A numerical study on heat transfer and MHD flow of a Newtonian through a porous channel—Local thermal nonequilibrium model

 Vast numbers of studies concentrate on the thermal equilibrium state whereas in many real-world applications the model exists in the nonequilibrium state. Also, local thermal non-equilibrium precisely represents the thermohydroflow characteristics. Therefore, the current study examines the heat transfer and fluid flow characteristics of the magnetohydrodynamic flow of a Newtonian fluid through a local thermal non-equilibrium (LTNE) porous channel in the presence of the induced magnetic field. The mathematical model of the prescribed flow encloses the coupled nonlinear equations which are difficult to approach analytically. Hence, they are solved numerically using the shooting method with the Newton–Raphson method. The implications of various physical parameters of the problem on fluid flow, induced magnetic field, current density, temperature profiles, and heat transfer are elucidated with the aid of plots and tables. From the examination, it is clear that the porous medium significantly influences the characteristics of the fluid flow. That is, the least value of the Darcy number is related to a higher momentum field. Another interesting phenomenon is that the induced magnetic field remarkably enhances when the Darcy number is high, whereas the process is contrary to the current density. The effect of LTNE on the flow characteristics and heat transfer ceases for higher values of inter-phase heat transfer coefficient and the ratio of thermal conductivities, which gives rise to the local thermal equilibrium (LTE) situation. Furthermore, the amount of heat transport is maximum in the LTE case compared to that of the LTNE case.     

Visualization of heat flow using Bejan’s heatline due to natural convection of water near 4 °C in thick walled porous cavity

A numerical study on natural convection heat transfer of cold water near 4 °C in a thick bottom walled cavity filled with a porous medium has been performed. It is assumed that the cavity is isothermally heated from the outside of the thick bottom wall and cooled from ceiling. The finite-difference method has been used to solve the governing partial differential equations of heat and fluid flow. Effects of thermal conductivity ratio, Rayleigh number and bottom wall thickness on heat transfer from the bottom to the ceiling have been studied. The heatline visualization technique has been used to demonstrate the path of heat transport through the enclosure. Moreover, streamlines and isotherms have been used to present fluid flow and temperature distributions. The obtained results show that multiple circulation cells are formed in the cavity and the local Nusselt numbers at the bottom wall and solid–fluid interface are highly affected by formed cells. The increase of Rayleigh number and thermal conductivity ratio increases heat transfer through the cavity. However, the increase of thickness of the bottom wall reduces the mean Nusselt number. Almost one-dimensional conduction heat transfer is observed in the solid bottom wall of the cavity.     

Natural convection in a shallow porous cavity—the Brinkman model

The natural convection in a shallow porous rectangular cavity with differentially heated sidewalls is examined using the Brinkman model. The heat transfer rate through the cavity is determined in terms of a Nusselt number, in the limit of vanishingly small aspect ratio. Two types of boundary conditions are considered. Case I deals with a cavity with all rigid boundaries so that the no-slip boundary conditions can be imposed. In case II, the cavity has a free upper surface. The present analysis shows that the Brinkman model and Darcy's law give virtually the same result for the heat transfer rate when the Darcy number, based on the depth of the cavity, is less than the order of 10⁻⁴. We also find that the presence of a free surface can significantly increase the heat transfer rate through the cavity, especially when the permeability of the medium is high.     

Transient and non-Darcian effects on natural convection flow in a vertical channel partially filled with porous medium: Analysis with Forchheimer–Brinkman extended Darcy model

The present study addresses the transient as well as non-Darcian effects on laminar natural convection flow in a vertical channel partially filled with porous medium. Forchheimer–Brinkman extended Darcy model is assumed to simulate momentum transfer within the porous medium. Two regions are coupled by equating the velocity and shear stress in the case of momentum equation while matching of the temperature and heat flux is taken for thermal energy equation. Approximate solutions are obtained using perturbation technique. Variations in velocity field with Darcy number, Grashof number, kinematic viscosity ratio, distance of interface and variations in temperature distribution with thermal conductivity ratio, distance of interface are obtained and depicted graphically. The skin-friction and rate of heat transfer at the channel walls are also derived and the numerical values for various physical parameters are tabulated.     

Investigation of buoyancy-driven flow and heat transfer in a trapezoidal cavity filled with water–Cu nanofluid

A numerical study is conducted to investigate the transport mechanism of free convection in a trapezoidal enclosure filled with water–Cu nanofluid. The horizontal walls of the enclosure are insulated while the inclined walls are kept at constant but different temperatures. The numerical approach is based on the finite element technique with Galerkin's weighted residual simulation. Solutions are obtained for a wide range of the aspect ratio (AR) and Prandtl number (Pr) with Rayleigh number (Ra = 10⁵) and solid volume fraction (? = 0.05). The streamlines, isotherm plots and the variation of the average Nusselt number at the left hot wall are presented and discussed. It is found that both AR and Pr affect the fluid flow and heat transfer in the enclosure. A correlation is also developed graphically for the average Nusselt number as a function of the Prandtl number as well as the cavity aspect ratio.     

