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

Abstract

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.

1. 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 CO₂ 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 CO₂ [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 CO₂ absorption. Karoor and Sirkar [3] used the hollow fiber membrane contactor to separate CO₂/N₂ 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 CO₂ 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 CO₂ 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 CO₂ [13]. Two conjugated governing equations for solving CO₂ 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 CO₂ 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.

2. Mathematical Formulations

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 $2\kappa R$ and $2(1-\kappa )R$, respectively. The hydrophobic microporous membrane with negligible thickness $\delta$ 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 ${\mathrm{C}}_{ai}$ 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.

2.1. 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.

$$\left[\frac{v_a{R}^2}{L{D}_a}\right]\frac{\partial {\psi }_a\left(\eta ,\xi \right)}{\partial \xi }=\frac{1}{\eta }\left[\frac{\partial }{\partial \eta }\left(\eta \frac{\partial {\psi }_a\left(\eta ,\xi \right)}{\partial \eta }\right)\right]$$

(1)

$$\left[\frac{v_b{R}^2}{L{D}_b}\right]\frac{\partial {\psi }_b\left(\eta ,\xi \right)}{\partial \xi }=\frac{1}{\eta }\left[\frac{\partial }{\partial \eta }\left(\eta \frac{\partial {\psi }_b\left(\eta ,\xi \right)}{\partial \eta }\right)\right]-k_{C{O}_2}{\psi }_b(\eta ,\xi )$$

(2)

$$v_a\left(\eta \right)=2\overline{v_a}\left[1-{\left(\frac{\eta }{\kappa }\right)}^2\right], 0\le \eta \le \kappa$$

(3)

$$v_b\left(\eta \right)=\frac{2\overline{v_b}}{\left[\frac{1-{\kappa }^4}{1-{\kappa }^2}-\frac{1-{\kappa }^2}{\ln (1/\kappa )}\right]}\left[1-{\eta }^2+\ln \left(\frac{1-{\kappa }^2}{\ln (1/\kappa )}\right)\ln \eta \right], \kappa \le \eta \le 1$$

(4)

in which

$$\eta =\frac{r}{R},\xi =\frac{z}{L},{\psi }_a=\frac{C_a}{C_{ai}},{\psi }_b=\frac{C_b}{C_{ai}},G{z}_a=\frac{\overline{v_a}{R}^2}{L{D}_a},G{z}_b=\frac{\overline{v_b}{R}^2}{L{D}_b}$$

(5)

The boundary conditions accompanied with the conjugated governing Equations (3) and (4) are

$${\psi }_a(\eta ,0)={\psi }_{ai}$$

(6)

$${\psi }_b(\eta ,0)={\psi }_{bi}$$

(7)

$$\frac{\partial {\psi }_a\left(0,\xi \right)}{\partial \eta }=0$$

(8)

$$\frac{\partial {\psi }_b\left(1,\xi \right)}{\partial \eta }=0$$

(9)

$$-\frac{\partial {\psi }_a\left(\kappa ,\xi \right)}{\partial \eta }=\frac{\epsilon R}{\delta }\left[{\psi }_a\left(\kappa ,\xi \right)-\frac{{K^{\prime }}_{ex}}{H}{\psi }_b\left(\kappa ,\xi \right)\right]$$

(10)

$$-D_a\frac{\partial {\psi }_a\left(\kappa ,\xi \right)}{\partial \eta }=D_b\frac{{K^{\prime }}_{ex}}{H}\frac{\partial {\psi }_b\left(\kappa ,\xi \right)}{\partial \eta }$$

(11)

in which the reduced equilibrium constant $K_{ex}^{\prime }=K_{ex}{\left[\mathrm{MEA}\right]/[\mathrm{H}}^{+}]$ and the equilibrium constant $K_{ex}={[\mathrm{MEACOO}}^{-}{\left]\right[\mathrm{H}}^{+}{]/\text{{}[\mathrm{CO}}_2\left]\right[\mathrm{MEA}]\text{}}=1.25\times {10}^{-5}$ at $T=298$ K [16] of the CO₂ absorption from gas phase to MEA aqueous solution, and the dimensionless Henry’s law constant H = 0.73 [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, ${\psi }_a$ and ${\psi }_b$, 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:

$${\psi }_a(\eta ,\xi )={\displaystyle \sum _{m=0}^{\infty }{S}_{a,m}{F}_{a,m}(\eta )G_m(\xi )}$$

