The heated, depressurized feed enters through a throttling valve; vapor V (enriched in the light component) leaves the top and liquid L (richer in the heavy component) leaves the bottom. The bars below show the compositions z, y and x.
$$y=\frac{\alpha\,x}{1+(\alpha-1)\,x}\qquad\text{and}\qquad z=\phi\,y+(1-\phi)\,x$$
The constant-volatility equilibrium relation and the component mass balance of the flash drum. α is the relative volatility (the ratio of how readily the two components evaporate) and φ = V/F is the vapor fraction (the fraction of the feed that becomes vapor).
$$(1-\phi)(\alpha-1)\,x^{2}+\bigl[(1-\phi)+\phi\alpha-z(\alpha-1)\bigr]x-z=0$$
The quadratic in the liquid composition x obtained by substituting the equilibrium relation into the mass balance. The physically valid root between 0 and 1 is taken as the solution.
$$V=\phi F,\qquad L=(1-\phi)F,\qquad \text{recovery}=\frac{V\,y}{F\,z}$$
Vapor flow V, liquid flow L, and the recovery of the light component to the vapor. F is the feed rate. The separation factor is given by y/x.