A Two-Equation Model for Heat Conduction in Porous Media (I: Theory)

A two-equation model is presented which describes the conservation of heat in each phase of a porous medium in which diffusion is the predominant means of heat transfer, and of which the phases are not in thermal equilibrium with each other. The model is derived using the method of local volume averaging. This formulation, together with the introduction of characteristic temperature distributions, yields the definition of an effective and a coupled thermal conductivity tensor.     

Effective Thermal Conductivity Modelling for Closed-Cell Porous Media with Analytical Shape Factors

This paper presents a fully analytical model for the effective thermal conductivity of two-phase porous media with two-/three-dimensional closed cells, applicable to honeycombs and closed-cell foams. The present model combines an existing analytical expression derived based on the Laplace heat conduction equation with an analytical shape factor which corrects the deviation caused from a non-circular (or non-spherical) pore inclusion. Results demonstrate the validity of the present model capable of analytically estimating the effective thermal conductivity of closed-cell porous media. The simple yet accurate model provides the physical mechanisms of how effective thermal conductivity depends upon the shape of pores.     

多层平行裂隙型多孔介质通道内流体流动传热特性

基于Brinkman-extended Darcy模型和局部热平衡模型, 对多层平行裂隙型多孔介质通道内的流动传热特性进行研究. 获得了多层平行裂隙型多孔介质通道内各区域的速度场、温度场、摩擦系数及努塞尔数解析解, 并分析了裂隙层数、达西数、空心率、有效热导率之比等对通道内流动传热特性的影响. 结果表明: 达西数较小时, 通道多孔介质层内会出现不随高度变化的达西速度, 此达西速度会随裂隙层数的增加而增大, 但却不受各裂隙层下多孔介质层位置变化的影响. 增加裂隙层数会减弱空心率对压降的影响, 会使通道内流体压降升高, 但升高程度会逐渐降低. 增大热导率之比或减小空心率会使多裂隙通道内出现阶梯式温度分布, 而在较小热导率之比或较大空心率时多裂隙情况下的温度分布曲线会趋于一致. 此外, 当热导率之比较小时, 多层裂隙通道内的传热效果在任何空心率下都要优于单裂隙情况, 当热导率之比较大时, 存在临界空心率使各裂隙层数通道内的传热效果相同, 且多裂隙通道内继续增加裂隙层数对传热强度影响不大.      

STUDY ON ANALYSIS METHOD OF “THREE BOX” IN HEAT AND MASS TRANSFER PROCESS OF RESERVOIR MEDIA

油藏多孔介质孔隙组成及结构变化多样,一些特性参数很难全部获得,精确描述和分析困难;另外,多孔介质内渗流过程水力条件和作用机理复杂,存在热流固耦合作用,目前的一些分析方法和研究模型具有一定的局限性.提出了油藏多孔介质的表征单元体（representative elementary volume,REV）描述表征方法;基于表征单元体建立了多孔介质的黑箱模型、灰箱模型和白箱模型,据此提出了多孔介质的“黑箱→灰箱→白箱”分析过程.基于黑箱模型和灰箱模型推导了REV导热系数计算公式、给出了REV热质传递过程的热平衡方程.结合中国油藏热采情况,对多孔介质导热系数变化规律和蒸汽驱热质传递特性进行了分析,得到了一些有意义的结果.该工作为多孔介质热质传递过程分析提供了新思路和新方法.     

A Highly Conductive Porous Medium for Solid–Gas Reactions: Effect of the Dispersed Phase on the Thermal Tortuosity

The heat transfer in a highly conductive material constituted by a graphite matrix in which a granular phase is dispersed is studied. The effective thermal conductivity of this anisotropic porous composite medium used in solid–gas reactors can vary largely with the component fractions. The effect of the dispersed grains on the deformable structure of the matrix is considered. A model developed on the basis of thermal tortuosity by analogy with mass transfer is adequately correlated with experimental results.     

Mixed Convection Heat Transfer in the Annulus Between Two Concentric Vertical Cylinders Using Porous Layers

A numerical investigation of the steady-state, laminar, axi-symmetric, mixed convection heat transfer in the annulus between two concentric vertical cylinders using porous inserts is carried out. The inner cylinder is subjected to constant heat flux and the outer cylinder is insulated. A finite volume code is used to numerically solve the sets of governing equations. The Darcy–Brinkman–Forchheimer model along with Boussinesq approximation is used to solve the flow in the porous region. The Navier–Stokes equation is used to describe the flow in the clear flow region. The dependence of the average Nusselt number on several flow and geometric parameters is investigated. These include: convective parameter, λ, Darcy number, Da, thermal conductivity ratio, K _r, and porous-insert thickness to gap ratio (H/D). It is found that, in general, the heat transfer enhances by the presence of porous layers of high thermal conductivity ratios. It is also found that there is a critical thermal conductivity ratio on which if the values of Kr are higher than the critical value the average Nusselt number starts to decrease. Also, it found that at low thermal conductivity ratio (K _r ≈ 1) and for all values of λ the porous material acts as thermal insulation.     

