Rocket Performance¶

Note

The equations presented here are derived for an isentropic rocket engine with constant-pressure combustion and steady, one-dimensional flow. For higher fidelity analysis, simulations with more realistic assumptions should be performed.

The Basic Things¶

Thermodynamic Relationships¶

Thermodynamic relationships have their foundation in gasses equations of state. I highly recommend going through the derivation to get to these equations. Shapiro’s The Dynamics and Thermodynamics of Compressible Fluid Flow has an excellent explanation and derivation.

\[ \begin{align}\begin{aligned}\frac{T_0}{T} = 1 + \frac{\gamma-1}{2}M^2\\\frac{p_0}{p}^{(\gamma-1)/\gamma} = \frac{\rho_0}{\rho}^{\gamma-1} = \frac{T_0}{T}\end{aligned}\end{align} \]

Thrust¶

The equation for thrust can be derived from the conservation of momentum by taking a control volume around the rocket. The result is a function of exhaust velocity (\(u_e\)), mass flow rate (\(\dot{m}\)), exit area (\(A_e\)), exit pressure (math:p_e), and ambient pressure (\(p_a\)).

\[T = \dot{m}u_e + (p_e - p_a)*A_e\]

Specific Impulse¶

A metric that describes the efficiency of the engine. Units of \(s\).

\[I_{sp} = \frac{T}{\dot{m}g_0}\]

Exhaust Velocity¶

\[u_e = \sqrt{2 c_p T_{02}\Big[1 - \Big(\frac{p_e}{p_a}^{(\gamma-1)/\gamma}\Big)\Big]}\]

Propellant Mass Flow Rate¶

\[\dot{m} = \frac{A^* p_{02}}{\sqrt{R T_{02}}}\sqrt{\gamma \Big(\frac{2}{\gamma + 1}\Big)^{(\gamma+1)/(\gamma-1)}}\]

Area Ratio¶

\[\frac{A}{A^*} = \frac{1}{M_e}\Big[\frac{2}{\gamma + 1}\Big(1 + \frac{\gamma - 1}{2}M_e^2\Big)\Big]^{\gamma/(\gamma-1)}\]

Characteristic Velocity and Thrust Coefficient¶

Characteristic Velocity¶

The characteristic velocity is a function of the combustion chamber properties. As stated below, it is a function of ratio of specific heats (\(\gamma\)), specific gas constant (\(R\)), and the chamber stagnation temperature (\(T_0\))

\[c^{*} = f(\gamma R T_0) = \frac{p_0 A}{\dot{m}}\]

Characteristic velocity can be written in a more verbose form,

\[c^{*} = \sqrt{\frac{1}{\gamma}\Big(\frac{\gamma+1}{2}\Big)^{(\gamma+1)/(\gamma-1)}R T_0}\]

Thrust Coefficient¶

The thrust coefficient is a performance metric used to describe nozzle.

\[C_T = \frac{T}{p_0 A}\]

Another form of the thrust coefficient makes the effect of nozzle performance abundantly clear.

\[C_T = \sqrt{\frac{2 \gamma^2}{\gamma-1}\Big(\frac{2}{\gamma+1}\Big)^{(\gamma+1)/(\gamma-1)}\Big[1 - \Big(\frac{p_e}{p_0}\Big)^{(\gamma-1)/\gamma}\Big]} + \frac{p_e - p_a}{p_0} \frac{A_e}{A^*}\]

Combining \(c^*\) and \(C_T\) yields an unsurprising result.

\[T = \dot{m}c^*C_T\]