The effect of three closures on critical conditions in two-phase flow with unequal phase velocities

Abstract

The paper develops an analytic approach to assess the effect of three closure conditions on the critical-flow state in two-phase flow. The approach is based on the geometrical-topological analysis of dynamical systems. The details of the analysis are developed in relation to a “general slip” model which is completed by three closure conditions: homogeneous flow, Bankoff's hypothesis and the drift-flux model. For the sake of simplicity, the study is restricted to adiabatic flows and thermodynamic equilibrium. These make it possible to reduce the model to a single quasi-linear differential equation and to parametrize the problem with respect to the mass-flow rate m ̇ and the total stagnation enthalpy h. The phase space is thus reduced to the P, z diagram. Detailed results are given for the flow of water/steam through a convergent-divergent nozzle discharging from a stagnation reservoir. All three closures lead to the same topological structure of the “potrait” of solutions in the phase space which incorporates a saddle point. The numerical results are summarized in table 1. This shows that the location of the critical cross-section (saddle point) is essentially unaffected by the closure chosen for the model; it is situated very close to the throat and downstream from it. The most significant critical-flow characteristics, such as mass-flow rate, critical pressure, critical velocity, critical void fraction and critical slip velocity differ markedly from those predicted by the homogeneous-flow assumption when the effects of slip are included. The differences between the two closures with the slip are significant but much smaller than between either of them and the no-slip model. All models calculate a very large increase in the void fraction as the critical state near the throat is approached. This is interpreted to mean that none of the three closures should be used to analyze critical flows, because their validity has been tested only for much lower values of void fraction. The critical velocity is a sole function of the local thermodynamic state. The two slip closures lead to considerable differences in this relationship when compared with the sound velocity under homogeneous-flow conditions.