As a means of enhancing heat transfer from high-performance logic chips, it is common to attach a heat sink to the chip surface in order to increase the surface area available for convection heat transfer. Because of the ease with which they are manufactured (by taking orthogonal saw cuts in a block of material), an attractive option is to use a heat sink consisting of an array of square fins of width w on a side, The spacing between adjoining fins s is uniform. Consider a square chip of width W. = 25 mm, The chip is cooled using a dielectric liquid with T. = 25o C and h = 1,500W/m2K. The heat sink is fabricated from copper (K = 400 WmK). lts base has negligible heat resistance and the contact resistance between the sink and the chip is also negligible. The width of each fin cannot be smaller than L mm or larger than 1.5 mm. The height of the fins, h1, due to space constraints cannot exceed 6 mm. Also, the spacing between fins cannot be smaller than l- mm, so that flow through the fins can be sustained at a sufficient rate so that the convective heat transfer coefficient can be maintained at the value stated.

Find the optimum number of fins needed to maximize the heat dissipated while minimizing the mass of the fins (which is proportional to its cost) by optimizing the spacing between the fhs. Notice that the surface of the chips cannot exceed T. = 85o C.