Perhaps someone can point me to a thread where scaling has already been hashed to death. If not, then consider the following:
Quoting from page 5 of the paper, EMC2-0491-03 Electron Recirculation in Electrostatic Multicusp Systems …, by Robert W. Bussard and Katherine E. King, approved for public release by DARPA, November 23, 1992.
QUOTE -Since fusion power output is proportional to the square of the core density, and since this is proportional to the surface density – and hence to the square of the B field – it is clear that increasing the B field strength has a strong effect on system fusion power generation. On this simple argument, the fusion power will vary as the fourth power of the B field, just as for conventional magnetic confinement machines.
END QUOTE
It seems self evident that the power out will be proportional to the volume of the fusion core, that is, proportional to the cube of the radius of something and since all the radii are proportional to each other, then I pick the radius of the MaGrid.
The above combined becomes the source of the R^7 scaling law. I frankly don’t have the focus to plow through all of the math and assumptions it takes to confirm this scaling law but I hope someone will pick up the torch and run with it. Maybe at least to the point where we can stop referring to scaling as some big questionable mystery and start referring to questionable details of the derivation.
OH, and the casual reader should note that there are four pages of ever more rigorous detail leading up to the quoted material, it doesn't just arrive in a vacuum.
The question of scaling
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Art Carlson
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Until the physics is understood well enough to think otherwise (and there could be a lot of reasons for this), it is accepted as a working hypothesis by all parties (at least, Nebel in one corner and me in the other), that the a polywell will have a beta limit (hopefully beta = 1, when properly defined), so that the fusion power output will be proportional to B^4*R^3. The additional assumption that B can scale as R is less firmly established, but provides a convenient benchmark. The more interesting question and the real fights occur over the scaling of the energy loss rate.
Art's right on this. The output power scaling for the Polywell is essentially the same scaling as for any magnetic confinement machine. The real physics issue is the input power required to keep the Beta constant. This has to balance the losses. In other words, the issue is how good the confinement is.
Thank you Dr.'s. Your posts have provided me with more enlightenment than a month of reading on this forum. You have answered the question that I didn't realize I had, "Why are we doing this?" I knew in general, but now I know better how the threads fit together.
we're trying to avoid the G^4 problem. Obviously R^7 for large enough R will trump any power loss mechanism we may have if we keep pumping in the power. Unless our reactor turns into a Gooey Glob of Glassy Gunk (G^4)
we're trying to avoid the G^4 problem. Obviously R^7 for large enough R will trump any power loss mechanism we may have if we keep pumping in the power. Unless our reactor turns into a Gooey Glob of Glassy Gunk (G^4) Aero
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Art Carlson
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I have suggested that the losses (at constant temperature/particle energy) could look like those from line cusps:
P_line ~ n*R*rho ~ R*B
I have suggested that the losses in Bussard's picture could look like those from point cusps:
P_point ~ n*rho^2 ~ const
Actually, I think Bussard had something else in mind, but I haven't been able to figure out what. I think something like classical cross-field diffusion. That would be a diffusive flux over the whole surface, so
P_cross ~ R^2*n*D/delta
where delta is the gradient scale length (possibly rho, possibly R), and the classical diffusivity in a highly magnetized plasma is D ~ rho^2*n*sigma*v ~ const. (Bohm diffusion would scale as D ~ B^-1.) That leads to
P_cross ~ R^2*B^2/delta
P_cross ~ R*B^2 if delta ~ R
P_cross ~ R^2*B^3 if delta ~ rho
If we make the B ~ R hypothesis, these reduce to
P_line ~ R^2
P_point ~ const
P_cross ~ R^3 if delta ~ R
P_cross ~ R^5 if delta ~ rho
That's quite a range of possibilities, and I don't see that it is very illuminating. We should be able to translate these scalings into something about the minimum size of a break-even device.
