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Aerospace
Gross axial momentum flux from a surveyed exhaust plane
Axial momentum flux through an exhaust plane that has been surveyed in cells, as the sum of density times axial velocity squared times area over the cells.
- Ref
- AER-47
- Rev
- v1
- Field
- Aerospace
- Result
- Jx [N]
The thrust from a jet comes from momentum flux. When the exhaust plane is surveyed in small cells, each with density ρi, axial velocity uxi and area ΔAi, the gross axial momentum flux is Jx = Σρiuxi²ΔAi. The velocity is the component normal to the plane, and outflow is positive. The momentum term is only part of the thrust: add the pressure force (pe − pa)A and subtract the inlet momentum for net thrust.
Axial momentum flux through an exhaust plane that has been surveyed in cells, as the sum of density times axial velocity squared times area over the cells. u is the axial velocity normal to the plane. Use positive outflow and enough cells. It is the momentum term only; add pressure forces and subtract inlet momentum for net thrust. Up to three cells here; leave unused ones at zero area.
Calculator
- Mass flow, Σρ u ΔA12.3 kg/s
- Mean axial velocity, Jx/ṁ461 m/s
- Total surveyed area0.05 m²
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- The surveyed exhaust carries 5.67 kN of axial momentum flux through 12.3 kg/s of gas, an average speed of 460.98 m/s.
- The momentum-averaged speed Jx/ṁ is the effective exhaust velocity; a non-uniform plane has a higher value than the mass-weighted mean, so cell resolution matters.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (N) | Local step | Change in result |
|---|---|---|---|---|---|---|
| u2 Axial velocity in cell 2 | 0 to 1,000 m/s | 1,920 | 16,920 | 15,000 | +10% | +13.9 % |
| ρ2 Density in cell 2 | 0.05 to 2 kg/m³ | 2,295 | 16,920 | 14,630 | +10% | +6.61 % |
| ΔA2 Area of cell 2 | 0 to 0.1 m² | 1,920 | 14,420 | 12,500 | +10% | +6.61 % |
| u1 Axial velocity in cell 1 | 0 to 1,000 m/s | 3,750 | 15,750 | 12,000 | +10% | +7.11 % |
| ΔA3 Area of cell 3 | 0 to 0.1 m² | 5,670 | 15,800 | 10,130 | +10% | 0 % |
| ΔA1 Area of cell 1 | 0 to 0.1 m² | 3,750 | 13,350 | 9,600 | +10% | +3.39 % |
| ρ1 Density in cell 1 | 0.05 to 2 kg/m³ | 3,910 | 10,150 | 6,240 | +10% | +3.39 % |
| ρ3 Density in cell 3 | 0.05 to 2 kg/m³ | 5,670 | 5,670 | 0 | +10% | 0 % |
| u3 Axial velocity in cell 3 | 0 to 1,000 m/s | 5,670 | 5,670 | 0 | +10% | 0 % |
Sweep of Axial velocity in cell 1
| Axial velocity in cell 1 (m/s) | Axial momentum flux (N) |
|---|---|
| 0 | 3,750 |
| 125 | 3,938 |
| 250 | 4,500 |
| 375 | 5,438 |
| 500 | 6,750 |
| 625 | 8,438 |
| 750 | 10,500 |
| 875 | 12,940 |
| 1,000 | 15,750 |
Sweep one variable
Variables and how each one contributes
Axial momentum flux
What the equation gives you. Everything else feeds it.
Density in cell 1
The air density in cell 1, in kg/m³. Raises that cell’s momentum flux in direct proportion. Raising it by 10 % (to 0.66) moves the result by +3.39 %, from 5,670 to 5,862 N.
Density in cell 2
The air density in cell 2, in kg/m³. Raises that cell’s momentum flux in direct proportion. Raising it by 10 % (to 0.55) moves the result by +6.61 %, from 5,670 to 6,045 N.
Density in cell 3
The air density in cell 3, in kg/m³. Raises that cell’s momentum flux in direct proportion. Raising it by 10 % (to 0.55) moves the result by +0 %, from 5,670 to 5,670 N.
Axial velocity in cell 1
The axial velocity in cell 1, in m/s. Raises that cell’s momentum flux with the square of the velocity. Raising it by 10 % (to 440) moves the result by +7.11 %, from 5,670 to 6,073 N.
Axial velocity in cell 2
The axial velocity in cell 2, in m/s. Raises that cell’s momentum flux with the square of the velocity. Raising it by 10 % (to 550) moves the result by +13.9 %, from 5,670 to 6,458 N.
Axial velocity in cell 3
The axial velocity in cell 3, in m/s. Raises that cell’s momentum flux with the square of the velocity. Raising it by 10 % (to 495) moves the result by +0 %, from 5,670 to 5,670 N.
Area of cell 1
The area of cell 1, in m² (zero if absent). Raises that cell’s share of the total in direct proportion. Raising it by 10 % (to 0.022) moves the result by +3.39 %, from 5,670 to 5,862 N.
Area of cell 2
The area of cell 2, in m² (zero if absent). Raises that cell’s share of the total in direct proportion. Raising it by 10 % (to 0.033) moves the result by +6.61 %, from 5,670 to 6,045 N.
Area of cell 3
The area of cell 3, in m² (zero if absent). Raises that cell’s share of the total in direct proportion. Raising it by 10 % (to 0) moves the result by +0 %, from 5,670 to 5,670 N.
Worked example with the values above
- Cell terms ρu²ΔA: cell 1 = 0.6 × 400² × 0.02 = 1,920 N; cell 2 = 0.5 × 500² × 0.03 = 3,750 N.
- Sum of the cell terms: 5,670 N.
- Jx = 5,670 N.
Assumptions and limits
- Steady flow, with each cell having uniform density and axial velocity over its area.
- Velocity measured normal to the plane, with outflow positive, and unused cells entered with zero area.
Common mistakes
- Using total velocity instead of the component normal to the plane.
- Entering density from stagnation instead of static conditions in each cell.
- Treating the result as net thrust without adding the pressure term and subtracting the inlet momentum.
Statistics and uncertainty
A survey of a few cells is a coarse integral. Real exhaust planes are non-uniform, and a 3-cell survey can miss thin boundary layers or hot streaks; the momentum flux then has an uncertainty of several per cent, and finer cells and measured density and velocity at each are needed for test-grade thrust.
Where it is used
- Sizing throats and predicting thrust of jet and rocket nozzles.
- Estimating mass flow through orifices, valves and venturis.
Maths you need
- Critical pressure ratio: (2/(γ+1))^(γ/(γ−1)), about 0.528 for air
- Momentum flux: ρu²A, the momentum carried by a stream
- Nozzle efficiency: actual divided by ideal kinetic energy