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Stress calculations and computer simulations of high pulsed field coils
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Physica B 211 (1995) 77-80
Stress calculations and computer simulations of high pulsed field coils F. Bolzoni*, I. Suberbielle lstituto Maspec, Via Chiavari 18/A, 43100 Parma, Italy
Abstract
We have studied a magnet model for 100 T field generation formed by two single-layer concentric coils. They are made of soft material and are mechanically independent. The containment is given by high strength glass fiber Composite jackets. The radial and longitudinal pressures on the wire are taken equal in order to keep in equilibrium the donductive turns (isostatic pressure).
1. Introduction
The use of intense magnetic fields in the 50 T region by pulsed techniques is now a well-established practice. The new science that could be achieved in high fields would allow new developments in numerous scientific areas such as biology, chemistry and solid state physics, The final goal concerns the generation of fields up to 100 T with duration of at least 1 s. It is clear that at these field levels a major limitation is imposed by the ability of the coil material to withstand the Lorentz forces generated within. That could be overcome since nonconductive composite materials exist suitable to contain the pressure generated in 100 T magnets. The calculations of the stress distribution in field producing coils have received considerable attention. For the coil with uniform current density, the stress at field level near 70 T exceeds the limits of presently available conductors with highest mechanical strength [1 2]. The situation is slightly better for coils with optimized current density distribution [3]. Similar results were given on coils with constant stress obtained by a proper reinforcement distribution [4].
The present work simulates by computer a coil, with dimensions allowing axial and radial pressures on turns to be equal (isostatic magnet), that would reach 100 T without failure.
2. Stress calculation
For the calculation of the axial and radial field at any point of a coil, a current sheet approximation iS used. The field axial and radial components at the point (z, r) are:
Hzlz,r)=
* Corresponding author. 0921-4526,/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0 9 2 1 - 4 5 2 6 ( 9 4 ) 0 0 9 4 8 - 1
f
RtR-rcosl
\LJ4Tc Jo R 2 + r 2 -- 2Rrcos~0 I
L - 2z
x iR 2 + r2 _ 2 R ~ c ~ s ~ - + (L/2 -- z)2) 1/2
L+2z
]
(R 2 + r z -- 2Rrcos ~0 + ( -- L/2 -- z)2) liz d~o, (1)
F. Bolzoni, 1. Suberbielle / Physica B 211 (1995) 77-80
78
ratio, respectively; the ratio dependence law is deduced from computer calculations:
,,,¢z, r)= \ L J 4~ Jo 2R cos ~o
x
[iR 2 + r 2 -
1
2Rrcos~o + ( -
Pa/P~= exp
L/2 - z)2)1/2
'
-(R2+r2_2Rrcos~o+(L/2_z)2)l/2
1 d~0,
(2)
where r and z are the radial and longitudinal coordinates, respectively, ~o the azimuthal angle, R the current sheet radius and L the coil length. In the following we consider a soft wire single-layer coil (Fig. 1) contained by a high strength jacket made of glass fiber composite. The radial magnetic pressure in the coil (in 1/#o units) is
NI fRt~H~(r, z) dr, P~(z) L(ge -- gi) i
(3)
where N is the number of turns, Re and Ri the outer and inner radii, respectively. The packing factor is included in the current 1 for simplicity. The axial pressure which direction is toward the magnet center, in the same units as before, is
gI Pa(z) L(R-~~- Ri)
f:
/2
(H~(z)) dz.
(4)
As the pressure into the layer is constant with respect to r except at the coil extremities, we had substituted H~(r, z) with the average field over the layer thickness (H~(z)). A computer program had been realized to calculate in any point of a particular magnet the radial and axial components of the magnetic field and of the pressure. The behavior of the axial to radial pressure ratio is obtained as a function of ~ and fl, which are the external to internal radii ratio, and the length to internal diameter
H
1
;wind
jacket
R ~
R
Fig. 1. Schematic diagram of a monolayer coil showing the involved pressure components.
- 1.
