If one makes a chart of compressor efficiency, you can start with one blade, then you can work upwards - where the eventual maximum efficiency is gained with an infinite number of blades (of infinitely small thickness).
A roots blower is (in some respects) a
bit like having a centrifugal blower with 4 blades (an absurdly low number) - strictly looking at things from a compression efficiency standpoint. The fact its also positive displacement (flow = volume x rpm), means you get gigantic pressure pulses, this is why on clever roots "style" blowers, you see people putting in angled entry and exit passages such that the rotor opens out into the port gradually, and also why you get people using screw designs to try to even out these pulses. Pressure pulses create flow unsteadiness and other undesirable characteristics which all create inefficiency, this inefficiency is mostly manifested in the form of frictional losses (in viscous form within the flow and from the fluid to the walls), this heats up the fluid, which makes it expand. Somewhat the reverse of the purpose of a compressor.
In reality the number of blades in centrifugal blowers is limited by manufacturing, weight, area-blockage and skin-friction losses. However people try to get in as many as reasonably possible. The more you have, the less freedom the air has to do unwanted things in the flow channels, like recirculate and so on.
Roots blowers only really work acceptably by having a tiny clearance at the tips, which is virtually impossible to maintain, as the thing starts off cold then gets hot - and then it eventually starts to wear out as bits of dirt and grit get in and get ground up between the rotors, and makes any sort of very high compression impossible, as the stuff just tries to flow back through the tip clearances, and very high pressures distort the casing around the rotors.
However its interesting to reflect that the roots blower by itself does not actually compress anything at all anyway ! Its only moving a blob of something from A > B. So it will only add pressure when the inlet passage is appreciably bigger than the outlet, which is another way of saying its sized to supply more air than the thing using it at the other end needs. Whereas a centrifugal blower generates a pressure difference even sitting in free air at both ends spinning, as the difference in pressure is purely a factor of the outer radius minis the inner radius of the impeller.
Once nice thing about roots blowers (like axial compressors) is that in a long & thin rotor format, they do lend themselves to being packaged conveniently in small spaces around engines. Wheras a sodding great big centrifugal blower, which needs a big diffuser to work well, is a very large package problem unless the entire car/plane has been designed around it in the first place. They are also used sometimes by very lazy engine designers who want to make a "hot-rod" engine, without doing any hard design work. for which a positive displacement pump is ideal as it works the second it starts spinning, and hence has no "lag". The fact its about 25% less efficient than a proper compressor is something they decide they dont care about, as often a sufficient performance boost can still be reached on the available fuel without detonation occuring. Hence pathetically one or two car-makers are still selling cars with roots blowers on, which is pretty embarrassing.
Generally, except for very low pressure oil pumps for oil-mist scavenging, where it is useful to have a positive displacement action (which works at low speeds), nobody with any sense uses roots blowers for anything these days. Numerous people do for historical reasons (top fuel dragsters etc), but from a thermodynamics perspective they are horrid useless things for pumping compressible fluids (like air).
I would expect to receive various incredulous replies to this comment from any American friends reading, but sadly the fact is they are just not very good, but easy to use....
The earliest report I`ve read where all the above was definitively stated regarding the two methods was 1931, although it is very likely that reports are available saying the same thing going back well into WW1.
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Sadly the earlier comment that two-stage centrifugal superchargers are more efficient than single-stage because they can operate each stage at lower tip speeds, is not the case. Up to the limit of the single-stage compressor, in terms of pressure ratio, it is always considerably more efficient to achieve the compression in a single stage, as the air-path between stages is invariably convoluted and causes significant dynamic pressure head losses. A two stage is simply needed above a certain pressure ratio, as you just cant get a single stage to do it. This is one reason the German engines in WW2 stuck to single-stage SC`s too long, as certain of their top scientists in compressors were obsessed with things being as efficient as possible (Others decided on more power ! - with a less "efficient" solution...)
You also cannot lower the tip speed of a two stage blower for a given flow rate relative to a single-stage SC of the same overall size , because the flow rate is limited by choke at the inlet eye diameter of whatever the first stage is. So for a given throughput, added stages only add pressure - not flow, because if the compressor was designed to be as small as possible in the first place, the flow the engine needs will already result in the first stage eye being supersonic just above whatever the max performance was designed to be anyway. (I`ve sort of explained that backwards, sorry....). This is because there are fairly strict design guidelines on the ratio of the inlet eye to the impeller tip diameter in order for the compressor to work well, and so if you have run out of packaging space, you basically cannot increase the diameter of the inlet pipe to the supercharger. Of course real life is a bit more complicated, because some compressors will be designed (by accident or otherwise) to choke at the outlet first, or be totally under or oversized in the first place anyway - in which case lots of tweaks can be made.
But essentially in "maths-world" that is the limit to it.