Transcript Slide 1

CERN Accelerator School
Superconductivity for Accelerators
Case study 1
Paolo Ferracin
([email protected])
European Organization for Nuclear Research (CERN)
Case study 1
Low-beta Nb3Sn quadrupoles for the HL-LHC
Introduction
LARGE HADRON COLLIDER (LHC) it will run at 6.5-7 TeV, providing 300 fb1 of integrated luminosity within the end of the decade.
After 2020, CERN is planning to have an upgrade of the LHC to obtain
ten times more integrated luminosity, i.e., 3000 fb-1 .
Part of the upgrade relies on reducing the beam sizes in the Interaction
Points (IPs), by increasing the aperture of the present triplets.
Currently, the LHC interaction regions feature NbTi quadrupole magnets
with a 70 mm aperture and a gradient of 200 T/m.
Goal
Design a Nb3Sn superconducting quadrupole with an 150 mm aperture
for the upgrade of the LHC interaction region operating at 1.9 K
Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013
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Case study 1
Low-beta Nb3Sn quadrupoles for the HL-LHC
Questions
1.
2.
Determine maximum gradient and coil size (using sector coil scaling laws)
Define strands and cable parameters
1.
2.
3.
4.
3.
Determine load-line (no iron) and “short sample” conditions
1.
4.
2.
6.
7.
8.
Compute jsc_ss , jo_ss , Iss , Gss , Bpeak_ss
Determine “operational” conditions (80% of Iss ) and margins
1.
5.
Strand diameter and number of strands
Cu to SC ratio and pitch angle
Cable width, cable mid-thickness and insulation thickness
Filling factor κ
Compute jsc_op, jo_op , Iop , Gop , Bpeak_op
Compute T, jsc , Bpeak margins
Compare “short sample”, “operational” conditions and margins if the same
design uses Nb-Ti superconducting technology
Define a possible coil lay-out to minimize field errors
Determine e.m forces Fx and Fy and the accumulated stress on the coil midplane in the operational conditions (80% of Iss )
Evaluate dimension iron yoke, collars and shrinking cylinder, assuming that
the support structure is designed to reach 90% of Iss
Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013
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Case study 1
Additional questions
Evaluate, compare, discuss, take a stand (… and justify it …)
regarding the following issues
High temperature superconductor: YBCO vs. Bi2212
Superconducting coil design: block vs. cos
Support structures: collar-based vs. shell-based
Assembly procedure: high pre-stress vs. low pre-stress
Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013
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Case study 1
Low-beta Nb3Sn quadrupoles for the HL-LHC
Questions
1.
2.
Determine maximum gradient and coil size (using sector coil scaling laws)
Define strands and cable parameters
1.
2.
3.
4.
3.
Determine load-line (no iron) and “short sample” conditions
1.
4.
2.
6.
7.
8.
Compute jsc_ss , jo_ss , Iss , Gss , Bpeak_ss
Determine “operational” conditions (80% of Iss ) and margins
1.
5.
Strand diameter and number of strands
Cu to SC ratio and pitch angle
Cable width, cable mid-thickness and insulation thickness
Filling factor κ
Compute jsc_op, jo_op , Iop , Gop , Bpeak_op
Compute T, jsc , Bpeak margins
Compare “short sample”, “operational” conditions and margins if the same
design uses Nb-Ti superconducting technology
Define a possible coil lay-out to minimize field errors
Determine e.m forces Fx and Fy and the accumulated stress on the coil midplane in the operational conditions (80% of Iss )
Evaluate dimension iron yoke, collars and shrinking cylinder, assuming that
the support structure is designed to reach 90% of Iss
Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013
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Case study 1 solution
Maximum gradient and coil size
The max. gradient that one could reach is almost 200 T/m
…but with a w/r = 2
150 mm thick coil!
Nb3Sn at 1.9 K
300
Gradient [T /m]
250
200
150
w/r=0.25
w/r=0.5
w/r=1
w/r=2
100
50
0
50
75
100
125
Aperture [mm]
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150
175
200
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Case study 1 solution
Maximum gradient and coil size
1.5
ISR MQ
SSC MQ
RHIC MQ
LHC MQM
LHC MQXA
l [adim]
1.4
1.3
TEV MQ
LEP I MQC
RHIC MQY
LHC MQY
HERA MQ
LEP II MQC
LHC MQ
LHC MQXB
1.2
1.1
current grading
1.0
0.0
0.5
1.0
1.5
aspect ratio w eq/r (adim)
2.0
Large aperture need smaller ratio w/r
For r=30-100 mm, no need of having w>r
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Case study 1 solution
Maximum gradient and coil size
We assume a value of w/r = 0.5
37 mm thick coil
We should get a maximum gradient around 170 T/m
Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013
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Case study 1
Low-beta Nb3Sn quadrupoles for the HL-LHC
Questions
1.
2.
Determine maximum gradient and coil size (using sector coil scaling laws)
Define strands and cable parameters
1.
2.
3.
4.
3.
Determine load-line (no iron) and “short sample” conditions
1.
4.
2.
6.
7.
8.
Compute jsc_ss , jo_ss , Iss , Gss , Bpeak_ss
Determine “operational” conditions (80% of Iss ) and margins
1.
5.
Strand diameter and number of strands
Cu to SC ratio and pitch angle
Cable width, cable mid-thickness and insulation thickness
Filling factor κ
Compute jsc_op, jo_op , Iop , Gop , Bpeak_op
Compute T, jsc , Bpeak margins
Compare “short sample”, “operational” conditions and margins if the same
design uses Nb-Ti superconducting technology
Define a possible coil lay-out to minimize field errors
Determine e.m forces Fx and Fy and the accumulated stress on the coil midplane in the operational conditions (80% of Iss )
Evaluate dimension iron yoke, collars and shrinking cylinder, assuming that
the support structure is designed to reach 90% of Iss
Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013
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Case study 1 solution
Cable and strand size
We assume a strand
diameter of 0.85 mm
Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013
We assume a pitch angle 
of 18
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Case study 1 solution
Cable and strand size
We assume
Thick. Comp. = -12 %
Width. Comp. = -3 %
40 strands
Ins. Thick. = 150 μm
We obtain
Cable width: 18 mm
Cable mid-thick.: 1.5 mm
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Case study 1 solution
Cable and strand size
Summary
Strand diameter = 0.85 mm
Cu to SC ratio = 1.2
Pitch angle  = 18
N strands = 40
Cable width: 18 mm
Cable mid-thickness: 1.5 mm
Insulation thickness = 150 μm
Area insulated conductor = 32.7 mm2
We obtain a filling factor
k = area superconductor/area insulated cable = 0.32
Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013
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Case study 1
Low-beta Nb3Sn quadrupoles for the HL-LHC
Questions
1.
2.
Determine maximum gradient and coil size (using sector coil scaling laws)
Define strands and cable parameters
1.
2.
3.
4.
3.
Determine load-line (no iron) and “short sample” conditions
1.
4.
2.
6.
7.
8.
Compute jsc_ss , jo_ss , Iss , Gss , Bpeak_ss
Determine “operational” conditions (80% of Iss ) and margins
1.
5.
Strand diameter and number of strands
Cu to SC ratio and pitch angle
Cable width, cable mid-thickness and insulation thickness
Filling factor κ
Compute jsc_op, jo_op , Iop , Gop , Bpeak_op
Compute T, jsc , Bpeak margins
Compare “short sample”, “operational” conditions and margins if the
same design uses Nb-Ti superconducting technology
Define a possible coil lay-out to minimize field errors
Determine e.m forces Fx and Fy and the accumulated stress on the coil midplane in the operational conditions (80% of Iss )
Evaluate dimension iron yoke, collars and shrinking cylinder, assuming that
the support structure is designed to reach 90% of Iss
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Case study 1 solution
Margins
Let’s work now on the load-line
2 jo0
The gradient is given by
sin 60ln1  w 
G

