Characteristic velocity

Template:Short description 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)

$c_{Theoretical}^{*}=\frac{I_{sp}{g}_0}{C_F}=\frac{c}{C_F}=\sqrt{\frac{R{T}_c}{\gamma }(\frac{\gamma +1}{2}{)}^{\frac{\gamma +1}{\gamma -1}}}$

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