Cryogenic thermo-acoustic oscillations highlight and study in the SPIRAL2 superconducting LINAC
Adnan Ghribi, Muhammad Aburas, Yoann Baumont, Pierre-Emmanuel Bernaudin, Stéphane Bonneau, Jean-François Leyge, Jean-Pierre Thermeau, Yann Thivel, Laurent Valentin, Adrien Vassal
CCryogenic thermo-acoustic oscillations highlight and study in the SPIRAL2superconducting LINAC
Adnan Ghribi,
1, 2, ∗ Muhammad Aburas,
1, 3
Yoann Baumont,
1, 3
Pierre-EmmanuelBernaudin,
1, 3
St´ephane Bonneau,
1, 2
Jean-Fran¸cois Leyge,
1, 3
Guillaume Lescali´e,
1, 3
Jean-Pierre Thermeau,
4, 2
Yann Thivel,
1, 3
Laurent Valentin,
1, 2 and Adrien Vassal
1, 5, 6, 7 Grand Acc´el´erateur National d’Ions Lourds (GANIL) Centre National de la Recherche Scientifique (CNRS - IN2P3) Commissariat `a l’Energie Atomique (CEA - IRFU) Laboratoire AstroParticule et Cosmologie Commissariat `a l’Energie Atomique (CEA/INAC-SBT) Univ. Grenoble Alpes Univ. Caen Basse Normandie (Dated: January 27, 2021)Cryogenic thermoacoustic oscillations is an area of interest of several studies. For superconduct-ing accelerators, it is an unwanted phenomena that we usually want to get rid off. The SPIRAL2superconducting accelerator had distributed Taconis all over its cryostats. This paper spans thedifferent steps from their first detection to their damping with a highlight on the methods and theinstrumentation that has been used. The presented study also sets the ground for a real life experi-mental investigation of thermo-ascoustics in complex geometries such as superconducting LINACs.With modern Big data analysis and simulation tools, it also sets the ground for the developmentsof new data linked codes that fit these complex situations.
I. INTRODUCTION
SPIRAL2 is a new generation heavy ions accelera-tor. Its heart is a superconducting linear accelerator(LINAC). 19 cryostats (called cryomodules) are spreadalong the beam line and encompass the acceleratingstructures (superconducting quater wave resonating cav-ities) [1–3]. They are connected with a cryodistributionline and fed with near atmospheric pressure liquid helium( 4 K) for the main cavities and 14 bars 60 K helium gasfor the thermal screens. The SPIRAL2 cryoplant cen-tred around an Air Liquide Helial LF cold box managesto furnish the necessary cooling power. It can manage120 g/s and 1100 W at 4 K [4, 5]. It is worth mentioninghere that the cryomodules are of two kinds/families. 12of them enclose a single cavity and are called type A. Theother 7 enclose two cavities and are called type B. Thevalves boxes that manage the fluids of these cryomodulesalso have some geometrical differences. One of the mainroles of the cryogenic system is to maintain stable condi-tions compatible with the operation constraints. That isto maintain all the cavities at a stable and uniform tem-perature (plunged in liquid helium) and the pressure vari-ation within ± ∗ Electronic address: [email protected] the cavity wouldn’t be matched anymore to its frequencyof operation. There are of course a number of correctionsto impedance or RF phase changes. For instance, theLow Level Radio Frequency (LLRF) system feeding theRF power to the cavity can manage a certain bandwidthat the cost of some power margin. This correction is fastand limited to small variations. The frequency tuningsystem can manage another correction, this time slow(more than one second) and adapted to large variationswithin its range of operation. The third and final way tolimit the impedance variations of the cavities is simplyby controlling the pressure in the phase separators. Thisstringent requirement for a biphasic cryogenic operationhas led to several model based developments of the wayinput and output valves can be controlled [6, 7].When we first fed RF power to the cavities, we no-ticed a strange behaviour in the amplitude and phase ofboth transmitted and reflected RF power. After deep in-vestigations, it turned out that what we were measuringwas due to cryogenic thermo-acoustic oscillations (TAO)spread all over the LINAC.This phenomenon named after K. W. Taconis who firstencountered it in 1945 [8] is know to occur in a cryogenicsystem for specific geometry, temperature and pressureconditions [9]. They have been thoroughly studied [10–14] but remain difficult to predict and to deal with.This paper relates the