The wall the suction puts in the curve

A jet pump is often chosen for a job because it has nothing in it to go wrong. It has no impeller to erode, no seal to leak and no shaft to align, and so it is sent down boreholes, into sumps and onto the suction side of other pumps. The essay on its losses found where such a machine is best — a nozzle a quarter of its throat, at a third efficiency — on the assumption that it can run wherever its characteristic says.

A liquid jet pump has one place where that assumption fails, and the failure is not a loss of performance. It is a change in what the characteristic is.

Where the pressure is lowest #

Two streams enter the throat side by side: the motive jet from the nozzle, and the entrained stream from the annulus around it. They share one static pressure at that section, because nothing separates them, and that pressure is the lowest anywhere in the machine.

It is lowest because of what the entrained stream has had to do to get there. It started at rest in the suction plenum and has been accelerated to the speed at which it enters the throat, and it paid for that speed with its own static pressure, plus a little extra lost at the inlet. The motive jet arrives at the same static pressure from much higher up. Downstream of the throat entry, every section is at a higher pressure: mixing raises it, as the momentum argument in mixing is a pump showed it must, and the diffuser raises it again.

In a liquid, that minimum has a floor. When the static pressure at the throat entry reaches the vapour pressure, the liquid there boils — not from heating, but because the pressure holding it liquid has been taken away — and a vapour cavity forms at the edge of the jet. Every argument that follows is about what the cavity does to the machine’s behaviour, and the first thing to see is that it can only depend on what happens upstream of the throat entry.

The flow ratio at which the wall arrives #

Write the pressures relative to the throat entry. The motive supply is the jet’s dynamic head plus the nozzle’s loss, and the suction plenum is the entrained stream’s dynamic head plus the inlet loss:

$$P_m - p_1 = \tfrac{1}{2}(1 + K_n)\,\rho V_n^2, \qquad P_s - p_1 = \tfrac{1}{2}(1 + K_s)\,\rho V_s^2.$$

The throat entry reaches vapour pressure when $p_1 = p_v$. The natural measure of how close a machine is to that is the cavitation parameter,

$$\sigma = \frac{P_s - p_v}{P_m - P_s},$$

which is the suction stream’s margin over boiling, measured against the pressure difference that drives the machine. Setting $p_1 = p_v$ in both relations, dividing one by the other, and using continuity through the annulus, $V_s/V_n = MR/(1 - R)$, gives the flow ratio at which the wall is reached:

$$M_c = \frac{1 - R}{R}\,\sqrt{\frac{\sigma\,(1 + K_n)}{(1 + K_s)(1 + \sigma)}}.$$

That expression was also found without the algebra, by stepping the flow ratio upward along the full operating model — momentum balance, all five losses, the pressures they produce — and stopping where the throat entry first touched the vapour pressure implied by σ. Across nine cases spanning area ratios from a tenth to a half and σ from 0.05 to 1, the march and the closed form agree to $2 \times 10^{-16}$.

Three things are absent from $M_c$, and each absence carries physics.

The throat friction and the diffuser are not in it. Both act downstream of the throat entry, on the mixed stream, and nothing downstream can change the pressure the entrained stream reaches on its way in. A better diffuser raises the head and leaves the wall exactly where it was.

The discharge pressure is not in it. The wall is a flow ratio, fixed by the suction margin and the inlet geometry, and the pressure the pump is working into does not appear. That is what turns the wall into a wall rather than into a lower curve.

The size of the machine is not in it. Everything is a ratio. A small jet pump and a large one with the same area ratio and the same σ cavitate at the same flow ratio — which is not quite true in practice, because cavitation begins in the small pressure dips of the shear layer before the mean pressure gets there, and those dips do not scale perfectly. The model’s wall is the mean-pressure wall, and a real machine meets cavitation somewhat earlier.

What the wall does to a machine that is running #

In a real installation the three pressures are given: a motive supply, a suction vessel, and a discharge line. Together they fix the head ratio $N$ the installation imposes, and the machine settles at whatever flow ratio its characteristic gives at that $N$. Lower the discharge pressure, and $N$ falls, and the flow ratio moves along the curve to a larger value.

Until it reaches $M_c$.

From a discharge of 294 kPa down to the suction pressure, the flow ratio is 0.675 exactly. Computed at four discharge pressures on the wall, it does not differ between them in a single bit, because once the throat entry is pinned at vapour pressure the entrained stream’s speed is pinned by the suction margin, and the motive jet’s speed by the motive margin, and neither of them reads the discharge.

