When the entrance Mach number is more
than 5, the increases in thermal efficiency are
insignificant. Therefore, there is no need for a
high entrance Mach number.
The conductivity to be used in the calculation
of output power density is that which is
determined on the basis of the theory of
magnetically induced ionization. This
conductivity depends on the velocity as well as
the usual parameters. All results obtained from
this study will be much more significant for
optimizing the efficiency of the MHD generator
in the future works.
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TAÏP CHÍ PHAÙT TRIEÅN KH&CN, TAÄP 17, SOÁ K1- 2014
Analyse the electrical characteristics of an
MagnetoHydroDynamic generator for
maximizing the thermal efficiency
• Le Chi Kien
Ho Chi Minh City University of Technology and Education
(Manuscript Received on February 10 th , 2014; Manuscript Revised August 13 th , 2014)
ABSTRACT:
In this study, a Faraday type maximizes the thermodynamic
MagnetoHydroDynamic (MHD) generator is efficiency, is independent of the
studied to consider the effect of electrical regenerator efficiency, but dependent on
characteristics to the thermal efficiency. The Mach number and the compressor
generator performance is specified by efficiency. It can also be seen that there
optimizing the cycle efficiency with respect to is no need for a high entrance Mach
the load parameter and by optimizing output number more than 5 because the
power density with respect to seed fraction increases in thermal efficiency are
and operating pressure. As the calculation insignificant.
results, the value of load parameter, which
Keywords: MHD generator, thermal efficiency, electrical characteristic, load parameter,
output power density.
1. INTRODUCTION electron temperature. Each of these studies
considers a particular noble gas and seed for
Techniques of Magnetohydrodynamic
which high conductivity was attained. In these
(MHD) power generation are being studied with
studies, however, no attempt has been made to
increasing interest for the development of high
consider the effect of electrical characteristics
temperature materials and high field strength
such as load parameter, electrical conductivity
magnets progresses. Devices using these
of MHD generator for a specified generator
techniques are to take the place of the turbo
operating under conditions appropriate for
generator in a conventional power generation
maximizing the thermal efficiency.
cycle. Several studies have been proposed that
combine Rankine, Brayton, or hybrid cycles In this study a constant area linear duct with
with liquids, vapors, and mixtures of these two segmented electrodes operating as a Faraday
as proposed working fluids [1-4]. Some of these type MHD generator is studied to consider the
studies may be used in a Brayton cycle where effect of electrical characteristics to the thermal
the working fluid is an alkali metal vapor seeded efficiency. The magnetic field is constant and
in a noble gas. These studies utilize the induced unaffected by the fluid. The current through
electric field of the plasma to increase the each pair of electrodes is adjusted so that the
Trang 37
SCIENCE & TECHNOLOGY DEVELOPMENT, Vol 17, No.K1- 2014
generated voltage is constant. The working fluid η
η = conv (6)
is a noble gas seeded with cesium, and the S 1−η
1− conv
2 1 γ
effects of viscosity and heat conduction are U ()γM 0 P
neglected. The comparison between different Heater
4
seeded noble gas working fluids will be
Compressor
examined for the optimum conditions to be 2′
obtained.
3
2. THERMODYNAMIC CYCLE
1 6 5′ MHDGenerator
EFIFICIENCY Cooler Regenerator
A Brayton cycle is considered with (a) Schematic diagram
temperatures defined as shown in figure 1. The T4
compressor efficiency, generator (isentropic)
T5′
efficiency, and the generator efficiency are T3
T5
defined as follows: Temperature
T2′
T − T =η (T ′ − T ) (1) T
2 1 comp 2 1 T2 6
ηS (T4 −T5 ) = T4 −T5′ (2)
T3 − T2′ = T5′ − T6 =ηreg (T5′ − T2′ ) (3)
T1
where the primed subscripts denote actual state Entropy
points in figure 1. It is of interest to relate the (b) Temperature-entropy diagram
generator efficiency to the variables defined in Figure 1. Brayton cycle temperature definitions
the text and to discuss some of the implications It should be noted that this isentropic
of the concept. The efficiency ηS can be efficiency is based on total properties. An
expressed in terms of the solution to the isentropic change in total enthalpy that is not
generator equations as follows: From the zero can occur if the work is being done. This
definition of ηS: can be illustrated as follows.
