Trevisan and Bejan (1987)

Trevisan and Bejan (1987) considered vapour transport across a plane vertical rectangular cavity at low mass transfer rates ( $\varPhi \rightarrow 0$). Uniform flux boundary conditions were imposed at the hot and cold walls. Exact solutions were reported for infinitely tall cavities ( $\mbox{$\mathcal A$}\rightarrow\infty$) at large combined Grashof numbers, $\mbox{\textit{Gr}}(1+N)$, for three limiting cases: $\mbox{\textit{Sc}}=\mbox{\textit{Pr}}$; $N=0$, $\mbox{\textit{Sc}}\gg\mbox{\textit{Pr}}$; and $\vert N\vert\rightarrow\infty$, $\mbox{\textit{Sc}}\ll\mbox{\textit{Pr}}$. The first of these is actually not quite exact, in that it assumes that there exists a stagnant stratified core between the boundary layers on the walls, so that the transverse length scale of the flow near the hot and cold walls is much smaller than the breadth of the cavity; it is therefore equivalent to Prandtl's (1952, p. 422; Elder 1965; Gill 1966) solution for heat transfer from a single wall to an unbounded pure fluid. A solution valid across the breadth of the cavity was given independently by Vest and Arpaci (1969) and Aung (1972). The paper also contains numerical solutions for $\mbox{$\mathcal S$}\rightarrow 0, 1\leq\mbox{$\mathcal A$}\leq 4$, $3.5\times10^5\leq\mbox{\textit{Gr}}\mbox{\textit{Pr}}\leq7\times10^6$, $-11\leq N\leq 9$, $1\leq(\mbox{\textit{Sc}}/\mbox{\textit{Pr}})\leq 40$ and $\mbox{\textit{Pr}}=0.7, 7$.

Trevisan and Bejan offered a simple scaling argument to determine when transpiration could be neglected for $N=0$ and $\mbox{\textit{Sc}}\gg\mbox{\textit{Pr}}\gg 1$. Their result was that this is safe when

$$\Delta m_{*r}\left(\frac{\mbox{\textit{Pr}}}{\mbox{\textit{Sc}}}\right)^{1/3}\ll 1.$$ (3.1)

This is more or less in agreement with the criterion obtained in chapter 6:

$$\varPhi \ll 1,$$ (3.2)

since $\varPhi$ approaches zero as the vapour mass fraction difference does. The inclusion of the Prandtl and Schmidt numbers in the criterion is incorrect, however, since although the transpiration-induced velocities decrease as the Schmidt number increases (2.59), this is balanced by the factor Sc in the advective flux (2.35). The influence of the Prandtl number is complicated by the associated effects of interdiffusion and the variability of the mixture specific heat. This matter is dealt with in more detail in chapter 6, in the light of the solutions including the effects of transpiration generated in chapters 4-5.