The Use of Porous Fins for Heat Transfer Augmentation in Parallel-Plate Channels

In this study, a steady, fully developed laminar forced convection heat augmentation via porous fins in isothermal parallel-plate duct is numerically investigated. High-thermal conductivity porous fins are attached to the inner walls of two parallel-plate channels to enhance the heat transfer characteristics of the flow under consideration. The Darcy–Brinkman–Forchheimer model is used to model the flow inside the porous fins. This study reports the effect of several operating parameters on the flow hydrodynamics and thermal characteristics. This study demonstrates, mainly, the effects of porous fin thickness, Darcy number, thermal conductivity ratio, Reynolds number, and microscopic inertial coefficient on the thermal performance of the present flow. It is found that the highest Nusselt number is achieved at fully filled porous duct which requires the highest pumping pressure. The results show that using porous fins requires less pumping pressure with comparable high heat augmentation weight against fully filled porous duct. It is found that higher Nusselt numbers are achieved by increasing the microscopic inertial coefficient (A), the Reynolds number (Re), and the thermal conductivity of the porous substrate k ₂. The results show that heat transfer can be enhanced (1) with the use of high thermal conductivity fins, (2) by decreasing the Darcy number, and (3) by increasing microscopic inertial coefficient.     

Effect of an Inserted Porous Layer Located at a Wall of a Parallel Plate Channel on Forced Convection Heat Transfer

A theoretical study is performed on heat and fluid flow in partially porous medium filled parallel plate channel. A uniform symmetrical heat flux is imposed onto the boundaries of the channel partially filled with porous medium. The dimensional forms of the governing equations are solved numerically for different permeability and effective thermal conductivity ratios. Then, the governing equations are made dimensionless and solved analytically. The results of two approaches are compared and an excellent agreement is observed, indicating correctness of the both solutions. An overall Nusselt number is defined based on overall thermal conductivity and difference between the average temperature of walls and mean temperature to compare heat transfer in different channels with different porous layer thickness, Darcy number, and thermal conductivity ratio. Moreover, individual Nusselt numbers for upper and lower walls are also defined and obtained. The obtained results show that the maximum overall Nusselt number is achieved for thermal conductivity ratio of 1. At specific values of Darcy number and thermal conductivity ratio, individual Nusselt numbers approach to infinity since the value of wall temperatures approaches to mean temperature.     

Thermal Analysis of Natural Convection Porous Fins

This work introduces a simple method of analysis to study the performance of porous fins in a natural convection environment. The method is based on using energy balance and Darcy’s model to formulate the heat transfer equation. The thermal performance of porous fins is then studied for three types of fins: long fin, finite-length fin with insulated tip and a finite-length fin with tip exposed to a known convection coefficient. It is found from the analysis that the effect of different design and operating parameters such as: Ra number, Da number, thermal conductivity ratio, Kr and length thickness ratio on the temperature distribution along the fin is grouped into one newly defined parameter called S_H. The effect of the variation of S_H on the porous fin thermal performance is established. The effect of varying the fin length and thermal conductivity ratio on the heat transfer rate from the fin is investigated and compared with that for a solid fin at certain conditions. It is found that the heat transfer rate from porous fin could exceed that of a solid fin. It is also found that increasing the fin length and effective thermal conductivity enhances the heat transfer from the fin up certain limit, where a further increase in these parameters adds no improvement to the fin performance. On Leave from Jordan University of Science and Technology, Irbid-Jordan     

空心材料导热性能预测研究

本文建立了一种预测空心材料导热性的方法。研究了空心材料的导热性。用柱形空心材料分析了体分比和孔洞的排列方式对整体材料导热性的影响，用圆柱形、方柱形空心材料和含裂纹材料，分析了空心形状对材料导热性的影响。同他人的实验结果和某些现有的理论模型比较表明，本文方法是有效的。本文的结果能够很好地解释实验结果。     

Thermal Non-equilibrium Natural Convection in a Square Enclosure with Heat-Generating Porous Layer on Inner Walls

Differentially heated enclosure with heat-generating porous layer on inner walls is studied computationally for non-Darcy flow and thermal non-equilibrium models. In this study, this problem is investigated for different internal and external Rayleigh numbers, Darcy numbers, porosity-scaled thermal conductivity ratio, solid-/fluid-scaled heat transfer coefficient and dimensionless thickness of the porous layer. The results indicate that the dimensionless thickness of the porous layer has an important effect on the heat transfer in the enclosure. It was found that the thermal non-equilibrium model is needed for small values of the porosity-scaled thermal conductivity ratio and the solid-/fluid-scaled heat transfer coefficient. It is shown that the convection of heat due to internal heat generation is increased in the enclosure when the ratio of internal Rayleigh number to external Rayleigh number is larger.     

