Hitoshi Koyama

A Numerical Study of Thermal Dispersion in Porous Media

Thermal dispersion in convective flow in porous media has been numerically investigated using a two-dimensional periodic model of porous structure. A macroscopically uniform flow is assumed to pass through a collection of square rods placed regularly in an infinite space, where a macroscopically linear temperature gradient is imposed perpendicularly to the flow direction. Due to the periodicity of the model, only one structural unit is taken for a calculation domain to resolve an entire domain of porous medium. Continuity, Navier-Stokes and energy equations are solved numerically to describe the microscopic velocity and temperature fields at a pore scale. The numerical results thus obtained are integrated over a unit structure to evaluate the thermal dispersion and the molecular diffusion due to tortuosity. The resulting correlation for a high-Peclet-number range agrees well with available experimental data.

Buoyancy-induced flow of non-Newtonian fluids over a non-isothermal body of arbitrary shape in a fluid-saturated porous medium

The buoyancy-induced flows of non-Newtonian fluids over non-isothermal bodies of arbitrary shape within saturated porous media have been treated using the boundary layer approximations and the power-law model to characterize the non-Newtonian fluid behavior. Upon introducing a general similarity transformation which considers both the geometrical effect and the wall temperature effect on the development of the boundary layer length scale, the governing equations for a non-isothermal body of arbitrary shape have been reduced to those for a vertical flat plate. The transformed equations reveal that a plane or axisymmetric body of arbitrary shape possesses its corresponding family of the wall temperature distributions which permit similarity solutions. Numerical integrations were carried out using the Runge-Kutta-Gill method, and the results of the heat transfer function were presented once for all plane and axisymmetric bodies. As illustrations, local wall heat flux distributions were discussed for wedges, cones, spheres, circular cylinders and other geometries. Furthermore, an approximate formula based on the Karman-Pohlhausen integral relation has been presented for speedy and sufficiently accurate estimation of heat transfer rates.

A general similarity transformation for combined free and forced-convection flows within a fluid-saturated porous medium

It is the purpose of the present paper to introduce a general transformation procedure appropriate to the problem of combined free and forced convection in a porous medium. It will be shown that particular transformations proposed in the previous papers by Cheng and Minkowycz and co-workers are simply the specific forms of the present general transformation. Pure forced convection will be treated first as a limiting case of combined free and forced convection. The analysis reveals that any two-dimensional or axisymmetric body of arbitrary shape possesses its corresponding class of wall temperature distributions which permit similarity solutions. Secondly, combined free and forced convection will be considered to seek similarity solutions. It is found that, unlike in pure forced convection, similarity solutions in mixed convection are possible only when the external free-stream velocity varies every where in proportion to the product of the streamwise component of the gravity force and the wall-ambient temperature difference.

Free Convective Heat Transfer Over a Nonisothermal Body of Arbitrary Shape Embedded in a Fluid-Saturated Porous Medium

The problem of free convective heat transfer from a nonisothermal two-dimensional ar axisymmetric body of arbitrary geometric configuration in a fluid-saturated porous medium was analyzed on the basis of boundary layer approximations. Upon introducing a similarity variable (which also accounts for a possible wall temperature effect on the boundary layer length scale), the governing equations for a nonisothermal body of arbitrary shape can be reduced to an ordinary differential equation which has been previously solved by Cheng and Minkowycz for a vertical flat plate with its wall temperature varying in an exponential manner. Thus, it is found that any two-dimensional or axisymmetric body possesses a corresponding class of surface wall temperature distributions which permit similarity solution. Furthermore, a more straightforward and yet sufficiently accurate approximate method based on the Karman-Pohlhausen integral relation is suggested for a general solution procedure for a Darcian fluid flow over a nonisothermal body of arbitrary shape. For illustrative purposes, computations were carried out on a vertical flat plate, horizontal ellipses, and ellipsoids with different minor-to-major axis ratios.

