A computer cooled by a fan contains eight printed circuit boards (PCBs), each dissipating \(10 \mathrm{~W}\) of power. The height of the PCBs is \(12 \mathrm{~cm}\) and the length is \(18 \mathrm{~cm}\). The clearance between the tips of the components on the \(P C B\) and the back surface of the adjacent \(P C B\) is \(0.3 \mathrm{~cm}\). The cooling air is supplied by a 10 -W fan mounted at the inlet. If the temperature rise of air as it flows through the case of the computer is not to exceed \(10^{\circ} \mathrm{C}\), determine (a) the flow rate of the air that the fan needs to deliver, \((b)\) the fraction of the temperature rise of air that is due to the heat generated by the fan and its motor, and ( \(c\) ) the highest allowable inlet air temperature if the surface temperature of the components is not to exceed \(70^{\circ} \mathrm{C}\) anywhere in the system. As a first approximation, assume flow is fully developed in the channel. Evaluate properties of air at a bulk mean temperature of \(25^{\circ} \mathrm{C}\). Is this a good assumption?

The total power dissipation of the system consists of 8 PCBs and the cooling fan: \(P_\text{total}=8\times10\,\text{W}+10\,\text{W}=90\,\text{W}\).

Using the relation: \(\dot{Q}=P_\text{total}\ /\ (\Delta T\cdot c_p)\), we can find the flow rate of the air. Before that, let's find the heat capacity of air at \(25^{\circ} \mathrm{C}\). At this temperature, we approximate air's heat capacity \(c_p\approx1005\,\text{J}/\text{kg}\cdot \text{K}\). Now, solving for the air flow rate, we get \(\dot{Q}=\frac{90\,\text{W}}{10\,\text{K}\cdot1005\,\text{J}/\text{kg}\cdot \text{K}}=0.00896\,\text{kg/s}\).

The fan generates \(10\,\text{W}\) of heat. Using the calculated air flow rate, we can determine the temperature rise due to the fan and motor: \(\Delta T_\text{fan}=\frac{10\,\text{W}}{0.00896\,\text{kg/s}\cdot 1005\,\text{J}/\text{kg}\cdot \text{K}}=1.11\,\text{K}\)

The total temperature rise of the system is \(10^{\circ} \mathrm{C}\). To find the fraction that corresponds to the temperature rise due to the fan, we have to divide the fan-induced temperature rise by the total temperature rise: \(\frac{\Delta T_\text{fan}}{\Delta T_\text{total}}=\frac{1.11\,\text{K}}{10\,\text{K}}=0.111\)

The critical surface temperature of the components is \(70^{\circ} \mathrm{C}\), and the maximum temperature rise is \(10^{\circ} \mathrm{C}\). Therefore, the highest allowable inlet air temperature must be \(70^{\circ} \mathrm{C} - 10^{\circ} \mathrm{C} = 60^{\circ} \mathrm{C}\).

We can evaluate the properties of air at a bulk mean temperature of \(25^{\circ} \mathrm{C}\) using tables or online resources. As stated in the question, we should verify if the flow is fully developed in the channel. Assuming the channel is horizontal, we can check if Reynolds number based on height \(h\) (clearance between the tips of the components) satisfies Laminar regime i.e., \(Re_h=\frac{4\dot{Q}}{\pi h \mu}<2300\), where \(\mu\) is the dynamic viscosity of air (approximately \(1.84\times10^{-5}\,\text{kg/m}\cdot \text{s}\) at \(25^{\circ} \mathrm{C}\) ). By plugging in the values and calculating, we get: \(Re_h=\frac{4\times0.00896\,\text{kg/s}}{\pi\times0.003\,\text{m}\times(1.84\times10^{-5}\,\text{kg/m}\cdot\text{s})}=863.9\). Since \(Re_h<2300\), our initial assumption that the flow is fully developed is valid. In conclusion, the fan must deliver air at a flow rate of \(0.00896\,\text{kg/s}\), the fraction of temperature rise due to the fan and motor is \(0.111\), and the highest allowable inlet air temperature is \(60^{\circ} \mathrm{C}\). The assumption of fully developed flow in the channel is valid, as confirmed by evaluating the properties of air at a bulk mean temperature of \(25^{\circ} \mathrm{C}\).