Magnetic field influence on nanofluid thermal radiation in a cavity with tilted elliptic inner cylinder

doi:10.1016/j.molliq.2016.12.024

Journal of Molecular Liquids

Volume 229, March 2017, Pages 137-147

https://doi.org/10.1016/j.molliq.2016.12.024 Get rights and content

Highlights

•
Lorentz forces effect on nanofluid free convection is investigated.
•
Thermal radiation effect is considered in energy equation.
•
CVFEM is selected for numerical procedure.
•
Nu_ave enhances with increase of ϕ, Ra and Rd

Abstract

In this attempt, influence of Lorentz forces on Fe₃O₄–water nanofluid is presented. Radiation source term is taken in to account in energy equation. Newly suggested model is imposed for viscosity of ferrofluid. Control Volume based Finite Element Method is selected to simulate this article. Graphs have been portrayed in order to explain the roles of Radiation parameter(Rd), inclination angle(ξ), Fe₃O₄-water volume fraction(ϕ), Hartmann(Ha) and Rayleigh (Ra)numbers. Obtained findings indicate that Nusselt number enhances with augment of inclination angle. Rate of heat transfer augments with enhance of buoyancy forces, radiation parameter but it reduces with rise of Lorentz forces.

Introduction

Newly, novel type of fluid should be replaced instead of common working fluid in heating system. Nano technology was offered as new method to augment heat transfer. Ayub et al. [1] investigated the electromagnet hydrodynamic boundary layer flow of nanofluid. They utilized shooting method in their paper. Wavy duct in existence of Brownian forces was investigated by Shehzad et al. [2]. They chose Nelder-Mead method to find the solution. Bhatti et al. [3] presented the entropy generation of nanofluid through a porous plate. They concluded that entropy generation enhances with rise of radiation and thermophoresis parameters. Sheikholeslami and Rokni [4] utilized the Buongiorno model for nanofluid flow in existence of induced magnetic field. Sheikholeslami and Ganji [5] applied semi analytical approach for nanofluid heat transfer under the impact of induced magnetic field. They considered Joule heating term in energy equation.

Bhatti and Rashidi [6] reported the influence of diffusion -thermo on Williamson nanofluid over a plate. Sheikholeslami and Ganji [7] gathered various application of nanofluid in an article. Thermal Radiation impact on MHD nanofluid entropy production over a plate was examined by Bhatti et al. [8]. Sheikholeslami and Chamkha [9] studied about impact of electric field on nanofluid flow and heat transfer. Chebyshev spectral collocation method has been selected by them. Zeeshan et al. [10] studied the magnetic dipole influence on flow in existence of radiative heat transfer. Rahman et al. [11] investigated the impacts of nanoparticles for the blood flow through an artery. They considered the slip impacts along with porous nature of the arterial wall. Ellahi et al. [12] analyzed free convection of carbon nanotubes over a cone. They considered the Lorentz forces impact in governing equations. Akbar et al. [13] investigated peristaltic Cu-water nanofluid flow in porous tube. They proved that as Cu concentration augments, pressure gradient decreases. Sheikholeslami and Shehzad [14] presented nanofluid convective heat transfer in a porous cavity. Ellahi et al. [15] illustrated the impact of particle shape on marangoni convection. They showed that sphere nanoparticles have maximum heat transfer rate. Sheremet et al. [16] reported the transient magnetohydrodynamic flow in an enclosure. Sheikholeslami [17] investigated the Lorentz forces impact on nanofluid flow in a permeable cylinder. Sheikholeslami and Ellahi [18] reported 3D simulation of Lorentz forces effect on nanofluid free convection. They showed that temperature enhances with rise of Lorentz forces. Sheikholeslami et al. [19] simulated the impact of inconstant Lorentz force on forced convection. They proved that higher Kelvin forces impact can be seen as Reynolds number enhances. In recent years, several researchers investigated about nanofluid thermal behaviors [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36].

The objective of present attempt is simulation of magnetohydrodynamic nanofluid flow and radiative heat transfer. By vorticity stream function approach, governing equations are solved computationally by using CVFEM. The influences of various physical fascinating parameters (Radiation parameter, Fe₃O₄-water volume fraction, Hartmann and Rayleigh numbers) on flow and heat transfer are plotted and considered through graphical illustrations.

Section snippets

Problem statement

Fig. 1 depicts the geometry, boundary condition and sample element. The formula of inner cylinder is: $b = \sqrt{1 - ε^{2}} . a$

where a , b , εare the major, minor axis of elliptic cylinder and eccentricity for the inner cylinder. The inner cylinder has constant heat flux condition.

