Drift-flux correlation disengagement models: Part I — Theory: Analytic and numeric integration details

Abstract

In the design of pressure relief systems for vessels containing liquid, the phase of the flow through the vent line is very important. Mounting the line on the top of the vessel does not necessarily guarantee all vapor flow. One must calculate whether vapor bubbles formed in the liquid will disengage before they reach the vent entrance. Disengagement can be predicted via an axial void fraction profile that is calculated based upon volumetric gas production. It is assumed that the liquid phase is continuous and that pseudo-steady state is reached. The disengagement model is based on a constant energy generation per unit mass of liquid. For non-foaming systems, one of two drift-flux correlations can be chosen on the basis of viscosity. The churn-turbulent drift-flux correlation is for low-viscosity systems, and the DIERS' viscous-bubbly drift-flux correlation is for high-viscosity system [1, 2]. This model reduces to a single ordinary differential equation (ode). Analytic integration results for this model are possible for constant cross-sectional area vessels (e.g., vertical cylinder) and non-unity distribution parameters Co [3–5]. If this calculation shows the bubbles do not disengage, either a partial differential equation model must be solved or the coupling equation must be used. The coupling equation uses the maximum void fraction (calculated from the ode) and ties together the vessel and vent models. The ode solutions relate the local and average void fractions to the dimensionless superficial vapor velocity. For the churn-turbulent drift-flux correlation, explicit relationships are presented for the first time. They validate the earlier approximation of Fauske et al. [6] (see also Ref. [3]). For the DIERS' viscous-bubbly drift-flux correlation, implicit relationships are presented for the first time in the open literature. The earlier approximation of Fauske et al. [6] (see also Ref. [7]) fit the data, but is different than these integration results. Further work is in progress to refit the data [8] and to clarify the best model to use. For non-constant cross-sectional area vessels (e.g., horizontal cylinders and spheres), the analytic integration is difficult, but numeric results have been presented [3, 7]. The details of the numeric integration are presented and discussed here. As the cross-sectional area converges (toward the top of the vessel), the vapor concentration (i.e., the void fraction) and velocity increase. The maximum local void fraction occurs at the top of the vessel. Numerical difficulties are encountered as a result of the cross-sectional area going to zero at the bottom and top of the vessel. The pseudo-steady-state model imposes void fractions of minimum and maximum at these extremes (i.e., 0 and 1/Co).