## 5 Illustrative Examples
### Mathematical Model
#### 5.1.2 Binary Distillation Column  
A distillation column is shown in Figure 14 with a feed flow \(F_{f}\)(mol/s) and its composition \(z_{f}\) in the center of the column. There are upper and lower products, \(x_{D}\)\(y\)\(x_{B}\). The system has two heat exchangers: a kettle at the bottom part that generates a rising steam \(V_{f}\) and a condenser in the upper part that cools the steam and generates the upper product and the reflux \(L_{n}\). Part of the liquid is sent to the reflux tank where there is a liquid retention of mass \(M_{D0}\)(kg) with a composition \(x_{D}\). The reflux is pumped to the top plate \(N_{T}\) at a rate of \(R_{0}\) and ejected at the rate of \(D_{f}\). The distillation column has a plate structure to optimize the heat transfer between the liquid and the steam by maximizing the contact surface between them. In the base, the heavy products are removed at the rate of \(B_{f}\) with a composition \(x_{B}\) and retention \(M_{B0}\)(kg). Boiling steam is generated in a reboiler at the rate of \(V_{0}\). * Equilibrium phases In this work, a binary system (two components) is chosen with a constant relative volatility throughout the column and trays without losses (100% efficient); that is, the steam that leaves the trap is in an equilibrium state with the liquid in the trap. \[Y_{n}=\frac{\alpha X_{n}}{1+X_{n}(\alpha-1)}\] (37) A simple relation vapor-liquid equilibrium of (37) can be used for each tray. where
* \(Y_{n}\): steam composition on plate \(n\);
* \(X_{n}\): liquid composition of plate \(n\);
* \(\alpha\): relative volatility. * Hydraulic balance For the hydraulic balance, the dump in Figure 15 is considered, which allows the liquid to overflow from each tray in the distillation column. Figure 14: Schematic diagram of the distillation column. The molar flowrate of the outgoing liquid will depend on the fluid mechanics of the tray; that is, it is related to the liquid trapped in the plate. \[L_{n}=\frac{2\sqrt{2g}}{3}\,c\cdot\rho\cdot L_{w}(\frac{M_{n}-M_{plate}}{\rho A _{plate}})^{\frac{3}{2}} \tag{38}\]  
where  
* \(L_{n}\): measured outlet flow (m\({}^{3}\)/s);
* \(c\): hydraulic discharge coefficient;
* \(\rho\): liquid density (kg/m\({}^{3}\));
* \(L_{w}\): weir length (m);
* \(M_{n}\): accumulation of liquid of the plate, \(n+1\);
* \(M_{plate}\): mass of the plate (kg);
* \(A_{plate}\): cross-sectional area of the plate (m\({}^{2}\)). To relate the accumulation of liquid in the tray (\(M_{n}\)) with the flow of liquid leaving the tray, (\(L_{n}\)) is used, as seen in (38). * Global mass balance and mass analysis by component. In a distillation column, a separation process stage is carried out in which, being a closed system, its mass will remain constant according to the law of conservation of mass. Each plate is considered as a stage in which the molar or mass component of the vapor and liquid flow is necessary to perform a mass and component balance. According to the schematic diagram (see Figure 16), the global mass analysis and mass analysis by component corresponds to  
Figure 16: Diagram of the dynamic balance of mass contact stage “n”. Figure 15: Schematic diagram of the distillation column. * Global mass analysis \[\frac{\mathrm{d}}{\mathrm{d}t}\,M_{n}=L_{n+1}+V_{n+1}-L_{n}-V_{n}\] (39) According to (39), each plate will receive a molar flow of liquid \(L_{n+1}\) from the immediate superior plate and a molar flow of steam \(V_{n-1}\) from the lower plate, and they will generate a molar flow of liquid of outlet \(L_{n}\) and a molar flow of steam \(V_{n}\). * Mass analysis by component \[\frac{\mathrm{d}}{\mathrm{d}t}\,M_{n}X_{n}=L_{n+1}X_{n+1}+V_{n-1}Y_{n-1}-L_{n} X_{n}-V_{n}Y_{n}\] (40) In the mass analysis by component according to (40), the respective concentrations of each of the flows must also be considered. Thus, the plate \"n\" receives the molar flow of the liquid \(L_{n+1}\) with a concentration \(X_{n+1}\) of the upper plate and a steam flow \(V_{n-1}\) with a concentration \(Y_{n-1}\) of the lower plate.