Magnetohydrodynamic Nanofluid Natural Convection in a Cavity under Thermal Radiation and Shape Factor of Nanoparticles Impacts: A Numerical Study Using CVFEM

Abstract

In this study, the natural convection of a magnetohydrodynamic nanofluid in an enclosure under the effects of thermal radiation and the shape factor of nanoparticles was analyzed numerically using the control-volume-based finite element method (CVFEM). Columns, spheres, and lamina are examples of the nanoparticle shapes used in the investigation. The study of nanofluid flow and heat transfer was accomplished with an extensive range of nanofluid volume fractions, radiation parameters, Hartmann numbers, Rayleigh numbers, and nanoparticle shape factors. Also, the correlation between the average Nusselt number and the influencing parameters of the current study was determined. The findings demonstrate that laminar nanoparticles have a more notable impact on the average and local Nusselt numbers than the other nanoparticle shapes.

1. Introduction

The inherently moderate thermal conductivity of regular fluids, such as water, oil, etc., causes a deterioration in the performance of engineering systems. To overcome this barrier, researchers add metallic nanoscale particles—owing to the fact that they have a higher thermal conductivity than regular fluids—to these fluids. This combination is called a nanofluid. Nanofluids have higher thermal conductivity in comparison with regular fluids. Most recently, a vast number of studies have been carried out concerning heat transfer and the flow of nanofluids [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24]. Entropy generation on a magnetohydrodynamic nanofluid toward a stagnation-point flow over a penetrable expanding wall was investigated by Bhatti and Rashidi [1]. They observed the increasing velocity of the fluid due to the pronounced effects of the magnetic field and the porosity parameter. Moreover, they noted that the entropy generation number increased with increasing Brinkman number and Reynolds number. Entropy generation for a non-Newtonian Eyring–Powell nanofluid through a penetrable expanding wall was analyzed by Bhatti et al. [2]. They deduced that the velocity profile rises owing to the high impact of the suction parameter. Moreover, they concluded that Brownian motion and thermophoresis raised the temperature profile remarkably. The impacts of the flow of an electromagnetohydrodynamic nanofluid over a Riga sheet were analyzed by Abbas et al. [3]. The convective Poiseuille boundary-layer flow of an ethylene glycol (C₂H₆O₂)-based nanofluid with suspended aluminum oxide (Al₂O₃) nanoparticles through a porous wavy channel was investigated by Zeeshan et al. [4]. In the presence of thermal radiation, the heat transfer and flow of a magnetohydrodynamic Go-water nanofluid between parallel surfaces were examined by Dogonchi et al. [5]. They determined that the Nusselt number and temperature profile both increased as the radiation parameter decreased and the solid volume fraction increased. Dogonchi and Ganji [6] examined nanofluid flow, mass, and heat transfer between non-parallel surfaces subject to a magnetic field. They concluded that concentration and temperature profiles along with the Nusselt number go up as the Schmidt number rises.

Recently, several researchers have considered the natural convection flow of a pure fluid and nanofluid in enclosures [25,26,27,28,29,30,31] due to the extensive scientific and engineering applications. Heat transfer and fluid flow in a wavy enclosure subject to a porous medium and nanoparticles were explored by Sheremet et al. [25]. They illustrated that an increasing Rayleigh number causes a secondary vortex and temperature stratification core in the left bottom corner and center of the cavity, respectively. In a wavy enclosure containing nanoparticles and a porous medium, natural convection and the influence of thermal dispersion was studied by Sheremet et al. [26]. They determined that the heat transfer was enhanced with the Rayleigh number, undulation number, and dispersion parameter. Sheremet et al. [27] analyzed entropy generation and nanofluid flow in an enclosure. The outcomes indicate that a higher wavenumber leads to greater mean entropy generation. Numerical and experimental investigations on natural convection in a cavity were conducted by Mahmoudinezhad et al. [28].

