The throat that stops listening

Attach a tank of compressed air to a hole and open a valve downstream. Lower the pressure on the far side and more air comes through, which is what anybody would expect.

Keep lowering it. At a specific pressure ratio the flow stops increasing — and it does not level off gracefully, it stops. Take the downstream pressure to a hard vacuum and not one extra gram per second comes through the hole.

The explanation is about information, not about resistance #

The instinct is to look for something that saturates — a pipe that cannot pass more, a friction that grows, a valve that has run out of travel. None of these is it. The duct is unchanged and the gas is unchanged.

What has happened is that the communication has failed.

Lowering the back pressure only increases the flow because the reduced pressure is felt upstream. The gas near the throat has to learn that conditions downstream have improved, and it learns this by a pressure signal travelling upstream at the speed of sound relative to the fluid. In a subsonic flow that signal makes progress: it moves upstream at $a - u$, which is positive.

At Mach one, $a - u = 0$. The signal is running upstream exactly as fast as the flow is carrying it downstream, so it stands still and never arrives. Everything upstream of the sonic throat is permanently ignorant of everything downstream of it.

That is choking. It is the same statement as the zone of silence applied to a duct instead of to a body, and it has the same cause: a signal speed that is finite and has been matched.

The panels also make clear why choking is a knife edge rather than a gradual loss. The upstream progress of a signal is $a - u$, which falls linearly to zero and then goes negative. There is no range over which the news arrives faintly. It arrives, or it does not.

The critical pressure ratio, computed #

The condition for choking is that the throat reaches Mach one, and the isentropic relations say exactly what pressure ratio that requires.

$$\frac{p^*}{p_0} = \left(\frac{2}{\gamma+1}\right)^{\frac{\gamma}{\gamma-1}} = 0.5283$$

for $\gamma = 1.4$. So a tank at any pressure whatever will choke through a hole as soon as the outside pressure is below 52.83 per cent of it — which for a tank at atmospheric pressure means an outside pressure below 0.535 bar, and which means that almost every compressed-gas device in ordinary use is choked. A car tyre being let down is choked. A cylinder of welding gas venting is choked. A domestic gas hob’s injector is choked, and that is why the flame does not change when the weather does.

Notice that the critical ratio depends on $\gamma$ and on nothing else. Not on the size of the hole, the temperature of the tank, the absolute pressure, or the gas’s molecular weight. Helium chokes at 0.487 and carbon dioxide at 0.546, and the difference between them is entirely the difference in how many ways their molecules can store energy.

The flow that gets through, computed #

Once choked, the mass flow is fixed by the throat area and the stagnation state:

$$\dot m = A^* p_0 \sqrt{\frac{\gamma}{R T_0}} \left(\frac{2}{\gamma+1}\right)^{\frac{\gamma+1}{2(\gamma-1)}}$$

which in the non-dimensional form this site uses is a flux of $0.6847\, p_0/\sqrt{R T_0}$ per unit throat area.

That expression is worth reading for what it contains. The mass flow is proportional to the stagnation pressure, so raising the tank pressure raises the flow without limit. It goes as the inverse square root of the stagnation temperature, so a hot gas flows more slowly by mass than a cold one at the same pressure — which is why the flow through a choked orifice is a thermometer as well as a flowmeter, and why cryogenic propellant is measured as carefully for temperature as for pressure.

And the back pressure does not appear at all.

Two routes to the same number #

The flux constant is checked here rather than quoted, because it is the sort of expression that is copied more often than derived.

assertChokedFlowDoesNotRespond computes it twice. Once from the closed form above, and once by assembling $\rho^* a^*$ out of the sonic state ratios — density $0.6339 \rho_0$ times the speed of sound at $0.8333 T_0$ — which uses none of the same algebra. The two agree to within $10^{-12}$.

It then sweeps a nozzle through eleven back pressures and requires every choked case to return a throat Mach number of exactly one and every unchoked case to return one below it. The rejection test hands it a “choked” nozzle whose throat is at Mach 0.94, which is a perfectly plausible-looking number and is a contradiction in terms, and requires the refusal.

How much gets through a hole, in practice #

Because the flux constant has no back pressure in it, a choked-flow calculation is unusually short, and it is worth doing once to see how little is needed.

