XYLENE POWER LTD.

PLASMA SPHEROMAKS

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

PLASMA SPHEROMAKS:

Semi-stable spheromaks can also form in plasmas.

A semi-stable plasma spheromak forms when there are plasma electrons spiraling around a quasi-toroidal shaped closed path.

The spheromak mathematical model accurately predicts the experimentally observed geometry of plasma spheromaks.

A plasma spheromak is also known as a toroidal plasma, a compact toroid or an electron spiral toroid.

Plasma spheromaks result from free electrons and ions following a stable three dimensional closed path that forms a sheet in the shape of an ellipsoid. This sheet, also known as the spheromak wall, has a net charge. The current path has both poloidal and toroidal motion components. The current makes Np revolutions around the main spheromak axis of symmetry and Nt revolutions around the toroidal axis before retracing its path. Plasma spheromaks have been generated and photographed in a laboratory. The shape of a laboratory plasma spheromak may be slightly distorted due to external electric and magnetic fields or due to the proximity of an enclosing vacuum chamber wall. The image below shows a plasma spheromak photograph made by General Fusion Inc.

This photograph shows that for this experimental spheromak the ratio of outside surface radius Rs to inside core radius Rc is about:
So^2 = (Rs / Rc) = 4.0
 

ADIABATIC COMPRESSION:
Adiabatic compression is an important step in raising the density and temperature of a deuterium-tritium random plasma to the levels required for thermal nuclear fusion to occur.

An adiabatic compression is a compression in which all of the energy delivered via the compression remains in the compressed substance. Adiabatic compression is a practical means of raising the thermal energy per particle (temperature) of a random plasma provided that the number of particles remains constant. The General Fusion MTF process and the Micro Fusion International PIF process both rely on adiabatic compression by high velocity spherically convergent liquid lead for second stage heating of a deuterium-tritium plasma.

This web page addresses three progressively more complex situations, adiabatic compression of a gas by an ideal wall, adiabatic compression ofa plasma by an ideal wall and adiabatic compression of a plasma by a real liquid lead wall.
 

DEFINITIONS:
Ekdf = deuterium ion kinetic energy at start of random plasma adiabatic compression;
Ekdg = deuterium ion kinetic energy at fusion ignition;
Ekdh = deuterium ion kinetic energy at end of random plasma adiabatic compression;
Rif = liquid lead sphere inside radius at start of adiabatic compression;
Rig = liquid lead sphere inside radius at fusion ignition;
Rik = liquid lead sphere inside radius at end of fusion;
Rii = liquid lead sphere inside radius when liquid lead has no radial velocity
Rof = liquid lead sphere outside radius at commencement of adiabatic compression
Rog - liquid lead sphere outside radius at fusion ignition;
Fr = fraction of plasma ions that has reacted during the current fusion energy pulse.

In the Plasma Impact Fusion (PIF) process the adiabatic compression of the random plasma starts at the largest practical value of Rif to maximize the plasma particle energy gain.
 

DEVELOPMENT OF RANDOM PLASMA COMPRESSION:
The derivation of the increase in particle kinetic energy resulting from adiabatic compression of a gas provides insight into the derivation of the increase in particle kinetic energy resulting from adiabatic compression of a random plasma. Hence we will start with the simpler case of a gas.
 

ADIABATIC COMPRESSION OF A GAS:
Consider Nn neutral particles each with mass Mn randomly bouncing around inside a container with rigid walls with wall area Av enclosing a volume Vol. The kinetic energy Ek of a particle is given by:
Ekn = (Mn / 2) (Vx^2 + Vy^2 + Vz^2)
where velocity components:
Vx, Vy, Vz
are orthogonal to one another.

Since the particle motion is random, on average:
(Ekn / 3) = (Mn / 2) Vx^2

The average number of particles per unit volume is:
Nn / Vol

Consider an element of volume dVol of thickness dRi located just inside the walls of the container. Hence:
dVol = Av dRi

The number of particles in volume dVol is:
(Nn / Vol)Av dRi

The number of wall strikes per unit time due to particles inside the element of volume dVol is:
(Nn / Vol) Av dRi(Vx / (2 dR))
where the divide by 2 arises because half the particles inside dVol are moving away from the wall.

