One number picks the machine

A hydroelectric scheme has a flow rate and a head. The generator has a shaft speed, fixed by the grid frequency and the number of poles. Those three numbers decide, before any drawing exists, which kind of turbine the scheme can use — and the decision is not a matter of taste or of what the supplier happens to make.

It comes out of dimensional analysis, from a group with no size in it at all.

Six quantities, rank three #

The variables a rotating machine’s performance can depend on are the shaft speed ω, the runner diameter D, the volume flow Q, the specific energy gH, the density ρ and the power P.

Buckingham’s theorem, worked as a rank computation, says how many independent dimensionless groups those six admit: six variables minus the rank of their dimension matrix. The rank is three — mass, length and time each appear, and no dimension is redundant — so there are three groups, and this site computes them rather than quoting them.

The three come out as the familiar coefficients:

$$\phi = \frac{Q}{\omega D^3},\qquad \psi = \frac{gH}{\omega^2 D^2},\qquad \Pi_P = \frac{P}{\rho\omega^3 D^5}$$

and each is checked here to lie in the computed null space rather than assumed to — a famous group is not the same thing as a dimensionless one, and the check costs four lines.

One modelling decision is buried in the variable list and it is worth exposing. The head appears as gH rather than as H, which is not a convenience. A head measured in metres is a length, and treating it as one would let a group cancel it against the diameter — putting gravity into a pump’s specific speed and then losing it again. What a machine actually cares about is the energy it adds per unit mass, which is gH, and which has the dimensions of a squared velocity.

The group with no diameter in it #

Three groups is more than a designer wants. What is wanted is a single number that characterises the duty without knowing the machine, and it exists because one particular combination of the three has a diameter exponent of exactly zero:

$$N_s = \frac{\phi^{1/2}}{\psi^{3/4}} = \frac{\omega\sqrt{Q}}{(gH)^{3/4}}$$

Eliminating D between the flow and head coefficients is the whole derivation. The site does it by requiring the exponent of the diameter to vanish and asserting the result — the returned vector’s diameter entry is 0 exactly, and the group is verified to lie in the null space.

A number with no size in it is a statement about shape. That is what makes it a machine-selection tool rather than a performance figure: two machines with the same specific speed are geometrically the same machine at different scales, whatever their diameters, whatever their speeds, whatever the flow.

The invariance that gives it meaning #

A claim of that kind is easy to assert and worth testing, so the site tests it directly.

Take a reference machine. Apply the affinity laws — Q ∝ ωD³, H ∝ ω²D², P ∝ ω³D⁵, which are what holding the three coefficients fixed means — to scale it to some quite different size and speed. Recompute the specific speed of the new duty from scratch.

That invariance is the licence to use the number for selection, and it is the same argument dynamic similarity makes for a model in a tunnel: a dimensionless group is worth having exactly when it is unchanged by the transformations one is allowed to make.

What the bands are, and what they are not #

Reading a runner type off the axis needs boundaries, and the boundaries are practice rather than physics. Nothing in the conservation laws says a Francis runner stops working at N_s = 2.2.

What is physics is why the bands are ordered as they are, and the reason can be read off the group. A high specific speed means a large flow at a small head: the machine must pass a great deal of water and add little energy to each kilogram, so it wants a large flow area and small turning — an axial propeller. A low specific speed means a small flow at a large head: the machine must add enormous energy to a trickle, which means a large blade speed and a large turning, and the extreme of that is a bucket struck by a jet at full head. A Pelton wheel is what a runner becomes when the specific speed is small enough that the flow no longer fills an annulus at all.

Everything between is a Francis runner, and the family shape changes continuously across the band: tall narrow runners at the low end, short wide ones at the high end, with the flow turning from radial to axial as the number rises.

A duty with no machine #

The most useful thing the number does is refuse.

A barrage duty — a hundred cubic metres a second at five metres of head — comes out at a specific speed of 42 on a 750 rev/min shaft, and there is nothing there. The bands stop at 6.

