optimizations were performed for the methane-selective and nitrogen-selective membranes, the permeances and selectivities of which are listed in Table 2. Furthermore, the feed gas stream was assumed to have a nitrogen mole fraction of 10 %, a methane mole fraction of 90 %, and a flow rate is 10 MMscfd at 500 psia. All membrane units were operated at 30 °C. Also, the total pressure of all permeate sides were set to be 100 psia.

Table 2. The permeances and selectivities of the nitrogen- and methane-selective membranes [31]. Membrane type Nitrogen-selective Methane-selective

Permeance (GPU) N2 CH4 50 20 50 150

Selectivity (-) CH4/N2 0.40 3.0

To demonstrate the power and versatility of the developed software product, OMPD, the product was tested on the following four cases: 

Case I: Find the optimal process configuration, which simultaneously maximizes product (CH4) purity and recovery, for two membrane units equipped with methaneselective membranes.



Case II: Find the optimal process configuration, which simultaneously maximizes the same objective functions as in Case I, for three membrane units equipped with methane-selective membranes.



Case III: Find the optimal process configuration, which, simultaneously maximizes the same objective functions as in Case I, for two membrane units equipped with nitrogenselective membranes.



Case IV: Find the optimal process configuration, which simultaneously maximizes the same objective functions as in Case I, for three membrane units equipped with nitrogen-selective membranes. 20

The purity is described in terms of the mole percent of methane in the product gas stream (Eq. (6)), and the methane loss (the fraction of methane leaving the separation unit through the waste gas stream) is utilized to determine the recovery (Eq. (7)).

5.2. Results and Discussion 5.2.1. Case I Figure 6 illustrates the optimization results for Case I. Contrary to single-objective optimizations, the multi-objective scenarios result in multiple solutions, known as Pareto optimal solutions. In this case, the intelligent optimizer has successfully suggested 7772 solutions satisfying the preset targets. In fact, 7772 members of the last population were placed in the first Pareto front as non-dominated solutions. As previously mentioned in the model development section, the initial population size was 10,000 members (potential solutions or process configurations) and the maximum number of iterations of the algorithm was set to be 1000. Hence, the developed optimizer has recalled the configuration generator and modifier 10,000,000 times in order to examine 10,000 potential configurations one-by-one at each epoch in an intelligent evolutionary manner. In the last epoch, 7772 out of 10,000 members of the last generation have successfully satisfied the primary criterion of the domination concept and placed in the first Pareto front. According to Figure 6, the obtained configurations were distributed between two extreme solutions specified by black dots. The former; i.e., either the Solution #1 or #2, is capable of achieving the maximum recovery (minimum methane loss: 0.00%), whilst the latter; i.e., either the Solution #3 or #4, represents the solution(s) resulted in the maximum attainable purity (minimum mole percent of nitrogen in product gas: 1.6539 mol%). It is clear that both extreme solutions are of significant importance from the theoretical point of view. From the practical standpoint, however, they are no useful solutions, because in the first case no separation takes place and in the second case no product is recovered.

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Figure 6. The optimal Pareto front reported by the developed auto-intelligent optimizer for nitrogen removal from natural gas by membrane separation process. The optimizer was only allowed to utilize two membrane units equipped with methane-selective membranes. The black dots are the extreme solutions while the red dots represent randomly selected solutions. The decoded versions (phenotypes) of two solutions (Solutions #6841 and #400) are shown as insets and at full scale in Figure 7.

To shed more light on the results obtained, ten randomly selected solutions are represented by the red dots. The details of the selected solutions are given in Table 3. In fact, the solutions located in the Pareto optimal front as phenotypes (Figure 6) are precisely interconnected to encoded process configurations as the corresponding genotypes (Table 3). The Excel worksheet entitled ‘Case I’ in the Supporting Information provides all details of 7772 solutions proposed by the intelligent optimizer. The solutions, located in the first Pareto front, were separated and stored because of being non-dominated. Apparently, all the solutions have been successful in simultaneous maximization of recovery and purity. The 1 st row of worksheet ‘Case I’ represents the solution number stored. The 2nd to 11th rows represent the encoded configurations proposed by the developed optimizer, i.e., the values of adjustable factors for each membrane unit described in previous sections including the direction and composition of each 22

retentate and permeate stream along with the membrane type and area for each membrane unit. Moreover, the flow rates of all retentate and permeate streams are reported in rows 12th to 15th. The 16th and 17th rows give the outputs of the process design for each configuration, i.e. methane loss (recovery) and mole percent of nitrogen in product gas (purity), respectively. Furthermore, the 18th and 19th rows host the values of the first and second criteria of the domination concept utilized by the intelligent optimizer. The former, entitled ‘Ranking’, is calculated based on the quality of the solutions, which classifies the solutions in different Pareto fronts. Being located in the first Pareto front, all solutions are acceptable and given a rank value of 1. The latter; i.e., C.D. row (Crowding Distance), represents the diversity of the solutions calculated and assigned based on the crowding distance equation for all solutions located in each Pareto front. The second criterion is capable of sorting the solutions isolated in different Pareto fronts separately. Mathematically, the solutions scoring higher crowding distance values are preferable. Obviously, Solutions #1 (or #2) and #3 (or #4) scoring the highest amounts of crowding distance, i.e. infinity (inf.), are ranked at the top. Interestingly, they are the solutions highlighted in black dots as extreme solutions in Figure 6, which are of scientific but not of practical interest.

