Question

Elementary Partial Differential Equations

Heat flow in a circular cylinder.

Consider a strand of heat-conducting material, homoge- neous with heat capacity c, thermal conductivity κ, and surface heat transfer coefficient μ. The strand is a right circular cylinder of radius R and height H. Unless otherwise indicated below, assume that no heat is being generated or destroyed inside the strand. For each of the following scenarios, set up the IBVP for the temperature distribution u in the strand. In each case, reduce the spatial dimension of the problem as far as possible, identify the independent variables (time t and a subset of the cylindrical coordinates r, θ, z), write all equations explicitly in terms of those variables, and indicate where exactly the equations are to hold. Also, whenever possible, set up the corresponding steady-state problem. If it reduces to an ODE, solve the steady-state problem and graph the solution.

(a) While the lateral surface and the top of the cylinder are perfectly insulated, the bottom is maintained at a constant temperature Tbot. The initial temperature distribution is a function f(z).(b) Same as in (a), except that the temperature at the bottom changes, at a constant rate and over a period of τ units of time, from an initial constant temperature T0 to a final constant temperature T∞, and is maintained at that final value ever after.
(c) Same as in (a), except that heat is exchanged across the top end, according to Newton’s law of cooling, with an external medium at constant temperature Text.
(d) Same as in (c), except that Newton’s law applies also on the lateral surface of the cylinder. (e) Same as in (d), except that top and bottom are perfectly insulated and the initial temperature
distribution is a function f(r).

(f) Same as in (e), except that the initial temperature distribution is a function f(r,θ) and that heat is generated inside the cylinder (for example, via Joule heating) at a constant rate G (heat units per unit time and unit volume).
(g) Same as in (f), except that heat is added also through the bottom of the cylinder, at a constant rate Q (heat units per unit time and unit area).

Answer 1