Introduction

A nozzle is a device that transforms pressure into thrust by restricting the expansion of a propellant in one direction, while leaving it free to expand in the other one. The unbalanced expansion creates thrust by pushing on the nozzle wall, and the nozzle pushes the vehicle. In a fusion starship, the propellant is an extremely hot conductive plasma. A magnetic nozzle, rather than a material one, is used because the conductive plasma can be restricted by the magnetic field so it doesn’t touch the actual physical elements of the nozzle. These would otherwise be destroyed by the high temperatures of the plasma.

The plasma expands from the heat generated within it by the fusion reaction. The expansion creates a pressure on the magnetic field, that transmits this pressure to the magnetic coils generating the field, that in turn push the nozzle structure, that pushes the starship in one direction, while the plasma leaves in the opposite direction.

Magnetic nozzle design is an active field or scientific research, notably for the Vasimir electric propulsion system(7) and some other forms of electric thrusters, but also for astronomy, fusion propulsion and nuclear fusion research₍₂₎. The Icarus Interstellar team carried out some research on nozzles: Hatcher and Stanic(a) used numerical simulations and the Nautilus code to simulate 2D behavior of the pulsed nozzle, confirming that the probable efficiencies were notably lower than the ones used for Daedalus, in the order of 70-80% at best. Perakis(1) studied the behavior of the fusion plasma in expansion using MHD equations. Although his results where somewhat inconclusive due to numerical modeling problems, the overall analysis is very interesting and complete. Analogous work has been done earlier by Hyde() and the Daedalus team(). NASA() has support a number of studies that have included fusion reaction simulations.

Nozzles are not an absolute requirement for a propulsion system. For example, the Orion nuclear pulsed fission rocket concept() used a flat plate and explosive units, that provided some directional control of the fission explosion, to push itself along. The Orion plate design was much less efficient that even the simplest of nozzle equipped drives, it’s just that the explosions provided so much power that the designers could afford the waste in exchange for simpler construction. The plate is also more appropriate for very low frequency pulsed propulsion, as we will see later.

The two types of fusion processes that can be used to propel starships require two different configurations of magnetic nozzles; the steady state fusion nozzle and the pulsed fusion nozzle.

Steady state magnetic nozzle

Continuous, or steady state, fusion takes place in a reaction chamber; be it a Z-pinch, a magnetic mirror or some form of tokamak like confinement system. The propellant is fed into the chamber, where it undergoes fusion and gains energy, transforming into a conductive plasma. The hot plasma leaves the chamber through an opening, the throat, and expands into a nozzle.

(-)(+)

both electrons(-) and ions(+) are involved in the steady state magnetic nozzle operation. The light electrons are accelerated much faster than the heavier ions when they expand into the nozzle. The rapidly moving electrons interact with the strong magnetic field that the superconducting magnets have created in the nozzle and an electromagnetic force is induced on them, that confines them to moving along the magnetic field lines of the nozzle magnets (along the axis of the nozzle). In turn, the moving electrons create a secondary electric field that confines the heavier ions axially, pulling them down and out of the nozzle and keeping them away from the nozzle ‘walls’. The magnetic forces pushing the ions away from the starship also push the vehicle forwards. As they are accelerated down the nozzle, the heavy ions gather sufficient inertia that they can detach from the magnetic field lines of the nozzle magnets, The departing ions, in turn, create an electromagnetic field that pulls the electrons along with them. This mechanism is called plasma detachment.

The electron flow from the nozzle to the leaving plasma stream is such that the net result is a neutral plasma, and therefore electrical charge does not accumulate on the ship.

Steady state nozzles have been developed for the Vasimir(7) electromagnetic thruster and for External field Magnetoplasmadynamic (EMPD) thrusters(). These are much smaller than what would be required for fusion but they have demonstrated the feasibility of the idea.

Inside the nozzle, the magnetic pressure is equal to the pressure from the plasma, containing it. By knowing the energy in the plasma, created by the fusion reaction, one can deduce the magnetic field intensity required to contain it, and the current in the superconducting magnets required to generate the magnetic field.

The energy of the expanding plasma is the sum of the driver energy and the fusion reaction energy, minus the neutron and x-ray losses. The volume of the plasma is the volume of the nozzle, minus the volume of the space we want to keep between the plasma and the wall.

Steady state magnetic nozzle equations:

To design the nozzle, the following approximations can be used. Detailed analysis requires requires more exact descriptions, than can be found at the following references()()().

