Liquefaction yield with separate warm-gas and liquid outlets ($y=(h_g-h_{\mathrm{in}}-q)/(h_g-h_l)$)

Apply the steady-flow energy balance to an entire liquefaction apparatus, allowing a warm gas outlet and a separate cold liquid outlet. With incoming heat $q$ per unit feed mass, no shaft work, and negligible changes in bulk kinetic energy and gravitational potential energy, conservation of mass and conservation of energy give $h_{\mathrm{in}}+q=y{h}_l+(1-y)h_g$. Solving this equation gives the displayed liquid mass fraction.

The two outlet streams need not share a temperature. For example, a counterflow heat exchanger can warm returning gas before it crosses the overall control volume boundary, while collected liquid leaves at the cold end. When the outlet temperatures differ, $h_g-h_l$ includes sensible heating as well as phase change; it is not simply a latent heat. A colder two-phase separator nested inside the apparatus has its own boundary states and a different local enthalpy balance.