An analysis on forced convection in a channel filled with a Brinkman-Darcy porous medium: Exact and approximate solutions

An analysis was made to investigate non-Darcian fully developed flow and heat transfer in a porous channel bounded by two parallel walls subjected to uniform heat flux. The Brinkmanextended Darcy model was employed to study the effect of the boundary viscous frictional drag on hydrodynamic and heat transfer characteristics. An exact expression has been derived for the Nusselt number under the uniform wall heat flux condition. Approximate results were also obtained by exploiting a momentum integral relation and an auxiliary relation implicit in the Brinkmanextended Darcy model. Excellent agreement was confirmed between the approximate and exact solutions even in details of velocity and temperature profiles.ZusammenfassungDiese Studie untersucht voll entwickelte „Non-Darcy“-Strömung und Wärmetransport in einem porösen Kanal, der durch zwei parallele Wände gebildet und einem einheitlichen Wärmestrom ausgesetzt wird. Das erweiterte „Brinkman-Darcy-Modell” wurde angewandt, um den Effekt des viskosen Reibungswiderstands der Wände auf die hydrodynamischen Charakteristiken und die Wärmeübertragung zu untersuchen. Ein exakter Ausdruck wurde für die Nusselt-Zahl unter einheitlichem Wandwärmestrom abgeleitet. Annähernde Ergebnisse wurden durch die Auswertung einer Impuls-Integral-Beziehung und einer im erweiterten „Brinkman-Darcy-Modell“ implizierten Hilfsfunktion erhalten. Ausgezeichnete Übereinstimmung zwischen den genäherten und den exakten Lösungen konnten sogar in den Geschwindigkeits- und Temperaturprofilen festgestellt werden.

Effect of thermal stratification on free convection within a porous medium

Free convection over a vertical flat plate embedded in a thermally stratified porous medium is analyzed by exploiting the similarity transformation procedure. Numerical integration results are presented for a series of wall and ambient temperature distributions which permit similarity solutions. The conjugate conduction connection problems of a free convection fin embedded in a thermally stratified porous medium is examined, and it is shown that the influence of the thermal stratification on the heat transfer is quite significant. 8 references.

Similarity solutions for buoyancy-induced flows over a non-isothermal curved surface in a thermally stratified porous medium

A generalized similarity transformation procedure was proposed for the analysis of buoyancy-induced flows over a curved heated surface embedded in a thermally stratified porous medium. The analysis considers two-dimensional and axisymmetric non-isothermal smooth bodies of arbitrary geometrical configuration. A generalized similarity variable which adjusts its vertical scaling according to the geometry as well as the surface temperature variation was introduced to show that, for any two-dimensional or axisymmetric smooth body shape, there exists a certain class of the surface and ambient temperature distributions which admit similarity solutions. Subsequently, the set of the governing partial differential equations were transformed into a single ordinary differential equation, which was, then, solved by a standard shooting procedure based on the Runge-Kutta method, for numerous sets of parameters. The results presented here may readily be translated for the problem of free convection over any particular two-dimensional or axisymmetric smooth body within a porous medium. The effects of the surface temperature and thermal stratification on the temperature profile and isotherms were also discussed in connection with the local surface heat flux.

Non-Darcian Boundary Layer Flow and Forced Convective Heat Transfer Over a Flat Plate in a Fluid-Saturated Porous Medium

The local similarity solution procedure was successfully adopted to investigate non-Darcian flow and heat transfer through a boundary layer developed over a horizontal flat plate in a highly porous medium. The full boundary layer equations, which consider the effects of convective inertia, solid boundary, and porous inertia in addition to the Darcy flow resistance, were solved using novel transformed variables deduced from a scale analysis. The results from this local similarity solution are found to be in good agreement with those obtained from a finite difference method. The effects of the convective inertia term, boundary viscous term, and porous inertia term on the velocity and temperature fields were examined in detail. Furthermore, useful asymptotic expressions for the local Nusselt number were derived in consideration of possible physical limiting conditions.

