Chapter 5, Problem 5.27P

A chip that is of length $L=5\text{mm}$ on a side and thickness $t=1\text{mm}$ is encased in a ceramic substrate, and its exposed surface is convectively cooled by a dielectric liquid for which $h=150{\text{W/m}}^{\text{2}}\cdot \text{K}$ and $T_{\infty }=20°\text{C}\text{.}$

In the off-mode the chip is in thermal equilibrium with the coolant $\left(T_i=T_{\infty }\right).$ When the chip is energized. however, its temperature increases until a new steady state is established. For purposes of analysis, the energized chip is characterized by uniform volumetric heating with $\stackrel{.}{q}=9\times {10}^6{\text{W/m}}^{\text{3}}.$ Assuming an infinite contact resistance between the chip and substrate and negligible conduction resistance within the chip, determine the steady-state chip temperature $T_f.$ Following activation of the chip, how long does it take to come within $1°\text{C}$ of this temperature? The chip density and specific heat are $\rho =2000{\text{kg/m}}^{\text{3}}$ and $c=700\text{J/kg}\cdot \text{K,}$ respectively.