Significance of Electron Recirculation Revisited
Significance of Electron Recirculation Revisited
My understanding of the contribution of electron recirculation in the operation of the Polywell is hopefully improved.
I have generally concieved that electron recirculation improves effective electron lifetimes by a factor of ~ 10, based on the 10 X improvement between WB4 and WB6. Thus WB4 has an assumed lifetime of 10,000 transits and WB6 has 100,000 transits before the electron is lost. In a 1 meter diameter wiffleball, and 10,000,000 meter/ second speeds this translates to ~ 1 millisecnd lifetimes for the nonrecirculated electron and ~ 10 ms for the recirculated electron. I'm assuming that the thermalization time is dependant on the unrecirculated lifetime, as recirculation resets the electron to the original energy and direction. In WB6, under these conditions ansd assumptions the unrecirculated electron lifetime would be ~ 0.2 milliseconds (10,000 transits of ~ 0.2 meter wide wiffleball at 10,000,000 m/s)).
But, it occured to me that I was probably assigning the wrong unrecirculated lifetime to WB4. 1000 transit wiffleball traping factors have been mentioned by Bussard, etel. So, 10,000 transits in WB4 actually includes ~ 10X recirculation factor. I had perhaps foolishly ignored this and calculated the lifetime of unrecirculated electrons as 10,000 transits. WB 4 does apparently have significant recirculation occuring, its just that WB6 was 10X better (100X overall) due to the design changes.
Transit lifetimes due to:
WB alone = 1,000
WB + WB4 recirculation = 10,000
WB + WB6 recirculation = 100,000
The significance of this is that the lifetimes of the unrecirculated electrons in a 1 meter machine would be 0.1 milliseconds, rather than 1 millisecond.
This provides only one tenth the time for thermalization to occur and relaxes the restraints on arguments that claim thermalization times exceeds the electrons unrecirculated lifetimes.
Dan Tibbets
I have generally concieved that electron recirculation improves effective electron lifetimes by a factor of ~ 10, based on the 10 X improvement between WB4 and WB6. Thus WB4 has an assumed lifetime of 10,000 transits and WB6 has 100,000 transits before the electron is lost. In a 1 meter diameter wiffleball, and 10,000,000 meter/ second speeds this translates to ~ 1 millisecnd lifetimes for the nonrecirculated electron and ~ 10 ms for the recirculated electron. I'm assuming that the thermalization time is dependant on the unrecirculated lifetime, as recirculation resets the electron to the original energy and direction. In WB6, under these conditions ansd assumptions the unrecirculated electron lifetime would be ~ 0.2 milliseconds (10,000 transits of ~ 0.2 meter wide wiffleball at 10,000,000 m/s)).
But, it occured to me that I was probably assigning the wrong unrecirculated lifetime to WB4. 1000 transit wiffleball traping factors have been mentioned by Bussard, etel. So, 10,000 transits in WB4 actually includes ~ 10X recirculation factor. I had perhaps foolishly ignored this and calculated the lifetime of unrecirculated electrons as 10,000 transits. WB 4 does apparently have significant recirculation occuring, its just that WB6 was 10X better (100X overall) due to the design changes.
Transit lifetimes due to:
WB alone = 1,000
WB + WB4 recirculation = 10,000
WB + WB6 recirculation = 100,000
The significance of this is that the lifetimes of the unrecirculated electrons in a 1 meter machine would be 0.1 milliseconds, rather than 1 millisecond.
This provides only one tenth the time for thermalization to occur and relaxes the restraints on arguments that claim thermalization times exceeds the electrons unrecirculated lifetimes.
Dan Tibbets
To error is human... and I'm very human.
Please define electron lifetime. Where do they come from. Where and how do they live. And where do they go at the end of this lifetime so that they avoid this thermalization you speak off.
Also based on lifetime and influx you should be able to calculate how many at any given moment are living inside or in reverse assuming how many there are (having a ballpark figure from electrostatic simulations and the Bussard's 1e6:1e6+1 ion/electron ratio should be easy for this)  what's the required influx/creation rate (lower limit).
