Stephan,
We will get electron cooling in the center once fuel atoms are added to the mix. Because of the positive charge on the grid positive ions will tend to be forced to the center. The cool ones will tend to stay there. They will attract cool electrons forming a double layer well. The Japanese IEC guys did the diagnostics on this in 1999.
For sure experiments will need to be done to get a clearer picture of how it all works.
Virtual Polywell
All I can say is "interesting!" My formulas show cold electrons are better than hot. But I've been wrong before!
What I see is basic physics. F = ma = q(E + v X B). If the grid is static, the electrons see a constant field and v is always increasing. You want cold electrons in the center so you have a high potential well for the ions. If you have hot electrons, they leave the center rapidly and there's no more well.
But my trivially simple model only helps me understand the basic forces. The next step is to look at a fluid model. I'm going to ignore waves and just look at electrostatics to start with, and most likely assume frozen ions in the center just to get a clue of what the math does.
Assuming a Maxwellian velocity distribution really helps the model a lot. I only need to track fluid density as a function of position since the assumption of Maxwellian velocity allows the formulas to be integrated (and differentiated) easily.
I'll write up the plasma model when I get a chance - I've got a touch more to do. It's just a model, like Bussard's radial power laws - the assumptions may well be wrong. But it sure is fun to do!
What I see is basic physics. F = ma = q(E + v X B). If the grid is static, the electrons see a constant field and v is always increasing. You want cold electrons in the center so you have a high potential well for the ions. If you have hot electrons, they leave the center rapidly and there's no more well.
But my trivially simple model only helps me understand the basic forces. The next step is to look at a fluid model. I'm going to ignore waves and just look at electrostatics to start with, and most likely assume frozen ions in the center just to get a clue of what the math does.
Assuming a Maxwellian velocity distribution really helps the model a lot. I only need to track fluid density as a function of position since the assumption of Maxwellian velocity allows the formulas to be integrated (and differentiated) easily.
I'll write up the plasma model when I get a chance - I've got a touch more to do. It's just a model, like Bussard's radial power laws - the assumptions may well be wrong. But it sure is fun to do!
Ah ok, I was refering to electrons in the wiffle ball in general, not just to those in the center. Of course cold electrons in the center we do want.
However cooling other electrons isn't a good idea (unless maybe if it could be limited to the transversal component). Specifically cooling those in the center probably isn't too, since they aren't likely to stay in there for long.
However cooling other electrons isn't a good idea (unless maybe if it could be limited to the transversal component). Specifically cooling those in the center probably isn't too, since they aren't likely to stay in there for long.
Basic physics says that energy conversation is maintained. The potential+kinetic energy of the charged particle is constant in a static system. This is about as fundamental as you can get.drmike wrote:What I see is basic physics. F = ma = q(E + v X B). If the grid is static, the electrons see a constant field and v is always increasing.
- Indrek
The physics is interesting and the assumptions made determine what you get.
For example, in the Vlasov equation the particle distribution is assumed to be a function of position and velocity. One term in that equation is v X B (dot) df/dv and if you take a Maxwellian distribution for f, df/dv ~ v*f. That makes v X B (dot) v*f = 0 and the B field makes no changes to the particle distribution function. But we know that the particles follow the B field lines from basic dynamics!
The whole point of IEC is to use a non-Maxwellian ion distribution. For electrons, Langmuir's paradox says we will get a Maxwellian distribution if there is not a large current to force it non-Maxwellian. Large currents take lots of power, so we get better efficiency in a power plant if they can be avoided.
Until I can see some data that explains why one model would be better than another, it will be fun to simply try various assumptions and see what happens. And fun is really what this all about. Until it works. Then it will get exciting
For example, in the Vlasov equation the particle distribution is assumed to be a function of position and velocity. One term in that equation is v X B (dot) df/dv and if you take a Maxwellian distribution for f, df/dv ~ v*f. That makes v X B (dot) v*f = 0 and the B field makes no changes to the particle distribution function. But we know that the particles follow the B field lines from basic dynamics!
The whole point of IEC is to use a non-Maxwellian ion distribution. For electrons, Langmuir's paradox says we will get a Maxwellian distribution if there is not a large current to force it non-Maxwellian. Large currents take lots of power, so we get better efficiency in a power plant if they can be avoided.
Until I can see some data that explains why one model would be better than another, it will be fun to simply try various assumptions and see what happens. And fun is really what this all about. Until it works. Then it will get exciting
Howdy Indrek,
I agree. So let's take a capacitor and charge it up. Hook the cap up to the MaGrid and let the system go. The potential energy stored in the cap will go into the kinetic energy of the electron. The cap eventually drains all of it's potential into the higher energy of the electron. Then the system is static.