P_line ~ n*R*rho ~ R*B
I have suggested that the losses in Bussard's picture could look like those from point cusps:
P_point ~ n*rho^2 ~ const
Actually, I think Bussard had something else in mind, but I haven't been able to figure out what. I think something like classical cross-field diffusion. That would be a diffusive flux over the whole surface, so
P_cross ~ R^2*n*D/delta
where delta is the gradient scale length (possibly rho, possibly R), and the classical diffusivity in a highly magnetized plasma is D ~ rho^2*n*sigma*v ~ const. (Bohm diffusion would scale as D ~ B^-1.) That leads to
P_cross ~ R^2*B^2/delta
P_cross ~ R*B^2 if delta ~ R
P_cross ~ R^2*B^3 if delta ~ rho
If we make the B ~ R hypothesis, these reduce to
P_line ~ R^2
P_point ~ const
P_cross ~ R^3 if delta ~ R
P_cross ~ R^5 if delta ~ rho
That's quite a range of possibilities, and I don't see that it is very illuminating. We should be able to translate these scalings into something about the minimum size of a break-even device.
Sounds like we are finally barking up the right tree. (At least with IEC machines) it's almost never a question of whether you can get some neutrons; it always seems to come down to a problem with Q due to poor confinement times.
After reading Dolan's review paper on electrostatically plugged cusps and magnetically shielded IEC machines (he claims they are equivalent), I think the major flaw with the polywell design is pretty much what Art has been saying for some time: that electrons will build up in the cusps and sheild the positive potential caused by the grid/coils, reducing the potential barrier for ions. In order to avoid that, electrons will either need to be removed (=high loss current) or the grid potential must be raised ridiculously high (=high loss voltage).
Another possible problem that Dolan points out is that the potential well will tend to trap impurities (for instance, atoms knocked of the chamber walls by electron beams coming out the cusps). This would aggravate the existing concerns with brems radiation.
After reading Dolan's review paper on electrostatically plugged cusps and magnetically shielded IEC machines (he claims they are equivalent), I think the major flaw with the polywell design is pretty much what Art has been saying for some time: that electrons will build up in the cusps and sheild the positive potential caused by the grid/coils, reducing the potential barrier for ions. In order to avoid that, electrons will either need to be removed (=high loss current) or the grid potential must be raised ridiculously high (=high loss voltage).
Another possible problem that Dolan points out is that the potential well will tend to trap impurities (for instance, atoms knocked of the chamber walls by electron beams coming out the cusps). This would aggravate the existing concerns with brems radiation.
Dr. Carlson: It would be nice if we could, but I frankly don't see a relationship. I see that I can write, in most simplistic terms,
Q-out = Konstant * R^7, and using numbers generally accepted on this forum, I can even write 100 megawatts = Konstant * 1.5 ^7 giving
Konstant = 5.852766347 * 10^6 .
That was fun, but now when I try to introduce the power losses, well, I can write down similar simplistic equations for the power loss choices you listed using other Konstants, but its not the same kind of power. Q-out is fusion power and the losses are what, grid power? I can't add or subtract them, they are apples and oranges. I cannot write: Q-out = Q - power loss
Of course, Konstant is not really a physical constant and maybe Q and power loss are related through their reality. But if so, I couldn't begin to write the relationship down.
Q-out = Konstant * R^7, and using numbers generally accepted on this forum, I can even write 100 megawatts = Konstant * 1.5 ^7 giving
Konstant = 5.852766347 * 10^6 .
That was fun, but now when I try to introduce the power losses, well, I can write down similar simplistic equations for the power loss choices you listed using other Konstants, but its not the same kind of power. Q-out is fusion power and the losses are what, grid power? I can't add or subtract them, they are apples and oranges. I cannot write: Q-out = Q - power loss
Of course, Konstant is not really a physical constant and maybe Q and power loss are related through their reality. But if so, I couldn't begin to write the relationship down.
Last edited by Aero on Mon Sep 01, 2008 9:34 am, edited 2 times in total.
Aero
Thanks Solo! I haven't seen that one before. That will take me a while to dig through.
I think the acknowledgments indicate there's a lot of history here. All the best people getting in their best shot. I'm betting there's a lot more history to come (i.e. many more research papers before fusion reactors are economic).
I think the acknowledgments indicate there's a lot of history here. All the best people getting in their best shot. I'm betting there's a lot more history to come (i.e. many more research papers before fusion reactors are economic).
