Chapter 6, Problem 6S.12P

A simple scheme for desalination involves maintaining a thin film of saltwater on the lower surface of two large (infinite) parallel plates that are slightly inclined and separated by a distance L.

A slow, incompressible, laminar airflow exists between the plates, such that the x velocity component is finite while the y- and z-components are zero. Evaporation occurs from the liquid film on the lower surface, which is maintained at an elevated temperature $T_0,$ while condensation occurs at the upper surface, which is maintained at a reduced temperature $T_L.$ The corresponding molar concentrations of water vapor at the lower and upper surfaces are designated as $C_{A,0}$ and $C_{A,L},$ respectively. The species concentration and temperature may be assumed to be independent of x and z.

1. Obtain an expression for the distribution of the water vapor molar concentration $C_A\left(y\right)$ in the air. What is the mass rate of pure water production per unit surface area? Express your results in terms of $C_{A,0},C_{A,L},L,$ and the vapor-air diffusion coefficient $D_{AB}.$
2. Obtain an expression for the rate at which heat must be supplied per unit area to maintain the lower surface at $T_0.$ Express your result in terms of $C_{A,0},C_{A,L},T_0,T_L,L,D_{\text{AB}},h_{fg}$(the latent heat of vaporization of water), and the thermal conductivity k.