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Effects of induced magnetic field on large scale pulsed MHD generator with two phase flow
Energy Conversion and Management 45 (2004) 707–724 www.elsevier.com/locate/enconman
Effects of induced magnetic field on large scale pulsed MHD generator with two phase flow M. Ishikawa *, Y. Koshiba, T. Matsushita Institute of Engineering Mechanics and Systems, University of Tsukuba, Tsukuba 305-8573, Japan Received 10 February 2003; accepted 5 July 2003
Abstract A large pulsed MHD generator ‘‘SAKHALIN’’ was constructed in Russia (the former Soviet-Union) and operated with solid fuels. The ‘‘SAKHALIN’’ with the channel length of 4.5 m could demonstrate the electric power output of 510 MW. The effects of induced magnetic field and two phase flow on the shock wave within the ‘‘SAKHALIN’’ generator have been studied by time dependent, one dimensional analyses. It has been shown that the magnetic Reynolds number is about 0.58 for Run No. 1, and the induced magnetic flux density is about 20% at the entrance and exit of the MHD channel. The shock wave becomes stronger when the induced magnetic field is taken into account, when the operation voltage becomes low. The working gas plasma contains about 40% of liquid particles (Al2 O3 ) in weight, and the present analysis treats the liquid particles as another gas. In the case of mono-phase flow, the sharp shock wave is induced when the load voltage becomes small such as 500 V with larger Lorentz force, whereas in the case of two phase flow, the shock wave becomes less sharp because of the interaction with liquid particles. 2003 Elsevier Ltd. All rights reserved. Keywords: Pulsed MHD generator; Induced magnetic field; Two phase flow; Strong MHD interaction; Effects of shock wave
1. Introduction Various pulsed MHD generators have been constructed and operated, mainly in Russia (the former Soviet-Union) [1]. Construction and operation of a large pulsed MHD generator named ‘‘SAKHALIN’’ with the electric power output of 510 MW were an important step in the history of MHD generator development. The self excited large MHD generator ‘‘SAKHALIN’’ was the
*
Corresponding author. Tel.: +81-298-53-5140; fax: +81-298-53-5207. E-mail address: [email protected] (M. Ishikawa).
0196-8904/$ - see front matter 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0196-8904(03)00182-1
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Faraday type with the active channel length of 4.5 m equipped with a pair of continuous electrodes. Cesium or potassium salt was added to solid propellants to achieve high electrical conductivity [2]. Analysis of the operating characteristics of ‘‘SAKHALIN’’ is an important research subject for understanding a very strong MHD interaction [3]. Since this MHD generator has a very strong MHD interaction and its operation magnetic flux density is comparatively low, the induced magnetic field cannot be neglected. ‘‘SAKHALIN’’ also has a special feature that liquid particles generated by combustion of the metal aluminum in the fuel cannot be neglected. The present paper intends to report the influences of the induced magnetic field and the effects of the two phase flow on the generator performance by using time dependent, one dimensional mono and two phase flow approximations. Our research group has conducted time dependent, one and two dimensional analyses with the assumption of mono and two phase flow and time dependent, three dimensional analyses with the assumption of mono phase flow for the Pamir-3U MHD generator with the electric power output of 10–20 MW. The shock wave, the boundary layer-shock wave interaction and others have been analyzed [4–9]. The present results are very useful for two and three dimensional analyses of ‘‘SAKHALIN’’ to understand strong MHD interactions and to design high performance MHD generators.
2. Operation analysis of ‘‘SAKHALIN’’ 2.1. Performance of ‘‘SAKHALIN’’ The large pulsed MHD generator was operated with solid fuels. The channel length is 4.5 m and the cross-section is 0.9 m (the direction of load current, y-direction) · 1.0 m (the direction of external applied magnetic field, z-direction) and 1.6 m (y-direction) · 1.0 m (z-direction) at the entrance and at the exit, respectively, where the height along the magnetic field is kept constant at 1.0 m. The nominal mass flow rate is 1000 kg/s. The combustor temperature is 3800–3900 K, while the combustor pressure is 4.0–5.6 MPa. The generator was quite powerful, light (only 50 tons) and compact, which could demonstrate the electric power output of 510 MW, the specific mass of 0.1 ton/MW and the specific volume of 0.3 m3 /MW. The air core magnet was activated to more than 2 T by the MHD generator itself, which is the Faraday type with a pair of continuous electrodes. Six experimental runs were conducted. The first two runs were operated with a resistance load, and the other runs were operated with the load of magnetic energy storage. The excitation current of a self excitation magnet was 52.5–190 kA, the magnetic flux density 1.5–2.2 T, the load voltage 1.41–2.6 kV, the load current 180–209 kA, the electric output 275–510 MW and the operation time was from 7.37–8.64 s [2]. 2.2. Induced magnetic field A preliminary analysis has indicated the magnetic Reynolds number is about 0.58 for Run No. 1, which is much different from many other MHD generators. The induced magnetic field, therefore, cannot be neglected. Then the one dimensional approximation has been applied to AmpereÕs law. Fig. 1 depicts distributions of the magnetic flux density for Run No. 1, where the solid line shows the externally applied field, the dashed line with dots shows the induced field and the dotted
M. Ishikawa et al. / Energy Conversion and Management 45 (2004) 707–724 2.5
0.4 0.3
2
0.2 1.5
0.1
1
0 -0.1
0.5 Bext Beff Bind
0
Bind (T)
Bext , Beff (T)
709
-0.2 -0.3
-0.5
-0.4 -2
-1
0
1
2
3
4
5
6
x (m) Fig. 1. Distributions of magnetic flux density (Run No. 1, solid line: external field, dashed line with dots: induced field, dotted line: effective field).
line shows the effective field. It is revealed that the induced magnetic flux density is about 20% at the entrance of the MHD channel, while it becomes about 19% at the exit for Run No. 1, being rather large. The magnetic Reynolds number Rm and induced magnetic field Bind are estimated as follows: ð1Þ Rm ¼ rl0 UL Z x Bind ¼ Bind ð0Þ l0 Jy dx ð2Þ 0
where r is the electrical conductivity, U the flow velocity, L the length, l0 ¼ 4p 107 [H/m] the vacuum permeability, Jy the load current density, x the distance from the entrance of channel. 2.3. Estimation of electrode voltage drop The value of electrode voltage drop can be automatically calculated in the case of multidimensional analyses, but the present one dimensional analysis requires estimation of the electrode voltage drop. It has been well known that the electrode voltage drop is nonlinear and shows a complicated behavior [10]. The present work, however, assumes rather a simple relation. At first, the electrode voltage drop is assumed to be constant without the induced magnetic field. It has been shown that the estimated constant electrode voltage drop ranges from 700 to 1200 V (except for Run No. 5), as listed below. Run No.
Average current (Jy , kA/m2 )
Electrode voltage (V)
1 2 3 4 5 6
44.5 42.89 43.33 46.44 40.00 44.44
1035 1199 1165 790 487 732
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M. Ishikawa et al. / Energy Conversion and Management 45 (2004) 707–724
Then, a simple linear relation to the local electric current density Jy is assumed as DV ¼ rdrp Jy þ DV0
ð3Þ
where DV is the electrode voltage drop and DV0 is the constant, and the value of the constant of the linear relation rdrp is estimated so as to agree with the electric output power, as shown below: Run No.
rdrp
Run No.
rdrp
1 3 5
0.0143 0.0175 0.0022
2 4 6
0.019 0.00836 0.0074
The value of DV0 is assumed to be 400 V, which must be estimated with experimental data and/or multi-dimensional analyses. This relation is used for all further calculations with or without induced magnetic field. 2.4. Nominal values of parameters of liquid phase Since measurements were difficult in the actual generator, details of the liquid phase are not clear, and a dusty gas flow is, therefore, assumed for spherical liquid particles. It is also assumed that the diameter of particles (d) is kept constant within the flow, and the ratio of velocities of gas and liquid particles (Slip0 ) and a temperature difference between them (DT0 ) are given at the nozzle entrance. The nominal values are given as follows: d ¼ 15 ½lm;
Slip0 ¼ 1:1;
DT0 ¼ 100 ½K
ð4Þ
where the subscript 0 stands for values at the nozzle entrance. Although the mass ratio of the liquid phase is about 30–40%, the exact value is not known, and then, the mass ratio is assumed to be 40% for all calculations.
