Momentum equation for two-phase flow

The terms represent the contributions from the total pressure gradient, the frictional drag of the pipe wall and the hydrostatic head of the two-phase mixture. 

For steady flow the net force acting on the fluid in the element is equal to the change of momentum flow rate  

Equation 7.10 shows that the total pressure gradient comprises three components that are due to fluid friction, the rate of change of momentum and the static head. The momentum term is usually called the accelerative component. Thus 

In principle, this is the same as for single-phase flow. For example in steady, fully developed, isothermal flow of an incompressible fluid in a straight pipe of constant cross section, friction has to be overcome as does the static head unless the pipe is horizontal, however there is no change of momentum and consequently the accelerative term is zero. In the case of compressible flow, the gas expands as it flows from high pressure to low pressure and, by continuity, it must accelerate. In Chapter 6 this was noted as an increase in the kinetic energy. 

It is convenient to work in terms of the void fraction a and the mass fraction of gas w, which is also known as the quality. The void fraction is defined as the time average fraction of the cross-sectional area through which the gas flows  

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The other specific feature of multiphase momentum equations is the term containing interphase momentum transport (Eq. (4.16)). The interphase momentum transport terms invariably contain the velocities of all interacting phases at that grid node. Typically, the discretized momentum equation for two-phase flows (for the node P) can be written ... 

Zhang DZ, VanderHeyden WB The effects of mesoscale structures on the macroscopic momentum equations for two-phase flows, Int J Multiphase Plow 28 805—822, 2002. 

A momentum balance for the flow of a two-phase fluid through a horizontal pipe and an energy balance may be written in an expanded form of that applicable to single-phase fluid flow. These equations for two-phase flow cannot be used in practice since the individual phase velocities and local densities are not known. Some simplification is possible if it... 

For two-phase flow, additional assumptions are made that thermodynamic phase equilibrium exists before and after the restriction (or expansion), and that no phase change occurs over the restriction. Romie (Lottes, 1961) wrote the equation for the momentum change across an abrupt expansion as... 

The basic discretization of the two-fluid model equations is similar to the approximations of the corresponding transport equations for single phase flow. A minor difference is that the two-fluid model equations contain the novel phase fraction variables that have to be approximated in an appropriate manner. More important, to design robust, stable and accurate solution procedures with appropriate convergence properties for the two-fluid model equations, emphasis must be placed on the treatment of the interface transfer terms in the phasic momentum, heat and mass transport equations. Because of these extra terms, the coupling between the different equations is even more severe for multiphase flows than for single phase flows. 

In this section, the hydrodynamics and heat transfer of the two-phase (liquid-gas) flow in porous media is addressed. First the volume-averaged momentum equation (for each phase) is considered. The elements of the hydrodynamics of three-phase systems (solid-liquid-gas) are discussed. Then the energy equation and the effective properties are reviewed. 

Two-fluid models presented earlier are applicable for two-phase flow of gas/Newtonian liquid systems. It has been successfully determined that the momentum balance equations for Newtonian fluids also apply to non-Newtonian fluids [52]. 

General equations of momentum and energy balance for dispersed two-phase flow were derived by Van Deemter and Van Der Laan (V2) by integration over a volume containing a large number of elements of the dispersed phase. A complete system of solutions of linearized Navier-Stokes equations... 

In this section we present the system of quasi-one-dimensional equations, describing the unsteady flow in the heated capillary tube. They are valid for flows with weakly curved meniscus when the ratio of its depth to curvature radius is sufficiently small. The detailed description of a quasi-one-dimensional model of capillary flow with distinct meniscus, as well as the estimation conditions of its application for calculation of thermohydrodynamic characteristics of two-phase flow in a heated capillary are presented in the works by Peles et al. (2000,2001) and Yarin et al. (2002). In this model the set of equations including the mass, momentum and energy balances is ... 

In reality, the slip velocity may not be neglected (except perhaps in a microgravity environment). A drift flux model has therefore been introduced (Zuber and Findlay, 1965) which is an improvement of the homogeneous model. In the drift flux model for one-dimensional two-phase flow, equations of continuity, momentum, and energy are written for the mixture (in three equations). In addition, another continuity equation for one phase is also written, usually for the gas phase. To allow a slip velocity to take place between the two phases, a drift velocity, uGJ, or a diffusion velocity, uGM (gas velocity relative to the velocity of center of mass), is defined as... 

Figure 7.6 illustrates a gas-liquid two-phase flow through an inclined pipe. For clarity the diagram is drawn for stratified flow but the equations to be derived are not limited to that flow regime. A momentum equation can be written for each phase but it will be sufficient for the present purposes to treat the whole flow. In this case the interfacial shear force SFs makes no direct contribution but it would have to be considered in writing the momentum equation for either of the phases individually. The net force acting in the positive x-direction is... 

Differential momentum, mechanical-energy, or total-energy balances can be written for each phase in a two-phase flowing mixture for certain flow patterns, e.g., annular, in which each phase is continuous. For flow patterns where this is not the case, e.g., plug flow, the equivalent expressions can usually be written with sufficient accuracy as macroscopic balances. These equations can be formulated in a perfectly general way, or with various limitations imposed on them. Most investigations of two-phase flow are carried out with definite limits on the system, and therefore the balances will be given for the commonest conditions encountered experimentally. 

Based on the above-mentioned assumptions, the mass, momentum and energy balance equations for the gas and the dispersed phases in two-dimensional, two-phase flow were developed [14], In order to solve the mass, momentum and energy balance equations, several complimentary equations, definitions and empirical correlations were required. These were presented by [14], In order to obtain the water vapor distribution the gas phase the water vapor diffusion equation was added. During the second drying period, the model assumed that the particle consists of a dry crust surrounding a wet core. Hence, the particle is characterized by two temperatures i.e., the particle s crust and core temperatures. Furthermore, it was assumed that the heat transfer from the particle s cmst to the gas phase is equal to that transferred from the wet core to the gas phase i.e., heat and mass cannot be accumulated in the particle cmst, since all the heat and the mass is transferred by diffusion through the cmst from the wet core to the surrounding gas. Based on this assumption, additional heat balance equation was written. 

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