MAGNETOHYDRODYNAMIC ION DRIVE AS A MAIN PROPULSION UNIT FOR A SPACE VEHICLE





Daniel Neuman

SFSU PHYSICS DEPT.

SPRING 97




ABSTRACT


For future interplanetary missions to Mars and beyond a new method of spacecraft propulsion must be implemented. Current chemical reaction based rockets will not be able to provide sufficient levels of performance to meet mission critical needs. This paper examines an alternative propulsion method, electric propulsion. Electric propulsion is the acceleration of propellant by electrostatic and electrodynamic body forces. The theory behind as well as the implementation and pros and cons will be discussed. It will be shown that electric propulsion is the only viable alternative to chemical rockets for large total impulse- high _ v missions.

INTRODUCTION

To understand the benefits of electric propulsion we must first understand some basic concepts of rocketry. The equation of motion of a spacecraft in a gravitational field is,[ ]

m = + Fg (I-1)

Where = acceleration of the rocket
= rate of change of rocket mass by exhaust of propellant
ue = exhaust velocity relative to rocket
Fg = local force of gravity
The term is the thrust T of the rocket,

T = (I-2)

The integral of the thrust over a complete trip is called the total impulse,

(I-3)

The specific impulse is defined to be the ratio of the thrust T to the rate of use of the sea level weight of propellant:

(I-4)

where g0 is the sea level acceleration of gravity. Specific impulse is essentially an indication of how much a rocket's velocity can be increased by a given amount of fuel[ ]. Specific impulse can be thought of as the gas mileage of the engine. From (I-3) we see that to obtain a large total impulse we can either have a large exhaust velocity or a large rate of change of the rocket mass. Equation (I-1) can be solved for the fraction of the original total rocket mass which can be accelerated through a velocity increment _ v

(I-5)

In the literature aerospace engineers are always talking about the `delta-v' of a particular propulsion system. Equation (I-1) can also be solved for delta-v, the change in velocity as a function of expended propellant mass.

_ v = ue ln (I-6)

Delta-v will tell an aerospace engineer how much mass needs to be expelled in order to bring about a desired change in velocity. Equation From (I-5) we see that ue has to be of the same order as _ v in order to bring a large fraction of the original mass to the final velocity. Here are some theoretically predicted values of _ v for impulsive missions over minimum propellant semiellipse trajectories[ ].

_ v values for a few long range missions

MISSION _ VELOCITY meters/sec
Earth orbit to Mars orbit and return 1.12 104
Earth surface to Mars surface and return 3.4 104
Earth orbit to Venus orbit and return 1.6 104
Earth orbit to Mercury orbit and return 3.1 104
Earth orbit to Jupiter orbit and return 6.4 104
Earth orbit to Saturn orbit and return 1.1 105

Now for comparison here are some exhaust velocities of a few chemical rockets[ ].

Exhaust velocities for chemical rockets

Propellant type ue meters/sec
Liquid mono-propellants 1.7- 2.9 103
Solid propellants 2.1- 3.2 103
Liquid bi-propellants 2.9 - 4.5 103
State-of-the-art space shuttle 5.0 103