Also from the image coil theory you can estimate the amount of charge needed to create the wiffleball. One more ballpark number to compare things with.
If you get sane numbers that add up with each other  I willl congratulate you. Write your reasoning down into a nice latex document and attach to some document library as pdf.
Or, well, you can just conceive and declare what all those numbers are.
Am I being too nosy?
Also based on lifetime and influx you should be able to calculate how many at any given moment are living inside or in reverse assuming how many there are (having a ballpark figure from electrostatic simulations and the Bussard's 1e6:1e6+1 ion/electron ratio should be easy for this)  what's the required influx/creation rate (lower limit).
Also from the image coil theory you can estimate the amount of charge needed to create the wiffleball. One more ballpark number to compare things with.
If you get sane numbers that add up with each other  I willl congratulate you. Write your reasoning down into a nice latex document and attach to some document library as pdf.
Or, well, you can just conceive and declare what all those numbers are.
Am I being too nosy?
Hi Indrek, long time no hear,
you said above;
you said above;
did you figure out how to do this? Did you use our pressure calculation (coil magnetic force) method or something else? I'd be interested to know how ... by email if necessary.Also from the image coil theory you can estimate the amount of charge needed to create the wiffleball
Nothing as complicated to be worthy of email. I assume the image coils produce a "perfect" wiffleball. Most of the counterfield is being produced by electrons. We know how much current is needed assuming some arbitrary ball size (which we think is reasonable). We can assume some sane distribution of the energy/speed of electrons (or we can even take the maximum at first to simply things). Now you have all the variables available to calculate the amount of electrons required to produce the current.
This is not terribly exact science, certainly, but I think it should give the right order of magnitude. And I can't see anything too wrong with this reasoning. It will give us some specific numbers to compare with. Certainly beats "conceiving" numbers. Anyways. If you think this is a bad idea, shoot away. I'm not too involved with this really.
This is not terribly exact science, certainly, but I think it should give the right order of magnitude. And I can't see anything too wrong with this reasoning. It will give us some specific numbers to compare with. Certainly beats "conceiving" numbers. Anyways. If you think this is a bad idea, shoot away. I'm not too involved with this really.
icarus wrote:Hi Indrek, long time no hear,
you said above;did you figure out how to do this? Did you use our pressure calculation (coil magnetic force) method or something else? I'd be interested to know how ... by email if necessary.Also from the image coil theory you can estimate the amount of charge needed to create the wiffleball
OK. Followed most of that but this bit I don't see.
and in this regard I'm thinking we only consider WBs with diameter less than 1/2 to 1/3 of coil diameter ... also just to be clear here, do we first regard only WBs made purely of electrons to begin with and then after that we go to WB that have electron/ion mixture but the same electronic charge excess as the pure electron WB?We know how much current is needed assuming some arbitrary ball size (which we think is reasonable).
Lets see... I think that WB6 was at ~ Beta= 1 with an electron current of ~ 40 amps. That would be a flow of ~ 6 x10^18 electrons/ coulumb X 40 Coulumbs / sec. (=40 amps) = 2.4 x 10^20 electrons per second. That was with a drive voltage of ~ 12,000 volts and a B field of ~ 0.1 Tesla.
I don't know what the average wiffleball diameter was. I'm assuming it was ~2/3 of the magrid diameter or ~20 cm. That would give a volume of ~ 8,000 cc or 8 liters.
If the electrons (with recirculation) lasted ~ 2 milliseconds, then the concentration of electrons at any time would be 2.4 x 10^20/sec x 0.002 sec= 4.8 X 10^17 electrons in a volume of 8 liters or ~ 6 X10^20 electrons per cubic meter.
Hopefully my math and assumptions are correct.