3. Mathematical model for gas–liquid two phase flow 3.1. Basic equations of gas phase The conservation of mass: oq þ r ðquÞ ¼ 0 ot The conservation of momentum: oðquÞ þ r ðquuÞ þ rp ¼ F þ CR ðul uÞ ot The conservation of energy: os þ r fðs þ pÞug ¼ Q þ CR ul ðul uÞ þ Ca ðTl T Þ ot
ð5Þ
ð6Þ
ð7Þ
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where q is the mass density, p the gas static pressure, u the gas velocity, ul the velocity of liquid particles, T the gas temperature, Tl the temperature of liquid particles, CR the coefficient of momentum transfer between the gas phase and liquid particles, F the force consisting of the viscosity 2 and the Lorentz force, s ¼ qðh p=q þ juj =2Þ, Ca the coefficient of heat transfer between the gas phase and the liquid phase, and Q the energy source consisting of the work done by the viscosity, the thermal conduction and the Joule heating. 3.2. Basic equations for liquid phase The conservation of mass: oql þ r ðql ul Þ ¼ 0 ot
ð8Þ
The conservation of momentum: oðql ul Þ þ r ðql ul ul Þ ¼ CR ðu ul Þ ot The conservation of energy: n o 2 ! ) ( 2 o ql cvl Tl þ ju2l j jul j ul ¼ CR ul ðu ul Þ þ Ca ðT Tl Þ þ r ql cvl Tl þ 2 ot where ql is the mass density of liquid phase, cvl the specific heat of liquid phase.
3 4p d ql ¼ q0 n 3 2
ð9Þ
ð10Þ
ð11Þ
where q0 is the mass density of Al2 O3 , d the diameter of liquid particles, and n the number density of liquid particles. 3.3. Quasi-one dimensional approximation The present study adopts the one dimensional approximation, resulting in the following equations for the gas and liquid phase: oðqAÞ oðquAÞ þ ¼0 ot ox
ð12Þ
oðquAÞ oðqu2 AÞ op þ þ A ¼ Jy BA f þ CR ðul uÞA ot ox ox
ð13Þ
oðs AÞ ofðs þ pÞuAg þ ¼ Jy Ey A q þ fCR ul ðul uÞ þ Ca ðTl T ÞgA ot ox
ð14Þ
oðql AÞ oðql ul AÞ þ ¼0 ot ox
ð15Þ
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oðql ul AÞ oðql u2l AÞ þ ¼ CR ðu ul ÞA ot ox n n o o u2 u2 o q cvl Tl þ 2l A o q cvl Tl þ 2l ul A ¼ fCR ul ðu ul Þ þ Ca ðT Tl ÞgA þ ox ot
ð16Þ
ð17Þ
where A is the cross-section area, J ¼ ðJx ; Jy ; 0Þ the electric current density, E ¼ ðEx ; Ey ; 0Þ the electric field, B the magnetic flux density of z-direction, f the friction loss, q the heat loss, u ¼ ðu; 0; 0Þ, and ul ¼ ðul ; 0; 0Þ. 3.4. Electrodynamics The basic equations of the electrodynamics are the Maxwell equations and the generalized OhmÕs law. The Maxwell equations: rE ¼0
ð18Þ
r J ¼0
ð19Þ
The generalized OhmÕs law: J ¼ rðE þ u BÞ
b ðJ BÞ jBj
ð20Þ
OhmÕs law is rewritten under the quasi-one dimensional approximation as follows: Jx þ bJy ¼ rEx DV bJx þ Jy ¼ r Ey uB þ hch
ð21Þ ð22Þ
where r is the electrical conductivity, b the Hall parameter, DV the electrode voltage drop, and hch the channel height in the y-direction. 3.5. Thermodynamical properties Thermodynamical properties of the working gas of ‘‘SAKHALIN’’ were not reported, and the present study, therefore, takes the values used for the analyses of Pamir-3U as: Gas constant [J/kg K] R ¼ Rg ðT Þ ¼ 359:875 þ 0:00345921T
ð23Þ
Enthalpy [J/kg] h ¼ hg ðT Þ ¼ 733630:0 þ 649:863T þ 0:310466T 2
ð24Þ
M. Ishikawa et al. / Energy Conversion and Management 45 (2004) 707–724
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Electrical conductivity [S/m] r ¼ expfð3:29702 þ 0:00140816T Þ=p 2:02024 þ 0:00212241T þ ð0:156856 þ 0:0000319339T Þpg=GF
ð25Þ
Hall parameter
pffiffiffiffi b ¼ ð0:393664 þ 0:00302522 T Þ=p=GF
ð26Þ
where p and T stand for the static pressure and the static temperature of the working gas, and GF the G-factor indicating the nonuniformity effect of the plasma, being assumed to be 1.1 in the present study. The mass density of liquid particles (Al2 O3 ) is assumed as q ¼ 5632:0 1:127T ½kg=m3
ð27Þ
The MacCormack method is used with the one dimensional approximation [3,4,11]. A total length of 710 cm is treated in the present calculations, with an acceleration nozzle of 130 cm, MHD power generation channel of 450 cm and diffuser of 130 cm. The numerical mesh is constant along the x direction with 711 points of 1 cm mesh. The electrical conductivity ranges from 120 to 130 S/m with the maximum value in the first half of the MHD channel, while the maximum value of the Hall parameter is 0.35, when the applied magnetic filed is strong (Runs 1 and 2). On the other hand, the electrical conductivity shows the peak value in the second half of the MHD channel, being 80 S/m or less, when the applied magnetic field is weak (Runs 3–6).
4. Effects of induced magnetic field (mono phase flow) This section examines the effects of the induced magnetic field when the working gas is assumed to be the mono phase flow. Fig. 2 depicts the distributions of the Mach number for Run No. 1, operated with the load voltage of 2550 V when the induced magnetic field is or is not taken into account, where the mono phase approximation is used. The solid line shows the case where the induced magnetic field is not taken into account, whereas the dotted line indicates the case where it is taken into account. On this load condition, the effective magnetic field Beff is smaller than the applied magnetic field Bext , about 40% at the generator entrance (x ¼ 0 m), and is larger than the applied magnetic field Bext , about 39% at the exit (x ¼ 4:5 m). The flow is, therefore, less decelerated along the first half of the MHD channel and is more accelerated along the second half when the induced magnetic field is considered. Fig. 3 shows the distributions of Mach number when the load voltage is decreased to be 1900 V and the larger current density is generated, together with stronger deceleration. Even when the supersonic flow can be maintained along the whole channel without the induced magnetic field, a shock wave is induced at the end region of the MHD channel when the induced magnetic field is considered. This is because the flow is less decelerated along the first half of the MHD channel since the induced magnetic field reduces the effective magnetic field there, whereas the flow experiences stronger deceleration along the second half of the MHD channel due to the enhanced magnetic field when the induced magnetic field is taken into account.
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M
2 1.8 1.6 1.4 1.2 1
-2
-1
0
1
2 x (m)
3
4
5
6
Fig. 2. Distributions of Mach number (mono phase) (2500 V; solid line: without induced magnetic field, dotted line: with induced magnetic field).
2.4 2.2 2 M
1.8 1.6 1.4 1.2 1 0.8 -2
-1
0
1
2
3
4
5
6
x (m) Fig. 3. Distributions of Mach number (mono phase) (1900 V; solid line: without induced magnetic field, dotted line: with induced magnetic field).
Fig. 4 depicts the distributions of Mach number, where the MHD generator is operated with the lower load voltage of 1600 V, showing that a slight shock wave is induced at the end region of the MHD channel, even when the induced magnetic field is not taken into account, whereas a stronger shock wave is produced and the position of the shock wave moves upwards when the induced magnetic field is taken into account. Fig. 5 shows the distributions of Mach number, when the MHD generator is operated with the load voltage of 1000 V, telling that a strong shock wave is induced, a subsonic region is produced in the second half region of the MHD channel, and the flow is again accelerated into supersonic flow along the diffuser even when the induced magnetic field is not taken into account. In summary, the flow field experiences the shock wave when the operation voltage becomes low in both cases, while the shock wave becomes stronger when the induced magnetic field is included. The flow is less decelerated along the first half of the channel, and the stronger shock wave is induced at the second half part of the channel, when the operation voltage becomes lower, such as 1000 V. Our experiences [7] tell that the strong shock wave always occurs together with boundary
M. Ishikawa et al. / Energy Conversion and Management 45 (2004) 707–724
715
2.4 2.2 2 M
1.8 1.6 1.4 1.2 1 0.8 -2
-1
0
1
2 x (m)
3
4
5
6
Fig. 4. Distributions of Mach number (mono phase) (1600 V; solid line: without induced magnetic field, dotted line: with induced magnetic field).
