I Radiometer rotation speed as function of pressure

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Summary
is there a known equation that describes the relationship between a crookes radiometer rotation speed and the pressure of its environment?
I am experimenting with using a radiometer as an approximate indicator of pressure in my homemade high vacuum system, running a small turbo pump. I am interested in the relationship between pressure and vane rotation speed, with light intensity being constant.

I have only been able to find vague numbers stating the the radiometric effect starts to occur at a few hundred Pa, has a maximum at 1Pa, and disappears around 10-4 Pa.


How may I develop a model of this to work off of?
 

Baluncore

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Crook's radiometer is a thermodynamic heat engine. It is not driven by radiation pressure.
https://en.wikipedia.org/wiki/Crookes_radiometer
https://en.wikipedia.org/wiki/Crookes_radiometer#Thermodynamic_explanation

The rate of rotation will depend on the insulation between the black and white surfaces of the vanes. It will be very difficult to generate a model that predicts RPM given gas pressure, for constant illumination.

You might get a better result if you used a small DC electric motor to spin clear vanes at a fixed speed. The motor current needed to maintain that speed would be proportional to aerodynamic drag, which is a function of air density.
 
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I have considered using drag to measure pressure as you describe, but I think at pressures as low as I’m working with, drag would be negligible compared to the torque produced by even a small motor.

I am interested in measuring in the range of 1Pa to 0.001Pa.
 

Baluncore

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drag would be negligible compared to the torque produced by even a small motor.
Torque is directly proportional to current. If you need less current to maintain the speed then the pressure is proportionally less. A minature electric motor reduces the complexity. Current can range over a factor of 10k. Your range of .001 Pa to 1 Pa is from 10uA to 10mA, which is easily measured.

The limit will be the motor bearing losses which will require a fixed current. Use magnetic bearings.

Crook's radiometer is a small motor driven by heated air. If the pressure is too great air drag and heat loss will stop it turning. If there is too little air it will not overcome the bearing friction and stop.
 
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At 0.001 Pa (and probably even at 1 Pa) I would expect the losses in the motor or the transmission to be dominant - no way to measure the tiny remaining drag.
 

Baluncore

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- no way to measure the tiny remaining drag.
Which is also then a problem with Crook's mill. The air is so thin that it provides very little thermal torque to rotate the mill. At the same time, there is very little air drag to slow the mill down. When the error bars in those two opposing terms overlap, the speed of rotation becomes meaningless.
 

Baluncore

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Maybe an induction motor?
Consider an aluminium foil disk with foil vanes, like Crook's mill, but all the same colour. Generate a two phase rotating magnetic field, at a fixed frequency, by driving two coils with regulated sine and cosine currents, external to the vacuum. You then have an induction motor.

Measure the rotation rate with an optical sensor, compute the slip as a percentage relative to the rotating field, which will be proportional to drag and therefore air density.

Make it work accurately over a wide range by having a step variable regulated drive current.
 
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Which is also then a problem with Crook's mill. The air is so thin that it provides very little thermal torque to rotate the mill. At the same time, there is very little air drag to slow the mill down. When the error bars in those two opposing terms overlap, the speed of rotation becomes meaningless.
Friction from the bearing will break the ambiguity. Would need a careful calibration, however.
 

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