Analyses on the effect of magnetic induction attenuation on the current distribution in a faraday MHD generator

TAẽP CH PHAT TRIEN KH&CN, TAP 16, SO K2- 2013

ANALYSES ON THE EFFECT OF MAGNETIC INDUCTION ATTENUATION ON
THE CURRENT DISTRIBUTION IN A FARADAY MHD GENERATOR
Le Chi Kien
Ho Chi Minh City - University of Technical Education
(Manuscript Received on December 24th, 2012, Manuscript Revised April 24th, 2013)

ABSTRACT: This paper examines the dependence of the attenuation of magnetic induction on
the current distribution etc. in the exit regions of the Faraday type non-equilibrium plasma MHD
generator by a two-dimensional calculation. The numerical analyses are made for an example of the
cesium-seeded helium. As a result, a reasonable magnetic induction attenuation can make the
distribution of current very uniform near the exit region of generator channel and has little influence on
the current distribution in the middle part of generator, and the output electrodes can be used without
great ballast resistors. Also the inside resistance of the exit region and the current concentration at the
exit electrode edges decrease with the attenuation of magnetic flux density. By the author's examination,
it is made clear that the exit electrodes of the diagonal Faraday type non-equilibrium plasma MHD
generator should be arranged in the attenuation region of the magnetic induction, since arranging them
in this region becomes useful for the improvement of the electrical parameters of generator.
Keywords: Numerical calculation, MHD generator, diagonal type, ballast resistance, twodimensional analysis.
1. INTRODUCTION

Accordingly,

by

a

two-dimensional

Already it has been ascertained that the

analysis, the author has investigated the

performance characteristics of a diagonal type

electrical characteristics in the central part of

non-equilibrium plasma generator can be well

the diagonal type non-equilibrium plasma

approached to those of the Faraday type one by

generator duct as described in [3], etc.

the quasi one-dimensional MHD theory [1,2].

Moreover, in the end regions of the MHD

However, though this theory is convenient for

generators there arise the so-called end effects,

us to grasp the outline of the generator

and

characteristics, it is very difficult to treat

characteristics of the generators.

they

degrade

the

total

electrical

accurately the effects of the spatial non-

Hence, up to now the end effects in the

uniformity of the working gas plasma in the

Faraday type generator have been analyzed in

generator duct cross section by the above

fair detail [4-6]. On the other hand, the end

theory.

effects in the diagonal type have been only a
little discussed [7-9]. Therefore, the author has
investigated

some

influences

of

the

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Science & Technology Development, Vol 16, No.K2- 2013
arrangement of the output electrodes and the

In order to evaluate the current distribution

attenuation of the magnetic induction along the

in the generator duct, we introduce the

generator duct on the current and potential

conventional stream function  defined by

distributions etc. near the entrance and exist of

J x   y , J y    x

(1)

the diagonal type MHD duct when the physical

quantities in the duct are assumed to be

where Jx and Jy are the x and y components of

uniform, and shown that the variation of the

current density, and the z component Jz is

arrangements of output electrodes has little

assumed not to exist.

effect on the current distribution etc. [10]
In this paper the author studies the end
plasma

generator

by

a

the boundary and subsidiary conditions are

A'1

Anode Ai

A'i

y

two-dimensional

analysis. In section 2, the basic equations and

A1

E2

E1

effects in the diagonal type non-equilibrium

Insulator

Rb

I

h

z
(B)



x
(u)

introduced, then, are shown configurations of
the gas velocity and the applied magnetic

C1

induction that are adopted in the present paper.

C'1

c

Ci

C'i

Cathode Cn

C'n

s

In section 3, by the numerical calculations are
investigated the influences of the attenuation of

Figure 1. Coordinate system and generator duct

the magnetic induction on the current and

geometry

potential distributions, the internal resistance

Then, it is assumed that the magnetic

etc. in the end regions of the generator.

induction and the gas velocity have only the z

2. BASIC EQUATIONS

component B and the x component u,

2.1. Basic equations for current distribution

respectively, from the Maxwell equations and

In the analysis of end effects in a diagonal
type MHD generator, it is assumed that 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 that the
gas velocity and temperature depend on only y
according to Eqs. (9) and (10) which will be
presented later, and that the pressure is kept
constant.