T −T η η
η = 4 5′ = conv = conv (4)
S T The momentum and energy equations of the
T4 −T5 5 1−Y
1− MHD generator are
T4
However, Y=( T /T )=( p /p )(γ-1)/ γ must also du dp
5 4 L H ρu + + jB = 0 (7)
be expressed in terms of the generator variables. dx dx
The ratio of total pressures pL/pH is expressed in
dh 2 du
terms of the dimensionless exit static pressure P, ρu + ρu − jE ⊥ = 0 (8)
dx dx
the exit gas velocity U, and the total temperature
Multiplying equation (7) by u and subtracting
ratio T5'/T4:
γ (γ −1) from the equation (8) yield
p PγM 2 T
L 0 5′
= γ γ −1 (5) 2
p 2 () T dh 1 dp j
H ()PγM 0U 4
ρu − = j()E⊥ + uB = (9)
dx ρ dx σ
so that
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TAÏP CHÍ PHAÙT TRIEÅN KH&CN, TAÄP 17, SOÁ K1- 2014
From the Second Law of Thermodynamics, The area Ar (radiator area), required for a fixed
however, the left side of this equation can be maximum temperature T 4 can be obtained from
written as
A T 4
ε eff σ SB r 4 1−ηth T4
2 = (16)
ds j W η T
ρuT = (10) th th ave
dx σ
where εeff is effective emissivity of radiator, σSB
so that a constant entropy process can occur if σ
is the Stefan-Boltzmann constant. Equation (15)
approaches infinity. Hence, for an MHD
is rewritten in terms of
generator, the isentropic efficiency compares the
actual generator to a generator using an a = (1−ηreg )[1− (1−Y )ηS ]
infinitely conducting working fluid.
ηreg (17)
b = []Yηcomp + ()1−Y
The parameters Y and Z (Y, Z ≤1) are defined as ηcomp
(γ − )1 γ 4
p T by using equation (15) to evaluate (T 4/T ave ) and
L 2
Y = , Z = (11) equation (2) to eliminate the temperature terms.
pH T4
Here, a, b are machine efficiency parameters,
The thermodynamic efficiency for zero pressure then the area per unit power output becomes
drops through the heater, regenerator, and cooler
ε σ A T 4 1−η 1
may be expressed in terms of these parameters eff SB r 4 = th .
W 3η a + ()b − Y Z
as th th (18)
1 1
η η − Z 1−Y . −
( S comp )( ) (12) 3 3 3
ηth = Y Z ()a + bZ
Z()()1−ηreg []1− 1−ηcomp Y +
ηcomp −
+ηreg ηcomp []1− ()1−Y ηS Differentiation with respect to Z produces the
following equation for Z, which minimizes A ,
If the cycle is to be used in the space r
in equation (15):
environment, then it is desirable to minimize
radiator area. The temperature ratio Z, which aY 3Z 3
Z 4 − 4 = 3ηSηcomp (19)
minimizes the area, can now be determined. The ()a + bZ − bY 3Z 4
heat radiated per unit electric power developed
can be expressed as The solution to this fifth-degree polynomial can
be obtained in two special cases. The parameters
Q 1−η
rad = th (13) ω and v are defined as
Wth ηth
YZ
ω =
where Wth is thermodynamic work delivered by
1− ()1− Y ηS
cycle, (20)
3YηSηcomp
v =
4 1− ()1− Y ηS
Qrad = ε eff σ SB ArTave (14)
and Equation (16) then becomes
3 3
4 3T6 T1
Tave = 2 2 (15)
T6 + T6T1 + T1
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SCIENCE & TECHNOLOGY DEVELOPMENT, Vol 17, No.K1- 2014
1.0
3 ηreg =0
()1−η reg ω 0.8
]
ω 4 − 4 = v (21) S
η
b b 4 )
1−η reg + ω − ω Y
Y Y 0.6 ηreg =1
/[1-(1-
It may be seen that when ηreg =1 0.4
YZ
=
v ω 0.2
ω = (22)
4
0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2
and when ηreg =0 (and b=0),
