Modeling and Simulation of the Absorption of CO 2 and NO 2 from a Gas Mixture in a Membrane Contactor

: The removal of undesirable compounds such as CO 2 and NO 2 from incineration and natural gas is essential because of their harmful inﬂuence on the atmosphere and on the reduction of natural gas heating value. The use of membrane contactor for the capture of the post-combustion NO 2 and CO 2 had been widely considered in the past decades. In this study, membrane contactor was used for the simultaneous absorption of CO 2 and NO 2 from a mixture of gas (5% CO 2 , 300 ppm NO 2 , balance N 2 ) with aqueous sodium hydroxide solution. For the ﬁrst time, a mathematical model was established for the simultaneous removal of the two undesired gas solutes (CO 2 , NO 2 ) from ﬂue gas using membrane contactor. The model considers the reaction rate, and radial and axial di ﬀ usion of both compounds. The model was veriﬁed and validated with experimental data and found to be in good agreement. The model was used to examine the e ﬀ ect of the ﬂow rate of liquid, gas, and inlet solute mole fraction on the percent removal and molar ﬂux of both impurity species. The results revealed that the e ﬀ ect of the liquid ﬂow rate improves the percent removal of both compounds. A high inlet gas ﬂow rate decreases the percent removal. It was possible to obtain the complete removal of both undesired compounds. The model was conﬁrmed to be a dependable tool for the optimization of such process, and for similar systems.


Introduction
Harmful gases are emitted into the atmosphere from industrial plants, because of the increase in the human population and the associated economic development, energy consumption, and the requirement of burning fossil fuels for water desalination and power generation purposes. Nitrogen dioxide (NO 2 ) is believed to be one of the gases that contributes to smog and acid rain and which is harmful to human and animal well-being. Accordingly, there is an obligation to capture and eliminate nitrogen dioxide and other harmful gases, such NO x , SO 2 , and CO 2 from industrial emission streams, proceeding to discharge into the atmosphere [1][2][3]. Various methods have been established for capturing the impurity of compounds such as physical and/or chemical absorption, adsorption, membrane technology, conversion to another compound, and condensation. There are various technologies available to remove CO 2 and NO x [3][4][5]. Physical absorption incorporates mass transport within the phases and mass transfer at the liquid-gas boundary. Operating conditions and gas solubility are the main factors affecting physical absorption. An example of physical absorption is the capture of CO 2 into liquid water using industrial absorption towers or gas-liquid membrane contactors. Chemical absorption is based on a chemical reaction between the absorbed substances and the liquid phase, such as the capture of CO 2 in amine solutions [4,6]. The most widely used commercial and economical method is the chemical absorption technique, used in the conventional or vice versa, in a co-current or countercurrent parallel flow. The pollutant gas compounds diffuse through the fiber walls towards the absorbent membrane-tube interface, as a result of the concentration gradient. Other gases are retained in the membrane pores because of their low diffusivity and low solubility in the liquid solvent. Figure 1 shows a schematic simplified geometry of the model domains representing the HFM module grounded on Happel's free surface [40]. The model considers the following three separate domains: the liquid phase in the tube side, the non-wetted membrane, and the gas phase in the shell side. The system is steady state, described by cylindrical coordinates, angular concentration gradients are neglected, and an asymmetrical approach is considered.
Processes 2019, 7, x 3 of 12 the fiber walls towards the absorbent membrane-tube interface, as a result of the concentration gradient. Other gases are retained in the membrane pores because of their low diffusivity and low solubility in the liquid solvent. Figure 1 shows a schematic simplified geometry of the model domains representing the HFM module grounded on Happel's free surface [40]. The model considers the following three separate domains: the liquid phase in the tube side, the non-wetted membrane, and the gas phase in the shell side. The system is steady state, described by cylindrical coordinates, angular concentration gradients are neglected, and an asymmetrical approach is considered. The sizes of the membrane used in the simulation are presented in Table 1. As seen from Table 1, the fiber length is 588 times longer than the fiber radius (effective module length is 200 mm and radius are 0.34 mm). Accordingly, the membrane length is scaled up so as to avoid an excessive number of elements and nodes and for a better appearance of the module in the simulation; therefore, a new scaled length is introduced by dividing the length by 100. The following assumptions were considered: The sizes of the membrane used in the simulation are presented in Table 1. As seen from Table 1, the fiber length is 588 times longer than the fiber radius (effective module length is 200 mm and radius are 0.34 mm). Accordingly, the membrane length is scaled up so as to avoid an excessive number of elements and nodes and for a better appearance of the module in the simulation; therefore, a new scaled length is introduced by dividing the length by 100. The following assumptions were considered:

•
The process is at isothermal and steady state conditions; • Gas phase is an ideal gas, and the liquid phase is incompressible and Newtonian; • Solubility is based on Henry's law at the liquid-gas interface.
The blended gas (CO 2 , NO 2 , and N 2 ) is transported in the shell side by convection and diffusion, whereas, in the membrane section, the only transport mechanism is diffusion. The liquid phase (NaOH + H 2 O) is transported in the lumen by diffusion and convection. The following mass transport equations are formulated to describe the chemical absorption system in the model domains (tube, membrane, and shell). The developed mass transport equations are presented as follows.