(5)
The relation is obtained in the interval 1.5 ~ fl ~ 5 where K is almost constant and found K = 0.75 + 0.0091ogfl. In particular for having Pa = P,,
(6)
3. lsostatic magnet Both optimized current density distribution and constant stress magnets would reach very large dimensions in the 100 T region. To generate this field a large amount of energy is needed which is not easily available. To reduce the stored energy in the magnet high conductivity wires making few layers have to be used. It is obvious that their strength is not sufficient to contain the magnetic pressure and of course a special strong jacket is necessary. This prevents the radial deformation of the turns, but the axial one is still allowed. To avoid it, a convenient choice of the magnet dimensions can give an axial contribution to the containment. For the soft wire case the radial and longitudinal pressures are taken equal on it in order to keep in equilibrium the conductive turns (isostatic pressure). The insulating layer between adjacent turns is subjected only to the axial pressure, to avoid its flowing a planar anisotropic insulation has to be used. The simulated magnets are basically constituted of few independent coils, each fed by a different pulsed source. As a first approach a high conductive wire with negligible yield strength is used, the containment is given by a high strength glass-fiber composite jacket. Many different ways were possible for the magnet geometry, among them two seem to be the most interesting: (i) all coils have the same r, thus same field homogeneity, and are independent of one another as far as axial pressure is concerned; (ii) all concentric coils have the same length, in this case the external coils increase substantially the axial force on the inner coils. To keep the pressure isostatic on turns, it is required to increase the wire thickness. Consequently the augmentation of the coil volume allows a longer pulse with the same final temperature, but the outer coil field homogeneity gets worse. We present simulations of a magnet that is constituted by two single-layer coils, mechanically decoupled, which have the same field homogeneity, i.e.
79
F. Bolzoni, L Suberbielle / Physica B 211 (1995) 77-80
220 180 -
~ ~ ' - . , .
,,.
140 -
\
10060-
i
20 0
5
lo
iI
15
20
25
225
30
Z (ram)
Fig. 3. Axial (solid line) and radial (dashed line)I pressure P (in l/N21211 o units) versus z-position on turns of the inner coil. P 12-
..............
..._.....!
108=
!
6~
""-,
420
Fig. 2. Schematic representation of the simulated magnet geometry. All dimensions are given in mm.
fl -- constant. The inner and the outer coils would generate fields up to 61 and 39 T, respectively. The coil dimensions are shown on Fig. 2. The axial (solid line) and radial (dashed line) pressures (in 1/N212#o units) on the inner coil are shown in Fig. 3. At the coil center the axial contribution is exactly equal to the radial one while the difference between them increases and reaches its maximum at the coil end. It is possible to increase the axial contribution to have a compromise in the whole range. The axial (solid line) and radial (dashed line) pressures (in 1/N2lZpo units) on the outer coil are shown in Fig. 4. The behavior of the two pressure components are different, in particular the m a x i m u m of the radial pressure is not at the coil center as before. Both components show inflection points corresponding to the inner coil end (vertical line). A compromise was adopted for the outer coil width. For the calculation of the radial (at) and tangential (ao) stress of the nonconductive jacket, the pressure vessel
0
2'0
40
i
i
fl
60
80
100
120
Z (mm)
Fig. 4. Axial (solid line) and radial (dashed linet pressure P (in 1/NZlZ#o units) versus z-position on turns of the outer coil; the vertical line corresponds to the inner coil end. approximation is used: O"r
P -- 1 + ( R J R i ) 2(1
(R~/R)2),
P a° = - l + ( R J R i ) z(1 + (Re~R)2)"
(7)
In particular, af = - P for R = Ri, while for calculating the width of the inner and outer jackets We assumed aom~x = 2 G P that is a good value for high strength glass fiber composite. The field pulse duration is limited by the maximum current density in coils causing nondamagirig adiabatic heating. Temperature rises from the liquid nihtrogen temperature in which the coils are immersed io the final allowed temperature, under adiabatic conditions, according to the action integral I-3]. The puise duration
80
F. BolzonL 1. Suberbielle / Physica B 211 (1995) 77-80
depends on the a m o u n t of the coil material at a constant current. In particular for the internal coil the rising time (180 las) is shorter than that of the outer one (28 ms) when final temperature is room temperature. Using one single power supply the magnet rising time is limited by the inner coil, while this limitation does not exist if two power sources with two different rising times are used. Practical realization of the magnets is actually in progress.
References [1] S. Foner, Appl. Phys. Lett. 49 (1986) 982. [2] R. Gersdorf, F.A. Muller and L.W. Roeland, Rev. Sci. Instrum. 36 (1965) 1100. [3] S. Askenazy, L. Von Bockstal, F. Hedach and H.-J. Schneider-Muntau, Meas. Sci. Technol. 4 (1993) 1058. [4] L. Van Bockstal, Physica B 177 (1992) 31.
