Conjugated Mass Transfer of CO2 Absorption through Concentric Circular Gas–Liquid Membrane Contactors

A new design of gas absorption that winds the permeable membrane onto an inner concentric tube to conduct a concentric circular gas–liquid membrane module has been studied theoretically in the fully developed region. An analytical formulation, referred to as conjugated Graetz problems, is developed to predict the concentration distribution and Sherwood numbers for the absorbent fluid flowing in the shell side and CO2/N2 gas mixture flowing in the tube side under various designs and operating parameters. The analytical solutions to the CO2 absorption efficiency were developed by using a two-dimensional mathematical modeling, and the resultant conjugated partial differential equations were solved analytically using the method of separation variables and eigen-function expansion in terms of power series. The predictions of CO2 absorption rate by using Monoethanolamide (MEA) solution in concentric circular membrane contactors under both concurrent- and countercurrent-flow operations are developed theoretically and confirmed with the experimental results. Consistency in both a good qualitative and quantitative sense is achieved between the theoretical predictions and experimental results. The advantage of the present mathematical treatment provides a concise expression for the chemical absorption of CO2 by MEA solution to calculate the absorption rate, absorption efficiency, and average Sherwood number. The concentration profiles with the mass-transfer Graetz number, inlet CO2 concentration, and both gas feed and absorbent flow rates are also emphasized. Both theoretical predictions and experimental results show that the device performance of the countercurrent-flow operation is better than that of the concurrent-flow device operation. The availability of such simplified expressions of the absorption rate and averaged Sherwood as developed directly from the analytical solutions is the value of the present study.


Introduction
Membrane contactor modules were applied to gas/liquid absorption process in aiming to avoid the existence of foaming, unloading and flooding in packed towers, bubble columns, and spray towers, which is performed in conventional gas absorption processes to remove CO 2 by absorbing with the gas mixtures dispersed into an aqueous amines solution. Using aqueous amines allows the simple heating process to regenerate the liquid absorbents. A substitutive gas absorption process has gained increasing attention in recent years as an alternative technology for capturing CO 2 [1]. Implementing hydrophobic microporous membranes [2,3] to overcome the disadvantage of non-dispersive contact allows the soluble gas to be absorbed on the membrane surface in the pore mouth adjacent to the solvent phase. The benefits of both membrane reactor and gas/liquid absorption processes were combined together in chemical absorption processes, which are widely utilized due to the high selectivity of amines towards CO 2 absorption. Karoor and Sirkar [3] used the hollow fiber membrane contactor to separate CO 2 /N 2 by using pure water as absorbents (physical absorption processes) or amine aqueous solutions (chemical absorption processes). The absorption efficiency of such membrane contactors is dependent on the distribution coefficient of gas solute between gas and absorbent phases. Numerous absorbent solutions [4,5] or hollow-fiber modules [6,7] used for CO 2 absorption improvement were further studied by many investigators. Polytetrofluoroethylene (PTFE) denotes the superior membrane material for absorption processes due to its extreme hydrophobicity [8] for the common amine solvents. Membrane gas absorption offers many advantages including the independent control of gas and absorbent flow rates, high contact surface area, and linear scale-up compared to traditional equipment [9].
The film theory has been described in a gas/liquid membrane contactor as a mass-transfer resistances in series model [10], in which the liquid can be contacted on the opposite side of the membrane and the gas/liquid interface is formed at each membrane pore entrance. The mass-transfer resistance in the absorbent solution was a dominant resistance when CO 2 diffused across the membrane and absorbed into the absorbent solution. Many mathematical models were developed [11,12] in order to study and evaluate the influences of absorption efficiency of CO 2 [13]. Two conjugated governing equations for solving CO 2 concentration distributions and the outlet concentrations were obtained by using the orthogonal technique and the method of separation of variables [14,15]. Theoretical investigation of two-dimensional concentration distributions in the gas/liquid concentric circular membrane extractor modules is the value of the present study. The purposes of this study are to develop the two-dimensional mathematical formulation of concentric circular gas/liquid membrane extractors, to solve the resultant partial differential equations analytically by using orthogonal expansion method, and to find the dimensionless outlet average concentrations for both gas and liquid streams. The operating conditions that affect device performance including gas and liquid flow rates and inlet CO 2 concentrations were investigated to examine the absorption rate in the concentric circular membrane contactor. The theoretical predictions of average Sherwood number are presented graphically with the mass-transfer Graetz number as the parameter. The theoretical results of absorption efficiency and absorption rate for concurrent-and countercurrent-flow patterns are compared to the experimental data to confirm the two-dimensional theoretical model in practical manners. Figure 1 shows a concentric circular gas/liquid membrane contactor that has a hydrophobic microporous permeable membrane which is wound in order to divide a circular tube into an inner tube and shell side with thicknesses of 2κR and 2(1 − κ)R, respectively. The hydrophobic microporous membrane with negligible thickness δ between gas feed and absorbent flow is passed through each channel individually. The volumetric flow rates Q a and Q b , and the inlet concentration C ai and C bi , are key for gas and liquid streams, respectively. The overall mass transfer resistance includes gas film resistance of transferring through the bulk gas phase, liquid film resistance of gas transferring into the bulk liquid phase, and the resistance diffusion through the membrane pores.