Tube Side
The mass balance equations for the gas components of CO 2 , NO 2 , and N 2 in the tube side can be stated, as per Equation (1), as follows: The subscripts in the material balance the following equations: t refers to tube side, m refers to membrane, and s refers to shell side, where C i,t refers to the concentration of component i in liquid moving in the tube side, i refers to the pollutant gas components: CO 2 , NO 2 , and R i is the rate of reaction of the species i. The length of the fiber is scaled to avoid excessive computation and to make the simulation result in a better profile. The scaling is performed as follows: let ξ = z/scale. The scaling factor is substituted in Equation (1). Consequently, the equation becomes Equation (2), as follows: where the velocity of liquid inside the hollow fiber (v z,t ) is described by the following parabolic equation: where Q t is the volumetric liquid flow rate in the tube side, and n is the number of hollow fibers. The appropriate set of boundary conditions are specified as follows: where m i is the dimensionless physical solubility of CO 2 and NO 2 in solvent, 0.82, 0.17, respectively. The values of the dimensionless physical solubility of CO 2 and NO 2 were calculated from Henry's constant (H): 0.034 kmol/m 3 atm, 0.007 kmol/m 3 atm [23], respectively, using the relation m i = RTxH. The liquid phase reactions between NO 2 and NaOH took several steps. First, NO 2 dissolved in the aqueous NaOH was reacted with H 2 O, then neutralized with sodium hydroxide [41]. The controlling liquid phase reactions are as follows: The overall reaction is designated, as per Equation (8), as follows: The general reaction rate is expressed in Equation (9), as follows: The reaction is the second order with a rate constant, k r,1 m 3 mol −1 s −1 = 1.0 × 10 5 , [1] The overall reaction of CO 2 and NaOH is represented by the following reaction [42].

Membrane Side
The transport of the solute gas (CO 2 and NO 2 ) components in the membrane section confined between r 1 and r 2 can be described by the steady state material balance equation (Equation (12)), where diffusion is the only transport mechanism in the membrane phase [34], as follows: The proper boundary settings are specified, as per Equation (13), as follows:

Shell Side
The steady state mass transport of solute gas (CO 2 and NO 2 ) components in the shell side (no chemical reaction occurs in the module shell zone) is expressed in Equation (14), as follows: The velocity profile in the shell side is described by Happel's free surface [40] and can be calculated as per Equation (15), as follows: The applicable boundary conditions are specified as follows: The radius of the free surface (r 3 ) can be determined as per Equation (17), as follows: The void fraction of the membrane module (ϕ) is calculated as per Equation (18): where R is the module inner radius, n is the number of fibers r 1 , and r 2 is the fiber outside radius. The parameters used in the model are shown in Table 2.

Numerical Solution
The model governing the partial differential and algebraic equations was solved simultaneously using COMSOL software version 5.4. The software uses a finite element method to solve the model equations.

