transient heat transfer model and stability analysis of he-cooled sc cables pier paolo granieri 1,2,...

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

31/10/2008 PP Granieri - Transient Heat Transfer and Stability of He-cooled SC Cables 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


Protection system (Beam Loss Monitor)

Beam loss:

quench of the magnetsInterruption of the LHC

BLM reaction thresholds:energy deposition

stability of the magnet

The estimation of the stability against transient distributed disturbances is the aim of this work

Beam Induced Quench


From 1-D to 0-D Model

1 L. Bottura, M. Calvi, A. Siemko, “Stability Analysis of the LHC cables”, Cryogenics, 46 (2006) 481-493. Presented at CHATS-2005, Enschede, Netherlands.

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.

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

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.

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

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0-D Model Description

)()()(

)()()(

)()(''

,,, ,,,

,,,,,,

,,,,

bibibisiisisHeiHeiHeii

iii

HeIIisHeiHeiHeiHeiHesHesHesHe

HeHeHe

isisisHesHesHesJouleexts

sss

TThpTThpTThpt

TCA

QppTThpTThpt

TCA

TThpTThpqqt

TCA

Cable cross-section: In the longitudinal direction:

micro-channel

He channel through the cable insulation

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.

3/1

3/1)(

1

h

b

T

Tt

HeII dTTfL

A

AQ

Subscripts : s – strands He : He in the cable i : insulation b : external He bath

p : contact (wetted) perimeterh : heat transfer coefficient

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Heat Transfer to Helium

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:

THe < Tλ

22hshsK TTTTh

Superfluid helium

6 S.W. Van Sciver, “Helium Cryogenics”, Clarendon Press, Oxford, 1986.

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Previous studies allowed determining that the most relevant parameter for stability calculations is the heat transfer coefficient between the strands and the helium filling the interstices among them.

Its theoretical evaluation represents a complex task. It depends on several parameters, which are often unknown and difficult to measure.

A careful research in the literature has been carried out to develop the heat transfer model. It consists of the composition of different terms related to the different He phases :

L. Bottura, M. Calvi, A. Siemko, “Stability Analysis of the LHC cables”, Cryogenics, 46 (2006) 481-493.



t

cKh hhhBL

When all the He reaches the Tλ, 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:

Tλ < THe < TSat

Normal helium He I

He II

He I

He II

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:

ss

BLK

BLKHeI h

hh

hhh ;max

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.

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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:

Heat transfer to helium

t

cKh hhhBL

The heat flux absorbed by the helium in the inlet surface is proportional to local temperature gradient:

t

eq dttt

CK

tt

jh

0

20201

0

xx

Kj

C

Kwhere

xt

2

2 00,x tt,0

t

dtxFt

tx0

,,

t

dt

txxtx

02/3

2 4/exp

2,

θ(x,t): He temperature profile in the cable’s transversal direction

Ф(t)=Ts(t)-Tλ: temp. of the strands – temp. at the λ transition

The implementation is ongoing…


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:

THe = TSat

Nucleate boiling

hs

mh

ms

boilingnucl TT

TTh

)(

.

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.


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t

t

hshshsfilm dTThptE*

,,

0

)()(

Efilm = Elim

Film boiling

hfilm boiling= 250 W/m²K

non ttE )( *

0lim

12 W.G. Steward. , “Transient helium heat transfer: phase I- static coolant”, International Journal of Heat and Mass Transfer, Vol. 21, 1978

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:

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.

14 M. Nishi et al., “Boiling heat transfer characteristics in narrow cooling channels”, IEEE Trans on Magnetics, Vol. Mag-19, No 3, May 1983

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.



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:

Energy deposited into the channel = Elat

Gas

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. 390–393, 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

, . .

K

HeI

s h nucl boil

film

gas

h

h

h h

h

h

lim

h

h Sat

h Sat

film

gas lat

T T

T T T

T T

E E

E E

He I

He II

He I

He II

He II

He I

NucleateBoiling

FilmBoiling

Gas


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 λ-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


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.

Impact of the Cooling SurfaceThe 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.

QHeII Pi,b

In the longitudinal direction:

micro-channel

He channel through the cable insulation

Pi,b effectiveness of the μ-channels

QHeII effectiveness of the He channels in the cable insulation


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


“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)”



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



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



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


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


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


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..

QHeII Pi,b Text bath

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transient heat transfer model and stability analysis of he-cooled sc cables pier paolo granieri 1,2,...

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