A hot surface at \(80^{\circ} \mathrm{C}\) in air at \(20^{\circ} \mathrm{C}\) is to be cooled by attaching 10 -cm-long and 1 -cm-diameter cylindrical fins. The combined heat transfer coefficient is \(30 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), and heat transfer from the fin tip is negligible. If the fin efficiency is \(0.75\), the rate of heat loss from 100 fins is (a) \(325 \mathrm{~W}\) (b) \(707 \mathrm{~W}\) (c) \(566 \mathrm{~W}\) (d) \(424 \mathrm{~W}\) (e) \(754 \mathrm{~W}\)

To calculate the heat transfer, we need to find the area of one fin. The fin is cylindrical with a length of 10 cm and a diameter of 1 cm. The surface area of a cylinder (excluding the top and bottom) can be found using the following formula: \(A_\text{fin} = 2 \pi r_\text{fin} L_\text{fin}\) where: \(A_\text{fin}\): the surface area of one fin \(r_\text{fin}= \dfrac{D_\text{fin}}{2}\): the radius of the fin \(L_\text{fin}\): the length of the fin We convert the length and diameter to meters and substitute the given values into the formula: \(r_\text{fin} = \dfrac{0.01 \, \text{m}}{2} = 0.005 \, \text{m}\) \(L_\text{fin} = 0.1 \, \text{m}\) \(A_\text{fin} = 2 \pi (0.005 \, \text{m}) (0.1 \, \text{m}) = 0.00314 \, \text{m}^2\)

Next, we need to find the temperature difference between the hot surface and the air. This is given by: \(\Delta T = T_\text{surface} - T_\text{air}\) Substituting the given values: \(\Delta T = 80^{\circ} \mathrm{C} - 20^{\circ} \mathrm{C} = 60 \mathrm{~K}\)

Now we can find the heat transfer rate of one fin using the formula: \(q_\text{fin} = h A_\text{fin} \eta_\text{fin} \Delta T\) where: \(q_\text{fin}\): the heat transfer rate of one fin \(h\): the combined heat transfer coefficient \(\eta_\text{fin}\): the fin efficiency \(\Delta T\): the temperature difference Substituting the given values and the values calculated in the previous steps: \(q_\text{fin} = (30 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}) (0.00314 \, \text{m}^2) (0.75) (60 \mathrm{~K}) = 4.243 \mathrm{~W}\)

Lastly, we multiply the heat transfer rate of one fin by the total number of fins to obtain the total heat transfer rate: \(q_\text{total} = N_\text{fins} q_\text{fin}\) Substituting the given number of fins and the value calculated in the previous step: \(q_\text{total} = 100 \; (4.243 \mathrm{~W}) = 424.3 \mathrm{~W}\) Now we can see that the answer is closest to option (d) \(424 \mathrm{~W}\).