This paper examines the effects of magnetic induction attenuation on current distribution in the exit regions of the Faradaytype, nonequilibrium plasma Magnetohydrodynamic (MHD) generator by numerical calculation using cesiumseeded helium. Calculations show that reasonable magnetic induction attenuation creates a very uniform current distribution near the exit region of generator channel. Furthermore, it was determined that the current distribution in the middle part of generator is negligible, and the output electrodes can be used without large ballast resistors. In addition, the inside resistance of the exit region and the current concentration at the exit electrode edges, both decrease with the attenuation of magnetic flux density. The author illustrates that the exit electrodes of the diagonal Faradaytype, nonequilibrium plasma MHD generator should be arranged in the attenuation region of the magnetic induction, in order to improve the electrical parameters of the generator.
1. Introduction
The quasi onedimensional MHD theory
[1

2]
tells us that performance characteristics of a diagonaltype, nonequilibrium MHD generator are similar to the performance characteristics of a Faradaytype MHD generator. However, the quasi onedimensional MHD theory is only relevant when talking about the boundary conditions of the working gas plasma, and does not accurately reflect the spatial nonuniformity effects in a crosssection of the generator’s channel. Moreover, socalled end effects arise in the exit regions of the MHD generators, degrading the total electrical performance.
Through numerical analysis, the author has investigated the electrical performance in the central part of the diagonal type nonequilibrium MHD generator channel as described in
[3]
.
In addition, while the end effects in the Faradaytype generator have been analyzed in fair detail
[4

