The question of scaling

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

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Aero
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Post by Aero »

Well, that has a dampening effect. Of course you are right. I can see where it might be a bit difficult to include R as a variable while empirically developing this loss equation, R being a constant for this machine. I wonder if there is any data available to us about this test as conducted on WB-2 and WB-4 which iirc have different values of R. Probably not applicable as the machines are of an evolving design and the cusp losses are different.

But I wonder why you suggest that the losses might scale as R^3 volume, rather than R^2, surface area?
Aero

Art Carlson
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Post by Art Carlson »

Aero wrote:Well, that has a dampening effect. Of course you are right. I can see where it might be a bit difficult to include R as a variable while empirically developing this loss equation, R being a constant for this machine. I wonder if there is any data available to us about this test as conducted on WB-2 and WB-4 which iirc have different values of R. Probably not applicable as the machines are of an evolving design and the cusp losses are different.
The tokamak community has spent a lot of effort to determine the scaling of energy confinement time with various parameters. For some parameters, like B, that can be varied significantly in a single machine, it is possible to derive a scaling exponent for each individual machine. You can also throw all the data together and look for a scaling across many machines. Even if each individual machine exhibits the same scaling tau ~ x^n, the lumped data might exhibit a different scaling tau ~ x^(n+delta). It is far from clear which one you really want to use to extrapolate to a new and bigger machine. I suspect the size variation in the WB machines is too small, and the variation in other design features too great, so that it will be impossible to determine an empirical size scaling with any degree of confidence.
But I wonder why you suggest that the losses might scale as R^3 volume, rather than R^2, surface area?
Just an example to illustrate the principle. If you look at the expressions I derived above, there are different scalings of the losses:
P_line ~ R
P_point ~ const
P_cross ~ R
if delta ~ R
P_cross ~ R^2 if delta ~ rho

rnebel
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Post by rnebel »

Art:

I have two favorite papers on Tokamak scaling. The first was by Taylor and Conner where they showed that all of the empirical tokamak scaling laws were inconsistent with any known plasma physics model. The second was a paper by Wayne Houlberg where he put the empirical tokamak scaling laws into his 1.5-D transport code to see if the code would give curves similar to the assumed scaling laws. When he plotted the results, it looked like you shot the page with a 12 gauge.

Aero
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Post by Aero »

Well, shucks. That's disheartening ... I don't suppose anyone has ever considered building a bigger machine so we could perhaps discover the answers to some of these questions and build up a theory, have they?

Like I don't know the answer to that one. :P
The alternative would be to build some bigger machines so we could develop the theory which would answer the questions.
Aero

TallDave
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Post by TallDave »

Sigh. Now I worry the funding team will want an intermediate-sized model to test scaling of power and losses before taking the plunge to WB-100.

Oh well, I guess we can survive a couple more years of waiting for the big one.
n*kBolt*Te = B**2/(2*mu0) and B^.25 loss scaling? Or not so much? Hopefully we'll know soon...

Jboily
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Post by Jboily »

Actually it is saying the losses scale as B^(1/4), since SQRT(n) should scale as B, and it makes no statement whatsoever about the scaling with R. If the complete scaling relation is, e.g. P_loss ~ B^(1/4)*R^3, we are out of luck again.
Art,

If the scaling of B grow with a factor of R (and possibly R^1.5 as a second scenario, depending on how the field is produced), would it mean the P_loss~R^(1/4), (and possible R^(0.375) in the second case).

The losses are associated to surfaces losses, this should indicated a R^2 relationship.

Now, the fusion power should grow at Pf~B^4*R^3 from simple arithmetic physic calculations according to Dr. Nebel and Dr. Bussard (and to my nuclear physic books), this would imply that the power ratio would grow roughly at Qr~R^(4.75)-Qo, (and possibly
Qr~ [R^(7-0.375)-Qo] in the second case), where Qo is a correction constant to adjust Qr when R=1.

We should have net power when R^(4.75) = Qo in the worst case.