investigation, discovery and sup-pression of these oscillations with a highlight on themethods and instrumentation used. In the first section,we introduce the first discovery of the TAO on the SPI-RAL2 cryogenic system. We then describe the experi-mental methods used to determine its frequency, ampli-tude and conditions of occurence. Finally we describe thetrials and experimental solutions tested, the final chosencorrection, the surveillance system that we put in place a r X i v : . [ phy s i c s . acc - ph ] J a n C a v iti e s li qu i d h e li u m b a t h p r e ss u r e [ m b a r] D e v i a ti on c oun t e r P c = m b a r ; D e lt a ( P ) = m b a r A A A A A A A A A A A A B B B B B B B Cryomodules
41 165 1825 53 68 72 315 365 264 98 507 10
FIG. 1: Top : pressure deviation counter with a threshold setto ± with its future evolution and numerical and theoreticalconsiderations on the subject. II. FIRST ENCOUNTERS
The first TAO that we discovered was in the main5000 L liquid helium dewar of the cryoplant in 2016.These oscillations were hard to miss : high thermal load,ice on the dewar neck, sometimes hearable continuousnoise and the fast displacement of the pointer needleof the pressure gauge. To better estimate and quantifythe phenomenon, we had to bypass the cold box Pro-grammable Logic Controller (PLC) acquisition systemthat was too slow to measure the occurring phenomenon.We tried several approaches to suppress the dewar TAO: investigation of all the room temperature output portswith sometimes long lines used for filling or supplying liq-uid helium (LHe) from or to small mobile dewars, shortcircuit between the vapour sky and the room temper-ature lines and buffer capacities of several dimensions.Finally the approach that worked the best was the use ofa 5 L buffer capacity in parallel with a micrometric valve[15]. This TAO was at 100 Hz and could not be seen withregular slow absolute pressure transmitters. However, itseffect could be seen also on cavities pressure. Even aftercorrection of this TAO, some behaviour in the LINACremained unexplained. This took the form of pressureglitches and pressure instabilities that propagated to allcavities with no prior notice. This behaviour made thepressure regulation very difficult, as if the absolute pres-sure sensors used by the valves to control the pressurewas not giving a reliable information. Figure 1 shows anexample of such behaviour. In July 2017, we began the first injection of RF lowpower in the cavities for their cold characterisation [16].We then noticed an amplitude modulation of the RFsignal. Frequencies ranged between 4 to 6 Hz. Themeasurements were consistent and repeatable. However,resonance frequencies seemed to depend on the cavitiespositions. Figure 2 shows an example of such a spec-trum. Figure 3 shows the diagram of the measurementtest setup. A low power RF frequency generator wasused to inject a signal locked at the resonance frequencyof the measured cavity (found by a self-oscillating closedloop circuit). All measurements were done directly in theLINAC tunnel, bypassing the LLRF system to avoid thehigh losses that would have blinded us from seeing thesignal we seek. The phase and the amplitude differencebetween the input and the output signal were extractedand measured thanks to a Yokogawa network analyser. . . . . . Frequency [MHz] +8 . × − − − − − A m p li t ud e [ d B ] ν ν ν − ν = 5.35 Hz FIG. 2: Amplitude modulation of the input/output signaldifference for cryomodule CMB07 as reported in [16].
MOD IQSynthesizer SIG GEN MOD IQSpectrum AnalyzerDAQ ɸ i npu t ɸ ou t pu t Δ(ɸ) sy n c i npu t Q W R c a v i t y
10 dB10 dB 10 dB
FIG. 3: Block diagram of the RF measurement setup for thelow power characterization the SPIRAL2 LINAC cavities.
The first procedure that has been applied was exter-nal vibration measurements with a triple axis piezoelec-tric accelerometer. Different locations have been mea-sured and monitored including : dewar pipes, valvesboxes pipes, pumps and cryomodules. Figure 4 showsa summary of these measurements. It has been shownthat there is no correlation between the frequency of theroughing pumps and the measured peaks. It has alsobeen found no peak bellow 10 Hz. However, peaks havebeen measured at 30 Hz and several of its harmonics onlyin the fluid direction (x direction in Fig. 4).
