A Simplified Kinetic Modeling of CO2 Absorption into Water and Monoethanolamine Solution in Hollow-Fiber Membrane Contactors

The absorption of CO2 from CO2-N2 gas mixtures using water and monoethanolamine (MEA) solution in polypropylene (PP) hollow-fiber membrane contactors was experimentally and theoretically examined. Gas was flowed through the lumen of the module, whereas the absorbent liquid was passed counter-currently across the shell. Experiments were carried out under various gas- and liquid-phase velocities as well as MEA concentrations. The effect of pressure difference between the gas and liquid phases on the flux of CO2 absorption in the range of 15–85 kPa was also investigated. A simplified mass balance model that considers non-wetting mode as well as adopts the overall mass-transfer coefficient evaluated from absorption experiments was proposed to follow the present physical and chemical absorption processes. This simplified model allowed us to predict the effective length of the fiber for CO2 absorption, which is crucial in selecting and designing membrane contactors for this purpose. Finally, the significance of membrane wetting could be highlighted by this model while using high concentrations of MEA in the chemical absorption process.


Introduction
Carbon dioxide has been proven to be the largest contribution of greenhouse gases, which results in the increase in the earth's surface temperature. It is also reported that half of CO 2 emissions are produced by power plants using fossil fuels [1][2][3]. Hence, the development of an efficient separation process is highly desired to remove CO 2 from the places where CO 2 is generated. In general, the bubble column, packed tower, venturi scrubber, and sieve tray column can be used for this purpose. The commercial process that is widely used for CO 2 separation is the packed column, but new technology is still required because the packed towers, for example, have many advantages such as channeling, flooding, and large-scale equipment. The gas absorption process using membrane units is considered as an alternative to recover CO 2 from waste gas streams. The hollow-fiber membrane contactor (HFMC) offers a much larger contact area per unit volume in comparison with packed and tray columns and has the advantages of no entrainment, no foaming, and no restrictions on operating flow rates [2][3][4][5][6][7][8][9].
Various kinetic models have been developed in the literature to follow the absorption of CO 2 in HFMCs by absorbent liquids [8,9,15,16,[18][19][20][21][22][23][24][25][26][27]. Most of them have used the complicated numerical methods to solve a set of differential mass balance equations to predict the change of CO 2 concentration along the hollow fibers. For example, Kim and Yang [18] have adopted a set of governing equations with common assumptions that the velocity in the lumen side can be described as fully developed parabolic profile and that the velocity through the shell side can be characterized by the free surface model of Happel [26]. Yeon et al. [15] have also employed differential mass balance equations to describe diffusion and forced convection in a medium that flows laminarly in the lumen side. They have assumed an irreversible reaction of CO 2 taking place with MEA and determined the mass transfer rates of CO 2 in PVDF and PTFE hollow fibers. Generally speaking, the mathematical complexity of numerical methods used for solving a set of differential mass balance equations makes the kinetic model somewhat practically inconvenient.
The absorption of CO 2 from a synthetic 15 vol% CO 2 -N 2 gas mixture by using water and MEA solutions was investigated in HFMCs. The effects of liquid-and gas-phase velocities as well as the pressure difference between liquid and gas phases on the flux of CO 2 absorption were explored. A simplified model involving mass balance equations-only was proposed to follow the absorption process of CO 2 and to estimate the effective length of the fiber, in which the non-wetting mode was assumed and the overall mass-transfer coefficient based on physical and chemical absorption was considered.

Determination of Overall Mass-Transfer Coefficient
The mass transfer between gas and liquid phases through HFMC occurs in three parts: stagnant gas film, the membrane itself, and stagnant liquid film [15]. The overall rate of CO 2 absorption, R A (mol s −1 ), is expressed by Equation (1): where K L is the overall mass-transfer coefficient based on liquid phase (m s −1 ); A T is the effective contact area (m 2 ); Q L is the volumetric flow rate of liquid phase (m 3 s −1 ); and C g,out and C g,in are the gas-phase concentrations of CO 2 in the outlet and inlet of the module, respectively (mol m −3 ). Additionally, ∆C lm is the logarithmic mean concentration difference, which is expressed by the following equation: Here, the value of H for CO 2 in water is adopted to be 2.916 kPa m 3 mol −1 [28], whereas it is taken to be 3.564 kPa m 3 mol −1 in 0.005-0.01 M of MEA solution and becomes 3.518 kPa m 3 mol −1 in 1.0 M of MEA solution [29].

