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Thrust coefficient

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From Wikipedia, the free encyclopedia
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  1. Thrust coefficient

Thrust coefficient or (sometimes ) is a dimensionless number that measures the performance of a nozzle, most commonly in a rocket engine, independent of combustion performance. It is often used to compare the performance of different nozzle geometries. After combining it with characteristic velocity , then an effective exhaust velocity and a specific impulse can be found to characterize the overall efficiency of a rocket engine design.1

This article is about rocket engines. For characteristic of wind turbine, see Wind-turbine aerodynamics § Thrust coefficient.

The thrust coefficient characterizes the supersonic flow in the expansion section downstream of the nozzle throat, in contrast to characteristic velocity which characterizes the subsonic flow in the combustion chamber and contraction section upstream of the throat.2

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Contents
  • (Top)
  • Physics and context
  • Formulas
  • References
  • Recent changes
  • Rabbit hole
Contents
Contents5 sections
  • (Top)
  • Physics and context
  • Industry examples
  • Formulas
  • Ideal nozzles
  • Corrections
  • References
  • Recent changes
  • Rabbit hole
Categories:
  • Rocketry
  • Engine technology

631 words · 11 references · revision 1341862027 · CC BY-SA 4.0

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Key concepts

  • Characteristic velocity2
  • Specific impulse2
  • Rocket engine nozzle1
  • Rocket engine1
  • Bell nozzle1
  • G. V. R. Rao1
  • Method of characteristics1
  • Rocketdyne F-11
  • RL101
  • RS-251

The articles this one links to most, weighted towards the introduction.

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Physics and context

Thrust coefficients characterize how well a nozzle will boost the efficiency of a rocket engine by expanding the exhaust gas and dropping its pressure before it meets ambient conditions. A of 1 corresponds to zero ambient pressure and no expansion at all; i.e. the throat exhausts straight to vacuum without any diverging nozzle at all. The effective exhaust velocity would then be equal to the characteristic velocity provided by the combustion chamber. Typical thrust coefficients seen in aerospace industry rocket engines vary between about 1.3 and 2.3 Virtually all large engines since the 1960s have used a bell nozzle geometry which optimizes for the highest thrust coefficient; this was first derived by Gadicharla V.R. Rao. using the method of characteristics.4

Industry examples

Rocket engineThrust coefficient at sea levelThrust coefficient in vacuumCitation
Rocketdyne F-11.591.825
Rocketdyne RL10A-12.056
Rocketdyne RS-25 / SSME1.531.917
Energomash RD-1201.958
Energomash RD-1701.711.868
Energomash RD-2531.651.838

Formulas

  • is the effective exhaust velocity.
  • is the characteristic velocity of the combustion.
  • is specific impulse.
  • is standard gravity.
  • is total thrust of the engine.
  • is chamber pressure.
  • is the area of the nozzle throat.

Ideal nozzles

An ideal nozzle has parallel, uniform exit flow; this is achieved when the pressure at the exit plane equals the ambient pressure. In vacuum conditions this means an ideal nozzle is infinitely long. The area ratio can be derived from isentropic flow, also given here:2

  • is the area of the nozzle exit plane.
  • is the ratio of specific heats of the exhaust gas.
  • is the ambient pressure of the surrounding atmosphere/vacuum.

The ideal thrust coefficient is then910

  • is the pressure of the exhaust gas at the exit plane. In the ideal case in vacuum (), the expression reduces to

.

For a diatomic gas (), . The absolute theoretical limit, corresponding to , is

Corrections

Various inefficiencies in a real nozzle design will reduce the overall thrust coefficient. Three major effects contribute as follows11

  • is the divergence loss efficiency (typically the most dominant inefficiency).

