Mixed convection in a vertical micro-porous channel with radiation and magnetic field. A thermal non-equilibrium perspective

This article presents a numerical study of the doublediffusive\nmixed convection in a vertical channel filled with porous\nmedium by using non-equilibrium model. The flow is assumed\nfully developed, uni-directional and steady state. The controlling\nparameters are thermal Rayleigh number (RaT ), Darcy number (Da),\nForchheimer number (F), buoyancy ratio (N), inter phase heat transfer\ncoefficient (H), and porosity scaled thermal conductivity ratio\n(γ). The Brinkman-extended non-Darcy model is considered. The\ngoverning equations are solved by spectral collocation method. The\nmain emphasize is given on flow profiles as well as heat and solute\ntransfer rates, when two diffusive components in terms of buoyancy\nratio are in favor (against) of each other and solid matrix and fluid\nare thermally non-equilibrium. The results show that, for aiding flow\n(RaT = 1000), the heat transfer rate of fluid (Nuf ) increases upto a\ncertain value of H, beyond that decreases smoothly and converges\nto a constant, whereas in case of opposing flow (RaT = -1000),\nthe result is same for N = 0 and 1. The variation of Nuf in (N,\nNuf )-plane shows sinusoidal pattern for RaT = -1000. For both cases\n(aiding and opposing) the flow destabilize on increasing N by inviting\npoint of inflection or flow separation on the velocity profile. Overall,\nthe buoyancy force have significant impact on the non-Darcy mixed\nconvection under LTNE conditions.

The effect of local thermal nonequilibrium (LTNE) on the entropy generation and heat transfer characteristics in the magnetohydrodynamic flow of a couple‐stress fluid through a high‐porosity vertical channel is studied numerically using the higher‐order Galerkin technique. The Boussinesq approximation is assumed to be valid and the porous medium is considered to be isotropic and homogeneous. Two energy equations are considered one each for solid and fluid phases. The term involving the heat transfer coefficient in both equations renders them mutually coupled. Thermal radiation and an internal heat source are considered only in the fluid phase. The influence of inverse Darcy number, Hartmann number, couple‐stress fluid parameter, Grashof number, thermal radiation parameter, and interphase heat transfer coefficient on velocity and temperature profiles is depicted graphically and discussed. The entropy generation, friction factor, and Nusselt number are determined, and outcomes are presented via plots. The effect of LTNE on the temperature profile is found to cease when the value of the interphase heat transfer coefficient is high, and in this case, we get the temperature profiles of fluid and solid phases are uniform. The physical significance of LTNE is discussed in detail for different parameters' values. It is found that heat transport and friction drag are maximum in the case of LTNE and minimum in the case of local thermal equilibrium. We observe that LTNE opposes the irreversibility of the system. The corresponding results of a fluid‐saturated densely packed porous medium can be obtained as a limiting case of the current study.

A thermal non-equilibrium perspective on mixed convection in a vertical channel

This article presents a numerical investigation of steady non-Darcy natural convection heat transfer in a square cavity filled with a heat-generating porous medium with partial cooling using a local thermal nonequilibrium (LTNE) model. Five different partial cooling boundary conditions and the fully cooled boundary condition are investigated under LTNE and local thermal equilibrium (LTE). The cooling portions of the left and the right sidewalls of the cavity are maintained at temperature T 0 while the enclosure's top and bottom walls, as well as the inactive parts of its sidewalls, are kept insulated. The simulation results show that the placement order of wall cooling has a significant effect on the flow pattern and heat transfer rate. Compared with the fully cooled wall, the partially cooled wall of the cavity yielded a higher local Nusselt number for both fluid and solid phases. Under the same boundary conditions, the LTNE and LTE models can demonstrate significant differences in flow patterns and temperature fields. The total heat transfer rate increases with both Darcy number and Rayleigh number. Enhancement of interphase heat transfer coefficient (H) reduces the impact of Darcy number on the heat transfer rate of a porous cavity. Also, the total heat transfer rate of the porous medium decreases steadily with thermal conductivity ratio γ and interphase heat transfer coefficient H.

This paper investigates the incompressible mixed convection flow of electrically conductive micropolar fluid with a thermal non-equilibrium condition that passes through the vertical circular (pipe) porous medium. The extension of the non-Darcy–Brinkman–Forchheimer model is considered to formulate the governing non-linear system of differential equations for the problem. Furthermore, the rigorous impact of different parameters such as thermal conductivity ratio ( γ ) , inter-phase heat transfer coefficient ( H ) , Darcy number ( D a ) , Grashof number ( G r ) , Eringen micro-polar parameter ( E r ) , Hartmann number ( H a ) , solid-heat generation ( β ) , and fluid heat generation parameter ( α ) on velocity profile ( f ∗ ) , micro-rotational (angular velocity) ( g ∗ ) , temperature of solid ( Θ f ∗ ) and fluid ( Θ s ∗ ) has been investigated by using the computational strength of artificial intelligence based Elman neural networks (ENN) and Levenberg–Marquardt algorithm (LMA). We have compared the solutions calculated by the designed ENN-LM algorithm with the Cuckoo Search Algorithm (CSA), Chebyshev spectral collocation method (CSCM), particle swarm optimization (PSO) algorithm, and Runge–Kutta method. The convergence rate and stability of the ENN-LM technique show that it can be applied to solving complex models involving partial and fractional differential equations.

