Isentropic Nozzle Choking

Theory

For a perfect gas (with specific heat ratio {gamma} = 1.4 and gas constant R = 287 J/kg/K) undergoing isentropic expansion from a stagnation state, the mass flow rate depends on the Mach number M and local area A. The normalized mass flux is given by M * (6 / (5 + M²))³. It reaches exactly 1 when the flow is choked (M = 1). For M > 0, the ratio of the local area to the critical throat area (A / A*) is simply the inverse of the normalized mass flux. The static temperature and pressure drop continuously as the flow accelerates according to isentropic relations. At M = 0, the flow is at the stagnation limit, meaning zero flow velocity and unbounded area ratio.

Experiment

Use the interactive module to observe how the mass flow and static properties change with the Mach number and stagnation conditions.

1. Select a Mach number of 1. Observe that the normalized mass flux reaches its peak of 1.0, and the area ratio is exactly 1.0. This corresponds to the sonic throat.
2. Change the Mach number to 0.5. Notice that the branch changes to "Subsonic", the normalized mass flux is lower, and the required area ratio is larger than 1.
3. Change the Mach number to 2. Observe that the branch becomes "Supersonic". The required area ratio is again larger than 1, but the static pressure drops significantly below the total pressure.
4. Adjust the total pressure to 200000 Pa while keeping Mach at 1, and note the mass flow rate doubles because it is proportional to total pressure.

Worked Example

Consider a flow with a Mach number of 0.5, total pressure p0 = 100000 Pa, total temperature T0 = 300 K, and local area A = 0.001 m².

1. Calculate the normalized mass flux: M * (6 / (5 + 0.5²))³ = 0.5 * (6 / 5.25)³ = 0.5 * (1.142857)³ ≈ 0.746355685.
2. Calculate the area ratio A/A*: 1 / 0.746355685 ≈ 1.3398437500000002.
3. Calculate the static temperature T = T0 / (1 + 0.2 * 0.5²) = 300 / 1.05 = 285.7142857142857 K.
4. Calculate the static pressure p = p0 / (1 + 0.2 * 0.5²)^(7/2) = 100000 / (1.05)^3.5 ≈ 84301.91754225531 Pa.
5. Calculate the mass flow rate: (0.001 * 100000 / sqrt(300)) * sqrt(1.4 / 287) * 0.5 * (1.05)^-3 ≈ 0.17416646982950257 kg/s.

Limitations

This module assumes steady, 1D, isentropic flow of an ideal gas with constant specific heats. It does not account for friction, heat transfer, shock waves, or 3D effects. Furthermore, the actual mass flow in a real nozzle will be lower due to the boundary layer displacement thickness.

References

For more details on isentropic mass flow and nozzle choking, consult the NASA Glenn Research Center's educational resources on the topic.

• NASA - Mass Flow Choking