Measuring vacuum w/ hot filament - calculation (pirani gauge)

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• Plat
In summary, the conversation discussed the challenges of measuring vacuum using a tungsten filament device and the relationship between current draw and pressure. The individual was seeking guidance on creating a mathematical model to determine the temperature of the filament based on ambient temperature, power consumption, and pressure. They also discussed the use of ethylene glycol and propylene glycol for calibration and the possible effects of gas density on heat loss. Ultimately, it was decided that adding more data points would improve accuracy and the mean free path of the gas could affect the measurement at lower pressures.
Plat
I am looking for guidance on how to successfully measure vacuum using a device similar to a pirani gauge, my device however is a tungsten filament which is heated electrically to about 90*C, and it's current draw is a measure of vacuum.The problem I am having is how to relate specifically the current draw to a particular pressure in the system. I know the current draw is inversely-proportional to the resistance, and the resistance varies with the filament temperature, and the temperature varies with gas pressure, warming as the pressure drops due to less thermal conduction and convection.

Running this in my hobby vacuum setup, it is amazingly sensitive. My particular filament running on 5v reads 46.7 milliamps at 28*C 100kPa, and ~38.1 milliamps as I approach 5pa. But I don't know this minimum pressure yet without an accurate gauge.

I really need a mathematical model than will tell me the temperature of a tungsten filament given the ambient temperature, power consumption, and pressure. Thanks for any help!Here is how the pressure and current relate at higher pressures:

This test was done with 3.3 volts, vs 5 volts above(corrected).

ma kPa
36.17 100
35.99 86.8
35.99 76.7
35.73 60.5
35.46 47.4
35.05 34.2
34.35 21.1
33.88 7.9
~32.2 nearly 0

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@Plat Hi
If you bought your device, then it should have a calibration chart wrt a certain gas showing the pressure/current curve which is not a straight line but has a few curves in it. It should show operating voltages, and ambient temperature recommendations.

The filament dissipates its heat in a manner of ways. Conduction through the ends of the wire, radiation to the envelope, conduction to the gas, convection to the gas.
Depending upon the density of the gas in the chamber, or pressure of the gas, the predominate method of heat loss varies.

With very, very low pressures radiation will predominate, along with conduction along the wire ends.
As the pressure of the gas increases, direct conduction will begin to predominate.
With more pressure, convection can begin to be more important, as can the fact that a heat envelope may form around the wire depending upon the mean free path of the gas molecules.

That's just my thoughts on the subject, if that is what you are after.

Thanks for responding, Actually this is completely homebrew, I am using a lightbulb filament that I have mounted inside the vacuum chamber. I was hoping for a way to extrapolate with the current draw known at STP for example, to any lower pressure. Assuming we are below pressures where convection plays a part, would the thermal energy loss due to conduction to surrounding gas be proportional to the density of that gas?

Just calibrate it yourself. Measure the current at a variety of known pressures and construct a calibration curve.

I tried doing this as well. I tried using boiling propylene glycol at known temperature to produce a known vapor pressure in the 5-50 Pa range, but I could not get it to boil consistently. I figured this was because of the pressure from he weight of the liquid in a container being too great for the vapor pressure to overcome.I then tried saturating a sponge in propylene glycol and inserting a thermometer into the sponge. This cools in the vacuum to a minimum of 11*C, which corresponds to a pressure of 3.4Pa. I find this suspicious because the pump is rated to 5Pa and I know the system isn't perfectly sealed. Do you think this is a valid result?

Note that you have some bad data at 76 & 86 kPa, the current is the same.

Here is the fitted equation for your data:
y = 34722600 + (4.229631 - 34722600)/(1 + (x/50.38422)^38.66706)

x = Pirani Current mAmp
y = Pressure KPa

Note that the "Y" axis is logarithmic in the graph.

Thanks so much, that's just what I needed. I can also add more data points to improve the accuracy. To my question above, when the temperature of the ethylene glycol drops as the pressure is reduced, this is because it is boiling and the pressure in the chamber will be equal to the vapor pressure of ethylene glycol at the particular temperature, correct?

