Three-Dimensional Flow and Heat Transfer within Highly Anisotropic Porous Media

CONTENTS 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 6.2 Volume-Averaged Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 6.3 Preliminary Consideration of Macroscopically Uniform

Flow Through an Isothermal Porous Medium . . . . . . . . . . . . . . . . . . . . . . . . . . 239 6.4 Periodic Boundary Conditions for Three-Dimensional Periodic

Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 6.5 Quasi-Three-Dimensional Numerical

Calculation Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 6.6 Method of Computation and Preliminary Numerical

Consideration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 6.7 Validation of Quasi-Three-Dimensional Calculation Procedure . . . . . 248 6.8 Determination of Permeability Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 6.9 Determination of Forchheimer Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 6.10 Determination of Interfacial Heat Transfer Coefficient . . . . . . . . . . . . . . . . 256 6.11 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

In order todesign efficient heat transfer equipment, onemust know thedetails of both flow and temperature fields within the equipment. Such detailed flow and temperature fields within a manmade assembly may be investigated numerically by solving the set of governing equations based on the first

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principles (i.e., continuity, momentum, and energy balance equations), so as to resolve all scales of flowandheat transfer in the system.However, in reality, it would hardly be possible to reveal such details even with the most powerful super computer available today. For example, a grid system, designed for a comparatively large scale of heat exchanger systems, would not be fine enough to describe the details of flow and heat transfer around a fin in a heat transfer element. It has been recently pointed out by DesJardin (personal communication,

2001) and many others [1,2] that the concept of local volume-averaging theory, namely, VAT, widely used in the study of porous media [3-5] may be exploited to investigate the flow and heat transfer within such a complex heat and fluid flow equipment. These complex assemblies usually consist of small-scale elements, such as a bundle of tubes and fins, which one does not want to grid. Under such a difficult situation, one may resort to the concept of VAT instead, so as to establish a macroscopic model, in which these collections of small-scale elements are treated as highly anisotropic porous media. There are a number of situations in which one has to introduce macroscopic models to describe complex fluid flow and heat transfer systems. Nakayama and Kuwahara [6] appealed to VAT and derived a set of mac-

roscopic governing equations for turbulent heat and fluid flow through an isotropic porous medium in local thermal equilibrium. The resulting set of governing equations was generalized by Nakayama et al. [7], to treat highly anisotropic porous media by integrating the microscopic governing equations, namely, the Reynolds averaged versions of continuity, Navier-Stokes, and energy equations. One can conveniently use thesemacroscopic equations designed for highly anisotropic porousmedia, to investigate the flow and heat transfer within complex equipment, since a single set of the volume-averagedgoverning equations canbe applied to the entire calculation domain within the complex heat transfer equipment consisting of both largeand small-scale elements. All that one has to do is to specify the spatial distributions of macroscopic model parameters such as porosity and permeability. The clear fluid flow regionwithout small-scale obstructions, for example, will be treated as a special case, as one sets the porosity for unitywith an infinitely large permeability. In order to utilize these macroscopic equations for such large-scale numer-

ical computations, one must close the macroscopic equations by modeling the flow resistance associated with individual subscale solid elements and also the heat transfer rate between the flowing fluid and the subscale elements, in terms of the macroscopic velocity vector and relevant geometrical parameters. Such subscale models can be established by carrying out direct numerical experiments at a pore scale for individual subscale elements. Since the subscale structure is often periodic, the numerical experiment can be performed economically, focusing on one structural unit and utilizing periodic boundary conditions there. The microscopic results, thus obtained, are processed to extract the macroscopic hydrodynamic and

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thermal characteristics, and eventually to determine the unknown model constants of the subscale models associated with permeability tensor, inertial (Forchheimer) tensor, and interfacial heat transfer coefficient. Kuwahara et al. [8], Nakayama and Kuwahara [9], Nakayama et al. [10], and De Lemos andPedras [11,12] have conducted suchmicroscopic computations successfully. The unknown model constants including the interfacial heat transfer coefficient, permeability, and Forchheimer constants were determined by carrying out exhaustive numerical experiments using a periodic array of square and circular cylinders. A review on the research towards this endeavor may be found in chapter 10 of the first edition of the handbook [13]. All these investigations, however, were limited to the cases of the cross-

flows over two-dimensional structures. In reality, allmanmade elements such as those in plate fin heat exchangers are three-dimensional in nature. Naturally, themacroscopic velocity vector is not always perpendicular to the axis of the cylinder. The deviating angle between the velocity vector and the plane perpendicular to the axis of the cylinder is called “yaw” angle. Thus, the three-dimensional yaw effects on the permeability tensor, inertial tensor, and interfacial heat transfer coefficient must be elucidated beforehand, in order to design such heat transfer elements and systems. Nakayama et al. [14] used a bundle of rectangular cylinders to describe such three-dimensional anisotropic porous media, and showed that, under macroscopically uniform flow, the three-dimensional governing equations reduce to quasi-threedimensional forms, in which all derivatives associated with the axis of the cylinder can be either eliminated or replaced by other determinable expressions. Thus, only two-dimensional storages are required for the dependent variables. This quasi-three-dimensional numerical calculation procedure has been exploited to investigate the three-dimensional effects on the permeability tensor, inertial tensor, and interfacial heat transfer coefficient, which are needed to close the proposed set of the macroscopic governing equations. In what follows, we shall review a series of extensive investigations on

three-dimensional flow and heat transfer within highly anisotropic porous media. A bank of long cylinders is considered as one of fundamental geometrical configurations often found in heat exchangers and many other manmade anisotropic porousmedia. Numerical determination of the important subscale model parameters, such as permeability tensor, inertial tensor, and interfacial heat transfer coefficient, will be described in detail, so as to elucidate the three-dimensional yaw effects on these macroscopic hydrodynamic and thermal parameters. The results are compared with available experimentaldata to substantiate thevalidityof thepresentmodeling strategy for three-dimensional flow and heat transfer within highly anisotropic porous media. Upon correlating these macroscopic results, a useful set of explicit expressions will be established for the permeability tensor, inertial Forchheimer tensor, and interfacial heat transfer coefficient, so as to characterize three-dimensional flow and heat transfer through a bank of infinitely long cylinders in yaw.