(12)

$$\psi b(\eta ,\xi )={\displaystyle \sum _{m=0}^{\infty }{S}_{b,m}{F}_{b,m}(\eta )G_m(\xi )}$$

(13)

We may assume the eigen-functions $F_{a,m}({\eta }_a)$ and $F_{b,m}({\eta }_b)$ to be expressed in polynomials without loss of generality in the following forms:

$$F_{a,m}(\eta )={\displaystyle \sum _{n=0}^{\infty }{d}_{mn}{\eta }^n},d_{m,0}=1\left(\mathrm{selected}\right),d_{m,1}=0$$

(14)

$$F_{b,m}(\eta )={\displaystyle \sum _{n=0}^{\infty }{e}_{mn}{\eta }^n},e_{m,0}=1\left(\mathrm{selected}\right),e_{m,1}=0$$

(15)

Substitutions of Equations (12) and (13) into Equations (3) and (4) and the boundary conditions in Equations (8)–(11) leads to

$$G_m(\xi )=e^{-{\lambda }_m\xi }$$

(16)

$$F_{a,m}^{″}(\eta )+\frac{1}{\eta }{F}_{a,m}^{\prime }(\eta )+[\frac{{\lambda }_m{R}^2{v}_a(\eta )}{L{D}_a}]F_{a,m}^{}(\eta )=0$$

(17)

$$F_{b,m}^{″}(\eta )+\frac{1}{\eta }{F}_{b,m}^{\prime }(\eta )+[\frac{{\lambda }_m{R}^2{v}_b(\eta )}{L{D}_b}]F_{b,m}^{}(\eta )-k_{C{O}_2}{F}_{b,m}^{}(\eta )=0$$

(18)

$$F_{a,m}^{\prime }(0)=0$$

(19)

$$F_{b,m}^{\prime }(1)=0$$

(20)

$$-S_{a,m}{F}_{a,m}^{\prime }(\kappa )=(\frac{\epsilon R}{\delta })[S_{a,m}{F}_{a,m}^{}(\kappa )-\frac{K_{ex}^{\prime }}{H}{S}_{b,m}{F}_{b,m}^{}(\kappa )]$$

(21)

$$-S_{a,m}{F}_{a,m}^{\prime }(\kappa )=(\frac{D_b}{D_a})\frac{{K^{\prime }}_{ex}}{H}{S}_{b,m}{F}_{b,m}^{\prime }(\kappa )$$

(22)

Equation (21) was rewritten to obtain the relationship between $S_{a,m}$ and $S_{b,m}$ as

$$S_{b,m}=\frac{H\left[\delta {F^{\prime }}_{a,m}\left(\kappa \right)+\epsilon R{F}_{a,m}\left(\kappa \right)\right]}{\epsilon R{K^{\prime }}_{ex}{F}_{b,m}\left(\kappa \right)}{S}_{a,m}$$

(23)

Moreover, combinations of Equations (21) and (22) with deleting $S_{a,m}$ and $S_{b,m}$ yields the following equations to calculate the eigenvalues (${\lambda }_1$, ${\lambda }_2$, …, ${\lambda }_m$, …)

$$-{F^{\prime }}_{a,m}\left(\kappa \right)=\frac{\epsilon R}{\delta }\left[F_{a,m}\left(\kappa \right)+\left(\frac{D_a}{D_b}\right)\frac{{F^{\prime }}_{a,m}\left(\kappa \right)F_{b,m}\left(\kappa \right)}{{F^{\prime }}_{b,m}\left(\kappa \right)}\right]$$

(24)

All the coefficients $d_{m,n}$ and $e_{m,n}$ may be expressed in terms of eigenvalues ${\lambda }_m$ 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 ${\lambda }_m$ as

$$\begin{array}{c}{d}_{m2}=\frac{G{z}_a{\lambda }_m}{2},\hfill \\ d_{m,n}=\frac{2{\lambda }_{m}G{z}_a}{\left[n\left(n-1\right)+n\right]}(d_{m(n-2)}-\frac{1}{{\kappa }^2}{d}_{m(n-4)}),n=4,5,6,\dots \hfill \end{array}$$

