The influence of synthesis parameters on the gas selectivity and permeability of carbon membranes: empirical modeling and process optimization using surface methodology

Abstract

One of the common polyimides, P84 HT (BTDA-TDI/MDI), was used to synthesize gas separating carbon molecular sieve (CMS) membranes and study their gas transport properties. This study explores the role of key variables such as blend composition and the pyrolysis conditions in synthesizing CMS membranes as well as the operating temperature and pressure on gas transport performance through modeling and optimization methods. An experimental design conducted through the implementation of D-optimal design took into account four main parameters to optimize gas permeability and ideal gas pair selectivity. The highest selectivity can be obtained from precursors carbonized at a higher temperature, but optimal values for both the selectivity and permeability were attained for those CMS membranes pyrolyzed at 760.2 °C. The optimum operating conditions for this membrane are 4.5 bar and 25 °C. Under these conditions, the selectivities of N₂/CH₄, CO₂/CH₄, O₂/N₂ and N₂/CO₂, were 3.65, 155.7, 28.4 and 49.6, respectively.

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1. Introduction

With extensive focus having been placed on reducing energy consumption in industry over the last few decades, chemical engineering has found a unique place.^1–3 O₂/N₂, CO₂/CH₄, N₂/CH₄ and CO₂/N₂ separation processes are used to purify gases in the refinery and fertilizer industries. Currently, commercially available separation techniques for this purpose are pressure swing adsorption, gas sweetening, amine absorption and cryogenic separation which are high energy and cost.^4–6 Industrial membrane separation technology is applied in the recycling of hydrogen gas, hydrocarbons, sweetening of sour gas, air separation and gas separation processing for the production of nitrogen passing, dehydration and, etc.^5,6

This has given an impetus to researchers in this field to explore novel materials and membrane structures such as CMS membranes, silica, zeolites and examine their prospects for gas separation applications.⁷

Among the membranes, CMS membranes are more remarkable and superior than other membranes due to high permeability and selectivity, time-independent permeability (productivity), high stability in the presence of organic vapors and solvents, high thermal stability and suitability for operation.^8,9

Producing a high-efficiency CMS is a complicated process due to the involvement of many steps needed to be designed and optimized properly.⁹ In other words, fabrication of the carbon membranes incorporates some important factors such as the polymer type and the morphology, microstructure and chemical formula of the CMS precursors, in addition to pyrolysis (carbonization) optimization (i.e. final temperature, ramp rate, thermal soak, atmosphere and pressure). Therefore, it is necessary to have a precise control on these factors to obtain a high performing carbon molecular membrane with desired characteristics.^10–18

Recently, many investigators have studied the gas separation performance of CMS membranes fabricated through the carbonization of various polymer precursors, such as polyimide (PI), resins, polyamides, polyacrylonitrile, polyacrylates, polypyrrolones and polyacrylates, etc.^5,6,19

Among the various polymeric precursors, polyimides have been identified as a remarkable precursor for CMS membrane preparation for the sake of their high performance with high glass transition temperature, high permselectivity and permeability, high melting point, and great structural and thermal stability.^20,21 Many types of polyimides with different dianhydrides, including those containing BTDA,^22,23 PMDA,^24–27 BPDA,^28–30 and 6FDA^31,32 have become commercially available.

Tin et al.³³ produced CMS membranes from BTDA-TDI/MDI through carbonizing dense films at temperatures of 550–800 °C under vacuum. Selectivity of the membranes was increased by raising the pyrolysis temperature of CO₂/CH₄ from 22 to 89 and membranes pyrolyzed at 800 °C have excellent properties with a permeability of 500 barrers.

Furthermore, many previous studies have revealed the outstanding advantages produced by the use of blending technologies for structural and thermal stability modification, and permselectivity and permeability enhancement of the membranes.²⁰ Blending techniques can help select suitable materials with several advantages such as reconciling classes of polymers with different gas separation properties and or structural characteristics.²⁰

In addition to the mentioned major factor for the manufacture of carbon molecular sieve membranes, conventionally the pyrolysis process is the key step used for the production of porous carbon. During the carbonization process, microporosity is formed in the carbon membrane.³⁴ Pyrolysis temperature is the most important factor affecting the membrane crystallinity, compactness, density, and the average inter-planar spacing between the layers.^20,34

Pyrolysis temperature is known to be the most important factor influencing the carbon membranes (CMS) considerably in terms of the structure of the membrane. Holistically, the separation performance, mechanism of gas transport and the experimental data demonstrate that the permeability of the membranes is enhanced as the pyrolysis temperature is increased from 325 to 475 °C for pure gases. However, the gases for gas permeability with smaller particles remain almost unchanged.³¹

Besides the prominent importance of the above parameters in synthesizing CMS membranes, it is also essential to consider the operating conditions (such as operating temperature and pressure) affecting the performance of carbon membranes.^35–38 Anderson et al.³⁷ found that the CO₂/CH₄ selectivity decreases as the operating temperature is increased. As the diffusion rate and the operating temperature are raised, the presence of CH₄ is increased. Furthermore, Azeman et al.³⁸ synthesized carbon membranes through the carbonization of asymmetric polyimide membranes. They realized that the efficiency of gas separation in carbon membranes is affected by the feed pressure.