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Art Carlson
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It shouldn't be so hard. In principle. Whether the scalings and numbers are correct is another story.Aero wrote:Dr. Carlson: It would be nice if we could, but I frankly don't see a relationship. I see that I can write, in most simplistic terms,
Q-out = Konstant * R^7, and using numbers generally accepted on this forum, I can even write 100 megawatts = Konstant * 1.5 ^7 giving
Konstant = 5.852766347 * 10^6 .
That was fun, but now when I try to introduce the power losses, well, I can write down similar simplistic equations for the power loss choices you listed using other Konstants, but its not the same kind of power. Q-out is fusion power and the losses are what, grid power? I can't add or subtract them, they are apples and oranges. I cannot write: Q-out = Q - power loss
Of course, Konstant is not really a physical constant and maybe Q and power loss are related through their reality. But if so, I couldn't begin to write the relationship down.
For the current experiments we have something like R = 0.1 m, V = 10 kV, and I = 100 A. (I think.) If the drive power scales as R^n, we can write
P_drive = (1 MW)*(R / 0.1 m)^n.
For the fusion power we can take your
P_fusion = (5.9 MW/m^7)*R^7 = (5.9e-7 MW)*(R / 0.1 m)^7.
(Is that supposed to be for D-T or p-B11, or something else?)
The fusion power will be converted to electricity at some efficiency, eta. Most people here probably dream of direct conversion at eta = 90%. I think a more realistic number of DC would be eta = 60%. You may also be stuck with thermal conversion at eta = 30%, expecially for D-T. You might also try to recover some of the drive power, but that will be more difficult. Some fraction (f_recirc) of the electric power produced will have to be used to drive the electrodes and injectors (and everything else in the plant). If f_recirc = 1, you have a self-sustaining reactor, but no electricity left over to sell. A typical goal is f_recirc = 0.20. The power available to drive the reactor is P_drive = P_fusion*eta*f_recirc, so Q = P_fusion/P_drive = 1/(eta*f_recirc). Q is usually required to be between 10 and 20, e.g. eta = 50% and f_recirc = 20% gives Q = 10.
Now we can write something like this:
P_fusion = Q * P_drive
(5.9e-7 MW)*(R / 0.1 m)^7 = 10*(1 MW)*(R / 0.1 m)^n
R = (0.1 m)*(1.7e7)^(1/(7-n))
If n = 0 (point-cusp-like losses), then a reactor can be built with a radius a bit over 1 m. If n = 5 (cross-field losses over a thickness scaling with rho), then the reactor would have to have a radius of 400 m.
Let's try this backwords now. Let's say we don't want our reactor to produce more than 10 GW of fusion. Then the radius cannot be over 2.9 m and the drive power required cannot be over 1 GW. Then the maximum possible value for n is given by
1000 = 29^n
n = 2
In summary, if you accept the thousand and one assumptions we have made along the way, an attractive reactor might be possible if you have point-cusp-like losses, a marginally acceptable reactor might be possible with line-cusp-like losses, but you can forget about it if the losses scale like cross-field diffusion.
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Barry Kirk
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Trying to get my head around this.
The fusion power may scale as R^7.
The loss gain is a real unknown, but would increase as R goes up.
If loss gain goes up slower than fusion power, then at a certain point, a power producing machine may be possible...
Part of the problem is that the ability to cool the machine is almost certain not to go up as R^7.
So, at a certain size as pointed out above, G4 issues, Gooey Glob of Glassy Gunk occur.
However, it may be possible to run a power positive machine in pulse mode and use the thermal mass of the reactor to absorb the pulse energy.
One of the drawbacks of running a machine in pulse mode, is that in addition to steady state losses, there would be additional losses for setting up the initial conditions for each pulse.
I guess a good question would be when can we expect to find out if a sweet spot exists where a economical net power production machine is possible?
The fusion power may scale as R^7.
The loss gain is a real unknown, but would increase as R goes up.
If loss gain goes up slower than fusion power, then at a certain point, a power producing machine may be possible...
Part of the problem is that the ability to cool the machine is almost certain not to go up as R^7.
So, at a certain size as pointed out above, G4 issues, Gooey Glob of Glassy Gunk occur.