v=3 YηSηcomp /[1-(1-Y)ηS]
ω(4 − ω 3 )= v (23)
Figure 2. Effect of regenerator efficiency ηreg
These two solutions, which are plotted in figure 3. ANALYSIS OF GENERATOR
2, are nearly the same for v≤2. As a matter of CHARACTERISTICS
fact, there is a condition for which the solutions
will all be the same, namely, when the second A linear MHD generator is analyzed using
term in the brackets of the equation (16) is small the fluid flow equations. The fluid is considered
compared to 4. It can be shown that if to be a perfect gas, and the effects of heat
conduction and viscosity are neglected. The
1−ηreg −ηconv electrical conductivity is to be calculated using
ηS ≥ηconv + (24)
ηcomp ()1+ηreg the concept of magnetically induced ionization
[5,6], which implies an elevated electron
where η (1− Y ) =η (25)
S conv temperature. This elevated temperature is the
then the second term will be less than 0.4. If result of an energy balance between the energy
ηreg =1, the inequality is always true. For the added to the electrons by the induced electric
remainder of the analysis, it will be assumed field and the energy lost by the electrons upon
that the parameters are chosen such that this collision with the other particles.
inequality is satisfied. Then, the value of Z that 3.1. Development of MHD Equations
minimizes Ar is
The continuity, momentum, energy, and
3 state equations for the MHD generator are the
Z = ηSηcomp (26)
4 following [7]:
and the thermodynamic cycle efficiency may be d
()ρu = 0 (28)
written as dx
1
η du dp
4 conv ρu + + jB = 0 (29)
ηth = (27) dx dx
ηreg ηconv + ()1−ηreg .
dh du
Y 2
3 ρu + ρu − jE ⊥ = 0 (30)
.1− ηconv 1+ηcomp dx dx
4 1− Y
γ p
h = (31)
γ −1 ρ
Trang 40
TAÏP CHÍ PHAÙT TRIEÅN KH&CN, TAÄP 17, SOÁ K1- 2014
where ρ is density, u is fluid velocity, p is equation (31) and the momentum and energy
pressure, j is current density, B is magnetic field equations. The resulting expression can be
strength, h is enthalpy, γ is ratio of specific heat, integrated to obtain the following relation
E⊥ is the transverse component of electric field. between the pressure and velocity:
The restriction imposed by Maxwell's Bw
()()ρu 2 + p − ρ u 2 + p = .
equation, curl E=-∂B/∂t, for a constant magnetic 0 0 0 V
(35)
field and a one-dimensional problem require u 3 ρ u 3
γ ρ γ 0 0
. up + − u0 p0 +
that be a constant, equal to -V/w (V is voltage γ −1 2 γ −1 2
and w is the distance between electrodes),
throughout the channel. This constant can be At this point, it is convenient to introduce
expressed as some fraction of the entrance open- the following non-dimensional variables and
parameters:
circuit field u0B as
V u γ −1 p
K = (32) U = , K L = K , P = ,
u γ ρ u 2
u0 Bw 0 0 0
2 1 ρ u 2
where K will be called the load parameter. M = 1− 1− , M 2 = 0 0 ,
L 2 0
γ +1 M 0 γp0
The generator is assumed to be segmented,
τ 2 = 1− K M − K
and the segments are assumed to be infinitely ( L )( L L )
thin, so that no axial currents flow. The proper
where U is non-dimensional fluid velocity, KL is
Ohm's Law is load voltage parameter, P is non-dimensional
V pressure, ML is Mach number parameter, M0 is
j = σ uB − (33)
w entrance Mach number, τ is a parameter.
