91Ó°ÊÓ

The Ideal Gas Law is a crucial concept in thermodynamics that helps us understand the relationship between pressure, volume, temperature, and amount of gas. The law can be expressed through the equation \( PV = nRT \), where:

• \( P \) is the pressure of the gas,
• \( V \) is the volume it occupies,
• \( n \) represents the number of moles of gas,
• \( R \) is the ideal gas constant,
• \( T \) is the absolute temperature of the gas.

This law assumes that the gas behaves ideally, meaning interactions between gas molecules are negligible and volume occupied by the gas molecules themselves is small compared to the volume of their container.

In practical applications like the heating system problem, the Ideal Gas Law can be used to find changes in gas behavior under varying conditions. For example, when a gas is heated in a duct, its temperature changes, which affects its pressure and volume proportionally. This principle allows us to relate initial and final states of the gas to determine volumetric flow rates and duct velocity.

Volumetric Flow Rate is a measure of the volume of fluid (or gas in this case) that passes through a given area per unit of time. It is usually denoted as \( Q \) and measured in units such as cubic feet per minute (cfm).

To determine the flow rate before and after heating in our problem, we used the Ideal Gas Law in a form suited for changes in state: \[ \frac{Q_{1}}{T_{1}} \cdot P_{1} = \frac{Q_{2}}{T_{2}} \cdot P_{2} \] where:

• \( Q_{1} \) = initial volumetric flow rate,
• \( T_{1} \) = initial temperature in Rankine,
• \( P_{1} \) = initial pressure,
• \( Q_{2} \) = final volumetric flow rate,
• \( T_{2} \) = final temperature in Rankine,
• \( P_{2} \) = final pressure.

By rearranging and solving this equation, you can find the final flow rate \( Q_{2} \), revealing how heating affects the volume of air delivered by the system.

Duct Velocity refers to the speed at which air travels through a duct. It is critical in HVAC (Heating, Ventilation, and Air Conditioning) applications to ensure adequate air delivery and uniform heating or cooling.

The velocity \( V \) can be calculated using the formula:\[ V = \frac{Q_{2}}{A} \] where \( Q_{2} \) is the volumetric flow rate we found earlier, and \( A \) is the cross-sectional area of the duct.

In the problem, we determine \( A \) for a square duct using the formula for an area of a square \( (\text{side})^2 \). Given a duct that is \( 0.5 \) ft by \( 0.5 \) ft, the area \( A \) equals \( 0.25 \, \text{ft}^2 \). Using this area, along with the volume flow rate post-heating \( Q_{2} \), allows us to solve for the duct velocity, telling us how fast the air is moving once it's been heated.

Temperature Conversion is essential when working with thermodynamic equations, as these often require temperatures to be in an absolute scale. For gas law scenarios, temperatures should be in Rankine (for use in US customary units) rather than degrees Fahrenheit.

The conversion from Fahrenheit to Rankine is straightforward and follows the formula:\[ T_{R} = T_{F} + 459.67 \] where:

• \( T_{R} \) is the temperature in Rankine,
• \( T_{F} \) is the temperature in Fahrenheit.

This conversion ensures that calculations involving the Ideal Gas Law and other thermal dynamics remain consistent and accurate. In our exercise, both the initial and final temperatures were converted to Rankine, which were essential for calculating changes in volumetric flow rate and ultimately, the duct velocity.