Mixed convection in porous media between vertical concentric cylinders

In this paper, the dimensionless parameters Gr^**/Re_p and H/d_p that govern mixed convection in a vertical annulus filled with porous media, are deduced through dimensional analysis, and the criteria for mixed convection is established thereby. The calculated results for detecting the non-Darcian effects are discussed. The predictions agree with the experiments. © 1999 Scripta Technica, Heat Trans Asian Res, 28(2): 95–101, 1999     

Free convection of composite porous medium in a vertical channel

A fully developed free convection flow of immiscible fluids in a vertical channel filled with a porous medium is analyzed in the presence of source/sink. The flow is modeled using the Darcy–Brinkman–Forchheimer equation model. The viscous and Darcy dissipation terms are included in the energy equation. The channel walls are maintained at two different constant temperatures. The transport properties of both fluids are assumed to be constant. Continuous conditions for velocity, temperature, shear stress, and heat flux of both fluids at the interface are employed. The resulting coupled nonlinear equations are solved analytically using regular perturbation method and numerically using finite difference method. The velocity and temperature profiles are obtained in terms of porous parameter, Grashof number, viscosity ratio, width ratio, conductivity ratio, and heat generation or heat absorption coefficient. It is found that the presence of porous matrix and heat absorption reduces the flow field. © 2011 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.20340     

Natural convection and its fractal for liquid freezing in a vertical cavity filled with porous medium

The numerical simulation for a freezing liquid-saturated porous media in a vertical cylindrical cavity under the third kind of thermal boundary condition is reported in this paper. It shows that the effect of natural convection in the liquid phase decreases the freezing layer thickness and the freezing front has a wave shape instead of a stable plane, with one or more pair of eddy cells. This indicates a fractal existence. © 1999 Scripta Technica, Heat Trans Asian Res, 28(3): 165–171, 1999     

Experimental study of natural convection heat transfer of air layers in vertical annuli under high rayleigh number conditions

In this paper experimental investigations of natural convection heat transfer of air layers in vertical annuli are presented. In these experiments, the surface of the inner cylinder is maintained at a constant heat flux condition and the outer cylinder is cooled in the atmosphere. In order to obtain the convective contribution, the overall heat transfer data are corrected for thermal radiation and axial conduction losses from the end plates in the annuli. Special emphasis, in these investigations, was placed on the high Rayleigh number region where no experimental data are available in the literature. Data were obtained for Rayleigh numbers greater than 10⁹. The radius ratios studied were 2.03 and 3.92, and the aspect ratios studied were 23.94 and 66.67. Heat transfer correlations for average Nusselt numbers were developed for different Rayleigh number regions. For the low Rayleigh number region the results of this paper agree with the correlations reported in the literature. Much needed data and correlations for the high Rayleigh number region are obtained for the first time. These results improve the predictive ability for the heat transfer characteristics in the high Rayleigh number region. ©1999 Scripta Technica, Heat Trans Asian Res, 28(1): 50–57, 1999     

Mathematical modeling of onset of convection in a porous layer with viscosity variation

IntroduthenIn a novel EOR (Enhanced Oil Recovery) method,diffusion-convechon may be employed as a drive toforce heavy, viscous oil to move. A ~ of gas isexpanded beneath oil, and gas is allowed tO diffuseupward. As oil dissolves gas, it becomes lighter.Therefore, an inverse density gradient is develOPed in oillayer gradually. When this density gradient exceeds acritical value, convechon is started. It may be concjudedthat develOPing a theory, which predictS the onset ofconvection, should…     

Natural convection along a vertical complex wavy surface

A previously proposed transformation has been applied to the natural-convection boundary layer along a complex vertical surface created from two sinusoidal functions, a fundamental wave and its first harmonic. The numerical results demonstrate that the additional harmonic substantially alters the flow field and temperature distribution near the surface. The conclusion that the averaged heat-transfer rate per unit-wetted wavy surface is less than that of a corresponding flat plate [L.S. Yao, Natural convection along a vertical wavy surface, ASME J. Heat Transfer 105 (1983) 465–468] has been confirmed for this more complex surface. On the other hand, the total heat-transfer rates for a complex surface are greater than that of a flat plate. The numerical results show that the enhanced total heat-transfer rate seems to depend on the ratio of amplitude and wavelength of a surface.     

