How does a gas behave when heat is added in a constant-area, frictionless duct? Adjust inlet Mach, temperature, pressure and heat input to see the exit Mach number and the margin to thermal choking update in real time. Build intuition for jet-engine combustors and ramjets.
Parameters
Inlet Mach number M₁
Mach number of the gas entering the heated section (subsonic)
Inlet temperature T₁
K
Inlet pressure P₁
kPa
Specific-heat ratio γ
Air γ = 1.40, hot combustion gas γ ≈ 1.33
Heat addition q
kJ/kg
Heat added per unit mass; equals fuel heating value times fuel-to-air ratio
Results
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Inlet stagn. temp. T₀₁ (K)
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Stagn. temp. after heating T₀₂ (K)
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Heat limit q_max (kJ/kg)
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Exit Mach number M₂
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Heat ratio q/q_max (%)
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Choking status
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Heated duct flow — acceleration animation
Subsonic gas enters a constant-area duct; heat addition lowers its density and accelerates it. Colour shows temperature (blue → orange → red); arrow length shows the local velocity (Mach number). A choking marker appears when M reaches 1.
The Rayleigh function. It gives the ratio of the stagnation temperature T₀ to the choked-state stagnation temperature T₀*, as a function of Mach number. It equals 1 at M = 1. Subsonic heat addition raises T₀ and drives M toward 1.
Definition of stagnation temperature and the maximum heat addition. c_p = γR/(γ−1) with R = 287 J/(kg·K). Thermal choking occurs exactly when q reaches q_max and M₂ = 1.
What is Rayleigh Flow (Heated Duct)?
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I have never heard of "Rayleigh flow". How is it different from ordinary pipe flow?
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Good question. Rayleigh flow is the idealised compressible flow of a gas through a constant-area duct where heat is added (or removed) along the way and friction is neglected. It is the textbook model of a jet-engine combustor or a ramjet heated duct. In real combustors the heat release dominates so heavily over friction that Rayleigh flow gives you most of the intuition for free.
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OK, but if you heat a gas, doesn't it just get hotter? In the simulator, when I raise the heat input q the exit Mach number M₂ also goes up, which really surprised me.
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That is the heart of Rayleigh flow. Add heat to a subsonic stream and the gas expands. The duct area is fixed, so mass conservation ρuA = const forces the velocity u to rise as the density ρ falls. Mach number M = u/√(γRT) does see a higher temperature, but in the subsonic branch the velocity gain wins over the speed-of-sound gain, so M climbs steadily toward 1. "Heating accelerates the flow" is the single most counter-intuitive result every student of compressible flow has to digest.
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Then if I just keep dumping more heat in, the flow keeps speeding up forever?
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No — there is a hard ceiling called "thermal choking". At a particular heat addition the exit Mach number hits exactly 1, and any more heat physically cannot pass through the same duct. The tool reports it as q_max. For the default condition (M₁ = 0.30, air) q_max ≈ 578 kJ/kg. What happens above q_max? The upstream pressure rises, the mass flow drops, or a shock appears, and the operating point shifts itself until q ≤ q_max. Real combustor designs limit fuel injection to roughly 60–70% of q_max for safety.
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Is the exit Mach M₂ obtainable in closed form? The Rayleigh equation is written in terms of T₀…
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Good observation. The Rayleigh function T₀/T₀* = (γ+1)M²(2+(γ−1)M²)/(1+γM²)² has fourth-order polynomials in M on both top and bottom, so you cannot solve it for M in closed form. This tool computes the post-heating ratio T₀₂/T₀* and then bisects on M in [M₁, 1] for 60 iterations. The Rayleigh function is monotonic in the subsonic branch, so the bisection converges cleanly. Classical gas-dynamics textbooks tabulated this as the "Rayleigh table", but with a few lines of code the answer is now instantaneous.
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Where does this show up in real hardware?
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The classic examples are jet-engine combustors, ramjets, scramjets, afterburners, and the heated duct downstream of a solid-rocket nozzle. All have a constant area into which a lot of heat is dumped. Ground-based industrial furnaces with long ducts and electric heaters also use the Rayleigh-flow view to estimate when "any extra heat will be cut off by mass-flow limiting" — i.e. when the heat input reaches q_max. Rayleigh flow is one of the closest one-dimensional idealisations to real hardware in compressible-flow textbooks.
Frequently Asked Questions
Rayleigh flow is the idealised model of an ideal gas streaming through a constant-area duct in which heat is added or removed, with friction neglected. It is the textbook picture of a jet-engine combustor where the heat input dominates over friction. Fanno flow is its twin: adiabatic (no heat in or out) but with friction, used to analyse losses in long pipes. Both are one-dimensional compressible flows at constant area; the driving energy term is the difference.
