RSM Modeling and Optimization of CO₂ Separation from High CO₂ Feed Concentration over Functionalized Membrane

Abstract

The challenges in developing high CO₂ gas fields are governed by several factors such as reservoir condition, feed gas composition, operational pressure and temperature, and selection of appropriate technologies for bulk CO₂ separation. Thus, in this work, we report an optimization study on the separation of CO₂ from CH₄ at high CO₂ feed concentration over a functionalized mixed matrix membrane using a statistical tool, response surface methodology (RSM) statistical coupled with central composite design (CCD). The functionalized mixed matrix membrane containing NH₂-MIL-125 (Ti) and 6FDA-durene, fabricated in our previous study, was used to perform the separation performance under three operational parameters, namely, feed pressure, temperature, and CO₂ feed concentration, ranging from 3.5–12.5 bar, 30.0–50.0 °C and 15–70 mol%, respectively. The CO₂ permeability and CO₂/CH₄ separation factor obtained from the experimental work were varied from 293.2–794.4 Barrer and 5.3–13.0, respectively. In addition, the optimum operational parameters were found at a feed pressure of 12.5 bar, a temperature of 34.7 °C, and a CO₂ feed concentration of 70 mol%, which yielded the highest CO₂ permeability of 609.3 Barrer and a CO₂/CH₄ separation factor of 11.6. The average errors between the experimental data and data predicted by the model for CO₂ permeability and CO₂/CH₄ separation factor were 5.1% and 3.3%, respectively, confirming the validity of the proposed model. Overall, the findings of this work provide insights into the future utilization of NH₂-MIL-125 (Ti)/6FDA-based mixed matrix membranes in real natural gas purification applications.

1. Introduction

The East Natuna Block, located in the Greater Sarawak Basin near Natuna Island, Indonesia, is the largest natural gas field in Southeast Asia. It contains 46 trillion cubic feet (Tcf) of natural gas, and is identified as a high carbon dioxide (CO₂) content gas field, with CO₂ concentrations up to 70 mol%. In addition, about 13 trillion cubic feet (Tcf) of high CO₂ gas fields in Malaysia remain unexploited, with CO₂ concentrations of more than 40 mol%. According to the requirements of the gas processing plant, the CO₂ content should be reduced to 8 mol% in order to comply with the criteria of the downstream sales gas process [1]. High CO₂ contents can reduce the heating value of natural gas [2] and cause corrosion issues in pipelines in the presence of water [3]. Thus, it is crucial to investigate technologies for separating CO₂ from methane at high CO₂ feed concentrations. Among conventional technologies such as absorption [4], adsorption [5], cryogenic distillation, and membrane separation [6], membrane separation has gained the most attention due to its small footprint, simple operation and energy efficiency [7].

The evolution of membrane materials in separation technology started with polymeric, followed by inorganic, mixed matrix and metal organic frameworks (MOFs), etc. Mixed matrix membranes comprised of polymeric material and inorganic filler were developed to address the trade-off in performance experienced by polymeric membranes [8] and reproducibility concerns regarding the fabrication of inorganic membranes [9]. Lately, incorporating MOF fillers in polymeric membranes for the formation of mixed matrix membranes has been the focus of many researchers due to the fact that the high porosity of MOF fillers can improve gas separation performance [10].

On top of that, numerous efforts have been made by researchers to further enhance the separation performance of mixed matrix membranes by functionalizing metal-organic frameworks (MOFs) fillers with -NH₂ functional groups. The presence of -NH₂ in the filler induces strong affinity toward CO₂ via hydrogen bonding, which enhances the interaction between CO₂ and the filler framework [11,12], and subsequently, improves the performance of the membrane in terms of CO₂ separation. Owing to its tunable pore size and affinity to CO₂ gas [13,14,15], NH₂-MIL-125 (Ti) has recently been studied as a filler for the formation of membranes for gas separation [3,16,17]. Previous studies reported that the separation performance of NH₂-MIL-125 (Ti)/polysulfone mixed matrix membranes was superior to that of neat polysulfone membranes [17]. This finding was attributed to the existence of porous fillers, which created an extra pathway for the penetration of CO₂ gas and promoted the diffusion of gas molecules through the membrane. Furthermore, in another work, a NH₂-MIL-125 (Ti)/Matrimid mixed matrix membrane was fabricated for CO₂/CH₄ separation and a CO₂/CH₄ gas pair selectivity of 50 was obtained when 15 wt% filler was loaded into the membrane [16]. This was mainly due to the small aperture size (6 Å) of NH₂-MIL-125 (Ti), which allowed CO₂ molecules to interact with the -OH groups in Ti clusters, as well as interacting with -NH₂ groups on the linker. Also, our previous study discovered that the inclusion of NH₂-MIL-125 (Ti) filler in 6FDA-durene polymer matrices significantly improved the CO₂ permeability and CO₂/CH₄ gas pair selectivity, i.e., by 119% and 331%, respectively, compared to neat 6FDA-durene membranes [3]. The improvement in performance was mainly attributed to the high porosity of NH₂-MIL-125 (Ti) fillers and the attraction of amine functional groups to CO₂ molecules. Although several studies have been published on the utilization of functionalized MIL-125 (Ti) as a filler for the preparation of mixed matrix membranes in CO₂/CH₄ separation, these studies mostly focused on single gas permeation, binary gas separation (50:50) and the effect of feed pressure on the membrane performance [16,17]. Hence, further research evaluating the separation performance of the membrane at high CO₂ feed concentrations and various temperatures and CO₂ feed pressures in binary gas mixtures is still crucial.

