J. Plasma Fusion Res. SERIES, Vol. 9 (2010)

Analysis of Disk AC MHD Generator Performance by Finite Element
MethodPattana INTANI, Toru SASAKI, Takashi KIKUCHI and Nob.HARADA
Department of Electrical Engineering, Nagaoka University of Technology,1603-1, Kamitomiokamachi, Nagaoka, Niigata 940-2188, Japan(Received: 1 November 2009 / Accepted: 16 March 2010)

This paper studies the possibility to generate alternating current (AC) electrical power by using disk MHD generator. Its phenomenon is performed by the finite element method (FEM). The configuration of generator consistsof a channel, an insulator and stators. The shape of channel is a flat disk, the liquid metal used as conductor flows
inside the channel. As a channel wall, an insulator is used to separate the metal fluid and stator. The stator winding
is designed as the coils on top and bottom of a channel. Under this condition, it can produce a magnetic field bymeans of time harmonics function. The interaction between the metal fluid and the electromagnetic wave can be
explained by Maxwell’s equations and Ohm’s law. As a result, the distribution of magnetic flux density throughout
the channel is evidently shown in two-dimensional space. The active power is limited by slip value and electricalconductivity. The optimized value of active and reactive power is suggested by small slip value as slip s-0.1. In
addition, mechanical power and power dissipation in the generator are reported in term of 1-s and s respectively.
Electrical efficiency is explained in term of 1/(1-s).
Keywords: MHD Generator

an AC MHD Generator

MHD induction generator

structures are considered with double-side exciting
winding to generate AC power. In addition, thequasi-one-dimensional technique also applied toinvestigate it.
This paper proposes the method to generate AC
power by the disk MHD generator. The performance ofthe generator is explained by finite element method. A
single-side exciting winding of MHD generator isconsidered. Its configuration contains a channel, an
insulator and stators. The shape of channel is a flat disk.
The liquid metal is acted as a conductor flowing inside
the channel. Channel wall acted as an insulator separates
metal fluid and stator coils. The stator winding isdesigned as the coils on top and bottom of channel. In
this condition, top stator winding can produce a magnetic
field by means of time harmonics function in the same
direction with metal fluid. On the contrary, bottom stator
winding is used as an inductive coil. The behavior ofinteraction between the metal fluid and electromagnetic
wave can be explained by Maxwell’s equations andOhm’s law based on the method of finite-elements. The
second order of triangular finite elements is utilized to
calculate the magnetic vector potential. The distributionof magnetic vector potential and magnetic flux density
throughout channel is evidently shown in two-dimension
space. Active power on top and the bottom of statorsurface is real part of the surface integrating Poynting’s

1. Introduction

Recently, Magnetohydrodynamics (MHD) generator
is effectively utilized to generate electrical energy
because of its simple structure and could directly converts
thermal energy into electrical energy. An MHD generator
can produce high power density with low environmental
issues. On the other hand, an AC MHD generator is one
type of MHD generator that generates an AC power
without an inverter. It performs under the interaction
between traveling wave and electrically conducting fluid.
The stator winding is served as a step-up transformer byincreasing terminal voltage and also reducing armature
current. As a result, operating characteristics and design
considerations of an AC MHD generator have been
proposed by W.D. Jackson and E.S. Pierson in 1965 [1,2].
An annular and a linear type of the generator are
considered under the condition of metal fluid andmagnetic field. The metal fluid is eutectic sodium
potassium (Nak). It has high electrical conductivity. W.D.
Jackson and E.S. Pierson suggested that the performanceof an AC MHD generator depends on magnetic Reynolds
number based on the wavelength and the value of slip.
Afterward, the theory and experiment of a linear AC
MHD generator using liquid metal has been proposed by
T.C. Wang and S.J. Dudzinsky in 1967 [3,4]. withdifferent approach for calculate. However, most of all
e-mail:intani@stn.nagaokaut.ac.jp580©2010 by The Japan Society of Plasma
Science and Nuclear Fusion Research

P. Intani et al., Analysis of Disk AC MHD Generator Performance by Finite Element Method

field in term of time harmonics function. Also, energy
from the interaction between magnetic field and metal
fluid can be extracted by inductive coils.

vector over the fluid surface. While reactive power isdefined by imaginary part of integrating Poynting’s
vector over the fluid surface. Finally, mechanical power,
power dissipation and electrical efficiency are consideredby induction machine equations.

