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Chemical absorption of carbon dioxide in a hollow fiber membrane module was studied theoretically in this work. Numerical simulation was performed by using CFD for separation of carbon dioxide from flue gas. Simulations were based on solving the conservation equations for carbon dioxide and absorbent in the membrane module. Laminar parabolic velocity distribution was used for the liquid absorbent flow in the tube side of module and the gas flow in the shell side was characterized by solving the Navier-Stocks equation. Axial and radial diffusion were considered in the mass transfer equations. The equations of model were solved by numerical method based on finite element method (FEM). The simulation results were compared with the experimental values reported by S. Yan et al. . They reported the experimental results for the separation of CO2 from flue gas by aqueous solutions using polypropylene membrane contactor. The simulation results were in good agreement with the experimental results for different values of process parameters. The simulation results showed that increasing liquid velocity in the membrane module increases the removal of carbon dioxide from flue gas. Also simulation results indicated that increasing of gas velocity reduces mass transfer of carbon dioxide. In the case of CO2 absorption in MEA the major mass transfer resistance is located in the gas phase.
Keywords: Parallel membrane module; Flue gas; CFD simulation; Polypropylene membrane; Finite element method
Expansion of industrial activities has caused the concentration of greenhouse gases to rise significantly in the atmosphere. This has contributed to global warming, which in turn has resulted in serious environmental problems . Carbon dioxide is representing about 80% of greenhouse gases and also, it is reported that half of the CO2 emissions are produced by industry and power plants using fossil fuels . Carbon dioxide is emitted from fossil fuel, natural and refinery off gases and many other sources. It is important to separate CO2 from gas mixtures to avoid the environmental problems to reach the carbon emission reduction targets set out by the Kyoto Agreement. Additionally, the CO2 concentrations are about 3-5% in gas-fired power plants and 13-15% in coal plants .
Present carbon dioxide separation processes are based on physical and chemical processes involving, absorption, adsorption, cryogenic and membrane processes . The separation of CO2 contains many problems in usual processes such as channeling, flooding, entraining, foaming, and also high capital and operating costs . Therefore, many researchers have studied the possibilities of enhancing the efficiency of these processes to reduce their problems.
Recently, the gas-liquid hollow fiber membrane contactors as gas absorption devices have been a subject of great interest. In these processes, the membrane contactor mainly acts as a physical barrier between two phases (gas and liquid) without any varieties in selectivity. Because of a very high surface/volume ratio, the hollow fiber membrane contactors (HFMCs) have a great potential for gas absorption.
In addition, in gas-liquid hollow fiber membrane contactors, the interfacial mass transfer area is much higher and better controlled, whereas the hydrophilic or hydrophobic nature of the membrane determines the position of the interface between the gas feed and the liquid absorbent .
The greatest interest of hollow fiber membrane contactors is that they allow a dispersion free contact. In addition, the velocities of both phases can be chosen independently, neither flooding nor unloading problems arise . Figure 1 shows a parallel hollow fiber membrane module.
Some experiments and theories about the hollow fiber membrane contactors had been done since Qi and Cussler first studied these devices . Using polypropylene hollow fiber membrane, Kreulen et al.  investigated the chemical absorption of CO2 into water/glycerol liquid mixtures as absorbent. They studied the hollow fiber membrane as gas-liquid contactors in the case of both physical and chemical absorption. The separation of CO2 from offshore gas using hollow fiber membrane contactors was studied by Falk-Pederson and Dannstrom , who optimized the process with respect to sizes, weight, and costs. Many authors have studied the use of hollow fiber membrane contactors for absorption of CO2 in a hydroxide solution , the CO2 removal in membrane using amino acid salts . Qi and Cussler  studied development of a theory of the operation of hollow fiber membrane contactors, and calculated mass transfer coefficients in liquid phase. They also obtained the overall mass transfer coefficients, including resistances in both liquid and membrane, and compared the performance of hollow fibers with that of packed towers.