According to the findings in the literature, operating temperature influences the CO₂ permeability of the membrane by affecting the mobility of the polymer chain and free volume [18] in the polymer, thus resulted in a change of CO₂ diffusivity [19]. In contrast, feed pressure may affect the gas solubility in the membrane, which affects the CO₂ permeability [11]. Furthermore, the CO₂ concentration in the feed can cause significant changes in CO₂ permeability through the membrane, which is mainly due to the higher sorption capability of CO₂ compared to CH₄. Hence, study of the correlation between the operational parameters and membrane separation performance is essential. In fact, several approaches, including response surface methodology (RSM), full factorial design and Taguchi methods, are available to investigate these questions.

For an experimental assessment and the optimization of the separation parameters, including significant responses and independent variables, comprehensive analytical tools and efficient optimization tools are practical. The multivariant method enabled the identification of the interaction between the variable parameters and responses. Furthermore, for our optimization study, the response surface methodology (RSM) was identified as the optimal tool. This technique combines the mathematical and statistical analyses by reducing the number of experiments. As a result, various researchers have reported the application of the RSM method in their studies.

Jusoh et al. [20] investigated the effect of pressure, temperature and CO₂ feed concentration on CO₂/CH₄ binary gas separation over a ZIF-8/6FDA-based mixed matrix membrane. They reported that the optimum parameter for CO₂/CH₄ separation performance was obtained at a feed pressure of 4.76 bar, a temperature of 30 °C, and a CO₂ feed concentration of 90 mol%. The latest study reported by Afarani et al. [18] utilized a polyurethane–zeolite 3A mixed matrix membrane to assess the impact of three independent variables, namely, zeolite content (0–24 wt%), operating temperature (25–45 °C), and operating pressure (0.2–0.1 MPa), on the single gas permeation performance. The optimum gas permeation performance of polyurethane–3A zeolite membrane was identified at a zeolite loading of 18 wt%, a temperature of 30 °C, and a pressure of 0.8 MPa. It can be observed that the RSM method has great potential for analyses of the relationship between the operational parameters and responses. Thus, the process of optimization of the separation performance over NH₂-MIL-125 (Ti)/6FDA-durene mixed matrix membranes by varying the operational parameters including feed pressure, temperature and CO₂ feed concentration using RSM method is presented in this study.

The 7.0 wt% NH₂-MIL-125 (Ti)/6FDA-durene mixed matrix membrane reported in our previous work [3] was used in this study, since the membrane demonstrated the best performance in CO₂ and CH₄ single gas permeation testing. First, the interaction of operational parameters, including feed pressure (A), temperature (B), and CO₂ feed concentration (C), toward the CO₂ and CH₄ permeability and CO₂/CH₄ separation factor were analyzed in detail using the RSM statistical method paired with the CCD optimization tool. Then, the significance of models was further evaluated by using analysis of variance (ANOVA). Next, 3D surface plots were used to visualize the linear, quadratic, and interaction between the operating parameters and significant responses. Subsequently, the optimum condition for the separation performance was generated and validated by experimental work.

2. Materials and Methods

2.1. Materials

Synthesis of NH₂-MIL-125 (Ti) fillers required trimethylacetic acid (99%), N,N-dimethylformamide (DMF), 2-aminoterephthalic acid (98%), terephthalic acid (98%), acetonitrile (99.8%) and tetraisopropyl ortthotitanate (97%). These chemicals were purchased from Sigma Aldrich (Missouri, MO, USA) and were used without further purification.

Synthesis of 6FDA-durene polyimide required two monomers, i.e., durene diamine (99%) and 6FDA dianhydride (99%). Durene diamine was purified through recrystallization using methanol, while 6FDA-dianhydride was purified via the vacuum sublimation technique. Vacuum distillation was used to purify N-methyl-2-pyrrolidone (NMP). Propionic anhydride (≥98%), triethylamine (≥99%), methanol (≥99.9%) and dichloromethane (≥99.8%) were purchased from Merck (Massachusetts, MA, USA) and were used as received.

The premixed gases used for the gas separation test were purchased from Air Products Sdn Bhd, (Kuala Lumpur, KL, Malaysia) with CO₂ compositions ranging from 15 mol% to 70 mol%.

2.2. Synthesis of 6FDA-Durene

In this study, 6FDA-durene polyimide was synthesized using a dual-step procedure reported in the literature [21]. First, equimolar amounts of durene diamine and 6FDA dianhydride monomers were dissolved in NMP for 24 h at room temperature under a nitrogen atmosphere. Next, polyimide was formed after adding propionic anhydride and triethylamine to the mixture solution. Then, the solution was precipitated in methanol and washed several times with methanol. The resulting polymer was dried under a vacuum for 24 h at 150 °C.

2.3. Synthesis of NH₂-MIL-125(Ti)

The NH₂-MIL-125 (Ti) filler was synthesized by following the method reported in our previous work [3]. Firstly, 17.5 g of pivalic acid was dissolved in a mixture solution containing 125 mL of acetonitrile and 5 mL of tetraisopropyl orthotitanate. Next, the mixture solution was placed in a Teflon-lined autoclave reactor and heated at 100 °C for 84 h to obtain white crystals. The white crystals were filtered and dried in an oven at 80 °C for 24 h. Then, 2.4 g of white crystals were dispersed in a mixture solution containing 50 mL methanol and 50 mL DMF. After that, the solution was added to a mixture solution containing 3.3 g of 2-aminoterephthalic acid and 75 mL of DMF. The final mixture was heated in an oven at 100 °C for 24 h. The resulting suspension was centrifuged at 7800 rpm for 5 min. Afterwards, the NH₂-MIL-125 (Ti) fillers were washed several times with methanol and DMF before being dried in an oven at 60 °C for 24 h.

