Consider a film of vapor in contact with a liquid. From a heat balance show that the mass flow rate in the vapor, \(\dot{m}\), per unit transfer area changes as

\[\hat{h}_{\lg } \frac{d \dot{m}}{d x}=\frac{k_{\mathrm{v}}}{\delta} \Delta T\]

where \(\delta\) is the local film thickness of the vapor film. The film thickness in turn is related to the mass flow rate from momentum-transfer considerations by

\[\dot{m}=\frac{g \delta^{3}\left(ho_{1}-ho_{\mathrm{v}}\right) ho_{\mathrm{v}}}{3 \mu_{\mathrm{v}}}\]

Combine these expressions and derive the formula given in the text for film thickness for the film boiling.