Other authors  investigated the separation of CO2 and SO2 from CO2/N2 and SO2/air gas mixtures, using water as an absorbent in a parallel module employing microporous polypropylene hollow fibers. A similar process has been recently studied by Zhang et al.  for co-current gas-liquid contact in a parallel hollow fiber module. In both studies, the authors assumed negligible axial diffusion, which may not be a good assumption, especially for low gas velocities. Kim and Yang  investigated the separation of CO2/N2 mixtures using hollow fiber membrane contactors theoretically and experimentally. Although there was an agreement between the model predictions with experimental results, the authors assumed a linear decrease of gas flow rate for the simulation purposes.
Al-Marzouqi et al.  developed a 2D mathematical model for simulation of CO2 separation from CH4 in a parallel hollow fiber membrane module. They studied hollow fiber membrane modules for physical absorption of carbon dioxide theoretically and experimentally. Recently Shirazian et al.  studied the simulation of gas absorption in hollow fiber membrane contactors for gas-liquid contacts. They simulated these devices by using CFD techniques and validated simulation results with the experimental data for the physical absorption of CO2 in pure water.
The main purpose of this study is to simulate separation of carbon dioxide from flue gas in a hollow fiber membrane module. Axial and radial diffusion inside the shell, through the membrane, and within the tube side of the membrane contactor are considered in the mass transfer equations. We also consider convection in the shell and tube as well as chemical reaction. The aim of the simulation is to predict the concentration of gas components in the membrane contactor. The influence of various process parameters on the mass transfer of CO2 is investigated. Chemical absorption for "non-wetted mode", where the membrane pores are filled with the gas mixture, is considered in this work. Chemical absorption is considered for absorption of CO2 in aqueous solutions of alkanolamines. The model is then validated using experimental data obtained from literature for the absorption of CO2 in amine aqueous solutions.
2.1. System to be studied
The mass transfer model was validated by comparing results of CO2 removal from flue gas by amines aqueous solutions using polypropylene hollow fibers obtained from experimental data reported by S. Yan et al. . Their experimental setup is shown schematically in Fig. 2. The feed gas (flue gas) was passed through the shell side, and the absorbent flowed counter currently through the inside of the hollow fibers. The gas inlet pressure maintained at 105 kPa while the liquid inlet pressure varied with the liquid flow rate by adjusting the liquid outlet valve. The gas compositions were measured by a gas analyzer. The material of the hollow fiber membrane was microporous hydrophobic polypropylene (PP). The parameters of the hollow fiber module are shown in Table 1 .
2.2. Formulation of mass transfer and absorption
A two-dimensional mathematical model was used for the transport of carbon dioxide through hollow fiber membrane contactors (HFMCs). This model first time was developed by Al-Marzouqi et al.  for modeling of CO2 absorption in hollow fiber membrane modules. In this work we study the separation of CO2 from flue gas using amines aqueous solutions as absorbents in a parallel flow hollow fiber membrane contactor. Fig. 1 shows this membrane contactor. The model was based on "non-wetted mode" in which the gas mixture filled the membrane pores for countercurrent gas-liquid contacts . The liquid absorbent flows from the tube side (Fluid #1), whereas the gas feed (flue gas) is fed to the shell side (Fluid #2) of membrane contactor. Laminar parabolic velocity distribution was used for the liquid flow in the tube side; whereas, the gas flow in the shell side was characterized by solving the Navier-Stocks equation. Axial and radial diffusion inside the shell, through the membrane, and within the tube side of the membrane contactor were considered in the model equations.
2.2.1. Model equations
Mass transfer model is used for a segment of a hollow fiber, as shown in Fig. 3b. The gas mixture (flue gas) flows with a fully developed laminar velocity in the shell side and the liquid absorbent flows with laminar flow in the tube side. Fig. 3c shows the cross sectional area of the hollow fiber membrane contactor. Based on Happel's free surface model , only portion of fluid surrounding the hollow fiber is considered and be approximated as circular cross section. Therefore, the hollow fiber membrane module consists of three sections: shell side, membrane, and tube side. The gas mixture is fed to the shell side (at z = L), while the absorbent is passed through the tube side (at z = 0). CO2 is separated from the flue gas by diffusing through the membrane and then is absorbed in the liquid absorbent.
