Problem

$4,28$

A thin $(1\mathrm{.}0-$mm$-$thick$)$ coat of fresh paint has just been sprayed over a $1\mathrm{.}5-m$ by $1\mathrm{.}5-m$ square steel body part, which approximates a flat surface. The paint contains a volatile solvent that initially constitutes $30wt\%$ of the wet paint. The initial density of the wet paint is $1\mathrm{.}5\frac{g}{c}{m}^3.$ The freshly painted part is introduced into a drying chamber. Air is blown into the rectangular drying chamber at a volumetric flow rate of $60\frac{m^3}{m}in,$ as shown in the figure below, which has dimensions $L=1\mathrm{.}5m,H=1\mathrm{.}0m,W=1\mathrm{.}5m.$ The temperature of the air stream and the steel body part are both maintained at $27C,$ and the total system pressure is $1\mathrm{.}0$atm. The molecular weight of the solvent is $78\frac{g}{g}$mole, the vapor pressure exerted by the solvent at $27C$ is $105mmHg,$ and the molecular diffusion coefficient of the solvent vapor in air at $27C$ and $1\mathrm{.}0$atm is $0\mathrm{.}097c\frac{m^2}{s}.$

a$)$ What is the Schmidt number and the average Sherwood number $(S{h}_1)$ for the mass$-$transfer process?

b$)$ What is the estimated solvent evaporation rate from the surface of the whole body part in units of $\frac{g}{m}in?$ It may be assumed that convection mass$-$transfer limits the evaporation rate, and that the concentration of solvent vapor in the bulk gas is finite, but can be approximated as $c_{A^{\text{'}}}$~~$0.$

c$)$ Using the results from part $($a$),$ how long will it take for the paint to completely dry?

d$)$ What are the hydrodynamic $()$ and concentration boundary layer $(c)$ thicknesses at $x=L=1\mathrm{.}5m?$ How does this compare to $H,$ the height of the drying chamber?

e$)$ What is the new required air flow rate $(\frac{m^3}{m}in)$ if the desired solvent mass$-$transfer rate is $150\frac{g}{m}in?$