Modeling and Simulation of the Absorption of CO₂ and NO₂ from a Gas Mixture in a Membrane Contactor

Abstract

The removal of undesirable compounds such as CO₂ and NO₂ 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 NO₂ and CO₂ had been widely considered in the past decades. In this study, membrane contactor was used for the simultaneous absorption of CO₂ and NO₂ from a mixture of gas (5% CO₂, 300 ppm NO₂, balance N₂) with aqueous sodium hydroxide solution. For the first time, a mathematical model was established for the simultaneous removal of the two undesired gas solutes (CO₂, NO₂) from flue gas using membrane contactor. The model considers the reaction rate, and radial and axial diffusion 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 effect 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 effect 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.

1. 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 (NO₂) 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, SO₂, and CO₂ 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 CO₂ 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 CO₂ 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 CO₂ in amine solutions [4,6]. The most widely used commercial and economical method is the chemical absorption technique, used in the conventional absorption packed bed towers employed in the absorption of CO₂, H₂S, and NO₂ from chimney gas and natural gas via alkanolamine solutions, where the flue gas and the absorbent liquid are in direct contact. The conventional scrubbing method requires a huge absorption column with excess liquid absorbent and a large cross-sectional area in order to prevent foaming and channeling. The large amount of absorbent liquid utilized in the absorption process (i.e., rich solvent) increases the operating and regeneration cost as more heat is required for the regeneration and pumping of the recycled lean solvent. The main disadvantages of conventional chemical absorption processes are channeling, foaming, corrosion, and a large space area required and hence high operating and capital cost. The idea of using membrane contactor was first proposed for the absorption of carbon dioxide in sodium hydroxide as a liquid absorbent utilizing a non-dispersive microporous membrane where gas and liquid phases are not dispersed in each other [7,8]. A hollow fiber membrane contactor (HFMC) provides a greater surface contact area per unit volume, more than that of a conventional absorption column [9,10,11,12,13]. In most cases of membrane contactor operation, gas flows in the shell side, and liquid absorbent flows in the tube side; vice versa is also possible [14]. The performance of a HFMC declines when the micropores of the membranes are wetted with a liquid solvent [3,11,15,16,17]. The advantages of membrane processes are as follows: Gas and liquid flow rates are independent, high ratio of surface area per volume, easy scale up and down, and no worry about flooding and channeling [18]. The membrane acts as an obstacle between gas and liquid and delivers an exchange surface zone for the two phases, without the dispersion of the gas phase in the liquid phase. In the non-wetted mode, the membrane pores are filled with gas, and by contrast, in the wetted mode, part of the membrane pores are filled with the liquid absorbent. The pollutant gas compounds’ amputation process occurs when the gas filling the membrane micropores diffuses from the gas stream and is absorbed by the liquid absorbent running in the membrane lumen ide [1,16,19]. The removal of the contaminant gas compounds from the gas stream depends on the solubility of the acid gas molecules in the absorbent liquid, and on the concentration incline among the gas stream absorbent solution. The interaction between the selected gas solute and the selective absorbent liquid defines the performance of the pollutant gas removal rate [20]. Membrane fouling and membrane wettability are the main drawbacks of the membrane contactor. The wetted portion of the membrane adds an additional mass transfer resistance. Accordingly, in order to avoid membrane wettability, researchers focused on the use of hydrophobic polymeric membranes, such those made from polypropylene (PP), polytetrafluoroethylene (PTFE), polyvinylidene fluoride (PVDF), and polyethylene (PE) [21,22,23,24]. The Gas Liquid Hollow Fiber Member Contactor (GLMC) processes were used for the removal of CO₂ from nitrogen, CO₂ from natural gas, and the instantaneous removal of CO₂ and SO₂ from combustion released gas [17,19,20,25,26,27,28,29,30,31,32,33]. The experimental simultaneous capture of CO₂ and NO₂ from a pretended flue gas mixture containing CO₂/NO₂/N₂ was first investigated using a commercial PTFE membrane by Ghobadi et al. [23]. The effects of the gas and liquid cross-flow velocity on the percent removal of these gasses were investigated. Various mathematical models were developed for the removal of carbon dioxide, sulfur dioxide, and hydrogen sulfide from simulated flue gas and natural gas streams [7,9,15,17,25,32,34,35,36,37,38,39]. To the extent of the author’s knowledge, so far, there is no mathematical model published to designate the synchronized capture of CO₂/NO₂ from a mixture of gas consisting of CO₂/NO₂/N₂.

