## 2 Convective Heat Transfer Coefficients
### Stator Vanes and Disc Blades  
In a first order approximation, the boundary layer over a stator vane or disc blade may be represented by the boundary layer over a flat plane. For a laminar flow the velocity and temperature distributions are given by the Blasius solution [15]. In this case the convective heat transfer correlation is  
\[h(x)=0.332\frac{k_{g}}{x}Re_{x}^{1/2}Pr^{1/3}, \tag{6}\]  
where \(x\) is the longitudinal distance from the leading edge, \(k_{g}\) is the gas thermal conductivity, \(Re\) and \(Pr\) are the nondimensional Reynolds and Prandtl numbers calculated by  
\[Re=\frac{\rho_{g}u_{g}x}{\mu_{g}}\ \text{and}\ Pr=\frac{\mu_{g}Cp}{k_{g}}. \tag{7}\]where \(\rho_{g}\) is the gas density, \(u_{g}\) is the freestream gas velocity at \(x\), \(\mu\) is the gas dynamic viscosity, and \(Cp\) is the gas specific heat at constant pressure. For turbulent flow the heat transfer coefficient is given by [16]. \[h(x)=0.0296\frac{k_{g}}{D}(Re_{x}^{0.8}Pr^{1/3}). \tag{8}\]  
At the leading edge, the heat transfer coefficient can be estimated based on the flow over a cylindrical surface  
\[h_{a}=a\bigg{[}1.14\frac{k_{g}}{D}(Re_{D})^{1/2}Pr^{0.4}\bigg{]}, \tag{9}\]  
where \(a\) is a turbulence enhancement factor, \(D\) is the leading edge diameter and \(Re_{D}\) is the Reynolds number based on the leading edge diameter and the approaching freestream velocity. The average heat transfer coefficient over the cylindrical surface at the leading edge is given by  
\[h_{av}=0.7h_{a}. \tag{10}\]  
The gas properties are computed as a function of temperature. In the present model the Prandtl number is assumed to be constant and equal to \(Pr=0.7\). The viscosity varies with temperature according to the power law [17],  
\[\mu_{g}=\mu_{0}\bigg{(}\frac{T_{g}}{T_{0}}\bigg{)}^{0.67}, \tag{11}\]  
where the reference viscosity of air is \(\mu_{0}=1.5\times 10^{-5}\) at the reference temperature of \(T_{0}=273\) (K). For combustion gases the reference viscosity is \(\mu_{g}=4.23\times 10^{-5}\) (kg/(m.s)) at the reference temperature of \(T_{0}=1100\) (K). The conductivity is computed from the definition of Prandtl number and the density is calculated by the perfect gas law  
\[k_{g}=\frac{\mu_{g}Cp}{Pr}\text{ and }\rho_{g}=\frac{P}{RT_{g}}. \tag{12}\]  
Equations (6), (8), and (10) are used to estimate the heat transfer coefficient distribution over the turbine blades and vanes. The same correlations apply to compressor blades and diffuser vanes. At the leading edge, the flow is always laminar and Equations (6) and (10) are used. Close to the trailing edge, the flow is usually turbulent and Equation (8) should be used. In gas turbine blades and vanes, the laminar and turbulent regions are separated by a transition zone that has a significant length compared to the chord. On the suction surface for both blades and vanes, there is a strong adverse pressure gradient, which hastens the transition to turbulence. In the present suction surface model, it is considered that the transition region starts at 10% of the chord, considering the leading edge as the starting point. It is also considered that the transition region ends at 40% of the chord. From this point on, the flow is assumed to be turbulent. On the pressure surface, the strong favorable pressure gradient close to the leading edge forces the flow to remain laminar. On the other hand, the freestream turbulence levels from the combustion chamber and the strong concave surface curvature try to transition the flow to a turbulent regime. These opposing effects result in a long transition region over the stator vanes. The assumed model in this work considers that on the pressure surface of the stator vanes, the transition starts at 20% of the chord length, starting from the leading edge. The flow is considered fully turbulent at 90% of the chord length. On rotor blades, the flow is considered fully turbulent at 30% of the chord.