Analytical and Numerical Study on Thermally Developing Forced Convective Flow in a Channel Filled with a Highly Porous Medium Under Local Thermal Non-Equilibrium

Transport in Porous Media - An analytical and numerical study was made on thermally developing forced convective flow in a channel filled with a fluid-saturated porous medium, subject to constant...     

Fully Developed Mixed Convection Flow Between Inclined Parallel Plates Filled with a Porous Medium

A fully developed mixed convection flow between inclined parallel flat plates filled with a porous medium is considered through which there is a constant flow rate and with heat being supplied to the fluid by the same uniform heat flux on each plate. The equations governing this flow are made non-dimensional and are seen to depend on two dimensionless parameters, a mixed convection parameter λ and the Péclet number Pe, as well as the inclination γ of the plates to the horizontal. The velocity and temperature profiles are obtained in terms of λ, Pe and γ when the channel is inclined in an upwards direction as well as for horizontal channels. The limiting cases of small and large λ and small Pe are considered with boundary-layer structures being seen to develop on the plates for large values of λ.     

Effect of Thermal Non-Equilibrium on Convective Instability in a Ferromagnetic Fluid-Saturated Porous Medium

The effect of local thermal non-equilibrium (LTNE) on the onset of thermomagnetic convection in a ferromagnetic fluid-saturated horizontal porous layer in the presence of a uniform vertical magnetic field is investigated. A modified Forchheimer-extended Darcy equation is employed to describe the flow in the porous medium, and a two-field model is used for temperature representing the solid and fluid phases separately. It is found that both the critical Darcy–Rayleigh number and the corresponding wave number are modified by the LTNE effects. Asymptotic solutions for both small and large values of scaled interphase heat transfer coefficient H _t are presented and compared with those computed numerically. An excellent agreement is obtained between the asymptotic and the numerical results. Besides, the influence of magnetic parameters on the instability of the system is also discussed. The available results in the literature are recovered as particular cases from the present study.     

Darcy Flow in a Wavy Channel Filled with a Porous Medium

Flow in channels bounded by wavy or corrugated walls is of interest in both technological and geological contexts. This paper presents an analytical solution for the steady Darcy flow of an incompressible fluid through a homogeneous, isotropic porous medium filling a channel bounded by symmetric wavy walls. This packed channel may represent an idealized packed fracture, a situation which is of interest as a potential pathway for the leakage of carbon dioxide from a geological sequestration site. The channel walls change from parallel planes, to small amplitude sine waves, to large amplitude nonsinusoidal waves as certain parameters are increased. The direction of gravity is arbitrary. A plot of piezometric head against distance in the direction of mean flow changes from a straight line for parallel planes to a series of steeply sloping sections in the reaches of small aperture alternating with nearly constant sections in the large aperture bulges. Expressions are given for the stream function, specific discharge, piezometric head, and pressure.     

Local Thermal Non-Equilibrium and Heterogeneity Effects on the Onset of Convection in a Layered Porous Medium

The effect of local thermal non-equilibrium on the onset of convection in a porous medium consisting of two horizontal layers is studied analytically. Linear stability theory is applied. Variations of permeability, fluid conductivity, solid conductivity, interphase heat transfer coefficient and porosity are considered. It is found that heterogeneity of permeability and fluid conductivity have a major effect, heterogeneity of interphase heat transfer coefficient and porosity have a lesser effect, while heterogeneity of solid conductivity is relatively unimportant.     

Fully Developed Forced Convection in a Parallel Plate Channel with a Centered Porous Layer

In this study, fully developed heat and fluid flow in a parallel plate channel partially filled with porous layer is analyzed both analytically and numerically. The porous layer is located at the center of the channel and uniform heat flux is applied at the walls. The heat and fluid flow equations for clear fluid and porous regions are separately solved. Continues shear stress and heat flux conditions at the interface are used to determine the interface velocity and temperature. The velocity and temperature profiles in the channel for different values of Darcy number, thermal conductivity ratio, and porous layer thickness are plotted and discussed. The values of Nusselt number and friction factor of a fully clear fluid channel (Nu _cl = 4.12 and fRe _cl = 24) are used to define heat transfer increment ratio (e_th = Nu_p/Nu_cl)({\varepsilon _{\rm th} =Nu_{\rm p}/Nu_{\rm cl})} and pressure drop increment ratio (e_p = fRe_p/fRe_cl )({\varepsilon_{\rm p} =fRe_{\rm p}/fRe_{\rm cl} )} and observe the effects of an inserted porous layer on the increase of heat transfer and pressure drop. The heat transfer and pressure drop increment ratios are used to define an overall performance (e = e_th/e_p)({\varepsilon = \varepsilon_{\rm th}/\varepsilon_{\rm p})} to evaluate overall benefits of an inserted porous layer in a parallel plate channel. The obtained results showed that for a partially porous filled channel, the value of e{\varepsilon} is highly influenced from Darcy number, but it is not affected from thermal conductivity ratio (k _r) when k _r > 2. For a fully porous material filled channel, the value of e{\varepsilon} is considerably affected from thermal conductivity ratio as the porous medium is in contact with the channel walls.     