I agree with you that this is rater simplistic, but it gives me hope :)

rnebel
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Post by rnebel »

Aero and TallDave:

My comments were primarily to give you an idea of the nature of the beast. The problem is that in order to experimentally study scaling you need to vary one parameter at a time. That is virtually impossible to do in a plasma. Secondly, all of the profiles have to be self-similar. That almost never happens either. Thirdly, all real physics depends on dimensionless parameters. You can't just scale with B, R, etc. The real physics should scale with parameters like Beta, the ratios of the thermal transit time to the Alven transit time, etc. If you don't know what physics to expect, then it's hard to pick the right set of dimensionless parameters. Finally, transport often depends on plasma turbulence. Turbulence isn't well understood even in incompressible fluids, let alone in plasmas.

It used to be the standing joke that the Tokamak community would come up with "the answer" to scaling and it always agreed with theory. But for some reason, both of them changed every year.

Art Carlson
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Post by Art Carlson »

Jboily, the power loss does not necessarily scale with R^2. A bigger radius, of course, implies a bigger area, but the energy flux per unit area may go down if the transport is diffusive, because the gradient scale length may also be proportional to the radius. If you have line cusps, the loss area is proportional to R (times the width of the cusp), not R^2, and the loss area of point cusps is not directly related to R at all.

hanelyp
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Post by hanelyp »

rnebel wrote:Finally, transport often depends on plasma turbulence. Turbulence isn't well understood even in incompressible fluids, let alone in plasmas.
My understanding is that the mean free path of particles in a polywell is comparable to the dimensions of the device, or larger. Is turbulence meaningful in such conditions? Mean free path, otoh, would enter strongly into cross field losses.

rnebel
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Post by rnebel »

Hanelp:

The collisionless limit is actually where turbulence is the most likely to dominate. In fluids, for instance, viscosity is what damps turbulence. The drivers are anything nonlinear. In fluids, that's usually convection. In plasmas, there are many nonlinearities. Any nonlinearity will cascade the turbulence to shorter wavelengths where it can be dissipated. The most likely candidate for generating turbulence in Polywells is the lower Hybrid Drift instability.

Rick

Art Carlson
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Post by Art Carlson »

rnebel wrote:The most likely candidate for generating turbulence in Polywells is the lower Hybrid Drift instability.
Funny thing, I was about to mention that, too. The good curvature of the polywell should suppress ballooning, but drift instabilities (I just realized) are not directly affected by curvature. Even linear instability theory is something you shouldn't be doing on napkins, and turbulence is a whole 'nother universe. If I recall, Krall did some work on the LHD instability (or am I thinking of Drake?). Is that one of the things he estimated for the polywell?

The Lower Hybrid Drift Instability happens to be a good friend of mine. I did an experimental dissertation on it. At the time it was the leading (practically only) candidate to explain transport in FRCs. The order of magnitude seemed about right, but the scaling was hard to establish empirically, so I set about to measure the fluctuations directly. They weren't there. Of course, a negative result like that is a tricky thing, but I think I did a good job of calibrating my setup and scanning parameter space. After the fact, I collected a number of effects that might be able to explain the suppression, but as far as I know, nobody has ever figured out for sure where the LHD fluctuations disappeared to, or what was causing the transport in their stead. It's a story I try to keep in mind when I criticize the polywell. Plasma behavior is hard to understand, and even a theory with a lot of supporting data can turn out to be wrong. Usually reality is worse than you expect, but (very) occasionally it is better.

TallDave
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Post by TallDave »

Cusp losses wouldn't be losses per se anyway, since in theory they only affect the density ratio between the inside and outside of the machine, though of course they do need to be limited enough to allow the 1000:1 or greater ratio Bussard talked about.

