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.     

Forced Convection Heat Transfer of Nanofluids in a Porous Channel

This article is concerned with the effects of flow and migration of nanoparticles on heat transfer in a straight channel occupied with a porous medium. Investigation of force convective heat transfer of nanofluids in a porous channel has not been considered completely in the literature and this challenge is generally considered to be an open research topic that may require more study. The fully developed flow and steady Darcy?CBrinkman?CForchheimer equation is employed in porous channel. The thermal equilibrium model is assumed between nanofluid and solid phases. It is assumed that the nanoparticles are distributed non-uniformly inside the channel. As a result the volume fraction distribution equation is also coupled with governing equations. The effects of parameters such as Lewis number, Schmidt number, Brownian diffusion, and thermophoresis on the heat transfer are completely studied. The results show that the local Nusselt number is decreased when the Lewis number is increased. It is observed that as the Schmidt number is increased, the wall temperature gradient is decreased and as a consequence the local Nusselt number is decreased. The effects of Lewis number, Schmidt number, and modified diffusivity ratio on the volume fraction distribution are also studied and discussed.     

Optimization of Forced Convection Heat Transfer in a Composite Porous Medium Channel

The optimization of heat transfer for forced convection in a composite porous channel was studied. We investigated the question where should one place, in the core or in the sheath, the material with high permeability and high-thermal conductivity and where should one place the material with low permeability and low-thermal conductivity, to maximize heat transfer from the walls. We also investigated the optimal heat transfer situation when one has the freedom to vary the relative volumes of the core and the sheath.     

Forced Convection Heat Transfer on a Flat Plate Embedded in Porous Media for Power-Law Fluids

The effect of the presence of an isotropic solid matrix on the forced convection heat transfer rate from a flat plate to power-law non- Newtonian fluid-saturated porous medium, has been investigated. Numerical results are presented for the distribution of velocity and temperature profiles within the boundary layer. The effects of the flow index, first-order and second-order resistance on the velocity, and temperature profiles are discussed. The missing wall values of the velocity and thermal functions are tabulated.     

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.     

Thermally Developing Forced Convection in a Porous Medium Occupied by a Rarefied Gas: Parallel Plate Channel or Circular Tube with Walls at Constant Heat Flux

An adaptation of the classical Graetz methodology is applied to investigate the thermal development of forced convection in a parallel plate channel or a circular tube filled by a porous medium saturated by a rarefied gas, with walls held at constant heat flux. The Brinkman model is employed. The analysis leads to expressions for the local Nusselt number Nu as functions of the dimensionless longitudinal coordinate and the Darcy number. It is found that an increase in the velocity slip coefficient generally increases Nu by a small or moderate amount (but the circular tube at large Darcy number is an exception) while an increase in the temperature slip coefficient reduces Nu by a more substantial amount. These trends are uniform as the longitudinal coordinate varies.     

Thermodynamically Consistent Limiting Forced Convection Heat Transfer in a Asymmetrically Heated Porous Channel: An Analytical Study

Fluid transport and the associated heat transfer through porous media is of immense importance because of its numerous practical applications. In view of the widespread applications of porous media flow, the present study attempts to investigate the forced convective heat transfer in the limiting condition for the flow through porous channel. There could be many areas, where heat transfer through porous channel attain some limiting conditions, thus, the analysis of limiting convective heat transfer is far reaching. The primary aim of the present study is focused on the limiting forced convection analysis considering the flow of Newtonian fluid between two asymmetrically heated parallel plates filled with saturated porous media. Utilizing a few assumptions, which are usually employed in the literature, an analytical methodology is executed to obtain the closed-form expression of the temperature profile, and in the following the expression of the limiting Nusselt numbers. The parametric variations of the temperature profile and the Nusselt numbers in different cases have been shown highlighting the influential role of different performance indexing parameters, like Darcy number, porosity of the media, and Brinkman number of forced convective heat transfer in porous channel. In doing so, the underlying physics of the transport characteristics of heat has been delineated in a comprehensive way. Moreover, a discussion has been made regarding an important feature like the onset of point of singularity as appeared on the variation of the Nusselt number from the consideration of energy balance in the flow field, and in view of second law of thermodynamics.     