The change in momentum per unit time (force) imparted to the wall by particles elastically bouncing off it is:
(Nn / Vol) Av dRi (Vx / (2 dRi))(2 Mn Vx)

The internal pressure P exerted on the wall is:
P = (Nn / Vol) Av dRi (Vx / (2 dRi))(2 Mn Vx)(1 / Av)
= (Nn / Vol) 2 (Mn Vx^2 / 2)
= (Nn / Vol) 2 (Ekn / 3)

By definition, during an adiabatic compression:
d(Nn Ekn) = - P dVol
= - (Nn / Vol) 2 (Ek / 3) dVol

Assume that Nn remains constant during the compression. Then:
dEkn / Ekn = - (2 / 3) (dVol / Vol)

Integrating from state "a" to state "b" gives:
Ln(Eknb / Ekna) = - (2 /3) Ln(Volb / Vola)
or
(Eknb / Ekna) = (Vola / Volb)^(2 / 3)

For the special case of a sphere:
Vola / Volb = Ria^3 / Rib^3
or
(Eknb / Ekna) = (Vola / Volb)^(2 / 3)
= (Ria^3 / Rib^3)^(2 / 3)
= (Ria / Rib)^2

Hence for the special case of a sphere containing a constant number of randomly moving neutral gas molecules the equation for adiabatic compression simplifies to:
(Ekn / Eknf) = (Rif / Ri)^2
where:
Ekn = particle kinetic energy after compression;
Eknf = particle kinetic energy before compression,
Rif = sphere radius before compression;
Ri = sphere radius after compression
 

ADIABATIC COMPRESSION OF A RANDOM PLASMA:
A plasma consists of a mixture of ions and free electrons. A random plasma consists of a volume of neutral plasma surrounded by a sheath with a positive space charge. Within the neutral plasma the number Ne of free electrons equals the number Ni of ions.

Assume that the volume of the neutral plasma is much greater than the volume of the charge sheath.

Define:
Ne = number of plasma free electrons;
Nd = number of plasma deuterium ions;
Nt = number of plasma tritium ions;
Me = electron rest mass;
Md = deuterium ion rest mass;
Mt = tritium ion rest mass;
Eke = kinetic energy of a free electron in the neutral plasma;
Ekd = kinetic energy of a deuterium ion in the neutral plasma;
Ekt = kinetic energy of a tritium ion in the neutral plasma;
Ve = velocity of a free electron in the neutral plasma;
Vd = velocity of a deuterium ion in the neutral plasma.

Hence, the total kinetic energy in the neutral plasma is:
[(Ne (Me / 2) Ve^2) + (Nd (Md / 2) Vd^2) + (Nt (Mt / 2) Vt^2)]

In the neutral plasma Ve is random. Hence:
Eke = Me Ve^2 / 2
= (Me / 2) (Vex^2 + Vey^2 + Vez^2)
= (Me / 2) (3 Vex^2)
or
Me Vex^2 = (2 Eke / 3)

In the neutral plasma Vd is random. Hence:
and
Ekd = Md Vd^2 / 2
= (Md / 2) (Vdx^2 + Vdy^2 + Vdz^2)
= (Md / 2) (3 Vdx^2)
or
Md Vdx^2 = (2 Ekd / 3)

In the neutral plasma Vt is random. Hence:
and
Ekt = Mt Vt^2 / 2
= (Mt / 2) (Vtx^2 + Vty^2 + Vtz^2)
= (Mt / 2) (3 Vtx^2)
or
Mt Vtx^2 = (2 Ekt / 3)

The density of ions in the neutral plasma is:
(Nd + Nt) / Vol

The density of electrons in the neutral plasma is:
Ne / Vol

Consider an element of volume dVol of area Av at the surface of the neutral plasma.
dVol = Av dR

The number of ions in element of volume dVol is:
((Nd + Nt) / Vol) Av dR

The number of ions entering the charge sheath per unit time is:
(Nd / Vol) Av dR (Vdx / 2 dR) + (Nt / Vol) Av dR (Vtx / 2 dR)

The momentum imparted to the charge sheath (and hence the liquid lead wall) per unit time is:
(Nd / Vol) Av dR (Vdx / 2 dR) (2 Md Vdx) + (Nt / Vol) Av dR (Vtx / 2 dR) (2 Mt Vtx)
= (Nd / Vol) Av Md Vdx^2 + (Nt / Vol) Av Mt Vtx^2

Hence the pressure Pi exerted by ions on the charge sheath is:
Pi = (Nd / Vol) Md Vdx^2 + (Nt / Vol) Mt Vtx^2
= (Nd / Vol) (2 Ekd / 3) + (Nt / Vol) (2 Ekt / 3)

Similarly, the pressure Pe exerted by electrons on the charge sheath is:
Pe = (Ne / Vol) Me Vex^2
= (Ne / Vol) (2 Eke / 3)

By definition of adiabatic compression:
d(Ne Eke + Nd Ekd + Nt Ekt) = - (Pe + Pd + Pt) dVol