where σ is electrical conductivity includes Hall Equation (35) may then be expressed as
effects and ion slip, and j is parallel to u×B . The 2
γ +1 τ
restriction that K is a constant places a γP =U − U − K L − (36)
2 U − K L
restriction on the load resistance RL: (Aej)RL =
(Aej)0RL,0 = constant Equation (36) represents the relation between
pressure and velocity. Since the duct is
where A is the electrode area, R is load
e L segmented with infinitely thin segments, the
resistance, and the subscript zero denotes
power developed in the generator can be
entrance values. If all electrodes are given the
obtained by integrating the product of voltage
same area A , the current can be eliminated as
e and current VjH dx over the length of the
follows:
generator:
u L
σ − K 1
Π = VjH dx = ρ u3KwH 1+ − ()U + P (37)
RL 0, j u0 ∫ 0 0 2
γM 0
= = (34) 0
RL j0 σ 0 ()1− K
where H is the height of electrodes.
To solve the system of equations (28) to (31),
the enthalpy h can be eliminated by using
Trang 41
SCIENCE & TECHNOLOGY DEVELOPMENT, Vol 17, No.K1- 2014
This power can be compared to the total 2
τ
enthalpy flux entering the generator: 1 1−
γ +1 U − K
ξ = ∫ L dU (44)
γ p 1 2γ ΩU ()U − K
0 2 (38) U
Total enthalpy flux = ρ0u0wH + u0
γ −1 ρ0 2
It is noticed that if the conductivity is constant
The ratio of these terms is called the ( =l), equation (44) can be integrated:
conversion efficiency ηconv and may be written
2
γτ 1 − K
as 1 − ln +
K U − K
K (1−U ) (U − M ) 2
η = L L (39) γ + 1 γτ 1 − K (45)
conv ξ = + ln L −
()U − K L M L
2γ K U − K L
2
The power output of a generator with a γτ K 1 1
− −
K γ 1 − K U − K
specified inlet condition can now be determined. L L
In order to calculate the output power density,
however, a relation between velocity and which is in agreement with the results of other
generator length must be determined. The two investigations [8,9].
variables, non-dimensional conductivity =σ/σ0 By using equation (44) for interaction length, it
and dimensionless interaction length ξ, defined is possible to express the output power density
by ℘ as follows:
2
σ 0 B x Π 2 2 K L (γ +1)(1−U )(U − M L )
ξ = (40) ℘= = σ 0u0 B (46)
ρ0u0 wHL 2(γ −1 )(U − KL )ξ
are introduced. Equation (29) can then be This is the power density for a constant-area
written as generator. It is of interest to gage the effect of
velocity variation as well as conductivity
d
()()U + P + Ω U − K = 0 (41) variation. The power density at the entrance to
dξ
the generator is
which can be expressed as
2 2
℘0 = σ 0u0 B K(1− K ) (47)
∂P
1 1+ The ratio of equation (46) to equation (47)
ξ = ∂U dU (42)
∫ ΩU ()U − K
U ℘ (γ +1)(1−U )(U − M )
= L (48)
℘ 2γξ U − K 1− K
Equation (42) provides a relation between U and 0 (L )( )
the interaction length. An expression for ∂P/∂U will be used for comparison. This ratio will be
can be obtained by differentiating equation (36): calculated for the constant conductivity case,
2 where ξconst is given by equation (45), and for ξ
∂P 1 γ +1 τ
= − 1+ (43) as determined from equation (44) by use of the
∂U γ 2γ U − K L
non-equilibrium conductivity.
