Keywords: two-phase flow, phase transition, hysteresis, finite element modelling.

In this paper a diffuse interface model of an annular-type two-phase flow with phase-change is presented. The model set-up is a horizontal steam tube with close to the saturation inlet conditions. At a point some distance downstream of the entrance, vapor begins to condense and a thin liquid film evolves on the interior wall of the tube. The rate of condensation is directly linked to the rate at which heat is transported across the film from the interface to the surface. Modeling the heat transfer and fluid flow associated with this liquid-vapor phase change process requires additional elements to the governing equations of the single-phase convective transport regarding to the non-equilibrium effects and dynamic interactions between the phases. The two-phase flow is assumed as a homogeneous phase mixture consists of a single fluid with continuous variation of thermodynamic state variables forming a diffuse interface [1]. To describe the phase transition it is necessary to select a so called order parameter (phi), which differs in the two phases. The energy balance equation involves the time derivative of the order parameter as a consequence of the dependence of the internal energy on phi. The evolution equation of phi is a hysteresis operator based on a Landau-Ginzburg free-energy density approximation [2].

A non-equilibrium assumption, i.e. the finite speed of phase transformation, enables some degree of supersaturation in the cluster without phase change. The upper limit of the acceptable supersaturation has to be determined. This limit has to be between the vapor spinodal and equilibrium curve. The theoretical limit, the spinodal of vapor supersaturation is proportional to p/p_c [3], where p_c is the critical pressure. In this work a T/T_c (T_c is the critical temperature) dependent supersaturation limit is introduced.

The applied hysteresis model of the vapor-liquid phase transition can improve other macroscale heat transfer models as well that are associated with boiling and condensation phenomena and are treated until now as virtually isothermal heat transfer processes. The presented model has been implemented into a finite element simulation and proved to be very effective and numerically stable.

1

Y. Sun, C. Beckermann, "Diffuse interface modeling of two-phase flows based on averaging: mass and momentum equatios", Phisica D,198, 281-308, 2004. doi:10.1016/j.physd.2004.09.003

2

G. Bertotti, I. Mayergoyz (Eds.) "Science of hysteresis", Vol.II, Elsevier, Amsterdam, 2005.

3

S.B. Kiselev, J.F. Ely, "Curvature effect on the physical boundary of metastable states in liquids", Physica A 299, 357-370, 2001. doi:10.1016/S0378-4371(01)00267-9

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