Mixing is a pump

A jet pump is a nozzle pointing down a parallel tube. Fast fluid comes out of the nozzle, slower fluid is dragged along beside it, the two mix over a metre or so of tube, and everything leaves at a higher pressure than the slow stream came in at. There is no impeller, no shaft, no seal and no bearing. The device is a piece of pipe with a hole in one end.

That such a thing can raise pressure at all is worth stopping over, because it appears to be forbidden. Nothing is doing work on the fluid — no surface is moving — and the flow is not decelerating in a diffuser. A Bernoulli account has nothing to offer: along any streamline in a steady inviscid flow the total head is constant, and here it rises for one stream and falls for the other.

The resolution is that this is a momentum device, and its energy comes from a loss.

Why the pressure has to rise #

Take a control volume round the mixing tube, with the nozzle stream and the entrained stream entering at one end and a single uniform stream leaving at the other. Momentum across it, with the pressure uniform on each face:

$$p_2 - p_1 = \frac{\rho}{A}\left(A_p V_p^2 + A_s V_s^2\right) - \rho V^2, \qquad V = \frac{A_pV_p + A_sV_s}{A}$$

The right-hand side is non-negative, always, and the reason is Cauchy–Schwarz: the momentum flux of a single uniform stream is always less than that of the two streams it was made from, with equality only when the two had the same speed to begin with. Mass and momentum are both conserved, the mixed stream carries the same mass at the mean velocity, and it therefore carries less momentum — so something has to make up the difference, and the only candidate is the pressure on the faces.

The pressure rise is not a design achievement. It is a theorem.

Where the energy comes from #

Momentum is conserved and energy is not, and the gap is the whole story.

Writing the energy flux in and out and subtracting gives the loss, and it has a closed form:

$$\text{loss} = \tfrac{1}{2}\left[\dot{m}_p(V_p - V)^2 + \dot{m}_s(V_s - V)^2\right]$$

which is the Borda–Carnot expression with two streams in it instead of one. Set the secondary flow to zero and it collapses to ½ṁ(V₁ − V₂)² exactly — the number a handbook prints beside a sudden enlargement, which is what an ejector entraining nothing is.

The same arithmetic is a waste in a pipe fitting and a pump here. That is the finding this essay exists for. A sudden enlargement is filed in every handbook as a loss to be avoided; run it with a secondary flow and the identical expression is the mechanism of a device that people buy.

What changed is not the physics but what is being paid for. In a pipe fitting, one stream loses head and nothing gains any. In an ejector, one stream loses a great deal and another gains some, and the gain is what the machine is for.

The pump curve nobody designed #

Sweeping the entrained stream’s speed at fixed geometry produces a characteristic, and it has the shape everybody recognises from a centrifugal pump’s data sheet: highest pressure at no flow, falling steadily, zero pressure at maximum flow.

That shape is usually explained by the impeller — blade angles, slip, recirculation at low flow. Here there is no impeller and the curve is the same. Shut-off occurs when the secondary stream is stationary and the whole momentum difference appears as pressure; free delivery occurs when the two streams already have the same speed and there is nothing left to transfer.

A pump characteristic, in other words, is not primarily a fact about impellers. It is a fact about what a momentum balance does when a flow rate is varied at fixed geometry, and the ejector is where that can be seen without any machinery in the way.

Two limits, and a wheel in disguise #

The operating point that matters is not the one with the best efficiency, and the distinction is a trap worth naming.

Efficiency here rises monotonically towards the point where the two streams already have the same speed — where nothing is happening, reversibly. That is perfectly true and perfectly useless. The quantity with an interior maximum is the delivered power, and the efficiency there is the number a designer wants.

As the nozzle area is made vanishingly small the two limits come out clean:

• most power at V_s = V_p/2, computed by search at 0.50001;
• efficiency there of exactly two thirds, computed at 0.66668.

The first is the Pelton wheel’s half-speed rule, arriving in a machine with nothing moving in it. That is not a coincidence and it is not quite the same statement either. A bucket takes most power when it runs at half the jet speed; a jet delivers most power to its surroundings when it drags them to half its own speed. Both are the same parabola — a transfer whose rate is the product of a force falling linearly with a speed and that speed itself — and both peak in the middle.

