Thermal Dispersion in High-Conductivity Porous Media
01 Jan 2012-pp 153-180
TL;DR: In this paper, the authors focus on methods for determining the thermal dispersion conductivity, as well as other effective properties, in high-conductivity porous materials, such as metal foams.
Abstract: Transport in high-conductivity porous media, such as metal foams, has many practical applications in the field of heat transfer. In order to numerically simulate the performance of devices incorporating such materials in a volume-averaged framework, it is necessary to have accurate estimates of all relevant effective properties, including the thermal dispersion conductivity. This chapter focuses on methods for determining the thermal dispersion conductivity, as well as other effective properties, in high-conductivity porous materials. Results are first presented for cylinder arrays with different particle shapes and arrangements. Following this, results for thermal dispersion are presented for an idealized graphite foam pore geometry and are used in volume-averaged simulations to evaluate the impact of the dispersion model on the overall heat transfer predictions. The overall finding of this review is that dispersion behaviour is complicated for all but the simplest pore geometries. Thus, any modelling efforts should consider the Reynolds and Prandtl numbers as separate influences, rather than lumping their effects into a relation based simply on the Peclet number.
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TL;DR: In this paper, metal foams are used within the structure of PEMFCs as an option that can potentially address cooling and flow distribution challenges associated with using conventional flow channels.
84 citations
Cites background from "Thermal Dispersion in High-Conducti..."
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TL;DR: In this paper, heat transfer in a random packed bed of monosized iron ore pellets is modelled with both a discrete three-dimensional system of spheres and a continuous Computational Fluid Dynamics (CFD) model.
Abstract: Heat transfer in a random packed bed of monosized iron ore pellets is modelled with both a discrete three-dimensional system of spheres and a continuous Computational Fluid Dynamics (CFD) model. Results show a good agreement between the two models for average values over a cross section of the bed for an even temperature profiles at the inlet. The advantage with the discrete model is that it captures local effects such as decreased heat transfer in sections with low speed. The disadvantage is that it is computationally heavy for larger systems of pellets. If averaged values are sufficient, the CFD model is an attractive alternative that is easy to couple to the physics up- and downstream the packed bed. The good agreement between the discrete and continuous model furthermore indicates that the discrete model may be used also on non-Stokian flow in the transitional region between laminar and turbulent flow, as turbulent effects show little influence of the overall heat transfer rates in the continuous model.
12 citations
TL;DR: In this article, the Fourier-Galerkin (FG) spectral element method is adapted to solve the coupled equations of Darcy's flow and heat transfer with a full velocity-dependent dispersion tensor, employing the stream function formulation.
Abstract: Natural convection in a porous enclosure in the presence of thermal dispersion is investigated. The Fourier–Galerkin (FG) spectral element method is adapted to solve the coupled equations of Darcy’s flow and heat transfer with a full velocity-dependent dispersion tensor, employing the stream function formulation. A sound implementation of the FG method is developed to obtain accurate solutions within affordable computational costs. In the spectral space, the stream function is expressed analytically in terms of temperature, and the spectral system is solved using temperature as the primary unknown. The FG method is compared to finite element solutions obtained using an in-house code (TRACES), OpenGeoSys and COMSOL Multiphysics®. These comparisons show the high accuracy of the FG solution which avoids numerical artifacts related to time and spatial discretization. Several examples having different dispersion coefficients and Rayleigh numbers are tested to analyze the solution behavior and to gain physical insight into the thermal dispersion processes. The effect of thermal dispersion coefficients on heat transfer and convective flow in a porous square cavity has not been investigated previously. Here, taking advantage of the developed FG solution, a detailed parameter sensitivity analysis is carried out to address this gap. In the presence of thermal dispersion, the Rayleigh number mainly affects the convective velocity and the heat flux to the domain. At high Rayleigh numbers, the temperature distribution is mainly controlled by the longitudinal dispersion coefficient. Longitudinal dispersion flux is important along the adiabatic walls while transverse dispersion dominates the heat flux toward the isothermal walls. Correlations between the average Nusselt number and dispersion coefficients are derived for three Rayleigh number regimes.
9 citations
TL;DR: In this article, the authors considered convective heat transport in a relatively thin porous layer of monosized particles and showed that a special correlation must be used when using a continuous model for flow perpendicular to a thin porous media to predict the dispersion in proper manner, especially in combination with higher velocities.
Abstract: Convective heat transport in a relatively thin porous layer of monosized particles is here modeled. The size of the particles is only one order of magnitude smaller than the thickness of the layer. Both a discrete three-dimensional system of particles and a continuous one-dimensional model are considered. The methodology applied for the discrete system is Voronoi discretization with minimization of dissipation rate of energy. The discrete and continuous model compares well for low velocities for the studied uniform inlet boundary conditions. When increasing the speed or for a thin porous layer however, the continuous model diverge from the discrete approach if a constant dispersion is used in the continuous approach. The new result is thus that a special correlation must be used when using a continuous model for flow perpendicular to a thin porous media in order to predict the dispersion in proper manner, especially in combination with higher velocities.
2 citations
TL;DR: In this paper , the authors use the volume averaging technique as a mathematical framework to convert the pointwise governing equations of a catalytic monolith into averaged equations for a porous domain.
