Bejan (1985)

Bejan (1985) extended the time-independent part of the scale analysis of Patterson and Imberger (1980) for the analogous single fluid heat transfer problem to binary mixtures. The finite mass transfer rate effects of transpiration and interdiffusion were neglected ( $\varPhi \rightarrow 0$). The scales for the mean Sherwood and Nusselt numbers are deduced from a simple model of the semi-infinite vertical plate boundary layer. The six possible permutations of $Pr_r$, $\mbox{\textit{Sc}}$ and unity in the relation $a\ll b \ll c$ are examined for $N\rightarrow 0$ and $N^{-1}\rightarrow 0$. Since $\nu$, $\lambda/\rho c_{pr}$ and $D$ are typically of a similar magnitude for gas-vapour mixtures; so that $\mbox{\textit{Pr}}_r\approx\mbox{\textit{Sc}}\approx 1$; none of these limiting cases are applicable. They are, however, of theoretical interest, and are important for amalgams and brines, for which $\mbox{\textit{Pr}}_r\ll 1\ll \mbox{\textit{Sc}}$ (Bergman & Hyun 1996) and $1\ll\mbox{\textit{Pr}}_r\ll\mbox{\textit{Sc}}$ (Hyun & Lee 1990), respectively.

Bejan verified his predictions by comparison with numerical solutions for the limiting case $N=0$ and $\mbox{\textit{Gr}}\rightarrow 0$ with $\mbox{\textit{Pr}}_r\sim O(1)$ and $\mbox{\textit{Sc}}^{-1}\sim O(\mbox{\textit{Gr}})$ in two-dimensional vertical cavities of height $\mbox{$\mathcal A$}=1,2$ and 4. Further tests of the predicted scales were undertaken by Béghein et al. (1992; §3.3.15).

A result relevant to the consideration of the conduction regime in chapters 4-5 is that the vertical velocity profile at midheight ( $y=\mbox{$\mathcal A$}/2$) depended only weakly on $\mathcal A$ for $\mbox{$\mathcal A$}\geq 2$.