(25)

and

$$\begin{array}{c}{e}_{m,2}=-\frac{1}{2}\frac{{\lambda }_{m}G{z}_b}{S}\left[1-\frac{3T}{2}\right]+\frac{k_{C{O}_2}}{4},e_{m3}=-\frac{4}{9}\frac{{\lambda }_{m}G{z}_{b}T}{S},\hfill \\ e_{m,n}=\frac{-{\lambda }_{m}Gz{}_b}{S\left[n\left(n-1\right)+n\right]}\left[\left(1-\frac{3}{2}T\right)e_{m,\left(n-2\right)}+2T{e}_{m,\left(n-3\right)}-\left(1+\frac{1}{2}T\right)e_{m,\left(n-4\right)}\right]\begin{array}{c}\\ \end{array}n=4,5,6,\dots \hfill \end{array}$$

(26)

where $T=\frac{1-{\kappa }^2}{\ln (1/\kappa )}$ and $S=\frac{1-{\kappa }^4}{1-{\kappa }^2}-\frac{1-{\kappa }^2}{\ln (1/\kappa )}$.

These eigenvalues ${\lambda }_m$ were calculated in Equation (24), requiring a negative set for both concurrent and countercurrent-flow operations.

Table 1. The dimensionless outlet concentrations and the associated eigenvalues and expansion coefficients.

m	λ0	λ1	λ2	λ3	λ4	λ5	S a,0	S a,1× 104	S a,2× 104	S a,3× 105	S a,4× 105	S a,5× 106	$\overline{{\psi }_{ae}}$
Concurrent-flow operations
3	0.0	−0.027	−1.316	−3.158	-	-	0.087	7.99	−3.86	8.82	-	-	0.5624
4	0.0	−0.027	−1.316	−3.158	−6.687	-	0.083	7.57	−3.41	4.42	−3.62	-	0.5791
5	0.0	−0.027	−1.316	−3.158	−6.687	−7.609	0.084	7.86	−3.19	4.63	−3.76	4.30	0.5791
Countercurrent-flow operations
3	0.0	−0.011	−1.487	−3.158	-	-	0.077	7.53	−1.37	8.63	-	-	0.5284
4	0.0	−0.011	−1.487	−3.158	0.022	-	0.074	6.69	−1.21	4.52	−3.53	-	0.5532
5	0.0	−0.011	−1.487	−3.158	0.022	3.621	0.074	6.41	−1.71	4.51	−3.61	4.43	0.5532

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 $Q_a=5.0\times {10}^{-6}{\mathrm{m}}^3/\mathrm{s}$ and $Q_b=6.67\times {10}^{-6}{\mathrm{m}}^3/\mathrm{s}$ under both concurrent- and countercurrent-flow operations as an illustration.

The orthogonality condition used in the concentric circular membrane contactor system of the case ${\lambda }_m\ne {\lambda }_n$ is verified, to solve for coefficients $S_{a,m}$ and $S_{b,m}$ as follows:

$$D_a{\displaystyle {\int }_0^{\kappa }\eta \left[\frac{R^2{v}_a(\eta )}{L{D}_a}\right]}{S}_{a,m}{S}_{a,n}{F}_{a,n}\left(\eta \right)F_{a,m}\left(\eta \right)d\eta +D_b\frac{K_{ex}^{\prime }}{H}{\displaystyle {\int }_{\kappa }^1\eta \left[\frac{R^2{v}_b(\eta )}{L{D}_b}\right]}{S}_{b,m}{S}_{b,n}{F}_{b,n}\left(\eta \right)F_{b,m}\left(\eta \right)d\eta =0$$

(27)

The dimensionless inlet and outlet concentrations can be obtained in the form of an infinite series by incorporating $\xi =0$ and $\xi =1$ into Equations (12) and (13) as

$${\psi }_a\left(\eta ,0\right)={\displaystyle \sum _{m=0}^{\infty }{S}_{a,m}{F}_{a,m}}\left(\eta \right)$$