How the electron density would increase with increasing B field is uncertain, though it should increase rapdly if the forth power scaling applies. Theoretically, density in the core could go way up above atmospheric pressures, but pratically the wiffleball traping facter of ~ 1000, or equivalent of 100,000 with recirculation (and the assumption that ion electrostatic confinement is greater than the electron magnetic confinement) limits the concentration due to arcing/ Pashin discharge if the gas/ charged particle concentration outside the magrid is too much. I suspect this external pressure limit is ~ 1 5 microns or ~ 0.00002 atm.
0.00002 X ~100,000 effective traping factor = 2 atm as the maximum obtainable internal density. I have heard mention of densities of ~ 1 X 10^22 particles per cubic meter being obtainable within the wiffleball. This would be well within my presumed pratical limits.
How the electrons are distributed within the wiffleball is another matter. A. Carlson assumes that they would be in a thermalized cloud (I think) with resultant square potential well. If they preserve some portion of thier radial motion over their lifetime there would be a greater time dependant concentration near the center (travelling slower there) and an elliptical potential well would result.
Also, the concentration of ions in various regions is dependant on the thermalization of the radial ions and subsequent periferal aneeling (if it exists). This would significantly effect the fusion rate if there is a tight (moderate?) focus of the ions towards the center. Then there are vertual central anodes, etc to consider.
Dan Tibbets
I don't know what the average wiffleball diameter was. I'm assuming it was ~2/3 of the magrid diameter or ~20 cm. That would give a volume of ~ 8,000 cc or 8 liters.
If the electrons (with recirculation) lasted ~ 2 milliseconds, then the concentration of electrons at any time would be 2.4 x 10^20/sec x 0.002 sec= 4.8 X 10^17 electrons in a volume of 8 liters or ~ 6 X10^20 electrons per cubic meter.
Hopefully my math and assumptions are correct.
How the electron density would increase with increasing B field is uncertain, though it should increase rapdly if the forth power scaling applies. Theoretically, density in the core could go way up above atmospheric pressures, but pratically the wiffleball traping facter of ~ 1000, or equivalent of 100,000 with recirculation (and the assumption that ion electrostatic confinement is greater than the electron magnetic confinement) limits the concentration due to arcing/ Pashin discharge if the gas/ charged particle concentration outside the magrid is too much. I suspect this external pressure limit is ~ 1 5 microns or ~ 0.00002 atm.
0.00002 X ~100,000 effective traping factor = 2 atm as the maximum obtainable internal density. I have heard mention of densities of ~ 1 X 10^22 particles per cubic meter being obtainable within the wiffleball. This would be well within my presumed pratical limits.
How the electrons are distributed within the wiffleball is another matter. A. Carlson assumes that they would be in a thermalized cloud (I think) with resultant square potential well. If they preserve some portion of thier radial motion over their lifetime there would be a greater time dependant concentration near the center (travelling slower there) and an elliptical potential well would result.
Also, the concentration of ions in various regions is dependant on the thermalization of the radial ions and subsequent periferal aneeling (if it exists). This would significantly effect the fusion rate if there is a tight (moderate?) focus of the ions towards the center. Then there are vertual central anodes, etc to consider.
Dan Tibbets
To error is human... and I'm very human.
I don't know, but that has never stoped me before. Bussard describes two methods of creating a wiffleball in one of his papers. In the WB6 expermants the controls were crude. First the magnets were turned on , then the electron guns were fired and the gas puffed. I think the (?) the electron guns were started before the gas puff. With the capaciter discharge powering the electron guns the current rapidly ramped up, aided by the progressive gas ionization providing a path to ground. The electron discharge was through tungsten(?) filiments from car light bulbs. As the tungsten heated up its resistance increased which would have limited electron flow some. Conversely, as the filiments heated up the thermal emmision of electrons would have increased. The picture is cloudy at best. In a graph in one of the papers the electron current started at a few amps, then quickly passed through 40 amps, then up to hundreds to ~ 1000 amps as the system shorted out. Apparently the time in wihch the current was ~ 40 amps lasted ~ 1/4th of a millisecond. Where the gas was introduced in the sequence is uncertain, but it must have been before the Beta=1 stage as the reported neutron rates apparently matched predictions.icarus wrote:OK. Followed most of that but this bit I don't see.and in this regard I'm thinking we only consider WBs with diameter less than 1/2 to 1/3 of coil diameter ... also just to be clear here, do we first regard only WBs made purely of electrons to begin with and then after that we go to WB that have electron/ion mixture but the same electronic charge excess as the pure electron WB?We know how much current is needed assuming some arbitrary ball size (which we think is reasonable).
