The specific speed a pump spends its life at

One number picks the machine evaluated $\omega\sqrt{Q}/(gH)^{3/4}$ at a duty and read the kind of machine off it. The duty that had no machine split a duty until the number fell into a band, and the group with no head in it showed that a second group, with the inlet margin in place of the head, can forbid the result. All three evaluated the number at one point, and the last of them ended by naming what that leaves out: a pump is chosen at its design duty and spends most of its life at another one.

A pump in a building’s cooling loop, a water main, an irrigation scheme or a process plant is asked for less than its design flow most of the time, and the flow is reduced in one of two ways — a valve that throttles it, or a drive that slows it. This essay follows the specific speed along both, and finds that the number does not stay where the pump was chosen, that where it goes is decided by the system rather than by the pump, and that the usual rule for what slowing a pump saves is a special case.

The pump is a single-stage machine drawn for 0.1 cubic metres a second against 40 metres at 1450 revolutions a minute: an impeller 0.1845 metres across, a specific speed of 0.545 at its best point, and 47.7 kilowatts of shaft power there at an efficiency of 0.82.

The number, written in the pump’s own coefficients #

A pump of fixed geometry has two dimensionless coefficients that describe any operating point: a flow coefficient $\varphi = Q/\omega D^3$ and a head coefficient $\psi = gH/\omega^2 D^2$. The dimensional analysis the whole argument rests on says that for a given geometry $\psi$ is a function of $\varphi$ alone, whatever the speed and whatever the size — that is the pump’s characteristic, and every operating point it can ever have lies on that one curve.

Substituting the two coefficients into the specific speed gives

$$n_s = \frac{\omega\sqrt{Q}}{(gH)^{3/4}} = \frac{\sqrt{\varphi}}{\psi^{3/4}}$$

exactly: the speed and the diameter cancel, which is the familiar statement that “the number contains no size”. Read this way round it says something that statement does not. The specific speed of a pump’s operating point is a coordinate along its own characteristic. The number quoted for a pump is the coordinate of its best point, and the pump has a different specific speed at every other flow it runs at.

The characteristic here is a stated parabola: a head coefficient that falls from a shut-off value of two thirds to 0.5 at the best point, so the shut-off head is four thirds of the design head, and an efficiency that is a parabola in the flow coefficient, peaking at 0.82 and vanishing at zero flow and at twice the design flow. Those shapes are chosen, not solved from an impeller, and every number in this essay that depends on them is labelled so. The identity does not depend on them at all.

What the identity makes plain is that the specific speed at an operating point moves only if the pump moves along its curve, and a pump moves along its curve only if its flow coefficient changes. Everything below is about what changes it.

The parabolas the number does not move along #

The affinity laws are the statement that a change of speed alone leaves both coefficients fixed: the flow scales as the speed, the head as its square. So on the head–flow plane a pump slowed or speeded up at a fixed flow coefficient moves along a parabola $H \propto Q^2$ through the origin, and those parabolas are the only lines along which the specific speed of an operating point is constant.

The system the pump delivers into has its own curve: the head it needs to pass a given flow. That head has two parts. One is friction in the pipework, which rises as the square of the flow in turbulent pipe flow. The other is a static lift — a height the water must be raised, or a pressure it must be delivered at — which does not depend on the flow at all. The system curve is therefore $H = H_{\text{static}} + KQ^2$, and here it is drawn through the design duty with the static lift a share $s$ of the design head.

With no static lift the system curve is itself a parabola through the origin, and in fact it is the parabola the best point sits on. Slowing the pump slides its operating point down that very parabola, the flow coefficient never changes, and the specific speed and the efficiency stay at their best-point values at every flow. That is the case the first figure’s flat line shows.

With a static lift the system curve starts at the lift and rises more gently, so it crosses the parabolas. A pump slowed to deliver less flow into it must still produce the lift, and producing a head at a lower speed needs a lower flow coefficient — a point further up its characteristic, towards shut-off. At 85 per cent speed the pump sits at 0.828 of its best flow coefficient; at 75 per cent, at 0.603. Its specific speed has moved with it.

A floor under the speed #

A static lift does something more drastic than moving the operating point: it sets a speed below which the pump delivers nothing.