Trang 64

the generalized Ohm's law which are given in
Eqs. (1) and (2) in Ref. [3], we can derive the

following partial differential equation:
 2   P   x  Q    y  R
(2)

where


TAẽP CH PHAT TRIEN KH&CN, TAP 16, SO K2- 2013
P / / / x / / y
Q / / / y / / x
R / { p e / y / en e / x

p e / x / en e / y

u B / x }

1 i


where E is the electric field intensity vector, ds
the line element vector of an optional integral
path from Ai to Ci, and Vi the potential
(3)

difference between Ai and Ci.
As the current which runs through an
arbitrary surface Si crossing the insulating wall

in which e is the electron charge, pe =nekTe the

surfaces A'i and C'i is equal to the load current

electron partial pressure, ne the electron

I, the second subsidiary condition is written as

density, k Boltzmann's constant, Te the electron

SJdS

I , i=1, 2, , n

(7)

i

temperature, the Hall parameter for electron,
i the Hall parameter for ion, and the scalar

where dS is the element vector of the surface

electrical conductivity of the plasma. In

Si.

addition, since , , ne and Te are given in Ref.

Lastly, let us assume that the electric

[3], we omit the explanation for them in this

quantities vary periodically in the period of the

paper.

electrode pitch s along the gas flow behind the

2.2. Boundary and subsidiary conditions

n-th electrode pair An and Cn. Then the

First, the boundary condition on the

by

electrode surfaces is
Ex 0

J (x s) J( x )

(4)

where Ex is the x component of electric field.
The one on the insulating wall surfaces is
Jy 0

(5)

(8)

By Eq. (1), the Eq. (8) is transformed into
( x s) ( x ) I (yn )

(8)

where I (yn ) is the current flowing into An.

Using Eq. (1), these conditions (4) and (5)
are transformed to

The current distributions in the diagonal
type generator can be found by numerically

/ y / x p e / x / en e 0 (4)

const

condition for the current density J(x) is given

solving Eq. (2) under the conditions (4)~(7)
and (8) (see section 3).

(5)

Next, in the diagonal type generator, the

2.3. Calculation of potential

potential difference must be zero between the

When Eq. (2) is numerically solved under

anode Ai and cathode Ci which are shorted each

the conditions (4)~(7) and (8), the electric

other as shown in Fig. 1. Therefore, the first

field E at the optional point can be evaluated by

subsidiary condition is obtained as

Eq. (1) and the generalized Ohm's law, with the

Vi

Ci

A

i

Eds 0 , i=1, 2, , n

(6)

obtained numerical solution of . Then the
potential at any point can be calculated by the
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Science & Technology Development, Vol 16, No.K2- 2013
numerical line integration of E along an

the end regions of the diagonal type generator,

arbitrary integral path from a reference point to

we assume that the intensity of B is constant in

the considered point.

the central region and decreases linearly from

2.4.

Gas

velocity

and

temperature

regions of the generator. In this connection in

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

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.
3.

relation [11]
u / u 0  4 y / h (1  y / h )m

NUMERICAL

METHOD

FOR

SUBSIDIARY CONDITIONS
(9)
In a diagonal generator, the solution of Eq.
n

( T  T w ) /( T0  Tw )  4 y / h (1  y / h )

(10)

respectively, where h is the duct height, u0 and
T0 are the gas temperature and velocity at the
center of flow, namely y=h/2 and Tw is the wall
temperature.

4

the left edge of the j-th electrode in the end

(3) is required to satisfy the two subsidiary
conditions (6) and (7). From Eqs. (1) and (7),
we can derive the following equation:
iA  iC  I w , i=1, 2,…, n (11)

where iA  and iC are the values of  on the
insulating

g=0.0

wall

surfaces

Ai

and

Ci

B [T]

respectively, and w is the duct width in the z

2.0

direction.

4.0

First, if the values of I and w are assumed

2

6.0
8.0

and iA  are given plausible values, the values

10.0

1 2 3 4 5 6 7 8
Electrodes

of iC are decided by Eq. (11). When Eq. (2)
is digitally solved with these values of iA
and iC and the appropriately assumed values

Figure 2. Configuration of applied magnetic
induction

of u,  and , we can obtain the numerical
solution of . By applying the solution to Eq.

2.5. Configuration of Applied Magnetic

(1) and the generalized Ohm's law, we can find

Induction

the values of Ex and Ey. Further, by substituting

For effective use of the applied magnetic

the values of Ex and Ey into the integral in Eq.

flux density B, the MHD generator duct may be

(6), we can decide the value of Vi. Then the

arranged in the attenuation region of B. So in

value of Vi obtained is not necessarily equal to

order to investigate the influence of the

zero.

configuration of B on the current distribution in
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TAẽP CH PHAT TRIEN KH&CN, TAP 16, SO K2- 2013
Let us consider the resistance between the

R i h

h 0 . 2 , s 0 . 1, w 0 . 1, c 0 . 06 m




(14)
u 0 2000 m / s , m n 1 7 , B 0 4 or 5 T

5, s 0 .3 %
T0 1800 K , Tw 1600 K , p 5 atm

electrodes Ai and Ci

i cw cos( ), i=1, 2,, n

(12)

where h, c and are the duct height, the
electrode width and the angle of inclination to
the y axis of the lines joining the equipotential
electrodes, respectively, and we assume that an
imaginary current defined by

where s is the seed fraction of Cs, B0 the
magnetic induction in the central region of
generator duct, and the collision loss factor.
These conditions are assumed with respect to a
generator of the pilot plant [13]. The load

