• Effect explanation
• Ion/air interaction by corona discharge in Lifter - thrust/current relation (~2 g/W !)
• Thrust/voltage relation in lifter
• Field intensity and charge density in corona discharge device (Strict formulation of corona discharge electrodynamics, for theory developers. Less prepared readers can skip)
• What it is not:
• Ionic jet thrust calculation (~10^-7N / 100W) 
• Radiation pressure (~10^-9 N / 100W)
• Dielecrophoresis
• Improving of thrust/power ratio by optimizing voltage/distance/corona wire radius
• Efficiency if distance and voltage are increased at fixed force
• Force if voltage and distance are increased while V/d is fixed, comparisson with Saviour's experimental data
• Experimental confirmation!
• Optimal wire spacing in 2-d mesh Lifter - the closer, the better! 5 g/100mm^2

If high voltage applied between two electrodes in dielectric medium, the only current flowing is a very small current due to few ions always present due to ionizing radiation. However, when voltage is increased above ionization threshold, some present ions are accelerated to speeds where each collision with neutral molecule result in its ionization. As result, an avalanche of ionizations occurs and whole medium becomes ionized. 

This high field density is difficult to achieve if electrodes are equally sized and far apart. However, if one of the electrodes is very small, thin or sharp (such as wire), field density near its surface is very high and ionization can occur at moderate voltages, while second electrode (collector) has no sharp edges and no ions are generated there. What happens then? 

Effect called corona discharge. Ions built near the surface of sharp electrode create a small ionized region (plasma, corona). Bulk medium can not become ionized outside this region as the field density there is not enough. Therefore when some ion leaves this area, it just keeps drifting to the opposite electrode following the electric field, without ionizing other molecules. 

Only ions with same sign to that of corona electrode can leave because ions with opposite sign are attracted to the corona electrode and become immediately neutralized. As result we have two regions:
1) thin neutral plasma region with both ion signs present (corona) and 
2) mono-polar region, with single sign ions present - having sign of corona electrode

One important result is that in mono-polar region all ions are moving _in same direction_ from corona wire to collector. At the other hand, every second ions undergo zillions of collisions with outside air so in fact they are moving only very slowly, much slower then if medium would not be present. 

In fact calculations shows that absolute most of momentum obtained by ions from electric field is given away to the medium. It is as if ions are "imbedded" in air as sticks in water, and electric field is "pooling" the entire device forward, "leaning" on this ions-sticks. 

In reality ions are not fixed in the air, they are moving a little, so whole picture is not as much as somebody pooling a rift leaning on a fixed in water stick, but more similar to a rower dragging an oar through water to "push" boat in opposite direction. Electric field drags ions from one electrode to another and due to their viscous interaction with air device is
pushed in opposite direction.

Some conclusion related to the form of electrodes of electric propulsion devices: 
1) Corona wire should be as thin as possible to provide highest field intensity near to it to provide more charge carriers. 

2) Collector should have a form which provides _minimal_ field intensity near surface, to prevent any ionization near to it resulting 
in "counter-current" which reduces the thrust. Collector should have no sharp edges and the best configuration is spherical or torroidal. 
Optimization can be made in electric-field simulators. 

3) At the other hand, it should not have too large surface departing from direct line between center of electrode and emitter, because flow 
lines deviating from this direct line contribute less to net thrust. From this point, conventional lifter configuration is best - because
all surface is on the axial line. 

But from point 2 conventional lifter could be improved to reduce counter-current from sharp edge of its collector. 

So what is the perfect lifter from this point? It appears to be wire-circle emitter placed co-axial with a toroidal collector of same 
diameter. 

See lots of more detailed explanations, equations and optimizations elsewhere in this site. 
 
 
 
 

Starting eqn. for force applied by electric field to medium with distributed charge q is:
F=q*E/d         (1)

where E is voltage applied between electrodes and d is lenght between wire and collector. This formula is strictly correct only for plate capacitor arrangement (where ion flight path is parallel to force) but difference due to different field form will be small. Obviously due to first law of mechanics, the same force is applied by medium to lifter, so this is our force of interest.