Model Development
A simplified mathematical model was developed based on the mass balance concept, combining process conditions, membrane properties, and module geometric characteristics. The following assumptions are made: (a) absorption of single component (CO 2 ) from a CO 2 -N 2 gas mixture flowing through the lumen side of the module into an aqueous solution flowing in the shell side; (b) steady state and isothermal operation; (c) Newtonian and (e) the applicability of Henry's law [8,9,18,19,21,22,[25][26][27].
As shown in Figure 1, the mass balance of CO 2 within the membrane contactor using water as absorbent liquid is described as follows: where J A is the flux of CO 2 absorption through the hollow fibers (mol m −2 s −1 ), which is calculated by the following equation: where C L is the concentration of CO 2 in liquid phase (mol m −3 ). A simplified mathematical model was developed based on the mass balance concept, combining process conditions, membrane properties, and module geometric characteristics. The following assumptions are made: (a) absorption of single component (CO2) from a CO2-N2 gas mixture flowing through the lumen side of the module into an aqueous solution flowing in the shell side; (b) steady state and isothermal operation; (c) Newtonian fluids with constant physical properties; (d) hydrophobic membrane with non-wetting; and (e) the applicability of Henry's law [8,9,18,19,21,22,[25][26][27].
As shown in Figure 1, the mass balance of CO2 within the membrane contactor using water as absorbent liquid is described as follows: where JA is the flux of CO2 absorption through the hollow fibers (mol m −2 s −1 ), which is calculated by the following equation: where CL is the concentration of CO2 in liquid phase (mol m −3 ). Since JA depends on the concentration of CO2, it must be calculated every time according to Equations (3) and (4) using the MATLAB program (MathWorks, Natick, MA, USA). The number of grids (Nz) in the computational domain of 500 (in the axial direction) was used. Therefore, the value of CL,out can be calculated by Equation (5): On the other hand, the mass balance within the membrane contactor using MEA solution as absorbent liquid is described as follows: where rA is the reaction rate between MEA and CO2 (mol m −3 s −1 ). In general, carbamate is formed when CO2 gas reacts with primary and secondary alkanolamines [13,30,31]: where R1 is an alkyl group, and R2 is H for primary amines and an alkyl group for secondary amines. The zwitterion mechanism has been commonly used in aqueous alkanolamine solutions [32,33]. The reaction steps successively involve the formation of a zwitterion, and the subsequent removal of the proton by a base B (base catalysis), Since J A depends on the concentration of CO 2 , it must be calculated every time according to Equations (3) and (4) using the MATLAB program (MathWorks, Natick, MA, USA). The number of grids (N z ) in the computational domain of 500 (in the axial direction) was used. Therefore, the value of C L,out can be calculated by Equation (5): On the other hand, the mass balance within the membrane contactor using MEA solution as absorbent liquid is described as follows: where r A is the reaction rate between MEA and CO 2 (mol m −3 s −1 ). In general, carbamate is formed when CO 2 gas reacts with primary and secondary alkanolamines [13,30,31]: where R 1 is an alkyl group, and R 2 is H for primary amines and an alkyl group for secondary amines. The zwitterion mechanism has been commonly used in aqueous alkanolamine solutions [32,33]. The reaction steps successively involve the formation of a zwitterion, and the subsequent removal of the proton by a base B (base catalysis), where B could be an amine, OH − , or H 2 O, although the contribution of OH − can be neglected because its concentration is very low, compared to that of the amine and H 2 O [14].
According to this zwitterion mechanism, the forward reaction rate equation for CO 2 at quasi-steady state, r A , has been derived by Danckwerts [34] as follows: So, in the MEA-CO 2 system we have where C w is equal to 55.5 M. The kinetic parameters of k 2,MEA , (k w /k −1 ), and (k MEA /k −1 ) adopted here at 25 • C are 6.358 m 3 mol −1 s −1 , 1.507 × 10 −6 m 3 mol −1 , and 2.485 × 10 −4 m 3 mol −1 , respectively [13,25]. In this case, the value of C L,out can be calculated by Equation (12):

Materials
MEA was purchased from Aldrich Chemicals Co. A Liqui-Cel microporous hollowfiber extra-flow 2.5 × 8 module, which was supplied from3M TM Separation and Purification Division. (Charlotte, NC, USA), was used as membrane contactor for CO 2 absorption in this work. The hollow fibers in this module were X-50 type and made of polypropylene (PP). The characteristics of the membrane contactor are listed in Table 1. Deionized water (Millipore, Milli-Q, Burlington, MA, USA) was used. All chemicals were used without any further purification.