The model is built considering the following assumptions:
(1) steady state and isothermal conditions.
(2) fully developed parabolic liquid velocity profile in the hollow fiber.
(3) ideal gas behavior is imposed.
(4) the Henry's law is applicable for gas-liquid interface.
(5) Laminar flow for gas and liquid flow in the contactor.
(6) non-wetted mode in which the gas mixture filled the membrane pores.
126.96.36.199. Shell side
The continuity equation for each species in a reactive absorption system can be expressed as :
Where and are the concentration, diffusive flux, reaction rate of species i, velocity and time, respectively. Either Fick's law of diffusion or Maxwell-Stefan theory can be used for the determination of diffusive fluxes of species i .
The continuity equation for steady state for CO2 in the shell side of membrane contactor for cylindrical coordinate are obtained using Fick's law of diffusion for the estimation of the diffusive flux:
We use the Navier-Stokes equations to characterize the shell side velocity. In laminar flow, the Navier-Stokes equations apply :
where ,, and denote, respectively, the velocity vector, the pressure, the density of the fluid and the dynamic viscosity.
The gas flow in the shell side of the membrane contactor can be configured as fluid envelope around the fiber (Fig. 3b) and there is no interaction between hollow fibers . The dimension of the free surface can be estimated by Happel's free surface model :
in which is the volume fraction of the void. It can be calculated as follows :
where is the number of fibers and is the module inner radius.
Boundary conditions for shell side are given as:
at z = L, CCO2-shell = CCO2-inlet = C0, (Inlet boundary) (6)
at r = r3, (Symmetry boundary), (No-slip condition) (7)
at r = r2, CCO2-shell = CCO2-membrane (Microporous membrane), (No-slip condition) (8)
The steady-state continuity equation for the transport of CO2 inside the membrane, which is considered to be due to diffusion alone, may be written as:
Boundary conditions are given as:
at r = r2, CCO2-membrane = CCO2-shell (Microporous membrane) (10)
at r = r1, CCO2-membrane = CCO2-tube/m (Base on Henry's law) (11)
where m is the solubility of CO2 in the solution.
188.8.131.52. Tube side
The steady-state continuity equation for the transport with chemical reaction of CO2 and absorbent in the tube side, where CO2 is absorbed and reacts with the absorbent may be written as:
The velocity distribution in the tube is assumed to follow Newtonian laminar flow :
where is average velocity in the tube side.
at z = 0, CCO2-tube = 0, Camine = Camine-inlet (Inlet boundary) (14)
at r = r1, CCO2-tube = CCO2-membrane m, (Non-wetted mode) (15)
at r = 0, (Symmetry boundary) (16)
184.108.40.206.1. Reaction rate for CO2 absorption into amine aqueous solution
Two typical amine solution of monoethanolamine (MEA) and methyldiethanol amine (MDEA) were employed as absorbent in the experiments reported by S. Yan et al. . The zwitterions-mechanism was adopted for the reaction of CO2 with primary or secondary alkanolamines :
where is an alkyl and is H for primary amines and an alkyl for secondary amines, is a base that could be an amine, OH, or H2O. Base on this mechanism, the reaction rate of CO2 with MEA can be expressed as follow :
The reaction kinetics for the reaction of CO2 with MDEA aqueous has been studied extensively. All the data for CO2 with MDEA are in agreement well with the pseudo-first- order reaction as follow [20, 22]:
The reaction kinetics for the reaction of CO2 with H2O can be expressed as follow :
The reaction of CO2 with H2O can be negligible due to the weak contribution .
