Transient Heat Transfer Model and Stability Analysis of He-Cooled SC Cables

1
Transient Heat Transfer Model and Stability
Analysis of He-Cooled SC Cables

• Pier Paolo Granieri 1,2, Marco Calvi 2,
  Panagiota Xydi 1
• Luca Bottura 1, Andrzej Siemko 1
• (1 CERN, 2 EPFL)

Acknowledgments M. Breschi , B. Baudouy, D.
Bocian
CHATS on Applied Superconductivity 2008 Workshop
KEK, Tsukuba, Japan, October 30 November 1,
2008
2
Outline

• Beam Induced Quenches
• Model Description
• Heat Transfer to helium
• Results
• Impact of the Model Parameters
• Systematic Calculation of the Stability Margins
• Conclusions Perspectives

3
Beam Induced Quench
Beam loss quench of the magnets Interruption of
the LHC
Protection system (Beam Loss Monitor)
BLM reaction thresholds energy deposition ??
stability of the magnet
The estimation of the stability against transient
distributed disturbances is the aim of this work
4
From 1-D to 0-D Model

• Follow-up of the work presented at CHATS 2005 1
• 1-D THEA model 2 (non-uniform T and I
  distribution)
• strands lumped into a single thermal component 3

(1)

• The longitudinal dimension can then be neglected
  for distributed
• energy depositions (e.g. beam loss) ? 0-D
  nimble model
• Large He contribution ? Particular focus on it

1 L. Bottura, M. Calvi, A. Siemko, Stability
Analysis of the LHC cables, Cryogenics, 46
(2006) 481-493. Presented at CHATS-2005,
Enschede, Netherlands.
2 L. Bottura, C. Rosso, M. Breschi, A general
Model for Thermal, Hydraulic, and Electric
Analysis of Superconducting Cables, Cryogenics,
vol. 40 (8-10), 2000, pp.617-626.
3 M. Breschi, P.P. Granieri, M. Calvi, M.
Coccoli, L. Bottura, Quench propagation and
stability analysis of Rutherford resistive core
cables, Cryogenics, 46 (2006) 606-614. Presented
at CHATS-2005, Enschede, Netherlands.
5
0-D Model Description
In the longitudinal direction
Cable cross-section
micro-channel
He channel through the cable insulation
(4)
Subscripts s strands He He in the cable i
insulation b external He bath
p contact (wetted) perimeter h heat transfer
coefficient
(5)
4 ZERODEE Software, CryoSoft, France, 2001.
5 B. Baudouy, M.X. Francois, F.P. Juster, C.
Meuris, He II heat transfer through
superconducting cables electrical insulation,
Cryogenics, 40 (2000) 127-136.
6
Heat Transfer to Helium
Superfluid helium
THe lt T?
At the beginning of the thermal transient the
heat exchange with He II is limited by the
Kapitza resistance at the interface between the
helium and the strands. The corresponding heat
transfer coefficient is
(6)
6 S.W. Van Sciver, Helium Cryogenics, Clarendon
Press, Oxford, 1986.
7
Heat Transfer to Helium
Normal helium
T? lt THe lt TSat
When all the He reaches the T?, the He I phase
starts growing and temperature gradients are
established in the He bulk. A thermal boundary
layer forms at the strands-He interface, where a
heat diffusion process takes place in a
semi-infinite medium
(7)
After the boundary layer is fully developed, the
heat transfer mechanism is driven by a steady
state heat transfer coefficient hss. The
transition is approximated in the following
empirical way
(8)
7 L. Bottura, M. Calvi, A. Siemko, Stability
Analysis of the LHC cables, Cryogenics, 46
(2006) 481-493.
8 V.D. Arp, Stability and thermal quenches in
forced-cooled superconducting cables, Proceeding
of 1980 superconducting MHD magnet design
conference, Cambridge (MA) MIT, 1980, pp.
142-157.
8
Heat transfer to helium
refers to a constant
temperature of the strands To generalize to the
case of an arbitrary temperature (or heat flux)
evolution at the strands surface, consider the
following heat conduction problem
?(x,t) He temperature profile in the cables
transversal direction ?(t)Ts(t)-T? temp. of
the strands temp. at the ? transition
The heat flux absorbed by the helium in the inlet
surface is proportional to local temperature
gradient
The implementation is ongoing
9
Heat Transfer to Helium
Nucleate boiling
THe TSat
Once the saturation temperature is reached, the
He enters the nucleate boiling phase. The
formation of bubbles separated from each other
occurs in cavities on the strands surface. The
transient heat exchange during this phase is very
effective
(9,10)
9 C. Schmidt, Review of Steady State and
Transient Heat Transfer in Pool Boiling He I,
Saclay, France Int.l Inst. of Refrigeration
Comm. A1/2-Saclay, 1981, pp. 17-31.
10 M. Breschi et al. Minimum quench energy and
early quench development in NbTi superconducting
strands, IEEE Trans. Appl. Sup., vol. 17(2), pp.
2702-2705, 2007.
10
Heat Transfer to Helium
13 C. Schmidt, Review of Steady State and
Transient Heat Transfer in Pool Boiling Helium I,
Saclay, France International Institute of
Refrigeration Commision A1/2-Saclay, 1981, pp.
17-31.
Film boiling
Efilm Elim
Afterwards the vapour bubbles coalesce forming an
evaporated helium thin film close to the strands
surface, which has an insulating effect. An
energetic criterion composed with a criterion
based on the critical heat flux has been used 11.
Then adapted to the case of narrow channels
(reduction of the critical heat flux with respect
to a bath). When the energy transferred to He
equals the energy needed for the film boiling
formation Elim, the heat transfer coefficient
drops to its film boiling value
hfilm boiling 250 W/m²K
14 M. Nishi et al., Boiling heat transfer
characteristics in narrow cooling channels, IEEE
Trans on Magnetics, Vol. Mag-19, No 3, May 1983
12 W.G. Steward. , Transient helium heat
transfer phase I- static coolant, International
Journal of Heat and Mass Transfer, Vol. 21, 1978
11 M. Breschi et al. Minimum quench energy and
early quench development in NbTi superconducting
strands, IEEE Trans. Appl. Sup., vol. 17(2), pp.
2702-2705, 2007.
11
Heat Transfer to Helium
Gas
Energy deposited into the channel Elat
The film thickness increases with the strands
temperature. Since the whole He in the channel is
vaporised (film thickness equal to the channel
width), the worsening of the heat transfer
observed in experimental works 15 is taken into
account allowing transition to a totally gaseous
phase. This is described by a constant heat
transfer coefficient extrapolated from
experiments
(16,17)
hgas 70 W/m²K
15 Y. Iwasa and B.A. Apgar, Transient heat
transfer to liquid He from bare copper surfaces
to liquid He in a vertical orientation I Film
boiling regime, Cryogenics, 18 (1978) 267-275.
16 M. Nishi, et al., Boiling heat transfer
characteristics in narrow cooling channel, IEEE
Trans. Magn., vol. 19, no. 3, pp. 390393, 1983.
17 Z. Chen and S.W. Van Sciver, Channel heat
transfer in He II steady state orientation
dependence, Cryogenics, 27 (1987) 635-640.
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Summary of the Heat Transfer Model
He II He I Nucleate Boiling Film Boiling Gas
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Impact of the Heat Transfer Model

