The theory forum has not had much input recently, so I'll start a new topic, expose my ignorance, and hopefull initiate some discussion and learn from it.
It seams the r^3 scaling is straight forward. So long as the 'reaction core' remains a nearly fixed percentage of the volume of the beast, the volume (reaction core/ aviable ions for collision and fusion) would go up as the third power of the linier dimension of the magrid.
The magnetic scaling is more of a mystery to me. If you keep the magrid tubes the same proportionate size relative to the linier diameter of the beast, then there would be more space inside the donut tube for more windings, so increased magnetic field strength at the same current (?). How this scales I'm uncertain. And, how the increased magnetic field strength effects the greater internal volume/ reaction density and aceleration (well depth) chariteristics is a mystery.
Keeping the same overall size, and only increasing the current through the magnetic windings would increase the magnetic field strength, presumably confine the vast majority of the electrons into a tighter ball. This would (I'm guessing) lead to a greater well depth (or at least steeper), lead to tighter focusing of the ions towards the center, and perhaps have some benificial effect on 'anealing' or slowing of thermalization. Being able to confine a certain number of electrons into a smaller volume would presumeably alternatly allow packing more electrons into the same volume, and thus increasing the number of electrostaticlly confined ions. This would alllow increased densities/ fusion rates in the same volume. Which is more attractive, shrinking the reaction volume ( with improved focus and steeper well?), or increasing it's density? Again, what is considered to be contributing to the claimed forth power scaling of the magnetic field effect?
The larger volume of the donut in larger machines would allow for more wires as I said, and give more room for cooling, shielding, etc. Is this increased size dependant winding#/ magnetic strength the whole story, a part, or only significant for engeenering concerns?
Dan Tibbets
Scaling laws...
Scaling laws...
To error is human... and I'm very human.
I have been looking on Google for some mention of 4th power scaling of fusion power output compared to magnetic field strength. Most end up talking about Tokamak's, as does the below article. I finally found this article that actually agrues the benifits of coper windings as opposed to superconducters for the magnets, but skimming the article, on page 2 -item (5), they mention the 4th power scaling. So, at least the scaling law claimed by Dr Bussard is not only his creation.
http://74.125.47.132/search?q=cache:PHJ ... d=52&gl=us
Dan Tibbets
http://74.125.47.132/search?q=cache:PHJ ... d=52&gl=us
Dan Tibbets
To error is human... and I'm very human.
-
Art Carlson
- Posts: 794
- Joined: Tue Jun 24, 2008 7:56 am
- Location: Munich, Germany
The B^4 scaling is nearly as straightforward as the R^3 scaling. Magnetic fields are proportional to currents, so the Lorentz force jXB scales as B^2. The plasma pressure nT is contained by the Lorentz force, so nT also scales as B^2. The temperature is essentially fixed by the fusion cross section, so the reaction rate scales as n^2. (Each ion has a chance to react, so you get one n, but it can react with any of the other ions, so you get a second n.) Therefore the reaction rate (or fusion power density) scales with n^2 and n scales with B^2, giving the final scaling with B^4.
More subtle is Bussard's contention that the maximum practical B scales with R, so that the total fusion power output scales with R^7. This is equivalent to an engineering limit on the current density in the conductors (I ~ B*R, so I/A ~ B/R. If I/A is constant, then B ~ R.) Sounds plausible, at least until you get into the multiple Telsa range, but it is hardly based on first principles. It might mean a constant power density from ohmic heating in the coils, but I would expect that power to be harder to remove as the coils get bigger. Unless you only want a one-bang experiment in which you let you magnets heat up until they don't quite melt.
More subtle is Bussard's contention that the maximum practical B scales with R, so that the total fusion power output scales with R^7. This is equivalent to an engineering limit on the current density in the conductors (I ~ B*R, so I/A ~ B/R. If I/A is constant, then B ~ R.) Sounds plausible, at least until you get into the multiple Telsa range, but it is hardly based on first principles. It might mean a constant power density from ohmic heating in the coils, but I would expect that power to be harder to remove as the coils get bigger. Unless you only want a one-bang experiment in which you let you magnets heat up until they don't quite melt.
Bussard's theory was that Amp turns go up with volume (for Cu coils).
And of course B field goes as 1/linear size with constant Amp-turns.
So B field with Cu coils scales linearly with size for constant intercept volume.
Of course you go with SC coils and the whole B field scaling with size goes out the window. I think that 20T fields with coils in the 2 m dia range is not too hard these days. And who knows? It might be up to 30 or 40 T in 5 years.
So far the best SCs can get into the 120T range at 0K. Less at higher temps.
And of course B field goes as 1/linear size with constant Amp-turns.
So B field with Cu coils scales linearly with size for constant intercept volume.
Of course you go with SC coils and the whole B field scaling with size goes out the window. I think that 20T fields with coils in the 2 m dia range is not too hard these days. And who knows? It might be up to 30 or 40 T in 5 years.
So far the best SCs can get into the 120T range at 0K. Less at higher temps.
