Question

The fan on a personal computer draws \(0.3 \mathrm{ft}^{3} / \mathrm{s}\) of air at 14.7 psia and \(70^{\circ} \mathrm{F}\) through the box containing the \(\mathrm{CPU}\) and other components. Air leaves at 14.7 psia and \(83^{\circ} \mathrm{F}\) Calculate the electrical power, in \(\mathrm{kW}\), dissipated by the \(\mathrm{PC}\) components.

Short answer

Answer: The electrical power dissipated by the PC components is approximately 0.000548 kW.

Step-by-step solution

Calculate the mass flow rate of air

To determine the heat carried away by the air flow, we first need to find the mass flow rate of air. We can calculate it using the given volumetric flow rate, and knowing the air's density at the inlet conditions. We have the following formula for the ideal gas: \(\rho = \frac{P}{R_{air}T}\) Where \(\rho\) is the air density, P is the pressure, and \(R_{air}\) and T are the air's specific gas constant and temperature respectively. Given the specific gas constant for air, \(R_{air} = 1716 \ \mathrm{ft.lb/(lb_m R)}\) and using the Rankine temperature scale (70°F + 460 = 530°R): \(\rho = \frac{14.7\mathrm{psia}}{1716\ \mathrm{ft.lb/(lb_m R)} * 530\mathrm{R}} = 0.002\mathrm{lb_m/ft^3}\) Now, we can find the mass flow rate by multiplying the density by the volumetric flow rate: \(\dot{m} = \rho \dot{V} = 0.002\mathrm{lb_m/ft^3} * 0.3\mathrm{ft^3/s} = 0.0006\,\mathrm{lb_m/s}\)

Calculate the heat carried away by the air

Now, we'll use the specific heat capacity of air at constant pressure (\(C_p\)) to find the heat carried away by the air flow. Given \(C_p\) of air is approximately \(0.240\,\mathrm{Btu/(lb_m°F)}\). The heat carried away by the air flow can be calculated as follows: \(Q_{out} = \dot{m} C_p (T_{out} - T_{in})\) Where \(Q_{out}\) is the heat carried away by the air flow, \(T_{out}\) and \(T_{in}\) are the outlet and inlet air temperatures, respectively. \(Q_{out} = 0.0006\,\mathrm{lb_m/s} \times 0.240\,\mathrm{Btu/(lb_m°F)} \times (83°F - 70°F) = 0.001872\,\mathrm{Btu/s}\)

Calculate the electrical power dissipated by the PC components

Using the energy conservation principle, we know that the electrical power dissipated by the PC components equals the heat carried away by the air flow. So, we have: \(P_{in} = Q_{out}\) However, power in Watts is given by: \(P_{in} = \frac{Q_{out}}{3.412\,\mathrm{Btu/(W.s)}}\) Now, we can calculate the electrical power dissipated by the PC components: \(P_{in} = \frac{0.001872\,\mathrm{Btu/s}}{3.412\,\mathrm{Btu/(W.s)}} = 0.000548\,\mathrm{kW}\) The electrical power dissipated by the PC components is approximately 0.000548 kW.

Key concepts

Mass Flow Rate

Mass flow rate is a crucial concept in thermodynamics that measures the amount of mass passing through a cross-sectional area per unit time. It is denoted by the symbol \( \dot{m} \). In the context of an air flow problem, the mass flow rate is critical to understand because it determines the quantity of air (usually in pounds per second in the Imperial system) that is capable of carrying away heat from a heat-generating device like a computer's CPU.

In the provided exercise, the mass flow rate is deduced from the volumetric flow rate (cubic feet per second) along with the air density. The air density is calculated using the ideal gas law, which links the pressure, temperature, and specific gas constant of air. Subsequently, the calculated density \( \rho \) is multiplied by the given volumetric flow rate \( \dot{V} \) to arrive at the mass flow rate. This step is pivotal because it lays the groundwork for determining how much heat is removed by the air flow—a key factor in assessing the cooling efficiency in systems like a personal computer.

Specific Heat Capacity

Specific heat capacity \( C_p \) refers to the amount of heat energy required to raise the temperature of a unit mass of a substance by one degree. It is usually expressed in terms of energy per mass per degree, such as BTU per pound-degree Fahrenheit in the Imperial system.

In our exercise, the specific heat capacity of air plays a central role in determining the heat carried away by the airflow. The formula \( Q_{out} = \dot{m} C_p (T_{out} - T_{in}) \) illustrates how to calculate the heat energy removed from the CPU, taking into account the temperature difference between the air entering and leaving the computer (\(T_{in}\) and \(T_{out}\), respectively) and the aforementioned mass flow rate. Understanding the specific heat capacity allows for an accurate calculation of this heat exchange, which directly affects the cooling performance and overall health of computer components.

Energy Conservation Principle

The energy conservation principle is a fundamental concept in physics and thermodynamics stating that energy cannot be created or destroyed, only transferred or converted from one form to another. In practical applications like the operation of a computer, this principle implies that the electrical power consumed by the computer components is converted into heat.

This heat must be adequately dissipated to prevent overheating; this is where the importance of airflow comes into play. The exercise demonstrates that the heat removed by the airflow \( Q_{out} \) from the PC components is equivalent to the electrical power \( P_{in} \) consumed, as per the energy conservation principle. By calculating the heat transfer, we are also indirectly determining the power dissipation of the CPU, enabling us to monitor the energy efficiency and thermal management of the computer system.

Ideal Gas Law

The ideal gas law is a powerful equation that relates the pressure \( P \), volume \( V \), temperature \( T \), and the amount of gas in moles \( n \) to a constant \( R \), which depends on the gas being considered. For any given gas, as long as the conditions are not too extreme, the law provides a good approximation to the behavior of the gas. The equation is given as \( PV = nRT \).

In the context of our exercise involving airflow within a computer, we're concerned with the derivative form that relates pressure, specific gas constant \( R_{air} \) for air, and temperature to density \( \rho \): \(\rho = \frac{P}{R_{air}T}\). By plugging in the values given for air pressure and temperature, we're able to calculate the density of the air, which is a necessary variable for further calculations, such as determining the mass flow rate. Understanding the ideal gas law is vital, as it allows us to predict and manipulate the properties of the gas, hence managing the thermal behavior of systems.