A simple mass and enthalpy balance on single stage distillation

 Here the case of distillation in a single stage is presented. This is a simple case corresponding to just boiling the mixture and then separating it in the column. Of course, neither plates nor packing is needed. Despite its simplicity it is worth to discuss the physics of the process.

Instantaneous evaporation

In sketch above a mixture feed $F$ is passed to a heat exchanger that boils the  liquid and then to a separator. In the separator, the vapor goes to the top while  the liquid goes to the bottom. A mass balance for this very schematic case is twofold: there is a global balance and a balance on the component of interest (which can be more than one). For the global mass balance we have,

$F=D+W$        Eq. (01)

and for the component of interest the balance is written as,

$Fz_F=Dy_D+Wx_W$        Eq. (02)

An energy balance based on the enthalpy of each stream can be written as,

$FH_D+Q=DH_D+WH_W$        Eq. (03)

where $Q$ is the heat supplied in the heat exchanger to boil the feed F. Equations (01-02) can be combined to give,

$\dfrac{W}{D}=\dfrac{y_D-z_F}{z_F-z_W}$        Eq. (04)

which can be a more useful relationship. Also, Eqs. (01,03) can also be combined. The result is,

$-\dfrac{W}{D}=\dfrac{H_D-\left( H_F+Q/F \right)}{H_W-\left( H_F+Q/F \right)}$        Eq. (05)

This is the end. Enjoy.