Volume-averaged Energy Equations Research Articles

Vegetation effects on radiation-induced natural convection in sloping water bodies

In this work, vegetation effects on the characteristics of radiation-induced natural convection (isotherms, circulation patterns, exchange flow rate) in sloping water bodies are investigated numerically. The water body consists of (a) a sloping vegetated region (with a bottom slope equal to 0.1) and (b) a deep region with a horizontal bottom. The vegetation of porosity 0.85 (typical of aquatic plants found in lakes) has a length equal to the length of the sloping region. It can block (totally or partially) the radiation and as a result a non-uniform (differential) heating is developed along the free surface of the water body. The Volume-Averaged Navier–Stokes equations together with the Volume-Averaged Energy equation are solved numerically in the vegetated region. The radiation-induced natural convection in a water body with only a sloping region (with no vegetation) is also considered for validation purposes since numerical and scaling analysis results are available in literature. The results indicate significant vegetation effects on the thermal and flow patterns especially for vegetation which blocks completely surface heating.

Diurnal heating and cooling of a sloping water body with vegetated shallow regions

In this work, diurnal heating, by solar radiation, and cooling in a sloping water body with vegetated shallow regions are investigated numerically. The sloping water body consists of a vegetated region with a bottom slope equal to 0.1 and a deep region with a horizontal bottom. The Volume-Averaged Navier-Stokes equations together with the Volume-Averaged Energy equation are solved numerically in the vegetated region. The latter has porosity equal to 0.85 (typical of aquatic plants found in lakes) and a length equal to the total length of the sloping region. At the top free surface, a time-dependent thermal forcing is applied, which is reduced in the vegetated region. Shallow water is considered with a maximum water depth being less than the penetration depth of the solar radiation. The non-vegetated sloping water body is also considered for validation purposes. The results (isotherms, streamlines, exchange flow rate), after ten full thermal forcing cycles, indicate significant vegetation effects on the daytime and night circulation and the exchange flow rates.

Natural convection due to surface cooling in sloping water bodies with vegetation

Natural convection induced by surface cooling in sloping water bodies with vegetation is investigated numerically. The sloping water body consists of a vegetated region with a bottom slope equal to 0.1 and a deep region with a horizontal bottom. The volume-averaged Navier–Stokes equations together with the volume-averaged energy equation are solved numerically in the vegetated region. Submerged vegetation has an equivalent porosity of 0.85. The vegetation length is either equal to the total or half the sloping region. A differential cooling is applied at the top free surface with a varying heat loss rate between the two regions. The non-vegetated sloping water body is also considered for validation purposes. The results indicate that, when cooling is reduced in the vegetated region (a) there is a time delay for the establishment of the quasi-steady regime and (b) the exchange flow rate and the gravity current's velocity decreases with increasing vegetation length. When cooling is completely blocked, different circulation patterns are developed.

Comparison of Pore-Level and Volume-Averaged Computations in Highly Conductive Spherical-Void-Phase Porous Materials

A study has been carried out to compare results obtained from pore-level simulations conducted on three-dimensional idealized spherical-void-phase geometric models to similar results obtained from a solver based on volume-averaging and local thermal non-equilibrium. The purpose of the comparison is to establish closure coefficients for the viscous and form drag terms in the volume-averaged momentum equations and the interstitial convective exchange coefficient required to couple the volume-averaged energy equations for the solid and fluid constituents. A method is also described for determining the solid-phase conduction shape factor, which is shown to be important for accurate volume-average simulation of highly conductive porous materials. The shape factor has been addressed in previous literature (using various terminology) and accounts in a bulk manner for resistance due to the elongated conduction path and for changes in the effective heat flow area along the conduction path. The conduction shape factor is a function of the geometry only and is found herein from a detailed comparison between pore-level and volume-averaged simulations of conjugate heat transfer. The conduction shape factor vastly improves volume-averaged predictions of the overall heat transfer and the temperature distributions in the porous material.