6]
, the end effects in the diagonaltype require further analysis
[7

9]
. Thus, the author has investigated influences of the output electrodes’ arrangement and the attenuation of the magnetic induction along the generator channel on the current, as well as potential distributions near the entrance and exist exit of the diagonaltype MHD channel when physical quantities in the channel are assumed to be uniform. The author’s examinations demonstrate that the variation of the arrangements of output electrodes has little effect on current distribution, etc.
[10]
In this paper the author studies the end effects in the diagonaltype, nonequilibrium MHD generator through numerical analysis. Section 2, introduces basic equations and boundary and subsidiary conditions, and then shows the configurations of gas velocity and applied magnetic induction adopted throughout the paper. In section 3, numerical calculations are made, to investigate the effect of the magnetic induction attenuation on current and potential distributions, internal resistance, etc., in the exit regions of the generator
2. Basic Equations
 2.1 Basic equations for current distribution
In the analysis of end effects in a diagonaltype MHD generator, the following assumptions are made: 1) the electric quantities, such as the current, electric field etc., vary with
x
and
y
, where
x
and
y
are the coordinates as shown in
Fig. 1
, and 2) the gas velocity and temperature depend only on
y
according to (9) and (10) which will be presented later, and 3) the pressure is kept constant.
In order to evaluate the current distribution in the generator channel, the author introduce the conventional stream function
Ψ
defined by:
where
J_{x}
and
J_{y}
are the
x
and
y
components of current density, and the
z
component
J_{z}
is assumed not to exist
Coordinate system and generator channel geometry
Then, it is assumed that the magnetic induction and the gas velocity have only the
z
component
B
and the
x
component
u
, respectively. From the Maxwell and the generalized Ohm's law Eqs. (1) and (2) in
[3]
, the following partial differential equation can be derived:
where,
in which
e
is the electron charge,
p_{e}
=
n_{e}kT_{e}
is the electron partial pressure,
n_{e}
is the electron density,
k
is the Boltzmann constant,
T_{e}
is the electron temperature,
β
is the Hall parameter for electron,
β_{i}
is the Hall parameter for an ion, and
σ
the scalar electrical conductivity of the plasma. In addition,
σ
=
μn_{e}e
,
β
=
eB
/
n_{e}m_{e}Qc_{e}
are explained in detail in
[3]
, where
μ
is electron mobility,
m_{e}
is electron mass,
Q
is the collision cross section, and
c_{e}
is the velocity of individual electron.
 2.2 Boundary and subsidiary conditions
First, the boundary conditions on the electrode surfaces is defined as:
where,
E_{x}
is the
x
component of the electric field.
The boundary condition on the insulating wall surface is defined as:
Using (1), these conditions (4) and (5) are transformed to:
Next, in the diagonaltype generator, the potential difference between the anode
A_{i}
and cathode
C_{i}
must be zero as shown in
Fig. 1.
Therefore, the first subsidiary condition is obtained as:
where,
E
is the electric field intensity vector, ds is the line element vector of an optional integral path from
A_{i}
to
C_{i}
, and
V_{i}
is the potential difference between
A_{i}
and
C_{i}
As the load current
I
runs through an arbitrary surface
S_{i}
, it crosses the insulating wall surfaces
A'_{i}
and
C'_{i}
, the second subsidiary condition is written as
where dS is the element vector of the surface
S_{i}
.
Finally, let us assume that the electric quantities vary periodically in the period of the electrode pitch
s
along the gas flow behind the
n
th electrode pair
A_{n}
and
C
_{n}
. Then the condition for the current density
J(x)
is given by
By (1, 8) is transformed into
where
is the current flowing into
A
_{n}
.
The current distributions in the diagonal type generator can be found by numerically solving (2) under the conditions (4′)~(7) and (8′) (see section 3).
 2.3 Calculation of potential
With the obtained numerical solution of
Ψ
, (2) can be solved numerically using conditions (4′)~(7) and (8′), and the electric field
E
at the optional point can be evaluated using (1) and the generalized Ohm's law. Then, the potential at any point can be calculated by the numerical line integration of
E
along an arbitrary integral path from a reference point to the point in consideration.
 2.4 Gas velocity and temperature distributions
As assumed in section 2, the velocity u has only the
x
component
u
, and
u
and
T
vary only in the
y
direction according to the following relation
[11]
where,
h
is the channel height,
u
_{0}
is the gas temperature and
T
_{0}
is the velocity at the center of flow, namely
y
=
h
/2 and
T_{w}
is the wall temperature.
 2.5 Configuration of applied magnetic induction
For effective use of the applied magnetic flux density
B
, the MHD generator channel should be arranged in the attenuation region of
B
. Thus, in order to investigate the influence of the configuration of
B
on the current distribution in the exit regions of the diagonal type generator, the intensity of
B
is assumed to be constant in the central region and decreases linearly from the left edge of the
j
th electrode in the exit regions of the generator. In this this numerical analysis, the author assumes the six configurations of
B
as plotted in
Fig. 2
, where
g
is the gradient of
B
and
j
=5.
Configuration of applied magnetic induction
3. Numerical Method for Subsidiary Conditions
In a diagonal generator, the solution of (3) is required to satisfy the two subsidiary conditions (6) and (7). From (1) and (7), the author can derive the following equation:
where
and
are the values of
Ψ
on the insulating wall surfaces
and
respectively, and
w
is the channel width in the
z
direction.
First, if the values of
I
and
w
are assumed and
are given plausible values, the values of
are decided by (11). When (2) is digitally solved with these values of
and
and using the appropriately assumed values of
u
,
σ
and
β
, the author can obtain the numerical solution of
Ψ
. By applying the solution to (1) and the generalized Ohm's law, the author can find the values of
E_{x}
and
E_{y}
. Furthermore, by substituting the values of
E_{x}
and
E_{y}
into the integral in (6), the author can decide the value of
V_{i}
. Then the value of
V_{i}
obtained is not necessarily equal to zero.
Let us consider the resistance between the electrodes
A_{i}
and
C_{i}
where
h
,
c
and
θ
are the channel height, the electrode width and the angle of inclination to the
y
axis of the lines joining the equipotential electrodes, respectively, and the author assume that an imaginary current defined by
flows through the resistance
R_{i}
′. To make
V_{i}
zero, it is needed to flow the inverse current –
I_{i}
through
R_{i}
. Then it is required to increase by –
I_{i}
the value of
w
(
), which gives the current running into the anode
A_{i}
Again beginning with the new modified values of
, the author must repeat the above mentioned calculation process. When
V_{i}
becomes adequately small after the many repetitions of the above mentioned process, at last the author can obtain the satisfactory numerical solution of
Ψ
.
In connection, the other parts of numerical calculation processes are explained in
[12]
.
4. Numerical Calculation
 4.1 Numerical conditions
Numerical analysis is carried out for the diagonal type MHD generator with the cesium seeded helium in nonequilibrium ionization using the following parameters:
where
ε_{s}
is the seed fraction of
C_{s}
,
B
_{0}
is the magnetic induction in the central region of generator channel, and
δ
is the collision loss factor. These conditions are assumed with respect to a generator of the pilot plant
[13]
. The load current
I
is assumed to flow equally into two output electrodes
E_{1}
and
E_{2}
through a ballast resistance
R_{b}
defined by (see
Fig. 1
)
 4.2 Calculation results
In
Figs. 3(a)
~
(c)
, the current distributions are plotted in the case of
g
=0, 6 and 10T/m, respectively,
B
_{0}
=4T and
I
=70A, where the contour interval of current streamlines is 1/20 of the load current
I
. In the figures, 〈
J
〉
_{el }
= 0.583A/cm
^{2}
, 〈
σ
〉=1.84 mho/m, 〈
β
〉=2.01 and
β_{crit}
=2.48, where 〈
J
〉
_{el}
is the average current density on the output electrodes, 〈
σ
〉 and 〈
β
〉 are the average electrical conductivity and Hall parameter in the center of flow, respectively, and
β
_{crit}
is the critical Hall parameter
[14]
.
Fig. 3(a)
shows that the current concentration at the edges of the output electrodes is very intensive when
B
does not attenuate. On the other hand,
Figs. 3(b)
and
(c)
indicate that the concentration weakens as the attenuation of
B
increases, since
β
becomes small in the area suffering a spatial reduction of
B
. Also it is seen that the current flowing into a diagonally connected electrode pair reduces with an increasing gradient of magnetic induction in the entrance region of channel. For instance, the currents of about 60, 25 and 15% of
I
flow into
C
, when
g
=0, 6 and 10T/m, respectively. Also the figures denote that the eddy current is not induced when the output electrodes are disposed in the region of the attenuating magnetic induction
[5]
, and that arranging the output electrodes within the attenuation region of
B
does not have a great influence on the current distribution in the central part of generator channel.
Current distributions
Next,
Fig. 4
shows the variation of the potential difference between the electrode pairs
A_{1}