I i Vi R i , i=1, 2,, n

(13)

current I is assumed to flow equally into two

flows through the resistance Ri. To make Vi

output electrodes E1 and E2 through a ballast

zero, it is needed to flow the inverse current Ii

resistance Rb defined by (see Fig. 1)

through R i. Then it is required to increase by Ii
the value of

w( iA1

iA ),

R

b



which gives the

E2

E

Eds

(I / 2)

(15)

1

4.2. Calculation Results

current running into the anode Ai.
Again beginning with the new modified

In Figs. 3a~c, the current distributions are

values of iA , we must repeat the above

plotted in the case of g=0, 6 and 10T/m,

mentioned calculation process.

When Vi

becomes adequately small after the many
repetitions of the above mentioned process, at
last we can obtain the satisfactory numerical
solution of .

respectively, B0=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/cm2, =1.84 mho/m, =2.01 and
crit=2.48, where Jel is the average current
density on the output electrodes, and are

In connection, the other parts of numerical
calculation processes are explained in Ref.
[12].
4. NUMERICAL CALCULATION
4.1. Numerical Conditions

the average electrical conductivity and Hall
parameter in the center of flow, respectively,
crit is the critical Hall parameter [14].
Figure

3(a)

shows

that

the

current

concentration at the edges of the output

Numerical analysis is carried out for the

electrodes is very intensive when B does not

diagonal type MHD generator with the cesium

attenuate. On the other hand, Figs. 3(b) and (c)

seeded helium in non-equilibrium ionization in

indicate that the concentration weakens as the

which

attenuation of B increases, since becomes
small in the area suffering a spatial reduction of
B. Also it is seen that the current flowing into a
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Science & Technology Development, Vol 16, No.K2- 2013
diagonally connected electrode pair reduces

electrodes are disposed in the region of the

with increasing the gradient of the magnetic

attenuating magnetic induction [5], and that

induction in the entrance region of duct, for

arranging the output electrodes within the

instance, the currents of about 60, 25 and 15%

attenuation region of B does not have a great

of I flow into C, when g=0, 6 and 10T/m,

influence on the current distribution in the

respectively. Also the figures denote that the

central part of generator duct.

eddy current is not induced when the output

Rb
E1

E2

C1

C2

A1

C3

A2

A3

A4

A5

A6

C4

C5

C6

C7

C8

(a) g=0, B0=4
Rb
E1

E2

C1

C2

A1

C3

A2

A3

A4

A5

A6

C4

C5

C6

C7

C8

(b) g=6, B0=4
Rb
E1

C1

E2

C2

A1

C3

A2

A3

A4

A5

A6

C4

C5

C6

C7

C8

(c) g=10, B0=4
Figure 3. Current distributions

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TAẽP CH PHAT TRIEN KH&CN, TAP 16, SO K2- 2013
4

g=0
2

[kV]

2

4
6
8

1

10
0

Ri/Ri0, Rb/Rb0, Jpeak/Jel(ì10)

1.2

3

1.0

Ri/Ri0

0.8
0.6

Jpeak/Jel
Rb/Rb0

0.4
0.2
0

E1 E2 A1 A2 A3 A4 A5 A6 A7 A8
Figure 4. Variation of potential difference

0

2

4
6
g [T/m]

8

10

Figure 5. Influence of g on Ri/R i0, Rb/Rb0 and J peak/J el
when B0 =4

for B0 =4

Next, Fig. 4 shows the variation of the

where V0 and V are no-load and load potential

potential difference between the electrode pairs

difference between the output electrode E1 and

A1-C 1~A8-C8 and the electrode E1. From the

the n-th electrode, respectively, and Jpeak is the

figures it is seen that the relatively large

maximum current density on the output

potential difference arises between the two

electrodes. In this connection, Jpeak/Jel=1

output electrodes E1 and E2 when B does not

means the state of no current concentration and

attenuate, namely g=0. On the other hand, the

Jpeak/Jel>>1

potential difference become smaller as g

concentration at an electrode edge.

becomes larger, it almost vanishes for g=6, and
the inverse difference appears for g>7. Also
Fig. 4 denotes that the potential differences in
the central part of generator duct are little
influenced by the decrease of the magnetic
induction.

does

the

intensive

current

Now Fig. 5 shows the variations of R i/Ri0,
Rb/Rb0 and Jpeak/Jel by g, where Ri0 and Rb0 are
Ri and Rb for g=0, respectively. From the
figure, it is seen that Ri decreases with g, for
instance the value of Ri for g=6.0 becomes
about 80% of the one of Ri0, and that Jpeak/Jel

