A microprocessor has an array of metal fins attached to it, whose purpose is to remove heat generated within the processor. Each fin may be represented by a long thin cylindrical copper rod with one end attached to the processor; heat received by the rod through this end is lost to the surroundings through its sides.
Show that the temperature T(x,t) at location x along the rod at time t obeys the equation
ρCp∂t∂T=κ∂x2∂2T−a2R(T),
where a is the radius of the rod, and R(T) is the rate of heat loss per unit area of surface at temperature T. The surroundings of the rod are at temperature T0. Assume that R(T) has the form of Newton's law of cooling, namely
R(T)=A(T−T0).
In the steady state:
(a) obtain an expression for T as a function of x for the case of an infinitely long rod whose hot end has temperature Tm;
(b) show that the heat that can be transported away by a long rod (with radius a) is proportional to a3/2, provided that A is independent of a.
In practice the rod is not infinitely long. What length does it need to have for the results above to be approximately valid? The radius of the rod, a, is 1.5mm.
[The thermal conductivity of copper is 380Wm−1K−1. The cooling constant A=250Wm−2K−1.]