Two-energy Equation Model Research Articles

Electromagnetic field effects on transport through porous media

The effect of an imposed electromagnetic field on forced convection in porous media is analyzed in this work. The transient Maxwell’s equations are solved to simulate the electromagnetic field inside the waveguide and within a porous medium. The Brinkman–Forchheimer extended Darcy (generalized model) equations are used to represent the flow fluid inside a porous medium. The local thermal non-equilibrium (LTNE) is taken into account by solving the two-energy equation model for fluid and solid phases. Computational domain is represented for a range of Darcy number from 10 −5 to 10 −7 and dimensionless electromagnetic wave power P ∗ from 0 to 1600, and dimensionless electromagnetic wave frequency f ∗ from 0 to 8. The effect of variations of the pertinent electromagnetic field parameters in affecting the flow and thermal fields and the Nusselt number are analyzed. This investigation provides the essential aspects for a fundamental understanding of forced convection in porous media while experiencing an applied electromagnetic field such as applications in the material-processing field.

Heat generation effects in natural convection inside a porous annulus

The interplay between internal heat generation and externally driven natural convection inside a porous medium annulus is studied in detail using numerical methods. The axisymmetric domain is bounded with adiabatic top and bottom walls and differentially heated side walls sustaining steady natural convection of a fluid with Prandtl number, Pr = 5, through a porous matrix of volumetric porosity, ϕ = 0.4. The generalized momentum equation with Brinkman–Darcy–Forchheimer terms and the local thermal non-equilibrium based two-energy equation model are solved to determine the flow and the temperature distribution. Beyond a critical heat generation value defined using an internal Rayleigh number, Ra I,cr ⁎, the convection transits from unicellular to bicellular mode, as the annulus T max becomes higher than the fixed hot-wall temperature. The Ra I,cr ⁎ increases proportionately when the permeability based external Rayleigh number Ra E ⁎ and the solid–fluid thermal conductivity ratio γ are independently increased. A correlation is proposed to predict the overall annulus Nu as a function of Ra E ⁎, Ra I ⁎, Da and γ. It predicts the results within ± 20% accuracy.

A macroscopic two-energy equation model for turbulent flow and heat transfer in highly porous media

In this paper, a model for turbulent flow and heat transfer in a highly porous medium is proposed and applied to a porous channel bounded by parallel plates. Macroscopic continuity, momentum and energy equations are presented. Local non-thermal equilibrium is considered by means of independent equations for the solid matrix and the working fluid. The numerical methodology used is based on the control-volume approach. The effects of thermal dispersion, Reynolds number, dimensionless particle diameter, thermal conductivity ratio and Darcy number, on the Nusselt number, are presented. For laminar and turbulent flows the thermal dispersion mechanism leads to larger local temperature differences. Increase in Re number causes values for Nu, of both phases, to increase. Porosity increase causes the solid phase Nusselt number to decrease whereas the fluid Nusselt number in augmented. In general, an increase in the particle diameter increases Nusselt number. Also, the thermal conductivity ratio causes the most pronounced effect on Nusselt numbers.

NUMERICAL MODELING OF EVAPORATION ENHANCEMENT OF AVIATION-GRADE KEROSENE SPRAY IN POROUS MEDIA COMBUSTORS

This article presents a numerical study of evaporation enhancement of aviation-grade kerosene spray in porous media combustors. A two-energy equation model was employed to predict porous medium and gas phase temperatures. Governing equations were solved on a two-dimensional axisymmetric computational domain using FLUENT. An airblast atomizer model was used to simulate injection of kerosene spray on the porous medium. Heat feedback rate to the porous medium from the combustion zone was simulated by defining a uniform volumetric heat source in the porous medium. A range of heat feedback rates varying from 0.8% to 1.1% of average heat input to the combustor was specified. At 1% heat feedback rate, the porous medium attained a peak temperature of 465 K. Surface temperature measurements indicate that the transverse distribution was uniform within ±1.5% of the mean value. Transverse vapor concentration results downstream of the porous medium showed that the distribution in the transverse direction was uniform within §5% of the mean value. This demonstrates that porous medium uniformly distributes the fuel vapor-air mixture, a desirable feature in combustors. Effects of heat feedback rate, porosity of the medium, fuel-air equivalence ratio, and flame temperature on transverse vapor concentration profiles were also numerically studied.

