The Bladeless Fan

The blades are in the base

A bladeless fan is a centrifugal fan wearing a disguise. Air comes in through a few hundred small holes round the bottom of the pedestal into a chamber under the impeller, and a bell mouth feeds it upward into the blades. The impeller is a mixed-flow type: a hub that flares as it rises, with nine blades wrapped round it, so the air leaves both upward and outward, part axial fan and part centrifugal blower, driven by a brushless DC motor above it (Gammack, Nicolas and Simmonds 2012). The air comes off the impeller spinning, so a ring of fixed vanes above it takes the swirl off and turns that motion back into pressure; the base then narrows into a neck that carries the air up into the hollow loop. Dyson's patent puts the flow at 20 to 30 litres a second, preferably about 27. That is the only air the motor ever touches, and the impeller is the only part of the fan that moves.

Inside the loop the air sits in a plenum, a channel running all the way round, at a pressure of a couple of hundred pascals. It has one way out.

Slot and surface

The way out is a slot about 1.3 mm wide running the whole circumference of the loop, on its inner face near the back. Squeezing 27 litres a second through a slot that thin and about 1.1 metres long makes a sheet of air moving at roughly 19 m/s. With the controls at their defaults the readout gives 216 Pa across the slot, which is also about what the impeller has to supply.

The sheet leaves the slot heading inward and meets a convex, rounded surface, and it follows that surface round the corner instead of carrying straight on. This is the Coanda effect, and there is nothing mysterious about it: a jet next to a wall entrains air from the gap between itself and the wall faster than that gap can be refilled, so the pressure there drops and the jet is pushed onto the surface. Once attached, the curved flow stays attached for the same reason a car stays on a banked curve: bending a stream of radius R needs a pressure difference across it of about ρU²h/R. For a 1.3 mm sheet at 19 m/s round a 10 mm radius that is some 56 Pa, the suction that holds the sheet to the metal.

After the curve comes a straight flared section, the diffuser, angled outward at about 15 degrees; the patent gives 7 to 20 as the working range. The sheet runs along it and leaves the trailing edge as a thin annular jet, a tube of fast air with a hole in the middle.

Where the air comes from

A turbulent jet cannot move through still air without dragging it along. Eddies at its edges fold slow air into the fast stream, the stream widens and slows, and its momentum is conserved while its volume flow grows. That growth is the whole trick. What the loop does is make the jet a thin sheet with a very long edge, because entrainment happens at edges.

A sheet has two faces. On the outside, air is drawn in from all around the rim: this is what Dyson calls entrainment. On the inside the sheet also entrains, but the only supply of air to the inner face is through the open middle of the loop, from behind. So the fan pulls a column of air through its own centre: inducement, in Dyson's word. At the default settings the model has that air crossing the loop at about 2 m/s.

For a plane turbulent jet from a slot of width B, the volume flow grows as the square root of distance, Q/Q₀ ≈ 0.58 √(x/B), a result that follows from momentum conservation plus the jet's measured spreading rate of about 0.11 (Gutmark and Wygnanski 1976). The thinner the slot, the faster the growth relative to what came out of it. Close to the loop each piece of the sheet behaves like that. Further out, the inner edges of the sheet meet on the axis, the core is used up, and the flow forgets it was ever annular: it grows like an ordinary round jet, linearly with distance, at the rate Ricou and Spalding measured in 1961. I put the hand-over where the two growth rates are equal, which turns out to be about 3.25 loop diameters for any slot. Nothing in the model is fitted to Dyson's numbers except the fan curve, and that only fixes how much air the motor sends.

Fifteen times what?

Dyson's headline figure is that the loop multiplies the motor's air about fifteen times. The model agrees, at the right distance: 16 times at one metre in front of a 350 mm loop. But the ratio is not a property of the fan. A jet keeps entraining for as long as it exists, so the figure depends on where you measure: 8 times at 25 cm, 16 at a metre, 31 at three metres, and it would keep climbing if the room let it.

It also depends on the slot, and that is where the second chart earns its place. Narrow the slot with the motor unchanged and three things happen together. Less air gets through, because the slot is a tighter restriction. What does get through goes faster. And the ratio climbs, from 16 at 1.3 mm to 23 at 0.6 mm, because the denominator shrank. The air actually moving past you at a metre goes the other way, from 433 to 368 litres a second. Widen the slot to 4 mm and the motor moves more air, but the jet is slow and the ratio falls to 9; delivered air falls too, to 343.

What you feel from a fan is set by the momentum it throws, and for a given impeller the jet's momentum peaks where the slot's resistance is matched to the fan, with half the fan's shut-off pressure across the slot. With the fan curve I assumed, that match falls at about 1.4 mm on a 350 mm loop, next to the 1.3 mm the patent prefers. I take that as the real reason the slot is about a millimetre wide. The multiplication ratio is a consequence of that choice, not the goal of it, and a design chasing the ratio alone would narrow the slot until the fan was starved.

The loop's diameter behaves differently. At a metre the ratio barely moves with it, since in the sheet regime it depends only on x/B. A bigger loop has a longer slot, so the same impeller pushes more air through it, and the sheet stays a sheet for longer before it becomes a round jet.

Against the data

A model this simple should be held to numbers it was not built from. There are three I could find.

The totals land within about fifteen per cent. The split is the model's weakest part: it lets the sheet entrain equally from both faces, where Jafari and colleagues found the inside face doing more of the work. That is what the Coanda surface is for. Their fan, like Dyson's, puts the curved surface on the inner side, and a jet bent round a convex wall entrains faster on that side than a straight free jet would.

What I could not settle

The fan curve. Dyson publishes the flow through the slot but not the impeller's shut-off pressure or free delivery. I chose a parabola through the patent's point with 400 Pa at shut-off and 40 l/s free. The matched slot width moves with that choice; the argument that a matched slot exists does not.

Laminar or turbulent. Based on the slot width, the sheet leaves at a Reynolds number of about 1,600, which is why some explanations call the flow laminar. The entrainment laws I use are for turbulent jets. A sheet that thin becomes unstable within a few centimetres and its edges roll up into the eddies that do the entraining, so I expect the turbulent laws to hold over most of the distance that matters, but I have not seen near-field measurements that show where the transition happens on a real loop.

Dyson's fifteen. The marketing figure never says at what distance it was measured. The patent's numbers make it about 15 to 18 at three loop diameters, which is where I have assumed it comes from.

The section. The cross-section in the lab is mine, drawn to the patent's slot width and flare angle. It is not a copy of any product's profile, and the lab does not depend on its exact curves: the numbers come from the slot, the loop and the fan. The base is the same kind of drawing: the order of its parts (perforated intake, bell mouth, mixed-flow impeller, motor, stator vanes, neck) follows the patent's figures, but the dimensions, the hole pattern and the finishes are my own, and none of them is a measurement of a real fan.