Fluid–structure interaction analysis of mixed convection heat transfer in a lid-driven cavity with a flexible bottom wall

A numerical investigation of steady laminar mixed convection heat transfer in a lid driven cavity with a flexible bottom surface is analyzed. A stable thermal stratification configuration was considered by imposing a vertical temperature gradient while the vertical walls were considered to be insulated. In addition, the transport equations were solved using a finite element formulation based on the Galerkin method of weighted residuals. In essence, a fully coupled fluid–structure interaction (FSI) analysis was utilized in this investigation. Moreover, the fluid domain is described by an Arbitrary-Lagrangian–Eulerian (ALE) formulation that is fully coupled to the structure domain. Comparisons of streamlines, isotherms, bottom wall displacement and average Nusselt number were made between rigid and flexible bottom walls. The results of this investigation revealed that the elasticity of the bottom wall surface plays a significant role on the heat transfer enhancement. Furthermore, the contribution of the forced convection heat transfer to that offered by natural convection heat transfer has a profound effect on the behavior of the flexible wall as well as the momentum and energy transport processes within the cavity. This investigation paves the road for future research studies to consider flexible walls when augmentation of heat transfer is sought.     

Analysis of conduction – natural convection conjugate heat transfer in the gap between concentric cylinders under solar irradiation

Although, in general, most of concentric cylinders used in engineering applications are heated non-uniformly, studies on the natural convection in the air layer between the cylinders have been performed only for the uniform heating so that it is difficult to apply the results to the real cases. In the present study, the effects of material property of outer cylinder and the air gap thickness between cylinders on the natural convection are investigated by computational fluid dynamics and experimental means. Namely, conduction-natural convection conjugated heat transfer in the concentric cylinders, gun and shield tube, under solar irradiation, is analyzed numerically and the various predictions on the resultant thermal deformations of gun are compared to the experimental data with variation of the gap thickness of cylinders and material properties of shield tube. Results show that larger the thermal conductivity of shield tube, smaller the Nusselt number variations of gun and shield tube surfaces. This is due to that balance of conduction in the cylinders and natural convection in the gap between inner and outer cylinders may increase the uniformity of the air layer temperature. High thermal conductivity of shield tube inhibits the natural convection in the air layer and there exists a gap thickness that exerts an influence critically on the thermal deformation of gun.     

Visualization of heat flow due to natural convection within triangular cavities using Bejan’s heatline concept

Natural convection and flow circulation within a cavity has received significant attention in recent times. The wide range of applicability of flow inside a cavity (food processing industries, molten metal industries, etc.) requires thorough understanding for cost efficient processes. This paper is based on comprehensive analysis of heat flow pattern using Bejan’s heatline concept. The key parameters for our study are the Prandtl number, Rayleigh number and Nusselt number. The values of Prandtl number (0.015, 0.026, 0.7 and 1000) have been chosen based on wide range of applicability. The Rayleigh number has been varied from 10² to 10⁵. Interesting results were obtained. For low Rayleigh number, it is found that the heatlines are smooth and perfectly normal to the isotherms indicating the dominance of conduction. But as Ra increases, flow slowly becomes convection dominant. It is also observed that multiple secondary circulations are formed for fluids with low Pr whereas these features are absent in higher Pr fluids. Multiple circulation cells for smaller Pr also correspond multiple cells of heatlines which illustrate less thermal transport from hot wall. On the other hand, the dense heatlines at bottom wall display enhanced heat transport for larger Pr. Further, local heat transfer (Nu_l, Nu_t) are explained based on heatlines. The comprehensive analysis is concluded with the average Nusselt number plots. A correlation for average heat transfer rate and Ra has been developed and the range of Rayleigh number is also found, to depict the conduction dominant heat transfer.     

MHD mixed convection of Cu–water nanofluid in a two-sided lid-driven porous cavity with a partial slip

ABSTRACT

A numerical simulation of magneto-hydrodynamic mixed convection flow and heat transfer of Cu–water nanofluid in a square cavity filled with a Darcian porous medium with a partial slip is numerically investigated. The left and right walls of the cavity are moving up with a constant speed in vertical direction, and the partial slip effect is considered along these walls. The top and bottom walls of the cavity are assumed to be adiabatic. The right vertical wall of the cavity is assumed to be kept at a lower temperature, while the left vertical wall is kept at a higher temperature. The developed equations of the mathematical model are nondimensionalized and then solved numerically subject to appropriate boundary conditions by the finite-volume method. A parametric study is performed and a set of graphical results is presented and discussed to demonstrate interesting features of the solution.     

