Textbook examples
Provide known stagnation temperatures/pressures. Leave fields blank to derive from efficiency defaults. Efficiencies will be back-calculated from the values you provide.
Station Properties
Convention (station number = station letter): 1(a) ambient · 2 inlet exit · 3 compressor exit · 4 turbine inlet · 5 turbine exit · 6 afterburner exit (shown only with reheat) · 7(e) nozzle exit.
| Station | (K) | (kPa) | (K) | (kPa) |
|---|---|---|---|---|
| 1(a) — Ambient / free stream | — | — | — | — |
| 2 — Inlet exit | — | — | — | — |
| 3 — Compressor exit | — | — | — | — |
| 4 — Turbine inlet (TIT) | — | — | — | — |
| 5 — Turbine exit | — | — | — | — |
| 7(e) — Nozzle exit | — | — | — | — |
Parametric Study
Single points answer questions; sweeps reveal structure. Vary one driving parameter across its range — flight Mach number or compressor pressure ratio — while every other input holds the value set in the dashboard, and the four trade curves of turbojet design appear.
Method and limitations of the parametric sweep
The sweep varies a single independent variable — flight Mach number or compressor pressure ratio — in uniform increments between the specified bounds. At each increment the remaining inputs are held at their dashboard values and the complete cycle is solved from intake to nozzle, including the afterburner when it is enabled. Each plotted point is an independent converged solution of the same governing equations used for the single-point analysis; no values are interpolated between points.
A single operating point gives one performance figure; a sweep resolves the trade-offs between figures. Raising the compressor pressure ratio, for example, increases thermal efficiency while reducing specific thrust, and the curves locate the resulting compromise. In parametric cycle analysis the trend across the range is the primary result rather than any individual value.
Limitations
- Single-variable variation. Only the swept parameter changes; all other quantities are fixed. An operating engine re-matches — shaft speed, bleed and nozzle area adjust together — so the curves are a controlled section through the design space, not measured engine behaviour.
- Model assumptions. The working fluid is treated as calorically perfect, with separate constant values of cₚ and γ for the cold and hot sections, and the flow as quasi-one-dimensional and steady. These assumptions lose accuracy toward the extremes of the range (high Mach number, very high pressure ratio) even where a curve is still drawn.
- Non-convergent points are omitted. Where a parameter value yields a non-physical cycle — for instance, insufficient turbine work to drive the compressor — the solver returns no result and the point is left blank rather than assigned an arbitrary value. A gap in a curve marks the boundary of the feasible cycle.
- Resolution. The number of points sets the resolution of the curve; too few may miss a local extremum. The 200-point limit bounds computation time.
- Intended use. Results are suitable for instruction and coursework. Trends are representative; absolute magnitudes are approximate and do not replace validated design tools.
Off-Design
Performance
Everything before this chapter solves the design point: you state a compressor pressure ratio and a mass flow, and the cycle tells you what they produce. Hardware does not work that way. Once an engine is built, the pressure ratio and the mass flow stop being inputs and become results — outcomes of the machine finding its own operating point at whatever altitude, flight speed and throttle setting it is given.
The bridge between the two is a single observation: the turbine inlet stator chokes, and stays choked over almost the whole useful range. A choked nozzle passes a fixed corrected flow, so is pinned at its design value, and with it the turbine temperature and pressure ratios. That one constraint, together with the shaft power balance, closes the problem — and it turns out to be exactly the same shaft balance the validated solver already uses, so the two agree to floating-point round-off.[3]
This module computes only those two numbers — the matched pressure ratio and the matched mass flow — and hands them to the untouched design-point solver. Every thrust, TSFC and station temperature below is that solver's output, so the off-design results inherit the same validation as the design point rather than asking you to trust a second, parallel implementation.
Off-design analysis needs a fixed reference: the point at which the turbine flow capacity, the turbine ratios and the compressor map scaling were all set. Solve whatever cycle you want in Section 0, then freeze it here. Everything below is that engine, flown elsewhere.
Freeze a design point and match an operating point to see the full matched state.
Assumptions and limitations of the off-design model
Off-design matching is the largest single addition in this build, and it is worth being precise about what it does and does not represent.
