Local Thermal Non-equilibrium and Heterogeneity Effects on the Onset of Convection in an Internally Heated Porous Medium

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

Free Convection Heat Transfer From A Sphere In A Porous Medium Using A Thermal Non-equilibrium Model

This work studies the free convection heat transfer from a sphere with constant wall temperature embedded in a fluid-saturated porous medium using a thermal non-equilibrium model. The governing equations are transformed into boundary-layer partial differential equations by the coordinate transform, and the obtained governing equations are then solved by the cubic spline collocation method. The temperature distributions for fluid and solid phases are shown for different values of the porosity scaled thermal conductivity ratio, the interphase heat transfer parameter, and the streamwise coordinate. The effects of the porosity scaled thermal conductivity ratio and the interphase heat transfer parameter between solid and fluid phases on the local Nusselt numbers for fluid and solid phases are examined. Results show the local Nusset number for the porous medium can be increased by increasing the porosity scaled thermal conductivity ratio. Moreover, the thermal non-equilibrium effect is more significant for low values of the porosity scaled thermal conductivity ratio or the interphase heat transfer parameter.     

Metal Foam Heat Exchangers for Heat Transfer Augmentation from a Cylinder in Cross-Flow

A numerical study has been conducted to examine the heat transfer from a metal foam-wrapped solid cylinder in cross-flow. Effects of the key parameters including the free stream velocity and characteristics of metal foam such as porosity, permeability, and form drag coefficient on heat and fluid flow are examined. Being a determining factor in pressure drop and heat transfer increment, the porous layer thickness is changed systematically to observe that there is an optimum layer thickness beyond which the heat transfer does not improve while the pressure drop continues to increase. This has been verified by the application of Bejan’s Intersection of Asymptotes method. Results have been compared to those of a finned-tube heat exchanger to observe much higher heat transfer rate with reasonable excess pressure drop leading to a higher area goodness factor for metal foam-wrapped cylinder.     

The Onset of Convection in an Anisotropic Porous Layer Using a Thermal Non-Equilibrium Model

The stability of a horizontal fluid saturated anisotropic porous layer heated from below and cooled from above is examined analytically when the solid and fluid phases are not in local thermal equilibrium. Darcy model with anisotropic permeability is employed to describe the flow and a two-field model is used for energy equation each representing the solid and fluid phases separately. The linear stability theory is implemented to compute the critical Rayleigh number and the corresponding wavenumber for the onset of convective motion. The effect of thermal non-equilibrium and anisotropy in both mechanical and thermal properties of the porous medium on the onset of convection is discussed. Besides, asymptotic analysis for both very small and large values of the interphase heat transfer coefficient is also presented. An excellent agreement is found between the exact and asymptotic solutions. Some known results, which correspond to thermal equilibrium and isotropic porous medium, are recovered in limiting cases.     

Stationary Fingering Instability in a Non-equilibrium Porous Medium with Coupled Molecular Diffusion

Finger type double diffusive convective instability in a fluid-saturated porous medium is studied in the presence of coupled heat-solute diffusion. A local thermal non-equilibrium (LTNE) condition is invoked to model the Darcian porous medium which takes into account the energy transfer between the fluid and solid phases. Linear stability theory is implemented to compute the critical thermal Rayleigh number and the corresponding wavenumber exactly for the onset of stationary convection. The effects of Soret and Dufour cross-diffusion parameters, inter-phase heat transfer coefficient and porosity modified conductivity ratio on the instability of the system are investigated. The analysis shows that positive Soret mass flux triggers instability and positive Dufour energy flux enhances stability whereas their combined influence depends on the product of solutal Rayleigh number and Lewis number. It also reveals that cell width at the convection threshold gets affected only in the presence of both the cross-diffusion fluxes. Besides, asymptotic solutions for both small and large values of the inter-phase heat transfer coefficient and porosity modified conductivity ratio are found. An excellent agreement is found between the exact and asymptotic solutions.     

