Free convection from a vertical plate with nonuniform surface heat flux and embedded in a porous medium

An analysis is presented for the calculation of heat transfer due to free convective flow along a vertical plate embedded in a porous medium with an arbitrarily varying surface heat flux. By applying the appropriate coordinate transformations and the Merk series, the governing energy equation is expressed as a set of ordinary differential equations. Numerical solutions are presented for these equations which represent universal functions and several computational examples are provided.     

Mixed convection of non-newtonian fluids from a vertical plate embedded in a porous medium

This paper investigates mixed free and forced convection of non-Newtonian fluids from a vertical isothermal plate embedded in a homogenous porous medium. A mathematical model is developed based on the modified Darcy's law and boundary-layer approximations, and the exact similarity solution is obtained as well as an integral solution. These two solutions agree within 3% for aiding flows and 10% for opposing flows. It is found that, non-Newtonian characteristics of fluids have appreciable influences on velocity profiles, temperature distributions and flow regimes.     

Effects of thermal dispersion and stratification on combined convection on a vertical surface embedded in a porous medium

The effects of thermal dispersion and thermal stratification on mixed convection about a vertical surface in a porous medium are studied. The conservation equations that govern the problem are reduced to a system of nonlinear ordinary differential equations. The resulting equations are solved on the basis of the local similarity approximation. The results indicate that both dispersion and stratification effects have considerable influence on the heat transfer rate.     

Natural convection of nanofluid over vertical plate embedded in porous medium: prescribed surface heat flux

The aim of the present paper is to analyze the natural convection heat and mass transfer of nanofluids over a vertical plate embedded in a saturated Darcy porous medium subjected to surface heat and nanoparticle fluxes. To carry out the numerical solution, two steps are performed. The governing partial differential equations are firstly simplified into a set of highly coupled nonlinear ordinary differential equations by appropriate similarity variables, and then numerically solved by the finite difference method. The obtained similarity solution depends on four non-dimensional parameters, i.e., the Brownian motion parameter (N _b), the Buoyancy ratio (N _r), the thermophoresis parameter (N _t), and the Lewis number (Le). The variations of the reduced Nusselt number and the reduced Sherwood number with N _b and N _t for various values of Le and N _r are discussed in detail. Simulation results depict that the increase in N _b, N _t, or N _r decreases the reduced Nusselt number. An increase in the Lewis number increases both of the reduced Nusselt number and the Sherwood number. The results also reveal that the nanoparticle concentration boundary layer thickness is much thinner than those of the thermal and hydrodynamic boundary layers.     

A boundary layer analysis of combined heat and mass transfer by natural convection from a concentrated source in a saturated porous medium

A boundary layer analysis was carried out to investigate the coupled phenomena of heat and mass transfer by natural convection from concentrated heat and mass sources embedded in saturated porous media. Both line and point source problems were treated. The boundary layer equations based on Darcy's law and Boussinesq approximation were solved by means of similarity transformation to obtain the details of velocity, temperature and concentration distributions above a concentrated heat source. Two important parameters, namely the Lewis number Le and the buoyancy ratioN were identified to conduct a series of numerical integrations. For the case of small Le, a substance diffuses further away from the plume centerline, such that the mass transfer influences both velocity and temperature profiles over a wide range. For large Le, on the other hand, the substance diffuses within a narrow range along the centerline. Naturally, the influence of mass transfer is limited to the level of the centerline velocity, so that a peaky velocity profile appears for positiveN whereas a velocity defect emerges along the centerline for negativeN. For such cases of large Le, the temperature profiles are found to be fairly insensitive to Le.     

Combined convection in power-law fluids along a nonisothermal vertical plate in a porous medium

The problem of combined convection from vertical surfaces in a porous medium saturated with a power-law type non-Newtonian fluid is investigated. The transformed conservation laws are solved numerically for the case of variable surface heat flux conditions. Results for the details of the velocity and temperature fields as well as the Nusselt number have been presented. The viscosity index ranged from 0.5 to 2.0.     

