Maximum size allowed by energy flux constraints

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

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TallDave
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Post by TallDave »

As always very interesting, thanks for sharing Rick. We (or I anyway) mostly stumble around in the dark here, and appreciate you flipping on the lights for a few minutes now and then.

Hope WB-7.1 is going well.

MSimon
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Post by MSimon »

KitemanSA wrote:So does any "reader" know how to determine this and would you be so kind as to tell me?

As to my "why bother with the electrons" question above, it finally filtered into my memory that folks have often said it takes 10,000 passes or more to fuse. And while 1000 is much better than the 100 or so from a typical fusor, it still isn't anywhere near 10,000. Oh well.

PS: Welcome back Dr. N!!
Don't confuse ions with alphas. The alphas are at higher energy (with a distribution). The ions should be roughly mono-energetic. Alphas not so much.
Engineering is the art of making what you want from what you can get at a profit.

Art Carlson
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Post by Art Carlson »

MSimon wrote:
KitemanSA wrote:As to my "why bother with the electrons" question above, it finally filtered into my memory that folks have often said it takes 10,000 passes or more to fuse. And while 1000 is much better than the 100 or so from a typical fusor, it still isn't anywhere near 10,000. Oh well.
Don't confuse ions with alphas. The alphas are at higher energy (with a distribution). The ions should be roughly mono-energetic. Alphas not so much.
The alphas have such a high energy that they shouldn't be affected much by the electric field between the plasma and the magrid, and their density is presumably too small to make much of an electric field on their own. Therefore their losses should be determined by single-particle orbits in the cusp magnetic field. The size of the hole will be determined by the thickness of the alpha sheath, i.e. an alpha gyro-radius, times the linear dimension of the device R (and some moderate numerical factor). That calculation is going to wind up closer to 10 passes than 1000 passes. Rick, can you tell us where your 1000 came from?

Art Carlson
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Re: time to shut up

Post by Art Carlson »

Art Carlson wrote:We are still struggling a bit with terminology. May I suggest:
R = the radius of the high beta, nearly spherical plasma ball
S = the length of the side of the cube, on whose faces the magrid coils lie
R_coil = the major radius of the toroidal magrid coils
r_coil = the minor radius of the toroidal magrid coils, including conductors, cooling, and shielding

My guess is:
S = (0.5-0.8)*S
R_coil = 0.2*S
r_coil = 0.1*S
What kind of garbage is that? "S = (0.5-0.8)*S"?!

Let's try R = (0.25-0.4)*S, so that the plasma diameter takes up a sizeable fraction of the cube, and the plasma volume is between 6.5% and 27% of the volume of the cube. (Yikes! I don't think I always properly accounted for this order of magnitude difference btween plasma volume and magrid volume.) Obvously, to produce more fusion power from a given device, you would want R/S to be bigger, but things like alpha Larmor radius and (possibly) form of the plasma volume will put an upper limit on it.

The coil dimensions are chosen so that the gap from the (imaginary) edge of the cube to the coil is 2r_coil, the diameter of the coil is 2r_coil, and the diameter of the hole in the middle of the coil is 2r_coil.

Art Carlson
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Post by Art Carlson »