Concurrent-Flow Operations
The mathematical formulations of the velocity profiles and mass conservation equations in describing the mass transfer behavior, as shown in Figure 1a, are derived after the following assumptions are made: (a) steady state and isothermal condition; (b) fully developed flow in both inner tube and shell side; (c) the physical properties of gas and liquid are constant; (d) negligible axial diffusion and conduction, entrance length, and end effects; (e) the applicability of Henry's law; (f) the applicability of thermodynamic equilibrium; (g) reaching the equilibrium state immediately with the assumed fast chemical reaction rate; (h) neglecting the membrane thickness compared to the circular-tube radius.

Concurrent-Flow Operations
The mathematical formulations of the velocity profiles and mass conservation equations in describing the mass transfer behavior, as shown in Figure 1a, are derived after the following assumptions are made: (a) steady state and isothermal condition; (b) fully developed flow in both inner tube and shell side; (c) the physical properties of gas and liquid are constant; (d) negligible axial diffusion and conduction, entrance length, and end effects; (e) the applicability of Henry's law; (f) the applicability of thermodynamic equilibrium; (g) reaching the equilibrium state immediately with the assumed fast chemical reaction rate; (h) neglecting the membrane thickness compared to the circular-tube radius.
in which The boundary conditions accompanied with the conjugated governing Equations (3) and (4) are ψ a (η, 0) = ψ ai (6) ∂ψ a (0, ξ) ∂η = 0 (8) Processes 2021, 9, 1580 (11) in which the reduced equilibrium constant K ex = K  [17]. The present work is actually the extension of our previous work [18] by following the similar general solution form except instead of concentric circular module, but the mathematical treatment is more complicated than that in the flat-plate module. The dimensionless concentration profiles of both phases, ψ a and ψ b , in such a conjugated system were obtained analytically with the use of an orthogonal expansion technique by the eigen-function expanding in terms of an extended power series [19,20], the separation variables are expressed in the form: We may assume the eigen-functions F a,m (η a ) and F b,m (η b ) to be expressed in polynomials without loss of generality in the following forms: Equation (21) was rewritten to obtain the relationship between S a,m and S b,m as Moreover, combinations of Equations (21) and (22) with deleting S a,m and S b,m yields the following equations to calculate the eigenvalues (λ 1 , λ 2 , . . . , λ m , . . . ) All the coefficients d m,n and e m,n may be expressed in terms of eigenvalues λ m after using Equations (19) and (20) by substituting Equations (14) and (15) into Equations (17) and (18), and can be expressed in terms of eigenvalue λ m as where T = 1−κ 2 ln (1/κ) and S = 1−κ 4 1−κ 2 − 1−κ 2 ln (1/ κ) . These eigenvalues λ m were calculated in Equation (24), requiring a negative set for both concurrent and countercurrent-flow operations. Table 1 shows that calculation results of the first five eigenvalues and their associated expansion coefficients are selected to meet the convergence requirement, and the dimensionless outlet concentration profiles of the gas feed with the terms n = 500 employed during the calculation procedure are acceptable due to the negligible truncation error of Q a = 5.0 × 10 −6 m 3 /s and Q b = 6.67 × 10 −6 m 3 /s under both concurrent-and countercurrent-flow operations as an illustration. The orthogonality condition used in the concentric circular membrane contactor system of the case λ m = λ n is verified, to solve for coefficients S a,m and S b,m as follows: The dimensionless inlet and outlet concentrations can be obtained in the form of an infinite series by incorporating ξ = 0 and ξ = 1 into Equations (12) and (13) as Processes 2021, 9, 1580 6 of 16 By following the similar derivation performed in the previous study [14], the expansion coefficient of S a,n and S b,n with the aid of Equation (23) are thus obtained using boundary conditions of Equations (28)-(31) as follows: Gz a κ 2 2 [δF a,n (κ)+εRF a,n (κ)][δF a,q (κ)+εRF a,q (κ)] Thus, the dimensionless averaged concentrations in axial direction were obtained from Equations (12) and (13) with the aid of Equation (23) integrating in gas and absorbent streams, respectively