However, it may be possible to run a power positive machine in pulse mode and use the thermal mass of the reactor to absorb the pulse energy.
One of the drawbacks of running a machine in pulse mode, is that in addition to steady state losses, there would be additional losses for setting up the initial conditions for each pulse.
I guess a good question would be when can we expect to find out if a sweet spot exists where a economical net power production machine is possible?
IIRC Bussard seemed to think losses scaled as roughly r^2, and gain scaled as roughly r^5 (and power as r^7 as mentioned above).
Then we're back to the questions of whether there are line cusp losses, what the losses are from the funny cusps, recirculation, density ratio, focusing, etc. Hopefully we'll get to see some more of the evidence for Bussard's belief, so we can figure out if a net power Polywell is 1.5m or 400m.
Then we're back to the questions of whether there are line cusp losses, what the losses are from the funny cusps, recirculation, density ratio, focusing, etc. Hopefully we'll get to see some more of the evidence for Bussard's belief, so we can figure out if a net power Polywell is 1.5m or 400m.
n*kBolt*Te = B**2/(2*mu0) and B^.25 loss scaling? Or not so much? Hopefully we'll know soon...
We've got two primary electron particle/energy loss paths: upscattering + collision with chamber walls, and downscattering and diffusion across the field lines. Dolan claims transport at ~1.5 times classical has been demonstrated in cusp machines.
Actually, here, let me write all of Dolan's eqns out for the energy confinement:
Actually, here, let me write all of Dolan's eqns out for the energy confinement:
For the spindle cusp, that R^4/L factor means R^3 scaling of power loss for our purposes, and the linear set of ring cusps gives R/rho^2=R*B^2=R^3 also. The linear set of ring cusps ought to behave somewhat like a whiffleball, in that it should have a large field-free plasma volume (larger than the spindle cusp), even ignoring any diamagnetic effects. But the polywell fields should have 'less cusp' than the ring cusp setup. So perhaps we are in the right ballpark for Q.paraphrase of Dolan wrote:1/tau_E_cross=1/(tau_conduction)+1/(tau_diff + tau_trap)where tau_trap is for trapping and detrapping between sheath and central plasma, and assume tau_cond ~=1.5 tau_diff
Assume tau_trap<<tau_diff, then tau_E_cross~=0.6*tau_diff
For classical transport, Pashtukov estimates:
tau_diff=2*tau_ei*a*V/(S*alpha*rho_a*rho_b)
where tau_ei is the electron-ion momentum exchange time, V is plasma volume, S is plasma surface area, rho_b is the electron Larmor radius in the sheath, rho_a is that in the cusp throat, and alpha is a parameter from 1.33 to 4.6 for T_i/T_e from 0 to 1.
Assuming ln(gamma) is 16, this reduces to tau_ie=9.4E14*T^(3/2)/n (what is n?) Then assuming a spindle cusp machine with radius R and length L, Lavrent'ev gets:
P_e=[phi+2T_e*(cosh(p)-1)*pi*R^4*D_0*n_0*p/(r_a^2*L*sinh(p))] where p=sqrt(tau_cross/tau_parallel), r_a is point cusp radius. This loss is least for p~5.
W/ a linear set of ring cusps, tau_diff=0.09*tau_ei*a*R/(rho_a*rho_b)
Dr. Carlson: The numbers, 100 Megawatts and 1.5 meter radius, are for pB11 fusion. They are extrapolations but I am not sure of their basis, or how they were originally derived. To correct that, I was looking for information to use in calculating “Konstant” from the WB-6 final report. Those tests used DD fusion. I did find some numbers, to many, as it happens, so I calculated values of
Konstant = 1.41 E+5 and 2.82 E+5 using radius=0.1m.
Are there one or two 17.6 MeV DD fusions per neutron?
One confusing factor is the definition of “Radius” for WB-6, (Paragraph 2, page 5 of the Final Report, “R is the radius of the device, from the center to the midplane of the field coils,”). It looks like WB-6 radius must be very close to 0.15 meters instead of the 0.1 meters used above. Using R=0.15m gives
Konstant = 8.24 E+3 or 1.65E+4.