3. Analysis model

The MHD schematic of the single-side disk AC
MHD generator is shown in Fig.2. The configuration of
disk channel is symmetry. Also, right hand side of disk
channel is investigated and applied to make the model of
AC MHD generator. The model used for numericalsimulation is illustrated in Fig.3. The area of metal fluid(I), channel wall (II) and stator (III) is designed in the
Cylindrical coordinate system. The metal fluid flows
along r-direction with constant velocity (Uf). Besides, theelectrical conductivity and permeability are replaced byconductivity V=Vf and permeability P=Pf, respectively.
Channel walls act as insulators separate the metal fluid
and the stator. Also, the electrical conductivity andpermeability are replaced by conductivity V=Vi andpermeability P=Pi, respectively. Electrical conductivity
and permeability of stator are defined by V=Vc and P=Pc,
respectively. The top stator winding is connected withthree phase system which produces traveling wave
velocity (Us) along r-direction. In this paper, the top
stator winding is replaced by surface current density. On
the other hand, energy from interaction between magnetic
field and metal fluid is transferred to electrical load byinductive coils that installed at bottom stator.

2. Basic of an AC MHD generation

The principle of an AC MHD generator operation is
the same as that of a polyphase induction generator. In an
induction generator, a rotating magnetic field (B) is
generated by a distributed polyphase winding. The
rotating field induces a voltage (E) in conductors
imbedded in the periphery of the cylindrical rotor. Hence,the current density in the conductor depends on electricalconductivity (V) of the conductor in term of VE. In
addition, induced voltage in conductor can be changed
with varying velocity (U) of conductor. Consequently, theinduced voltage in the conductor is explained by crossproduct of velocity of rotor and magnetic flux density(U×B). The current density in conductor is replaced interms of V(U×B). Therefore, the induced voltage in the
conductor due to moving conductor in time varied field is
replaced by Eind=E+U×B. The current density inconductor is explained by Ohm law in term of
Jind=V(E+U×B). The energy in the conductor istransferred into electrical load by connecting the
conductor.

IB

IB

E ind

E U u B

IC

IA

IB

IC

IC

IB IA

IC

IA

IA

Z

r

I

U

B

U
E ind

E U uB

U

E ind

B
E U uB

B

Fig.1

Basic of AC MHD generation.
Fig.2

Figure 1 shows the basic of linear AC MHD generator.
The stator of AC MHD generator similarly acts as in thegeneral induction generator. However, its windings are
distributed in a flat magnetic structure. Therefore, themotion of the field is a straight line rather than rotating as
in a generator. The rotor is replaced by a layer of metal
fluid confined in an envelope of rectangular cross sectionof the channel. Induced current in the metal fluid depends
on the current density in terms of Jind=V(E+U×B).
Therefore, the induced current generates the magnetic

Configuration of the disk AC MHD generator.
OJs

J m e j Zs t

Us

P

z

V
Uf

Fig.3

581

r

The model of an AC MHD generator.

Pc ,V
Vi,P

Vc
Pi

V

V f ,P

Pf

P

Pc ,V

Vc

P. Intani et al., Analysis of Disk AC MHD Generator Performance by Finite Element Method

A(r,z,t)=rAm(z)exp(jZst)aI where r is radius of disk
generator. Am is magnitude of magnetic vector potential,
aI is unit vector along I-direction, Zs is angular velocity.
In addition, an electromagnetic source generating by thestator winding is replaced in term of Js=Jmexp(jZst)aI.
Also, equation (9) can be described by

The phenomenon of AC MHD generator is explainedby Maxwell’s equation. It can be evaluated by thefollowing formulas, while a displacement current density
is neglected:
 u E w B / wt
uH J B 0J 0
where
E is electric field intensity
H is magnetic field intensity
B is magnetic flux density
J is current density

(1)
(2)
(3)
(4)

ȝıU f ·§¸ Am
 2 Am  ¨¨ jȝıZ s 
r ¸¹
©
§1
· wA
 ¨  ȝıU f ¸ m  J m
r
©
¹ wr

ı E U f u B .

