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MHD pressure drop in ferritic pipes of fusion blankets
Fusion Engineering and Design 69 (2003) 309 /313 www.elsevier.com/locate/fusengdes
MHD pressure drop in ferritic pipes of fusion blankets J. Reimann, Leo Bu¨hler *, K. Messadek, R. Stieglitz Forschungszentrum Karlsruhe, Postfach 3640, 76021 Karlsruhe, Germany
Abstract Magnetohydrodynamic flows in pipes of ferromagnetic material is an important issue for liquid metal blanket concepts using MANET as wall material. Fusion relevant magnetic fields of 4 /8 T cause high pressure drop in the blanket header where a massive structure of ferromagnetic material exists. It is briefly outlined that in the blanket the reduction of pressure drop due to magnetic shielding is limited to about 10%. Remarkable reduction of pressure drop is possible by means of electrical insulation that prevents currents from short-circuiting through the very thick walls of the headers. Direct contact of the insulating material with the liquid metal is excluded by using metallic liners. Results are reported on the fabrication of such a test section and corresponding pressure drop measurements confirm the effective contribution of the electrical decoupling. # 2003 Elsevier Science B.V. All rights reserved. Keywords: MHD; Thick walls; Ferromagnetic walls; Electric insulation
1. Introduction The major part of the pressure drop in the European water cooled lead lithium (WCLL) blanket arises in the headers, which supply the liquid breeder through small circular access tubes to the breeding zone that consists of large poloidal containers. Therefore, these parts are of prior importance. The fact that the structural material (MANET or EUROFER) has ferromagnetic properties up to mw /400 [1] reduces partly magnetohydrodynamic pressure losses due to magnetic shielding. However, since the magnetic fields in fusion applications are much higher than
* Corresponding author. Tel.: /49-7247-82-4397; fax: /497247-82-4837. E-mail address: [email protected] (L. Bu¨hler).
the magnetic saturation of the wall material this reduction is expected to be small. The ferromagnetic effect due to shielding of the MHD flow is associated with an effective magnetic flux field inside the pipe. If that field is known, classical solutions [2 /6] apply for flows in insulating or conducting pipes. The pressure drop in pipes is dominated by a balance with the Lorentz force exerted on the moving electrically conducting fluid, which is exposed to the effective magnetic field inside the pipe. Perfect electrical insulation at the pipe wall would exclude the additional path for currents through the highly conducting wall and minimize the current in the fluid and the associated MHD pressure drop. Unfortunately, there exists up to now no insulating layer that retains its electrical insulation when being exposed to flowing lead lithium alloys
0920-3796/03/$ - see front matter # 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0920-3796(03)00361-2
310
J. Reimann et al. / Fusion Engineering and Design 69 (2003) 309 /313
at high temperatures. For that reason a technique is selected were the insulation layer is separated by a thin sheet of metal from a direct contact with the fluid. If the metal sheet is kept thin its electric resistance is still high so that MHD pressure losses remain small. To demonstrate the feasibility of this approach a test section has been fabricated and tested in the MEKKA laboratory of the Forschungszentrum Karlsruhe [7]. In this paper we outline briefly the physical problem, describe the design and fabrication technique for the insulating test section and finally show experimental and theoretical results for the influence of ferromagnetism on pressure drop in different types of ducts.
2. Formulation The steady state fully developed inductionless MHD flow of a fluid with constant density r , conductivity s, and kinematic viscosity n through a pipe with inner radius L is governed by the nondimensional momentum equation: 1 2 9 vjB kx; ˆ Ha2
(1)
where v u(y; z) x; ˆ B, j, k , are the fluid velocity, magnetic field, current density, and magnitude of pressure gradient scaled by the average velocity v0, the magnitude of the externally applied magnetic field B0, the quantities sv0B0, and sv0B20, respectively. Ohm’s law gives the coupling with the fluid motion for a solenoidal current density with 9 ×/ j/0 as: jsr (9fvB):
(2)
In Eq. (2) sr is the conductivity of the (wall) material scaled by the conductivity s of the fluid and f stands for the electric potential scaled by Lv0B0. In the following we have sr /1 in the fluid and sr /0 in insulating materials or in the insulating atmosphere around the pipe. For the wall we generally have different values and denote them as sr /sw .