Steady state equilibrium P_b=P_t

Magnetic Pressure= P_b (Pa)

P_b= B²/2·u₀

Where:

B= Magnetic field intensity (Tesla)

u₀= permeability of the vacuum= 1.26e-6 (J/mA2 = H/m)

Plasma thermodynamic pressure = P_t (Pa)

P_t=E/V (J/m3=N/m2=Pa)

Where:

E= Energy in the plasma (J) (reaction energy or a different number, energy of the plasma actually in the nozzle??)

V= Volume of the plasma (m3)

Time in the nozzle

t=2s/Vn

Where

Vn = η·Ve exhaust velocity (m/s)

l= length of nozzle (m)

η= nozzle efficiency

E=l·ḿ·Vn (J) Energy in the plasma

B=u₀·N·I/l (Magnetic field in a solenoid, an approximation)

N= Number of magnetic spires

I = Current (A) in the solenoid

The temperature of the plasma can be deduced from the perfect gas law

P_t=2·n·kB·T

where:

P_t= Plasma thermodynamic pressure = P_t (Pa)

n= m/V·A·m_u

kB= Boltzmann constant = 1.38e-23 (J/K)

m_u= Atomic mass unit = 1,66E-27 (kg)

A= Atomic mass number =2 for mostly deuterium plasma

m= t·ḿ (kg) mass of plasma in the nozzle

V= Nozzle volume (m3)

T= Plasma Temperature (K)

Pulsed magnetic nozzle

For the pulsed fusion magnetic nozzle, the nozzles serves both as reaction chamber and as a nozzle. The pulsed operation system goes through a number of states: In its initial phase, the nozzle is filled by a relatively light magnetic field induced by superconducting magnetic coils, know as seed coils. The fuel pellet is then injected into the nozzle area, where a driver system compresses it to fusion ignition, and it expands explosively as a conductive plasma. The expansion of the conductive plasma in the magnetic field induces a current in the plasma that creates a second magnetic field, that in turn induces a current in a secondary set of coils in the nozzle, the thrust coils. This new current increases the existing magnetic field in the nozzle while the plasma expands, eventually reaching a stagnation point where the magnetic pressure is equal to the expansion pressure of the plasma.

As the rear part of the nozzle is wide open, plasma is leaving the nozzle area at a high rate; the plasma bubble ‘deflates’ (its density goes down) and the magnetic field pushes the remaining plasma backwards, propelling the starship at the same time. When the plasma leaves, the induced magnetic field disappears and the nozzle returns to its original state.

Daedalus used a thrust shell, rather than thrust coils, for a similar mode of operation.

The equations for the continuous nozzle and the pulsed nozzle are mostly the same. However, the pulsed nozzle operates for very short periods and therefore endures much higher peak loads.

The strength of the seed magnetic field is a function of the secondary magnetic field that needs to be induced to contain the rapidly expanding plasma.

The plasma front expanding into the nozzle is decelerated to a standstill, then accelerated rearwards by the magnetic field, eventually reaching exhaust velocity and leaving the nozzle area.

Since the nozzle is relatively small, the plasma leaves it quickly. In fact, the nozzle will spend most of its time empty of plasma. Increasing frequency increases the usage factor of the nozzle, and reduces the strain on the nozzle components.

Pulsed nozzle equations

Time in the nozzle :

t=2l/Ve

Where:

Ve = exhaust velocity (m/s)

l= radius of nozzle (m)

For an exhaust velocity of 10e6 m/s, and a nozzle radius of 5m, the plasma stays in the nozzle for about 1e-6s and at a frequency of 24 pellets per second, the nozzle has material in it less than 0,002% of the time.

F=ḿ·Ve

Force, energy and structure

The average force exerted on the magnetic nozzle is not particularly high. The thrust of a single Space shuttle nozzle main engine was about 2000 kN (200 tonnes). The thrust of the Icarus Firefly nozzle would be 600 kN (60 tonnes). So the structure of the nozzle doesn’t need to be very strong. Nozzle elements are typically under tension, due to the pressure from the plasma and the magnetic field, while the elements that link it to the starship are under compression.

However, nozzles used in pulsed propulsion need to be designed to absorb the energy from the pulse and redistribute it over time. For the same thrust, the higher the frequency of pulses the less energy needs to be stored and the smaller and lighter the thrust structure needs to be. Looking back at Orion, the design frequency was about 1 explosion per second and this required a large shock absorption system. Daedalus, on the other hand, was planned with a frequency of 250 pulses per second, and the double walls built into the starship’s nozzles were sufficient to absorb the shock energy. A full numerical analysis of the Daedalus nozzle was done by Reddy(8) showing it could operate as originally planned.

Steady state nozzles do not need to take shock into account, and therefore can be lighter than the equivalent pulsed nozzle.