2.4. Fabrication of Dense Membrane

In our previous work [3], the 7.0 wt% NH₂-MIL-125 (Ti) /6FDA-durene mixed matrix membrane showed the highest performance in terms of CO₂ and CH₄ single gas permeation; therefore, this membrane was utilized to obtain the optimum condition in CO₂/CH₄ binary gas separation. The mixed matrix membrane was prepared by dispersing fillers and dissolving the polymer in DCM separately. After stirring, the suspension was sonicated to disperse the fillers in DCM. Then, the priming step was introduced by adding 10 wt% of the polymer solution into the filler suspension under stirring to induce the polymer–filler interface [22] before the suspension was sonicated again. The remaining polymer solution was added to the suspension, which was then sonicated. Subsequently, the suspension was vigorously stirred for 1 h before being cast onto a petri dish. The petri dish was then covered with perforated aluminum foil for solvent evaporation at room temperature for 24 h. After complete evaporation, the membrane film was carefully removed from the petri dish and dried in an oven at 60 °C. Then, the membrane was further dried under vacuum at 60 °C for 24 h, followed by thermal annealing at 250 °C for 24 h.

2.5. Binary Gas Separation Measurement

Binary gas separation testing was carried out by using a custom-made permeation test rig as described in our previous work [3]. The compositions of gases in the retentate and permeate gas streams were determined using a gas analyzer (Fuji, NDIR Gas Analyzer ZPAJ). Then, the permeability of the gases was calculated using Equations (1) and (2), as follows [20]:

$$P_{CO2}=\frac{V_P{y}_{CO2}t}{A_m\left(p_h{x}_{CO2}-p_l{y}_{CO2}\right)}$$

(1)

$$P_{CH4}=\frac{V_P{y}_{CH4}t}{A_m\left(p_h{x}_{CH4}-p_l{y}_{CH4}\right)}$$

(2)

where $P_{CO2}$ and $P_{CH4}$ are the permeability of CO₂ and CH₄ (Barrer, 1 Barrer = 1 × 10⁻¹⁰ cm³(STP).cm/s·cm²·cmHg), V_p is the volumetric flowrate (cm³(STP)/s), A_m and t are the operational area (cm²) and the thickness (cm) of the mixed matrix membrane, respectively, p_h and p_l are the pressure of feed and permeate side, respectively (cmHg), and x and y represent the volume fraction of the component in retentate and permeate streams by assuming ideal gas behaviors. The CO₂/CH₄ separation factor was calculated using Equation (3), as follows [20]:

$${\alpha }_{C{O}_2/C{H}_4}=\frac{y_{C{O}_2}/y_{C{H}_4}}{x_{C{O}_2}/x_{C{H}_4}}$$

(3)

where ${\alpha }_{CO2/CH4}$ indicates the CO₂/CH₄ separation factor, and x and y represent the volume fraction of the component in the retentate and permeate streams, respectively.

2.6. Optimization of Membrane Separation Performance

The optimization of CO₂ separation from CH₄ was conducted based on the experimental runs generated by Design-Expert software V9.0 (Stat-Ease Inc., Minneapolis, MN, USA). The central composite design (CCD) available in the software was selected due to its various advantages, including its flexibility, efficiency, and continuous run. In the current study, three operational parameters were selected, namely, feed pressure (A), temperature (B), and CO₂ feed concentration (C). Meanwhile, three significant responses were determined, i.e., CO₂ permeability, CH₄ permeability and CO₂/CH₄ separation factor. All parameters were tested at three measurement levels from low to high range. Then, the alpha (α) was fixed at 1, which is considered to be a face-centered design. The range and level of the operational parameters are shown in

Table 1. Experimental range and levels of the operational parameters.

Operational Parameters	Units	Coded Measurement Levels
		−1 (Low)	0 (Center)	1 (High)
Feed Pressure (A)	Bar	3.5	8.0	12.5
Temperature (B)	°C	30.0	40.0	50.0
CO2 Feed Concentration (C)	(mol%)	15.0	42.5	70.0

.

The interaction between the operational parameters, i.e., feed pressure, temperature and CO₂ feed concentration, and responses, i.e., the permeability of the gases and the CO₂/CH₄ separation factor, were further analyzed via analysis of variance (ANOVA). The experimental results were fitted into empirical models (second-order polynomial function) expressed in Equation (4) in order to correlate the corresponding responses over the operational parameters [18].

$$Y=B_0+{{\displaystyle \sum }}_{i=1}^3{B}_i{x}_i+{{\displaystyle \sum }}_{i=1}^2{{\displaystyle \sum }}_{j=i+1}^3{B}_{ij}{x}_i{x}_j+{{\displaystyle \sum }}_{i=1}^3{B}_{ii}{x}_i{}^2$$

(4)

where Y is the corresponding response, B₀ is a fixed term, and B_i, B_ij and B_ii are linear, interaction and quadratic terms, respectively. Meanwhile, x_i and x_j represent the coded terms for operational parameters.