2.4 2.2 2
M
1.8 1.6 1.4 1.2 1 0.8 0.6 -2
-1
0
1
2
3
4
5
6
x (m) Fig. 5. Distributions of Mach number (mono phase) (1000 V; solid line: without induced magnetic field, dotted line: with induced magnetic field).
layer separation, indicating that two and three dimensional analyses are required to understand the MHD interaction in detail. Fig. 6 depicts the voltage–current characteristics of the large MHD generator under the operation conditions of Run No. 1. The electric power output results in 511 MW when the induced magnetic field is not included, while it becomes 509 MW when the induced magnetic field is taken into account. The effects of the induced magnetic field are cancelled out with each other between the first and the second half parts of the MHD channel.
5. Effects of liquid particles (two phase flow) 5.1. Comparison of mono and two phase Figs. 7 and 8 show the distributions of the gas velocity with the mono phase approximation and the two phase approximation, respectively, where the nominal values of the liquid phase
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M. Ishikawa et al. / Energy Conversion and Management 45 (2004) 707–724 5000 4500 4000
V (V)
3500 3000 2500 2000 1500 1000 500 0
50000 100000 150000 200000 250000 300000 350000
I (A)
Fig. 6. Voltage–current characteristics (mono phase) (conditions of Run No. 1; solid line: without induced magnetic field, dotted line: with induced magnetic field).
2000 1800 1600
u x (m/s)
2000
V=2550 V=2000 V=1500 V=1000 V= 500
1800 1600
1400
1400
1200
1200
1000
1000
800
800
600
-2
-1
0
1
2
3
4
5
6
600
x (m)
Fig. 7. Distributions of velocity (mono phase approximation).
2000 1800 1600
u x (m/s)
2000
V=2550 V=2000 V=1500 V=1000 V= 500
1800 1600
1400
1400
1200
1200
1000
1000
800
800
600
-2
-1
0
1
2
3
4
5
6
600
x (m)
Fig. 8. Distributions of velocity (two phase approximation).
M. Ishikawa et al. / Energy Conversion and Management 45 (2004) 707–724
717
2400 2200 2000 1800 1600 1400 1200 1000 800 600 -2
one-gas two-gas two-liq
-1
0
1
2
3
4
5
6
2400 2200 2000 1800 1600 1400 1200 1000 800 600
ul (m/s)
u(m/s)
parameters are used. The induced magnetic field is taken into account in all calculations of this section. The shock wave becomes less sharp when the two phase flow is assumed, as shown in Fig. 8, although the tendency is the same as Fig. 7. Since the liquid particles cannot follow the sudden change of gas plasma velocity, and they pass through the shock wave at a velocity much higher than the gas velocity, the momentum of the liquid particles is transferred into the gas phase by the interaction between the liquid and gas phases. Fig. 9 compares the distributions of the gas velocity in the mono phase approximation and the gas phase and liquid phase velocities in the two phase approximation when the generator is operated with the load voltage of 500 V. The two phase approximation leads to the observation that the liquid velocity is higher than the gas velocity before the shock wave and the liquid particles drag the gas phase, and the shock wave is pushed down stream-ward. Fig. 10 shows the distributions of the gas pressure with the mono phase approximation. The pressure decreases rapidly along the acceleration nozzle where the flow is accelerated into supersonic flow and the thermal energy is converted into the kinetic energy. When the load voltage becomes small, the Lorentz force becomes strong, and it becomes impossible to maintain the supersonic flow over the whole channel. The shock wave is, therefore, generated, as shown in
x(m)
Fig. 9. Comparison of flow velocity distributions (500 V).
30
30 V=2550 V=2000 V=1500 V=1000 V= 500
P(atm)
25
25
20
20
15
15
10
10
5
5
0
0 -2
-1
0
1
2
3
4
5
6
x(m)
Fig. 10. Distributions of pressure (mono phase approximation).
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Fig. 7, and the sudden rise of pressure occurs when the load voltage becomes less then 1500 V. It turns out that the pressure rise becomes large, and the shock wave moves to the generator entrance if the load voltage becomes lower. Fig. 11 shows the variation of the pressure distributions when the alumina liquid particles are taken into consideration. Although the tendency is the same as Fig. 10, the discontinuous rise of pressure is no longer seen. This is because the momentum of the liquid particles is transferred into the gas, and the shock wave becomes less sharp, as in the case of Figs. 7 and 8. Fig. 12 depicts the gas temperature distributions with the mono phase approximation. As the flow is accelerated along the acceleration nozzle, the thermal energy is converted into kinetic energy, and therefore the temperature decreases rapidly. When the load voltage becomes small and the Lorentz force becomes strong, it turns out that the kinetic energy is converted again into thermal energy without or with the shock wave. Fig. 13 indicates that the temperature distribution changes when the alumina liquid particles are taken into account. Although the tendency is the same as Fig. 12, the rise of temperature becomes less sharp even at the shock wave position. This is caused by the interaction of the liquid particles and the gas.
30
30 V=2550 V=2000 V=1500 V=1000 V= 500
P(atm)
25
25
20
20
15
15
10
10
5
5
0
-2
-1
0
1
2
3
4
5
6
0
x(m)
Fig. 11. Distributions of pressure (two phase approximation).
3700 3500
T (K)
3700
V=2550 V=2000 V=1500 V=1000 V= 500
3600
3600 3500
3400
3400
3300
3300
3200
3200
3100
3100
3000
3000
2900
2900
2800
-2
-1
0
1
2
3
4
5
6
2800
x (m)
Fig. 12. Distributions of temperature (mono phase approximation).
M. Ishikawa et al. / Energy Conversion and Management 45 (2004) 707–724 3700 3500
T(K)
3700
V=2550 V=2000 V=1500 V=1000 V= 500
3600
3600 3500
3400
3400
3300
3300
3200
3200
3100
3100
3000
3000
2900
2900
2800 -2
719
2800 -1
0
1
2
3
4
5
6
x(m)
Fig. 13. Distributions of temperature (two phase approximation).
one phase two phases
Power (MW)
500
500
400
400
300
300
200
200
100
100
0
0
-100
-100
500 1000 1500 2000 2500 3000 3500 4000 4500 5000
V (V)
Fig. 14. Comparison of electric power output.
Fig. 14 compares the electric power output obtained with the mono phase approximation and with the two phase approximation. The electric power output becomes smaller compared with the case of mono phase flow, when the two phase approximation is used. This is because the kinetic energy of the liquid particles cannot be directly converted into electric energy. When the load voltage is large, the difference is large, while the difference becomes small when the load voltage is small, where the effect of the shock wave dominates. 5.2. Effects of diameter of liquid particles The diameter of the liquid particles (d) is assumed to be 15 lm until now, although it must have a distribution [6], because there are no experimental data. The diameter of the liquid particles is changed and the influence is examined now. Fig. 15 shows the distributions of the gas velocity and liquid particles velocity with the load voltage of 2550 V (experimental operation condition) and 500 V (shock wave condition) when the diameter of the liquid particles is assumed to be 5 lm. Since the diameter of the liquid particles is small and the interaction between the gas and liquid particles becomes strong, the liquid particles
M. Ishikawa et al. / Energy Conversion and Management 45 (2004) 707–724 2400
2400
u ,V =2550 u,V= 500 u ,V =2550 l ul ,V = 500
2200
u(m/s)
2000
2200 2000
1800
1800
1600
1600
1400
1400
1200
1200
1000
1000
800
800
u l (m/s)
720
600
600 -2
-1
0
1
2
3
4
5
6
x (m)
Fig. 15. Comparison of flow velocity distributions (d ¼ 5 lm).