This study focusses on the development of a numerical model for the capture of CO₂/NO₂ from a gas containing: 5% CO₂, 300 ppm NO₂, and the balance is N₂. The principal model equations were solved using COMSOL Multiphysics (Version 5.4, Comsol Inc., Zürich, Switzerland). The developed model was used to predict the influence of various operating parameters on the percent removal and molar flux of the acid gas components. The model was verified and validated with experimental data from literature [23].

2. Model Development

The gas-liquid membrane process consists of many hollow fibers assembled in a module, where the liquid solvent flows inside the membrane lumen, and the gas flows in the shell sideways, 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.

The sizes of the membrane used in the simulation are presented in

Table 1. Membrane module dimensions [23].

Property	Value
Inner fiber radius (mm)	0.34
Outer fiber radius (mm)	0.60
Number of fibers	590
Module outer radius (mm)	25.4
Module effective length (mm)	200

.

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 (CO₂, NO₂, and N₂) 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 + H₂O) 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.

2.1. Tube Side

The mass balance equations for the gas components of CO₂, NO₂, and N₂ in the tube side can be stated, as per Equation (1), as follows:

$$D_{i,t}\frac{1}{r}\left(\frac{\partial }{\partial r}r\left(\frac{\partial C_{i,t}}{\partial r}\right)\right)+D_{i,t} \frac{{\partial }^2{C}_{i,t}}{\partial z^2}=v_{z,t}\left(\frac{\partial C_{i,t}}{\partial z}\right)+R_i$$

(1)

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: CO₂, NO₂, 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 $\xi =z/scale$. The scaling factor is substituted in Equation (1). Consequently, the equation becomes Equation (2), as follows:

$$D_{i,t}\frac{1}{r}\left(\frac{\partial }{\partial r}r\left(\frac{\partial C_{i,t}}{\partial r}\right)\right)+\frac{D_{i,t}}{scal{e}^2}\frac{{\partial }^2{C}_{i,t}}{\partial {\xi }^2}=\frac{v_{z,t}}{scale}\left(\frac{\partial C_{i,t}}{\partial \xi }\right)+R_i$$

(2)

where the velocity of liquid inside the hollow fiber ($v_{z,t}$) is described by the following parabolic equation:

$$v_{z,t}=\frac{2Q_t}{n\pi r_1^2}\left(1-{\left(\frac{r}{r_1}\right)}^2\right)$$

(3)

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:

$$\begin{array}{lll}\mathrm{at}z=0,& C_{i,t}=0& \left(\mathrm{a}\right)\\ \mathrm{at}z=H,& \frac{{\partial }^2{C}_i,t}{\partial z^2}=0& \left(\mathrm{b}\right)\\ \mathrm{at}r=0,& \frac{\partial C_i,t}{\partial r}=0& \left(\mathrm{c}\right)\\ \mathrm{at}r=r_1,& C_{i,t}=m_i{C}_{i,m}& \left(\mathrm{d}\right)\end{array}$$

(4)

where m_i is the dimensionless physical solubility of CO₂ and NO₂ in solvent, 0.82, 0.17, respectively. The values of the dimensionless physical solubility of CO₂ and NO₂ were calculated from Henry’s constant (H): 0.034 kmol/m³ atm, 0.007 kmol/m³ atm [23], respectively, using the relation m_i = RTxH. The liquid phase reactions between NO₂ and NaOH took several steps. First, NO₂ dissolved in the aqueous NaOH was reacted with H₂O, then neutralized with sodium hydroxide [41]. The controlling liquid phase reactions are as follows:

$$2{\mathrm{NO}}_2+{\mathrm{H}}_2\mathrm{O}\to {\mathrm{HNO}}_2+{\mathrm{HNO}}_3$$

(5)

$${\mathrm{HNO}}_2+\mathrm{NaOH}\to {\mathrm{NaNO}}_2+{\mathrm{H}}_2\mathrm{O}$$

(6)

$${\mathrm{HNO}}_3+\mathrm{NaOH}\to {\mathrm{NaNO}}_3+{\mathrm{H}}_2\mathrm{O}$$

(7)

