Continuous time random walks and heat transfer in porous media

An approach to describe heat transfer in porous media is presented on the basis of the continuous time random walk (CTRW) framework. CTRW is capable of quantifying both local equilibrium and non-equilibrium heat transfer in heterogeneous domains, and is shown here to match published experimental data of non-equilibrium thermal breakthrough. It is argued that CTRW will be particularly applicable to the quantification of heat transfer in naturally heterogeneous geological systems, such as soils and geothermal reservoirs.     

Convective heat transfer in a parallel plate channel with porous lining

The paper proposes a theoretical model for the study of flow and heat transfer in a parallel plate channel, one of whose walls is lined with non-erodible porous material, both the walls being kept at constant temperatures. The analysis uses Brinkman model in the porous medium and employs the velocity and temperature slips at the interface (the so called nominal surface). The influence of the thickness as well as the permeability of the porous medium on the flow field and Nusselt numbers at the walls is investigated.

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Konvektive Wärmeübertragung in einem Parallelplattenkanal mit porösem Überzug

Zusammenfassung Die vorliegende Arbeit befaßt sich mit dem Vorschlag eines theoretischen Modells, um die Wärmeübertragung in einem Parallelplattenkanal mit unauswaschbarem porösem Überzug zu studieren. Die Strömung innerhalb des porösen Überzugs ist mit Hilfe der Brinkmannschen Gleichung analysiert. An der Grenze (der sogenannten Nominalfläche) zwischen dem Überzug und der freien Strömung sind die Geschwindigkeitsgleitung und die Temperaturgleitung benutzt. Die Beeinflussung des Geschwindigkeitsfelds und die Nusseltschen Zahlen an den Wänden in Abhängigkeit von der Dicke und der Durchlässigkeit des porösen Überzugs ist untersucht.

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Nomenclature u streamwise velocity in Zone 1 (Fig. 1) - û streamwise velocity in Zone 2 (Fig. 1) - p pressure - coefficient of viscosity of the fluid - k absolute permeability of the material used for lining - density of the fluid - R Reynolds number - the average velocity in Zone 1 (Fig. 1) - T temperature in Zone 1 (Fig. 1) - T temperature in Zone 2 (Fig. 1) - K thermal conductivity in Zones 1 and 2 (Fig. 1) - M ₁ non-dimensional mass flow rate in Zone 1 (Fig. 1) - M ₂ non-dimensional mass flow rate in Zone 2 (Fig. 1) - (Nu)_U Nusselt number at the upper plate (Fig. 1) - (Nu) _L Nusselt number at the lower plate (Fig. 1) - _E experimental value of the temperature in the channel (with porous lining) at a specified point - _E/* experimental value of the temperature in the channel (without porous lining) at a specified point     

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.     

Heat transfer in an unevenly heated porous layer

For a uniform saturated porous layer heated from below, the dependence of the quantity of heat transferred on the distribution of the heat source is investigated. It is found, using perturbation methods and numerical techniques, that very small nonuniformities in the heat source having the same wavelength as the preferred convection mode significantly reinforce natural convection.     

Finite element analysis of convective heat transfer in porous media

The finite element method is used to analyse convective heat transfer in a porous medium. Convection past a vertical surface embedded in the medium and convection in a confined porous medium enclosure are analysed using the above method. The results are compared with those available in the literature and the agreement is found to be good. The method is applicable for two-dimensional analysis in a porous body of any arbitrary shape. The restriction of the boundary layer assumption is relaxed.     

Convective heat transfer of nanofluids with correlations

To investigate the convective heat transfer of nanofluids, experiments were performed using silver–water nanofluids under laminar, transition and turbulent flow regimes in a horizontal 4.3 mm inner-diameter tube-in-tube counter-current heat transfer test section. The volume concentration of the nanoparticles varied from 0.3% to 0.9% in steps of 0.3%, and the effects of thermo-physical properties, inlet temperature, volume concentration, and mass flow rate on heat transfer coefficient were investigated. Experiments showed that the suspended nanoparticles remarkably increased the convective heat transfer coefficient, by as much as 28.7% and 69.3% for 0.3% and 0.9% of silver content, respectively. Based on the experimental results a correlation was developed to predict the Nusselt number of the silver–water nanofluid, with ±10% agreement between experiments and prediction.     