The two-thirds is the ejector’s own. It is the price of transferring momentum by mixing rather than by turning: a bucket at its optimum leaves the water dead and takes everything, whereas a mixing tube at its optimum still has two streams at different speeds inside it, and what they do to each other on the way to equality is what the third goes on.

The design sweep, which has no single answer #

The ends of that sweep are two different products.

A narrow nozzle entrains enormously and lifts almost nothing. That is a ventilation ejector, a fume extractor, a laboratory water aspirator: the purpose is to move a large volume of something undesirable, and the pressure rise needed is a few millibar.

A wide nozzle entrains little and lifts a lot. That is a steam-jet ejector on a condenser, or the eductor in a firefighting foam system: the purpose is to reach a vacuum or a head, and the flow rate is secondary.

The same equations, the same machine, and a factor of a thousand between the two applications’ duties.

The machine at a different scale #

Changing the motive speed changes everything about the numbers and nothing about the shape, which is what a dimensionless account of the machine would predict and is worth seeing directly.

That invariance is a hint about how the device should be described, and the hint is taken seriously two rungs later. Everything in this essay could be written in terms of the area ratio and the velocity ratio alone, with the fluid, the size and the speed scaled out — which is the same move the dimensional essay makes for a body in a stream and the same one a rotating machine’s specific speed makes.

There is also a lesson here about what a control volume can and cannot see. The balance is blind to the length of the mixing tube, so the model has no way to say whether the mixing has finished — and if it has not, every number above is optimistic. A method that refuses to look inside cannot report that the inside was too short, and this is the one place in the field where that refusal costs something real.

Why anyone uses a device this inefficient #

Sixty-eight per cent is the model’s ceiling; a real jet pump reaches twenty to thirty. Against a centrifugal pump’s eighty that looks indefensible, and it is chosen anyway, for reasons that are all about what the machine does not have.

It has no moving parts, so it cannot wear out, and it needs no lubrication or alignment. Nothing in it can cavitate destructively in the way a pump impeller does, because the low-pressure region is a free jet rather than a blade surface, and a collapsing bubble in midstream damages nothing. It can pass solids, slurries and fibres that would destroy an impeller. It works submerged, buried, or inside a vessel nobody can open. It can be made entirely of glass, of PTFE, of graphite — anything that can be formed into a tube — so it will handle fluids that attack every pump material there is. And it can be driven by a fluid rather than by electricity, which matters where electricity is the thing that must not be present.

Efficiency is a currency, and it is not always the scarcest one. That is a general point about this whole field: a control volume gives the ceiling on what a machine can do, and whether the ceiling is worth reaching is a question the control volume does not answer.

What is not in the model #

No mixing length. The balance assumes the two streams have become one by the outlet. A real mixing tube needs six to ten diameters to do it, and if it is shorter the exit is not uniform, the momentum flux out is larger than the model’s, and the pressure rise is smaller. This is the single biggest gap between the arithmetic here and a real device.

No friction on the tube wall, and no diffuser. Both are real losses and both are outside the control volume’s faces. A real ejector has a diffuser after the mixing tube to recover the mixed stream’s velocity head, and the diffuser’s efficiency is a large part of the machine’s.

The nozzle is ideal. The motive stream is assumed to arrive at V_p having expanded from a plenum without loss. A real nozzle is 95 per cent efficient at best, and the loss is squared into the velocity.

Nothing here is a solved field. No streamline is drawn anywhere in this essay, because the mixing region is exactly the sort of shear flow this site’s turbulence field declines to compute: two parallel streams at different speeds are the unstable configuration par excellence, and a picture of the eddies would be an invention. What is drawn is a box, its faces, and the numbers crossing them.

No compressibility. Steam-jet ejectors — by far the commonest industrial application — run the motive stream supersonically and the analysis needs the compressible machinery rather than this. Everything above is the incompressible case, which is the water-driven eductor and the ventilation jet.

The characteristic a steam ejector actually has #

The compressible case is excluded above, and it is worth one section, because the commonest industrial ejector by a wide margin is a steam-jet one and its characteristic is not the smooth curve this essay draws.