Absorption Experiments
The experimental setup for CO 2 removal is schematically illustrated in Figure 2. The gas containing 15 vol% of CO 2 (balance N 2 ) was passed upstream in the lumen side of the module, and the absorbent liquid was supplied downstream in the shell side. The absorbent liquids used in this study were deionized water and the aqueous solution containing 0.005-1.0 M MEA, which had a volume of 1 L otherwise stated elsewhere. The gas-phase velocity u g was varied in the range of 0.041-0.124 m s −1 , and the liquid-phase velocity u L changed in the range of 0.008-0.02 m s −1 . The pressure difference of the liquid and gas phases was changed in the range of 15-85 kPa by a needle valve to form the stable gas-liquid interface within the module. The gases coming from the absorption were sampled and analyzed by TCD-GC (Shimadzu, GC-14B, Kyoto, Japan) at pre-set time intervals. After each run, the deionized water (2 dm 3 ) was poured into both sides of the membrane contactor to remove the absorbent liquids. Then, ethanol (0.5 dm 3 ) was flowed through both sides of the module to remove water in the pores of the membrane. Finally, N 2 gas went through both sides of the module for 30 min.
Membranes 2023, 13, x FOR PEER REVIEW 5 of 16 velocity ug was varied in the range of 0.041-0.124 m s −1 , and the liquid-phase velocity uL changed in the range of 0.008-0.02 m s −1 . The pressure difference of the liquid and gas phases was changed in the range of 15-85 kPa by a needle valve to form the stable gasliquid interface within the module. The gases coming from the absorption were sampled and analyzed by TCD-GC (Shimadzu, GC-14B, Kyoto, Japan) at pre-set time intervals. After each run, the deionized water (2 dm 3 ) was poured into both sides of the membrane contactor to remove the absorbent liquids. Then, ethanol (0.5 dm 3 ) was flowed through both sides of the module to remove water in the pores of the membrane. Finally, N2 gas went through both sides of the module for 30 min.

Effect of Fluid Velocity on KL without Absorbent Recycling
The measured changes of CO2 concentrations in both phases between the inlet and outlet of the module were used for the calculation of the rates of CO2 absorption RA and the overall mass-transfer coefficients KL by Equation (1). Figures 3 and 4 show the influences of fluid velocities on the KL values in PP hollow fibers using water and 0.005 M of MEA solution as absorbent liquids. As expected, KL increases initially with increasing gasphase velocity ug but then decreases ( Figure 3). The decreased KL is likely because the retention time of gas reduces when ug is increased. It is noted that the flux of CO2 absorption, JA, at higher ug is still larger than that at low ug. On the other hand, the KL value always increases with increasing liquid-phase velocity uL using both water and MEA solutions as absorbent liquids as shown in Figure 4. Furthermore, the influence of gas-phase flow rate in chemical absorption (i.e., MEA) is more significant than that in physical absorption (i.e., water).