If Nebel, etel. tested WB7 with ion guns and a more controllable electron current I assume they obtained cleaner, and more percise data.
PS: Guessing from the glow discharge picture of WB7 that was on EMC2's web site, I would guess the wiffleball would have been a little larger  perhaps 1/2 to 2/3 the diameter of the magrid.
Dan Tibbets
To error is human... and I'm very human.
I posit that the WB is mainly generated by electrons (B is proportional to speed (v) of particles, v of ions is much smaller than that of electrons, say 60x for deuterons). Ions are in there but don't really take significant part in forming the countering magnetic fields. Actually. If you run the numbers you'll see that you can't have electrononly plasma generated wiffleball. As it'd blow itself electrostatically apart.
So I assume all the current in image coils is electrons.
Anyways. This all is to get ballpark numbers, not an exact result.
So I assume all the current in image coils is electrons.
Anyways. This all is to get ballpark numbers, not an exact result.
icarus wrote:OK. Followed most of that but this bit I don't see.and in this regard I'm thinking we only consider WBs with diameter less than 1/2 to 1/3 of coil diameter ... also just to be clear here, do we first regard only WBs made purely of electrons to begin with and then after that we go to WB that have electron/ion mixture but the same electronic charge excess as the pure electron WB?We know how much current is needed assuming some arbitrary ball size (which we think is reasonable).
Seems noone volunteered. Big surprise there, huh. Here's the basic calculation from the image coil theory.
http://www.mare.ee/indrek/ephi/tmp/imag ... deriv.pdf
I got 6.4e16. This is the lower bound. And assuming smaller energies, bigger ball  you can get that number slightly bigger.
Now Dibbler got 4.8e17 from his conceiving. I haven't checked those calcs though. But hey. These two numbers are pretty close. Amazing, huh.
Now I gave you three different ways to calculate the same number. We more or less have two done. Now please hack away at the third. I don't have the time.
http://www.mare.ee/indrek/ephi/tmp/imag ... deriv.pdf
I got 6.4e16. This is the lower bound. And assuming smaller energies, bigger ball  you can get that number slightly bigger.
Now Dibbler got 4.8e17 from his conceiving. I haven't checked those calcs though. But hey. These two numbers are pretty close. Amazing, huh.
Now I gave you three different ways to calculate the same number. We more or less have two done. Now please hack away at the third. I don't have the time.
Indrek: not sure that you are showing anything useful here, the lower bound for electron number in the WB maybe something ...
We had some other threads a while back that discussed the possible electron sheath properties on the surface of the plasma. The surface currents generated in the sheath are expected to be a mechanism that can exclude the magnetic field from the plasma, i.e. socalled field "pushback". Lets assume that most all electrons that leave the plasma as losses or to be "recirculated" will originate from the sheath region.
Dan T: In my mind, recirculation has not been welldefined but is a vague, catchall term for unspecified physics that might be helpful in avoiding electron losses. To be more specific lets consider recirculation as the following multistep process:
i) figure the electron flux out of the plasma "ball" and into the sheath (crossfield diffusion times surface area of plasma ball?)
ii) figure the proportion of electrons going into the sheath that then stream along field lines out to the cusps (losscone angle?)
iii) figure proportion of electrons escaping and traversing the cusps that either have their energy electrically recaptured or are redirected back into the plasma via the magnetic field paths.
Now reasoning/equations/experiments and numbers for steps i)iii) are 'all' that's needed to put a face on the recirculation saving angel.