Analytical Solution for Forced Convection in a Semi-Circular Channel Filled with a Porous Medium

Fully developed laminar forced convection inside a semi-circular channel filled with a Brinkman-Darcy porous medium is studied. Analytical solutions for flow and constant flux heat transfer are found using a mixture of Cartesian and cylindrical coordinates. The problem depends on a parameter s, which is proportional to the inverse square of the Darcy number. Velocity boundary layers exist when s is large. Both friction factor-Reynolds number product and Nusselt number are determined. Closed form expressions for the clear fluid () limit are found. Rare analytical solutions not only describe fundamental channel flows, but also serve as a check for more complicated numerical solutions.     

Analysis of Laminar Flow and Forced Convection Heat Transfer in a Porous Medium

The flow of an incompressible Newtonian fluid confined in a planar geometry with different wall temperatures filled with a homogenous and isotropic porous medium is analyzed in terms of determining the unsteady state and steady state velocities, the temperature and the entropy generation rate as function of the pressure drop, the Darcy number, and the Brinkman number. The one-dimensional approximate equation in the rectangular Cartesian coordinates governing the flow of a Newtonian fluid through porous medium is derived by accounting for the order of magnitude of terms as well as accompanying approximations to the full-blown three-dimensional equations by using scaling arguments. The one-dimensional approximate energy and the entropy equations with the viscous dissipation consisting of the velocity gradient and the square of velocity are derived by following the same procedure used in the derivation of velocity expressions. The one-dimensional approximate equations for the velocity, the temperature, and the entropy generation rate are analytically solved to determine the velocity, the temperature, and the entropy distributions in the saturated porous medium as functions of the effective process parameters. It is found that the pressure drop, the Darcy number, and the Brinkman number affect the temperature distribution in the similar way, and besides the above parameters, the irreversibility distribution ratio also affects the entropy generation rate in the similar way.     

Effects of Viscous Dissipation and Flow Work on Forced Convection in a Channel Filled by a Saturated Porous Medium

Fully developed forced convection in a parallel plate channel filled by a saturated porous medium, with walls held either at uniform temperature or at uniform heat flux, with the effects of viscous dissipation and flow work included, is treated analytically. The Brinkman model is employed. The analysis leads to expressions for the Nusselt number, as a function of the Darcy number and Brinkman number.     

On Temporal Stability Analysis for Hydromagnetic Flow in a Channel Filled with a Saturated Porous Medium

In this paper, a linear stability analysis is presented to trace the time evolution of an infinitesimal, two-dimensional disturbance imposed on the base flow of an electrically conducting fluid in a channel filled with a saturated porous medium under the influence of a transversely imposed magnetic field. An eigenvalue problem is obtained and solved numerically using the Chebyshev collocation spectral method. The critical Reynolds number Re _c, the critical wave number α _c and the critical wave speed c _c are obtained for a wide range of the porous medium shape factor parameter S and Hartmann number H. It is found that an increase in the magnetic field intensity and a decrease in porous medium permeability have a stabilizing effect on the fluid flow.     

Flow of a Weakly Conducting Fluid in a Channel Filled with a Porous Medium

We investigate the fully developed flow in a fluid-saturated porous medium channel with an electrically conducting fluid under the action of a parallel Lorentz force. The Lorentz force varies exponentially in the vertical direction due to low fluid electrical conductivity and the special arrangement of the magnetic and electric fields at the lower plate. Exact analytical solutions are derived for fluid velocity and the results are presented in figures. All these flows are new and are presented for the first time in the literature.     

Forced Convection with Laminar Pulsating Flow in a Saturated Porous Channel or Tube

A perturbation approach is used to obtain analytical expressions for the velocity, temperature distribution, and transient Nusselt number for the problem of forced convection, in a parallel-plates channel or a circular tube occupied by a saturated porous medium modeled by the Brinkman equation, produced by an applied pressure gradient that fluctuates with small amplitude harmonically in time about a non-zero mean. It is shown that the fluctuating part of this Nusselt number alters in magnitude and phase as the dimensionless frequency increases. The magnitude increases from zero, goes through a peak, and then decreases to zero. The height of the peak decreases as the modified Prandtl number increases. The phase (relative to that of the steady component) decreases from π/2 to − π/2. The height of the peak at first increases, goes through a maximum, and then decreases as the Darcy number decreases.     

