Molecular investigation into the effect of carbon nanotubes interaction with CO2 in molecular separation using microporous polymeric membranes

The use of nanofluids has been recently of great interest to separate acidic contaminants such as CO2. The main objective of this research is to assess the influence of carbon nanotubes (CNTs) addition to distilled water on enhancing the CO2 molecular separation through a porous membrane contactor (PMC). For this aim, a comprehensive model is developed based on non-wetted and counter-current operational modes to evaluate the principal mass and momentum transport equations in tube, membrane and shell compartments of PMC. Consequently, a CFD-based axisymmetrical simulation is implemented according to finite element technique (FET) to prognosticate the results. It is found from the results that the addition of 0.1 wt% carbon nanotubes (CNTs) particles to water significantly enhances the mass transfer and consequently the CO2 molecular separation efficiency from 38 to 63.3%. This considerable enhancement can be justified due to the existence of two momentous phenomena including Brownian motion and Grazing effect, which enhance the mass transport of CO2 molecules in the PMC. Moreover, the effect of CNTs concentration, some membrane's parameters such as number of hollow fibers and porosity and also some module's design parameters including module radius and length on the CO2 separation performance are investigated in this paper as another highlight of the current work.

www.nature.com/scientificreports/ aforementioned drawbacks in traditional ones. Porous membrane contactors (PMC), as one of the well-known membrane separation techniques for capturing CO 2 molecules, have been widely studied by researchers [7][8][9][10] . The gas-liquid interfacial area in PMC systems is definite and constant. Due to the laminar regime of liquid fluid inside the fibers, the hydrodynamics of the liquid side is well-known. Thus, the diffusivity of the fiber side can be simply estimated from basic principles. With no changes in the interfacial area, the gas-liquid exposure time in PMC systems can be varied independently 11,12 . Thus, owing to these characteristics, PMC can be proposed as an appropriate and promising device for sequestration of CO 2 molecules. It is very important to select proper absorbents in MGS systems. There are several criteria in selecting absorbents, such as chemical compatibility, viscosity, surface tension, and absorption/regeneration capability. Nanofluid, as one of the absorbents which possesses the mentioned criteria, has currently attracted tremendous attention from researchers. Nanofluids consist of nanoparticles (NPs) such as Fe 3 O 4 , Al 2 O 3 , SiO 2 , and CNTs, dispersed in base fluids like water and amine solutions [13][14][15] .
It has been reported that the absorption rate of the gas solutes in MGS systems can be improved by using nanofluids. For example, Peyravi et al. experimentally investigated the CO 2 absorption enhancement by introducing various nanofluids (Fe 3 O 4 , CNTs, Al 2 O 3 , and SiO 2 ) using a HFMC 16 . The authors found that Al 2 O 3 , CNTs, Fe 3 O 4 , and SiO 2 nanoparticles could increase the absorption rate of CO 2 by 44, 38, 25, and 3%, respectively. Golkhar et al. studied CO 2 capture from CO 2 and air gaseous mixture by applying water-based SiO 2 and CNTs nanofluids through a porous polypropylene membrane 17 . They perceived that CNTs, due to their hydrophobicity and high adsorption capacity, had better separation performance compared to SiO 2 nanoparticles and improved the CO2 adsorption rate by 40%.
Currently, the application of computational fluid dynamics (CFD) approach has been of significant interest among various researchers all over the world to solve the principal momentum/mass transfer partial differential equations (PDEs) and analyze the removal efficiency of major greenhouse contaminants like CO 2 , NO 2 , SO 2 and H 2 S from different gaseous flows [18][19][20][21][22][23]  This paper aims to implement a comprehensive modeling and its corresponding CFD-based axisymmetrical simulation to evaluate the influence of CNTs addition to distilled water on improving the molecular separation of CO 2 molecules. Mechanistic model is developed to analyze the governing equations inside the principal domains of PMC related to mass and momentum transport processes and CFD-based axisymmetrical simulation is assembled to predict the results. As another highlight, this work aims to assess the effect of CNTs concentration, membrane's design parameters (i.e., number of hollow fibers and porosity) and also some module's design parameters (i.e., module radius/length) on the CO 2 removal performance in order to optimize the separation process. Moreover, accurate prediction of CO 2 sequestration percentage from CO 2 /N 2 gaseous flow using distilled water and CNTs water-based nanofluid procedure is considered as another highlight of this investigation. Although computational simulations of membrane systems have been studied in literature, a thorough understanding the mass transfer at molecular level is required for design of this novel systems, which is addressed in the current study using a comprehensive multidimensional mechanistic model.

