MiniBoone-like horn - dynamic response to
magnetic and thermal pulses
Cracow University of Technology
Institute of Applied Mechanics
Piotr Cupial, EUROν Annual Meeting, Rutherford Appleton
Laboratory, 18-21 January 2011
1/23
Outline of the talk
•Response to a sequence of magnetic pulses
•Joule heating of the horn
•Response to a sequence of Joule heat pulses
•Power deposition in the horn due to secondary particles
•Response to secondary particle pulses
Piotr Cupial, EUROν Annual Meeting, Rutherford Appleton
Laboratory, 18-21 January 2011
2/23
Four-horn configuration
1
1
VOLUMES
MAT
APR 14 2010
21:50:11
NUM
VOLUMES
MAT
APR 14 2010
21:52:12
NUM
Z
Y
X
Z
Y
X
4 Horns
4 Horns
The proton beam is split into four beams and each horn is pulsed
with the frequency 12.5 Hz.
Piotr Cupial, EUROν Annual Meeting, Rutherford Appleton
Laboratory, 18-21 January 2011
3/23
Geometry of the axisymmetric horn
The horn configuration to be used with emgedded solid target is being
studied. The horn inner radius is 3 cm (higher than for the
configuration with integrated horn studied by B.Lepers).
Piotr Cupial, EUROν Annual Meeting, Rutherford Appleton
Laboratory, 18-21 January 2011
4/23
Magnetic pressure applied to the horn
µ0 I 2
p= 2 2
8π R
The formula is
applicable to cylinders,
and approximatly
applicable to conical
sections.
Piotr Cupial, EUROν Annual Meeting, Rutherford Appleton
Laboratory, 18-21 January 2011
5/23
Response to magnetic pulses
Maximum von Mises stress due to magnetic pulses = 18 MPa (at 300 kA)
= 24.5 MPa (at 350 kA)
Piotr Cupial, EUROν Annual Meeting, Rutherford Appleton
Laboratory, 18-21 January 2011
6/23
Response to magnetic pulses
Response to a single pulse
Response to a sequence of twentyfive pulses
Piotr Cupial, EUROν Annual Meeting, Rutherford Appleton
Laboratory, 18-21 January 2011
7/23
Approxiamte calculation of Jule heating due
to a sequence of half-sine pulses
In the simplified analysis it is assumed that the skin depth is
calculated for harmonic excitation with frequency equal to
1/(2*pulse duration) (J.M.Maugain, S. Rangod „CERN-NUFACT
Note 80; B.Lepers)
2
Pav = RI RMS
I RMS =
R = ρL /( 2πRm δ)
τ 2
( )
2 T
I0
δ = ρ /( πfµ 0 µ r )
(skin depth)
−8
ρ
=
4
⋅
10
Ω⋅m
For an alluminium alloy with
and f=5000 Hz: δ=1.4 mm
Piotr Cupial, EUROν Annual Meeting, Rutherford Appleton
Laboratory, 18-21 January 2011
8/23
Fourier spectrum of the sequence of halfsine pulses
The spectra shown correspond to: Imax=300 kA, pulse duration
100 µs, pulse repetition frequency 12.5 Hz
Envelope of the sprctrum over
the frequency range [0,50 kHz]
Detail showinig the discrete
nature of the spectrum –
frequency spacing equals to
12.5 Hz
Piotr Cupial, EUROν Annual Meeting, Rutherford Appleton
Laboratory, 18-21 January 2011
9/23
Avarage power dissipated in the horn waist
– approximate vs. Fourier analysis
Average power dissipated in a cyliner 78 cm long, with
inner diameter 3 cm. The skin depth cannot exceed the
horn thickness (3 mm).
I0=300 kA
I0=350 kA
Approximate
analysis
6.5 kW
8.9 kW
Fourier analysis
5.5 kW
7.5 kW
Approximate formula gives reasonable results, with higher
values than the Fourier analysis (is more conservative). It has
been used in the calculation of the Joule losses in the horn.
Piotr Cupial, EUROν Annual Meeting, Rutherford Appleton
Laboratory, 18-21 January 2011
10/23
Dynamic stress due to a sequence of Joule
heating pulses
Pimpulse
T
= Pav
τ
Power per rectangular impulse with duration τ
I0=300 kA
I0=350 kA
Approximate
analysis
5.2*106 W
7.1*106 W
Fourier analysis
4.4*106 W
6.0*106 W
Piotr Cupial, EUROν Annual Meeting, Rutherford Appleton
Laboratory, 18-21 January 2011
11/23
Joule heating losses in the Mini-Boone-like
horn
Estimates of Joule losses for half-sine pulses of duration τ=100µs,
repeated with frequency 12.5 Hz
Average Joule heating:
Inner cond. cylinder 1 (waist)
Inner cond. conical 1
Inner cond. cylinder 2
Inner cond. conical 2
Inner cond. cylinder 3
Outer conductor
Front face
Exit face
Average Joule heating for
max. current = 300 kA
4.94 kW
1.32 kW
0.69 kW
1.35 kW
0.09 kW
0.93 kW
1.03 kW (scaled from
Benjamin’s result)
2.94 kW (scaled from
Benjamin’s result)
Average Joule heating for
max. current = 350 kA
6.72 kW
1.81 kW
0.93 kW
1.83 kW
0.12 kW
1.27 kW
1.4 kW (B. Lepers)
4.0 kW (B. Lepers)
Piotr Cupial, EUROν Annual Meeting, Rutherford Appleton
Laboratory, 18-21 January 2011
12/23
Joule heating losses in the Mini-Boone-like
horn
Power per equivalent rectangular pulse:
Inner cond. cylinder 1 (waist)
Inner cond. conical 1
Inner cond. cylinder 2
Inner cond. conical 2
Inner cond. cylinder 3
Outer conductor
Front face
Exit face
Joule heat per pulse for
max. current = 300 kA
3.95*106 W
1.06*106 W
0.55*106 W
1.08*106 W
0.073*106 W
0.75*106 W
0.82*106 W
2.35*106 W
Joule heat per pulse
max. current = 350 kA
5.38*106 W
1.45*106 W
0.75*106 W
1.47*106 W
0.1*106 W
1.02*106 W
1.12*106 W
3.2*106 W
Piotr Cupial, EUROν Annual Meeting, Rutherford Appleton
Laboratory, 18-21 January 2011
13/23
Dynamic stress due to a sequence of Joule
heating pulses
Maximum dynamic stress due to Joule heating is 2.2 MPa (at 300 kA)
and about 3 MPa (at 350 kA).