Performance analysis of incorporating phase change materials in asphalt concrete pavements

The use of phase change materials (PCMs) in asphalt pavement mixtures has the potential to be an effective method of regulating extreme temperatures. In this article, the primary objective is to understand conditions for which PCMs mitigate extreme temperatures in a flexible pavement system. The volume-averaged energy equation with phase dependent thermal properties is used to analyze the heat transfer process in the integrated PCM pavement system. Our results show that (i) a pavement system consisting of a layer of PCM that is directly below the asphalt-concrete (ac)-layer yields higher surface temperatures than a system without the PCM layer and (ii) a pavement system in which PCM is embedded in the ac-layer with varying volume fraction has lower surface temperature values than that of the pavement without PCM when the PCM volume fraction is below a critical volume fraction, 60% for the conditions used; otherwise higher surface temperature values exist. Finally, we show the effective thermal conductivity has a strong effect on the temperature distribution throughout the surface layer of PCM-embedded pavement system.

Numerical Results for the Effective Flow and Thermal Properties of Idealized Graphite Foam

To simulate the heat transfer performance of devices incorporating high-conductivity porous materials, it is necessary to determine the relevant effective properties to close the volume-averaged momentum and energy equations. In this work, we determine these effective properties by conducting direct simulations in an idealized spherical void phase geometry and use the results to establish closure relations to be employed in a volume-averaged framework. To close the volume-averaged momentum equation, we determine the permeability as defined by Darcy’s law as well as a non-Darcy term, which characterizes the departure from Darcy’s law at higher Reynolds numbers. Results indicate that the non-Darcy term is nonlinearly related to Reynolds number, not only confirming previous evidence regarding such behavior in the weak inertia flow regime, but demonstrating that this is generally true at higher Reynolds numbers as well. The volume-averaged energy equation in the fluid phase is closed by the thermal dispersion conductivity tensor, the convecting velocity, and the interfacial Nusselt number. Overall, it has been found that many existing correlations for the effective thermal properties of graphite foams are oversimplified. In particular, it has been found that the dispersion conductivity is not well characterized using the Péclet number alone, rather the Reynolds and Prandtl numbers must be considered as separate influences. Additionally, the convecting velocity modification, which is not typically considered, was found to be significant, while the interfacial Nusselt number was found to exhibit a nonzero asymptote at low Péclet numbers. Finally, simulations using the closed volume-averaged equations reveal significant differences in heat transfer when employing the present dispersion model in comparison to a simpler dispersion model commonly used for metallic foams, particularly at high Péclet numbers and for thicker foam blocks.

Multi-component gas transport in micro-porous domains: Multidimensional simulation at the macroscale

For decades, models based on the Maxwell–Stefan diffusion equations have been used to predict mass transfer of multi-component gases in porous media. However, the validity of these models is restricted to transport processes that are dominated by diffusion and wall-friction. Therefore, a new computational model is presented, which is based on a comprehensive theory of multi-component gas transport. It comprises conservation equations of mass and momentum for each species contained in the gas mixture together with an energy balance for the mixture. The model is intended to be used for investigations on the macroscopic scale. Thus, the local volume averaging technique is applied and friction coefficients, which encompass the full range of Knudsen numbers, are used to describe the drag forces exerted by the pore structure. In the volume averaged energy equation the gas and solid phases are considered a pseudo-homogeneous medium. The resulting set of highly coupled transport equations is implemented in a commercial computational fluid dynamics program. For validation purposes, the mass transfer model is first used to simulate the isothermal diffusion problem in a Loschmidt tube. Then, the simultaneous mass and heat transfer in a porous duct with cooled walls is investigated. In both cases, the numerical results are in very good agreement with experimental data.

Closure of non-equilibrium volume-averaged energy equations in high-conductivity porous media

A formulation for closure of the volume-averaged energy equations under local thermal non-equilibrium conditions has been proposed, providing a general method for obtaining the relevant effective properties for high-conductivity porous materials. The method requires numerical solution of the flow field, as well as closure problems, over a representative volume of porous media. This study considers an array of highly conductive circular cylinders for which all relevant effective properties are obtained. Comparisons are made to results in literature for low Reynolds numbers, while results at higher Reynolds numbers are presented to provide insight into the effects of inertia on the effective properties.