The escape is arithmetic rather than ingenuity. Specific speed goes as √Q, so splitting the flow between n identical machines divides it by √n: sixteen units at the same head bring 42 down to 10.6, and slowing the shaft brings it down further. Real low-head schemes have many units for exactly this reason, and the alternative — one enormous slow machine — is what a bulb turbine is.

That is the shape of a great many engineering decisions and it is worth naming: the constraint is a dimensionless number, the fix is to change which physical quantities are in it, and the arithmetic says how many machines are needed before any of them is designed.

The linearity is worth reading off. Doubling the shaft speed doubles the specific speed; quadrupling the flow doubles it; and raising the head by a factor of sixteen halves it. Those three exponents — one, one half, minus three quarters — are the entire content of the group, and they came out of a matrix of integers rather than out of any experiment.

The same duty on a slower shaft #

Shaft speed is the one variable in the group a designer usually cannot choose freely, because a synchronous generator’s speed is the grid frequency divided by a whole number of pole pairs. That makes the selection a search over a short list rather than over a continuum.

Reading that backwards is how the decision is actually made. The duty is fixed by the site; the candidate speeds are fixed by the generator; the specific speed at each is a single division; and the runner type falls out. A scheme that wants a Francis runner and has only slow generators available is a scheme that needs a gearbox or a different runner, and the arithmetic that says so takes a minute.

There is an echo here of the affinity laws’ other use: the same relations that let a model predict a full-scale machine are the ones that let a duty predict a runner, because both are statements that a family of machines is one machine at different scales.

What the number does not know #

It says nothing about efficiency. Two machines at the same specific speed are the same shape; one may be beautifully made and the other dreadful. Selection charts often show a peak efficiency against specific speed, and that is an empirical envelope of what has been achieved rather than a statement about what the number permits.

It says nothing about cavitation. A runner that is correct on this axis may still sit low enough in its own pressure field to tear the water it is pumping, and the quantity that decides that is a different dimensionless group — the suction specific speed, built from the net positive suction head rather than from the working head. A pump can be right on one axis and impossible on the other.

It says nothing about the flow inside. A specific speed of 1.14 says a Francis runner; it says nothing about the blade angles, the number of vanes, the draught tube, or whether the flow will separate on the suction side of a blade at part load. Every hard question about a turbine remains after the number has chosen its family.

It is not the only convention. Specific speed appears in the literature in at least four forms: the dimensionless one used here, and three dimensional ones in which the head is in metres or feet and the speed in revolutions per minute, differing by factors of up to a thousand. A number quoted without its convention is unusable, and a large part of the confusion around this quantity is that.

It says nothing about what the fluid is. Density has cancelled, so a pump moving mercury and one moving petrol are selected identically. That is correct and slightly startling, and it is what a dimensionless account of a machine ought to do.

The size that cancelled, and what it took with it #

The group’s virtue is that the diameter divides out, and the price of that virtue is worth stating, because it is paid every time a machine is bought.

Two machines at the same specific speed are the same shape. They are not equally efficient, and the larger one is reliably better. Three things happen when a runner is scaled up geometrically and none of them is in the group.

The Reynolds number rises, so the friction coefficient on every wetted surface falls. The relative roughness falls, because a casting is finished to a tolerance set by the process rather than by the part’s size — a millimetre of surface texture is a large fraction of a model’s passage and a negligible one of a prototype’s. And the relative tip clearance falls, for the same reason: the gap a machinist can hold is an absolute length, so it is a smaller fraction of a bigger blade.

Every one of the three moves in the same direction, which is why the effect is large and reliable rather than a wash. A model turbine tested at metre scale may reach ninety per cent where its prototype reaches ninety-five, and the difference is not a defect of the model.

The industry’s response is a step-up formula: the inefficiency is taken to scale as a small negative power of the size, so that

$$\frac{1-\eta_2}{1-\eta_1} = \left(\frac{D_1}{D_2}\right)^{n}$$

with $n$ around a fifth. Moody proposed it in the 1920s and something of the same shape is written into the acceptance standards under which large turbines are bought and sold — because a machine too big to test is accepted on the basis of a model test plus this correction, and the correction is worth a substantial fraction of the contract.

