Question

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?

Short answer

Answer: The fan must deliver air at a flow rate of 0.00896 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°C.

Step-by-step solution

1. Calculate total power dissipation

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

2. Determine the air flow rate

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 \(\dot{Q}=\frac{90\,\text{W}}{10\,\text{K}\cdot1005\,\text{J}/\text{kg}\cdot \text{K}}=0.00896\,\text{kg/s}\).

3. Calculate the temperature rise due to the fan and motor

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: \(\Delta T_\text{fan}=\frac{10\,\text{W}}{0.00896\,\text{kg/s}\cdot 1005\,\text{J}/\text{kg}\cdot \text{K}}=1.11\,\text{K}\)

4. Calculate the fraction of temperature rise due to the fan

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: \(\frac{\Delta T_\text{fan}}{\Delta T_\text{total}}=\frac{1.11\,\text{K}}{10\,\text{K}}=0.111\)

5. Calculate the highest allowable inlet air temperature

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 \(70^{\circ} \mathrm{C} - 10^{\circ} \mathrm{C} = 60^{\circ} \mathrm{C}\).

6. Evaluate the properties of air and verify the initial assumption

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 \(25^{\circ} \mathrm{C}\).

Key concepts

PCB Power Dissipation

When it comes to cooling electronic systems such as computers, understanding PCB power dissipation is fundamental. Each PCB in the computer system generates heat due to electrical power being converted into thermal energy. The rate at which a PCB dissipates power is quantified in watts (W) and can be affected by various factors, including the efficiency of the electrical components and their operating voltage and current.

In our example, eight PCBs, each dissipating an individual power of 10 W, result in a significant thermal load that must be managed to prevent overheating. The cooling fan adds to this thermal load, with its operation also dissipating power. The total heat load to be removed therefore includes not only the heat from the PCBs but also from the cooling fan. This is generally summarized as the total power dissipation of the system.

Air Flow Rate Calculation

The air flow rate calculation is critical for designing an effective cooling system. The flow rate refers to the volume of air that must be moved through the system per unit of time to achieve sufficient cooling. The air flow rate is determined by the total power dissipation and the permissible temperature rise of the air.

Specifically, the calculation uses the equation \[\begin{equation}\dot{Q} = \frac{P_{\text{total}}}{\Delta T \cdot c_p}\end{equation}\]\br where \(\dot{Q}\) is the air flow rate in kg/s, \(P_{\text{total}}\) is the total power dissipation in watts, \(\Delta T\) is the allowable temperature increase in kelvins, and \(c_p\) is the specific heat capacity of air at constant pressure. By computing this, technicians can ensure that the fan chosen for the system is capable of delivering the required flow rate.

Temperature Rise due to Cooling Fan

The cooling fan does not just facilitate cooling; it also introduces additional heat to the system since it consumes power to operate. The temperature rise due to the cooling fan can be calculated by determining how much the fan's power consumption heats the air passing through the computer case.

The temperature increase specifically attributable to the cooling fan is found with the equation \[\begin{equation}\Delta T_{\text{fan}} = \frac{P_{\text{fan}}}{\dot{Q} \cdot c_p}\end{equation}\]\br where \(P_{\text{fan}}\) is the power dissipation of the fan and \(\dot{Q}\) and \(c_p\) retain their previous meanings. Acknowledging this aspect of cooling system design is essential because it influences the total permitted temperature rise within the computer case.

Max Allowable Inlet Air Temperature

Managing the max allowable inlet air temperature is crucial to prevent damage from excessive heat. For a computer's cooling system to function effectively, the air entering the system must be below a certain temperature to ensure the components operate within safe thermal limits.

The highest allowable inlet air temperature is determined by subtracting the permissible temperature rise from the maximum operating temperature of the components. Using the formula \[\begin{equation}\text{Highest Inlet Temperature} = T_{\text{max components}} - \Delta T_{\text{total}}\end{equation}\]\br ensures that the heat added by the PCBs and the fan will not push the component temperatures beyond safe operating levels. Maintaining proper inlet air temperature is, therefore, a balancing act between external temperatures, heat generated by the system, and the effectiveness of the cooling process.

Evaluation of Air Properties

To conduct precise calculations in the design of cooling systems, evaluation of air properties at different temperatures is essential. Properties such as specific heat capacity (\(c_p\))) and dynamic viscosity (\(\mu\))) of air are temperature-dependent and crucial for determining flow rates and assessing whether the flow is laminar or turbulent.

In our example, properties of air at a bulk mean temperature of 25°C were used for calculations. This evaluation is justified by the temperature conditions expected during the system's operation. However, this is an approximation, and in a more detailed analysis, these properties might be evaluated at several points to account for the air's temperature increase as it absorbs heat from the components. Overall, the assumption of fully developed flow should be verified against the Reynolds number to ensure the cooling system's design is based on accurate airflow dynamics.