CMACMBConnection boxDewar feedlineValves Boxes A
Frequency [Hz]
Valves Boxes B X axisY axis
FIG. 4: Accelerometer vibration measurements at differentlocations of the LINAC. A and B valves boxes are the valvesboxes of cryomodules types A (CMA) and B (CMB). The Xdirection is the direction of the pipes. The Y direction is ahorizontal orthogonal direction. The Z direction is not shownhere but has not exhibit any peak. The peaks are normalizedto an amplitude of G = 0.1 are shown with the same scale.
III. FURTHER EXPERIMENTALINVESTIGATIONSA. Single cryomodule experiment Q W R c a v i t y Valves Box DAQ
PT PTPT absolutepressurePT
EPICSIOC ANALYSISPIPELINE
CHLTabsolute pressureheaterliquid helium level piezoelectric relative pressurepiezoelectric relative pressureΔ(ɸ)Acc accelerometer vibration piezoelectric measurements
FIG. 5: Block diagram of the 2018 measurement setup.
In 2018, a different measurement setup has been in-stalled. It included simultaneous measurements of abso-lute pressures, relative pressures, RF phase shifts, liquidhelium level, heaters and accelerometers at different lo-cations of a target cryomodule. A block diagram of suchsetup is shown in Figure 5. We measured both the ab-solute and the fast relative pressure directly in the cry- omodule phase separator and in the corresponding valvesbox return line. We used piezoelectric sensors (PCB113B28) for relative measurements and metallic processisolating diaphragm sensors (Cerabar PMP71) for abso-lute pressure measurements. We also measured vibra-tions with a tri-axis piezoelectric accelerometer. Thesame setup shown in Figure 3 has been used to extractthe phase shift between the RF input and output sig-nals. A National Instrument Compact DAQ centralisedthe fast acquisition with a 3 channels 2.8 kS/s/ch 24-Bitanalog modules for the IEPE signals (Integrated Elec-tronics Piezo-Electric) and a universal analog module forthe RF signal. The NI DAQ was driven by an exter-nal laptop running Labview. Other data such as ab-solute pressures, heater power, liquid helium level andtemperatures were measured through our regular PLCand archived with an Experimental Physics and Indus-trial Control System (EPICS) Input/Output Controller(IOC). A Python analysis pipeline assembled fast andslow acquisitions together with other correlation factorsand a common clock. The results showed (see Figure 6)a direct correlation between the pressure measurementsand the RF measurements. As a reminder, the purposeof this experiment is not only to understand but to atten-uate this phenomena down to an acceptable level. TheRF amplitude and phase variation were used as a proxyto determine the acceptable damping amplitude of thepressure oscillations. Given the margin on the RF am-plifiers, the limit has been set to 20 dB in RF amplitudemodulation which corresponds to roughly 10 mbars fastpressure variation. T i m e [ hh : mm : ss ] Pressure [mbar] . . . . . Phase shift [deg]
FIG. 6: Overnight simultaneous measurements of pressureoscillations and RF phase shift modulations for cryomoduleCMA04 (2018-11-24).
