## 2 Materials and Methods  
### Governing Equations and Solution Method  
This numerical study was based on finite volu me analysis software and user-defined functions. In the fluid region, the conservation equations of mass, momentum, energy, and species mass fractions were numerically solved to study the heat transfer and endothermic fuel pyrolysis of the China RP-3 aviation kerosene at a supercritical pressure.  
\[\
abla\cdot(\rho\overset{\rightarrow}{U})=0, \tag{1}\]  
\[\
abla\cdot(\rho\overset{\rightarrow}{U}\overset{\rightarrow}{U}\overset{ \rightarrow}{U})-\
abla P-\
abla\cdot\tau=S_{M}, \tag{2}\]  
\[\
abla\cdot(\rho\overset{\rightarrow}{U}\overset{\rightarrow}{H})-\
abla \cdot\lambda_{eff}\
abla T-\
abla\cdot\left(\tau_{eff}\overset{\rightarrow} {U}\right)=0, \tag{3}\]  
\[\
abla\cdot\left(\rho\overset{\rightarrow}{U}\overset{\rightarrow}{Y}_{i} \right)-\
abla\cdot\rho D_{i,m,eff}\
abla Y_{i}=R_{i}, \tag{4}\]  
where \(\rho\), \(\overset{\rightarrow}{U}\), \(p\), \(\tau\), \(H\), \(T\) and \(D\) are the density, velocity, pressure, viscous stress, enthalpy, temperature, and diffusion coefficient of the fluid, respectively. \(S_{M}\) is the source term of the chemical reaction, \(\lambda\) is the thermal conductivity, \(Y_{i}\) is the mass fraction of specie \(i\), and \(R\) is the molar gas constant. The subscripts \(m\) and \(\mathit{eff}\) represent the mixture and effective parameter.  
In the solid region, the heat conduction equation is numerically solved:  
\[\
abla\cdot(\lambda\
abla T)=0, \tag{5}\]  
where \(\lambda\) is the thermal conductivity of the solid wall. In addition, the temperature and heat flow boundary conditions at the interface between fluid domain and solid domain were set as follows to meet the continuity conditions:  
\[\begin{array}{c}T_{w,s}=T_{w,f}\\ q_{w,s}^{\prime}=q_{w,f}^{\prime}\end{array}, \tag{6}\]  
For the mass diffusion coefficient, [PERSON]'s empirical formula was used at first to obtain the binary mass diffusion coefficient \(D_{ij}\) at low pressure:  
\[D_{ij}=\frac{0.00143T^{1.75}}{PM_{ij}^{1/2}\left[(\sum v)_{i}^{1/3}+(\sum v)_{ j}^{1/3}\right]}, \tag{7}\]  
where \(M_{ij}\) is the mixed molar mass of species \(i\), and species \(j\); \(v\) is the diffusion volume.  
Then, the effect of pressure was considered through the corresponding-states method suggested by [PERSON], as shown below:  
\[\frac{D_{ij}p}{\left(D_{ij}p\right)^{+}}=f(T_{r},p_{r}), \tag{8}\]  
where superscript + indicates the value under low pressure; \(T_{r}\) and \(p_{r}\) are the comparison temperature and pressure, respectively.  
Finally, based on the binary diffusion coefficient, the effective diffusion coefficient was calculated by the following formula:  
\[D_{i,m}=\frac{1-x_{i}}{\sum\limits_{j\
eq i}x_{i}/D_{ij}}, \tag{9}\]  
where \(x_{i}\) is the mole fraction of species \(i\).  
In this paper, \(\kappa\)-\(\omega\)\(SST\) turbulent model was adopted to calculate the internal turbulent flow and heat transfer for its high accuracy and good numerical stability to the boundary [29; 30]. The fuel density was calculated with the Peng-Robinson equation of state because the physical properties of fuel change dramatically with temperature under supercritical conditions [31; 32]:  
\[P=\frac{RT}{V-b}-\frac{a}{V+2 bV-b^{2}}, \tag{10}\]  
where \(V\) is the volume, and \(a\), \(b\) are parameters.  
The empirical method proposed by [PERSON] was selected to estimate viscosity and thermal conductivity of hydrocarbon fuel [33]. The calculation method of the above thermophysical properties was entered into the commercial CFD software FLUENT through the user-defined function UDF. In the numerical calculation, the solver was based on the finite-volume scheme, and the SIMPLEC algorithm.