# Stress Corrosion Cracking Analysis of a Hot Blast Stove Shell with an Internal Combustion Chamber
## 2 Numerical Modelling
### Theoretical Corrosion Rate
#### 2.2.1 Theoretical Water Condensation Rate  
A direct relationship between the mass transfer coefficient of water, represented by \(k_{\rm H_{2}O}\), and the concentration difference between the bulk flow and the interface is often used to estimate the water vapour condensation rate. The water vapour condensation rate \(\dot{m}_{\rm cd,H_{2}O}\) is calculated by the following ordinary differential equation:  
\[d\dot{m}_{\rm cd,H_{2}O}=k_{H_{2}O}(y_{H_{2}O}-y_{i})dA \tag{3}\]  
where \(y_{H_{2}O}\) and \(y_{i}\) are the mole fractions of the water vapour in the flue gas and at the interface, respectively, and \(A\) is the heat transfer area. By employing the Lewis relation, the mass diffusion of water and heat transfer were correlated, leading to the calculation of the mass transfer coefficient of the water, \(k_{H_{2}O}\). The mass transfer coefficient of the water is calculated as follows:  
\[k_{H_{2}O}=\frac{h_{g}M_{H_{2}O}}{C_{p,g}M_{g}y_{\rm lm}\rm Le_{H_{2}O-gas}^{2/ 3}} \tag{4}\]  
where \(h_{g}\) is the heat transfer coefficient of the wet flue gas side, \(M_{H_{2}O}\) and \(M_{g}\) are the molecular weights of the water and the wet flue gas, respectively, \(C_{p,g}\) is the specific heat of wet flue gas, \(y_{\rm lm}\) is the log-mean mole fraction difference of non-condensable gas, and \(\rm Le_{H_{2}O-gas}\) is the Lewis number of the water vapour in the wet flue gas. \(y_{\rm lm}\) is expressed as the following Equation (5):  
\[y_{\rm lm}=\frac{y_{\rm ni}-y_{\rm nb}}{\ln(y_{\rm ni}-y_{\rm nb})} \tag{5}\]  
where \(y_{\rm ni}\) and \(y_{\rm nb}\) are the mole fractions of non-condensable gas at the interface and bulk flow, respectively. The convective heat transfer was estimated with the analytic adiabatic Y-plus method [32]. The Lewis number of the water vapour in the wet flue gas \(\rm Le_{H_{2}O-gas}\) is expressed as Equation (6):  
\[\rm Le_{H_{2}O-gas}=\frac{\alpha_{g}}{D_{H_{2}O-gas}} \tag{6}\]  
where \(D_{H_{2}O-gas}\) is the mass diffusivity of the water vapour in the wet flue gas, and \(\alpha_{g}\) is the thermal diffusivity of the wet flue gas. Typically, the Lewis number for gases has the order of unity. From the given statement, it can be concluded that changes in the structural and thermal profiles occur at similar rates in flue gases that experience both heat transfer and mass diffusion processes simultaneously. Thus, it is possible to estimate mass diffusivity by utilising the following equation based on other properties:  
\[D_{H_{2}O-gas}=\left(\frac{\alpha_{g}}{\alpha_{\rm air}}\right)D_{H_{2}O-air}. \tag{7}\]  
where \(\alpha_{\rm air}\) is the thermal diffusivity of air, and \(D_{H_{2}O-air}\) is the mass diffusivity of the water vapour in the air. \begin{table}
\begin{tabular}{c c c} \hline
**Compound** & **Temperature Range (\({}^{\circ}\)C)** & **Reaction Mechanism** \\ \hline \multirow{4}{*}{Sulfuric acid} & \(T>1200\) & \(S(\rm g)+O_{2}(\rm g)\rightarrow\rm SO_{2}(\rm g)\) \\  & \(400<T\leq 1200\) & \(\rm SO_{2}(\rm g)+\frac{1}{2}O_{2}(\rm g)\rightleftarrows SO_{3}(\rm g)\) \\  & \(120\leq T\leq 400\) & \(\rm SO_{3}(\rm g)+H_{2}O(\rm g)\to H_{2}SO_{4}(\rm g)\) \\  & \(T<T_{\rm dew,H_{2}SO_{4}}\) & Condensation of \(\rm H_{2}SO_{4}\) \\ \hline \multirow{4}{*}{Nitric acid} & \(650<T<1200\) & \(\rm N_{2}(\rm g)+O_{2}(\rm g)\to NO(\rm g)\) \\  & \(150<T\leq 650\) & \(\rm 2 NO(\rm g)+O_{2}(\rm g)\rightleftarrows 2 NO_{2}(\rm g)\) \\ \cline{1-1}  & \(120\leq T\leq 400\) & \(\rm 2 NO(\rm g)+O_{2}(\rm g)\to 2 NO_{2}(\rm g)\) \\ \cline{1-1}  & \(T<T_{\rm dew,HNO_{3}}\) & \(\rm NO_{2}(\rm g)+H_{2}O(\rm g)\to 2 HNO_{3}(\rm 1)+NO(\rm g)\) \\ \cline{1-1}  & & Condensation of \(\rm HNO_{3}\) \\ \hline \end{tabular}
\end{table} 
Table 3: Reaction mechanisms considered in the CFD analysis. A vapour pressure equation based on the Antoine equation, which explains the correlation between vapour pressure and temperature of pure components, was used to determine the interfacial mole fraction of water vapour.