(5)
Us

The relationship between magnetic flux density andvector magnetic potential is explained byB

 u A.

The components of magnetic flux density (B) in the
Cylindrical coordinates, are replaced by Br=-A/z,
Bz=-(rA)/rr and BI=0. Also, the electric field intensity
is expressed as
E

s

(7)

§  wA·ȝ0 ı ¨ U f u  u A ¸.
© wt
¹

0.

Us  U f
Us.

ȝıs 1 - s Z s ·§ 2 Am  ¨ jȝıs Z s 
¸ Am
r
©
¹
w
A
1
§
·
 ¨  ȝıs 1 - s Z s ¸ m  J m 0.
©r
¹ wr

(8)

(12)

(13)

From (13), the magnetic vector potential is a function of
the slip value. The standard algorithm [5] of finite
element analysis is referred in this paper.

From (8), D is displacement current as zero when Шк D,and Ш(Шк A)=0, is obtained that§ wA·
U f u  u A¸
 2 A  ȝ0 ı ¨
© wt
¹

(11)

The appearance of different frequencies in exciting
winding and metal fluid is explained by Zf=sZs. Also,
substituting (11) and (12) into (10), then a magnetic
vector potential is defined bywhere A is magnetic vector potential. Substituting (6) and(7) into (5), then a current density is defined by Ohm’s
law as follow

uuǹ

ZsO
.
2S

where O is wavelength. Also, the relationship between
wave velocity (Us) and metal fluid velocity (Uf) isrepresented by the slip value (s) as

(6)

wA
,
wt

(10)

where Zs=2Sf, f is the electromagnetic wave frequency, j
is imaginary part of complex number and Jm is anamplitude current density at stator winding. The
relationship between angular velocity and wave velocity
is defined as

By Ohm’s law neglecting Hall effect, the current density
equation is given byJ0,

5. Power flow in AC MHD generator(9)

The power flow in AC MHD generator is evaluatedby using electrical power. The mechanical power is thepressure gradient through the channel by applying
electromagnetic force. Furthermore, an electrical power iscalculated by integrating Poynting’s vector over the fluid
surface (Ș). The active power is evaluated by

4. Finite element formulation

The AC MHD equation is evaluated in finite element
method. The magnetic vector potential (A) is defined by582

P. Intani et al., Analysis of Disk AC MHD Generator Performance by Finite Element MethodP§1· Re¨¨ ³ E z H * d* ¸¸,©2*¹parameters in the simulation model are listed in Table 1.
This model is divided into 792 elements and 1665
triangular element nodes. The second-order is consideredin this paper. In case of boundary condition, the magnetic
vector potential at the edges of bottom and top stator isdefined by zero (A=0)(14)

where d* is channel surface and H* is complex
conjugate of magnetic field intensity. The reactive powerof generator is defined by imaginary part of integrating
Poynting’s vector over the fluid surface asQ·§1
 Im¨¨ ³ E z H * d* ¸¸.
¹
©2*

A 0

(15)

Uf

The mechanical power is the relationship between
electrical power and slip. It is calculated byJs

11.52 u 106 e

Pf

P0 , V f

P

2000P 0 , V

Pi

P0 , V i

Vc

jZ s t U s

1u 106

0

A 0

Pm

·
§1
 Re¨¨ ³ E z H * d* ¸¸(1  s ).
2
¹
© *

(16)

Fig.4

Table 1.

Moreover, the power dissipation from generator can be
defined by

Pd

·
§1
 Re¨¨ ³ E z H * d* ¸¸ s ,
¹
©2*

(17)

where Pm is the mechanical power in fluid based onelectromagnetic force. Pd is the power dissipation in thegenerator. Thus, electrical efficiency of induction
generator is given by

K

P Pm

1 1 s .

Configuration of AC MHD generator consideration.
Parameter for consideration.

