Overall vapour and energy transfer rates

The definition of the Sherwood number for the cuboid, (6.21), is here modified for the sphere:

$$\mbox{\textit{Sh}}= \,\mathrm{sgn}\, x\;\mbox{\boldmath$\hat... ...cdot\mbox{\boldmath$\nabla$}m, \quad (r=\mbox{$\frac{1}{2}$}).$$ (8.76)

The net flux through the hemispherical surface $\{r=1/2, \cos\phi>0\}$ is

$$\left(\mbox{$\frac{1}{2}$}\right)^2\int_{-\pi/2}^{\pi/2}\int... ...{\textit{Sh}}\sin\theta \,\mathrm{d}\theta\,\,\mathrm{d}\phi,$$ (8.77)

which is equal to the net flux through the disk, $\cos\phi=0$ (i.e. the $x=0$ plane) or the other hemispherical surface: $r=1/2, \cos\phi<0$. The Sherwood number is averaged by dividing the common net flux through these surfaces by the projection of their area in the $yz$-plane,

$$\left(\mbox{$\frac{1}{2}$}\right)^2\int_{-\pi/2}^{\pi/2}\int... ...n\theta \,\mathrm{d}\theta\,\,\mathrm{d}\phi = \frac{\pi}{4};$$ (8.78)

thus

$$\overline{\mbox{\textit{Sh}}} = \frac{1}{\pi}\int_{-\pi/2}^{... ...x{\textit{Sh}}\sin\theta \,\mathrm{d}\theta\,\,\mathrm{d}\phi.$$ (8.79)

To first order in Gr,

$$\overline{\mbox{\textit{Sh}}} \sim \overline{\mbox{\textit{Sh}}_0} + \mbox{\textit{Gr}}(1+N)\overline{\mbox{\textit{Sh}}_1} + O(\mbox{\textit{Gr}}^2)$$ (8.80)

$$\sim 1 + 0 + O(\mbox{\textit{Gr}}^2).$$ (8.81)

The first order correction to the mean Sherwood number vanishes because $m_1$ is an odd function of $\phi$. The increased diffusion in the quadrants $xy>0$ is balanced by the lower rate in the other two quadrants. The first order correction to the Sherwood number may also have been expected to vanish from more general symmetry considerations: $\overline{\mbox{\textit{Sh}}}$ should be an even function of $\mbox{\textit{Gr}}$ (§2.6.3).

The results for the energy transfer rate are completely analogous.