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 of 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: $$ [1] \quad F = B_{final} + D_{total} $$ and the balance for the more volatile component is: $$ [2] \quad F x_{F} = B_{final} x_{B, final} + D_{total} x_{D,avg} $$ 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: $$ [3] \quad x_{D,avg} = \sum_{ i = 1 }^{ n } \left( \frac{ D_{i} x_{D,i} + D_{i + 1} x_{D, i+1} ... D_{n} x_{D, n} }{ D_{total} } \right) $$ where the total number of collection stages is \( n \) (equivalent to the total number of distillate collection flasks). Total distillate is given by: $$ [4] \quad D_{total} = \sum_{ i = 1 }^{ n } \left( D_{i} + D_{ i + 1 } ... D_{ n } \right) $$ 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 compositions change with time, except when a maximum temperature azeotrope is reached. At equilibrium, $$ [5] \quad y_{B} = f(x_{B}) $$ For an ideal system, Raoult's law is used to relate \( y_{B} \) to \( x_{B} \). The \( x_{B,i+1} \) composition is calculated by integration, where \( y_{B} \) is given by equation [5]. $$ [6] \quad \int_{B_{i}}^{B_{i + 1}} \frac{ 1 }{ B } dB = \int_{x_{B, i}}^{ x_{B, i + 1} } \frac{ 1 }{ y_{B} - x_{B} } dx_{B} $$ The distillate composition is determined from a mass balance: $$ [7] \quad x_{ D, i + 1 } = \frac{ B_{i} x_{B, i} - B_{ i + 1 }{ x_{ B, i + 1 } } }{ B_{i} - B_{i + 1} } $$

References:

1. [1] P. C. Wankat, "Chapter 9: Batch Distillation," Separation Process Engineering: Includes Mass Transfer Analysis, 3rd ed., Upper Saddle River, NJ: Prentice Hall, 2012 pp. 329-347.