This concise review illustrates that it is still worthwhile to allocate more effort to the study of natural convection in enclosures using a combination of metallic nanoscale particles and normal fluids. Therefore, this work analyzed magnetohydrodynamic nanofluid natural convection in an enclosure and the impact of the shape factor of the nanoparticles and thermal radiation using control-volume finite element method (CVFEM) and also obtained a correlation with the average Nusselt number. The flow and heat transfer effects of diverse control parameters were explored.

2. Problem Description

The natural convection of a magnetohydrodynamic nanofluid in an enclosure subject to the effects of thermal radiation and the shape factor of nanoparticles is analyzed. A sample grid distribution and a physical model of this work are displayed in Figure 1. We assume the nanofluid to be Newtonian and incompressible, while we assume that the flow is laminar. Under the above-mentioned suppositions and the Boussinesq approximation, the energy, momentum, and continuity equations for the current problem can be defined as:

$$\frac{\partial v}{\partial y}+\frac{\partial u}{\partial x}=0$$

(1)

$$v\frac{\partial u}{\partial y}+u\frac{\partial u}{\partial x}=-\frac{1}{{\rho }_{nf}}\frac{\partial p}{\partial x}+\frac{{\mu }_{nf}}{{\rho }_{nf}}\left(\frac{{\partial }^{2}u}{\partial y^2}+\frac{{\partial }^{2}u}{\partial x^2}\right)$$

(2)

$$v\frac{\partial v}{\partial y}+u\frac{\partial v}{\partial x}=-\frac{1}{{\rho }_{nf}}\frac{\partial p}{\partial y}+\frac{{\mu }_{nf}}{{\rho }_{nf}}\left(\frac{{\partial }^{2}v}{\partial y^2}+\frac{{\partial }^{2}v}{\partial x^2}\right)+{\beta }_{nf}g\left(T-T_c\right)-\frac{{\sigma }_f{B}_0^2}{{\rho }_{nf}}v$$

(3)

$$u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\frac{k_{nf}}{{\left(\rho C_p\right)}_{nf}}\left(\frac{{\partial }^{2}T}{\partial x^2}+\frac{{\partial }^{2}T}{\partial y^2}\right)-\frac{1}{{\left(\rho C_p\right)}_{nf}}\left(\frac{\partial q_{x,rad.}}{\partial y}+\frac{\partial q_{y,rad.}}{\partial x}\right)$$

(4)

In these equations v, u, B₀, P, σ_f, T, and q_rad denote the velocity in the y-direction, velocity in the x-direction, magnetic field, pressure, electric conductivity, and temperature, respectively, with the last variable representing radiative heat flux.

For radiation, by applying the Rosseland approximation, we have:

$$\begin{array}{l}{q}_{x,rad.}=-\left(4{\sigma }^{\ast }/3k_{nf}^{\ast }\right)\frac{\partial T^4}{\partial x},q_{y,rad.}=-\left(4{\sigma }^{\ast }/3k_{nf}^{\ast }\right)\frac{\partial T^4}{\partial y}\\ T^4\cong -3T_{\infty }^4+4T_{\infty }^{3}T\end{array}$$

(5)

So, Equation (4) has been altered to

$$u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\frac{k_{nf}}{{\left(\rho C_p\right)}_{nf}}\left(\frac{{\partial }^{2}T}{\partial x^2}+\frac{{\partial }^{2}T}{\partial y^2}\right)+\frac{16{\sigma }^{\ast }{T}_{\infty }^3}{3k_{nf}^{\ast }{\left(\rho C_p\right)}_{nf}}\left(\frac{{\partial }^{2}T}{\partial x^2}+\frac{{\partial }^{2}T}{\partial y^2}\right)$$

(6)