Effect of Fluid Velocity on K L without Absorbent Recycling
The measured changes of CO 2 concentrations in both phases between the inlet and outlet of the module were used for the calculation of the rates of CO 2 absorption R A and the overall mass-transfer coefficients K L by Equation (1). Figures 3 and 4 show the influences of fluid velocities on the K L values in PP hollow fibers using water and 0.005 M of MEA solution as absorbent liquids. As expected, K L increases initially with increasing gas-phase velocity u g but then decreases ( Figure 3). The decreased K L is likely because the retention time of gas reduces when u g is increased. It is noted that the flux of CO 2 absorption, J A , at higher u g is still larger than that at low u g . On the other hand, the K L value always increases with increasing liquid-phase velocity u L using both water and MEA solutions as absorbent liquids as shown in Figure 4. Furthermore, the influence of gas-phase flow rate in chemical absorption (i.e., MEA) is more significant than that in physical absorption (i.e., water).
Yeon et al. [15] have ever investigated the absorption of CO 2 in PVDF and PTFE hollow fiber modules using single MEA solution. They also found that the flux of CO 2 increases with an increase in liquid-phase velocity, and that initially increases with an increase in gas-phase velocity. However, it has been reported according to theoretical analysis that the flux of CO 2 absorption by MDEA solution is virtually unaffected by liquid-phase velocity [16]. This is likely due to the variations of membrane configuration and the range of gas-phase velocity. Under the present conditions studied, the difference of K L scales by 8 times between the systems using water and MEA solution is due to the characteristics of physical and chemical absorption.  Yeon et al. [15] have ever investigated the absorption of CO2 in PVDF and PTFE hollow fiber modules using single MEA solution. They also found that the flux of CO2 increases with an increase in liquid-phase velocity, and that initially increases with an increase in gas-phase velocity. However, it has been reported according to theoretical analysis that the flux of CO2 absorption by MDEA solution is virtually unaffected by liquidphase velocity [16]. This is likely due to the variations of membrane configuration and the range of gas-phase velocity. Under the present conditions studied, the difference of KL scales by 8 times between the systems using water and MEA solution is due to the characteristics of physical and chemical absorption.
It is expected that the experimental KL values evaluated by Equation (1) should vary with MEA concentration in chemical absorption processes. This fact makes the proposed kinetic model, Equation (12), rather complicated and practically unpromising. For sim-  Yeon et al. [15] have ever investigated the absorption of CO2 in PVDF and PTFE hollow fiber modules using single MEA solution. They also found that the flux of CO2 increases with an increase in liquid-phase velocity, and that initially increases with an increase in gas-phase velocity. However, it has been reported according to theoretical analysis that the flux of CO2 absorption by MDEA solution is virtually unaffected by liquidphase velocity [16]. This is likely due to the variations of membrane configuration and the range of gas-phase velocity. Under the present conditions studied, the difference of KL scales by 8 times between the systems using water and MEA solution is due to the characteristics of physical and chemical absorption.
It is expected that the experimental KL values evaluated by Equation (1) should vary with MEA concentration in chemical absorption processes. This fact makes the proposed kinetic model, Equation (12), rather complicated and practically unpromising. For sim- It is expected that the experimental K L values evaluated by Equation (1) should vary with MEA concentration in chemical absorption processes. This fact makes the proposed kinetic model, Equation (12), rather complicated and practically unpromising. For simplicity, accordingly, the value of K L evaluated from the physical absorption process is applied throughout this work for model predictions, regardless of physical or chemical absorption processes.

Effect of Operation Parameters on CO 2 Removal with Absorbent Recycling
The time changes of the concentrations of CO 2 in the outlet of the module under various gas-and liquid-phase velocities using different absorbent liquids are shown in Figures 5 and 6. At a specific time (that is, a specific position along the axial direction in HFMCs), the steady state is assumed to be calculated J A from Equations (3) and (4) in the case of using water as well as Equations (6) and (11) in the case of using MEA solution using the MATLAB program. Then, we can calculate the exit concentrations of CO 2 in the water and MEA solution, C L,out by Equations (5) and (12), respectively. The exit concentration of CO 2 in the gas can obtained using Henry's law, P g,out = C L,out × H, and ideal gas law. At the next time, similar procedures are repeated. Consequently, we can obtain the modeled results of C g,out at different times.  It is found that CO2 can be removed by pure water only within 4 min, and the time elapsed when Cg,out/C0 reaches unity becomes shorter with increasing ug. It is expected that the absorption of CO2 by recycling absorbent liquids is faster at a higher ug. This phenomenon is more obvious in the case of aqueous MEA solution (Figure 5b,c). The time required when Cg,out = C0 for MEA solution is much longer than that for water. This is because the CO2 loading of the MEA solution, particularly at 1 M of MEA, is larger than that of water. It is found that CO 2 can be removed by pure water only within 4 min, and the time elapsed when C g,out /C 0 reaches unity becomes shorter with increasing u g . It is expected that the absorption of CO 2 by recycling absorbent liquids is faster at a higher u g . This phenomenon is more obvious in the case of aqueous MEA solution (Figure 5b,c). The time required when C g,out = C 0 for MEA solution is much longer than that for water. This is because the CO 2 loading of the MEA solution, particularly at 1 M of MEA, is larger than that of water.
However, the effect of liquid-phase velocity u L on CO 2 removal is not so apparent, as shown in Figure 6a,b. Figure 6c shows that the time required for C g,out to reach C 0 decreases with increasing u L at high MEA concentrations, which is mainly a result of a larger mass-transfer coefficient at higher u L . The loading of CO 2 of absorbent liquids is depleted more quickly when the rate of absorption increases. Furthermore, it is inferred that the resistance of liquid phase mass transfer is always important in chemical absorption, particularly at high MEA concentrations.

