$(25$ points$)$ see Figure $2-$ To improve the air quality in a residence warmed with a fireplace, a

homeowner boils water in a cast iron pan set atop a wood burning stove. The pan is $1\mathrm{.}5$ cm

thick $($t$)$ and presents $650c{m}^2$ of surface area to the stove top. Under boiling conditions, the

convection heat transfer coefficient from the pan to the water is $6\mathrm{.}2k\frac{W}{m^2}*K.$ In your analysis,

use the HDE in Cartesian coordinates, assume a uniform heat flux into the pan bottom of $47$

$k\frac{W}{m^2},$ use pure iron material properties at $300$ K for the pan, neglect radiation heat transfer,

and assume steady state conditions.

a$.$ Simplify the HDE for conduction through the pan bottom and state the boundary

conditions.

b$.$ Solve for the temperature profile, $T(x)[K],$ through the bottom of the pan.

c$.$ Find the temperature $[[$:$C\}$ of the pan surface opposite the stove, i$.$e$.$ at $x=1\mathrm{.}5cm.$

d$.$ Assuming the water is at $100C,$ how long will it take $[$ min $]$ to boil off $4$ liters of water?

$($Hint: Use the heat of vaporization $(h_{fg})$ and density for water at $100C)$

water

water

Figure $1$