The 3.4Pa vs 5Pa I would consider possible, if only because the pump mfg. may have conservatively rated the pump to allow for both degradation during the warranty period and for manufacturing variations.

As for the "sponge and propylene glycol", I have zero experience with such things and leave it to others to make a reasonable assessment. Sounds like an appropriate and clever approach though! (as long as it doesn't contaminant the vacuum system.)

Thanks for your thoughts, I was hoping I had found a reliable way produce very low reference pressures.

I added the data points for the region below 20Pa and there seems to be a huge change in the graph somewhere between 20Pa and 7.9kPa. At or above 7.9kPa pressure changes very quickly with current draw, and at 20Pa and below, it changes very, very slowly. I'm guessing the transition between cooling largely by conduction and radiation occurs in that gap.

OK, I'll speculate. How does the mean free path of the gas compare with the sensor size around 20Pa to 7.9kPa?

I suspect the mean free path becomes longer than the filament diameter in the range. I hadn't considered that affect. But I think it is still useful for a sensor because the graph has a one-to-one correlation everywhere.

Looks like 1.2E-6 meters at 7.9kPa and 4.7E-4 meters or 0.47mm at 20 Pa. That is certainly larger than the filament diameter.

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A few thoughts.

A possible further investigation would be find the breakpoint in the calibration curve then use a much larger sensor and see if the breakpoint moves to a lower pressure.

To get higher sensitivity, using a thermistor may help. Outgassing may be a problem that could be avoided by purchasing one sealed in a glass probe. That would increase the response time though.

In any case, since the calibation curve changes shape at lower pressure, you will probably need a higher order formula than mycurvefit.com supplied. There should be other sites that can fit to third or fourth order equations, it's just a matter of finding them.

Please keep us updated on your results; that way we too can learn. (and it makes us feel good too.)

Very interesting..

I wonder if the thermal element from an automotive mass airflow sensor might work.

Tom.G said:
Please keep us updated on your results; that way we too can learn. (and it makes us feel good too.)
Seconded !

I vote for splunking down some money and buy an ion gauge and be done with.

Z.

nsaspook
I found a polynomial regression calculator that can be set to any order here: http://arachnoid.com/polysolve/

It looks like 3rd order fits best, I don't think higher orders can be right because the function should be continuously increasing, and have no point where the slope is zero. I will test the 600Pa to 7.9kPa range with water of various temperatures and 20Pa to 600Pa with the propylene glycol and fill in that gap.

Jim, that would be interesting to play with. Sounds good with the hot wire and temperature sensor in one package. Actually never knew how those worked before I searched for it just now.

Plat said:
I suspect the mean free path becomes longer than the filament diameter in the range. I hadn't considered that affect. But I think it is still useful for a sensor because the graph has a one-to-one correlation everywhere.

Looks like 1.2E-6 meters at 7.9kPa and 4.7E-4 meters or 0.47mm at 20 Pa. That is certainly larger than the filament diameter.
I'll have to take your word for the mean free path for ethylene glycol.
For nitrogen, though it should be something like a meter or so, and 0.0001 m, I think.

Larger filament diameter will move the bottom flatness of the curve slightly upwards due to more radiation heat transfer.
It shouldn't affect the loss throught the filament end wires if they stay the same diameter.
Both radiation and end effects should be fairly constant with change in pressure.

You want a large enough mean free path so that molecular collisions are minimized. Ideally for the gas conduction to be linear, the molecule should grab the heat from the filament and directly transfer it to the vacuum casing and then come back for more. Collisions are what develops the heat envelope around the filament.
Colliding molecules exchange energy, so there begins to form a dT/dx around the filament as the mean free path decreases with increased density or pressure.
At some point gravity puts its nose into the mix, with the resulting bouyancy effect producing convection, and the curve should show an "improved" heat transfer, leveling off again as that effect saturates.

