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

The spherical liquid lead shell is formed within a 4.4 m ID spherical pressure vessel and is the result of joining at the edges of about 212 disk shaped liquid lead slugs that are shot at 300 m / s from flywheel guns. The liquid lead shell surface segments form a closed radially shrinking sphere that is used to confine the spheromaks and the injected neutral deuterium-tritium gas so that deuterium-tritium atoms/ions can not leak out during the subsequent random plasma formation and adiabatic compression.

The liquid lead slugs, which are accelerated by the flywheel guns, must all arrive at Ri = Rid at nearly exactly the same instant in time and with nearly exactly the same negative radial velocity. Practical realization of this liquid lead slug position and velocity matching is a technical challenge but is not beyond the scope of presently available technology.

A related significant issue is synchronizing the liquid lead shell surface formation with the port valves used for spheromak injection into the spherical pressure vessel so that the liquid lead shell forms and closes around the spheromak as soon as possible after spheromak injection and so that the neutral deuterium-tritium gas injection occurs as soon as possible after complete liquid lead shell closure. For the process to work as contemplated the injected spheromak lifetime must be longer than the liquid lead shell closure time (~ 1.833 mS).

In order to minimize the liquid lead shell closure time at the instant of spheromak injection into the spherical pressure vessel the liquid lead slugs must already be in motion at the maximum negative radial velocity of 300 m / s and must at their respective gun barrel muzzles.

The liquid lead flywheel guns have overall barrel lengths of about 6 m. The barrel is not used for liquid lead acceleration. The purpose of the barrel is off set the flywheel guns from the spherical pressure vessel enough to provide the space around the outside ends of the gun barrels required to accommodate the flywheel guns, which are each about 1 m in outside diameter.

The liquid lead is accelerated by the inertia of the flywheels which operate in a vacuum at a temperature above the melting point of liquid lead. Liquid lead is mechanically injected into the hub of a spinning flywheel, collects in a near hub chamber, and then is released by an electric valve and is accelerated while passing through a curved internal channel within the flywheel. As the liquid lead flows along the internal channel from the flywheel's near hub chamber to the flywheel perimeter the flow channel back pressure is adjusted via a change in the flow channel cross sectional area so that the liquid lead remains as a slug and does not become an extended stream.

This flywheel design is achieved by fabricating the flywheel from three sandwitched discs. the two outer disks are uniform whereas the inner disk has a curved slot which provides the liquid lead flow channel.

Between successive flywheel gun discharges each flywheel is accelerated to the required exact angular velocity and required exact angular synchronization by a high efficiency stepper motor.

The stepper motor control electronics rely on position feedback from marks at about 1 mm intervals along the outer rim of the flywheel. A phase lock loop amplifies the pulse output frequency by about a factor of 16 to give an average operating pulse output frequency of about:
(300 m / s) X (1000 marks / m) X 16 = 4.8 MHz.

A major advantage of flywheel guns over pneumatic guns is that fly wheel guns do not inject propulsion gas into the vacuum chamber. Further, the magnetic coupling between the stepper motor and the rotating flywheel armature allows the electromagnetic coils of the strpper motor to be located in a low temperature environment outside the vacuum chamber. Further, with this equipment arrangement the overall efficiency of conversion of electric energy into liquid lead kinetic energy is relatively high. This efficiency is critical to the success of the PIF energy process.

The flywheel guns are designed to provide a muzzle discharge velocity of about 300 m / s and hence require a flywheel perimeter tangential velocity slightly in excess of 300 m / s.

The flywheel must be fabricated from a material that has a low density but high yield stress while operating at a temperature of about 350 degrees C, above the melting point of lead. Possible candidate materials for flywheel fabrication include titanium, beryllium, silica and carbon fiber.

The internal stress within the flywheel can be estimated using the assumptions that:
Rfb = flywheel outside radius = 0.33 m;
Rfa = flywheel inside radius = 0.066 m;
Rhof = density of flywheel material;
Tf = thickness of flywheel;
Wf = angular frequency of flywheel

the maximum internal stress Si in the flywheel material is approximately given by:
Si = Integral from Rfa to Rfb of:
(dM V^2 / R) / [2 (Rfb - Rfa) T]
= Integral from Rfa to Rfb of:
(dR 2 Pi R T Rhof) (R Wf)^2 / [R 2 (Rfb - Rfa) T]
= Integral from Rfa to Rfb of:
(dR Pi Rhof) (R Wf)^2 / [2 (Rfb - Rfa)]
= Pi Rhof (Rfb^3 - Rfa^3) Wf^2 / [6 (Rfb - Rfa)]

