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ared with the pore dimensions, but small compared with the dim That is, microstructural details are ignored, but variations in v and e considered. The pore structure enters only through its effect or experiment. age porosity ɛ (volume fraction of pores), use Darcy's law to ob vation of mass in the fluid phase, in integral form and then in dif ompressible fluid, the differential equation from part (a) reduces porous material consisting of straight, cylindrical pores of diam res are parallel to the x, y, or z axes and that they intersect at po cubic lattice of dimension e > d. The overall dimensions of a sa of the order of L, where L> e. The volume flow rate Q in a sin bed by Poiseuille's law,

ared with the pore dimensions, but small compared with the dim That is, microstructural details are ignored, but variations in v and e considered. The pore structure enters only through its effect or experiment. age porosity ɛ (volume fraction of pores), use Darcy's law to ob vation of mass in the fluid phase, in integral form and then in dif ompressible fluid, the differential equation from part (a) reduces porous material consisting of straight, cylindrical pores of diam res are parallel to the x, y, or z axes and that they intersect at po cubic lattice of dimension e > d. The overall dimensions of a sa of the order of L, where L> e. The volume flow rate Q in a sin bed by Poiseuille's law,

Elements Of Electromagnetics
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
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2-3. Flow in Porous Media: Darcy's Law A relationship used to model flow in isotropic porous media is Darcy's law, к K v = --(VP - pg) -VP where k is the Darcy permeability (units of m2), g is the gravitational acceleration, and P is the dynamic pressure (Chapter 6). The velocity and pressure in Darcy's law are each averaged over a length scale that is large compared with the pore dimensions, but small compared with the dimen- sions of the system of interest. That is, microstructural details are ignored, but variations in v and P at a more macroscopic level are considered. The pore structure enters only through its effect on K, which is usually determined by experiment. (a) For a material of average porosity ɛ (volume fraction of pores), use Darcy's law to obtain expressions for conservation of mass in the fluid phase, in integral form and then in differ- ential form. (b) Show that, for an incompressible fluid, the differential equation from part (a) reduces to V-P = 0. (c) Consider an idealized porous material consisting of straight, cylindrical pores of diameter d. Assume that all pores are parallel to the x, y, or z axes and that they intersect at points described by a simple cubic lattice of dimension e>» d. The overall dimensions of a sam- ple of this material are of the order of L, where L> l. The volume flow rate Q in a single pore segment is described by Poiseuille's law, 128μ/ where |AP is the pressure drop per segment. Evaluate k for this material.
Transcribed Image Text:2-3. Flow in Porous Media: Darcy's Law A relationship used to model flow in isotropic porous media is Darcy's law, к K v = --(VP - pg) -VP where k is the Darcy permeability (units of m2), g is the gravitational acceleration, and P is the dynamic pressure (Chapter 6). The velocity and pressure in Darcy's law are each averaged over a length scale that is large compared with the pore dimensions, but small compared with the dimen- sions of the system of interest. That is, microstructural details are ignored, but variations in v and P at a more macroscopic level are considered. The pore structure enters only through its effect on K, which is usually determined by experiment. (a) For a material of average porosity ɛ (volume fraction of pores), use Darcy's law to obtain expressions for conservation of mass in the fluid phase, in integral form and then in differ- ential form. (b) Show that, for an incompressible fluid, the differential equation from part (a) reduces to V-P = 0. (c) Consider an idealized porous material consisting of straight, cylindrical pores of diameter d. Assume that all pores are parallel to the x, y, or z axes and that they intersect at points described by a simple cubic lattice of dimension e>» d. The overall dimensions of a sam- ple of this material are of the order of L, where L> l. The volume flow rate Q in a single pore segment is described by Poiseuille's law, 128μ/ where |AP is the pressure drop per segment. Evaluate k for this material.
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