Plat said:
I added the data points for the region below 20Pa and there seems to be a huge change in the graph somewhere between 20Pa and 7.9kPa. At or above 7.9kPa pressure changes very quickly with current draw, and at 20Pa and below, it changes very, very slowly. I'm guessing the transition between cooling largely by conduction and radiation occurs in that gap.
At the 20Pa you may be hitting the radiation/end effect plateau, and at 7.9kPa the beginning of significant convection currents.

Now that you mention it, I completely overlooked the difference in mean free path for ethylene glycol. I just used an online calculator and the molecular diameter of 0.3nm as would be for air. I don't know how to calculate mfp correctly for ethylene glycol. Sorry, I keep confusing ethylene and propylene glycol. I had planned on using propylene and ended up using ethylene instead, having trouble keeping them straight in my head.

Added another data point at 1.45kPa. Here the curve is becoming very steep and starting to bridge the previously posted graph with the very flat low pressure region.

Just out of curiosity I looked for pricing of ion gauges; USD 130 to 650 just for the sensor. The controller/electronics are sold separately. The sensor itself looks moderately simple, perhaps an electronic vacuum tube with its envelope removed could be used. You would still need power supplies of a couple hundred volts, the readout is just a milliamp meter. This could be a project in itself though.

https://en.wikipedia.org/wiki/Hot-filament_ionization_gauge
http://mmrc.caltech.edu/Vacuum/Teledyne Hastings/Ion Gauges.pdf

When convenient, could you post either you current calibration readings or the curve & fitted equation?

I might have to start over and do something different in the way I am creating the reference pressures. Using water, I get significantly different results when I use the sponge vs letting the water boil in a beaker. Even then, the water boils in fits and starts. For some reason, the reading of current draw is much higher when using the sponge, given the water is at the same temperature. I don't understand what could cause this discrepancy. I will be sure to post it up when I think I have good data.I'm not sure about the ion gauge, it looks like I would be operating at the very high end of their pressure range and could easily burn the filament. I would love to try an old electron tube if I had a power supply in the 150-200v range.

For higher pressures than the heated ion gauges like, there are cold cathode gauges without a filament. The drawback is they need a few kilovolts to operate. If you decide to 'brew your own', a fluorescent lamp ballast is a readily available high voltage transformer (one of the old fashioned magnetic ballasts, not a solid state one.) I'm not really suggesting a cold cathode gauge, just pointing out their existence!

Commercial vacuum equipment often uses two gauges; one during initial roughing pumpdown, then power on the the low-pressure sensitive ones after a decent vacuum is reached.

Plat said:
...if I had a power supply in the 150-200v range.
You could build one for $50 or under. If your line voltage is fairly stable you wouldn't need regulation, else add$10 or so for a Zener regulator.

It takes a transformer ($21), bridge rectifier (under$1), filter capacitor 5-10uF, 250V-400V ($0.50 to$2). Since you will also need a supply around 50v, add a second bridge and a 100v capacitor. And the inevitable power cord, board or chassis to build it on, connectors, etc, and of course tax and shipping.

Still working on this, trying to get reliable and repeatable results when gathering data at various pressures. I settled on using water and propylene glycol boiling, but not a rolling boil. I know they are boiling because the temperature steadily drops. A rolling boil is much too violent and the readings are very erratic. I have also decided to pulse the gauge on and off, instead of running it continuously. This eliminates creep in the readings as when run continuously, the filament warms it's surroundings and causes the reading to drop due to increased temperature. Hopefully can post the graph up tonight.

ZapperZ said:
I vote for splunking down some money and buy an ion gauge and be done with.

Z.
One problem with an ion gauge: It doesn't start working till you get to about 1/10th of a millitorr or less than 1E-3 torr or so. The pressures here are in the torr range, WAY too high for an ion gauge.