 r
2
So, for a Jsc= 1600 A/mm
jo = jsc * k = 512 A/mm2
G = 142 T/m
Bpeak = G * r * λ = 142 * 75e-3 * 1.15 = 12.2 T
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Case study 1 solution
Margins
Nb3Sn parameterization
Temperature, field, and strain dependence of Jc is given by Summers’
formula
2
2 2

C Nb Sn   

 T  
B
J C B , T ,   
1  B T ,    1   T    
B 
C2
   C 0  
3
2
2
 T  
 T  
BC 2 T ,     T   


  1  0.31
 1  1.77ln
 
 1  
BC 20
  TC 0     
 TC 0    
 TC 0    



C Nb3Sn    C Nb3Sn ,0 1   Nb3Sn 

1  
BC 20    BC 20 m 1   Nb3 Sn 
TC 0    TC 0m
Nb3 Sn


1.7 1 / 2


1.7
1.7 1 / 3
where Nb3Sn is 900 for  = -0.003, TCmo is 18 K, BCmo is 24 T, and CNb3Sn,0 is a
fitting parameter equal to 60800 AT1/2mm-2 for a Jc=3000 A/mm2 at 4.2 K
and 12 T.
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Case study 1 solution
Margins
Nb-Ti parameterization
Temperature and field dependence of BC2 and TC are provided by
Lubell’s formulae:
  T 1.7 
 