(28)

$${\psi }_b\left(\eta ,0\right)={\displaystyle \sum _{m=0}^{\infty }{S}_{b,m}{F}_{b,m}}\left(\eta \right)$$

(29)

$${\psi }_a\left(\eta ,1\right)={\displaystyle \sum _{m=0}^{\infty }{S}_{a,m}{F}_{a,m}\left(\eta \right)e^{-{\lambda }_m}}$$

(30)

$${\psi }_b\left(\eta ,1\right)={\displaystyle \sum _{m=0}^{\infty }{S}_{b,m}{F}_{b,m}\left(\eta \right)e^{-{\lambda }_m}}$$

(31)

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:

$$\begin{array}{c}\left(\mathrm{i}\right)n=0\hfill \\ \hspace{1em}\frac{G{z}_a{\kappa }^2{\psi }_{ai}}{2}+\frac{G{z}_b\left(1-{\kappa }^2\right)D_b{\psi }_{bi}}{2D_a}\hfill \\ \hspace{1em}\hspace{1em}={\displaystyle \sum _{q=0}^{\infty }{S}_{a,q}{e}^{-{\lambda }_q}\{{\displaystyle {\int }_0^{\kappa }\eta \left[\frac{v_a{R}^2}{L{D}_a}\right]F_{a,q}d\eta }}+\frac{D_b}{D_a}\frac{{K^{\prime }}_{ex}}{H}{\displaystyle {\int }_{\kappa }^1\eta \left[\frac{v_b{R}^2}{L{D}_b}\right]\left(\frac{H\left[\delta {F^{\prime }}_{a,q}\left(\kappa \right)+\epsilon R{F}_{a,q}\left(\kappa \right)\right]}{\epsilon R{K^{\prime }}_{ex}{F}_{b,q}\left(\kappa \right)}\right)F_{b,q}}d\eta \}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=S_{a,0}\left(\frac{G{z}_a{\kappa }^2}{2}+\frac{G{z}_b\left(1-{\kappa }^2\right)D_b}{2D_a}\right)-{\displaystyle \sum _{q=1}^{\infty }{S}_{a,q}\frac{\kappa e^{-{\lambda }_q}}{{\lambda }_q}\left\{F_{a,q}^{\prime }\left(\kappa \right)-\frac{D_b\left[\delta {F^{\prime }}_{a,q}\left(\kappa \right)+\epsilon R{F}_{a,q}\left(\kappa \right)\right]}{D_a\epsilon R}\frac{F_{b,q}^{\prime }\left(\kappa \right)}{F_{b,q}\left(\kappa \right)}\right\}}\hfill \end{array}$$

(32)

$$\begin{array}{c}\left(\mathrm{ii}\right)n\ne 0,n=q\hfill \\ \hspace{1em}-\frac{{\psi }_{ai}}{{\lambda }_n}\kappa F_{a,n}^{\prime }\left(\kappa \right)+\frac{D_b}{D_a}\frac{{K^{\prime }}_{ex}}{H}\frac{{\psi }_{bi}\kappa H\left[\delta {F^{\prime }}_{a,n}\left(\kappa \right)+\epsilon R{F}_{a,n}\left(\kappa \right)\right]F_{b,n}^{\prime }\left(\kappa \right)}{\epsilon R{K^{\prime }}_{ex}{\lambda }_n{F}_{b,n}\left(\kappa \right)}\hfill \\ \hspace{1em}\hspace{1em}={\displaystyle \sum _{q=1}^{\infty }{S}_{a,q}\kappa \{\left[\frac{\partial F_{a,n}^{\prime }}{\partial {\lambda }_n}\left(\kappa \right)F_{a,n}\left(\kappa \right)-\frac{\partial F_{a,n}}{\partial {\lambda }_n}\left(\kappa \right)F_{a,n}^{\prime }\left(\kappa \right)\right]}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\frac{D_b}{D_a}\frac{{K^{\prime }}_{ex}}{H}{\left[\frac{H\left[\delta {F^{\prime }}_{a,n}\left(\kappa \right)+\epsilon R{F}_{a,n}\left(\kappa \right)\right]}{\epsilon R{K^{\prime }}_{ex}{F}_{b,n}^{}\left(\kappa \right)}\right]}^2\left[\frac{\partial F_{b,n}^{\prime }}{\partial {\lambda }_n}\left(\kappa \right)F_{b,n}\left(\kappa \right)-\frac{\partial F_{b,n}}{\partial {\lambda }_n}\left(\kappa \right)F_{b,n}^{\prime }\left(\kappa \right)\right]\}\hfill \end{array}$$