Linear and non-linear double diffusive convection in a rotating porous layer using a thermal non-equilibrium model

Double diffusive convection in a fluid-saturated rotating porous layer heated from below and cooled from above is studied when the fluid and solid phases are not in local thermal equilibrium, using both linear and non-linear stability analyses. The Darcy model that includes the time derivative and Coriolis terms is employed as momentum equation. A two-field model that represents the fluid and solid phase temperature fields separately is used for energy equation. The onset criterion for stationary, oscillatory and finite amplitude convection is derived analytically. It is found that small inter-phase heat transfer coefficient has significant effect on the stability of the system. There is a competition between the processes of thermal and solute diffusions that causes the convection to set in through either oscillatory or finite amplitude mode rather than stationary. The effect of solute Rayleigh number, porosity modified conductivity ratio, Lewis number, diffusivity ratio, Vadasz number and Taylor number on the stability of the system is investigated. The non-linear theory based on the truncated representation of Fourier series method predicts the occurrence of subcritical instability in the form of finite amplitude motions. The effect of thermal non-equilibrium on heat and mass transfer is also brought out.     

Theoretical Analysis of Natural Convection in an Enclosure Filled with Disconnected Conducting Square Solid Blocks

Two different approaches have been implemented to interpret the existing data in Merrikh and Lage (2005a, in: Pop, Ingham, Transport Phenomena in Porous Media, Elsevier, Oxford) pertinent to the pore-scale simulation of natural convection in a laterally heated square enclosure filled with a fluid bathing discrete, disconnected and conducting solid blocks. Mathematical analysis of the problem based on the least squares method leads us to a Nu-Ra correlation with the solid-to-fluid thermal conductivity ratio and the porous medium permeability as the controlling parameters. In this study, while the porosity is fixed, the permeability is varied by changing either the block size or the number of blocks. Hence, three independent variables N, Ra, and κ being the number of blocks, Rayleigh number, and the conductivity ratio between the solid and the fluid, respectively, affect the overall heat transfer process. Based on a simplified thermal resistance approach, an alternative correlation is proposed to predict the Nusselt number as a function of the aforementioned parameters. Detailed analysis of the results and the expected errors are presented.     

Effect of temperature gradient within a solid particle on the rotation and oscillation modes in solid-dispersed two-phase flows

Liquid–solid two-phase flow with heat transfer is simulated, and the effect of temperature gradient within a solid particle on the particle behaviour and heat transfer is studied. The interaction between fluid and particles is considered with our original immersed solid approach on a rectangular grid system. The local heat flux at the fluid–solid interface is described with an anisotropic heat conductivity matrix, and the governing equation of temperature is time-updated with an implicit treatment for the diffusion term. The method is applied to a 2-D natural convection flow of a relatively low Rayleigh number including multiple particles. Heat transfer and particle behaviours are studied for different solid heat conductivities (ratio to the fluid conductivity ranging between 10⁻³ and 10³) and solid volume fractions. Under a condition of relatively low heat conductivity ratio, the particles show a simple circulating flow. By increasing the heat conductivity ratio, a transition of the particulate flow is observed to oscillation mode around the domain centre due to the buoyancy force as a restitution force. The oscillation period is found to vary with the heat conductivity ratio, and it is related to the time scales for the heat transfer via fluid and solid.     

Pore-Scale Numerical Experiment on the Effect of the Pertinent Parameters on Heat Flux Splitting at the Boundary of a Porous Medium

Pore-scale heat and fluid flow simulation through reconstructed porous media is presented with the aim of investigating the physics of heat flux splitting at the boundary of porous media. As such, the effects of the solid to fluid thermal conductivity ratio, porosity, pore-scale Reynolds number, Prandtl number and heat conduction within the solid matrix are investigated. The results of the present study for heat transfer coefficient and pressure drop are compared with available experimental data and good agreement was observed. The validated results are then used to investigate the validity of the existing volume-averaged models. It was observed that while results based on the volume-averaged models are reasonably close to current predictions for $\varepsilon \le 0.7$ , the discrepancy between the two becomes notable for higher porosities. While existing models rely exclusively on porosity and thermal conductivity ratio, our newly proposed correlations show the effects of Reynolds number on the heat split mechanism for high porosities. On the other hand, the Prandtl number, at least for the range of parameters studies here, is found to be less influential on the boundary heat split mechanism.     

