Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
Series

Specific speed — the series

7 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 1.139 asks for a Francis. The specific-speed axis, with the four machine types on it and one duty marked: 3 m³/s at 60 m, on a shaft turning at 750 rev/min. The number is 1.1387, and the choice of runner follows from it before any blade has been drawn. What the number contains is a ratio of flow to head; what it does not contain is any size at all, which is why one axis serves a garden pump and a gigawatt turbine.

    One number picks the machine

    A flow rate, a head and a shaft speed contain exactly one dimensionless combination with no size in it. That combination decides whether a duty wants an impulse wheel, a Francis runner or a propeller — before anything has been drawn, sized, or costed.

    part 1 · applied
  2. Two ways to move a duty, along one axis, in opposite directions. A duty at a specific speed of 0.01 and what splitting it does. Dividing the head between stages in series multiplies each stage's specific speed by the number of stages to the three-quarter power, because the group carries (gH)^(−3/4); dividing the flow between units in parallel divides it by the square root of the number of units, because the group carries √Q. Both exponents are read off the group rather than remembered, and they are why the two operations are not interchangeable: three stages buy a factor of 2.28 and three units cost a factor of 0.58. The bands are drawn in the colour this site reserves for a borrowed claim, because where a Francis runner stops is practice rather than a result.

    The duty that had no machine

    Some duties have a specific speed outside every band, and no runner will do them at any size because the number contains no size. That is true and it is not the end. The same group says how to split the duty until it fits, and the two ways of splitting move it along the same axis in opposite directions, by exponents read straight off it.

    part 2 · applied
  3. The shaft speed is a window, and cavitation closes the top of it. Two groups against shaft speed for the same duty: the specific speed, which must be inside a band for a runner to exist, and the suction specific speed, which must be below about 3 for the impeller not to cavitate. Both rise with the shaft speed, so raising it to reach a band is also raising it towards the cavitation limit. The window here runs from 274.98 rpm to 962.31 rpm and cavitation sets its top. A duty whose window is empty needs something other than a different machine — a booster, a lower installation, or an inducer.

    The group with no head in it

    The number that picks a machine says nothing about whether the machine can exist. A second group formed from the same variables, with the delivered head replaced by the margin available at the inlet, decides that — and the delivered head has left the expression entirely, so how far a pump lifts is irrelevant to whether it tears the liquid apart at its own entrance.

    part 3 · applied
  4. A pump slowed into a system with a static lift leaves its own specific speed. The specific speed of the operating point, as a share of its value at the best point, against the fraction of the design flow delivered, for a pump drawn for 0.1 m³/s against 40 m at 1450 rpm, specific speed 0.545 at its best point. Under speed control into a system whose static lift is 0 per cent, 30 per cent, 60 per cent, 90 per cent of the design head (lines), and under a throttle at design speed into the 60 per cent system (dashed). With no static lift the speed-controlled pump stays exactly at its best point and its specific speed never moves. At half the design flow it has fallen to 1.000, 0.800, 0.708, 0.652 of the design value as the static share rises, and to 0.598 under the throttle.

    The specific speed a pump spends its life at

    A pump is chosen by its specific speed at its best point and then run somewhere else. Written in the pump's own coefficients the number is √φ/ψ^¾, a position along its characteristic, and a variable-speed drive keeps it there only when the system it pumps into has no static lift. Every metre of lift moves a slowed pump along its own curve, towards shut-off, and a throttle moves it further.

    part 4 · applied
  5. One fast pump or two slow ones. Pump efficiency against the flow asked for, for two pumps into a system with 60 per cent static lift: one pump under speed control, up to the flow at which it reaches full speed; both under speed control; and the best number running with the pumps held at full speed and a valve taking up the difference. Below 0.601 of the design flow one pump is better than two, and above it two are better — before the single pump has run out of speed at 0.65. The throttle is far worse at every flow.

    One fast pump or two slow ones

    A pumping station with several identical pumps in parallel has a control that a single pump does not: how many of them are running. With no static lift the answer is all of them, always, slowed together. With lift it is a number that falls with the flow and is almost never whole — the flow times the number of pumps, over the square root of the head the system asks. Staging rounds it, and the right moment to round up comes before the running pumps have run out of speed.

    part 5 · applied
  6. Below a specific speed of a half, a pump's losses are its discs and its narrow passages. Each loss as a share of the shaft power against the specific speed, for pumps of 40 m head at 1450 rpm, so that every one has the same impeller diameter and only the flow changes. At Ωₛ = 1.0 the fixed share is most of the loss and the efficiency is 0.902; at 0.3, disc friction is 7.65% and passage friction 5.32%, and the efficiency 0.777; at 0.15 the two together take 37.9%.

    Two small pumps carry two big discs

    Split a pumping duty between two pumps of the same head and speed and each one keeps the big pump's impeller diameter, because the head fixes the tip speed. So the pair spins two full-size discs in their casings for the flow one disc used to carry, and the efficiency penalty that a station comparison assumed to be two points is four. That moves the lift at which splitting saves energy from 44 per cent to 65, and below a specific speed of about a half it never does.

    part 6 · applied
  7. At one specific speed a bigger pump is a better one, and by less as it grows. Best efficiency against impeller radius for geometrically similar pumps of 40 m head at three specific speeds; the shaft speed falls as the radius grows, so the tip speed is the same at every size. At Ωₛ 0.55 a 25 mm impeller reaches 0.797, a 0.2 m one 0.869 and a 1.6 m one 0.887: 7.17 points for the first factor of eight, 1.84 for the second.

    Moody's fifth root is true at one size

    A model pump is less efficient than the full-size machine, and since 1925 the gap has been bridged with a fifth root of the size ratio. Built from its losses, a pump's inefficiency scales three ways at once — leakage as the inverse size where a clearance cannot shrink, friction as a fifth root until the passages go rough, and a fixed share not at all — so the exponent runs from a third down to a twentieth, and the fifth root is right only near one size. The same arithmetic says a duty should be split by speed, not by width.

    part 7 · applied

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