Saturday, October 22, 2011

Plasma: G-force Fuel

CONTENT


Thought Experiment (TE) uses plasma as fuel for a g-force spaceship. TE's notional g-force vessel continually expels specified quantity of charged particles (ions) at a very high speeds for an extremely favorable momentum exchange.

Plasma is an amorphous mass of charged particles which can be contained and moved by electromagnetic fields. Plasmas are freely moving charged particles, i.e., electrons and ions. Formed at high temperatures when electrons are stripped from neutral atoms, plasma is by far the most common state in nature. This might prove counter intuitive because humans seldom notice plasmas close up. However, stars are predominantly plasma, and stars contain the vast majority of mass in the universe.
As the "Fourth State of Matter", plasmas are distinct from solids, liquids and gases. Plasma's numerous charged particles (ions) can channel via magnetic waveguides into particle accelerators to increase particle velocity to near light speeds.

For all rockets, a stream of high speed, exhaust particles propels a very large vehicle to a slightly higher speed in the opposite direction. Momentum is mass times velocity; in a closed system, momentum exchanges are equal. TE considers a spaceship in space to be a closed system.

Momentum
Exchange
In a closed system, total momentum (mass times velocity) remains the same.
Consider spaceship propulsion.
(In space, this is a closed system.)

Mship*Vship= mExh*vExh
LEFT SIDE MOMENTUM
=RIGHT SIDE MOMENTUM
ship's large
mass
*
small increase in
ship's vel.
=
small  mass
plasma particles
*
very high
exhaust velocities

Divide each side by one second.
Mship * Vship
1 sec
=
mExh * vExh
1 sec
Mship * 
Vship

1 sec
=
mExh 
1 sec
* vExh
Redistribute terms as shown.
9.80665 m/sec
sec
= g =
Vship
1 sec
For each second of  powered flight, let ship's velocity increase additional 9.80665 meters per second. This is same acceleration experienced by a freely falling object near Earth's surface.
TE Assumes: Einstein's "Equivalence Principle" applies, and spaceship occupants experience " virtual gravity".
Mship * 
g
=
ffsec
* vExh
Define one second's fuel flow as mass quantity of plasma particles per second which exit ship's exhaust.
mExh
sec
 ffsec
Mix Units of Mass
Express ffsec in either grams or even kilograms (per second).

Express Mship in metric Tonnes (mTs) or even kiloTonnes (kmT)

CONVERSION FACTOR: It's possible Mship could be expressed as Mega-ffsec (disregard "per sec" for now).
Solve for Mship.
Mship=
ffsec * vExh
g
TE chooses to apply conversion factor at a later step.
Express fuel particle's exhaust velocity  as  dcc, decimal light speed.
 (Light speed, c, is 299,792,460 m/s.)
dc (= vExh/c ) is decimal component of fuel particle exhaust speed.
vExh = dc c
Mix Units of Speed
g = 9.80665 meters per sec per sec
often expressed as 9.8 m/sec2
Substitute dcc for vExh and regroup terms as shown.
Mship=ffsec*dc*
c
g
Mship=ffsec*dc*
30.57×106sec
c
c
g
=
299,792,460 m/sec

9.80665 m/sec2
=
30.57×106sec
mT
=
106 gm
ffsec
*dc*30.57×106sec*
mT
10
6 gm
1 mT ÷ 106 gms =1, and any expression times one remains equivalent.

106 factor now appears in both numerator and denominator;
thus, 106 ÷ 106 equals one and cancels out.
Term "ffsec" is normally understood to be grams of mass consumed  per second;
TE finds expedient to reexpress as the ffsec value times 1 gm per sec.
gm
sec
ffsec *dc*30.57sec*
mT
gm
Terms, gm and sec, appear in both numerator and denominator.
Thus, they become "one equivalents" and can be removed.