多相材料传热微结构的多目标优化设计

提出基于散热弱度的材料微结构热传导性能的预测方法,分别从理论和数值上验证该方法与均匀化方法的等效性;推导出微结构等效热传导系数的灵敏度计算格式,建立传热微结构拓扑优化的数学模型.以二维、三维多相材料等效热传导系数的加权组合为目标,采用凸规划对偶优化算法和二次型周长约束进行材料微结构的设计和材料分布的棋盘格控制.数值算例表明基于散热弱度的传热材料微结构设计是可行、有效的,可以为实际的材料设计提供依据.     

Influence of Nonlinear Local Properties on Effective Transport

The assumption of constant local coefficients is one of the first restrictions in most of the smoothing theories for transport in porous media. In this paper we present a formal analysis of the effects produced by nonconstant local transport coefficients on the nonlinear behavior of the effective transport properties. In particular, we use the volume averaging method to study heat transport in a two-component system considering the local thermal conductivities as analytical functions of the temperature. Within this approach we obtain a general expression for the effective nonlinear thermal conductivity dependence on the averaged temperature gradient. The important result is that the effective conductivity is obtained by a linearly bounded problem (the closure problem), just as if the conductivities were constants, by replacing the constant conductivities by the actual temperature dependent ones. As an example, we model the porous medium as cylindrical inclusions in a periodic array and solve the closure problem for the case of the one-equation model. We analyze the values of the second derivative of the thermal conductivity with respect to the temperature to establish the range where the nonlinear corrections must be considered to correctly describe the effective transport.     

Effects of thermal radiation on MHD viscous fluid flow and heat transfer over nonlinear shrinking porous sheet

This paper investigates the effects of thermal radiation on the magnetohy-drodynamic (MHD) flow and heat transfer over a nonlinear shrinking porous sheet. The surface velocity of the shrinking sheet and the transverse magnetic field are assumed to vary as a power function of the distance from the origin. The temperature dependent viscosity and the thermal conductivity are also assumed to vary as an inverse function and a linear function of the temperature, respectively. A generalized similarity transformarion is used to reduce the governing partial differential equations to their nonlinear coupled ordinary differential equations, and is solved numerically by using a finite difference scheme. The numerical results concern with the velocity and temperature profiles as well as the local skin-friction coefficient and the rate of the heat transfer at the porous sheet for different values of several physical parameters of interest.     

油藏多孔介质热质传递“三箱”分析模型研究

油藏多孔介质孔隙组成及结构变化多样,一些特性参数很难全部获得,精确描述和分析困难;另外,多孔介质内渗流过程水力条件和作用机理复杂,存在热流固耦合作用,目前的一些分析方法和研究模型具有一定的局限性.提出了油藏多孔介质的表征单元体（representative elementary volume,REV）描述表征方法;基于表征单元体建立了多孔介质的黑箱模型、灰箱模型和白箱模型,据此提出了多孔介质的“黑箱→灰箱→白箱”分析过程.基于黑箱模型和灰箱模型推导了REV导热系数计算公式、给出了REV热质传递过程的热平衡方程.结合中国油藏热采情况,对多孔介质导热系数变化规律和蒸汽驱热质传递特性进行了分析,得到了一些有意义的结果.该工作为多孔介质热质传递过程分析提供了新思路和新方法.      

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.     

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.     

Steady and Unsteady Heat Transfer in a Channel Partially Filled with Porous Media Under Thermal Non-Equilibrium Condition

Steady and pulsatile flow and heat transfer in a channel lined with two porous layers subject to constant wall heat flux under local thermal non-equilibrium (LTNE) condition is numerically investigated. To do this, a physical boundary condition in the interface of porous media and clear region of the channel is derived. The objective of this work is, first, to assess the effects of local solid-to-fluid heat transfer (a criterion indicating on departure from local thermal equilibrium (LTE) condition), solid-to-fluid thermal conductivity ratio and porous layer thickness on convective heat transfer in steady condition inside a channel partially filled with porous media; second, to examine the impact of pulsatile flow on heat transfer in the same channel. The effects of LTNE condition and thermal conductivity ratio in pulsatile flow are also briefly discussed. It is observed that Nusselt number inside the channel increases when the problem is tending to LTE condition. Therefore, careless consideration of LTE may lead to overestimation of heat transfer. Solid-to-fluid thermal conductivity ratio is also shown to enhance heat transfer in constant porous media thickness. It is also revealed that an increase in the amplitude of pulsation may result in enhancement of Nusselt number, while Nusselt number has a minimum in a certain frequency for each value of amplitude.     

Effective thermal conductivity of wire-woven bulk Kagome sandwich panels

Thermal transport in a highly porous metallic wire-woven bulk Kagome (WBK) is numerically and analytically modeled. Based on topology similarity and upon introducing an elongation parameter in thermal tortuosity, an idealized Kagome with non-twisted struts is employed. Special focus is placed upon quantifying the effect of topological anisotropy of WBK upon its effective conductivity. It is demonstrated that the effective conductivity reduces linearly as the porosity increases, and the extent of the reduction is significantly dependent on the orientation of WBK. The governing physical mechanism of anisotropic thermal transport in WBK is found to be the anisotropic thermal tortuosity caused by the intrinsic anisotropic topology of WBK.