so that equation (42) becomes The cycle thermodynamic efficiency may be
conveniently expressed in terms of a generator
Trang 42
TAÏP CHÍ PHAÙT TRIEÅN KH&CN, TAÄP 17, SOÁ K1- 2014
(isentropic) efficiency. This efficiency, which is 1.0
defined as the actual change in total enthalpy of
K
the working fluid in the generator compared to 0.8
the change in total enthalpy for an isentropic
Kmax
process between the same total pressure
0.6
conditions, is derived in section 2 above. The
Load parameter, parameter, Load
thermodynamic cycle efficiency for the Brayton K∞
0.4
cycle under conditions appropriate for space 0 1 2 3 4 5 6 7 8 9
application is also calculated in section 2. Entrance Mach number, M0
Certain limiting values for ηconv , however, can Figure 3. Load parameters K (ratio of voltage to
be obtained without specifying the conductivity. open-circuit voltage) for maximum thermal efficiency
and infinite choking length for initial compressor
3.2. Limiting Case efficiency of 0.8
From equation (42) it can be seen that, as U 0.28
ηreg =0.99
approaches K, ξ will approach infinity; 0.24
th
obviously, this is a limiting value for U. This η 0.20
ηreg =0.9
situation represents the maximum interaction 0.16
ηreg =0.8
length and, consequently, the maximum amount 0.12
of energy that can be taken from the fluid. In
0.0 8
Thermal efficiency, efficiency, Thermal η =0
some cases, however, the interaction length reg
0.0 4
cannot become indefinitely large. It is limited by
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
the phenomenon called “choking”, which can be Load parameter, K
(a) Entrance Mach number of 2.0
characterized by the criterion that the local
Mach number reaches 1. In the dimensionless 0.14
symbols defined previously, this condition is 0.12
50.0
th
η 0.10 10.0
equivalent to 3.0
0.08
U = γP (49)
0.06 2.0
This condition, when substituted into the 0.0 4
Thermal efficiency, efficiency, Thermal
equation (36), leads to the following 1.5
0.0 2
specification of U at choking: M0=1.1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Load parameter, K
U = K +τ (50)
ch L (b) Regenerator efficiency of 0.0
It is noticed that this is the value of the
velocity for which the integrand in equation (44)
is zero; that is, Uch is the condition that makes
∂ξ/∂U= 0.
Trang 43
SCIENCE & TECHNOLOGY DEVELOPMENT, Vol 17, No.K1- 2014
0. 28 The quantity ηconv can be calculated from
th η =0.9
0. 24 reg
η equation (39) and ηS from section 2 for a
0. 20 0.8 specified γ and Mach number as a function of K.
0. 16
Therefore, the thermal efficiency ηth can be
0. 12 0.0 calculated by means of equation (27) for a
0. 08 specified compressor efficiency and regenerator
Thermal efficiency, efficiency, Thermal 0.04 efficiency. In figure 4(a) this efficiency is
0 1 2 3 4 5 6 7 8 9 plotted for γ=5/3, M0=2.0, and ηcomp =0.8 with
Entrance Mach number, M0 regenerator efficiency as a parameter. Two
(c) Maximum load parameter
items should be noted: first, the efficiency has a
Figure 4. Thermal efficiency for limiting solution maximum at some values of K, and second, this
with compressor efficiency of 0.8
value of K is independent of ηreg even though
Two different operation limits have been the efficiency varies with ηreg (this is true for all
described: first, when U=K and the duct is supersonic Mach numbers). The value of K also
infinitely long, and second, when U=Uch and the depends on ηcomp but that dependency will not
duct is choked. For any generator operation the be investigated.
proper limiting value can be determined by
In figure 4(b), the efficiency is plotted again as a
considering the case where the duct is choked at
function of K with γ=5/3 and ηcomp =0.8, but with
infinity. Formally, this occurs when Uch =K. This
ηreg =0 and Mach number as the parameter. It can
condition can be substituted into equation (50)
be seen that the K for the optimum efficiency
and the K for which this occurs (call it K ) can
∞ does depend on the Mach number. The value of
be determined from the following:
K for which the thermodynamic efficiency is
γ −1 optimized is called Kmax and is shown in figure 3.
K = K +τ (51)
∞ γ ∞
In figure 4(c), the efficiency at K=Kmax and
which may be written as ηcomp =0.8 is plotted as a function of Mach
number with regenerator efficiency as a
2 2
()()γ −1 1− M L (γ −1 )(1+ M L )
+ M L − parameter. It can be seen that when M0>5, the
K = γ 4 2 (52)
∞ 1− ()γ −1 2 increases in thermal efficiency are insignificant.