Validity of the Proposed Kinetic Model
Figures 5 and 6 also show the calculated results (solid and dashed curves) in the present PP hollow-fiber module. It is found that the measured results agree reasonably well with the calculated ones using water and low MEA concentrations as shown in Figures 5a,b and 6a,b. The standard deviation (SD) is less than 11% (mostly 7%), which is defined by where the subscripts 'calc' and 'expt' are the calculated and measured values, respectively, and N is the number of data points. The agreement is also acceptable using 0.05 M of MEA solution, where SD is less than 15%. At higher MEA concentrations (e.g., 1.0 M of MEA in Figures 5c and 6c), the large SD (more than 100%) is likely attributed to the ignorance of membrane wetting effect in this model. However, the fact that the predicted lines still approximately pass through the "center of symmetry" of the measured curves, as shown in Figures 5c and 6c, can be understood by a uniform distribution of pore wetting within the membrane. Although the large SD values found in Figures 5c and 6c, this simple model would still reveal the time required for C g,out to reach C 0. It is recognized that the wetting ratio of the membrane is a crucial factor on the mass transfer resistance during the absorption of gases in HFMCs. In general, the wetting ratio is dominantly affected by the pressure difference between the gas-and liquid-phases and hydrophobicity of absorbent. Experiments with various liquid-phase pressures P L at a fixed gas-phase pressure (20 kPa) were also conducted in this work to understand such a factor on the rate of CO 2 absorption. It is found from Figure 7 that the effect of such pressure differences on CO 2 removal can be neglected using 0.005 M of MEA solution as absorbent liquid under the conditions investigated. However, the effect of liquid-phase velocity uL on CO2 removal is not so apparent, as shown in Figure 6a,b. Figure 6c shows that the time required for Cg,out to reach C0 decreases with increasing uL at high MEA concentrations, which is mainly a result of a larger masstransfer coefficient at higher uL. The loading of CO2 of absorbent liquids is depleted more quickly when the rate of absorption increases. Furthermore, it is inferred that the resistance of liquid phase mass transfer is always important in chemical absorption, particularly at high MEA concentrations.