Nonsimilar solutions for mixed convection of water at 4°C over a vertical surface with a convection boundary condition in a porous medium

A boundary layer analysis has been presented for the mixed convection of water at 4°C over a vertical plate embedded in a porous medium. The Robin or convective boundary condition at the surface has been considered where the heat lost from the surface is the product of a heat transfer coefficient and the temperature difference between the surface and the free stream. The governing non‐similar boundary layer equations for both the forced and free convection dominated regimes were solved numerically by means of an implicit finite difference method. The friction factor and dimensionless heat transfer rate (Nusselt number) are presented for several values of the dimensionless heat transfer coefficient and buoyancy parameter. © 2012 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library (wileyonlinelibrary.com/journal/htj). DOI 10.1002/htj.21022     

Effects of buoyancy assisting and opposing flows on mixed convection boundary layer flow over a permeable vertical surface

The present study deals with new similarity solution of steady mixed convection boundary layer flow over a permeable surface for convective boundary condition. It has been shown that a self similar solution is possible when the mass transfer velocity at the surface of the plate varies like x^−1/2, where x is the distance from the leading edge of the solid surface. Two point boundary value problem governed by non-linear coupled ordinary differential equations have been solved numerically using implicit finite difference scheme in combination with the quasi-linearization technique. It is interesting to note that dual solutions exist for buoyancy assisting flow, besides that usually reported in literature for buoyancy opposing flow. Further, the buoyancy assisting force causes considerable overshoot in the velocity profile and the Prandtl number strongly affects the thermal boundary layer thickness including the surface heat transfer rate.     

Transient free convection in cold water past an infinite vertical porous plate

A finite-difference solution of the transient free convection flow of water near 4°C past an infinite vertical porous plate has been carried out. The most appropriate relation for the density in terms of the temperature, as derived by Gebhart and Mollendorf, is employed. It has been observed that, when the buoyancy force is in the upward direction, the velocity profiles are positive and opposite to those in the case where the buoyancy force is in the downward direction. Also, the skin friction is more when the buoyancy force is in the upward direction than when the buoyancy force is in the downward direction.     

Natural convection heat transfer of high temperature gas in an annulus between two vertical concentric cylinders

Water cooling panels have been adopted as the vessel cooling system of the High Temperature Engineering Test Reactor (HTTR) to cool the reactor core indirectly by natural convection and thermal radiation. In order to investigate the heat transfer characteristics of high temperature gas in a vertical annular space between the reactor pressure vessel and cooling panels of the HTTR, we carried out experiments and numerical analyses on natural convection heat transfer coupled with thermal radiation heat transfer in an annulus between two vertical concentric cylinders with the inner cylinder heated and the outer cylinder cooled. In the present experiments, Rayleigh number based on the height of the annulus ranged from 2.0 × 10⁷ to 5.4 × 10⁷ for helium gas and from 1.2 × 10⁹ to 3.5 × 10⁹ for nitrogen gas. The numerical results were in good agreement with the experimental ones regarding the surface temperatures of the heating and cooling walls. As a result of the experiments and the numerical analyses, the heat transfer coefficient of natural convection coupled with thermal radiation was obtained as functions of Rayleigh number, radius ratio, and the temperatures and emissivities of the heating and cooling wall surfaces. © 2005 Wiley Periodicals, Inc. Heat Trans Asian Res, 34(5): 293–308, 2005; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/htj.20070     

Double diffusive natural convection along an inclined wavy surface in a porous medium

This paper studies the double diffusive natural convection near an inclined wavy surface in a fluid saturated porous medium with constant wall temperature and concentration. A coordinate transformation is employed to transform the complex wavy surface to a smooth surface, and the obtained boundary layer equations are solved by the cubic spline collocation method. Effects of angle of inclination, Lewis number, buoyancy ratio, and wavy geometry on the heat and mass transfer characteristics are studied. Results show that a decrease in the angle of inclination leads to a greater fluctuation of the local Nusselt and Sherwood numbers. Moreover, increasing the angle of inclination tends to increase the total heat and mass transfer rates.     