Second law analysis of porous thermoacoustic stack systems

Abstract The objective of this paper is to analyze the irreversibility of the porous thermoacoustic stacks in terms of entropy generation. Thermoacoustic stacks are modeled in this paper as porous channels having wall with finite thickness. The flow field in the porous channel is described by the Darcy momentum equation. The thermal field inside the porous stack is modeled by volume averaged energy equation with local thermal equilibrium assumption. While the thermal field inside the solid wall is modeled by the heat conduction equation. The entire problem is treated as a conjugate heat transfer problem. The governing momentum equation inside the porous stack and the energy equations inside both porous stack and solid wall are simplified and linearized using a first order perturbation expansion in the low Mach number limit. The linearized governing differential equations are solved using the complex exponential method. Velocity and temperature results thus obtained are applied for subsequent entropy generation analysis. Entropy generation rate is used as a measure of irreversibilities associated with viscous and heat transfer effects in the porous stack region. For the specific thermoacoustic situation considered in this paper, a time-averaged entropy generation rate followed by local and global entropy generation rates are calculated and graphically presented. The entropy generation distribution in the stack thus enables the designers to find and modify the parameters producing high energy losses characterized by large entropy production rates. A limited experimental work is performed to measure the temperature difference generated across the stack ends. A fair qualitative agreement between the experimental and analytical results is obtained.

THREE-DIMENSIONAL SIMULATION OF A POROUS THERMAL ENERGY STORAGE SYSTEM USING SOLID-LIQUID PHASE CHANGE MATERIAL

The objective of this study is to analyze the behavior of an energy storage system which is made of a horizontal channel crossed by a fluid and whose walls contain a porous medium saturated with phase change material (PCM). We developed a three-dimensional model to study the two processes: thermal energy charging (during PCM melting) and thermal energy discharging (during PCM solidification). The volume averaged energy equation with phase change in the porous medium is discretized by the control volume finite element method (CVFEM). The resulting system of algebraic equations was solved by the bi-conjugate gradient-stabilized iterative solver. A series of numerical tests are then undertaken to assess the effects of matrix porosity and PCM type on the rate of charged and released thermal energy.

NUMERICAL STUDY OF BÉNARD CONVECTION WITH TEMPERATURE-DEPENDENT VISCOSITY IN A POROUS CAVITY VIA LATTICE BOLTZMANN METHOD

In this paper, the heat transfer characteristics of a two-dimensional steady Bénard convection flow with a temperature-dependent viscosity are studied numerically by the lattice Boltzmann method (LBM). The double-distribution model for LBM is proposed, one is to simulate incompressible flow in porous media and the other is to solve the volume averaged energy equation. The method is validated by comparing the numerical results with those existing literature. The effect of viscosity dependent on temperature is investigated. The average Nusselt numbers for the cases of exponential form of viscosity-temperature and effective Rayleigh number based on average temperature (T ref = 0.5 (Th +Tc)) are compared. A new formula of reference temperature (T ref = Tc +f (b) (Th -Tc)) is proposed and the numerical results show that the average Nusselt numbers predicted by this method have higher precision than those obtained by average temperature.

A rigorous derivation of the bioheat equation for local tissue heat transfer based on a volume averaging theory

A general three-dimensional bioheat equation for local tissue heat transfer has been derived with less assumptions, exploiting a volume averaging theory commonly used in fluid-saturated porous media. The volume averaged energy equations obtained for the arterial blood, venous blood and tissue were combined together to form a single energy equation in terms of the tissue temperature alone. The resulting energy equation turns out to be remarkably simple as we define the effective thermal conductivity tensor, which accounts not only for the countercurrent heat exchange mechanism but also for the thermal dispersion mechanism. The present equation for local tissue heat transfer naturally reduces to the Weinbaum-Jiji equation for the unidirectional case.

A232 動脈と静脈の対向を考慮に入れた生体伝熱の式(OS-3:多孔質体内の伝熱(III)・一般講演)

A general bioheat equation for local tissue heat transfer has been derived exploiting a volume averaging theory. The volume averaged energy equations obtained for the arterial blood, venous blood and tissue were combined together to form a single energy equation in terms of the tissue temperature alone. The resulting energy equation turns out to be remarkably simple as we define the effective thermal conductivity tensor, which accounts for the countercurrent heat exchange mechanism. This bioheat equation can be coupled with the energy equations of large artery and vein in order to investigate the temperature field in a deep tissue layer "core tissue layer" in a human arm. Such a multiscale numerical analysis is described in detail.

