## 4 Mathematical Models of Simulation Used During the Search for the Optimal Solution
### Mathematical Models  
Equations to allow developing simple mathematical models for simulating the vapour-liquid equilibrium models.  
Total Mass (Molar) Balance:  
\[F=L+V \tag{1}\]  
Mass (Molar) Balance for Component:  
\[Ff_{j}=Ll_{j}+Vv_{j}\qquad j\in\{1,2\} \tag{2}\]  
\(L/F\) Ratio:  
\[\phi=\frac{L}{F} \tag{3}\]  
Unitary Mass (Molar) Balance:  
\[\sum_{j=1}^{2}v_{j}=1\quad\text{and}\qquad\sum_{j=1}^{2}l_{j}=1 \tag{4}\]  
Phase Equilibrium Relationship:  
\[v_{j}=K_{j}l_{j}\quad j\in\{1,2\}\qquad\text{or}\qquad l_{j}=v_{j}/K_{j}\quad j \in\{1,2\} \tag{5}\]  
\(K\)-value (the vapour-liquid equilibrium is established by [PERSON]'s law [PERSON] (1960)):  
\begin{table}
\begin{tabular}{c l} \hline \hline
**Variable** & **Definition** \\ \hline \(F\) & Molar flow rate of the feed to the flash distillation vessel. \\ \(L\), \(V\) & Liquid or vapour molar flow rates leaving the flash distillation vessel. \\ \(f_{j}\) & Overall molar composition of the compound \(j\) in the binary system \\  & or in the feed to the flash distillation vessel. \\ \(l_{j}\), \(v_{j}\) & Molar fractions of the chemical species \(j\) in the liquid or vapour of the binary system, \\  & or in the liquid or vapour stream that leaves the flash distillation vessel. \\ \(K_{j}\) & Vapour-liquid saturation ratio or \(K\)-value of compound \(j\). \\ \(p_{j}^{*}\) & Vapour pressure of chemical species \(j\). \\ \(p\) & Total system pressure. \\ \(T\) & System temperature. \\ \(A_{j}\), \(B_{j}\), \(C_{j}\) & Constants A, B and C of [PERSON]’s equation [PERSON] (1998) \\  & for vapour pressure calculation of the chemical species \(j\). \\ \hline \hline \end{tabular}
\end{table}
Table 1: Variables considered in the model.  
\[K_{j}=P_{j}^{*}/P\qquad j\in\{1,2\} \tag{6}\]  
Physical Properties: Vapour pressure of chemical species \(j\) at a given temperature \(T\):  
\[P_{j}^{*}=\exp[A_{j}-B_{j}/(T+C_{j})]\qquad j\in\{1,2\} \tag{7}\]  
Equation (8) must be satisfied at the bubble point to calculate its temperature \(T_{BP}\) and the molar fraction of the first vapour produced, i.e., \(v_{j}\). In these conditions, the vapour produced is in equilibrium with the liquid that has a composition \(l_{j}=f_{j}\). This equation is built by combining (4)-(7).  
\[\sum_{j=1}^{2}v_{j}=1=\sum_{j=1}^{2}(K_{j}l_{j}) \tag{8}\]  
Similarly, the condition in (9) must be satisfied at the dew point to calculate its temperature \(T_{DP}\) and the composition of the first drop of liquid produced, i.e., \(l_{j}\). In these situations, the liquid produced is in equilibrium with the vapour phase, which has a molar composition \(v_{j}=f_{j}\). This equation is generated by combining (4)-(7).  
\[\sum_{j=1}^{2}l_{j}=1=\sum_{j=1}^{2}(v_{j}/K_{j}) \tag{9}\]  
Equations (8) and (9) are solved for temperature \(T\), the bubble point or dew point, using the bisection numerical method [41].  
Finally, for the flash distillation calculations, the condition in (10) must be satisfied in order to calculate the \(\phi=L/F\) ratio and the molar fraction of the liquid formed, i.e., \(l_{j}\). Under these conditions, the vapour phase formed is in equilibrium with the liquid phase which has a molar fraction \(l_{j}\). Equation (10) is described by combining (1)-(7). It is solved for \(\phi\), also using the bisection numerical method [41].  
\[\sum_{j=1}^{2}l_{j}=1=\sum_{j=1}^{2}\frac{f_{j}}{\phi+(1-\phi)K_{j}} \tag{10}\]