Ideal Rocket

assumptions

We adopt the classical ideal model for analytic clarity:

• steady, 1-D, adiabatic, inviscid flow through chamber–throat–nozzle;
• calorically perfect gas (constant $\gamma ,R$), no boundary layers or shocks inside the nozzle;
• frozen chemistry (composition fixed through nozzle), no multiphase;
• nozzle axis aligned with thrust (no divergence loss) and perfect expansion when $p_e=p_a$.

Symbols: chamber/stagnation $(\cdot {)}_0$; exit $(\cdot {)}_e$; throat area $A_t$; exit area $A_e$; expansion ratio $\epsilon \equiv A_e/A_t$; ambient pressure $p_a$.

rocket engine analysis (thrust, $I_{sp}$, $c^{\ast }$, $C_f$)

Thrust (control-volume momentum + pressure):

$$\boxed{T=\dot{m}{v}_e+(p_e-p_a)A_e}.$$

Specific impulse (effective exhaust velocity $c$):

$$I_{sp}=\frac{T}{\dot{m}{g}_0},c\equiv I_{sp}{g}_0=\frac{T}{\dot{m}}.$$

Characteristic velocity (chamber performance):

$$\boxed{c^{\ast }\equiv \frac{p_0{A}_t}{\dot{m}}=\sqrt{\frac{R{T}_0}{\gamma }}{\left(\frac{\gamma +1}{2}\right)}^{\frac{\gamma +1}{2(\gamma -1)}}}.$$

Thrust coefficient (nozzle performance):

$$\boxed{C_f\equiv \frac{T}{p_0{A}_t}=C_{f,\mathrm{ideal}}+\frac{(p_e-p_a)A_e}{p_0{A}_t}},$$

with the ideal (perfectly expanded) piece

$$C_{f,\mathrm{ideal}}=\sqrt{\frac{2{\gamma }^2}{\gamma -1}{\left(\frac{2}{\gamma +1}\right)}^{\frac{\gamma +1}{\gamma -1}}\left[1-{\left(\frac{p_e}{p_0}\right)}^{\frac{\gamma -1}{\gamma }}\right]}.$$

Linking the pieces: $c=C_f{c}^{\ast }$ and (for $p_e=p_a$) $T=\dot{m}c$.

nozzle / exit-flow analysis (isentropic)

For isentropic steady flow of a calorically perfect gas:

$$\frac{T}{T_0}={\left(1+\frac{\gamma -1}{2}{M}^2\right)}^{-1},\frac{p}{p_0}={\left(1+\frac{\gamma -1}{2}{M}^2\right)}^{-\frac{\gamma }{\gamma -1}},$$ $$\frac{\rho }{{\rho }_0}={\left(1+\frac{\gamma -1}{2}{M}^2\right)}^{-\frac{1}{\gamma -1}},v=M\sqrt{\gamma RT}.$$

Area–Mach relation:

$$\boxed{\frac{A}{A_t}=\frac{1}{M}{\left[\frac{2}{\gamma +1}\left(1+\frac{\gamma -1}{2}{M}^2\right)\right]}^{\frac{\gamma +1}{2(\gamma -1)}}}.$$

Given $\epsilon =A_e/A_t$, solve for $M_e>1$, then obtain

$$\frac{p_e}{p_0}={\left(1+\frac{\gamma -1}{2}{M}_e^2\right)}^{-\frac{\gamma }{\gamma -1}},v_e=\sqrt{\frac{2\gamma }{\gamma -1}R{T}_0\left(1-{\left(\frac{p_e}{p_0}\right)}^{\frac{\gamma -1}{\gamma }}\right)}.$$

Choked mass flow ( $M_t=1$ ):

$$\boxed{\dot{m}=p_0{A}_t\sqrt{\frac{\gamma }{R{T}_0}}{\left(\frac{2}{\gamma +1}\right)}^{\frac{\gamma +1}{2(\gamma -1)}}}.$$

matching the exit to ambient (perfect/under/over expansion)

perfectly expanded

$$\boxed{p_e=p_a}\Rightarrow T=\dot{m}{v}_e,\text{maximum ideal nozzle efficiency for given }{p}_0,T_0,\epsilon .$$

underexpanded flow

$$p_e>p_a.$$

The jet continues expanding outside the nozzle: Prandtl–Meyer fans originate at the lip; shock-cell (Mach diamond) structure forms downstream as the plume adjusts toward ambient. More expansion potential remains → a larger $\epsilon$ (or lower $p_a$) would raise $v_e$ and $C_f$.

overexpanded flow

$$p_e<p_a.$$

The external environment compresses the jet; an internal compression shock can form near/inside the nozzle. If sufficiently overexpanded, separation may occur (adverse side-load risk, loss of performance). Designers avoid severe overexpansion at expected $p_a$ or adopt altitude-compensating contours (e.g., aerospike, expansion–deflection).

Practical note: ascent engines at sea-level often accept mild off-design operation; upper-stage nozzles (large $\epsilon$) target near-vacuum $p_a$ and operate greatly underexpanded at low altitude (but are fired in vacuum).

quick recipes

quick sizing (ideal, given $p_0,T_0,\gamma ,R,\epsilon ,p_a$)

1. Solve $M_e$ from $A_e/A_t=\epsilon$ using the area–Mach relation.
2. Compute $p_e/p_0,T_e/T_0$.
3. Get $v_e$ from isentropic energy.
4. If $p_e\ne p_a$, include pressure term $(p_e-p_a)A_e$ in $T$.
5. Use $\dot{m}(p_0,T_0,A_t)$ and $c^{\ast },C_f$ to report $I_{sp}$.

design heuristics

• Sea-level engines: choose $\epsilon$ so $p_e\approx p_a$ near peak static thrust; limit overexpansion to avoid separation.
• Vacuum engines: maximize $\epsilon$ (geometrically feasible) for higher $v_e$; accept strong underexpansion at sea level (usually irrelevant operationally).
• Altitude compensating nozzles: mitigate off-design losses across $p_a$ sweep.