A rain droplet with a radius of $2$ mm falls down with a velocity of $7$ m s$1$

$.$ The droplet

evaporates with a mass transfer coefficient of $0\mathrm{.}14$ m s$1$

$.$ The droplet is initially at $15$

$$C$.$ The air has a temperature of $15$C and a relative humidity of $50\%.$

Calculate the temperature profile in the droplet at different times resulting from

evaporative cooling as the droplet falls down.

Use the following assumptions:

$$ The heat transfer flux at the droplet surface equals the mass transfer flux of

water vapor around the droplet multiplied by the heat of evaporation $(45,000$ J

mol$1$

$).$

$$ The water vapor concentration at the droplet surface can be calculated from

the ideal$-$gas law, assuming that the water partial pressure is the vapor

pressure at the temperature of the droplet surface. The water partial pressure

in the bulk air equals the water vapor pressure at $15$C multiplied by the

relative humidity $(0\mathrm{.}5).$ The water mass transfer is driven by this concentration

difference.

$$ The vapor pressure of water is given by the following equation:

v $()()$

$6757\mathrm{.}426$ p T Pa exp $58\mathrm{.}593654\mathrm{.}891124$ln

T

$$

$=$

$,$ where T is the

temperature in K$.$ The equation is valid in the $273$ K $303$ K range.

Once the evaporation heat flux is known, it can be used as a heat flux boundary at

the outside radius of the droplet, with a virtual point boundary condition similar to the

convection heat flux boundary condition used in the steel ball example.

You can use an existing code $($e$.$g$.,$ the steel ball code$)$ as a basis for your code.

Submit a brief $(2$ pages$)$ report with your results, as well as the Matlab$/$Octave code of your simulation