can follow the change of gas velocity, and the gas velocity and liquid particles velocity become almost the same. The phenomena becomes similar to those with the mono phase approximation, and the sharp shock wave has arisen. Fig. 16 depicts the velocity distributions of the gas and the liquid particles with the load voltage of 2550 and 500 V when the liquid particles diameter is 15 lm. When the diameter of the liquid particles becomes large, the interaction of the gas and liquid particles becomes weaker, the liquid particles cannot follow the change of gas velocity, and thus, the shock wave becomes less sharp as already explained. Fig. 17 also compares the velocity distributions of the gas and liquid particles with the load voltage of 2550 and 500 V when the liquid particles diameter is 25 lm. The diameter of the liquid particles becomes larger, the interaction of the gas and liquid particles becomes weaker further, and the liquid particles, therefore, cannot follow the change of gas velocity, leading to the shock wave becoming smooth further. Fig. 18 depicts the temperature distributions of the gas and the liquid particles with the load voltage of 2550 and 500 V when the liquid particles diameter is 25 lm. The weaker interaction of the gas and liquid particles results in the smooth temperature rise even at the shock wave position. Fig. 19 shows the effects of the load voltage on the distributions of the load current density (Jy ) in the two phase flow analysis (d ¼ 15 lm). If the load voltage becomes small, the load current
1800
u (m/s)
2000
u,V=2550 u,V= 500 ul,V=2550 ul,V= 500
1800
1600
1600
1400
1400
1200
1200
1000
1000
800
800
u l (m/s)
2000
600
600 -2
-1
0
1
2
3
4
5
6
x(m)
Fig. 16. Comparison of flow velocity distributions (d ¼ 15 lm).
M. Ishikawa et al. / Energy Conversion and Management 45 (2004) 707–724
2200 2000
u (m/s)
2400
u,V=2550 u,V= 500 ul,V=2550 ul,V= 500
2200 2000
1800
1800
1600
1600
1400
1400
1200
1200
1000
1000
800 600 -2
u l (m/s)
2400
721
800 600 -1
0
1
2
3
4
5
6
x (m)
Fig. 17. Comparison of flow velocity distributions (d ¼ 25 lm).
3600
T(K)
3800
T,V=2550 T,V= 500 Tl,V=2550 Tl,V= 500
3600
3400
3400
3200
3200
3000
3000
2800 -2
-1
0
1
2
3
4
5
6
T l (K)
3800
2800
x (m)
Fig. 18. Comparison of temperature distributions (d ¼ 25 lm).
20000
20000 V=2550 V=2000 V=1500 V=1000 V= 500
0
Jy (A/m 2)
-20000
0 -20000
-40000
-40000
-60000
-60000
-80000
-80000
-100000
-100000
-120000
-120000 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
x (m)
Fig. 19. Distribution of load current density (Jy ) (d ¼ 15 lm).
density becomes large. When the shock wave is induced, the velocity and electromotive force decrease largely. As a result, the change of Jy by the loading becomes small in the second half of the generator.
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Fig. 20 shows the effects of the load voltage on the distributions of the Hall current density (Jx ) in the two phase flow analysis (d ¼ 15 lm). The Hall current is quite large in any operation conditions. The sharp drop with the load voltage of 500 V corresponds to the shock wave. It can be expected that the secondary flow grows on the generator cross-section as a result of the three dimensional MHD interaction, indicating the necessity of three dimensional analyses. Fig. 21 depicts the distributions of the electrical conductivity in the two phase flow analysis (d ¼ 15 lm). Near the experimental load condition, the electric conductivity does not change much, whereas the electric conductivity becomes more than 200 S/m, since the gas temperature raises in the second half of the generator if the load voltage decreases. Fig. 22 shows the effects of the load voltage on the distributions of the Hall parameter in the two phase flow analysis (d ¼ 15 lm). The Hall parameter is 0.15–0.4. The shock wave decreases the Hall parameter sharply with the load voltage of 500 V. 5.3. Effects of velocity slip and temperature slip
Jx (A/m 2 )
Since there are no experimental data of Slip and DT , we have assumed the nominal values of Slip and DT to be 1.1 and 100 K, respectively, which are taken from other studies [4,6]. The effects 30000
30000
25000
25000
20000
20000
15000
15000
10000
10000 V=2550 V=2000 V=1500 V=1000 V= 500
5000 0 -5000 0
0.5 1
5000 0
1.5 2
2.5 3
3.5 4
-5000 4.5
x (m)
Fig. 20. Distribution of Hall current density (Jx ) (d ¼ 15 lm).
250 200
200 150
σ
150
250 V=2550 V=2000 V=1500 V=1000 V= 500
100
100
50
50
0 0
0.5
1
1.5
2
2.5
3
3.5
4
0 4.5
x (m)
Fig. 21. Distributions of electrical conductivity (d ¼ 15 lm).
β
M. Ishikawa et al. / Energy Conversion and Management 45 (2004) 707–724 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0
V=2550 V=2000 V=1500 V=1000 V= 500
0.5
1
1.5
2
2.5
3
3.5
4
723
0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 4.5
x(m)
Fig. 22. Distributions of Hall parameter (d ¼ 15 lm).
of Slip and DT have been examined, and it has been found that there is no large influence of Slip and DT on the generator performance. 6. Concluding remarks Two phase flow analyses have been performed using the one dimensional time dependence approximation with and without the induced magnetic field in order to study the influence of the induced magnetic field and liquid particles on the generator performance of the large pulsed MHD generator ‘‘SAKHALIN’’. The main results are as follows: (1) Since the ‘‘SAKHALIN’’ MHD generator has a very strong MHD interaction and the magnetic Reynolds number is about 0.58 for Run No. 1, the induced magnetic field cannot be neglected. The induced magnetic flux density is about 20% at the entrance and about 19% at the exit of the MHD channel. (2) If the induced magnetic field is taken into account, the magnetic field decreases and the deceleration of the flow decreases in the first half of the generator. In the second half, the magnetic field becomes large, the flow is decelerated strongly and the stronger shock wave is generated. (3) If alumina liquid particles are taken into account, the momentum and thermal energy are transferred from the liquid particles into the gas, and the shock wave becomes less sharp. (4) When the diameter of the alumina liquid particles is small (d ¼ 5 lm), the interaction of the liquid particles and gas becomes strong and almost the same behavior appears as for the mono phase flow. (5) Consideration of the alumina liquid particles reduces the electric power output about 5% (d ¼ 15 lm). This is because the energy of the liquid particles cannot be converted directly into electric energy. Acknowledgements Discussions with Prof. V.P. Panchenko, TRINITI, were very useful. KDK computers, Kyoto University were used for the numerical analyses in part.
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References [1] Zeigarnik VA, Ishikawa M. Proc of Int Conf on MHD Power Generation and High Temperature Technologies 1999;III:775–92. [2] Velikhov EP, et al. Proc of Int Conf on MHD Power Generation and High Temperature Technologies 1999;II:387– 98. [3] Ishikawa M, Koshiba Y. Proc of Sym on Advanced Research of Energy Technology 2001. p. 251–58. [4] Matsuo T, Sugita H, Ishikawa M. Trans IEE Jpn 1998;118-B(2):199–204 [in Japanese]. [5] Sugita H, Matsuo T, Inui Y, Ishikawa M. Trans IEE Jpn 1998;118-B(12):1374–9 [in Japanese]. [6] Sugita H, Matsuo T, Inui Y, Ishikawa M. Trans IEE Jpn 1999;119-B(6):682–8 [in Japanese]. [7] H. Sugita, T. Matsuo, Y. Inui, M. Ishikawa. 30th AIAA Plasmadynamics and Lasers Conference, AIAA 99-3483; 1999. [8] Matsuo T, Sugita H, Ishikawa M. Trans IEE Jpn 1999;119-B(6):690–5 [in Japanese]. [9] Tadamatsu A, Matsuo T, Ishikawa M. Technical Report FTE-00-8, New Energy and Environment TC, IEE Japan; 2000. p. 1–6 [in Japanese]. [10] Yakushev AA, Panchenko VP, et al. Proc of Int Conf on MHD Power Generation and High Temperature Technologies 1999;II:423–34. [11] Ishikawa M. Technical Report of Institute of Atomic Energy, Kyoto University, No. 192; 1982.
