User:Tokamac/Electrothermal instability

The electrothermal instability (also known as the ionization instability or Velikhov instability in the literature) is a magnetohydrodynamic (MHD) instability appearing in magnetized non-thermal plasmas used in MHD converters. It was first theoretically discovered in 1962 and experimentally measured into a MHD generator in 1964 by Evgeny Velikhov.

Physical explanation and characteristics

Evolution of the electrothermal instability in a Faraday MHD converter. Electric current lines.

This instability is a turbulence of the electron gas in a non-equilibrium plasma (i.e. where the electron temperature T_e is greatly higher than the overall gas temperature T_g). It arises when a magnetic field powerful enough is applied in such a plasma, reaching a critical Hall parameter β_cr.

Locally, the number of electrons and their temperature fluctuate (electron density and thermal velocity) as the electric current and the electric field. The physical explanation of this instability comes from a heat transfer rate increase to free electrons through Joule effect, which overtakes the time needed to evacuate this heat surplus through collisional, rotational, vibrational, radiative or conductive energy transfers.

The electrothermal instability occurs extremely quickly, in a few microseconds. The plasma becomes non-homogeneous, transformed into alternating layers of high free electron and poor free electron densities. Visually the plasma appears stratified, as a "pile of plates".

Hall effect in plasmas

The Hall effect in ionized gases has nothing to do with the Hall effect in solids (where the Hall parameter is always very inferior to unity). In a plasma, the Hall parameter can take any value.

The Hall parameter β in a plasma is the ratio between the electron gyrofrequency Ω_e and the electron-heavy particles collision frequency ν:

$\beta \,=\,{\frac {\Omega _{e}}{\nu }}\,=\,{\frac {e\ B}{m_{e}\ \nu }}$

where

e is the electron charge (1.6 × 10^-19 coulomb)

B is the magnetic field (in teslas)

m_e is the electron mass (0.9 × 10^-30 kg)

The Hall parameter value increases with the magnetic field strength.

Physically, when the Hall parameter is low, the trajectories of electrons between two encounters with heavy particles (neutral or ion) are almost linear. But if the Hall parameter is high, the electron movements are highly curved. The current density vector J is no more colinear with the electric field vector E. The two vectors J and E make the Hall angle θ which also gives the Hall parameter:

$\ \beta \,=\,\tan \theta$

Plasma conductivity and magnetic fields

In a non-equilibrium ionized gas with high Hall parameter, Ohm's law,

$\mathbf {J} =\sigma \mathbf {E}$

where σ is the electrical conductivity (in siemens per metre),

is a matrix, because the electrical conductivity σ is a matrix:

$\sigma =\sigma _{s}{\begin{Vmatrix}{\dfrac {1}{1+\beta ^{2}}}&{\dfrac {-\beta }{1+\beta ^{2}}}\\{\dfrac {\beta }{1+\beta ^{2}}}&{\dfrac {1}{1+\beta ^{2}}}\end{Vmatrix}}$

σ_S is the scalar electrical conductivity:

$\sigma _{s}={\frac {n_{e}\ e^{2}}{m_{e}\ \nu }}$

where n_e is the electron density (number of electrons per cubic meter).

The current density J has two components:

$J_{\parallel }={\frac {n_{e}\ e^{2}}{m_{e}\ \nu }}\ {\frac {E}{1+\beta ^{2}}}\qquad {\text{and}}\qquad J_{\perp }={\frac {-n_{e}\ e^{2}}{m_{e}\ \nu }}\ {\frac {\beta \ E}{1+\beta ^{2}}}$

Therefore

$J_{\perp }=J_{\parallel }\ \beta$

The Hall effect makes electrons "crabwalk".

When the magnetic field B is high, the Hall parameter β is also high.

So when β is high, ${\frac {1}{1+\beta ^{2}}}\ll 1$

Thus both conductivities σ_⊥ and σ_// become weak, therefore the electric current cannot flow in these areas. This explains why the electron current density is weak where the magnetic field is the stongest.

Critical Hall parameter

The electrothermal instability occurs in a plasma at a (T_e > T_g) regime when the Hall parameter is higher that a critical value β_cr.

We have

$f={\frac {\left({\frac {\delta \mu }{\mu }}\right)}{\left({\frac {\delta n_{e}}{n_{e}}}\right)}}$

where μ is the electron mobility (in m²/(V·s))

and

$s={\frac {2\ k\ T_{e}^{2}}{E_{i}\;(T_{e}-T_{g})}}\times {\frac {1}{1+{\dfrac {3}{2}}\ {\dfrac {k\;T_{e}}{E_{i}}}}}$

where E_i is the ionization energy (in electron volts) and k the Boltzmann constant.