Abstract: Catalytic monoliths are being explored in conventional catalytic processes for their ability to achieve process intensification. Scientific computing can play an essential role in this exploration. The high computational cost of the first-principles models has led to modeling these reactors as a porous medium. However, this modeling strategy is not performed in a mathematically rigorous manner. We use the volume averaging technique as a mathematical framework to convert the pointwise governing equations for a monolith into averaged equations for a porous domain. These averaged equations require the closure of several unclosed terms, which are neglected in the classical porous medium (CPM) assumption. We discuss these unclosed terms and their impact on model predictions. We show that except in the limit of negligible Damköhler number the treatment of the catalytic reactions in CPM leads to significant errors. We propose a technique to accurately calculate the catalytic reaction rates in the volume-averaging-based porous medium (VAPM) model developed here. This technique is valid for a wide range of Damköhler numbers for both linear and nonlinear kinetics. Moreover, to calculate the effective properties of the porous medium, such as thermal conductivity, we employ asymptotic averaging (numerical homogenization) that can be used for any arbitrary channel shape and size. Predictions of the proposed VAPM model are assessed against three-dimensional multichannel monolith simulations, resolving the solid and fluid phases, for elementary and complex kinetics. In addition, VAPM is validated against the experiments of steam methane reforming in a catalytic monolith. The developed methodology reduces the computational cost by 3 orders of magnitude while maintaining the accuracy of the detailed multichannel simulations.
References
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TL;DR: In this paper, it was shown analytically that the distribution of concentration produced in this way is centred on a point which moves with the mean speed of flow and is symmetrical about it in spite of the asymmetry of the flow.
Abstract: When a soluble substance is introduced into a fluid flowing slowly through a small-bore tube it spreads out under the combined action of molecular diffusion and the variation of velocity over the cross-section. It is shown analytically that the distribution of concentration produced in this way is centred on a point which moves with the mean speed of flow and is symmetrical about it in spite of the asymmetry of the flow. The dispersion along the tube is governed by a virtual coefficient of diffusivity which can be calculated from observed distributions of concentration. Since the analysis relates the longitudinal diffusivity to the coefficient of molecular diffusion, observations of concentration along a tube provide a new method for measuring diffusion coefficients. The coefficient so obtained was found, with potassium permanganate, to agree with that measured in other ways. The results may be useful to physiologists who may wish to know how a soluble salt is dispersed in blood streams.
4,530 citations
Book•
01 Oct 1991
TL;DR: In this article, the authors identify the principles of transport in porous media and compare the available predicted results, based on theoretical treatments of various transport mechanisms, with the existing experimental results, and the theoretical treatment is based on the volume-averaging of the momentum and energy equations with the closure conditions necessary for obtaining solutions.
Abstract: Although the empirical treatment of fluid flow and heat transfer in porous media is over a century old, only in the last three decades has the transport in these heterogeneous systems been addressed in detail. So far, single-phase flows in porous media have been treated or at least formulated satisfactorily, while the subject of two-phase flow and the related heat-transfer in porous media is still in its infancy. This book identifies the principles of transport in porous media and compares the available predicted results, based on theoretical treatments of various transport mechanisms, with the existing experimental results. The theoretical treatment is based on the volume-averaging of the momentum and energy equations with the closure conditions necessary for obtaining solutions. While emphasizing a basic understanding of heat transfer in porous media, this book does not ignore the need for predictive tools; whenever a rigorous theoretical treatment of a phenomena is not available, semi-empirical and empirical treatments are given.
2,551 citations
TL;DR: In this paper, an experimental and numerical study of forced convection in high porosity (e∼0.89-0.97) metal foams was conducted using air as the fluid medium.
Abstract: We report an experimental and numerical study of forced convection in high porosity (e∼0.89-0.97) metal foams. Experiments have been conducted with aluminum metal foams in a variety of porosities and pore densities using air as the fluid medium. Nusselt number data has been obtained as a function of the pore Reynolds number. In the numerical study, a semi-empirical volume-averaged form of the governing equations is used. The velocity profile is obtained by adapting an exact solution to the momentum equation. The energy transport is modeled without invoking the assumption of local thermal equilibrium. Models for the thermal dispersion conductivity, k d , and the interstitial heat transfer coefficient, h sf , are postulated based on physical arguments. The empirical constants in these models are determined by matching the numerical results with the experimental data obtained in this study as well as those in the open literature. Excellent agreement is achieved in the entire range of the parameters studied, indicating that the proposed treatment is sufficient to model forced convection in metal foams for most practical applications
911 citations
TL;DR: In this article, the dispersion equation for a single, nonreacting, nonadsorbing species is derived for incompressible, laminar flow in anisotropic porous media.
Abstract: The dispersion equation for a single, nonreacting, nonadsorbing species is derived for incompressible, laminar flow in anisotropic porous media. Direct integration of the appropriate differential equations gives rise to a dispersion vector ψi and a tortuosity vector τi, both of which must be evaluated experimentally. For the dispersion vector, this is conveniently done by representing ψi in terms of the velocity and gradients of the velocity and concentration. The experimental determination of τi is not straightforward except for the case of pure diffusion. The analysis yields a result which contains all the features of previously presented dispersion equations without making any assumptions as to the nature of the flow, that is, bypassing, cell mixing, etc., except that it be laminar. Attacking the dispersion problem in terms of the differential diffusion equation provides a rational basis for the correlation of experimental data and illustrates the connection between the microscopic and macroscopic equations.
716 citations
TL;DR: In this paper, open-cell metal foams with an average cell diameter of 2.3 mm were manufactured from 6101-T6 aluminum alloy and were compressed and fashioned into compact heat exchangers.
681 citations