Transient non-Darcy free convection between parallel vertical plates in a fluid saturated porous medium

Transient non-Darcy free convection between two parallel vertical plates in a fluid saturated porous medium is investigated using the generalized momentum equation proposed by Vafai and Tien. The effects of porous inertia and solid boundary are considered in addition to the Darcy flow resistance. Exact solutions are found for the asymptotic states at small and large times. The large time solutions reveal that the velocity profiles are rather sensitive to the Darcy number Da when Da<1. It has also been found that boundary friction alters the velocity distribution near the wall, considerably. Finite difference calculations have also been carried out to investigate the transient behaviour at the intermediate times in which no similarity solutions are possible. This analytical and numerical study reveals that the transient free convection between the parallel plates may well be described by matching the two distinct asymptotic solutions obtained at small and large times.Nomenclature C empirical constant for the Forchheimer term - f velocity function for the small time solution - F velocity function for the large time solution - g acceleration due to gravity - Gr* micro-scale Grashof number - H a half distance between two infinite plates - K permeability - Nu Nusselt number - Pr Prandtl number - t time - T temperature - u, v Darcian velocity components - x, y Cartesian coordinates - effective thermal diffusivity - coefficient of thermal expansion - porosity - dimensionless time - similarity variable - dimensionless temperature - viscosity - kinematic viscosity - density - the ratio of heat capacities     

Mixed convection boundary-layer flow over a vertical surface embedded in a porous medium

In this paper we have numerically investigated the existence and uniqueness of a vertically flowing fluid passed a model of a thin vertical fin in a saturated porous media. We have assumed the two-dimensional mixed convection from a fin, which is modelled as a fixed, semi-infinite vertical surface, embedded in a fluid-saturated porous media under the boundary-layer approximation. We have taken the temperature, in excess of the constant temperature in the ambient fluid on the fin, to vary as  , where is measured from the leading edge of the plate and λ is a fixed constant. The Rayleigh number is assumed to be large so that the boundary-layer approximation may be made and the fluid velocity at the edge of the boundary-layer is assumed to vary as . The problem then depends on two parameters, namely λ and , the ratio of the Rayleigh to Péclet numbers. It is found that when λ>0 (<0) there are (is) dual (unique) solution(s) when is grater than some negative values of (which depends on λ). When λ<0 there is a range of negative value of (which depends on λ) for which dual solutions exist and for both λ>0 and λ<0 there is a negative value of (which depends on λ) for which there is no solution. Finally, solutions for 0<1 and 1 have been obtained.     

On natural convection from a vertical plate with a prescribed surface heat flux in porous media

This paper presents a theoretical and numerical investigation of the natural convection boundary-layer along a vertical surface, which is embedded in a porous medium, when the surface heat flux varies as (1 +x ²)), where is a constant andx is the distance along the surface. It is shown that for > -1/2 the solution develops from a similarity solution which is valid for small values ofx to one which is valid for large values ofx. However, when -1/2 no similarity solutions exist for large values ofx and it is found that there are two cases to consider, namely < -1/2 and = -1/2. The wall temperature and the velocity at large distances along the plate are determined for a range of values of .Notation g Gravitational acceleration - k Thermal conductivity of the saturated porous medium - K Permeability of the porous medium - l Typical streamwise length - q _w Uniform heat flux on the wall - Ra Rayleigh number, =gK(q _w /k)l/(v) - T Temperature - Too Temperature far from the plate - u, v Components of seepage velocity in the x and y directions - x, y Cartesian coordinates - Thermal diffusivity of the fluid saturated porous medium - The coefficient of thermal expansion - An undetermined constant - Porosity of the porous medium - Similarity variable, =y(1+x ) ^/3/x ^1/3 - A preassigned constant - Kinematic viscosity - Nondimensional temperature, =(T – T )Ra^1/3 k/qw - Similarity variable, = =y(log_e x)^1/3/x ^2/3 - Similarity variable, =y/x ^2/3 - Stream function     

Effect of the source term on steady free convection boundary layer flows over a vertical plate in a porous medium. Part II

The problem of steady free convection boundary layer over a vertical isothermal impermeable flat plate which is embedded in a fluid-saturated porous medium with volumetric heat generation or absorption is studied in this paper using the Darcy equation model. The case of the externally prescribed source terms S = S(x,y) is considered in this paper. It is shown that the corresponding boundary value problem depends on the sign of the plate temperature, which implies that the source term breaks the usual upflow or downflow symmetry of the free convection problem. Looking for similarity solutions, analytical and numerical solutions of the transformed boundary value problem are obtained for several values of the problem parameters. It is also shown that, contrary to the widely spread opinion, the exponential form of the internal heat generation term is not a necessary requirement of similarity reduction.     

Non-Darcy free convective flow along a vertical cylinder embedded in a porous medium with surface mass flux

The nonsimilar non-Darcian free-convection flow about a vertical cylinder with impermeable surface embedded in a saturated porous medium, where surface temperature of the cylinder varies as x^m, a power function of distance from the leading edge, has been studied by employing the implicit finite-difference method together with the Newton's quasilinearization technique. In the present investigation, effects of the surface mass flux together with the inertial effects on the rate of heat transfer at the surface, on the velocity distribution, and on the temperature distribution are shown graphically.     