CONVERSION CONSTANT is 30.57
Mship
ship's mass in metric Tonnes.
ffsec
mass of exhaust plasma particles (aka "fuel flow per second") is in grams per second.
dc
decimal component of c, light speed. (Velocity of exhaust plasma particles is expressed as decimal light speed.)
ffsec*dc*30.57mT= Mship

Since humans are familiar with solids, liquids and gases, and much less familiar with plasma, plasma is often called the "fourth state of matter". However, plasma is definitely the first and foremost state of matter. All matter was plasma long before it coalesced and became gaseous, liquid or solid, the more likely "fourth state".

A plasma contains charged particles, positive ions and negative electrons. It is similar to gas in which a certain portion of the particles are ionized. Plasma constitutes the matter found in stars and thus constitutes about 99% of all matter in the universe.
Table 1: Non-Relativistic Momentum

Non-Relativistic Momentum

Mship=
30.57
*dc*ffsec
Mshipdcffsec
3.057 mT0.1 c1.0 gm
6.114 mT0.1 c2.0 gm
9.171 mT0.1 c3.0 gm
0.1 c = dc = 29,979,246 m/s
For Table 1, TE assumes neglible relativistic effects for particle exhaust speeds up to 10% light speed, c. Table 1 shows extremely simple case where ship size correlates to flow fuel quantity, a linear relationship.

One way to vary ship size: adjust quantity of exhaust particles.

At particle exhaust velocity ≈ .1 c, propulsion might be enough to g-force a spacecraft the minimum distance from Earth to Mars (approximately .5 AU).

Relativistic Growth of Exhaust Particles
Lorentz Transform (LT )
quantifies relativistic growth.

ffExh=
ffsec 
√(1-(vExh/c)2)
Express particle exhaust speed as
vExh=dc* c
vExh2/c2 reduces to dc2.
dc2=

(dc*c)2

c2


 Term, ffExh , fuel flow at exhaust speed, quantifies mass increase of fuel flow per second consumed at relative rest.

Thus, TE uses "n" as relativistic growth factor. 
Exhaust fuel flow, ffExh, reflects relativistic grow of ffsec due to very high speed achieved in particle accelerator, heart of TE's propulsion system.
ffExh=
ffsec

√(1-dc2)
=n * ffsec
where n = 1/√(1-dc2)
ffExhdcffsec
1.005 gm0.1 c1.0 gm
1.020 gm0.2 c1.0 gm
1.048 gm0.3 c1.0 gm
1.091 gm0.4 c1.0 gm
For clarity, above table uses constant unity for fuel consumed (ffsec). 
ffsec could be any value.

Thus, TE assumes that fuel flow per second (ffsec) is good descriptor of fuel consumption which subsequently reduces ship's mass. However, TE further assumes that Exhaust fuel flow (ffExh) better describes particles' collective contribution to momemtum exchange which propels ship.
To generate plasma on a spaceship, one might heat water until it becomes a gas and then further heat it to dissociate molecular bonds and obtain atoms of hydrogen and oxygen. Further heating drives electrons out of atoms (ionization) and water finally becomes an amorphous mass of ions (ie. "plasma").

Since ions have charges, plasma is electrically conductive and responds strongly to electromagnetic fields. Plasma contains ions and electrons in about equal numbers so that the resultant space charge is very small.

Like gas, plasma does not have a definite shape or volume unless enclosed in a container; unlike gas, plasma can be directly manipulated by a magnetic field, it may form structures such as filaments, beams and double layers. A common use of Earth bound plasmas is neon signs.

Plasma in Space. Plasmas are by far the most common state of matter in the universe, both by mass and by volume. Stars are made of plasma, and even the space between the stars is filled with a sparse amount of plasma. Very small grains within a gaseous plasma will also pick up a net negative charge, so that they in turn may act like a very heavy negative ion component of the plasma (see dusty plasmas).


Definition of a Plasma
Plasma is loosely described as an electrically neutral medium of positive and negative particles (i.e. the overall charge of a plasma is roughly zero). It is important to note that although they are unbound, these particles are not ‘free’. Electro-magnetic fields can control them and govern their collective behavior with many degrees of freedom.