Findings by the above investigations and many other studies on CMS fabrication and the performance of gas separation reveal that the efficiency of CMS membranes is affected by various variables. Statistical modeling is recognized as a practical tool for process development and optimizing complex variables. Modeling can effectively figure out the relationship between one or more variables and a set of quantitative experimental parameters in the field of gas separation and membrane technology.^39–41

He et al.⁴² also used an orthogonal experimental design (OED) approach to explore the gas transport properties of membranes through making modifications to the carbonization trend. The obtained results uncovered the significance of the carbonization parameters being studied on the function of the CMS membrane finding this order: purge gas > final carbonization temperature > heating rate > final soak time.

Statistical modeling and optimization studies were contemplated with respect to improvement of the experimental investigations. The major objective of this work was to explore the influences of the different precursor structures and the pyrolysis temperature in membrane fabrication improvement and the effects of operating conditions on gas selectivity and permeability. Additionally, another point of this study was to use D-optimal design to identify the unilateral and mutual effects of the experimental parameters to obtain optimum factors and conditions for the preparation of high performance CMS membranes and utilization of the optimum operating conditions. These CMS membranes possess various compositions and chemical structures and they are prepared by using different pyrolysis temperatures tested at different operating temperatures and pressures. Optimization of the main contributing factors such as the percentage of polyimides in PBI, final pyrolysis temperature, and the operating temperature and pressure was applied. Eventually, to our best knowledge, this work is the first experimental, modeling and optimization study on the effect of four effective parameters (simultaneously) on gas transport properties addressing the relationship between precursor structure and pyrolysis and operating conditions by applying the statistical technique for CMSMs derived from polymeric blend precursors. The experimental results of this study can prove that the type of precursor and blend composition of polyimides will significantly effect the gas separation performance compared with the operating conditions and pyrolysis temperature. Moreover, the results specify that feed pressure and temperature, known as effective parameters, should also be taken into account in the design of membrane modules and the industrial applications of carbon molecular sieve membranes. The investigation of the operating temperature and feed pressure with other CMS preparation parameters is the novelty of this work and the CMS membranes developed in this research can be exploited as appropriate and optimized membranes with effective productivity and efficiency performance for the main gases (CO₂, CH₄, O₂ and N₂) with potential industrial applications.

2. Materials and experimental methodology

2.1. Materials

P84 HT (copolyimide of 3,3′4,4′-benzophenone tetracarboxylic dianhydride and 80% methylphenylene-diamine + 20% methylene diamine, BTDA-TDI/MDI) was supplied by Lenzing. Polybenzimidazole (PBI) was provided by the Aldrich Chemical Company Inc. (Milwaukee, USA) as the main polymer with a high T_g, outstanding chemical resistance and excellent thermal stability. The measured density of P84 HT and PBI are about 1.071 g cm⁻³ and 1.311 g cm⁻³, respectively. Besides, the glass transition temperature (T_g) of the P84 HT and PBI were determined using differential scanning calorimetry (DSC) with a heating rate of 10 °C min⁻¹, and found to be 223.2 °C and 425 °C, respectively. Both powders were dried overnight at 120 °C under vacuum prior to testing. N-Methyl-2-pyrolidone (NMP) was offered by Merck and used as a solvent for the polymers.

2.2. Preparation of the CMS membranes and gas property measurements

Precursors were synthesized as films from the blending of PBI and P84 HT in three compositions, 25, 50 and 75 wt%. Fig. 1 shows the structures of P84 HT and PBI. The detailed preparation methods and steps including polymer blending and dissolution, degassing, casting steps and evaporation are reported elsewhere.²⁸ All of the homogenous membrane dense films fabricated from the blends of P84 HT and PBI are displayed in Fig. 2. These blended and pure precursors were pyrolyzed in a vacuum furnace equipped with rotary and diffusion pumps manufactured by Azar Co. Two final pyrolysis temperatures, 580 and 800 °C under vacuum pressures of 10⁻⁷ Torr were applied. The pyrolysis protocol with a final pyrolysis temperature of 800 °C displayed in Fig. 3 was used in preparing the molecular sieve membranes. In this protocol, temperature was increased over time with the temperature rising from 25 to 800 °C (vacuum pressure of 10⁻⁷ Torr). The temperature was increased by adjusting the rate of heating, starting at 7.5 °C min⁻¹, and then at the lower rates of 3, 1.5 and 0.15 °C min⁻¹. The membranes were cooled gradually to room temperature at the end of each protocol. The detailed calculation procedures and equations of gas permeability and selectivity for CMS membranes are reported elsewhere.²⁸ Fig. 4 illustrates a schematic diagram and the lab-scale design setup for measuring the gas separation properties. The permeability and selectivity values of pure gas feed at various feed temperatures and pressures were measured by using Gas Chromatography (GC) and other facilities. The thickness of the membranes used for separation performance measurements was about 60 μm.

