Film cooling is used in many applications to reduce convective heat transfer to a surface. Examples are the film cooling of gas turbine combustion chambers, vanes and blades which are subjected to high heat transfer from combustion gases [Metzger et al. (1993)]. Gas which is cooler than the freestream is passed onto the external surface via small slots or rows of holes within the surface. The aim is to introduce the coolant into the boundary layer without significantly increasing turbulence and entraining additional hot freestream gas. There are three temperatures in this problem: the freestream temperature, the coolant temperature and the wall temperature. For incompressible flow with constant fluid properties, the heat transfer rate to the surface, , may be expressed in the following form:
h is a heat transfer coefficient. Taw and Tw are the adiabatic wall temperature, which is now different from the freestream temperature, and wall temperature, respectively. Alternatively, the relationship may be given in terms of the known temperature differences as,
α and β are solely functions of the flowfield. The film cooling effectiveness, η, is thus given as,
(as can be seen if in Equation (1) is made equal to zero).
Simple modelling of the process assumes that the coolant injection does not disturb the boundary layer and as a result entrainment takes place as for a turbulent boundary layer. The gas temperature within the boundary layer is determined by an enthalpy balance for coolant and entrained mainstream giving the local adiabatic wall temperature. Due account is taken of mass addition and nonuniform temperature through the boundary layer to give predictions for film cooling effectiveness at a distance x downstream from a slot of width s of the form below:
ρ, c, μ, and Pr are density, specific heat at constant pressure, viscosity and Prandtl Number, respectively. The suffices c and g refer to coolant and freestream, E is a factor which gives an increase in entrainment associated with angled injection. This formula gives satisfactory predictions for slots and low injection rates as shown in Figure 1.
Figure 1. A comparison of the predicted effectiveness with experimental results from Hartnett (1985).
Further models which are applicable to higher injection rates employ wall jet entrainment. The coolant from discrete holes often penetrate the boundary layer at high injection rates and may increase the heat transfer above the value in the absence of film cooling. Film cooling holes may be shaped so as to diffuse the coolant as it enters the boundary layer and multiple rows of holes are often employed. Due to the wide range of geometries used in film cooling, a large number of empirical correlations are present in the literature. Coolant exit mass flux, momentum flux, and velocity ratios with respect to the local freestream values are parameters which have been used in correlating experimental results. Freestream pressure gradients, turbulence and wall curvature and roughness are all factors which influence film cooling performance. An article by Hartnett (1985) gives a comprehensive review of this work and builds on an earlier article by Goldstein (1971). LeFebvre (1983) deals with combustion chamber film cooling.
Boundary layer numerical computations are also employed to predict the cooling process. For example, Crawford et al. (1980) distribute the coolant locally within the boundary layer according to prescribed rules dependent on the injection rate and angle. The local mixing length is also altered to take into account enhanced turbulence.
In recent years, the entire flowfield from the coolant plenum has been computed using the three-dimensional, Navier-Stokes equations [e.g., Fougeres and Heider (1994) and Garg and Gaugler (1994)]. The separation in the coolant holes is simulated as is the complex vortex structures which develop in the coolant jet as it turns on encountering the mainstream. Deficiencies in the turbulence modeling limit these predictions, however, such methods do indicate the way forward in design.
Crawford, M. E., Kays, W. M., and Moffat, R. J. (1980) Full Coverage Film Cooling on Flat, Isothermal Surfaces: A Summary Report on Data and Predictions, NASA CR-3219.
Fougeres, J. M. and Heider, R. (1994) Three-Dimensional Navier-Stokes Prediction of Heat Transfer with Film Cooling, ASME Paper 94-GT-14, The Hague.
Garg, V. K. and Gaugler, R. E. (1994) Prediction of Film Cooling on Gas Turbine Airfoils, ASME Paper 94-GT-16, The Hague.
Goldstein, R. J. (1971) Film Cooling, Advances in Heat Transfer. T. F. Irvine, Jr. and J. P. Hartnett), Eds., Vol. 8, Academic, New York.
Hartnett, J. P. (1985) Mass Transfer Cooling, Handbook of Heat Transfer Applications, 2nd Edition, W. M. Rohsenow, J. P. Hartnett and E. N. Ganic, Eds., McGraw-Hill.
LeFebvre, A. H. (1983) Gas Turbine Combustion, Hemisphere, New York. DOI: 10.1016/0142-727X(84)90057-2
Metzger, D. E., Kim, Y. W. and Yu, Y. (1993) Turbine Cooling: An Overview and Some Focus Topics, Proc 8th Int. Symp. Transport Phenomena in Thermal Engineering, Seoul.