A pump’s highest head at any speed is its shut-off head, $\psi_0\omega^2D^2/g$, and it can lift nothing into a system whose static lift is larger. Setting the two equal gives the minimum speed as a share of design,

$$\frac{\omega_{\min}}{\omega_d} = \sqrt{\frac{s\,\psi^*}{\psi_0}},$$

which for this characteristic is $\sqrt{0.75\,s}$: 0.474 at 30 per cent lift, 0.671 at 60, 0.822 at 90.

Just above the floor the pump is running near shut-off, where its characteristic is flat, so a small increase in speed raises the head only slightly above the lift but lets a large increase in flow through. A drive controlling a pump against a high static lift is therefore working in a narrow band of speed — from 82 to 100 per cent at 90 per cent lift — and the control is correspondingly sensitive. Near the floor the pump is also at a very low flow coefficient, which is where the flow at its inlet stops following the blades and recirculates, a condition real pumps are protected from by a minimum-flow bypass that a stated parabola knows nothing about.

The efficiency is the same distance #

Because the efficiency was stated as a function of the flow coefficient, its fall along each line is exactly the distance the operating point has moved along the characteristic — the same distance the specific speed records.

At half the design flow a pump against 90 per cent static lift is running at 0.672 efficiency where the same pump against no lift runs at 0.82. Nothing about the pump differs; it is being asked to sit at a different place on the same curve.

The throttle is worse at every flow, and the reason is geometric rather than hydraulic. A throttle holds the speed and slides the operating point straight along the design-speed characteristic, so at half the flow the flow coefficient is exactly half its best value. Speed control against a system with lift also moves the flow coefficient down, but by less, because slowing the pump lowers the head it makes and so moves it down the system curve as well as along its own.

The throttle’s other cost is outside the pump. The valve dissipates the difference between what the pump makes and what the system needs, and that dissipation is a loss with no viscosity in it — the momentum of the flow through a restriction, thrown away. At half flow into the 60 per cent system the pump makes 50.0 metres of head at design speed where the system asks for 28.0, and the valve takes the other 22.0.

The cube law has a hypothesis #

The saving a variable-speed drive offers is usually quoted from the affinity laws as the cube of the flow: half the flow, an eighth of the power. The figure shows where that comes from and what it assumes.

Power is flow times head over efficiency. Under the affinity laws with the flow coefficient fixed, head falls as the square of flow and efficiency does not change, so power falls as the cube: that is exact, and the model reproduces it to nine decimal places. But fixing the flow coefficient is the no-static-lift case. With a lift the head cannot fall with the square of the flow, because a floor of it is needed however little flows, and the efficiency falls because the pump has left its best point.

The consequence is large. Slowing the pump to half its flow saves 87.5 per cent of the design power with no lift, 75 per cent with 30 per cent lift, 60 per cent with 60, and 44 per cent with 90. A heating loop with no lift and a boiler-feed or high-rise supply with a great deal of it are not the same case, and a drive justified by the cube law in the second has been justified by the arithmetic of the first.

The throttle’s figure is the other end of the same argument. At half flow into the 60 per cent system the throttled pump still draws 83 per cent of its design power, because it is running at full speed near shut-off and the valve is converting most of its work into heat.

Over a working life #

A pump does not spend its life at one flow, and what matters for its energy bill is an average over the flows it actually runs at.

Two things stand out, and neither is visible in a calculation at the design point.

Speed control is better at every lift, and by much less than the cube law says when the lift is high. With no lift it keeps the pump at its best efficiency throughout the duty while the throttle delivers under a third of the shaft energy to the system; at 90 per cent lift the gap has shrunk from 0.52 to 0.17. The throttle’s figure rises with the lift because the system itself asks for most of the pump’s head, so less of it is wasted in the valve.

And the pump was chosen at the wrong point. A pump chosen with its best point at full flow, and run for three quarters of its life below 80 per cent of it, spends most of its hours away from the specific speed and efficiency on its data sheet. Against a system with lift, a pump whose best point sits nearer the flow it most often delivers — or two smaller pumps, one of which is switched off at low demand — keeps the operating point nearer the best-point parabola, which is the only line along which the number stays still.

The inlet group moves the other way #

The essay below this one added a second group to the first, with the inlet margin in place of the delivered head: $S = \omega\sqrt{Q}/(g\,\mathrm{NPSH})^{3/4}$, which must stay under a practical limit if the liquid is not to tear at the impeller’s entrance. Evaluated with the margin the plant supplies, which does not depend on the flow, it contains only $\omega\sqrt{Q}$ of the operating point, and both ways of reducing the flow reduce that.

