Integrated ionic liquid and rate-based absorption process design for gas separation: Global optimization using hybrid models

A new method for integrated ionic liquid (IL) and absorption process design is proposed where a rigorous rate-based process model is used to incorporate absorption thermodynamics and kinetics. Different types of models including group contribution models and thermodynamic models are employed to predict the process-relevant physical, kinetic, and thermodynamic (gas solubility) properties of ILs. Combining the property models with process models, the integrated IL and process design problem is formulated as an MINLP optimization problem. Unfortunately, due to the model complexity, the problem is prone to convergence failure. To lower the computational difficulty, tractable surrogate models are used to replace the complex thermodynamic models while maintaining the prediction accuracy. This provides an opportunity to find the global optimum for the integrated design problem. A pre-combustion carbon capture case study is provided to demonstrate the applicability of the method. The obtained global optimum saves 14.8% cost compared with the Selexol process.


| INTRODUCTION
Gas separation is an inevitable process in many industries. Several technologies such as absorption, adsorption, and membrane separation are available for gas separation. Among these existing technologies, solventbased absorption is a mature and easy-to-operate one. Its industrial applications include carbon capture, natural gas sweetening, dehydration, and so on. 1 It is known that the core for employing absorption processes lies in the use of high-performing solvents, the search for new advanced solvents is essential to enhance the performance of absorption processes. 2 Ionic liquids (ILs) are regarded as potential alternatives to organic solvents for gas separation due to their superior gas solubility, low volatility as well as chemical and thermal stability. 3,4 However, the number of ILs is almost infinite when considering the large number of anions, cations, and substituent groups. Clearly, using the traditional trial-and-error method is inefficient for selecting optimal ILs. 5 So far, many efforts have been made on computational IL screening. For the purpose of IL absorbent screening, various predictive models such as ab initio calculations, equations of state (EOS), and activity coefficient models (e.g., COSMO-RS) have been utilized to estimate the thermodynamic properties of ILs (mainly gas solubility). [5][6][7][8][9] Although these approaches can efficiently screen ILs with desired properties, they are limited by the number of available IL candidates in the databases. To further expand the search space and find new promising ILs, a systematic approach for IL design is highly necessary.
Computer-aided design methods have been developed and successfully applied for both organic solvent design [10][11][12][13] and IL solvent design. [14][15][16][17] For computer-aided ionic liquid design (CAILD), an IL molecule is decomposed into different structural groups and its properties are calculated using quantitative structure-property relationship models (e.g., group contribution models). With these, the IL molecular structure is optimized to achieve desired properties through the formulation and solution of a mixed-integer optimization problem. 17 For predicting IL physical properties (e.g., melting point and boiling point), simple linear group contribution (GC) models are suitable and can be easily applied. [18][19][20] Besides, for predicting thermodynamic properties (e.g., activity coefficient and solubility) of IL systems, the nonlinear GC-based UNIFAC models are often used. 21,22 In order to make accurate predictions, the involved UNIFAC model parameters for IL systems have been extensively fitted from experimental data. 23,24 No matter which type of solvent (organic or IL) is used, it ultimately serves a specific process and the process performance depends on the solvent selection and process operations. 25 Given the strong interdependency between these two issues, integrated solvent and process design is always preferred for enhancing the overall process efficiency. [26][27][28] For absorption processes, the left of Figure 1 shows that such an integrated design problem needs to consider the variations at the interlinked molecular, phase, and process levels. In the literature, a few efforts have been made on the integrated solvent and absorption process design. [28][29][30][31] However, the existing studies only considered the absorption thermodynamics in the phase level so that relatively simple absorption process models can be applied. In fact, absorption kinetics can dominate over the thermodynamics. For instance, the key role of absorption kinetics over thermodynamics in IL selection for CO 2 capture has been revealed. [32][33][34] Ignoring kinetics can lead to sub-optimal or even poor solutions for absorption process development. To identify truly optimal solvent and absorption process, a rigorous rate-based absorption process model that incorporates absorption thermodynamics and kinetics should be used. In this work, a new computer-aided IL and process design (CAILPD) method is developed based on the rigorous rate-based absorption process model where the effects of ILs on absorption thermodynamics and kinetics are simultaneously considered. This approach can theoretically provide a better and more comprehensive design on gas absorption processes.
Note that for a given solvent, the optimization of rigorous ratebased absorption processes has been studied. 35 However, no one has used the rate-based process model for integrated solvent and absorption process design. The main reason is that such design problems usually lead to challenging mixed-integer nonlinear programming (MINLP) problems, especially when complex thermodynamic models are used. 25,36 Thus, an efficient solution strategy is highly desired to solve the complex integrated design problems. Recently, surrogate models that are constructed from reliable data to substitute complicated physical models have been widely adopted in process optimization. 37  model is used to capture both effects. In addition, the equipment sizes and process costs can be calculated. Such a multilevel design problem is formulated as an MINLP problem. The objective is to minimize the process cost while fulfilling multiple constraints. Traditionally, the thermodynamic property (i.e., equilibrium gas solubility) is calculated by GC-based predictive models such as UNIFAC-IL and Peng-Robinson (PR). 9,39 Our idea is to construct a mathematically simple but reliable surrogate model for predicting the gas solubility based on existing experimental data. Along with other physical models, a hybrid model-based design formulation is formed. This enables us to pursue global optimization of the resulting MINLP problem, which can provide a better and more comprehensive absorption process design. The detailed modeling framework is described below. Note that both the two treatments of thermodynamics are included for better comparison and demonstration of the advances of our proposed method. On the other hand, gas fugacity and activity coefficients in the vapor and liquid phases can be predicted by traditional thermodynamic models and the equilibrium gas solubility is then calculated through the solution of the vapor-liquid equilibrium (VLE) equation. Alternatively, the thermodynamic equilibrium solubility can be directly predicted by surrogate models. Substituting the mass transfer coefficient and thermodynamic driving force into the rate-based absorption model, the corresponding process performance can thus be evaluated for a given IL and process condition. After completing the forward step, the best IL structure and optimal process conditions can be reversely identified by solving an MINLP problem where the process performance is optimized considering all the models or equations as well as IL structural constraints. The proposed method and framework are illustrated using a pre-combustion carbon capture example.

