Chapter 4, Problem 4.58P

Consider an aluminum heat sink $\left(k=240\text{W/m}\cdot \text{K}\right),$ such as that shown schematically in Problem 4.28. The inner and outer widths of the square channel are $w=20\text{mm}$ and $W=40\text{mm,}$ respectively, and an outer surface temperature of $T_s=50°\text{C}$ is maintained by the array of electronic chips. In this case. it is not the inner surface temperature that is known. but conditions $\left(T_{\infty },h\right)$ associated with coolant flow through the channel, and we wish to determine the rate of heat transfer to the coolant per unit length of channel. For this purpose consider a symmetrical section of the channel and a two-dimensional grid with $\Delta x=\Delta y=5\text{mm}\text{.}$

1. For $T_{\infty }=20°\text{C}$ and $h=5000{\text{W/m}}^{\text{2}}\cdot \text{K,}$ determine the unknown temperatures, $T_1,..,T_7,$ and the rate of heat transfer per unit length of channel, $q\text{'}.$
2. Assess the effect of variations in h on the unknown temperatures and the heat rate.