Electron thermalization time versus confinement time.

Discuss how polywell fusion works; share theoretical questions and answers.

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D Tibbets
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Electron thermalization time versus confinement time.

Post by D Tibbets »

This topic is presented to hopefully consolidate various arguments into one thread.
A. Carlson has recently commented on the EMC2 target of matching the electron confinement time to the thermalization time. I would rephrase it to suggest that they recognize there is no advantage to have electron confinement times longer that the thermalization times within the Polywell system.

What is the thermalization time? I posted in the past a study that gave thermalization time of ~25 ms for a small low voltage mirror machine. A. Carlson shot that down as it was evidentially not applicable.

So some questions:

1) what is a reasonable electron thermalization time in a WB6 type Polywell? For that matter what degree of thermalization will the machine tolerate- 20%, 90% ?
If there was a pulse of radial mono energetic electrons injected once, how long would it take for them to thermalize. How would the continuous injection of new mono energetic electrons to replace the more thermalized escaping electrons modify the process? Where would these competing processes reach an energy loss limit (for the system to work)? ie- How much electron current is needed to replace the thermalizing and escaping electrons such that some needed thermalization limit is not exceeded? Is that compatible with breakeven?

2) What knobs can be used to modify this competition? Some possibilities- microwave resonate heating of selected cold electrons, increased loss rates for hot electrons. What are the energy penalties incurred ?

3) Which should be considered as the balance point? Is it the Wiffleball confinement time versus the thermalization time, or is it the Wiffleball confinement time x the recirculation factor?

In the WB6 I estimate that the Wiffleball trapping for the electrons resulted in a confinement time of ~ 0.5 ms or less*. How does that compare with the expected thermalization time under those conditions?. With recirculation added, the confinement time effectively increases to 5.0 ms. But, I believe these 9 out of 10 electrons that are recirculated match the mono energetic and near radial directions of new injected electrons (without the energy cost). Ideally those 1 out of 10 electrons that escape recirculation are upscattered electrons (electrons spreading into the hot side of the thermalization graph) so there is some advantageous bias in the resulting thermalization curve.
Also, consider that the upscattered electrons that do escape give back a portion of their energy to the magrid, so there is less of an energy penalty for removing them from the system .

I'm guessing that the recirculation increases the effective confinement time while maintaining (or even improving) the limiting thermalization time.
This would represent a 10 fold gain or more in the energy balance (input energy needed to maintain a desired condition).

There are a lot of complex tradeoffs and dynamic considerations that need to be accounted for to derive a final answer.

To confuse the issue even more, with increasing size and magnetic field strength I'm assumeing the confinement times will increase. If there is indead a favorable balance in the small machines, will scaling to larger sizes and strengths crowed the limits ? Or do the +/- effects scale equally?


* Electron confinement time derived from some numbers and assumptions. Bussard said the electrons traveled ~ 1 billion cm per second in WB6. I don't know if this was the injection speed or the average speed. I think it is the injection speed (10,000 eV) so I took an average speed of 500 million cm/s. I assume the Wiffleball diameter is ~ 20 cm (or less?), so there would be ~ 25 million passes per second (assuming the electrons retain most of their radial directions, some limited thermalization would blur this number some). WB trapping was reported as ~ 10,000 passes. 10,000 passes divided by 25 million passes per second = 0.5 ms

Dan Tibbets
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Art Carlson
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Post by Art Carlson »

If I may quote myself,
Art Carlson wrote:
D Tibbets wrote:
Art Carlson wrote:...To maintain a non-maxwellian distribution will require electrons to be removed at one energy and replaced at another energy. At least one of those populations of electrons will have to have an energy near E_e... [/b]
Isn't that percisely what occurs in the Polywell? The electrons are acellerated by the pos magrid to ~ 12,000 eV, to provide an internal well depth of ~ 10,000eV (WB6 example). They will bounce around inside, tending to thermalize with time, but the lifetime is less than the thermalization time (relaxation time?). Befor they can thermalize fully they escape the system through a cusp, and are replaced by new mono-energetic electrons (12,000 eV).
Right. This is the way Nebel looks at it and it is definitely one way - perhaps the easiest way - to maintain a non-maxwellian distribution.

The trouble is that pouring electrons in and letting them run out costs power. I showed above that the simplest model, where the electrons take most of their hard-won energy with them when they leave, can only work at energies above 3 MeV. You don't want to go there, for reason we could discuss. A better way to look at the numbers might be to fix E_e at 10 to 20 keV and look at the times involved. If we assume that when an electron passes through the system, it loses a fraction f of the energy E_e, then we can show that f has to be less than about 1/2400 for practical D-T fusion. In other words, your set-up has to recover 99.96% of the energy your accelerator gives to your electrons. (We've only talked about physics so far, but you have a technological problem too: Your direct conversion of fusion power to electrical power needs to be 99.96% efficient, too.)