To better understand these measurements, a pipingand instrumentation diagram of a type B cryomodulewith its valves box is shown in Figure 7. The magentacoloured circles show the position of the two main pro-cess valves. The latters control the liquid helium leveland the pressure in the cavity phase separator. Know-ing that thermoacoustic oscillations usually appear forcertain conditions with high temperature gradients, weinvestigated all room temperature ports with a cold end.We identified the red line (see Figure 7) that links theliquid helium bath to the security and purge ports of thevalves box as a possible candidate. In order to investigatethe possible start conditions of the oscillations (pressure,valves positions, thermal load, liquid helium level), weinstalled a couple of pressure sensors (absolute and rela-tive) directly at the outlet of the security port of the liq-uid helium phase separator of the cryomodule (port X1 ).We installed another one at the outlet of the candidateroom temperature port (port X 2 ). We found that thepressure oscillations start for a combination of pressure,liquid helium level and thermal load when the absolutepressure difference between X1 and X2 is around 1 mbar.We also noticed that when the TAO is present, the tem-perature of the return line increases. Thermodynamicmodelling of the cryomodule and its valves box using theSimcryogenics library [7] confirmed the consistency of themeasurements with an increase of the thermal load in thereturn line that fitted the temperature difference of thereturn line before and after a TAO [6].We then tried several methods documented in the lit-erature [17–20] to suppress or damp the TAO :1. Short circuit between the phase separator vapor skyand the return line (see Fig. 8) : Here we linked thetwo ports X1 and X2 with several pipes of severallength and cross-sections.2. Buffer (see Fig 9) : Here we connected severalbuffers of different volumes to the port X2.3. Piston (see Fig. 10) : Here we inserted a piston inthe port X2 and we monitored the behaviour of thesystem for several insertion depths.For every solution tried, we spanned all operating con-ditions to determine the most suitable solution for ourcase. In order to compare the results, we used an effi-ciency criteria defined as : ζ = P offbath P onbath (1)where P offbath is the amplitude of the pressure oscillationswith no TAO correction and P onbath is the amplitude of thepressure oscillations with the considered TAO correction.For every applied correction, we did see an effecton TAO damping but no total suppression has beenachieved. An example of efficiency of every correctionis shown in Figure 11 for different pressures and liq-uid helium operating conditions. It appeared that themost efficient solution for every case is the line short cir-cuit correction. This solution was efficient enough to bedeployed in all LINAC. The TAO correction efficiencyreached 1000 for some cases, damping the oscillationsfrom [10-100] mbars to les than 0.1 mbar. The short cir-cuit line was sufficient to recover the pressure balance TAO suspect line (cold GHe return)Absolute pressure sensorRelative pressure sensor Temperature sensor
Vavlves boxCryomoduleCavityLHe phaseseparator
LHe inlet G H e ou t l e t FIG. 7: P&ID of a type B cryomodule and its valves box. Thered line is the low pressure return line from the LHe bath tothe room temperature passive heater. Magenta circles arethe main process valves. Yellow circles are the rapid relativepiezoelectric pressure sensors. Blue circles are the absoluteprocess pressure sensors. The green circle is the position ofthe temperature sensor on the return line. between port X1 and port X2, therefore avoiding the ap-pearance of high amplitude TAO oscillations. However,the flow rate was so important sometimes that it frozepart of the line and the upper neck of the cryomoduleor resulted in condensation in the same locations. Wethen deployed it to all cryomodules with permanent linesterminated by a on/off hand valve for one end, a micro-metric hand control valve for the other hand and pressuresecurity valve in between. The micro-metric valve al-lowed us to control the flow in the correction line in orderto avoid water condensation or ice. The on/off valve al-lowed us to suppress any flow through the correction line.This was useful especially when cooling down the LINAC.This on/off valve allowed us also to de-activate or activatethe TAO correction for a more thorough investigation ofTAO amplitudes and frequencies cross-couplings.
Flexible line
FIG. 8: Vapor sky short circuit line solution setup - zoom inFig. 7.
Buffer capacity
FIG. 9: Buffer volume solution setup - zoom in Fig. 7.
B. Multiple cryomodules investigation
While it might seem obvious, for an isolated system,that thermoacoustic oscillations occur because of localconditions, the answer is not clear for coupled or con-nected systems. For the LINAC, we are in the complexcase of interconnected cryogenic clients with several roomtemperature ports with cold ends. It is therefore un-clear wether the amplitude and frequency dependance ofthe oscillation is dominated by local effects or global ef-fects. Observed transient pressure fluctuations with sud-den changes in frequency of the TAO could be due to
Piston
FIG. 10: Piston solution setup - zoom in Fig. 7.
90 95 L H e b a t h a b s o l u t e o p e r a t i n g p r e ss u r e [ m b a r ] Line correction 85 90 95Liquid helium level [%]
Buffer correction 25 85
Piston correction ζ FIG. 11: TAO correction efficiency ζ as a function of thepressure (PT001) and the liquid helium level (LT200) for thethree considered experimental corrections : line (vapor skyshort circuit), buffer and piston for cryomodules CMA05 andCMB03. such connections.