Results and Discussion
The accuracy of the mathematical model was checked prior to using the model for studying the effect of the various operating parameters on the percent deletion of CO 2 and NO 2 from the imitated flue gas. The model was authenticated with experimental data for the simultaneous absorption of an NO 2 and CO 2 from gas mixture in a PTFE polymeric gas-liquid hollow fiber membrane [23]. The percent removal of CO 2 and NO 2 was calculated as per Equation (19), as follows: where F g,in , F g,out , C i,in , and C i,out are the inlet gas flow rate, outlet gas flow rate, inlet gas concentration of component i, and outlet gas concentration of component i, respectively. The 2D surface plot of the CO 2 and NO 2 concentration profile throughout the model domains are shown in Figure 2. The figure reveals that, even though the solubility of CO 2 (0.75) is higher than NO 2 (0.17) in the aqueous NaOH solution, the removal rate of nitrogen dioxide is much higher than that of carbon dioxide, which is attributed to the high reaction rate of NO 2 -NaOH.   The predicted results were also authenticated with the experimental investigations for the case of the effects of the variable liquid velocities on the percent removal of NO2 and CO2 ( Figure 4) at a fixed gas cross-flow velocity of 2.11 m/s, 0.5 M NaOH, 2% CO2, 300 ppm NO2, and the balance was nitrogen. The simulation results matched the experimental data, to a certain extent [23]. The results revealed that the increase in solvent velocity increased the percent removal of CO2 and NO2 sharply at low liquid velocities (below 0.05 m/s); as the liquid velocity increased further, there was a slight increase in the percent simultaneous removal of CO2 and NO2 from the gas mixture. The insignificant increase in liquid velocity higher than 0.05 m/s was attributed to a decrease in the residence time. The removal flux was calculated as per Equation (20):     The predicted results were also authenticated with the experimental investigations for the case of the effects of the variable liquid velocities on the percent removal of NO2 and CO2 ( Figure 4) at a fixed gas cross-flow velocity of 2.11 m/s, 0.5 M NaOH, 2% CO2, 300 ppm NO2, and the balance was nitrogen. The simulation results matched the experimental data, to a certain extent [23]. The results revealed that the increase in solvent velocity increased the percent removal of CO2 and NO2 sharply at low liquid velocities (below 0.05 m/s); as the liquid velocity increased further, there was a slight increase in the percent simultaneous removal of CO2 and NO2 from the gas mixture. The insignificant increase in liquid velocity higher than 0.05 m/s was attributed to a decrease in the residence time. The removal flux was calculated as per Equation (20):  [23] for the influence of the inlet gas velocity on the simultaneous percent removal of CO 2 and NO 2 (velocity of liquid: 0.05 m/s; solvent concentration: 0.5 M NaOH; inlet gas composition: 2% CO 2 ; 300 ppm NO 2 ; the balance is N 2 ).
The predicted results were also authenticated with the experimental investigations for the case of the effects of the variable liquid velocities on the percent removal of NO 2 and CO 2 ( Figure 4) at a fixed gas cross-flow velocity of 2.11 m/s, 0.5 M NaOH, 2% CO 2 , 300 ppm NO 2 , and the balance was nitrogen. The simulation results matched the experimental data, to a certain extent [23]. The results revealed that the increase in solvent velocity increased the percent removal of CO 2 and NO 2 sharply at low liquid velocities (below 0.05 m/s); as the liquid velocity increased further, there was a slight increase in the percent simultaneous removal of CO 2 and NO 2 from the gas mixture. The insignificant increase in liquid velocity higher than 0.05 m/s was attributed to a decrease in the residence time. The removal flux was calculated as per Equation (20):  The influence of the speed of the gas on the molar flux of CO2 and NO2 is illustrated in Figure 5. The figure reveals that there was noticeable increase in the removal flux of CO2 with the gas velocity; by contrast, the removal flux of NO2 was insignificant because of its lower inlet concentration in the gas stream (300 ppm), compared with the CO2 inlet concentration (2%). When the velocity of gas was increased from 1.05 m/s to 2.11 m/s, the removal flux increased from 0.003 to 0.0038 mol/m 2 ·s; at a high gas velocity, the increase was insignificant, from example, with the increase in gas velocity from 4.21 to 6.32 m/s, the increase in molar flux was very small. This was attributed to a decrease in residence time, as well as the insufficient amount of solvent available for the excess amount of CO2 and NO2 components associated with the increase in gas stream volumetric feed rate. The influence of the speed of the gas on the molar flux of CO 2 and NO 2 is illustrated in Figure 5. The figure reveals that there was noticeable increase in the removal flux of CO 2 with the gas velocity; by contrast, the removal flux of NO 2 was insignificant because of its lower inlet concentration in the gas stream (300 ppm), compared with the CO 2 inlet concentration (2%). When the velocity of gas was increased from 1.05 m/s to 2.11 m/s, the removal flux increased from 0.003 to 0.0038 mol/m 2 ·s; at a high gas velocity, the increase was insignificant, from example, with the increase in gas velocity from 4.21 to 6.32 m/s, the increase in molar flux was very small. This was attributed to a decrease in residence time, as well as the insufficient amount of solvent available for the excess amount of CO 2 and NO 2 components associated with the increase in gas stream volumetric feed rate.  Figure 6 demonstrates the effect of the change in the inlet CO2 mole fraction at a fixed inlet concentration of NO2 (300 ppm) on the component's molar flux. The CO2 molar flux increased significantly when its concentration increased, which was expected, because as the amount of CO2 increased in the inlet gas stream, more CO2 was being absorbed, and hence the CO2 removal molar flux increased (molar flux: moles gas removed per area per time). By contrast, because of the fixed low concentration of NO2 in the feed stream, its removal flux was insignificant compared to that of CO2.    Figure 6 demonstrates the effect of the change in the inlet CO 2 mole fraction at a fixed inlet concentration of NO 2 (300 ppm) on the component's molar flux. The CO 2 molar flux increased significantly when its concentration increased, which was expected, because as the amount of CO 2 increased in the inlet gas stream, more CO 2 was being absorbed, and hence the CO 2 removal molar flux increased (molar flux: moles gas removed per area per time). By contrast, because of the fixed low concentration of NO 2 in the feed stream, its removal flux was insignificant compared to that of CO 2 .  Figure 6 demonstrates the effect of the change in the inlet CO2 mole fraction at a fixed inlet concentration of NO2 (300 ppm) on the component's molar flux. The CO2 molar flux increased significantly when its concentration increased, which was expected, because as the amount of CO2 increased in the inlet gas stream, more CO2 was being absorbed, and hence the CO2 removal molar flux increased (molar flux: moles gas removed per area per time). By contrast, because of the fixed low concentration of NO2 in the feed stream, its removal flux was insignificant compared to that of CO2.    Figure 7 explains the effect of change in the inlet NO 2 mole fraction in the feed gas stream at a fixed concentration of CO 2 (2%) on the removal flux of CO 2 and NO 2 . The predicted results are in the range of the experimental data [23] under the same conditions. The effect of change in the inlet mole fraction of NO 2 on the CO 2 removal flux was insignificant, the CO 2 removal flux was kept around 0.004 mol/m 2 ·s and was not influenced by the change of the NO 2 inlet mole fraction. By contrast, there was a slight increase in the removal flux of NO 2 which caused an increase form 3 × 10 −5 to 15 × 10 −5 mol/m 2 ·s. This was attributed to the low inlet concentration of NO 2 (in ppm) compared with the CO 2 inlet concentration (2%), and consequently, the amount absorbed from CO 2 and NO 2 did not change significantly.
Processes 2019, 7, x 10 of 12 range of the experimental data [23] under the same conditions. The effect of change in the inlet mole fraction of NO2 on the CO2 removal flux was insignificant, the CO2 removal flux was kept around 0.004 mol/m 2 ·s and was not influenced by the change of the NO2 inlet mole fraction. By contrast, there was a slight increase in the removal flux of NO2 which caused an increase form 3 × 10 −5 to 15 × 10 −5 mol/m 2 ·s. This was attributed to the low inlet concentration of NO2 (in ppm) compared with the CO2 inlet concentration (2%), and consequently, the amount absorbed from CO2 and NO2 did not change significantly.