If the number itself doesn't scare you, we could go into details about why you can't do it. If you inject electrons at an energy E_e and siphon them off at E_e*(1+f), then you have a steep gradient in the distribution in velocity space. This will make the time scale for relaxation of the distribution even shorter than the value I used.

Your own calculation pretty much misses the point on two counts. For one, it doesn't address the question of the value needed for the Lawson product.

The other is that the paper you site actually confirms my calculation. They measure a significant change in the distribution on a time scale not much shorter than the electron collision time they calculate using the same formula I use. The reason they deal with milliseconds is that they have a wimpy plasma, only 3e9 cm^-3.
So I guess I already did this calculation once.

What conclusion should we draw from this calculation? The weakest conclusion is that the polywell can't work ( = would be extremely difficult) the way Bussard and Nebel desribe it. Could we possibly live with the thermalized electrons? One of the troubles with that (another would be shielding of potentials) is the loss of upscattered electrons. I think you might be able to reduce that to an acceptable level if you run (somehow) with a magrid potential several times deeper than the electron temperature. This is analogous to the requirement, which will still hold, that the central well depth has to be several times deeper than the required ion temperature Since these requirements are multiplicative, you will have to have some heavy duty voltages (a few hundred kV), even for D-T, which will fold in to the requirement for low losses to other channel besides upscattering.

"Other channels" would bring us back to the cusps. Under the conditions I just described, the electrons leaving through the cusps should (indeed) be electrostatically trapped (although I'm not sure that an increasing population of electrons outside the magrid won't cause you - possibly fatal - headaches). The electron beams filling the cusps will give the ions the opportunity to tunnel through the electrostatic barrier, leaving their raised bowl to roll down the steep slope into the wall. Just like I've been saying all along. I still see ion loss through the cusps as the unfixable problem.

alexjrgreen
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Post by alexjrgreen »

The cusps are spherically symmetrical.

Doesn't that mean that the ions see a point negative charge in the centre, as indicated by Gauss's Law?
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Art Carlson
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Post by Art Carlson »

alexjrgreen wrote:The cusps are spherically symmetrical.

Doesn't that mean that the ions see a point negative charge in the centre, as indicated by Gauss's Law?
The cusps are not spherically symmetrical. They have a finite rotational symmetry.

alexjrgreen
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Post by alexjrgreen »

Art Carlson wrote:
alexjrgreen wrote:The cusps are spherically symmetrical.

Doesn't that mean that the ions see a point negative charge in the centre, as indicated by Gauss's Law?
The cusps are not spherically symmetrical. They have a finite rotational symmetry.
Correction accepted.

So lets break this down: For an ion in one of the cusps, the sphere of the wiffleball represents a point negative charge by Gauss's Law, the cylinders of the four sideways facing central cusp jets cancel out sideways, adding to the apparent central charge. The cusp jet ahead of the ion contributes much more than the cusp jet opposite, so the central charge is balanced by the cusp jet.

Since there are many more electrons in the wiffleball than in the cusp jet, and a net negative charge overall, the ion is attracted less strongly than before to a point between the cusp and the centre of the wiffleball, but much closer to the centre.

Just as chrismb said.
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D Tibbets
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Post by D Tibbets »

Art Carlson wrote:...To maintain a non-maxwellian distribution will require electrons to be removed at one energy and replaced at another energy. At least one of those populations of electrons will have to have an energy near E_e...
This is the way Nebel looks at it and it is definitely one way - perhaps the easiest way - to maintain a non-maxwellian distribution.
...

The trouble is that pouring electrons in and letting them run out costs power. I showed above that the simplest model, where the electrons take most of their hard-won energy with them when they leave, can only work at energies above 3 MeV. You don't want to go there, for reason we could discuss. A better way to look at the numbers might be to fix E_e at 10 to 20 keV and look at the times involved. If we assume that when an electron passes through the system, it loses a fraction f of the energy E_e, then we can show that f has to be less than about 1/2400 for practical D-T fusion. In other words, your set-up has to recover 99.96% of the energy your accelerator gives to your electrons. (We've only talked about physics so far, but you have a technological problem too: Your direct conversion of fusion power to electrical power needs to be 99.96% efficient, too.)

