Pressure Effects

Control Volume Analysis

Take a steady, 1-D control volume around the nozzle. Neglect gravity and assume axisymmetric, adiabatic, inviscid flow, with the exit plane perpendicular to the axis and negligible inlet momentum flux (chamber nearly stagnant).

Thrust (streamwise):

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

where $v_e$ is the axial exit speed, $p_e$ the exit static pressure, $p_a$ ambient pressure, and $A_e$ exit area.

• First term $\dot{m}{v}_e$: momentum thrust.
• Second term $(p_e-p_a)A_e$: pressure (over/under-expansion) thrust.

IMPORTANT

If the nozzle has finite divergence half-angle $\theta$, an idealized correction is $v_{e,\text{axial}}\approx v_e\cos \left(\theta \right)$ (a divergence loss); we assume $\theta \approx 0$ unless noted.

Characteristic Velocity

The chamber performance metric $c^{\ast }$ is independent of $p_a$:

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

with $p_0,T_0$ the chamber (stagnation) pressure and temperature, $A_t$ the throat area, and $\gamma ,R$ the effective (equilibrium/frozen) gas constants.

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

Thrust Coefficient

Define the 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 isentropic piece (perfectly expanded surrogate):

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

Linking parts: $T=(p_0{A}_t)C_f=\dot{m}c$ with the effective exhaust velocity $c=C_f{c}^{\ast }$.

Expansion

• Let $\epsilon \equiv A_e/A_t$. For isentropic, calorically perfect flow:

• Area–Mach:

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

• Exit ratios:

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

Full Expansion

“full” (a.k.a. perfectly expanded) means $\boxed{p_e=p_a}$.

• Pressure term cancels: $T=\dot{m}{v}_e$.
• For a given $p_0,T_0$ and ambient $p_a$, “full” corresponds to a unique ${\epsilon }^{\star }$ satisfying $p_e({\epsilon }^{\star })=p_a$.

Optimal Expansion

Design sense: choose $\epsilon$ so that at the design ambient $p_a^{†}$,

$$\boxed{p_e({\epsilon }_{\mathrm{opt}})=p_a^{†}}$$

which maximizes $C_f$ (and $T$) at that ambient with no separation.

Design Considerations

NOTE

• Sea-level engines: pick ${\epsilon }_{\mathrm{opt},\mathrm{SL}}$ to avoid strong over-expansion $(p_e<p_a)\Rightarrow$ internal shock/separation risk.
• Vacuum engines: large $\epsilon$ to raise $v_e$; at sea level they’d be under-expanded $(p_e>p_a)$.