How to calculate the wind speeds produced by an ionic thruster

[Jon Doe]

Homework Statement

Find the speed produced by an ionic thruster functioning at the atmospheric level (thus drag on the sped-up particles can not be ignored). The thruster is operating with 712 volts and the gap between the electrodes is 20mm (the electrodes form a uniform electric field).

Relevant Equations

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I have tried using the equations for the mobility of ions in liquids as that was the only thing I could find that addressed the issue. My explanation of the calculations can be found down below:

I think all this makes sense but then I add values to it and the following results are produced:

The issue is that the teacher says this is wrong but I am not sure where...

Any help or attempt will be appreciated!

 

[Steve4Physics]

Hi @Jon Doe and welcome to PF.

Here are a few issues to consider...

Your solution states: “Let us consider the above stated thruster …”. But we (well, me anyway) need a description/diagram to understand what the setup is. I’m no expert but I’m guessing the question is about a ‘gridded ion thruster’. You need to clarify this.

You stated: “When any two electrodes are connected to a DC power source, a uniform electric electric field is produced”. That’s incorrect, e.g. consider two point electrodes. However, you may have a (roughly) uniform field inside your thruster, depending on the design.

In the presence of the atmosphere, the speed of the ions would (I’d guess) rapidly reduce to zero (or, to be more precise, random thermal motion) once emitted from the thruster. Are you asking about the speed of the ions while they move between the electrodes or after the ions have left?

Your table contains a unitless quantity called the “Weighted distribution of particles”. What does that mean? Is it related to the ion’s mass (which presumably is also needed)?

You are using a value for ionic charge of $4.454 \times 10^{-19} C$, but this is not an integer multiple of the elementary charge ($e = 1.602 \times 10^{-19}C$).

You have forgotten to include the unit for electric field strength in your table.

If you can clarify some of the above issues, you are more likely to get replies.

 

[Jon Doe]

Hi @Steve4Physics Thank you for your reply!

Firstly, thank you for the pointers that will help me out here cuz this was my first post on this forum...

Anyways, I really don't have a diagram but I do have a subpar image of the contraption

These are the 2 electrodes. The one on the left of the image is the positive electrode and is formed of straight thin copper wires that are negatively charged whereas the one on the right is formed with straight copper tubes which are considerably thicker than the wires (as they are tubes).

Due to the design, it creates a section of uniform electric field in between the two electrodes (at least with my understanding but I am not very confident).

Your speculation of the speed of the ions reaching zero as soon as they exit may be true however whilst the ions are still in the field, they are continuously moving across it, and while on the way they collide with neutral particles which may be the wind that I am experiencing. And I am certain about the wind as you can also see the hanging foam ball being acted upon by a force that is pushing it to the right which is also the direction of the produced winds (I can also feel them when I put my hand in between the ball and thruster).

The weighted distribution of particles is where my approach is a bit unsure. This is because, after a lot of research, I was not sure what the charges of the ions would be so I said that as there is 78% Nitrogen and 22% Oxygen (I know its 21% however for the simplicity of the model I assumed it to be 22%) and then also assumed the nitrogen would have a charge of -3 and oxygen will have a charge of -2. With this, I then "found" the charge distribution by the following calculation:

Charge distribution = (78*3+22*2)/100

I then multiplied this with the elementary charge e = 1.602*10^-19 C to get 4.454*10^-19 C

I apologize for not including the electric field strength unit, it is V/m or Vm^-1 (where both mean the same thing).

Again, thank you very much for the reply and the pointers to make this a better question!

 

[Steve4Physics]

Anyways, I really don't have a diagram but I do have a subpar image of the contraption

View attachment 328835.

First impression: there is lot of bare metal in the apparatus – and exposure to 712V! So I hope you have some appropriate current-limiting device or there are serious safety issues.

I guess that the thin wires are producing some sort of field-emission/coronal discharge - but the voltage doesn’t sound high enough. Maybe some other members can shed light on this.

Due to the design, it creates a section of uniform electric field in between the two electrodes (at least with my understanding but I am not very confident).

You might approximate the field as uniform over much of the gap. But note that it is far from uniform near the thin wires. The thin wires will have a high local electric field (which presumably is what ionises air).

Your speculation of the speed of the ions reaching zero as soon as they exit may be true however whilst the ions are still in the field, they are continuously moving across it, and while on the way they collide with neutral particles which may be the wind that I am experiencing. And I am certain about the wind as you can also see the hanging foam ball being acted upon by a force that is pushing it to the right which is also the direction of the produced winds (I can also feel them when I put my hand in between the ball and thruster).

So you want the steady-state ‘drift’ speed of the ions in-between the two electrodes.