BC 2 T   BC 20 1  
T
  C 0  
TC B 
1 / 1.7
  B 1/1.7 
 
 TC 0 1  
  BC 20  
where BC20 is the upper critical flux density at zero temperature
(~14.5 T), and TC0 is critical temperature at zero field (~9.2 K)
Temperature and field dependence of Jc is given by Bottura’s
formula
J C B, T  C NbTi

J C , ref
B
 B 
 B (T ) 
 C2

 NbTi

B 
1

 B (T ) 
C2


 NbTi
  T 1.7 
 
1  
T
  C 0  
 NbTi
where JC,Ref is critical current density at 4.2 K and 5 T (~3000
A/mm2) and CNb-Ti (27 T), Nb-Ti (0.63), Nb-Ti (1.0), and Nb-Ti (2.3) are
fitting parameters.
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Case study 1 solution
Margins Nb3Sn
Let’s assume  = 0.000
The load-line intercept the
critical (“short-sample”
conditions) curve at
jsc_ss = 1970 mm2
jo_ss = jsc_ss * k = 630 mm2
Iss = jo_ss * Ains_cable= 20600 A
Gss = 175 T/m
Bpeak_ss = 15 T
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Case study 1 solution
Margins Nb3Sn
The operational conditions
(80% of Iss)
jsc_op = 1580 mm2
jo_op = jsc_op * k = 505 mm2
Iop = 16480 A
Gop = 140 T/m
Bpeak_op = 12.1 T
Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013
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Case study 1 solution
Margins Nb3Sn
In the operational
conditions (80% of Iss)
4.6 K of T margin
(4000-1580) A/mm2 of jsc
margin
(15.8-12.1) T of field margin
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Case study 1 solution
Margins Nb-Ti
“Short-sample” conditions
jsc_ss = 1350 mm2
jo_ss = jsc_ss * k = 430 mm2
Iss = jo_ss * Ains_cable= 14060 A
Gss = 119 T/m
Bpeak_ss = 10.3 T
The operational conditions
(80% of Iss)
jsc_op = 1080mm2
jo_op = jsc_op * k = 344 mm2
Iop = 11250 A
Gop = 95 T/m
Bpeak_op = 8.2 T
2.1 K of T margin
(2370-1080) A/mm2 of jsc
margin
(11-8.2) T of field margin
Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013
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Case study 1
Low-beta Nb3Sn quadrupoles for the HL-LHC
Questions
1.
2.
Determine maximum gradient and coil size (using sector coil scaling laws)
Define strands and cable parameters
1.
2.
3.
4.
3.
Determine load-line (no iron) and “short sample” conditions
1.
4.
2.
6.
7.
8.
Compute jsc_ss , jo_ss , Iss , Gss , Bpeak_ss
Determine “operational” conditions (80% of Iss ) and margins
1.
5.
Strand diameter and number of strands
Cu to SC ratio and pitch angle
Cable width, cable mid-thickness and insulation thickness
Filling factor κ
Compute jsc_op, jo_op , Iop , Gop , Bpeak_op
Compute T, jsc , Bpeak margins
Compare “short sample”, “operational” conditions and margins if the same
design uses Nb-Ti superconducting technology
Define a possible coil lay-out to minimize field errors
Determine e.m forces Fx and Fy and the accumulated stress on the coil midplane in the operational conditions (80% of Iss )
Evaluate dimension iron yoke, collars and shrinking cylinder, assuming that
the support structure is designed to reach 90% of Iss
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Case study 1 solution
Coil layout
One wedge coil sets to zero b6 and b10 in quadrupoles
~[0°-24°, 30°-36°]
~[0°-18°, 22°-32°]
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0
y (mm)
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Some examples
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Case study 1
Low-beta Nb3Sn quadrupoles for the HL-LHC
Questions
1.
2.
Determine maximum gradient and coil size (using sector coil scaling laws)
Define strands and cable parameters
1.
2.
3.
4.
3.
Determine load-line (no iron) and “short sample” conditions
1.
4.
2.
6.
7.
8.
Compute jsc_ss , jo_ss , Iss , Gss , Bpeak_ss
Determine “operational” conditions (80% of Iss ) and margins
1.
5.
Strand diameter and number of strands
Cu to SC ratio and pitch angle
Cable width, cable mid-thickness and insulation thickness
Filling factor κ
Compute jsc_op, jo_op , Iop , Gop , Bpeak_op
Compute T, jsc , Bpeak margins
Compare “short sample”, “operational” conditions and margins if the same
design uses Nb-Ti superconducting technology
Define a possible coil lay-out to minimize field errors
Determine e.m forces Fx and Fy and the accumulated stress on the coil midplane in the operational conditions (80% of Iss )
Evaluate dimension iron yoke, collars and shrinking cylinder, assuming that
the support structure is designed to reach 90% of Iss
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Case study 1 solution
E.m. forces and stresses
For a quadrupole sector coil, with an inner radius a1, an
outer radius a2 and an overall current density jo , each block
(octant) see
Horizontal force outwards
Fx  
2 0 J 02