(33)

$$\begin{array}{c}\left(\mathrm{iii}\right)n\ne 0,n\ne q\hfill \\ \hspace{1em}-\frac{{\psi }_{ai}}{{\lambda }_n}\kappa F_{a,n}^{\prime }\left(\kappa \right)+\frac{D_b}{D_a}\frac{{K^{\prime }}_{ex}}{H}\frac{{\psi }_{bi}\kappa H\left[\delta {F^{\prime }}_{a,n}\left(\kappa \right)+\epsilon R{F}_{a,n}\left(\kappa \right)\right]F_{b,n}^{\prime }\left(\kappa \right)}{\epsilon R{K^{\prime }}_{ex}{\lambda }_n{F}_{b,n}\left(\kappa \right)}\hfill \\ \hspace{1em}\hspace{1em}=-S_{a,0}\frac{\kappa e^{{\lambda }_n}}{{\lambda }_n}\left\{F_{a,n}^{\prime }\left({\eta }_i\right)+\frac{D_b}{D_a}\frac{{K^{\prime }}_{ex}}{H}\left[\frac{\left[\delta {F^{\prime }}_{a,n}\left(\kappa \right)+\epsilon R{F}_{a,n}\left(\kappa \right)\right]}{\epsilon R{K^{\prime }}_{ex}}\right]\frac{F_{b,n}^{\prime }\left(\kappa \right)}{F_{b,n}\left(\kappa \right)}\right\}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\displaystyle \sum _{q=1}^{\infty }{S}_{a,q}\frac{\kappa e^{{\lambda }_n-{\lambda }_q}}{{\lambda }_q-{\lambda }_n}\left\{\left[F_{a,n}^{\prime }\left(\kappa \right)F_{a,q}\left(\kappa \right)-F_{a,n}\left(\kappa \right)F_{a,q}^{\prime }\left(\kappa \right)\right]+H\frac{D_b}{D_a}\frac{\left[\delta {F^{\prime }}_{a,n}\left(\kappa \right)+\epsilon R{F}_{a,n}\left(\kappa \right)\right]\left[\delta {F^{\prime }}_{a,q}\left(\kappa \right)+\epsilon R{F}_{a,q}\left(\kappa \right)\right]}{{\epsilon }^2{R}^2{K^{\prime }}_{ex}}\right\}}\hfill \end{array}$$

(34)

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

$$\overline{{\psi }_a}\left(\xi \right)=\frac{{\displaystyle {\int }_0^{\kappa }{v}_{a}2\pi R^2\eta {\psi }_a\left(\eta ,\xi \right)d\eta }}{{\displaystyle {\int }_0^{\kappa }{v}_{a}2\pi R^2\eta d\eta }}=S_{a,0}-\frac{2}{G{z}_a\kappa }{\displaystyle \sum _{m=1}^{\infty }\frac{1}{{\lambda }_m}{S}_{a,m}{F}_{a,m}^{\prime }\left(\kappa \right)e^{-{\lambda }_m\xi }}$$

(35)

$$\overline{{\psi }_b}\left(\xi \right)=\frac{{\displaystyle {\int }_{\kappa }^1{v}_{b}2\pi R^2\eta {\psi }_b\left(\eta ,\xi \right)d\eta }}{{\displaystyle {\int }_{\kappa }^1{v}_{b}2\pi R^2\eta d\eta }}=S_{b,0}-\frac{2\kappa }{G{z}_b\left(1-{\kappa }^2\right)}{\displaystyle \sum _{m=1}^{\infty }\frac{1}{{\lambda }_m}{S}_{b,m}{F}_{b,m}^{\prime }\left(\kappa \right)e^{-{\lambda }_m\xi }}$$

(36)