ρ_nf, μ_nf, (ρCp)_nf are defined as follows:

$${\rho }_{nf}=\varphi {\rho }_s+\left(1-\varphi \right){\rho }_f,{\mu }_{nf}={\left(1-\varphi \right)}^{-2.5}{\mu }_f,{\left(\rho C_p\right)}_{nf}=\varphi {\left(\rho C_p\right)}_s+\left(1-\varphi \right){\left(\rho C_p\right)}_f$$

(7)

Considering the influences of nanoparticle shape, the nanofluid’s thermal conductivity (k_nf) is characterized as follows:

$$\frac{k_{nf}}{k_f}=\frac{\left(m-1\right)\varphi \left(k_s-k_f\right)+k_s-\left(1-m\right)k_f}{k_s-\left(1-m\right)k_f-\varphi \left(k_s-k_f\right)}$$

(8)

The shape factor is represented here as m. The shape factor for diverse particle shapes is illustrated in

Table 1. Shape factor values for various nanoparticle shapes [32].

Particle Shapes	Sphere	Column	Lamina
m	3	6.3698	16.1576

[32]. Further, the thermophysical features of the nanofluid are listed in

Table 2. Thermophysical properties of water and nanoparticle [15].

	$\mathit{\rho }\left(\mathbf{k}\mathbf{g}/{\mathbf{m}}^3\right)$	${\mathit{C}}_{\mathit{p}}\left(\mathbf{J}/\mathbf{k}\mathbf{g}\mathbf{K}\right)$	$\mathit{k}\left(\mathbf{W}/\mathbf{m}\mathbf{K}\right)$
Copper (Cu)	8933	385	401
Pure water	997.1	4179	0.613

[15].

The vorticity and stream function are stated as:

$$u=\frac{\partial \psi }{\partial y},v=-\frac{\partial \psi }{\partial x},\omega =\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}$$

(9)

We define the following dimensionless variables:

$$Y=\frac{y}{L},X=\frac{x}{L},\mathsf{\Omega }=\frac{\omega L^2}{{\alpha }_f},\mathsf{\Psi }=\frac{\psi }{{\alpha }_f},V=\frac{uL}{{\alpha }_f},U=\frac{vL}{{\alpha }_f},\theta =\frac{T-T_c}{T_h-T_c}$$

(10)

Subject to Equation (14), the governing equations are changed to the following non-dimensional forms:

$$\frac{\partial \mathsf{\Omega }}{\partial X}\frac{\partial \mathsf{\Psi }}{\partial Y}-\frac{\partial \mathsf{\Psi }}{\partial X}\frac{\partial \mathsf{\Omega }}{\partial Y}=\frac{{\mu }_{nf}}{{\mu }_f}\frac{{\rho }_f}{{\rho }_{nf}}\mathrm{Pr}\left(\frac{{\partial }^2\mathsf{\Omega }}{\partial X^2}+\frac{{\partial }^2\mathsf{\Omega }}{\partial Y^2}\right)+\frac{{\beta }_{nf}}{{\beta }_f}Ra\mathrm{Pr}\frac{\partial \theta }{\partial X}-\frac{{\rho }_f}{{\rho }_{nf}}H{a}^2\mathrm{Pr}\frac{\partial V}{\partial X}$$

(11)

$$\frac{\partial \mathsf{\Psi }}{\partial Y}\frac{\partial \theta }{\partial X}-\frac{\partial \mathsf{\Psi }}{\partial X}\frac{\partial \theta }{\partial Y}=\frac{k_{nf}}{k_f}\frac{{\left(\rho C_p\right)}_f}{{\left(\rho C_p\right)}_{nf}}\left(1+\frac{4}{3}N\right)\left(\frac{{\partial }^2\theta }{\partial X^2}+\frac{{\partial }^2\theta }{\partial Y^2}\right)$$

(12)

$$\frac{{\partial }^2\mathsf{\Psi }}{\partial X^2}+\frac{{\partial }^2\mathsf{\Psi }}{\partial Y^2}=-\mathsf{\Omega }$$

(13)