The Hartmann number: sffiffiffiffiffiffi s :103 104 Ha LB0 rn
(3)
accounts for the importance of electromagnetic forces compared with viscous friction. In the equations shown above it is assumed that the magnetic field is known. In fact we can determine the magnetic field a priori from the magnetic potential as B /9 /A. For our applications the magnetic potential has only a single component A A(y; z)xˆ along the duct axis which is determined by: 1 9× 9A 0; (4) mr with boundary conditions @ z A /1, @ y A /0 as y , z 0/. The relative magnetic permeability mr depends on the magnetization curve of the wall material. An analysis using magnetic coordinates [8] yields the major result that the velocity u depends on the distribution of electric and magnetic potential as: u
k B2
@f @A
:
(5)
The variable f is determined by the conductivity of the wall while A fixes the magnetic coordinates. A normalization of the velocity determines finally the pressure drop. The lowest values of pressure drop are obtained for an insulating layer between the fluid and the wall. For that case the pressure gradient becomes: k ak
3p 8Ha
:
(6)
For ak /1 we recover the well known relation for pressure drop in non-ferromagnetic insulating circular pipes. In ferromagnetic pipes the effective magnetic field is reduced by a factor ak compared with non-ferromagnetic materials, i.e. Beff /ak . For protection against the corrosive attack of the liquid metal the insulating layer is separated by a thin metal sheet of nondimensional thickness o from a direct contact with the fluid. Such an
J. Reimann et al. / Fusion Engineering and Design 69 (2003) 309 /313
arrangement changes the dependence of the pressure drop to values as: 1 R2 1 1 k a2k 1 ; (7) sw R2 1 where R /1/o stands for the outer radius of the conducting liner. The pressure gradient scales now proportional to a2k while for perfect insulation the variation with ak is linear. Small values for the pressure gradient can be obtained if R is close to unity, i.e. for o /1. In the experiment described below we have o /0.11. For cases without insulation R is the outer radius of the ferromagnetic pipe.
3. Test section A sketch of the test section is shown in Fig. 1. In the following we describe briefly how the test section was fabricated: tubes were cut from raw material and plates from EUROFER were machined; the tubes (upright) and plates annealed; five tubes selected for grinding to final diameter, the remaining wall thickness was 0.6 mm; three tubes selected for plasma spraying; 8 mm molybdenum primer applied; plasma sprayed with 300
Fig. 1. Drawing of the test section. Circular pipe with insulating layer of Al2O3 (gray) between two plates of steel.
311
mm Al2O3; grounded to final diameter; thickness of insulating layer tested with ultrasonic device; only one tube met the specification concerning straightness with a layer thickness of 235 mm; ends of tubes enlarged to minimize gap between tube and plates; helium leak test performed; plates machined without oil or coolant; semicircular grooves machined to the tube size; plates and tubes cleaned with ultra pure acetone; all pieces joined and fixed; evacuated in EB welding machine for more than 10 h; outer seams welded; tested; diffusion welded in HIP facility at 980 8C and 50 MPa; ultrasonic tests. Finally both ends of the test section were machined until the insulating layer was visible in order to avoid a current path at the ends of the insulating layer into the massive structure. Then the test section was welded to tubes with pressure tabs in the liquid metal circuit. For a comparison of results additional test sections without insulation (ferromagnetic and non-ferromagnetic) have been installed in the liquid metal loop and tested simultaneously in the large dipole magnet of the MEKKA facility [7].