Assume that the reaction chamber is fully closed before commencement of
adiabatic compression so that during the adiabatic compression:
Ne = Nd + Nt = constant
Then:
d(Eke + Ekd + Ekt)
= - [(1/ Vol) (2 Eke / 3) + (1 / Vol) (2 Ekd / 3) + (1 / Vol) (2 Ekt / 3) ] dVol
= - [(2 / 3 Vol) (Eke + Ekd + Ekt)]dVol

Rearranging gives:
d(Eke + Ekd + Ekt) / (Eke + Ekd + Ekt) = - [2 dVol / 3 Vol]

Let:
Vol = (4 / 3) pi R^3
and
dVol = 4 Pi R^2 dR

Hence:
d(Eke + Ekd + Ekt) / (Eke + Ekd + Ekt) = - [2 dVol / 3 Vol]
= - (2 / 3) [(4 Pi R^2 dR) / ((4 / 3) Pi R^3)]
= - 2 dR / R

Hence:
(Ekeh + Ekdh + Ekth) / (Ekef + Ekdf + Ektf) = (Rf / Rh)^2
= (Ekh / Ekf)

VELOCITY RELATIONSHIP:
Recall that:
Ektd = (Mt / 2) Vtd^2
and
Ekth = (Mt / 2) Vth^2
giving:
Ekth / Ektd = (Vth / Vtd)^2

Thus for a random plasma compressed under adiabatic conditions:
Ekh / Ekd = (Rid / Rih)^2 = (Vth / Vtd)^2
or
(Rid / Rih) = (Vth / Vtd)
and
Ekd = (Rh / Rd)^2 Ekh
= (Vtd / Vth)^2 Ekh
 

LIMITATIONS ON ADIABATIC COMPRESSION OF A RANDOM DEUTERIUM-TRITIUM PLASMA:
There are two fundamental limitations on adiabatic compression of a plasma with a liquid lead wall. The first limitation relates to the negative radial velocity of the liquid lead wall required to prevent loss of plasma energy to the wall. The second limitation relates to the deuterium ion kinetic energy that causes sputtering of lead atoms into the plasma.
 

LIQUID LEAD RADIAL WALL VELOCITY Vl:
A key issue is the inward radial velocity that the liquid lead wall must have at the instant of spheromak randomization. A spheromak can exist in an enclosure with cool walls but a random plasma will rapidly cool unless the inward wall velocity (-dRi / dT) is sufficient to provide adiabatic conditions.

The minimum liquid lead wall velocity requirement is set by the maximum particle mass of the particles being compressed, which is the tritium ion mass.

In order for the compression to be adiabatic every collision between a lead atom and a tritium ion must transfer momentum and hence energy from the lead atom to the tritium ion, not vice versa. In order for that energy transfer to occur, as a result of a tritium ion-lead atom collision the lead atom must always transfer inward radial momentum to the deuterium ion. Define parameters as follows:
Ml = mass of a lead atom
Mt = mass of a tritium atom
Vl = inward radial velocity of lead atoms
Vtw = maximum radial velocity of a tritium ion at the liquid lead wall

The law of conservation of linear momentum gives the condition for inward radial momentum transfer and hence adiabatic compression of a random plasma as:
Ml Vl > Mt Vtw
or
Vl > (Mt / Ml) Vtw

Assume that the ions and electrons in the neutral plasma are in energy equilibrium. At the liquid lead wall the kinetic energy of a tritium ion is the sum of the tritium ion kinetic energy Ekt in the neutral plasma plus another Ekt that the the ion acquires in traversing the plasma charge sheath. Hence the maximum kinetic energy Ektw of a tritium ion at the liquid lead wall is given by:
Ektw = 2 Ekt
= (Mt / 2)(Vtw^2)

Rearranging this equation gives:
Vtw = (4 Ekt / Mt)^0.5

Recall that for adiabatic compression:
Vl > (Mt Vtw / Ml)

Substitution of Vtw into this inequality gives for lead atoms compressing tritium ions:
Vl > (Mt / Ml)(4 Ekt / Mt)^0.5
= (2 / Ml) (Ekt Mt)^0.5
= (2 / Ml)(Mt^2 Vt^2 / 2)^0.5
= (Mt / Ml)(2^0.5) Vt
where:
Mt = 3.0 X 1.67 X 10^-27 kg
and
Ml = 207.2 X 1.67 X 10^-27 kg

However:
Vl = (-dRi / dT)
giving:
(-dRi / dT) > (2 / Ml)(Ekt Mt)^0.5

This inequality imposes constraints on (-dRi / dT) both at the commencement of the adiabatic compression and at the end of the adiabatic compression.
 

