AAWSAP DIRD, MHD Air Breathing Propulsion and Power for Aerospace Applications, November 2010

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Similarly, if a voltage source (e.g., a battery) is connected to the electrodes
placed on either side of the flow in such a way that the applied e.m .f. acts
against the induced Faraday e.m.f., then the current will flow in the direction
opposite to the Faraday current, and the jxB force will be in the direction of the
flow. This will be an MHD accelerator that converts the battery-supplied electrical
energy partially into enthalpy of the flow and partially into Joule dissipation in
the circuit. The corresponding load factor, k, defined as the ratio of the applied
electric field E and the product uB, k=E/uB, is greater than 1 in this accelerator
configuration.
In generator and accelerator devices, the interaction between the induced
motion of electric charges across the B field and that field results in an e.m.f.
induced along the flow. This secondary e.m.f. is called the "Hall e.m.f., " and the
magnitude of Hall effect increases with the ratio of electron-cyclotron frequency,
ws=eB/m, and the electron collision frequency, v. This ratio is called the electron
Hall parameter, ne. As the Hall parameter approaches 1, the Hall current
directed along the flow increases at the expense of the transverse Faraday
current, resulting in reduction of the jxB force. To reduce or eliminate the Hall
current, the electrodes placed on either side of the flow are normally segmented
and thus form multiple pairs. Each electrode pair has a proper resistor and/or
battery in its circuit. Theoretically, this segmented-electrode Faraday
configuration enables the performance equal to that without the parasitic Hall
effect. However, as the Hall parameter increases, so does the voltage fall
between the adjacent electrode segments, so that eventually arcing between the
segments starts, effectively negating the advantage of segmenting.
A better (and more "natural") MHD configuration at high values of the Hall
parameter is the one where each electrode pair (with the electrodes on either
sides of the flow and right across each other) is shorted, and the voltage is
either extracted (in the generator case) or applied (in the accelerator case) along
the flow, between the first and the last electrode pair. This is called the "Hall
configuration ."
A useful dimensionless parameter reflecting the strength of MHD interaction is
called the MHD interaction parameter, or the Stuart number, and it represents
the ampere body force effect relative to the flow momentum flux:
S= aBL
pu
In this equation, cr is the electrical conductivity of the fluid, B is the magnetic
field strength, L is the characteristic linear dimension, p is the fluid density, and
u is the velocity.
Therefore, for a significant MHD effect in high-speed, high dynamic pressure flow
(e.g., in hypersonics), the conductivity and the B field strength must be high .
Herein lies the principal problem for aeronautical MHD applications. Indeed,
normal air is not an electrical conductor. Air does become ionized and thus
electrically conducting as it is heated to very high temperatures (3,000- 10,000K
or higher), such as those achieved in shock and boundary layers around reentry
vehicles, at Mach numbers M= 12-25 or so. The ionization fraction then reaches
10·5-10·2, and the conductivity from 100 to about 3,000 mho/m ensues. Seeding
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