Effect of Heat Generation on Forced Convection Through a Porous Saturated Duct

The effect of heat generation on the flow characteristics of the fully developed forced convection through a porous duct is investigated analytically on the basis of Brinkman?CForchheimer model. The duct is bounded by two isoflux plates. For solving momentum equation a regular asymptotic expansion method is used for hyper-porous materials and a matched asymptotic expansion method is used for low-porous materials. This solution permits a uniform solution for the energy equation to find the temperature distribution as well as Nusselt number. A numerical solution is found here to check the accuracy of the asymptotic one.     

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.     

Forced Convection Heat Transfer from a Finite-Height Cylinder

This paper presents a large eddy simulation of forced convection heat transfer in the flow around a surface-mounted finite-height circular cylinder. The study was carried out for a cylinder with height-to-diameter ratio of 2.5, a Reynolds number based on the cylinder diameter of 44 000 and a Prandtl number of 1. Only the surface of the cylinder is heated while the bottom wall and the inflow are kept at a lower fixed temperature. The approach flow boundary layer had a thickness of about 10% of the cylinder height. Local and averaged heat transfer coefficients are presented. The heat transfer coefficient is strongly affected by the free-end of the cylinder. As a result of the flow over the top being downwashed behind the cylinder, a vortex-shedding process does not occur in the upper part, leading to a lower value of the local heat transfer coefficient in that region. In the lower region, vortex-shedding takes place leading to higher values of the local heat transfer coefficient. The circumferentially averaged heat transfer coefficient is 20 % higher near the ground than near the top of the cylinder. The spreading and dilution of the mean temperature field in the wake of the cylinder are also discussed.     

Analytical Study of Fluid Flow and Heat Transfer during Forced Convection in a Composite Channel Partly Filled with a Brinkman–Forchheimer Porous Medium

In this paper, the problem of fully developed forced convection in a parallel-plate channel partly filled with a homogeneous porous material is considered. The porous material is attached to the walls of the channel, while the center of the channel is occupied by clear fluid. The flow in the porous material is described by a nonlinear Brinkman–Forchheimer-extended Darcy equation. Utilizing the boundary-layer approach, analytical solutions for the flow velocity, the temperature distribution, as well as for the Nusselt number are obtained. Dependence of the Nusselt number on several parameters of the problem is extensively investigated.     

Effect of Internal Heat Generation on the Onset of Marangoni Convection in a Fluid Layer Overlying a Layer of an Anisotropic Porous Medium

Linear stability analysis has been performed to investigate the effect of internal heat generation on the criterion for the onset of Marangoni convection in a two-layer system comprising an incompressible fluid-saturated anisotropic porous layer over which lies a layer of the same fluid. The upper non-deformable free surface and the lower rigid surface are assumed to be insulated to temperature perturbations. The fluid flow in the porous layer is governed by the modified Darcy equation and the Beavers–Joseph empirical slip condition is employed at the interface between the two layers. The resulting eigenvalue problem is solved exactly. Besides, analytical expression for the critical Marangoni number is also obtained by using regular perturbation technique with wave number as a perturbation parameter. The effect of internal heating in the porous layer alone exhibits more stabilizing effect on the system compared to its presence in both fluid and porous layers and the system is least stable if the internal heating is in fluid layer alone. It is found that an increase in the value of mechanical anisotropy parameter is to hasten the onset of Marangoni convection while an opposite trend is noticed with increasing thermal anisotropy parameter. Besides, the possibilities of controlling (suppress or augment) Marangoni convection is discussed in detail.     