Countercurrent-Flow Operations
The modeling equations of mass transfer for both gas and absorbent streams may also be obtained in the similar forms as referred to Equations (1) and (2), as shown in Figure 1b The results of all coefficients may be obtained as follow: Similarly, the dimensionless radially averaged concentrations of both absorbent and gas streams are

Absorption Efficiency
The CO 2 absorption rate was calculated as follows: The absorption efficiency I M was illustrated by calculating the percentage of CO 2 absorbed in the concentric circular membrane contactor module as Furthermore, the local mass-transfer coefficient in the absorbent solution plays an important role to dominate absorption efficiency and could be obtained by The local Sherwood number is defined by where D b and D eq,b are the diffusivity of CO 2 and the equivalent diameter in the liquid stream, and the averaged Sherwood number is thus obtained by integrating the local Sherwood number within the conduit length

Experimental Apparatus
The experimental setup of the membrane absorption of CO 2 by using MEA absorbent flowing into the concentric circular gas-liquid membrane contactor is illustrated by Figure 2. A hydrophobic polytetrafluoroethylene (PTFE) composite membrane of a nominal pore size of 0.2 µm, a porosity of 0.72 and a thickness of 130 µm with supported polypropylene (PP) net is used in the experiments for its superior chemical resistance and thermal stability. The hydrophobic PTFE/PP composite membrane (manufactured by ADVANTEC, Japan, J020A330R) is inserted for most chemically aggressive solvents, strong acids and bases, and thermostable up to 100 • C with an operating temperature range of −35 • C~130 • C. Figure 2 illustrates the schematic configurations of the concentric circular gas/liquid membrane contactor module, in which the MEA solution passes through the shell side and the gas feed flows into the tube side. A photo of the operating circular-tube experimental apparatus is shown in Figure 3. The photo of a more detailed configuration of the concentric-tube membrane contactor module is represented by     The inner concentric ring tube was made of stainless-steel wire matrix (No. with 1.5 cm inside diameter shown in Figure 4a, which was formed with perforated of 1.26 mm × 1.26 mm square to allow gas to diffuse through. A hydrophobic memb PTFE/PP was wound around the circumference of the surface of the circular wire m ring tube, and routed with a 0.2 mm nylon fiber on the membrane surface on the ou of the inner tube shown in Figure 4c. The empty lumen channel in concentric cir membrane contactor tube is constructed by using an effective 0.2 m long concentric t lar acrylic tube of outer diameter 3.0 cm shown in Figure 4d. A gas mixture conta CO2/N2 from the gas mixing tank, and the MEA solution from a reservoir in therm (G-50, 60L, 3500W, DENG YNG, New Taipei, Taiwan) were selected as the feed gas the absorbent solution, respectively. The various operation conditions of flow rate fo uid absorption with various flow rates (5.0, 6.67, 8.33, and 10.0 cm 3 /s) into the lumen c nel were regulated by a flow meter (MB15GH-4-1, Fong-Jei, New Taipei, Taiwan), w keeping a gas stream containing CO2/N2 from the gas mixing tank (EW-06065-02, by using the mass flow controller (S/N:12031501PC-540, Brooks Instrument, Protec, Hatfield, PA, USA). The 30 wt% MEA solution was prepared to conduct experimental runs for various inlet CO2 concentrations (30, 35 and 40%). The outlet CO2 concentrations flowing out from the inner tube were then monitored and measured with gas chromatography (Model HY 3000 Chromatograp, China Corporation). The specifications and parameters of the experimental runs are summarized in Table 2.  The experimental results deviate from the theoretical predictions calculating by the following definition of the accuracy deviation [21] Table 3. The accuracy deviation of experimental results from theoretical predictions is within expectation and goes reasonably well, at  The inner concentric ring tube was made of stainless-steel wire matrix (No. 3016) with 1.5 cm inside diameter shown in Figure 4a, which was formed with perforated holes of 1.26 mm × 1.26 mm square to allow gas to diffuse through. A hydrophobic membrane PTFE/PP was wound around the circumference of the surface of the circular wire matrix ring tube, and routed with a 0.2 mm nylon fiber on the membrane surface on the outside of the inner tube shown in Figure 4c. The empty lumen channel in concentric circular membrane contactor tube is constructed by using an effective 0.2 m long concentric tubular acrylic tube of outer diameter 3.0 cm shown in Figure 4d. A gas mixture containing CO 2 /N 2 from the gas mixing tank, and the MEA solution from a reservoir in thermostat (G-50, 60L, 3500W, DENG YNG, New Taipei, Taiwan) were selected as the feed gas and the absorbent solution, respectively. The various operation conditions of flow rate for liquid absorption with various flow rates (5.0, 6.67, 8.33, and 10.0 cm 3 /s) into the lumen channel were regulated by a flow meter (MB15GH-4-1, Fong-Jei, New Taipei, Taiwan), while keeping a gas stream containing CO 2 /N 2 from the gas mixing tank (EW-06065-02, Cole Parmer company, Chicago, IL, USA) with a fixed flow rate of 5.0 cm 3 /s into the inner tube by using the mass flow controller (S/N:12031501PC-540, Brooks Instrument, Protec, Hatfield, PA, USA). The 30 wt% MEA solution was prepared to conduct experimental runs for various inlet CO 2 concentrations (30, 35 and 40%). The outlet CO 2 concentrations flowing out from the inner tube were then monitored and measured with gas chromatography (Model HY 3000 Chromatograp, China Corporation). The specifications and parameters of the experimental runs are summarized in Table 2. The experimental results deviate from the theoretical predictions calculating by the following definition of the accuracy deviation [21] where N exp , ω theo,i and ω exp,i are the number of experimental runs, theoretical predictions and experimental results of absorption rates, respectively. The accuracy deviations of both concurrent-and countercurrent-flow operations are shown in Table 3. The accuracy deviation of experimental results from theoretical predictions is within expectation and goes reasonably well, at 6.63 × 10 −2 ≤ E ≤ 7.74 × 10 −2 .

