Home | Energy | Nuclear | Electricity | Climate Change | Lighting Control | Contacts | Links |
---|

**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**

**LEAD VOLUME:**

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.

Conservation of lead volume gives:

**(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

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.

Home | Energy | Nuclear | Electricity | Climate Change | Lighting Control | Contacts | Links |
---|