Engineering is the art of making what you want from what you can get at a profit.
And ohmic heating in copper coils scales linear with wire lenght. Wire lenght scales up r² if you grow coil volume too.
That make ohmic coils quite unpractical.
Art can maybe do some calculation how much it take power to feed say 6pcs 1m coils if coil diameter is 10cm and B=1T. Assume that coil material is copper at 20C.
--Eerin
That make ohmic coils quite unpractical.
Art can maybe do some calculation how much it take power to feed say 6pcs 1m coils if coil diameter is 10cm and B=1T. Assume that coil material is copper at 20C.
--Eerin
</ Eerin>
Art already answered this, but what the hell, I like going through this.
Since the Polywell operates at beta=1, where electron pressure balances against the magnetic field, the electron density is dependent on B and scales as B^2. Since the Polywell as a whole is quasineutral (but not always locally quasineutral,
hi Art!), the ion density is dependent on the electron density and also scales as B^2. Since fusion power is (roughly, for the cross-sections involved) ion density squared, fusion power scales as B^4.
The real question, of course, is how the losses will scale. A lot of people don't like Bussard's claim that there are only electron losses (and those only to cross-field diffusion and losses to unshielded surfaces, since cusp "losses" allegedly recirculate/oscillate) and they scale at a measly r^2.
Since the Polywell operates at beta=1, where electron pressure balances against the magnetic field, the electron density is dependent on B and scales as B^2. Since the Polywell as a whole is quasineutral (but not always locally quasineutral,
hi Art!), the ion density is dependent on the electron density and also scales as B^2. Since fusion power is (roughly, for the cross-sections involved) ion density squared, fusion power scales as B^4.The real question, of course, is how the losses will scale. A lot of people don't like Bussard's claim that there are only electron losses (and those only to cross-field diffusion and losses to unshielded surfaces, since cusp "losses" allegedly recirculate/oscillate) and they scale at a measly r^2.
Copper wire length grows as 2pi *r (circumfrence). So a doubling in size results ( example 15 cm to 30 cm) in 2*15*pi=94 vs 2*30*pi=188cm or a doubling of copper wire length and resistance. But, if the same coil cross section/ machine diameter is maintained, the cross section increases in the same proportion (example 2 cm to 4 cm crossection of round coil) results in the same aspect ration of coil cross section diameter to total magrid length. Area of that cross section is r^2 pi, or in the example 12.5 vs 50 cm^2.eros wrote:And ohmic heating in copper coils scales linear with wire lenght. Wire lenght scales up r² if you grow coil volume too.
That make ohmic coils quite unpractical.--Eerin
So keeping the same aspect ratio would allow for 4 times as many amp turns with the same gauge of wire at the expense of twice the resistance. Going to wire with twice the cross section area would allow twice as many amp turns at the same resistance. By my reasoning, this would allow an ideal doubling of the magnetic field strength for each doubling of magrid diameter with the same total resistance.
So, increasing the diameter of the Polywell from 0.3 m to 3 m would allow a 10 fold increase in the magnetic field at with the same total resistance of appropiatly sized copper wire. Using liquid nitrogen to carry away the Ohmic heating and to decreas the resistance of the copper might allow up to ~ 80 fold increase in the magnetic strength at the cost of ~ 8 fold increase in current, but total resistance similar to the original 30 cm example. So long as the liquid nitrogen could keep up with this Ohmic heating the magnets should be able to run indefinatly (ignoring external heating from the plasma, fusion products, etc). At least that is my hopefully acurate calculation.
What I don't know, is how much the magnetic field strength needs to increase to keep the appropriate containment properties as the linier dimension of the structure increases. Is it 1:1, squared, cubed...?
For a research machine like WB 6, Bussard stated that the magnets could be run for at most a couple of seconds. This makes me wounder if the ~0.25 ms runs were limited by the power supply and/ or arching as opposed to magnet heating concerns, ie- if Nebel's team were able to improve WB 7 power supply, and gas puffing (soon to be ion gun) have (could) they obtained longer runs well above 1 ms that gives more confident data. And, for that matter, at what drive energeies have been used? Some data/ detailed results would sure be interesting!
Dan Tibbets
To error is human... and I'm very human.
Beacon Power has tested out their "Smart Energy Matrix" flywheel storage system for power grid frequency regulation. It stores ~5MWhr and can dump it at ~20MW. Perhaps EMC2 (or us, if we actually decide to build one of these puppies) should contact BePo and see if we could somehow use excess night time capacity. Hmmm?D Tibbets wrote:For a research machine like WB 6, Bussard stated that the magnets could be run for at most a couple of seconds. This makes me wounder if the ~0.25 ms runs were limited by the power supply and/ or arching as opposed to magnet heating concerns, ie- if Nebel's team were able to improve WB 7 power supply, and gas puffing (soon to be ion gun) have (could) they obtained longer runs well above 1 ms that gives more confident data. And, for that matter, at what drive energeies have been used? Some data/ detailed results would sure be interesting!