On the entropy generation for a porous enclosure subject to a magnetic field: Different orientations of cardioid geometry

doi:10.1016/j.icheatmasstransfer.2020.104712

International Communications in Heat and Mass Transfer

Volume 116, July 2020, 104712

https://doi.org/10.1016/j.icheatmasstransfer.2020.104712 Get rights and content

Abstract

Natural or free convection is one mode of convection heat transfer where temperature changes leads to density variation. For example, free convection can happen in an enclosure that has two walls with different temperatures. In the present study, the heat transfer behavior of a cardioid shaped porous cavity is investigated. Regarding the situation of the inner cardioid, four modes can be considered for the problem that is called cases A, B, C and D. The Cu-water nanofluid has been selected as working fluid and the flow is affected by an external magnetic field. Non-dimensional governing equations (stream-vorticity functions) are solved numerically using the control volume finite element method (CVFEM). The ecological coefficient of performance (ECOP) as a criterion of cavity performance is evaluated. The effects of decision variables such as Darcy number, Hartmann number, Rayleigh number, and nanoparticle volume fraction on the entropy generation number, the ECOP and the average Nusselt number are studied. The results show that case C is better rather than other cases from the view point of the second law of thermodynamics.

Introduction

Convective heat transfer (convection) is the study of heat transport processes affected by the fluids flow [1]. Convection can be divided into two parts that are called forced and free (natural) convection (the combination of these is called mixed convection). One of the most important and interesting issues of engineering is the investigation of free convection in enclosures. Heat transfer analysis issues in enclosures can be divided at least from six aspects:

1.
geometry of enclosure: it can be simple (such as circular, rectangular, triangular, and so on) or to be complex geometry (such as non-rectangular cavities with the inclined, wavy and curved side walls) [[2], [3], [4], [5]].
2.
Working fluid: the enclosure can be filled by a pure fluid (such as water, mineral oil, ethylene glycol, and so on) or by a nanofluid (Cu- water, Al₂O₃ – water, and so on) [[6], [7], [8]].
3.
Governing equations: energy, continuity, and momentum balance equations are written for the problem depending on the assumptions (such as compressible or non-compressible flow, steady or unsteady flow, laminar or turbulent flow and so on). Also, external fields (such as magnetic or electrical fields) have effect on the governing equations (especially on the momentum equation). Internal heat generation or thermal radiation can effect on the energy equation, too [[9], [10], [11]].
4.
Boundary conditions: the boundary conditions can be assumed to be constant (such as constant wall temperature, constant heat flux or adiabatic) or to be depending on the time (such as sinusoidal wall temperature) [[12], [13], [14]].
5.
Method of solution: In general, differential equations are solved in three analytical, semi-analytical and numerical methods. Numerical methods employ a wider range of equations. Numerical methods can be the Euler method, Hun method, Taylor method, Rang-Kutta method, and so on. Also, cited network methods such as finite volume method (FVM), finite difference method (FDM), finite element method (FEM), CVFEM, lattice Boltzmann method (LBM) and non-network methods belong to the numerical methods [[15], [16], [17], [18], [19]].
6.
Results presentation: in many published papers, the streamlines and isotherms for fluid flow in an enclosure have been presented as a part of results. Also, the maximum absolute value of stream function and average Nusselt number have been reported. But, recently in new papers, the entropy generation analysis has been investigated too. In this case, the average Bejan number and entropy generation number have been reported as the main results [[20], [21]].

Natural convection analyses for enclosures have been performed by many researchers. In 2017, a good review paper has been published by Das et al. [22] that it describes and classifies the important previous works in the field of free convection in enclosures. After this, in 2018, the free convection analysis for a porous cavity studied by Dogonchi et al. [23]. They assumed copper-water nanofluid to be working fluid. They studied the effects of the Hartmann number, Rayleigh number, Darcy number, shape factor and nanoparticles percentage on the flow. They solved governing equations by CVFEM. Their results discovered that the Nusselt number has a direct relationship with all parameters except that the magnetic field angle and Hartmann number. In 2018, Izadi et al. [24], investigated free convection in a C-shaped enclosure loaded with a nanofluid using the LBM. In 2018, the entropy generation number were calculated within an enclosure in the presence of a heated plate by Sivaraj and Sheremet [25]. They discovered that the entropy generation and convective flow are suppressed by the magnetic field. In 2018, Dogonchi et al. [26] studied the free convection flow for a triangular cavity with Cu–H₂O nanofluid. In 2018, Waqas et al. [27] investigated the flow of stratified medium subjected to gyrotactic micro-organisms, heat generation, and variable conductivity. In 2018, the entropy generation number for a porous cavity was calculated by Rashad et al. [28]. They assumed that an external magnetic field has effect on the inclined square cavity loaded with a nanofluid. They investigated the size and location impacts for a heat source and a heat sink. In 2018, Dogonchi and Ganji [29] studied radiation effect on nanoliquid flow using the revised Fourier formula. In 2019, the effect of the nanoparticle shapes on entropy generation for a semi-annulus nanofluid-filled enclosure was studied by Seyyedi et al. [30]. Their results discovered that the entropy generation number enhances by ascending the Rayleigh number, the nanoparticle percent and the turn angle for the cavity. In 2019, Chamkha et al. [31] studied hybrid nanofluid flow in a rotation system between two surfaces. They considered terms of Joule heating and thermal radiation in their work. In 2019, the free convection flow in an enclosure with an elliptical-shaped inclined heater was performed by Dogonchi et al. [32]. They studied fluid flow considering a magnetic field and discovered the impacts of shape factor on the average Nusselt number. In 2019, Dogonchi et al. [33] scrutinized the heat transfer analysis using CVFEM for an annulus with considering thermal radiation. They considered Fe₃O₄-H₂O nanofluid as working fluid. In 2019, the entropy generation and the natural convective flow inside a semi-annulus cavity were examined by Seyyedi et al. [34,35]. They uncovered the impact of magneto-hydrodynamic (MHD) on the entropy generation number. Alamri et al. [36] proposed the modeling of magneto-second-grade liquid using modified Fourier formula for stretchable cylinder. In 2019, Dogonchi et al. [37] conducted the investigation of free convection subject to magnetic field for a cavity with a wavy cylindrical heater with nanofluid.