Numerical Modeling of Multidirectional Flow and Heat Transfer in Graphitic Foams

To investigate the feasibility of the use of foams with an interconnected spherical pore structure in heat transfer applications, models for heat transfer and pressure drop for this type of porous materials are developed. Numerical simulations are carried out for laminar multidirectional thermofluid flow in an idealized pore geometry of foams with a wide range of geometry parameters. Semiheuristic models for pressure drop and heat transfer are developed from the results of simulations. A simplified solid-body drag equation with an extended high inertia term is used to develop the hydraulic model. A heat transfer model with a nonzero asymptotic term for very low Reynolds numbers is also developed. To provide hydraulic and heat transfer models suitable for a wide range of porosity, only a general form of the length-scale as a function of pore structure is defined a priori, where the parameters of the function were determined as part of the modeling process. The proposed ideal models are compared to the available experimental results, and the source of differences between experimental results and the ideal models is recognized and then calibrated for real graphitic foam. The thermal model is used together with volume-averaged energy equations to calculate the thermal dispersion in graphitic foam. The results of the calculations show that the linear models for thermal dispersion available in literature are oversimplified for predicting thermal dispersion in this type of porous material.

Analysis of Solid–Liquid Phase Change Under Pulsed Heating

Solid/liquid phase change occurring in a rectangular container with and without metal foams subjected to periodic pulsed heating is investigated. Natural convection in the melt is considered. Volume-averaged mass and momentum equations are employed, with the Brinkman–Forchheimer extension to Darcy’s law used to model the porous resistance. A local thermal nonequilibrium model, assuming equilibrium melting at the pore scale, is employed for energy transport through the metal foams and the interstitial phase change material (PCM). Separate volume-averaged energy equations for the foam and the PCM are written and are closed using a heat transfer coefficient. The enthalpy method is employed to account for phase change. The governing equations for the PCM without foam are derived from the porous medium equations. The governing equations are solved implicitly using a finite volume method on a fixed grid. The coupled effect of pulse width and natural convection in the melt is found to have a profound effect on the overall melting behavior. The influence of pulse width, Stefan number, and Rayleigh number on the temporal evolution of the melt front location and the melting rate for both the cases with and without metal foams is investigated.

Porous Medium Interconnector Effects on the Thermohydraulics of Near-Compact Heat Exchangers Treated as Porous Media

A novel approach of treating near-compact heat exchangers (NCHX) (surface to volume ratio, α=100-300m2∕m3 with hydraulic diameter DM∼6mm) as a “global” porous media, whose thermohydraulic performance is being influenced by the presence of “local” tube-to-tube porous medium interconnectors, connecting the in-line arrangement of tubes (D=2mm) having square pitch of XT=XL=2.25, is investigated in this study using numerical methods. The thermohydraulics of the global porous media (NCHX) are characterized by studying the effect of transverse thickness (δ) and permeability (represented by Dai) of the local metal foam type porous medium interconnectors on the global heat transfer coefficient (Nu) and nondimensional pressure drop (ξ). The fluid transport in the porous medium interconnectors is governed by the Brinkman–Darcy flow model while the volume averaged energy equation is used to model energy transport, with the tube walls kept at constant temperature and exchanging heat with the cooling fluid having Pr=0.7 under laminar flow (10<Re<100). For the chosen NCHX configuration, ξ and Nu increases for an increase in Re and also with an increase in the thickness (δ) of the interconnecting porous medium. However, as the local Darcy number (Dai) of the interconnecting porous medium increases, the ξ decreases but the Nu increases. Treating the heat exchanger as a global porous media this result translates to an increase in the ξ and Nu as the global permeability (represented by Dag) decreases, where the decrease in Dag is because of either an increase in δ or a decrease in Dai. Separate correlations predicting ξ and Nu as a function of Re and Dag (which in turn is correlated to δ and Dai) have been developed for the chosen NCHX configuration, both of which predict the numerical data with ±20% accuracy.