Adding heat makes the gas expand, so its density falls. Because the duct area is constant, mass conservation rho*u*A = constant forces the velocity u to rise. Mach number is M = u / sqrt(gamma R T): the temperature also rises, but in the subsonic branch the velocity increase dominates over the speed-of-sound increase, so M climbs toward 1. This counter-intuitive result is the signature of Rayleigh flow.
If you keep adding heat to a subsonic Rayleigh flow, the exit Mach number eventually hits exactly 1 and no more heat can fit through the same duct. This is thermal choking. This tool reports q_max, the maximum heat that fits. The three escape routes are: (1) reduce the upstream mass flow, (2) increase the duct area, or (3) accept the upstream pressure rising until the operating point shifts. Real combustors limit the fuel injected and use a wide combustor to lower the inlet Mach number.
The Rayleigh function T0/T0* = (gamma+1)M^2(2+(gamma-1)M^2)/(1+gamma*M^2)^2 is a monotonic function of M in the subsonic branch and equals 1 at M = 1. After computing the post-heat stagnation temperature T02, this tool solves T02/T0* = rayleighFunc(M2) for M2 by bisection on [M1, 1] with 60 iterations. When the heat input exceeds q_max the target becomes greater than 1 and M2 is clamped to 1 with a thermal-choking warning.
Real-World Applications
Jet-engine combustors: Turbojet and turbofan combustion chambers are direct Rayleigh-flow targets. Air is compressed, fuel is injected and burned, and several hundred to a thousand kJ/kg of heat is released. Designers size the combustor so the inlet Mach is held to 0.2–0.3, keeping q comfortably below q_max and avoiding the pressure loss and temperature runaway that thermal choking would cause. Push M₁ above 0.5 in this tool and increase q to feel how aggressive q_max becomes.
Ramjets and scramjets: In these engines the inlet air is decelerated (ramjet) or held supersonic (scramjet) and then burned. Inside the ramjet combustor the flow is subsonic and follows Rayleigh flow exactly. Scramjets keep the flow supersonic to avoid thermal choking, exploiting the fact that supersonic Rayleigh flow also drives M toward 1 — but from above, so it slows the flow down on heat addition.
Afterburners (reheat): The bright blue tail of a fighter jet under reheat is a textbook Rayleigh-flow device: more fuel is dumped into the turbine exhaust to gain a 50% thrust boost. Afterburners operate close to the choking limit, so the fuel controller juggles q at 80–90% of q_max with tight closed-loop feedback to avoid both choking and combustor blowout.
Pre-study for CFD analysis: A one-dimensional Rayleigh model like this tool gives a first-order sanity check before running detailed CFD (chemistry + turbulence + 3-D geometry) of a real combustor. If the CFD exit Mach number or pressure ratio differs from the Rayleigh estimate by orders of magnitude, suspect boundary conditions or combustion-model settings. Undergraduate and graduate compressible-flow courses always pair Rayleigh flow with the h–s "Rayleigh line" diagram.
Common Misconceptions and Pitfalls
The biggest misconception is "heating a gas only raises its temperature". In a closed vessel that is true, but in an open duct flow the gas expands, the density falls, the velocity rises and — for subsonic Rayleigh flow — the Mach number climbs toward 1. Supersonic Rayleigh flow is the mirror image: heat addition slows the flow down, but again drives M toward 1. The single correct statement is that "heat addition always pushes M toward 1, from either side". The "heating speeds the flow up" result is specific to the subsonic branch.
Next is "what happens if I try to inject more heat than q_max?". Physically, no such steady state exists: the upstream pressure rises (back-pressure propagates), the mass flow drops, or a shock appears, until the operating point relaxes to q ≤ q_max. This tool clamps M₂ at 1 and reports "thermal choking" when q exceeds q_max, but in real hardware this regime shows up as pumping (mass-flow oscillation) and combustion instability, both very dangerous. The standard design rule is q ≤ 0.6–0.7 × q_max.
Finally, do not forget that Rayleigh flow neglects friction. In real long combustors and heated ducts both friction and heating act together, and the actual exit Mach number is slightly higher than the pure Rayleigh prediction (friction also drives M toward 1 in subsonic flow). Short, high-heat-load combustors are well represented by Rayleigh; long, low-heat-load ducts need a combined Fanno + Rayleigh analysis or a generalised one-dimensional method. Use this tool as the idealised model of a "short section dominated by heat addition".