The best one for these pressure ranges are TC gauges, Kurt Lesker Co, sells them and you can get a Hastings gauge readout box for about 100 bucks on Ebay.
TC gauges work like his homemade device does except it has a thin wire with current in it and it gets heated to some temperature but there is a thermocouple added to the wire so it measures the temperature very accurately and that corrolates to the pressures. Most work from about 1 torr to about 1 millitorr.

litup said:
One problem with an ion gauge: It doesn't start working till you get to about 1/10th of a millitorr or less than 1E-3 torr or so. The pressures here are in the torr range, WAY too high for an ion gauge.

I'm well aware of that. The message that I was trying to convey was "why are we reinventing the wheel?"

The TC gauge that you suggested certainly will work better than the ion gauge.

Z.

This is what I came up with. When I mentioned earlier the sudden change between 20Pa and 7.9kPa, that looks to be due to a mismatch in data between using the sponge and not. For this data, I used the rough vacuum gauge, boiling ethylene glycol, water, and 70% by volume isopropyl alcohol (conscious of fire hazard). For the alcohol, I calculated the vapor pressure of the mixture using the mole fraction and Raoult's law(although I know there is some error as this deviates from an ideal mixture). The different substances seem to agree pretty well, but the reading for the alcohol seemed to result at slightly lower pressure vs current draw. Maybe this is due to a difference in thermal conductivity as a gas? Anyway, I decided to leave in data points that seem contradictory (water vs alcohol), thinking the function would be more accurate with all of them considered.

On second thought, I should remove the alcohol data points, I think there is too much error due to this not being an ideal mixture.

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Tom.G
Inadvertently, I have found another interesting thing about gas at low pressure. There looks to be a specific pressure below which thermal conductivity of air decreases suddenly and dramatically. In the chamber, I was running a small fan with a piece of tissue paper taped to the front, so I could observe the movement of the tissue from airflow at various pressures. The moving air from the fan was also cooling the filament and increasing the current draw substantially. This affect is almost constant above about 3kPa, and decreases to zero very quickly thereafter as pressure continues to drop.

I have also finalized the graph, it looks very similar to the one posted above, except that the conflicting points from the isopropyl alcohol have been removed. Turns out, they seemed out of place because the alcohol was boiling at a higher pressure than I had calculated it should, due to the mixture not being close to an ideal one.

1. How does a hot filament pirani gauge measure vacuum?

A hot filament pirani gauge works by using a heated filament to measure the thermal conductivity of the surrounding gas. As the gas molecules collide with the hot filament, they transfer heat away from the filament, causing it to cool down. The amount of cooling is directly related to the pressure of the gas, allowing for an indirect measurement of vacuum.

2. What is the principle behind the calculation for measuring vacuum with a hot filament pirani gauge?

The principle behind the calculation involves measuring the change in resistance of the heated filament as it cools down due to gas molecules colliding with it. This change in resistance is then correlated to the pressure of the gas, which can be converted to a vacuum measurement using the appropriate calibration curve.

3. What factors can affect the accuracy of vacuum measurements using a hot filament pirani gauge?

The accuracy of vacuum measurements using a hot filament pirani gauge can be affected by factors such as the temperature and geometry of the filament, the composition of the gas being measured, and the quality of the vacuum system. Additionally, changes in pressure due to temperature fluctuations or gas leaks can also impact the accuracy of the measurement.

4. How is the sensitivity of a hot filament pirani gauge determined?

The sensitivity of a hot filament pirani gauge is determined by the amount of change in resistance for a given change in pressure. This can be calculated by dividing the change in resistance by the change in pressure and is typically expressed in units of ohms per millitorr.

5. Are there any limitations to using a hot filament pirani gauge for vacuum measurement?

Yes, there are some limitations to using a hot filament pirani gauge for vacuum measurement. This type of gauge is not suitable for measuring very low pressures (below 1 millitorr) or high pressures (above 1000 torr). Additionally, it is not suitable for measuring corrosive or reactive gases, as they can damage the filament. It is also important to regularly calibrate the gauge and ensure proper handling and maintenance to ensure accurate measurements.

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