Chapter 3, Problem 3.33P

Consider a power transistor encapsulated in an aluminum case that is attached at its base to a square aluminum plate of thermal conductivity $k=240\text{W/m}\cdot \text{K,}$ thickness $L=6\text{mm,}$ and width $W=20\text{mm}\text{.}$ The case is joined to the plate by screws that maintain a contact pressure of 1 bar, and the back surface of the plate transfers heat by natural convection and radiation to ambient air and large surroundings at $T_{\infty }=T_{\text{sur}}=25°\text{C}\text{.}$ The surface has an emissivity of $\epsilon =0.9,$ and the convection coefficient is $h=4{\text{W/m}}^{\text{2}}\cdot \text{K}\text{.}$ The case is completely enclosed such that heat transfer may be assumed to occur exclusively through the base plate.

1. If the air-filled aluminum-to-aluminum interface is characterized by an area of $A_c=2\times {10}^{-4}{\text{m}}^{\text{2}}$ and a roughness of $10\text{μm,}$ what is the maximum allowable power dissipation if the surface temperature of the case, $T_{s,c}$ is not to exceed 85°C?
2. The convection coefficient may be increased by subjecting the plate surface to a forced flow of air. Explore the effect of increasing the coefficient over the range $4\le h\le 200{\text{W/m}}^{\text{2}}\cdot \text{K}\text{.}$