4. Results Before connecting the test sections to the liquid metal loop they were installed inside the magnet. A magnetic sensor has been inserted into the bore and the magnetic field on the axes of the test sections has been recorded for various externally applied magnetic fields B0. The quantity a /B/B0 has been evaluated and plotted in Fig. 2. The results show that due to ferromagnetic properties of the test sections the field inside the pipes is reduced especially for magnetic fields below the magnetic saturation saturation Ms. We observe for the EUROFER test section a ratio of the field inside the pipe to the externally applied field of a /B /B0 :/0.62, which corresponds to a value of relative magnetic permeability of mr :/4.5. This value is far below the value published in [1] where values up to mr /400 are reported. The big difference can be explained by the fact that the material had been exposed to a high temperature near 980 8C for a certain period of time during the
312
J. Reimann et al. / Fusion Engineering and Design 69 (2003) 309 /313
Fig. 2. Ratio of the magnetic field inside the pipe to the externally applied field, a /B /B0.
fabrication of the test section. Nevertheless, weak ferromagnetic properties are still present and their influence can be investigated. For a comparison the measured value of a for a ferromagnetic test section (not exposed to a heat treatment during manufacturing) has been added to the figure. For the latter test section shielding is very efficient for B0 /Ms . For larger values of B0 shielding is weaker and the curve approaches that for EUROFER. For B0 /Ms the value of a for both test sections will approach unity. For fusion applications the reduction of the field in the test sections is estimated to be less than 5/10%. Results for pressure gradient in several types of ferromagnetic test sections had been calculated [8]. They are summarized in Fig. 3 and compared with experimental data. Instead of plotting directly the pressure gradients, the more general value ak is shown. Using this representation allows us to compare different conditions like the perfectly insulating case according to Eq. (6) or the conducting test section with and without the liner technique for which Eq. (7) applies. The measured values compare well with the theoretical predictions for the ferromagnetic (a /ak B/1) and for the non-ferromagnetic (a /ak /1) material. Unfortunately it was not possible to obtain direct values for pressure drop from the insulating test section since any pressure tap would have destroyed the insulation at least locally. For that
Fig. 3. Magnitude of effective magnetic field inside the ferromagnetic pipe depending on B0/Ms .
reason the pressure difference was measured between two points which are located close to the ends of the ferromagnetic block but outside. As a results the pressure difference Dp shown in Fig. 4 accounts for the pressure drop through the insulating test section of total length lins /433 mm plus that in the short pipes of total length (both ends together) lpipes /55 mm. The experimental results may also contain a certain fraction of pressure drop that is associated with 3D effects when the fluid enters or leaves the ferromagnetic part (fringing field). The latter part has not been considered for the theoretically obtained values shown in Fig. 4. The value of ak /a has been used in the pressure drop correlation (7) for the
Fig. 4. Pressure drop in the insulating test section. Open symbols denote experimental data.
J. Reimann et al. / Fusion Engineering and Design 69 (2003) 309 /313
ferromagnetic part, where a was used according to the data shown in Fig. 2. The agreement is qualitatively good. While for smaller magnetic fields the theoretical value slightly overestimates the experiment we find the opposite tendency for stronger fields. The differences between theoretical and experimental values seems acceptable concerning the large uncertainties that arise especially in this experiment (possibility that the insulation is good but not perfect, unknown 3D effects, etc.). The aim of the experiment was to show the capability of an insulating layer to reduce the pressure drop in massive structures. In fact, calculations show that the pressure drop for a similar test section without insulation would be at least higher by a factor of about 5 compared with the experimental or theoretical predictions shown in Fig. 4. Therefore, the observed reduction of pressure corresponds well to that expected for a perfect insulation between the liner and the thick wall.
5. Conclusions Theoretical and experimental investigations of MHD flows in pipes made of ferromagnetic material have been performed with respect to applications of such materials in liquid metal blankets for fusion reactors. The experimental data confirms the theoretical scaling laws which predict pressure drops proportional to a2k , where ak stands for the ratio of effective magnetic field inside the pipe to the externally applied field B0. For fusion relevant magnetic fields the shielding of the field inside ferromagnetic pipes is small. A reduction of pressure drop due to shielding is therefore, not significant. The pressure drop can be reduced more efficiently by an electrical decoupling of the liquid metal and the thick conducting walls. A technically feasible solution is the use of an insulating layer, which is separated from the fluid by a thin steel
313
liner. This technique does not reduce the pressure drop to the minimum value expected for an insulating layer in direct contact with the fluid. The big advantage, however, is that the problem of compatibility of the fluid and the layer is no longer a severe issue and even small cracks in the insulation are not crucial, since the liquid metal can not penetrate into the gaps. The technology for the fabrication of such a test section is not trivial but has been successfully tested. The pressure drop is reduced for the present case by a factor of about 5, which is expected for perfect insulation between the liner and the thick wall test section. For applications in fusion even higher values are possible if the diameter of the pipes are larger or the thickness of the liner is chosen smaller compared with the present experiment.
