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Turbojet Thrust Equation

General thrust equation for an airbreathing jet engine including momentum and pressure-imbalance terms.

Written by the Vorzep Team

PropulsionJet EnginesFundamental

Turbojet Thrust Equation Calculator

F = ṁ · (Ve − V0) + Ae · (pe − p0)
Solve for F, ṁ, Ve, V0, Ae, pe, or p0
F ⟷ṁ, Ve, V0, Ae, pe, p0
N
kg/s
m/s
m/s
m²
Pa
Pa
Solve for:
Result —
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Thrust vs. Mass Flow Rate F(ṁ) for fixed Ve, V0, Ae, pe, p0
F(ṁ) for fixed parameters Computed point
F = ṁ·(Ve−V0) + Ae·(pe−p0) • All values in SI units

Interpretation

Turbojet thrust: F = ṁ·(V_e − V_0) + A_e·(p_e − p_0), where ṁ is mass flow, V_e exit velocity, V_0 flight speed, A_e exit area, p_e exit pressure, p_0 ambient. It is the general thrust equation. Example: ṁ=50 kg/s, V_e=400 m/s, V_0=250 m/s, p_e=1 atm, p_0=0.5 atm, A_e=0.3 m² → F = 50×150 + 0.3×(101325−50662.5) = 7500 + 15199 ≈ 22,699 N.

F = mdot * (V_e - V_0) + A_e * (p_e - p_0)
Turbojet Thrust Equation

Variables

SymbolQuantityUnit
FNet thrustN
mdotMass flow ratekg/s
V_eExhaust velocitym/s
V_0Freestream velocitym/s
A_eExit aream2
p_eExit static pressurePa
p_0Ambient pressurePa

What it means

The turbojet thrust equation is similar to the rocket equation but accounts for the incoming air momentum. It shows that thrust is produced by the change in momentum of the airflow plus the pressure thrust at the nozzle exit. For a jet engine, the mass flow includes air and fuel. This equation is used to calculate thrust for given engine conditions and flight speed. It is fundamental for engine performance analysis and for sizing the engine for a given aircraft. Understanding this equation is essential for propulsion engineers.

Worked example

Turbojet Thrust – Two Examples

Real‑World
Scenario: ṁ = 50 kg/s, V_e = 600 m/s, V₀ = 230 m/s, fully expanded (p_e = p₀). Find thrust.
ParameterValue
ṁ50 kg/s
V_e600 m/s
V₀230 m/s
1F = ṁ·(V_e - V₀) = 50 × (600-230) = 50 × 370 = 18,500 N
Result 18.5 kN ✓ Moderate
Scenario: ṁ = 80, V_e = 650, V₀ = 250, A_e = 0.6, p_e = 101,500, p₀ = 101,325. Find thrust.
ParameterValue
ṁ80
V_e650
V₀250
A_e0.6
p_e-p₀175
1F = 80×(650-250) + 0.6×175 = 80×400 + 105 = 32,105 N
Result 32.1 kN ✓ Higher
Key insight: Turbojet thrust = momentum thrust + pressure thrust – net thrust.

Common mistakes

  • Turbojet thrust equation: F = ṁ·(V_e − V₀) + A_e·(p_e − p₀).
  • ṁ: Mass flow rate (kg/s).
  • V_e: Exit velocity (m/s).
  • V₀: Freestream velocity (m/s).
  • A_e: Exit area (m²).
  • p_e: Exit pressure (Pa).
  • p₀: Ambient pressure (Pa).
  • If nozzle is perfectly expanded, pressure term is zero.

Applications

The turbojet thrust equation, F = ṁ·(V_e − V_0) + A_e·(p_e − p_0), is the general thrust expression for a turbojet, accounting for momentum and pressure thrust. It is used to design and evaluate turbojet engines, to size nozzles, and to compute thrust at different flight conditions. Engineers use this to integrate the engine with the airframe, to assess installation effects, and to predict performance. By applying this equation, aerospace engineers can optimise engine design for specific mission profiles, achieving the required thrust with minimal fuel consumption.

  • Turbojet engine design and performance prediction
  • Nozzle sizing and expansion ratio optimisation
  • Engine‑airframe integration and installation losses
  • Thrust measurement and validation in test cells
  • Performance analysis of gas turbine engines

Frequently Asked Questions

Q01What is the Turbojet Thrust Equation used for?
A01

It is the general thrust equation for an airbreathing jet engine including momentum and pressure‑imbalance terms.

Q02What do the variables ṁ, Ve, V0, Ae, pe, and p0 represent?
A02

ṁ = mass flow rate through the engine (kg/s)
Ve = exhaust velocity (m/s)
V0 = freestream velocity (m/s)
Ae = nozzle exit area (m²)
pe = static pressure at exit (Pa)
p0 = ambient pressure (Pa)

Q03Why is the pressure‑imbalance term important?
A03

It accounts for the thrust contribution from the difference between exit and ambient pressures. For an ideally expanded nozzle (pe=p0), it is zero.

Q04What are common mistakes when using this equation?
A04

  • Dropping the pressure‑imbalance term entirely; it is small but not always negligible, especially for underexpanded nozzles.
  • Using the wrong mass flow (e.g., core flow only vs. total flow).
  • Confusing exit velocity with effective exhaust velocity.

Q05Give a worked example.
A05

ṁ = 50 kg/s, Ve = 600 m/s, V0 = 200 m/s, Ae = 0.5 m², pe = 1.1 atm, p0 = 1 atm. F = 50×(600−200) + (1.1−1)×101325×0.5 = 50×400 + 0.1×50662.5 = 20000 + 5066.25 = 25066.25 N.

Q06How does the thrust vary with flight speed?
A06

As V0 increases, the momentum term decreases, reducing net thrust (for a fixed mass flow and exhaust velocity).

Q07What is the effect of afterburner on thrust?
A07

Afterburning increases Ve (by adding heat) and often increases mass flow, boosting thrust significantly, but at the cost of increased TSFC.

Q08How does the nozzle pressure ratio affect thrust?
A08

A higher pressure ratio (pc/p0) increases Ve and the pressure term, increasing thrust.

Q09What is the difference between this equation and the ideal thrust equation?
A09

The ideal equation often neglects the pressure term and assumes fully expanded flow.

Q10How do you measure the exhaust velocity?
A10

Exhaust velocity is difficult to measure directly; it is usually inferred from thrust and mass flow measurements.