If the electron hits the wall because its velocity is too high, the cap will stop draining.
We are both saying the same thing really. Total energy is conserved. What I am saying is that with a static voltage on the MaGrid, electron kinetic energy will increase due to accelerating fields. If you put a capacitor bank on the MaGrid, the system isn't really "static", but for short time scales it can look that way to first approximation.
I agree. So let's take a capacitor and charge it up. Hook the cap up to the MaGrid and let the system go. The potential energy stored in the cap will go into the kinetic energy of the electron. The cap eventually drains all of it's potential into the higher energy of the electron. Then the system is static.
If the electron hits the wall because its velocity is too high, the cap will stop draining.
We are both saying the same thing really. Total energy is conserved. What I am saying is that with a static voltage on the MaGrid, electron kinetic energy will increase due to accelerating fields. If you put a capacitor bank on the MaGrid, the system isn't really "static", but for short time scales it can look that way to first approximation.
I have put up some notes on my web page that describe a purely analytical solution to a free particle in a polywell type field. It can be found on my web page here. The general solution is a huge mess, and it requires a lot of checking to ensure tracking which parts are real and which are imaginary. The purpose of putting up the notes is so my comments here are backed up with something other people can check. I won't say "easily check", because it's pretty messy.
drmike,
I would expect at start up that a lot of energy would drain from the system due to the starting conditions being Maxwellian.
Also if the wiffle ball doesn't form in your system I could see problems.
I'll have a look at your stuff and see if I can make heads or tails of it.
I would expect at start up that a lot of energy would drain from the system due to the starting conditions being Maxwellian.
Also if the wiffle ball doesn't form in your system I could see problems.
I'll have a look at your stuff and see if I can make heads or tails of it.
Engineering is the art of making what you want from what you can get at a profit.
I looked. Haven't done that kind of math in 40 years. I usually let my mathematician turn it into algebra and then I try to turn it into something that works. Division of labor.MSimon wrote:drmike,
I would expect at start up that a lot of energy would drain from the system due to the starting conditions being Maxwellian.
Also if the wiffle ball doesn't form in your system I could see problems.
I'll have a look at your stuff and see if I can make heads or tails of it.
In other words I didn't find any thing obviously wrong.
I agree that brute force is probably the better way to go. Closer to first principles with fewer assumptions. Harder work for the computer. But that is why they were invented.
==
Fixed typo
Last edited by MSimon on Thu Jan 17, 2008 3:06 pm, edited 2 times in total.
Engineering is the art of making what you want from what you can get at a profit.
True, but knowing the calculation is correct requires having at least one simulation on a known answer. If you don't have theory, you can't check the code.
A pure electron fluid should just explode, but it should be fun to watch! If I can work out that theory, adding in ions will be a cake walk. I'm also remembering stuff from 25 years ago, and it's coming back a lot faster than it went in the first time!
A pure electron fluid should just explode, but it should be fun to watch! If I can work out that theory, adding in ions will be a cake walk. I'm also remembering stuff from 25 years ago, and it's coming back a lot faster than it went in the first time!
So one step I took over the past couple of evenings is to look at the E field from the point of view of potential. Since we don't have the MaGrid B field changing in time for steady state operation (yet) I can work with electrostatic potential rather than E field directly. For plasma particle distribution this is important since the probability of finding a particle at any given energy depends on it having enough energy to be there (with classical physics anyway).
There are several ways to approach the problem, but all involve integrating over all the charges in the system. The force on the plasma depends on the E field, but the probability distribution depends on the potential. The E field can either be computed as the differential of the potential, or it can be computed from integrating over all the charges in the system (again, but with a 1/r^2 times the same integral of potential.
From a purely computational problem, what is better: Integrations over the whole volume of interest for both potential and E field, or integration over potential and then computing differences on that to get E field?
My gut feeling is that integration is more noise immune, but it probably takes a lot more time. This is one place where the argument over precision of the compute engine matters - it may well be worth while to have slower (or more power hungry) computers with more precision so the need to do double integrals is eliminated.
but then again, you might gain it back with higher speed lower precision since a few tricks might allow the double integral to be simply computed simultaneously since the integrands are so similar.
I'm gonna grind on theory, I'd love to know what you guys (any gals out there???) think about how to grind it out.
There are several ways to approach the problem, but all involve integrating over all the charges in the system. The force on the plasma depends on the E field, but the probability distribution depends on the potential. The E field can either be computed as the differential of the potential, or it can be computed from integrating over all the charges in the system (again, but with a 1/r^2 times the same integral of potential.
From a purely computational problem, what is better: Integrations over the whole volume of interest for both potential and E field, or integration over potential and then computing differences on that to get E field?