If the number itself doesn't scare you, we could go into details about why you can't do it. If you inject electrons at an energy E_e and siphon them off at E_e*(1+f), then you have a steep gradient in the distribution in velocity space. This will make the time scale for relaxation of the distribution even shorter than the value I used.

Your own calculation pretty much misses the point on two counts. For one, it doesn't address the question of the value needed for the Lawson product.


I have several problems
The other is that the paper you site actually confirms my calculation. They measure a significant change in the distribution on a time scale not much shorter than the electron collision time they calculate using the same formula I use. The reason they deal with milliseconds is that they have a wimpy plasma, only 3e9 cm^-3
...So I guess I already did this calculation once.

What conclusion should we draw from this calculation? The weakest conclusion is that the polywell can't work ( = would be extremely difficult) the way Bussard and Nebel desribe it. Could we possibly live with the thermalized electrons? One of the troubles with that (another would be shielding of potentials) is the loss of upscattered electrons. I think you might be able to reduce that to an acceptable level if you run (somehow) with a magrid potential several times deeper than the electron temperature. This is analogous to the requirement, which will still hold, that the central well depth has to be several times deeper than the required ion temperature Since these requirements are multiplicative, you will have to have some heavy duty voltages (a few hundred kV), even for D-T, which will fold in to the requirement for low losses to other channel besides upscattering.
...
"Other channels" would bring us back to the cusps. Under the conditions I just described, the electrons leaving through the cusps should (indeed) be electrostatically trapped (although I'm not sure that an increasing population of electrons outside the magrid won't cause you - possibly fatal - headaches). The electron beams filling the cusps will give the ions the opportunity to tunnel through the electrostatic barrier, leaving their raised bowl to roll down the steep slope into the wall. Just like I've been saying all along. I still see ion loss through the cusps as the unfixable problem.
I have several problems with your numbers, hoprfully not entirely due to my ignorance or stupidity.

I don't know where your 3 MeV potential well comes from. Even Rider only needed 900 KeV for his P-B11 calculation. I'll guess it may come from complexity in using Maxwellian distributions for some part of the potential well depth (eg- using the energetic tail of the Maxwellian distribution as the target injection energy to maintain the average temperature of the vastly larger population of particles near the average temperature, or concersly to drag them up to the tail energy). Short of such convoluted reasoning on my part I cannot see why a potential well could not be maintained with an injection energy near the average energy of the potential well- which will presumable have smaller excursion from the average energy due to the more 'monoenergtic ' nature of the beast. I could see the electron current going way up to maintain the desired conditions, but I don't see why the voltage would need to increase.
How much does the shape of the potential well effect your conclusions.

Your discription of the electron loss ratios make sense, and I'll even accept your needed 99.96% percent energy efficiency. But, I'm not sure this is such a daunting number.
Some perhaps reasonable numbers, using the WB6 example again.
Cost of injecting a new electron = 12 KeV
Cost of recirculating 9 out of 10 escaping electrons = 0 KeV
Cost of losing an upscattered electron = 1 to a few thousand eV.

The recirculation automatically gives you a 10X advantage, and without any penalty in the thermalization time.

Your empasis on the cost of replacing the upscattered electrons may be overemphasized. The accelerating potential was 12.000 volts, and the potential well was 10,000 volts. Any electron above 10,000 eV could be concidered upscattered but it has to reach 12,001 eV before it escapes zero cost recirculation. So the question becomes what average upscattered eV energies do the escaping electrons have. If most of them are only a few hundred eV then the energy loss per electron is small. It would depend on how fast the average upscattered electron could gain energy before it hit a cusp. Remember that (I claim) upscattered electrons are travelling faster so they leave the cusps at a faster rate. I'm guessing this will moderate how much excess energy they can accumulate.
Assume a 10X advantage from recirculation, and an average excess energy of 100eV (12,100 eV upscattered electron - 12,000 V potential on the magrid). This would give you an efficieny of 99.9%

This may be consistant with Dr Nebel's concern about losses to the interconnects/ nubs between the magnets. It may not be much, but significant because of the high efficiency needed.

For arguments sake abandon your contention that cusp flows are ambipolar. How does the numbers change if the ions escape at a rate 1/10th of the electrons, 1/1000th of the electrons? Thermalization of the ions over these longer containment times is not allowed due to effective annealing (acepted for arguments sake).

The need for an excessive well depth required to contain most of the upscattered ions is not applicable. Bussard stated that upscattered ion loses are not a significant energy loss mechanism because they are near the top of the potential well, and because they are far fewer of them escaping per unit of time compared to the electrons(?). His contention was that ion losses are primarily a concern in maintaining the required density inside to produce usefull fusion, while preventing accumulation outside (along with electrons?) that could lead to arcing.