The fitted model for each corresponding response was analyzed in terms of the statistical significance of the operational parameters and interactions using the F- and p-values. The F-value is defined as the ratio of the mean square model to the mean square error. The mean square model is the sum of squares divided by the number of degrees of freedom. The p-value indicates the significance of the model; therefore, p < 0.05 shows that the regression model is relevant, and the null hypothesis (H_o): p ≤ 0.5 is rejected. In addition, the consistency of the models was quantified by the coefficient of determination (R²). Residual is the unexplained variation by the fitted model. ‘Lack of fit’ denotes that the lack of fit is insignificant in comparison to the pure error.

3. Results and Discussion

3.1. Characteristics of Membrane

The physicochemical properties, morphology and filler distribution, thermal stability, fractional free volume (FFV) analysis as well as CO₂ and CH₄ single gas performance of the NH₂-MIL-125 (Ti)/6FDA-durene membranes at various filler loadings were discussed in detail in our previous work [3]. As noted in our previous work, the 7.0 wt% NH₂-MIL-125 (Ti)/6FDA-durene mixed matrix membrane exhibited the highest CO₂ permeability and CO₂/CH₄ gas pair selectivity, i.e., 1115.7 Barrer and 37.1, respectively, at a feed pressure of 3.5 bar and at room temperature. In this work, the performance of this membrane in mixed gas separation with various operating parameters, particularly at high CO₂ feed concentration, was investigated using a statistical approach consisting of a central composite design (CCD) paired with a response surface methodology (RSM). Figure S1 and Table S1 display the characterization results comprising the XRD, FESEM, EDX and FFV for the 7.0 wt% NH₂-MIL-125 (Ti)/6FDA-durene mixed matrix membrane. In reference to Figure S1a, the significant peaks observed at 6.9° and 9.8° corresponded to the X-ray reflection planes (011) and (002) of the NH₂-MIL-125 (Ti) fillers. Meanwhile, the fillers were encapsulated and uniformly dispersed in the 6FDA-durene polymer matrix, as shown in Figure S1b. On the other hand, Table S1 compares the CO₂ and CH₄ single gas permeation performance and FFV values of pure and 7.0 wt% NH₂-MIL-125 (Ti)/6FDA-durene mixed matrix membrane.

3.2. Central Composite Design (CCD)

From CCD, 20 trial experiments, including eight factorial points, six axial points, and six central point replicates in a randomized sequence, were generated by the DoE software. The replicated center points were used to estimate the pure error for the lack of fit test. The randomized sequence for the experimental run was intended to limit the impact of uncontrollable factors. Therefore, experimental 20 runs were conducted and three significant responses, namely, CO₂ permeability, CH₄ permeability, and CO₂/CH₄ separation factor, were measured.

Table 2. CCD design matrix of 2³ factorial generated by DOE and the obtained significant responses.

Run	Operational Parameters			Significant Responses
	A: Pressure (Bar)	B: Temperature (°C)	C: CO2 Concentration (mol%)	CO2 Permeability (Barrer)	CH4 Permeability (Barrer)	CO2/CH4 Separation Factor
1	3.5	40.0	42.5	569.6	58.3	9.9
2	8.0	40.0	42.5	569.6	58.3	9.9
3	3.5	50.0	70.0	326.2	52.3	5.5
4	12.5	30.0	70.0	609.1	81.9	7.3
5	8.0	40.0	70.0	450.2	92.7	6.0
6	8.0	50.0	42.5	504.0	39.9	12.8
7	3.5	30.0	70.0	448.2	33.9	13.0
8	12.5	50.0	70.0	293.2	36.2	7.3
9	12.5	30.0	15.0	506.6	83.2	6.1
10	8.0	40.0	15.0	419.7	39.7	8.6
11	12.5	40.0	42.5	794.4	147.5	5.3
12	8.0	40.0	42.5	442.9	39.5	8.6
13	3.5	50.0	15.0	502.3	47.4	10.8
14	8.0	40.0	42.5	321.5	28.7	11.4
15	3.5	30.0	15.0	464.4	58.4	8.1
16	8.0	30.0	42.5	650.2	97.0	8.4
17	8.0	40.0	42.5	520.3	55.5	9.3
18	12.5	50.0	15.0	567.5	74.3	7.5
19	8.0	40.0	42.5	520.3	55.5	9.3
20	8.0	40.0	42.5	510.1	91.0	6.1

shows the condition of the operational parameters and significant responses of the membrane separation performance observed in the experiments. From Table 2, it can be observed that the permeability of CO₂ and CH₄ were in the range of 293.2 to 794.4 Barrer and 28.7 to 147.5 Barrer, respectively. The percentage errors between the actual and predicted values of membrane separation performances are listed in Table S2.

Meanwhile, the CO₂/CH₄ separation factor obtained ranged from 5.6 to 13.0. The margin of error of the experimental results was ±5%. Furthermore, compared to single gas permeation testing [3], the performance of the membrane in binary gas separation demonstrated lower values. These difference could be explained by the competitive sorption between CO₂ and CH₄ gas molecules [23]. The presence of a competitive gas may substantially affect the gas penetration over the membrane [24].