Free convection in a shallow cavity with variable properties—2. Porous media

Free convection in a porous shallow cavity with differentially heated end walls has been studied. The governing differential equations are analytically solved by applying the method of asymptotic expansions. The results show that the constant-property solution (Boussinesq approximation) deviates approximately 3% from the variable-property solution, if the properties in the Boussinesq solution are taken as the arithmetic mean between the hot and cold end wall temperature.     

Hydromagnetic natural cooling of a triangular heat source in a triangular cavity with water–CuO nanofluid

This paper presents a numerical analysis of natural cooling of a right triangular heat source by a water–CuO nanofluid in a right triangular cavity that is under the influence of a horizontal magnetic field. A computational domain is defined and a numerical scheme based on the control volume formulation using the SIMPLE algorithm is developed. The convection–diffusion terms are discretised using a power-law scheme. The effects of the Rayleigh number, the solid volume fraction, the Hartmann number and the heat source position in the cavity on the heat transfer performance of the cavity are examined. The thermal performance of the cavity is enhanced as the Rayleigh number increases, the Hartmann number decreases and the distance of the heat source with the cold walls decreases. An optimum solid volume fraction is found that maximises the heat transfer at high Rayleigh numbers.     

Numerical study of mixed convection within porous square cavities using Bejan’s heatlines: Effects of thermal aspect ratio and thermal boundary conditions

The present numerical study deals with mixed convection flows within square enclosures filled with porous media. The influence of various thermal boundary conditions on bottom and side walls based on thermal aspect ratio (A) is investigated for a wide range of parameters (1 ? Re ? 100, 0.015 ? Pr ? 7.2, 10^?5 ? Da ? 10^?3 and 10³ ? Gr ? 10⁵). A penalty finite element method with bi-quadratic elements has been used to investigate the results in terms of streamlines, isotherms and heatlines and average Nusselt numbers. Lid driven effect is dominant at low Darcy number (Da = 10^?5), whereas buoyancy driven effect is dominant at high Darcy numbers (Da = 10^?4 and Da = 10^?3) for Re = 1. Asymmetric pattern is observed in isotherms and heatlines for Re = 100. It is found that thermal gradient is high at the center of the bottom wall for A = 0.1 due to large dense heatlines at that zone and that is low for A = 0.9 irrespective of Re, Pr and Gr. Overall heat transfer rates are higher for A = 0.1 compared to other thermal aspect ratios (A = 0.5, A = 0.9) irrespective of Darcy number, Prandtl number and Reynolds number.     

Effect of nanofluid variable properties on natural convection in enclosures filled with a CuO–EG–Water nanofluid

This work focuses on the study of natural convection heat transfer characteristics in a differentially-heated enclosure filled with a CuO–EG–Water nanofluid for different published variable thermal conductivity and variable viscosity models. The problem is given in terms of the vorticity–stream function formulation and the resulting governing equations are solved numerically using an efficient finite-volume method. Comparisons with previously published work are performed and the results are found to be in good agreement. Various results for the streamline and isotherm contours as well as the local and average Nusselt numbers are presented for a wide range of Rayleigh numbers (Ra = 10³–10⁵), volume fractions of nanoparticles (0 ≤ φ ≤ 6%), and enclosure aspect ratios (½ ≤ A ≤ 2). Different behaviors (enhancement or deterioration) are predicted in the average Nusselt number as the volume fraction of nanoparticles increases depending on the combination of CuO–EG–Water variable thermal conductivity and viscosity models employed. In general, the effects the viscosity models are predicted to be more predominant on the behavior of the average Nusselt number than the influence of the thermal conductivity models. The enclosure aspect ratio is predicted to have significant effects on the behavior of the average Nusselt number which decreases as the enclosure aspect ratio increases.     

Unsteady fluid mechanics and heat transfer study in a double-tube air–combustor heat exchanger with porous medium

Fluid mechanics and heat transfer are studied in a double-tube heat exchanger that uses the combustion gases from natural gas in a porous medium located in a cylindrical tube to warm up air that flows through a cylindrical annular space. The mathematical model is constructed based on the equations of continuity, linear momentum, energy and chemical species. Unsteady fluid mechanics and heat transfer by forced gas convection in the porous media, with combustion in the inner tube, coupled to the forced convection of air in the annular cylindrical space are predicted by use of finite volumes method. Numerical simulations are made for four values of the annular air flow Reynolds number in the range 100 ? Re ? 2000, keeping constant the excess air ψ = 4.88, the porosity ε = 0.4, and the air–fuel mixture inlet speed Uo = 0.43 m/s. The results obtained allow the characterization of the velocity and temperature distributions in the inner tube and in the annular space, and at the same time to describe the displacement of the moving combustion zone and the annular porous media heat exchanger thermal efficiency. It is concluded that the temperature increase is directly related to the outer Reynolds number.     

