# XYLENE POWER LTD.

## SPHERICAL COMPRESSION PART A

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

SPHERICAL COMPRESSION:
Spherical compression is the technique that is used to achieve the large transient pressure and temperature that are required for triggering thermo nuclear fusion. Spherical compression was originally perfected during WWII as a method of triggering plutonium type atomic bombs.

Spherical compression is used in the General Fusion Magnetized Target Fusion (MTF) process and in the Micro Fusion International Plasma Impact Fusion (PIF) process because liquid lead behaves as an incompressible fluid above its own speed of sound.

Consider a liquid lead shell of outside radius Ro containing a smaller plasma sphere of radius Ri. The two spheres share a common center. The radius values when the liquid lead has formed a closed shell and is ready for deuterium fuel injection are:
Ri = Rid = 1.45 m
and
Ro = Rod

Spheromak injection occurs when:
Ri = Ric

Spherical compression of random plasma occurs when the inside sphere radius Ri shrinks from Rif to Rii, where:
Rif > Ri > Rii

The volume of liquid lead Voll is given by:
Voll = (4 Pi / 3)(Ro^3 - Ri^3)

The volume of lead is unchanged through the compression cycle. Hence:
Voll = (4 Pi / 3)(Rod^3 - Rid^3)
= (4 Pi / 3)(Roi^3 - Rii^3)

However:
Rii ~ 0
which gives:
Roi^3 = Rod^3 - Rid^3

Recall that:
Voll = (4 Pi / 3)(Ro^3 - Ri^3)
Rearrangement of this equation gives:
(3 Voll / 4 Pi) = Ro^3 - Ri^3
or
Ro^3 = (3 Voll / 4 Pi) + Ri^3
or
Ro = [(3 Voll / 4 Pi) + Ri^3]^0.333
Hence:
Rod = [(3 Voll / 4 Pi) + Rid^3]^0.333
and
Roi = [(3 Voll / 4 Pi)]^0.333

KINETIC ENERGY IN THE LIQUID LEAD:
An element of liquid lead volume 4 Pi R^2 dR at radius R has a kinetic energy dEkl given by:
dEkl = [(Rhol 4 Pi R^2 dR) / 2] (dR / dT)^2
where:
Rhol = density of liquid lead.

(dR / dT) = (Ri / R)^2 (dRi / dT)

Substitution of (dR / dT) into the expression for dEkl gives:
dEkl = [(Rhol 4 Pi R^2 dR) / 2] (dR / dT)^2
= Rhol 2 Pi R^2 dR [(Ri / R)^2 (dRi / dT)]^2
= Rhol 2 Pi (Ri^4 / R^2) (dRi / dT)^2 dR.

The kinetic energy of the liquid lead at T = Td, Ri = Rid, Ro = Rod is given by:
Ekld = Integral from R = Rid to R = Rod of:
Rhol 2 Pi (Rid^4) (dRid / dT)^2 dR / R^2
= Rhol 2 Pi Rid^4 (dRid / dT)^2 [(1 / Rid) - (1 / Rod)]
= Rhol 2 Pi Rid^3 (dRid / dT)^2 [(Rod - Rid) / Rod]

In general:
Ekl = Rhol 2 Pi Ri^3 (dRi / dT)^2 [(Ro - Ri) / Ro]
which has important cases:
Ekld = Rhol 2 Pi Rid^3 (dRid / dT)^2 [(Rod - Rid) / Rod]
and
Eklg = Rhol 2 Pi Rig^3 (dRig / dT)^2 [(Rog - Rig) / Rog]

At near fusion conditions:
Rog >> Rig
giving:
Eklg ~ Rhol 2 Pi Rig^3 (dRig / dT)^2

Hence near fusion conditions:
(dRig / dT)^2 ~ Eklg / Rhol 2 Pi Rig^3
or
(dRig / dT) ~ - [Eklg / Rhol 2 Pi Rig^3]^0.5
Notice that near fusion conditions the liquid lead wall velocity is proportional to:
(1 / Ri)^1.5
whereas the ion velocity is proportional to:
(1 / Ri).
Hence if the requirements for adiabatic compression are met at a larger radius they are also met at a smaller radius until Ekl rapidly decreases at Ri = Rih.