Thermal control

The nozzle is subjected to radiation from the fusion reaction and the exhaust plume. This radiation takes the form of high velocity neutrons (14.07 MeV for tritium-deuterium reactions, 2.45 MeV for deuterium-deuterium reactions), some thermalized neutrons that escape the outer layers of the reactions and of x-rays from Bremsstrahlung radiation. The superconducting magnets used to create the magnetic field must be protected from this radiation, or they will heat up and lose their superconductivity under load, leading to their quasi instantaneous destruction. This protection is relatively simple for fusion designs that produce low levels of radiation, but very difficult for those that produce high ones. A detailed description of the nozzle cooling systems can be found in the Thermal Control page.

Nozzle efficiency

The lower bound of nozzle efficiency (η) would be 65% for a single ring nozzle used for pulsed propulsion₍₃₎, but 80% seems achievable with simple geometries ₍₄₎ and some low powered firings of Vasimir achieved 95%₍₅₎

Daedalus used a nozzle efficiency of 95%. A study by Perakis(1) proposes this should be reduced to about 80% for the 4 ring configuration of Daedalus, but more efficient arrangements may be possible.

The efficiency of the nozzle is mostly a factor of beam divergence. The fusion reaction efficiency is explored in other chapters. Losses due to induction currents and the generation of the seed currents are small compared to the overall power and are not taken into account in most first level analysis, although they would need to be included in a more detailed analysis.

Energy recapture

The nozzle integrates energy recapture elements through direct energy conversion. See the Direct energy Conversion page for details. This energy is used to power ship systems, and particularly the fusion driver system.

Testing and development path

A facility capable of testing both steady state and pulsed plasma propulsion systems would greatly help in the design of magnetic nozzles, both for electric and fusion propulsion. A GigaWatt class installation in a vacuum environment would be ideal. This could be accomplished in Earth orbit, or on the Moon, with solar panels and a new space station dedicated to this type of research. In the interim, a larger installation with a bigger vacuum chamber and more capacitors should be built, using the existing capabilities of the Gigawatt class Godzilla installation at Ohio State University or similar installations, allowing for experiments lasting more than 1 or 2 milliseconds and providing better understanding of steady state operations of these types of equipment.

References

(a) PROJECT ICARUS: ANALYSIS OF MAGNETIC NOZZLE DESIGN FOR PULSED-FUSION PROPULSION SYSTEM Richard Hatcher,University of Alabama in Huntsville, AL, USA, rbh0008@uah.edu,Milos Stanic,University of Alabama in Huntsville, AL, USA, mds0005@uah.edu, Jason Cassibry,University of Alabama in Huntsville, AL, USA, jason.cassibry@uah.edu, John Loverich, TechX Corporation, Boulder, CO, USA, loverich@txcorp.com, Madhusudhan Kundrapu, TechX Corporation, Boulder, CO, USA, madhusnk@txcorp.com

(1) Modeling and Simulation of Fusion Plasma Thrusters with Magnetohydrodynamics, Nikolaos Perakis, Bachelor Thesis, Technische Universitat Munchen, Max Planck Institute for Plasma Physics, Department of Numerical Methods in Plasma Physics, 2015.

(2) AIAA 99-2703

High-Energy Space Propulsion based on Magnetized Target Fusion

Y. C. F. Thio, B. Freeze, R. C. Kirkpatrick(1), B. Landrum(2), H. Gerrish, G. R. Schmidt NASA Marshall Space Flight Center Huntsville, Alabama

(3) Sakaguchi, Nobuyasu, Yoshihiro Kajimura, and Hideki Nakashima. "Thrust efficiency calculation for magnetic nozzle in laser fusion rocket." Transactions of the Japan Society for Aeronautical and Space Sciences 48.161 (2005): 180-182.

One ring nozzle at 65% and two ring 75%.

(4) Mikellides, Ioannis G., et al. "Design of a fusion propulsion system-Part 2: Numerical simulation of magnetic-nozzle flows." Journal of propulsion and power 18.1 (2002): 152-158.

70-75%, much of the energy lost in detachment

(5) Williams, Craig H., et al. "Realizing" 2001: A space odyssey": Piloted spherical torus nuclear fusion propulsion." Journal of spacecraft and rockets39.6 (2002): 874-885.

80% efficiency hypothesis, 3 ring notional concept for the design of Discovery II.

(6) Simple Thrust Formula for an MPD Thruster with Applied Magnetic Field from Magnetic Stress Tensor M. Coletti * University of Southampton, Southampton, United Kingdom

(7) Vasimir papers

(8) Reddy, S.K and H.Benaroya,

Next up : Direct energy conversion