Conclusions
Model equations based on material balance were utilized to describe and study the simultaneous detention of NO2 and CO2 with aqueous NaOH solution in a membrane module. The hollow fiber membranes were fabricated from PTFE polymer. The model equations were solved, and the model predicted results were compared with data from experimental investigation available in literature. The model was found to be in good agreement with the experimental findings. The mathematical model was then employed to study the influence of the inlet flow rate of gas and liquid, concentration of CO2 and NO2 in the feed stream on their percent removal and molar flux. The results revealed that the increase in CO2 inlet mole fraction and gas cross-flow velocity shows a strong impact on the molar flux. By contrast, the change in the NO2 inlet concentration showed insignificant influence on the CO2 removal flux. Figure 7. Effect of NO 2 mole fraction in the feed gas stream on the removal molar flux of CO 2 (left) and NO 2 (right) at other fixed parameters (liquid velocity: 0.05 m/s; gas velocity: 2.11 m/s; 0.5 M NaOH; 100 to 500 ppm NO 2 ; 2% CO 2 ; the balance is N 2 ).

Conclusions
Model equations based on material balance were utilized to describe and study the simultaneous detention of NO 2 and CO 2 with aqueous NaOH solution in a membrane module. The hollow fiber membranes were fabricated from PTFE polymer. The model equations were solved, and the model predicted results were compared with data from experimental investigation available in literature. The model was found to be in good agreement with the experimental findings. The mathematical model was then employed to study the influence of the inlet flow rate of gas and liquid, concentration of CO 2 and NO 2 in the feed stream on their percent removal and molar flux. The results revealed that the increase in CO 2 inlet mole fraction and gas cross-flow velocity shows a strong impact on the molar flux. By contrast, the change in the NO 2 inlet concentration showed insignificant influence on the CO 2 removal flux.

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