Elementary Partial Differential Equations

Heat flow in a circular cylinder.

Consider a strand of heat-conducting material, homoge- neouswith heat capacity c, thermal conductivity κ, and surface heattransfer coefficient μ. The strand is a right circular cylinder ofradius R and height H. Unless otherwise indicated below, assumethat no heat is being generated or destroyed inside the strand. Foreach of the following scenarios, set up the IBVP for thetemperature distribution u in the strand. In each case, reduce thespatial dimension of the problem as far as possible, identify theindependent variables (time t and a subset of the cylindricalcoordinates r, θ, z), write all equations explicitly in terms ofthose variables, and indicate where exactly the equations are tohold. Also, whenever possible, set up the correspondingsteady-state problem. If it reduces to an ODE, solve thesteady-state problem and graph the solution.

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

(f) Same as in (e), except that the initial temperaturedistribution is a function f(r,θ) and that heat is generated insidethe 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 thebottom of the cylinder, at a constant rate Q (heat units per unittime and unit area).