We had some other threads a while back that discussed the possible electron sheath properties on the surface of the plasma. The surface currents generated in the sheath are expected to be a mechanism that can exclude the magnetic field from the plasma, i.e. socalled field "pushback". Lets assume that most all electrons that leave the plasma as losses or to be "recirculated" will originate from the sheath region.
Dan T: In my mind, recirculation has not been welldefined but is a vague, catchall term for unspecified physics that might be helpful in avoiding electron losses. To be more specific lets consider recirculation as the following multistep process:
i) figure the electron flux out of the plasma "ball" and into the sheath (crossfield diffusion times surface area of plasma ball?)
ii) figure the proportion of electrons going into the sheath that then stream along field lines out to the cusps (losscone angle?)
iii) figure proportion of electrons escaping and traversing the cusps that either have their energy electrically recaptured or are redirected back into the plasma via the magnetic field paths.
Now reasoning/equations/experiments and numbers for steps i)iii) are 'all' that's needed to put a face on the recirculation saving angel.
I have seen a number of simulations of gridded devices and they all show oscillation  bunched electrons leaving and entering the reaction area. Not only that  the electrons seem to self coordinate so that the bunches leave an enter the device in synchronization. It seems to be a self ordering property.
This is the actual mechanism of "recirculation".
This is the actual mechanism of "recirculation".
Engineering is the art of making what you want from what you can get at a profit.

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"Actual" seems like a mouthful. And even if something like this is someday shown to happen, you are using the word "recirculation" different than everyone starting with Bussard does.MSimon wrote:I have seen a number of simulations of gridded devices and they all show oscillation  bunched electrons leaving and entering the reaction area. Not only that  the electrons seem to self coordinate so that the bunches leave an enter the device in synchronization. It seems to be a self ordering property.
This is the actual mechanism of "recirculation".
I assume you are talking about electrons "leaving" to spend some time somewhere between the core and the wall, from whence they "enter" again, that is, not following some closed path around the coils or through solid conductors. And I assume you are talking about densities relevant to power production with at least DT. If so, then you better be sending ions in and out with your electrons, or you will be violating Maxwell's equations somewhere.
I dunno. Field lines that intersect with a wall or other structure are not going to recirculate anything  I believe you have come to that conclusion yourself."Actual" seems like a mouthful. And even if something like this is someday shown to happen, you are using the word "recirculation" different than everyone starting with Bussard does.
And there was a long discussion of the term recirculation at NASA Spaceflight (I think around May of 2007) about the meaning of "recirculate". I didn't like it. I assumed (without evidence at the time) that following the field lines was unlikely. The conclusion of that discussion was that oscillation was a better term.
Confirmed by the subsequent simulations I saw.
I think Doc B did not use the best term for what actually happens. He was not always right about everything. He planned for 1 MW/m^2 of heat removal for alpha impingement in a 100 MW pB11 machine. Rick pointed out last year that if the gyroradius of the alphas was small enough there would be no impingement. I have done some BOE calculations on that and have come to the conclusion Rick is correct. Of course it won't be zero. But I think at worst it will be 10% of what Doc B expected. And it might be as little as .01% possibly even less.
I think Doc B was on the right track. I think it is unwarranted to assume that he got everything right.
Engineering is the art of making what you want from what you can get at a profit.
To what degree would the positive HV electric field from the coils balance those electrons, rather than needing ions to do it?Art Carlson wrote:I assume you are talking about electrons "leaving" to spend some time somewhere between the core and the wall, from whence they "enter" again, that is, not following some closed path around the coils or through solid conductors. And I assume you are talking about densities relevant to power production with at least DT. If so, then you better be sending ions in and out with your electrons, or you will be violating Maxwell's equations somewhere.
In theory there is no difference between theory and practice, but in practice there is.
I've been rude, again. My apologies. I'll try to be a better person in the future.
