## 2 Methodology
### Mathematical Formulation  
As shown in Figure 2b, the relational formula for the heat plate temperature and the spar-cap upper surface temperature for each section separated by the thickness offset was established. Figure 3 is a graph using \(\overline{T}_{S,N}\), which is the average temperature of the spar-cap top surface of each section from case 1 to case 6, and \(\overline{T}_{H,N}\), which is the average temperature of the heat panel. \(\overline{T}_{S,N}\) and \(\overline{T}_{H,N}\) are the average values for each section of the temperature distribution calculated by the heat transfer simulation. The obtained formula for each section shown in Figure 3 is a linear equation as with Equation (1). This equation can be used to determine the \(T_{H,N}\) that can derive the desired spar-cap top face temperature for each section. \(B_{N}\), which is a coefficient of \(T_{M,N}\), and \(D_{N}\), which is a constant, can be found in the process of establishing a trend equation. \[T_{S,N}=B_{N}T_{H,N}+D_{N} \tag{1}\]  
Figure 1: Method to improve temperature unevenness due to shape and thickness of spar-cap using reference models. Figure 3: Obtained formula between heat panel temperature and top surface temperature of the spar-cap: (**a**) aluminum mold; (**b**) GFRP mold. Figure 2: Spar-cap modeling for 20 kW wind turbine: (**a**) full-scale geometry model of heat transfer simulation; (**b**) schematic diagram of spar-cap modeling that defines dimensions and sections. The governing equation for numerical analysis of heat transfer is a complex heat transfer consisting of heat conduction, convection, and radiation. The heat transfer simulation is performed by Equations (2)-(3) and (5)-(6). Equation (2) is a typical equation used to calculate the heat conduction inside the material. This calculates the heat conduction from the heat panel to the spar-cap top face. Thermal conductivity \(k\), specific heat C, and density \(\rho\) are thermophysical properties of the material, and \(\dot{e}_{gen}\) is the amount of heat generated from the heat panel. Assuming a heat panel is a volumeless surface, \(\dot{e}_{gen}\) is defined as heat [W/m\({}^{2}\)]. \[\
abla^{2}T+\frac{\dot{e}_{gen}}{k}=\frac{k}{C\rho}\frac{\partial T}{ \partial t} \tag{2}\]  
Equation (3) can calculate the heat loss due to convection heat transfer on the spar-cap surface. For the heat transfer coefficient \(h\), a method of calculating the total heat transfer coefficient of the spar-cap surface through a temperature measurement experiment may be common. However, calculating the heat transfer coefficient through experiments requires a large amount of equipment and human resources in a highly controlled environment. In addition, the size of the spar-cap is very large and there are many unpredictable environment variables, so \(h\) uses generalized values in natural convection conditions. \[\dot{Q}_{conv}=hA_{s}\ (T_{s}-T_{\infty}) \tag{3}\]  
Equation (4) serves to obtain the heat transfer coefficient \(h_{r}\) that determines the heat flux at the contact interface between the mold and the spar-cap. If the thickness and thermal conductivity of the epoxy coated on the contact interface are known, the heat transfer coefficient \(h_{r}\) of the contact interface can be obtained [22]. \[h_{r}=k_{r}\ /\ \delta \tag{4}\]  
Radiation is often omitted in the process of simplifying the heat transfer simulation. However, in this model, the temperature difference between the spar-cap surface and the shielding film surface made of vinyl (which can conduct heat with the outside) is large, so the radiation is too large to ignore. Therefore, radiation calculates the mutual heat exchange between the spar-cap surface and the shield surface via Equation (5) called the view factor (VF) method. In Equation (5), the emissivity \(\varepsilon_{1}\), \(\varepsilon_{2}\), and temperature \(T_{1}\), \(T_{2}\) are the inherent characteristics of the object, then the calculation of radiation by distance and direction is performed.