Accurate calculation of the pressure and temperature of water, steam, and ice: Formulation for CFD

Abstract

An accurate approach is proposed for calculating the thermodynamic properties of water in three states: liquid, steam and ice, and the transitional states among them. The formulation is expressed in terms of quantities that are naturally used in Computational Fluid Dynamics (CFD), namely the specific volume (v) and specific internal energy (u), through the use of Gibbs free energy. The Gibbs free energy formula proposed by IAPWS, formulated as a function of pressure and temperature, is used as a basis in our calculations. The Jacobian matrix resulting from the transformation between sets of variables (p, T) and (v, u) are derived for each phase; the Newton-Raphson method is used to iteratively solve the nonlinear equations. Numerical calculations have been carried out for the entire phase diagram covering all three phases. The numerical results are compared with the original data of IAPWS and the associated errors are analyzed. It is confirmed that the pressure and temperature are accurately calculated, with largest relative error on the order of 10−7 in the ice phase. Hence, other thermodynamic properties are also obtained within the same level of accuracy. The method proposed in this paper for calculating pressure and temperature, variables needed in CFD, is reliable and can be applied to the numerical simulation of multiphase flows, including phase changes.