Parameters

Value

Power line frequency (Hz)
Permeability of metal fluid (H/m)
Permeability of iron core (H/m)
Permeability of insulator (H/m)
Electrical conductivity of metal fluid (S/m)
Current density (A/m2)
Wave length (m)Wave velocity (m/s)
Magnetic Reynolds number50
4S×10-7
2000× 4S×10-7
4S×10-7
1×106
11.52x106
0.154
7.7
0.24

The numerical simulation is investigated the modelof AC MHD generator which is produced AC power frominteraction between traveling wave and metal fluid. The
power flow in AC MHD generator is evaluated under theratio of the metal fluid velocity and the electromagnetic
wave velocity which is the slip value in term ofs=(Us-Uf)/Us. In this paper, the range of the slip value isdefined by -1s1. In case of the negative slip value, thevelocity of metal fluid is higher than two time of thevelocity of electromagnetic wave. The maximum active
power of -2.54kW at s=-0.6 with Vf=1×106S/m is shownin Fig.5. The negative active power is expressed bygenerating electrical power in the MHD channel. On thecontrary, if the velocity of traveling wave is higher thanmetal fluid velocity, the slip value is positive. Thisphenomenon acts as an accelerator. The metal fluid isaccelerated by electromagnetic force. The maximum
power used for propulsion is 1.77kW at slip s=0.7 withelectrical conductivity Vf=1×106S/m. Nevertheless, in therange of the slip value -0.1s0.1, this area has low
active power; the generator is acted as accelerator. As aresult, the AC MHD generator is designed with small slip

(18)

Therefore, the electrical efficiency (Ș) is determined byonly the fluid electrical losses in channel.

6. Numerical results and discussions

The AC MHD generator configuration is designed byconditions of constant fluid velocity throughout thechannel illustrated in Fig. 4. The electrical conductivity
and permeability are replaced by electrical conductivityof a eutectic sodium potassium as Vf=1×106S/m and apermeability of free space as Pf̓ P0=4S×10-7H/m,
respectively. The channel walls are electrical insulator,
conductivity and permeability are defined by Vi=0 andPi̓ P0, respectively. Electrical conductivity andpermeability of stators as laminated steel are defined by
Vc=0 and Pc̓ 2000P0 respectively. The stator on top is an
exciting winding with traveling wave Js=11.52exp(jZst)
A/m2 along r-direction with O=0.154m. Physical583

P. Intani et al., Analysis of Disk AC MHD Generator Performance by Finite Element Method

value s0.1. In the same way, if generator is acted as anaccelerator. It is designed with high slip value s0.1.
3

Accelerator area

1.72kW

2
1
0
-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-1
-2

-2.75kW

-3

slip

Fig.5

The relationship between active power and slip.

Figures 6 and 7 show magnetic flux density distribution at s=-0.6 and s=0.7, respectively. The magnetic field
not only distributes in the channel but also induces highmagnetic flux density at the metal surface.

Accelerator area

40
35

29.32kVAR

30

24.22kVAR

25
20
15
10
5
0

-1

Fig.8

Fig.6

45

Generator area
Reactive power [kVAR]

Active power [kW]Generator areafrom the reactive power in application. As a result, themaximum reactive power as 39kVAR is around the slip
value at the origin. This slip value should not be used generator and accelerator. For maximum active power, maximum reactive power in case of generator and accelerator is
24.22kVAR and 29.32kVAR, respectively. Also, the reactive power should be compensated by connecting a capacitor at the input terminal.

-0.8

-0.6

-0.4

-0.2

0

0.2

slip

0.4

0.6

0.8

1

The relationship between reactive power and slip.

Figure 9 shows the relationship between mechanicalpower and slip. The power used to moving the metal fluid
into the channel is defined by mechanical power. It is calculated by the induction machine equations based on theslip. In case of positive mechanical power, the phenomenon acts as generator, the mechanical power is increases
with 1-s time of the active power. The mechanical power is
4kW with s=0.6. However, In case of negative mechanicalpower, the phenomenon acts as an accelerator. Metal fluid
is also accelerated by the electromagnetic force. The mechanical power is reduces with 1-s time of the active power.
The power is -0.52kW with s=0.7. It suggests that the mechanical power of the generator is higher than the accelerator with constant active power at metal surface.