C_{1}
~
A_{8}

C_{8}
and the electrode
E_{1}
. From the figures it is seen that the relatively large potential difference arises between the two output electrodes
E_{1}
and
E_{2}
when
B
does not attenuate, namely
g
=0. On the other hand, the potential difference become smaller as g becomes larger, it almost vanishes for
g
=6, and the inverse difference appears for
g
>7. Also
Fig. 4
demonstrates that the potential differences in the central part of generator channel are influenced little by the decrease of the magnetic induction.
Next, for estimation of the end effects of the generator, the author evaluates the internal resistance
R_{i}
of the exit regions and the grade of the current concentration on the output electrodes given by the relations where
V
_{0}
and
V
are noload and load potential difference between the output electrode
E_{1}
and the
n
th electrode, respectively, and
J
_{peak}
is the maximum current density on the output electrodes. In this connection,
J_{peak}
/〈
J
〉
_{el}
=1 means the state of no current concentration and
J_{peak}
/〈
J
〉
_{el}
>>1 does the intensive current concentration at an electrode edge.
Variation of potential difference for B_{0}=4
Influence of g on R_{i}/R_{i0}, R_{b}/R_{b0} and J_{peak}/〈J〉_{el} when B_{0}=4
Now
Fig. 5
shows the variations of
R_{i}
/
R
_{i0}
,
R_{b}
/
R
_{b0}
and
J_{peak}
/〈
J
〉
_{el}
by
g
, where
R_{i0}
and
R_{b0}
are
R_{i}
and
R_{b}
for
g
=0, respectively. From the figure, it is seen that
R_{i}
decreases with
g
, for instance the value of
R_{i}
for
g
=6.0 becomes about 80% of the one of
R
_{i0}
, and that
J_{peak}
/〈
J
〉
_{el}
decreases from
g
=0 to 8T/m, reaches the minimum value 1.90 and increases again. This fact shows that the current concentration at the edges of the output electrodes is almost diminished when
g
=8T/m. Accordingly, arranging the output electrodes in the attenuation area of the magnetic flux density is useful to guard the output electrodes. Also
Fig. 5
tells that
R_{b}
/
R
_{b0}
decreases with
g
, becomes almost zero for
g
=6.5 and then increases with
g
. Therefore, it is shown that many output electrodes will require large ballast resistors when
B
does not attenuate or exceeds 8, but they can be used without large ballast resistors in the range of
g
=6~7T/m.
In
Fig. 6
, the current distribution is plotted when
g
=6T/m,
B
_{0}
=5T and
I
=150A, 〈
J
〉
_{el}
=1.25A/cm
^{2}
, 〈
σ
〉=2.85 mho/m, 〈
β
〉=2.48 and
β_{crit}
=1.90. The figure indicates that the streamer is induced in the central part of generator, while the current distribution becomes successively uniform as
B
attenuates along the generator channel and the current concentration is almost swept away near the output electrodes. This demonstrates that arranging the output electrodes within the attenuating region of
B
is effective for the case where the streamer is generated in the central region of generator channel, too.
Current distribution for g=6 and B_{0}=5
5. Conclusions
The main conclusions derived from the above described numerical calculation are as follows:
A suitable distribution of the magnetic flux density can make the current distribution very uniform near the exit region of generator channel, both when the streamer is not induced and when it is induced in the central region.
Disposing the output electrodes within the attenuation area of magnetic flux density has little influence on the current distribution in the central part of generator channel.
When the output electrodes are disposed in the region with a suitably reduced magnetic flux density, the potential difference and the ballast resistance between two output electrodes become very small. Accordingly it is thought that many output electrodes can be used without large ballast resistors.
The internal resistance in the exit region of the generator channel decreases as the magnetic flux density attenuates.
The current concentration at the edges of output electrodes can be fairly eliminated by attenuating magnetic flux density.
As mentioned above, it is made clear that the output electrodes of the diagonal type nonequilibrium plasma MHD generator should be arranged in the region of the attenuating magnetic flux density, since arranging them in the region of the decreasing magnetic flux density become useful for the improvement of the electrical performance of the generator
BIO
Le Chi Kien He received B.S degree in Electrical Engineering from HCMCUniversity of Technical Education, Vietnam in 1997. He received his M.S. and Ph.D. degree from Nagaoka University of Technology, Japan in 2002 and 2005. Since 2005, he is lecturer at HCMCUniversity of Technical Education, Faculty of Electrical and Electronics Engineering. His research interests are high efficiency power generation systems, Magnetohydrodynamics and combined systems.
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