Under speed control with no static lift the speed and the flow both halve at half flow, so the group falls to 0.354 of its design value — the three-halves power of the flow fraction. Against 60 per cent lift the pump is still turning at 0.766 of design speed at half flow and the group falls only to 0.542; against 90 per cent lift, to 0.615. Under a throttle the speed does not fall at all, and the group falls as the square root of the flow, to 0.707.

So part flow never makes the inlet group worse, which is the reassuring half of the statement and the one a data sheet implies. The other half is that the group was built for flow arriving onto the blades at their design angle. At the flow coefficients a throttled pump or a slowed pump against a high lift reaches — 0.50 and 0.57 of the best value at half flow — the flow at the impeller’s eye no longer follows the blades, a ring of it turns back into the suction pipe, and the cavitation that follows is set by that recirculation rather than by the margin in the group. The group is right about the thing it measures and silent about a failure that moves the other way along the same line.

The same drift as a compressor’s stages #

A fixed geometry driven along its own curve by something outside it has been met before. Matched at one speed and at no other found each stage of a multistage compressor leaving its design flow coefficient as the shaft slowed, because the density the air reached no longer matched the areas the stages were cut for.

The pump’s drift has the same structure with a different cause. Each machine has one best flow coefficient, fixed by its blades. Each is driven away from it by something the blades cannot see: in the compressor, the density ratio the upstream stages delivered; in the pump, the static lift of the system downstream. And each has a special case in which the drift vanishes — a compressor whose density does not rise, a system whose head falls with the square of the flow — which in both essays turns out to be the incompressible, similarity-preserving case the affinity laws were built for.

The general statement is that similarity is a property of the machine and the conditions together. The affinity laws say how a pump’s operating point moves when only its speed changes; they say nothing about whether the thing it is connected to lets only its speed change.

What the stated characteristic leaves out #

The characteristic is a parabola and the efficiency a parabola. A real pump’s head curve is steeper or flatter depending on its design, often has a slight droop near shut-off, and its efficiency curve is not symmetric about the best point. The identity and the geometry of the level lines are exact; how far a given control moves the efficiency depends on the real curves.

The drive and the motor have efficiencies too. A variable-frequency drive and an induction motor both lose efficiency at low load, so the shaft power computed here is not the electrical power drawn, and the saving at low flow is smaller again.

Low flow is not only inefficient. Well below its best flow a centrifugal pump develops recirculation at its inlet and outlet, with vibration and, at the inlet, cavitation that the net positive suction head of the essay below does not predict; above its best flow the head required at the inlet rises. Both limit how far along its curve a pump may be run, and neither is in a stated characteristic.

The system curve is fixed. Real systems change: tank levels move the static lift through a day, valves elsewhere change the friction coefficient. The operating line is then a band rather than a curve.

Who worked it out #

The similarity rules for pumps — flow with speed, head with its square, power with its cube — were in use in the nineteenth century and are older than the dimensional analysis that later explained them. The observation that a pump should be matched to its system curve rather than to a duty point is as old as the centrifugal pump itself.

What changed the practical importance of this essay’s argument was the variable-frequency drive, which from the 1980s made slowing a pump cheap enough to replace a throttle almost anywhere. The cube law went with it into sales literature; the caveat about static lift went more slowly into design guidance, and the industry guides on variable-speed pumping of the early 2000s spend a good deal of their length on exactly the difference between the flat line and the falling lines in the first figure.

Still open: how many pumps should be running #

Every figure here had one pump. A great many installations have several in parallel, and the number running is itself a control variable: at low demand one pump runs near its best point where two would both run far from theirs. Switching pumps on and off moves the operating point between curves in a way neither a throttle nor a drive can, and it interacts with speed control — two slowed pumps against a high lift can sit further from their best point than one pump at full speed.

Finding the number of pumps and the speed that together keep an installation nearest its best-point parabola, over a duty rather than at a point, is a question with a definite answer for a stated system, and it is the calculation this one leaves open. Beside it is the efficiency correlation the first essay on specific speed set aside: where the best efficiency itself sits as a function of the best-point specific speed, which decides how much each pump in such an arrangement is worth running at all.