3.3. CO₂ Permeability

The quadratic polynomial model suggested by the DoE software for CO₂ permeability is shown in Equation (5) in the form of a coded value.

$$\begin{array}{cc}C{O}_{2}Permeabilit{y}_{coded}=& 518.31-67.77A+36.63B+98.10C-72.10AB-33.77AC+16.25BC+\\ & \hspace{1em}\hspace{1em}\hspace{1em}28.97A^2-117.49B^2+50.49C^2\end{array}$$

(5)

where A, B and C denote the feed pressure (bar), temperature (°C) and CO₂ feed concentration (mol%), respectively.

Table 3. Analysis of variance (ANOVA) for CO₂ permeability.

Source	Sum of Squares	Degree of Freedom	Mean Square	F-Value	Prob > F
Model	2.471 × 105	9	27,456.76	27.11	<0.0001 a
A-Pressure	45,922.31	1	45,922.31	45.34	<0.0001 a
B-Temperature	13,418.30	1	13,418.30	13.25	0.0045 a
C-Concentration	96,228.25	1	96,228.25	95.00	<0.0001 a
AB	41,584.40	1	41,584.40	41.05	<0.0001 a
AC	9123.30	1	9123.30	9.01	0.0133 a
BC	2111.85	1	2111.85	2.08	0.1794 b
A2	2307.53	1	2307.53	2.28	0.1621 b
B2	37,959.26	1	37,959.26	37.47	0.0001 a
C2	7009.65	1	7009.65	6.92	0.0251 a
Residual	10,129.32	10	1012.93
Lack of Fit	5282.07	5	1056.41	1.09	0.4636 b
R2	0.96

^a statistically significant, ^b statistically not significant.

presents the findings of the ANOVA and regression analysis for CO₂ permeability over the membrane. As shown in Table 3, model F and p values of 27.11 and <0.05 were achieved, respectively, suggesting that the model is statistically significant. In this case, A, B, C, AB, AC, B², C² are statistically significant model terms. It can be observed from Table 3 that an R² value of 0.96 was achieved, which validated the accuracy of the model for CO₂ permeability.

Figure 1a–c display a 3D plot of the effect of different operational parameters on CO₂ permeability. The increment of temperature at a feed pressure of 3.5 bar and a fixed CO₂ feed concentration of 42.5 mol% resulted in higher CO₂ permeability, as observed in Figure 1a. This could be due to the higher CO₂ diffusivity [19] caused by the increment of polymer chain mobility and free volume [18]. Meanwhile, the increment of feed pressure at temperatures between 30 to 40 °C resulted in a slight improvement in CO₂ permeability due to the increase in gas solubility as a consequence of thermodynamic promotion [11]. On the other hand, a decreasing trend of CO₂ permeability can be seen for temperatures ranging from 40 to 50 °C, mainly because of the CO₂ sorption isotherm, which follows the dual-mode sorption mechanism [25]. Generally, higher temperatures lead to higher CO₂ permeability; however, in this study, we observed a different trend. This might have been related to the lower solubility of gas molecules with increasing temperature. The adverse impact of temperature on sorption enthalpy would have affected the transport and sorption behavior of gases over the membrane [26]. From the obtained results, a maximum CO₂ permeability of 609.1 Barrer was achieved at a feed pressure of 3.5 bar and a temperature of 40 °C. Meanwhile, the lowest CO₂ permeability, i.e., 321.5 Barrer, was achieved at a feed pressure of 8 bar and a temperature of 30 °C.

Figure 1b shows the effect of CO₂ feed concentration and feed pressure on CO₂ permeability at a constant temperature of 40 °C. It can be seen in Figure 1b that the lowest CO₂ permeability, i.e., 470.7 Barrer, was achieved at a CO₂ feed concentration of 15 mol% and a feed pressure of 8 bar. In contrast, the highest CO₂ permeability, 792.2 Barrer, was obtained at a feed pressure of 3.5 bar and a CO₂ feed concentration of 70 mol%. Additionally, at constant temperature, increasing the CO₂ feed concentration in the feed caused significant improvement in CO₂ permeability; this was mainly due to the higher sorption of CO₂ than CH₄ over the membrane. Furthermore, the interaction between CO₂ molecules and -NH₂ elements also contributed to the improvement of CO₂ diffusion through the membrane [27]. On the other hand, the CO₂ permeability decreased slightly with increasing feed pressure. This result was attributed to the saturation of available sorption sites, which resulted in a lower solubility coefficient [28]. Furthermore, this trend was also related to the dual-mode sorption and diffusion mechanism on gas transport behavior over the membrane.

Figure 1c illustrates the effect of CO₂ feed concentration and temperature on CO₂ permeability at a fixed feed pressure of 8 bar. As shown in Figure 1c, increasing the CO₂ feed concentration led to an increase in CO₂ permeability due to the higher CO₂ sorption effect relative to CH₄, which, subsequently, enhanced the sorption of CO₂ over CH₄ gas molecules in the membrane. A similar incremental trend of CO₂ permeability was observed under fixed feed pressure conditions, as shown in Figure 1b. When the temperature increased from 30 °C to 40 °C at a constant feed pressure of 8 bar, the CO₂ permeability improved due to the increase in movement and flexibility of polymer chains, as well as the kinetic energy of gas molecules [26]. Furthermore, it could be that the impact of temperature on the gas transport and the sorption behavior compensated for the adverse effect of temperature on gas solubility. However, a slight decrease in CO₂ permeability was discovered after increasing the temperature from 40 °C to 50 °C. This was mainly due to the sorption competition between the CO₂ and CH₄ gases [25]. The reduction in gas solubility caused by the reduced interaction of CO₂ gas with the polymer matrix was attributed to the increment of adsorption energy with temperature; consequently, the thermodynamic effect overcame the kinetic effect during gas penetration [29]. From Figure 1c, it may be seen that the maximum CO₂ permeability, i.e., 650.2 Barrer, was achieved at a feed pressure of 8 bar, a CO₂ feed concentration of 70 mol% and a temperature of 40 °C. A parity plot for CO₂ permeability is shown in Figure S2a. It can be seen that the actual and predicted values of the responses were scattered near to the 95% prediction limits.

