Consider the transient performance of the fin in Problem 8.12 when the microprocessor is cycled with a

Question:

Consider the transient performance of the fin in Problem 8.12 when the microprocessor is cycled with a frequency $ω_{0}$ .

a. What is the differential equation describing the transient situation?

b. The differential equation can be solved by assuming the transient response solution can be written as:

$θ = T - T_{\infty} = θ_{o} (x) e^{i ω_{o} t}$

The transient solution must obey the following conditions:

$\begin{array}{ll} x = 0 & - \frac{k}{L} \frac{d θ}{d x} = q_{o}^{''} e^{i ω_{o} t} \\ x = L & θ = 0 t = 0 θ = 0 \end{array}$

where we assume that the fin behaves as if it is infinitely long.

c. Using the information in part (b), solve the differential equation.

Problem 8.12

A microprocessor has an aluminum heat sink attached that consists of a series of parallel, rectangular fins. Each fin is $2 m m$ thick, $25 m m$ long, $25 m m$ high, and spaced on $6 m m$ centers. When the microprocessor is turned on it begins producing heat at a rate of $1 m W /$ $c m 2$ of surface area. A fan within the computer circulates air at ${????}_{\infty} = 25 \circ C$ through the machine providing a heat transfer coefficient of $150 W / m 2 K$ . The fin tip can be assumed to be exposed to the convecting fluid.