Begell House
Journal of Porous Media
Journal of Porous Media
1091-028X
2
1
1999
Tree Networks for Flows in Composite Porous Media
This article reports a numerical study of the geometric minimization of the resistance to Darcy flow between a finite-size volume and one point. The volume is two dimensional and contains materials with several permeabilities. The optimization starts with the smallest volume subsystem, and proceeds toward larger subsystems (assemblies) until the given volume is covered. It is shown that at every scale the geometric shape of the subsystem can be optimized. This principle allows us to construct the volume-to-point flow path by using assemblies of previously optimized building blocks, hence the “constructal” name for the associated theory of shape and structure formation in natural flow systems. The optimized flow architecture is such that the regions of relatively high permeability form a tree network that is completely deterministic. Every feature of this architecture is the result of a single optimization principle: the geometric minimization of flow resistance subject to constraints.
M. R.
Errera
Department of Mechanical Engineering and Materials Science, Duke University, Box 90300, Durham, North Carolina 27708-0300, USA
Adrian
Bejan
Department of Mechanical Engineering and Materials Science, Duke University, Box 90300, Durham, NC 27708-0300, USA
1-17
The Effective Stagnant Thermal Conductivity of Porous Media with Periodic Structures
Existing analytical models for predicting the effective stagnant thermal conductivity of fluid-saturated spatially periodic media are summarized. These include the analytical solutions given by Maxwell (1873) and Rayleigh (1892), and the early models proposed by Deissler and Eian (1952), Kunii and Smith (1960), and Zehner and Schlunder (1970). Recent models such as the area-contact model, the phase-symmetry model, the single-scale lumped parameter model, and the multiscale lumped parameter model are emphasized. Simple algebraic expressions for the effective stagnant thermal conductivity of a number of geometries based on these recent models are presented. These include two-dimensional geometries such as square, circular, and elliptic cylinders and composite materials consisting of fiber bundles, as well as three-dimensional geometries such as ellipsoids, cubes, and wire screens. The effects of porosity, shape, and arrangement of the solid phases, as well as the solid-to-fluid thermal conductivity ratio are illustrated. The effects of point and finite contacts between the solid phase on the effective thermal conductivity of the porous medium are discussed. Comparisons of the analytical expressions with numerical solutions and experimental data are made whenever possible.
Ping
Cheng
Mechanical Engineering Department, The Hong Kong University of Science and Technology; School of Mechanical and Power Engineering, Shanghai Jiaotong University
Chin-Tsau
Hsu
Department of Mechanical Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
19-38
Free Convective Boundary Layer Flow from a Heated Surface in a Layered Porous Medium
We examine theoretically the steady free convective flow from an isothermal vertical flat plate embedded in a saturated porous medium. We consider in detail the effect of parallel layering on the flow and rate of heat transfer. The layering arises from discrete changes in either the permeability or the diffusivity of the porous medium. Mathematically, the presence of layering causes the boundary layer equations to be non-similar, and these equations are solved numerically using the Keller-box method. The numerical work is supplemented by an asymptotic analysis of the flow in the far-downstream limit. Where there is a finite number of sublayers sandwiched between the heated surface and the remaining isotropic medium, detailed results are limited by the appearance of eigensolutions in the asymptotic expansion. When the medium is composed of alternating sublayers an asymptotic analysis yields an equivalent homogeneous medium.
D. Andrew S.
Rees
Department of Mechanical Engineering, University of Bath, Claverton Down, Bath BA2 7AY, United Kingdom
39-58
Forced Convection of a Power-Law Fluid in a Porous Channel—Integral Solutions
The integral method based on the boundary layer analysis is used to conduct a theoretical study of the primary hydrodynamic and heat transfer mechanisms for forced convection of a power-law fluid in a porous channel for both cases of uniform heat flux (UHF) and uniform wall temperature (UWT) boundary conditions. The flow in the porous medium is modeled using the modified Brinkman—Forchheimer-extended Darcy model for power-law fluids. The results indicate that for a high-permeability porous medium, the thickness of the momentum boundary layer depends on the Darcy number, inertia parameter, and power law index, but for a low-permeability porous medium it depends only on the Darcy number. Consequently in the non-Darcy regime, the effects of power law index on hydrodynamic and heat transfer behaviors in the porous channel are significant, whereas in the Darcy regime, the effects of Darcy number are predominant. Also the hydrodynamic behavior of shear thickening fluids (n > 1.0) is more sensitive to the Darcy number whereas the behavior of shear thinning fluids (n < 1.0) is more sensitive to the inertia parameter in the non-Darcy regime. Based on the fully developed Nusselt number solutions, the valid region for the applicability of the present integral method is illustrated graphically. The valid region of the integral solutions covers the Darcy regime for any value of the power law index within the range considered, while in the non-Darcy regime, the valid region for a shear thinning fluid becomes smaller than that for a shear thickening fluid as the inertia parameter decreases.
