91Ó°ÊÓ

The Ideal Gas Law is a useful equation that describes the state of an ideal gas. It is written as:
\[ P V = n R T \]
where:

• P is the pressure of the gas
• V is the volume of the gas
• n is the amount of substance (in moles)
• R is the universal gas constant
• T is the absolute temperature

In many engineering applications, it's often more practical to use the mass form of the ideal gas law:
\[ P V = m R T \]
where m is the mass of the gas and R is the specific gas constant. For our problem, we need to find the density ÒÏ to later find the area.
Using the ideal gas law and rearranging for density \( \rho \):
\[ \rho = \frac{P}{RT} \]
This relationship helps us express density in terms of pressure, temperature, and the specific gas constant. This will be crucial when we use the continuity equation to determine the exit area.

Enthalpy is a measure of the total energy of a thermodynamic system. For an ideal gas, the enthalpy (\( h \)) can be related to temperature (\( T \)) using the specific heat at constant pressure (\( c_p \)). The equation is:
\[ h = c_p T \]
In the context of our nozzle problem, we use the steady-state energy equation:
\[ \frac{V_1^2}{2} + h_1 = \frac{V_2^2}{2} + h_2 \]
Given:

• \( V_1 = 20 \) m/s
• \( h_1 = c_p T_1 = 1.039 \times 340 \)
• \( V_2 = 478.8 \) m/s
• \( h_2 = c_p T_2 \)

By substituting these values into the equation, we solve for the exit temperature \( T_2 \). This approach ensures that we account for the changes in kinetic and thermal energy of the nitrogen gas as it passes through the nozzle.

The continuity equation is fundamental to fluid dynamics. It expresses the conservation of mass in a fluid flow system. The general form is:
\[ \dot{m} = \rho A V \]
where:
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\( \dot{m} \) is the mass flow rate

\( \rho \) is the fluid density

\( A \) is the cross-sectional area

\( V \) is the flow velocity

For our problem, we solve for the cross-sectional exit area (\( A_2 \)) using the known values for the mass flow rate (3 kg/s), the exit velocity (478.8 m/s), and the density at the exit (\( \rho_2 \)). Applying the ideal gas law helps in finding \( \rho_2 \), then the continuity equation guides us to determine the necessary exit area for the gas flow.

The mass flow rate (\( \dot{m} \)) is a critical concept when analyzing fluid systems. It is defined as the mass of substance passing through a given surface per unit time. For a given conduit, it can be written as:
\[ \dot{m} = \rho A V \]
In the nozzle problem, the mass flow rate is constant and is given as 3 kg/s. This value is important because it helps establish the relationship between the variables of interest in both the inlet and outlet conditions. Knowing \( \dot{m} \), we can further calculate exit parameters such as area and velocity.

To solve for A₂ (exit area):

• Calculate density \( \rho_2 \) using the ideal gas law
• Substitute \( \rho_2 \), A₂ and V₂ in the continuity equation

Hence, knowing and understanding the mass flow rate is crucial to solving fluid flow problems involving changes in flow area and velocity.