Laminar heat transfer in a porous channel simulated with a two-energy equation model

Laminar heat transfer in a porous channel is numerically simulated with a two-energy equation model for conduction and convection. Macroscopic equations for continuity, momentum and energy transport for the fluid and solid phases are presented. The numerical methodology employed is based on the control volume approach with a boundary-fitted non-orthogonal coordinate system. Fully developed forced convection in a porous channel bounded by parallel plates is considered. Solutions for Nusselt numbers along the channel are presented for laminar flows. Results simulate the effects Reynolds number Re, porosity, particle size and solid-to-fluid thermal conductivity ratio on Nusselt sumber, Nu, which is defined for both the solid and fluid phases. High Re, low porosities, low particle diameters and low thermal conductivity ratios promote thermal equilibrium between phases leading to higher values of Nu.

Computation of turbulent heat transfer in a moving porous bed using a macroscopic two-energy equation model

Heat transfer between phases in a moving porous bed is analyzed. This work proposes a set of transport equations for solving problems involving turbulent flow and heat transfer in a moving bed equipment. The device is modeled as a saturated porous matrix in which the solid phase moves with a steady imposed velocity. Additional drag terms appearing the momentum equation, as well as interfacial heat transfer between phases, are assumed to be a function of the relative velocity between the fluid and solid phases. Turbulence transport equations are here also dependent on the speed of the solid material. Results indicate that, as the phases attain velocities of equal order, turbulence in damped and heat transfer between solid and fluid occurs mainly by conduction mechanism.

A general bioheat transfer model based on the theory of porous media

A volume averaging theory (VAT) established in the field of fluid-saturated porous media has been successfully exploited to derive a general set of bioheat transfer equations for blood flows and its surrounding biological tissue. A closed set of macroscopic governing equations for both velocity and temperature fields in intra- and extravascular phases has been established, for the first time, using the theory of anisotropic porous media. Firstly, two individual macroscopic energy equations are derived for the blood flow and its surrounding tissue under the thermal non-equilibrium condition. The blood perfusion term is identified and modeled in consideration of the transvascular flow in the extravascular region, while the dispersion and interfacial heat transfer terms are modeled according to conventional porous media treatments. It is shown that the resulting two-energy equation model reduces to Pennes model, Wulff model and their modifications, under appropriate conditions. Subsequently, the two-energy equation model has been extended to the three-energy equation version, in order to account for the countercurrent heat transfer between closely spaced arteries and veins in the circulatory system and its effect on the peripheral heat transfer. This general form of three-energy equation model naturally reduces to the energy equations for the tissue, proposed by Chato, Keller and Seiler. Controversial issues on blood perfusion, dispersion and interfacial heat transfer coefficient are discussed in a rigorous mathematical manner.

Interfacial and capillary pressure effects on the thermal performance of wax/foam composites

A numerical investigation study was performed to study the phase change behavior of wax/foam composite encapsulated in an aluminum casing. Two types of foam materials, namely, aluminum and carbon, were infiltrated with paraffin wax. The progress of melt interface and temperature distribution within the encapsulated composite was analyzed using computational fluid dynamics software (CFD). A two-energy equation model was implemented in the CFD software through the use of user defined function (UDF). Interfacial effects influencing the heat transfer process at the casing-composite junction and between the wax-foam surfaces within the composite were addressed through the use of separate UDF. In addition, the effect of capillary pressure developed within the foam matrix was incorporated using an area ratio parameter. The contact resistance at the foam-casing interface and the capillary pressure had a major influence on the thermal behavior of the system. These two factors lowered down the heat transfer rate considerably and the melting area was reduced by more than 30%. The temperature profiles for the foam material showed a different pattern as compared to the temperature within the wax, which was due to the effect of thermal nonequilibrium.