Therefore, there is no need for a high entrance
The criterion for distinguishing between the Mach number more than 5.
two limiting cases may therefore be stated as
For the limiting values of U, ηconv in equation
follows: For K>K∞, the duct will not choke and
(39) becomes
U will approach K, while for K<K∞, the duct
will be choked and U will approach Uch . The γ −1
η = (1− K )(K − M )K>K (53)
duct is infinitely long and choked for K=K . conv max max L ∞
∞ M L
When γ=5/3, K∞, is as shown in figure 3. It is
or
noted that for M0<1 the duct will always choke,
if sufficiently long, since K must be less than 1;
K L (1−U ch )(U ch − M L )
ηconv = K<K∞ (54)
whereas, as shown in equation (52) K∞, must be
()U ch − K L M L
greater than unity.
Trang 44
TAÏP CHÍ PHAÙT TRIEÅN KH&CN, TAÄP 17, SOÁ K1- 2014
4. CONCLUSIONS load parameter, but this value of load parameter
is independent of the regenerator efficiency
In conclusion, it may be stated that a value
even though the thermal efficiency varies with
of the load parameter which maximizes the
the regenerator efficiency. From the calculations,
thermodynamic efficiency of the limiting
the load parameter for the optimum thermal
solution has been calculated. This value is
efficiency clearly depends on the Mach number.
independent of the regenerator efficiency, but
dependent on Mach number, and the compressor When the entrance Mach number is more
efficiency (assumed to be 0.8 for all calculations than 5, the increases in thermal efficiency are
presented herein). insignificant. Therefore, there is no need for a
high entrance Mach number.
For the limiting solutions the efficiency is
independent of the form of the electrical The conductivity to be used in the calculation
conductivity. Of course, the electrical of output power density is that which is
conductivity of the plasma is of great practical determined on the basis of the theory of
importance in that it largely determines the magnetically induced ionization. This
generator length required to extract power, conductivity depends on the velocity as well as
which in turn determines the output power the usual parameters. All results obtained from
density of the generator. It is natural, then, to this study will be much more significant for
use the generator output power density as a optimizing the efficiency of the MHD generator
means of comparing the usefulness of various in the future works.
working fluids (the larger the better, of course).
It is concluded that, if the duct is sufficiently
long, for the entrance Mach numbers smaller
than 1, the duct will always choke. The thermal
efficiency has a maximum at some values of
Phân tích các ñ c tính ñi n c a máy phát
t th y ñ ng ñ c c ñ i hi u su t nhi t
• Lê Chí Kiên
Tr ư ng ð i h c S ư ph m K thu t TP.HCM
TÓM T T
Bài báo này nghiên c u máy phát T nhi t c a h th ng. Ho t ñ ng c a máy
th y ñ ng lo i Faraday và xem xét nh phát T th y ñ ng ñư c ch rõ b ng
hư ng c a các thu c tính ñi n ñ n hi u su t cách t i ưu hóa hi u su t nhi t có xét
Trang 45
SCIENCE & TECHNOLOGY DEVELOPMENT, Vol 17, No.K1- 2014
ñ n tham s t i và t i ưu hóa m t ñ công nhi t nh ưng l i ph thu c vào s Mach
su t phát ra có xét ñ n t l ch t c y và áp và hi u su t máy nén khí. K t qu cũng
su t làm vi c. Theo k t qu phân tích, giá tr cho th y r ng không c n thi t s Mach
tham s t i mà làm c c ñ i hi u su t nhi t, c a vào l n h ơn 5 vì khi ñó hi u su t
không ph thu c vào hi u su t b tái sinh nhi t t ăng không ñáng k .
T khóa: Máy phát MHD, hi u su t nhi t, ñ c tính ñi n, tham s t i, m t ñ công su t.
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Trang 46
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