One fast pump or two slow ones

The specific speed a pump spends its life at follows one pump along the line its control system drives it on. Its central finding is that a pump’s specific speed is a coordinate along its own characteristic, that the affinity laws keep it fixed only along parabolas through the origin, and that a system with static lift — a head that has to be supplied however little flows — crosses those parabolas, so a pump slowed to deliver less flow into it slides towards shut-off and away from its best efficiency. A throttle slides it further. Over a working life at part load the difference between the two is large, and at high lift it is much smaller than the cube law promises.

It closes on the arrangement a pumping station actually has. Water supply, drainage, district heating and cooling circuits are almost always served by several identical pumps side by side, discharging into a common header, and the number running is itself a control. This essay asks what that control is worth: what a second pump adds, how many pumps should run at a given flow, when to start the next one, and whether a station of several smaller pumps beats one large pump over a duty.

What a second pump adds #

Pumps in parallel share a head and add their flows. Two identical pumps at full speed therefore behave as one pump with the same shut-off head and twice the flow at every head, and a station designed to meet its duty with both running puts the design point where that doubled curve crosses the system curve.

The first figure is the picture every pump handbook draws, with the numbers put in. The pump characteristic is the one the earlier essay stated: a head coefficient falling as a parabola from a shut-off value four thirds of the design head, and an efficiency parabola peaking at 0.82. Each pump is sized for half the design flow at the design head. Switch one off and the survivor does not deliver half the flow. The system asks for less head at lower flow, so the running pump moves out along its curve until the two meet: with no static lift at 76 per cent of the design flow, with 30 per cent lift at 71, and with 60 per cent at 65.

Read the other way, the second pump adds a third, two-fifths and a half of the first pump’s flow, not as much again. The more of the system’s head is friction, the less a second pump adds, because the friction it has to overcome rises as the square of the flow it adds. Static lift, which does not rise, makes the second pump more valuable. A station built for a friction-dominated system that expects to double its output by starting a second pump will be disappointed by a third, for the reason a loss with no viscosity in it gives every fitting in the line: its loss grows as the square of what flows through it.

The pump left running alone is the more serious consequence. It sits at 1.51 times its best flow coefficient with no lift and 1.30 with 60 per cent, well out towards run-out, at an efficiency of 0.61 and 0.75, and it draws 17 and 10 per cent more than its own design power. A motor sized for the pump’s best point overloads, and the inlet of a pump near run-out needs more suction head than at its best point, which is where the group with no head in it says cavitation starts. Pumps in parallel are sized for the case of one running alone as often as for the case of all running together, and that is why.

Every pump at its best point #

Now put the pumps under speed control, all running ones at a shared speed. The earlier essay’s characteristic makes the question close in one line. With $n$ of $N$ pumps running, each carries $x = qN/n$ of its own design flow when the station delivers a fraction $q$ of its design flow, and the system asks for $h = s + (1 - s)q^2$ of the design head, with $s$ the static share. Matching the pump’s head to the system’s at a speed fraction $\nu$ gives

$$\nu^2 = (1 - c)\,h + c\,x^2,$$

where $c = \tfrac14$ is the characteristic’s curvature — the droop in head that the impeller’s change of swirl suffers as the flow through it grows — and the pump’s flow coefficient relative to its best is $r = x/\nu$. The pump is exactly at its best point when $r = 1$, which happens when $x^2 = h$ — and then its speed is $\nu = \sqrt{h}$, whatever the shape of the characteristic.

So the number of pumps that would keep every running pump exactly at its best point is

$$n^* = \frac{qN}{\sqrt{s + (1 - s)q^2}}.$$

With no static lift it is $N$ at every flow: all the pumps should always run, slowed together, and each stays at its best point at every flow, the station behaving as one large pump that obeys the affinity laws perfectly. That is the case in which the cube law holds, and in it staging can only do harm; switching a pump off at any flow moves the survivors to a flow coefficient 1.51 times their best, and the arithmetic confirms that the power is then higher at every flow.

With lift the second figure shows $n^*$ falling with the flow, and falling faster the more lift there is. At 60 per cent lift and 40 per cent of the design flow it is 0.98: one pump, slowed to 81 per cent speed, sits almost exactly at its best point, where the earlier essay’s single large pump, slowed to deliver the same flow into the same system, was dragged to 55 per cent of its best flow coefficient and an efficiency of 0.65. That is the whole case for staging in one comparison: the same flow, the same system, and 0.82 against 0.65, because the small pump is running at the specific speed it was designed for and the large one is not.