Rfb Wf = 300 m / s
Wf = (300 m / s) / 0.33 m
= 1000 rad / s

Si = Pi Rhof (Rfb^3 - Rfa^3) Wf^2 / [6 (Rfb - Rfa)]
= Rhof Pi [(0.333 m)^3 - (0.066 m)^3] 10^6 (rad / s)^2 / [6 (0.33 m - 0.066 m)]
= Rhof Pi [.0370 m^3 - 0.0003 m^3] 10^6 (rad / s)^2 / [1.584 m]
= Rhof [72788 m^2 / s^2]

for titanium:
Rhof = 4.506 X 10^3 kg / m^3
Si = (4.506 X 10^3 kg / m^3) X (72788 m^2 / s^2)
= 328 X 10^6 kg / m s^2
= 328 X 10^6 Pa

The yield stress for titanium below 400 degrees C is about 434 X 10^6 Pa. Hence in principle the required flywheel could be fabricated from titanium. The flywheel thickness may have to taper with increasing radius to equally distribute stress through the flywheel material.

The exact acceleration characteristic of a liquid lead gun is dependent on the liquid lead slug mass. Hence during gun discharge the flywheel electric drive must be precisely controlled to compensate for small uncontrolled variations in the timing of the liquid lead valve release near the flywheel hub and small uncontrolled variations in the injected liquid lead slug mass. This precise real time control of the flywheel angular velocity and angular acceleration should be possible with modern electronic technology.

Additional fine control of the liquid lead flow through the flywheel can be obtained by controlled application of a transverse magnetic field across the flywheel. If the flywheel material is electrically conductive this transverse field will act as an eddy current brake for the entire fly wheel. If the flywheel material is not electrically conductive this transverse field will act as an eddy current brake to the lead flow. In this respect high purity silica is a superior fly wheel material.

The timing accuracy of liquid lead release from the flywheel is important. To avoid the liquid lead hitting the inside walls of the gun barrels near the gun muzzles the liquid lead slug positioning accuracy must be about +/- .025 m at the gun muzzle. Let DeltaT be the flywheel gun discharge timing accuracy. Then:
Wf (6 m) DeltaT = .025 m
DeltaT = .025 m / [Wf 6 m]
= .025 m / [ 1000 rad / s X 6 m]
= (25 / 6) X 10^-6 s
= 4.2 microsecond

A major issue in liquid lead flywheel gun design is design of the electric near hub valve. It is anticipated that this valve will rely on induced high frequency electric current to control the exact timing of the liquid lead release into the flywheel acceleration channel. This electric valve needs only operate intermittantly because the injection of liquid lead into the flywheel hub can be reliably mechanically achieved a few ms before the near hub electric valve has to open. The mechanical injector, which is stationary, establishes the mass of the liquid lead slug. There should be a electromagnetic liquid lead brake about half way along the liquid lead channel to achieve precise control of the liquid lead discharge timing and discharge velocity.

In order to achieve the desired liquid lead projectile shape each flywheel gun has a carefully engineered discharge nozzle on the rim of the flywheel. The discharge nozzle effectively breaks the liquid lead slug into a pattern of smaller liquid lead spheres that during their flight reposition themselves with respect to each other so as to have the desired slug shape when these liquid lead spheres reach Rd, the liquid lead shell closure radius.

The liquid lead injector is located within the vacuum chamber and loads the flywheel gun with a precise mass of liquid lead a few ms before the near hub electric valve opens. These parameters must be correct so that all of the liquid lead slugs arrive at Ri = Rid at the same instant in time.

In state a the liquid lead flywheel gun is loaded and ready to fire.

In state c the liquid lead has moved through the gun barrel and is at the gun barrel muzzle.

In state d the liquid lead shell has formed and has an inside diameter of about 2.9 m.

The time required to close the liquid lead shell is the time (Td - Tc) required to move the liquid lead radially 0.55 m from state c at the instant of spheromak injection to state d when the liquid lead shell surface is fully closed.

Let Z designate the relative negative radial movement of the liquid lead with respect to:
Ri = 8 m.

Let a designate the initial state when there is no liquid lead radial motion.
In state a:
T = Ta
Z = Za = 0 m

Let c designate the state when:
T = Tc
Z = Zc = 6.0 m
and the liquid lead is at the specified gun muzzle velocity of 300 m / s but is not obstructing spheromak injection.

Let d designate the state when:
T = Td
Z = Zd = 6.55 m
and the liquid lead spherical shell is closed.

Note that (Zd - Zc) = 0.55 m is constrained by practical pressure vessel size considerations.

Evaluation of (Td - Tc):
(Td - Tc) = 0.55 m / (300 m / s)
= 1.833 ms

Hence the time between liquid lead exiting the flywheel and liquid lead being in position to form a liquid lead shell is about:
(6.55 m) / (300 m / s) = 21.83 ms

Clearly the liquid lead shell assembly and closure time and the related flywheel gun control is a key issue. The first step in realizing this apparatus is to develop a prototype liquid lead flywheel gun complete with a suitable mechanical liquid lead injector and the related near hub electric valve control system.

This web page last updated April 6, 2015.

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