3.4. CH₄ Permeability

Equation (6) depicts the quadratic polynomial model for CH₄ permeability in terms of coded value.

$$\begin{array}{cc}C{H}_{4}Permeabilit{y}_{coded}=& 55.48-23.89A+13.76B+10.52C-12.10AB-7.01AC+7.08BC-0.15A^2-\\ & \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}24.03B^2+36.76C^2\end{array}$$

(6)

where A, B and C denote the feed pressure (bar), temperature (°C) and CO₂ feed concentration (mol%), respectively.

Table 4. Analysis of variance (ANOVA) for CH₄ permeability.

Source	Sum of Squares	Degree of Freedom	Mean Square	F-Value	Prob > F
Model	15,005.47	9	1667.27	29.95	<0.0001 a
A-Pressure	5708.28	1	5708.28	102.54	<0.0001 a
B-Temperature	1892.55	1	1892.55	34.00	0.0002 a
C-Concentration	1106.91	1	1106.91	19.88	0.0012 a
AB	1171.28	1	1171.28	21.04	0.0010 a
AC	393.68	1	393.68	7.07	0.0239 a
BC	400.73	1	400.73	7.20	0.0230 a
A2	0.066	1	0.066	1.180 × 10−3	0.9733 b
B2	1587.90	1	1587.90	28.52	0.0003 a
C2	3716.16	1	3716.16	66.75	<0.0001 a
Residual	556.69	10	55.67
Lack of Fit	429.72	5	85.94	3.38	0.1035 b
R2	0.96

^a statistically significant, ^b statistically not significant.

demonstrates the ANOVA and regression analysis of CH₄ permeability. As shown, F and p values of 29.95 and < 0.05, respectively, were achieved, indicating that the model terms were statistically significant. Additionally, A, B, C, AB, AC, BC, B² and C² were statistically significant model terms in this context. In addition, an R² value of 0.96 was obtained, confirming the accuracy of the model for CH₄ permeability.

Figure 2a–c display a 3D plot of the CH₄ permeability over the membrane. The increment of temperature from 30 °C to 40 °C resulted in higher CH₄ permeability, as observed in Figure 2a. The increase in chain mobility and free volume resulted in higher diffusivity, and thus, increased the penetration of CH₄ molecules through the membrane [30]. The relationship between temperature and permeability can be defined using Arrhenius equation, where CH₄ molecules exhibit higher activation energy than CO₂ molecules, and therefore, promote the diffusion of nonpolar gases over the glassy polymer. Meanwhile, a reduction of CH₄ permeability was found with the increment of temperature from 40 °C to 50 °C. This was ascribed to the reduction in solubility, which subsequently enhanced the restriction of CH₄ gas permeation through the membrane [31]. Meanwhile, increasing the feed pressure caused a slight reduction in CH₄ permeability, mainly due to the decrease in the dual-sorption properties for CH₄ gas [19], as well as the high compressibility of CH₄ gas molecules [32] in the membrane.

Figure 2b shows the effect of CO₂ feed concentration and feed pressure on CH₄ permeability at a constant temperature of 40 °C. Referring to Figure 2b, the CH₄ permeability obtained at feed pressures ranging from 3.5 bar to 12.5 bar demonstrated a similar trend to the results presented in Figure 2a. On the other hand, an increase in CO₂ feed concentration from 15 mol% to 42.5 mol% led to a reduction of CH₄ permeability, due to the lower sorption of CH₄ compared to CO₂, as well as the slower diffusion of CH₄ gas over the membrane [20]. However, increasing the CO₂ feed concentration from 42.5 mol% to 70 mol% enhanced the permeability of CH₄. The increasing trend in CH₄ permeability was because of the swelling of polymer chain packing at higher CO₂ feed concentrations, and therefore, increased the segmental mobility [19]. On the other hand, the lowest CH₄ permeability, i.e., 33.9 Barrer, was seen with a CO₂ feed concentration of 42.5 mol% and a feed pressure of 12.5 bar. In contrast, the maximum CH₄ permeability, 133.3 Barrer, was achieved at a CO₂ feed concentration of 70 mol%, a temperature of 40 °C and a feed pressure of 3.5 bar (Figure 2b).