Effects of induced magnetic field on large scale pulsed MHD generator with two phase flow M. Ishikawa *, Y. Koshiba, T. Matsushita Institute of Engineering Mechanics and Systems, University of Tsukuba, Tsukuba 305-8573, Japan Received 10 February 2003; accepted 5 July 2003
Abstract A large pulsed MHD generator ‘‘SAKHALIN’’ was constructed in Russia (the former Soviet-Union) and operated with solid fuels. The ‘‘SAKHALIN’’ with the channel length of 4.5 m could demonstrate the electric power output of 510 MW. The effects of induced magnetic field and two phase flow on the shock wave within the ‘‘SAKHALIN’’ generator have been studied by time dependent, one dimensional analyses. It has been shown that the magnetic Reynolds number is about 0.58 for Run No. 1, and the induced magnetic flux density is about 20% at the entrance and exit of the MHD channel. The shock wave becomes stronger when the induced magnetic field is taken into account, when the operation voltage becomes low. The working gas plasma contains about 40% of liquid particles (Al2 O3 ) in weight, and the present analysis treats the liquid particles as another gas. In the case of mono-phase flow, the sharp shock wave is induced when the load voltage becomes small such as 500 V with larger Lorentz force, whereas in the case of two phase flow, the shock wave becomes less sharp because of the interaction with liquid particles. 2003 Elsevier Ltd. All rights reserved. Keywords: Pulsed MHD generator; Induced magnetic field; Two phase flow; Strong MHD interaction; Effects of shock wave
1. Introduction Various pulsed MHD generators have been constructed and operated, mainly in Russia (the former Soviet-Union) [1]. Construction and operation of a large pulsed MHD generator named ‘‘SAKHALIN’’ with the electric power output of 510 MW were an important step in the history of MHD generator development. The self excited large MHD generator ‘‘SAKHALIN’’ was the
*
Corresponding author. Tel.: +81-298-53-5140; fax: +81-298-53-5207. E-mail address: [email protected] (M. Ishikawa).
0196-8904/$ - see front matter 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0196-8904(03)00182-1
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Faraday type with the active channel length of 4.5 m equipped with a pair of continuous electrodes. Cesium or potassium salt was added to solid propellants to achieve high electrical conductivity [2]. Analysis of the operating characteristics of ‘‘SAKHALIN’’ is an important research subject for understanding a very strong MHD interaction [3]. Since this MHD generator has a very strong MHD interaction and its operation magnetic flux density is comparatively low, the induced magnetic field cannot be neglected. ‘‘SAKHALIN’’ also has a special feature that liquid particles generated by combustion of the metal aluminum in the fuel cannot be neglected. The present paper intends to report the influences of the induced magnetic field and the effects of the two phase flow on the generator performance by using time dependent, one dimensional mono and two phase flow approximations. Our research group has conducted time dependent, one and two dimensional analyses with the assumption of mono and two phase flow and time dependent, three dimensional analyses with the assumption of mono phase flow for the Pamir-3U MHD generator with the electric power output of 10–20 MW. The shock wave, the boundary layer-shock wave interaction and others have been analyzed [4–9]. The present results are very useful for two and three dimensional analyses of ‘‘SAKHALIN’’ to understand strong MHD interactions and to design high performance MHD generators.
2. Operation analysis of ‘‘SAKHALIN’’ 2.1. Performance of ‘‘SAKHALIN’’ The large pulsed MHD generator was operated with solid fuels. The channel length is 4.5 m and the cross-section is 0.9 m (the direction of load current, y-direction) · 1.0 m (the direction of external applied magnetic field, z-direction) and 1.6 m (y-direction) · 1.0 m (z-direction) at the entrance and at the exit, respectively, where the height along the magnetic field is kept constant at 1.0 m. The nominal mass flow rate is 1000 kg/s. The combustor temperature is 3800–3900 K, while the combustor pressure is 4.0–5.6 MPa. The generator was quite powerful, light (only 50 tons) and compact, which could demonstrate the electric power output of 510 MW, the specific mass of 0.1 ton/MW and the specific volume of 0.3 m3 /MW. The air core magnet was activated to more than 2 T by the MHD generator itself, which is the Faraday type with a pair of continuous electrodes. Six experimental runs were conducted. The first two runs were operated with a resistance load, and the other runs were operated with the load of magnetic energy storage. The excitation current of a self excitation magnet was 52.5–190 kA, the magnetic flux density 1.5–2.2 T, the load voltage 1.41–2.6 kV, the load current 180–209 kA, the electric output 275–510 MW and the operation time was from 7.37–8.64 s [2]. 2.2. Induced magnetic field A preliminary analysis has indicated the magnetic Reynolds number is about 0.58 for Run No. 1, which is much different from many other MHD generators. The induced magnetic field, therefore, cannot be neglected. Then the one dimensional approximation has been applied to AmpereÕs law. Fig. 1 depicts distributions of the magnetic flux density for Run No. 1, where the solid line shows the externally applied field, the dashed line with dots shows the induced field and the dotted
M. Ishikawa et al. / Energy Conversion and Management 45 (2004) 707–724 2.5
0.4 0.3
2
0.2 1.5
0.1
1
0 -0.1
0.5 Bext Beff Bind
0
Bind (T)
Bext , Beff (T)
709
-0.2 -0.3
-0.5
-0.4 -2
-1
0
1
2
3
4
5
6
x (m) Fig. 1. Distributions of magnetic flux density (Run No. 1, solid line: external field, dashed line with dots: induced field, dotted line: effective field).
line shows the effective field. It is revealed that the induced magnetic flux density is about 20% at the entrance of the MHD channel, while it becomes about 19% at the exit for Run No. 1, being rather large. The magnetic Reynolds number Rm and induced magnetic field Bind are estimated as follows: ð1Þ Rm ¼ rl0 UL Z x Bind ¼ Bind ð0Þ l0 Jy dx ð2Þ 0
where r is the electrical conductivity, U the flow velocity, L the length, l0 ¼ 4p 107 [H/m] the vacuum permeability, Jy the load current density, x the distance from the entrance of channel. 2.3. Estimation of electrode voltage drop The value of electrode voltage drop can be automatically calculated in the case of multidimensional analyses, but the present one dimensional analysis requires estimation of the electrode voltage drop. It has been well known that the electrode voltage drop is nonlinear and shows a complicated behavior [10]. The present work, however, assumes rather a simple relation. At first, the electrode voltage drop is assumed to be constant without the induced magnetic field. It has been shown that the estimated constant electrode voltage drop ranges from 700 to 1200 V (except for Run No. 5), as listed below. Run No.
Average current (Jy , kA/m2 )
Electrode voltage (V)
1 2 3 4 5 6
44.5 42.89 43.33 46.44 40.00 44.44
1035 1199 1165 790 487 732
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M. Ishikawa et al. / Energy Conversion and Management 45 (2004) 707–724
Then, a simple linear relation to the local electric current density Jy is assumed as DV ¼ rdrp Jy þ DV0
ð3Þ
where DV is the electrode voltage drop and DV0 is the constant, and the value of the constant of the linear relation rdrp is estimated so as to agree with the electric output power, as shown below: Run No.
rdrp
Run No.
rdrp
1 3 5
0.0143 0.0175 0.0022
2 4 6
0.019 0.00836 0.0074
The value of DV0 is assumed to be 400 V, which must be estimated with experimental data and/or multi-dimensional analyses. This relation is used for all further calculations with or without induced magnetic field. 2.4. Nominal values of parameters of liquid phase Since measurements were difficult in the actual generator, details of the liquid phase are not clear, and a dusty gas flow is, therefore, assumed for spherical liquid particles. It is also assumed that the diameter of particles (d) is kept constant within the flow, and the ratio of velocities of gas and liquid particles (Slip0 ) and a temperature difference between them (DT0 ) are given at the nozzle entrance. The nominal values are given as follows: d ¼ 15 ½lm;
Slip0 ¼ 1:1;
DT0 ¼ 100 ½K
ð4Þ
where the subscript 0 stands for values at the nozzle entrance. Although the mass ratio of the liquid phase is about 30–40%, the exact value is not known, and then, the mass ratio is assumed to be 40% for all calculations.