The number is almost never whole, and that is where the control problem is. At 60 per cent lift and 60 per cent flow it is 1.39; at 80 per cent flow 1.73. A station cannot run 1.39 pumps. It has to choose between one pump pushed out towards run-out and two pumps held back towards shut-off, and the answer depends on which is further from its best point in efficiency.

When to start the second pump #

The third figure draws the choice for 60 per cent lift. One pump under speed control peaks at 0.82 near 40 per cent of the design flow and falls as it is pushed further; two pumps under speed control rise towards 0.82 at the design flow. The curves cross at 60.1 per cent of the design flow, and that is where the second pump should be started — at equal shaft power, not at the flow where the running pump reaches full speed, which for this system is 65.0 per cent. A station that starts its second pump only when the first is flat out spends the band between 60 and 65 per cent running one pump at up to 1.3 times its best flow coefficient when two would be closer to theirs; at 62 per cent flow the difference is 0.759 against 0.774.

The throttled curve is the comparison the earlier essay drew for one pump, and it is no better here. Holding the pumps at full speed and closing a valve to meet the system, choosing the best number to run, gives 0.47 at the jump to two pumps and never catches either speed-controlled curve. Staging recovers part of what a throttle wastes — the station-wide duty efficiency with two throttled pumps staged is 0.630 at 60 per cent lift, against 0.474 for one large throttled pump — but speed control recovers far more.

The fourth figure follows the switch across every system. With no lift there is no switch at all: two pumps are cheaper at every flow, and the curves run down to zero. As the lift rises the switch moves to higher flows — 0.470 of the design flow at 30 per cent lift, 0.601 at 60 — and it meets the dashed full-speed line at about 70 per cent lift. Above that the switch is forced: the running pump reaches full speed before a second would pay, and the usual rule of adding a pump when the running ones are flat out becomes, for once, the right one. For three pumps the picture repeats with two switches, from one to two at 0.28 and 0.38 of the design flow at 30 and 60 per cent lift, and from two to three at 0.61 and 0.73.

That rule’s own conditions are worth stating. It is right when most of the system’s head is static, as in a borehole, a high-rise building or a boiler feed, and it is wrong, by a band of flow that grows as the lift falls, in a circulating loop where most of the head is friction. The earlier essay found the same division in the cube law, which is exact with no lift and optimistic with a lot, and the two findings are one: static lift is what stops the affinity laws from holding a slowed pump at its best point, and a smaller pump is what puts it back.

Over a duty, and what a smaller pump costs #

A pumping station’s energy is spent over a year of flows, not at one of them. The earlier essay used a duty of ten per cent of the time at full flow, a quarter at 80 per cent, 35 per cent at 60 and 30 per cent at 40, and it found that one large pump under speed control delivers 0.769 of its shaft energy to the water at 60 per cent lift. Split into two half-size pumps, staged at their best switch, the station delivers 0.800; into three, 0.810.

The fifth figure follows the comparison across the lift. With no lift the lines coincide at 0.82, as they must: every arrangement runs all its pumps at their best point. With lift the single pump falls away and the staged stations hold up, three pumps better than two, so the more lift a system has, the more a station gains from being split. The steps near 80 per cent lift are the points of the duty at which one pump can no longer carry the flow and a second has to be started early.

The dashed line is the honest correction. Two pumps each sized for half the flow at the same head and speed are not scaled copies of the large one; they have half its specific speed, narrower passages and relatively larger clearances, and pumps of low specific speed and small size are measurably less efficient — the dependence that the first essay on specific speed set aside. How much less depends on the sizes, and the calculation here does not model it; it asks only what a penalty of two points in best efficiency does. At 60 per cent lift the staged pair still wins, 0.781 against 0.769. At 30 per cent it loses, 0.791 against 0.802. The two cross at 44 per cent lift. Below that, a station would do better with one large pump and a drive than with two smaller pumps staged perfectly, and the case for splitting rests on standby and redundancy rather than on energy.

Many small pumps, and the limit they approach #

If two pumps beat one and three beat two, the obvious question is where it stops. With $N$ pumps staged, the number running can be rounded to the nearest $1/N$ of the station, and as $N$ grows the rounding error in $n^*$ shrinks: every running pump is held closer to its best point at every flow. Over the same duty at 60 per cent lift the energy-weighted efficiency is 0.769 for one pump, 0.800 for two, 0.810 for three, 0.816 for four and 0.820 for six, which is the best efficiency the characteristic has. At 90 per cent lift, where one pump manages only 0.733, it takes ten pumps to reach 0.820. The limit is a station that runs every pump at its best point all the time, and the only thing between a real station and that limit is how finely it can divide its flow.