| Representation of ionic liquids
The first step for designing ILs is to decompose them into different building blocks. ILs can be represented in different ways. In this work, an IL is decomposed into an anion, a cation core, and substituents linked to the cation core. This representation can provide large design space and flexibility. 40 Taking 1-propyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide [C 3 mim][Tf 2 N] as an example (see Figure 3), this IL is constituted by an anion "Tf 2 N," a cation core "MIm," and 3 substituent groups including 1 "aN_CH 2 ," 1 "CH 2 ," and 1 "CH 3 ." In the present work, 23 anions, 12 cation cores, and 17 cation substituent groups are considered for the IL design, as listed in  Table S1. In addition, their contributions to properties (e.g., molecular weight, melting point, etc.) and the upper bounds in the CAILPD program are listed in Table S1 as well. These contributions are considered as parameters in the following sections. Note that the nomenclature is presented at the end of the Appendix S1.

| Ionic liquids structural constraints
To generate a feasible IL, certain structural constraints must be satisfied. A feasible IL consists of only one anion and one cation (Equations 1 and 2). Equation (3) shows that the number of each constituent group should be non-negative and less than its upper bound n upp i . The total number of constituent groups including cation, anion, and substituent groups is between 2 and 8.
Since cation substituents in a normal IL are acyclic, the simple octet rule in Equation (5) Based on the chemical intuitions, two functional groups are generally not linked to one single carbon atom. Thus, Equation (7) means that the total number of functional substituent groups is no more than that of the alkyl substituent groups. G fg and G ag are the sets of functional and alkyl substituent groups, respectively. The detailed classification of cation substituent groups are given in Table 2.
Considering the structural characteristics of normal ILs, the structural complexity of ILs is refined using Equations (8)- (11). Equations (8) and (9) ensure that the total number of ether and hydroxyl groups (G eh ) is less than 2 and less than the total number of non-CH 3 alkyl groups (G nCH3 ), respectively. In addition, the total number of fluorized alkyl groups (G fag ) is less than 2 (Equation 10). Ether and hydroxyl groups and fluorized alkyl groups cannot exist simultaneously (Equation 11). Group not directly linked to cation (G NDC ) Equations (12)- (16) ensure that the cation is properly linked to the corresponding substituent groups. G aN and G cycN are the sets of alkyl groups linked to the aromatic and cyclic nitrogen, respectively. G N and G P are the alkyl groups linked to acyclic nitrogen and acyclic phosphorous, respectively. Based on the cation structures (Table S1), any alkyl group linked to Im13, MIm, MMIM, Py, or MPy is directly linked to an aromatic nitrogen. With this, the set G aN is defined. In addition, the first part of Equation (12) ensures that if any of these cations is selected, some alkyl groups in G aN should be selected and the number of the selected G aN groups must match the cation valency. Meanwhile, Equation (13) ensures that the alkyl groups in the other sets (i.e., G cycN , G N , and G P ) are not selected. On a similar manner, the other parts of Equation (12) and Equations (14)- (16) are proposed.

