Fluid Flow & Aerodynamics
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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.
14 min read 8 figures One number decides the regimeA smooth picture proves nothingTaught wrongly, everywhere

Worth reading first: Two small pumps carry two big discs · One number picks the machine.

Two small pumps carry two big discs built a centrifugal pump’s losses from its specific speed and found that splitting a duty between pumps of the same head and shaft speed costs four points of efficiency, not two, because the head fixes the tip speed and so each smaller pump keeps the large pump’s impeller diameter and its disc friction. It ended by noting a way out the station had not taken: run the smaller pumps faster. Two pumps of half the flow at 2\sqrt2 times the speed have the whole duty’s specific speed and are geometrically similar to the large pump at 1/21/\sqrt2 of its size. Whether that recovers the four points depends on one thing — how a pump’s efficiency changes with its size when its shape is held — and the standard answer to that is a formula from 1925.

Lewis Moody proposed that the inefficiency of geometrically similar turbines scales as the inverse fifth root of their size: (1−η2)/(1−η1)=(D1/D2)1/5(1 - \eta_2)/(1 - \eta_1) = (D_1/D_2)^{1/5}. The fifth root is the Reynolds-number exponent of turbulent skin friction, and the formula follows if every loss in the machine is skin friction. This computes what the exponent actually is when they are not.

Where a fifth root comes from

The exponent has a pedigree worth stating before testing it. A turbulent boundary layer’s skin-friction coefficient falls roughly as the inverse fifth power of its Reynolds number, which is the Blasius-era fit that the cost of going turbulent prices for a plate. In a machine of fixed shape every velocity scales with the tip speed and every length with the size, so the Reynolds number of every surface scales with the size at a fixed tip speed, and every friction loss scales as the size to the minus one-fifth. If all of a machine’s inefficiency were friction, the whole of it would scale that way, and Moody’s formula would be exact.

That premise is what the budget lets be tested, because the budget knows which of its losses are friction. Two of the four are: the disc torque, whose coefficient falls as the fifth root of the disc’s Reynolds number, and the passage friction, whose factor falls roughly as the fourth root in the smooth range and then not at all. The other two are not, and the question is how much of the total they are at each size — which, as the Reynolds number, and the length in it puts it for a single number, depends on which length the loss actually lives on.

A family of one shape

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.
Fig. 1 Best efficiency against impeller radius for geometrically similar pumps of 40 metres head at three specific speeds; the shaft speed falls as the radius grows, so the tip speed is the same at every size.

The pumps here are the same loss budget as the essay below — enclosed-disc friction on both shrouds, leakage through a wear ring, friction in the impeller passages and the volute, and a fixed share for mixing, incidence, diffusion and bearings — swept over size with the specific speed and the head held. Holding the head holds the tip speed, so every velocity in the pump scales with it and every velocity ratio is the same at every size: the family is exactly similar, its outlet width and eye radius the same fractions of the impeller at fifty millimetres and at eight hundred. What changes is the Reynolds number, which grows in proportion to the radius; the roughness of the passages relative to their size; and the clearance of the wear ring relative to the ring.

At a specific speed of 0.55 a pump with a 25-millimetre impeller reaches 0.797. At 0.2 metres it reaches 0.869, and at 1.6 metres 0.887. The first factor of eight in size is worth 7.2 points and the second 1.8. At the lower specific speed of 0.3 the gain is larger at every size, from 0.643 to 0.827, and at 1.0 smaller, from 0.869 to 0.910. A bigger pump is a better pump, and by less and less as it grows.

Three losses, three laws

Leakage is a small pump's loss, and the fixed share a large one's. Each loss as a share of the shaft power against impeller radius at Ωₛ 0.55 and 40 m head. Leakage falls fastest, from 7.81% to 0.935%, because a clearance cannot be made smaller than its floor however small the ring. Disc and passage friction fall slowly with the Reynolds number. The fixed share does not fall at all, 6.38% to 7.1%, and by 0.2 m it is most of the loss.
Fig. 2 Each loss as a share of the shaft power against impeller radius at a specific speed of 0.55 and 40 metres head.

Split into its parts the reason is plain: the four losses obey different size laws, and which one dominates changes with size.

Leakage falls fastest, from 7.8 per cent of the shaft power at 25 millimetres to 0.94 per cent at 1.6 metres. A wear ring’s clearance is specified as a floor plus a fraction of the ring’s size — an eighth of a millimetre here plus a thousandth of its radius — because a ring cannot be machined, aligned and run with a gap below some absolute value however small it is. On a small pump the floor dominates, the relative gap grows as the inverse size, and the leakage share with it. Disc friction and passage friction fall slowly, as the fifth root of the Reynolds number, which is Moody’s law. The fixed share does not fall at all — 6.4 per cent at the smallest size, 7.1 per cent at the largest, rising only because the total it is a share of is smaller — and by an impeller of 0.2 metres it is most of what is left.

