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Granieri_2013_Cryogenics

Thermo-electric analysis of the interconnection of the LHC main superconducting bus bars

P.P.Granieri a ,b ,?,M.Breschi c ,M.Casali c ,L.Bottura a ,A.Siemko a

TE Department,CERN,Geneva,Switzerland

Swiss Federal Institute of Technology (EPFL),Particle Accelerator Physics Laboratory (LPAP),Lausanne,Switzerland c

University of Bologna,Department of Electrical Engineering (DIE),Italy

a r t i c l e i n f o Article history:

Available online 17June 2012Keywords:

Accelerator magnets Interconnections LHC

Superconducting bus bar

a b s t r a c t

Spurred by the question of the maximum allowable energy for the operation of the Large Hadron Collider (LHC),we have progressed in the understanding of the thermo-electric behavior of the 13kA supercon-ducting bus bars interconnecting its main magnets.A deep insight of the underlying mechanisms is required to ensure the protection of the accelerator against undesired effects of resistive transitions.This is especially important in case of defective interconnections which can jeopardize the operation of the whole LHC.

In this paper we present a numerical model of the interconnections between the main dipole and quad-rupole magnets,validated against experimental tests of an interconnection sample with a purposely built-in defect.We consider defective interconnections featuring a lack of bonding among the supercon-ducting cables and the copper stabilizer components,such as those that could be present in the machine.We evaluate the critical defect length limiting the maximum allowable current for powering the magnets.We determine the dependence of the critical defect length on different parameters as the heat transfer towards the cooling helium bath,the quality of manufacturing,the operating conditions and the protec-tion system parameters,and discuss the relevant mechanisms.

1.Introduction

A few days after its start-up in September 2008,the particle accelerator Large Hadron Collider (LHC)at CERN experienced an incident causing considerable damages and delaying the restart of the machine by more than 1year.The incident was initiated by an electrical fault due to a defective interconnection (IC)be-tween two adjacent main dipole magnets [1,2].The fault consisted of a bad soldering between the two superconducting (SC)cables in the bus bar and between the cables and the copper stabilizer,along with a lack of longitudinal continuity in the stabilizer,whose role is to guarantee the protection of the circuit.Furthermore,the sensi-tivity of the bus bar quench detection system was not suf?cient to detect the voltage rise of the resistive zone,because the inter-vention threshold was set too high.

Large efforts were undertaken to measure the resistance of the interconnections in the machine both at warm and at cold [3].They included a calorimetric technique and electrical measurements performed using either invasive or non-invasive diagnostics.The invasive technique requires direct access to a single interconnec-tion:it consists of room temperature electrical resistance measure-ment allowing to quantify the longitudinal continuity of the copper stabilizer across it.The non-invasive electrical resistance measure-ments were carried out at room and cryogenic temperatures exploiting the magnet voltage taps.They covered therefore a bus bar segment containing more than one interconnection,and the interpretation depended on the Residual Resistivity Ratio (RRR )of the copper bus bar.Furthermore,gamma-radiography was used to visualize the internal volume of the interconnections.This mea-surements campaign allowed determining the presence of defects in the machine,as well as their dimension.The largest defects were repaired [4]before the LHC restart in November 2009.However,a number of low quality interconnections could still be present in the accelerator.For this reason the decision was taken to limit the maximum current in the main circuits,until the full consolida-tion campaign [5]that will take place in 2013–2014.A new quench detection system across the interconnections is implemented with a reduced sensitivity threshold,as well as other remedies [2].

In this paper we present a numerical thermo-electrical model of the LHC superconducting bus bars and interconnections,both for the Main Bending (MB)dipole and Main Quadrupole (MQ)mag-nets.The model aims at estimating the critical length of the inter-connection defect.It is validated by reproducing experimental tests of a defective interconnection sample.We report investiga-

?Corresponding author at:TE Department,CERN,Geneva,Switzerland.Tel.:+41

227676277.

E-mail address:pier.paolo.granieri@cern.ch (P.P.Granieri).

tions of the heat transfer towards the cooling helium bath in the interconnection and bus bar region,and evaluate its impact on the calculations.The in?uence of the quality of manufacturing, operating conditions and protection system parameters is analyzed as well.Parametric analyses are presented,in adiabatic and non-adiabatic conditions,as a function of the defect dimension,current decay time constant,RRR of the copper of the SC cable matrix and of the stabilizer,spatial distribution of the defect.

2.The LHC main superconducting bus bars and interconnections

The13kA bus bar connecting the LHC MB dipole and MQ mag-nets is made of type02[6]superconducting cable embedded in a hollow copper stabilizer piece[7].The cable can be seen in the left and right extremities of Fig.1,where the copper stabilizer is cut to show the bus bar cross-section.The void space inside the stabilizer must be?lled with tin–silver alloy to provide good thermal and electrical contact between the bus bar constituents.