The onset of transient convection in bottom heated porous media

The theory of transient convection in bottom heated porous media under constant heat flux (CHF) condition or fixed surface temperature (FST) condition is advanced and verified by computational fluid dynamics (CFD) simulations. The use of κ∗, instead of κ_m tends to artificially inflate the value of Rayleigh number by about 30%. A new transient Rayleigh number for unsteady-state heat conduction was defined to predict the onset of transient convection in porous media, which were successfully simulated. The critical transient Rayleigh number from the simulation for CHF was about 29.60, which is close to the theoretical value of 27.1 calculated by Ribando and Torrance in 1976. In the case of FST, the critical transient Ra_c was found to be 30.9, which is close to the theoretical value of 32.3. The critical times of onset for simulations were predicted with good accuracy. The prediction of the critical wavelengths of the emerging plumes were fair for the 2D simulations. Any experiment to verify the linear stability analysis for thermal instability must simultaneously concur in the three eigenvalue parameters, namely the Biot number, the critical wavenumber and the corresponding critical Rayleigh number, apart from the physical boundaries. The average maximum transient Nusselt number was found to be 3.41 for CHF and 3.5 for FST respectively.     

Natural convection heat and mass transfer along a vertical wavy surface

A numerical study of natural convection heat and mass transfer along a vertical wavy surface has been performed. The wavy surface is maintained at uniform wall temperature and constant wall concentration. A simple coordinate transformation is employed to transform the complex wavy surface to a flat plate. A marching finite-difference scheme is used for the analysis. The buoyancy ratio N, amplitude-wavelength ratio α, and Schmidt number Sc are important parameters for this problem. The numerical results, including the developments of skin-friction coefficient, velocity, temperature, concentration, Nusselt number as well as Sherwood number along the wavy surfaces are presented. The effects of the buoyancy ratio N and the dimensionless amplitude of wavy surface on the local Nusselt number and the local Sherwood number have been examined in detail.     

Effect of variable viscosity on mixed convection boundary layer flow over a vertical surface embedded in a porous medium

The steady mixed convection boundary layer flow over a vertical impermeable surface embedded in a porous medium when the viscosity of the fluid varies inversely as a linear function of the temperature is studied. Both cases of assisting and opposing flows are considered. The transformed boundary layer equations are solved numerically by a finite difference method. Numerical results for the flow and heat transfer characteristics are obtained for various values of the mixed convection parameter ε and the variable viscosity parameter θ_e. It has been found that in the opposing flow case, dual solutions exist and boundary separation occurs.     

MHD free convection flow along a vertical wavy surface with heat generation or absorption effect

We formulate the problem of free convection from a vertical wavy surface embedded in a uniform porous medium in the presence of an external magnetic field and internal heat generation or absorption effects. Using the appropriate transformations, the boundary layer equations are reduced to non-linear partial differential equations. The transformed boundary layer equations are solved numerically using Runge–Kutta integration scheme with the shooting technique. We have focused our attention on the evaluation of the local Nusselt number Nu_x, dimensionless velocity, f′, and temperature, θ. The governing parameters are the amplitude of the waviness of the surface, a, ranging from 0.0 (flat plate) to 0.3, and the heat generation absorption parameter Q ranging from − 0.25 to 0.25, and magnetic parameter Mn, ranging from 0.0 to 2.0. The effect of all these parameters are discussed and plotted.     

