A total of 10 rectangular aluminum fins \((k=\) \(203 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) are placed on the outside flat surface of an electronic device. Each fin is \(100 \mathrm{~mm}\) wide, \(20 \mathrm{~mm}\) high and \(4 \mathrm{~mm}\) thick. The fins are located parallel to each other at a center- tocenter distance of \(8 \mathrm{~mm}\). The temperature at the outside surface of the electronic device is \(60^{\circ} \mathrm{C}\). The air is at \(20^{\circ} \mathrm{C}\), and the heat transfer coefficient is \(100 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Determine \((a)\) the rate of heat loss from the electronic device to the surrounding air and \((b)\) the fin effectiveness.

First, we need to calculate the fin thickness (\(t_f\)) and fin area (\(A_f\)) using the given values of fin width (100 mm), height (20 mm), and thickness (4 mm). Fin thickness, \(t_f = 4 \times 10^{-3} ~m\) (converting from mm to m) Fin area, \(A_f = (100 \times 10^{-3}) (20 \times 10^{-3}) ~m^2\) (converting from mm to m)

Now we will calculate the heat transfer rate per unit length (\(q''\)) using the heat transfer coefficient (\(h\)) and the temperature difference between the electronic device (\(T_s\)) and air (\(T_\infty\)). Equation for heat transfer rate per unit length is: \(q'' = h A_f (T_s - T_\infty)\). Plug in the known values: \(h = 100 ~W/m^2K\) \(A_f = (100 \times 10^{-3})(20 \times 10^{-3})~m^2\) \(T_s = 60^\circ C\) \(T_\infty = 20^\circ C\) \(q'' = 100 (100 \times 10^{-3})(20 \times 10^{-3})(60-20) ~W/m\)

To find the total heat transfer rate (\(Q\)) for all the fins, we have to multiply the heat transfer rate per unit length (\(q''\)) by the total length of the fins (\(L_{fins}\)). Since there are 10 fins and each has a center-to-center distance of 8 mm, the total length can be calculated as follows: \(L_{fins} = 10 \times 8 \times 10^{-3} ~m\) Now, we can calculate the total heat transfer rate: \(Q = q'' L_{fins} = (100 (100 \times 10^{-3})(20 \times 10^{-3})(60-20))(10 \times 8 \times 10^{-3}) ~W\)

Finally, we will calculate the fin effectiveness (\(\varepsilon\)), which is defined as the ratio of the actual heat transfer rate of the finned surface to the heat transfer rate of an equivalent unfinned surface. First, we'll find the heat transfer rate for an unfinned surface. The unfinned surface area (\(A_u\)) is the same as a single fin area as per the given width and height. \(A_u = (100 \times 10^{-3}) (20 \times 10^{-3}) ~m^2\) Now, calculate the heat transfer rate for the unfinned surface (\(q''_u\)) with the same heat transfer coefficient and temperature difference: \(q''_u = h A_u (T_s - T_\infty) = 100 (100 \times 10^{-3})(20 \times 10^{-3})(60-20) ~W/m\) Finally, calculate the fin effectiveness: \(\varepsilon = \frac{Q}{q''_u L_{fins}}\) Plug in the calculated values for \(Q\) and \(q''_u L_{fins}\).