2.2. 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, except replacing the velocity distribution of Equation (3) and boundary condition of Equation (11) by Equations (37) and (38), respectively

$$v_a\left(\eta \right)=-2\overline{v_a}\left[1-{\left(\frac{\eta }{\kappa }\right)}^2\right], 0\le \eta \le \kappa$$

(37)

$${\psi }_a(\eta ,1)={\psi }_{ai}\mathrm{at}\xi =1$$

(38)

The results of all coefficients may be obtained as follow:

$$\begin{array}{c}\left(\mathrm{i}\right)n=0\hfill \\ \hspace{1em}-\frac{G{z}_a{\kappa }^2{\psi }_{ai}}{2}-\frac{G{z}_b\left(1-{\kappa }^2\right)D_b{\psi }_{bi}}{2D_a}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=S_{a,0}\left(-\frac{G{z}_a{\kappa }^2}{2}-\frac{G{z}_b\left(1-{\kappa }^2\right)D_b}{2D_a}\right)-{\displaystyle \sum _{q=1}^{\infty }{S}_{a,q}\frac{\kappa }{{\lambda }_q}\left\{F_{a,q}^{\prime }\left(\kappa \right)-\frac{D_b{e}^{-{\lambda }_q}\left[\delta {F^{\prime }}_{a,q}\left(\kappa \right)+\epsilon R{F}_{a,q}\left(\kappa \right)\right]}{D_a\epsilon R}\frac{F_{b,q}^{\prime }\left(\kappa \right)}{F_{b,q}\left(\kappa \right)}\right\}}\hfill \end{array}$$

(39)

$$\begin{array}{c}\left(\mathrm{ii}\right)n\ne 0,n=q\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-\frac{{\psi }_{ai}}{{\lambda }_n}\kappa F_{a,n}^{\prime }\left(\kappa \right)-\frac{D_b}{D_a}\frac{{K^{\prime }}_{ex}}{H}\frac{{\psi }_{bi}\kappa H\left[\delta {F^{\prime }}_{a,n}\left(\kappa \right)+\epsilon R{F}_{a,n}\left(\kappa \right)\right]F_{b,n}^{\prime }\left(\kappa \right)}{\epsilon R{K^{\prime }}_{ex}{\lambda }_n{F}_{b,n}\left(\kappa \right)}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}={\displaystyle \sum _{q=1}^{\infty }{S}_{a,q}\kappa \{\left[\frac{\partial F_{a,n}^{\prime }}{\partial {\lambda }_n}\left(\kappa \right)F_{a,n}\left(\kappa \right)-\frac{\partial F_{a,n}}{\partial {\lambda }_n}\left(\kappa \right)F_{a,n}^{\prime }\left(\kappa \right)\right]}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-\frac{D_b}{D_a}\frac{{K^{\prime }}_{ex}}{H}{\left[\frac{H\left[\delta {F^{\prime }}_{a,n}\left(\kappa \right)+\epsilon r_f{F}_{a,n}\left(\kappa \right)\right]}{{K^{\prime }}_{ex}\epsilon R{F}_{b,n}^{}\left(\kappa \right)}\right]}^2\left[\frac{\partial F_{b,n}^{\prime }}{\partial {\lambda }_n}\left(\kappa \right)F_{b,n}\left(\kappa \right)-\frac{\partial F_{b,n}}{\partial {\lambda }_n}\left(\kappa \right)F_{b,n}^{\prime }\left(\kappa \right)\right]\}\hfill \end{array}$$

(40)