Fig. 1 Chemical structures of (a) P84 HT and (b) polybenzimidazole (PBI).

Fig. 2 The homogenous membranes prepared from the PBI-P84 HT blends; no indication of phase separation was detected by visual observation.

Fig. 3 Pyrolysis protocol and vacuum furnace used in the preparation of carbon molecular sieve membranes.

Fig. 4 Designed and created setup for testing the gas separation properties: (a) schematic diagram and (b) lab-scale.

2.3. Statistical analysis

In this research, a combined method of D-optimal experiments design (design model: 2FI) and statistical analysis were employed in order to obtain the valuable models with an optimized number of experiments for the investigation of CMS membranes. One of the main advantages of D-optimal design is that it is able to analyze all possible combinations and interactions of the levels across the present variables in comparison to the conventional statistical approach. D-optimal is a functional statistical technique for experimental design, model development, evaluation of the effects of multiple variables, and finding the optimum conditions for applicable functions. For statistical calculations, the experimental parameters X_actual have been coded as X_coded based on the following equation:where, X_coded refers to the dimensionless coded value of the variables (X_actual); X̄ stands for the mean value of them (X_actual); and d is the absolute difference between X_actual and X̄. The response as a function of variables can be written by performing multiple regressions using the least squares method to fit the equation below:

Y = b₀ + b₁X₁ + b₂X₂ + b₃X₃ + b₄X₄ + b₁₂X₁X₂ + b₁₃X₁X₃ + ... (2)

Eqn (2), X₁, X₂, X₃ and X₄ signify the model variables, b₀ is the general coefficient; b_i denotes the coefficient of linear impacts and b_ii is the coefficient of quadratic impacts. An interaction between the individual variables is written as the combinations of variables (e.g., X₁X₂). In the current work, four key variables used for factorial analysis and the corresponding values are presented in Table 1. These variables and their codes are as follows: percentage of precursor (X₁), pyrolysis temperature (X₂), feed pressure (X₃) and feed temperature (X₄). On the other hand, O₂ permeability (P_O2), N₂ permeability (P_N2), CH₄ permeability (P_CH4), CO₂ permeability (P_CO2), in addition to N₂/CH₄ selectivity (α_O2/N2), CO₂/CH₄ selectivity (α_CO2/CH4), O₂/N₂ selectivity (α_O2/N2) and CO₂/N₂ selectivity (α_CO2/N2) were chosen as functions (response). All of the variables of the membrane precursor material are nominal (non-categorical) and dimensional, so it has only numerical coded levels.

Table 1 General factorial experiments design matrix with variables and levels

Variable	Type	Level
		Actual	Coded
Percentage of precursor	X1	0	−1
		0.25	−0.5
		0.5	0
		0.75	0.5
		1	1
Pyrolysis temperature	X2	580	−1
		800	1
Feed pressure	X3	3	−1
		6	1
Feed temperature	X4	25	−1
		75	1

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Design-Expert (Version 7.0.0, 2005; Stat-Ease, Minneapolis, MN, USA) was employed for both regression analysis of the data and prediction of the equation coefficients. The best developed model was validated based on the significant terms (p < 0.05), the least significant lack of fit, the coefficient of variation (CV), the multiple correlation factor (R²), and the adjusted multiple correlation coefficient (adjusted R²) and many tables and diagrams will be discussed in detail in the following sections.

3. Results and discussion

3.1. Effect of pyrolysis and operating conditions on the performance of CMS membranes

Table 2 represents the experiment results, parameters and corresponding gas separation properties as variables and responses. The data provides the different gas performance selectivity and permeability tendencies for CMS membranes derived from P84 HT-PBI precursors under different conditions. It is possible to observe different effects of the variables in this table. According to the results, it is clear that selectivity increases and permeability decreases along with increasing pyrolysis temperature. From another point of view, the increment in pyrolysis temperature was accompanied by enhancements in ideal gas pair selectivity and this trend is valid for all of the CMS membranes. The increase in this temperature also creates a carbon membrane with higher density and crystallinity, and smaller average interplanar graphite layer spacing. As a general rule, the optimum pyrolysis temperature varies with the precursor type. Holistically, the optimum pyrolysis temperature is different with different precursors. However, it is concluded that as the pyrolysis temperature is increased, the selectivity diminishes. For example, it is observed that the increase of pyrolysis temperature in the carbon membrane PBI-P84 HT (50%) causes the permeability of O₂ and N₂ to reduce from 126 to 34.8 and 64.6 to 91.7, respectively. It is concluded that a higher pyrolysis temperature provides a CMS membranes with a lower permeability; the lowest belonging to the membrane carbonized at 800 °C. The selectivity for O₂/N₂ of this membrane improves and rises from 1.95 to 7.71. A similar trend of changing the temperature from 580 to 800 °C is observed with different weight percents (such 25/75 etc.). According to the table, increasing the pressure of the feed raises both the permeability and selectivity; but it doesn’t have any significant effect on the efficiency of the membrane, however, it can increase both the permeability and selectivity if it is accompanied with high pyrolysis temperature. For example, the permeability of O₂ and N₂ in the carbon membrane derived from PBI-P84 HT (50%) with a pyrolysis temperature of 580 °C and a feed temperature of 25 °C, and increasing the feed pressure from 3 to 6 bar, changes from 126 to 172, and 64.6 to 91.7 barrer respectively, while the selectivity of O₂/N₂ changes from 1.95 to 1.88. But both the permeability and selectivity increase for the conditions of a pyrolysis temperature of 800 °C (with the same weight percent), a feed temperature of 25 °C, and by increasing the feed pressure from 3 to 6 bar. Hence, at a decomposition temperature of 800 °C and a feed pressure of 6 bar, the permeability of O₂ and N₂ are raised from 61.2 to 69.8 and from 5.6 to 8 barrel, respectively, through increasing the feed temperature from 25 to 75 °C. This is due to the direct impact of pressure on the absorption of gases with different mechanisms contributing to gas passage through the membrane. Quite similar trends could be noticed for N₂ permeability. In fact, the enhancement is remarkably observed for N₂ and CH₄, while such enhancement is negligible for O₂ and CO₂. It seems that this was because of the molecular size ratio of these gases. It should be considered that applying a higher feed pressure can be directed into either reducing or stabilizing gas pair ideal selectivity for all of the gas pairs studied with CMS membranes produced at lower pyrolysis temperatures. Contrariwise, using higher pressure increases the gas pair ideal selectivity for all of the gas pairs studied with CMS membranes produced at higher pyrolysis temperatures. In total, the highest improvement in the gas pair selectivity was observed for CO₂/CH₄ and CO₂/N₂. This is probably due to higher porosity of the membranes for higher pyrolysis temperatures which consequently effected on the passage of gas at high feed pressures.