Model development
A mathematical modeling and its corresponding computational axisymmetrical simulation is aimed to be implemented considering a cylindrical coordinate system to numerically/computationally study the molecular removal of CO 2 applying distilled water and CNTs water-based nanofluid as absorbents in the PMC. The schematic design of CO 2 mass transport and geometrical structure of a PMC is presented in Fig. 1.
The PMC comprised of three domains including shell, tube, and membrane. The mass transfer of CO 2 is the consequence of the CO 2 molecular diffusion from the gas mixture, which circulates inside the shell section (from top to bottom), to the pores of membrane wall and the chemical capture via distilled water and CNTs waterbased nanofluid as absorbents flowing counter-currently through the PMC tube section (from bottom to top). Figure 2 shows the depiction of the contactor module's cross section and Happel's model. The latter deals with the influence of surrounding shell's void fraction on the fluid stream around each hollow fiber and is applied to prognosticate the assumptive effective radius (diameter) of shell (r 3 ) around each fiber necessary for gas-liquid operational contact 26 . Table 1 presents the characteristics and dimensions of CNTs existed in the CNTs water-based nanofluid considered in the simulations.
COMSOL software version 5.2, which functions on the basis of Finite Element Technique (FET), is used as an authentic commercial package to solve partial differential equations (PDEs) associated with the mass and momentum transport in the abovementioned sections of PMC. The UMFPACK is an eligible numerical solver among the extensive range of solvers due to its significant characteristics such as robustness and ability of solving stiff and non-stiff boundary problems. The utilized working conditions, the detailed specifications of employed PMC and physicochemical parameters of CO 2 , distilled water and CNTs water-based nanofluid are listed in Table 2.
There are two major mechanisms for increasing the mass transport rate in the presence of nanofluids, which can be described by Brownian motion and the Grazing effect. The first mechanism is random movements of nanoparticles, which can cause an increase in the velocity and induce micro-convection around the nanoparticles Scientific RepoRtS | (2020) 10:13285 | https://doi.org/10.1038/s41598-020-70279-5 www.nature.com/scientificreports/ and ultimately lead to mass diffusion flux enhancement as well as improvement of diffusion coefficient in the mass transfer domain 34 . The second mechanism expresses the gas adsorption in the presence of particles at the liquid-gas interface 35 . The velocity of distilled water and CNTs water-based nanofluid in the tube section is assumed to be fully developed, which may be justified by the ignorance of end effects and the particles impacts due to their low concentration. The gas flow in the shell side is described by the Happel's model. The simplifying assumptions considered in the model development are as below 36-41 :   www.nature.com/scientificreports/ • Isothermal process and steady state circumstance; • It is assumed that CNTs are spherical and homogeneous; • The gas phase through the shell follows the ideal behavior; • Non-wetted condition in the micropores; • The assumption of incompressible and Newtonian fluid flow of the CNTs water-based nanofluid; • It is assumed that radial convection can be negligible; • Henry's law is employed to interpret the gas phase-nanofluid equilibrium; Continuity equation for CO 2 molecular mass transfer in the PMC's shell compartment can be attained by using Fick's law for prediction of diffusive flux as below 42-45 : where, V z,s , D CO 2 ,s and z are the z-direction's axial velocity in the shell part, CO 2 diffusion coefficient inside the shell and distance along the fiber length, respectively. The gas phase velocity profile in the shell part is elucidated by