Subject to the boundary conditions:

$$\begin{array}{ll}\theta =1& \mathrm{left}\mathrm{wall}\\ \theta =0& \mathrm{right}\mathrm{wall}\\ \frac{\partial \theta }{\partial Y}=0& \mathrm{top}\mathrm{and}\mathrm{bottom}\mathrm{walls}\\ \mathsf{\Psi }=0& \mathrm{all}\mathrm{walls}\end{array}$$

(14)

In these equations, Pr, Ra, N, and Ha denote the Prandtl number, Rayleigh number, radiation parameter, and Hartmann number, respectively, which are defined as:

$$\begin{array}{l}\mathrm{Pr}={\nu }_f/{\alpha }_f,Ra=g{\beta }_f\left(T_h-T_c\right)L^3/{\alpha }_f{\nu }_f,N=4{\sigma }^{\ast }{T}_{\infty }^{\ast }/k_{nf}{k}_{nf}^{\ast }\\ Ha=B_{0}L\sqrt{{\sigma }_f/{\mu }_f}\end{array}$$

Along the hot wall, the local and average Nusselt number can be determined by:

$$N{u}_{loc.}=\frac{k_{nf}}{k_f}\left(1+\frac{4}{3}N\right)\frac{\partial \theta }{\partial X},N{u}_{ave.}={\displaystyle {\int }_0^{1}N{u}_{loc.}dY}$$

(15)

3. Numerical Solution and Validation

We solved Equations (11)–(13) subject to the conditions in Equation (14) via the control-volume-based finite element method (CVFEM) [33,34,35,36]. To ensure mesh independence, Nu_ave was obtained for diverse mesh sizes.

Table 3. Impact of grid size on Nu_avg when Ra = 10⁵, Ha = 50, $\varphi$ = 0.04, N = 4 and m = 16.1576.

Grid Dimension (X × Y)	Nuavg
61 × 61	8.892393
81 × 81	8.848909
101 × 101	8.821577
121 × 121	8.802795
141 × 141	8.789614
161 × 161	8.784837

indicates that a grid of 141 × 141 should be applied to the present problem. Further, we measured our FORTRAN code outcomes against other works in the literature to prove the numerical solution under study [37,38]. They are in excellent accordance, as is exhibited in

Table 4. Comparison between present results and other works for Nu_avg.

Ra	Present Work	Khanafer et al. [37]	De Vahl Davis [38]
103	1.1307	1.118	1.118
104	2.2674	2.245	2.243
105	4.5851	4.522	4.519
106	8.8341	8.826	8.799

.

4. Results and Discussion

In this research, a numerical study was carried out to analyze the influences of the Hartmann number (Ha = 0–50), Rayleigh number (Ra = 10³, 10⁴, and 10⁵), radiation parameter (N = 1–3), volume fraction of the nanofluid (ϕ = 0–0.04), and shape factor (m = 3, 6.3698, and 16.1576) on magnetohydrodynamic nanofluid natural convection in a cavity. The influences of thermal radiation and the shape factor of nanoparticles were determined.

The impacts of the Ra and Ha on isotherms and streamlines are illustrated in Figure 2 and Figure 3, respectively. As shown by the increasing Ra, the stream function’s maximum absolute value increases; hence, the flow moves more quickly in the cavity. This phenomenon is due to the intensity of the natural convection, which has a direct relationship with the stream function’s maximum absolute value. On the other hand, a higher Ra leads to a distortion of the isotherms and consequently causes the vertical stratification of isotherms, particularly at lower Ha and higher Ra. That is, the thermal boundary layer develops along two side walls. This is because, at a higher Ra and a lower Ha, the prevailing mode of heat transfer is convection. Moreover, the power of natural convection and, consequently, the stream function’s maximum absolute value both diminish as Ha goes up. In addition, as Ha goes up, the central streamline expands, and the core’s temperature stratification diminishes. So, the two walls’ thermal boundary layers will disappear. In fact, in this case, the isotherms are parallel to each other. Since an increasing Ha leads to a decreased buoyancy force, the conduction mode of heat transfer dominates over the convection mode, especially at higher Ha.