Moody’s exponent, measured locally

Moody's fifth root is right at one size, and a different size for each pump. The local exponent of the loss, −d ln(1 − η)/d ln r₂, against impeller radius at three specific speeds, beside Moody's constant 1/5. Small pumps' losses fall faster than the fifth root, because leakage against a clearance floor falls as the inverse size; large pumps' fall far slower, because the fixed share does not fall. The exponent passes 1/5 at a radius of 104 mm at Ωₛ 0.3, 72.3 mm at 0.55 and 38 mm at 1.0.
Fig. 3 The local exponent of the loss, −dln⁡(1−η)/dln⁡r2-d\ln(1-\eta)/d\ln r_2, against impeller radius at three specific speeds, beside Moody’s constant one-fifth.

The fifth root is a claim about a slope, and the budget’s slope can be read at every size. The local exponent of the inefficiency is 0.33 at a 25-millimetre impeller and falls continuously, to 0.24 at fifty millimetres, 0.12 at two hundred and 0.044 at 1.6 metres, at a specific speed of 0.55. It passes Moody’s one-fifth at a radius of 72 millimetres. At a specific speed of 0.3, where disc friction is a larger share and falls with the fifth root, it passes one-fifth at 104 millimetres; at 1.0, where the fixed share dominates sooner, at 38.

Read that way, Moody’s formula is not wrong so much as local. Its exponent is the exponent of one loss, and a pump’s total loss has that exponent only at the size where the faster-falling leakage and the non-falling fixed share happen to average to it. For small pumps the inefficiency falls faster than the fifth root; for large ones much slower, and the gains of scale-up that the formula promises at the top of the range are not there to be had.

Stepping a model up

A model test stepped up by the fifth root over-promises the full-size pump. A model of 50 mm impeller radius at Ωₛ 0.55 reaching 0.834, and the full-size efficiency predicted from it against the scale ratio: by the loss budget, by Moody's fifth root and by Ackeret's half-scaling rule. At ten times the size the budget gives 0.88; Moody gives 0.895, 1.5 points too high, and Ackeret 0.864, 1.5 too low.
Fig. 4 A model of 50 millimetres impeller radius at a specific speed of 0.55, and the full-size efficiency predicted from it against the scale ratio by the loss budget, by Moody’s formula and by Ackeret’s.

The practical use of a step-up formula is a model test. A model pump or turbine is built at a fraction of the prototype’s size, its efficiency is measured, and the prototype’s is predicted. A 50-millimetre model at a specific speed of 0.55 reaches 0.834. Ten times larger, the budget gives 0.880. Moody’s fifth root gives 0.895 — one and a half points too optimistic, which on a large pump is the difference a contract guarantee is written about. Ackeret’s later rule, which lets only half the loss scale with the fifth root, gives 0.864, one and a half points too pessimistic: the right fraction is not a half either, because the leakage part scales faster than the fifth root and pulls the true curve above Ackeret’s at moderate scale ratios.

This is the same difficulty the model that cannot be matched describes for a ship or an aircraft in a tunnel: one dimensionless group can be held and the others cannot, and a scale-up rule is a guess about how the unmatched ones move. For a pump the unmatched quantities are the Reynolds number, the relative roughness and the relative clearance, and they move at three different rates. A step-up formula with one exponent can only be right where one of them dominates.

The error also depends on the specific speed, which is why the modern codes make their split a function of it. Repeat the tenfold step-up from a 50-millimetre model at a specific speed of 0.3 and the budget gives 0.805 from a model at 0.707; Moody gives 0.815, a point high, and Ackeret 0.761, four and a half points low, because at that specific speed most of the loss is disc friction and passage friction, which do scale, and Ackeret’s fixed half is far too large. At 1.0 the budget gives 0.907 from 0.887; Moody gives 0.929, two points high, because there most of the loss is the fixed share, which does not scale; Ackeret, at 0.908, is almost exactly right. Each formula is the right one for a particular balance of losses, and the balance moves with both the size and the shape.