Temperature profiles of local thermal nonequilibrium for thermal developing forced convection in porous medium parallel plate channel

Based on the two-energy equation model, taking into account viscous dissipation due to the interaction between solid skeleton and pore fluid flow, temperature expressions of the solid skeleton and pore fluid flow are obtained analytically for the thermally developing forced convection in a saturated porous medium parallel plate channel, with walls being at constant temperature. It is proved that the temperatures of the two phases for the local thermal nonequilibrium approach to the temperature derived from the one-energy equation model for the local thermal equilibrium when the heat exchange coefficient goes to infinite. The temperature profiles are shown in figures for different dimensionless parameters and the effects of the parameters on the local thermal nonequilibrium are revealed by parameter study.

A Correlation for Interfacial Heat Transfer Coefficient for Turbulent Flow Over an Array of Square Rods

Interfacial heat transfer coefficients in a porous medium modeled as a staggered array of square rods are numerically determined. High and low Reynolds k-ϵ turbulence models are used in conjunction of a two-energy equation model, which includes distinct transport equations for the fluid and the solid phases. The literature has documented proposals for macroscopic energy equation modeling for porous media considering the local thermal equilibrium hypothesis and laminar flow. In addition, two-energy equation models have been proposed for conduction and laminar convection in packed beds. With the aim of contributing to new developments, this work treats turbulent heat transport modeling in porous media under the local thermal nonequilibrium assumption. Macroscopic time-average equations for continuity, momentum, and energy are presented based on the recently established double decomposition concept (spatial deviations and temporal fluctuations of flow properties). The numerical technique employed for discretizing the governing equations is the control volume method. Turbulent flow results for the macroscopic heat transfer coefficient, between the fluid and solid phase in a periodic cell, are presented.

Numerical Study of Conjugate Heat Transfer in a System Consisting of Porous Media and Bounding Walls.

A numerical scheme based on the control volume method to solve conjugate heat transfer problems in a system consisting of porous media and bounding walls has been developed using a two-energy equation model for porous insert. Appropriate wall heat flux expressions for the fluid phase and solid phase were obtained from the proposed heat transfer model at the interface between porous media and solid wall. The heat transfer model clarifies physics of the known thermal boundary condition at the interface. Predicted interface temperature distribution and Nusselt number show very good agreement with the available experimental data for the case of using metal foams as the porous insert. Effect of wall thickness on heat transfer to the porous media has been investigated and the importance of considering heat conduction in the wall as well as heat transfer in porous media was demonstrated.

NATURAL CONVECTION IN POROUS ENCLOSURES: THE APPLICATION OF THE TWO-ENERGY EQUATION MODEL

The present investigation is concerned with the numerical simulation of a fluid-saturated porous medium emplaced inside a square enclosure. The flow field is modeled using the generalized form of the momentum equation by accounting for the viscous effects and the inertial effects as well. In addition, the local fluid and solid temperatures were both accounted for separately by employing the two-energy equation model. The sensitivity of the Darcy number and Grashof number values on the momentum and energy transport processes are highlighted in the range of 10 -5 h Da h 10 -1 and 10 4 h Gr h 10 6 , respectively. Furthermore, Nusselt number formulation, based on the overall energy balance, was developed for the two-energy equation model. Finally, an assessment of the local thermal equilibrium condition was carried out to weigh its significance over the range of flow conditions and porous configurations considered in the analysis.

A two-energy equation model for conduction and convection in porous media

A two-energy equation model is proposed for analyzing conduction and convection within porous media in local thermal non-equilibrium. Hsu's closure model for the tortuosity effect has been extended to treat not only conduction but also convection in porous media. The two energy equations for the individual phases are combined into a fourth-order ordinary differential equation, so as to treat one-dimensional steady-state problems. Then, the exact solutions are obtained for two fundamental cases, namely, one-dimensional steady conduction in a porous slab with internal heat generation within a solid, and also thermally developing unidirectional flow through a semi-infinite medium.