Now, lets calculate amount of charge q distributed between electrodes at any time when corona discharge is on.
i=q*w/d 
so 
q=i*d/w            (2)

i is current and w is drift velocity of ions .  Ions are moving not on straight path because electrically induced motion is overlayed by thermal mothion. During this whole motion ions interract with molecules of enviroment. So drift rate is a net velocity of ions in direction from corona to collector.

Now, drift velocity is related to fild strengh as
w=k*E/d            (3)

Here k is mobility coefficient of ion in air. From the web-site of the institute of electrostatic technology (http://fee.mpei.ac.ru/elstat/lect  in russian) I have values for mobility of positive ions (if wire is positive) and negative ions (if wire negative). k (+) = 2.1 cm^2/volt*sec    k(-) 2.24 cm^2/volt*sec
I will use k(+) for calculations below. 

Now bringing it all together, w from (3) into (2) and than q from (2) into (1) we have our force:

F=i*d*E/(w*d) 
F=i*d*E/(k*E*d/d)

Simplifying we get
F=i*d/k 
**********

Now to be totally exact we have to substract the momentum which ions retain when they hit the collector. F_lost = m`*w. We can easilty calculate mass flow m`=dm/dt knowing current and molecular weight of ions, which is M=19gm/mole. 

m` = M*i/(e*Na)
Na is Avogadro number = 6.0221367E+23 mole^-1, e - electron charge = 1.6*10^-19
so (substituting w from (3))

F_lost = M*i*k*E/(d*e*Na)

F_total =  F-F_lost

*************************************
F_total=i*d/k  - M*i*k*E/(d*e*Na)
****************************************

Wow! while current certainly depends on voltage, the result is that at given current thrust does not depend on voltage. At given current thrust increases with lenght between electrodes.
Later we will see that F_lost is negligibly small, so good equation to work with is simply  i*d/k

Now the ultimate test - I compare the predicton of this equation with real experimentally observed thrust for lifters 1-4 (data on current, config and experimentaly observed thrust from table in http://jnaudin.free.fr/html/lifter4.htm).

First lifter 1:
d= 30mm
k=k (+) = 2.1 cm^2/volt*sec
i=450uA

F1=0.064 N
F_lost = 1.8*10^-7N (so will be neglected in all other calculations)
experimental F1exp=2.3+1gm*g =0.032N

WOW! the prediction and experiment differ only 2 times! 

Remember that we considered the straight fligh path of ions, but actually it is curved so in reality less force is applied because not all force is directed parallely to wire/collector line. Additional power loss can happen due to "counter-current" of ions with opposite sign because of small corona formation at collector (if edge is too sharp)
More exact calculation considering exact field configuration might come even closer, but hey - we have our raw equation! Du you think numbers are so close just by chance!?

Anyway, lets see other lifters. 
Lifter2:
d = 30mm
i=570uA

F2=0.077 N
F2exp=6.6+3gm*g=0.094N

Lifter3:
d=30mm
i=2.46mA
F3=0.351 N
F3exp=16+4gm*g= 0.196 N

Lifter4:
d=40mm
i=2.01mA
F4=0.383 N
F4exp=32+4gm*g=0.353 N

Wow! This is realy close. Note that lifter 4 used rounded-up top of collector to minimize counter-current, so it achieved higher lift efficiency vs. theoretical equation compared to other lifters.

To summarize - Prediction falls quite near to experimental results for all sizes of lifter, and the closes result is in case where minimal counter-current can be expected. Finaly you have an equation to judge lifter efficiency, and additional proof for ion-propulsion mechanism.

Later I will investigate relation between voltage and current.  Anway, the main point in improvement of lifter's force/power ratio - how to increase the current and distance between wire and collector without increasing voltage. 