The weighted distribution of particles is where my approach is a bit unsure. This is because, after a lot of research, I was not sure what the charges of the ions would be so I said that as there is 78% Nitrogen and 22% Oxygen (I know its 21% however for the simplicity of the model I assumed it to be 22%) and then also assumed the nitrogen would have a charge of -3 and oxygen will have a charge of -2.

You seem to be assuming that each atom loses all of its valance electrons. But note that atmospheric oxygen and nitrogen are molecular. As a first approximation, I’d assume that, when ionisation occurs, each ion is a singly charged molecule.

Can you measure the current? If you know the current you might be able to use it in the calculation.

 

[Jon Doe]

Thank you for the reply!

I have looked into it and I agree, initially the voltage did seem too low for anything to happen however there s corona discharge at the thin wire which is visible in a dark room.

So you want the steady-state ‘drift’ speed of the ions in-between the two electrodes.

What I am looking for is the speed of particles (molecules or ions) right when they exit the thruster which I would assume is almost the same as the speed between them so what you are saying is right.

As a first approximation, I’d assume that, when ionisation occurs, each ion is a singly charged molecule.

Right, that makes sense however that further reduces the theoretical speed which seems a bit unrealistic seeing that the small hanging ball moves a considerable amount when doing the experiment with the same distance between the electrodes, voltage, etc.

Can you measure the current?

I do have the current which I measured to be 8.4 mA but I am not sure how that will help.

Thank you again!

 

[Steve4Physics]

I have looked into it and I agree, initially the voltage did seem too low for anything to happen however there s corona discharge at the thin wire which is visible in a dark room.

Surprising. I would have thought a few kV would be needed. But there you go!

What I am looking for is the speed of particles (molecules or ions) right when they exit the thruster which I would assume is almost the same as the speed between them so what you are saying is right.

Ok. So in effect, you're looking for the drift speed (terminal velocity) between the electrodes.

Right, that makes sense however that further reduces the theoretical speed which seems a bit unrealistic seeing that the small hanging ball moves a considerable amount when doing the experiment with the same distance between the electrodes, voltage, etc.

This is a complex system and is difficult to model accurately. I would have thought that coming up with an answer correct within a factor of 2 or 3 would be quite reasonable. Do you know what your teacher expects?

I do have the current which I measured to be 8.4 mA but I am not sure how that will help.

Sorry. A bit of a red herring. If you knew the charge-density (ions per unit volume) it would be useful. But you don't.

An alternative approach might be to use the mobility of air ions. The drift velocity ($v_d$) of an ion in an electric field ($E$) is given by $v_d = \mu E$ where $\mu$ is called the mobility. You can easily look up/read about mobility – e.g. search for ‘electrical mobility’.

You know $E$. If you have a figure for $\mu$ for atmospheric ‘air ions’, you can calculate $v_d$.

When I Googled ‘mobility of air ions’ I was told:
“The mobility of positive ions in dry air, which is constant in the range E/p = 45 - 70 V cm⁻¹ torr⁻¹ and is equal to 2.2 × 10³ cm² V⁻¹ sec⁻¹ (at 1 torr and at 20 °c), gradually decreases at higher E/p values.”

You could try $\mu = 2.2 × 10^3$ cm²V⁻¹s⁻¹ (sort the units) and see what answer it gives. If promising, try to find a better/referencable source for the value of $\mu$.

 

[Jon Doe]

Thank you for the reply!

μ=2.2×103 cm²V⁻¹s⁻¹

with this, I do get the best answer which is very close to the experimental value of the wind and I have consulted with my professor about it. He recommends we try to derive anything we can which is why I think it would be best for me to find a way to get the ion mobility value (though I am not sure if it is an experimental value or not). The reason for the requirement of the deviation is that we are writing a paper as practice and he recommends that doing so will benefit us.

I have tried searching how to calculate it but I have not found anything yet...

 

[Steve4Physics]

with this, I do get the best answer which is very close to the experimental value of the wind and I have consulted with my professor about it. He recommends we try to derive anything we can which is why I think it would be best for me to find a way to get the ion mobility value (though I am not sure if it is an experimental value or not). The reason for the requirement of the deviation ...

You mean 'derivation'!

I presume there are both experimental and theoretical ways to find mobilities. Unfortunately it's not an area I'm familiar with.

... we are writing a paper as practice and he recommends that doing so will benefit us.

I suspect the derivation is quite demanding. Out of interest, what level are you? High school? Uni', Post-grad'?

I have tried searching how to calculate it but I have not found anything yet...

I did a quick Google search using 'derive ion mobility'. Itgave few links., including this one:

This appears to be about a software package for calculating ion mobilities in gas. It gives some potentially useful background in the introductory text which could help you find further reading.

Sorry I can't help more.