Characteristic velocity, denoted c ∗ {\displaystyle c^{*}} (pronounced 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 p = c g 0 {\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 n v {\displaystyle n_{v}} , and is calculated by dividing c A c t u a l ∗ {\displaystyle c_{Actual}^{*}} with c T h e o r e t i c a l ∗ {\displaystyle c_{Theoretical}^{*}} . Standard values for n v {\displaystyle n_{v}} range from 0.85 to 1.03.
Formula
c A c t u a l ∗ = p c A t m ˙ {\displaystyle c_{Actual}^{*}={\frac {p_{c}A_{t}}{\dot {m}}}}
c ∗ {\displaystyle c^{*}} is the characteristic velocity (m/s, ft/s).
p c {\displaystyle p_{c}} is the chamber pressure (Pa, psi).
A t {\displaystyle A_{t}} is the area of the throat (m2, in2).
m ˙ {\displaystyle {\dot {m}}} is the mass flow rate of the engine (kg/s, slug/s).
c T h e o r e t i c a l ∗ = I s p g 0 C F = c C F = R T c γ ( γ + 1 2 ) γ + 1 γ − 1 {\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:
c T h e o r e t i c a l ∗ = I s p g 0 C F = c C F = g 0 γ R T c γ 2 γ + 1 γ + 1 γ − 1 {\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}}}}}}}
I s p {\displaystyle I_{sp}} is the specific impulse(s).
g 0 {\displaystyle g_{0}} is the gravitational acceleration at sea-level (m/s2).
C F {\displaystyle C_{F}} is the thrust coefficient.
c {\displaystyle c} is the effective exhaust velocity (m/s).
γ {\displaystyle \gamma } is the specific heat ratio for the exhaust gases.
R {\displaystyle R} is the gas constant per unit weight (J/kg-K).
T c {\displaystyle T_{c}} is the chamber temperature (K).
See also Characteristic length (L*) Thrust coefficient (CF)
References