Aerospike vs Bell Nozzle — Altitude Compensation

Ascent Profile

Altitude 0.0 km

Nozzle Design

Bell expansion ratio ε = A_e/A_t 25.0

Chamber pressure P_c 10.0 MPa

Live Readouts

— perfectly expanded —

Ambient P_a

101.3 kPa

Bell exit P_e

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C_F bell nozzle

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C_F aerospike

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Thrust-coefficient gain (aerospike vs bell)

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C_F vs Altitude

How it works

A converging-diverging (de Laval) nozzle's exit pressure P_e is fixed once its geometric expansion ratio ε = A_e/A_t is cut into the bell — it cannot change in flight. Ambient pressure P_a, however, falls roughly exponentially with altitude h:

P_a(h) ≈ P0 · exp(−h / H),   P0 = 101.3 kPa,  H ≈ 7.64 km

Isentropic nozzle relations (γ ≈ 1.2 for combustion gas):
  A/A* = (1/M) [ (2/(γ+1))(1 + (γ−1)/2 M²) ]^((γ+1)/(2(γ−1)))
  Pe/Pc = (1 + (γ−1)/2 Me²)^(−γ/(γ−1))

Thrust coefficient:
  CF = √[ (2γ²/(γ−1))(2/(γ+1))^((γ+1)/(γ−1)) (1 − (Pe/Pc)^((γ−1)/γ)) ]
       + (Pe − Pa)/Pc · ε

A bell nozzle is only perfectly expanded (P_e=P_a, maximum C_F) at one design altitude. Lower down it is over-expanded (P_e<P_a) — the plume necks inward and, once P_e/P_a drops below roughly 0.4, the flow risks separating from the wall before reaching the exit (Summerfield criterion). Higher up it is under-expanded (P_e>P_a) and the plume flares outward, wasting some of the pressure energy that never got converted to axial thrust.

An aerospike (plug) nozzle has no confining outer wall — the exhaust expands freely against the ambient air along the outside of a tapered spike, so the effective expansion ratio adjusts itself to track close to the ideal P_e≈P_a condition at every altitude, up to a maximum set by how far the spike is truncated. This model caps that effective ratio at 1.6× the bell's design ε (a real, compact linear-aerospike spike cannot be arbitrarily long), so the two curves converge again at very high altitude once the spike's own limit is reached — matching flight data from tested aerospike engines (e.g. the X-33 J-2T / XRS-2200 program), which showed the biggest sea-level-to-vacuum C_F advantage low in the atmosphere.

Altitude — moves the rocket up the exponential atmosphere; watch both plumes and the graph respond.
Bell expansion ratio ε — re-designs the bell for a different target altitude; a high ε over-expands badly at sea level.
Chamber pressure P_c — scales how much the ambient-pressure mismatch matters relative to the momentum term.

altitude 0.0 km

Bell nozzle (fixed ε) Aerospike (self-compensating)

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