多孔介质中二维对流传热的分叉

本文利用分叉理论研究了流体饱和的二维多孔介质从底部加热所引起的自然对流,用有限差分方法确定对流的分叉进程;揭示其模式转换机理及分叉对非正常流动图象形成的影响;同时确定了矩形截面宽高比与临界端利数的关系。还提出了一个判别分支稳定笥的简明方法。     

Convective heat transfer in laser-irradiated graphite

The gas dynamic and thermal processes developing near the surface of graphite after exposure to a 20-nsec laser pulse with an energy E- 0.1–1 J and a wavelength of 0.6943 m are investigated experimentally and by mathematical modeling. The times required for the shock wave to degenerate into an acoustic wave are also considered. Typical density profiles over the axial section of the inhomogeneity are presented for various moments of time. It is noted that the rate of ascent of the thermal inhomogeneity is much higher than the free convection velocity. The convective heat-transfer processes are studied in detail through numerical solution of the system of two-dimensional Navier-Stokes equations. The results of the calculations are in satisfactory agreement with the experimental data.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 180–182, May–June, 1989.     

Freezing heat transfer in water-saturated porous media in a vertical rectangular vessel

Numerical and experimental study is performed for the freezing of water-saturated porous media in a vertical rectangular vessel. The governing equations are solved by using a variable transformation and employing a finite difference scheme. The SOR method is utilized to solve numerically the equations. Different size and types of spherical beads are used as the porous media. The temperature of cold wall is kept at –10°C, while that of the hot wall is varied from 2°C to 22°C. Comparisons of the analytical results with the experimental ones are made. The effects of the Stefan number, the Darcy number, the modified Prandtl number and the ratio of the temperature of cold wall to the temperature of hot wall are discussed for freezing of the water-saturated porous media.Diese numerische und experimentelle Untersuchung das Gefrieren in mit Wasser gesättigten porösen Medien in einem vertikalen rechteckigen Behälter. Die Erhaltungsgleichungen werden mit einem Variablen-Transformations-Verfahren und einem Differenzen-Verfahren gelöst. Zur numerischen Lösung der Differentialgleichung wird die SOR Methode verwendet. Als poröses Medium wurden verschiedene Typen und Größen von kugelförmigen Perlen benutzt. Die kalte Wand wurde auf –10°C gehalten, die Temperatur der warmen Wand lag zwischen 2 und 22°C. Die errechneten Ergebnisse wurden mit den experimentell Ermittelten verglichen. Die Einflüsse der Stefan-, Darcy- und Prandtl-Zahlen sowie des Verhältnisses der Temperatur der warmen zur kalten Wand auf das Gefrieren in gesättigten porösen Medien wurden diskutiert.     

Convective mass transfer across fluid interfaces in straight angular pores

Steady convective mass transfer to or from fluid interfaces in pores of angular cross-section is theoretically investigated. This situation is relevant to a variety of mass transport process in porous media, including the fate of residual non-aqueous phase liquid ganglia and gas bubbles. The model incorporates the essential physics of capillarity and solute mass transfer by convection and diffusion in corner fluid filaments. The geometry of the corner filaments, characterized by the fluid–fluid contact angle, the corner half-angle and the interface meniscus curvature, is accounted for. Boundary conditions of zero surface shear (‘perfect-slip’) and infinite surface shear (‘no-slip’) at the fluid–fluid interface are considered. The governing equations for laminar flow within the corner filament and convective diffusion to or from the fluid–fluid interface are solved using finite-element methods. Flow computations are verified by comparing the dimensionless resistance factor and hydraulic conductance of corner filaments against recent numerical solutions by Patzek and Kristensen (J. Colloid Interface Sci 236, 305–317 2001). Novel results are obtained for the average effluent concentration as a function of flow geometry and pore-scale Peclet number. These results are correlated to a characteristic corner length and local pore-scale Peclet number using empirical equations appropriate for implementation in pore network models. Finally, a previously published “2D-slit” approximation to the problem at hand is checked and found to be in considerable error.     

