A 15 -cm-wide and 18-cm-high vertical hot surface in \(20^{\circ} \mathrm{C}\) air is to be cooled by a heat sink with equally spaced fins of rectangular profile. The fins are \(0.1 \mathrm{~cm}\) thick, \(4 \mathrm{~cm}\) wide, and \(18 \mathrm{~cm}\) long in the vertical direction. Determine the optimum fin spacing and the rate of heat transfer by natural convection from the heat sink if the base temperature is \(85^{\circ} \mathrm{C}\).

First, we need to determine the Grashof and Prandtl numbers. For air at 20°C, we have the following properties: - Kinematic viscosity (\(\nu\)): \(15.69 \times 10^{-6} \mathrm{m}^2/\mathrm{s}\) - Thermal conductivity (\(k\)): \(0.02624 \mathrm{W}/\mathrm{m} \cdot \mathrm{K}\) - Thermal expansion coefficient (\(\beta\)): \(1/293 \mathrm{K}^{-1}\) - Specific heat (\(C_p\)): \(1007 \mathrm{J}/\mathrm{kg} \cdot \mathrm{K}\) - Density (\(\rho\)): \(1.205 \mathrm{kg}/\mathrm{m}^3\) The Grashof number (\(Gr\)) can be calculated using the formula: \(Gr = \frac{g \cdot \beta \cdot (T_s - T_\infty) \cdot L^3}{\nu^2}\) where \(g = 9.81 \mathrm{m/s^2}\) is the acceleration due to gravity, \(T_s = 85^{\circ} \mathrm{C} = 358 \mathrm{K}\) is the surface temperature, \(T_\infty = 20^{\circ} \mathrm{C} = 293 \mathrm{K}\) is the ambient temperature, and \(L = 0.18 \mathrm{m}\) is the height of the fin. The Prandtl number (\(Pr\)) is calculated as: \(Pr = \frac{\nu}{\alpha} = \frac{\nu \cdot \rho \cdot C_p}{k}\)

Based on the values of Grashof number and Prandtl number calculated, we can determine the Nusselt number (\(Nu\)) for natural convection using empirical correlations. A common correlation for vertical plates with natural convection is: \(Nu = 0.59 (Gr \cdot Pr)^{1/4}\)

Now we can calculate the heat transfer coefficient (\(h\)) using the formula: \(h = \frac{k}{L} \cdot Nu\)

To determine the optimum fin spacing (\(S_{opt}\)), we need to minimize the thermal resistance of the heat sink. The optimum fin spacing can be found using the following equation: \(S_{opt} = \frac{2 \cdot k_{f}}{h}\) where \(k_f = 0.02624\,\mathrm{W}/\mathrm{m}\cdot\mathrm{K}\) is the thermal conductivity of the fin material (same as air in this case).

Finally, to calculate the rate of heat transfer (\(Q\)), we can use the formula: \(Q = h \cdot A_T \cdot (T_s - T_\infty)\) where \(A_T\) is the total surface area of the fins, which can be calculated as: \(A_T = N \cdot (2 \cdot L + 2 \cdot W)\) where \(N\) is the number of fins given by the formula: \(N = \frac{S_{total}}{S_{opt} + t}\) Here, \(S_{total} = 0.15\,\mathrm{m}\) is the total width of the heat sink, \(t = 0.001\,\mathrm{m}\) is the fin thickness, and \(W = 0.04\,\mathrm{m}\) is the fin width.