$(60$ pts$.)$ Consider water in a $25-mm-$diameter test tube evaporating into air outside the tube. The

outside air is at $15C$ and $1$ atm and has a relative humidity of $40$ percent. The distance between

the water$-$air interface and the top of the tube is $L=12cm.$ The binary diffusivity for water vapor

in air is $2\mathrm{.}6*{10}^{-5}\frac{m^2}{s}.$ Take the water temperature to be $15C.$ The saturation pressure of water

at $15C$ is $1\mathrm{.}7051$ kPa $,$ and the enthalpy of vaporization is $40,010k\frac{J}{k}g.$

a$)$ Determine the mass evaporation rate of the water.

b$)$ Determine the water vapor mass fraction at $x=\frac{L}{2}.$

c$)$ Determine the fraction of the water mass flow that is contributed by bulk flow and the fraction

contributed by diffusion at $x=\frac{L}{2}.$

d$)$ Determine the rate of heat transfer $($in $W)$ required to maintain the liquid water at $15C.$