Characteristic velocity

Characteristic velocity or ${\displaystyle c^{*}}$, 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^[1][2]. 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: ${\displaystyle I_{sp}={\frac {c}{g_{0}}}}$. 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 ${\displaystyle n_{v}}$, and is calculated by dividing ${\displaystyle c_{Actual}^{*}}$ with ${\displaystyle c_{Theoretical}^{*}}$. Standard values for ${\displaystyle n_{v}}$ range from 0.85 to 1.03.

${\displaystyle c_{Actual}^{*}={\frac {p_{c}A_{t}}{\dot {m}}}}$

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

${\displaystyle c_{Theoretical}^{*}={\frac {I_{sp}g_{0}}{C_{F}}}={\frac {c}{C_{F}}}={\sqrt {{\frac {RT_{c}}{\gamma }}{\Bigl (}{\frac {\gamma +1}{2}}{\Bigr )}^{\frac {\gamma +1}{\gamma -1}}}}}$

Alternative Imperial Form:

${\displaystyle c_{Theoretical}^{*}={\frac {I_{sp}g_{0}}{C_{F}}}={\frac {c}{C_{F}}}={\frac {\sqrt {g_{0}\gamma RT_{c}}}{\gamma {\sqrt {{\frac {2}{\gamma +1}}^{\frac {\gamma +1}{\gamma -1}}}}}}}$

• ${\displaystyle I_{sp}}$ is the specific impulse (s)
• ${\displaystyle g_{0}}$ is the gravitational acceleration at sea-level (m/s²)
• ${\displaystyle C_{F}}$ is the thrust coefficient
• ${\displaystyle c}$ is the effective exhaust velocity (m/s)
• ${\displaystyle \gamma }$ is the specific heat ratio for the exhaust gases
• ${\displaystyle R}$ is the gas constant per unit weight (J/kg-K)
• ${\displaystyle T_{c}}$ is the chamber temperature (K)

• Characteristic Length (L*)
• Thrust Coefficient (C_F)