Relaxation Model

3. GOVERNING EQUATIONS

In this paper, relaxation model is employed to account for thermal non-equilibrium between the phases. This model is presented by Downar-Zapolski et al. [16] and is employed by several authors to numerically simulate steady flashing flows [17-19]. Barret et al. [20] used this model to compute unsteady flashing flows in variable cross section ducts.

Relaxation model consists of three conservation laws (mass, momentum, energy) for the two-phase mixture in addition to the mass balance law for the vapor phase: ∂U/∂t+∂F/∂z=S where, U=[■(ρ@ρu@■(E@ρx))],F=[■(ρu@ρu^2+p@■(u(E+p)@ρux))],S=[■(0@(-2fρu^2)⁄d@■((4h_TP (T_w-T_L ))⁄d@(-ρ(x-x ̅ ))⁄θ))] where, ρ, p, u, x, x ̅, θ, f and hTP represent respectively mixture density,

Indeed, the mass conservation law for the vapor phase accounts for the delayed liquid–vapor transition by a relaxation toward thermodynamic equilibrium. In other words, a very high value of the relaxation time (O(θ)≈1) corresponds to a very small inter-phase mass transfer and the model behaves as the frozen model. However, in the case of very low values of the relaxation times (O(θ)≈〖10〗^(-3)), the time required to achieve equilibrium becomes very low and the model approaches the equilibrium model. The relaxation time can be obtained from the correlations for relaxation time as suggested by Downar–Zapolski et al.[16] which gives a separate correlation for high pressure and low pressure flow. In this paper, to avoid discontinuity at the transition pressure; the correlation of Gopalakrishnan [17] which is a combination of two suggested correlations of Downar-Zapolski is employed: θ=6.51×〖10〗^(-4) α^(-0.54) [(p_s (T_in )-p)/(p_s (T_in )

To close the system of equations of Eq. (1) an accurate equation of state as well as proper relations for the void fraction, fanning friction factor and the heat transfer coefficient should be provided. Details of which are presented in the following subsections.

3.1 CONSTITUTIVE EQUATIONS FOR THE VOID FRACTION

Original relaxation model utilizes a simple relation for the void fraction: α= xρ/ρ_sv

According to the study of Bhagwat et. al. [13], the constitutive equations for the void fraction presented by Rouhani et. al. [11], Choi et. al. [12] are among the most accurate equations especially in the range of low and moderate values of void fraction. Both of these correlations are based on drift flux. The general form of void fraction correlations based on drift flux is: α= ρux/(C_o ρu[x+ρ_sv/ρ_ml (1-x) ]+ρ_sv V_j )

Where, Co and Vj are respectively the distribution parameter and the drift velocity. The relations given by Rouhani et. al. [11] for Co and Vj are: C_o=1+0.12(1-x)

V_j=1.18∜(gσ(ρ_sv-ρ_ml )/〖ρ_ml〗^2 )