Table 2 The experimental values of gas permeability and selectivity for PBI-P84 HT

Run	Variable				Response
					Permeability (barrer)				Selectivity
	X1	X2	X3	X4	PO2	PN2	PCH4	PCO2	αN2/CH4	αCO2/CH4	αO2/N2	αCO2/N2
1	0	580	3	25	111.4	4.4	1.4	138.8	3.121	98.2	25.23	31.4
2				75	131.4	6.1	1.7	172.1	3.531	99.8	21.57	28.3
3			6	25	155.9	5.4	1.6	183.8	3.311	113.7	29.14	34.4
4				75	177.7	7.6	2.0	191.2	3.822	96.2	23.39	25.2
5		800	3	25	11.1	0.6	0.1	27.6	3.808	187.9	19.79	49.3
6				75	12.5	0.7	0.2	31.8	3.808	163.7	16.94	43.0
7			6	75	20.9	1.0	0.2	45.1	3.933	182.1	21.48	46.3
8	0.25	580	3	25	89.4	47.8	21.0	396.1	2.277	18.9	1.87	8.3
9				75	91.6	62.6	27.1	475.3	2.313	17.5	1.46	7.6
10			6	25	123.6	61.3	25.7	503.1	2.384	19.6	2.02	8.2
11				75	126.1	74.1	32.4	543.3	2.289	16.8	1.70	7.3
12		800	3	25	26.2	2.6	1.1	86.5	2.407	81.7	10.29	33.9
13				75	30.2	4.0	1.6	92.6	2.538	59.4	7.64	23.4
14			6	25	44.0	3.7	1.4	123.6	2.615	86.4	11.76	33.0
15				75	50.2	4.9	1.9	128.5	2.596	67.6	10.16	26.0
16	0.5	580	3	25	126.0	64.6	28.1	578.4	2.302	20.6	1.95	9.0
17			6	25	172.4	91.7	39.1	730.8	2.347	18.7	1.88	8.0
18		800	3	25	34.8	4.5	1.9	114.2	2.321	58.7	7.71	25.3
19			6	25	61.2	5.6	2.3	179.2	2.476	79.4	10.91	32.1
20				75	69.8	8.0	3.1	183.5	2.565	58.7	8.70	22.9
21	0.75	580	3	75	205.3	135.6	78.4	807.1	1.729	10.3	1.51	6.0
22			6	25	249.9	138.4	63.3	891.7	2.187	14.1	1.81	6.4
23		800	3	25	49.9	6.3	3.1	139.2	2.041	45.0	7.90	22.0
24			6	75	93.2	11.5	4.7	252.3	2.412	53.2	8.14	22.0
25	1	580	3	75	260.8	166.6	125.4	1036.8	1.329	8.3	1.56	6.2
26			6	25	341.2	142.8	88.9	1291.1	1.606	14.5	2.39	9.0
27		800	3	25	70.5	11.7	7.5	241.0	1.558	32.1	6.03	20.6
28			6	75	144.1	21.8	13.6	423.9	1.594	31.1	6.62	19.5

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Also from experimental data in this table, it can be comprehended that increasing the precursor to 75% causes rising of O₂/N₂ selectivity. Remarkably, it enhances selectivity up to 75% (except for N₂/CH₄). Therefore, it is necessary to obtain the optimized pyrolysis conditions, operating conditions and other CMS fabrication variables after the modeling development. This optimization and the co-efficiencies of regression are obtained by using the DX package. These results are presented as empirical data and figures that will be explained in the next parts.