Note that I have listed 4 different values for one number that represents a different machine (DD fusion) than do your numbers given previously. I think perhaps the 1.65E+4 is the most nearly sound of the four.
But I got excited about what I found next, and posted below.
Konstant = 1.41 E+5 and 2.82 E+5 using radius=0.1m.
Are there one or two 17.6 MeV DD fusions per neutron?
One confusing factor is the definition of “Radius” for WB-6, (Paragraph 2, page 5 of the Final Report, “R is the radius of the device, from the center to the midplane of the field coils,”). It looks like WB-6 radius must be very close to 0.15 meters instead of the 0.1 meters used above. Using R=0.15m gives
Konstant = 8.24 E+3 or 1.65E+4.
Note that I have listed 4 different values for one number that represents a different machine (DD fusion) than do your numbers given previously. I think perhaps the 1.65E+4 is the most nearly sound of the four.
But I got excited about what I found next, and posted below.
Aero
I also found this near the bottom of page 21 of the WB-6 Final Report.
QUOTE
Procedure, Beta=1 tests
Several beta=1 tests were run during October, 2005, to try and determine the value of Kj, the MaGrid transport equation coefficient, which sets the loss rates for cross-field losses of electrons. This coefficient and the equation itself have been studied extensively over the preceding 7 years of experimental work, and are documented in several EMC2 Technical Reports.
END QUOTE
Back on page 17, we see “the simplistic single-term electron cross-field transport equation, as derived from tests stretching from WB-2 through WB-4.” (That is, an empirical equation.)
It is:
Ie = Kj(Ei)SQRT(n)/B^(3/4)
Then, back at the top of page 23, we find a value,
Kj = 2-4 E-12. from 2E-12 to 4E-12.
This is exciting to me because the report claims, and the test methodology supports that this equation accounts for ALL losses, cross field transport and cusp losses.
Oh, yea,
Ie is the electron current in amps,
Ei is electron injection energy or drive voltage/energy.
I think this equation is saying that losses scale as B^(-3/4) - If so, we can deal with it.
QUOTE
Procedure, Beta=1 tests
Several beta=1 tests were run during October, 2005, to try and determine the value of Kj, the MaGrid transport equation coefficient, which sets the loss rates for cross-field losses of electrons. This coefficient and the equation itself have been studied extensively over the preceding 7 years of experimental work, and are documented in several EMC2 Technical Reports.
END QUOTE
Back on page 17, we see “the simplistic single-term electron cross-field transport equation, as derived from tests stretching from WB-2 through WB-4.” (That is, an empirical equation.)
It is:
Ie = Kj(Ei)SQRT(n)/B^(3/4)
Then, back at the top of page 23, we find a value,
Kj = 2-4 E-12. from 2E-12 to 4E-12.
This is exciting to me because the report claims, and the test methodology supports that this equation accounts for ALL losses, cross field transport and cusp losses.
Oh, yea,
Ie is the electron current in amps,
Ei is electron injection energy or drive voltage/energy.
I think this equation is saying that losses scale as B^(-3/4) - If so, we can deal with it.
Aero
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Art Carlson
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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.Aero wrote:Back on page 17, we see “the simplistic single-term electron cross-field transport equation, as derived from tests stretching from WB-2 through WB-4.” (That is, an empirical equation.)
It is:
Ie = Kj(Ei)SQRT(n)/B^(3/4)
Then, back at the top of page 23, we find a value,
Kj = 2-4 E-12. from 2E-12 to 4E-12.
This is exciting to me because the report claims, and the test methodology supports that this equation accounts for ALL losses, cross field transport and cusp losses.
Oh, yea,
Ie is the electron current in amps,
Ei is electron injection energy or drive voltage/energy.
I think this equation is saying that losses scale as B^(-3/4) - If so, we can deal with it.
Even considering just the scaling with B, without a theoretical underpinning we must be very skeptical of extrapolating the empirical scaling outside the range of parameters for which it has been tested. Suppose there are two loss mechanisms, one of which is independent of B and the other of which scales with some power, e.g. B^2, and suppose at the conditions of our experiments the first is rather larger than the second. Our data may well show losses scaling with B^(1/2), but when we raise the field substantially, the B^2 term rises up and clobbers us.