## 2 Computational Models
### Chemical Mechanisms  
Figure 2 illustrates the chemical reactions during the desulfurization process. In the upper left corner of Figure 2, it indicates that the SO\({}_{2}\) gas dissolves into liquid phase and hydrates to form SO\({}_{2}\)-H\({}_{2}\)O. One proton is lost and HSO\({}_{3}\)\({}^{-}\) is generated because of the dissociation, and HSO\({}_{3}\)\({}^{-}\) has subsequent dissociation to lose another proton and form SO\({}_{3}\)\({}^{-2}\):  
\[ \begin{array}{l}\text{SO}_{2(aq)}+\text{H}_{2}\text{O}\rightleftharpoons \text{H}^{+}+\text{HSO}_{3}^{-}\\ \text{HSO}_{3}^{-}\rightleftharpoons\text{H}^{+}+\text{SO}_{3}^{2-}\end{array}  \tag{6}\]  
The subscript \"\((aq)\)\" in Equation (6) denotes the aqueous phase. The Mg(OH)\({}_{2}\) solution is a strong electrolyte, which can dissociate completely to release magnesium ions (Mg\({}_{2}\)\({}^{+}\)) and OH\({}^{-}\) ions. The neutralization takes place between OH\({}^{-}\) ions from Mg(OH)\({}_{2}\) and H\({}^{+}\) ion from Equation (6):  
\[\text{H}_{2}\text{O}\rightleftharpoons\text{H}^{+}+\text{OH}^{-} \tag{7}\]  
The nonlinear algebraic equations for Equations (6) and (7) can be listed as below:  
\[ \begin{array}{l}C_{l}^{\text{HSO}_{3}^{-}}C_{l}^{\text{H}^{+}}/C_{l}^{\text {SO}_{2}}=K_{1}\qquad K_{1}=5.78\times 10^{-3}\\ C_{l}^{\text{SO}_{3}^{-}}C_{l}^{\text{H}^{+}}/C_{l}^{\text{ISO}_{3}^{-}}=K_{2} \qquad K_{2}=4.82\times 10^{-8}\\ C_{l}^{\text{H}^{+}}C_{l}^{\text{OH}^{-}}=K_{3}\qquad K_{3}=8.88\times 10^{-14} \end{array}  \tag{8}\]  
In Equation (8), \(K_{1}\), \(K_{2}\), and \(K_{3}\) are the equilibrium constants for the corresponding reactions based on 58 \({}^{\circ}\)C [5]. The symbol \(C_{l}^{\text{H}+}\) corresponds to the molar concentration (mole/m\({}^{3}\)) of \(H^{+}\) in the liquid phase \(l\), which can also be applied to the symbols for other species. The sulfarous species, such as HSO\({}_{3}\)\({}^{-}\), SO\({}_{3}\)\({}^{2-}\), and SO\({}_{2}\) in the liquid phase, are usually added together and regarded as a single variable, as denoted by\(C_{l}^{Slurry}\)[5; 9]:  
\[C_{l}^{Slurry}=C_{l}^{\text{HSO}_{3}^{-}}+C_{l}^{\text{SO}_{3}^{2-}}+C_{l}^{ \text{SO}_{2}} \tag{9}\]  
Moreover, Equation (10) is the charge balance:  
\[C_{l}^{\text{H}^{+}}+2C_{l}^{\text{Mg}^{2+}}=C_{l}^{\text{HSO}_{3}^{-}}+C_{l} ^{\text{OH}^{-}}+2C_{l}^{\text{SO}_{3}^{2-}} \tag{10}\]  
Figure 2: This schematic illustrates the chemical reactions during the desulfurization process. As the Mg(OH)\({}_{2}\) solution has complete dissociation, the initial value \(C_{l}^{\text{OH}-}\) from the spray nozzles is double that of the \(C_{l}^{\text{Mg}2+}\). Additionally, the value of pH from the nozzles has always been kept as 7.7 in the facility. The value of \(C_{l}^{\text{Mg}2+}\) can be calculated:  
\[2\times 10^{-7.7}\times C_{l}^{\text{Mg}2+}=K_{3} \tag{11}\]  
In Equation (11), \(10^{-7.7}\) is the concentration of H\({}^{+}\) ion from the inlet of nozzle sprays, and \(K_{3}\) is equilibrium constant for neutralization as shown in Equations (7) and (8). \(C_{l}^{\text{Mg}2+}\) is calculated as 3.08 \(\times\)\(10^{-7}\) mole/m\({}^{3}\) from Equation (13). Since Mg\({}^{2+}\) does not participate in the chemical reactions, this value is regarded as constant within the flow domain and only has influence on the balance of the electric charge in Equation (10). Therefore, from Equations (8) to (10), there are six unknowns, \(C_{l}^{\text{H}^{+}}\), \(C_{l}^{\text{OH}^{-}}\), \(C_{l}^{\text{HSO}_{3}^{-}}\), \(C_{l}^{\text{SO}_{3}^{-}}\), \(C_{l}^{\text{SO}_{3}^{-}}\), \(C_{l}^{\text{SO}_{3}^{-}}\), \(C_{l}^{\text{SO}_{3}^{-}}\), and \(C_{l}^{slurry}\), within five equations. The nonlinear algebraic sets of Equations (8)-(10) can be solved numerically once \(C_{l}^{Slurry}\) is known. This value can be calculated in Equation (12):  
\[C_{l}^{Slurry}=\frac{\rho_{l}\alpha_{l}Y_{Slurry}}{W_{Slurry}} \tag{12}\]  
The molecular weight of the slurry phase \(W_{slurry}\) (kg/mole) is 80 in Equation (12) [5; 10; 33].