Natural convection in a porous medium coupled across an impermeable vertical wall with film condensation

This paper presents a theoretical study of thermofluid interaction between natural convection in fluid-saturated porous medium and film condensation, coupled through an impermeable vertical wall. The two heat transfer modes are analyzed separately. The solutions are matched on the wall. The complexion of this two-fluid problem is governed by a dimensionless interaction parameter which relates the heat transfer effectiveness of the two heat transfer mechanisms. The effect of this parameter on the flow and heat transfer is documented. Results regarding the overall heat transfer coefficient are obtained for a wide range of the independent parameters. Received on 19 January 1998     

Thermal interaction between natural convection on one side of a vertical wall and forced convection on the other side

An analysis is made for the conjugate heat transfer problem of natural convection on one side of a vertical wall and forced convection on the other side. The natural convection mode is treated analytically by employing the Oseen linearization approach developed by Gill. The forced convection boundary layer is analyzed on the basis of the integral technique. The two solutions are matched on the separating wall so as to satisfy the continuity of heat flux between the two fluids. The analysis shows that the complexion of this two-fluid problem is governed by a dimensionless conjugate parameter, R, which relates the heat transfer effectiveness of forced convection mode to that of free convection mode. The boundary conditions at the wall are not prescribed in the analysis in advance, rather, determined among the results. The heat transfer and flow characteristics in the two counter-flowing boundary layers are presented graphically. Heat transfer results of engineering importance are determined as a function of the conjugation parameter. Received on 19 August 1998     

Subcooled forced convection film boiling over a vertical flat plate embedded in a fluid-saturated porous medium

The problem of subcooled forced convection film boiling on a vertical flat plate embedded in a porous medium was attacked exploiting similarity transformations on the governing equations and boundary conditions in both vapor and liquid layers. Similarity solutions were obtained to investigate the effects of the vapor super-heating and liquid subcooling. The heat transfer groupingNu _x /Ra _x ^1/2 is expressed in terms of a function of three parameters associated with the degree of liquid subcooling (Sub), the degree of vapor superheating (Sup) and the vapor buoyancy effect relative to the liquid forced convection effect (R). It is found that the level ofNu _x /Ra _x ^1/2 increases asSup orR decreases and asSub increases. Furthermore, asymptotic expressions were reduced considering the physical limiting conditions, namely, thin and thick vapor films.     

Natural convection near a vertical plate with ramped wall temperature

The unsteady natural convective flow of an incompressible viscous fluid near a vertical plate has been considered. It is assumed that the bounding plate has a ramped temperature profile. The exact solutions of the energy and momentum equations, under the usual Boussinesq approximation, have been obtained in closed form. There are two different solutions for the fluid velocity—one valid for the fluids of Prandtl numbers different from unity, and the other for which the Prandtl number is unity. The variations of the fluid temperature, velocity as well as the Nusselt number and wall skin friction have been presented graphically. The natural convection near a ramped temperature plate has also been compared with the flow near a plate with constant temperature.     

Natural convection in vertical slots with wall temperature oscillation

Natural convection in vertical slots has been studied experimentally with oscillation of a cold wall temperature. It is found that at sufficiently large Rayleigh numbers travelling-wave instability occurs, but only in the region close to the cold wall when the hot wall is maintained at the initial temperature, and appears in both cold and hot regions when both hot and cold wall temperatures are changed simultaneously and symmetrically with the initial temperature value, although the hot wall temperature is kept constant after the change. The observed instability may be attributed to the leading-edge effect induced by the cold wall temperature oscillation, and selectively amplified by the natural convection flow.The author would like to thank Professor R. S. Tankin for his encouragement and assistance throughout this study. The financial support of the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged.     

Transient natural convection in a rectangular enclosure with one heated side wall

This paper describes a numerical and theoretical study of the transient natural convection heating of a two-dimensional rectangular enclosure filled with fluid. The heating is applied suddenly along one of the side walls, while the remaining three walls are maintained insulated. It is shown that the process has two distinct phases, an early period dominated by conduction and a late period dominated by convection. The scaling laws for the heat transfer rate and the effectiveness (energy storage fraction) are determined based on scale analysis. These theoretical results are confirmed by numerical experiments conducted in the domain Ra = 10³−10⁶, Pr = 7, A = 1, where Ra is the Rayleigh number based on height and initial temperature difference, Pr is the Prandtl number, and A is the height/length ratio of the enclosure. Correlations for heat transfer rate and effectiveness are constructed by comparing the theoretical scaling laws with the numerical results.     

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.     

Effects of anisotropy on the free convection from a vertical plate embedded in a porous medium

The effects of anisotropy on the steady laminar boundary-layer free convection over a vertical impermeable surface are analysed by using the method of integral relations. If the permeability in the direction orthogonal to the plate is greater than the permeability along the plate, then there is an increase in the temperature field.     

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.     

Mixed convection flow over a vertical wedge embedded in a highly porous medium

 The steady mixed convection flow over a vertical wedge with a magnetic field embedded in a porous medium has been investigated. The effects of the permeability of the medium, surface mass transfer and viscous dissipation on the flow and temperature fields have been included in the analysis. The coupled nonlinear partial differential equations governing the flow field have been solved numerically using the Keller box method. The skin friction and heat transfer are found to increase with the parameters characterizing the permeability of the medium, buoyancy force, magnetic field and pressure gradient. However the effect of the permeability and magnetic field on the heat transfer is very small. The heat transfer increases with the Prandtl number, but the skin friction decreases. The buoyancy force which assists the forced convection flow causes an overshoot in the velocity profiles. Both the skin friction and heat transfer increase with suction and the effect of injection is just the reverse. Received on 21 May 1999     

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.     

