Library / Propulsion / PRP-02
Propulsion
Rocket thrust
Thrust from the momentum of the exhaust plus a pressure term at the nozzle exit.
- Ref
- PRP-02
- Rev
- v1
- Field
- Propulsion
- Result
- F [kN]
ṁ is the propellant mass flow rate in kg/s, v_e the exhaust speed relative to the rocket in m/s, p_e the gas pressure at the nozzle exit and p_a the ambient pressure, both in kPa, and A_e the exit area in m². The first term is the momentum thrust and the second the pressure thrust. The answer is in kilonewtons.
Thrust from the momentum of the exhaust plus a pressure term at the nozzle exit. Thrust rises as ambient pressure falls.
Calculator
- Momentum thrust ṁ ve125 kN
- Pressure thrust (pe − pa)Ae9.74 kN
- Specific impulse F/(ṁ g₀)274.8 s
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.
- At sea level the defaults give 135 kN, of which 9.7 kN is pressure thrust because the exit pressure (150 kPa) is above ambient (101.3 kPa).
- Thrust rises as the rocket climbs: in vacuum, with pa = 0, this engine would give 155 kN, 15 % more.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (kN) | Local step | Change in result |
|---|---|---|---|---|---|---|
| ṁ Propellant flow rate | 0.1 to 5,000 kg/s | 9.99 | 12,510 | 12,500 | +10% | +9.28 % |
| Ae Nozzle exit area | 0.001 to 10 m² | 125 | 612 | 487 | +10% | +0.723 % |
| ve Exhaust speed | 500 to 5,000 m/s | 34.74 | 259.7 | 225 | +10% | +9.28 % |
| pe Nozzle exit pressure | 0 to 1,000 kPa | 104.7 | 304.7 | 200 | +10% | +2.23 % |
| pa Ambient pressure | 0 to 101.3 kPa | 134.7 | 155 | 20.26 | +10% | −1.5 % |
Sweep of Propellant flow rate
| Propellant flow rate (kg/s) | Thrust (kN) |
|---|---|
| 0.1 | 9.99 |
| 0.3867 | 10.71 |
| 1.495 | 13.48 |
| 5.782 | 24.2 |
| 22.36 | 65.64 |
| 86.47 | 225.9 |
| 334.4 | 845.7 |
| 1,293 | 3,242 |
| 5,000 | 12,510 |
Sweep one variable
Variables and how each one contributes
Thrust
What the equation gives you. Everything else feeds it.
Propellant flow rate
The mass of propellant burned each second, in kg/s. Almost proportional at the default: the momentum term is 93 % of the total, so a 10 % rise in ṁ raises F by about 9 %.
Exhaust speed
The speed of the exhaust gas leaving the nozzle relative to the rocket. Almost proportional, as for ṁ: a 10 % rise in ve raises F by about 9 %.
Nozzle exit pressure
The gas pressure across the nozzle exit plane, in kPa. Raises thrust through the pressure term only: a 10 % rise (15 kPa) adds about 3 kN.
Ambient pressure
The ambient pressure outside the nozzle, in kPa (101.3 at sea level, 0 in vacuum). Lowers thrust as ambient pressure rises. Climbing to vacuum adds (pa × Ae) to the thrust.
Nozzle exit area
The area of the nozzle exit, in m². Raises thrust through the pressure term only while pe exceeds pa. If the nozzle is over-expanded (pe below pa), a larger exit area reduces thrust.
Worked example with the values above
- Momentum thrust: ṁ ve = 50 × 2,500 = 125,000 N.
- Pressure thrust: (pe − pa)Ae = (150 − 101.3) kPa × 0.2 m² = 9.74 kN.
- F = 125 + 9.74 = 134.7 kN.
Assumptions and limits
- Steady one-dimensional flow with an effective exhaust speed ve.
- Pressures measured at the exit plane and the ambient pressure uniform over the outside of the nozzle.
Common mistakes
- Entering the pressures in Pa when the boxes expect kPa, which makes the pressure term 1000 times too large.
- Ignoring the pressure term. A nozzle with a large exit area can lose thrust at low altitude when pe is below pa.
Statistics and uncertainty
The momentum term dominates, so the errors in ṁ and ve pass straight through. The pressure term is a product of a difference of two pressures and the area, and it is the less certain: an error of 10 kPa in pe is 2 kN, more than a 1 % error in the momentum term. Specific impulse is usually the number quoted, since it takes out the flow rate.
Where it is used
- Estimating thrust at sea level and in vacuum.
- Checking that a nozzle is well matched to the altitude it will fly at.
Maths you need
- Momentum: force equals rate of change of momentum
- Pressure and area: force = pressure × area