Rough passages stop improving

A large pump's passages are rough, and their friction stops scaling. The passage-friction share against impeller radius at Ωₛ 0.55 for hydraulically smooth passages, cast passages of 25 µm roughness and rough ones of 150 µm. Smooth, the share keeps falling, to 1.33% at 1.6 m; cast, it flattens to 1.53%, and rough to 1.98%. A large passage at a large Reynolds number is fully rough, where the friction factor no longer depends on the Reynolds number at all, only on the roughness over the passage's size — and that falls far more slowly with size than the Reynolds number rises.
Fig. 5 The passage-friction share against impeller radius at a specific speed of 0.55, for hydraulically smooth passages, cast passages of 25 micrometres roughness and rough ones of 150.

Even the friction part stops following the fifth root at large sizes. A hydraulically smooth passage’s friction factor keeps falling with the Reynolds number, and its share of the shaft power falls to 1.3 per cent at 1.6 metres. A cast passage of 25 micrometres roughness flattens to 1.5 per cent, and a rough one of 150 micrometres to 2.0. At the Reynolds numbers of a large pump — tens of millions on the discs — the viscous sublayer is thinner than the roughness, the passage is fully rough, and its friction factor depends on the roughness over the passage’s size and not on the Reynolds number at all. The roughness a wall cannot feel is the smooth regime of the same story: a small pump’s sublayer hides roughness that a large pump’s exposes. So a large pump’s friction gain from size is set by how its surface finish compares with its passages, which is a matter of casting and machining rather than of similarity, and why large pumps are often polished or coated where a small one would gain nothing.

The clearance a ring cannot shed

Without a clearance floor, leakage would scale away with the pump. The leakage share against impeller radius at Ωₛ 0.55, with the stated clearance of 0.125 mm plus a thousandth of the ring radius, and with the proportional part alone. Proportional, the share is nearly the same at every size, 0.785% to 0.812%; with the floor it is 7.81% at 25 mm. The clearance a ring cannot be made below is what makes leakage a small pump's loss.
Fig. 6 The leakage share against impeller radius at a specific speed of 0.55, with the stated clearance rule and with only its proportional part.

The leakage law is entirely the floor. With only the proportional part of the clearance — a gap that is the same fraction of the ring at every size — the leakage share is the same at every size to within a few per cent, 0.79 to 0.81 per cent, because the ring is then an exactly similar orifice whose discharge coefficient is only the shape of the hole. With the floor it is ten times larger on the smallest pump. That is the loss a small pump suffers for being small and not for being slow, and it is the one an engineer can attack directly: tighter rings, labyrinths, or ring materials that tolerate running closer. It is also why the smallest commercial pumps are so often made with open impellers or as positive-displacement machines, which have different leakage paths altogether.

Split by speed, not by width

Split a duty by speed, not by width, and the penalty is the size effect alone. The station of the earlier essays over its duty against static lift: one pump under speed control; two half-flow pumps at the same shaft speed, which keep its impeller and narrow it; and two and three pumps at √2 and √3 times the speed, which keep its specific speed and shrink it. The fast pair gives up 0.63 points against the slow pair's 4.22, and saves energy above 27.2% lift instead of 65.4%; three fast pumps above 28.7%.
Fig. 7 The station of the earlier essays over its duty against static lift: one pump under speed control; two half-flow pumps at the same shaft speed; and two and three pumps at 2\sqrt2 and 3\sqrt3 times the speed.

The question the essay below left is now an exercise in reading the size curve. Its station takes 0.1 cubic metres a second at 40 metres and 1450 revolutions a minute, a specific speed of 0.545, with a 369-millimetre impeller. Two pumps of half the flow at the same speed have a specific speed of 0.385 and the same 369 millimetres, and give up 4.2 points. Two pumps of half the flow at 2051 revolutions a minute — 2\sqrt2 times as fast — have the duty’s own specific speed and a 261-millimetre impeller, 1/21/\sqrt2 of the size, and give up only the size effect between 261 and 369 millimetres: 0.63 points.

Run through the station’s duty with that penalty, the split by speed beats one pump above 27 per cent static lift, where the split by width needed 65. Three pumps at 3\sqrt3 times the speed, a fifth smaller again, give up a little more and beat one pump above 29 per cent. So the right way to divide a duty between pumps, if energy is the measure, is to keep its specific speed and let the pumps shrink, and the price of doing so is set by the slope of the size curve where the pumps sit — which at this size and specific speed is gentle.

In a year, on the duty of the earlier essays at 30 per cent static lift, the station delivers 142 megawatt-hours to the water. One pump draws 177.4 megawatt-hours, the slow pair 184.5 and the fast pair 176.8 — slightly less than one pump, with a standby unit included. The seven megawatt-hours a year that splitting by width cost are recovered by spinning the same two motors’ worth of pumps faster, and the station gets its redundancy for nothing.