Magnetic flux density distribution at s=-0.6.

Mechanical power [kW]Generator area
4kW

5

Accelerator area

4
3
2
1
0

-1

-0.8

-0.6

-0.4

-0.2

-1

0

-2

0.2

0.4

0.6

0.8

1

-0.52kW

Slip

Fig.9
Fig.7

Magnetic flux density distribution at s=0.7.

The relationship between mechanical power and slip.

Loss in the generator is explained by the power dissipation and slip in Fig. 10. The power dissipation is evaluated by product of slip value with an active power. In caseof generator, the maximum power dissipation increases

Figure 8 shows the relationship between reactive
power and slip. The optimized range of slip is evaluated584

P. Intani et al., Analysis of Disk AC MHD Generator Performance by Finite Element Method

Power dissipation [kW]Generator area

Accelerator area

2

40
30

1

20
10

0

1.5kW

1.5

1.2kW

Fig.12

0.5

-0.6

-0.4

-0.2

0

Slip

0.2

0.4

0.6

0.8

Efficiency [%]Accelerator area

100
80

62.5%

60

30%

40
20

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

1.E+08The relationship between active power, reactive powerand electrical conductivity.

8. References0-1

1.E+07This paper has proposed the possibility to produce an AC
power by disk MHD generator with a single-side exciting.
Its performance is evaluated by finite element method.
The result confirms that the distribution of magnetic fieldappears at the metal surface and high magnetic flux density. The active power of interaction between travelingmagnetic field and metal fluid is limited by the slip value
and electrical conductivity of the metal fluid. The optimized value of an active and reactive power of the generator is suggested by small slip value as s-0.1. Maximum power in this system is suggested with active power(-2.54kW), reactive power (24.22kVAR), mechanicalpower (4kW), power dissipation (1.5kW) and efficiency(62.5%). In addition, active power is function of electricalconductivity. However, AC MHD generator is adjusted toaccelerator by s0.1. Maximum power in this system isperformed by active power (1.72kW), reactive power(29.32kVAR), mechanical power (-0.52kW), power dissipation (1.2kW) and efficiency (30%).

The electrical efficiency is evaluated only under the
fluid electrical losses. Figure 11 shows the electrical efficiency of AC MHD generator and accelerator depend onvalue of 1/(1-s) and (1-s), respectively. In case of generator,
the maximum active power with -2.54kW at s=-0.6 andVf=1×106S/m is shown in Fig. 6. At this point, electricalefficiency is 62.5%. In case of accelerator, the maximum
active power with 1.72kW at s=0.7 and Vf=1×106S/m isshown in Fig. 5. At this point, electrical efficiency is 30%.

120

1.E+06

7. Conclusions

1

The relationship between power dissipation and slip.

Generator area

1.E+05

Electrical conductivity(S /m)

1

-0.8

0

1.E+04

0

0.8

[1]

1

Slip

Fig.1160

2-1

Fig.10

Reactive power

50

Active power [kW]2.5

Active power
3

Reactive power [kVAR]

with small slip value. Since, high magnetic flux density
appears at the surface of metal fluid near exciting winding.
Power dissipation is 1.5kW with s=-0.6. In case of an accelerator, the maximum power dissipation is increases withincreasingly value of slip. The maximum power dissipationis 1.2kW with s=0.7.

[2]

The relationship between electrical efficiency and slip.

The relationship between absolute maximum active
power and electrical conductivity of the metal fluid isshown in Fig. 12. The optimum value of the electricalconductivity is defined by Vf=5×106S/m for active power
2.7kW. However, electrical conductivity is less thanVf=5×105S/m, a maximum active power decreases. As aresult, the generator has high power dissipation. Moreover, an electrical conductivity is higher thanVf=5×107S/m, a maximum active power decreases. Thus,an electromagnetic induction appears on the metal surface.
However, a maximum reactive power is constant

[3]

[4]

[5]

585

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pp. 107-112 (1967).
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