the assumption of HFSM and laminar flow pattern through the following equation 40,43,46 : In Eq. (2), − V s and r 2 stand for the average velocity of shell side (gas phase) and the PMC's outer fiber radius, respectively. Besides, r 3 describes the shell side's assumptive radius, which is computed as 28,47,48 : In Eq. (3), the packing density in the PMC is explained by (1 − ω) and is computed as follows 47-49 : In Eq. (4), R 2 and n represent the module radius and the number of fibers embedded in the module, respectively. Moreover, by mixing the two previous equations (Eqs. 3 and 4), r 3 is determineded as 0.0005 m. The utilized boundary conditions in the shell section are given as below: CNTs true density 2.2 g cm −3 17 Membrane tortuosity ( τ) www.nature.com/scientificreports/ It is perceived that the addition of optimum amount of particles is able to increase the molecular mass transfer process of CO 2 molecules through the PMC due to increasing the gas-liquid operational interface. The first mechanism proposed for justifying this behavior is Brownian motion. The emergence of velocity disturbance field due to particles micro-convection may enhance the diffusivity of nanofluid. For this case, no mathematical/ theoretical equation exists but currently, an experimental-based relationship has been rendered in some literature. Based on these investigations, the CNTs water-based nanofluid's diffusion coefficient is derived as follows 14,50 : In the abovementioned equation, m 1 = 1,650, m 2 = 0.039, m 3 = − 1.064 and m 4 = 0.203 14,50 . In this equation, ∅ , Sc and Re are respectively denoted as volume fraction, Schmidt and Reynolds numbers for the Brownian motion of CNT. The amount of Re dimensionless number can be calculated using the following equation 51 : In this equation, K , T , ρ , ρ p , d p and µ are respectively interpreted as the Boltzmann constant, temperature, density of carrier fluid, density of CNTs particles, particles diameter, and carrier fluid's viscosity. Grazing effect (mechanism of shuttle effect in gas-liquid systems) is considered as the second principle mechanism to justify the enhancement of the CO 2 molecular mass transfer process 17,52,53 . The Grazing effect is applied to describe the gas transport process from the liquid-gas interface to the bulk of liquid phase. The Grazing effect may be investigated by dividing the liquid phase into two separate phases including solid and liquid phases. Therefore, the mass transfer equation (continuity equation) must be derived for both liquid (distilled water) and solid (CNTs) phases. Continuity equation for CO 2 in the solid (CNTs) is derived by the following equation 17,34,52,53 : In Eq. (11), k p and a p stand for the mass transfer coefficient between CNTs and distilled water (liquid phase) and specific surface area of CNTs. The value of k p is obtained via the following equation 34 : Langmuir isothermal adsorption model (LIAM) is applied to calculate q (amount of CO 2 molecules adsorbed by CNTs) as follows 34 : In the above equation, q m , k d and C s are respectively expressed as the highest amount of adsorption using CNTs, Langmuir constant and CO 2 molecular concentration at the interface of liquid-solid. Application of mass balance equation for CO 2 molecules in the tube section of PMC and considering CO 2 adsorption on the surface of CNTs results in the appearance of C S . The fundamental mass transfer equation based on the steady state and non-wetted conditions for component CO 2 molecules inside the PMC tube side is gained as below 23,28,42,44,48 : where, D CO 2 ,tube stands for the diffusion coefficients of CO 2 molecules inside the PMC's tube compartment. Moreover, R i and V z,tube state the reaction rate and axial velocity, respectively. The flow regime inside the tube is assumed to be Newtonian laminar flow. Accordingly, the axial velocity distribution can be defined by the following equation 28,43 : In Eq. (15), − V t , r and r 1 