• Different heat transfer models have been
  considered
• I. an unrealistic one only using the Kapitza heat
  transfer coefficient
• II. the full model as previously described
• III. the full model without the external
  reservoir and without boiling phases

• Time scales of the heat transfer mechanisms
• 400 µs Kapitza regime
• then ?-transition ( boiling)
• 5 s heat transfer through the insulation
• The enthalpy reserve (between Tbath and Tcs) of
  the cable components gives the lower and upper
  limits of the stability margin
• Approach based on the response of the components,
  not just on their enthalpy

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Impact of the Cooling Surface
The influence of the He micro-channels located
between the cables in the coil and of the He
channels network through the cable insulation is
unknown. This parametric study allows to
investigate the effectiveness of these heat
transfer mechanisms.

• The wetted perimeter pi,b between the insulation
  and the external bath is important for long
  heating times (pi,b 35 mm is an unrealistic
  case corresponding to an entirely wetted
  insulation 8.2 mm corresponds to the case of
  effective µ-channels, while 2.6 mm corresponds to
  µ-channels not effective at all)
• The actual size of the He channels network
  (represented by G) starts playing a role for
  values above 10-5 m5/3
• In the calculations a conservative case has been
  considered, where pi,b 2.6 mm and G 0 m5/3.

Pi,b?? effectiveness of the µ-channels
QHeII
Pi,b
QHeII?? effectiveness of the He
channels in the cable insulation
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Systematic Calculation of the Stability Margins
Stability margins of cable 1 (inner layer of the
main dipoles) as a function of heating time, for
different current levels

• Slower is the process, higher is the impact of
  helium
• For low currents and short heating times the
  stability margin curve gets flat, since the
  decision time is much greater than the heating
  time

I
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Decision Time
The decision time (time the system waits before
deciding to quench or not) is noticeably longer
at low current regimes
decision time (I 1850 A)
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Systematic Calculation of the Stability Margins
Stability margins of cable 1 as a function of
magnetic field (ranging from 0.5 T to the peak
field for a given current) and for several
current levels

• The stability margin for a given cable has been
  estimated as a function of the heating time,
  current and magnetic field. This allows
  interpolating the results at any field for any
  cable in the magnet cross section

I
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Systematic Calculation of the Stability Margins
Stability margins of all the CERN LHC cables
working at 1.9 K as a function of heating time

• The stability of all the CERN LHC superconducting
  cables working at an operational temperature of
  1.9 K has been numerically computed with respect
  to the actual range of beam loss perturbation
  times

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Systematic Calculation of the Stability Margins
The stability margin and the quench power needed
to quench the cable are estimated for different
heating times up to the steady-state value

• The model developed links the transient to the
  steady state regime
• for long heating times the power needed to
  quench the cable decreases and reaches
  asymptotically the steady state value

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Conclusions

• 0-D thermal model ? stability of superconducting
  magnets
• Complete heat transfer model ? considers all
  helium phases following a transient perturbation
• Concept of decision time ?? behavior of a cable
  against transient disturbances
• Sensitivity analysis with respect to the
  parameters of the model
• Parametric analysis of the stability of the LHC
  magnets with respect to heating time, current,
  magnetic field

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Perspectives

• Refinement of the model are ongoing
• Modeling of heat transfer through the insulation
  (pressure, insulation scheme) will be soon
  available (measurements on the drainable heat
  through the cable insulation, for the LHC upgrade
  phase I)
• Dedicated experiments are foreseen to validate
  the described model, as well as the building of
  an instrumented LHC magnet
• Increase of the complexity of the 0-D approach
  (e.g. a multi-strand approach with periodical
  boundary conditions), to better describe the
  internal structure of the cable and the predicted
  shape of a beam loss energy deposit

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Comparison with a steady-state experiment
18 D. Bocian, B. Dehning, A. Siemko, Modelling
of quench limit for steady state heat deposits in
LHC magnets, to be published on IEEE Trans.
Appl. Sup..