Next, for estimation of the end effects of

decreases from g=0 to 8T/m, reaches the

the generator, the author evaluates the internal

minimum value 1.90 and increases again. This

resistance Ri of the end regions and the grade

fact shows that the current concentration at the

of the current concentration on the output

edges of the output electrodes is almost

electrodes given by the relations

diminished

R i V0 V I
J peak

J

el

1

(16)
(17)

arranging

when
the

g=8T/m.

output

Accordingly,

electrodes

in

the

attenuation area of the magnetic flux density is
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Science & Technology Development, Vol 16, No.K2- 2013
useful to guard the output electrodes. Also Fig.

electrodes will require large ballast resistors

5 tells that Rb/Rb0 decreases with g, becomes

when B does not attenuate or exceeds 8, but

almost zero for g=6.5 and then increases with

they can be used without large ballast resistors

g. Therefore, it is shown that many output

in the range of g=6~7T/m.

Rb
E1

E2

C1

C2

A1

A2

A3

A4

A5

A6

C3

C4

C5

C6

C7

C8

Figure 6. Current distribution for g=6 and B0=5

In Fig. 6, the current distribution is plotted
when

g=6T/m,

B0=5T

and

distribution very uniform near the end

I=150A,

region of generator duct, both when the

Jel=1.25A/cm , =2.85mho/m, =2.48 and

streamer is not induced and when it is

crit=1.90. The figure indicates that the streamer

induced in the central region.

2

is induced in the central part of generator,
while

the

current

distribution

2.

becomes

Disposing the output electrodes within the
attenuation area of magnetic flux density

successively uniform as B attenuates along the

has

generator duct and the current concentration is

distribution in the central part of generator

almost swept away near the output electrodes.

duct.

Therefore it is seen that arranging the output

3.

little

influence

on the

current

When the output electrodes are disposed

electrodes within the attenuating region of B is

in the region with a suitably reduced

effective for the case where the streamer is

magnetic flux density, the potential

generated in the central region of generator

difference and the ballast resistance

duct, too.

between two output electrodes become

5. CONCLUSIONS

very small. Accordingly it is thought that
many output electrodes can be used

The main conclusions derived from the

without large ballast resistors.

above described numerical calculation are as
follows:
1.

4.

The internal resistance in the end region

A suitable distribution of the magnetic

of the generator duct decreases as the

flux density can make the current

magnetic flux density attenuates.

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TẠP CHÍ PHÁT TRIỂN KH&CN, TẬP 16, SỐ K2- 2013
5.

The current concentration at the edges of

equilibrium plasma MHD generator should be

output electrodes can be fairly eliminated

arranged in the region of the attenuating

by attenuating magnetic flux density.

magnetic flux density, since arranging them in

As mentioned above, it is made clear that

the region of the decreasing magnetic flux

the output electrodes of the diagonal type non-

density become useful for the improvement of
the electrical characteristics of the generator.

PHÂN TÍCH ẢNH HƯỞNG CỦA SỰ SUY GIẢM CẢM ỨNG TỪ ĐẾN SỰ PHÂN BỐ
DỊNG ĐIỆN TRONG MÁY PHÁT ĐIỆN TỪ THỦY ĐỘNG LOẠI FARADAY
Le Chi Kien
Ho Chi Minh City - University of Technical Education

TĨM TẮT: Bài báo nghiên cứu ảnh hưởng của sự suy giảm của cảm ứng từ đến sự phân bố
dòng điện trong vùng phía cuối của máy phát điện Từ thuỷ động loại điện cực chéo dùng plasma khơng
cân bằng bằng phân tích hai chiều. Những tính tốn số đã được thực hiện cho trường hợp khí làm việc
hê-li được cấy thêm xê-zi. Kết quả là một sự suy giảm phù hợp của cảm ứng từ có thể tạo ra sự phân bố
dòng điện rất đồng nhất gần khu vực cuối của ống dẫn máy phát điện, và ảnh hưởng nhỏ đến phân bố
dòng điện ở khu vực giữa, và điện cực đầu ra có thể được dùng mà khơng cần điện trở cân bằng lớn.
Điện trở nội của vùng cuối và sự tập trung dòng điện tại điện cực đầu ra cũng giảm cùng với sự suy
giảm của mật độ từ thơng. Theo như khảo sát được từ bài báo, rõ ràng là điện cực đầu ra của máy phát
điện Từ thuỷ động loại điện cực chéo dùng plasma khơng cân bằng nên được sắp xếp trong khu vực suy
giảm của cảm ứng từ bởi vì việc sắp xếp này sẽ trở nên hữu ích trong việc cải thiện những thuộc tính
điện của máy phát.

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