rnebel wrote:Art et. al.:
A few comments on scaling laws….
To a certain extent we are in the same boat as everyone else as far as the previous experiments go since Dr. Bussard’s health was not good when we started this program and he died before we had a chance to discuss the previous work in any detail. Consequently, we have had to use our own judgement as to what we believe from the earlier experiments and what we think may be questionable. Here’s how we look at it:
rnebel wrote:1. We don’t rely on any scaling results from small devices. The reason for this is that these devices tend to be dominated by surface effects (such as outgassing) and it’s difficult to control the densities in the machines. This is generally true for most plasma devices, not just Polywells.
Good call.
rnebel wrote:2. Densities for devices prior to the WB-7 were surmised by measuring the total light output with a PMT and assuming that the maximum occurred when beta= 1. We’re not convinced that this is reliable. Consequently, we have done density interferometry on the WB-7. We chose this diagnostic for the WB-7 because we knew through previous experience that we could get it operational in a few months (unlike Thomson scattering which by our experience takes more than a man-year of effort and requires a laser which was outside of our budget) and density is always the major issue with electrostatic confinement. This is particularly true for Polywells which should operate in the quasi-neutral limit where Debye lengths are smaller than the device size.
Another good call. Very good. I can't wait to see the results.
rnebel wrote:3. As discussed by several people earlier, power output for a constant beta device should scale like B**4*R**3. All fusion machines scale this way at constant beta. Input power scales like the losses. This is easy to derive for the wiffleball, and I’ll leave that as an “exercise to the reader”. This is the benchmark that we compare the data to.
The fusion power has always been the easy part.
  • Do you go on Bussard's assumption that the losses are primarily cross-field (by no means "easy to derive"!) or that the losses are dominated by the cusps?
  • For cusp confinement, do you, like Bussard, take some sort of credit (he took infinity) for "recycling" of particles lost through the cusps? (If not, then we can probably agree on an energy loss rate something like p*v*A, where p is the plasma pressure, v is either an average velocity of one species or the speed of sound, and A is the area of the "hole" through the cusps.)
  • Do you agree with Bussard, that the effective hole size scales like the square of the sheath thickness, that is, without taking either line cusps or sheath thickening due to flux conservation into account?
  • Do you assume the sheath thickness is close to the electron gyro-radius, the ion gyro-radius, or the hybrid gyro-radius?
This reader has done the exercise and shown his work, but I don't know if it's the same as the answer in the back of the book.
rnebel wrote:4. As for Mr. Tibbet’s questions relating to alpha ash, these devices are non-ignited (i.e. very little alpha heating) since the alpha particles leave very quickly through the cusps. If you want to determine if the alphas hit the coils, the relevant parameter is roughly the comparison of the alpha Larmor radius to the width of the confining magnetic field layer. I’ll leave that as an “exercise to the reader” as well.
This is related to my guess that R would be chosen to be (0.25-0.4) times the length of the side of the magrid cube. As long as they remember to take the alpha radius into account, I will leave the detailed choice to the engineers.

Art Carlson
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Post by Art Carlson »

Art Carlson wrote:
  • Do you go on Bussard's assumption ...
  • For cusp confinement, do you, like Bussard, ...
  • Do you agree with Bussard, ...
  • Do you assume ...
The easy way to answer all these question (without violating your NDA, I am sure) is to just give us the formula you use for your benchmark.

rnebel
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Post by rnebel »

Loss fraction = (summation (pi*rl**2))/(4*pi*R**2) where rl is the electron gyroradius and R is the coil radius. The summation is a summation over each of the point cusps. If you calculate rl from one of the coil faces, then there are "effectively" ~ 10 point cusps (fields are larger in the corners than the faces). The factor that your observed confinement exceeds this model is then lumped together as the cusp recycle factor.

The other model is to look at mirror motion along field lines. For this model you look at loss cones and assume that the electrons effectively scatter every time they pass through the field null region. This model describes the confinement which was observed on the DTI machine in the late 80s.

I don't know how to predict cross-field diffusion on these devices. The gradient scale lengths of the magnetic fields are smaller than the larmor radii and the electrostatic fields should give rise to large shear flows. On top of that, the geometry is 3-D.

sdg
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Loss Fraction

Post by sdg »

Fascinating!

If I understand this correctly, you are benchmarking that line or "corner" cusps have approximately half the loss of point cusps. Do I have this right? (6 point + 8 * .5 corners) = 10.

And, dare I ask (please ignore this second question only, if NDA applies), is this in the ballpark of what you are observing in WB-7?

Art Carlson
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Post by Art Carlson »

rnebel wrote:Loss fraction = (summation (pi*rl**2))/(4*pi*R**2) where rl is the electron gyroradius and R is the coil radius. The summation is a summation over each of the point cusps. If you calculate rl from one of the coil faces, then there are "effectively" ~ 10 point cusps (fields are larger in the corners than the faces). The factor that your observed confinement exceeds this model is then lumped together as the cusp recycle factor.
Thanks. I'll have to chew on that.
rnebel wrote:The other model is to look at mirror motion along field lines. For this model you look at loss cones and assume that the electrons effectively scatter every time they pass through the field null region. This model describes the confinement which was observed on the DTI machine in the late 80s.
Does this model result in a simple formula? If you have a volume where the field vanishes, how do you handle the infinite mirror ratio?
rnebel wrote:I don't know how to predict cross-field diffusion on these devices. The gradient scale lengths of the magnetic fields are smaller than the larmor radii and the electrostatic fields should give rise to large shear flows. On top of that, the geometry is 3-D.
If you want a house number, you could use Bohm diffusion: D_Bohm = (1/16) kT_e / eB, but I wouldn't trust it farther than I could throw it. How do you know "The gradient scale lengths of the magnetic fields are smaller than the larmor radii"? That's hard to measure, and it contradicts Dolan.