Forced Convection Heat Transfer from a Heated Cylinder in an Axial Background Flow in a Porous Medium: Three Exactly Solvable Cases

Assuming a background flow of velocity U = U(x) in the axial direction x of a circular cylinder with surface temperature distribution T _w = T _w (x) in a saturated porous medium, for the temperature boundary layer occurring on the cylinder three exactly solvable cases are identified. The functions {U(x), T _w (x)} associated with these cases are given explicitly, and the corresponding exact solutions are expressed in terms of the modified Bessel function K ₀ (z), the incomplete Gamma function Γ (a, z) and the confluent hypergeometric function U(a, b, z), respectively. The correlation between the Nusselt number and the Péclet number as well as the curvature effects on the heat transfer are discussed in all these cases in detail. Some “universal” features of the exponential surface temperature distribution are also pointed out.     

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.     

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.     

Nonsimilar Boundary Layer Analysis of Free Convection Heat Transfer Over a Vertical Cylinder in Bidisperse Porous Media

This work presents a boundary layer analysis for the free convection heat transfer from a vertical cylinder in bidisperse porous media with constant wall temperature. A boundary layer analysis and the two-velocity two-temperature formulation are used to derive the nonsimilar governing equations. The transformed governing equations are solved by the cubic spline collocation method to yield computationally efficient numerical solutions. The effects of inter-phase heat transfer parameter, modified thermal conductivity ratio, and permeability ratio on the heat transfer and flow characteristics are studied. Results show that an increase in the modified thermal conductivity ratio and the permeability ratio can effectively enhance the free convection heat transfer of the vertical cylinder in a bidisperse porous medium. Moreover, the thermal nonequilibrium effects are strong for low values of the inter-phase heat transfer parameter.     

A Two-Velocity Two-Temperature Model for a Bi-Dispersed Porous Medium: Forced Convection in a Channel

A two-velocity two-temperature model for bi-dispersed porous media is formulated. Using the model, an analytic solution is obtained for the problem of forced convection in a channel between parallel plane walls that are held either at uniform temperature or uniform heat flux. In each case, Nusselt number values are given as functions of a conductivity ratio, a velocity ratio, a volume fraction, and an internal heat exchange parameter.     

Onset of Convection in a Porous Rectangular Channel with External Heat Transfer to Upper and Lower Fluid Environments

The conditions for the onset of convection in a horizontal rectangular channel filled with a fluid saturated porous medium are studied. The vertical sidewalls are assumed to be impermeable and adiabatic. The horizontal upper and lower boundary walls are considered as impermeable and subject to external heat transfer, modelled through a third-kind boundary condition on the temperature field. The external fluid environments above and below the channel, kept at different temperatures, provide the heating-from-below mechanism which may lead to the onset of the thermal instability in the porous medium. The linear response of the fluid saturated porous channel, in a basic motionless state, is tested with respect to three-dimensional normal mode disturbances of the temperature field and of the pressure field. The linearised disturbance equations are solved analytically leading to an implicit-form expression of the neutral stability condition, formulated as a functional relationship between the Darcy?CRayleigh number and the continuous longitudinal wave number of the normal modes, for any assigned aspect ratio of the cross-section and for any given Biot number. The analysis of the neutral stability is carried out. The analysis is extended to the case of a channel with a finite length in the longitudinal direction, and with adiabatic and impermeable capped ends.     

Forced Convection Flow of Power-Law Fluids Over a Flat Plate Embedded in a Darcy-Brinkman Porous Medium

The steady forced convection flow of a power-law fluid over a horizontal plate embedded in a saturated Darcy-Brinkman porous medium is considered. The flow is driven by a constant pressure gradient. In addition to the convective inertia, also the “porous Forchheimer inertia” effects are taken into account. The pertinent boundary value problem is investigated analytically, as well as numerically by a finite difference method. It is found that far away from the leading edge, the velocity boundary layer always approaches an asymptotic state with identically vanishing transverse component. This holds for pseudoplastic (0 < n < 1), Newtonian (n = 1), and dilatant (n > 1) fluids as well. The asymptotic solution is given for several particular values of the power-law index n in an exact analytical form. The main flow characteristics of physical and engineering interest are discussed in the paper in some detail.