Chapter 2, Problem 2.27P

A pan is used to boil water by placing it on a stove, from which heat is transferred at a fixed rate $q_0.$ Thre are two stages to the process. In Stage 1, the water is taken from its initial (room) temperature $T_i$ to the boiling point, as heat is transferred from the pan by natural convection. During this stage, a constant value of the convection coefficient h may be assumed. while the bulk temperature of the water increases with time, $T_{\infty }=T_{\infty }\left(t\right).$ In Stage 2, the water has come to a boil, and its temperature remains at a fixed value, $T_{\infty }=T_b,$ as heating continues. Consider a pan bottom of thickness L and diameter D. with a coordinate system corresponding to $x=0$ and $x=L$ for the surfaces in contact with the stove and water, respectively.

1. Write the form of the heat equation and the boundary/initial conditions that determine the variation of temperature with position and time, $T\left(x,t\right),$ in the pan bottom during Stage 1. Express your result in terms of the parameters $q_O,$ D, L, h, and $T_{\infty },$ as well as appropriate properties of the pan material.
2. During Stage 2, the surface of the pan in contact with the water is at a fixed temperature, $T\left(L,t\right)=T_L>T_b.$ Write the form of the heat equation and boundary conditions that determine the temperature distribution $T\left(x\right)$ in the pan bottom. Express your result in terms of the parameters $q_o,$ D. L and $T_L,$ as well as appropriate properties of the pan material.