My gut feeling is that integration is more noise immune, but it probably takes a lot more time. This is one place where the argument over precision of the compute engine matters - it may well be worth while to have slower (or more power hungry) computers with more precision so the need to do double integrals is eliminated.
but then again, you might gain it back with higher speed lower precision since a few tricks might allow the double integral to be simply computed simultaneously since the integrands are so similar.
I'm gonna grind on theory, I'd love to know what you guys (any gals out there???) think about how to grind it out.
Assuming you want to use a method running in O(n log n) or better computing the differences of the potential probably is the easiest way.
Integrating over the charges to calculate the potential is the same as solving the poisson equation over the system. There are good algorithms to do that. Also you wouldn't have to worry about the charges on the coils and shell, but could simply set the potential on them as boundary conditions.
If you want a more precise E-field you might want to use the fast multipole method for both the potential and the E-field. However this requires you to know the charges on the coils and the shell.
Implementing the FMM seems to be more difficult to me than just a poisson solver, but I guess there should be code for both available somewhere.
It's difficult to tell what is faster, the computing time in both methods also depends on the precision required.
Integrating over the charges to calculate the potential is the same as solving the poisson equation over the system. There are good algorithms to do that. Also you wouldn't have to worry about the charges on the coils and shell, but could simply set the potential on them as boundary conditions.
If you want a more precise E-field you might want to use the fast multipole method for both the potential and the E-field. However this requires you to know the charges on the coils and the shell.
Implementing the FMM seems to be more difficult to me than just a poisson solver, but I guess there should be code for both available somewhere.
It's difficult to tell what is faster, the computing time in both methods also depends on the precision required.
I was looking at pricing for the NVIDIA Tesla online... about $2k/copy for 1 TFLOP, if you like your FLOPs in single-precision. The link you posted to the 64-bit thread about using multiple 32-bit floats to synthesize double precision seems like a good place to start until they get to full 64-bit precision (and hopefully 80-bit past that). I'll spend some time reading up on this stuff -- way beyond anything I ever did as an undergraduate.drmike wrote:I'm gonna grind on theory, I'd love to know what you guys (any gals out there???) think about how to grind it out.
All those nice number crunchers are starting to sound very, very useful. I'm trying to come up with a really simple analytical formula for an electron distribution inside the polywell. I'm just looking for some particular distribution that kind of has the 1/r^2 distribution, but is flat in the center - and a nice formula is
1
-------------------------
exp(-a*r^2) + b*r^2
as r -> 0, the exponential takes over and you get 1. As r -> infinity, you get 1/r^2 (and the 1/r^2 comes from one of Bussard's papers).
The plot of this is very nice. If a > b it gives a double hump (double in the sense that +/- r each have a hump away from r = 0), and that looks like a well. Compare this to the plots of electron distributions in the Farnsworth fusor, it's very interestingly close.
I then started looking at what the potential will be for this distribution. To make a long story shorter - I have to find a potential which equals the integral of itself over all space, for each point in space! This is a form of self consistent integral equation that I haven't worked on in a couple of decades. It will take a few days to get back up to speed
But darn this is fun!!
1
-------------------------
exp(-a*r^2) + b*r^2
as r -> 0, the exponential takes over and you get 1. As r -> infinity, you get 1/r^2 (and the 1/r^2 comes from one of Bussard's papers).
The plot of this is very nice. If a > b it gives a double hump (double in the sense that +/- r each have a hump away from r = 0), and that looks like a well. Compare this to the plots of electron distributions in the Farnsworth fusor, it's very interestingly close.
I then started looking at what the potential will be for this distribution. To make a long story shorter - I have to find a potential which equals the integral of itself over all space, for each point in space! This is a form of self consistent integral equation that I haven't worked on in a couple of decades. It will take a few days to get back up to speed
But darn this is fun!!
I've been having fun with math and plotting, but it's not really accomplishing much. I kept getting 0.0 (or 1e-16 which is close enough to 0.0) for potentials. This morning I realized why and came up with a nice theoretical expansion. Unfortunately the expansion is far more complicated than simply computing things using brute force!!!
The advantage of a theoretical expansion is that I can trade off cpu time for memory storage and get more accuracy since there is no interpolation. But when the trade off adds more time to a calculation it's the wrong way to go.
One of these days I should plot the messy functions - I bet they look really pretty. For now, it's back to brute force.
The advantage of a theoretical expansion is that I can trade off cpu time for memory storage and get more accuracy since there is no interpolation. But when the trade off adds more time to a calculation it's the wrong way to go.
One of these days I should plot the messy functions - I bet they look really pretty. For now, it's back to brute force.