References

  1. 1Heister, Stephen D.; Anderson, William E.; Pourpoint, Timothée; Cassady, Joseph (2018). Rocket propulsion (First ed.). New York: Cambridge University Press. ISBN 978-1-108-42227-7.
  2. 2Rao, G. V. R. (November 1961). "Recent Developments in Rocket Nozzle Configurations". ARS Journal. 31 (11): 1488–1494. doi:10.2514/8.5837. ISSN 1936-9972.
  3. 3Huzel, Dieter K.; Huang, David H. (2000). Modern Engineering for Design of Liquid-Propellant Rocket Engines. Progress in Astronautics and Aeronautics. Reston: American Institute of Aeronautics and Astronautics. ISBN 978-1-56347-013-4.
  4. 4Rao, G. V. R. (June 1958). "Exhaust Nozzle Contour for Optimum Thrust". Journal of Jet Propulsion. 28 (6): 377–382. doi:10.2514/8.7324. ISSN 1936-9980.
  • is the two-phase flow loss efficiency.
  • is the skin friction loss efficiency (typically about 0.99).
  • Conical nozzles

    Source:2

    • is the half-angle of the conical nozzle.

    Annular nozzles

    Source:2

    These nozzles are typically found in aerospike engines or in jet engines.

    • is the half-angle of the outer wall of the nozzle (rad).
    • is the (positive) half-angle of the inner wall of the plug inside the nozzle (rad).

    Generalized contour nozzles

    There are no simple relations for divergence inefficiency for a more general nozzle contour, such as a bell nozzle. Instead the thrust coefficient must be integrated directly, assuming pressure variation across the nozzle exit plane has already been found:2

    • is the inner radius of the nozzle at the exit plane. In an annular nozzle it is the distance between the outer wall and the plug at the exit plane.
    • is the distance from the central axis to the point of interest. The relationship assumes radial symmetry of all properties.
    • is the pressure of the exhaust gas at the exit plane at a given .
    • is the density of the exhaust gas at the exit plane at a given .
    • is the speed of the exhaust gas at the exit plane at a given .
    • is the angular direction of the exhaust gas velocity at the exit plane at a given .
    5
    "F-1". www.astronautix.com. Retrieved 2023-08-31.
  • 6"RL-10A-1". www.astronautix.com. Retrieved 2023-08-31.
  • 7"SSME". www.astronautix.com. Retrieved 2023-08-31.
  • 8Sutton, George Paul; Biblarz, Oscar (2017). Rocket propulsion elements (Ninth ed.). Hoboken, New Jersey: John Wiley & Sons Inc. ISBN 978-1-118-75388-0.
  • 9"Modeling of rocket nozzles; effects of nozzle area ratio" (PDF). MIT OpenCourseWare. 2012.
  • 10"Rocket Thrust Coefficient Tables (Gamma = 1.2)". engineering.purdue.edu. Retrieved 2025-06-19.
  • 11Mace and Parkinson (November 1980). "ON THE CALCULATION OF THRUST COEFFICIENT" (PDF). Propellants, Explosives and Rocket Motor Establishment.
  • Created
    29 Aug 2023
    History
    40 edits
    15 contributors
    No edits in the last 30 days
    Last edited
    5 Mar 2026

    40 revisions in total · 15 contributors

    1. 5 Mar 2026, 14:30 UTCEvgeny§ Ideal nozzles Quote specific values for a diatomic gas and the absolute limit+215diff
    2. 5 Mar 2026, 14:20 UTCEvgeny§ Ideal nozzles Clarify that the ideal case is in vacuum+14diff
    3. 5 Mar 2026, 14:17 UTCEvgeny§ Ideal nozzles Show the actual ideal-case (p_e = p_a) expression+150diff
    4. 5 Mar 2026, 10:01 UTCEvgeny§ Formulas Remove units: The expressions remain valid independently of the unit system used.−134diff
    5. 3 Jan 2026, 16:01 UTCJessicapiercemminor copy edits+64diff
    6. 26 Dec 2025, 15:25 UTCRJFJR§ Annular Nozzles move ref out of heading+9diff
    7. 23 Nov 2025, 16:22 UTCRJFJR§ Generalized Contour Nozzles move ref out of heading0diff
    8. 23 Nov 2025, 16:21 UTCRJFJR§ Conical Nozzles move ref out of heading+9diff

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    Created
    29 Aug 2023
    History
    40 edits
    15 contributors
    No edits in the last 30 days
    Last edited
    5 Mar 2026

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