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**LIQUID LEAD SPHERE FORMATION:**

The choice of initial liquid lead shell inside diameter of **2.9 m** and initial liquid lead shell wall thickness of **0.0052 m** arises from simultaneously meeting the minimum liquid lead shell size necessary for deuterium-tritium fusion while maintaining pressure vessel component sizes that are practical to transport.

Practical issues related to performance of the required fusion energy pulse containment spherical pressure vessel set its inside diameter at **4.4 m** and its outside diameter at about **4.8 m**. Larger pressure vessels can in principle be fabricated but there are numerous practical transportation constraints imposed by bridges, tunnels and highway overpasses.

The radially converging hollow liquid lead shell is realized by using **212 synchronized liquid lead guns**, each **6.0 m** long, that are positioned in a spherical array around the spherical pressure vessel. These guns shoot **0.4 m diameter** (~ 16" diameter) disk shaped liquid lead projectiles on a radially converging path. The average length of each projectile is **5.2 mm.**

To ensure positive liquid lead gravity drainage along the gun barrel tubes there are no gun barrels on the equatorial plane. The guns are arranged in rows like lines of latitude. The north and south hemispheres are mirror images of each other. With reference to the northern hemisphere the row closest to the equator has **25 guns**. The next row has **24 guns**. The next row has **21 guns**. The next row has **17 guns**. The next row has **12 guns**. The next row has **6 guns**. Then there is **1 composite gun** made from 4 little guns that form the pole plug. This pattern repeats in the southern hemisphere for a grand total of **212 equivalent full size guns**. Due to the composite guns there are actually **218** gun control mechanisms.

The liquid lead projectiles form a closed spherical shell, initially **2.9 m** inside diameter, that causes spherically symmetric compression of the contained plasma. It is essential that the spheromak/FRC lifetime **Ts** be longer than the liquid lead shell formation time. To minimize the liquid lead shell formation time the distance between the gun muzzles and the initial inside position of the liquid lead sphere is set at its minimum practical value of **0.55 m**. Due to spheromak injection requirements, a smaller distance between the pneumatic gun muzzles and the liquid lead shell is not practical. Note that the gun muzzles project about **0.2 m** into the pressure vessel sphere beyond its inner wall to provide a recess for the lithium containing blanket and liquid lead circulation and drainage.

Due to the axial spheromak insertion ports there are no gun barrels on the pressure vessel sphere axis. To provide the pole caps for the liquid lead sphere the liquid lead used to form the pole caps is propelled at an angle off the polar radial path. In order to maintain the required spherical radial velocity component the muzzle velocity of the composite pole cap forming guns must be higher than the muzzle velocity of the other guns. Hence the pole cap forming gun barrels should be longer than the other gun barrels. The extended pole cap forming barrels give the finished pressure vessel assembly the appearance of being slightly ellipsoidal. A pole cap is formed by directing four 0.200 m diameter slugs of liquid lead that converge at the equivalent gun muzzle position. These guns are designed so that at the equivalent shell position the net tangential liquid lead momentum is zero and the net radial liquid lead momentum is equal to a standard gun.

**GUN DETAIL:**

The 6.0 m long gun barrels are field assembled to the pressure vessel sphere. The guns are designed to minimize the transient pressure stress on the walls of the spherical pressure vessel.

Each gun contains 631 0.5 inch OD tubes. Each gun uses a fixed amount of compressed inert gas to accelerate the liquid lead through the tubes.

After gun discharge the gun tubes assist in dissipation of the liquid lead vapor pressure.

Holes through the 16 inch diameter cap behind each gun allow for: liquid lead speed loading, rapid and precise measurement of liquid lead axial position, evacuation of the cavity behind each gun and controlled application of propulsion pressure.

There is an automatic apparatus for speed loading a precise quantity of liquid lead into each gun tube an instant before the gun fires.

**FUSION ENERGY PULSE:**

The fusion energy release is violent. The fusion energy pressure pulse must be safely contained. The inner portion of the liquid lead shell wall will vaporize due to high energy alpha particle impact and the shell will expand radially. Liquid lead will fly everywhere. The hot lead vapor must be cooled and condensed.

After the lead vapor condenses the remaining gas molecules must be vacuum extracted, separated and recycled. The heat due to high energy neutron scattering must be removed by pumpingliquid lead through an isolated heat exchanger.

The pressure vessel metallurgy is complicated by high temperature high vacuum sealing requirements, neutron irradiation, wear resistance, and potential hydrogen cracking.

**PRESSURE CALCULATIONS:**

The pressure vessel volume is the sum of the contained volume of the pressure vessel sphere, the contained volume of the spheromak injection tubes and the contained volume of the gun tubes.

The nuclear reaction together with the liquid lead compression generates about 600 MJ of high pressure lead vapor.

The contained volume of the pressure vessel, including the gun tubes, is given by:

**(4 / 3) Pi (2.2 m)^3 + (212) (631) (6.93681953 X 10^-5 m^2) (6 m) + (2) Pi (.3 m)^2 (3 m)
= 44.602 m^3 + 55.677 m^3 + 1.696 m^3]
= 101.975 m^3**

Hence the peak pressure following a fusion energy pulse is:

665.5 X 10^6 J / 101.975 m^3

= 6.5261 X 10^6 Pa

= **64.61 bar**

**ENERGY RECOVERY:**

The presently planned maximum gross thermal output is **665.5 MJ** per fusion pulse.