Figure 2c demonstrates the effect of CO₂ feed concentration and temperature on CH₄ permeability at a constant feed pressure of 8 bar. As shown, the CH₄ permeability gradually improved with increasing the temperature, owing to the enhanced mobility and flexibility of the polymer chains [26], as well as to the higher activation energy of CH₄ [33]. Meanwhile, a reduction in CH₄ permeability was observed at CO₂ feed concentrations ranging from 15 mol% to 42.5 mol%, and an improvement in CH₄ permeability was found when the CO₂ feed concentration increased from 42.5 mol% to 70 mol%. This trend was identical to that shown in Figure 2b. Apart from that, the highest CH₄ permeability, i.e., 106.1 Barrer, was obtained with a CO₂ feed concentration of 70 mol%, a feed pressure of 8 bar and a temperature of 40 °C (Figure 2c); this was associated with concentration polarization effects [34]. Meanwhile, the lowest CH₄ permeability, 28.7 Barrer, was observed at a CO₂ feed concentration of 42.5 mol% and a temperature of 30 °C. The increase in CH₄ permeability with the increase in temperature and CO₂ feed concentration was mainly due to the increase in polymer free volume caused by the alteration of polymer chain packing and intersegmental motion [33], as well as the swelling effect induced by CO₂ gas. The presence of high CO₂ feed concentrations caused the rapid diffusion of CH₄ gas through the membrane [35], because CH₄ gas molecules are more accessible to a swollen polymer matrix. A parity plot for CH₄ permeability is shown in Figure S2b. As shown, very few data points fall outside the 95% prediction limits.

3.5. CO₂/CH₄ Separation Factor

The quadratic polynomial model for CO₂/CH₄ separation factor is defined in Equation (7) in the form of coded values.

$$\begin{array}{cc}Separation Facto{r}_{coded}=& 9.93+1.82A-1.05B+0.84C-0.54AB+0.46AC-0.80BC+0.30A^2+\\ & \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}0.11B^2-2.70C^2\end{array}$$

(7)

where A, B and C denote the feed pressure (bar), temperature (°C) and CO₂ feed concentration (mol%), respectively.

The ANOVA analysis of the CO₂/CH₄ separation factor is presented in

Table 5. Analysis of variance (ANOVA) for CO₂/CH₄ separation factor.

Source	Sum of Squares	Degree of Freedom	Mean Square	F-Value	Prob > F
Model	91.12	9	10.12	18.04	<0.0001 a
A-Pressure	33.27	1	33.27	59.29	<0.0001 a
B-Temperature	10.96	1	10.96	19.54	0.0013 a
C-Concentration	7.12	1	7.12	12.69	0.0052 a
AB	2.38	1	2.38	4.23	0.0666 b
AC	1.71	1	1.71	3.05	0.1113 b
BC	5.18	1	5.18	9.24	0.0125 a
A2	0.25	1	0.25	0.44	0.5229 b
B2	0.036	1	0.036	0.064	0.8057 b
C2	20.06	1	20.06	35.75	0.0001 a
Residual	5.61	10	0.56
Lack of Fit	3.43	5	0.69	1.57	0.3168 b
R2	0.94

^a statistically significant, ^b statistically not significant.

. It can be seen that model F and p values of 18.04 and <0.05 were achieved, respectively, indicating that the model terms are statistically significant, while A, B, C, BC and C² are statistically significant model terms. Furthermore, an R² value of 0.94 was achieved, showing the accuracy of the model for CO₂/CH₄ separation factor.

Figure 3 displays 3D plots of the effect of feed pressure, temperature, and CO₂ feed concentration on the CO₂/CH₄ separation factor over the membrane. As shown, increasing the feed pressure resulted in a higher CO₂/CH₄ separation factor owing to the higher CO₂ adsorption capacity compared to CH₄, as well as the good compatibility of the polymer-filler, which was induced by the presence of -NH₂ groups in the membrane [27]. The enhanced CO₂/CH₄ separation factor was related to the improvement of CO₂ permeability and the reduction of CH₄ permeability observed in Figure 1a and Figure 2a. In contrast, increasing the temperature revealed a negligible impact on CO₂/CH₄ separation factor, since the contribution of solubility and diffusivity is interchangeable. Additionally, as shown in Figure 3a, the highest CO₂/CH₄ separation factor, i.e., 13.0, was obtained at a temperature of 30 °C and a feed pressure of 12.5 bar. In comparison, the lowest CO₂/CH₄ separation factor, 7.3, was observed at a feed temperature of 50 °C and a feed pressure of 3.5 bar.

Figure 3b illustrates the effect of CO₂ feed concentration and feed pressure on the CO₂/CH₄ separation factor at a constant temperature of 40 °C. It can be observed that the CO₂/CH₄ separation factor increased with increasing feed pressure. This was mainly due to the higher sorption and permeation of CO₂ molecules over the membrane compared to CH₄ [33], which resulted in higher CO₂ permeability and lower CH₄ permeability, as observed in Figure 1b and Figure 2b, respectively. The higher CO₂ permeability also contributed to the increase in CO₂ solubility and the reduction of CH₄ solubility, and thus, improved the CO₂/CH₄ solubility selectivity [36]. Furthermore, the CO₂/CH₄ separation factor displayed an increasing trend with increasing CO₂ feed concentration up to 42.5 mol%, and then decreased when the CO₂ feed concentration was increased further to 70 mol%. The increment of CO₂ feed concentration caused a reduction of CH₄ permeability and led to an increase in CO₂/CH₄ separation factor, owing to the smaller kinetic diameter of CO₂ compared to CH₄. Meanwhile, increasing the CO₂ feed concentration further from 42.5 mol% to 70 mol% caused an improvement of CH₄ permeability, but a reduction in the CO₂/CH₄ separation factor was observed due to the early stage of CO₂-induced plasticization behavior [33,37]. The highest separation factor, i.e., 13.0, was achieved at a feed pressure of 12.5 bar and CO₂ a feed concentration of 42.5 mol%. Meanwhile, the lowest CO₂/CH₄ separation factor, 6.0, was observed at a feed pressure of 3.5 bar and a CO₂ feed concentration of 15 mol%, mainly resulting from the presence of a larger amount of CH₄ gas (85 mol%) in the feed mixture.

