JxB Hartmann Flow experiment
Cremona, September 2020.
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This page describes a simple experiment of a so-called Hartmann (plasma) flow.
By creating a DC electrical discharge in one direction (between two long electrodes, at high voltage) and placing some
magnets as to create a magnetic field that is perpendicular to the direction of the discharge current, one gets a bulk force that
accelerates the plasma out of the channel.
Therefore, this configuration could actually be seen as the simplest plasma thruster ever, and is pretty related to
MPD (MagnetoPlasmaDynamic) thruster configurations.
Our configuration could be seen as a simple "applied-field MPD thruster".
Here's a schematic of the configuration and my simple experimental setup.
"B" is the magnetic field, and in my setup, I am using some tiny neodymium magnets, placed on the two sides of the channel.
The magnetic field should be somewhere around 100 - 1000 Gauss inside the channel.
For comparison, the magnetic field at the Earth's surface is somewhere around 0.5 Gauss.
In the figure, "J" is the current density, and flows from the positive electrode on the top, to the negative one at the
bottom (actually, inverted in my setup).
This crossed current and magnetic field configuration creates a volume force on the plasma, that can be computed as
F = J x B
where bold characters stand for vectors and "x" is the cross product.
If you use the "right hand rule", you will see that the force is directed as to push the plasma out of the channel.
You can see that my setup is pretty rough.
It's basically made of plexiglass and held together with hot glue.
In the theoretical Hartmann flow, the magnetic field is supposed to be uniform across the channel, and this allows to obtain nice analytical results.
My small magnets for sure fail to meet this condition, but the concept works nonetheless.
Indeed, if you switch on the plasma, you can see pretty clearly that it is being pushed in one direction!
In the previous figure, I have plugged to the electrodes a flyback high voltage generator (connected to a ZVS driver), that generates somewhere between 10 and 50 kV.
The total current through the plasma is around 10 mA.
Note that the results are strongly dependent on the background gas pressure.
In this plot, the pressure was around 0.5 Torr (65 Pascal roughly).
At higher pressures, you don't have such a beautiful one-sided expansion.
Check out the next image.
At "high" pressures (say, above 10 Torr) the collisions of electrons with the background gas are pretty frequent, and one can compute that
the mean free path is smaller than some 50 micrometers.
Electrons (and ions) in such conditions are scattered very frequently by neutrals, and they cannot easily accelerate by effect of the JxB force.
This situation changes at lower pressures, and starting somewhere around 2 to 4 Torr you start noticing the acceleration clearly.
In the next image you can see a couple random pictures taken during the pump down.
Note that colors are not very indicative, as the camera did some authomatic color balancing.
Some back-of-the-envelope calculations
It is quite interesting to perform a couple of calculations for estimating the thrust force that such an improvised thruster would generate.
As mentioned previously, the force per unit volume exerted on the plasma is equal to:
F = J x B
where all bold quantities are vectors and by "x" we denote the cross product.
Since we can assume the current density J and the magnetic field to be kind of perpendicular (see diagram above), the cross
product transforms in the simple relation:
F = J*B
This may be not exact, but this is kind of reasonable in this configuration, and this is OK for getting an estimation of the quantities involved.
So, bear with me.
Now, this is the force per unit volume that the fields (and therefore the magnets and the whole support structure in the end)
apply to the plasma.
This force eventually accelerates it outwards.
To be precise, this force depends on the position, since the current density and the magnetic field are local quantities (the current could be more intense in some part of the domain and
dimmer in others, and the magnetic field is not really uniform).
Therefore, the total force on the plasma bulk is obtained by summing up, point by point, all forces inside the plasma chamber.
In other words, one can get the total force by integrating the JxB force all over the volume.
But since we are only trying to obtain some estimation of the result, we can just approximate this integral and compute the simple product
of the chamber volume "V" times the JxB force.
Let us call T the total force (also equal to the thrust that this configuration generates).
The total force reads:
T = J*B*V
Let us plug in the numbers.
In my case the length of the chamber is around L = 7.5 cm = 0.075 meters, the height is around h = 1 cm = 0.01 m and the width is also around w = 1 cm.
This gives a volume V = 7.5e-6 m^3.
The side area of an electrode plate is A = L*w = 7.5e-4 m^2, and the
current density is therefore J = I/A = 26.6 A/m^2.
And finally, we can compute the total force (equal to the thrust):
T = J*B*V = 0.00001 N
which is equivalent to 1 milligram of weight.
Ridiculously low, right?
Well, yes it's pretty low by itself, we should really increase the current (and maybe the magnetic field) if we want some reasonable thrust...
but let me take this chance to speak a bit about electric propulsion.
About the thrust force in electric propulsion devices
So, you may have heard that plasma (electric) thrusters are the future of space exploration, right?
Well, you should keep in mind that the whole concept of electric space propulsion trades thrust for efficiency.
Electric thrusters almost always have a ridiculously low thrust, but do have a very high efficiency, meaning that they consume very little fuel.
The way this is achieved is by using a lower mass flow rate, but have a much higher exhaust velocity if compared to classical (chemical) rocket engines.
Anyway, the low thrust may appear a strong disadvantage at a first glance, but... not quite.
If you think about it, space travel is a pretty slow activity.
Once you get out of the Earth (or whatever planetary space), it takes quite a bit of time to reach the final destination, at least with present-day technology.
To get to the Moon it takes days.
Reaching Mars takes months.
So, keeping this in mind, a low-thrust (but high-efficiency) propulsion system is not really a disadvantage,
since the cumulative effect of keeping an electric thruster active for some days
easily outperforms the strong boost of a chemical rocket (which on the other hand would only last for a few seconds).
For further readings, let me suggest the book: Physics of Electric Propulsion by Jahn.
Acknowledgements
Thanks a lot to Elena Capelli for helping out building the setup and running the facility!
Where to go now?
Check out the VACUUM DIARIES page
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...or you can GO BACK TO THE HOME PAGE,
or to the EXPERIMENTATION PAGE.
Cheers,
Stefano