A numerical study of interfacial convective heat transfer coefficient in two-energy equation model for convection in porous media

A numerical experiment has been conducted to determine the interfacial convective heat transfer coefficient in the two-energy equation model for convection in porous media, which is needed when the local thermal equilibrium between the fluid and solid phases breaks down. The similarity of periodically fully developed temperature profiles allows one to perform a numerical experiment using only a single structural unit for determining the fully developed heat transfer coefficient without any empiricism. A universal correlation for the Nusselt number, which agrees well with available experimental data, has been established using the results obtained for a wide range of porosity, Prandtl and Reynolds numbers.

221 多孔質体内熱流動の界面熱伝達率に関する数値実験

A numerical experiment has been conducted to determine the interfacial convective heat transfer coefficient in the two-energy equation model of porous media, which is needed when the assumption of the local thermal equilibrium between the fluid and solid phases is not valid. Three fundamental case, in which the assumption of local equilibrium fails, were treated using structural models of porous media, to determine the interfacial heat transfer coefficient without any empiricism. A universal correlation for the Nusselt number, which agrees very well with the experimental data, has been established using the results obtained for a wide range of porosity, Prandtl and Reynolds numbers.

A Numerical Study of Interfacial Convective Heat Transfer Coefficient in Two-Energy Equation Model of Porous Media.

A numerical experiment has been conducted to determine the interfacial convective heat transfer coefficient in the two-energy equation model of porous media, which is needed when the assumption of the local thermal equilibrium between the fluid and solid phases is not valid. The similarity of the fully developed temperature profile allows one to perform a numerical experiment using only a single structural unit, for determining the fully-developed heat transfer coefficient without any empiricism. A universal correlation for the Nusselt number, which agrees very well with the experimental data, has been established using the results obtained for a wide range of porosity, Prandtl and Reynolds numbers.

Transient response of non-thermal equilibrium packed beds

This paper reports the results concerning the dynamic response of the temperature distribution in a semi-infinite porous medium, with a time-varying fluid inlet temperature and arbitrary initial condition for the solid temperature distribution. A two-energy equation model is employed, the solution gives the fluid and solid temperatures in the porous medium as a function of time and axial distance. The results are obtained by a direct mathematical attack on the governing differential equations derived from the appropriate physical laws. The general solutions, presented as converging series of integrals of modified Bessel functions on space and time, are accurately computed on a common personal computer with very low computational time.

Transient analysis of incompressible flow through a packed bed

The present investigation aims at numerically simulating the temporal energy transport for incompressible flows through a packed bed. The governing equations are formulated according to the volume averaging method. The generalized momentum equation which accounts for the inertial and viscous forces was used to carry out this research, whereas the two-energy equation model was used for simulating the energy transport. The implications of the non-Darcian terms and the thermal dispersion effects on the transient energy transport were explored. Error maps were introduced for a wide range of flow conditions and bed configurations. In addition, the investigation also tackles the applicability of the local thermal equilibrium condition and the one-dimensional approach for studying the thermal response in packed bed.

MODEL FOR HEAT AND MOISTURE TRANSFER IN ARBITRARILY SHAPED TWO-DIMENSIONAL POROUS MEDIA

A model was developed to predict heat and moisture transfer due to natural convection and diffusion in arbitrarily shaped two-dimensional porous media. Boundary conditions were diurnally varying ambient temperature on the outside of walls with moderate Biot number. Other important boundary conditions were developed for typical storage and transportation situations, A two-energy equation model was used to allow for the difference between the fluid and solid temperatures and its effect on mass transfer in the porous medium. The governing equations were solved with a finite-difference method in a generalized coordinate system using a stream function formulation. It was found that the energy and moisture transport equations were best solved using a modified Crank-Nicolson method that was developed to control the tendency for instability caused by the source terms in these equations. All of the boundary conditions that were developed worked satisfactorily The two-energy equation model predicted small differences between the fiuid and solid particle temperatures and natural convection only impacted the temperature solution significantly in the upper comers of the porous media.