# Optimization Design of Lattice Structures in Internal Cooling Channel of Turbine Blade
## 2 Functions of Heat Transfer and Mechanical Performances
### The Function of Heat Transfer Performance  
In the analysis of cooling performance, boundary condition of a constant heat flux was employed [37], which was consistent with the boundary condition of the target surface of the LS selected. In order to characterize the heat transfer performance of LS, the \(Nu\) formula was derived according to the fin method [38]. \(Nu\) is used to evaluate heat transfer performance. Functional relationship between geometric parameters of LS and \(Nu\) is established:  
\[Nu=\frac{hD_{h}}{k_{f}} \tag{1}\]  
\[h=\frac{q}{T_{w}-T_{in}} \tag{2}\]  
\[D_{h}=\frac{2S_{y}H}{S_{y}+H} \tag{3}\] \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Parameter** & **Value** & **Parameter** & **Value** \\ \hline Material (used in simulation setting) & & Material (used in experimental validation, printed) & \\ Density & IN 718 & Density & PLA \\ Young’s modulus & 8240 kg/m\({}^{3}\) & Young’s modulus & 1240 kg/m\({}^{3}\) \\ Poisson’s ratio & 199.9 GPa & Poisson’s ratio & 4000 MPa \\ Geometric parameters & 0.3 & Geometric parameters & 0.3 \\ Thickness of core walls & 2 mm & Core height (\(H\)) & 40 mm (initial) \\ Core length (\(S_{y}\)) & 80 mm & Diameter of member (\(D\)) & 2 mm (initial) \\ Core width (\(S_{x}\)) & 80 mm & Inclination angle of member (\(\omega\)) & 45\({}^{\circ}\) (initial) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Geometric parameters of LS core and its material information. Figure 2: PLS and XLS used in internal cooling channels of blade. where \(h\) is the overall convective heat transfer coefficient of structure, \(D_{h}\) is hydraulic diameter, \(k_{f}\) is the thermal conductivity of air, \(q\) is the average heat flux of the target surface, \(T_{w}\) is the average temperature of the target surface, and \(T_{in}\) is the average inlet temperature of air. Heat transfer of each member in LS channels was considered as the same, so a single member in LS was analyzed. The temperature control equation is as follows:  
\[\frac{d^{2}T_{s}(\eta,n)}{d\eta^{2}}-\frac{h_{\infty}P}{k_{s}S}\Big{(}T_{s}( \eta,n)-T_{f}(n)\Big{)}=0 \tag{4}\]  
where \(\eta\) is the coordinates along member direction; \(n\) is the number of LS cores along the flow direction in LS channel; \(T_{f}(\eta,n)\) is the temperature of the \(\eta\) coordinate in member of the \(n\)th row of LS core; \(T_{f}(n)\) is the average temperature of air in the \(n\)th row of LS core; \(h_{\infty}\) is the heat transfer coefficient between member surface, and according to the research in literature [39], \(h_{\infty}=1.19(Re)^{0.6}\); \(P\) is the section perimeter of the member; \(S\) is the cross-sectional area of the member; \(k_{s}\) is the thermal conductivity of the member. Initial heat flux (\(q_{0}\)) flowed into member from the target surface is calculated:  
\[-k_{s}\frac{dT_{s}}{d_{\eta}}\Big{|}_{\eta=0}=q_{0} \tag{5}\]  
where \(q_{0}\) is the heat flux coming from the member in the place of \(\eta=0\). By combining Equations (4) and (5):  
\[T_{s}(\eta,n)-T_{f}(n)=\frac{q_{0}}{k_{s}m}\cdot\frac{\cosh(m(l-\eta))}{\sinh( ml)} \tag{6}\]  
where \(l\) is the length of a single member; \(l=H/\sin\omega\), and \(H\) is the height of the LS channel; \(m=2\sqrt{h_{\infty}/(k_{s}D)}\), with the dimension of L\({}^{-1}\). When \(\eta=0\), the temperature of the member is equal to that of the wall, i.e., \(T_{s}(\eta,n)=T_{f}(0,n)=T_{w}(n)\), according to Equation (6):  
\[T_{w}(n)-T_{f}(n)=\frac{q_{0}}{k_{s}m}\cdot\frac{\cosh(ml)}{\sinh(ml)} \tag{7}\]  
From the law of conservation of energy, the following function is obtained:  
\[\rho_{f}S_{y}Hu_{m}C_{p}\Big{[}T_{f}(n)-T_{f}(0)\Big{]}=qS_{y}S_{x}n \tag{8}\]  
where \(\rho_{f}\) is air density, \(u_{m}\) is the average velocity of air, \(u_{m}=Re\mu/\rho_{f}D_{h}\); \(Re\) is Reynolds number, \(\mu\) is dynamic viscosity of air, and \(C_{P}\) is the specific heat capacity of air.