| Ionic liquid physical and kinetic property prediction
Physical and kinetic properties of ILs are required for the rate-based process modeling.  Table S1).
IL molar volume (MV) depends on the temperature (T) and pressure (P) in Equation (24). The molar volume MV 0 ð Þ at T 0 ¼ 298:15 K and P 0 ¼ 1:0 bar is calculated using the GC approach in Equation (25). PMV i is the ith group's contribution to molar volume (see Table S1). 44 MV An artificial neural network (ANN) based GC model has been developed to predict IL viscosity. 45 The model consists of three layers (i.e., input layer, hidden layer, and output layer). Based on the ANN-GC model, the viscosity (μ) of ILs can be expressed as The detailed model equations are given in Appendix S2.

| Correlation models to physical and kinetic properties
The density (ρ) and heat capacity (C p ) of ILs are calculated using the following correlation functions. 43,46 ρ ¼ 1000 Â MW MV ð27Þ Moreover, when ILs are used for CO 2 absorption in a packed column, it can be assumed that mass transfer resistance occurs in the vicinity of gas-liquid interface. With this, the mass transfer coefficient (k) can be estimated using the Onda' correlation. 47,48 k ρ 1000 Â μ Á g 1 where, g is the gravity constant. a p and d p are the specific packing surface area and diameter, respectively. Sc and D mass are the Schmidt number and the mass diffusivity coefficient, respectively. The detailed models to calculate these two variables are given in Appendix S2.

| Rigorous thermodynamic model
Based on thermodynamics, the vapor-liquid equilibrium (VLE) of CO 2 determines its solubility in ILs. At equilibrium, the molar fraction of CO 2 in ILs (x eq CO2 ) can be expressed as Psat CO2 ¼ 10 Â e 12:3312À 4759:46 where, y CO2 is the molar fraction of CO 2 in the gas phase. Psat CO2 is the saturated pressure of CO 2 (in bar), estimated by the extrapo- Specifically, the CO 2 equilibrium molar fraction x eq CO2 is calculated by the following equations.
x eq CO2 ¼ where, the superscripts 1 and 2 denote the hidden layer and output layer, respectively. Subscripts t and i are the neuron and IL group indices, respectively. NW, TW, and PW are weighting factors to IL group numbers, temperature, and pressure, respectively. bW represents the bias. All these parameters can be found in Ref. 49.

| Absorber
The rate-based model is used in the absorber. The column is presumably isothermal and T AB should be larger than the melting temperature to ensure that the ILs are in liquid. In addition, the column is divided into NT ¼ 20 sections. In each section, the amount of CO 2 absorbed from gas to ILs (pern) is equal and decided from the mass balance.
where, y feed CO2 and θ are the CO 2 molar fraction in the feed gas and the percentage of CO 2 to be absorbed, respectively. If the solubility of other gases in ILs is negligible, the liquid and vapor molar fractions of CO 2 in the nth section (n ¼ 1, …, NT from the bottom up) can be calculated by where, x FT out,CO2 is the molar fraction of CO 2 in the ILs regenerated from the flash tank (FT). In addition, the following summation equations must be satisfied for each section.
where, the subscript c denotes the gas and liquid components. Moreover, the height of the nth section (h n ) is calculated in Equation (43) and the height of the absorber H AB is obtained by Equation (44). 48 pern Á MV IL where, MV IL denotes the molar volume of IL in the absorber (see Equations 24 and 25). Void p is the void fraction of packing. x eq n,CO2 represents the equilibrium CO 2 solubility in the nth column section. If rigorous thermodynamic models are used, it is calculated using Equations (30)- (33) with known y n,CO2 . Alternatively, this equilibrium solubility can be directly calculated by Equations (34)- (36) if the ANN-based surrogate model is applied. In addition, Equation (45) ensures a positive driving force for CO 2 absorption.

| Flash tank
The heated CO 2 -loaded ILs are fed into the flash tank for solvent regeneration. The temperature in the flash tank cannot exceed the boiling point of IL to prevent its vaporization. The CO 2 molar fraction in the lean ILs x FT out,CO2 is estimated by the short-cut model in Equation (47). 50 x FT out,CO2 ¼ 1 À In order to facilitate the computation in the absorber, x FT out,CO2 is fixed to 0.02. In this case, the operating pressure of the flash tank P FT depends on the temperature T FT . Moreover, the flash tank is presumably operated on a half-full basis and the total volume of ILs in the flash tank is equal to the volume of 5 min IL flows. 51 Thus, the volume of the flash tank (V FT ) is expressed as where, MV FT is the IL molar volume at T FT and P FT .