We see that for chemical rockets their exhaust velocities are not on the same order of magnitude as _ v. Chemical rockets can not bring a large fraction of their original mass up to a desired final velocity. Chemical rockets get around this limitation by using a staged propulsion system. As each stage is used up it is jettisoned and the total mass of the rocket decreases. But the staged system itself adds mass and complexity and is more expensive to build than a single stage system.
For long duration missions such as interplanetary travel or satellite station keeping where the total impulse and _ v are large, chemical rockets will not work efficiently.
The exhaust velocity of a chemical rocket is limited by several factors: (1) the total amount of energy available from a chemical reaction, (2) how much thermal stress the engine itself can withstand, and (3) the energy lost in internal modes. With all of these intrinsic limiting factors working against you, has to be very large to obtain large total impulse. Correspondingly their specific impulse is small. Typical values are on the order of a few kilograms/sec[ ]. Having large is very disadvantageous because the spacecraft will have to carry a large amount of fuel. With a larger mi the rocket will have to expend more fuel to escape earth's gravity well and there will be less available space for payload. Conventional rockets change their velocity by having a very large amount of thrust over a short time. What if you could really increase ue? If ue could be made large enough then would not have to be very large and you avoid the large penalties. You could build a spacecraft that can be smaller and lighter and obtain a higher final velocity. Figures(1-a,b,c) show the difference, in some critical mission parameters between, small ue chemical rockets and a large ue propulsion system. It is seen that with high ue and large specific impulse the trip time to Mars is drastically reduced.
Electric propulsion, the acceleration of propellant by electrostatic and electrodynamic body forces, is the answer. Typical ue and values are 3.0 104 m/s and a few milligrams/sec respectively[ ], numbers that are good enough to take us to the surface of Mars and back with a single stage. Electric propulsion devices do not suffer from the same limitations as chemical rockets. There is not such a strict limit to their ue, and there is no heating of the engine walls. They do have limitations, fundamentally different from chemical rockets. These limitations will be discussed in a later section of this paper.
Ion drives are a subfield of the larger electric propulsion device group. In a nutshell, ion drives work by ionizing a gaseous fuel such as Xenon and then accelerating it in an electric field and ejecting it out the back of the spaceship. One scheme, called the electron-bombardment method, ionizes the Xenon by bombarding it with energetic (~40eV) electrons. The ions and electrons are then accelerated, in different directions, through a potential difference. The accelerated ions leaving the unit provide the thrust. To keep the exhaust neutral, electrons need to be added to the exhaust stream shortly after leaving the thruster. This can be done by placing a thermionic cathode on the periphery of the ion beam. The thrust available using this method depends only on the exhaust speed, on the mass of the ion, and on the total ion flux that can be accommodated by the source-accelerator-neutralizer system.
Ion drives are a good idea because they can provide high specific impulse and very low thrust. With the ion drives high it can obtain the same or greater total impulse as a chemical rocket by thrusting for a longer period of time. The ion drives small allows it to obtain this velocity with a small amount of propellant mass ejected. This can translate into smaller, lighter propulsion systems. From a business point of view ion drives are cost-effective because with less space being taken up by engines there is more room for commercial payloads and thus more profit.
Historically the theory of electric propulsion has been around since 1906. Robert H. Goddard is considered the inventor. During the great space race of the 50's and 60's scientists knew about the benefits of electric propulsion but could not implement them for a few technical reasons. First, where was the electricity going to come from to accelerate the propellant? The thousand volt potential difference needed for acceleration of propellant could not be generated with the technology of the time. A small compact, light-weight power source was needed. Small nuclear reactors could power such a system but environmental concerns effectively grounded a nuclear fission powered rocket. Not until the invention of efficient solar cells and modern batteries-fuel cells can we have practical electric propulsion. Even so, electric propulsion powered by solar cells will only work out to about 3 AU's because of the decrease of solar flux with distance.

THEORY

The basic theory of how ion engines work is almost embarrassingly simple. The equations of electrodynamics concerned with the motion of a charged particle in a constant electric and magnetic field are all we need. The equation of motion of a particle with charge q and mass m in a region of space with constant E and B is[ ],

 

The particle will undergo a circular orbit at a frequency called the cyclotron frequency.

With a radius, called the larmor radius given by,

 

An ion engine can be thought of as a cylindrical plasma where the E field and B field are in the same direction say the z direction(see figure(3)). If an ion was given an initial velocity purely in the z direction then the equation of motion simplifies to,

 

That is to say, if we can keep the component of velocity perpendicular to the direction of B and the magnitude of B very small the propellant ions will basically move in a straight trajectory. The electrons, on the other hand, are injected into the plasma nearly perpendicular to B. The electrons will follow a helical orbit in the opposite direction from the ions. The B field is of such strength that rl is equal to the radius of the plasma. Even when the ion's velocity has a perpendicular component, B is so small and their mass is so large compared with electrons that they still travel in basically a straight line. The end result is that the ions are accelerated out of the rocket by the electrostatic E field and electrons follow a much longer helical path to the anode.
We have learned about why we need electric propulsion and some of the general theory now lets get more specific.

ELECTRON BOMBARDMENT ION ENGINE

Space charge limited flows:

The electron bombardment Thruster can be seen in figures(2-a,b,c).
The propellant velocity is given by the equation for a charged particle that is accelerated through a potential difference.

 

One of the parameters that effect the thrust that an ion engine can produce is the ion flux that the engine can accommodate. Child's law[ ],

 

Where M = mass of the ion
q= charge of the ion
= Potential at ion source
= distance between accelerating grids
originally determined for electron current in a vacuum diode, represents one fundamental limit on the current that can be drawn across a given plane gap by a given potential difference. This is called the space-charge limited current. With this limit on the ion density there will be a corresponding limit on the thrust density. The limit on thrust density can be found from the relations for velocity, current and thrust[ ].