And it is contested, for a principled reason. Part of the loss is friction, which scales with the Reynolds number in a way that can be estimated; part is separation, secondary flow and leakage, which does not scale that way at all. Applying one exponent to their sum is a convenience. So the number that decides which machine to build is exact and dimensionless, and the number that decides what it will be worth is a fitted power law with a standards committee attached.

Why this belongs in a field about control volumes #

Every other rung in this field draws a box round a machine and refuses to look inside. This one does not draw a box at all: it is dimensional analysis, and its argument is prior to any conservation law.

It belongs here because it answers the question the boxes cannot. A control volume tells what a machine of a given kind can do — how much a disc can extract, how much a rotor’s swirl is worth, how much an ejector can raise — and it is silent on which kind to build. The specific speed answers exactly that and nothing else.

The two methods have the same virtue, which is why the field ends here. Both survive with the machine unbuilt and unspecified: a box drawn round it with nothing assumed about the inside, and a matrix of exponents with nothing assumed about the mechanism. What a machine may do, and what shape it must be, are both knowable before it exists.

What the same argument does for a pump #

Everything above was written for a turbine and applies unchanged to a pump, with one sign reversed and one extra constraint that is worth naming because it is where the analogy stops.

A pump’s specific speed is the same group with the head it produces rather than the head it consumes, and the bands are the same in kind: low specific speed gives a radial impeller with a large diameter and a narrow passage, high specific speed gives an axial propeller. A domestic circulator, a fire pump and a cooling-water pump sit at three different points on the same axis for the same reasons.

The extra constraint is on the inlet. A pump must draw fluid in before it can raise its pressure, and the pressure at its inlet is below the ambient one — so it is at risk of tearing the liquid apart in a way a turbine is not. The quantity that governs that is the suction specific speed, built from the same flow and shaft speed but from the net positive suction head rather than the working head.

A pump therefore has to satisfy two dimensionless conditions at once, and they can conflict: a duty whose specific speed asks for a fast, small impeller may have a suction specific speed forbidding it, and the resolution is a slower machine, a larger one, or an inducer stage ahead of the impeller. Two axes, one machine, and the interval between them can be empty — which is the shape the similarity essay describes for a wind-tunnel model and which recurs wherever a design has more dimensionless requirements than free parameters.

Who found it, and when #

The affinity laws were in use by pump makers through the nineteenth century as scaling rules of thumb. Camerer and Rateau put them on a dimensional footing around 1900, and specific speed as a selection tool dates from the same decade — Moody’s turbine work in the 1920s established the bands still drawn today. Buckingham’s theorem, which explains why any of it works, was published in 1914 and therefore after the practice it justifies.

That order is common in this subject and worth noticing. The dimensional argument did not produce the selection chart; the selection chart existed, and the dimensional argument explained why a chart with one axis could exist at all. What the theorem added was the knowledge that there is exactly one such axis, that its form is forced, and that nothing else about a machine can be scaled away.

The number as a shape, drawn #

One last way of reading the group, which is the way a designer holds it.

Write the specific speed as φ^½/ψ^¾ and both of its ingredients are shape statements. The flow coefficient φ = Q/ωD³ is how much passage area the runner devotes to flow; the head coefficient ψ = gH/ω²D² is how hard the blades work per unit of blade speed. A machine with a large φ and a small ψ is open and lightly loaded; one with a small φ and a large ψ is narrow and heavily loaded.

Specific speed is the ratio that survives when the size is divided out, and what is left is precisely the proportion of a runner: how tall its passage is against its diameter, how far the flow turns, whether the water enters radially and leaves axially or does neither. Photographs of runners laid out in order of specific speed show a continuous morphological series, from a bucket wheel through tall narrow radial runners to squat propellers, and the series is a plot of one number.

Where the field goes next #

The machines are finished: a disc, a pipe, a channel, a ball, a sail, an artery, a bubble, a rotor, a jet, a meter, a slam, a ram, an ejector, and the number that chooses between them. What remains in this field is the flow that has no single body in it at all — where the obstacle is everywhere at once, and the only thing left to describe is an average.