The objective of this problem is to analyze the flow of air, assumed to behave as an ideal gas, through a converging diverging nozzle. The transition point from the converging to the diverging section represents the "critical point" in this flow. The pressure and temperature at this critical point are referred to as the critical pressure and critical temperature, respectively.
First, determine where the nozzle narrows down and where it begins to widen again. The following assumptions are made to simplify the problem:
Air behaves as an ideal gas: $P v = R T$
There is no entropy change throughout the flow (an isentropic process).
As the flow progresses, the pressure continuously decreases while the entropy of the air remains constant.
Given Data:
Inlet pressure: $P_i = 1500 kPa$
Inlet temperature: $T_i = 600 K$
Mass flow rate: $\dot{m} = 10 kg/s$ (constant along the flow)
Inlet velocity: $V_{in} = 10 m/s$
$c_p = 1.0 kJ/(kg \cdot K)$ (constant along the process)
$\gamma = 1.38$ (constant along the process)
Molar mass of air: 29 g/mol
The outlet pressure of the nozzle is $P_{out} = 100 kPa$. Reduce the pressure in steps of 100 kPa from $P_i$ to $P_{out}$, and for each pressure value, determine:
1. Nozzle cross-sectional area
2. Pressure of air
3. Temperature of air
4. Density of air
5. Row velocity of air
6. Mach number ($M_a$)
Assumptions:
The mass flow rate is constant.
Energy is conserved $h + \frac{V^2}{2}$.
Entropy is conserved ($s$).
For each pressure value, determine the temperature assuming isentropic conditions. Using the known pressure and temperature, calculate the density of the air. From the temperature, calculate enthalpy and flow velocity using energy conservation. Using mass conservation and flow velocity, determine the nozzle cross sectional area. Finally, calculate the local speed of sound and determine the Mach number.
Questions:
1. What is the smallest cross sectional area along the flow? (It may not correspond to one of the 100 kPa steps).
2. If the nozzle operates between two tanks with an inlet pressure of 1500 kPa and an outlet pressure of 50 kPa, what would the mass flow rate of air be?
3. For the same nozzle, if no supersonic flow occurs anywhere ($M_a < 1$), what should the outlet pressure be set to when the inlet pressure is 1500 kPa?