An analysis for friction and heat transfer characteristics of power-law non-Newtonian fluid flows past bodies of arbitrary geometrical configuration

A general integral solution procedure has been suggested for the analysis of the forced convective heat transfer to the power-law-non-Newtonian fluids from bodies of arbitrary geometrical configuration. Both the free stream velocity and wall temperature are allowed to vary in arbitrary fashion. The set of governing equations has been eventually reduced to a pair of characteristic equations: one first order ordinary differential equation and another integral equation, which can readily be solved, once the power-law exponent, body geometry, external velocity distribution and wall temperature distribution are specified. Comparison of the calculated results with available experimental data and series expansion solutions suggests an excellent performance of the present approximate solution procedure.ZusammenfassungEs wird eine generelle Methode für eine integrale Lösung zur Analyse des Wärmeübergangs bei Zwangskonvektion für strukturviskose, nicht Newtonsche Fluide vorgeschlagen, die Körper unterschiedlicher geometrischer Konfiguration umströmen. Sowohl die Freistrahlgeschwindigkeit als auch die Wandtemperatur können beliebig variieren. Der die Vorgänge beschreibende Gleichungssatz wurde schließlich auf ein Paar charakteristische Gleichungen reduziert, nämlich eine gewöhnliche Differentialgleichung erster Ordnung und eine Integralgleichung, die sofort gelöst werden können, wenn der Exponentialansatz für das strukturviskose Verhalten die Geometrie des umströmten Körpers, die Verteilung der Anströmgeschwindigkeit und die Verteilung der Wandtemperatur spezifiziert sind. Ein Vergleich der berechneten Ergebnisse mit verfügbaren Messungen und anderen Lösungen zeigt ein ausgezeichnetes Verhalten der vorliegenden Näherungslösungs-Methode.

An analysis for forced convection heat transfer from external surfaces to non-Newtonian fluids

A theoretical analysis has been proposed for the forced convection heat transfer from external surfaces immersed in non-Newtonian fluids of the power-law model. The integral treatment previously introduced for Newtonian fluids has been successfully extended to the non-Newtonian fluids over a flat plate and a wedge of an arbitrary included angle. The integral momentum and energy equations are transformed into a pair of characteristic equations, which can readily be solved for the velocity shape factor and the boundary layer thickness ratio, once the exponents in the expressions for the power-law model, free stream velocity and wall temperature variation are specified. It has been also found that an asymptotic expression derived under the assumption of large Prandtl number, is valid practically for all power-law fluids, and hence, can be used for a speedy, and yet accurate estimation of the local heat transfer to non-Newtonian fluids.ZusammenfassungEs wird eine theoretische Analyse für den Wärmeübergang bei Zwangskorivektion von äußeren in nicht-Newtonsche Fluide eintauchende Oberfläche unter Benutzung des Exponentialansatzes vorgeschlagen. Dabei wurde die früher für Newtonsche Fluide eingeführte Integralbehandlung erfolgreich auf nicht-Newtonsche Fluide erweitert, die über eine ebene Platte und einen Keil beliebigen Winkels strömen. Die integralen Impuls- und Energiegleichungen werden in ein Paar charakteristischer Gleichungen transformiert, die für den Geschwindigkeits-Formfaktor und für das Grenzschicht-Dickenverhältnis leicht gelöst werden können, wenn die Exponenten in den Ausdrücken für den Potenzansatz die Anströmgeschwindigkeit und die Änderung der Wandtemperatur spezifiziert sind. Es stellte sich heraus, daß eine asymptotische Näherung, die unter der Annahme großer Prandtl-Zahl abgeleitet wurde, für Fluide beliebigen Potenzansatzes praktisch gültig ist und dadurch für eine rasche und trotzdem genaue Abschätzung des örtlichen Wärmeübergangs an nicht-Newtonsche Fluide benützt werden kann.