Plasma has three characteristics:
1. Collective effects: Charged particles are close; so close, that each particle influences many nearby charged particles, not just the closest particle (such collective effects are a distinguishing feature of a plasma).
2. Bulk interactions: Ion interactions in the bulk of the plasma are much more numerous; thus, more pertinent than the much fewer interactions at the plasma edge which are affected by boundary effects.
3. Plasma frequency: The electron plasma frequency (electron oscillations) greatly exceeds the electron-neutral collision frequency (collisions between electrons and neutral particles). I.e, electrostatic interactions overcome ordinary gas kinetics.
Table 2: Low Relativistic Momentum

Relativistic Impact on Ship Size

Let ffsecstay at a constant 1.0 kilograms to more readily observe results of increasing particles' exhaust speed.
Mship=
30,570*dc*ffsec
√[1-dc2]
Mshipdc cffSec
3,072 mT0.1 c1.0 kg
6,240 mT0.2 c1.0 kg
9,614 mT0.3 c1.0 kg
13,341 mT0.4 c1.0 kg
17,650 mT0.5 c1.0 kg
22,927 mT0.6 c1.0 kg
A more accurate momentum transfer equation uses the mass that exits the ship, ffExh (=ffsec/√[1-dc2]).
Due to relativistic effects, exhaust particle mass (ffExh) is larger then original fuel flow mass (ffsec). For particle speeds up to .6c, mass growth is fairly low; up to 25%
High speed exhaust particles contribute to ship's momentum:
---Sheer velocity of the exhaust particles, dc c.
---Relativistic mass increase of the exhaust particle ffExh due to near light speed.
TE increases fuel quantities by a factor of 1,000 by using kilograms versus grams.
Reason: one gram of fuel can impart g-force to a 3 Tonne vessel; this is roughly the size of an automobile. For a vessel to accomplish interplanetary voyages, we want a much bigger vessel, perhaps a 1,000 times larger. Since a kilogram is a thousand times larger then a gram, TE now proposes 1 kg for fuel consumed per second (ffsec). This is a lot of fuel (>86 mTs/day); but such quantities of water are available in Earth's oceans (for starters) and in numerous comets for longer term.

If exhaust particles achieve low relativistic velocities (.2 c ≤ dc c ≤ .5 c), propulsion might be sufficient to g-force spacecraft minimum distance to Ceres (approximately 2.8 AU).

Table 3: Mid Relativistic Momentum

Relativistic Impact on Ship Size

n=
1 

√(1-dc2)
let ffsec = 1.0 kg
Mship=
30,570
*dc*n*ffsec
Mshipdcn
9,614 mT0.3 c1.05
13,342 mT0.4 c1.09
17,650 mT0.5 c1.15
22,928 mT0.6 c1.25
29,965 mT0.7 c1.40
40,760 mT0.8 c1.67
TE arbitrarily picks particle velocity range  from .3c to .8c.
ffExh=

ffsec

√(1-dc2)
Recall definition of Exhaust fuel flow.

n=
1

√(1-dc2)
Recall definition of relativity growth factor.

ffExh=
n*ffsec
Redefine Exhaust fuel flow.

If exhaust particles achieve mid relativistic velocities
.5 c ≤ dc ≤ .7 c
 propulsion might be enough for spacecraft to travel to Uranus (approximately 20 AU)
Man Made Plasma for Ship's Particle Stream
Mankind has several methods to make plasma; however, they all require considerable energy input to attain then sustain plasma state.

Prior to the powered flight, initial plasma supply might be generated by an electrical current  applied across a dielectric gas or fluid (an electrically non-conducting material). As the voltage increases, the current stresses the material (by electric polarization) beyond its dielectric limit (termed strength) into a stage of electrical breakdown, marked by an electric spark, where the material transforms from an insulator to a conductor (as it becomes increasingly ionized).

This plasma could be the source of particles for the initial propulsion stream in the ship's particle accelerator.

After g-force propulsion stream is flowing, longer term plasma generation could divert some high speed particles from that propulsion stream to collide with the water awaiting for plasma transformation.