3.2. Model development

Tables 3 and 4 show the variance analysis (ANOVA) for the response surface model and the least-squares fit and parameters of O₂ permeability, respectively. According to these tables, the degrees of freedom (DOF) and sum of squares (SOS) signify the sum of squared deviations from the mean for each model and the number of terms in the model, respectively. The significance of the quadratic terms is checked through using the mean square. The SOS for each model is calculated for the F-value after elimination of the average, block and linear effects. Based on the results summarized in Table 4, cubic models are found to be significant, as suggested by the small probability value and F-value test. According to these tables, the total deviation from the mean, number of added phrases to the model, and the magnitude of the F-value are studied respectively with the total squares, and degrees of freedom of the quadratic model. The magnitude of the P-value is used to contemplate the importance of each coefficiency. In cases where the magnitude of the P value is lower, the importance of those coefficiencies are more remarkable. It is clear that the quadratic model is very important and the cube model is well coordinated. By checking the model significance and applying multiple regression analysis, the following models were obtained for the O₂ permeability and the O₂/N₂ selectivity:

P_O2 = 92.35 + 60.23X₁ − 32.35X₂ + 10.94X₃ − 5.23X₄ + 11.95X₂X₄ − 13.02X₃X₄ + 53.32X₁² + 21.85X₁X₂X₄ − 12.8X₁X₃X₄X₄² − 36.27X₁²X₂ − 2.71X₁²X₃ + 11.16X₁²X₄ (3)

α_O2/N2 = 1.23 + 0.26X₁ + X₂ + 0.023X₃ − 0.091X₄ + 0.21X₁X₂ − 0.13X₁X₃ + 0.11X₁X₄ − 0.024X₂X₃ + 0.093X₂X₄ + 0.94X₁² − 0.092X₁X₃X₄ − 0.12X₂X₃X₄ − 0.59X₁²X₂ + 0.16X₁²X₃ − 0.096X₁²X₄ − 1.08X₁³ (4)

Table 3 The results of the analysis of variance (ANOVA) for the response surface model for the O₂ permeability of membranes

Model	Sum of squares	Degrees of freedom	Mean square	F-value	p-value, prob > F	Status
Mean vs. total	339 084	1	339 084
Linear vs. mean	149 440	4	37 359.9	26.27	<0.0001	Suggested
2FI vs. linear	9148	6	1525	1.1	0.4021
Quadratic vs. 2FI	10 568	1	10 568	13.01	0.0024	Aliased
Cubic vs. quadratic	9539	8	1192	2.758	0.0864	Aliased
Residual	3459	8	432.4
Total	521 237	28	18 615.6

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Table 4 The least-squares fit and parameters (for the O₂ permeability response)

Source	Sum of squares	Degrees of freedom	Mean square	F-value	p-value, prob > F	Status
Model	176 495.3	12	14 707.9	39.0	<0.0001	Significant
X1	44 415.9	1	44 415.9	117.7	<0.0001
X2	9303.7	1	9303.7	24.7	0.0002
X3	1063.2	1	1063.2	2.8	0.1139
X4	220.2	1	220.2	0.6	0.4567
X2X4	2351.5	1	2351.5	6.2	0.0247
X3X4	2793.0	1	2793.0	7.4	0.0158
X1^2	11 933.6	1	11 933.6	31.6	<0.0001
X1X2X4	5187.2	1	5187.2	13.8	0.0021
X1X3X4	1778.2	1	1778.2	4.7	0.0464
X1^2X2	5252.9	1	5252.9	13.9	0.0020
X1^2X3	29.3	1	29.3	0.1	0.7844
X1^2X4	486.2	1	486.2	1.3	0.2741
Residual	5658.2	15	377.2
Cor total	182 153.6	27

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The models indicate that X₄ has the most effect on the selectivity of O₂/N₂. Positive co-efficiencies X₁, X₂ and X₃ have linear effects on increasing the selectivity of O₂/N₂, with X₂ having the most effect, while the negative co-efficiency X₄ has a linear effect on decreasing the selectivity of O₂/N₂. X₂X₃ and X₂X₄ are not ineffectual, while X₁X₂ has an important impact on the selectivity of O₂/N₂. X₁³ and X₁² are important phrases in the model. However, action and reaction among these four variables have significant effects on the final result. Using these analytical methods and techniques (such as eqn (3) and (4)) resulted in various 2^nd and 3^rd order polynomial models for the permeability values of N₂, CH₄, and CO₂, and the selectivity of the values of N₂/CH₄, CO₂/CH₄ and CO₂/N₂ which are shown in Table 5 and their codes are mentioned in Table 1.