3
12
 1 12a 24  36a14  a1 1  3 
  ln
 a1 

a2
 72
 a 2 3  
Vertical force towards the mid-plan
Fy  
2 0 J 02

3
2
 5  2 3 3 1 a14 2  3 a1 3 1  3
 3

a1 
a


ln
a


2

2
1

 
36
12
a
6
a
9
2

2
2

 
In case of frictionless and “free-motion” conditions, no shear, and
infinitely rigid radial support, the forces accumulated on the midplane produce a stress of
  _ mid  plane 
 /6
 f rd  
0
Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013
20 J 02

3 
a2 r 4  a14 

r  r ln 
8 
r
4r 3 
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Case study 1 solution
E.m. forces and stresses
In the operational conditions (140 T/m)
Fx (octant) = +1.90 MN/m
Fy (octant) = -4.02 MN/m
The accumulates stress on the coil mid-plane is
Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013
Case study 1
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Case study 1
Low-beta Nb3Sn quadrupoles for the HL-LHC
Questions
1.
2.
Determine maximum gradient and coil size (using sector coil scaling laws)
Define strands and cable parameters
1.
2.
3.
4.
3.
Determine load-line (no iron) and “short sample” conditions
1.
4.
2.
6.
7.
8.
Compute jsc_ss , jo_ss , Iss , Gss , Bpeak_ss
Determine “operational” conditions (80% of Iss ) and margins
1.
5.
Strand diameter and number of strands
Cu to SC ratio and pitch angle
Cable width, cable mid-thickness and insulation thickness
Filling factor κ
Compute jsc_op, jo_op , Iop , Gop , Bpeak_op
Compute T, jsc , Bpeak margins
Compare “short sample”, “operational” conditions and margins if the same
design uses Nb-Ti superconducting technology
Define a possible coil lay-out to minimize field errors
Determine e.m forces Fx and Fy and the accumulated stress on the coil midplane in the operational conditions (80% of Iss )
Evaluate dimension iron yoke, collars and shrinking cylinder, assuming
that the support structure is designed to reach 90% of Iss
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Case study 1 solution
Dimension of the yoke
The iron yoke thickness can be estimated with
r 2G
~ t iron B sat
2
Therefore, being
G = 156 T/m (at 90% of Iss )
r = 75 mm and
Bsat = 2 T
we obtain
tiron = ~220 mm
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Case study 1 solution
Dimension of the support structure
We assume a 25 mm thick collar
Images not in scale
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Case study 1 solution
Dimension of the support structure
We assume that the shell will
close the yoke halves with the
same force as the total horizontal
e.m. force at 90% of Iss
Fx_total = Fx_quadrant * 2 * sqrt(2) = +6.8
MN/m
Assuming an azimuthal shell
stress after cool-down of
shell = 200 MPa
The thickness of the shell is
tshell = Fx_total /2/1000/ shell ~ 17 mm
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Case study 1 solution
Magnet cross-section
Coil inner radius: 75 mm
Coil outer radius: 112 mm
The operational conditions (80% of Iss)
jsc_op = 1580 mm2
jo_op = jsc_op * k = 505 mm2
Iop = 16480 A
Gop = 140 T/m
Bpeak_op = 12.1 T
Collar thickness: 25 mm
Yoke thickness: 220 mm
Shell thickness: 17 mm
OD: 748 mm
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Comparison
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