Impellers for vacuum

Three bench instruments for the centrifugal fan inside a vacuum cleaner: what the blade angle does, how each type behaves against a throttle, and why motors stack stages. Every number shown comes from the same model that moves the parts.

1. One wheel, three types

A vacuum's impeller is a centrifugal fan: air enters at the eye, the blades fling it outward, and the pressure it gains is the suction. The types differ in one number, the blade angle at the rim β₂.

u: rim speedw: air relative to the blade, leaving at β₂c = u + w: what the air really doesRuler: the swirl c_u, 10 m/s per tickA 110 mm wheel with 9 blades at 30 000 rpm: the rim moves at 173 m/s. The triangle is drawn for 35 L/s and ideal blades (no slip, no losses), so Δp is Euler's ρ·u₂·c_u. The blade angle runs linearly from 30° at the eye to β₂ at the rim, 25° to 155°. Bending blades are the demonstrator's licence: a real wheel is one of these shapes, riveted.

Euler's equation
The wheel can only raise pressure by giving the air swirl. With no swirl at the eye, Δp = ρ·u₂·c_u2: rim speed times the tangential part of the air's absolute velocity. Everything about a blade shape acts through c_u2.
The blade angle
Air leaves along the blade, so c_u2 = u₂ − c_m·cot β₂. Backward blades (β₂ < 90°) subtract from the rim speed, a radial tip leaves it alone, forward blades (β₂ > 90°) add to it. More flow means a larger c_m and a stronger effect.
Pressure or speed
The reaction R = 1 − c_u2 / 2u₂ is the share of the rise made as pressure inside the wheel. Forward blades fall below one half: they hand the casing fast air, and turning speed into pressure there is where the losses are.

2. Against a throttle

A fan never works alone: the hose, bag and filter resist the flow, and the fan settles where the pressure it makes equals the pressure they need. Close the gate and each blade set answers differently.

The wheel, slowed about 1 700 times so the blades can be seenGate: lifted is openDials: 0–30 kPa and 0–3 kW; the flap swings out with flowThe same 110 mm wheel at a fixed 30 000 rpm, now with Wiesner's slip and three losses: 45 % of the outlet's velocity head in the casing, friction in the blade passages and shock at the eye away from the design flow. The gate is an orifice up to 25 mm across. The loss coefficients are plausible, not measured, and a real motor would slow under the forward set's load rather than hold its speed.

Backward-curved
Pressure falls gently as flow rises and the shaft power levels off, so the motor cannot be overloaded by a split hose or an empty bag. It is the most efficient of the three, and what nearly every vacuum motor uses.
Radial-tip
Straight tips are strong and shed dust, so they suit dirty-air and industrial blowers. Pressure stays nearly flat and power rises in proportion to flow.
Forward-curved
The most pressure and flow from a given diameter and speed, which is why quiet ventilation fans use them at low speed. At a vacuum's speed the power runs away as the gate opens, and much of the rise is velocity the casing wastes.

3. Stages in series

Suction grows with the square of the rim speed, and rim speed is limited by noise, bearings and the strength of a thin stamped wheel. To pull harder, a motor passes the same air through several wheels one after another.

Liquid columns: what each stage adds, and the totalShrouded wheels: the seams and rivets trace the blades beneathStator shell: return vanes turn the air back to the next eyeBackward-curved 110 mm wheels at 20 000 rpm, breathing through a 12.5 mm test orifice (not drawn). The board has a rule every 5 kPa. Each stage after the first is taken to deliver 88 % of the first one's rise, the rest lost in the return channel. The air is treated as incompressible, which flatters three stages by roughly a tenth. A stage sitting on its stand adds nothing.

Pressures add
In series every stage sees the same flow and adds its own rise: the columns for the stages stack up into the total. Two stages give nearly twice the sealed suction of one, three nearly three times.
More flow too
Through a fixed restriction, more pressure pushes more air: flow grows roughly with the square root of the suction. Shaft power grows fastest of all, with flow times pressure.
Why not one big wheel
The same rise needs 1.7 times the rim speed from a single wheel, by diameter or by rpm. Stages keep the motor slim and the wheels lightly stressed, at the price of the return-channel loss. Wet-and-dry and central vacuums use two or three.

1. One wheel, three types

Euler's equation

The blade angle

Pressure or speed

2. Against a throttle

Backward-curved

Radial-tip

Forward-curved

3. Stages in series

Pressures add

More flow too

Why not one big wheel