As soon as a quench is detected in a magnet,its coil is immedi-ately heated and electrically by-passed by means of a cold diode,so that the coil current decays to almost zero in less than1s.Mean-while the bus bar by-passing the quenched magnet,as well as the other magnets of the sector,all powered in series,still carry the full current which is decaying almost exponentially with a time constant of tens of seconds[8].The bus bar was therefore designed with a copper cross-section suf?cient to safely carry the current during discharge,in case its superconducting cable undergoes a resistive transition.The design of the MB and MQ bus bars is iden-tical except for the copper cross-section[7]that is larger in case of the MB bus bars because of the larger current decay time constant (initial time constant of around104s for MB vs.37s for MQ at the nominal beam energy of7TeV).

The exploded view of the bus bar interconnection between two adjacent MB or MQ magnets is schematically shown in Fig.1.It is composed of a120mm long?at and a150mm long U-shaped cop-per pro?le that enclose the120mm long overlapping zone of the two superconducting cables coming from the right and left side magnets.A soft soldering technique[9]based on tin–silver alloy is used to splice the superconducting cables between them and to the interconnection stabilizer,as well as to connect the inter-connection stabilizer to the bus bar copper pro?le.A well soldered interconnection,where all the void inner spaces are?lled with sol-der,looks like a continuation of the bus bar as shown in Fig.2.This [10].The bus bar is inserted in a horizontal pipe and cooled by sta-tic pressurized He II at1.9K.

In order to limit the heat dissipation in the interconnection due to Joule effect,the electric resistance between superconducting cables is speci?ed to be lower than0.6n X[9].The measured resis-tances on samples during production were below this design value. However,the calorimetric technique developed after September 2008allowed determining the actual electric resistance associated to the measured temperature increase,corresponding to a value between180and260n X for the interconnection initiating the incident[2].This high value was identi?ed as the cause of the quench.Note that a resistive transition of a bus bar interconnection can also be induced by external heating,for instance due to warm helium coming from a nearby quenching magnet.

A bus bar quench is not a fault condition,given the presence of a large copper stabilizer cross-section.However,it can trigger a ther-mal runaway in case of the concurrent presence of two manufac-turing defects[11,12]:lack of transverse contact between cable and copper stabilizer(Fig.3)and lack of longitudinal continuity in the copper stabilizer.The scheme of a defective interconnection featuring such defects is shown in Fig.4.The combination of the two defects prevents current redistribution between cable and sta-bilizer:the total operating current is forced to?ow through the superconducting cable along the defect length,which is the bus bar length where at least one of the two defects is present.The interconnection can therefore experience a thermal runaway, depending on the length of the defect zone.This length determines whether or not the dissipation by Joule heating is balanced by the heat evacuated towards the coolant and by longitudinal solid con-duction through the bus bar.

3.Model

3.1.Description

The multi-physics model adopted for the analysis is the Cryo-Soft code THEA,described in detail in[13],that allows taking into account the material properties dependence on temperature,cur-rent and magnetic?eld.THEA is based on the assumption that the components length is much larger than their transverse dimen-sion,so that all phenomena can be analyzed with a1-D approxima-tion.However,considerable efforts are made for the de?nition of the transverse heat transfer towards the cooling bath.The longitu-dinal length is discretized with a non-uniform mesh in the frame of a?nite element approach.

The bus bar is modeled as a single homogeneous thermal ele-ment with uniform temperature over the cross-section.The total cross-section A b of the bus bar(subscript b)is obtained as sum of the partial cross-sections of the constituents,i.e.the superconduc-ting?laments,the copper stabilizer matrix of the strands and, when present,the copper stabilizer of the bus bar.The homoge-nized density q b and thermal conductivity k b are obtained weight-ing on the area,whereas the homogenized speci?c heat C b is obtained weighting on the mass of the constituents.The bus bar thermal capacity,the heat transport by conduction,the heat ex-change at the interface with the cooling medium and the Joule heat generation_q0

Joule;b

are described by the following thermal equation: A b q b C b

@T b

A b k b

@T b

tp b He HTCeT bàT HeT?_q0

Joule;b

:e1T

T b and T He are the bus bar and helium temperature,p b_He and HTC represent the wetted perimeter and the heat transfer coef?cient, respectively.The temperature of the helium hydraulic component T He is calculated by the equations that model compressible?ow in a1-D pipe,detailed in[13],that complement Eq.(1).Considering

Fig.1.View of the bus bar interconnection constituents.