The reaction rate of CO2 with hydroxyl ion can be described as :
where is the CO2 loading in amine solution. The value of and is given in Table 2.
2.3. Numerical solution of equations
The dimensionless model equations related to shell, membrane and tube side with the appropriate boundary conditions were solved using COMSOL software, which uses finite element method (FEM) for numerical solutions of model equations. The finite element analysis is combined with adaptive meshing and error control using numerical solver of UMFPACK . This solver is an implicit time-stepping scheme, which is well suited for solving stiff and non-stiff non-linear boundary value problems. We used an IBM-PC-Pentium4 (CPU speed is 2800 MHz) to solve the set of equations. The computational time for solving the set of equations was about 40 minutes.
Fig. 4 shows a segment of the mesh used to determine the gas transport behavior in hollow fiber membrane contactor (HFMC). It should be pointed out that the COMSOL mesh generator creates tetrahedral that are isotropic in size. A large number of elements are then created with scaling. A scaling factor of 800 (the fiber length is 800 mm) has been employed in z-direction due to large difference between r and Z. COMSOL automatically scales back the geometry after meshing. This generates an anisotropic mesh around 1215 elements.
3. Results and discussion
In this process, the CO2 removal efficiency (Î·) and mass transfer rate (JCO2) were used to describe the process as follow :
where Î· is the CO2 removal efficiency, %; JCO2 is the mass transfer rate of CO2, mol/ (m2h); and are the gas flow rates at the inlet and the outlet, respectively, m3/h; and are CO2 volumetric concentrations in the gas phase at the inlet and outlet, respectively; is the real temperature of the flue gas, K; is the gas-liquid interfacial area, m2 . is calculated by integrating the local concentration at the outlet of shell side (z=0):
3.1. Concentration distribution of CO2 in the membrane module
Fig. 5 presents the concentration distribution of CO2 in the shell, membrane and tube side of the membrane contactor. The gas mixture (flue gas) flows from one side of the contactor (z = L) where the concentration of CO2 is the highest (C0), whereas the solvent flows from the other side (z = 0) where the concentration of CO2 is assumed to be zero. As the gas flows through the shell side, it moves to the membrane due to the concentration difference, and then it is absorbed by the moving solvent. The parameters used in the simulations are the same as those in the S. Yan et al. 's experiments.
3.2. Effect of axial diffusion
The effect of axial diffusion in the mass transfer equations is presented in Fig. 6. The figure indicates the average CO2 concentration along the length of the membrane contactor for different values of Peclet number (Pe = (Ugas L) / Dgas) . The Peclet number was changed by changing the gas flow rate with the gas-liquid volumetric flow rate ratio kept constant at 0.5. Neglecting axial diffusion in the mass transfer equations results in a much smaller carbon dioxide concentration along the length of the membrane contactor and thus higher removal rate at low values of Pe number (Pe = 15). This is because at low Pe number, fast axial diffusion results in more uniform concentration distribution along the membrane contactor length . However, this difference becomes less as the value of Pe number increases (Pe = 60).
4. Model validation
The model was validated using the results obtained experimentally by S. Yan et al. . They reported experimental results for separation of CO2 from flue gas by membrane contactor. In this section we compare the simulation results with experimental values to validate the simulation results.
4.1. Effect of gas and liquid flow rates
The mass transfer rate of CO2 along the contactor for different values of liquid flow rates (the effect of convection term) is presented in Fig. 7. As expected, increasing liquid flow rate increases the mass transfer rate. As the solvent moves faster, the gas concentration at the inner surface of the fiber along the length of the contactor becomes less, resulting in higher concentration gradient at the interface and thus higher CO2 mass transfer rate . The CO2 removal efficiency along of the contactor for different values of gas flow rates is presented in Fig. 8. The increase in the gas flow rate reduces the residence time of gas phase in the membrane contactor and reduces the removal rate of carbon dioxide . It can be seen from figures that absorption of CO2 in MEA is higher than MDEA because of high solubility and reaction rate of CO2 with MEA.