What is assumed
- The turbine stator chokes and stays choked. This fixes the turbine's corrected flow, and therefore the turbine temperature and pressure ratios, at their design values. It is a good assumption over the normal operating range and a poor one at very low power, where the turbine unchokes and the running line relaxes — which this model does not reproduce.
- The shaft balance omits the fuel-addition term. The shipped solver writes the power balance as without the factor on the turbine side, and this module omits it identically. Adding the fuller form here would make off-design disagree with the design point the project is validated against, and internal consistency is worth more than a marginal completeness gain.
- The compressor map has the right shape, not real data. It is an analytic map scaled onto your design point by the standard three factors — flow, pressure ratio and efficiency. Speed lines steepen toward surge and flatten toward choke, and there is an efficiency island around the design point. Every number it produces is indicative. Measured map data is a drop-in replacement, and until it is dropped in, surge margins here indicate a trend rather than a certification.
- Nozzle area. With both the turbine and the nozzle choked, the required nozzle area comes out constant automatically — which is a consistency check rather than a coincidence, and the single-point read-out reports the ratio so you can see it hold.
What the closed-form and map fidelities differ on
The closed-form fidelity holds every component efficiency at its design value and always produces an answer. The map fidelity lets the compressor's efficiency fall away from the design point, and can report that no self-consistent operating point exists — which is what a surge limit looks like from inside the algebra. Those points are still shown, flagged, using the unconstrained cycle numbers, because deleting them would hide the most important thing the map has to say.
The Design Space,
Two Axes at a Time
The parametric sweep in Section 1 varies one input and draws four curves. That is the right first tool, and it conceals something important: the optimum compressor pressure ratio is not a number. It is a function of turbine inlet temperature and flight Mach, and a one-dimensional sweep takes a single section through that surface without telling you the section moved.
Evaluate the cycle over a grid of two parameters instead and the surface appears. The ridge line marking maximum specific thrust and the valley marking minimum fuel consumption are never the same line — the gap between them is the design trade, drawn rather than argued.[3]
Evaluate a grid to see the optima, the feasible fraction and the trade.
How Much of
the Answer Is Known
The dashboard reports thrust to three decimal places. That precision is real, in the sense that the arithmetic is exact — and misleading, in the sense that nobody knows a compressor's isentropic efficiency to better than about a point and a half. A number carried to 0.001 kN out of inputs known to 1.5% invites a confidence the inputs do not support.
So put error bars on the inputs and see what comes out. Each sample below is a complete, ordinary run of the same validated solver on a perturbed input set — no linearisation, no surrogate model. That matters, because the cycle is not linear near its limits: the same tolerance that moves thrust by a per cent at the design point can push the cycle over a feasibility boundary somewhere else, and only a real solve finds that out. Samples that fail to converge are counted and reported, never quietly dropped.
The sampler is seeded and never touches the system random number generator, so a quoted interval can be reproduced exactly by someone else. That is the whole point of quoting one.
Defaults describe a cycle deck whose component levels are known from experience rather than a clean sheet. Every figure is editable; none of them is a physical constant. Sampling is truncated at the solver's own validated input domain, so the study measures the physics rather than the validator.
Run the study to put an interval on every headline number.
Rank correlation between each sampled input and the output, computed on the samples already drawn. Unlike the tornado it measures influence with everything else moving too. Where the two rankings disagree, the disagreement is the finding: it means the input interacts, or its effect is asymmetric about the nominal.
Run the study to rank the drivers.
The CFD
and CAD Bridge
A cycle deck and a three-dimensional CFD run need each other in opposite directions. The deck knows the stagnation state at every station and nothing about geometry; CFD needs exactly those states as boundary conditions and cannot start without a flow path to put them in. Coming back the other way, CFD measures component efficiencies far better than a cycle deck can assume them, and those measurements belong back in the deck.
Three things live in this chapter: the solved station states written out in the form a solver actually reads, a preliminary annulus sizing from continuity, and a route for measured efficiencies to come home. The distinction that matters throughout is between what was computed and what was assumed: the stagnation states carry the solver's validation, and every static value and every radius rests on an assumed axial Mach number and hub/tip ratio. Both exports say so in their own headers, so nothing downstream can mistake a sizing assumption for a result.
Solve the cycle in Section 0, then build the boundary conditions.