108P.P.Granieri et al./Cryogenics53(2013)107–118

all the bus bar components lumped into a unique element does not allow catching the details of the current distribution phenomena,which would require the de?nition of the cable and bus bar as sep-arate thermal and electrical elements.On the other hand,the one element approach was demonstrated to be appropriate when con-sidering heating sources distributed across the whole cross-section [14].

The concurrent presence of transverse and longitudinal lack of solder in the interconnection,as described in the previous section,is considered in the calculations.This makes the model more com-plete than previous studies that had taken into account either a lo-cal resistive heating due to faulty splices assuming a longitudinal continuity of the bus bar stabilizer [15],or the only reduction of the stabilizer cross-section in the longitudinal direction.The com-plex geometry of a defective interconnection is simulated by the

halving it (as shown in Fig.5c)and saving a of computation time.

It is worth noticing that the defect length tional to R add ,the additional resistance with value of a good interconnection at room temperature.tances of good interconnections are 10–12q el is the resistivity of the SC cable copper matrix at room temper-ature,n st is the number of strands in the cable,A st is the strand cross section and A Cu and A SC are the strand copper and superconducting area,respectively.3.2.Implementation

As previously stated,the manufacturing imperfections can be dangerous in case of a resistive transition of the bus bar.As long as the cable is in the superconducting state the transport current ?ows in the Nb–Ti ?laments without heat dissipation,except for the dissipation due to the inter-cables contact resistance.As soon as a quench occurs the current starts ?owing from the supercon-ducting ?laments to the strand copper matrix,then possibly to the bus bar copper stabilizer.The parametric studies presented consider a quench already developed over the sample:an initial temperature of 10K is assumed,slightly above the critical temper-Fig.2.Left:3-D sketch of a well soldered interconnection.Right:longitudinal cross-section of its right-hand side.

of the interconnection,assumed to occur at 500K.A trial length is selected and the transient temperature pro?le is observed.The de-fect length is then adjusted according to the maximum tempera-ture reached,and a new transient is simulated.This trial-and-error procedure is repeated until the difference between the largest defect leading to a controlled quench and the smallest defect lead-ing to a thermal runaway is equal to 1mm.

In all simulations with a single defect a 2m long sample is con-sidered (with half the defect length),whereas a 4m long sample is considered for the case of double defects (total defect length).The bus bar cross section,and consequently its properties,is a function of position.The Nb–Ti cross-section is constant all over the sample because the presence of one single superconducting cable is con-sidered.The copper cross-section is equal to the sum of the copper stabilizer of the superconducting cable and of that of the bus bar outside the defect,whereas it is equal to the only copper of the superconducting cable in the defect.Since the RRR of the two types of copper differ because of different manufacturing processes,the value considered in the calculations is the average RRR weighted on the area outside the defect,or the RRR of the SC cable copper in-side it.Because of the uncertainty of the RRR values of the acceler-ator components [17],a parametric study is performed and shown in next section:the cable RRR is varied between 80and 160,whereas the bus bar RRR is varied between 100and 220.Default values are set to the conservative 80and 100,respectively.The SnAg solder is not considered to avoid unnecessary complication of the model.The polyimide electrical insulation is not de?ned as an additional thermal component.Its presence is taken into ac-count in the de?nition of the heat transfer coef?cient to helium,as reported in Section 3.3.The geometric parameters considered for calculations are reported in Table 1.

The transport current is varied between 3and 13kA,correspond-ing to beam energies between 1.8and 7.7TeV,respectively.The bus

bar is subjected to its magnetic self-?eld,ranging between 0.12and 0.52T depending on the current.The current is kept constant until the detection time s det is reached.Then it decays exponentially with the time constant s dump .The detection time was determined calcu-lating the voltage developed per meter of MB bus bar from the cop-per resistivity at room temperature q el ,the RRR bus here assumed to be 215[7],the bus bar copper cross-section A Cu and the current I .Considering a voltage threshold V th of 0.3mV at 7TeV of the new quench detection system [2]and the measured quench propagation velocity v q =0.4m/s [18],s det for I =13kA can be calculated as:

s det ?

V th

v q á

q el áI

RRR bus áA Cu

?0:2s :e3T

For simplicity and in order not to affect the parametric studies,the same detection time is considered for the MB and MQ bus bar,for any current.The time constant of the current decay depends on the set-up of the dumping resistor in parallel to the magnets circuit that is activated in case of a magnet quench,as well as on the cur-rent (or beam energy)level.The s dump is varied in our simulations to perform the parametric analyses reported in Section 5.Unless speci?ed otherwise,its value is taken equal to 100and 20s for the MB and MQ circuit,respectively.