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Table 3. The details of ten randomly selected solutions (see Figure 6) reported by the proposed intelligent optimizer for nitrogen removal from natural gas applying two membrane units equipped with methane-selective membranes. All reported compositions represent nitrogen mole percents in retentate and/or permeate streams. Unit 1st unit

187

217

352

400

6202

6325

6634

6841

7021

7384

R1 Direction W/P

W/P

W/P

W/P

W/P

W/P

W/P

F

F

F

XR1

26.81

34.72

11.63

22.42

13.84

13.76

17.21

17.58

27.25

20.85

P1 Direction

F

F

F

F

F

F

F

W/P

W/P

W/P

XP1

7.61

9.26

4.49

6.72

4.97

4.95

5.66

5.74

7.7

6.4

B

B

W/P

B

B

W/P

B

W/P

W/P

W/P

XR2

10

10

5.17

10

10

6.94

10

29.26

58.12

40.51

P2-Dir.

W/P

W/P

W/P

W/P

W/P

W/P

W/P

W/P

W/P

W/P

XP2

3.59

3.97

1.92

3.37

2.93

2.36

3.11

10.59

22.94

14.65

Area - Unit1

2412.58

8518.91

262.43

1596.93

575.11

495.75

959.62

757.26

1099.04

903.29

Area - Unit2

820.2

916.68

52.04

736.37

395.05

205.42

575.29

290.9

166.15

263.9

QR1 QP1 QR2

2.76

1.96

7.72

3.48

6.48

5.74

4.89

3.6

1.18

2.49

19.42

65.36

2.28

13.16

4.95

4.26

8.12

6.4

8.82

7.51

12.18

57.32

1.8

6.64

1.43

2.41

3.01

1.35

0.14

0.6

QP2

7.24

8.04

0.47

6.52

3.52

1.86

5.11

2.25

1.03

1.89

Obj1: Recovery

22.45

14.22

94.83

29.99

62.04

79.87

44.95

10.59

0.67

3.95

Obj2: purity

3.59

3.97

1.92

3.37

2.93

2.36

3.11

7

9.3

8.06

Ranking

1

1

1

1

1

1

1

1

1

1

C.D.

0.00112

0.00108

0.00098

0.00095

0.00005

0

0

0

0

0

Solution No.

2nd Unit R2-Dir.

To visualize the proposed optimal process configuration(s), two out of ten specified solutions were selected (Solutions #6841 and #400) and the process configuration for the selected solutions was depicted (see Figure 7 and also insets in Figure 6). The former; i.e., Solution #6841, resulted in purity of 93.00 mol% (7.00 mol% of nitrogen in product gas), whilst the latter; i.e., Solution #400, resulted in recovery of 70.00% (methane loss of 30.00%).

24

Figure 7. Two optimal configurations randomly selected from the obtained optimal Pareto front (A: Solution #6841 and B: Solution #400) for nitrogen removal from natural gas applying two membrane units equipped with methane-selective membranes. All reported compositions represent nitrogen mole percent.

As can be observed in Figure 7A (Solution #6841), the retentate stream of the first membrane unit is forwarded to be considered as the feed for the second membrane unit. In addition, both permeate streams along with the retentate stream of the second membrane unit are neither forwarded nor sent back. In fact, both permeate streams have been combined to form the product gas and the retentate of the second membrane unit is considered as the waste gas. The membrane area for the first and second membrane units is predicted to be 757 and 291 m2, respectively. Also, the maximum recovery and purity for this case are 89.41% (i.e. methane loss of 10.59%) and 93 mol% (i.e. 7.00 mol% of nitrogen in product gas), respectively. 25

In contrast, in the process configuration depicted based on the codified Solution #400, the permeate stream of the first unit is forwarded and the retentate of the same membrane unit is considered as the waste gas stream (see Figure 7B). Moreover, the permeate stream of the second unit is considered as the product gas, while its retentate stream is sent backward to be combined with the feed stream which enters the previous membrane unit. The reported membrane areas for the first and second membrane units are 1597 and 736 m2, respectively. This configuration results in maximum recovery and purity of 70.00% (i.e. methane loss of 30%) and 96.63 mol% (i.e. 3.37 mol% of nitrogen in product gas), respectively. Figure 8 presents the optimal Pareto fronts found by OMPD, where the objective functions are the purity and recovery.

5.2.2. Case II In Case II, in contrast to Case I, the optimal Pareto front has considerably shifted downwards and leftwards, which implies that for the same membrane type, higher values of purity and recovery can be reached. In other words, increasing the number of membrane units, the optimizer can indeed find and report better solutions that are closer to the ideal goal of the membrane gas separation process.

26

Figure 8. The optimal Pareto fronts proposed by the developed auto-intelligent optimizer for nitrogen removal from natural gas applying different numbers of membrane units and different membrane types. Case II: Three membrane units equipped with methane-selective membranes;

27

Cases III and IV: Two and three membrane units equipped with nitrogen-selective membranes, respectively.