Results and Discussion
The procedure for calculating the theoretical values of the dimensionless outlet average concentration, absorption rate and absorption efficiency are described as follows. First, the eigenvalues in the membrane contactor are solved from Equation (24), the associated eigen-functions obtained from Equations (25) Figure 5a,b shows the dimensionless averaged CO 2 concentration profiles along the axial direction in both gas and liquid phases for various inlet CO 2 concentrations under both concurrent-and countercurrent-flow operations. Notice that the axial dimensionless averaged concentration distribution in the absorbent solution, ψ b (ξ), increases as the inlet CO 2 concentration decreases from 45% to 30%. The results show that a higher driving-force concentration gradient is kept between two phases under a larger inlet CO 2 concentration, which turns out to have a higher absorption rate, leading to a smaller dimensionless averaged outlet CO 2 concentration for both flow patterns. Figure 5a,b shows the dimensionless averaged CO2 concentration profiles along the axial direction in both gas and liquid phases for various inlet CO2 concentrations under both concurrent-and countercurrent-flow operations. Notice that the axial dimensionless averaged concentration distribution in the absorbent solution, , increases as the inlet CO2 concentration decreases from 45% to 30%. The results show that a higher drivingforce concentration gradient is kept between two phases under a larger inlet CO2 concentration, which turns out to have a higher absorption rate, leading to a smaller dimensionless averaged outlet CO2 concentration for both flow patterns.   Comparisons were made of both theoretical predictions and experimental results of dimensionless outlet CO 2 concentrations for both concurrent-and countercurrent-flow operations, demonstrated in Figure 6. It can be observed from Figure 6 that the dimensionless outlet CO 2 concentration increases with the magnitude of inlet CO 2 concentration. The theoretical predictions of dimensionless outlet CO 2 concentrations are consistent with experimental results. The comparison reveals that the higher the absorbent flow rate, the lower the dimensionless outlet CO 2 concentration found in both flow patterns, which is also as expected. Comparisons were made of both theoretical predictions and experimental results of dimensionless outlet CO2 concentrations for both concurrent-and countercurrent-flow operations, demonstrated in Figure 6. It can be observed from Figure 6 that the dimensionless outlet CO2 concentration increases with the magnitude of inlet CO2 concentration. The theoretical predictions of dimensionless outlet CO2 concentrations are consistent with experimental results. The comparison reveals that the higher the absorbent flow rate, the lower the dimensionless outlet CO2 concentration found in both flow patterns, which is also as expected. Exp. Theo.   Figure 7, where the averaged mass-transfer Sherwood number Sh in countercurrentflow operations is higher than that in concurrent-flow operations. This result confirms The local mass-transfer Graetz number symbolizes the ratio of mass transfer coefficient in radial direction versus diffusion in axial direction, say Sh ξ = k bξ D eq,b /D b . The averaged mass-transfer Sherwood number Sh plays a significant role in determining the CO 2 absorp-tion rate in considering the mass transfer behavior. The theoretical prediction Sh which varies with the mass-transfer Graetz number of the absorbent solution (absorbent flow rates Q b ) for both concurrent-and countercurrent-flow operations is presented in Figure 7, where the averaged mass-transfer Sherwood number Sh in countercurrent-flow operations is higher than that in concurrent-flow operations. This result confirms that the higher mass transfer coefficient is obtained in countercurrent-flow operations that come up with a higher absorption rate due to lower outlet CO 2 concentrations, as demonstrated in Figure 6. It's reasonable to conclude that the mass transfer rate represented by Sh increases in a linear relationship the smaller the mass-transfer Graetz number of the absorbent solution. Theoretical predictions and experimental results of CO2 absorption rate that vary with absorbent flow rate and inlet CO2 concentration in both concurrent-and countercur rent-flow operations, as indicated in Figure 8 for comparison. The results show that th CO2 absorption rate increases with the absorbent flow rates. Meanwhile, the CO2 absorp tion rate is higher in countercurrent-flow operations than that in concurrent-flow opera tions, as shown in Figure 8. The higher absorbent flow rate and inlet CO2 concentration obtain a higher CO2 absorption rate as predicted above accordingly.   Theoretical predictions and experimental results of CO 2 absorption rate that vary with absorbent flow rate and inlet CO 2 concentration in both concurrent-and countercurrentflow operations, as indicated in Figure 8 for comparison. The results show that the CO 2 absorption rate increases with the absorbent flow rates. Meanwhile, the CO 2 absorption rate is higher in countercurrent-flow operations than that in concurrent-flow operations, as shown in Figure 8. The higher absorbent flow rate and inlet CO 2 concentration obtain a higher CO 2 absorption rate as predicted above accordingly. Figure 9 illustrates the deviation between the theoretical predictions and experimental results of CO 2 absorption efficiency that vary with absorbent flow rate and inlet CO 2 concentrations for both flow patterns. Notice that the higher inlet CO 2 concentration results in the higher CO 2 absorption rate but the lower absorption efficiency, as shown in Figure 8 and Figure 9, respectively. In addition, the CO 2 absorption efficiency is found to be increased with increasing absorbent flow rate. One may find the experimental result is most consistent with the theoretical prediction at the lower absorbent flow rate. The discrepancy between theoretical predictions and experimental result is increasing as the absorbent flow rate increases.
Theoretical predictions and experimental results of CO2 absorption rate that vary with absorbent flow rate and inlet CO2 concentration in both concurrent-and countercurrent-flow operations, as indicated in Figure 8 for comparison. The results show that the CO2 absorption rate increases with the absorbent flow rates. Meanwhile, the CO2 absorption rate is higher in countercurrent-flow operations than that in concurrent-flow operations, as shown in Figure 8. The higher absorbent flow rate and inlet CO2 concentration obtain a higher CO2 absorption rate as predicted above accordingly.  Figure 9 illustrates the deviation between the theoretical predictions and experimental results of CO2 absorption efficiency that vary with absorbent flow rate and inlet CO2 concentrations for both flow patterns. Notice that the higher inlet CO2 concentration results in the higher CO2 absorption rate but the lower absorption efficiency, as shown in Figures 8 and 9, respectively. In addition, the CO2 absorption efficiency is found to be increased with increasing absorbent flow rate. One may find the experimental result is most consistent with the theoretical prediction at the lower absorbent flow rate. The discrepancy between theoretical predictions and experimental result is increasing as the absorbent flow rate increases.  The effect of the operating parameters investigated on the absorption efficiency includes gas and absorbent flow rates, inlet CO2 concentration, and flow patterns. Figure 10 illustrates the CO2 absorption efficiency M I varies with mass-transfer Graetz number b Gz with inlet CO2 concentrations and gas feed rates as parameters under both concurrent-and countercurrent-flow operations. The lower gas flow rate results in the higher absorption efficiency due to having a longer resident time in contact with the membrane surface. The CO2 absorption efficiency increases with decreasing the inlet CO2 concentration and mass-transfer Graetz number a Gz due to balancing the chemical reaction equilibrium and saturated CO2 concentration. The theoretical results indicate that the CO2 absorption efficiency in countercurrent-flow operation is higher than that in concurrent-flow The effect of the operating parameters investigated on the absorption efficiency includes gas and absorbent flow rates, inlet CO 2 concentration, and flow patterns. Figure 10 illustrates the CO 2 absorption efficiency I M varies with mass-transfer Graetz number Gz b with inlet CO 2 concentrations and gas feed rates as parameters under both concurrent-and countercurrent-flow operations. The lower gas flow rate results in the higher absorption efficiency due to having a longer resident time in contact with the membrane surface.
The CO 2 absorption efficiency increases with decreasing the inlet CO 2 concentration and mass-transfer Graetz number Gz a due to balancing the chemical reaction equilibrium and saturated CO 2 concentration. The theoretical results indicate that the CO 2 absorption efficiency in countercurrent-flow operation is higher than that in concurrent-flow operations, as illustrated in Figures 7-10, because the driving-force of the CO 2 concentration gradient between both gas and absorbent phases is greater in countercurrent-flow operations than that in concurrent-flow operations, as confirmed in Figure 5.

