How the demos compute it

Six engines, one solver. Every number in the jet demos — each temperature, velocity, efficiency bar and sweep curve — comes from the ideal cycle equations of this course, evaluated in a few hundred lines of framework-free code. This closing article maps the course onto that code and draws the honest boundary around what it can claim.

The pipeline

Every engine page runs the same five steps at every input change:

• 1. Atmosphere. Altitude → $T_0$, $p_0$, $\rho$, $a_0$ via the two-layer ISA of the atmosphere article; then $V_0 = M_0\,a_0$.
• 2. Ratios. $\tau_r = 1 + \tfrac{\gamma-1}{2}M_0^2$, $\tau_c = \pi_c^{(\gamma-1)/\gamma}$, $\tau_\lambda = T_{t4}/T_0$, plus $\tau_f$ for the turbofan.
• 3. Work balance. The one line that distinguishes the families:
  $$\tau_t = 1 - \frac{\tau_r}{\tau_\lambda}\Bigl[(\tau_c - 1) + \alpha(\tau_f - 1)\Bigr]$$
  with $\alpha = 0$ for the turbojet, the full form for the turbofan, no turbine at all for the ram family, and an oversized extraction (the free power turbine) for the shaft engines.
• 4. Nozzles. Each stream’s exit velocity from its total temperature and cumulative ratio chain — the $V^2/2 = c_p T_t(1 - 1/\tau_{\text{chain}})$ step of the turbojet article, applied per stream.
• 5. Assembly. Specific thrust averages the streams, $f = c_p T_0(\tau_\lambda - \tau_r\tau_c)/h_{PR}$, TSFC divides them, and the three efficiencies come out of the same velocities. Shaft engines additionally report $P/\dot m$ and PSFC.

$\gamma = 1.4$

“cold-air standard”: one γ everywhere, even in the burner [–]

$c_p = 1005$

constant specific heat of air [J/(kg·K)]

$h_{PR} = 43\ \text{MJ/kg}$

jet-fuel heating value [–]

The feasible flag

The solver refuses to pretend. A parameter set is marked infeasible — and the demo dims its readouts — when the bookkeeping breaks:

• $\tau_t \le$ its floor: the turbine cannot extract enough work to spin the compressor (and fan) you asked for at this $T_{t4}$;
• $\tau_\lambda \le \tau_r\tau_c$: burner entry is already hotter than the turbine-inlet limit — no room to add fuel;
• specific thrust ≤ 0: the exhaust leaves slower than flight speed.

These are not arbitrary guards — each is one of the course’s equations reaching its own boundary, which is why dragging a slider to the infeasible zone is itself instructive.

The sweep charts

The performance chart on each page simply re-runs the whole pipeline at 40-odd Mach numbers across the engine’s envelope and plots the result — nothing is precomputed or fitted. That is why the curves kink exactly where the feasibility conditions bite, and why the figures in this course (drawn by the same solver) agree with the demos to the pixel.

What the model leaves out — deliberately

• Component losses. Real compressors and turbines are 85–92% efficient, inlets and burners lose total pressure, nozzles under-expand. Rule of thumb: real TSFC runs 20–40% worse than the ideal numbers shown.
• Hot-gas properties. Burned gas has $\gamma \approx 1.30$ and a higher $c_p$; two-γ analyses split the engine at the burner. The demo keeps one γ for transparency.
• Off-design behaviour. The model solves a design point at every slider position — as if the engine were rebuilt for each input. Real engines have fixed geometry, spool dynamics, surge margins and control laws.
• Installation. No nacelle drag, no inlet spillage, no bleed air, no afterburner. Thrust here is uninstalled engine thrust.
• Real gas at hypersonics. Dissociation above ~2000 K is ignored, flattering the ramjet and scramjet at their hottest points (flagged in that article).