Dan Tibbets
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Art Carlson
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Post by Art Carlson »

D Tibbets wrote:I don't know where your 3 MeV potential well comes from.
From this calculation. The idea is that the thermalization rate drops rapidly with increasing temperature, but not rapidly enough to allow D-T fusion with non-Maxwellian electrons.
D Tibbets wrote:How much does the shape of the potential well effect your conclusions.
The shape of the potential well does not enter in to these arguments.
D Tibbets wrote:Your discription of the electron loss ratios make sense, and I'll even accept your needed 99.96% percent energy efficiency. But, I'm not sure this is such a daunting number.
Some perhaps reasonable numbers, using the WB6 example again.
Cost of injecting a new electron = 12 KeV
Cost of recirculating 9 out of 10 escaping electrons = 0 KeV
Cost of losing an upscattered electron = 1 to a few thousand eV.

The recirculation automatically gives you a 10X advantage, and without any penalty in the thermalization time.
A recirculating electron will come back in with the same energy it left with. The thermalization goes on unaffected by recirculation. When it is finally lost over the top of the well, some of the energy can be recovered. Exactly how much depends on some details of the distribution. My picture is that the electrons mostly bounce around inside the polywell until they first thermalize and then escape through a cusp. In that case, the spread in energies will be roughly equal to the temperature, and this is the energy that cannot be recovered.
D Tibbets wrote:Your empasis on the cost of replacing the upscattered electrons may be overemphasized. The accelerating potential was 12.000 volts, and the potential well was 10,000 volts. Any electron above 10,000 eV could be concidered upscattered but it has to reach 12,001 eV before it escapes zero cost recirculation. So the question becomes what average upscattered eV energies do the escaping electrons have. If most of them are only a few hundred eV then the energy loss per electron is small. It would depend on how fast the average upscattered electron could gain energy before it hit a cusp. Remember that (I claim) upscattered electrons are travelling faster so they leave the cusps at a faster rate. I'm guessing this will moderate how much excess energy they can accumulate.
I know it sounds a bit strange that the escaping electrons take ~kT_e of energy with them when going over the top. After all, they have to lose a lot of energy to climb up the side of the well. But those without enough energy never make an appearance on top of the wall, and those that do show up there are a mixture of some with just barely enough energy and other with more than enough.
D Tibbets wrote:For arguments sake abandon your contention that cusp flows are ambipolar. How does the numbers change if the ions escape at a rate 1/10th of the electrons, 1/1000th of the electrons? Thermalization of the ions over these longer containment times is not allowed due to effective annealing (acepted for arguments sake).
I am not contending that the cusp flows are necessarily ambipolar, only that the cusp plasma is quasi-neutral. Like with a Langmuir probe, the electron flow can be adjusted by choosing the voltage. In contrast, it is hard to make the ion flux anything other than n*c_s.
D Tibbets wrote:The need for an excessive well depth required to contain most of the upscattered ions is not applicable. Bussard stated that upscattered ion loses are not a significant energy loss mechanism because they are near the top of the potential well, and because they are far fewer of them escaping per unit of time compared to the electrons(?). His contention was that ion losses are primarily a concern in maintaining the required density inside to produce usefull fusion, while preventing accumulation outside (along with electrons?) that could lead to arcing.
Bussard's argument only apply to ions leaving "over the top". I am arguing that the interaction of the electrons and ions bores a hole through the potential otherwise confining the ions, all the way from the plasma ball to the wall.

alexjrgreen
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Post by alexjrgreen »

Art Carlson wrote:I am arguing that the interaction of the electrons and ions bores a hole through the potential otherwise confining the ions, all the way from the plasma ball to the wall.
How about showing us a picture? So far you haven't demonstrated that the ions feel anything other than a variable attraction to the central sphere.
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Art Carlson
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Post by Art Carlson »

alexjrgreen wrote:
Art Carlson wrote:I am arguing that the interaction of the electrons and ions bores a hole through the potential otherwise confining the ions, all the way from the plasma ball to the wall.
How about showing us a picture? So far you haven't demonstrated that the ions feel anything other than a variable attraction to the central sphere.
A picture is an excellent idea. 2 problems, though: (1) I need to find a bit of time. I won't do it right away, but I should be able to do a sketch in the next couple days. (2) I need to find a place to post the picture, so that I can link it here. I'll bet somebody here would volunteer to do that for me, wouldn't you?

