# Stress Corrosion Cracking Analysis of a Hot Blast Stove Shell with an Internal Combustion Chamber
## Appendix A Computational Fluid Dynamics Model of the Hot Blast Stove  
Here is a brief description of the computational fluid dynamic model of the hot blast stove. For a detailed description, refer to the authors' previous publication [28]. The CFD model consisted of five models, which are the fluid flow, species transport, chemical reaction, radiation, and porous media models. There are three assumptions in the fluid flow model: (a) the fluid is Newtonian fluid, (b) it follows an incompressible ideal gas law, and (c) turbulent flow regime. Based on the assumptions, the continuity, linear momentum, and energy equations are derived as follows:  
\[\frac{\partial\rho}{\partial t}+\frac{\partial\rho u_{i}}{\partial x_{i}}=0 \tag{1}\]  
\[\frac{\partial(\rho u_{i})}{\partial t}+\frac{\partial(\rho u_{i}u_{j})}{ \partial x_{j}}=-\frac{\partial p}{\partial x_{i}}+\frac{\partial}{\partial x _{j}}\Bigg{[}\mu\Bigg{(}\frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial u _{j}}{\partial x_{i}}\Bigg{)}-\rho\overline{u_{i}^{\prime}u_{j}^{\prime}} \Bigg{]}+\rho g_{i} \tag{2}\]  
\[\frac{\partial(\rho c_{p}T)}{\partial t}+\frac{\partial(\rho c_{p}u_{i}T)}{ \partial x_{i}}=\frac{\partial}{\partial x_{i}}\bigg{(}\lambda\frac{\partial T }{\partial x_{i}}-\rho c_{p}\overline{u_{i}^{\prime}T^{\prime}}\bigg{)} \tag{3}\]  
where \(t\) is time, \(\rho\) is the fluid density, \(u_{i}\) is the velocity component, \(x_{i}\) is the Cartesian coordinate component, \(p\) is the pressure, \(\mu\) is the dynamic viscosity, \(g_{i}\) is the gravity component, \(T\) is the fluid temperature, \(c_{p}\) is the fluid specific capacity, and \(\lambda\) is the fluid thermal conductivity. The standard \(k\)-\(\varepsilon\) model with the standard wall function was utilised for the turbulent flow of the hot blast stove. The transport equations are described as follows:\[\frac{\partial(\rho k)}{\partial t}+\frac{\partial(\rho ku_{i})}{ \partial x_{i}}=\frac{\partial}{\partial x_{i}}\bigg{[}\bigg{(}\mu+\frac{\mu_{t} }{\sigma_{k}}\bigg{)}\frac{\partial k}{\partial x_{i}}\bigg{]}+G_{k}+G_{b}-\rho\varepsilon \tag{10}\] \[\frac{\partial(\rho\varepsilon)}{\partial t}+\frac{\partial(\rho \varepsilon u_{i})}{\partial x_{i}}=\frac{\partial}{\partial x_{i}}\bigg{[} \bigg{(}\mu+\frac{\mu_{t}}{\sigma_{\varepsilon}}\bigg{)}\frac{\partial \varepsilon}{\partial x_{i}}\bigg{]}+C_{1\varepsilon}\frac{\varepsilon}{k}(G_ {k}+C_{3\varepsilon}G_{b})-C_{2\varepsilon}\rho\frac{\varepsilon^{2}}{k} \tag{11}\]  
where \(k\) and \(\varepsilon\) are the turbulent kinetic energy and its dissipation, \(C_{1\varepsilon}\), \(C_{2\varepsilon}\), and \(C_{3\varepsilon}\) are the model constants. \(\sigma_{k}\) and \(\sigma_{\varepsilon}\) are the turbulent Prandtl numbers for \(k\) and \(\varepsilon\). \(\mu_{t}\) is the turbulent viscosity, respectively. \(G_{k}\) and \(G_{b}\) are the generation of turbulence kinetic energy due to the mean velocity and buoyancy, respectively. \(C_{3\varepsilon}\) determines the buoyancy effect on \(\varepsilon\). \(\mu_{t}\), \(G_{k}\), \(G_{b}\), and \(C_{3\varepsilon}\) are defined as follows:  
\[\mu_{t} =\rho C_{\mu}\frac{k^{2}}{\varepsilon} \tag{12}\] \[G_{k} =-\rho\overline{u_{i}^{\prime}u_{j}^{\prime\prime}}\frac{\partial u _{j}}{\partial x_{i}}\] \[G_{b} =-g_{i}\frac{\mu_{t}}{\rho\text{Pr}_{t}\frac{\partial\rho}{ \partial x_{i}}}\] \[C_{3\varepsilon} =\tanh\abs{u_{g}/u_{gp}}\]  
where \(C_{\mu}\) is the model constant, and \(\text{Pr}_{t}\) is the turbulent Prandtl number for energy. \(u_{g}\) and \(u_{gp}\) are the velocity components parallel and perpendicular to the gravitational vector, respectively. The species transport model can analyse the mixing characteristics between the fuel gas and air.