3. Mathematical model for gas–liquid two phase flow 3.1. Basic equations of gas phase The conservation of mass: oq þ r ðquÞ ¼ 0 ot The conservation of momentum: oðquÞ þ r ðquuÞ þ rp ¼ F þ CR ðul uÞ ot The conservation of energy: os þ r fðs þ pÞug ¼ Q þ CR ul ðul uÞ þ Ca ðTl T Þ ot
ð5Þ
ð6Þ
ð7Þ
M. Ishikawa et al. / Energy Conversion and Management 45 (2004) 707–724
711
where q is the mass density, p the gas static pressure, u the gas velocity, ul the velocity of liquid particles, T the gas temperature, Tl the temperature of liquid particles, CR the coefficient of momentum transfer between the gas phase and liquid particles, F the force consisting of the viscosity 2 and the Lorentz force, s ¼ qðh p=q þ juj =2Þ, Ca the coefficient of heat transfer between the gas phase and the liquid phase, and Q the energy source consisting of the work done by the viscosity, the thermal conduction and the Joule heating. 3.2. Basic equations for liquid phase The conservation of mass: oql þ r ðql ul Þ ¼ 0 ot
ð8Þ
The conservation of momentum: oðql ul Þ þ r ðql ul ul Þ ¼ CR ðu ul Þ ot The conservation of energy: n o 2 ! ) ( 2 o ql cvl Tl þ ju2l j jul j ul ¼ CR ul ðu ul Þ þ Ca ðT Tl Þ þ r ql cvl Tl þ 2 ot where ql is the mass density of liquid phase, cvl the specific heat of liquid phase.
3 4p d ql ¼ q0 n 3 2
ð9Þ
ð10Þ
ð11Þ
where q0 is the mass density of Al2 O3 , d the diameter of liquid particles, and n the number density of liquid particles. 3.3. Quasi-one dimensional approximation The present study adopts the one dimensional approximation, resulting in the following equations for the gas and liquid phase: oðqAÞ oðquAÞ þ ¼0 ot ox
ð12Þ
oðquAÞ oðqu2 AÞ op þ þ A ¼ Jy BA f þ CR ðul uÞA ot ox ox
ð13Þ
oðs AÞ ofðs þ pÞuAg þ ¼ Jy Ey A q þ fCR ul ðul uÞ þ Ca ðTl T ÞgA ot ox
ð14Þ
oðql AÞ oðql ul AÞ þ ¼0 ot ox
ð15Þ
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oðql ul AÞ oðql u2l AÞ þ ¼ CR ðu ul ÞA ot ox n n o o u2 u2 o q cvl Tl þ 2l A o q cvl Tl þ 2l ul A ¼ fCR ul ðu ul Þ þ Ca ðT Tl ÞgA þ ox ot
ð16Þ
ð17Þ
where A is the cross-section area, J ¼ ðJx ; Jy ; 0Þ the electric current density, E ¼ ðEx ; Ey ; 0Þ the electric field, B the magnetic flux density of z-direction, f the friction loss, q the heat loss, u ¼ ðu; 0; 0Þ, and ul ¼ ðul ; 0; 0Þ. 3.4. Electrodynamics The basic equations of the electrodynamics are the Maxwell equations and the generalized OhmÕs law. The Maxwell equations: rE ¼0
ð18Þ
r J ¼0
ð19Þ
The generalized OhmÕs law: J ¼ rðE þ u BÞ
b ðJ BÞ jBj
ð20Þ
OhmÕs law is rewritten under the quasi-one dimensional approximation as follows: Jx þ bJy ¼ rEx DV bJx þ Jy ¼ r Ey uB þ hch
ð21Þ ð22Þ
where r is the electrical conductivity, b the Hall parameter, DV the electrode voltage drop, and hch the channel height in the y-direction. 3.5. Thermodynamical properties Thermodynamical properties of the working gas of ‘‘SAKHALIN’’ were not reported, and the present study, therefore, takes the values used for the analyses of Pamir-3U as: Gas constant [J/kg K] R ¼ Rg ðT Þ ¼ 359:875 þ 0:00345921T
ð23Þ
Enthalpy [J/kg] h ¼ hg ðT Þ ¼ 733630:0 þ 649:863T þ 0:310466T 2
ð24Þ
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713
Electrical conductivity [S/m] r ¼ expfð3:29702 þ 0:00140816T Þ=p 2:02024 þ 0:00212241T þ ð0:156856 þ 0:0000319339T Þpg=GF
ð25Þ
Hall parameter
pffiffiffiffi b ¼ ð0:393664 þ 0:00302522 T Þ=p=GF
ð26Þ
where p and T stand for the static pressure and the static temperature of the working gas, and GF the G-factor indicating the nonuniformity effect of the plasma, being assumed to be 1.1 in the present study. The mass density of liquid particles (Al2 O3 ) is assumed as q ¼ 5632:0 1:127T ½kg=m3
ð27Þ
The MacCormack method is used with the one dimensional approximation [3,4,11]. A total length of 710 cm is treated in the present calculations, with an acceleration nozzle of 130 cm, MHD power generation channel of 450 cm and diffuser of 130 cm. The numerical mesh is constant along the x direction with 711 points of 1 cm mesh. The electrical conductivity ranges from 120 to 130 S/m with the maximum value in the first half of the MHD channel, while the maximum value of the Hall parameter is 0.35, when the applied magnetic filed is strong (Runs 1 and 2). On the other hand, the electrical conductivity shows the peak value in the second half of the MHD channel, being 80 S/m or less, when the applied magnetic field is weak (Runs 3–6).
4. Effects of induced magnetic field (mono phase flow) This section examines the effects of the induced magnetic field when the working gas is assumed to be the mono phase flow. Fig. 2 depicts the distributions of the Mach number for Run No. 1, operated with the load voltage of 2550 V when the induced magnetic field is or is not taken into account, where the mono phase approximation is used. The solid line shows the case where the induced magnetic field is not taken into account, whereas the dotted line indicates the case where it is taken into account. On this load condition, the effective magnetic field Beff is smaller than the applied magnetic field Bext , about 40% at the generator entrance (x ¼ 0 m), and is larger than the applied magnetic field Bext , about 39% at the exit (x ¼ 4:5 m). The flow is, therefore, less decelerated along the first half of the MHD channel and is more accelerated along the second half when the induced magnetic field is considered. Fig. 3 shows the distributions of Mach number when the load voltage is decreased to be 1900 V and the larger current density is generated, together with stronger deceleration. Even when the supersonic flow can be maintained along the whole channel without the induced magnetic field, a shock wave is induced at the end region of the MHD channel when the induced magnetic field is considered. This is because the flow is less decelerated along the first half of the MHD channel since the induced magnetic field reduces the effective magnetic field there, whereas the flow experiences stronger deceleration along the second half of the MHD channel due to the enhanced magnetic field when the induced magnetic field is taken into account.
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M. Ishikawa et al. / Energy Conversion and Management 45 (2004) 707–724 2.4 2.2
M
2 1.8 1.6 1.4 1.2 1
-2
-1
0
1
2 x (m)
3
4
5
6
Fig. 2. Distributions of Mach number (mono phase) (2500 V; solid line: without induced magnetic field, dotted line: with induced magnetic field).
2.4 2.2 2 M
1.8 1.6 1.4 1.2 1 0.8 -2
-1
0
1
2
3
4
5
6
x (m) Fig. 3. Distributions of Mach number (mono phase) (1900 V; solid line: without induced magnetic field, dotted line: with induced magnetic field).
Fig. 4 depicts the distributions of Mach number, where the MHD generator is operated with the lower load voltage of 1600 V, showing that a slight shock wave is induced at the end region of the MHD channel, even when the induced magnetic field is not taken into account, whereas a stronger shock wave is produced and the position of the shock wave moves upwards when the induced magnetic field is taken into account. Fig. 5 shows the distributions of Mach number, when the MHD generator is operated with the load voltage of 1000 V, telling that a strong shock wave is induced, a subsonic region is produced in the second half region of the MHD channel, and the flow is again accelerated into supersonic flow along the diffuser even when the induced magnetic field is not taken into account. In summary, the flow field experiences the shock wave when the operation voltage becomes low in both cases, while the shock wave becomes stronger when the induced magnetic field is included. The flow is less decelerated along the first half of the channel, and the stronger shock wave is induced at the second half part of the channel, when the operation voltage becomes lower, such as 1000 V. Our experiences [7] tell that the strong shock wave always occurs together with boundary
M. Ishikawa et al. / Energy Conversion and Management 45 (2004) 707–724
715
2.4 2.2 2 M
1.8 1.6 1.4 1.2 1 0.8 -2
-1
0
1
2 x (m)
3
4
5
6
Fig. 4. Distributions of Mach number (mono phase) (1600 V; solid line: without induced magnetic field, dotted line: with induced magnetic field).