Also the Figs. 7 & 8 indicate that liquid flow rate dose not greatly affect the CO2 removal in the membrane contactor. In the case of CO2 absorption in MEA aqueous solution the gas velocity changes has high effect on the mass transfer of carbon dioxide because the reaction rate of CO2 with MEA is very higher than MDEA and absorption of CO2 in MEA is controlled by gas phase. For the absorption of CO2 in MEA aqueous solution the mass transfer resistance in the membrane and liquid phase is very small compared with resistance in the gas phase.
4.2. Effect of absorbent concentration on the mass transfer rate of CO2
The dependence of mass transfer rate of CO2 on initial absorbent concentration is illustrated in Fig. 9. It is clearly shown that the mass transfer rate generally increases with the absorbent concentration. This is because that the active component absorbing CO2 in the liquid boundary layer increases with the absorbent concentration, which results in higher CO2 solubility and lower liquid flow rate. As far as the higher CO2 removal efficiency and the lower liquid flow rate are concerned, it is worthwhile to increase the initial absorbent concentration . Fig. 9 also shows that the mass transfer rate will be eventually saturated at a certain concentration value. When the absorbent concentration increased from 1 mol/L to 3 mol/L, the mass transfer rate only increased 1.11% using MEA and 2.2% using MDEA, respectively.
4.3. Effect of liquid temperature
Fig. 10 shows the mass transfer rate of CO2 as a function of liquid temperature. The mass transfer rate increases with the liquid temperature. When liquid temperature increased from 30°C to 50°C, the mass transfer rate increased from 0.59 mol/ (m2 h) to 1.15 mol/ (m2 h) using MDEA, and the mass transfer rate increases from 2.09 mol/ (m2 h) to 2.12 mol/ (m2 h) using MEA. It is known that temperature increase favored reaction rate according to the Arrhenius expression of reaction rate constant, and diffusion. But temperature increase also resulted in a decrease in CO2 solubility and an increase in evaporation of absorbent, which are not conducible to absorption. When increasing the temperature of amine solutions, the effect of temperature on reaction rate and diffusion rate perhaps is higher than that on CO2 solubility .
4.4. Effect of CO2 volume fraction at the feed gas inlet
Effect of CO2 volume fraction at the inlet flue gas on the mass transfer rate of CO2 is illustrated in Fig. 11. It is clearly shown that the inlet CO2 volume fraction has a significant effect on the mass transfer. Increasing of CO2 volume fraction at the feed gas increases driving force of mass transfer in the contactor, which results in higher mass transfer rate of CO2 along the contactor.
Numerical simulation of CO2 absorption in hollow fiber membrane modules was studied in this work. Simulations were performed to study the gas transport in gas-liquid hollow fiber membrane contactors. The simulation predicts the steady state absorbent and CO2 concentrations in the membrane contactor by solving the conservation equations for CO2 and absorbent. The model equations were based on non-wetting conditions, taking into consideration axial and radial diffusion in the shell, membrane and tube sides of the contactor. The simulation results were validated using the experimental data obtained from CO2 removal from flue gas by amine aqueous solutions as the liquid solvent by S. Yan. et al.  and were in good agreement with the experimental data for different values of gas and liquid velocities. The simulation results for the absorption of CO2 in liquid solvent indicated that the removal of CO2 increased with increasing liquid velocity in the tube side. On the other hand, increasing gas velocity in the shell side has an opposite effect.
The results in this study show that the detailed numerical simulation is able to predict the performance of the gas-liquid hollow fiber membrane modules. The numerical simulation can take into account complex chemical reaction schemes. Thus the model can be used to predict the mass transfer performance of the gas-liquid hollow fiber membrane contactors for other reactive as well as non-reactive systems simply by changing the physical and kinetic parameters. Furthermore, this numerical method represents a design and optimization tool for multi-component membrane gas absorption processes.
The authors would like to thank Arak University, for the financial supports of this project.