Darcy's Law for Immiscible Two-Phase Flow: A Theoretical Development

Darcy's Law for Immiscible Two-Phase Flow: A Theoretical Development

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Journa l of Porous Media 8 ( 6 ) , 557–567 ( 2005 ) 'DUF\·V± /DZ± IRU± ,PPLVFLEOH± 7ZR²3KDVH± )ORZ³ $± 7KHRUHWLFDO± 'HYHORSPHQW Francisco J. Valdés-Parada and Gilberto Espinosa-Paredes Area de Ingeniería en Recursos Energéticos, Universidad Autónoma Metropolitana-Iztapalapa Apartado Postal 55-534, 09340 México D.F., México * E-mail: [email protected] ABSTRACT The aim of this paper is to show how the method of volume averaging can be used to obtain a closed set of averaged equations for bubbling flow. The Navier–Stokes equations are considered as the starting point for the volume-averaging method. The closure was formulated as an associ- ated problem with the deviations around averaged values of the local variables. When the traditional length-scale restrictions are imposed, the volume-averaged momentum equation can be given by v k k v m m = K k ( p k k + ρ k g k ) , which is equivalent to Darcy’s law. The tensor K k is determined by closure problems that must be solved using a spa- tially periodic model of a two-phase flow medium. 557 Received January 13, 2004; Accepted April 23, 2004 Copyright © 2005 Begell House, Inc . Electronic Data Center, http://edata-center.com Downloaded 2006-2-8 from IP 67.87.127.223 by Dahlia De Jesus
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INTRODUCTION The process of two-phase flow occurs in a wide variety of technological applications such as chemical plants, nuclear reactors, oil and geothermal pipelines and wells, evapora- tors, and condensers, among others. However, the mecha- nisms of mass, momentum, and energy transfer through the two-phase interface are extremely complicated due to the existence of interfaces and their associated discontinuities so that solutions via numerical models largely depend on relevant empirical correlations for a wide range of operat- ing and design conditions. 558 Valdés-Parada and Espinosa-Paredes NOMENCLATURE a k , a m , p pressure, Pa d k , d m auxiliary vectors defined in Eq. (87) r k position vector that locates any point for the k and m phases, Pa s/m in the k th phase, m km interface area, m 2 r 0 radius of the averaging volume, m b j , C j closure variables, Pa s/m ( j = m , k ) t characteristic time, s B k , B m closure variables, dimensionless t time, s A k , A m , T stress tensor, Pa D k , D m auxiliary tensors defined in Eq. (87) averaging volume, m 3 for the k and m phases, dimensionless V k volume of the k th phase, m 3 g gravity vector, m/s 2 v velocity vector, m/s g k , g m auxiliary vectors defined in Eq. (67) w speed of displacement of the surface for the k and m phases, Pa s/m vector, m/s G k , G m auxiliary tensors defined in Eq. (67) x position vector that locates the centroid for the k and m phases, dimensionless of the averaging volume, m h k , h m auxiliary vectors defined in Eq. (77) y k relative position vector that locates for the k and m phases, Pa s/m points in the k th phase relative H k , H m auxiliary tensors defined in Eq. (77) to the centroid of , m for the k and m phases, dimensionless Z m auxiliary tensor defined in Eq. (55) I identity tensor, dimensionless for the m phase, diensionless K k tensor defined in Eq. (35), m
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