Chapter 3, Problem 3.27P

Approximately ${10}^6$ discrete electrical components can be placed on a single integrated circuit (chip), with electrical heat dissipation as high as $30,000{\text{W/m}}^{\text{2}}.$ The chip, which is very thin. is exposed to a dielectric liquid at its outer surface, with $h_o=1000{\text{W/m}}^{\text{2}}\cdot \text{K}$ and $T_{\infty ,o}=20°\text{C,}$ and is joined to a circuit board at its inner surface. The thermal contact resistance between the chip and the board is ${10}^{-4}{\text{m}}^{\text{2}}\cdot \text{K/W,}$ and the board thickness and thermal conductivity are $L_b=5\text{mm}$ and $k_b=1\text{W/m}\cdot \text{K,}$ respectively. The other surface of the board is exposed to ambient air for which $h_i=40{\text{W/m}}^{\text{2}}\cdot \text{K}$ and $T_{\infty ,i}=20°\text{C}\text{.}$

1. Sketch the equivalent thermal circuit corresponding to steady-state conditions. In variable form, label appropriate resistances, temperatures, and heat fluxes.
2. Under steady-state conditions for which the chip heat dissipation is $q_c^n=30,000{\text{W/m}}^{\text{2}},$ what is the chip temperature?
3. The maximum allowable heat flux, $q_{c,m}^n,$ is determined by the constraint that the chip temperature must not exceed 85°C. Determine $q_{c,m}^n,$ for the foregoing conditions. If air is used in lieu of the dielectric liquid, the convection coefficient is reduced by approximately an order of magnitude. What is the value of $q_{c,m}^n$ for $h_o=100{\text{W/m}}^{\text{2}}\cdot \text{K?}$ With air cooling, can significant improvements be realized by using an aluminum oxide circuit board and/or by using a conductive paste at the chip/board interface for which $R_{t,c}^n={10}^{-5}{\text{m}}^{\text{2}}\cdot \text{K/W}?$