characteristic velocity

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Characteristic velocity or $c^*$ , or C-star is a measure of the combustion performance of a rocket engine independent of nozzle performance, and is used to compare different propellants and propulsion systems. It is independent of the nozzle, making it a useful metric for evaluating propellant combustion alone. c* should not be confused with c, which is the effective exhaust velocity related to the specific impulse by: $I_s = \frac{c}{g_0}$ . Specific impulse and effective exhaust velocity are dependent on the nozzle design unlike the characteristic velocity, explaining why C-star is an important value when comparing different propulsion system efficiencies. c* can be useful when comparing actual combustion performance to theoretical performance in order to determine how completely chemical energy release occurred, or the combustion efficiency. This is known as c*-efficiency, or $n_v$ , and is calculated by dividing $c^*_{Actual}$ with $c^*_{Theoretical}$ . Standard values for $n_v$ range from 0.85 to 1.03.

Formula

$c^*_{Actual} = \frac{p_c A_t }{\dot{m}}$

$c^*$ is the characteristic velocity (m/s, ft/s)
$p_c$ is the chamber pressure (Pa, psi)
$A_t$ is the area of the throat (m², in²)
$\dot{m}$ is the mass flow rate of the engine (kg/s, slug/s)

$c^*_{Theoretical} = \frac{I_{sp} g_0}{C_F} = \frac{c}{C_F} =\sqrt{\frac{RT_c}{\gamma} \Bigl(\frac{\gamma+1}{2}\Bigr)^{\frac{\gamma+1}{\gamma-1}}}$

Alternative Imperial Form:

$c^*_{Theoretical} = \frac{I_{sp} g_0}{C_F} = \frac{c}{C_F} =\frac{\sqrt{g_0 \gamma R T_c}}{\gamma \sqrt{{\frac{2}{\gamma +1}}^\frac{\gamma +1}{\gamma -1}}}$

$I_{sp}$ is the specific impulse (s)
$g_0$ is the gravitational acceleration at sea-level (m/s²)
$C_F$ is the thrust coefficient
$c$ is the effective exhaust velocity (m/s)
$\gamma$ is the specific heat ratio for the exhaust gases
$R$ is the gas constant per unit weight (J/kg-K)
$T_c$ is the chamber temperature (K)

References

Rocket Propulsion Elements, 7th Edition by George P. Sutton, Oscar Biblarz
Rocket Propulsion Elements, 9th Edition by George P. Sutton, Oscar Biblarz
Modern Engineering for Design of Liquid-Propellant Rocket Engines, Volume 147 by Dieter K. Huzel, David H. Huang

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Category:Rocket propulsion

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