Figure 3c shows the effect of CO₂ feed concentration and temperature on the CO₂/CH₄ separation factor at a pressure of 8 bar. From Figure 3c, increasing the temperature at a fixed feed pressure of 8 bar caused a slightly drop in the CO₂/CH₄ separation factor. This reduction was mainly due to the improvement of CO₂ and CH₄ permeability, as explained in Section 3.3 and Section 3.4. Furthermore, at a fixed feed pressure of 8 bar, the increase of CO₂ feed concentration from 15 mol% to 42.5 mol% resulted in the enhancement of the CO₂/CH₄ separation factor, which was consistent with the increase of CO₂ permeability, as shown in Figure 1c. On the other hand, increasing the CO₂ feed concentration from 42.5 mol% to 70 mol% led to a slight drop in the CO₂/CH₄ separation factor due to the saturation of CO₂ gas molecules inside the polymer voids [11]. A parity plot for CO₂/CH₄ separation factor is shown in Figure S2c. Overall, the actual and predicted values of the responses were scattered near the 95% prediction limits. As shown in Figure S2c, very few data points fall outside the 95% prediction limits.

3.6. Optimization of CO₂/CH₄ Separation Performance

The primary objective of this work is to identify the optimal operational conditions for membrane separation in terms of CO₂ permeability and CO₂/CH₄ separation factor. Therefore, the optimization conditions included maximizing the CO₂ feed concentration as an operational parameter, as well as CO₂ permeability and CO₂/CH₄ separation factor. Additionally, the desirability function (DF) was used in RSM to optimize a sequence of quadratic models. The geometric mean of individual desirability (d) was used to calculate the total desirability (D), as expressed in Equation (8) [38]:

$$D_d={\left(d_1\times d_2\times \dots d_n\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$n$}\right.}$$

(8)

where D_d is the total desirability and d_n is the nth desirability, n = 1, 2, …, n. The total desirability was measured from 0 to 1, where 0 represents the most undesirable response and 1 the most desirable.

The optimum solution and its desirability generated by the DOE software is listed in

Table 6. Experimental conditions generated by the DOE software and responses.

Pressure (bar)	Temperature (°C)	Concentration (mol%)	CO2 Permeability (Barrer)	CH4 Permeability (Barrer)	CO2/CH4 Separation Factor	Desirability
12.5	34.7	70.0	571.9	60.4	11.9	0.8

in term of actual values. The optimum operational parameters suggested for CO₂ feed concentration, feed pressure, and temperature were 70 mol%, 12.5 bar and 34.7 °C, respectively, which yielded optimal CO₂ permeability and separation factor values, i.e., 571.9 Barrer and 11.9, respectively.

3.7. Validation of the Optimum Condition

In order to validate the optimum conditions and predicted responses shown in Table 6, three repeated experiments were conducted based on the suggested conditions; the results are presented in

Table 7. Validation of optimal condition for membrane separation.

Run	CO2 Permeability			Separation Factor
	Actual (Barrer)	Predicted (Barrer)	Error (%)	Actual	Predicted	Error (%)
1	609.3	571.9	6.5	11.6	11.9	2.5
2	592.7	571.9	3.6	11.8	11.9	0.8
3	605.3	571.9	5.8	11.3	11.9	5.0
Average (%)			5.3	Average (%)		2.8
Standard deviation			1.5	Standard deviation		2.1

. The percentage (%) error for CO₂ permeability ranged from 3.6% to 6.5% relative to the standard deviation of 1.5%. On the other hand, the percentage (%) error for CO₂/CH₄ separation factor ranged from 0.8% to 5.0% with a standard deviation of 2.1%. Meanwhile, the average errors obtained for CO₂ permeability and CO₂/CH₄ separation factor were 5.3% and 2.8%, respectively. Overall, the average error for the experimental and predicted values were within 5%, indicating that the model validity reached 95% of the prediction interval. Consequently, the model was successfully validated, and as such, the optimization of operational parameters was achieved utilizing the RSM approach.

4. Conclusions

In conclusion, the effects of operational parameters, i.e., feed pressure, temperature and CO₂ feed concentration, were identified as dominant factors influencing the separation performance of a NH₂-MIL-125 (Ti)-6FDA/durene membrane. CO₂ feed concentration demonstrated a significant effect on CO₂ permeability, whereas feed pressure was the primary parameter influencing CH₄ permeability and CO₂/CH₄ separation factor, with F-values of 102.5 and 59.3, respectively. The R² values obtained ranged from 0.94 to 0.96, indicating that the regression models were statistically significant. The optimum operational parameters, i.e., a feed pressure of 12.5 bar, a temperature of 34.7 °C and a CO₂ feed concentration of 70 mol%, yielded the maximum CO₂ permeability, i.e., 609.3 Barrer, and a CO₂/CH₄ separation factor of 11.6. The average errors for CO₂ permeability and CO₂/CH₄ separation factor were 5.3% and 2.8%, respectively, suggesting that the model was 95% reliable. Overall, the experimental findings show that the RSM paired with the CCD method is a better strategy to obtain optimal CO₂/CH₄ separation performance with a NH₂-MIL-125 (Ti)/6FDA-based mixed matrix membrane. The present research provides an experimental reference for further improvements and scale-ups of membranes in CO₂/CH₄ separation with high CO₂ feed concentrations.