Fully Developed Forced Convection Heat Transfer in a Porous Channel with Asymmetric Heat Flux Boundary Conditions

An analytical study is performed on steady, laminar, and fully developed forced convection heat transfer in a parallel plate channel with asymmetric uniform heat flux boundary conditions. The channel is filled with a saturated porous medium, and the lower and upper walls are subjected to different uniform heat fluxes. The dimensionless form of the Darcy–Brinkman momentum equation is solved to determine the dimensionless velocity profile, while the dimensionless energy equation is solved to obtain temperature profile for a hydrodynamically and thermally fully developed flow in the channel. Nusselt numbers for the lower and upper walls and an overall Nusselt number are defined. Analytical expressions for determination of the Nusselt numbers and critical heat flux ratio, at which singularities are observed for individual Nusselt numbers, are obtained. Based on the values of critical heat flux ratio and Darcy number, a diagram is provided to determine the direction of heat transfer between the lower or upper walls while the fluid is flowing in the channel.     

Steady and Unsteady Heat Transfer in a Channel Partially Filled with Porous Media Under Thermal Non-Equilibrium Condition

Steady and pulsatile flow and heat transfer in a channel lined with two porous layers subject to constant wall heat flux under local thermal non-equilibrium (LTNE) condition is numerically investigated. To do this, a physical boundary condition in the interface of porous media and clear region of the channel is derived. The objective of this work is, first, to assess the effects of local solid-to-fluid heat transfer (a criterion indicating on departure from local thermal equilibrium (LTE) condition), solid-to-fluid thermal conductivity ratio and porous layer thickness on convective heat transfer in steady condition inside a channel partially filled with porous media; second, to examine the impact of pulsatile flow on heat transfer in the same channel. The effects of LTNE condition and thermal conductivity ratio in pulsatile flow are also briefly discussed. It is observed that Nusselt number inside the channel increases when the problem is tending to LTE condition. Therefore, careless consideration of LTE may lead to overestimation of heat transfer. Solid-to-fluid thermal conductivity ratio is also shown to enhance heat transfer in constant porous media thickness. It is also revealed that an increase in the amplitude of pulsation may result in enhancement of Nusselt number, while Nusselt number has a minimum in a certain frequency for each value of amplitude.     

The Convective Boundary-Layer Flow on a Reacting Surface in a Porous Medium

The convective boundary-layer flow on an impermeable vertical surface in a fluid-saturated porous medium is considered where the flow results from the heat released by an exothermic catalytic reaction on the surface converting a reactive component within the convective fluid to an inert product. The reaction is modelled by first-order kinetics with an Arrhenius temperature dependence. Numerical solutions of the governing equations are obtained for a range of parameter values. These show, for large activation energies, that localized rapid changes in wall temperature and localized high reaction rates occur a little way from the leading edge. Asymptotic expansions, valid at large distances from the leading edge, are derived, the form that these expansions take is qualitatively different depending on whether or not reactant consumption is included in the model.     

A Note on Local Thermal Non-Equilibrium in Porous Media Near Boundaries and Interfaces

Recent work on what has been called the phenomenon of heat flux bifurcation, that occurs at a boundary of a porous medium, or at an interface with a fluid clear of solid material, when a two-temperature model for the porous medium is employed, is discussed. An alternative interpretation of the situation, one in which the physics of the problem is emphasized, is presented.     

Buoyant Flow with Viscous Heating in a Vertical Circular Duct Filled with a Porous Medium

Combined, forced, and free flow in a vertical circular duct filled with a porous medium is investigated according to the Darcy–Boussinesq model. The effect of viscous dissipation is taken into account. It is shown that a thermal boundary condition compatible with fully developed and axisymmetric flow is either a linearly varying wall temperature in the axial direction or, only in the case of uniform velocity profile, an axial linear-exponential wall temperature change. The case of a linearly varying wall temperature corresponds to a uniform wall heat flux and includes the uniform wall temperature as a special case. A general analytical solution procedure is performed, by expressing the seepage velocity profile as a power series with respect to the radial coordinate. It is shown that, for a fixed thermal boundary condition, i.e., for a prescribed slope of the wall temperature, and for a given flow rate, there exist two solutions of the governing balance equations provided that the flow rate is lower than a maximum value. When the maximum value is reached, the dual solutions coincide. When the flow rate is higher than its maximum, no axisymmetric solutions exist. E. Magyari is on leave from the Institute of Building Technology, ETH—Zürich.     

Chaos in the Convective Flow of a Fluid Mixture in a Porous Medium

The results of the study of the global behaviour of the convective flow of a binary mixture in a porous medium are presented. Bifurcation diagram, fixed points, periodic, chaotic solutions, stable and unstable manifolds, and basins of attraction have been calculated. Different behaviours (chaos, undecidable behaviour, etc.) have been found.     

Thermal Analysis of Flow in a Porous Medium Over a Permeable Stretching Wall

This work presents a similarity solution for boundary layer flow through a porous medium over a stretching porous wall. Two considered wall boundary conditions are power-law distribution of either wall temperature or heat flux which are general enough to cover the isothermal and isoflux cases. In addition to momentum, both first and second laws of thermodynamics analyses of the problem are investigated. Independent numerical simulations are also performed for verification of the proposed analytical solution. The results, from the two independent approaches, are found to be in complete agreement. A comprehensive parametric study is presented and it is shown that heat transfer and entropy generation rates increase with Reynolds number, Prandtl number, and suction to the surface.