The pressure difference the characteristic would have used to drive more flow is taken up inside the machine instead. A vapour cavity forms at the throat entry and grows along the edge of the jet, and the mixed stream recovers from a lower pressure than the momentum balance assumed. When the discharge rises back above 294 kPa, the cavity collapses and the characteristic resumes. Collapsing cavities are what erode the throat wall a few diameters downstream of the entry — which is where worn jet pumps are worn, and where the collapse argument puts the damage in any other machine.

Reading σ as two margins #

The cavitation parameter is easier to reason with once it is read as what it is: two pressure differences, both measured from the suction plenum, one downwards and one upwards.

The numerator is how far the entrained liquid is from boiling before it has been accelerated at all. For the installation above, drawing from a vessel at 60 kPa absolute, that is 60 less 2.34, or 57.7 kPa. The denominator is how much pressure the motive stream brings to the throat above the suction — 800 less 60, or 740 kPa. Their ratio is 0.078.

The throat entry has to sit below the suction plenum by the entrained stream’s velocity head, and the motive jet sets how fast that stream is dragged in. So the ratio says how much of the motive stream’s driving pressure the entrained stream can be allowed to convert into speed before it boils. At 0.078 the answer is not much, and it shows: a nozzle 0.28 of its throat is most efficient at a flow ratio of 0.905, and this installation holds it to 0.675. On that wall its best available efficiency is 31.3 per cent rather than 33.4, reached at a head ratio of 0.46 rather than 0.37 — the machine has been pushed back up its own curve towards more head and less flow, by the suction alone.

The same reading explains why σ, and not the suction pressure on its own, is the number to quote. A jet pump drawing from 60 kPa with a 2-bar supply has σ = 0.41, a wall at a flow ratio of 1.36, and no cavitation problem anywhere near its best point. The identical machine with a 20-bar supply has σ = 0.030 and a wall at 0.43, less than half the flow ratio it is best at. The suction vessel did not change. What changed is how hard the machine was asked to pull on it.

The same wall, in a pipe with nothing mixed in it #

This behaviour has already been met once, in a device with a single stream. The venturi that stops listening downstream has a throat whose pressure falls as the downstream pressure is lowered, until it reaches vapour pressure; below that the flow rate is fixed, and the venturi has become a flow limiter.

The jet pump is the same mechanism with a second stream. What freezes is not a flow rate but a flow ratio, because the motive stream is also pinned — its speed set by its own margin to the same vapour pressure at the same section. And the same thing is true in both devices about what the frozen state means for the flow behind it: the throat has stopped transmitting information upstream, just as a choked gas nozzle stops transmitting it once its throat is sonic. In the gas the reason is that the throat velocity has reached the speed of sound. In the cavitating liquid it is that the throat pressure has reached a floor, and the vapour-laden mixture there has a sound speed so low that the throat is, in that sense too, supersonic.

The distinction worth keeping is where each wall sits on the characteristic. A choked gas ejector has its flat part at the high-flow end and its failure — the sudden collapse of entrainment — at a critical back pressure above it. A cavitating liquid jet pump has its flat part at the high-flow end too, but nothing collapses above it; it is simply a pump curve that has had its right-hand end cut off vertically.

When the wall arrives before the best point #

The loss analysis put the most efficient jet pump at a nozzle 0.275 of its throat, running at a flow ratio of 0.924. That flow ratio is a fixed number, and the wall is a moving one, and whether the machine can reach its own best point is a question about which is larger.

At the efficiency optimum the two meet at σ = 0.148. Above that the machine can run at its best point and the wall is irrelevant to it. Below that it cannot, and the question becomes what the best reachable efficiency is — over every area ratio, with the flow ratio held to the wall.

The shape of that curve divides the design problem into two regimes with different masters.

With a generous suction margin, the losses decide. The ceiling is flat at 33.4 per cent and the borrowed loss coefficients set its height. A better diffuser raises it; a better suction pressure does nothing.

With a short margin, the suction decides. The ceiling falls slowly at first — less than a point by σ = 0.1, under three by 0.07 — and then steeply, losing twelve points between σ = 0.05 and 0.01. The losses barely matter there, because the machine is not operating where they would. Here a better diffuser does nothing and a few kilopascals more at the suction are worth several points of efficiency.

The nozzle that reaches the constrained optimum is narrower, which is at first surprising: a narrower nozzle has its wall at a larger flow ratio, since $M_c$ goes as $(1 - R)/R$, and so it can run further along its own curve before cavitating. It trades some of its intrinsic efficiency for room, and at σ = 0.02 the best trade is a nozzle 0.188 of its throat, running on its wall at a flow ratio of 0.59.

The cure that makes it worse #

The cavitation parameter has the motive pressure in its denominator. Raising the motive pressure lowers σ, and so brings the wall closer.