The bus bar is in contact through the electrical insulation with the static He II coolant ?lling the pipe.No helium is assumed to be inside the bus bar or inside the interconnection.The considered operating conditions of 1.9K and 0.1MPa are imposed as initial conditions,as well as boundary conditions for the coolant.In order to assess the impact of the cooling,other heat transfer modes are also considered,namely He I and the adiabatic case.The heat trans-fer coef?cients HTC s are detailed in Section 3.3.The operating parameters used for the calculations are reported in Table 2.

As far as the simulation parameters are concerned,convergence studies were performed to obtain con?dence in the results and set appropriate values of the space and time step to be used for the numerical integration.Calculations were repeated varying the number of elements in the mesh as well as the time step.As for the mesh,if on the one hand the system sensitivity to the defect length requires short elements,on the other hand a minimum length of the bus bar is needed to avoid that the boundary condi-tions affect the results.Ful?lling both requirements using a uni-form mesh would result in a very large number of elements (at least 0.5mm long elements over a 2m long bus bar,thus 4000ele-ments),meaning long computation times.A static optimized mesh was therefore developed,subdividing the physical domain into two different parts as shown in Fig.5c:

àa ?ne mesh in the interconnection region,including the defect;àa coarse mesh zone at the edge,far from the defect zone.The convergence studies,carried out for different defect lengths,investigated the element size inside the ?ne zone and in-side the coarse zone,as well as the length of the ?ne zone.They showed that,for defects smaller than 3cm,the elements length in the defect region should be smaller than 0.5mm to catch the solution features.For defects larger than 3cm,a uniform mesh with 2.5mm long elements is suf?cient for the scope of the analy-ses and does not require long computation times.As for the time step,the convergence study indicated that time steps shorter than 10ms at the beginning of the transient and limited to a maximum of 0.1s are suitable.

3.3.Heat transfer to the helium cooling bath

The mechanisms of heat transfer from bus bar and interconnec-tion towards the helium bath are of crucial importance for the modeling,as it will be shown.A dedicated test provided the

heat

Fig.5.From top to bottom:scheme of an interconnection with a single defect;modeling concept;model with half the defect length assumed for the calculations.Not to scale.

53(2013)107–118

transfer coef?cient through the bus bar electrical insulation (HTC bus)in He II and He I bath for a limited range of temperatures. The data analysis allowed extending the results to higher temper-atures.The results of this study,which can be found in[19],are summarized in Fig.6.The HTC features larger values in He I than in He II bath,because the dominant heat extraction mechanism is thermal conduction through the polyimide insulation.In these experimental and theoretical studies,a constant bath temperature was considered.However,the validity of such hypothesis has not been demonstrated in the LHC conditions.In a?rst approximation of heat deposit localized in the middle of the defect and assuming turbulent He II heat transport law,an axial cooling power of around100W would be provided along the sample,while main-taining the super?uid helium state.This would allow extracting the heat deposited in an interconnection experiencing a runaway in nominal operating conditions,and maintaining the mentioned HTC.In case of He I bath,the latent heat of the helium volume sur-rounding the bus bar would be greater than the heat deposited for the typical duration of a controlled quench or runaway.That would result also in this case in a constant bath temperature.

While a good knowledge of the heat transfer coef?cient through the bus bar insulation(HTC bus)was obtained,a complete descrip-tion of the heat transfer coef?cient through the interconnection insulation(HTC ic)would require further investigations.Fig.7sche-matically shows the regions of the two heat transfer coef?cients. HTC ic can be derived from the investigation of experimental tests reproducing defective interconnections[20],thus allowing further improving the reliability of the calculations.An analysis based on local heat transfer coef?cients has provided information in this direction[21]and the main results are summarized in the next sec-tion.For the parametric analyses reported in Section5,two differ-ent approaches are considered:either the same bus bar heat transfer coef?cient is assumed everywhere,that is HTC ic=HTC bus, or the conservative assumption of an adiabatic interconnection is made(HTC ic=0).In the?rst case,the bus bar wetted perimeter is assumed over the whole sample length.Unless speci?ed other-wise,the cooling bath is considered at the nominal temperature of1.9K.

4.Model validation

Prior to the presentation of the parametric analyses,we report in this section the validation of the model,which was performed by simulating experimental tests of an interconnection sample fea-turing a purposely built-in defect.The setup details and experi-mental results are described in[20].The analyses here reported refer to tests in He I bath without external magnetic?eld.The con-sidered instrumented sample features a35mm long defect located on one side of the interconnection,corresponding to an additional resistance R add of42l O.Fig.8shows a sketch of the sample longi-tudinal cross-section,where the defect is created by polyimide layers around the superconducting cables and between interconnection and bus bar stabilizer.Thermofoil heaters are placed in contact with the interconnection(heater W)and the bus bar stabilizer(heater M)to start the normal zone.The ther-mo-couple junctions P2,U and M1are located inside the left bus bar,interconnection and right bus bar Cu stabilizer,respectively.