A thermal non-equilibrium model for 3D double diffusive convection of power-law fluids with chemical reaction in the porous medium

Effects of temperature-dependent viscosity on natural convection in a porous cavity with a circular cylinder under local thermal non-equilibrium condition

Buoyancy-driven flow inside a superposed enclosure filled with composite porous-hybrid nanofluid layers was investigated numerically using a local thermal nonequilibrium model for the heat transfer between the fluid and the solid phases. The bottom wall of the enclosure was partly heated to provide a heat flux, while the other parts of the wall were thermally insulated. The top and vertical walls of the enclosure were maintained at constant cold temperatures. The Darcy-Brinkman model was adopted to model the flow inside the porous layer. The Galerkin finite element method was used to solve the governing equations using the semi-implicit method for pressure linked equations algorithm. The selected parameters are presented for the Rayleigh number (Ra), 103 ≤ Ra ≤ 107, the Darcy number (Da), 10−7 ≤ Da ≤ 1, the porous layer thickness (S), 0 ≤ S ≤ 1, the modified conductivity ratio (γ), 10−1 ≤ γ ≤ 104, the interphase heat transfer coefficient (H), 10−1 ≤ H ≤ 1000, the heat source length (B), 0.2, 0.4, 0.6, 0.8 and 1, and the nanoparticle volume fraction (ϕ), 0 ≤ ϕ ≤ 0.2. It has been concluded that the rate of heat transfer of hybrid nanofluid (Cu−Al2O3/water) is higher than with the pure fluid. Furthermore, at Ra ≤ 105, the heat transfer rate maintains its maximum value when S reaches the critical value (S = 0.3). The values of S, Da, and B were found to have a significant effect on the heat removal from the heat source. Increasing the values of γ and H can strongly enhance the heat transfer rate and satisfy the thermal equilibrium case.

This present work reports the fully developed hydromagnetic mixed-convection nanofluid flow in a vertical channel teeming by porous media with variable thermal and electrical wall conductivities and thermal non-equilibrium condition is taken into account. The fluid is assumed electrically conducted and taking as a mixture of base fluid (water) and three different metallic nanoparticles copper, alumina and titanium dioxide. The non-Darcy-Brinkman-Forchheimer extended model has been contemplated and solved governed differential equation by analytically as well as by numerically. Special attention is given to understand the effect of solid volume friction of the nanofluid (ψ) and wall thermal conductivity (τ) parameters for both buoyancy assisted as well as opposed cases when the interval of inter-phase heat transfer coefficient H is taken from [1, 500]. It is observed that the point of inflection and flow separation are appeared in the velocity profile for both cases and disappeared slowly from the profile as both parameters ψ and τ increases and it stabilize the system. There exists a minimum value H0 of H for both case when τ ≤ 0.5 the heat transfer rate Nunf of the fluid increases at the wall. Further, for τ > 0.5 Nunf suddenly decreases and converge asymptotically in the case of buoyancy assisted flow. It is also observed that there exist two type interval of H in the buoyancy opposed for different value of τ. In first interval [0, H0], when τ ≤ 0.5 profile is same as in buoyancy assisted case but as soon as τ increases heat transfer rate first decreases upto a threshold value of H and after that in increases rapidly. Over all for both the cases system become stable and non thermal equilibrium condition convert into thermal equilibrium when H, τ and Φ increases.

Natural convection inside an enclosure partly filled with a porous slab saturated with a nanofluid has been investigated numerically using various thermal boundary conditions. The Galerkin finite element method was used to solve the governing equations. Four different scenarios were modelled. Firstly, two-dimensional laminar natural convection in a vertical or a horizontal alignment to the porous-nanofluid layers was investigated with a linearly heated left-hand side enclosure wall. At low values of the thermal conductivity ratio and Darcy number, the heat transfer rate was higher for the horizontal alignment compared to the vertical alignment and vice versa at a high value of the Darcy number. Secondly, the same geometry was studied with a sinusoidally heated left-hand side enclosure wall. It was found that the temperature amplitude and wave number of the sinusoidally heated wall significantly affected the heat transfer rate. At the thermal conductivity ratio < 1 and the Darcy number ≥ 10−3, the heat transfer rate increased in the vertical alignment of the porous-nanofluid layers compared to the horizontal alignment. In both of these scenarios, the porous slab direction inside the enclosure played a significant role on the heat transfer. Thirdly, two-dimensional laminar natural convection of a hybrid nanofluid inside the porous-nanofluid layers using a thermal nonequilibrium model has been simulated. It was found that increasing the modified thermal conductivity ratio and interphase heat transfer coefficient values strongly enhanced the heat transfer rate and satisfy the thermal equilibrium case. Finally, the amplitude and the wave number of the corrugated wall have a significant role on the turbulent natural convection in a three-dimensional enclosure partly filled with porous slab saturated with a hybrid nanofluid. For all scenarios, the lower thickness of the porous slab using the nanofluid predicted a new trend of the fluid flow and heat transfer compared to the porous enclosure.

