INTEGRAL ANALYSES OF THE CONVECTIVE HEAT TRANSFER AROUND ICE PROTECTED AIRFOILS WITH NON-ISOTHERMAL SURFACES

The heated airfoil operating under icing conditions has some important characteristics that differentiates the problem from the case of adiabatic airfoil subjected to ice growth. In presence of thermal ice protection, the hypothesis of flow over isothermal surfaces, which is assumed by most classic icing codes, may not represent the operation satisfactorily. The present paper implemented successfully two modeling strategies that considers streamwise surface temperature variations in thermal boundary-layer evaluation by integral procedure: 1) solution of the approximated enthalpy thickness integral equation assuming flow over a non-isothermal surface; 2) application of superposition principle to thermal boundary-layer solutions to represent the effects of flow over a surface with non-uniform temperature distribution. The numerical results were compared with isothermal model results as well as the experimental results of flat plate in a wind tunnel and two NACA aifoils, the 0012 and the 652-0016, operating in icing tunnel under clear air and icing conditions. The streamwise surface temperature gradient, water evaporation rate variation and the presence of laminar-turbulent transition, when occurring within the protected area, are effects that are represented adequately by the mathematical models. INTRODUCTION Some authors [Gelder and Lewis, 1951, Sogin, 1954, Gent et al., 2003] have observed that the convective heat transfer coefficient is one of of most important and difficult parameter in aircraft wing anti-ice thermal performance estimation. This relevance was also observed in ice shape prediction [Gent et al., 2000, Stefanini et al., 2007]. Most works, found in present research, applied boundary layer integral analysis to icing cylinders [Makkonen, 1985] and airfoils [Wright, 1995, Guffond and Brunet, 1988, Gent, 1990, Wright et al., 1997, Gent et al., 2000], where the main objective is to evaluate ice growth and predict its shape. Those papers use mathematical models that assumes laminar and turbulent flows over isothermal, fully rough icing surface with moderate pressure gradient and no evaporation effects. The laminarturbulent transition is considered to occur abruptly, i.e., the flows goes from fully laminar to fully turbulent at the onset position. Few works applied the momentum and thermal boundary layer integral models to anti-ice simulation satisfactorily. Morency et al. [1999b] developed the numerical code CANICE A that evaluates the heat transfer coefficient considering laminar flow over isothermal surface[Smith and Spalding, 1958], turbulent flow over smooth and non-isothermal surface[Ambrok, 1957] and abrupt laminar-turbulent transition. Same authors developed other version of the code, CANICE B, and use the experimental data of heat transfer coefficient . Only this second code represented satisfactorily the surface temperature numerical results. The transient ice protection code developed in Henry [1989] applied the Makkonen [1985] boundary-layer model that is used in ONERA2D icing code[Guffond and Brunet, 1988]. Al-Khalil et al. [2001] employed experimental data in his ANTICE code simulations. Neither LEWICE [Wright, 1995] nor ANTICE heat transfer coefficients were used in validation process. Gent et al. [2003] reported difficulties when applying TRAJICE2 boundary layer model[Gent, 1990, Gent et al., 2000, Makkonen, 1985] to rotorcraft ice protection systems. The authors got overestimated results and recommended more research about heat transfer coefficient evaluation[Gent et al., 2003]. Other works tried differential modelling strategies to improve predictions accuracy since integral analyses appeared to show significant limitations. So did Henry [1992], who used a two-dimensional finite difference code to evaluate heat transfer in ice protection transient operation. In the same way, Morency et al. [1999a] published the CANICE FD version that evaluates hair distribution with Cebeci’s finite difference code[Cebeci and Bradshaw, 1984]. Those works provided numerical results with smaller deviation to experimental data than their integral counterparties. This fact may indicate that finite difference strategies could represent more adequately momentum, heat and mass transfer phenomena around heated airfoils operating under icing conditions. A three-dimensional finite element flow solver has been developed by Croce et al. [2002] to estimate conjugated conduction and convection heat transfer on thermally protected aircraft wings. OBJECTIVE The objective of the present paper is to evaluate the convective heat transfer around ice protected airfoils by adopting different types for momentum and thermal boundary-layer analysis: 1) nonisothermal integral procedures as developed by Ambrok [1957] for laminar and turbulent boundarylayers, which has been applied to aircraft anti-icing by Silva [2002], Silva et al. [2003]; 2) superposition of isothermal integral boundary layer evaluations in laminar [Lighthill, 1950] and turbulent [Reynolds et al., 1958a] regimes. The numerical results are compared with boundary-layer isothermal model results as well as experimental data to assess applicability of the modeling strategies proposed herein. AIRFOIL ANTI-ICE MATHEMATICAL MODEL This paper uses the anti-ice thermal model developed by Silva et al. [2007a,b], whom briefly described the mathematical model, presented some numerical code results and compared with experimental data as well as other codes results. The anti-ice system operation simulation applies the First Law of Thermodynamics to liquid water flow and airfoil surface and also the Conservation of Mass and Momentum to liquid water flow. The wetness factor estimation, by water film breakdown and rivulets formation, was based in other work [Silva et al., 2006] plus the assumption of constant rivulets spacing. Solvers The anti-ice simulation problem requires the solution in a sequence of steps : (step 1) velocity and pressure fields around the airfoil; (step 2) droplet trajectories; (step 3) momentum and thermal boundary layers to obtain the coupled heat and mass transfer over the airfoil solid surface and liquid water flow; (step 4) First Law of Thermodynamics to the liquid water and airfoil solid surface plus the Conservation of Mass and Momentum to the liquid water flow (film and rivulets) over the airfoil. Both flow field around airfoil and local collection efficiency data were provided by external numerical codes (steps 1 and 2). The momentum and thermal boundary-layer are evaluated (step 3) in order to estimate the heat and mass transfer around airfoil over non-isothermal and transpired surfaces with a smooth laminar-turbulent transition occurrence. With data from previous steps, the anti-ice mathematical model (step 4) is able to predict operational parameters like solid surface temperatures, runback mass flow rate and convection heat transfer coefficient distributions along the airfoil solid surface. The present paper presents modeling strategies for the thermal boundary-layer (step 3). All the other models (steps 1,2 and 4 and also momentum boundary-layer) are kept as presented in previous works and are not included herein [Silva, 2002, Silva et al., 2003, 2006, 2007a]. Domains Figure 1 shows the coordinates system and the five domains used in the present anti-ice mathematical model, which are: I) free stream flow; II) gaseous flow; III) momentum or thermal boundary layers; IV) water film flow; V) solid surface. By using this strategy for domain division, the mathematical model can be organized and simplified. Only the convective heat transfer estimation is discussed herein, since the objective of the present paper is to assess alternatives for modeling strategies of the domain (III). Airfoil solid surface (V) Water film flow (IV) y