Table 5 The polynomial models obtained for the permeability and selectivity of various investigated gases using statistical significance checks and multiple regression analysis

PN2 = 1/(0.33 − 0.29 × X1 + 0.018 × X2 − 0.21 × X1X2 + 0.11 × X1X3 − 3.39 × 10^−3 × X2X3 − 0.062 × X2X4 + 0.36 × X1^2 + 0.093 × X1X2X3 − 0.037 × X1X2X4 − 0.048 × X1X3X4 + 0.31 × X1^2X2 − 0.091 × X1^2X4)

PCH4 = 1/(0.18 + 0.69 × X1 − 0.24 × X2 + 0.012 × X3 − 1.07 × X1X2 + 0.11 × X2X3 − 0.18 × X2X4 + 0.068 × X3X4 + 1.29 × X1^2 + 0.4 × X1X2X3 + 1.42 × X1^2X2 − 0.19 × X1^2X3 − 0.43 × X1^2X4 − 2.13 × X1^3)

PCO2 = 433.4 + 274.77 × X1 − 262.38 × X2 − 150.61 × X1X2 + 75.57 × X1X2X4

αN2/CH4 = exp(0.84 + 3.33 × 10^−3 × X1 + 0.046 × X2 + 0.031 × X3 − 0.011 × X4 − 0.046 × X1X2 − 9.8 × 10^−3 × X1X3 − 0.014 × X2X3 + 0.021 × X2X4 − 0.023 × X3X4 + 0.02 × X1X2X3 + 0.027 × X2X3X4 − 0.025 × X1^2X3 − 0.4 × X1^3)

αCO2/CH4 = 22.76 − 48.83 × X1 + 26.43 × X2 + 5.94 × X3 − 10.91 × X4 − 9.14 × X1X2 + 53.89 × X1^2 + 6.22 × X1X2X3 − 3.48 × X1X2X4 − 7.74 × X1X3X4 + 1.36 × X2X3X4 + 3.95 × X1^2X2 − 0.54 × X1^2X3 + 3.73 × X1^2X4

αCO2/N2 = 14.37 − 1.14 × X1 + 9.17 × X2 + 1.15 × X3 − 2.65 × X4 − 2.44 × X1X2 − 0.92 × X1X3 + 1.98 × X1X4 + 0.43 × X2X3 − 0.37 × X2X4 + 11.11 × X1^2 − 10.78 × X1^3

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Fig. 5 and 6 display the performance of carbon membranes compared to the trade-off line for the gas pair for CO₂/CH₄ and O₂/N₂ separation, respectively. It can be found that the separation performance of the developed CMS membranes are very distinguished particularly for O₂/N₂ with selectivities of as high as 10–20 could be obtained with these membranes. As illustrated in these figures, carbon membranes derived from P-84 HT/PBI precursors stand well above the trade-off lines for CO₂/CH₄ and O₂/N₂ separation. These CMSMs have rigid and dense structures with pore sizes that approach the gas molecule sizes. The high gas selectivity, combined with relatively high gas permeability, allow many of these carbon membranes to operate above the upper bound. Thus, membranes from the blends of P84 HT and PBI as high performance polymers made according to the recommended procedures and having appropriate compositions can be regarded as potential breakthroughs for oxygen and nitrogen purification and CO₂ removal from methane gas, which can bring substantial savings in energy and the economic impact of natural gas prices.

Fig. 5 Performance of CMS membranes prepared and operated under various conditions for CO₂/CH₄ separation with respect to a revised Robeson trade-off line.

Fig. 6 Performance of CMS membranes prepared and operated under various conditions for O₂/N₂ separation with respect to a revised Robeson trade-off line.

3.3. Estimated gas transport properties and surface area analysis

Models of gas transport properties could be utilized to estimate any response (e.g., P_CH4, P_O2, P_N2, P_CO2, α_CO2/CH4, α_N2/CH4, α_CO2/N2 and α_O2/N2) corresponding to the specific value of the variables. This evaluation is generally applicable for data located within the range of the analysis, and one should be careful about the extrapolation beyond this range. Fig. 7 shows the presence of a subtle conformity between the data acquired from the experiments and predictions obtained using the models for both the permeability of O₂ (P_O2) and the selectivity of O₂/N₂ (α_O2/N2) plotted for eqn (3) and (4). It seems that they are equal to real values especially between the limits of 11 to 341 barrer for O₂ permeability, and also under 1 up to 29 for O₂/N₂ selectivity.

Fig. 7 Estimated (a) O₂ permeability and (b) O₂/N₂ selectivity versus actual experimental data obtained under optimum conditions.

Fig. 8 and 9 feature the 3D responses manifesting detailed descriptions of the independent variables from the gas transport and separation models and provide a better arrangement of the impact of the variables. Typically, an elliptical shape of the contour plots can specify the interactions among the independent variables. These figures present the effects of X₁ and X₄ on the O₂ permeability (X₂ and X₃ are constant) and the impacts of X₃ and X₄ on the O₂/N₂ selectivity (X₁ and X₂ are constant).

Fig. 8 The combined effects of the pyrolysis temperature and feed pressure on the O₂ permeability. (a) Surface plot and (b) contour plot.

Fig. 9 The combined effects of the percentage of precursor and pyrolysis temperature on the O₂/N₂ selectivity. (a) Surface plot and (b) contour plot.

