Multiphase flows consist of interacting phases that are dispersed randomly in space and in time. An additional complication arises from the fact that the flow region of interest often contains irregularly shaped structures. While, in principle, the intraphase conservation equations for mass, momentum, and energy, and their initial and boundary conditions can be written, the cost of detailed fluid flow and heat transfer analysis with explicit treatment of these internal structures with complex geometry and irregular shape often is prohibitive, if not impossible. In most engineering applications, all that is required is to capture the essential features of the system and to express the flow and temperature field in terms of local volume-averaged quantities while sacrificing some of the details. The present study is an attempt to achieve this goal by applying time averaging after local volume averaging.
Local volume averaging of conservation equations of mass, momentum, and energy for a multiphase system yields equations in terms of local volume-averaged products of density, velocity, energy, stresses, and field forces, together with interface transfer integrals. These averaging relations are subject to the following length scale restrictions:
where d is a characteristic length of the pores or dispersed phases, a characteristic length of the averaging volume, and L is a characteristic length of the physical system. Solutions of local volume-averaged conservation equations call for expressing these local volume-averaged products in terms of products of averages. In nonturbulent flows, this can be achieved by expressing the "point" variable as the sum of its intrinsic volume average and a spatial deviation. In turbulent flows, the same can be achieved via subsequent time averaging over a duration T such that
where Ο HF is a characteristic time of high-frequency fluctuation and Ο LF is a characteristic time of low-frequency fluctuation. In this case, and instantaneous "point" variable Ο k of phase k is decomposed into a low-frequency component Ο kLF and a high-frequency component Ο k , similar to Reynolds analysis of turbulent flow. The low-frequency component consists of the sum of the local intrinsic volume average 3i Ο k LF and its local spatial deviation ΟkLF . Time averaging then reduces the volume-averaged products to products of averages plus terms representing eddy and dispersive diffusivities of mass, Reynolds and dispersive stresses, and eddy and dispersive conductivities of heat, etc. These terms arise from both high-frequency fluctuations and local spatial deviations. This procedure of time averaging after local volume averaging leads to a set of differential-integral equations of conservation for multiphase flow. This set of multiphase flow conservation equations is particularly suitable for numerical analysis with staggered grid computational systems.
Attention is focused on multiphase flow in a region containing fixed and dispersed heat-generating and absorbing solid structures. The novel porous media formulation employs the concept of volume porosity, directional surface porosities, distributed resistance and distributed heat source Abbreviations: EPYTI, interfacial enthalpy transfer integral; HTI, interfacial heat transfer integral; IETI, interfacial internal energy transfer integral; MTI, interfacial mass transfer integral; MMTI, interfacial momentum transfer integral; PTI, interfacial pressure transfer integral; PWI, interfacial pressure work integral (PWI) (h) is associated with enthalpy production (PWI) (u) is associated with internal energy production; TETI, interfacial total energy transfer integral; VDI, interfacial viscous dissipation integral; VSTI, interfacial viscous stress transfer integral; VWI, interfacial viscous stress work integral * Corresponding author.