Mass and energy fluxes at the vertical walls

In the narrow cavity limit, sufficiently far from the horizontal surfaces, the dimensionless mass transfer rates, at the hot and cold walls are obtained by substituting the horizontal velocity (4.23) into (2.62)

$$\mbox{\textit{Sh}}_0=\mbox{\textit{Sh}}_1=-\frac{\mbox{\text... ...frac{-\varPhi }{\mbox{\textit{Gr}}(1+N)\mbox{\textit{Sc}}} = 1$$ (4.39)

or the mass fraction field (4.24) into (2.63)

$$\mbox{\textit{Sh}}_0 = \frac{1-\mathrm{e}^{-\varPhi }}{\varPhi }\times \frac{\varPhi }{1-\mathrm{e}^{-\varPhi }} = 1$$ (4.40)

$$\mbox{\textit{Sh}}_1 = \frac{1-\mathrm{e}^{-\varPhi }}{\varPhi \mathrm{e}^{-\varPhi }}\times \frac{\varPhi \mathrm{e}^{-\varPhi }}{1-\mathrm{e}^{-\varPhi }} = 1.$$ (4.41)

The two methods of calculating the mass transfer rate--from the transpiration velocity or the mass fraction gradient--are in agreement, as expected. That the Sherwood numbers at the hot and cold walls are equal means that the rates of evaporation and condensation are equal, so that no net mass enters the cavity at any level where the fully developed solution applies.

The Nusselt number is also a constant (independent of position on the wall, and the same for each wall) but does depend on a fixed combination of $\varPhi _T$ and $\varLambda$. It is given by equations (2.66) and (4.26) as

$$\mbox{\textit{Nu}}= 1 + [1-\exp(-\varPhi _T)]\varLambda .$$ (4.42)