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Aerospace 2020, 7, 123 5 of 15 law of the ice and water and the equation of phase change at the interface between the ice and water [31]. In the EMM, the temperature profile is not calculated explicitly and the heat transfer is computed based on the assumption of a linear temperature profile for ice and water. The present MEMM is in accordance with this assumption. Figure 2 shows the EMM is widely used in icing simulations and the framework of the thermodynamics model of the MEMM. Here, there are four heat-receiving terms (aerodynamic heating Qa, kinetic energy of incoming droplets Qk, heat brought in by runback Qin, and latent heat release (only for the rime ice condition) Ql) and four heat-releasing terms (convection Qc, cooling by the incoming droplets Qd, evaporation and sublimation Qes, and radiation Qr). In the EMM model, the ambient temperature Tair and freezing temperature Tf are used as the wall surface and water film temperatures, respectively. In contrast, in the modified EMM model developed in the present study, the contact temperature Tc is used as the wall surface and water film temperatures Tw in order to consider in the heating effect from the wall. The contact temperature Tc at the interface between wall and the water film is evaluated using the impinging droplet and runback water temperatures, as Tw = m ̇imTair+m ̇inTin, (7) m ̇ i n + m ̇ i n ρwcwkwTw +ρwallcwallkwallTheater Tc= ρck+ρck , (8) w w w wall wall wall respectively, where m ̇ , ρ, c, k, and T are the mass flow rate, density, specific heat, heat conductivity, and temperature, respectively, and the subscripts im, in, air, w, heater, and wall denote the impingement, runback in, air, water, heater, and wall, respectively. Accordingly, the wall surface temperature is Tc. If Tc is lower than the freezing point, icing (phase change) occurs, and the EMM is used to calculate ice layer growth; if Tc is higher than the freezing point, icing does not occur. The runback water temperature was considered as Tc in order to model heat conduction from the heater, where it was estimated by substituting Theater into Tair in the unheated region. On the other hand, the runback water temperature is assumed to be equal to the freezing point in the EMM. For the ice thickness computation, essentially the EMM is employed, although the energy brought in by runback water Qin is estimated using Tc of the upstream neighboring computational cell, as follows; Qin = m ̇ inCw(Tc − Tf ). (9) We then reconstruct the grids and compute the flow without ice-shape smoothing because the smoothing artificially affects the icing location. Note that the one-shot computation estimated ice shape. Figure 2. Framework of thermodynamics model used in this study. The arrows represent the energy balance in the wall cell. The red and blue arrows indicate the energy into and out of the cell, respectively. )UDPHZRUN RI ([WHQGHG 0HVVLQJHU 0RGHO Air Water (Tw) Tc Heating Wall Heater (Theater) PDF Image | Anti-Icing Electric Heaters for Icing on the NACA 0012 Airfoil
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