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XYLENE POWER LTD.

PLASMA SPHEROMAK:

By Charles Rhodes, P.Eng., Ph.D.

PLASMA SPHEROMAK ENERGY:
Recall that:
Efs = (Bpo^2 / 2 Mu) (K Ro)^3 Pi^2
and
Ett = Ke Efs
= Ke (Bpo^2 / 2 Mu) (K Ro)^3 Pi^2
= Ke (Bpo^2 / 2 Mu) (Rs Rc)^1.5 Pi^2

Subject to determination of Ke as a function of So this equation can be used to accurately evaluate the recoverable energy in a plasma spheromak via measurements of Bpo, Rs and Rc.
 

CONSIDER LINEAR COMPRESSION OF A PLASMA SPHEROMAK:
Consider linear compression of a spheromak from Roa to Rob:
(Ettb / Eta) = [(Bpob)^2 (Rsb Rcb)^1.5] / [(Bpoa)^2 (Rsa Rca)^1.5]
= [(Bpob / Bpoa)^2 (Rob / Roa)^3]
 

EXPERIMENTAL PLASMA SPHEROMAK DATA:
General Fusion has reported spheromak free electron kinetic energies ranging from 20 eV - 25 eV for low energy density spheromaks at the spheromak generator to 400 ev - 500 eV for higher energy density spheromaks at the downstream end of the conical plasma injector. General Fusion reports a spheromak linear size reduction between these two positions of between 4X and 5X. The corresponding observed apparent electron densities rise from 2 X 10^14 cm^-3 to 2 X 10^16 cm^-3. The corresponding observed magnetic field increases from .12 T to 2.4 T to 3 T. At this time this author does not know for certain: where on the spheromak the electron kinetic energy was measured, where on the spheromak the apparent electron density was measured, where on the spheromak the magnetic field was measured or the absolute dimensions of the measured spheromaks and their enclosure.

Hence:
16 < [Ekeb / Ekea] < 25
4 < [Ekeb / Ekea]^0.5 < 5
20 < (Bpob / Bpoa) < 25
400 < (Bpob / Bpoa)^2 < 625
4 < (Roa / Rob) < 5
16 < (Roa / Rob)^2 < 25
64 < (Roa / Rob)^3 < 125
[(Nea / Roa^3) / (Neb / Rob^3)]^2 = 10^-2

[(Bpob / Bpoa)^2 (Rob / Roa)^3]min < (Ettb / Etta) < [(Bpob / Bpoa)^2 (Rob / Roa)^3]max
or
[400 / 125] < (Ettb / Etta) < [625 / 64]
or
3.2 < (Ettb / Etta) < 9.76

This is the possible range of plasma energy gain via spheromak compression realized by General Fusion Inc. as indicated by the above described experimental results.
 

ELECTRON KINETIC ENERGY OF A PLASMA SPHEROMAK:
The web page CHARGE HOSE PROPERTIES shows that for a plasma spheromak:
(Ni - Ne)^2 C^2 = (Ne Ve)^2
where Ve = electron velocity.

The kinetic energy Eke of a free electron with mass Me is given by:
Eke = (Me / 2) Ve^2

Hence:
(Ni - Ne)^2 C^2 = Ne^2 (2 Eke / Me)
or
Qs = Q (Ni - Ne)
= Q (Ne / C) (2 Eke / Me)^0.5

Consider a plasma spheromak compressed from state "a" to state "b".
(Qsb / Qsa) = (Neb / Nea)(Ekeb / Ekea)^0.5
= (Neb / Rob^3) (Roa^3 / Nea) (Rob / Roa)^3 (Ekeb / Ekea)^0.5

During spheromak compression Qsb = Qsa, giving:
(Neb / Rob^3) (Roa^3 / Nea) (Rob / Roa)^3 (Ekeb / Ekea)^0.5 = 1

This equation indicates that there is something mildly wrong with the General Fusion Inc. measurements of free electron concentration before and/or after spheromak compression. The problem is likely due to the non-uniform spacial free electron concentration within the spheromak.
 