The corona onset voltage V0 is given by Peek's equation (links to some chapters of Peek's book are in http://www.ee.vill.edu/ion/p61.html):

V0=g0*r*ln(d/r)
where
g0=30*kV/cm*delta*(1+0.301/sqrt(delat*r))

where r is radius of corona-wire in cm, d distance between wires and 
delta is a factor depending on air pressure and temperature as
delta=3.92b/(273+t )
where b is pressure in cm of barometric pressure and
t is temperature in degree C.

At d=30mm  and r=0.5 mm  we get V0=14.4 kV

Anyway, in my derivation the field strengh E has canceled out because it at one side it increases thrust, and the other side decreases it by decresing number of carriers (charge) inside the interval. So there is no voltage in the equation, only current.

If we want voltage/thrust dependence, it also can be done. 
The current/voltage characteristic of flat collector / wire combination 
is derived by Copperman (see http://www.me.umn.edu/courses/me5115/notes/ESPnotes.pdf
for details). 

It has general form:
***********
I = k*G*V(V-V0)             eqn. 1
***********
where k ion mobility coeficient and V voltage and  G depends on particular electrode configuration.

For the case of wire/parallel flat plate electrode configuration
G = 2*pi*e0*L/(d^2*ln(f_geo/r))

              e0 - dielectric permittivity of air
                r is the wire radius;
                d is the wire-plate spacing;
                W plate width
                L  plate lenght (should be >> W) 
f_geo is the characteristic length of particular electrode geompetry 
      (1)       f_geo=4d/pi for 2*d/W<= 0.6, and 
      (2)      f_geo=W/(2*pi exp(pi*d/W))   for 2*d/W>=2.0 
First is more near to lifter case, but maybe second case with some effective W can also be used.

Unfortunately wire/paralel plate approximation is not very good for Lifter.
For example for Lifter 1 at 40kV and d=30mm we get 
with f_geo(1) and r (30 gauge)=0.1275 mm
I= 1.8mA whereas experiment shows 450uA.

If we use f_geo (2) it is not clear what to put as plate witdh W. Intuitivelly it should be less that foil hight h=40mm (because foil is not parallel to wire) but much more then thickness of the foil. I found that using empyrical "effective width" h/7 gives current
I=480 uA which is near to experimental so eqn. can be used in this form.

Anyway, derivation of strict eqn. for G for lifter electrode configuration is still open.

So what about thrust? From above eqn and using my previous eqn. for thrust F=i*d/k
we have voltage/thrust relation:

****************
F=2*pi*e0*L*V(V-V0)/(d*ln(f_geo/r)))                   eqn.2
****************
Remarkalbe thing is that k canceled out and that d went into denominator which
indicates that it should be kept as small as possible because it decreases current as d^2 but increases thrust only as d^1.

Using f_geo (2) with W=h/6 we get for Lifter 1 where 
L=200*3=600mm
V=40kV
r= (assuming 50 gauge = 0.255 mm diameter) = 0.1275 mm

F=0.069 N  which is about twice of experimentally observed 0.032 N probably due to some counter-current. Anyway it is not bad for a raw assesment and considering that counter-current should reduce the thrust.

Let's see what we would get using 50 gauge wire as Tom Ventura recently 
tried (quite a cool experiment considering how brittle it should be)
r= (30 gauge = 0.025 mm diameter) = 0.01275
I obtain using eqn. 1
i= 51 uA
F=0.072 N at 40kV

So with decreasing wire thickness we get thrust increase of 4.3%.

I will explore later how to optimize power/thrust relation based on this eqn., and to find form of G which corresponds exactly to lifter electrode configuration.
 