1. Experiment description
To have a better picture, we repeated the experimentdescribed in subsection III A and deployed the setup de-scribed in Figure 5 in all the LINAC. We deployed 19piezoelectric relative pressure sensors (one for every cry-omodule) in the LINAC and one at the central cold boxreturn line. Acquisition was made by the same DAQpreviously described with seven 24-Bit analog modulescontrolled with a Labview program. As previously, allfast acquisition data were treated with a python pipelineanalysis that combined PLC slow sensors and NI DAQfast sensors. This time, the pipeline allowed automaticpeak extraction and TAO detection. This allowed to havea much better view on what was going on in the LINAC.Several configurations have been tested :Configuration 1
Full LINAC OFF
In this configuration, no TAO correction has beenapplied. This configuration was meant to see thedistribution of amplitudes and frequencies of theTAO in the LINAC with mixing individual cry-omodules contributions as well as crosstalks.Configuration 2
Full LINAC OFF-1
In this configuration, TAO correction has been ap-plied to only one cryomodule at a time. That meansthat we closed all short circuit lines valves in allthe LINAC, disabling the TAO correction but inone cryomodule. We then spanned this configura-tion over all cryomodules. This setup was meantto see the effect of other cryomodules that had anactivated TAO on a cryomodule with an activatedcorrection.Configuration 3
Full LINAC ON
In this configuration, we activated the TAO cor-rection in all cryomodules at the same time. Thiswas meant to observe if we had other fast transientpressure fluctuations.Configuration 4
Full LINAC ON-1
In this configuration, TAO correction was activatedin all LINAC but in one cryomodule at a time. Wethen spanned this configuration to all cryomodules.This was meant to observe the frequency and am-plitude of a local TAO correction and if it had aneffect on other cryomodules. [] P r e ss u r e [ m b a r ] Time domain [] P r e ss u r e [ m b a r ] (5.3 Hz, 31.4 mbar) Frequency domain : : : : : : : : Time [hh:mm] R F ph a s e s h i f t [ d e g r ee ] Frequency [Hz] R F ph a s e s h i f t [ d e g r ee ] (5.3 Hz, 14.6 deg) FIG. 12: Example of time and frequency domain data ex-tracted and calculated with the pipeline analysis for CMA11[2018-11-11].
2. Results
As described previously the pipeline allowed automaticextraction of both frequency and amplitudes of the TAO oscillations. All data were gathered in time domain. Thereference of a relative pressure sensor is always zero whichmeans that the correct amplitude is found by offsettingthe relative data by the absolute pressures measured withthe CERABAR pressure sensors. The amplitude of theoscillations was found by enveloppe calculations withinthe considered time window using its Hilbert transform.For the frequency peak detection, we first applied a Four-rier transform on the data in a time window of 4 minutesin order to have a high resolution. We then applied a highpass filter to avoid the 1/f noise bellow 1 Hz and a lowpass filter to avoid high order harmonics. We finally com-puted the centroid of the resulting spectrogram to extractthe frequency peak. An example of a time and frequencydomain extracted data is shown in Figure 12. Thanks tothe extracted data, Parzen-Rozenblatt kernel density es-timations have been computed for both frequencies andamplitudes of TAO oscillations for every considered con-figuration. The analysis of 10,787 datasets allowed thecomparison of the different configurations :
Full LINAC OFF VS FULL LINAC OFF-1
Fig. 13
Frequency
For the configuration 1, with no TAO correc-tion, we see that most cryomodules have avery sharp resonance frequency around 5 Hz.There are several exceptions such as CMA02,CMA04, CMA10 and CMB03 that exhibitmultiple peaks. When the single cryomoduleTAO correction is applied (configuration 2) wesee a frequency shift for most cryomodules.
Amplitude
For the configuration 1, we see that the TAOamplitudes are distributed unevenly. Thewider distributions concerns most type A cry-omodules. We also see that the single cry-omodule TAO correction is very efficient asall amplitudes drop to almost 0 mbar. No ef-fect of the remaining TAO oscillations is visi-ble here.
Full LINAC ON VS FULL LINAC ON-1
Fig. 14
Frequency
This time, when only one cryomodule at atime has an activated TAO, the oscillationfrequencies seem un-correlated. When thecorrection is applied to all the LINAC, wesee many low amplitude oscillation frequen-cies with a distribution that seem commonto the type A cryomodules and a distributionthat seem common to the type B cryomodules.This suggests the native presence of multiplelow amplitude and geometry dependent oscil-lations occurring in the LINAC.