Annulus areas follow from continuity at the assumed axial Mach numbers; radii follow from an assumed hub/tip ratio through the cold section and a constant mean radius aft of the compressor exit. There is no stage count, no blade row, no stress check and no cooling flow anywhere in this. It is the shape a first layout starts from and nothing more.
Paste the component efficiencies a CFD run or rig test measured. Every value is range-checked against the solver's validated input domain before it is accepted, and anything rejected is named with the reason. A silent partial import would be worse than none: the deck would run on a mixture of measured and assumed values with no record of which was which.
Four Engines,
One Kernel
Everything above this chapter is a single-spool turbojet. That was the roadmap's starting point and it is the only cycle here with published worked examples behind it. This chapter adds the other three the roadmap called for — turbofan, ramjet, scramjet — and the interesting question is not whether they run. It is what backs each of them up.
Two of them are anchored to the validated solver by an exact reduction. Set a turbofan's bypass ratio to zero and its fan pressure ratio to one and it is the turbojet, so it must return the turbojet's answer to the last floating-point digit. Set a compressor pressure ratio to one and the turbojet is a ramjet, for the same reason. Those two edges of the input space turn the validated solver into a test oracle for the new ones, and the button below runs both checks live against whatever is currently in the dashboard.
The scramjet has no such edge. It is not a stagnation-state Brayton cycle at all: the flow stays supersonic through the burner, so velocity is the physics rather than a boundary condition, and the treatment is a stream-thrust analysis.[8] It is held up instead by its own conservation laws — momentum flux across the burner, stagnation energy, continuity at every station — and by refusing to run wherever those assumptions stop holding.
Choosing an engine here re-points the off-design, design-space, uncertainty and CFD-bridge chapters at it. Each engine carries its own inputs, its own sensible ranges and its own failure modes.
The claim two of these four engines rest on, executed rather than asserted. Both new solvers are driven to the corner of their input space where they must become the validated turbojet, and the answers are laid against each other quantity by quantity. A difference at the last floating-point digit is the expected result. Anything larger means one of the two has moved, and that is exactly what this is here to catch.
Run it against the current dashboard cycle.
Each engine at its own design point, because they do not share a flight condition and could not — a turbofan at Mach 8 and a scramjet at Mach 0.8 are both nonsense. What the table shows is the shape of the trade across the whole family.
Solve each engine at its design point and lay them side by side.
What each engine assumes, and where it stops
Turbofan — separate exhaust, one fan, one burner
- The core flow passes through the fan and then the compressor, so π_c is measured from the fan exit and the overall pressure ratio is π_f · π_c.
- Specific thrust is per unit total air, which is the turbofan convention and the only one in which the bypass trade reads directly.
- The shaft balance omits the (1+f) fuel term, identically to the shipped solver. Adding it here alone would break the reduction that anchors the file.
- Off-design holds the fan/compressor work split at its design value. That is a single-spool idealisation; a real two-spool engine re-splits the work, and the model stops closing once the inlet air is much warmer than at the design point. It says so and names the remedy rather than returning a number.
Ramjet — no rotating machinery
- Two inlet models. The efficiency form is the one the turbojet uses, and it is what makes the exact reduction possible — but it is a subsonic-intake idealisation and flatters a ramjet badly above about Mach 2. The MIL-E-5008B schedule is what a supersonic intake actually achieves, and it is the default.
- A ramjet cannot produce static thrust. The solver refuses at low Mach and says why, rather than failing later with a message about nozzle expansion.
- Off-design is not a matching problem: nothing inside has a speed that has to settle. Mass flow follows from the fixed capture area, ρ∞·V∞·A, and everything else is the cycle re-solved.
Scramjet — supersonic combustion
- Constant-pressure combustion in a diverging duct. The wall pressure force cancels the two end-plane terms exactly, so momentum flux is conserved and adding fuel slows the flow down: V₄ = V₃/(1+f). That closure is derived, not assumed, and the burner exit Mach number is a result.
- If that exit Mach comes out subsonic the flow has thermally choked and the model's own assumption has been violated. It refuses, and points at dual-mode ramjet operation, which is not modelled here.
- Fuel is metered by equivalence ratio, not burner temperature. Above about Mach 8 the air arrives at four thousand kelvin stagnation before a drop of fuel is added, and stagnation temperature stops being a usable control variable. The temperature form is still offered, and still refuses up there — correctly.