References [1] P. Ruatto, Entwiuklung einer Methode zur Berechnung der elektromagnetischen Kra˚fte durch Magnetfelda˚nderungen in ferromagntischen Strukturen und Anwendung dieser Methode auf den Plasmaabbruch in einem Tokamakreaktor. Technical Report FZKA 5683, Forschungszentrum Karlsruhe, 1996. [2] J.A. Shercliff, Steady motion of conducting fluids in pipes under transverse magnetic fields, Proc. Camb. Phil. Soc. 49 (1953) 136 /144. [3] R.R. Gold, Magnetohydrodynamic pipe flow. Part 1, J. Fluid Mech. 13 (1962) 505 /512. [4] J.A. Shercliff, The flow of conducting fluids in circular pipes under transverse magnetic fluids, J. Fluid Mech. 1 (1956) 644 /666. [5] J.A. Shercliff, Magnetohydrodynamic pipe flow. Part 2. High Hartmann number, J. Fluid Mech. 13 (1962) 513 /518. [6] C. Chang, S. Lundgren, Duct flow in magnetohydrodynamics, Zeitschrift fur angewandte Mathematik und Physik, XII (1961) 100 114. [7] L. Barleon, K.-J. Mack, U. Stieglitz, The MEKKA-facility */a flexible tool to investigate MHD-flow phenomena, Technical Report FZKA 5821, Forschungszentrum Karlsruhe, 1990. [8] L. Bu¨hler, K. Messadek, R. Stieglitz, Magnetohydrodynamic flow in ferromagnetic pipes, Fusion Eng. Des. 63 /64 (2002) 353 /359.
MHD pressure drop in ferritic pipes of fusion blankets J. Reimann, Leo Bu¨hler *, K. Messadek, R. Stieglitz Forschungszentrum Karlsruhe, Postfach 3640, 76021 Karlsruhe, Germany
Abstract Magnetohydrodynamic flows in pipes of ferromagnetic material is an important issue for liquid metal blanket concepts using MANET as wall material. Fusion relevant magnetic fields of 4 /8 T cause high pressure drop in the blanket header where a massive structure of ferromagnetic material exists. It is briefly outlined that in the blanket the reduction of pressure drop due to magnetic shielding is limited to about 10%. Remarkable reduction of pressure drop is possible by means of electrical insulation that prevents currents from short-circuiting through the very thick walls of the headers. Direct contact of the insulating material with the liquid metal is excluded by using metallic liners. Results are reported on the fabrication of such a test section and corresponding pressure drop measurements confirm the effective contribution of the electrical decoupling. # 2003 Elsevier Science B.V. All rights reserved. Keywords: MHD; Thick walls; Ferromagnetic walls; Electric insulation
1. Introduction The major part of the pressure drop in the European water cooled lead lithium (WCLL) blanket arises in the headers, which supply the liquid breeder through small circular access tubes to the breeding zone that consists of large poloidal containers. Therefore, these parts are of prior importance. The fact that the structural material (MANET or EUROFER) has ferromagnetic properties up to mw /400 [1] reduces partly magnetohydrodynamic pressure losses due to magnetic shielding. However, since the magnetic fields in fusion applications are much higher than
* Corresponding author. Tel.: /49-7247-82-4397; fax: /497247-82-4837. E-mail address: [email protected] (L. Bu¨hler).