Bussard seems to have believed that besides arcing, the only electron losses would be to the machine via cross-field transport and to unshielded portions of the machine, which would increase with the surface area (thus r^2).
Fusion power scales as the fourth power of the B field and the cube of the size, thus Pf = (k1)B4R3, while the unavoidable electron injection drive power loss scales as the surface area of the machine, thus is proportional to R2. Assuming the use of super-conductors for the magnetic field drive coils, the electron losses are the only major system losses. Then, the ratio of these two power parameters is the gain (Qf), which is thus seen to scale as Qf = (k2) B4R3/R2 = (k2) B4R.
Plasma behavior is hard to understand, and even a theory with a lot of supporting data can turn out to be wrong. Usually reality is worse than you expect, but (very) occasionally it is better.
Hehe, life in general does seem to work that way.
n*kBolt*Te = B**2/(2*mu0) and B^.25 loss scaling? Or not so much? Hopefully we'll know soon...

Art Carlson
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Post by Art Carlson »

TallDave wrote:Cusp losses wouldn't be losses per se anyway, since in theory they only affect the density ratio between the inside and outside of the machine, though of course they do need to be limited enough to allow the 1000:1 or greater ratio Bussard talked about.
Says who? Bussard? He doesn't even seem to understand the difference between neutral density and electron density. Give me a break. Better yet, explain to me how to reflect electrons back into the machine with an electric field without sucking ions out.
Bussard seems to have believed that besides arcing, the only electron losses would be to the machine via cross-field transport and to unshielded portions of the machine, which would increase with the surface area (thus r^2).
Fusion power scales as the fourth power of the B field and the cube of the size, thus Pf = (k1)B4R3, while the unavoidable electron injection drive power loss scales as the surface area of the machine, thus is proportional to R2. Assuming the use of super-conductors for the magnetic field drive coils, the electron losses are the only major system losses. Then, the ratio of these two power parameters is the gain (Qf), which is thus seen to scale as Qf = (k2) B4R3/R2 = (k2) B4R.
This is almost embarassing. Is this a physicist who so blithly says P_loss ~ R^2. Period. Without so much as a mention of the loss mechanism??? No room for additional geometric effects like gradient lengths? No role at all for the magnetic field? Please, spare me.

TallDave
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Post by TallDave »

neutral density and electron density.
Wouldn't Paschen arcing depend on both? Without sufficient electrons, there's nothing to arc. (I'm assuming we are talking about the cloud of exterior electrons arcing to the wall, not the Magrid arcing to the wall).
Better yet, explain to me how to reflect electrons back into the machine with an electric field without sucking ions out.


Heh, I think we have a thread on that already.
Without so much as a mention of the loss mechanism???
This part is a summary. He mentions mechanisms earlier:
1. Direct MG transport through the B-shielded surfaces,
2. Electron losses to poorly shielded or unshielded
metal surfaces, and
3. Losses due to local arcing.
He claims they have an empirically-derived model for #1.
Says who? Bussard? He doesn't even seem to understand the difference
Well, he did found the U.S. tokamak program, so we should be skeptical of all his claims. ;)
Last edited by TallDave on Wed Sep 03, 2008 10:51 pm, edited 3 times in total.
n*kBolt*Te = B**2/(2*mu0) and B^.25 loss scaling? Or not so much? Hopefully we'll know soon...

icarus
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Post by icarus »

The collisionless limit is actually where turbulence is the most likely to dominate. In fluids, for instance, viscosity is what damps turbulence. The drivers are anything nonlinear. In fluids, that's usually convection. In plasmas, there are many nonlinearities. Any nonlinearity will cascade the turbulence to shorter wavelengths where it can be dissipated. The most likely candidate for generating turbulence in Polywells is the lower Hybrid Drift instability.
With all due respect, a lot of what has been written above is sweeping generalisations and extending into misleading gross speculation. I could take it apart line by line but almost each one would be a new topic in itself.

I hope this has been a "hand-wavvy lapse" by Dr. Nebel and not the usual order of business for Polywell research or else I wouldn't be holding out much hope for success.

Give me non-peer-reviewed technical reports with hard numbers, facts and results over paragraphs of pontification and circumlocution any day.

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