COMMENCEMENT OF ADIABATIC COMPRESSION:
At the commencement of the adiabatic compression soon after randomization the injected plasma will interact with the liquid lead wall and tend to take on the ion Ekt value defined by:
(-dRi / dT) = (2 / Ml)(Ekt Mt)^0.5
or
Ekt = (Ml^2 / 4 Mt) (-dRi / dT)^2
= [(207.2 X 1.67 X 10^-27 kg)^2 / (4 X 3.0 X 1.67 X 10^-27 kg)] [300 m / s]^2
= [(207.2^2 X 1.67 X 10^-27 kg) / (4 X 3.0)] [300 m / s]^2
= 5.377213 X 10^-19 J X (1 eV / 1.602 X 10^-19 J)
= 3.3565 eV

Thus for:
Rif = 1.0 m
the tritium ion kinetic energy at:
Rih = .0053329 m
is limited to:
3.3565 eV X (1.00 m / .0053329)^2 = 118,021 keV
 

END OF ADIABATIC COMPRESSION:
At the end of the adiabatic compression range at fusion conditions (- dRi / dT) goes to zero. Hence the fusion end of the adiabatic compression range is at:
(-dRi / dT) = (2 / Ml)(Ekt Mt)^0.5
or
(dRi / dT)^2 = (2 / Ml)^2 (Ekt Mt)

From the web page titled: SPHERICAL COMPRESSION A the liquid lead kinetic energy Ekl near fusion is given by:
Ekl = (dRi / dT)^2 [Rhol 2 Pi Ri^3]
or
(dRi / dT)^2 = Ekl / [Rhol 2 Pi Ri^3]

Equating these two expressions for (dRi / dT)^2 gives:
Ekl / [Rhol 2 Pi Ri^3] = (2 / Ml)^2 (Ekt Mt)
or
Ekl = (2 / Ml)^2 (Ekt Mt)[Rhol 2 Pi Ri^3]

However:
Ekl = Ekld - Ep
= Ekld - Epf (Rif / Ri)^2
and Eph = Ekld / 2 = Epf(Rif / Rih)^2

Thus:
Epf = (Ekld / 2)(Rih / Rif)^2
giving:
Ekl = Ekld - (Ekld / 2)(Rih / Rif)^2 (Rif / Ri)^2
= Ekld - (Ekld / 2)(Rih / Ri)^2
= Ekld [1 - (1 / 2) (Rih / Ri)^2]

Equating the two expressions for Ekl gives at the fusion end of the adiabatic compression region:
Ekld [1 - (1 / 2) (Rih / Ri)^2] = (2 / Ml)^2 (Ekt Mt)[Rhol 2 Pi Ri^3]

Within the adiabatic compression range:
Ekt = Ekth (Rih / Ri)^2

Thus at the fusion end of the adiabatic compression range:
Ekld [1 - (1 / 2) (Rih / Ri)^2] = (2 / Ml)^2 (Ekth (Rih / Ri)^2 Mt)[Rhol 2 Pi Ri^3]
= (2 / Ml)^2 (Ekth Rih^2 Mt Rhol 2 Pi Ri)
= 8 Pi (Mt / Ml) (Ekth Rih^2 Rhol Ri) / Ml

At Ri = Rih:
LHS = (Ekld / 2)
= 65.5 MJ / 2
= 32.75 MJ and
RHS = 8 Pi (Mt / Ml) (Ekth Rih^3 Rhol) / Ml
= 8 Pi (3 / 207.2)(120,000 eV X 1.602 X 10^-19 J / eV)(.0053329 m)^3 X 10.66 X 10^3 kg / m^3
/ (207.2 X 1.67 X 10^-27 kg)
= 0.1130999 X 10^-16 J kg / 346.02 X 10^-27 kg
= 3.26859 X 10^-4 X 10^11 J
= 32.6859 X 10^6 J

Thus the compression is adiabatic in the region:
Rih < Ri < Rif
 

POTENTIAL PROBLEMS WITH THE ADIABATIC COMPRESSION:
A major issue with the adiabatic compression is that it will not work if the number of confined particles increases as the compression progresses. Hence at the commencement of the compression the plasma must be fully ionized and there must be no exposed lithium atoms at the liquid lead surface which can sputter off the lead and increase the number of plasma particles. Hence lithium cannot be alloyed into the liquid lead to breed tritium. Tritium for the fusion reaction must be realized by a different means.
 

This web page last updated January 23, 2015.