Conclusions
The mathematical formulations of CO2 absorption through a laminar concentric circular gas-liquid membrane contactor under both concurrent-and countercurrent-flow operations have been studied and examined theoretically and experimentally. The results of conjugated equations are solved analytically by using the orthogonal technique to expand eigen-function in terms of a power series. The dimensionless outlet concentration profiles in both inner tube and shell side, absorption rate, absorption efficiency, and averaged Sherwood number are calculated and represented graphically for comparisons in this study. The theoretical predictions of the CO2 absorption rate and absorption efficiency are confirmed and corroborated quantitatively by the experimental results, which implies a satisfactory consistency in matching theory with experiment. A good approximation is achieved by selecting only first five eigenvalues in the eigen-function expansion procedure. The device performance primarily examined by the CO2 absorption rate in the concentric circular membrane contactor is examined with various absorbent flow rates, gas feed rates, inlet CO2 concentrations, and flow patterns which are treated as key parameters. The value of the mathematical modeling in the present study is to calculate the absorption rate and absorption efficiency as well as the averaged Sherwood number directly from the analytical solutions. The theoretical predictions show that the CO2 absorption efficiency increases with the liquid absorbent flow rate but decreases with the gas feed flow rate in the concentric circular gas-liquid membrane contactor.
It is apparent that the mathematical treatments developed in this study with concentric circular membrane contactors are only conducted in a chemical absorption sense with MEA absorbent solution. This comprehensive theory with orthogonal expansion techniques could also be applied to other conjugated Graetz problems with various separation technologies.