Validity of the Proposed Kinetic Model
Figures 5 and 6 also show the calculated results (solid and dashed curves) in the present PP hollow-fiber module. It is found that the measured results agree reasonably well with the calculated ones using water and low MEA concentrations as shown in Figures  5a,b and 6a,b. The standard deviation (SD) is less than 11% (mostly 7%), which is defined by where the subscripts 'calc' and 'expt' are the calculated and measured values, respectively, and N is the number of data points. The agreement is also acceptable using 0.05 M of MEA solution, where SD is less than 15%. At higher MEA concentrations (e.g., 1.0 M of MEA in Figures 5c and 6c), the large SD (more than 100%) is likely attributed to the ignorance of membrane wetting effect in this model. However, the fact that the predicted lines still approximately pass through the "center of symmetry" of the measured curves, as shown in Figures 5c and 6c, can be understood by a uniform distribution of pore wetting within the membrane. Although the large SD values found in Figures 5c and 6c, this simple model would still reveal the time required for Cg,out to reach C0. It is recognized that the wetting ratio of the membrane is a crucial factor on the mass transfer resistance during the absorption of gases in HFMCs. In general, the wetting ratio is dominantly affected by the pressure difference between the gas-and liquid-phases and hydrophobicity of absorbent. Experiments with various liquid-phase pressures PL at a fixed gas-phase pressure (20 kPa) were also conducted in this work to understand such a factor on the rate of CO2 absorption. It is found from Figure 7 that the effect of such pressure differences on CO2 removal can be neglected using 0.005 M of MEA solution as absorbent liquid under the conditions investigated.  It has been reported that neglecting axial diffusion will result in a much smaller CO 2 concentration along the length of the fiber and, thus, a higher rate of absorption. This is the case at low Peclet numbers (=u g L/D A,g ), where D A,g is the diffusivity of CO 2 in gas phase (= 1.51 × 10 −5 m 2 s −1 ) [25]; However, such an effect becomes less when the Peclet number increases (for example, >50) [26]. In the present system, the axial diffusion plays a negligible role in kinetic modeling because the Peclet number is larger than 500.
Once the validity of the proposed simplified model was confirmed, an attempt was made to understand the role of the size of hollow fibers. Figure 8 shows the calculated results in a large HFMC, extra-flow 4 × 28 module (fiber length 620 mm, contact area 20 m 2 , fiber inner diameter 0.22 mm), whose characteristics are listed in Table 1. In contrast to the case of extra-flow 2.5 × 8 module (fiber length 190 mm, contact area 1.4 m 2 , fiber inner diameter 0.22 mm), the effect of u L on the time required for C g,out reaching C 0 in a larger module is more significant (Figure 8a,b). It is noted that the volume of absorbent liquid was kept the same (14.3 L) in both modules. Moreover, the use of larger amount of absorbent liquid leads to a higher efficiency of CO 2 absorption, as compared in Figure 6b vs. Figures  8b and 6c vs. Figure 8c. It has been reported that neglecting axial diffusion will result in a much smaller CO2 concentration along the length of the fiber and, thus, a higher rate of absorption. This is the case at low Peclet numbers (=ugL/DA,g), where DA,g is the diffusivity of CO2 in gas phase (= 1.51 × 10 −5 m 2 s −1 ) [25]; However, such an effect becomes less when the Peclet number increases (for example, >50) [26]. In the present system, the axial diffusion plays a negligible role in kinetic modeling because the Peclet number is larger than 500.
Once the validity of the proposed simplified model was confirmed, an attempt was made to understand the role of the size of hollow fibers. Figure 8 shows the calculated results in a large HFMC, extra-flow 4 × 28 module (fiber length 620 mm, contact area 20 m 2 , fiber inner diameter 0.22 mm), whose characteristics are listed in Table 1. In contrast to the case of extra-flow 2.5 × 8 module (fiber length 190 mm, contact area 1.4 m 2 , fiber inner diameter 0.22 mm), the effect of uL on the time required for Cg,out reaching C0 in a larger module is more significant (Figure 8a,b). It is noted that the volume of absorbent liquid was kept the same (14.3 L) in both modules. Moreover, the use of larger amount of absorbent liquid leads to a higher efficiency of CO2 absorption, as compared in Figure 6b vs. Figure 8b and Figure 6c vs. Figure 8c. Given the absence of experimental data using a larger extra-flow 4 × 28 module, this note is necessarily highly prospective. Additionally, smaller modules were used in the literature and in this work; they could deliver good agreement between the measured and predicted results.

Determination of Effective Fiber Length
On the other hand, the present model enables us to predict the concentration change of CO2 along the fiber from the gas inlet to the outlet. It is found from Figure 9a Figure 9c. We call this the "effective" fiber length, Leff. It is also found that Leff remarkably reduces when more concentrated MEA solution is used as an absorbent liquid. Zhang et al. [27] have actually reported that CO2 absorption by aqueous DEA solution in HFMC (Celgard MiniModule ® 0.75 × 5 module, X-50 type fibers; fiber length 113 mm, contact area 0.09 m 2 ) is mainly conducted in the front segments near the inlet. For example, 20 vol% CO2 in CO2-N2 mixture (ug = 0.032 m s −1 ) is mainly absorbed by 2 M of DEA (uL = 0.15 m s −1 ) in the segments up till z/L = 0.6 while CO2 absorption is negligible in the rest of the segments. Although increasing ug in the range 0.032-0.090 m s −1 can increase Leff, approximately 20% of the total length has little absorption capacity. They thus concluded that increasing the length of the module is not an effective way to enhance CO2 absorption when the module is longer than Leff. Given the absence of experimental data using a larger extra-flow 4 × 28 module, this note is necessarily highly prospective. Additionally, smaller modules were used in the literature and in this work; they could deliver good agreement between the measured and predicted results.