Amplitude
When only one TAO is activated at a time,pressure amplitudes seem less widely dis-tributed than in the configuration 1. This sug-gests mixed contributions of distributed TAOoscillations in the configuration 1. Once again,amplitudes seem to be geometry dependent asmore similarities arise among type A cryomod-ules and among type B cryomodules. The effi-ciency of the TAO correction (configuration 3)is noticeable as we see no high amplitude TAOoccurence when the correction is activated forall LINAC.
Full LINAC OFF-1 VS FULL LINAC ON
Fig. 15This comparison asks specifically the question ofthe effect of neighbouring or distant TAO oscilla-tions on a cryomodule that has an activated TAOcorrection.
Frequency
When other TAO oscillations are present,most of the time, we see clear sharp single ordouble peak frequencies. These are signaturesof distributed resonances and do not include alocal resonance (damped by the short circuitline). When the remaining TAO are damped,frequencies distributions take a wider shape.The latter conclusion is certainly a generalisa-tion that does not include CMA10 et CMB03that exhibit double resonance like behaviour.Nor does it include CMA06 and CMA07 andCMA09 that keep a sharp single peak reso-nance.
Amplitude
The observations of the frequency distribu-tions are confirmed by the amplitudes distri-butions. The latters drastically drop whenall TAO corrections are activated. CMA07and CMA09 are exceptions that remain andshow both sharp frequency peaks and notice-able resonance amplitudes (can reach 1 mbar).This is not critical as the phase modulation orrather RF impedance modulation resulting inthese oscillations is not significant and can becompensated by the RF amplifiers power mar-gin.
IV. ANALYTICAL INVESTIGATION
Given the known inner dimensions of the cryogenicpipes, the volumes and the temperatures, we tried ananalytical investigation of a localised thermoacoustic os-cillation in a cryomodule. We used Rott’s theory [21] toinvestigate criteria that drives the appearance of ther-moacoustic oscillations. We now know that being able todescribe these phenomena heavily depends on our under-standing of the viscous actions of the high density heliumin the inner tubes surfaces. We also know that thesedepend on several geometric parameters such as tubesdiameters, lengths, thicknesses as well as materials andtemperature gradients. To better represent the bound-ary conditions of appearance of the TAO, Rott has intro-duced a dimensionless quantity Y c that represents the ra- Cryomodule F r e qu e n c y [ H z ] TAO correction
Full LINAC OFF Full LINAC OFF - 1 C M A C M A C M A C M A C M A C M A C M A C M A C M A C M A C M A C M A C M B C M B C M B C M B C M B C M B C M B Cryomodules positions P r e ss u r e [ m b a r] FIG. 13: Full LINAC OFF vs Full LINAC OFF-1 : kerneldistributions half violin plots normalized at widths for com-parison between configuration 1 and configuration 2.