$$\begin{array}{c}\left(\mathrm{iii}\right)n\ne 0,n\ne q\hfill \\ \hspace{1em}-\frac{{\psi }_{ai}}{{\lambda }_n}\kappa F_{a,n}^{\prime }\left(\kappa \right)+\frac{D_b}{D_a}\frac{{K^{\prime }}_{ex}}{H}\frac{{\psi }_{bi}\kappa H\left[\delta {F^{\prime }}_{a,n}\left(\kappa \right)+\epsilon R{F}_{a,n}\left(\kappa \right)\right]F_{b,n}^{\prime }\left(\kappa \right)}{{K^{\prime }}_{ex}{\lambda }_n\epsilon R{F}_{b,n}\left(\kappa \right)}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=-S_{a,0}\frac{\kappa e^{{\lambda }_n}}{{\lambda }_n}\left\{F_{a,n}^{\prime }\left({\eta }_i\right)-\frac{D_b}{D_a}\frac{{K^{\prime }}_{ex}}{H}\left[\frac{\left[\delta {F^{\prime }}_{a,n}\left(\kappa \right)+\epsilon R{F}_{a,n}\left(\kappa \right)\right]}{\epsilon R{K^{\prime }}_{ex}}\right]\frac{F_{b,n}^{\prime }\left(\kappa \right)}{F_{b,n}\left(\kappa \right)}\right\}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\displaystyle \sum _{q=1}^{\infty }{S}_{a,q}\frac{\kappa e^{{\lambda }_n-{\lambda }_q}}{{\lambda }_q-{\lambda }_n}\left\{\left[F_{a,n}^{\prime }\left(\kappa \right)F_{a,q}\left(\kappa \right)-F_{a,n}\left(\kappa \right)F_{a,q}^{\prime }\left(\kappa \right)\right]-\frac{D_b}{D_a}\frac{H\left[\delta {F^{\prime }}_{a,n}\left(\kappa \right)+\epsilon R{F}_{a,n}\left(\kappa \right)\right]\left[\delta {F^{\prime }}_{a,q}\left(\kappa \right)+\epsilon R{F}_{a,q}\left(\kappa \right)\right]}{{\epsilon }^2{R}^2{K^{\prime }}_{ex}}\right\}}\hfill \end{array}$$

(41)

Similarly, the dimensionless radially averaged concentrations of both absorbent and gas streams are

$$\overline{{\psi }_a}\left(\xi \right)=S_{a,0}+\frac{2}{G{z}_a\kappa }{\displaystyle \sum _{m=1}^{\infty }\frac{1}{{\lambda }_m}{S}_{a,m}{F}_{a,m}^{\prime }\left(\kappa \right)e^{-{\lambda }_m\xi }}$$

(42)

$$\overline{{\psi }_b}\left(\xi \right)=S_{b,0}-\frac{2\kappa }{G{z}_b\left(1-{\kappa }^2\right)}{\displaystyle \sum _{m=1}^{\infty }\frac{1}{{\lambda }_m}{S}_{b,m}{F}_{b,m}^{\prime }\left(\kappa \right)e^{-{\lambda }_m\xi }}$$

(43)

2.3. Absorption Efficiency

The CO₂ absorption rate was calculated as follows:

$$\omega =Q_a\left(\overline{C_{ai}}-\overline{C_{ae}}\right)$$

(44)

The absorption efficiency $I_M$ was illustrated by calculating the percentage of CO₂ absorbed in the concentric circular membrane contactor module as

$$I_M=\frac{\overline{C_{ai}}-\overline{C_{ae}}}{\overline{C_{ai}}}\times 100\%$$

(45)

Furthermore, the local mass-transfer coefficient in the absorbent solution plays an important role to dominate absorption efficiency and could be obtained by

$$k_{b\xi }=\frac{D_b}{R}\frac{\partial {\psi }_b(\kappa ,\xi )/\partial \eta }{{\psi }_b(\kappa ,\xi )-\frac{{K^{\prime }}_{ex}}{H}{\overline{\psi }}_b(\xi )}$$

(46)

The local Sherwood number is defined by

$$S{h}_{\xi }=\frac{k_{b\xi }{D}_{eq,b}}{D_b}$$

(47)

where $D{}_b$ and $D_{eq,b}$ are the diffusivity of CO₂ 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

$$\overline{Sh}={\displaystyle {\int }_0^{1}S{h}_{\xi }d\xi }={\displaystyle {\int }_0^1\frac{2\left(1-\kappa \right){\displaystyle \sum _{m=1}^{\infty }{S}_{b,m}{F^{\prime }}_{b,m}\left(\kappa \right)e^{-{\lambda }_m\xi }}}{{\displaystyle \sum _{m=1}^{\infty }{S}_{b,m}\left[F_{b,m}\left(\kappa \right)-\frac{{K^{\prime }}_{ex}}{H}\frac{2\kappa }{G{z}_b\left(1-{\kappa }^2\right){\lambda }_m}{F}_{b,m}^{\prime }\left(\kappa \right)\right]e^{-{\lambda }_m\xi }}}}d\xi$$

(48)

3. Experimental Apparatus

The experimental setup of the membrane absorption of CO₂ 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$\mathsf{\mu }\mathrm{m}$, a porosity of 0.72 and a thickness of 130$\mathsf{\mu }\mathrm{m}$ 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 Figure 4.