Determination of Effective Fiber Length
On the other hand, the present model enables us to predict the concentration change of CO 2 along the fiber from the gas inlet to the outlet. It is found from Figure 9a,b that the CO 2 concentration in the outlet of the module, C g,out , is higher when liquid-phase velocity u L is smaller. In addition, C g,out becomes zero for all u L values tested when u g = 0.124 m s −1 using 0.05 M MEA solution as an absorbent liquid (Figure 9c). For example, the dimensionless length of the fiber (z/L) needed for nearly complete removal of 15% CO 2 is 0.2 (whereas z/L = 0.06 when C g /C 0 = 0.01) using 0.05 M of MEA solution as u g = 0.124 m s −1 and u L = 2.0 × 10 −2 m s −1 , as shown in Figure 9c. We call this the "effective" fiber length, L eff . It is also found that L eff remarkably reduces when more concentrated MEA solution is used as an absorbent liquid. Zhang et al. [27] have actually reported that CO 2 absorption by aqueous DEA solution in HFMC (Celgard MiniModule ® 0.75 × 5 module, X-50 type fibers; fiber length 113 mm, contact area 0.09 m 2 ) is mainly conducted in the front segments near the inlet. For example, 20 vol% CO 2 in CO 2 -N 2 mixture (u g = 0.032 m s −1 ) is mainly absorbed by 2 M of DEA (u L = 0.15 m s −1 ) in the segments up till z/L = 0.6 while CO 2 absorption is negligible in the rest of the segments. Although increasing u g in the range 0.032-0.090 m s −1 can increase L eff , approximately 20% of the total length has little absorption capacity. They thus concluded that increasing the length of the module is not an effective way to enhance CO 2 absorption when the module is longer than L eff .  Table 2. It is evident that Leff decreases with increasing liquid-phase velocity uL and absorbent concentration but increases with increasing gas-phase velocity ug. Similarly, Table  3 shows the results of Leff in extra-flow 4 × 28 module.  Table 2. It is evident that L eff decreases with increasing liquid-phase velocity u L and absorbent concentration but increases with increasing gas-phase velocity u g . Similarly, Table 3 shows the results of L eff in extra-flow 4 × 28 module. Evidently, such a simplified model can predict a suitable membrane length of the fiber for desired CO 2 removal or estimate an appropriate absorbent concentration for fixed length of the fiber. Generally speaking, the knowledge of effective fiber length is important for economical applications of HFMC processes. The relationship among effective fiber length, gas-phase flow rate, and absorbent concentration for fixed emissive CO 2 concentration is crucial for the HFMC process to scale up.

Conclusions
The absorption of CO 2 from CO 2 -N 2 mixtures using water and monoethanolamine (MEA) solution as absorbent liquids in hollow-fiber polypropylene membrane contactors has been experimentally and theoretically investigated. Through the use of the overall masstransfer coefficients purely evaluated from physical absorption (i.e., water), a simplified model that considers mass balance equations and membrane non-wetting acceptably followed chemical absorption process (i.e., MEA); this is particularly true for a concentration of MEA lower than 0.05 M, where the standard deviation was less than 15%. The time changes of the gas-phase CO 2 concentration in the outlet of the module, as well as the gas-phase CO 2 concentration along the fiber within the module, could be obtained by iterative calculation. Model prediction of the effective length of the fiber, defined as the length from the inlet where the gas-phase CO 2 concentration reduced to zero, depended not only on the contact area of the module but also on the total length of the fiber. Although membrane wetting was more serious at higher MEA concentrations (e.g., 1.0 M), on average, model predictions still adequately described the dynamics of CO 2 absorption process in hollow-fiber membrane contactors.

Data Availability Statement:
The data presented in this study are available on request from the corresponding author.

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

List of Symbols
A T = effective contact area (m 2 ) C = concentration of components (mol m −3 ) C 0 = initial CO 2 concentration in gas phase (mol m −3 ) H = Henry's law constant of CO 2 (m 3 Pa mol −1 ) J A = flux of CO 2 (mol m −2 s −1 ) K L = overall mass-transfer coefficient based on liquid phase (m s −1 ) k -1 = first-order rate constant for the reverse reaction defined in Equation (8)