Skipjack
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Post by Skipjack »

Art, Rick, it is soooo awesome to watch you two talk! Dont let that die!
It makes me thirst for more!

rnebel
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Post by rnebel »

Art:

The mirror model is a bit of a handwaving model that I believe Nick Krall came up with. The mirror ratio is calculated from the field where the electron Larmor radius is on the order of the device size. Any smaller field than that will not have adiabatic motion. If particles enter the field null region, it is assumed that they effectively scatter. I believe that Dave Anderson at LLNL did a fair amount of particle tracing calculations for FRMs in the late 70s, and not surprisingly saw jumps in the adiabatic invariants when moving through field null regions. I presume similar behavior was observed on FRC simulations. Anyway, it's a ballpark model.

My other comment was related to electrons trapped in the wiffleball. Over most of their orbit there is little or no magnetic field (i.e. Larmor radius bigger than the device size) with the electrons turning when they hit the barrier magnetic field. The electron behavior is stochastic since there are no invariants. We don't have any direct measure of the internal magnetic fields, but we do know the density and have a pretty good idea what the electron energy is. High beta discharges should expel the magnetic field. The vacuum fields should be in a mirror regime (as was the DTI device) while the wiffleball fields should transition to better confinement. There is about 3 orders of magnitude difference in the predicted confinement times so it's pretty easy to see which regime the device operates in (unless, of course, the cusp recycle is truly enormous).

As you suggest, Bohm diffusion is kind of a catch-all for any kind of confinement you don't understand. We hope we don't end up there, and so far we're OK.

chrismb
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Post by chrismb »

Art Carlson wrote: The alphas have such a high energy that they shouldn't be affected much by the electric field between the plasma and the magrid, and their density is presumably too small to make much of an electric field on their own.
A p11B fusion device will emit 70 amps of alpha particles per 100MW of alpha emissions. So a 500MW device after a conversion of 40% would be outputting some 1kA. Similarly, the charge left behind would need 1kA of other positive ions to flow inwards to ever stabilise the e-field (presumably??), yet they are reciprocating around the centre so do not get to stabilise it. So would a large negative charge not build up in the centre? I think the total flux of of alpahs would, indeed, generate electric fields, and I also think they would enter various oscillatory [stable or unstable?] states by interactions with the other charged particles in the device.

Incidentally, I think this flux of 3E21 alpha particles would require a pumping speed of 120 billion litres per second to maintain a 1E-9 torr vacuum. That's about 50,000 olympic sized swimming pools of volume - per second. Those will sure be some cool vacuum pumps!!.... Where [on earth] does one buy billion litre/sec vacuum pumps?? Whoever makes them is gonna have a lot of business because we'll need over a hundred of them for each 500MW BFR!

JohnP
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Post by JohnP »

Incidentally, I think this flux of 3E21 alpha particles would require a pumping speed of 120 billion litres per second to maintain a 1E-9 torr vacuum.
You may wish to check your math.

chrismb
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Post by chrismb »

JohnP wrote:
Incidentally, I think this flux of 3E21 alpha particles would require a pumping speed of 120 billion litres per second to maintain a 1E-9 torr vacuum.
You may wish to check your math.
approx...

3E21particles/s = 1E21(p11Breactions)/s = 8.68E6 * 1.6E-19 * 1E21 = 1.4GW alphas, at 40% efficiency ~=500MW

density at 1E-9 torr = 3E10 particles/litre.

[3E21 particles/s]/[3E10particles/litre]=1E11 litres/s

?

duly checked......

bcglorf
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Problem

Post by bcglorf »

chrismb wrote:
JohnP wrote:
Incidentally, I think this flux of 3E21 alpha particles would require a pumping speed of 120 billion litres per second to maintain a 1E-9 torr vacuum.
You may wish to check your math.
approx...

3E21particles/s = 1E21(p11Breactions)/s = 8.68E6 * 1.6E-19 * 1E21 = 1.4GW alphas, at 40% efficiency ~=500MW

density at 1E-9 torr = 3E10 particles/litre.

[3E21 particles/s]/[3E10particles/litre]=1E11 litres/s

?

duly checked......
Ow, my head hurts. I'm merely a software engineer but unless I know even less about physics than I think I do that's NOT how to calculate the pumping requirements.

My uneducated guess is that what's important is the mass of your alpha's per second. Assuming your 3E21/s we get:
3E21 particles/s at 6.6E-27 kg/particle = 19.8mg/s

Meaning your pumps need to move that much mass/second at our desired pressure.

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