express the average velocity in the tube side of fiber, radial direction and the inner radius of fibers, respectively. The boundary conditions employed in the tube section are interpreted as below: (6) at r = r 3 : ∂C CO 2 ,shell /∂r = 0 (7) at z = 0: Convective flux (8) at z = L: C CO 2 ,shell = C initial www.nature.com/scientificreports/ The continuity equation based on steady state and non-wetted operating modes inside the PMC's membrane segment is given by Eq. 20 28,38,42,43,48 . Non-wetting assumption caused that the fiber pores are only filled with the gas molecules. Hence, the principle mass transport mechanism through the membrane micropores is diffusion of CO 2 molecules inside the gas: In the abovementioned equation, C CO 2 ,mem and D CO 2 ,mem stand for the amount of CO 2 concentration and molecular diffusion coefficient in the membrane pores, respectively. D CO 2 ,mem posesses a direct relationship with membrane porosity ( ε ) and inverse relationship with membrane tortuosity ( τ ) which can be derived as 28,40,43,44 : In Eq. 21, CO 2 molecular diffusion coefficient in the shell of PMC is defined by D CO 2 ,shell . Employed boundary conditions in the membrane section are given as below:

Results and discussion
Validation of modeling/simulation results. To verify the validity of modeling/simulation results, they are compared with existed experimental data reported in literature. Indeed, the CFD-based results of CO 2 sequestration percentage using CNTs water-based nanofluid are compared with measured data provided by Peyravi et al. in various gas flow rates (Q g ) 16 . The sequestration yield of CO 2 molecules applying distilled water and CNTs water-based nanofluid is presented in the following equation, where C and Q are respectively expressed as the concentration of CO 2 in the gas phase and volumetric gas flow rate 44,48,54 : Figure 3 implies the presence of a great agreement between the measured data and modeling/simulation results for CO 2 molecular separation percentage using CNTs water-based nanofluid with an average deviation (AD) of about 6%.
Influence of the CNTs addition on CO 2 concentration distribution and molecular sequestration. Figure 4a,b demonstrate the modeled surface plots of CO 2 concentration distribution in all main domains of PMC applying distilled water and CNTs water-based nanofluid as absorbents, respectively. An inlet gaseous flow containing CO 2 and N 2 enters the PMC from the top side (at z = L) to the bottom side (at z = 0), while the absorbents including distilled water and CNTs water-based nanofluid enter the tube side of the PMC from the bottom side (at z = 0) to the top side (at z = L) in a counter-current operational mode. Diffusion process of CO 2 molecules takes place due to the concentration gradient from the gas mixture in the shell side through the membrane wall pores and its simultaneous removal by absorbents in the tube side. The capability of N 2 as a carrier gas to dissolve in distilled water and CNTs water-based nanofluid is negligible compared to CO 2 . Both surface plots illustrate that the CO 2 molecular concentration in the tube pathway of PMC is much lower than the membrane and shell side, which is attributed to the diffused CO 2 sequestration by the distilled water and CNTs water-based nanofluid as liquid absorbents. Figure 5 illustrates the influence of the CNTs addition to distilled water on improving the molecular sequestration of CO 2 pollutant inside the PMC. As can be observed, addition of 0.1 wt% CNTs to water dramatically decreases the CO 2 concentration at the outlet of shell pathway from 0.62 to 0.367, which implies a major enhancement in the molecular separation efficiency of CO 2 acidic gas from 38 to 63.3%. This increment can be well justified due to the existence of Brownian motion in the nanoparticles that eventuates in increasing the CO 2 molecular diffusion and consequently its mass transfer inside the CNTs water-based nanofluid.