A tank of air at 10 bar and 288 K, venting through a 5 mm hole. The stagnation density is $p_0/RT_0 = 12.1$ kg/m³; the throat area is $1.96 \times 10^{-5}$ m²; the sonic density and temperature are 0.634 and 0.833 of the stagnation values, giving a throat density of 7.67 kg/m³ and a sonic speed of $\sqrt{1.4 \times 287 \times 240} = 310$ m/s. The mass flow is the product: 0.047 kg/s.

Nothing about the room the gas is venting into entered that. The same hole at the same tank conditions passes 0.047 kg/s into a laboratory, into a partial vacuum, or into orbit.

What choking is used for #

Being unable to increase the flow sounds like a limitation. It is more often a feature, and several pieces of everyday hardware exist only because of it.

A flow standard. A choked orifice passes a mass flow that depends only on upstream conditions, so it is a metering device that is immune to everything downstream of it. Critical-flow venturis are used as primary standards for gas flow for exactly this reason.

A safety valve. A vessel rupturing through a hole cannot lose gas faster than the choked rate, which makes the worst case computable rather than open-ended — a bound of the same practical kind as the one a solved boundary layer puts on friction drag, in that it converts an open question into an arithmetic one.

A rocket engine’s stability. The chamber of a rocket is separated from the outside world by a choked throat, so what happens in the exhaust plume — including at what altitude the engine is flying — cannot propagate back into the combustion. Chamber pressure is set by the propellant flow and the throat area, and by nothing outside.

A wind tunnel’s independence. A supersonic tunnel’s test section conditions cannot be affected by the diffuser or the exhaust, because the flow passed through a sonic throat on the way in.

What happens downstream instead #

The flow does not stop responding to the back pressure. It stops responding upstream of the throat. Downstream, the response is dramatic and is the subject of the next rung.

The convergent half being identical in every choked case is the strongest visual statement of the argument. Five different back pressures, five different flows, and one shared upstream solution.

Choking without any walls #

The word “throat” suggests hardware, and the phenomenon needs none. What is required is a cross-section of the flow at which the local Mach number reaches one, and a stream tube can supply that on its own.

This is why a wing can choke. The flow accelerating over a thick section is passing through a stream tube that narrows and then widens, and if the free-stream Mach number is high enough that stream tube goes sonic at its narrowest point. Everything downstream of the sonic line is then cut off from everything upstream, in mid-air, with no duct anywhere — and the reconciliation between the two regions is a shock standing on top of the wing.

The same effect limits ventilation ducts, sets the maximum flow through a partially closed valve, and puts a hard ceiling on the throughput of a wind tunnel’s own working section — one more way in which a model test can fail to represent the flight it stands for. In each case the sonic cross-section is somewhere the designer did not put a throat.

The number that decides is not a pressure difference #

A distinction that trips up a great deal of practical work: choking depends on the pressure ratio, not on the pressure difference.

A tank at 10 bar venting to 6 bar has a difference of 4 bar and a ratio of 0.6 — unchoked. A tank at 2 bar venting to 1 bar has a difference of 1 bar and a ratio of 0.5 — choked. The larger difference is the subsonic case.

This follows from the same dimensionless structure as everything else in this field: the isentropic relations are functions of Mach number alone, and the pressures enter only as ratios to a stagnation value. An absolute pressure difference is not a dimensionless quantity and therefore cannot be the criterion for anything.

What it feels like from upstream #

There is a way of putting this that makes the strangeness land, and it is worth stating because the mathematics can be followed without ever noticing how odd the conclusion is.

Consider the gas sitting in the convergent section of a choked nozzle. It is accelerating, it has a perfectly definite pressure and speed at every station, and those values are fixed by the stagnation conditions and the local area — both roots of which are properties of the duct alone. Now open the far end of the apparatus to space.

Nothing happens. Not “a small change”; nothing. That gas has no mechanism by which the existence of space could be communicated to it, so its state is exactly what it was, and it will remain so whatever is done downstream of the throat forever.

The consequence for measurement is worth having: a pressure tapping anywhere in the convergent section of a choked nozzle is a useless instrument for anything downstream. It reads the same value at every back pressure below critical. Diagnosing a nozzle from upstream taps alone cannot distinguish a shock inside the divergent section from a perfectly expanded supersonic exhaust, and several generations of test engineers have had to learn that the hard way.