That limit is worth naming because it is where the gain comes from, and it is not from the pumps being small. It is from the number running being a free variable. A station of many pumps turns a problem with one control — the speed — into a problem with two, and a second control is exactly what a system with static lift requires, since that lift is what the first control cannot follow on its own. The same result could in principle be had with one pump and a second continuous control, such as variable inlet guide vanes or an adjustable impeller, which move the characteristic itself rather than sliding along it. In water pumping the second control is usually the number running, because pumps are cheap to duplicate and duplication also buys the standby capacity that a supply cannot do without.

In practice the gain flattens long before the limit. Beyond three or four pumps the improvement is a fraction of a point of efficiency, against the cost of the extra units, their valves and their controls, and against the efficiency every halving of pump size gives up, which the duty comparison above priced at two points. The optimum number of pumps is set by that trade and not by the hydraulics alone.

The closed form, checked #

The sixth figure lists the numbers and what each was tested against. The shared speed, which the whole argument uses in its closed form, was found a second way for every combination of lift, flow and number running: by bisecting on the speed at which the pump’s own stated characteristic, evaluated at the flow coefficient that speed implies, makes the head the system asks for. The two agree to $2\times10^{-16}$, which is rounding. The no-lift case was checked to put every pump at exactly its best point and to cost more power with a pump switched off at every flow tested. And one pump alone at full speed was checked to meet the system head exactly at the flows quoted, so that the second pump’s contribution is read off the same curves the first figure draws.

The same arithmetic in a compressor #

The problem of several machines sharing one duty, each able to run only near its own best point, is not peculiar to pumps. Matched at one speed and at no other finds its axial version in a multistage compressor, whose stages are in series rather than in parallel: each stage’s flow coefficient is set by the density the stages before it delivered, the stages agree only at the design speed, and at part speed the front stages are pushed towards stall while the rear ones choke. In parallel the pumps agree with each other by construction — they share a head and a speed — and what drifts is their common operating point against the system. In series the machines drift against each other. Both are cases of a set of machines held at one operating point by a constraint that holds only at design, and both are handled in practice by the same kind of intervention: variable geometry or bleed in the compressor, a variable number of running units in the station.

What the picture cannot show #

A stated characteristic. The head and efficiency parabolas are chosen shapes, not the curves of a particular pump, and the switch flows depend on them. The closed form for the shared speed, and the rule that a pump is at its best when $qN/n = \sqrt{h}$, do not: the speed at the best point is $\sqrt{h}$ for any characteristic that obeys the affinity laws.

No motor, drive or header losses. A variable-speed drive loses a few per cent at part load, more at low speed; the motor’s own efficiency falls at light load; and the header and non-return valves add a friction loss that grows with the number of pumps running. All three shift the switch, usually towards running fewer pumps.

Identical pumps at one shared speed. Stations sometimes run one pump at full speed and trim with a second on a drive, or mix sizes; the best mixed arrangement is a different optimisation and is not computed here.

Quasi-steady switching. Starting and stopping a pump is a transient, and in a long rising main a pump that trips is a water-hammer event; the switch points here assume the station sits at each flow long enough for the transient not to matter, and hysteresis between starting and stopping is left out.

The convention the numbers depend on #

Flows are fractions of the station’s design flow, heads fractions of the design head, and efficiencies are hydraulic power delivered to the system — flow times the system’s head, not the pump’s — over shaft power, so a throttle’s valve loss counts against it. Static lift $s$ is the share of the design head that the system needs at zero flow; the rest of the system head rises as the square of the flow, as friction in a fully rough pipe does. The switch is placed where the two arrangements draw equal shaft power at the same flow.

Who worked it out #

The rule that pumps in parallel add flows at a common head is as old as centrifugal pumps, and the diagram of the first figure has been in pump handbooks since the early twentieth century. Staging pumps against demand was practised long before variable- speed drives existed, as the only way to vary output without throttling. The combination of staging with speed control became routine with cheap inverter drives from the 1980s onwards, and the rule of switching at equal power rather than at full speed is the result of the optimisation studies that followed, carried out for water utilities and building services more than in the academic literature.

Still open: where the efficiency penalty really comes from #

The duty comparison turned on a two-point penalty that was stated rather than derived, and the conclusion flipped at 44 per cent lift because of it. The penalty has a physical source: halving the flow at a fixed head and speed halves the specific speed, and the leakage past the impeller’s wear rings, the friction on its discs and the losses in its narrower passages all grow as a share of the power as the specific speed falls. The next calculation builds that loss budget — leakage, disc friction and passage friction, each scaled from the similarity that fixes a pump’s shape — and uses it to put a number on the efficiency a pump of half the specific speed gives up, and so to say for which systems splitting a duty between smaller pumps saves energy and for which it only buys redundancy.