The modeling approach presented in the previous section is used,accounting in this case also for the details of the interconnec-tion region and focusing on the de?nition of local heat transfer

Table2

Operating parameters of the MB and MQ bus bar considered for the calculations.

MB MQ

Nominal current(kA)11.85

Nominal beam energy(TeV)7

Current range(kA)3–13

Beam energy range(TeV) 1.8–7.7

Nominal magnetic self-?eld(T)0.47

Magnetic self-?eld range(T)0.12–0.52

Detection time s det(s)0.2

Nominal decay time constant s dump(s)10020 s dump range(s)50–10010–30 Initial Bus Bar temperature(K)10

SC cable critical temperature T c(K)9.01

Nominal He initial temperature T He in(K) 1.9

T He in range(K) 1.9–4.25

He initial pressure P He in(MPa)0.1

He mass?ow(g/s)0

Bus bar boundary conditions Adiabatic

He boundary conditions T=T He in,P=P He in

coef?cients.The extensive description of the model and of the re-sults is reported elsewhere[21].In particular,the superconducting cables and the Cu stabilizer are de?ned as separate thermal and electric elements,whereas the heaters constitute the third thermal element.The thermal resistances among the elements are de?ned by solid conduction through polyimide and SnAg layers.The heat transfer coef?cient between the Cu stabilizer and the bath in the bus bar region is the one reported in Fig.6.For the interconnection region,the heat transfer coef?cient is given by solid conduction through the polyimide and?berglass layers,whether the stabilizer temperature is below6.5K.The?berglass is used to model the VP310material.Above6.5K the thermally insulating?lm boiling layer is considered,described by a heat transfer coef?cient of 250W/m2K.The threshold temperature represents a rough model to describe?lm boiling formation in the interconnection.It is cal-culated from the heat?ux at which,according to[19],?lm boiling formation is estimated to occur around the bus bar.As for the heat transfer from stabilizer and cables to the He?lling the void spaces inside interconnection and bus bar,it is described by the Kapitza resistance Cu–He before the He vaporizes and by the?lm boiling heat transfer coef?cient afterwards.The geometric and operating parameters are set equal to the measured values.

Fig.9reports measured and calculated temperatures for a hea-ter calibration test:only heater M was turned on for30s with a power of18W,without any current?owing in the sample.The steady-state temperature,reached after few seconds,is higher for the sensor located below the heater M than for the sensors on the other side of the defect.

The calculations are performed using both a simpli?ed model featuring an adiabatic interconnection and the above mentioned complete model.The?rst model,assuming no heat transfer to-wards the He bath and no He?lling the void spaces inside inter-connection and bus bar,does not catch the features of the measurements.The calculated steady-state temperatures are high-er than the measured ones,the transient states are also very differ-ent and the measured time delays are not predicted.The complete model instead well simulates the test.The heat transfer through the interconnection insulation lowers the steady-state tempera-tures that are correct within0.2K.The He inside the interconnec-tion and the bus bar allows reproducing the transient features and the initial time delays.It is worth noticing that the changes of slope of the calculated curves,which re?ect those of the measured ones, are associated to boiling of the He inside interconnection and bus bar.In particular the?rst change of slope of the calculated U curve occurring at2s is associated to the end of boiling of the He close to the defect.The second change of slope at2.4s corresponds to the start of boiling of the He located at the left interconnection extrem-ity.The end of boiling of this He occurs at2.7s when the U curve features the last change of slope and the P2sensor,located just on top of this He,starts heating up.

In the tests with current the heat sources are both external and internal,respectively due to heaters M and W both turned on and to Joule heating in the resistive components.Fig.10reports the time evolution of the three thermo-couple junctions.The current is ramped up to the constant level of11kA and no temperature in-crease or voltage is detected in the?rst second,before the heaters are?red.The heat pulses of15W for heater M and14W for heater W last between1and2.2s.Due to larger heat generation and less ef?cient heat extraction,the temperature increase in the defect zone is larger than in the left side of the interconnection.As soon as the voltage threshold for switching off the power supply is reached,the current is shut down with a dump time of few hun-dreds of milliseconds and the sample recovers the superconducting state.