(17) at r = r 1 : C CO 2 ,tube = m CO 2 C CO 2 ,mem (18) at z = 0: C CO 2 ,tube = 0, C solution,tube = C initial (19) at z = L: Convective flux of CNTs concentration on the molecular separation performance of CO 2 is rendered in Fig. 6. According to derived Eqs. (9) and (11), increment in the CNTs concentration directly influences both Brownian motion and Grazing effect. Additionally, particles concentration possesses significant effect on the particles size and nanofluids stability 55 . Following the presented explanation, enhancement of the CNTs concentration improves the   www.nature.com/scientificreports/ agglomeration feasibility of particles. The CNTs agglomeration declines the micro-convection process in the proximity of particles, which eventuates in decreasing the effective surface area, declining mass transport rate and therefore, decreasing CO2 molecular removal performance. Additionally, the demonstrated deterioration in the CO2 removal efficiency is also associated with the decrement in the CO 2 concentration at the solid-liquid interface that is the consequence of more particles addition. According to the Grazing effect, by declining the value of CO 2 concentration, it's adsorption rate on the particles decreases, which possesses negative influence on the separation percentage of CO 2 molecules. As may be observed, increase in the CNTs concentration from 0.08 to 0.2 wt% decreases the separation efficiency of CO 2 from 65.8 to 61.7%.
Influence of the membrane's design parameters on the CO 2 separation efficiency. Number of porous hollow fibers and porosity are considered as the momentous membrane's design parameters, which their impacts on the separation efficiency of CO 2 molecules are aimed to be investigated in this study. By increasing the number of porous hollow fibers in the contactor, the contact area of CO 2 /N 2 gaseous mixture and liquid absorbing agents (distilled water and CNTs water-based nanofluid) increases, which provides more suitable opportunity for operational mass transfer process of CO 2 molecules inside the PMC from the shell side to the fiber pores, which improves the CO 2 molecular separation. Table 3 renders the percentage of CO 2 molecular separation in different number of porous hollow fibers using employed distilled water and CNTs water-based nanofluid absorbing agents. D CO 2 ,mem as the effective diffusion coefficient of porous membrane possesses a direct relationship with the fiber porosity. Enhancement of the fiber porosity results in improving D CO 2 ,mem parameter and consequently the mass transport in the membrane side of PMC, which is the main reason of increasing the CO 2 molecular separation percentage. Table 4 represents the percentage of CO 2 molecular separation in different values of porosity using employed distilled water and CNTs water-based nanofluid absorbing agents.
Influence of the module's design parameters on the CO 2 separation efficiency. Module radius and module length are regarded as important module-related design parameters, which their impression on the molecular separation performance of CO 2 acidic is shown in Tables 5 and 6, respectively. As can be seen, enhancement of the module length significantly increases the duration of liquid-gas contact and also their resi-  www.nature.com/scientificreports/ dence time inside the PMC, which encourages the molecular separation efficiency of CO 2 . Table 5 lists the percentage of CO 2 molecular separation in different values of module length using distilled water and CNTs water-based nanofluid absorbing agents. Table 6 indicates the impact of module radius on the efficiency of CO 2 separation. According to Eq. 4, increase in the module radius (R) declines the amount of packing density ( 1 − ϕ) inside the PMC, which eventuates in decreasing the gas-liquid contact area and consequently the molecular mass transfer of CO 2 molecules and CO 2 molecular separation inside the PMC.

conclusions
In an endeavor to evaluate the influence of CNTs addition to distilled water on the removal performance of CO 2 molecules, a mechanistic modeling and a computational 2D simulation are developed. Solution of principle equations in tube, membrane and shell compartments of PMC related to mass and momentum transport process are conducted using CFD, which operates based on finite element technique (FET). The validity of developed modeling and simulation results are corroborated by comparing them with obtained experimental data and a reasonable agreement is found with an average deviation of nearly 6%. It is understood from the simulation results that the addition of 0.1 wt% CNTs to distilled water enhances the molecular separation of CO 2 acidic gas from 38 to 63.3%, which implies about 40% enhancement in the separation efficiency of CO 2 . This increment may be justified due to the existence of Brownian motion in the nanoparticles that eventuates in increasing the CO 2 molecular diffusion and consequently its mass transfer inside the CNTs water-based nanofluid. It is identified from the results that the addition of CNTs is considered as a promising technology to improve the molecular separation efficiency of CO 2 molecules inside membrane contactors.
Received: 22 May 2020; Accepted: 27 July 2020 Table 6. Percentage of CO 2 molecular separation in different amount of module radius.