Possible wiffle-ball analytical solution

[kcdodd]

What are A, R, and S?

[Indrek]

What are A, R, and S?

A is the ball radius. R is the planar radius of the physical coils. S is the spacing between the physical coils. See http://www.mare.ee/indrek/ephi/images.pdf for reference. 'i' indicates inverse/image coils. Field value is in teslas of course.

- Indrek

[kcdodd]

Mmk, with the same parameters I get a much lower bfield at the surface, like half. So a lot of field is bypassing the points which are solved for.

[Solo]

Hmkay, I have a question: I know the Poisson eqn. gives the potential caused by a charge distribution, but what's the analogous eqn to find the magnetic field produced by a plasma? (I'm trying to understand how to model the diamagnetic B-field reduction.) Would it work to use the eqn for the field of a moving point charge, and then apply that to the avg. velocity times density at a given point, and then sum those contributions at each point in space?

[rcain]

Indrek - Dr Nebel has gone very quiet - it seems it is all up to you now

is your model complete and accurate yet?

[drmike]

Solo - If you have a current, you can model the B field from it using Biot-Savart law. You can then use that B field to model how your current moves using F = J X B. As a bulk operation it is a crude but effective estimate.

[rcain]

there are a couple of interesting papers i came across here:

http://www.ornl.gov/sci/fed/Theory/publ ... t7b13e.pdf

and

http://fusion.gat.com/THEORY/images/a/a ... sher02.pdf


whilst they're not directly modeling our configuration, i think some of the techniques might be quite useful - in particular:

the first approach goes into Laplace and Fourier space, then uses Random Walk with Probability Density Function (PDF) spaces - could save a lot of processing time.

the second also uses time-orthogonal space and uses 'blending functions' - might be useful for the 'wiffle-ball' wall and diamagnetic effects.

[icarus]

Here's some more results following from using the image system of a spherical wiffle-ball.

http://www.mare.ee/indrek/ephi/invwb/Pwb.pdf

Here we used Indrek's code for calculating forces on the physical coils, with and without the wiffle-ball present. Then summing the difference of the reaction forces and dividing by wiffleball surface area, we get an approximation for plasma pressure at the wiffle-ball surface as a function of wiffleball radius.

I've done the plot for 5 different coil currents (100kAmp to 500 kAmp) to get an idea of how that affects the curve also. The physical coils are the 0.15 [m] radius, 0.08 [m] spacing Polywell configuration.

Depending on accuracy, it maybe a useful tool for "eye-balling" the plasma wiffle-ball diameter, e.g. through a portal in the reaction chamber, and getting a rough estimate of pressure.

NB: Has been recently discussed on this thread here
viewtopic.php?t=939&start=45
but I thought it better to keep it together with other results here, derived from the spherical wiffle-ball analytical solution

[icarus]

Another result graph generated by Indrek, has useful Log scale for low pressure wiffleballs. (scroll down to bottom of page)

http://www.mare.ee/indrek/ephi/invwb/

[icarus]

Anybody got a possible number of the coil minor diameter for a reactor size device?

I'm going to work with the following unless anybody has a good reason why not (wiffle-ball diameter is not a fixed input):

% set up polywell geometry, current, images, etc
global R S CURRENT B_max mu_0 iR iS iCURRENT
%
R = 1; % major coil radius [m]
S = 0.08; % edge spacing between coils [m]
A = 1; % wiffleball radius [m]
B_max = 10; % maximum mag field at coil face center [T]
mu_0 = 4*pi*1e-7; % magnetic constant [T.m/A]

% get current from mag field max at cusp
%
CURRENT =2*B_max*R/mu_0 ;

I'm going to rework these simulations for the larger machine and see what kind of spherical wiffle-ball fields we are looking at ... already I think I see some noticeable changes because of the coil spacing versus coil-diameter ratio. A coil minor diameter will give me a handle on edge spacing plus/minus a gyro-radius or two.

[rcain]

Anybody got a possible number of the coil minor diameter for a reactor size device?

I'm going to work with the following unless anybody has a good reason why not (wiffle-ball diameter is not a fixed input):

% set up polywell geometry, current, images, etc
global R S CURRENT B_max mu_0 iR iS iCURRENT
%
R = 1; % major coil radius [m]
S = 0.08; % edge spacing between coils [m]
A = 1; % wiffleball radius [m]
B_max = 10; % maximum mag field at coil face center [T]
mu_0 = 4*pi*1e-7; % magnetic constant [T.m/A]

% get current from mag field max at cusp
%
CURRENT =2*B_max*R/mu_0 ;

I'm going to rework these simulations for the larger machine and see what kind of spherical wiffle-ball fields we are looking at ... already I think I see some noticeable changes because of the coil spacing versus coil-diameter ratio. A coil minor diameter will give me a handle on edge spacing plus/minus a gyro-radius or two.

how about r= 3 Nautical Miles?

(lets test the scaling while we ae at it )

seriously, i would be very interested in low flux, high charge models around corner cusps, just inside and outside cube (coils). seeing what it looked like, i mean.

ps. how long does your model take to run?

rob

pps. i believe A and B are derived/generated quantities. ie. not something we control directly. they are coupled geometrically with machine geometry and initial conditions.

ppps. that http://www.mare.ee/indrek/ephi/invwb/ is quite beautiful

[icarus]

Test:

octave:109>A=0.850000
R=1.000000, S=0.200000, B_max=10.000000
octave:111>



Numeric scale on colour-bar denotes magnetic pressure.

[rcain]

cubical, but in a different way. did you change the orientation?

wonderful stuff keep it up

ps. can you also give us an illuminated view of the inside of the WB surface, viewed from the centre of the device outwards?

per chance..

strike that. its pretty much the same i think. but i'm going there. some real-time 3d tomography would help.... a slice through the middle perhaps?

[D Tibbets]

Anybody got a possible number of the coil minor diameter for a reactor size device?

I'm going to work with the following unless anybody has a good reason why not (wiffle-ball diameter is not a fixed input):

% set up polywell geometry, current, images, etc
global R S CURRENT B_max mu_0 iR iS iCURRENT
%
R = 1; % major coil radius [m]
S = 0.08; % edge spacing between coils [m]
A = 1; % wiffleball radius [m]
B_max = 10; % maximum mag field at coil face center [T]
mu_0 = 4*pi*1e-7; % magnetic constant [T.m/A]

% get current from mag field max at cusp
%
CURRENT =2*B_max*R/mu_0 ;
...

Minor diameter of coils was 10% in WB 6, and ~ 25% in WB4 (based on measurements from a picture). I assume a larger minor diameter is an advantage within limits as it provides for more windings and insulating, and cooling layers. Of course weather copper wire or superconductors are used will affect the volume that is dedicated to the various elements.

I have no idea weather there is an ideal size from a magnetic field geometry standpoint. A thicker crossection might 'crowd' the center cusp more making it smaller (?). Also, the cusps might be longer, allowing for more tolorance to ion travel into the cusps befor they see the positive charge. Would the cusp throats be tighter or more open? Would there be more transport losses to the magnetic cases? Would a realatively smaller wiffleball result? Would the convex wiffleball surface be closer to a spikey sphere?


Dan Tibbets