Nonlinear convection in a non-Darcy porous medium

In this paper, the natural convection in a non-Darcy porous medium is studied using a temperature-concentration-dependent density relation. The effect of the two parameters responsible for the nonlinear convection is analyzed for different values of the inertial parameter, dispersion parameters, Rayleigh number, Lewis number, Soret number, and Dufour number. In the aiding buoyancy, the tangential velocity increases steeply with an increase in the nonlinear temperature parameter and the nonlinear concentration parameter when the inertial effect is zero. However, when the inertial effect is non-zero, the effect of the nonlinear temperature parameter and the nonlinear concentration parameter on the tangential velocity is marginal. The concentration distribution varies appreciably and spreads in different ranges for different values of the double dispersion parameters, the inertial effect parameter, and also for the parameters which control the nonlinear temperature and the nonlinear concentration. Heat and mass transfer varies extensively with an increase in the nonlinear temperature parameter and the nonlinear concentration parameter depending on Dacry and non-Darcy porous media. The variation in heat and mass transfer when all the effects, i.e., the inertial effect, double dispersion ef- fects, and Soret and Dufour effects, are simultaneously zero and non-zero. The combined effects of the nonlinear temperature parameter, the nonlinear concentration parameter and buoyancy are analyzed. The effect of the nonlinear temperature parameter and the nonlinear concentration parameter and also the cross diffusion effects on heat and mass transfer are observed to be more in Darcy porous media compared with those in non- Darcy porous media. In the opposing buoyancy, the effect of the temperature parameter is to increase the heat and mass transfer rate, whereas that of the concentration parameter is to decrease.     

Boundary-layer non-Newtonian flow over vertical plate in porous medium saturated with nanofluid

The free convective heat transfer to the power-law non-Newtonian flow from a vertical plate in a porous medium saturated with nanofluid under laminar conditions is investigated. It is considered that the non-Newtonian nanofluid obeys the mathematical model of power-law. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. The partial differential system governing the problem is transformed into an ordinary system via a usual similarity transformation. The numerical solutions of the resulting ordinary system are obtained. These solutions depend on the power-law index n, Lewis number Le, buoyancy-ratio number N _r, Brownian motion number N _b, and thermophoresis number N _t. For various values of n and Le, the effects of the influence parameters on the fluid behavior as well as the reduced Nusselt number are presented and discussed.     

Forced convection from an isolated heat source in a channel with porous medium

A numerical study is made of flow and heat transfer characteristics of forced convection in a channel that is partially filled with a porous medium. The flow geometry models convective cooling process in a printed circuit board system with a porous insert.The channel walls are assumed to be adiabatic. Comprehensive numerical solutions are acquired to the governing Navier-Stokes equations, using the Brinkman-Forchheimer-extended Darcy model for the regions of porous media. Details of flow and thermal fields are examined over ranges of the principal parameters; i.e., the Reynolds number Re, the Darcy number Da (≡K/H²), the thickness of the porous substrate S, and the ratio of thermal conductivities R_k (≡k_eff/k). Two types of the location of the porous block are considered. The maximum temperature at the heat source and the associated pressure drop are presented for varying Re, Da, S, and R_k. The results illustrate that as S increases or Da decreases, the fluid flow rate increases. Also, as R_k increases for fixed Da, heat transfer rates are augmented. Explicit influences of Re on the flow and heat transport characteristics are also scrutinized. Assessment is made of the utility of using a porous insert by cross comparing the gain in heat transport against the increase in pressure drop.     

MHD natural convection in porous media-filled enclosures

In this work, the magnetohydrodynamics （MHD） natural convection heat transfer problem inside a porous medium filled with inclined rectangular enclosures is investigated numerically. The boundary conditions selected on the enclosure are two adiabatic and two isothermal walls. The governing equations, continuity, and Forchheimer extension of the Darcy law and energy are transformed into dimensionless forms by using a set of suitable variables, and then solved by using a finite difference scheme. The governing parameters are the magnetic influence number, the Darcy Rayleigh number, the inclination angle, and the aspect ratio of the enclosure. It is found that the magnetic influence number and the inclination angle have pronounced effects on the fluid flow and heat transfer in porous media-filled enclosures.     