About the third way to calculate the number of electrons in the system. Enter electrostatics. Actually. I have already done all the work we need years ago, so all we need is to get results from
http://www.mare.ee/indrek/ephi/pef8/
Here we depict the net charge as 0.1m radius uniform ball of charge in the middle. We don't know the exact distribution but this is a start. Looking at the potential wellcharge graph.
Suppose we get 1/3 the effective (for ions so that they can't leak out) potential well of the electron speed. That is 4.5e8C by the graph. We have 12KV not 10KV, so lets correct this by factor of 1.2: q=5.4e8. The Bussard provided Tambov constant is 1e6, so the total amount of electrons is q=5.4e2C. To get the number of electrons divide this by 1.6e19. We get N=3.4e17.
So we have three numbers representing three models for total electron count in that experiment using numbers that Bussard provided: 4.8e17, 6.4e16, 3.4e17. All were derived by additional guestimates and assumptions, but it seems their order of magnitude is a match. Changing any of the guestimates won't significantly change the result.
What does this tell us? Firstly, the numbers Bussard gave us fit into various mathematical models  they make sense. Does this prove polywell? No. I bet several of the numbers (like the 1e6) actually come from similar reasoning with similar models. We used them here as inputs  so in reverse.
Now there has been talk about this allelectron wiffleball. With a description: We send in electrons. Then we puff some deuterum in it and it starts to fuse. How large wiffleball can you have with just that charge that made up the potential well. 5.4e8C. If you work my image coil derivation in reverse you'll see the wiffleball we would get would be the size of I dunno, less than a millimeter? And that's absurd.
And if we added all those electrons in, all ~1e17 of them, with no ions, we would get a potential well of billion volts. With 12kV electrons. If that worked, Maxwell would start spinning so fast in his grave we could attach magnets to his corpse and solve all the world's energy problems. So can't do that.
Now how large is the wiffleball. I'd say it is rather larger than smaller. My argument is following. In some of the first pictures I generated,
http://www.mare.ee/indrek/ephi/polywell ... ld_log.png
This depicts the bfield magnitude. Here in places of green to black the bfield is so low, the electrons almost can't see it. It's not strong enough to turn them back. Now how can in those places be this edge of a ball of 12kV electrons (as you said 1/31/2 of coil diameter)? It can't. So it must be larger. Actually one could trace the magnitude of the bfield and based on that make a good numeric lower bound (I love lower bounds it seems;) guestimate where that edge might be. The octave scripts are out there, it's a simple matter if you are interested.
About recirculation. As said we can speak of oscillations which is hinted by both field line topology and individual electron movement simulations. As pointed out by Art, there can't be that much of it or Maxwell would get mad again. Well. maybe there is very little but that little is significant  it's a matter of crude electrostatic simulation to determine the higher bound on how much could possibly be out there. And then see if it tracts with any numbers we have. If anyone wants to try it, go ahead. The alternative (and probably more likely scenario) is of course that ions are there too, but that complicates things quite a bit.
About true recirculation. Making rounds around coils. This can happen if the elctrons penetrate very deep into the magnetic field. But as I see it they get stuck there and thermalize. Form a sheath over the coils, destroy the potential well (it gets flattened down at the sides  coil faces), ions leak out and it's game over for fusion. This is part of the reasons why I can't see polywell working.
This is actually why I entered my very first question  what is the true lifecycle of electrons. How do they leave the system so that they can avoid this "thermalization" everyone seems to be obsessed about? Now in our pipe dreams they hit the coils and go away before bad things(tm) happen. But coils are protected by a powerful jedi forceshield.
So they thermalize, destroy the effective potential well and no fusion can happen. Maybe there is salvation, but I can't see it.
Sorry for the long comment. I know noone wants to read other's comments and only write their own. I'm guilty of that too.
About the third way to calculate the number of electrons in the system. Enter electrostatics. Actually. I have already done all the work we need years ago, so all we need is to get results from
http://www.mare.ee/indrek/ephi/pef8/
Here we depict the net charge as 0.1m radius uniform ball of charge in the middle. We don't know the exact distribution but this is a start. Looking at the potential wellcharge graph.