Fully-developed conjugate heat transfer in porous media with uniform heating

We propose a computational method for approximating the heat transfer coefficient of fully-developed flow in porous media. For a representative elementary volume of the porous medium we develop a transport model subject to periodic boundary conditions that describes incompressible fluid flow through a uniformly heated porous solid. The transport model uses a pair of pore-scale energy equations to describe conjugate heat transfer. With this approach, the effect of solid and fluid material properties, such as volumetric heat capacity and thermal conductivity, on the overall heat transfer coefficient can be investigated. To cope with geometrically complex domains we develop a numerical method for solving the transport equations on a Cartesian grid. The computational method provides a means for approximating the heat transfer coefficient of porous media where the heat generated in the solid varies “slowly” with respect to the space and time scales of the developing fluid. We validate the proposed method by computing the Nusselt number for fully developed laminar flow in tubes of rectangular cross section with uniform wall heat flux. Detailed results on the variation of the Nusselt number with system parameters are presented for two structured models of porous media: an inline and a staggered arrangement of square rods. For these configurations a comparison is made with literature on fully-developed flows with isothermal walls.     

The Effect of Heterogeneity on the Onset of Double-Diffusive Convection Induced by Internal Heating in a Porous Medium: A Layered Model

In this paper we have investigated the effects of both weak and strong heterogeneity on the onset of double-diffusive convection which is induced by combined effects of internal heating and solutal gradient. To make analytical progress, we considered a composite porous medium consisting of two horizontal layers. We investigated the effects of heterogeneities in permeability, thermal conductivity, volumetric heat source strength, and porosity. The solutal diffusivity, which becomes effective when a vertical salinity gradient is imposed, is affected by variation of porosity. We found that the effect of solutal diffusivity is stabilizing when the porosity increases upwards.     

Linear and nonlinear double diffusive convection in a rotating sparsely packed porous layer using a thermal non-equilibrium model

Double diffusive convection in a fluid-saturated rotating porous layer is studied when the fluid and solid phases are not in local thermal equilibrium, using both linear and nonlinear stability analyses. The Brinkman model that includes the Coriolis term is employed as the momentum equation. A two-field model that represents the fluid and solid phase temperature fields separately is used for the energy equation. The onset criterion for stationary, oscillatory, and finite amplitude convection is derived analytically. It is found that small inter-phase heat transfer coefficient has significant effect on the stability of the system. There is a competition between the processes of thermal diffusion, solute diffusion, and rotation that causes the convection to set in through either oscillatory or finite amplitude mode rather than stationary. The effect of solute Rayleigh number, porosity modified conductivity ratio, Lewis number, diffusivity ratio, Vadasz number, and Taylor number on the stability of the system is investigated. The nonlinear theory based on the truncated representation of Fourier series method predicts the occurrence of subcritical instability in the form of finite amplitude motions. The effect of thermal non-equilibrium on heat and mass transfer is also brought out. Some of the convection systems previously reported in the literature is shown to be special cases of the system presented in this study.     

Numerical Study of Heat Transfer in a Rectangular Channel with Porous Covering Obstacles

A numerical study is performed to analyze steady laminar forced convection in a channel in which discrete heat sources covered with porous material are placed on the bottom wall. Hydrodynamic and heat transfer results are reported. The flow in the porous medium is modeled using the Darcy–Brinkman–Forchheimer model. A computer program based on control volume method with appropriate averaging for diffusion coefficient is developed to solve the coupling between solid, fluid, and porous region. The effects of parameters such as Reynolds number, Prandtl number, inertia coefficient, and thermal conductivity ratio are considered. The results reveal that the porous cover with high thermal conductivity enhances the heat transfer from the solid blocks significantly and decreases the maximum temperature on the heated solid blocks. The mean Nusselt number increases with increase of Reynolds number and Prandtl number, and decrease of inertia coefficient. The pressure drop along the channel increases rapidly with the increase of Reynolds number.     

Numerical Simulation of Natural Convection in Wavy Porous Cavities Under the Influence of Thermal Radiation Using a Thermal Non-equilibrium Model

Continuum equations governing thermal non-equilibrium modeling of steady natural convection inside wavy enclosures with the effect of thermal radiation are developed. These equations account for such effects as the inter-phase heat transfer coefficient effect, the thermal radiation effect, the modified conductivity ratio effect and the Rayleigh number effect. Finite difference method is employed to solve these equations and comparisons between previous published works are presented. Numerical results for the flow and heat transfer for the fluid and solid phases are obtained for various combinations of the physical parameters. Graphical and tabular results illustrating interesting features of the physics of the problem are presented and discussed.     