2.4 2.2 2
M
1.8 1.6 1.4 1.2 1 0.8 0.6 -2
-1
0
1
2
3
4
5
6
x (m) Fig. 5. Distributions of Mach number (mono phase) (1000 V; solid line: without induced magnetic field, dotted line: with induced magnetic field).
layer separation, indicating that two and three dimensional analyses are required to understand the MHD interaction in detail. Fig. 6 depicts the voltage–current characteristics of the large MHD generator under the operation conditions of Run No. 1. The electric power output results in 511 MW when the induced magnetic field is not included, while it becomes 509 MW when the induced magnetic field is taken into account. The effects of the induced magnetic field are cancelled out with each other between the first and the second half parts of the MHD channel.
5. Effects of liquid particles (two phase flow) 5.1. Comparison of mono and two phase Figs. 7 and 8 show the distributions of the gas velocity with the mono phase approximation and the two phase approximation, respectively, where the nominal values of the liquid phase
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M. Ishikawa et al. / Energy Conversion and Management 45 (2004) 707–724 5000 4500 4000
V (V)
3500 3000 2500 2000 1500 1000 500 0
50000 100000 150000 200000 250000 300000 350000
I (A)
Fig. 6. Voltage–current characteristics (mono phase) (conditions of Run No. 1; solid line: without induced magnetic field, dotted line: with induced magnetic field).
2000 1800 1600
u x (m/s)
2000
V=2550 V=2000 V=1500 V=1000 V= 500
1800 1600
1400
1400
1200
1200
1000
1000
800
800
600
-2
-1
0
1
2
3
4
5
6
600
x (m)
Fig. 7. Distributions of velocity (mono phase approximation).
2000 1800 1600
u x (m/s)
2000
V=2550 V=2000 V=1500 V=1000 V= 500
1800 1600
1400
1400
1200
1200
1000
1000
800
800
600
-2
-1
0
1
2
3
4
5
6
600
x (m)
Fig. 8. Distributions of velocity (two phase approximation).
M. Ishikawa et al. / Energy Conversion and Management 45 (2004) 707–724
717
2400 2200 2000 1800 1600 1400 1200 1000 800 600 -2
one-gas two-gas two-liq
-1
0
1
2
3
4
5
6
2400 2200 2000 1800 1600 1400 1200 1000 800 600
ul (m/s)
u(m/s)
parameters are used. The induced magnetic field is taken into account in all calculations of this section. The shock wave becomes less sharp when the two phase flow is assumed, as shown in Fig. 8, although the tendency is the same as Fig. 7. Since the liquid particles cannot follow the sudden change of gas plasma velocity, and they pass through the shock wave at a velocity much higher than the gas velocity, the momentum of the liquid particles is transferred into the gas phase by the interaction between the liquid and gas phases. Fig. 9 compares the distributions of the gas velocity in the mono phase approximation and the gas phase and liquid phase velocities in the two phase approximation when the generator is operated with the load voltage of 500 V. The two phase approximation leads to the observation that the liquid velocity is higher than the gas velocity before the shock wave and the liquid particles drag the gas phase, and the shock wave is pushed down stream-ward. Fig. 10 shows the distributions of the gas pressure with the mono phase approximation. The pressure decreases rapidly along the acceleration nozzle where the flow is accelerated into supersonic flow and the thermal energy is converted into the kinetic energy. When the load voltage becomes small, the Lorentz force becomes strong, and it becomes impossible to maintain the supersonic flow over the whole channel. The shock wave is, therefore, generated, as shown in
x(m)
Fig. 9. Comparison of flow velocity distributions (500 V).
30
30 V=2550 V=2000 V=1500 V=1000 V= 500
P(atm)
25
25
20
20
15
15
10
10
5
5
0
0 -2
-1
0
1
2
3
4
5
6
x(m)
Fig. 10. Distributions of pressure (mono phase approximation).
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M. Ishikawa et al. / Energy Conversion and Management 45 (2004) 707–724
Fig. 7, and the sudden rise of pressure occurs when the load voltage becomes less then 1500 V. It turns out that the pressure rise becomes large, and the shock wave moves to the generator entrance if the load voltage becomes lower. Fig. 11 shows the variation of the pressure distributions when the alumina liquid particles are taken into consideration. Although the tendency is the same as Fig. 10, the discontinuous rise of pressure is no longer seen. This is because the momentum of the liquid particles is transferred into the gas, and the shock wave becomes less sharp, as in the case of Figs. 7 and 8. Fig. 12 depicts the gas temperature distributions with the mono phase approximation. As the flow is accelerated along the acceleration nozzle, the thermal energy is converted into kinetic energy, and therefore the temperature decreases rapidly. When the load voltage becomes small and the Lorentz force becomes strong, it turns out that the kinetic energy is converted again into thermal energy without or with the shock wave. Fig. 13 indicates that the temperature distribution changes when the alumina liquid particles are taken into account. Although the tendency is the same as Fig. 12, the rise of temperature becomes less sharp even at the shock wave position. This is caused by the interaction of the liquid particles and the gas.
30
30 V=2550 V=2000 V=1500 V=1000 V= 500
P(atm)
25
25
20
20
15
15
10
10
5
5
0
-2
-1
0
1
2
3
4
5
6
0
x(m)
Fig. 11. Distributions of pressure (two phase approximation).
3700 3500
T (K)
3700
V=2550 V=2000 V=1500 V=1000 V= 500
3600
3600 3500
3400
3400
3300
3300
3200
3200
3100
3100
3000
3000
2900
2900
2800
-2
-1
0
1
2
3
4
5
6
2800
x (m)
Fig. 12. Distributions of temperature (mono phase approximation).
M. Ishikawa et al. / Energy Conversion and Management 45 (2004) 707–724 3700 3500
T(K)
3700
V=2550 V=2000 V=1500 V=1000 V= 500
3600
3600 3500
3400
3400
3300
3300
3200
3200
3100
3100
3000
3000
2900
2900
2800 -2
719
2800 -1
0
1
2
3
4
5
6
x(m)
Fig. 13. Distributions of temperature (two phase approximation).
one phase two phases
Power (MW)
500
500
400
400
300
300
200
200
100
100
0
0
-100
-100
500 1000 1500 2000 2500 3000 3500 4000 4500 5000
V (V)
Fig. 14. Comparison of electric power output.
Fig. 14 compares the electric power output obtained with the mono phase approximation and with the two phase approximation. The electric power output becomes smaller compared with the case of mono phase flow, when the two phase approximation is used. This is because the kinetic energy of the liquid particles cannot be directly converted into electric energy. When the load voltage is large, the difference is large, while the difference becomes small when the load voltage is small, where the effect of the shock wave dominates. 5.2. Effects of diameter of liquid particles The diameter of the liquid particles (d) is assumed to be 15 lm until now, although it must have a distribution [6], because there are no experimental data. The diameter of the liquid particles is changed and the influence is examined now. Fig. 15 shows the distributions of the gas velocity and liquid particles velocity with the load voltage of 2550 V (experimental operation condition) and 500 V (shock wave condition) when the diameter of the liquid particles is assumed to be 5 lm. Since the diameter of the liquid particles is small and the interaction between the gas and liquid particles becomes strong, the liquid particles
M. Ishikawa et al. / Energy Conversion and Management 45 (2004) 707–724 2400
2400
u ,V =2550 u,V= 500 u ,V =2550 l ul ,V = 500
2200
u(m/s)
2000
2200 2000
1800
1800
1600
1600
1400
1400
1200
1200
1000
1000
800
800
u l (m/s)
720
600
600 -2
-1
0
1
2
3
4
5
6
x (m)
Fig. 15. Comparison of flow velocity distributions (d ¼ 5 lm).