To compare the optimization results, Solution #3991 is selected from the optimal Pareto front of Case II (see Figure 9A). Although Solution #400 (Case I) and Solution #3991 (Case II) have resulted in process configurations leading to recovery of 70%, the reported purity for the latter configuration is much higher than the former case. In other words, the maximum achieved purities for the final product gas by Solution #3991 (Case II) and Solution #400 (Case I) are 97.97 mol% and 96.63 mol%, respectively.

5.2.3. Cases III and IV The optimization results for Cases III and IV are also shown in Figure 8. As in Cases I and II, the optimizer was capable of effectively exploring the preset search space and discovering all the potential solutions for the scenarios with nitrogen-selective membranes. As can be seen, the optimal Pareto front has moved downwards and leftwards in case of the last scenario (i.e., Case IV) which clearly denotes the possibility of enhancing the quality of solutions by increasing the number of membrane units. As discussed above, from the practical point of view these configurations will need to be optimized also in economic terms, because of additional costs related to recompression and membrane investment. To quantitatively compare the obtained results, the configurations leading to purity of 95 mol% (i.e. 5.00 mol% of nitrogen in product gas) were selected and specified in Figure 8 for Cases III and IV. Even though both selected solutions, i.e. Solution #190 (Case III) and #998 (Case IV), have resulted in exactly the same purity value, the latter is more preferable as it has been able to return the mentioned purity at lower amount of methane loss. The reported methane loss for Solution #190 and Solution #998 is 28.64% and 17.02%, respectively. The decoded versions of the specified solutions, i.e. the corresponding process configurations, are depicted and presented in Figure 9. This figure visually reports the optimal sample solutions including the direction, composition, and the flow rate of each retentate and permeate stream, along with the membrane area for each membrane unit.

28

Figure 9. Visualization of three optimal configurations randomly selected from the obtained optimal Pareto fronts for nitrogen removal from natural gas. A, B, and C are the decoded versions of Solutions #3991, #190, and #998 selected as samples from Cases II, III, and IV, respectively (see Figure 8). All reported compositions represent nitrogen mole percent.

To investigate the influence of the number of membrane units on the performance of membrane gas separation process, the optimization of nitrogen removal from natural gas considering one to five membrane units equipped with methane-selective membrane was performed. The optimal Pareto fronts obtained for the mentioned optimization Cases are illustrated in Figure 10A. According to the results obtained, both recovery and purity are considerably improved by increasing the number of consecutive membrane units from 1 to 5. The variation of the maximum attainable purity at a preset methane loss of 30% versus the number of consecutive member units equipped with methane-selective membrane is depicted in 29

Figure 10B. As can be observed, the preset number of membrane units has considerable impacts on the process performance. In fact, the developed intelligent optimizer will have more flexibility and access to a larger search space and also a greater variety of process configurations by increasing the number of membrane units, which makes it possible to expect more qualified solutions.

Figure 10. A: The comparison of optimal Pareto fronts proposed by OMPD for nitrogen removal from natural gas applying different numbers of membrane units equipped with methane-selective membranes; B: The variation of the maximum achievable purity at a preset methane loss of 30% calculated for configurations with 1-5 membrane units equipped with methane-selective membranes. 30

Although simultaneous maximization of purity and recovery is assumed as the optimization objectives in the current study, the established auto-intelligent optimizer is capable of considering more targets/constraints of different types as the optimization goals especially the economic aspects, including the costs of recompression and the required membrane area. Taking into account these factors, the algorithm is capable of minimizing environmental footprint, energy and raw material consumption and simultaneously maximizing/improving the productivity and product quality. Obviously, considering the economic aspects as extra optimization targets forces the optimizer to simultaneously take into account both the product quality and the economic aspects; i.e., the techno-economic characteristics. It definitely confines the available search domain of the automatic configuration generator. Future research in this area will deal with the application of the proposed intelligent optimizer to practically manage a real membrane gas separation process design, taking both technical and economic aspects into account as optimization targets, i.e. minimizing the cost of separation while achieving the separation goals.

31

6. Conclusions This article presents a software product, called OMPD, for the optimal design of membrane gas separation processes. The software product first generates a pool of practicallymeaningful process configurations and then solves a multi-objective optimal problem to arrive at optimal design specifications, including a membrane process configuration and the design specifications of each membrane unit. The application and performance of the software product was shown by applying it to case studies. The results obtained clearly indicate that the developed software is capable of effectively cracking the complexities of membrane gas separation process design, precisely mapping the preset search space, and also computing/reporting the optimal solution(s) for predefined optimization cases with any number of objectives and constraints. It was also shown that there exists no limitation on the type and number of objectives and constraints which should be mutually satisfied. This allows for including, for instance, capital costs for the membrane units and operational costs for (re)compression, thus enabling an optimization of the process performance also in economic terms, which will be the subject of future studies. Moreover, this computational framework can be extended to handle optimal process design projects in case of other membrane-based separation processes including reverse osmosis and pervaporation.

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