According to Fig. 8, there is no significant change in O₂ permeability through increasing the operating temperature and enhancing the percentage of precursor from 0 to 0.5. But the O₂ permeability is increased at the limit up to 0.5 barrer by increasing the P84 HT weight percent. O₂ permeability increases slightly through enhancing the feed temperature at one percent precursor although it doesn’t have any tangible effect on the permeability in general. The effect of the feed pressure is clearly observable in this figure, and the selectivity of O₂/N₂ as well as O₂ permeability are increased when it is raised. Fig. 10a and b show the selectivity of two other gas pairs (CO₂/N₂ and N₂/CH₄). The effects of feed and pyrolysis temperature are investigated in Fig. 10a, increasing the feed temperature and pyrolysis temperature causes the selectivity of N₂/CH₄ to increase. It is clear in Fig. 10 that the selectivity of CO₂/N₂ is increased through increasing the pyrolysis temperature and is reduced relatively through increasing the feed temperature. Fig. 8–10 demonstrate that the intensification of P84 HT in the blend with PBI beyond 0.5 increased the permeability of O₂. This behavior could be explicated through the chemical structure of the membrane and the gas type. This suggests that achieving such good selectivity and lower permeability can possibly be due to the contribution of rigid PBI chains with higher density (1.311 g cm⁻³) than P84 HT (1.071 g cm⁻³) that can have large effects on the chain configurations within the membrane context in addition to the pore formation process during pyrolysis. This could be due to the fact that compared to P84 HT, higher carbon elements are present in the repeating unit of PBI. This can assist in retaining the original stability of the membrane during the pyrolysis process. Therefore, increasing the percentage of the P84 HT blend within the tested range was advantageous to the permeability, while it has a negative effect on the selectivity. The results also demonstrate that at the maximum final pyrolysis temperature (800 °C), the N₂/CH₄ separation performance of the membrane is slightly improved along with the increase in the operating temperature, in contrast, the performance of membrane is improved with the decrease in the operating temperature.

Fig. 10 Response surface plot for the (a) N₂/CH₄ and (b) CO₂/N₂ selectivity of the combined effects of the percentage of precursor, pyrolysis temperature and feed pressure.

3.4. Adequacy evaluation of the model

It is indispensable to either confirm the fitted model in order to give a guarantee or provide an adequate approximation to the real system. The residuals from fitting of the least squares play a key role in judging the accuracy of the model. Through a normal probability plot of the residuals, some evaluations were made based on routine assumptions (Fig. 11). This assumption was met as the residual plot approximated along a straight line and it was satisfied. Fig. 12 shows a diagram of the remainder. In contrast, the estimated response and general results show the random mode of the remaining dispersion and represent that the variance of the main observation is positive for all of the variables. It is observed that this diagram offers an appropriate empirical model both for permeability and selectivity. It suggests that the variance of the original observation is constant for all values of the responses. Both Fig. 11 and 12 are satisfactory; it is shown that the gas transport models are suitable to determine the gas permeability and selectivity using the developed models. Also, the accuracy of all of the developed models was checked using various factors including the coefficient of variation (CV), correlation coefficient (R) and determination coefficient (R²). Here, a lower value of CV (17.65 and 34.17 for O₂ permeability and O₂/N₂ selectivity) obtained in our analysis suggested a superior reliability and accuracy of the experiments. Furthermore, R² values for each model denote sample variations of 96.89, 92.24, 95.92 and 90.54 for the O₂, N₂, CH₄ and CO₂ permeability, respectively, as well as 95.5 and 91.55 for the O₂/N₂ and CO₂/N₂ selectivity, respectively.

Fig. 11 The plots of normal probability vs. internal residuals for the (a) O₂ permeability and (b) O₂/N₂ selectivity of membranes.

Fig. 12 The plots of internal residuals vs. estimated response for the (a) O₂ permeability and (b) O₂/N₂ selectivity of membranes.

3.5. Optimization

All of the gas transport and separation models were used for optimization. For the optimization procedure, as shown in Table 6, lower and upper values of the parameters were specified with a goal being set for each model to obtain the optimal conditions. Frequently, selectivity and permeability tests were done to manifest an optimized method under favorable conditions which are shown in this table. Results from this table recommended that high-performance CMS membranes should be pyrolyzed at a final pyrolysis temperature between 728.7 and 800 °C. According to the data of this table, the optimum operating conditions for gas separation were between 4.5 to 6 bar for the feed pressure and between 25 to 30 °C for the feed temperature. The best values of selectivity for CH₄/CO₂, N₂/CH₄, O₂/N₂ and CO₂/N₂ were 155.7, 3.65, 28.64 and 49.6, respectively, as well as permeability values for CH₄, CO₂, N₂ and O₂ of 41.983, 135.5, 0.874 and 0.24 barrer, respectively. Maximum responses were achieved through CMS membranes produced from PBI/P84 HT pyrolyzed at 760 °C and tested at an operating pressure and temperature of 4.5 bar and 25 °C, respectively. For the validation of the optimization procedure, repeat gas permeability and selectivity experiments under the above optimal conditions were carried out. The permeation data in Table 7 reveal that the prediction data from our models were in good agreement with experimental data values and have approximately 2% error. This negligible error and discrepancy might be due to the slight variation in the setup of the device and experimental conditions. Consequently, the models and corresponding responses indicate tolerable coordination between estimated and experimental data.