I'm not sure what you think I have or haven't demonstrated, or even what you consider to be a demonstration. What I think I have shown (or at least can show) by physical arguments, some of them quantitative, is this:
  • The cusp fan/jet, even if it is only rho_e thick, must be quasi-neutral at magnetic fields, particle densities, and electron energies relevant for fusion power.
  • Following the standard derivation, if the potential of the wall is more negative than the space potential near the throat of the cusp, there will be
    • a "pre-sheath" potential drop in the plasma ball accelerating the ions to a speed near c_s = sqrt(kT_e/m_i) and reducing the density to about half the maximum value, and
    • a sheath potential drop in the last few Debye lengths in front of the wall.
  • Depending on the potentials and driving fluxes, the flux of electrons to the wall could be much less than, comparable to, or much greater than the flux of ions to the wall.
Is there one of these points that you think has not yet been sufficiently discussed?

Art Carlson
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Post by Art Carlson »

Art Carlson wrote:Following the standard derivation, if the potential of the wall is more negative than the space potential near the throat of the cusp, there will be
  • a "pre-sheath" potential drop in the plasma ball accelerating the ions to a speed near c_s = sqrt(kT_e/m_i) and reducing the density to about half the maximum value, and
  • a sheath potential drop in the last few Debye lengths in front of the wall.
Let me ponder this a while. The requirement that the ion velocity be >= c_s holds only at the wall, not necessarily at the cusp throat. If the magnetic field expands significantly from the cusp throat to the wall, there might be acceleration there, which could lead to a much smaller flux. Or else not. I'll report back when I think I know the answer.

alexjrgreen
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Post by alexjrgreen »

Art Carlson wrote:if the potential of the wall is more negative than the space potential near the throat of the cusp
I certainly wasn't expecting this to be true. Why do you suppose it?
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TallDave
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Post by TallDave »

Art wrote:I know it sounds a bit strange that the escaping electrons take ~kT_e of energy with them when going over the top. After all, they have to lose a lot of energy to climb up the side of the well. But those without enough energy never make an appearance on top of the wall, and those that do show up there are a mixture of some with just barely enough energy and other with more than enough.
You've confused the top of the well with the bottom. For electrons, the Magrid is the bottom and the center of the plasma is the top.

The electrons that leave are leaving at a point near the bottom. They will tend to be less energetic because the less energetic electrons get stuck at the bottom.
I am arguing that the interaction of the electrons and ions bores a hole through the potential otherwise confining the ions, all the way from the plasma ball to the wall.
This argument is not at all persuasive. The densities will fall off dramatically through the cusp, and it's hard for any ions not already in the cusp to get to the negative potential at the end of the cusp anyway, even assuming the potential is within the Debye sphere of any significant number of ions.
Last edited by TallDave on Mon Dec 14, 2009 3:12 pm, edited 3 times in total.

Art Carlson
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Post by Art Carlson »

alexjrgreen wrote:
Art Carlson wrote:if the potential of the wall is more negative than the space potential near the throat of the cusp
I certainly wasn't expecting this to be true. Why do you suppose it?
Because everybody says so?

It's the basis of so many statements, both true and false, like electrons get turned around and recirculate when they leave the cusp, like electrons are inserted into the system with low energy near the wall and accelerated toward the magrid, like the enery of electrons leaking out of the system can be recovered because their kinetic energy is turned into electric energy as they move toward the wall.

alexjrgreen
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Post by alexjrgreen »

Art Carlson wrote:
alexjrgreen wrote:
Art Carlson wrote:if the potential of the wall is more negative than the space potential near the throat of the cusp
I certainly wasn't expecting this to be true. Why do you suppose it?
Because everybody says so?

It's the basis of so many statements, both true and false,
I was expecting the wall to be at ground, the magrid to be at a positive potential V and the well depth in the wiffleball to be 0.8V. The space potential near the throat of the cusp I was expecting to be slightly negative.
Art Carlson wrote:like electrons get turned around and recirculate when they leave the cusp,
They're decelerated and then re-accelerated by the magrid, which is more positive than the wall.
Art Carlson wrote:like electrons are inserted into the system with low energy near the wall and accelerated toward the magrid,
They're accelerated by the magrid, which is more positive than the wall.
Art Carlson wrote:like the enery of electrons leaking out of the system can be recovered because their kinetic energy is turned into electric energy as they move toward the wall.
They're decelerated by the magrid, which is more positive than the wall.
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Art Carlson
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Post by Art Carlson »

???

I said
Art Carlson wrote:if the potential of the wall is more negative than the space potential near the throat of the cusp
You say
alexjrgreen wrote: I was expecting the wall to be at ground, the magrid to be at a positive potential
and (three times)
alexjrgreen wrote: the magrid, which is more positive than the wall.
where is the contradiction?

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