91Ó°ÊÓ

The First Law of Thermodynamics is a principle of conservation of energy. It states that the energy in an isolated system is constant. Energy cannot be created or destroyed, only transformed from one form to another. In this problem, the electronic components and fan generate electrical power. This power is converted into thermal energy, which must be balanced within the system to avoid overheating.

The formula embodying the First Law for a control volume, assuming steady state, is: \[ \text{Total Energy In} = \text{Total Energy Out} \] In our case: \[ Q_{in} = P_{total} \] Understanding this concept is crucial because it tells us the total energy input of the system must equal the energy taken away by the airflow.

Energy balance involves making sure that the total energy entering a system equals the total energy leaving it. This ensures there is no accumulation of unaccounted energy within the system.

In the given problem, the electronic components consume 0.1 kW of power, and the fan adds 0.025 kW. Summing these gives the total energy input: \[P_{total} = 0.1 \text{ kW} + 0.025 \text{ kW} = 0.125 \text{ kW} \]

This total energy input must be equal to the rate of energy change in the air, which is calculated using the specific heat capacity and temperature difference of the air in and out of the enclosure.

The Ideal Gas Law relates the pressure, volume, and temperature of an ideal gas with the amount of gas present: \[ PV = nRT \] Where:

• P is pressure
• V is volume
• n is the number of moles
• R is the universal gas constant
• T is temperature in Kelvin

In this exercise, it's used to convert mass flow rate to volumetric flow rate. We rearrange the ideal gas law to solve for the volumetric flow rate \[ \frac{{\text{Mass Flow Rate} \times R \times T_{\text{in}}}} {P_{\text{atm}}} \] Here, we must ensure temperature is in Kelvin and pressure in Pascals. This allows us to find the volumetric flow rate in m³/s, which is crucial for understanding how much air is flowing through the system.

Specific heat capacity (\text{cp}) is a material's ability to absorb heat energy. It is the amount of energy required to raise the temperature of 1 kg of a substance by 1 degree Celsius. For air, the specific heat capacity is typically 1.005 kJ/kg·K.

In our problem, we use \text{cp} to relate the input energy to the temperature change of air flowing into and out of the system: \[ Q_{in} = \text{Mass Flow Rate} \times \text{cp} \times \text{Temperature Change} \] Rearranging this equation allows us to solve for the mass flow rate: \[ \text{Mass Flow Rate} = \frac{Q_{in}}{\text{cp} \times (T_{\text{out}} - T_{\text{in}})} \] Understanding \text{cp} is essential because it tells us how much energy is required to achieve the desired cooling effect.