Cryomodule F r e qu e n c y [ H z ] TAO correction
Full LINAC ON - 1 Full LINAC ON C M A C M A C M A C M A C M A C M A C M A C M A C M A C M A C M A C M A C M B C M B C M B C M B C M B C M B C M B Cryomodules positions P r e ss u r e [ m b a r] FIG. 14: Full LINAC ON vs Full LINAC ON-1 : kernel distri-butions half violin plots normalized at widths for comparisonbetween configuration 3 and configuration 4. tio of the inner tube radius to the Stokes boundary layerthickness. Y c also drives the amplitudes of the Taconisoscillations. The Rott’s approach is a stepped stabilityanalysis with a discontinuity model where a temperaturejump happens somewhere in the tube where the oscilla-tion occurs [12]. The position of the temperature jumpis described as follows : ξ = L − l c l c (2)with L the length of the tube between the cold end andthe warm end and l c the length at which the temperature Cryomodule F r e qu e n c y [ H z ] TAO correction
Full LINAC OFF - 1Full LINAC ON C M A C M A C M A C M A C M A C M A C M A C M A C M A C M A C M A C M A C M B C M B C M B C M B C M B C M B C M B Cryomodules positions P r e ss u r e [ m b a r] TAO correction
Full LINAC OFF - 1Full LINAC ON
FIG. 15: Full LINAC OFF-1 vs Full LINAC ON : kernel distri-butions half violin plots normalized at widths for comparisonbetween configuration 2 and configuration 3. jump occurs (from the cold end). Y c is then given by : Y c = 2 Dα β a + ξ − + λ c ξ (3)with α being the ratio between the temperature of thewarm end T H and the cold end T C : α = T H T C (4)Here, additional constraints apply on α depending on ξ .When 0 < ξ < α = 0 . ξ Y . c . When ξ > alpha = 0 . Y . c . λ c is a dimensionless frequencyparameter defined by : λ = ω la c (5)where ω is the angular frequency, a c is the speed of soundin the gas at the cold end, l is the tube cold length.Usually, an assumption is taken that ξ = 1 meaning thatthe temperature jump occurs in the middle. In our casewe will consider all the solutions with ξ varying from 0to 1. To search for a solution for λ c , we can consider theequation from [22] : Y c λ = (cid:18) a c λ c (cid:19) r l − (6)where ν c is the kinematic viscosity of the the fluid. Thisleads to : Y c λ c = r (cid:18) a c Lν c ( ξ + 1) (cid:19) (7)For helium at such temperatures, we consider a c =1 . cm.s − . Given Rott’s analysis, D and α are constants with values that depend on the intermolecularforce field [22]. For helium between 300 K and 4 . K theyare set to D = 1 .
19 and β = 0 . r = 9 mm .These equations lead us the two branches of the Rott’sstability curve shown in Figure 16. This plot spans allcases of ξ from 0 to 1. The outer surface of the tracerepresents the stability region whereas the inner surfacerepresents the region where a thermoacoustic oscillationcan occur. Given the experimental observations, we alsoconsider that T C = 5 K and T H = 300 K , which leads to α = 60. This makes the SPIRAL2 case fall at the edgeof the stability curve implying that the overall length ofthe line is smaller than one meter. Given the observa-tions (frequencies and geometries), it is safe to say thatthe estimations of the location of the SPIRAL2 case inthe Rott’s stability curve is not correct. Several assump-tions can lead to this mistake. In fact, the geometry ofthe cryomodule/valves box ensemble is not as simple as asimple pipe. For example depending on the position, theinner tube has different diameters. The immediate envi-ronment also has an effet on the conditions of oscillationskick-off. It is therefore necessary to have more detailedapproach that fits the complex situation of inclusive sys-tems where cryogenic thermoacoustic oscillations occur. Yc10203040506070 α SPIRAL2 . . . . . . . . . . ξ FIG. 16: Rott stability curve and the case of SPIRAL2.
For the moment, let us consider that we don’t knowanything about the effective geometric parameters thatwould allow us to make a Rott’s analysis. Let us consider,as did Rayleigh in 1877 [23], the system as an oscillator(which in fact it is). In this situation, the gas in the crit-ical volume V is the spring mass and the gas in the tubeis the proving mass. The frequency of the oscillation, canbe directly derived from eq. 3.48 in [21] as : ω = (cid:18) πr l c V (cid:19) / (8)If we consider that the oscillating mass is the mass inthe tube, with V = πr ( L − l c ) and ξ = ( L − l c ) /l c , thenit comes : l c = a c πνξ − (9)We have observed that when the single oscillators arecoupled, we are dominated by a central frequency lyingaround 5 Hz (see Figure 13). If we consider this as aninput and vary the effective cold end length and the tem-perature of the cold end as a function ξ , we see (Figure17) that, in order to have small cold ends, we need ξ ≥ ξ = 1 (which is usually consideredfor this kind of analysis), we see from Figure 18 that theeffective cold end length should be somewhere between2 m and 15 m . If we consider an electromagnetic analogyand given the low frequencies observed (with respect toacoustic frequencies), this