Single-stage Batch Distillation

In batch distillation of a binary mixture, a fixed amount of the feed mixture, \(F\) (mol), with initial mole fraction, \(x_F\), evaporates into a distillate collection flask; \(D_i\) (mol) is the mass distillate collected into flask \(i\), and \(x_{D,i}\) is the mole fraction within distillate flask \(i\). This process is repeated to fill several distillate collection flasks. The overall mass balance is: 

\[F = B_{final} + D_{total} \hspace{5 cm} (1)\] and the balance for the more volatile component is: 

\[F\,x_F = B_{final}\, x_{B,final} + D_{total}\, x_{D,avg} \hspace{3 cm} (2)\] where \(x_{D,avg}\) is the average mole fraction within the distillate collection flasks and \(B_{final}\) and \(x_{B,final}\) are the final mass and mole fraction within the bottom (boiler) vessel. The value of \(x_{D,avg}\) can be calculated as: 

\[x_{D,avg} = \sum\limits_{i=1}^n (D_i x_{D,i}) /D_{total} \hspace{2 cm} (3)\] where the total number of collection stages is \(n\) (equivalent to the total number of distillate collection flasks).

Total distillate is given by:

\[D_{total} = \sum\limits_{i=1}^n D_i \hspace{4 cm} (4)\] Because the saturated vapor is in thermodynamic equilibrium with the saturated liquid in the vessel (and there is only one equilibrium stage: the boiler/bottom vessel), the composition of the vapor \(y_B\) is a function of the composition in the bottom vessel. During evaporation, both composition change with time, except when the composition of the liquid is an azeotrope. An equilibrium function is:

\[y_B = f(x_B) \hspace{4 cm} (5)\] Numerical approximations such as Antoine’s equation are often used to model \(f(x_B)\) using experimental data. Other cases, such as constant relative volatility, may also be used to approximate this relationship.

Composition of \(x_{B, i+1}\) cam be calculated by integration. Equation (5) must be substituted for \(y_B\).

\[\int\limits_{B_i}^{B_{i+1}} \frac{1}{B} dB = \int\limits_{x_{B,i}}^{x_{B,i+1}} \frac{1}{y_B -x_B} dx_B \hspace{3 cm} (6)\] Finally, a mass balance can be used to solve for the distallate composition, \(x_{D, i+1}\):

\[x_{D,i+1} = \frac{B_i\,x_{B,i} – B_{i+1}\,x_{B,i+1}}{B_i – B_{i+1}} \hspace{3 cm} (7)\]

Multi-stage Batch Distillation

Multi-stage batch distillation can obtain much higher product purity than simple batch distillation. The overall mass balance is: 

\[F = W_{final} + D_{total}\] where \(F\) is the initial molar quantity of the feed mixture, \(W_{final}\) is the molar quantity of the bottoms vessel (the “waste” product), and \(D_{total}\) is the total molar quantity of collected distillate. The component mass balance is:

\[F\,x_F = W_{final}\,x_{W,final} + D_{total}\,x_{D,avg}\] where \(x_F\) is the initial mole fraction of component B in the feed, \(x_{D,avg}\) is the average mole fraction of component B within the collected distillate, and \(x_{W,final}\) is the final mole fraction within the bottoms vessel.

Each tray in the column is assumed to be in vapor-liquid equilibrium:

\[y = f(x)\] The function \(y = f(x)\) forms an equilibrium curve, plotted on an x-y diagram. The composition of each stage lies somewhere upon this curve. An operating line relates the composition of a stage to the adjacent stage composition. The operating line equation is:

\[y_n = \frac{R}{R+1} x_{n+1} + \left(1 – \frac{R}{R+1} \right) x_D\] where \(y_n\) is the vapor mole fraction of B at stage\(n\), \(R\) is the reflux ration, and \(x_{n+1}\) is the liquid mole fraction of B at stage \(n+1\) The reflux ration is:

\[R = \frac{L}{D}\] where \(L\) is the flow rate of liquid returning from the condenser to the column, and \(D\) is the flow rate of distillate being collected. The Rayleigh Equation can be used to solve for the final composition of distillate and bottoms, 

\[ln\left(\frac{W_{final}}{F}\right) = \int\limits_{x_F}^{x_{W,final}} \frac{1}{x_D – x_W} dx_W\] but the relationship between distillate composition and bottoms composition, \(f(x_W) = x_D\) is complicated with multi-stage columns, and the mole fractions are obtained numerically.