Mixed convection heat transfer from a horizontal plate with variable surface heat flux in a porous medium

In this article nonsimilarity solution for mixed convection from a horizontal surface in a saturated porous medium was obtained for the case of variable surface heat flux. The entire mixed convection regime, ranging from pure forced convection to pure free convection, is considered by introducing a single nonsimilarity parameter. Heat transfer results are predicted by employing four different flow models, namely, Darcy's law, the Ergun model, and the Brinkman-Forchheimer-extended Darcy model with constant and variable porosity. The variable porosity effect is approximated by an exponential function. Effects of transverse thermal dispersion are taken into consideration in the energy equation, along with variable stagnant thermal conductivities. The formulation of the present problem shows that the flow and heat transfer characteristics depend on five parameters, that is, the power in the variation of surface heat flux, the nonsimilarity mixed-convection parameter, the inertia effect parameter, the boundary effect parameter, and the ratio of thermal conductivity of the fluid phase to that of the solid phase. Numerical results for the local Nusselt number variations, based on the various flow models, are presented for the entire mixed convection regime. The impacts␣of different governing parameters on the heat transfer results are thoroughly investigated. Received on 7 August 1997     

Numerical characterization of micro heat exchangers using experimentally tested porous aluminum layers

A microporous heat exchanger device is being developed for cooling high-power electronics. The device uses a mechanically compressed aluminum porous layer to improve the heat transfer at the coolant/solid interface and to provide more uniform cooling of the electronics. The hydraulic characteristics (porosity, permeability, and Forchheimer coefficient) of nine distinct compressed layers are obtained experimentally. These layers have porosity from 0.3 to 0.7 and permeability from 1.8 × 10⁻¹⁰ m2 to 1.2 × 10⁻⁹ m2. The inertia coefficient varies from 0.3 to 0.9. These hydraulic characteristics are used in the numerical simulations of a real microporous heat exchanger for cooling phased-array radars in development. Thermal and hydraulic performances are illustrated in terms of total pressure drop across the heat exchanger, maximum temperature difference in the direction transverse to the electronic modules, and maximum temperature within the coolant passage. Results indicate that the proposed design is capable of achieving a maximum transverse temperature difference of 2°C using polyalphaolephin as coolant.     

Mathematical simulation of radial heat transfer in packed beds by pseudohomogeneous modeling

Uniform flow regime and constant effective thermal conductivity inside packed beds are commonly accepted in the evaluation of the fluid dynamics and heat transfer in such systems.However,several authors have confirmed the presence of an oscillatory velocity profile caused by the effective contribution of porosity profile in the fluid dynamic behavior of packed beds,which directly influences the heat transfer inside the beds.This paper describes the application of a pseudo-homogeneous mathematical ...     

热局部非平衡下一维半无限长饱和多孔柱体的温度场分析

对于端部受温度载荷的一维半无限长多孔介质柱体,给出了热局部非平衡下固相和流相温度场在Laplace变换域中的解析表达式.对于冲击温度载荷的情况,获得了温度场在短时间内的Laplace逆变换渐近解析解.数值分析了流、固两相热扩散系数之比以及热交换系数对固相和流相温度场的影响,比较了热局部非平衡下加权温度与热局部平衡下温度之间的差别.     

Heat Transfer and Particle Behaviours in Dispersed Two-Phase Flow with Different Heat Conductivities for Liquid and Solid

Liquid-solid two-phase flow with heat transfer is directly simulated, to investigate the effects of the ratios of heat conductivities (solid to liquid) and bulk solid volume fraction from dense to dilute situations. The interaction between fluid and particles is solved by our original immersed solid approach on a rectangular grid system. A discrete element method with a soft-sphere collision model is applied for particle-particle and particle-wall interactions. Governing equation of temperature is time-updated with an implicit treatment for the diffusion term, which enables robust simulation with particles of very high/low ratios of heat conductivities (from 1/1000 to 1000) to the fluid. The local heat flux at the fluid-solid interface is modelled by a new flux decomposition technique, and incorporated into the implicit scheme of the temperature. The method is applied to a 2-D particulate flow in a natural convection in a square domain at a relatively low Rayleigh number. In the dense condition, for the cases with high ratios of heat conductivity, the heat transfer is promoted by strong convection, while the particles of low ratios of heat conductivity tend to hinder the development of the temperature rise in the flow field, causing a weak convection and low Nusselt number. Under a condition of relatively low solid volume fraction, fixed particles only depress the heat convection as the number of particles and heat conductivity ratio increase. For the cases with freely-moving particles, on the other hand, heat conductivity of particles has a stronger influence on the heat transfer of the system than the number of particles. The above simulation results highlight the effect of temperature distributions within the particles and liquid.