The catch is the one the specific speed was invented to show. A faster pump at the same head needs more suction margin, since the group with no head in it rises with the shaft speed, and a station that runs its small pumps at 2051 rather than 1450 revolutions a minute needs a deeper sump or a lower suction setting to keep them from cavitating. The energy argument and the suction argument pull in opposite directions, and the budget here prices only the first.

How the family was checked

What the size series was checked against. The checks: geometric similarity across a factor of sixteen in size, the friction factor's limits, and the two step-up formulas reduced to identities at a scale ratio of one.
Fig. 8 Geometric similarity across a factor of sixteen in size, the friction factor’s laminar limit, and the two step-up formulas at a scale ratio of one.

The family’s similarity is a check rather than an assumption: the outlet width and eye radius as fractions of the impeller, and the specific speed, are the same at fifty millimetres and at 0.8 metres to rounding. The friction factor returns 64/Re64/Re below 2300 exactly. Both step-up formulas reduce to identities at a scale ratio of one. The local exponent is a centred difference over a tenth of a per cent of the size, small enough that halving it changes the third figure of nothing quoted. The budget’s own checks — the disc share against its closed form, the geometry against its duty — are in the essay below.

The convention: similar pumps at one head

Every pump in a family here has the same specific speed and the same head, so the same tip speed, and is a scaled copy of the others; its size is its impeller radius. Efficiencies are best-point efficiencies. Moody’s formula is written for the inefficiency 1−η1-\eta against the diameter ratio, and Ackeret’s lets half of 1−η1-\eta scale with the Reynolds number to the minus one-fifth and holds the other half; both are quoted in the forms the step-up literature uses. A model test at a lower head would change the Reynolds number again, through the tip speed, and that variant is not computed.

What the size curve cannot show

The fixed share is stated, not computed. Mixing, incidence and diffusion losses do change with size a little, through the boundary layers on the vanes, and a budget that gave them a weak Reynolds dependence would bend the large-size end of every curve down a little and move the exponent’s crossing outwards. Clearances in service open as rings wear, so a pump’s leakage law is a property of its age as well as its size. Surface finish was held at one absolute roughness across sizes, as a casting process would give; a large pump that is machined or coated sits nearer the smooth curve. And the shape is held: a real small pump is usually not a scaled copy of a large one, having thicker vanes, fewer of them and a cruder volute for reasons of manufacture, which is a separate penalty from the size effect and often a larger one.

Step-up rules and what they averaged over

Moody’s formula of 1925 was proposed for hydraulic turbines, where the losses of a large machine are dominated by friction in long passages, and it fitted the turbine data of its time. Ackeret, Hutton and others proposed rules splitting the loss into a scalable and a non-scalable part in the decades that followed, because Moody’s over-predicted for large machines, and the International Electrotechnical Commission’s model-test codes for hydraulic machines now step up efficiency with an explicit split of that kind, with the scalable fraction a function of the specific speed. The budget reproduces the reason for that history in one figure: an exponent that varies with size by a factor of seven cannot be captured by a single constant, and the split that the codes adopted is a two-term version of the three terms here.

Still open: the loss that rises with the specific speed

Every result here, and in the essay below, sits on the low side of the specific-speed range, where disc friction, narrow passages and leakage grow as the specific speed falls. The other side has its own losses: as an impeller widens towards mixed flow and then axial, its tip clearance becomes a leakage path across every blade rather than one ring, its secondary flows strengthen, and its best efficiency levels and falls. The next calculation adds a tip-clearance loss and a blade-loading loss that grow with specific speed, finds where the efficiency curve turns over, and asks whether the measured maximum near a type number of 40 to 60 — the Francis-to-mixed-flow range that the first essay on specific speed placed its bands around — is where the two families of loss cross, which would make the best machine’s position a result rather than a summary of practice.

Where this goes next

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. Reynolds number: one number, four different flows. Reynolds number is inertia ÷ viscosity. It is not a property of the fluid or of the shape but of the combination, and crossing a threshold changes the physics rather than the magnitude. The model that cannot be matched A scale model behaves like the real thing when its dimensionless numbers agree. With one number that is a matter of choosing the tunnel speed. With two it is usually impossible, and every wind-tunnel result ever published has been obtained in spite of that. 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.

Shares its objects with

Essays naming at least two of the same things, that neither author linked.

Named objects

A dashed tag is an object no other essay names yet.

EfficiencyModel limitPump characteristicReynolds numberRoughnessSimilaritySpecific speedThreshold