Physica B 211 (1995) 77-80
Stress calculations and computer simulations of high pulsed field coils F. Bolzoni*, I. Suberbielle lstituto Maspec, Via Chiavari 18/A, 43100 Parma, Italy
Abstract
We have studied a magnet model for 100 T field generation formed by two single-layer concentric coils. They are made of soft material and are mechanically independent. The containment is given by high strength glass fiber Composite jackets. The radial and longitudinal pressures on the wire are taken equal in order to keep in equilibrium the donductive turns (isostatic pressure).
1. Introduction
The use of intense magnetic fields in the 50 T region by pulsed techniques is now a well-established practice. The new science that could be achieved in high fields would allow new developments in numerous scientific areas such as biology, chemistry and solid state physics, The final goal concerns the generation of fields up to 100 T with duration of at least 1 s. It is clear that at these field levels a major limitation is imposed by the ability of the coil material to withstand the Lorentz forces generated within. That could be overcome since nonconductive composite materials exist suitable to contain the pressure generated in 100 T magnets. The calculations of the stress distribution in field producing coils have received considerable attention. For the coil with uniform current density, the stress at field level near 70 T exceeds the limits of presently available conductors with highest mechanical strength [1 2]. The situation is slightly better for coils with optimized current density distribution [3]. Similar results were given on coils with constant stress obtained by a proper reinforcement distribution [4].
The present work simulates by computer a coil, with dimensions allowing axial and radial pressures on turns to be equal (isostatic magnet), that would reach 100 T without failure.
2. Stress calculation
For the calculation of the axial and radial field at any point of a coil, a current sheet approximation iS used. The field axial and radial components at the point (z, r) are:
Hzlz,r)=
* Corresponding author. 0921-4526,/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0 9 2 1 - 4 5 2 6 ( 9 4 ) 0 0 9 4 8 - 1
f
RtR-rcosl
\LJ4Tc Jo R 2 + r 2 -- 2Rrcos~0 I
L - 2z
x iR 2 + r2 _ 2 R ~ c ~ s ~ - + (L/2 -- z)2) 1/2
L+2z
]
(R 2 + r z -- 2Rrcos ~0 + ( -- L/2 -- z)2) liz d~o, (1)
F. Bolzoni, 1. Suberbielle / Physica B 211 (1995) 77-80
78
ratio, respectively; the ratio dependence law is deduced from computer calculations:
,,,¢z, r)= \ L J 4~ Jo 2R cos ~o
x
[iR 2 + r 2 -
1
2Rrcos~o + ( -
Pa/P~= exp
L/2 - z)2)1/2
'
-(R2+r2_2Rrcos~o+(L/2_z)2)l/2
1 d~0,
(2)
where r and z are the radial and longitudinal coordinates, respectively, ~o the azimuthal angle, R the current sheet radius and L the coil length. In the following we consider a soft wire single-layer coil (Fig. 1) contained by a high strength jacket made of glass fiber composite. The radial magnetic pressure in the coil (in 1/#o units) is
NI fRt~H~(r, z) dr, P~(z) L(ge -- gi) i
(3)
where N is the number of turns, Re and Ri the outer and inner radii, respectively. The packing factor is included in the current 1 for simplicity. The axial pressure which direction is toward the magnet center, in the same units as before, is
gI Pa(z) L(R-~~- Ri)
f:
/2
(H~(z)) dz.
(4)
As the pressure into the layer is constant with respect to r except at the coil extremities, we had substituted H~(r, z) with the average field over the layer thickness (H~(z)). A computer program had been realized to calculate in any point of a particular magnet the radial and axial components of the magnetic field and of the pressure. The behavior of the axial to radial pressure ratio is obtained as a function of ~ and fl, which are the external to internal radii ratio, and the length to internal diameter
H
1
;wind
jacket
R ~
R
Fig. 1. Schematic diagram of a monolayer coil showing the involved pressure components.
- 1.