The overall reaction is designated, as per Equation (8), as follows:

$$2{\mathrm{NO}}_2+2\mathrm{NaOH}\to {\mathrm{NaNO}}_2+{\mathrm{NaNO}}_3+{\mathrm{H}}_2\mathrm{O}$$

(8)

The general reaction rate is expressed in Equation (9), as follows:

$$r_{N{O}_2-NaOH}=k_{r,1}\left[N{O}_2\right]\left[NaOH\right]$$

(9)

The reaction is the second order with a rate constant, $k_{r,1}\left({\mathrm{m}}^3{\mathrm{mol}}^{-1}{\mathrm{s}}^{-1}\right)=1.0\times {10}^5$, [1]

The overall reaction of CO₂ and NaOH is represented by the following reaction [42].

$${\mathrm{CO}}_2+2\mathrm{NaOH}\to {\mathrm{Na}}_2{\mathrm{CO}}_3+{\mathrm{H}}_2\mathrm{O}$$

(10)

The rate of the reaction is determined, as per Equation (11), as follows:

$$r_{C{O}_2-NaOH}=k_{r,2}\left[C{O}_2\right]\left[NaOH\right]$$

(11)

The reaction rate constant (in Equation (11)) is $k_{r,2}=8.37 ({\mathrm{m}}^3{\mathrm{mol}}^{-1}{\mathrm{s}}^{-1})$ [23].

2.2. Membrane Side

The transport of the solute gas (CO₂ and NO₂) 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:

$$D_{i,m}\frac{1}{r}\left(\frac{\partial }{\partial r}r\left(\frac{\partial C_{i,m}}{\partial r}\right)\right)+\frac{D_i,m}{scal{e}^2}\frac{{\partial }^2{C}_{i,m}}{\partial {\xi }^2}=0$$

(12)

The proper boundary settings are specified, as per Equation (13), as follows:

$$\begin{array}{lll}\mathrm{at}z=0,& \frac{\partial C_i,m }{\partial z}=0& \left(\mathrm{a}\right)\\ \mathrm{at}z=H,& \frac{\partial C_i,m }{\partial z}=0& \left(\mathrm{b}\right)\\ \mathrm{at}r=r_1,& D_{i,m}\frac{\partial C_i,m }{\partial r}=D_{i,t}\frac{\partial C_i,t }{\partial r}& \left(\mathrm{c}\right)\\ \mathrm{at}r=r_2& C_{i,m}=C_{i,s}& \left(\mathrm{d}\right)\end{array}$$

(13)

2.3. Shell Side

The steady state mass transport of solute gas (CO₂ and NO₂) components in the shell side (no chemical reaction occurs in the module shell zone) is expressed in Equation (14), as follows:

$$D_{i,s}\frac{1}{r}\left(\frac{\partial }{\partial r}r\left(\frac{\partial C_{i,s}}{\partial r}\right)\right)+\frac{D_{i,s}}{scal{e}^2}\frac{{\partial }^2{C}_{i,s}}{\partial {\xi }^2}=\frac{v_{z,s}}{scale}\left(\frac{\partial C_{i,s}}{\partial \xi }\right)$$

(14)

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:

$$v_{z,s}=v_{z,ma\mathrm{x}}\left\{1-{\left(\frac{r_2}{r_3}\right)}^2\right\}\left\{\frac{{\left(\frac{r}{r_3}\right)}^2-{\left(\frac{r_2}{r_3}\right)}^2-2ln\left(\frac{r}{r_2}\right) }{3+{\left(\frac{r_2}{r_3}\right)}^4-4{\left(\frac{r_2}{r_3}\right)}^2+4\ln \left(\frac{r_2}{r_3}\right)}\right\}$$

(15)

The applicable boundary conditions are specified as follows:

$$\begin{array}{lll}\mathrm{at}z=H,& C_{i,s}=C_{i,0}& \left(\mathrm{a}\right)\\ \mathrm{at}z=0,& \frac{{\partial }^2{C}_i,s}{\partial z^2}=0& \left(\mathrm{b}\right)\\ \mathrm{at}r=r_2,& D_{i,s} \frac{\partial C_i,s }{\partial r}=D_{i,m} \frac{\partial C_i,m }{\partial r}& \left(\mathrm{c}\right)\\ \mathrm{at}r=r_3,& \frac{\partial C_i,s }{\partial r}=0& \left(\mathrm{d}\right)\end{array}$$

(16)

The radius of the free surface ($r_3$) can be determined as per Equation (17), as follows:

$$r_3=r_2{\left(\frac{1}{1-\phi }\right)}^{0.5}$$

(17)

The void fraction of the membrane module ($\phi$) is calculated as per Equation (18):

$$\phi =\frac{R^2-n r_2^2}{R^2}$$

(18)

where R is the module inner radius, n is the number of fibers r₁, and r₂ is the fiber outside radius. The parameters used in the model are shown in

Table 2. Physical properties used in the model.

