A pan is used to boil water by placing it on a stove, from which heat is transferred at a fixed rate qo. There are two stages to the process. In Stage I, the water is taken from its initial (room) temperature T; 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_∞ = T_∞ (t). In Stage 2, the water has come to a boil and its temperature remains at a fixed value, T_∞ = 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.

(a) Write the form of the heat equation and the boundary/initial conditions that determine the variation of temperature with position and time, T(x, t), in the pan bottom during Stage 1. Express your result in terms of the parameters qo, D, L, h, and T_∞, as well as appropriate properties of the pan material.

(b) During Stage 2, the surface of the pan in contact with the water is at a fixed temperature, T (L, t) =

T_L > T_b Write the form of the heat equation and boundary conditions that determine the temperature distribution, T(x), in the pan bottom, Express your result in terms of the parameters qo, D, L, and T L , as well as appropriate properties of the pan material.