Piotr Cupial, EUROν Annual Meeting, Rutherford Appleton
Laboratory, 18-21 January 2011
14/23
Dynamic stress due to a sequence of Joule
loss pulses
Response to a single pulse
Response to a sequence of
twenty-five pulses
Piotr Cupial, EUROν Annual Meeting, Rutherford Appleton
Laboratory, 18-21 January 2011
15/23
Power deposited in the horn by secondary
particles
According to the simulations by A. Longhin and C.Bobeth, for a horn with
an embedded graphite target.
Inner cond. cylinder 1 (waist)
Inner cond. conical 1
Inner cond. cylinder 2
Inner cond. conical 2
Inner cond. cylinder 3
Outer conductor
Front face
Exit face
Average power from
deposition of secondary
particles
5.5 kW
1.76 kW
1.12 kW
0.58 kW
0.36 kW
3.34 kW
0.76 kW
0.48 kW
Power per equivalent
rectangular pulse
88*106 W
28.1*106 W
17.9*106 W
9.28*106 W
5.76*106 W
53.4*106 W
12.2*106 W
7.7*106 W
The pulses due to secondary particles have 5µs duration.
Piotr Cupial, EUROν Annual Meeting, Rutherford Appleton
Laboratory, 18-21 January 2011
16/23
Dynamic stress resulting from a sequence of
pulses due to secondary particle heating
Maximum dynamic stress due to secondary particles is 3 MPa
Piotr Cupial, EUROν Annual Meeting, Rutherford Appleton
Laboratory, 18-21 January 2011
17/23
Dynamic stress resulting from a sequence of
pulses due to secondary particle heating
Response to a sequence of twenty-five secondary particle pulses at two
selected points.
Piotr Cupial, EUROν Annual Meeting, Rutherford Appleton
Laboratory, 18-21 January 2011
18/23
Summary of dynamic stress for the
MiniBoone like horn
at 300 kA
Stress due to magnetic pressure
Stress due to Joule heating
Stress due to secondary particles
Overall
at 350 kA
18 MPa
24.5 MPa
2 MPa
3 MPa
3 MPa
3 MPa
23 MPa
30.5 MPa
Piotr Cupial, EUROν Annual Meeting, Rutherford Appleton
Laboratory, 18-21 January 2011
19/23
Fatigue analysis – SN curve (M.Kozien)
Piotr Cupial, EUROν Annual Meeting, Rutherford Appleton
Laboratory, 18-21 January 2011
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Fatigue analysis – formulas (Morrow,Goodman)
σa
σ ar =
, σm > 0
σm
1−
σu
σ ar = σ a , σ m ≤ 0
EFFECTIVE STRESS AMPLITUDE (von Mises)
σ a = σ aHMH =
1
=
2
(σ
1
2
(σ 1a − σ 2 a ) + (σ 1a − σ 3a ) + (σ 2 a − σ 3a ) =
2
2
2
2
2
2
−
σ
+
σ
−
σ
+
σ
−
σ
+
τ
+
τ
+
τ
6
(
)
)
(
)
(
xa
ya
xa
za
ya
za
xya
xza
yza )
2
2
EFFECTIVE MEAN STRESS (Sines)
σ m = a ⋅ ( σ 1m + σ 2 m + σ 3 m
a =1
)
2
ULTIMATE STRESS
σ u = 315 MPa , alloy 6082
Piotr Cupial, EUROν Annual Meeting, Rutherford Appleton
Laboratory, 18-21 January 2011
21/23
Fatigue analysis – Miniboone horn estimation
SINES
STATIC
STRESS
[MPa]
102.5
VON MISES
DYNAMIC
STRESS
[MPa]
26.2
MORROW
FATIGUE
STRESS
[MPa]
38.9
CRITICAL
VALUE
[MPa]
100
Piotr Cupial, EUROν Annual Meeting, Rutherford Appleton
Laboratory, 18-21 January 2011
22/23
Conclusions and further studies
•Dynamic response of the horn to magnetic and thermal shock has been
done which are needed to estimate the fatigue life of the horn.
•Magnetic forces result in higher stress levels than thermal shock.
•The first estimate of fatigue life are optimistic (but more studies are
required to study stress concentration, welds, corrosion etc.).
•The embedded target will add to the thermal load on the horn. This will be
included in the simulation studies when the target baseline is established.
Piotr Cupial, EUROν Annual Meeting, Rutherford Appleton
Laboratory, 18-21 January 2011
23/23