A study on interstitial heat transfer in consolidated and unconsolidated porous media

The Nusselt number expressions are presented for the interstitial heat transfer coefficients for both consolidated and unconsolidated porous media. A boundary layer consideration is made for unconsolidated porous media to derive a general Nusselt number correlation, which shows its square root dependence of the Reynolds number, and matches fairly well with existing experimental data and correlations. As for low density consolidated porous media, rigorous mathematical arguments based on the volume averaging theory are provided, so as to explain the reason why the Reynolds number exponent of the Nusselt number expression for the case of low density consolidated porous media is much greater than that of unconsolidated porous media. The resulting expressions are compared against available experimental data and empirical correlations, and found to be in good accord with them.     

A generalized variational formulation for convective heat transfer

The Scope of this paper is to develop the basic equations for a variational formulation which can be used to solve problems related to convection and/or diffusion dominated flows. The formulation is based on the introduction of a generalized quantity defined as the hear displacement. The governing equation is expressed in terms of this quantity and a variational formulation is developed which leads to a system of equations similar in form to Lagrange's equations of mechanics. These equations can be used for obtaining approximate solutions, though they are of particular interest for application of the finite element method. As an example of the formulation two finite element models are derived for solving convectiondiffusion boundary value problems. The performance of the two models is investigated and numerical results are given for different cases of convection and diffusion with two types of boundary conditions. The applications of the developed formulations are not limited to convection-diffusion problems but can also be applied to other types of problems such as mass transfer, hydrodynamics and wave propagation.     

Numerical experiments on convective heat transfer in water-saturated porous media at near-critical conditions

Fluid and heat flow at temperatures approaching or exceeding that at the critical point (374 °C for pure water, higher for saline fluids) may be encountered in deep zones of geothermal systems and above cooling intrusives. In the vicinity of the critical point the density and internal energy of fluids show very strong variations for small temperature and pressure changes. This suggests that convective heat transfer from thermal buoyancy flow would be strongly enhanced at near-critical conditions. This has been confirmed in laboratory experiments. We have developed special numerical techniques for modeling porous flow at near-critical conditions, which can handle the extreme nonlinearities in water properties near the critical point. Our numerical simulations show strong enhancements of convective heat transfer at near-critical conditions; however, the heat transfer rates obtained in the simulations are considerably smaller than data reported from laboratory experiments by Dunn and Hardee. We discuss possible reasons for this discrepancy and develop suggestions for additional laboratory experiments.     

Unsteady flow and heat transfer of porous media sandwiched between viscous fluids

The problem of unsteady oscillatory flow and heat transfer of porous medin sandwiched between viscous fluids has been considered through a horizontal channel with isothermal wall temperatures. The flow in the porous medium is modeled using the Brinkman equation. The governing partial differential equations are transformed to ordinary differential equations by collecting the non-periodic and periodic terms. Closed-form solutions for each region are found after applying the boundary and interface conditions. The influence of physical parameters, such as the porous parameter, the frequency parameter, the periodic frequency parameter, the viscosity ratios, the conductivity ratios, and the Prandtl number, on the velocity and temperature fields is computed numerically and presented graphically. In addition, the numerical values of the Nusselt number at the top and bottom walls are derived and tabulated.     

Convective heat transfer in a rotating annulus device

Convective heat transfer in a horizontal annulus device rotating around its horizontal axis has been examined. The results show that heat transfer in the annulus depends on the rotational speed. At a certain value of the rotational speed there is only conduction in the annulus. A criterium is given to calculate this rotational speed from the physical properties of the liquids. For the calculation of the heat transfer in the standstill of the annulus two equations are proposed.

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Konvektiver Wärmeübergang in einem rotierenden Ringspalt

Zusammenfassung Der Wärmeübergang bei Konvektion in einem um seine Horizontalachse rotierenden Ringspalt wurde untersucht. Wie die Ergebnisse zeigen, hängt der Wärmeübergang von der Drehzahl ab. Ab einer bestimmten Drehzahl wird Wärme nur noch durch Leitung übertragen. Es wird ein Kriterium angegeben, diese Drehzahl aus den physikalischen Daten der Flüssigkeiten zu berechnen. Zur Berechnung des Wärmeübergangs im Stillstand werden zwei Gleichungen vorgeschlagen.