- The calorically-perfect-gas assumption is the weakest link. At these temperatures oxygen dissociates and the real specific heat climbs well away from the constant used here. Results are flagged wherever that bites, and they are indicative rather than predictive.
- Specific thrust here includes the fuel momentum, unlike the other three. At Mach 8 the fuel stream carries a tenth or more of the thrust; omitting it would not be a convention, it would be an error. Both figures are always shown.
Textbook Validation
A calculator is only as trustworthy as the published answers it can reproduce. The three preset buttons in the dashboard load worked examples 7.1 and 7.2 from El-Sayed — every input, every component efficiency, the book's own rounded sea-level atmosphere — so the solver's output can be laid directly against the printed result.[6]
Below, the full station-by-station derivation of the current dashboard state, rendered live. Every line is the exact expression the solver evaluated, with the numbers it produced. Change any input above and this ledger rewrites itself.
Calculate the cycle first to see the worked solution.
Assumptions
- Steady, one-dimensional flow throughout the engine.
- Calorically semi-perfect gas: three constant-property zones — cold (a→comp exit), hot (turbine inlet→turbine exit), and afterburner (post-AB→nozzle exit when the afterburner is active).
- Mach correction applied at free stream: , .
- Diffuser (inlet): adiabatic — stagnation temperature conserved. Real mode: stagnation pressure recovery via .
- Compressor: isentropic efficiency on stagnation enthalpy basis.
- Combustor: constant total pressure (ideal) or with fractional drop (real). Energy balance yields fuel/air ratio .
- Single-spool shaft power balance: (compressor work equals turbine work divided by mechanical efficiency).
- Turbine: isentropic efficiency ; exit stagnation pressure derived from shaft-balance exit temperature.
- Nozzle: fully expanded to ambient () unless choked — if critical ratio, , pressure-thrust term added.
- ISA standard atmosphere used for and from altitude input.
- No afterburner, no cooling bleed, no inlet distortion.
Input Constraints & Validity Checks
- Flight Mach number: 0 – 5
- Altitude: 0 – 40 000 m (ISA model)
- Air mass flow: 0.1 – 1000 kg/s
- Compressor pressure ratio : 1 – 60
- Turbine inlet temperature : 800 – 2200 K; must exceed compressor exit temperature
- All component efficiencies : 0.5 – 1.0; combustor pressure loss : 0 – 0.3
- Specific heat ratio : 1.2 – 1.5; : 0.8 – 1.5 kJ/kg·K; LHV: 20 000 – 120 000 kJ/kg
- Computed fuel/air ratio must lie in (0, 0.10]
- Nozzle exit temperature must exceed ambient temperature
The Physics
of Thrust
A turbojet is a momentum machine. It swallows a river of still air, pays for it in fuel, and throws it backwards faster than it arrived. Newton settles the account: the reaction to that rearward acceleration of gas is the forward push on the airframe.[1]
Drawing a control volume around the whole engine and applying the integral momentum theorem gives the uninstalled thrust of a single-stream engine. Two terms appear: a momentum-flux term, and a pressure term that survives only when the nozzle cannot expand the flow all the way down to ambient pressure.[2]
Dividing by the air mass flow yields specific thrust — the figure of merit ThrustLab reports first, because it separates the thermodynamic quality of the cycle from the sheer size of the engine. A second figure, thrust-specific fuel consumption, measures what the thrust costs in fuel per hour.[3]
Here is the total fuel-to-air ratio — the main-burner fuel-air ratio plus the afterburner fuel-air ratio, . With the afterburner off, so .