can follow the change of gas velocity, and the gas velocity and liquid particles velocity become almost the same. The phenomena becomes similar to those with the mono phase approximation, and the sharp shock wave has arisen. Fig. 16 depicts the velocity distributions of the gas and the liquid particles with the load voltage of 2550 and 500 V when the liquid particles diameter is 15 lm. When the diameter of the liquid particles becomes large, the interaction of the gas and liquid particles becomes weaker, the liquid particles cannot follow the change of gas velocity, and thus, the shock wave becomes less sharp as already explained. Fig. 17 also compares the velocity distributions of the gas and liquid particles with the load voltage of 2550 and 500 V when the liquid particles diameter is 25 lm. The diameter of the liquid particles becomes larger, the interaction of the gas and liquid particles becomes weaker further, and the liquid particles, therefore, cannot follow the change of gas velocity, leading to the shock wave becoming smooth further. Fig. 18 depicts the temperature distributions of the gas and the liquid particles with the load voltage of 2550 and 500 V when the liquid particles diameter is 25 lm. The weaker interaction of the gas and liquid particles results in the smooth temperature rise even at the shock wave position. Fig. 19 shows the effects of the load voltage on the distributions of the load current density (Jy ) in the two phase flow analysis (d ¼ 15 lm). If the load voltage becomes small, the load current
1800
u (m/s)
2000
u,V=2550 u,V= 500 ul,V=2550 ul,V= 500
1800
1600
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1400
1400
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1000
1000
800
800
u l (m/s)
2000
600
600 -2
-1
0
1
2
3
4
5
6
x(m)
Fig. 16. Comparison of flow velocity distributions (d ¼ 15 lm).
M. Ishikawa et al. / Energy Conversion and Management 45 (2004) 707–724
2200 2000
u (m/s)
2400
u,V=2550 u,V= 500 ul,V=2550 ul,V= 500
2200 2000
1800
1800
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1600
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1000
1000
800 600 -2
u l (m/s)
2400
721
800 600 -1
0
1
2
3
4
5
6
x (m)
Fig. 17. Comparison of flow velocity distributions (d ¼ 25 lm).
3600
T(K)
3800
T,V=2550 T,V= 500 Tl,V=2550 Tl,V= 500
3600
3400
3400
3200
3200
3000
3000
2800 -2
-1
0
1
2
3
4
5
6
T l (K)
3800
2800
x (m)
Fig. 18. Comparison of temperature distributions (d ¼ 25 lm).
20000
20000 V=2550 V=2000 V=1500 V=1000 V= 500
0
Jy (A/m 2)
-20000
0 -20000
-40000
-40000
-60000
-60000
-80000
-80000
-100000
-100000
-120000
-120000 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
x (m)
Fig. 19. Distribution of load current density (Jy ) (d ¼ 15 lm).
density becomes large. When the shock wave is induced, the velocity and electromotive force decrease largely. As a result, the change of Jy by the loading becomes small in the second half of the generator.
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Fig. 20 shows the effects of the load voltage on the distributions of the Hall current density (Jx ) in the two phase flow analysis (d ¼ 15 lm). The Hall current is quite large in any operation conditions. The sharp drop with the load voltage of 500 V corresponds to the shock wave. It can be expected that the secondary flow grows on the generator cross-section as a result of the three dimensional MHD interaction, indicating the necessity of three dimensional analyses. Fig. 21 depicts the distributions of the electrical conductivity in the two phase flow analysis (d ¼ 15 lm). Near the experimental load condition, the electric conductivity does not change much, whereas the electric conductivity becomes more than 200 S/m, since the gas temperature raises in the second half of the generator if the load voltage decreases. Fig. 22 shows the effects of the load voltage on the distributions of the Hall parameter in the two phase flow analysis (d ¼ 15 lm). The Hall parameter is 0.15–0.4. The shock wave decreases the Hall parameter sharply with the load voltage of 500 V. 5.3. Effects of velocity slip and temperature slip
Jx (A/m 2 )
Since there are no experimental data of Slip and DT , we have assumed the nominal values of Slip and DT to be 1.1 and 100 K, respectively, which are taken from other studies [4,6]. The effects 30000
30000
25000
25000
20000
20000
15000
15000
10000
10000 V=2550 V=2000 V=1500 V=1000 V= 500
5000 0 -5000 0
0.5 1
5000 0
1.5 2
2.5 3
3.5 4
-5000 4.5
x (m)
Fig. 20. Distribution of Hall current density (Jx ) (d ¼ 15 lm).
250 200
200 150
σ
150
250 V=2550 V=2000 V=1500 V=1000 V= 500
100
100
50
50
0 0
0.5
1
1.5
2
2.5
3
3.5
4
0 4.5
x (m)
Fig. 21. Distributions of electrical conductivity (d ¼ 15 lm).
β
M. Ishikawa et al. / Energy Conversion and Management 45 (2004) 707–724 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0
V=2550 V=2000 V=1500 V=1000 V= 500
0.5
1
1.5
2
2.5
3
3.5
4
723
0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 4.5
x(m)
Fig. 22. Distributions of Hall parameter (d ¼ 15 lm).
of Slip and DT have been examined, and it has been found that there is no large influence of Slip and DT on the generator performance. 6. Concluding remarks Two phase flow analyses have been performed using the one dimensional time dependence approximation with and without the induced magnetic field in order to study the influence of the induced magnetic field and liquid particles on the generator performance of the large pulsed MHD generator ‘‘SAKHALIN’’. The main results are as follows: (1) Since the ‘‘SAKHALIN’’ MHD generator has a very strong MHD interaction and the magnetic Reynolds number is about 0.58 for Run No. 1, the induced magnetic field cannot be neglected. The induced magnetic flux density is about 20% at the entrance and about 19% at the exit of the MHD channel. (2) If the induced magnetic field is taken into account, the magnetic field decreases and the deceleration of the flow decreases in the first half of the generator. In the second half, the magnetic field becomes large, the flow is decelerated strongly and the stronger shock wave is generated. (3) If alumina liquid particles are taken into account, the momentum and thermal energy are transferred from the liquid particles into the gas, and the shock wave becomes less sharp. (4) When the diameter of the alumina liquid particles is small (d ¼ 5 lm), the interaction of the liquid particles and gas becomes strong and almost the same behavior appears as for the mono phase flow. (5) Consideration of the alumina liquid particles reduces the electric power output about 5% (d ¼ 15 lm). This is because the energy of the liquid particles cannot be converted directly into electric energy. Acknowledgements Discussions with Prof. V.P. Panchenko, TRINITI, were very useful. KDK computers, Kyoto University were used for the numerical analyses in part.
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References [1] Zeigarnik VA, Ishikawa M. Proc of Int Conf on MHD Power Generation and High Temperature Technologies 1999;III:775–92. [2] Velikhov EP, et al. Proc of Int Conf on MHD Power Generation and High Temperature Technologies 1999;II:387– 98. [3] Ishikawa M, Koshiba Y. Proc of Sym on Advanced Research of Energy Technology 2001. p. 251–58. [4] Matsuo T, Sugita H, Ishikawa M. Trans IEE Jpn 1998;118-B(2):199–204 [in Japanese]. [5] Sugita H, Matsuo T, Inui Y, Ishikawa M. Trans IEE Jpn 1998;118-B(12):1374–9 [in Japanese]. [6] Sugita H, Matsuo T, Inui Y, Ishikawa M. Trans IEE Jpn 1999;119-B(6):682–8 [in Japanese]. [7] H. Sugita, T. Matsuo, Y. Inui, M. Ishikawa. 30th AIAA Plasmadynamics and Lasers Conference, AIAA 99-3483; 1999. [8] Matsuo T, Sugita H, Ishikawa M. Trans IEE Jpn 1999;119-B(6):690–5 [in Japanese]. [9] Tadamatsu A, Matsuo T, Ishikawa M. Technical Report FTE-00-8, New Energy and Environment TC, IEE Japan; 2000. p. 1–6 [in Japanese]. [10] Yakushev AA, Panchenko VP, et al. Proc of Int Conf on MHD Power Generation and High Temperature Technologies 1999;II:423–34. [11] Ishikawa M. Technical Report of Institute of Atomic Energy, Kyoto University, No. 192; 1982.