the magnetic saturation of the wall material this reduction is expected to be small. The ferromagnetic effect due to shielding of the MHD flow is associated with an effective magnetic flux field inside the pipe. If that field is known, classical solutions [2 /6] apply for flows in insulating or conducting pipes. The pressure drop in pipes is dominated by a balance with the Lorentz force exerted on the moving electrically conducting fluid, which is exposed to the effective magnetic field inside the pipe. Perfect electrical insulation at the pipe wall would exclude the additional path for currents through the highly conducting wall and minimize the current in the fluid and the associated MHD pressure drop. Unfortunately, there exists up to now no insulating layer that retains its electrical insulation when being exposed to flowing lead lithium alloys
0920-3796/03/$ - see front matter # 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0920-3796(03)00361-2
310
J. Reimann et al. / Fusion Engineering and Design 69 (2003) 309 /313
at high temperatures. For that reason a technique is selected were the insulation layer is separated by a thin sheet of metal from a direct contact with the fluid. If the metal sheet is kept thin its electric resistance is still high so that MHD pressure losses remain small. To demonstrate the feasibility of this approach a test section has been fabricated and tested in the MEKKA laboratory of the Forschungszentrum Karlsruhe [7]. In this paper we outline briefly the physical problem, describe the design and fabrication technique for the insulating test section and finally show experimental and theoretical results for the influence of ferromagnetism on pressure drop in different types of ducts.
2. Formulation The steady state fully developed inductionless MHD flow of a fluid with constant density r , conductivity s, and kinematic viscosity n through a pipe with inner radius L is governed by the nondimensional momentum equation: 1 2 9 vjB kx; ˆ Ha2
(1)
where v u(y; z) x; ˆ B, j, k , are the fluid velocity, magnetic field, current density, and magnitude of pressure gradient scaled by the average velocity v0, the magnitude of the externally applied magnetic field B0, the quantities sv0B0, and sv0B20, respectively. Ohm’s law gives the coupling with the fluid motion for a solenoidal current density with 9 ×/ j/0 as: jsr (9fvB):
(2)
In Eq. (2) sr is the conductivity of the (wall) material scaled by the conductivity s of the fluid and f stands for the electric potential scaled by Lv0B0. In the following we have sr /1 in the fluid and sr /0 in insulating materials or in the insulating atmosphere around the pipe. For the wall we generally have different values and denote them as sr /sw .
The Hartmann number: sffiffiffiffiffiffi s :103 104 Ha LB0 rn
(3)
accounts for the importance of electromagnetic forces compared with viscous friction. In the equations shown above it is assumed that the magnetic field is known. In fact we can determine the magnetic field a priori from the magnetic potential as B /9 /A. For our applications the magnetic potential has only a single component A A(y; z)xˆ along the duct axis which is determined by: 1 9× 9A 0; (4) mr with boundary conditions @ z A /1, @ y A /0 as y , z 0/. The relative magnetic permeability mr depends on the magnetization curve of the wall material. An analysis using magnetic coordinates [8] yields the major result that the velocity u depends on the distribution of electric and magnetic potential as: u
k B2
@f @A
:
(5)
The variable f is determined by the conductivity of the wall while A fixes the magnetic coordinates. A normalization of the velocity determines finally the pressure drop. The lowest values of pressure drop are obtained for an insulating layer between the fluid and the wall. For that case the pressure gradient becomes: k ak
3p 8Ha
:
(6)
For ak /1 we recover the well known relation for pressure drop in non-ferromagnetic insulating circular pipes. In ferromagnetic pipes the effective magnetic field is reduced by a factor ak compared with non-ferromagnetic materials, i.e. Beff /ak . For protection against the corrosive attack of the liquid metal the insulating layer is separated by a thin metal sheet of nondimensional thickness o from a direct contact with the fluid. Such an
J. Reimann et al. / Fusion Engineering and Design 69 (2003) 309 /313
arrangement changes the dependence of the pressure drop to values as: 1 R2 1 1 k a2k 1 ; (7) sw R2 1 where R /1/o stands for the outer radius of the conducting liner. The pressure gradient scales now proportional to a2k while for perfect insulation the variation with ak is linear. Small values for the pressure gradient can be obtained if R is close to unity, i.e. for o /1. In the experiment described below we have o /0.11. For cases without insulation R is the outer radius of the ferromagnetic pipe.