NUMERICAL EVALUATION OF FIELD ENERGY FOR A TYPICAL UNCOMPRESSED PLASMA SPHEROMAK:
Typically:
(Rs / Rc) = 4
Rc = 1.5 m / (2.71 X 4) = 0.1384 m
Q = 1.602 X 10^-19 coulombs
Eke = 30 eV = 30 X 1.602 X 10^-19 J = 48.06 X 10^-19 J
Nr = (3 / 5) = 0.6
Mu = 4 Pi X 10^-7 T^2 m^3 / J
Me = 9.1 X 10^-31 kg
Ne = 10^17 free electrons

Evaluating terms:
(Lp / Lt)
= 2 Pi [Rc + (Rs - Rc) / 2] / Pi (Rs - Rc)
= 2 [5 Rc / 2] / (3 Rc)
= 5 / 3
= 1.66666

{(Nr Lp / Lt)^2 / [(Nr Lp / Lt)^2 + 1]}^2
= {(1)^2 / [(1)^2 + 1]}^2
= {1 / 2}^2
= .25

(Rs / Rc)^1.5 [(Rs + Rc) / (Rs - Rc)]^2 [(1 + (Rc / Rs)^2]^2
= 8 X 2.77777 X 1.1289
= 25.087

Ne^2 (Eke / Me) {(Mu Q^2) / (4 Pi^2 Rc)}
= 10^34 (48.06 X 10^-19 J / 9.1 X 10^-31 kg) X 10^-7 T^2 m^3 / J X 2.5664 X 10^-38 coul^2 / [3.14159 (0.1384 m)]
= 31.173 X 10 J coul^2 T^2 m^3 /J kg m
= 311.73 coul^2 T^2 m^2 / kg
= 311.73 (kg / s)^2 m^2 / kg
= 311.73 kg m^2 / s^2
= 311.73 J

Hence the corresponding value of Efs is given by:
Efs = {(Nr Lp / Lt)^2 / [(Nr Lp / Lt)^2 + 1]}^2
X (Rs / Rc)^1.5 [(Rs + Rc) / (Rs - Rc)]^2 [(1 + (Rc / Rs)^2]^2
X Ne^2 (Eke / Me) {(Mu Q^2) / (4 Pi^2 Rc)}

= 0.25 X 25.087 X 311.73 J
= 1955 J

Clearly Ne ~ 10^17 free electrons in order to get ~ 1955 joules of energy into a practical uncompressed plasma spheromak with:
Rc = 0.1384 m.
and
Rs = 0.5536 m

Note that for the PIF process to operate as designed this spheomak must be compressed to deliver a field energy of at least 3000 J.

Hence a spheromak can carry sufficient energy but not sufficient D-T ions for the subsequent PIF process steps. Neutral D-T gas injection is required after spheromak injection to increase the number of D-T ions available for fusion.
 

FIND Bpo:
Recall that:
Efs = Uo Rc^3 (Rs / Rc)^1.5 Pi^2
and
Uo = Bpo^2 / 2 Mu

Hence:
Efs = (Bpo^2 / 2 Mu) Rc^3 (Rs / Rc)^1.5 Pi^2

Rearranging gives for a typical plasma spheromak:
Bpo^2 = [2 Mu Efs] / [Rc^3 (Rs / Rc)^1.5 Pi^2]
= [2 X 4 Pi X 10^-7 T^2 m^3 / J X 1955 J] / [(0.1384 m)^3 X 8 Pi^2]
= [10^-7 T^2 m^3 / J X 1955 J] / [2.651 X 10^-3 m^3 X Pi]
= 1.955 X 10^-4 T^2 m^3 / 8.328 X 10^-3 m^3
= .02443 T^2

Hence:
Bpo = 0.1563 T

There may be an additional concern with the radial electric field magnitude at the plasma injector wall.
 

This web page last updated September 1, 2016.

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