# Stress Corrosion Cracking Analysis of a Hot Blast Stove Shell with an Internal Combustion Chamber
## 2 Numerical Modelling
### Theoretical Corrosion Rate
#### 2.2.2 Theoretical Acid Condensation Rate  
Equation (9) provides the derivation of the mass transfer coefficient for acid condensation. \[k_{\mathrm{acid}}=\frac{h_{g}M_{\mathrm{acid}}}{C_{p,g}M_{g}y_{lm}\mathrm{Le }_{\mathrm{acid-gas}}^{2/3}} \tag{9}\]  
where \(M_{\mathrm{acid}}\) is the molecular weight of the acid, and \(\mathrm{Le}_{\mathrm{acid-gas}}\) is the Lewis number of the acid in the wet flue gas. Note that the non-condensable gas fraction contains the water vapour mole fraction until the point of water vapour condensation. As described in the previous section, two kinds of acids, sulfuric and nitric acids, were considered in this study. The sulfuric and nitric acids dew points were calculated based on the correlations from [PERSON] and Banchero [34] and Kiang [35], respectively, as follows:  
\[y_{i,\mathrm{H_{2}SO_{4}}} =\exp\left(\frac{\left[1/T_{\mathrm{dew,H_{2}SO_{4}}}\right]-2.27 6\times 10^{-3}+2.943\times 10^{-5}\ln P_{\mathrm{H_{2}O}}}{6.2\times 10^{-6} \ln P_{\mathrm{H_{2}O}}-8.58\times 10^{-5}}\right)\bigg{/}P_{\mathrm{tot}} \tag{10}\] \[y_{i,\mathrm{HNO_{3}}} =\exp\left(\frac{\left[1/T_{\mathrm{dew,HNO_{3}}}\right]-3.6614 \times 10^{-3}+1.446\times 10^{-4}\ln P_{\mathrm{H_{2}O}}}{7.56\times 10^{-6} \ln P_{\mathrm{H_{2}O}}-8.27\times 10^{-5}}\right)\bigg{/}P_{\mathrm{tot}} \tag{11}\]  
where the subscripts \(\mathrm{H_{2}SO_{4}}\) and \(\mathrm{HNO_{3}}\) denote sulfuric and nitric acids, respectively, \(T_{\mathrm{dew}}\) is the dew point temperature of the acids in K, \(P_{\mathrm{H_{2}O}}\) and \(P_{\mathrm{acid}}\) are partial pressures of water vapour and each acid in mmHg, respectively. By inverting the equations mentioned above, one can obtain an expression for the acid mole fraction at the interface, denoted by \(y_{i,\mathrm{acid}}\), based on the interfacial temperature \(T_{i,\mathrm{H_{2}O}}\) and the partial pressure \(P_{i,\mathrm{H_{2}O}}\) of the water vapour. The mole fraction at the interface of each acid is expressed as follows:  
\[\frac{1000}{T_{\mathrm{dew,H_{2}SO_{4}}}} =2.276-2.943\times 10^{-2}\ln P_{\mathrm{H_{2}O}}-8.58\times 10^{- 2}\ln P_{\mathrm{H_{2}SO_{4}}} \tag{12}\] \[\quad+6.2\times 10^{-3}\ln P_{\mathrm{H_{2}O}}\ln P_{\mathrm{H_{2}SO _{4}}}\] \[\frac{1000}{T_{\mathrm{dew,HNO_{3}}}} =3.6614-1.446\times 10^{-1}\ln P_{\mathrm{H_{2}O}}-8.27\times 10^{- 2}\ln P_{\mathrm{HNO_{3}}}\] (13) \[\quad+7.56\times 10^{-3}\ln P_{\mathrm{H_{2}O}}\ln P_{\mathrm{HNO _{3}}}\]  
where \(y_{i,\mathrm{acid}}\) is the mole fraction of each acid in %vol wet, \(T_{i,\mathrm{H_{2}O}}\) is in K, and \(P_{i,\mathrm{H_{2}O}}\) and \(P_{\mathrm{tot}}\) are in mmHg. It is important to note that, until water vapour starts condensing, its mole fraction is considered part of the non-condensable gas fraction. The Lewis number of the acid for the wet flue gas can be expressed as follows:  
\[\mathrm{Le}_{\mathrm{acid-gas}}=\frac{\alpha_{g}}{D_{\mathrm{acid-gas}}} \tag{14}\]  
where \(D_{\mathrm{acid-gas}}\) is the mass diffusivity of the acid for the wet flue gas. The fact that the order of the Lewis number of the gas is unity also can be applied to the wet flue gases with acids. It makes it possible for the mass diffusivity of the acid for the wet flue gas to be expressed as the following equation similar to the water vapour:  
\[D_{\text{acid-gas}}=\frac{\alpha_{g}}{\alpha_{\text{air}}}D_{\text{acid-air}} \tag{15}\]  
where \(D_{\text{acid-air}}\) is the mass diffusivity of the acid in the air. The mass diffusivities of sulfuric and nitric acids in the air are expressed as the following equations [36]:  
\[D_{\text{H2 SO4-gas}} =5.0032\times 10^{-6}+1.04\times 10^{-8}T+1.64\times 10^{-11}T^{2}- 1.566\times 10^{-14}T^{3} \tag{16}\] \[y_{i,\mathrm{H_{2}SO_{4}}} =\exp\left(\frac{\left[1/T_{\mathrm{dew,H_{2}SO_{4}}}\right]-2.27 6\times 10^{-3}+2.943\times 10^{-5}\ln P_{\mathrm{H_{2}O}}}{6.2\times 10^{-6} \ln P_{\mathrm{H_{2}O}}-8.58\times 10^{-5}}\right)\bigg{/}P_{\mathrm{tot}} \tag{10}\]0  
where \(T\) is in K and \(D_{\text{acid-air}}\) is in m\({}^{2}\)/s.