| Process economics
The performance of the IL-based carbon capture process is evaluated using the total annualized cost (TAC).
where, CRF is the capital recovery factor. C cap is the summation of the capital costs of all the equipment. C ope is the annual operating cost that accounts for the consumption of utilities (i.e., steam, electricity, and cooling water) and other operating costs including labor, maintenance, and IL losses. The detailed calculation of the capital and operating costs is presented in Appendix S2.

| RESULTS AND DISCUSSION
As a major CO 2 emission resource, the pre-combustion flue gas is usually produced in an integrated gasification combined cycle (IGCC) based power plant. The feedstock such as natural gas is reacted with oxygen under high temperature and pressure to produce synthesis gas consisting of CO, H 2 , and CO 2 . Through a water-gas shift reaction, the CO is converted into CO 2 and the pre-combustion flue gas comprising mainly H 2 and CO 2 is formed. The CO 2 must be removed from the H 2 before power generation. In this work, the CAILPD framework is applied for pre-combustion carbon capture. The flue gas is assumed to be at 313.15 K and 20 bar with a molar flowrate of 10 kmol/s. The compositions of CO 2 and H 2 are set to 0.4 and 0.6, respectively. With an assumption that H 2 is not soluble in the ILs, the goal is to design a cost-effective IL-based absorption process for capturing no less than 90% CO 2 from the flue gas.

| First trail using rigorous thermodynamic model
In the first trial, the CAILPD problem is formulated using the classic thermodynamic models (i.e., UNIFAC and PR models) to predict the CO 2 equilibrium solubility. An MINLP problem is formed and summarized below. The objective is to minimize the process TAC while fulfilling the carbon capture requirement stated above. The design variables (degrees of freedom) consist of the discrete variable n i and the continuous variables P AB and T FT . Equality constraints include IL structural constraints, property and process models as well as process economics. The specific feed gas conditions as well as process and costing parameters are listed in Table 3   The DEPG-based Selexol process is widely used for pre-combustion carbon capture. As a benchmark, the economic performance of the Selexol process is evaluated. The flowsheet of the Selexol process is shown in Figure S1. It is simulated in Aspen Plus V8.8 according to literature reports 53,54 and the detailed process specifications are listed in Table S2. For consistency, the process economics is assessed with the same cost models used in this work. Figure 6 compares the cost breakdown of the two processes. As indicated, although a much larger amount of steam and cooling water are consumed in the IL-based process, the electricity cost of the Selexol process is much higher than that of the IL-based process. This is because a higher pressure (30 bar) is needed in the absorber of the Selexol process to meet the CO 2 capture requirement, which leads to a larger electricity consumption for gas compressing. In contrast, our designed IL has a higher CO 2 solubility than DEPG, making the absorption operable at a lower pressure (21.5 bar). In total, the IL-based process can achieve 14.8% total cost reduction compared with the benchmark Selexol process for precombustion carbon capture, which demonstrates the significance and large benefit of integrated IL and process design. Selexol process, the optimal IL-based process can achieve a better economic performance for the investigated carbon capture task.
To the best of our knowledge, this work is the first attempt in the global optimization of an integrated IL and process design problem where rigorous rate-based process model is employed. This study can be extended in several ways. For instance, the method can be applied to handle different CO 2 emission sources. In addition, the IL-based absorption process can be compared with other processes (e.g., adsorption and membrane) to identify the most efficient technology for gas separation problems. Despite the large progress, limitations should not be neglected. First, a simple inequality constraint is added to prevent unreasonable extrapolation of the ANN-based solubility model. In the future work, more advanced methods (e.g., convex hull 55 and topological data analysis 56 ) can be used to confine the results into the validity region. Second, the current work cannot distinguish between structural isomers. To do so, higher-order GC models incorporating the group connectivity information are required, which may significantly increase the computational demand. A more realistic strategy is to investigate the practical performance of isomers in a post-design step, for example, by experimental studies. Thirdly, the uncertainty associated with property prediction can directly affect the quality of the optimal design. This issue is worth to be explicitly considered in the computer-aided design stage for better reliability. Efforts in these directions are underway. resources.

DATA AVAILABILITY STATEMENT
The data that support the findings of this study are available from the corresponding author upon reasonable request.

SUPPORTING INFORMATION
Additional supporting information may be found online in the Supporting Information section at the end of this article.
How to cite this article: Zhang X, Ding X, Song Z, Zhou T, Sundmacher K. Integrated ionic liquid and rate-based absorption process design for gas separation: Global optimization using hybrid models. AIChE J. 2021;e17340. https://doi.org/10.1002/aic.17340