Bejan’s heatline analysis of natural convection in right-angled triangular enclosures: Effects of aspect-ratio and thermal boundary conditions

Natural convection in right-angled triangular enclosures with various top angles ($\phi$=15°, 30°, 45°) is studied in detail via heat flow analysis for various uniform isothermal and linear isothermal heating thermal boundary conditions. Detailed analysis on the effects of aspect-ratio and thermal boundary conditions on the fluid and heat flow inside the triangular enclosures have been carried out for a range of fluids (Pr = 7.2, 1000, 0.015) within Ra = 10³–10⁵. Interesting features of heat flow patterns under various thermal boundary conditions are ‘visualized’ by heatlines. The effect of increase in φ of triangular enclosures is such that the maximum heat flux at the top vertex decreases and the thermal mixing in cavity increases with the increase in $\phi$. It is found that, the fluid in the lower corners is adequately heated in presence of hot right wall compared to that in left wall heating cases. Further, the heat transfer characteristics, in terms of local and average Nusselt numbers, indicate that isothermal heating cases exhibit exponential decrease in Nu_l whereas linear heating cases interestingly show local intermediate maxima. Also, various qualitative and quantitative features of Nu and $\overline{Nu}$ are adequately explained based on heatlines. Finally, the correlations for $\overline{N{u}_l}$ and Ra are obtained for various fluid with all heating situations.     

Numerical study on the convective heat transfer of nanofluid in a triangular minichannel heat sink using the Eulerian–Eulerian two-phase model

In this paper, the laminar forced convection heat transfer of the water-based nanofluid inside a minichannel heat sink is studied numerically. An Eulerian two-fluid model is considered to simulate the nanofluid flow inside the triangular heat sink and the governing continuity, momentum, and energy equations for both phases are solved using the finite volume method. Comparisons of the Nusselt number predicted by the Eulerian–Eulerian model with the experimental data available in the literature demonstrate that the simulation results are in excellent agreement with the experimental data and the maximum deviation from experimental data is 5%. The results show that the heat sink with nanofluid has a better heat transfer rate in comparison with the water-cooled heat sink. Also, the heat transfer enhancement increases with an increase in Reynolds number and nanoparticle volume concentration. In addition, the friction factor increases slightly for nanofluid-cooled minichannel heat sink.     

Numerical simulations of a coupled radiative–conductive heat transfer model using a modified Monte Carlo method

Radiative–conductive heat transfer in a medium bounded by two reflecting and radiating plane surfaces is considered. This process is described by a nonlinear system of two differential equations: an equation of the radiative heat transfer and an equation of the conductive heat exchange. The problem is characterized by anisotropic scattering of the medium and by specularly and diffusely reflecting boundaries. For the computation of solutions of this problem, two approaches based on iterative techniques are considered. First, a recursive algorithm based on some modification of the Monte Carlo method is proposed. Second, the diffusion approximation of the radiative transfer equation is utilized. Numerical comparisons of the approaches proposed are given in the case of isotropic scattering.     

Numerical study of coupled double-diffusive natural convection and radiation in a square cavity filled with A N2–CO2 mixture

The coupling of transient double diffusive convection with radiation is investigated numerically in a square cavity filled with a mixture of N₂ and CO₂. Horizontal temperature and concentration gradients are prescribed and the temporal evolution of the flow has been computed in both aiding and opposing situations. The numerical results show that gas radiation modifies the structure of the velocity and thermal fields and accelerates the convergence to steady state in aiding case, while it favours the generation of instabilities and delays the arrival to a stable solution in opposing one.     

Viscous Joule heating MHD–conjugate heat transfer for a vertical flat plate in the presence of heat generation

The problem of steady conjugate heat transfer through an electrically-conducting fluid for a vertical flat plate in the presence of transverse uniform magnetic field taking into account the effects of viscous dissipation, Joule heating, and heat generation is formulated. The general governing equations which include such effects are made dimensionless by means of an apposite transformation. The ultimate resulting equations obtained by introducing the stream function with the similarity variable are solved numerically using the implicit finite difference method for the boundary conditions based on conjugate heat transfer process. A representative set of numerical results for the velocity and temperature profiles, the skin friction coefficients as well as the rate of heat transfer coefficient and the surface temperature distribution are presented graphically and discussed. A comprehensive parametric study is carried out to show the effects of the magnetic parameter, viscous dissipation parameter, Joule heating parameter, conjugate conduction parameter, heat generation parameter and the Prandtl number on the obtained solutions.