This is an "avalanching" ionization process, where ions collide with neutral gas atoms to create more ions and electrons.

 After the first collision, the number of charged particles rapidly increases rapidly to "millions after "only about 20 successive sets of collisions”, mainly due to a small mean free path (average distance between collisions).

With ample current density and ionization, collision chain forms a luminous electric arc (essentially lightning) between the electrodes. Electrical resistance along this arc creates heat, which ionizes more gas molecules (where degree of ionization is determined by temperature), and as per the sequence: solid-liquid-gas-plasma, the gas gradually turns into a thermal plasma.

A thermal plasma is in thermal equilibrium; temperature is relatively homogeneous throughout the heavy particles (i.e. atoms, molecules and ions) and electrons. When thermal plasmas are generated, electrical energy goes to electrons. Since electrons are very mobile and very numerous, they rapidly disperse this energy to heavy particles via elastic collisions.


Relativistic Growth Factor
Re-express Decimal Component
New Product
n =
1 

√(1-dc2)
dc =
√(n2-1) 

n
n*dc =
√(n2-1)

dc cn
0.6 c1.25
0.7 c1.40
0.8 c1.67
0.866 c2.00

dc cn
0.866 c2
0.943 c3
0.968 c4
0.980 c5

ndcc√(n2-1)
20.866 c
1.73
30.943 c
2.83
40.968 c
3.87
50.980 c
4.90

Independently increase particle velocity, dc .

Observe corresponding increase in relativistic growth factor, n.

Note nonlinear relationship.
Solve for dc .

Independently increase n.

Combine to get:
√(n2-1)

n=
1

√(1-dc2)
Mshipffsecn
52,949 mT1.0 kg
2
86,465 mT1.0 kg
3
118,397 mT1.0 kg
4
149,762 mT1.0 kg
5
180,855 mT1.0 kg
6
211,795 mT1.0 kg
7

Relativistic Impact on Ship Size

Mship=30,5700*√(n2-1)*ffsec
If able to achieve consistent near light speed velocities, TE chooses to use relativistic growth factor, n , as independent variable and observe results for dependent variable, ship mass (Mship ).

If exhaust particles achieve high enough velocities for significant relativistic growth (2 ≤ n ≤ 4), propulsion might be enough for spacecraft to travel to exoplanetary distances (beyond 20 AU).
NOTE: TE artificially assumes 100% efficiency for particle acceleration and subsequent propulsion. Adjustments will be made in later chapters.
Examples of industrial/commercial plasma
With their sizable temperature and density ranges, plasmas now find many uses in research, technology and industry. These include: industrial and extractive metallurgy, surface treatments such as thermal spraying (coating), etching in microelectronics, metal cutting and welding; as well as in everyday vehicle exhaust cleanup and fluorescent/luminescent lamps, while even playing a part in supersonic combustion engines for aerospace engineering.

Thought Experiment (TE) built a notional spaceship which continually expels desired quantity of charged particles (ions) at a relativistic speed (expressed as decimal light speed, dc). This constant propulsive force of exhaust particles produces a 1-g force upon the contents (pax and payload) to simulate Earth's gravity throughout powered flight.

This chapter considered relationships between the exhaust particles and and the spacecraft with regards to momentum exchange.

Summary

Non-Relativistic
Mship=
30.57
*dc*ffsec
Mshipdcffsec
mTdec. cgm
3.057.1 c1.0
6.114.1 c2.0
9.171.1 c3.0
To Mars
Low Relativistic
Mship=
30,570
*dc*ffExh
MshipdcffExh
mTdec. ckg
6,240.2 c1.021
9,614.3 c1.048
13,341.4 c1.091
To Ceres
Mid Relativistic
Mship=
30,570
*dc*n*ffsec
Mshipdcn
mTdec. c
22,928.6 c 1.25
29,965.7 c1.40
40,760.8 c 1.67
To Uranus
High Relativistic Mship=30,570√(n2-1) ffsec
Mshipnffsec
mTkg
52,94921.0
86,46531.0
118,39741.0
Exoplanetary

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