Figure 4 demonstrates the influences of N and ϕ on the local and average Nusselt number. Based on Equation (15), the multiplication of three terms, i.e., (k_nf/k_f), ((4/3)N + 1), and $\partial \theta /\partial X$, will create a non-dimensional number called the local Nusselt number. By increasing ϕ, the growth of the ratio (k_nf/k_f) will overcome a smaller $\partial \theta /\partial X$ for a specified N, so the local Nusselt number will rise as ϕ is increased. Conversely, by increasing N, the increased ((4/3)N + 1) term will counteract the reduced value of $\partial \theta /\partial X$ for a given ϕ, which will result in a higher Nu_loc. However, the average Nu number behaves similarly to the local Nu number.

Figure 5 and Figure 6 illustrate the impacts of Ra, Ha, and m (sphere, column, and lamina) on the local and average Nusselt number, respectively. According to Equation (8) and Table 1, for a specified nanofluid (for example, Cu–water), as the shape factor increases, (k_nf/k_f) also increases. For instance, for a Cu–water nanofluid with ϕ = 2%, the spherical nanoparticle (m = 3) results in the lowest thermal conductivity rate (k_nf/k_f = 1.0609), while the laminar nanoparticle (m = 16.1576) yields the highest thermal conductivity rate (k_nf/k_f = 1.3216) of the studied various nanoparticle shapes in this work. Hence, based on Equation (15), by increasing the shape factor, the increased ratio (k_nf/k_f) will overcome the reduction in $\partial \theta /\partial X$ for a specified ϕ, so the local Nusselt number increases with an increased shape factor. One can easily surmise, then, that both the local and average Nusselt numbers increase with increasing Ra and decreasing Ha. These fluctuations of Nu_ave. and Nu_loc. with Ha and Ra can be explained as follows: lower values of Ra lead to a weakened buoyancy force in the system. So, in this case, the conduction mode of heat transfer governs the system. However, the convection mode dominates over the conduction mode at higher Ra because of the increased buoyancy force. Furthermore, introducing a magnetic field can lessen the power of the buoyancy force in the cavity. So, at higher Ha, the conduction mode will govern the system.

Finally, we defined the relationship for the average Nusselt number and the influencing parameters in the current study. The relationship is defined as follows:

$$\begin{array}{ll}N{u}_{ave}& =2.18015+5.79302\times {10}^{-5}\times Ra-0.067176\times Ha\\ & +1.37548\times N+6.56049\times \varphi -1.34977\times {10}^{-6}\times Ra\times Ha\\ & +6.23829\times {10}^{-6}\times Ra\times N-0.024939\times Ha\times N\\ & +1.49480\times {10}^{-3}\times H{a}^2\end{array}$$

(16)

This correlation’s R-squared equals 0.9927. The above correlation is valid for Ha = 10–30, ϕ = 0.01–0.03, N = 1–3, and m = 3.

5. Conclusions

The natural convection of a magnetohydrodynamic nanofluid in an enclosure subject to the effects of thermal radiation and the shape factor of nanoparticles was examined numerically using CVFEM. The investigation of nanofluid heat transfer and flow was conducted using an extensive range of Rayleigh numbers, radiation parameters, nanofluid volume fractions, Hartmann numbers, and nanoparticle shape factors. The outcomes indicate that the local and average Nusselt numbers increase with increasing Rayleigh number, radiation parameter, and nanofluid volume fraction, while they decrease with decreasing Hartmann number. Moreover, the laminar nanoparticle has a higher local Nusselt number than the other nanoparticle shapes. Also, in this study, a correlation for the average Nusselt number and the other factors studied was obtained with an R-squared of 0.9927.