Suppose we get 1/3 the effective (for ions so that they can't leak out) potential well of the electron speed. That is 4.5e8C by the graph. We have 12KV not 10KV, so lets correct this by factor of 1.2: q=5.4e8. The Bussard provided Tambov constant is 1e6, so the total amount of electrons is q=5.4e2C. To get the number of electrons divide this by 1.6e19. We get N=3.4e17.
So we have three numbers representing three models for total electron count in that experiment using numbers that Bussard provided: 4.8e17, 6.4e16, 3.4e17. All were derived by additional guestimates and assumptions, but it seems their order of magnitude is a match. Changing any of the guestimates won't significantly change the result.
What does this tell us? Firstly, the numbers Bussard gave us fit into various mathematical models  they make sense. Does this prove polywell? No. I bet several of the numbers (like the 1e6) actually come from similar reasoning with similar models. We used them here as inputs  so in reverse.
Also as the image coil model gave us such good (for the scale of assumptions made) numbers it gives us confidence that we are on the right track with it. This model is useful. There may be more treasures in it we just have to estimate out. Also Dan T here can back up his electron lifetime estimates using various mathematical models. That is useful and gives them credibility.icarus wrote:Indrek: not sure that you are showing anything useful here, the lower bound for electron number in the WB maybe something ...
Now there has been talk about this allelectron wiffleball. With a description: We send in electrons. Then we puff some deuterum in it and it starts to fuse. How large wiffleball can you have with just that charge that made up the potential well. 5.4e8C. If you work my image coil derivation in reverse you'll see the wiffleball we would get would be the size of I dunno, less than a millimeter? And that's absurd.
And if we added all those electrons in, all ~1e17 of them, with no ions, we would get a potential well of billion volts. With 12kV electrons. If that worked, Maxwell would start spinning so fast in his grave we could attach magnets to his corpse and solve all the world's energy problems. So can't do that.
Now how large is the wiffleball. I'd say it is rather larger than smaller. My argument is following. In some of the first pictures I generated,
http://www.mare.ee/indrek/ephi/polywell ... ld_log.png
This depicts the bfield magnitude. Here in places of green to black the bfield is so low, the electrons almost can't see it. It's not strong enough to turn them back. Now how can in those places be this edge of a ball of 12kV electrons (as you said 1/31/2 of coil diameter)? It can't. So it must be larger. Actually one could trace the magnitude of the bfield and based on that make a good numeric lower bound (I love lower bounds it seems;) guestimate where that edge might be. The octave scripts are out there, it's a simple matter if you are interested.
About recirculation. As said we can speak of oscillations which is hinted by both field line topology and individual electron movement simulations. As pointed out by Art, there can't be that much of it or Maxwell would get mad again. Well. maybe there is very little but that little is significant  it's a matter of crude electrostatic simulation to determine the higher bound on how much could possibly be out there. And then see if it tracts with any numbers we have. If anyone wants to try it, go ahead. The alternative (and probably more likely scenario) is of course that ions are there too, but that complicates things quite a bit.
About true recirculation. Making rounds around coils. This can happen if the elctrons penetrate very deep into the magnetic field. But as I see it they get stuck there and thermalize. Form a sheath over the coils, destroy the potential well (it gets flattened down at the sides  coil faces), ions leak out and it's game over for fusion. This is part of the reasons why I can't see polywell working.
This is actually why I entered my very first question  what is the true lifecycle of electrons. How do they leave the system so that they can avoid this "thermalization" everyone seems to be obsessed about? Now in our pipe dreams they hit the coils and go away before bad things(tm) happen. But coils are protected by a powerful jedi forceshield.
So they thermalize, destroy the effective potential well and no fusion can happen. Maybe there is salvation, but I can't see it.
Sorry for the long comment. I know noone wants to read other's comments and only write their own. I'm guilty of that too.