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**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

=

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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