During the high pressure transient accompanying a fusion energy pulse, part of the fusion energy release is captured by pistons and argon reservoir to assist with powering the next liquid lead wall acceleration cycle.

**GUN BARRELS:**

The gun barrels, which have a material yield stress rating of at least 30,000 psi and an outside diameter of 16 inches, have a wall thickness of 0.656 inches which can safely withstand repeated:

14.688 inch X 1 inch X P = 10,000 psi X 2 X .656 inch^2

or

P = (10,000 psi X 2 X .656) / 14.688

= 893.25 psi

= **60.76 bar** internal pressure pulses. However, the pressure pulses on the gun barrels are mitigated by the 631 internal 0.5 inch OD tubes. The manner of attaching of these tubes to each other is of great importance. A high temperature solder or like material is required.

The gun barrels are sloped and are kept at a temperature well above the melting point of the liquid lead to ensure positive drainage of liquid lead from every gun barrel after each fusion energy pulse and to maintain the desired low liquid lead viscosity between the liquid lead and the tube. The gun barrel temperature must be regulated to control the liquid lead alloy viscosity.

**PRESSURE VESSEL SPHERE WALL THICKNESS:**

A major issue in this equipment design is the pressure vessel sphere wall thickness. This wall is pierced with about 210 X 0.4064 m diameter holes for the gun barrels and 2 X 0.6 m diameter holes for spheromak injection and liquid lead pole cap injection. These holes greatly weaken the sphere wall.

If the spherical pressure vessel wall had no holes the required wall thickness **Tw** would be given by:

**6.5261 X 10^6 Pa X Pi X (2.2 m)^2 = Pi X 4.4 m X Tw X 10^4 psi X 1.01 X 10^5 Pa / 14.7 psi**

or

**Tw = [6.5261 X 10^6 Pa X Pi X (2.2 m)^2 X 14.7 psi] / [Pi X 4.4 m X 10^4 psi X 1.01 X 10^5 Pa]
= [6.5261 X 10 X 1.1 m X 14.7] / [10^4 X 1.01]
= 0.10448 m**

However, the total area of the holes through the wall is:

**(212 X Pi X (.2159 m)^2) + (2 Pi X (0.3 m)^2)
= Pi (9.882 m^2 + .18 m^2)
= 31.61 m^2**

The ideal sphere inside surface area is given by:
**4 Pi (2.2 m)^2 = 60.82 m^2**

Hence as a minimum the wall thickness must be increased by a factor of at least:

**60.82 / (60.82 - 31.61 m^2)
= 60.82 / 29.21
= 2.08**

Hence the absolute minimum pressure vessel sphere wall thickness is given by:

2.08 X .10448 m = .217 m.

To allow for casting issues, weak points (such as the spheromak injection ports), machining, erosion, neutron damage consider increasing this value.

Allowing for gun flange fittings approximately equal in volume to the cut out discs, the volume of steel in the basic 0.22 m thick machined spherical pressure vessel is given by:

(4 / 3) Pi [(2.42 m)^3 - (2.2 m)^3]

= (4 / 3) Pi [(14.172 m^3) - (10.648 m^3)]

= **14.763 m^3**

The density of steel is about 7850 kg / m^3. Hence the basic spherical pressure vessel has a mass of about:

14.763 m^3 X 7850 kg / m^3 = 115,892 kg

= **115.89 tons**

To permit transport by railway this weight needs to be reduced to about 100 tons. Hence the maximum sphere wall thickness becomes:

(100 tons / 115.89 tonnes ) X 0.22 m = **0.1898 m**

This thickness may need further adjustment depending on the fitting geometry.

**AXIAL VALVES:**

The pressure vessel has connected to it two rapid acting full port 0.6 m ball valves to isolate the spheromak generator and high vacuum pumps from the pressure vessel. These full port valves must move from being fully open for spheromak injection to being fully closed against liquid lead spraying in about **7 ms**. Practical realization of these valves is a major project.

The plasma injector neck inside diameter, downstream from the axial valves, must be **0.6 m** to minimize field emission current that shortens spheromak lifetime. The wall thickness of the neck pipes will have to be at least 1.00 inch to withstand the pressure pulses.

**THERMAL SHOCK:**

It will require careful pressure vessel design to accommodate the large liquid lead temperature swings without thermal shock damage.

**CYCLE TIME CONSTRAINTS:**

Each fusion energy pulse results in a transient high lead vapor pressure inside the pressure vessel. Practical issues related to lead vapor condensation and subsequent vacuum pump out of leakage argon and other gases from the pressure vessel before the next fusion pulse limit the fusion pulse rate.

Sufficient time must pass for completion of lead vapor condensation before the pressure vessel axial ports are opened. Vacuum extraction of unwanted gas molecules from the apparatus, draining of the liquid lead, gun barrel draining and gun reloading must be complete before formation of new spheromaks is initiated.

This author believes that the target fusion pulse rate of one pulse per second originally contemplated by General Fusion is over optimistic.

**RISK:**

A source of risk that has not yet been addressed is erosion due to liquid lead alloy kinetic energy acquired due to vaporization of liquid lead during the fusion energy pulse. It may difficult to prevent high velocity liquid lead alloy droplets from gradually eroding internal surfaces of the pressure vessel.

Another potential source of risk is problems related to maintaining pressure and vacuum seals in the presence of the high temperatures, high temperature swings and high impulse mechanical stresses prevailing in this energy system.

This web page last updated January 26, 2015.

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