The Two Fluid Model
SAM-N – In the short history of computation physics there have been a few singular efforts that in hindsight can be considered disruptive innovations. Such innovations are distinguished by capturing complexity at a level not previously conceived. The modeling and simulation of two-phase flow has gone through at least three such generations, beginning with the three-equation Homogeneous-Equilibrium (HEM) model (a.k.a. the mixture or diffusion model), exemplified by the FLASH model. This was followed by the four- or five-equation Separated Flow model (where applications of the Drift Flux model were found to be the most successful). The third was the six-equation Two-Fluid model, introduced by Mamoru Ishii in the mid-1970s. The governing equations of the today’s large thermal-hydraulic system codes are based on the Two-Fluid model.
The two-fluid model is distinguished from the HEM and separated flow models by considering mass, momentum, and energy transport between the phases. In the equations below the dependent variables are void fraction (α), pressure (P), liquid-phase internal energy (U(f)), and vapor-phase internal energy (U(g)), liquid velocity (V(f)) and vapor velocity (V(g)). The Ishii-inspired phasic continuity equations as they appear in the RELAP5 code are
where ρ(f) and ρ(g) are liquid and vapor density, respectively, and A is flow area. The vapor generation (or condensation) term contains two components: wall and interface, i.e., Γ(g)= Γ(gi)+ Γ(w). Assuming no mass sources or sinks, continuity yields that the liquid generation term is the negative of the vapor generation, i.e., Γ(f) =- Γ(g)
The RELAP5 phasic momentum equations are
The terms on the right hand side of the momentum equations are respectively: momenta convection (i.e., acceleration), the pressure gradient, the body force (B), wall friction, momenta due to interphase mass transfer, interphase frictional drag, and force due to virtual mass. The terms FWG, FWF, FIG, and FIF are the wall friction and interphase friction coefficients, respectively for vapor and liquid phases. FWG and FWF relate to the wall friction factor for vapor and liquid, respectively while FIG and FIF relate to interphase frictional for vapor and liquid, respectively. The C and λ terms are ad hoc dimensionless factors designed to ensure that the mass and momentum exchange processes are always dissipative (contributes to numerical stability).
The RELAP5 phasic energy equations are
The terms on the right hand side of the energy equations account for energy contribution from energy convection, flow work, wall heat transfer, interfacial heat transfer, interfacial mass transfer, and dissipation. The phasic enthalpies [h(g)*,h(f)*] associated with interphase mass transfer are defined in such a way that the interface energy jump conditions at the liquid-vapor interface are satisfied. In particular, h(g)* and h(f)* are chosen to be h(g)s and h(f) respectively for the case of vaporization and h(g) and h(f)s respectively for the case of condensation [s is an abbreviation for “saturation”]. DISS(g) and DISS(f) are the phasic energy dissipation terms and are sums of wall friction and pump effects.
While Ishii’s model has the advantage that the actual transport processes can be rigorously defined, its disadvantage is that detail related to these kinetic processes is required, implying a much greater depth of experimental data and insight. As such, the form credited in RELAP5 and like systems codes represents a simplification, primarily necessitated by the sensitivity of two-phase flow dynamics to the topology of the fluid stream. Another key distinction between the RELAP5 form and Ishii’s model relates to how turbulence is addressed (typically represented by shear stress terms using the greek letter tau, τ). Instead of an explicit closure relation, turbulence effects are treated as wall and interface dissipative terms, derived by single-phase empirical correlations that are corrected for two-phase condition.
With the redirection of priorities following the ECCS Hearings and the TMI-2 accident of the 1970s, investment in progressively more accurate system prediction has been sustained by the USNRC and others. Nonetheless, broad acceptance of the two-fluid model has not come easy. Early technical audits sponsored by the USNRC of their principal thermal-hydraulic system codes, RELAP5 and TRAC, frequently cited incompleteness of their model closure packages. In particular, several correlations were identified as simply being based on engineering judgment as a workaround for knowledge and data gaps. In the time since Ishii’s original derivation much of the incompleteness has been resolved through experimental programs in the US and around the world, transforming the two-fluid model into a mature technology that provides the backbone to safety analysis methods.
Postscript: Currently investment in two-fluid computation fluid dynamics is being made internationally in both the public and private sectors, suggesting that the leading edge of this science is in the midst of transitioning to the fourth generation!