The aim of the current study is the investigation both entropy generation analysis and natural convection analysis for a complex geometry (cardioid shaped enclosure) subject to a magnetic field. As we know, electronic equipment and hydrocarbon reservoirs are constructed similar to a cardioid. Cu-H2O nanofluid has been selected as a working fluid and the effects of decision variables such as the volume fraction of nanoparticles, the Darcy number, the Rayleigh number, and the Hartmann number on the fluid flow are investigated. The privilege of the current work versus many of previous studies can be stated as follows:

1.
The geometry of the enclosure is complex i.e. cardioid enclosure.
2.
Entropy generation number is evaluated in this study.
3.
The governing equations are solved by a powerful numerical method that is called CVFEM.
4.
The ECOP, a rather new criterion for evaluation of cavity thermal performance, is computed.
5.
Two formulas are proposed for computation of the entropy generation number and average Nusselt number.

Section snippets

Physical model and boundary conditions

A cardioid shaped cavity is enclosed within a cylinder. Regarding to the situation of the inner cardioid, four modes can be considered for the problem that are called case A, B, C and D. Fig. 1 presents the cardioid shapes and their mathematical formula. The schematic diagram and the grid of the cardioid-shaped enclosure are presented in Fig. 2. The inner cardioid-shaped and outer cylinder are held at fixed temperatures T_h and T_c (T_c < T_h), respectively. An external uniform magnetic field (B₀

Governing equations

The governing equations in the non-dimensional form can be rewritten as follows [30,35]: $\frac{\partial^{2} Ψ}{\partial X^{2}} + \frac{\partial^{2} Ψ}{\partial Y^{2}} = - Ω$ $[\frac{∂Ψ}{∂Y} \frac{∂Ω}{∂X} - \frac{∂Ψ}{∂X} \frac{∂Ω}{∂Y}] = \frac{μ_{nf} / μ_{f}}{ρ_{nf} / ρ_{f}} Pr (\frac{\partial^{2} Ω}{\partial X^{2}} + \frac{\partial^{2} Ω}{\partial Y^{2}}) + \frac{β_{nf}}{β_{f}} Ra Pr \frac{∂θ}{∂X} - \frac{μ_{nf} / μ_{f}}{ρ_{nf} / ρ_{f}} \frac{Pr}{Da} Ω + \frac{σ_{nf} / σ_{f}}{ρ_{nf} / ρ_{f}} {Ha}^{2} Pr ({cos}^{2} λ \frac{\partial^{2} Ψ}{\partial X^{2}} + {sin}^{2} λ \frac{\partial^{2} Ψ}{\partial Y^{2}} + 2 sin λ cos λ \frac{\partial^{2} Ψ}{∂X∂Y})$ $[\frac{∂Ψ}{∂Y} \frac{∂θ}{∂X} - \frac{∂Ψ}{∂X} \frac{∂θ}{∂Y}] = \frac{α_{nf}}{α_{f}} (\frac{\partial^{2} θ}{\partial X^{2}} + \frac{\partial^{2} θ}{\partial Y^{2}})$

In the Eq. (2), Ra, Pr, Da, and Ha are the Rayleigh number, Prandtl number, Darcy number and, the Hartmann number respectively, which are expressed as follows: $Ha = B_{0} L \sqrt{\frac{σ_{f}}{ρ_{f} ν_{f}}}, Ra = \frac{g β_{f} L^{3} (T_{h} - T_{c})}{ν_{f} α_{f}}, Pr = \frac{ν_{f}}{α_{f}}, Da = \frac{K}{L^{2}}$ where L=r_out − r_in.

Study of mesh independence and code validation

In order to solve the governing equations, CVFEM has been applied for writing a FORTRAN code.

Results and discussion

In the present work, the entropy generation and free convection heat transfer in cardioid shaped enclosure loaded with a copper-water nanofluid are numerically investigated using CVFEM. In this study, the values of m and λ are assumed to be 3 and 0, respectively. The thermophysical properties of the based fluid and Cu-H₂O nanoparticles are displayed in Table 3.

Fig. 3 depicts the streamlines and isotherms for cases A, B, C and D at Ra = 10⁵, Pr = 6.2, ϕ = 0.04, and Ha = 0. As it can be seen

Conclusion

Both magneto natural convection and second law of thermodynamics analyses were performed for cardioid shaped enclosures (four cases) occupied by cu-water nanofluid. For solving the governing equations, the CVFEM was employed. The streamlines and the isotherms were plotted in some cases. The impacts of active parameters on the ECOP, the N_gen and the Nu_avewere studied. The major outcomes can be cited as below:

1.
The minimum and maximum values of |Ψ_max|, is corresponding to cases C and D,

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.