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 water-based 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 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^{36,37,38,39,40,41}:

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^{42,43,44,45}:

$${D}_{text{CO}_{2}, s}left[frac{{partial }^{2}{C}_{text{CO}_{2},s}}{partial {r}^{2}}+frac{1}{r}frac{partial {C}_{text{CO}_{2},s}}{partial r}+frac{{partial }^{2}{C}_{text{CO}_{2},s}}{partial {z}^{2}}right]= {V}_{z,s}frac{partial {C}_{text{CO}_{2},s}}{partial z}$$

(1)

where, ({V}_{z,s}), ({D}_{text{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}:

$${V}_{z,s}=2stackrel{-}{{V}_{s}}left[1-{left(frac{{r}_{2}}{{r}_{3}}right)}^{2}right]times left[frac{({r/{r}_{3})}^{2}-({{r}_{2}/{r}_{3})}^{2}+2mathit{ln}left({r}_{2}/rright)}{3+({{r}_{2}/{r}_{3})}^{4}-4({{r}_{2}/{r}_{3})}^{2}+4mathit{ln}left({r}_{2}/{r}_{3}right)}right]$$

(2)

In Eq. (2), ({stackrel{-}{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}:

$${r}_{3}={r}_{2}{left(frac{1}{1-omega }right)}^{0.5}$$

(3)

In Eq. (3), the packing density in the PMC is explained by ((1-omega )) and is computed as follows^{47,48,49}:

$$1-omega =frac{n{r}_{2}^{2}}{{R}^{2}}$$

(4)

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:

$$at;; r={r}_{2}{:}; { C}_{text{CO}_{2},shell}={C}_{text{CO}_{2},mem}$$

(5)

$$at;; r={r}_{3}{:}; partial {C}_{text{CO}_{2},shell}/partial r=0$$

(6)

$$at ;;z=0{:}; Convective;flux$$

(7)

$$at ;;z=L{:}; {C}_{text{CO}_{2},shell}={C}_{initial}$$

(8)

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}:

$${D}_{nf}= {D}_{bf}left(1+{m}_{1}{Re}^{{m}_{2}}{Sc}^{{m}_{3}}{varnothing }^{{m}_{4}}right)$$

(9)

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, (varnothing ), *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}:

$$Re= sqrt{frac{18KT{rho }^{2}}{pi {d}_{p}{rho }_{p}{mu }^{2}}}$$

(10)

In this equation, (K), (T), (rho ), ({rho }_{p}), ({d}_{p}) and (mu ) 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}:

$$varnothing {rho }_{p}{V}_{z}frac{partial q}{partial z}= {k}_{p}{a}_{p}({C}_{text{CO}_{2},tube}-{C}_{s})$$

(11)

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}:

$$Sh= frac{{k}_{p}{d}_{p}}{{D}_{text{CO}_{2}}}=2$$

(12)

Langmuir isothermal adsorption model (LIAM) is applied to calculate *q* (amount of CO_{2} molecules adsorbed by CNTs) as follows^{34}:

$$q={q}_{m} frac{{k}_{d}{C}_{s}}{1+{k}_{d}{C}_{s}}$$

(13)

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}:

$${D}_{text{CO}_{2},tube}left[frac{{partial }^{2}{C}_{text{CO}_{2},tube}}{partial {r}^{2}}+frac{1}{r}frac{partial {C}_{text{CO}_{2},tube}}{partial r}+frac{{partial }^{2}{C}_{text{CO}_{2},tube}}{partial {z}^{2}}right]= {V}_{z,tube}frac{partial {C}_{text{CO}_{2},tube}}{partial z}+frac{{k}_{p}{a}_{p}}{1-varnothing }({C}_{text{CO}_{2},tube}-{C}_{s})$$

(14)

where, ({D}_{text{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}:

$${V}_{z,tube}=2overline{{V}_{t}}left[1-{left(frac{r}{{r}_{1}}right)}^{2}right]$$

(15)

In Eq. (15), (stackrel{-}{{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:

$$at;; r=0{:}; partial {C}_{text{CO}_{2},tube}/partial r=0$$

(16)

$$at;; r={r}_{1}{:}; { C}_{text{CO}_{2},tube}={{m}_{text{CO}_{2}}C}_{text{CO}_{2},mem}$$

(17)

$$at;; z=0{:}; {C}_{text{CO}_{2},tube}=0 , ;;;;{ C}_{solution,tube}= {C}_{initial }$$

(18)

$$at ;;z=L{:}; Convective ;flux$$

(19)

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:

$${D}_{text{CO}_{2},mem}left[frac{{partial }^{2}{C}_{text{CO}_{2},mem}}{partial {r}^{2}}+frac{1}{r}frac{partial {C}_{text{CO}_{2},mem}}{partial r}+frac{{partial }^{2}{C}_{text{CO}_{2},mem}}{partial {z}^{2}}right]= 0$$

(20)

In the abovementioned equation, ({C}_{text{CO}_{2},mem}) and ({D}_{text{CO}_{2},mem}) stand for the amount of CO_{2} concentration and molecular diffusion coefficient in the membrane pores, respectively. ({D}_{text{CO}_{2},mem}) posesses a direct relationship with membrane porosity ((varepsilon )) and inverse relationship with membrane tortuosity ((tau )) which can be derived as^{28,40,43,44}:

$${D}_{text{CO}_{2},mem}=frac{{varepsilon D}_{text{CO}_{2},shell}}{tau }$$

(21)

In Eq. 21, CO_{2} molecular diffusion coefficient in the shell of PMC is defined by ({D}_{text{CO}_{2},shell}). Employed boundary conditions in the membrane section are given as below:

$$at;; r={r}_{1}{:}; { C}_{text{CO}_{2},mem}={C}_{text{CO}_{2},tube}/{m}_{text{CO}_{2}}$$

(22)

$$at ;;r={r}_{2}{:}; {C}_{text{CO}_{2},mem}={C}_{text{CO}_{2},shell}$$

(23)

$$at ;;z=0{:}; Insulated$$

(24)

$$at;; z=L{:}; Insulated$$

(25)