3. Test section A sketch of the test section is shown in Fig. 1. In the following we describe briefly how the test section was fabricated: tubes were cut from raw material and plates from EUROFER were machined; the tubes (upright) and plates annealed; five tubes selected for grinding to final diameter, the remaining wall thickness was 0.6 mm; three tubes selected for plasma spraying; 8 mm molybdenum primer applied; plasma sprayed with 300
Fig. 1. Drawing of the test section. Circular pipe with insulating layer of Al2O3 (gray) between two plates of steel.
311
mm Al2O3; grounded to final diameter; thickness of insulating layer tested with ultrasonic device; only one tube met the specification concerning straightness with a layer thickness of 235 mm; ends of tubes enlarged to minimize gap between tube and plates; helium leak test performed; plates machined without oil or coolant; semicircular grooves machined to the tube size; plates and tubes cleaned with ultra pure acetone; all pieces joined and fixed; evacuated in EB welding machine for more than 10 h; outer seams welded; tested; diffusion welded in HIP facility at 980 8C and 50 MPa; ultrasonic tests. Finally both ends of the test section were machined until the insulating layer was visible in order to avoid a current path at the ends of the insulating layer into the massive structure. Then the test section was welded to tubes with pressure tabs in the liquid metal circuit. For a comparison of results additional test sections without insulation (ferromagnetic and non-ferromagnetic) have been installed in the liquid metal loop and tested simultaneously in the large dipole magnet of the MEKKA facility [7].
4. Results Before connecting the test sections to the liquid metal loop they were installed inside the magnet. A magnetic sensor has been inserted into the bore and the magnetic field on the axes of the test sections has been recorded for various externally applied magnetic fields B0. The quantity a /B/B0 has been evaluated and plotted in Fig. 2. The results show that due to ferromagnetic properties of the test sections the field inside the pipes is reduced especially for magnetic fields below the magnetic saturation saturation Ms. We observe for the EUROFER test section a ratio of the field inside the pipe to the externally applied field of a /B /B0 :/0.62, which corresponds to a value of relative magnetic permeability of mr :/4.5. This value is far below the value published in [1] where values up to mr /400 are reported. The big difference can be explained by the fact that the material had been exposed to a high temperature near 980 8C for a certain period of time during the
312
J. Reimann et al. / Fusion Engineering and Design 69 (2003) 309 /313
Fig. 2. Ratio of the magnetic field inside the pipe to the externally applied field, a /B /B0.
fabrication of the test section. Nevertheless, weak ferromagnetic properties are still present and their influence can be investigated. For a comparison the measured value of a for a ferromagnetic test section (not exposed to a heat treatment during manufacturing) has been added to the figure. For the latter test section shielding is very efficient for B0 /Ms . For larger values of B0 shielding is weaker and the curve approaches that for EUROFER. For B0 /Ms the value of a for both test sections will approach unity. For fusion applications the reduction of the field in the test sections is estimated to be less than 5/10%. Results for pressure gradient in several types of ferromagnetic test sections had been calculated [8]. They are summarized in Fig. 3 and compared with experimental data. Instead of plotting directly the pressure gradients, the more general value ak is shown. Using this representation allows us to compare different conditions like the perfectly insulating case according to Eq. (6) or the conducting test section with and without the liner technique for which Eq. (7) applies. The measured values compare well with the theoretical predictions for the ferromagnetic (a /ak B/1) and for the non-ferromagnetic (a /ak /1) material. Unfortunately it was not possible to obtain direct values for pressure drop from the insulating test section since any pressure tap would have destroyed the insulation at least locally. For that
Fig. 3. Magnitude of effective magnetic field inside the ferromagnetic pipe depending on B0/Ms .
reason the pressure difference was measured between two points which are located close to the ends of the ferromagnetic block but outside. As a results the pressure difference Dp shown in Fig. 4 accounts for the pressure drop through the insulating test section of total length lins /433 mm plus that in the short pipes of total length (both ends together) lpipes /55 mm. The experimental results may also contain a certain fraction of pressure drop that is associated with 3D effects when the fluid enters or leaves the ferromagnetic part (fringing field). The latter part has not been considered for the theoretically obtained values shown in Fig. 4. The value of ak /a has been used in the pressure drop correlation (7) for the
Fig. 4. Pressure drop in the insulating test section. Open symbols denote experimental data.
