# Conjugate Heat Transfer Advancements and Applications in Aerospace Engine Technology
## 3 Case Study of Conjugate Heat Transfer
### Flow Field over the C3X Vane  
The blade investigated is the NASA-C3X turbine vane, for which empirical data were sourced from the study by [PERSON] et al. [44]. The schematic of the C3X vane, accessible to the public domain, is presented in Figure 9, serving as a visual reference for the geometric configuration under examination, where the axial chord length is 0.07816 m.  
The NASA-C3X turbine vane is equipped with a strategic layout featuring ten radial cooling holes. The detailed configuration of these holes for the C3X vane is delineated in  
Figure 8: Procedure scheme. Figure 9: C3X vane sketch [44]. reference [44]. Leveraging the experimental specifications of this vane, a comprehensive computational model encompassing both the fluid dynamics and solid structures of the C3X vane has been devised, as depicted in Figure 10.  
In order to validate mesh independence, four sets of fluid grids with varying densities and one set of solid grids were generated. The solid grid was composed entirely of unstructured tetrahedral elements. The fluid grid incorporated boundary layers at both the coupling surfaces and the fluid boundary walls, ensuring that the y+ \(<\) 1 of the first layer of cells had a growth rate of 1.2, resulting in an unstructured hybrid mesh configuration. For the fluid-solid coupling interface, it is not necessary for the grids to align; instead, a correspondence relationship is established during the computation based on the \"supermesh\" methodology. The mesh distribution of the solid domain is shown in Figure 11. Four sets of fluid meshes are shown in Figure 12. The overall computational time for the model with the maximum number of grids is approximately 50 h on Hygon C86 7185 processors. In the fluid computational model, periodic boundary conditions are imposed to replicate the conditions within a single blade passage. At the inlet, the flow field is defined by the total temperature, total pressure, and the incoming flow velocity vector. Conversely, at the exit, an average static pressure boundary condition is prescribed, while all the other boundaries are presumed to behave as viscous walls. For thermophysical considerations, the model treats the working fluid as an ideal compressible gas, wherein the specific heat ratio \(\gamma\) varies, in a linear fashion, with temperature, adhering to the relationship \(\gamma=-7\times 10^{-5}\times T+1.4231\), where \(T\) denotes the temperature in Kelvin. Furthermore, both the molecular viscosity \(\mu\) and the thermal conductivity \(K\) of the fluid are articulated as temperature-dependent functions, utilizing the Sutherland's formula, as encapsulated in Equations (3) and (4):  
\[\mu(T)=\mu_{0}\bigg{(}\frac{T}{T_{0}}\bigg{)}^{\frac{3}{2}}\frac{T_{0}+S}{T+S} \tag{3}\]  
Figure 11: Solid domain mesh distribution. Figure 10: Computational model: (**a**) C3X blade model and (**b**) flow field model. \[K(T)=\lambda_{0}\bigg{(}\frac{T}{T_{0}}\bigg{)}^{\frac{3}{2}}\frac{T_{0}+S}{T+S} \tag{4}\]  
where \(\mu_{0}\) = 1.7894 \(\times\) 10\({}^{-5}\) Pa\(\cdot\)s; \(T_{0}\) = 273.11 K; \(\lambda_{0}\) = 0.0261 W/(m\(\cdot\)K); and \(S\) = 110.56. Fitting the specific heat capacity of gas \(C_{p}\) using a temperature polynomial:  
\[C_{p}=a_{0}+a_{1}T+a_{2}T^{2}+a_{3}T^{3}+a_{4}T^{4} \tag{5}\]  
where \(a_{0}\) = 957.110256; \(a_{1}\) = 0.2365234; \(a_{2}\) = 5.141 \(\times\) 10\({}^{-6}\); \(a_{3}\) = -3.39 \(\times\) 10\({}^{-9}\); and \(a_{4}\) = \(-\)6.09 \(\times\) 10\({}^{-12}\). The interface demarcating the external surfaces of the turbine blades and the inner walls of the cooling passages is classified as a coupled boundary, facilitating an integrated analysis of conjugate heat transfer. This technique allows for a precise depiction of the thermal interplay between the fluid and solid constituents, yielding a more authentic appraisal of the cooling efficacy afforded to the turbine blades.