The motive steam leaves a converging–diverging nozzle at Mach two, three or four. The entrained gas is accelerated in the annular passage that narrows around that jet, and if it accelerates far enough it reaches Mach one there — not against any wall, but against the jet itself, which acts as the inner boundary of a converging duct. The secondary stream is then choked, in exactly the sense a throat is choked: nothing downstream can reach back past the sonic section to tell it anything.

The consequence is a characteristic of a completely different shape. In this doubly-choked mode the entrainment ratio does not depend on the back pressure at all — the curve is horizontal, and raising or lowering the vacuum being pulled changes nothing about how much gas the ejector moves. That is a very good property for a vacuum device, and it is why steam ejectors hold their capacity all the way down to their design suction pressure instead of sagging as a fan would.

And it ends abruptly. Above a critical back pressure the secondary flow can no longer stay choked, the shock system inside the mixing tube moves upstream, and the entrainment collapses over a small range of pressure — sometimes reversing, so that the machine blows backwards through its own suction line. There is no gentle degradation: a steam ejector is specified by its critical back pressure, and operating above it is a failure rather than a loss of efficiency.

So the incompressible model’s smooth falling curve and the compressible machine’s flat-then-cliff are not the same object approximated differently. The choking has replaced the characteristic.

The Bernoulli story, and why it is not enough #

The usual explanation of a jet pump is that the fast jet has a low pressure, and the low pressure sucks the secondary fluid in.

The first half is true and useful: the pressure at the nozzle exit is below ambient, that is what starts the secondary flow, and it is why an aspirator on a tap can pull a partial vacuum. The second half of the sentence is where it fails, and the failure is instructive rather than pedantic.

Bernoulli’s equation forbids what this machine does. Along a streamline in a steady inviscid flow the total head is constant; here the entrained stream’s total head rises between inlet and outlet. No Bernoulli argument can produce that, because Bernoulli is a statement about a flow with no losses in it, and this machine’s entire mechanism is a loss. An explanation built on it can account for the suction at the nozzle and cannot account for the pressure at the outlet — which is the part somebody is paying for.

This is the same failure mode the site’s suction essay diagnoses in a different setting: a real low pressure, correctly identified, given a causal role it cannot carry. The correct statement is a momentum balance across the whole tube, and it is two lines long.

The device seen as a momentum transformer #

There is a compact way to say what an ejector is, and it puts the machine beside two others in this collection.

A pump adds energy to a fluid through a moving surface. An ejector transfers momentum between two streams with no moving surface at all — and momentum, unlike energy, can be transferred without anything doing work, because a momentum flux is carried by the flow itself.

The price of that convenience is set by a simple asymmetry. Momentum is conserved in the exchange and energy is not, so a transfer that conserves the first must lose some of the second. The loss is the sum of ½ṁ(V_i − V)² over the two streams, which is zero only when the streams already match and grows as the square of their mismatch — so an ejector that entrains a great deal from a very fast jet is necessarily inefficient, and one that entrains a little from a barely-faster jet is efficient and useless.

Two other machines here run on the same trade. A hydraulic ram transfers energy between a large flow and a small one through a pressure pulse rather than a shaft, and its ceiling is an audit. An actuator disc transfers momentum between a stream and a shaft, and its ceiling is Betz’s fraction. In each case the machine’s simplicity is bought with a bound, and the bound comes out of the same box-drawing method.

Who found it, and when #

James Thomson — Kelvin’s brother — described the jet pump in 1852, and Henry Giffard’s injector of 1858 is the same device solving a problem that looks impossible: feeding water into a steam boiler using steam from that boiler, at a pressure higher than the steam supplying it. That it works astonished contemporaries, and the explanation is exactly the arithmetic above with a condensing motive stream.

The Borda–Carnot loss the machine lives on is a century older, and was derived to explain why a pipe that widens abruptly wastes head. The gap between the two — a loss described in 1766 and turned into a pump in 1852 — is the sort of interval this field keeps producing: the conservation laws were finished long before anybody thought to run them backwards.

Where the ladder goes next #

Three machines have now been analysed by drawing a box round them and refusing to look inside. What none of those boxes can say is which kind of machine a duty needs, and that question has an exact answer too — from dimensional analysis rather than from conservation, and from a group with no size in it at all.