Natural convection and radiation heat transfer in a vertical porous layer with a hexagonal honeycomb core (Part 1: numerical analysis)

Combined heat transfer characteristics were obtained numerically for three-dimensional natural convection and thermal radiation in a long and wide vertical porous layer with a hexagonal honeycomb core. We assumed that the porous layer was both homogeneous and isotropic. The pure Darcy law for fluid flow and Rosseland's approximation for radiation were used. The numerical methodology was based on an algebraic coordinate transformation technique and the transformed governing equations were solved using the SIMPLE algorithm. The effect of radiation on the heat transfer characteristics was investigated over a wide range of radiation numbers and temperature ratios for two Darcy-Rayleigh number values (Ra^* = 100 and 1000) and for a fixed aspect ratio of H/L = 1. The results are presented in the form of combined radiation and convection heat transfer coefficients and are compared with the corresponding values for pure natural convection. © 1999 Scripta Technica, Heat Trans Asian Res, 28(4): 278–294, 1999     

Fluid flow and heat transfer of natural convection around array of vertical heated plates

Natural convective flows around an array of vertical heated plates were investigated experimentally. Main concerns were directed to the influences of plate numbers on the heat transfer characteristics of the plates. Both surfaces of the test plates were heated with constant and equal heat fluxes and their local heat transfer coefficients were measured. The results showed that the coefficients of the surfaces of the array facing outward became higher than those facing inward. The flow fields around the bottom of the plate array were visualized with smoke. The result showed that the ambient flow is directed from the sides to the center of the array and enters the parallel channel obliquely. These flows cause the above difference in the coefficients. While the difference gradually diminished in between the plates placed in the central section of the array, their coefficients asymptotically approach those of the analytical correlation that assumed a uniform velocity at the channel inlet. © 2008 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/htj.20232     

Free convection to power‐law fluids from a vertical cone of variable surface temperature

Steady free convection laminar boundary‐layer flow about a heated vertical cone immersed in a quiescent power‐law fluid is investigated. Surface temperature is assumed to have a power‐law variation with the distance. Similarity solutions are obtained numerically for the boundary‐layer velocity and temperature profiles. Magnitudes of the velocity and temperature profiles are found to decrease with increasing values of the flow behaviour index and the surface temperature exponent. The effects of flow behaviour index and the generalized Prandtl number on the surface shear stress and heat flux are determined. Copyright © 2001 John Wiley & Sons, Ltd.     

Prandtl number dependence of unsteady natural convection along a vertical plate in a stably stratified fluid

The Prandtl number dependence of unsteady laminar natural convection along an infinite vertical plate in a thermally stratified fluid is investigated. Flows are induced by an impulsive (step) change in plate temperature and by a suddenly imposed plate heat flux. Analytical solutions of the viscous equations of motion and thermodynamic energy are obtained for Prandtl numbers near unity by the method of Laplace transforms and a regular perturbation expansion. The zeroth-, first- and second-order terms in the expansion are obtained for an impulsive change in plate temperature, while the zeroth- and first-order terms are obtained for a sudden application of a plate heat flux. The developing boundary layers are thicker, more vigorous, and more sensitive to the Prandtl number at smaller Prandtl numbers (<1) than at larger Prandtl numbers (>1). The analytical results are confirmed and extended with results from numerical simulations for Prandtl numbers strongly deviating from unity.     

On the analytical solution of the Bejan‐Khair equation arising in free convection in porous media

This paper revisits the fundamental problem of free convection heat and mass transfer over a heated vertical surface embedded in a porous medium using analytical techniques. An integral procedure is applied to the boundary layer similar equation for the combined heat and mass transfer from a vertical surface in a porous medium. The analytical approximations have covered a wide range of governing parameters of the problem. The results for both Nusselt and Sherwood numbers agree well with numerical results published in the literature. © 2010 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/htj.20297