Improvements of Critical Heat Flux Models Based on the Viscous Potential Flow Theory

The absence of fluid viscosities in most existing models may be attributed to the fact that inviscid flow analyses are performed for the model development. For example, the hydrodynamic theory and macrolayer dryout models rely on the Rayleigh-Taylor, Kelvin-Helmholtz, and capillary instabilities for inviscid fluids. However, as the viscosities of two fluids become closer, none of them cannot be neglected. Moreover, the gas viscosity effect cannot be neglected on the condition that the gas layer is thin. Nevertheless, the previous studies neglected the viscous effect. Recently, Kim et al. showed that for the model development of critical heat flux and minimum film boiling, the Rayleigh-Taylor instability should be analyzed with a thin layer of viscous gas instead of a thick layer of inviscid gas. The decrease of the most unstable wavelength was shown to improve the prediction accuracy of critical heat flux models for various fluids, particularly at elevated pressures. In addition, the most dangerous wavelength and the most rapid growth rate for viscous thin films are shown to be applicable to the minimum heat flux condition. Kim et al. touch only the most unstable wavelength for developing critical heat flux models. The critical heat flux is inversely proportional to the square root of the most unstable wavelength (Zuber, Guan et al). Here, we notice that the existing critical heat flux models make use of the Kelvin-Helmholtz instability of inviscid flows. The Kelvin-Helmholtz instability determines the maximum vapor escape velocity (Zuber) and the initial liquid macrolayer thickness (Haramura and Katto). Therefore, there is a room for improving the prediction accuracy by the help of the Kelvin-Helmholtz instability of viscous fluids. The Kelvin-Helmholtz instability arises when the different fluid layers are in relative motion. Usually, a uniform flow is considered in each fluid layer, allowing a velocity discontinuity at the interface. Therefore, in general, the Kelvin-Helmholtz instability is analyzed based on a potential flow of inviscid fluids. However, if the viscosity effect is taken into consideration, a nonuniform flow occurs due to the shear stress at the interface. The idea to incorporate the effects of fluid viscosities into the Kelvin-Helmholtz instability can be found in the viscous potential flow theory. Joseph and Liao showed that the potential (irrotational) flow of viscous fluids satisfies the Navier- Stokes equation. For the potential flow, since the vorticity is identically zero, the viscous term vanishes in the Navier-Stokes equation; the motion of fluid is governed by the Bernoulli equation. However, the viscous stresses do not vanish in general. Therefore, the viscous pressure is entered through the normal stress balance at the interface. In the viscous potential flow, the shear stress is neglected at the interface and wall, and thus there is a velocity slip at the interface. These treatments are consistent with the fact that the interface waves are induced more by pressure than by shear force. Funada and Joseph presented a viscous potential flow analysis of the Kelvin-Helmholtz instability. Funada et al. carried out a stability analysis of a circular fluid jet into another fluid. Funada and Joseph considered the capillary instability. The viscous potential flow analysis is more accurate than the inviscid flow analysis in terms of the growth rate. Therefore, the critical condition of the Kelvin-Helmholtz instability can be predicted more accurately than the inviscid flow analysis. In this study, the interfacial instabilities of viscous potential flows are applied to critical heat flux models for saturated pool boiling on infinite horizontal surfaces, with the aim of including the effects of fluid viscosities. The critical conditions of the circular jet and Kelvin- Helmholtz instabilities are incorporated into the hydrodynamic theory model and liquid macrolayer dryout model. Circular jet instabilities is incorporated into the hydrodynamic theory model. For water, the circular jet instability of an inviscid potential flow does not show any improvement of the prediction accuracy over the existing model, whereas the circular jet instability of a viscous potential flow shows a considerable improvement particularly at elevated pressures. There is one unknown variable in the revised hydrodynamic model. Interestingly, the variable is shown to vary little from fluid to fluid, except for water