Recall that when the liquid lead shell forms:
Ekld = Rhol 2 Pi Rid^3 (dRid / dT)^2 [(Rod - Rid) / Rod]
or
[(Rod - Rid) / Rod] = Ekld / [Rhol 2 Pi Rid^3 (dRid / dT)^2]

For the case of Ekld = 65.5 MJ, (dRid / dT) = 300 m / s, Rid = 1.45 m:
[(Rod - Rid) / Rod]
= 65.5 X 10^6 J / [10.66 X 10^3 kg / m^3 X 2 Pi X (1.45 m)^3 X (300 m / s)^2]
= .003564167
= 1 - (Rid / Rod)

Thus:
(Rid / Rod) = 1 - .003564167
or
Rod = Rid / (1 - .003564167)
or
Rod - Rid = Rid[(1 / (1 - .003564167)) - 1]
= Rid[.003564167 / (1 - .003564167)]
= 1.45 m [.003564167 / .9643583273]
= .005359 m
~ 5.359 mm

QUANTIFICATION OF Voll AND Roi:
The liquid lead kinetic energy is given by:
Ekld = [(Rhol Voll) / 2] (dRid / dT)^2

Rearranging this equation gives:
Voll = 2 Ekld / [Rhol (dRid / dT)^2]
= (2 X 65.5 X 10^6 J) / [10.66 X 10^3 kg / m^3 X (300 m / s)^2]
= 0.1365 m^3

The corresponding value of Roi is given by:
Voll = (4 / 3) Pi Roi^3
or
Roi = (3 Voll / 4 Pi)^0.333
= (3 X 0.1365 m^3 / 4 Pi)^0.333
= (.032587 m^3)^0.333
= (32.587 X 10^-3)^0.333
= 0.3194 m

CONVERGENCE TIME:
Recall that:
Ekl = Rhol 2 Pi Ri^4 (dRi / dT)^2 [(1 / Ri) - (1 / Ro)]
or
(dRi / dT)^2 = Ekl / {Rhol 2 Pi Ri^4 [(1 / Ri) - (1 / Ro)]}
= Ekl / {Rhol 2 Pi Ri^3 [1 - (Ri / Ro)]}
= [Ekl / (Rhol 2 Pi Ri^3)] [Ro / (Ro - Ri)]

or
(dRi / dT) = [Ekl / {Rhol 2 Pi Ri^3]^0.5 [Ro / (Ro - Ri)]^0.5

Separation of variables gives:
dT = dRi {[Rhol 2 Pi Ri^3 / Ekl]^0.5} [(Ro - Ri) / Ro]^0.5

At Ri = Rii:
Roi = 0.3194 m
and
[(Rob - Rib) / Rob]^0.5 = 1

Recall that at Ri = Rid = 1.45 m:
[(Rod - Rid) / Rod]^0.5
= [.003564167]^0.5
= .059700

Recall that:
Voll = (4 Pi / 3)(Ro^3 - Ri^3)
or
Ro^3 = [3 Voll / 4 Pi] + Ri^3
or
Ro = {[3 Voll / 4 Pi] + Ri^3}^0.333
or
[(Ro - Ri) / Ro]
= ({[3 Voll / 4 Pi] + Ri^3}^0.333 - Ri) / {[3 Voll / 4 Pi] + Ri^3}^0.333

Hence:
dT = dRi {[Rhol 2 Pi Ri^3 / Ekl]^0.5} [(Ro - Ri) / Ro]^0.5
= dRi {[Rhol 2 Pi Ri^3 / Ekl]^0.5} [({[3 Voll / 4 Pi] + Ri^3}^0.333 - Ri) / {[3 Voll / 4 Pi] + Ri^3}^0.333]^0.5

This differential equation is difficult to solve except numerically. However, close to fusion conditions:
Ri << Ro
which simplifies the differential equation to:
dT = dRi {[Rhol 2 Pi Ri^3 / Ekl]^0.5}
or
Eklh^0.5 = (dRih / dT) {[Rhol 2 Pi Rih^3]^0.5}
or
Eklh = (dRih / dT)^2 [Rhol 2 Pi Rih^3]

The significance of this equation is that at fusion conditions (dRih / dT) must be sufficient to satisfy the requirement for adiabatic heating of the plasma, which sets a minimum value on (dRih / dT) and hence a minimum value on Eklh.

Eklh is an important parasitic energy load that must be supplied by Ekld in addition to the energy required to heat the plasma.

This web page last updated January 18, 2015.