Natural convection from a vertical cylinder in a thermally stratified porous medium

The problem of non-Darcy natural convection adjacent to a vertical cylinder embedded in a thermally stratified porous medium has been analyzed. Nonsimilarity solutions are obtained for the case that the ambient temperature increases linearly with height of the cylinder. A generalized flow model was used in the present study to include the effects of the macroscopic viscous term and the microscopic inertial force. Also, the thermal dispersion effect is considered in the energy equation. Thus, the main aim of this work is to examine the effects of thermal stratification and non-Darcy flow phenomena on the free convection flow and heat transfer characteristics. It was found that the present problem depends on six parameters, namely, the local thermal stratification parameter ξ, the boundary effect parameter Bp, the modified Grashof number Gr^*, wall temperature exponent m, the curvature parameter ω, and the modified Rayleigh number based on pore diameter Ra _d . The impacts of these governing parameters on the local heat transfer parameter are discussed in great detail. Also, representative velocity and temperature profiles are presented at selected values of the thermal stratification parameter. In general, the local heat transfer parameter is increased with increasing the values of m, ω, and Ra _d ; while it is decreased with increasing the values of ξ, Bp, and Gr^*. Received on 19 May 1998     

The inertial effect on the natural convection flow within a fluid-saturated porous medium

The general momentum equation for fluid flow within a porous medium is supposedly valid for any fluid-porous medium configuration. One of the main concerns of using the general equations refers to the inclusion of both inertia terms, namely, the convective inertia term and the Forchheimer term. In this study, we go beyond the important discussion about the correctness of including both terms in the general momentum equations by focusing upon the effect of the convective inertia term on the heat transfer results. The fluid-porous medium system considered here is a cavity bounded by solid surfaces with vertical walls maintained at constant but different temperatures. The natural convection problem is solved numerically, and the results are compared with a general theory developed by using the method of scale analysis. It is demonstrated that the convective inertia term effect upon the heat transfer results is minor for 0.01 ≤ Pr ≤ 1, 10 ≤ Ra_D ≤ 10⁴, 10⁻⁸ ≤ Da ≤ 10⁻², and porosities 0.4 and 0.8. It is also shown that, contrary to the general belief, the convective inertial effect upon the heat transfer within the cavity is minimized when the Prandtl number is reduced.     

Natural convection in a vertical strip immersed in a porous medium

In this work, the conjugated heat transfer characteristics of a thin vertical strip of finite length, placed in a porous medium has been studied using numerical and asymptotic techniques. The nondimensional temperature distribution in the strip and the reduced Nusselt number at the top of the strip are obtained as a function of the thermal penetration parameter s, which measures the thermal region where the temperature of the strip decays to the ambient temperature of the surrounding fluid. The numerical values of this nondimensional parameter permits to classify the different physical regimes, showing different solutions: a thermally long behaviour, an intermediate transition and a short strip limit.     

Thermal convection in a porous layer: Effects of anisotropy and surface boundary conditions

The theory describing the onset of convection in a homogeneous porous layer bounded above and below by isothermal surfaces is extended to consider an upper boundary which is partly permeable. The general boundary condition p + λ ∂p/∂n = constant is applied at the top surface and the flow is investigated for various λ in the range 0 ⩽ λ < ∞. Estimates of the magnitude and horizontal distribution of the vertical mass and heat fluxes at the surface, the horizontally-averaged heat flux (Nusselt number) and the fraction of the fluid which recirculates within the layer are found for slightly supercritical conditions. Comparisons are made with the two limiting cases λ → ∞, where the surface is completely impermeable, and λ = 0, where the surface is at constant pressure. Also studied are the effects of anisotropy in permeability, ξ = K _H /K _V , and anisotropy is thermal conductivity, η = k _H /k _V , both parameters being ratios of horizontal to vertical quantities. Quantitative results are given for a wide variety of the parameters λ, ξ and η. In the limit ξ/η → 0 there is no recirculation, all fluid being converted out of the top surface, while in the limit ξ/η → ∞ there is full recirculation.     

Numerical investigation of Dufour and Soret effects on unsteady MHD natural convection flow past vertical plate embedded in non-Darcy porous medium

The Dufour and Soret effects on the unsteady two-dimensional magnetohydro-dynamics(MHD) double-diffusive free convective flow of an electrically conducting fluidpast a vertical plate embedded in a non-Darcy porous medium are investigated numeri-cally.The governing non-linear dimensionless equations are solved by an implicit finitedifference scheme of the Crank-Nicolson type with a tridiagonal matrix manipulation.The effects of various parameters entering into the problem on the unsteady dimension-less velocity,temperature,and concentration profiles are studied in detail.Furthermore,the time variation of the skin friction coefficient,the Nusselt number,and the Sherwoodnumber is presented and analyzed.The results show that the unsteady velocity,tem-perature,and concentration profiles are substantially influenced by the Dufour and Soreteffects.When the Dufour number increases or the Soret number decreases,both the skinfriction and the Sherwood number decrease,while the Nusselt number increases.It isfound that,when the magnetic parameter increases,the velocity and the temperaturedecrease in the boundary layer.