Chapter 3, Problem 3.144P

An isothermal silicon chip of width $W=20\text{mm}$ on a side is soldered to an aluminum heat sink $\left(k=180\text{W/m}\cdot \text{K}\right)$ of equivalent width. The heat sink has a base thickness of $L_b=3\text{mm}$ and an array of rectangular fins, each of length $L_f=15\text{mm}\text{.}$ Airflow at $T_{\infty }=20°\text{C}$ is maintained through channels formed by the tins and a cover plate, and for a convection coefficient of $h=100{\text{W/m}}^{\text{2}}\cdot \text{K,}$ a minimum fin spacing of 1.8 mm is dictated by limitations on the flow pressure drop. The solder joint has a thermal resistance of $R_{t,c}^{"}=2\times {10}^{-6}{\text{m}}^{\text{2}}\cdot \text{K/W}\text{.}$

1. Consider limitations for which the array has $N=11$ tins. in which case values of the tin thickness $t=0.182\text{mm}$ and pitch $S=1.982\text{mm}$ are obtained from the requirements that $W=\left(N-1\right)S+t$ and $S-t=1.8\text{mm}$ If the maximum allowable chip temperature is $T_c=85°\text{C,}$ what is the corresponding value of the chip power $q_c?$An adiabatic tin tip condition may be assumed. and airflow along the outer surfaces of the heat sink may be assumed to provide a convection coefficient equivalent to that associated with airflow through the channels.
2. With $\left(S-t\right)$ and h fixed at 1.8 mm and $100{\text{W/m}}^{\text{2}}\cdot \text{K,}$ respectively, explore the effect of increasing the tin thickness by reducing the number of fins. With $N=11$ and $S-t$ fixed at 1.8 nun, but relaxation of the constraint on the pressure drop, explore the effect of increasing the airflow, and hence the convection coefficient.