 

Speed of single charged particle with mass m and charge e accelerated between two electrodes with voltage V is: v= sqrt(2*V*e/m)
At the other hand, momentum of single particle is p=m*v = m*sqrt(2*V*e/m)

To calcualte number of particles flying at given power in unit time, we divide total passed charge by the charge of single particle
n=Q/(e*t)=i/e
current i we obtain from power E as i=E/V so n=E/(V*e)

Total force applyed is equal to total exchanged momentum per unit time (assuming all momentum of accelerated particle is used for propulsion, which is maximal possible estimate).
As result we have:
F=n*p=m*sqrt(2*V*e/m)*E/(e*V) 

If you put there m=me=9.1093897E-31kg (electron mass) that gives ~10^-6 N at
E=100Watt, V=30 000Volt.
In experiment with lifter 4 they 
http://jnaudin.free.fr/html/lifter4.htm) observed thrust 0.4N at 100W, so number above is way to small. 

But if you put m=14*2=28*mp (N2) or m=16*2=32*mp (02), where mp is mass of
proton=1.6726231E-27kg, we have better picture (10^-4N) but still way too small
compared to 0.4N. Whatever ions are used in this "ion wind", these are not ions taken from the air...

Mass of particle required to achieve such thrust is 
m = F^2*V*ee/(2*E^2). For F=0.4N and above voltage and power we have:
m~2*10^7*mp (where mp is proton mass) or 3.69*10^-20 kg.
This mass is not too large to be practical - it could be clusters of the electrode material or electrode coating brocken off the surface under very high stress applied by the voltage.
 
 

That is the easy one. Electromagnetic radiation has momentum,
it has been shown theoretically by Maxwell and experimentally 
measured by Lebedev.
Momentum of electromagnetic wave, p = E/c

To calculate the force, F = p/t = W/c. 
For radiation power W=100Watt we have force W/c ~10-9 N. 

Recalculating this into "equivalent mass" we get ~10^-5 gm...
too small... indeed, electrons moving with curved trajectories and
speeds around 1/3 c (like they do in lifters)should radiate 
microwaves. There are devices called reltrons, used to generate
directed high power microwave impulses, which have exactly same 
principle, and look very much like lifter immerced in a vacuum tube.

However, generated momentum of microwaves is just too small...
 
 

I have taken more close look at it as described here
http://www.ibmm.informatics.bangor.ac.uk/pages/science/dep.htm
and suddently realized that it can not possiblly have to do anything with 
discussed effect. Why? - because dielectrophoresis is a _transient_ effect. 
 It is redistribution of particles in inhomogenous electric field depending on their polarizability. Once the redistribution is finished, material flow is also finished, therefore dielectrophoresis is a _transient_ effect resulting in change of material state from "Voltage OFF" to "Voltage ON", it can not have a continuous component.

1) There can not be any continuous flow of electrons between electrodes due to this effect, because there is no way how they can pass electrode/dielectric boudary. When voltage is switched on, there will be a transient current due to change of capacitance due to dielectric material redistribution. When redistribution is finished, capacitance becomes constant and current stops.
 However, electric current (as well as thrust) in lifter are observed continuously as long as voltage is on.

2) Correspondingly, from 1 follows that can not be any continous flow of dielectric material, which shows that dielectrophoresis can apply to lifters only for short moment when voltage is switched on. 

Now some points not related to lifters (which are DC), but could be relevant to AC-based propulsion devices:

3) When voltage is switched off, material flow is reverse to original one and the momentum of outflowing material is same as momentum of inflowing material therefore making net effect of one voltage pulse equal to zero. 
 This is true unless there is a difference in medium friction coefficient at high and low flow speed, which could result in some momentum retained by electrode arrangement. This point needs to be investigated separately.

4) AC-voltage can be represented as a series of interchanging positive and negative voltage pulses, therefore point 3) applies to them as well. There is no net material flow except that due to differences in friction of in/outflowing material.

That is why application of electrophoresis require some additional material flow to separate more and less polarizable particles, as they say in above cited web-site:

"Selective separation can thus be achieved by applying an additional force such as gravity or fluid flow".