In such case the temperature and voltage responses of the mod-el are sensitive to the parameters that drive the Joule effect inside the interconnection.The tuning of the thermal and electrical resis-tances between the Cu stabilizer and the superconducting cables is therefore necessary,to take into account the contact resistances in addition to the SnAg bulk heat and current transport properties. The ratio of the tuned thermal and electrical conductivity yields a value of3.5,which agrees with the a coef?cient included in the Wiedemann–Franz law in[22]to account for impure materials. The computed and experimental temperatures presented in Fig.10are in good agreement,except for the initial transient state of the U temperature sensor.The corresponding voltage traces re-ported in[21]show as well very good agreement for different transport currents.

5.Parametric analyses

This section reports the parametric studies performed to assess the impact of several parameters on the interconnection critical defect length,assuming both adiabatic and non-adiabatic condi-tions.We used the model described in Section3,which is simpli-?ed with respect to the one mentioned in the previous section while maintaining the fundamental physics.The critical defect length leading to thermal runaway is determined as a function of the current decay time constant s dump,RRR of the SC cable copper matrix,RRR of the bus bar stabilizer and spatial distribution of the defect.The major effect of the heat transfer coef?cient in the bus bar and in the interconnection region is investigated.

5.1.Parametric studies in adiabatic conditions

The?rst sets of simulations were performed in adiabatic condi-tions,thus neglecting the last term at left hand side of Eq.(1)con-

Measured vs.calculated temperatures in case of no current and only switched on.

cerning the heat exchange with helium.The temperature evolution as a function of time at different positions is shown in Fig.11for two defect lengths of the MB bus bar interconnection.Fig.11(left) and(right)refer to the case of a7and8mm long single defect, respectively.They constitute the maximum defect length allowing recovery and the minimum one leading to the interconnection burn out.The middle of the defect represents the hot spot,which in our calculations is located at the right extremity of the analyzed domain and corresponds to the longitudinal coordinate x of2m. The temperature obviously decreases towards the bus bar inlet, but cannot decrease down to the initial temperature because adia-batic conditions were de?ned along the cable as well as at the boundaries.The longitudinal temperature gradient is located with-in1m from the defect,since the temperature pro?le is in practice the same at the coordinates0.5and1m.

After15s of almost uniform temperature rise all along the sam-ple,the defect zone starts heating up faster than elsewhere.The larger Joule heating is due to the smaller available copper cross-section.The change of slope corresponds to that of the copper elec-trical resistivity vs.temperature.In case of Fig.11(left)the tem-perature remains always below400K.A recovery is observed because the rate of heat removal by conduction along the bus bar is greater than the rate of heat addition by the Joule effect. Therefore there is no further temperature increase.In case of Fig.11(right),in the same operating conditions but with a1mm longer defect,the behavior of the sample drastically changes. Although the situation is not different during the?rst15s,a ther-mal runaway occurs in the defect because the heat generated is not compensated anymore by the heat conduction mechanism.The hot spot temperature exceeds500K in about25s,which is referred to as burn out time.The sample integrity is not guaranteed any longer.

Based on these considerations,a curve can be traced reporting the minimum defect length leading to thermal runaway for differ-ent current levels,i.e.different beam energies.Such current will be referred to as limiting current,which leads to an interconnection hot spot temperature of500K.Two space regions are identi?ed: a region of controlled quench corresponding to operating condi-tions located below this limiting current curve,and a region of thermal runaway above the curve.The difference between the largest defect leading to a controlled quench and the smallest de-fect leading to a thermal runaway is always of1mm,as in the case just mentioned.This difference is comparable to the uncertainty in the interconnection defect measurement in the LHC[4].

Fig.12reports the limiting current curves calculated for the MB (left)and MQ(right)interconnection in case of different current decay time constants s dump:50and100s for MB;10,20and30s for MQ.They correspond to the most probable s dump in case of a 3.5and7TeV operation.Note that the same s dump is here assumed for any current/energy level.Although this is an approximation that does not re?ect reality,it is useful to assess the in?uence of this parameter.The MQ interconnection exhibits a higher limiting current with respect to the MB one because of the shorter current decay time constant.The reduction of s dump allows signi?cantly increasing the critical defect length in the whole current range. For MB it becomes between20%and50%larger when reducing s dump from100to50s,depending on the current.As for MQ,its in-crease is more relevant when reducing s dump from20to10s than from30to20s.s dump has such an impact on the critical defect length because it is comparable to the burn out time.A s dump much larger than the burn out time would feature a considerably smaller effect,as it will be the case in non-adiabatic conditions(Sec-tion5.3).These results obtained in adiabatic conditions are in good agreement with other models[17],with differences within10%.