Natural convection in vertical cavities with internal heat generating porous medium

Steady natural convection heat transfer in a two dimensional cavity filled with a uniform heat generating, saturated porous medium has been studied. The boundary conditions were: Two isothermal walls at different temperatures, two horizontal adiabatic walls. The aspect ratio was varied from 0.1 to 10 and the Rayleigh number from 10⁰ to 10⁸. The results are presented in terms of the isotherms and stream functions, the temperature variation and maximum temperature in the cavity and heat transfer from the vertical walls. The study indicates that asymmetric vertical boundary conditions with _h >0 has an important effect on the temperature and flow fields as well as on the heat transfer characteristics of the cavity with highly asymmetric results. Various heat transfer modes are identified dependent on the Rayleigh number and the aspect ratio.Es wurde stetiger Wärmeübergang durch freie Konvektion in einen zweidimensionalen Hohlraum, gefüllt mit gleichmäßig wärmeerzeugendem porösen Medium, untersucht. Dabei wurden folgende Randbedingungen festgelegt: zwei isotherme Wände unterschiedlicher Temperatur, zwei horizontale adiabate Wände. Das Verhältnis vonH/L wurde zwischen 0,1 und 10, die Rayleigh-Zahl zwischen 10⁰ und 10⁸ variiert. Die Ergebnisse werden durch Terme der Temperaturverläufe, der maximalen Temperatur im Hohlraum, der Isothermen und der Strömungsfunktion, sowie der Wärmeübertragung der vertikalen Wände dargestellt. Die Studie zeigt, daß asymmetrische Randbedingungen mit _h >0 einen großen Einfluß auf Temperatur- und Strömungsfelder, als auch auf die Wärmeübertragungskennzahlen des Hohlraumes mit stark asymmetrischen Ergebnissen haben. Die verschiedenen Arten der Wärmeübertragung werden in Abhängigkeit von der Rayleigh-Zahl und dem Verhältnis vonH/L beschrieben.     

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     

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.     

Conjugate free convection over a vertical slender hollow cylinder embedded in a porous medium

A numerical study of the steady conjugate free convection over a vertical slender, hollow circular cylinder with the inner surface at a constant temperature and embedded in a porous medium is reported. The governing boundary layer equations for the fluid-saturated porous medium over the cylinder along with the one-dimensional heat conduction equation for the cylinder are cast into dimensionless form, by using a non-similarity transformation. The resulting non-similarity equations with their corresponding boundary conditions are solved by using the Keller box method. Emphasis is placed on the effects caused by the wall conduction parameter, p, and calculations have covered a wide range of this parameter. Heat transfer results including the temperature profiles, the interface temperature profiles and the local Nusselt number are presented. Received on 17 November 1997     

Natural convection flow with solidification between two vertical plates filled with a porous medium

Quasi-steady solidification between two vertical flat plates filled with a saturated porous medium has been investigated. The medium is homogeneous and isotropic. The convection flow of liquid takes place in the porous medium in the variable space between the two walls. One of the vertical walls is set to a temperature lower than the solidification temperature of the medium and therefore a frozen crust is formed on this wall. The second wall has a high temperature then the fusion temperature of the medium. The problem has been simplified by assuming laminar flow and the Brinkman and the Oberbeck–Bousinesq’s approximations. The results are presented in terms of the velocity for different properties of the porous medium. Various velocities are displayed in dependence of the Rayleigh and Darcy numbers. The study indicates that asymmetric boundary conditions have an important effect on the temperature and flow field. In addition, the growth of the thickness of the frozen layer with time has been derived from a simple analytical solution of the interface energy equation.     

The effect of thermal dispersion on free convection film condensation on a vertical plate with a thin porous layer

An analytical solution to the problem of condensation by natural convection over a thin porous substrate attached to a cooled impermeable surface has been conducted to determine the velocity and temperature profiles within the porous layer, the dimensionless thickness film and the local Nusselt number. In the porous region, the Darcy–Brinkman–Forchheimer (DBF) model describes the flow and the thermal dispersion is taken into account in the energy equation. The classical boundary layer equations without inertia and enthalpyterms are used in the condensate region. It is found that due to the thermal dispersion effect, the increasing of heat transfer is significant. The comparison of the DBF model and the Darcy–Brinkman (DB) one is carried out.     

Mixed convection film condensation from downward flowing vapors onto a sphere with variable wall temperature

A model is developed for the study of mixed- convection film condensation from downward flowing vapors onto a sphere with variable wall temperature. The model combined natural convection dominated and forced convection dominated film condensation, concerning effects of pressure gradient (P), interfacial vapor shear drag and non-uniform wall temperature variation (A), has been investigated and solved numerically. The effect of pressure gradient on the dimensionless mean heat transfer, NuˉRe^−1/2 will remain almost uniform with increasing P until for various corresponding available values of F. Meanwhile, the dimensionless mean heat transfer, NuˉRe^−1/2 is increasing significantly with F for its corresponding available values of P. Although the non-uniform wall temperature variation has an appreciable influence on the local film flow and heat transfer; however, the dependence of mean heat transfer on A can be almost negligible. Received on 10 October 1996