Chapter 7, Problem 7.29P

An array of electronic chips is mounted within a sealed rectangular enclosure, and cooling isimplemented by attaching an aluminum heat sink $\left(k=180\text{W/m}\cdot \text{K}\right).$ The base of the heat sink has dimensions of $w_1=w_2=100\text{mm,}$ while the 6 fins are of thickness $t=10\text{mm}$ and pitch $S=18\text{mm}\text{.}$ The fin length is $L_f=50\text{mm,}$ and the base of the heat sink has a thickness of $L_b=10\text{mm}\text{.}$

If cooling is implemented by water flow through the heat sink, with $u_{\infty }=3\text{m/s}$ and $T_{\infty }=17°\text{C,}$ what is the base temperature $T_b$ of the heat sink when power dissipation by the chips is $P_{\text{elec}}=1800\text{W?}$ The average convection coefficient for surfaces of the fins and the exposed base may be estimated by assuming parallel flow over a flat plate. Properties of the water may be approximated as $k=0.62\text{W/m}\cdot \text{K,}\rho =995{\text{kg/m}}^{\text{3}},c_p=4178\text{J/kg}\cdot \text{K,}v=7.73\times {10}^{-7}{\text{m}}^{\text{2}}\text{/s,}$ and $\mathrm{Pr}=\mathrm{5.2.}$