Physical Property	Value	Reference
Diffusivity of CO2 in shell, ${\mathrm{D}}_{{\mathrm{CO}}_2,\mathrm{s}}$	1.855 × 10−5 m2/s	[9]
Diffusivity of CO2 in tube, ${\mathrm{D}}_{{\mathrm{CO}}_2,\mathrm{t}}$	1.92 × 10−9 m2/s	[19]
Diffusivity of CO2 in membrane, ${\mathrm{D}}_{{\mathrm{CO}}_2,\mathrm{m}}$	${\mathrm{D}}_{{\mathrm{CO}}_2,\mathrm{s}}$ × ε/τ	[37]
Diffusivity of NO2 in shell, ${\mathrm{D}}_{{\mathrm{NO}}_2,\mathrm{s}}$	1.54 × 10−5 m2/s	[43]
Diffusivity of NO2 in tube, ${\mathrm{D}}_{{\mathrm{NO}}_2,\mathrm{t}}$	1.4 × 10−9 m2/s	[43]
Diffusivity of solvent in tube, Ds,t	0.5 × ${\mathrm{D}}_{{\mathrm{CO}}_2,\mathrm{t}}$	Estimated
Diffusivity of NO2 in membrane, ${\mathrm{D}}_{{\mathrm{NO}}_2,\mathrm{m}}$	${\mathrm{D}}_{{\mathrm{NO}}_2,\mathrm{s}}$ × ε/τ	[37]
Porosity, ε	0.52	[23]
Tortuosity, τ	(2−ε)2/ε	[9]

.

3. 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.

4. 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 CO₂ and NO₂ from the imitated flue gas. The model was authenticated with experimental data for the simultaneous absorption of an NO₂ and CO₂ from gas mixture in a PTFE polymeric gas–liquid hollow fiber membrane [23]. The percent removal of CO₂ and NO₂ was calculated as per Equation (19), as follows:

$$\%\mathrm{Removal}=\frac{F_{g,in} C_{i,in}-F_{g,out }{C}_{i,out}}{F_{g,in }{C}_{i,in}}\times 100$$

(19)

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 CO₂ and NO₂ concentration profile throughout the model domains are shown in Figure 2. The figure reveals that, even though the solubility of CO₂ (0.75) is higher than NO₂ (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 NO₂-NaOH.

Figure 3. illustrates the association of this model’s predictions relative to the experimental results for the effect cross-flow velocity of the feed gas on the simultaneous percent removal of CO₂ and NO₂ with fixed other parameters. A comparison of the percent removal of NO₂ and the experimental data obtained from the literature was in good agreement; by contrast, there is a slight variance in the case of CO₂. The removal flux decreased with the increased inlet gas velocity, attributable to the low residence time.

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):

$$J_i=\frac{\left(y_{i,in}{Q}_{in}-y_{i,out}{Q}_{out}\right)\times 273.15\times 1000}{22.4\times T_g\times A}$$

(20)

where J_i is the removal molar flux of component i (mol/m²·s), y_i,in, y_i,out are the inlet and outlet molar fraction of component i in the gas phase, Q_in, Q_out are the inlet and outlet gas volumetric flow rate (m³/s), respectively, in gas molar volume (liter/mol) at standard conditions (1 atm and 0 °C) is 22.4; A is the membrane surface area (m²); 1000 is the conversion factor (liter/m³); Tg is the gas temperature in K. The 273.15 is the temperature at 0 °C (273.15 K).

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/m²·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.

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 NO₂ mole fraction in the feed gas stream at a fixed concentration of CO₂ (2%) on the removal flux of CO₂ and NO₂. 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 NO₂ on the CO₂ removal flux was insignificant, the CO₂ removal flux was kept around 0.004 mol/m²·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/m²·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.

5. 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.