Consider an aluminum heat sink (k = 240 W/m ∙ K), such as that shown schematically in Problem 4.28. The inner and outer widths of the square channel are w = 20mm and W = 40mm, respectively, and an outer surface temperature of Ts = 50°C is maintained by the array of electronic chips. In this case, it is not the inner surface temperature that is known, but conditions (T∞, h) 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 ∆x = ∆y = 5 mm.

(a) For T∞ = 20°C and h = 5000W/m2 ∙ K, determine the unknown temperatures, T1, T7, and the rate of heat transfer per unit length of channel, q'.

(b) Assess the effect of variations in h on the unknown temperatures and the heat rate.

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Heat sink, k •T3 Coolant, T,h