As stated in Section3.1,the defect length is directly propor-tional to R add.The highest measured R add of60l X[16],which was repaired after the incident,is reported in Fig.12(left).This va-lue is larger than the R add of10l X that should not be exceeded to allow operating the LHC at full energy.This is the reason for ini-tially running the LHC at the limited energy of3.5TeV per beam with the current interconnection design,whereas a consolidation intervention is needed to reach the nominal beam energy of7TeV.

Fig.13presents the results obtained for different values of cable and bus bar copper Residual Resistivity Ratios(RRR s).The RRR is de?ned as the ratio of the copper resistivity at room temperature over its value at10K,which is just above the superconductor crit-ical temperature T c.The variation ranges were?xed according to measured values that were summarized in[17]:between80and 160for RRR cable and between100and220for RRR bus.For clarity, simulations carried out with the only boundary values are reported here.The effect of larger RRR values,permitting larger heat evacu-ation through longitudinal copper conduction,is only evident at small current levels.The cable copper has a larger impact than the bus bar copper.This is because the cable copper is in contact with the superconducting?laments,also inside the defect.For in-stance at3500A the RRR cable has a two to three times greater effect than the RRR bus on the critical defect length,when both are varied between the boundary values of their relative ranges.

So far the case of a single defect located on one side of the inter-connection was considered.The defect can also be split in two parts,as shown in Fig.14a.The additional resistance measure-ments performed throughout the whole interconnection do not al-low distinguishing between single or double defect,which provide the same additional resistance R add in case of equal total length. The double defect case was analyzed according to the model shown in Fig.14b,whose length is4m instead of2.In order to ob-tain a direct comparison to the single defect case,in the following we will refer to the length of a double defect as the sum of its two parts.The investigated proportions between them are50–50%and 75–25%.

Fig.15shows the temperature pro?les along the4m long con-sidered sample at different times.The double defect(50–50%)has a total length of8mm.A single defect of the same length in the same operating conditions leads to a thermal runaway,as it was shown in Fig.11b.In this case instead,a recovery can be observed after a temperature rise up to$200K,where the peaks correspond to the middle points of the two defect parts.The split defect exhibits therefore higher limiting current with the same total length, thanks to larger longitudinal heat conduction from the hot spots.

Limiting current curves corresponding to double defects are presented in Fig.16.The50–50%case is reported,as well as the 75–25%.The latter has a smaller limiting current than the equally split defect because of the more severe conditions of its largest part.The splitting of the defect provides major increase of the

crit-53(2013)107–118113

ical defect length at all current levels:it is of the order of60%and 80%when going from a single defect(100–0%)to a50–50%double one for the MB and MQ,respectively.

5.2.Impact of the heat transfer mechanisms

The previous results were obtained in the conservative hypoth-esis of adiabatic conditions.It is not the case of the LHC bus bars, which are immersed in a helium bath at1.9K.The present subsec-tion analyzes the effects of helium cooling.Fig.17(left)reports the hot spot temperature evolution as a function of time for several de-fect lengths of the MB bus bar interconnection.With respect to the adiabatic case of Fig.11,in the same operating conditions,the transverse heat transfer allows considerably increasing the critical defect length from8to24mm.The burn out time is consequently shorter,from25to8s,due to the larger associated heat generation that results from the increased R add.As far as defects leading to controlled quench are concerned,larger defects feature higher peak temperatures and longer recovery times.A sudden tempera-ture drop can be observed when the Joule heating is interrupted due to the resistive-to-superconducting transition of the whole sample.However,the transition does not occur at once:the discon-tinuity preceding the temperature drop corresponds indeed to the recovery of the region at the sample inlet.

Fig.17(right)summarizes the main features of simulations per-formed with various defect lengths:defects leading to controlled quench are characterized by the maximum hot spot temperature reached before recovery(left y-axis),whereas defects leading to thermal runaway are characterized by the burn out time(right y-axis).If the defect length is such that the hot spot temperature

Temperature evolution in time for a MB defect leading to a controlled quench(left)or to a thermal runaway(right),in adiabatic Limiting current curve as a function of the current dump time for MB(left)and MQ(right)interconnection,in adiabatic conditions.

Limiting current curve as a function of the cable and bus bar RRR for MB(left)and MQ(right)interconnection,in adiabatic conditions.

exceeds$35K,the upper temperature limit is reached within0.5–8s.Because of the larger Joule heating,the burn out time decreases with increasing defect length.The vertical line dividing the two zones corresponds to the critical defect length.This last one will be the only information reported in the following?gures.

