Strict formulation of corona discharge electrodynamics.
Solution for field intensity, charge density and force.

On-Line Lecture
original in http://sudy_zhenja.tripod.com/lifter_theory

Evgenij Barsoukov
11/26/02


In my derivations of force/voltage and force/current equations,
well known to the readers, I used knowledge about corona
discharge kinetics, obtained from textbooks and literature on the
subject. However, I tried to avoid any "technical" issues and
mathematics for the merit of not-prepared reader, which in some cases
resulted in the feeling that the more deep explanation does not exist
at all and it is all just an empyrical "scalling theory".

However, in recent times the quest of Lifter-design has been
jouned by some research labs (such as Army lab) and individuals
with sufficient math. knowledges to pursue own theoretical development.

Therefore I feel a need to give more deep insite into "how" the equations
I was using (like Cooperman's equation) were derived so anybody
can involve into similar derivation (particularly numerical modeling)
having a clean start with the best available today scientific basis.
Note that everything in this section is well known to specialists.
Nevertheless I feel that it is good to have it here in compact form.
Most information is taken from the web-site of Moscow institute
of energetics, department of technology and electro-physics
of high voltages: http://fee.mpei.ac.ru/elstat/


What are we dealing with - basic starting points:

- plasma is formed by lavine ionization electric field near narrow electrode. This
electrode if further called "corona wire".

- plasma formation is taken place when Peek's condition for electric field
is satisfied at the surface of corona wire. Peek's condition for wire with
radius r is given as:

Eqn 0

Peek's condition is found empyrically. Theoretically same value is found
using the formalizm known as Townsend avalanche, but this part of plasma
physics is outside the scope here.

- Once plasma is formed around the corona wire, ions of the same sign as
the wire move to the opposite electrode. The area between plasma and
opposite electrode has ions of only one side, therefore we will call it unipolar
region. Ions in unipolar region are interacting with air and transfer momentum
to it.

- Because amount of momentum transferred to to air is much larger that
own momentum of ions, it is a good approximation to consider that electric
field is applied directly to air, which is "carrying" certain charge density produced
by ions. Therefor force applied to a basic volume of air will be
!dF(x,y,z)=!dE(x,y,z)*r(x,y,z)
where r is charge density. (!) indicates vector. Therefore to find total force
we need to integrate vector !dF through entire monopolar region. For this we need
to wind dE(x,y,z) and r(x,y,z) unipolar region

- To find the needed density and field we need to solve the complete system
of vector field equations describing electrodynamics of the process. We need to
consider the following:

1) Puasson equation describing connection between electric field and
charge density

Eqn. 1

2) Dependence of the field instensity from potential gradient

Eqn. 2

3) Continuity of the charge flow (charge concervation)

Eqn. 3

4) Charge flow density itself is defined by mobility of ions and charge density

Eqn. 4

Note that "mobility" of ions is a macroscipic characteristic describing migration
of ions in electric field in given medium. It is a result of process of multiple collisions
of ions accelerated by external field, resulting in macroscopically constant ion velocity
k*E. k can be obtained theoretically using Maxwell-Bolztman theory for any pure gas
and is therefore a well understood quantity, even if in some cases it is easier to use
empyrically measured value. Additionaly, it is worth noting that k depends on temperature
and pressure as k=k0/delta (same delta as in Peek's eqn (0)).
To be completely precise, it should be noted that k depends on flight time. For
example at flight time <0.5msec +ion mobility is 2.1 cm^2/V*sec, but for large
flight time neutral molecules get attached to ion, resulting in average value of 1.4 cm^2/V*sec.

System of equations 1-4 gives a complete description of uni-polar region of corona
discharge. However, to solve it, we need to know boundary conditions at all electrodes.
They are given as follows:

1) Potential at corona electrode is U
2) Potential at collector is 0
3) Potential gradient at the surface of corona electrode is equal to Peek's critical
field density

Eqn. 5

It is quite simple to justify the condition 3 by considering that if potential gradient
becomes larger then -E[0], ionisation rate increases which in turn increase
the volume of the plasma. Increased volume of the plasma has larger radius,
which in turn result in increase of E[0] value. Increase of E[0] value results in
decrease of the ionization rate, which therefore shrinks the plasma back to original
size. It can be seen that due to this "active feed-back" condition 3 is always satisfied.

Now, anybody who knows how to solve vector field differential equations is welcome
to apply above to absolutely any configuration of electrodes, solve it of r and !E and
calculate force as mentioned above. Obviously, analytical
solution will be possible only in some rare cases where geometry can be reduced
to 1 dimensional . However, possibilities for numberical solutions are not restricted,
and there are both commpercial and free 3d-vector field solvers which can be applied here. That is where anybody doing lifters as a full-time job, particularly one getting taxpayers money, should look at.

Now, some examples how this is applied and what kind of field and charge densities
we would get as result. One case which can be solved analytically is the cylindrical
configuration - corona wire inside and coaxial to a cylindrical collector.

In this case problem can be expressed in cylindrical coordinates as one-dimensional and all parameters depend only on radius. The system of equaitions 1-4 will take the form:

Eqn 1'

Eqn 2'

using current per unit lengh
of corona wire A we get for 3:

Fig. 1 Field density with corona ON (2, red line) and
corona OFF (2, black line), using eqn 8

Eqn 3'

Eqn 4'

Substituting 3 into 4 we get charge density as

Eqn, 6

Now we can put r inot Eqn 1' to get:

Eqn.7

To find E, integrating this equation from r0 to R

giving

Eqn. 8

We can calculate field intensity at any point using this equation.
Corespondingly, substituting it into eqn 6 we get charge density

Eqn. 9

Current A can be found by integrating eqn. 2', with substituted E found in eqn.8

As result we get

where

and

Now, by using this A in equations 8 and 9 we can find profiles E(r) and r(r) for any given
voltage U between electrodes.

Such profiles for E(r) are given in at Fig. 1 showing field with (large U) and
without (U<U0) corona discharge. It can be seen that the field in presence of
discharge becomes constant at certain distance from electrode, and is lower near corona electrode then in case without discharge (result of shielding by space-charge).

Because from 6 charge is inversely proportional to field and electrode distance
it is clear that it will have maxmim near the corona (small r) but then gradualy
decreases with constant E and increasing r. Resulting contribution to force (E*r) will
decrease with r in this case.

It is worth noting that while all calculatinos were made with disregard of vector
directions, we should always remember about them when speaking about force.
For example, in case of cylindrical geometry, the direction of !E around the wire
undergo complete 360 degree change therefore integral of !E*r is zero, resulting
in zero overal force.

Quite different situation is observed in wire-plane geometry, There field density
is in fact increasing with distance to corona electrode near collector, as can be seen in Fig 2 (result of numerical solutin for wire/plate geometry). The charge density first decreases and then stabilizes. As result, field increase counterweights the decrease of charge density and results to constant contribution to force from certain distance.

Fig. 2 Distribution of electric field (1-without corona discharge,
2- with corona discharge) and charge density (3) in case of
wire/plane geometry. r[0] is position of corona electrode and h
is position of collector.

Well, I hope I gave enough information here for everybody to have fun with
your own derivations for any complex geometries. Obvious qualitative result here is that contribution to the force is smoothly changing all over the monopolar region and is not going to zero in any place of it. The change itself is wildly dependend on particular geometry and has to be looked at quantitatively in any case.

Finaly, for those who is interested in actual force/voltage characteristics of
lifter, Cooperman equation derived using similar way as above can be used directly to find force with reasonable precision (using simplified assumptions about charge density instead of integration). See "force/voltage" section on my web-site for more details on that.