That runs against intuition, because a stronger jet ought to pull harder. It does pull harder — the jet is faster and the suction stream it entrains at a given flow ratio is faster with it — but the suction stream’s speed is what lowers the throat-entry pressure, and the suction plenum’s pressure has not risen to pay for it. The flow ratio at the wall depends on the ratio of the two margins, and only the motive margin grew.

The place this matters most is the shallow-well jet pump: a jet pump mounted above the water, pulling it up a suction pipe. Its suction pressure is atmospheric less the lift, so every metre of lift lowers $P_s$ by about 9.8 kPa, and σ falls with it.

At 6 bar that machine can run at its best point only up to a lift of 2.3 metres. At 4 bar it manages 4.9 metres. At 10 bar it cannot run at its best point with any lift, because σ at atmospheric suction is already below 0.148. More motive pressure buys more head at the discharge, and it pays for it at the suction.

This is why the jet pumps built for deep wells do not sit at the surface. A deep-well jet pump puts the jet assembly down the borehole, below the water level, and drives it with a pipe of motive water from the surface. Its suction pressure is then atmospheric plus the submergence, σ is large, the wall is far away, and the machine can run at its efficiency optimum however deep the water table is. The surface pump and the down-hole pump are the same device placed on opposite sides of the one number that decides whether its best point exists.

A ledger of the wall #

The two checks that carry the argument are the first two rows. The closed form was derived by setting one pressure equal to another and solving; the march never used that algebra, only the full operating model and a comparison at each step, and the two agree to rounding. And the flow ratio on the wall does not move by one bit between discharge pressures of 280 and 70 kPa, which is the “stops listening” statement made exact rather than approximate.

What the one-dimensional machine cannot show #

Incipient cavitation comes first. The wall here is where the mean static pressure at the throat entry reaches vapour pressure. The shear layer at the edge of the jet is full of eddies whose cores are at a lower pressure than the mean, and bubbles appear there first — at a σ that measurements put somewhat above the mean-pressure value. The model has a factor for that and sets it to one, because the right value is a property of a particular nozzle’s shear layer and is borrowed when it is known.

Dissolved gas. Water that has stood open to the air carries dissolved gas, and a falling pressure brings it out of solution before vapour pressure is reached. That makes a gas cavity rather than a vapour one — it does not collapse violently, and it does not erode — but it takes up throat area and lowers the performance in much the same way, starting at a higher pressure.

The cavity’s length and its effect on mixing. Once the wall is reached, the model says only that the flow ratio is frozen. It does not say how long the cavity is, where it collapses, or how much the head falls as the discharge pressure is lowered below the knee. Those depend on the two-phase flow in the throat and on how the cavity disturbs the mixing, which is a problem with no one-dimensional answer.

The suction pipe. The lift figure treats the suction line as frictionless. A real suction line has its own loss, which subtracts from $P_s$ exactly as lift does, and a long or narrow one can take a metre or two off every number drawn.

Temperature. Vapour pressure rises steeply with temperature — 2.3 kPa at 20 °C, 12.3 at 50, 47 at 80 — and it enters σ directly. Hot water brings the wall much closer, which is why jet pumps handling condensate or hot process liquids are specified with generous suction heads.

Who found it #

The cavitation limit of the liquid jet pump was worked out in the form used here by R. G. Cunningham and his collaborators at Pennsylvania State University in the 1960s, who measured the vertical cut-off in the characteristic, defined the cavitation parameter against the motive-to-suction difference, and showed that the limiting flow ratio followed the square-root dependence above. It is recognisably the same analysis that had been applied to venturi throats and to pump inlets since the 1920s, and it is the jet-pump version of what pump engineers call the available net positive suction head.

What is not in that literature in so many words, and follows immediately once the closed form is on the page, is that the motive pressure sits in the denominator — that the natural way to push a struggling jet pump harder is the one that makes its cavitation worse.

Still open: the gas machine, where the wall is choking #

A liquid jet pump meets a floor in pressure. A gas ejector meets no such floor, and its limit arrives in a different variable: the entrained gas, accelerated in the narrowing annulus beside a supersonic jet, reaches the speed of sound. Once it has, the entrainment is fixed and independent of the back pressure — the same flat part this essay drew, arrived at by choking instead of boiling — and it stays fixed up to a critical back pressure, above which it collapses.

The next calculation carries the momentum balance into compressible flow: a supersonic primary nozzle, a secondary stream that chokes at a throat formed by the jet itself, a normal shock in the mixing tube, and a diffuser. It should say where the flat part ends, why the entrainment on it trades directly against the critical back pressure, and what that trade means for how many stages a steam ejector needs to reach a vacuum.