The mixture that chokes at walking pace #

The criterion for choking is a Mach number of one, and every number above assumes the speed of sound is the gas’s own. There is a common industrial fluid for which it is neither the gas’s nor the liquid’s but far below both, and the consequence is a choking condition that arrives at speeds nobody would think to check.

Bubbles in a liquid. The mixture’s density is dominated by the liquid, because the gas weighs almost nothing; its compressibility is dominated by the gas, because the liquid barely compresses at all. A sound speed is built from one over the product of the two, so the mixture inherits the worst of each — heavy and squashy at once — and the result is much slower than either constituent.

The arithmetic is worth doing. For air and water at atmospheric pressure in equal volumes, the mixture density is about 500 kg/m³ and its compressibility is essentially the air’s diluted by a half, and the resulting sound speed is about 24 metres a second. Air alone carries sound at 340 and water at 1,500; their mixture carries it at less than a tenth of the slower of the two, and the minimum sits near equal volumes rather than at either end.

So a bubbly stream moving at thirty metres a second is supersonic, and a two-phase flow through a restriction chokes at a velocity that a single-phase calculation would call negligible.

That is not a curiosity. It is the reason relief systems for flashing liquids are sized by a different calculation from gas systems. A hot pressurised liquid vented through an orifice begins to boil as its pressure falls, so what passes through the throat is a mixture rather than a liquid, its sound speed collapses, it chokes, and the mass flow that gets out is far less than a liquid-flow calculation predicts. A relief valve sized as though the fluid stayed liquid is undersized, and the vessel it was protecting is not protected.

Everything else in this essay survives the change. The mechanism is still that a signal cannot travel upstream against a flow moving at the signal’s own speed, the criterion is still a local Mach number of one, and the flow upstream of the sonic section still knows nothing of what is beyond it. What has moved is the number the speed is compared against, and it has moved by a factor of fifty.

Where the model stops #

Friction is absent. A long pipe rather than a short nozzle is dominated by friction, and friction also drives the flow towards Mach one — from below in a subsonic pipe, and from above in a supersonic one. The exit of a long choked pipe is sonic for a rather different reason than the throat of a nozzle is, and the mass flow is lower than the ideal expression gives.

The boundary layer takes some of the throat. The effective throat area is the geometric one minus the displacement thickness of the layer on its walls, so the real mass flow is a per cent or two below prediction. Discharge coefficients exist to carry that, and they are close to but never equal to one.

Unsteadiness is out of scope. Everything here is a steady solution. The transient as a valve opens involves waves running both ways along the duct, and during it the flow does things no steady solution describes.

The gas is perfect and $\gamma$ is fixed. Choking of a real gas near its critical point, or of a gas whose composition changes as it expands, needs a different equation of state.

What the picture cannot show #

The choking figure is a plot of a quantity against a control, and the thing it hides is where in the duct the interesting event happens.

The curve goes flat, and nothing in it says that the reason is a single cross-section — the narrowest one — having reached a particular speed. A reader could take away that the nozzle as a whole saturates. It does not: the throat does, and every other station is a passenger.

The figure also cannot show the signal that fails to arrive, which is the actual mechanism. There is no way to draw an absence of communication, and the flat line is the consequence rather than the cause. The nearest this site comes to drawing the cause is the wavefronts of a source at Mach one, where every pulse piles up on a single plane and gets no further forward.

Who found it, and when #

Choking was observed before it was understood, as usual. Steam engineers in the nineteenth century knew that orifices reached a maximum discharge, and Saint-Venant and Wantzel published a nozzle theory in 1839 which predicted the flow correctly up to the critical ratio and then, notoriously, predicted that it would fall again as the pressure was lowered further.

That prediction is wrong and the error is instructive: their theory assumed the exit pressure always equalled the back pressure, which is true only until the throat goes sonic. Beyond that the exit pressure detaches from the back pressure and stays at the critical value, and the flow stays flat instead of falling. The experiments showing the plateau rather than the fall took decades to be believed, because the theory was elegant and the measurement was awkward.

Where the ladder goes next #

Something has been passed over. The area–Mach relation gives two Mach numbers at every station of the divergent section, and this essay has quietly used the subsonic one in some places and the supersonic one in others without saying which the nozzle actually picks.

Nothing local decides it. What decides it is the back pressure, acting through the only route it has left — and where the two solutions cannot be reconciled, a shock appears inside the duct to join them.