“The engine does not push against the air behind it. It pushes against the air it carries within.” — folklore of the propulsion lab
The deepest tension in jet propulsion lives inside Eq. 1.2. A large velocity excess buys thrust, but the kinetic energy left in the jet is wasted. Propulsive efficiency — the fraction of mechanical power that actually propels the aircraft — falls as the jet gets hotter and faster:
Everything the dashboard above computes — every station temperature, every pressure ratio — exists to evaluate these three numbers honestly. Change the compressor ratio in Section 0 and watch climb while specific thrust eventually sags: the trade is structural, not accidental.[4]
The Cycle,
Read on a T-s Plane
The turbojet runs an open Brayton cycle: compress, burn at nearly constant pressure, expand. On temperature–entropy axes the ideal cycle is a clean quadrilateral bounded by two isentropes and two isobars. The real cycle — the gold curve in Figure 0.1 — leans to the right, because every component manufactures entropy.[5]
Read the live diagram station by station. From a to 2 the inlet recovers ram pressure; adiabatic, so stagnation temperature is conserved while a real diffuser loses some stagnation pressure. From 2 to 3 the compressor climbs steeply — a real machine needs more temperature rise than the isentropic minimum to reach the same pressure:
From 3 to 4 the combustor adds heat at essentially constant pressure; the great rightward sweep of entropy on the diagram is this heat addition. From 4 to 5 the turbine extracts exactly the shaft work the compressor demands — the single-spool power balance that closes the cycle algebraically. Whatever stagnation enthalpy remains above ambient is spent in the nozzle as kinetic energy.
Show full derivation — ideal-cycle thermal efficiency
With the afterburner lit, a second constant-pressure heat addition appears after the turbine — the red segment from 5 to 6. It is thermodynamically crude (heat added at low pressure converts poorly to work) but operationally decisive: thrust rises steeply while TSFC roughly doubles. Toggle it in the dashboard and watch both diagrams restructure.[6]
Station Analysis
The solver walks the gas path in the canonical station numbering of Babu Ch. 7 — the same sequence the schematic above wears as numbered chips. Each station is a bookkeeping plane: a place where stagnation temperature and stagnation pressure are tallied before the next component takes its cut.[2]
Hover any station chip in the dashboard and the probe reports all four properties — stagnation and static — straight from the last solve. The colour field inside the annulus is the same data, rendered as temperature.
What Is Proven,
and What Is Not
This page is the turbojet module in full — the shipment-ready analyzer plus every capability the roadmap places on top of it. That makes it more useful and less finished than index.html, and the difference is worth stating plainly rather than leaving a reader to work out which numbers carry evidence.
The rule the project has held since its first week applies here too: the thermodynamics, solver logic, station numbering and efficiency calculations are never modified. Every chapter above computes a small number of matched or perturbed inputs and hands them to the same untouched solver. Nothing on this page can change a design-point answer, which is why the validation in Section 6 still stands for all of it.
- Design-point cycle. Against El-Sayed Ch. 7 worked examples 7.1 and 7.2, dry and with reheat: thrust within 0.035%, exit area within 0.11%, TSFC within 0.18%. TSFC is the loosest because the printed example rounds its own intermediate steps; the published figures are themselves quoted to 3–4 significant figures.
- 149 automated assertions. Physics, boundaries, error guards, numerical stability, parametric sweep, UI contract.
- ISA atmosphere. US Standard layers 1–5, correct to 47 km, proven bit-identical below 20 km against the original two-layer model.
- Input hardening. No path produces a non-finite displayed value.
- Off-design matching. Algebraically consistent with the validated solver — it reproduces the design point to round-off — but no measured engine data has been compared against it.
- Compressor map. Correct in shape and scaled properly onto the design point. It is not any real compressor, so surge margins indicate a trend rather than a limit.
- Uncertainty bands. The propagation is exact for the stated input spreads. Whether those spreads are the right ones is an engineering judgement, not a computation.
- Flow-path sizing. Areas are continuity; radii are convention. No mechanical, stress or cooling constraint has been applied.
- Any engine other than a single-spool turbojet. The registry below declares the family; only one member has a cycle behind it.
- Transient behaviour. Every point on this page is a steady state. Acceleration, surge recovery and thermal soak are absent.
- Comparison against established codes. GasTurb, NPSS and PyCycle would each be a stronger check than another textbook.
- Turbine and nozzle maps. Only the compressor is mapped; the turbine is represented by its choked flow capacity alone.
The architecture is written for a family of engines rather than one. Members without a validated cycle behind them are declared here and refuse to compute rather than returning a plausible number — a tool that is confidently wrong is worse than a tool that says it cannot help.
The honest critical path is unchanged: breadth of engine types is the visible progress, but off-design capability and independent validation evidence are what would make a serious engineering organisation take this seriously. Section 2 is the first of those two. The second is still open.
References
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