J. Reimann et al. / Fusion Engineering and Design 69 (2003) 309 /313
ferromagnetic part, where a was used according to the data shown in Fig. 2. The agreement is qualitatively good. While for smaller magnetic fields the theoretical value slightly overestimates the experiment we find the opposite tendency for stronger fields. The differences between theoretical and experimental values seems acceptable concerning the large uncertainties that arise especially in this experiment (possibility that the insulation is good but not perfect, unknown 3D effects, etc.). The aim of the experiment was to show the capability of an insulating layer to reduce the pressure drop in massive structures. In fact, calculations show that the pressure drop for a similar test section without insulation would be at least higher by a factor of about 5 compared with the experimental or theoretical predictions shown in Fig. 4. Therefore, the observed reduction of pressure corresponds well to that expected for a perfect insulation between the liner and the thick wall.
5. Conclusions Theoretical and experimental investigations of MHD flows in pipes made of ferromagnetic material have been performed with respect to applications of such materials in liquid metal blankets for fusion reactors. The experimental data confirms the theoretical scaling laws which predict pressure drops proportional to a2k , where ak stands for the ratio of effective magnetic field inside the pipe to the externally applied field B0. For fusion relevant magnetic fields the shielding of the field inside ferromagnetic pipes is small. A reduction of pressure drop due to shielding is therefore, not significant. The pressure drop can be reduced more efficiently by an electrical decoupling of the liquid metal and the thick conducting walls. A technically feasible solution is the use of an insulating layer, which is separated from the fluid by a thin steel
313
liner. This technique does not reduce the pressure drop to the minimum value expected for an insulating layer in direct contact with the fluid. The big advantage, however, is that the problem of compatibility of the fluid and the layer is no longer a severe issue and even small cracks in the insulation are not crucial, since the liquid metal can not penetrate into the gaps. The technology for the fabrication of such a test section is not trivial but has been successfully tested. The pressure drop is reduced for the present case by a factor of about 5, which is expected for perfect insulation between the liner and the thick wall test section. For applications in fusion even higher values are possible if the diameter of the pipes are larger or the thickness of the liner is chosen smaller compared with the present experiment.
References [1] P. Ruatto, Entwiuklung einer Methode zur Berechnung der elektromagnetischen Kra˚fte durch Magnetfelda˚nderungen in ferromagntischen Strukturen und Anwendung dieser Methode auf den Plasmaabbruch in einem Tokamakreaktor. Technical Report FZKA 5683, Forschungszentrum Karlsruhe, 1996. [2] J.A. Shercliff, Steady motion of conducting fluids in pipes under transverse magnetic fields, Proc. Camb. Phil. Soc. 49 (1953) 136 /144. [3] R.R. Gold, Magnetohydrodynamic pipe flow. Part 1, J. Fluid Mech. 13 (1962) 505 /512. [4] J.A. Shercliff, The flow of conducting fluids in circular pipes under transverse magnetic fluids, J. Fluid Mech. 1 (1956) 644 /666. [5] J.A. Shercliff, Magnetohydrodynamic pipe flow. Part 2. High Hartmann number, J. Fluid Mech. 13 (1962) 513 /518. [6] C. Chang, S. Lundgren, Duct flow in magnetohydrodynamics, Zeitschrift fur angewandte Mathematik und Physik, XII (1961) 100 114. [7] L. Barleon, K.-J. Mack, U. Stieglitz, The MEKKA-facility */a flexible tool to investigate MHD-flow phenomena, Technical Report FZKA 5821, Forschungszentrum Karlsruhe, 1990. [8] L. Bu¨hler, K. Messadek, R. Stieglitz, Magnetohydrodynamic flow in ferromagnetic pipes, Fusion Eng. Des. 63 /64 (2002) 353 /359.