Governing formulation

2D steady convective flow of nanofluid in a porous media is considered in existence of constant magnetic field. The PDEs equations are: $\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0,$ $(v \frac{\partial u}{\partial y} + \frac{\partial u}{\partial x} u) = [- σ_{nf} B_{y}^{2} u + σ_{nf} B_{x} B_{y} v + (\frac{\partial^{2} u}{\partial y^{2}} + \frac{\partial^{2} u}{\partial x^{2}}) μ_{nf} - \frac{\partial P}{\partial x}] {(ρ_{nf})}^{- 1},$ $\begin{array}{l} ρ_{nf} (\frac{\partial v}{\partial x} u + \frac{\partial v}{\partial y} v) = + μ_{nf} (\frac{\partial^{2} v}{\partial x^{2}} + \frac{\partial^{2} v}{\partial y^{2}}) - \frac{\partial P}{\partial y} + B_{y} σ_{nf} B_{x} u - B_{x} σ_{nf} B_{x} v \\ + (T - T_{c}) β_{nf} g ρ_{nf}, \\ B_{x} = B_{o} cos λ, B_{y} = B_{o} sin λ, \end{array}$ $\begin{array}{l} {(ρ C_{p})}_{nf} (v \frac{\partial T}{\partial y} + u \frac{\partial T}{\partial x}) = k_{nf} (\frac{\partial^{2} T}{\partial x^{2}} + \frac{\partial^{2} T}{\partial y^{2}}) - \frac{\partial q_{r}}{\partial y}, \\ [q_{r} = - \frac{4 σ_{e}}{3 β_{R}} \frac{\partial T^{4}}{\partial y}, T^{4} ≅ 4 T_{c}^{3} T - 3 T_{c}^{4}] . \end{array}$

(ρC_p)_nf , (ρβ)_nf, ρ_nf , k_nfand σ_nf are defined as: ${(ρ C_{p})}_{nf} = {(ρ C_{p})}_{f} (1 - ϕ) + {(ρ C_{p})}_{s} ϕ,$ ${(ρβ)}_{nf} = {(ρβ)}_{f} (1 - ϕ) + {(ρβ)}_{s} ϕ,$ $ρ_{nf} = ρ_{f} (1 - ϕ) + ρ_{s} ϕ,$ $k_{n f} = k_{f} (\frac{k_{s} + 2 k_{f} + 2 ϕ (k_{s} - k_{f})}{k_{s} - ϕ (k_{s} - k_{f}) + 2 k_{f}}),$ $\frac{σ_{nf}}{σ_{f}} = 1 +$

Grid independent test and validation

To reach the grid independent results, several grids are tested. As shown in Table 2, a grid size of 81 × 241 can be selected to complete this simulation. Fig. 2 demonstrates the accuracy of FORTRAN code for nanofluid natural convection heat transfer ([38], [39]). Table 3 depicts that our code has good accuracy for magnetohydrodynamic flow [40].

Results and discussion

CVFEM is utilized to simulation of magnetic field impact on nanofluid free convection. Radiation source term is taken into account. MFD viscosity is considered. Results are reported for different amount of Radiation parameter (Rd = 0to0.8), volume fraction of Fe₃O₄-water (ϕ = 0 to 0.04), Rayleigh number (Ra = 10³ , 10⁴and10⁵), inclination angle (ξ = 0^∘ and90^∘) and Hartmann number (Ha = 0to40).

Fig. 3 illustrates the impact of radiation parameter on nanofluid hydrothermal treatment. Radiation parameter can

Conclusions

CVFEM is applied to investigate the effect of Lorentz forces on ferrofluid free convection in presence of thermal radiation. MFD viscosity is considered for ferrofluid. Streamlines, Nusselt number and isotherms are reported for different Fe₃O₄-water volume fraction, inclination angle, Rayleigh number, Hartmann number and radiation parameter. Results indicated that the inner wall temperature reduces with rise of Rayleigh number but it augments with rise of Rd , Ha. Nusselt number enhances with

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Magnetic field influence on nanofluid thermal radiation in a cavity with tilted elliptic inner cylinder

Highlights

Abstract

Introduction

Section snippets

Problem statement

Governing formulation

Grid independent test and validation

Results and discussion

Conclusions

J. Mol. Liq.

Int. J. Heat Mass Transf.

J. Mol. Liq.

J. Mol. Liq.

J. Taiwan Inst. Chem. Eng.

J. Mol. Liq.

J. Mol. Liq.

Comput. Methods Prog. Biomed.

Int. J. Heat Mass Transf.

J. Taiwan Inst. Chem. Eng.

J. Mol. Liq.

Int. J. Heat Mass Transf.

Int. J. Heat Mass Transf.

Int. J. Heat Mass Transf.

Appl. Math. Comput.

Int. J. Heat Mass Transf.

J. Taiwan Inst. Chem. Eng.

J. Mol. Liq.

Int. J. Heat Mass Transf.

J. Mol. Liq.