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 CO₂/N₂ 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 cm³/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 CO₂/N₂ from the gas mixing tank (EW-06065-02, Cole Parmer company, Chicago, IL, USA) with a fixed flow rate of 5.0 cm³/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 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. Specifications and parameters of the experimental runs.

Fixed Parameters		Valuable Parameters
Outer diameter of shell (mm)	28	MEA absorbent flow rates (cm3/s)	5.0~10.0
Inner diameter of shell (mm)	17	CO2/N2 gas flow rate (cm3/s)	5.0 cm
Effective length of tube (cm)	20	Inlet CO2 concentrations (%)	30, 35, 40
Outer diameter of tube (mm)	15	MEA solution (wt %)	30
Membrane pore size (μm)	0.2	Henry’s law constant H	0.73
Membrane porosity	0.72	Diffusion coefficient Da (cm2/s)	1.67 × 10−3
Membrane thickness (μm)	98 (PTFE)	Diffusion coefficient Db (cm2/s)	5.0 × 10−5
	32 (PP)	Equilibrium constant Kex(298 K)	1.25 × 10−5

.

The experimental results deviate from the theoretical predictions calculating by the following definition of the accuracy deviation [21]

$$E(\%)=\frac{1}{N_{\exp }}{\displaystyle \sum _{i=1}^{N_{\exp }}\frac{\left|{\omega }_{theo,i}-{\omega }_{\exp ,i}\right|}{{\omega }_{\exp ,i}}}\times 100$$

(49)

where N_exp, ${\omega }_{theo,i}$ and ${\omega }_{\exp ,i}$ 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 between theoretical predictions and experimental results.

Cai (%)	Concurrent Flow	Countercurrent Flow
	E (%)	E (%)
30	6.63	7.74
40	6.89	7.71
45	6.92	6.58

. The accuracy deviation of experimental results from theoretical predictions is within expectation and goes reasonably well, at $6.63\times {10}^{-2}\le E\le 7.74\times {10}^{-2}$.

4. 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) and (26). Next, the expansion coefficients $S_{a,n}$ are calculated from Equations (32)–(34) for the concurrent-flow operations and from Equations (39)–(41) for countercurrent-flow operations. Thus, the expansion coefficients $S_{b,n}$ are obtained from Equation (23). Last, the dimensionless averaged outlet concentrations are calculated from Equations (35) and (36) for concurrent-flow operations and from Equations (42) and (43) for countercurrent-flow operations. The absorption rate, absorption efficiency, mass transfer coefficient, and averaged Sherwood number are calculated from Equations (44), (45) and (48), respectively.

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, $\overline{{\psi }_b}\left(\xi \right)$, increases as the inlet CO₂ 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 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 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.

The local mass-transfer Graetz number symbolizes the ratio of mass transfer coefficient in radial direction versus diffusion in axial direction, say $S{h}_{\xi }=k_{b\xi }{D}_{eq,b}/D_b$. The averaged mass-transfer Sherwood number $\overline{Sh}$ plays a significant role in determining the CO₂ absorption rate in considering the mass transfer behavior. The theoretical prediction $\overline{Sh}$ 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 $\overline{Sh}$ 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 CO₂ concentrations, as demonstrated in Figure 6. It’s reasonable to conclude that the mass transfer rate represented by $\overline{Sh}$ 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 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 Figure 8 and Figure 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 $I_{\mathrm{M}}$ varies with mass-transfer Graetz number $G{z}_b$ 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 $G{z}_a$ 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 operations, as illustrated in Figure 7, Figure 8, Figure 9 and Figure 10, because the driving-force of the CO₂ 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.

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.