A Two-Temperature Model for Solid-Liquid Phase Change in Metal Foams

Transient solid-liquid phase change occurring in a phase-change material (PCM) embedded in a metal foam is investigated. Natural convection in the melt is considered. Volume-averaged mass and momentum equations are employed, with the Brinkman-Forchheimer extension to the Darcy law to model the porous resistance. Owing to the difference in the thermal diffusivities between the metal foam and the PCM, local thermal equilibrium between the two is not assured. Assuming equilibrium melting at the pore scale, separate volume-averaged energy equations are written for the solid metal foam and the PCM and are closed using an interstitial heat transfer coefficient. The enthalpy method is employed to account for phase change. The governing equations are solved implicitly using the finite volume method on a fixed grid. The influence of Rayleigh, Stefan, and interstitial Nusselt numbers on the temporal evolution of the melt front location, wall Nusselt number, temperature differentials between the solid and fluid, and the melting rate is documented and discussed. The merits of incorporating metal foam for improving the effective thermal conductivity of thermal storage systems are discussed.

A Two-Temperature Model for the Analysis of Passive Thermal Control Systems

Abstract Passive control of steady and unsteady thermal loads using effective thermal conductivity enhancers, such as metal foams, internal fins and metal filler particles, is being explored for a variety of electronics applications. The interstices are filled with air, phase change materials, or other fluids. Local thermal equilibrium between the solid filler and the matrix is not ensured in such systems since their thermal diffusivities are frequently very different. The use of a single volume-averaged energy equation for both the phases cannot be justified in such situations. A two-medium approach is used in the present work to account for the local thermal non-equilibrium. Separate energy equations are written for the solid and fluid respectively, and are closed using a steady-state interphase heat transfer coefficient between the two phases. A general momentum equation which includes the Brinkman-Forchheimer extension to Darcy flow is employed. The resulting equations are solved implicitly using a fully transient method on fixed orthogonal co-located finite volumes. Unsteady natural convection in a metal-foam filled cavity is computed. The influence of various parameters such as the ratios of solid-to-fluid thermal conductivities and heat capacities, Rayleigh number, Prandtl number and Darcy number on the thermal and flow fields is investigated. The results illustrate that local thermal equilibrium is not assured, either during the transient or at steady state for the range of parameters considered. Furthermore, even if the steady-state solid-to-fluid temperature differences are small, large temperature differences are seen during the unsteady response.

A theory to describe heat transfer during laminar incompressible flow of a fluid in periodic porous media

In this paper, a theory is formulated that describes the heat transfer during the laminar flow of an incompressible fluid within a rigid periodic porous medium. To describe the macroscopic temperature distribution in porous media, the volume average concept is employed to derive the volume averaged energy equation. Previous researchers have used either the one-equation model with the local thermal equilibrium assumption between the various phases or the complex two-equation model to deal with the average temperatures of the solid phase and the fluid phase. Both types of models are derived and applied in a stationary observation frame. In this paper, we propose a generalized one-equation model by relaxing the local thermal equilibrium assumption between the phases and invoking the frame invariant principle. The three new terms arise due to the microscopic transient, convective, and conductive effects. By taking advantage of the objectivity of the generalized model, it is shown that the microscopic temperature distribution in a periodic unit cell results in a periodic heat flux boundary condition. The direct temperature solution is obtained in the periodic unit cell by solving the generalized volume averaged energy equation. A modified effective thermal conductivity tensor is calculated which encompasses the micro-convective effects encountered in porous media. An analytic study of a non-isothermal flow through two parallel thick plates is performed to explore the role of each new term in the generalized model.

Non-isothermal liquid flow and heat transfer in sintered metallic porous media

Abstract This paper presents an experimental and analytical study of non-isothermal liquid flow in sintered fiber metallic porous media. The purpose of the paper is to present a self-consistent procedure for determining heat-transfer characteristics of porous metals. The analysis uses the volume averaged conservation equations. Darcy's law is assumed to hold and local thermal equilibrium is assumed to exist between the solid and liquid phases. The coupled conservation equations are solved for one and two dimensional temperature distributions using finite difference schemes. Different flow and thermal boundary conditions are used. The experiments involve the measurement of temperature distributions and effective thermal conductivities of water saturated copper and nickel sintered fiber metal wicks. The effective thermal conductivities are measured using the steady-state method of comparison and the resulting values are used in the volume averaged energy equation. The measured temperature distributions and those predicted by the analysis show good agreement. Thus the procedure presented in this paper for predicting heat transfer seems to be valid.