3. Thrust and Performance¶

3.1 Thrust¶

The force exerted by the rocket on its vehicle is derived by applying the linear momentum equation to the nozzle control volume (see §1.8 for the full derivation):

\[ F = \dot{m}\,v_e + (P_e - P_\text{ext})\,A_e = C_f\,P_0\,A_t \]

Implemented in get_thrust_from_thrust_coefficient.

3.2 Total Impulse¶

Total impulse is the time integral of thrust over the entire burn:

\[ I_{tot} = \int_0^{t_{burn}} F\,dt \]

For practical computation, the discrete thrust-time trace is integrated numerically. The closed-form equivalent for a constant average thrust is:

\[ \boxed{I_{tot} = F_{avg}\cdot t_{burn}} \]

Implemented in get_total_impulse.

3.3 Specific Impulse¶

Specific impulse \(I_{sp}\) measures the impulse delivered per unit weight of propellant consumed — the rocket analogue of fuel efficiency:

\[ \boxed{I_{sp} = \frac{I_{tot}}{m_{prop}\,g_0}} \quad [\text{s}] \]

where \(g_0 = 9.80665\ \text{m/s}^2\) is the standard gravitational acceleration. The unit seconds is universal: it does not depend on the unit system used for thrust or mass.

Implemented in get_specific_impulse.

3.4 Characteristic Velocity¶

The characteristic velocity \(c^*\) isolates the combustion quality from the nozzle performance. Rearranging the choked mass flow (§1.7):

\[ \dot{m} = \frac{P_0\,A_t}{c^*} \implies \boxed{c^* = \frac{P_0\,A_t}{\dot{m}} = \frac{\sqrt{R T_0 / k}}{\left(\dfrac{2}{k+1}\right)^{(k+1)/[2(k-1)]}}} \quad [\text{m/s}] \]

A high \(c^*\) indicates high flame temperature and/or low molecular weight of the products — properties determined solely by the propellant chemistry (computed by CEA and stored in machwave.models.propellants).

3.5 Effective Exhaust Velocity¶

Combining the thrust coefficient (§1.8) with \(c^*\):

\[ F = C_f\,P_0\,A_t = C_f\,\dot{m}\,c^* \implies c_{eff} = \frac{F}{\dot{m}} = C_f\,c^* \]

The specific impulse can therefore also be written as:

\[ I_{sp} = \frac{c_{eff}}{g_0} = \frac{C_f\,c^*}{g_0} \]

This factorisation shows that nozzle efficiency (\(C_f\)) and combustion efficiency (\(c^*\)) contribute independently to \(I_{sp}\).

References¶

Sutton, G. P., & Biblarz, O. (2017). Rocket Propulsion Elements (9th ed.). Wiley. Ch. 2, 3.
Huzel, D. K., & Huang, D. H. (1992). Modern Engineering for Design of Liquid-Propellant Rocket Engines. AIAA Progress in Astronautics and Aeronautics, Vol. 147. Ch. 1.