The growth rate of the instability is

$g={\frac {\sigma \ E^{2}}{n_{e}\;\left(E_{i}+{\frac {3}{2}}k\;T_{e}\right)\;\left(1+\beta ^{2}\right)}}\;(\beta -\beta _{cr})$

And the critical Hall parameter is

$\beta _{cr}=1.935f+0.065+s~$

The critical Hall parameter β_cr greatly varies according to the degree of ionization α :

$\alpha ={\frac {n_{i}}{n_{n}}}$

where n_i is the ion density and n_n the neutral density (in particles per cubic metre).

The electron-ion collision cross section Q_ei is much bigger than the electron-neutral collision cross section Q_en:

$Q_{ei}=10^{4}\;Q{en}$

Therefore with only α = 10^-4, the electron-ion collision frequency ν_ei equals the electron-neutral collision frequency ν_en.

• For a partially or weakly ionized gas (non-Coulombian plasma, when ν_ei < ν_en ):

$\beta _{cr}\approx (s^{2}+2s)^{\frac {1}{2}}$

• For a fully ionized gas (Coulombian plasma, when ν_ei > ν_en ):

$\beta _{cr}\approx (2+s)$

NB: The term "fully ionized gas" has been imposed by Lyman Spitzer but a "fully" ionized plasma can have a degree of ionization as low as 0.01%.

Technical problems and solutions

The unexpected large and quick drop of current density due to the electrothermal instability ruined many MHD projects worldwide, while previous calculation envisaged energy conversion efficiencies over 60% within these devices. This prevented further developments of non-equilibrium MHD generators and compelled most engaged countries to cancel their MHD power plants programs and to retire completely from this research field in the early 1970's, because this technical problem was considered as an impassable stumbling block in these days.

However experimental studies about the growth rate of the electrothermal instability and the critical conditions showed a stability region exists for high electron temperatures. The stabiliy is given by Coulombian oscillations that bring the Hall parameter down, under the critical value. This shows an MHD generator can be used at (T_e > T_g) regime when the gas is "fully ionized", with a critical Hall parameter β_cr < 2.

This data would imply only Faraday generators could be used with that solution, while the Hall generators would be condemned because in these systems only a high Hall parameter gives a good efficiency. Studies were made with the two configurations in non-equilibrium regimes and confirmed this problem.

Coulombian regime can be achieved through a preionization and a special gas compound with low ionization potential to reach a very high ionization rate, greater than the growth rate of the instability. The plasma is then stabilized by the ionization mechanism itself. A solution with a helium-argon mix and 2% addition of carbon dioxide enables a quick absorbtion of electron energy thanks to the large CO₂ cross section. The stability range occurs for T_e = 10,000 kelvins and T_g = 4,000 K. But such an electrothermal control do not allow to go below T_g = 3,000 K and the converter regime is near short circuit, so such a solution is not practical for any industrial energy conversion.

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However some discoveries hinted at some possible technological mastering of the instability:

1. Special gas compound with very low ionization potential seed. Cools down electrons and overtake the electrothermal instability, with a ionization rate fast enough to decrease the Hall parameter under the critical threshold. For example, 2% addition of carbon dioxide enables a quick absorbtion of elecron energy thanks to the large CO₂ cross section. Such a method allows to control the electrothermal instability in a non-equilibrium MHD generator along a stability range of T_e = 10,000 Kelvin and T_i = 4,000 K (until 3,000 K, limit where the instability hands over the reins again).
2. Dynamic stabilization by oscillating fields. Oscillation of the electric field or of an additional RF electromagnetic field modify the Hall parameter.
3. Non-homogeneous magnetic fields with magnetic pressure gradients. Lowest magnetic pressure areas are distributed in order to create magnetic confinement potential wells between opposite electrodes and canalize the current as several fine streamers without turbulence. Some fluctuation of the B-field over space creates high B and low B areas. In the high B areas the Hall parameter is high which increases the local resistivity of the plasma, so the electrons will tend to flow along the minimum B pathes. The local current density J is increased along these pathes. There the electron density and temperature are subsequently rised, such as some Coulomb regime is locally reached, that decreases the local Hall parameter value. Where the value of this non-uniform Hall parameter plasma field is lower than the critical value, the streamers are stable in the minimum B pathes, surrounded by unstable high Hall parameter area. This method allows a Faraday MHD generator with segmented electrodes to work with high Hall parameter.
4. Laser guidance. Another alternative confinement method, similar to the magnetic gradient effect, would be to force each streamer to follow a direct path between a pair of electrodes thanks to a laser, which would locally increase the ionization rate. But this is still a theoretical idea.

See also

• Magnetohydrodynamics
• MHD generator
• Evgeny Velikhov

External links

• M. Mitchner, C.H. Kruger Jr., Two-temperature ionization instability: Chapter 4 (MHD) - Section 10, pp. 230–241. From the plasma physics course book Partially Ionized Gases, John Wiley & Sons, 1973 (reprint 1992), Mechanical Engineering Department, Stanford University, CA, USA. ISBN 0-471-61172-7