Conclusions
The mathematical formulations of CO 2 absorption through a laminar concentric circular gas-liquid membrane contactor under both concurrent-and countercurrent-flow operations have been studied and examined theoretically and experimentally. The results of conjugated equations are solved analytically by using the orthogonal technique to expand eigen-function in terms of a power series. The dimensionless outlet concentration profiles in both inner tube and shell side, absorption rate, absorption efficiency, and averaged Sherwood number are calculated and represented graphically for comparisons in this study. The theoretical predictions of the CO 2 absorption rate and absorption efficiency are confirmed and corroborated quantitatively by the experimental results, which implies a satisfactory consistency in matching theory with experiment. A good approximation is achieved by selecting only first five eigenvalues in the eigen-function expansion procedure. The device performance primarily examined by the CO 2 absorption rate in the concentric circular membrane contactor is examined with various absorbent flow rates, gas feed rates, inlet CO 2 concentrations, and flow patterns which are treated as key parameters. The value of the mathematical modeling in the present study is to calculate the absorption rate and absorption efficiency as well as the averaged Sherwood number directly from the analytical solutions. The theoretical predictions show that the CO 2 absorption efficiency increases with the liquid absorbent flow rate but decreases with the gas feed flow rate in the concentric circular gas-liquid membrane contactor.
It is apparent that the mathematical treatments developed in this study with concentric circular membrane contactors are only conducted in a chemical absorption sense with MEA absorbent solution. This comprehensive theory with orthogonal expansion techniques could also be applied to other conjugated Graetz problems with various separation technologies. Funding: The authors wish to thank the Ministry of Science and Technology (MOST) of the Republic of China (Taiwan) for its financial support.

Acknowledgments:
The administrative and technical supports provided by Tamkang University are greatly acknowledged.

Conflicts of Interest:
The authors declare no conflict of interest.