A numerical investigation of the natural convection heat transfer in a rectangular cavity filled with a heat-generating porous medium by adopting the local thermal nonequilibrium (LTNE) model is reported in this paper. The top and bottom walls of the enclosure are adiabatic, the left wall is linearly cooled, and the right wall is cooled by a linear or uniform temperature profile. The results show that the isotherms for the fluid and solid phases become similar with the increase of the interphase heat transfer coefficient H; the increasing heat transfer between the two phases brings their temperatures closer to each other and thus the solid and fluid phases are in a state of the thermal equilibrium at higher values of H. For case A, the interphase heat transfer coefficient has little influence on the heat transfer rate of the solid phase of the porous cavity and the heat transfer profile of the solid phase (Nusy) is symmetrical with respect to the center point of line Y = 0.5. For case B, the interphase heat transfer coefficient H has a significant effect on that at the right wall, and the total heat transfer of the heat-generating porous cavity is implemented by the right-side wall. The total heat transfer rate Q of case B is higher than that of case A at a high thermal conductivity ratio γ (γ = 1, 10).

This article is related with the investigation of heat transfer in an annular cylinder containing porous medium. The outer surface of the vertical annulus is heated to hot temperature and inner surface is cooled. The simulation is carried out by considering the thermal non equilibrium between the solid and fluid phases of the medium. Finite element method is used to solve the equations. The work is an extension of Part I of this paper that considered physical parameters such as inter-phase heat transfer coefficient, thermal conductivity ratio and Rayleigh number. However, this article elaborates the effect of geometric parameters along with radiation parameter.This article is related with the investigation of heat transfer in an annular cylinder containing porous medium. The outer surface of the vertical annulus is heated to hot temperature and inner surface is cooled. The simulation is carried out by considering the thermal non equilibrium between the solid and fluid phases of the medium. Finite element method is used to solve the equations. The work is an extension of Part I of this paper that considered physical parameters such as inter-phase heat transfer coefficient, thermal conductivity ratio and Rayleigh number. However, this article elaborates the effect of geometric parameters along with radiation parameter.

This article is related with the investigation of heat transfer in an annular cylinder containing porous medium. The outer surface of the vertical annulus is heated to hot temperature and inner surface is cooled. The simulation is carried out by considering the thermal non equilibrium between the solid and fluid phases of the medium. Finite element method is used to solve the equations. The work is an extension of Part I of this paper that considered physical parameters such as inter-phase heat transfer coefficient, thermal conductivity ratio and Rayleigh number. However, this article elaborates the effect of geometric parameters along with radiation parameter.This article is related with the investigation of heat transfer in an annular cylinder containing porous medium. The outer surface of the vertical annulus is heated to hot temperature and inner surface is cooled. The simulation is carried out by considering the thermal non equilibrium between the solid and fluid phases of the medium. Finite element method is used to solve the equations. The work is an extension of Part I of this paper that considered physical parameters such as inter-phase heat transfer coefficient, thermal conductivity ratio and Rayleigh number. However, this article elaborates the effect of geometric parameters along with radiation parameter.

The local thermal nonequilibrium (LTNE) condition is the temperature difference between the base fluid and the nanoparticle. In the present research article, linear analysis was done to know about the onset of convection in the system, and a weakly nonlinear stability analysis was done to know about heat and mass transport in the system for both the unsteady and steady case. Here we have taken temperature to be constant and nanoparticle flux to be zero at the upper and lower boundaries of the system. The normal mode technique is used for linear analysis, and the truncated Fourier series method is used for nonlinear analysis; plot streamlines, isotherms, and isohaline are used to visualize the conduction, convection, and steady state. We found that the behavior of Hele-Shaw cell is the same in the case of LTNE and local thermal equilibrium (LTE). The effect of Hele-Shaw number, interphase heat transfer coefficient, modified thermal capacity ratio, thermal diffusivity ratio, amplitude, and frequency of modulation on the onset of convection, heat, and mass transfer are depicted graphically. We found that the effect of LTNE can be seen only for the intermediate values of the interphase heat transfer coefficient, and this region is called the LTNE region. We also discuss the result of thermal Nusselt number, streamlines, and isothermals of fluid and particle phase for steady case and plot the graphs with respect to the Hele-Shaw cell Rayleigh number. Rate of heat transfer for the particle phase is higher than the fluid phase for both the unsteady and steady state. In this research paper we find the result for both LTE and LTNE conditions with unsteady and steady cases, while in the previous study we analyzed only for the LTE condition.

Effect of asymmetrical wall heat flux and wall temperature ratio on mixed convection in a vertical micro-porous-channel with internal heat generation