(5)
The relation is obtained in the interval 1.5 ~ fl ~ 5 where K is almost constant and found K = 0.75 + 0.0091ogfl. In particular for having Pa = P,,
(6)
3. lsostatic magnet Both optimized current density distribution and constant stress magnets would reach very large dimensions in the 100 T region. To generate this field a large amount of energy is needed which is not easily available. To reduce the stored energy in the magnet high conductivity wires making few layers have to be used. It is obvious that their strength is not sufficient to contain the magnetic pressure and of course a special strong jacket is necessary. This prevents the radial deformation of the turns, but the axial one is still allowed. To avoid it, a convenient choice of the magnet dimensions can give an axial contribution to the containment. For the soft wire case the radial and longitudinal pressures are taken equal on it in order to keep in equilibrium the conductive turns (isostatic pressure). The insulating layer between adjacent turns is subjected only to the axial pressure, to avoid its flowing a planar anisotropic insulation has to be used. The simulated magnets are basically constituted of few independent coils, each fed by a different pulsed source. As a first approach a high conductive wire with negligible yield strength is used, the containment is given by a high strength glass-fiber composite jacket. Many different ways were possible for the magnet geometry, among them two seem to be the most interesting: (i) all coils have the same r, thus same field homogeneity, and are independent of one another as far as axial pressure is concerned; (ii) all concentric coils have the same length, in this case the external coils increase substantially the axial force on the inner coils. To keep the pressure isostatic on turns, it is required to increase the wire thickness. Consequently the augmentation of the coil volume allows a longer pulse with the same final temperature, but the outer coil field homogeneity gets worse. We present simulations of a magnet that is constituted by two single-layer coils, mechanically decoupled, which have the same field homogeneity, i.e.
79
F. Bolzoni, L Suberbielle / Physica B 211 (1995) 77-80
220 180 -
~ ~ ' - . , .
,,.
140 -
\
10060-
i
20 0
5
lo
iI
15
20
25
225
30
Z (ram)
Fig. 3. Axial (solid line) and radial (dashed line)I pressure P (in l/N21211 o units) versus z-position on turns of the inner coil. P 12-
..............
..._.....!
108=
!
6~
""-,
420
Fig. 2. Schematic representation of the simulated magnet geometry. All dimensions are given in mm.
fl -- constant. The inner and the outer coils would generate fields up to 61 and 39 T, respectively. The coil dimensions are shown on Fig. 2. The axial (solid line) and radial (dashed line) pressures (in 1/N212#o units) on the inner coil are shown in Fig. 3. At the coil center the axial contribution is exactly equal to the radial one while the difference between them increases and reaches its maximum at the coil end. It is possible to increase the axial contribution to have a compromise in the whole range. The axial (solid line) and radial (dashed line) pressures (in 1/N2lZpo units) on the outer coil are shown in Fig. 4. The behavior of the two pressure components are different, in particular the m a x i m u m of the radial pressure is not at the coil center as before. Both components show inflection points corresponding to the inner coil end (vertical line). A compromise was adopted for the outer coil width. For the calculation of the radial (at) and tangential (ao) stress of the nonconductive jacket, the pressure vessel
0
2'0
40
i
i
fl
60
80
100
120
Z (mm)
Fig. 4. Axial (solid line) and radial (dashed linet pressure P (in 1/NZlZ#o units) versus z-position on turns of the outer coil; the vertical line corresponds to the inner coil end. approximation is used: O"r
P -- 1 + ( R J R i ) 2(1
(R~/R)2),
P a° = - l + ( R J R i ) z(1 + (Re~R)2)"
(7)
In particular, af = - P for R = Ri, while for calculating the width of the inner and outer jackets We assumed aom~x = 2 G P that is a good value for high strength glass fiber composite. The field pulse duration is limited by the maximum current density in coils causing nondamagirig adiabatic heating. Temperature rises from the liquid nihtrogen temperature in which the coils are immersed io the final allowed temperature, under adiabatic conditions, according to the action integral I-3]. The puise duration
80
F. BolzonL 1. Suberbielle / Physica B 211 (1995) 77-80
depends on the a m o u n t of the coil material at a constant current. In particular for the internal coil the rising time (180 las) is shorter than that of the outer one (28 ms) when final temperature is room temperature. Using one single power supply the magnet rising time is limited by the inner coil, while this limitation does not exist if two power sources with two different rising times are used. Practical realization of the magnets is actually in progress.
References [1] S. Foner, Appl. Phys. Lett. 49 (1986) 982. [2] R. Gersdorf, F.A. Muller and L.W. Roeland, Rev. Sci. Instrum. 36 (1965) 1100. [3] S. Askenazy, L. Von Bockstal, F. Hedach and H.-J. Schneider-Muntau, Meas. Sci. Technol. 4 (1993) 1058. [4] L. Van Bockstal, Physica B 177 (1992) 31.