Table 6 Minimum and maximum limits and the set goal for each response to generate optimal conditions

Name	Goal	Lower limit	Upper limit	Lower weight	Upper weight	Importance
Precursor material (X1)	Is in range	0	1	1	1	3
Blend composition (X2)	Is in range	580	800	1	1	3
Pyrolysis temperature (X3)	Is in range	3	6	1	1	3
Pyrolysis vacuum pressure (X4)	Is in range	25	75	1	1	3
PCH4	Maximize	11.08	341.20	3	1	2
PN2	Minimize	0.56	166.64	1	3	2
PO2	Minimize	0.15	125.42	1	3	3
PCO2	Maximize	27.63	1291.10	3	1	3
αN2/CH4	Maximize	1.33	3.93	3	1	4
αCO2/CH4	Maximize	8.27	187.86	5	1	5
αO2/N2	Maximize	1.46	29.14	4	1	5
αCO2/N2	Maximize	5.95	49.33	3	1	4

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Table 7 Optimum process conditions and their results for validation of the model

Number	X1	X2	X3	X4	PCH4	PN2	PO2	PCO2	αN2/CH4	αCO2/CH4	αO2/N2	αCO2/N2	Desirability
1	0	760.2	4.5	25.0	41.983	0.874	0.20	135.5	3.65	155.7	28.64	49.6	0.181
2	0	767.4	3.8	25.0	34.210	0.778	0.19	133.1	3.60	161.6	25.22	49.3	0.177
3	0	753.5	6.0	25.5	54.105	1.116	0.23	136.9	3.78	146.7	37.27	51.1	0.171
4	0	728.7	6.0	64.3	70.710	1.405	0.31	104.1	3.78	143.8	25.28	41.0	0.122
5	0.92	800.0	6.0	25.0	121.983	2.173	1.00	211.4	1.93	68.0	11.19	24.2	0.021
6	0.94	799.1	6.0	27.1	125.735	2.068	1.02	222.4	1.87	68.3	10.55	23.6	0.020
7	0.97	800.0	6.0	25.0	128.556	1.685	0.98	217.0	1.73	74.3	9.77	22.8	0.020
8	0.76	799.5	6.0	25.0	105.400	4.816	1.45	198.3	2.29	58.1	12.49	26.2	0.016
9	0.92	800.0	5.3	25.0	106.931	2.697	1.13	211.6	1.88	60.8	10.13	23.8	0.016
Repeat test	0	760.2	4.5	25.0	41.100	0.869	0.21	138.4	3.69	151.5	28.00	50.4
Error percent × 100					−2.1	−0.5	2.8	2.1	1.0	−2.7	−2.2	1.5

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4. Conclusions

Developed and optimized CMS membranes were synthesized from blends of PBI and P84 HT. The effect of various blend compositions and pyrolysis temperatures, as well as the operating pressure and temperature, on the performance of these CMS membranes for industrial gas separation applications were investigated. For finding the effects of the mentioned variables on the gas separation performance and obtaining the optimum conditions, experimental design, and modeling of the gas separation properties were performed by using D-optimal design to detect the effect of the eight cubic models for gas pair selectivity, and gas permeability responses were derived. Our findings specify that CMS membranes produced from PBI-P84 HT (higher than 90 wt% PBI) exhibited a higher gas separation performance than the other blends. It could be identified that a higher final pyrolysis temperature prepared CMS membranes with a lower permeability and higher selectivity; the CMS membrane carbonized at 760 °C for all of the gases. High N₂/CH₄, CO₂/CH₄, O₂/N₂ and CO₂/N₂ ideal selectivities of 3.65, 155.7, 28.64 and 49.6 were obtained for carbonized PBI-P84 HT at 760 °C with an operating temperature and pressure of 25 °C and 4.5 bar, respectively, whose values are the highest among all of the resultant membranes. The CMS membranes fabricated and optimized in this research possess high performance properties for gas pair separation compared to the Robeson trade-off line for O₂/N₂ and CO₂/CH₄ separation. Thus, carbon membranes developed from technical design of the fabrication and operation according to the recommended procedures having a suitable percentage of the precursor, pyrolysis temperature, feed pressure and temperature can be considered as potential breakthroughs for oxygen and carbon dioxide removal from nitrogen and natural gas, respectively, and can bring substantial savings in energy and the economic impact of natural gas prices.

Nomenclature

b0, b1, …, bn	Regression coefficients
DOF	Degrees of freedom
F-value	Ratio of variances, computed value
i and j	Subscripts (integer variables)
n	Number of factors (variables)
P-value	Statistical criterion
CV	Coefficient of variation
R	Correlation coefficient
R^2	Coefficient of multiple determinations
Radj^2	Adjusted statistic coefficient
X1, X2, X3	Coded variables
X̄	Mean value of variables
Y	Response

Acknowledgements

V. Pirouzfar would like to acknowledge the student financial aid provided by the National Iranian Gas Company (NIGC).

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