5.3.Parametric studies in non-adiabatic conditions

Besides the assessment of the major improvement of the critical defect length due to helium cooling,we determined the effect of the parameters considered in Section5.1also in non-adiabatic con-

The helium cooling enhances the cable copper RRR effect,as shown in Fig.20.The critical defect length roughly doubles when increasing RRR cable from80to160.Such a behavior,although con-tradictory at a?rst glance,is due to the better heat conduction associated to a higher RRR.The heat generated in the hot spot spreads more easily in the longitudinal direction thus increasing the heat exchange surface to the bath.The effect is evident at all current levels.On the contrary the bus bar copper RRR has only a minor impact,as it was the case in adiabatic conditions.

As far as the splitting of the defect in cooled conditions is con-cerned,it permits a critical defect length improvement of the same order of magnitude as in the adiabatic case.Fig.21reports the rel-evant calculations.6.Conclusions

A thermo-electric analysis of the bus bar interconnection of the LHC main dipole and quadrupole magnets was performed,follow-ing the2008incident,to estimate the critical length of potential interconnection manufacturing defects.A1-D model was devel-oped with particular care in the de?nition of transverse local heat transfer coef?cients towards the cooling helium bath.Such ap-proach was supported by dedicated measurements and analyses of heat transfer through the bus bar electrical insulation.

The model was validated by comparison to experimental tests of defective interconnections performed in He I bath.The calcula-tions addressed the heat transfer coef?cient through the intercon-

Limiting current curve as a function of the current dump time for MB(left)and MQ(right)interconnection,in non-adiabatic conditions.

nection insulation,showing that the interconnection cannot be considered adiabatic.However,further investigations would be needed to completely describe the thermal mechanisms occurring in the interconnection.Therefore we made various hypothesis of cooling,which allowed identifying the predominant impact of the helium contribution on the limiting current,compared to the other parameters analyzed.The closest to reality assumption of He II cooled bus bar and adiabatic interconnection can be seen as a reasonably conservative lower limit.Such cooling model provides an increase of the critical defect length by a factor2with respect to the most pessimistic hypothesis of fully adiabatic conditions.The most optimistic hypothesis of He II bus bar heat transfer coef?cient assumed for the whole sample length allows an increase by a factor 3.

The effect of parameters related to the interconnection manu-facturing quality,operating conditions and protection system was estimated through sensitivity analyses.The cooling mode ap-pears to strongly modify their in?uence on the limiting current.In particular:

àthe impact of the current discharge time is signi?cant in adia-batic conditions and negligible in non-adiabatic conditions.This is due to the different burn out time,comparable to the dis-charge time in the?rst case and much smaller in the second case.

àthe cable RRR has a greater impact in cooled than in adia-batic conditions.The reason is the larger spread of heat in longitudinal direction,which increases the heat exchange surface to the bath.The bus bar RRR exhibits a low impact both in adiabatic and cooled case.

àthe defect spatial distribution has a considerable effect both in adiabatic and non-adiabatic conditions.

The model complemented other evaluations used for the best estimates of interconnection critical additional resistances,prepar-ing for LHC2012operation at the highest possible beam energies. Calculations were performed assuming the actual current dump time constants or possible longer values for beam energies be-tween3.5and4.5TeV.The most pessimistic(adiabatic),most opti-mistic(full cooling)and most likely(partial cooling)conditions were considered,by varying cable/bus bar RRR and space distribu-tion of the defect.These calculations showed that:

-MB and MQ limits obtained in adiabatic conditions,low cable/ bus bar RRR(80/100),for single-sided defect are comparable to those quoted in previous analyses(e.g.R add=43l X for MB at4TeV,s dump of52s);

-an increase of the cable/bus bar RRR(160/220)to the values measured(bus)or expected(cable)in the LHC,and considering

a soldering defect distributed in the interconnection(50–50%),

yields an increase of the acceptable value of R add by more than

a factor2(R add=97l X for MB at4TeV,s dump of52s);

-assuming conservative values for the cable/bus bar RRR(80/ 100)and single-sided defect,but considering realistic values for the heat transfer from bus bar to helium,yields a50% increase of the acceptable value of R add(R add=66l X for MB at4TeV,s dump of52s).

Acknowledgments

The authors wish to thank M.Bianchi who performed the sim-ulations of the experimental tests during an internship at CERN. They are also grateful to F.Bertinelli,P.Fessia and D.Richter for the fruitful comments and suggestions,as well as to K.Dahlerup-

Limiting current curve as a function of the defect spatial distribution for MB(left)and MQ(right)interconnection,in non-adiabatic Limiting current curve as a function of the cable and bus bar RRR for MB(left)and MQ(right)interconnection,in non-adiabatic

Petersen,A.P.Verweij and G.P.Willering for the useful information and data provided.

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