Hamid
Hadim
Stevens Institute of Technology, 1 Castle Point Terrace, Hoboken, NJ 07030, USA
G.
Chen
Department of Mechanical Engineering, Stevens Institute of Technology, Hoboken, New Jersey 07030, USA
59-69
Thermal Protection System with Use of Porous Media for a Hypersonic Reentry Vehicle
The effectiveness of a transpiration-cooled thermal protection system for a hypersonic reentry vehicle consisting of porous material is numerically analyzed with the new concept of coupling of outer free stream and inner flow within the porous matrix. This analysis is more effective in estimation of surface temperature and heat flux to the system compared with previous methods. The transpiration of coolant fluid has great cooling effectiveness as a result of decrease of surface temperature and heat flux, as transpired fluid injected into the outer flow cools the surface as well as creates a moderate temperature gradient near the surface. Increase of back face pressure corresponding to increase of transpiration mass flow causes the lower surface temperature and smaller heat flux. The cooling effectiveness per mass transpiration rate based on decrease of surface temperature is improved by smaller porosity, which shows the superiority of smaller porosity material for practical use.
Hirotoshi
Kubota
Department of Aeronautics and Astronautics, Faculty of Engineering, University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113, Japan
Shunichi
Uchida
Department of Aeronautics and Astronautics, University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-8656, Japan
71-85
Magnetohydrodynamic Mixed Convection from a Rotating Cone Embedded in a Porous Medium with Heat Generation
A similarity transformation is employed to convert the partial differential equations governing steady, laminar, hydromagnetic, mixed convection from a rotating non-isothermal and permeable cone embedded in a uniform non-Darcian porous medium into ordinary ones. The coupled nonlinear equations are solved numerically by an iterative implicit finite-difference method. Comparisons with previously published work are performed and found to be in excellent agreement. Graphical results are presented to show the influence of the possible presence of wall mass transfer, heat generation effects, magnetic field, and porous media.
Ali J.
Chamkha
Mechanical Engineering Department, Prince Mohammad Bin Fahd University, Al-Khobar 31952,
Kingdom of Saudi Arabia; Prince Sultan Endowment for Energy and Environment, Prince Mohammad Bin Fahd University, Al-Khobar 31952, Saudi Arabia; RAK Research and Innovation Center, American University of Ras Al Khaimah, P.O. Box
10021, Ras Al Khaimah, United Arab Emirates
87-105
Deep Hydrothermal Systems: Mathematical Modeling of Hot Dense Brines Containing Noncondensible Gases
Quantitative description of deep hydrothermal systems requires mathematical modeling of the heat and mass transfer associated with the motion of multicomponent fluids in high-temperature high-pressure environments within porous rock structures. In this article, earlier work investigating models that describe the behavior of brine systems (where the fluid is represented by water + sodium chloride) at high temperatures and pressures is extended to include the presence of noncondensible gases. It is assumed in the model equations that the gases are represented by carbon dioxide. The consequent H2O−NaCl−CO2 system is modeled as a brine with a noncondensible gas component added. The phase-space of this ternary system is four dimensional; however, three-dimensional (3D) “cross-sections,” imagined as cuts by surfaces of given CO2 concentration, can be used to aid in visualizing this. The resulting cross-section is a 3D brine T−p−X phase subspace (T is the temperature, p is the pressure, and X is the mass fraction of chloride within the brine component) which is regarded as a perturbation of the T−pb−X phase-space for the H2O−NaCl system [pb is the (partial) pressure of the brine which has salinity X]. The characteristics of the various regions of the latter 3D phase-space (McKibbin and McNabb, 1993) are presumed to still apply, but the boundaries are slightly altered in shape owing to the presence of the noncondensible gas. Conservation equations, together with various thermodynamic relationships and gas laws, are solved for some simple steady vertical flows. The example results provide some insights into the complex relationships between the concentrations and distribution of the various components.
Robert
McKibbin
Institute of Information and Mathematical Sciences, Massey University at Albany, Auckland, New Zealand
Alex
McNabb
Department of Mathematics, The University of Auckland, Private Bag 92 019, Auckland, New Zealand
107-126