fits much better with a longwavelength. The problem that arises again here is thatthe effective impedance of an interconnected system isnot easy to estimate. This is critical in order to be ablede design a proper damper or any other solution to thiskind of problem. It is worth mentioning here that, inthese considerations, the dynamic behaviour of He hasbeen taken into account with respect to its temperatureand the pressure of the process (1300 mbars). Calcu-lations have shown that there is very little dependancebetween the frequency of the oscillations and the pres-sure of the process. However, experiments have showeda high dependance between the pressure and the start upconditions of the TAO. . . . . . . . . l c [m]5101520253035404550 T [ K ] . . . . variations of ξ [0 →
1] for ν = 5 Hz FIG. 17: Contours of the cold effective cold end length as afunction of the cold temperature value for different values of ξ and ν = 5 Hz. . . . . . . . . l c [m]5101520253035404550 T [ K ] . . . . . . . variations of ν [Hz] for ξ = 1 FIG. 18: Contours of the cold effective cold end length as afunction of the cold temperature value for different values of ν and ξ = 1. V. DAMPER DESIGN
The correction that proved efficient in subsectionIII B 2 has few drawbacks. It creates a helium flow inthe pipe passage where the liquid helium level sensorstands. This modifies the cold helium stratification shapein the liquid helium pipe volume and hence affects its be-haviours. It manifests as a plateau in the liquid heliumreadout. This effect has been observed several times dur-ing operation. The plateau’s location was different forevery cryomodule but it was usually between 89% and97%. Such plateau is a non linear behaviour that can’tbe handled by the PID controller in charged of the levelcontrol. In the end the global stability of the liquid levelis affected by those plateau. One explanation is that, inthe upper part of the level probe, the additional heliumflow causes a sub-cooling of the level probe superconduct-ing wire, which maintains the wire in the superconduct-ing state independent of the helium level. Tuning thecurrent and the boost current of the liquid helium levelsensors proved inefficient to solve the problem. We there-fore spanned a range of set-points for liquid helium levelcontrol in order to find a suitable set-point compromise.Although this compromise has been found for our cur-rent operation conditions, it is suitable to find a differentsolution that does not affect the behaviour of a criticalcomponent of the cryostats. In subsection III B 2, weused what we had in hand to solve quickly and efficientlythe problem. A proper design of a suitable damper wouldsolve the problem.In [20], instructions are given on how to design such adamper. Let’s explore the buffer reservoir solution. If weconsider the incoming wave impedance Z i = ρaA (10)0, the reservoir impedance Z r = − iρa ωV (11)and the orifice impedance (valve) : Z = √ µρω (1 + h/ r + ∆) + iρω ( h + 1 . r ) r π (12)the wave is damped when : Z i = Z r + Z (13)Obviously, the solution depends on the temperatureof the gas and its pressure as well as the wave parame-ters. Figure 19 shows a design compromise contour fordifferent gas temperatures and pressures for the geomet-rical parameters of the SPIRAL2 valves boxes and thedimensions of the port considered for the correction. . . . . . .
5P [bar]5101520253035404550 T [ K ] Ideal damper sphere radius for ν = 5 Hz Sph e r e r a d i u s [ c m ] FIG. 19: Damper ideal sphere diameter as a function of thetemperature and the pressure of He gas for an oscillation at5 Hz.
VI. CONCLUSION
Investigating cryogenic thermoacoustic oscillation oc-curring in a tube in a lab experiment and a real life TAOin a working superconducting LINAC is a completely dif-ferent thing. Although the two share the same physicalprinciples, in an accelerator, we have to deal with manydependencies and subsystems. This paper is a result ofseveral years investigations ending up in only few months of dedicated experimentation. In fact, since the discov-ery of the problem in 2017, the prime purpose was tofind a quick, easy and efficient solution, to approve itand deploy it. This has been done with success and al-lowed the commissioning of the LINAC and its currentramp up. However, we know now that our operationconditions can change (thermal load, pressure, liquid he-lium level) and that for some operation condition, theTAO wave can appear again and cause pressure instabil-ities. We also know that the chosen solution is not per-fect because of the liquid helium level observed plateau.After deploying the vapour sky short circuit line, ourpriority was to be able to monitor this phenomena thatremained unseen for years, hence the deployment of asimultaneous acquisition of piezo-electric pressure sen-sors at different locations. The acquisition system is nowevolving from the NI-DAQ solution to an integrated pro-grammable ARM DAQ (MSX-E3601) compatible withModBus TCP communication protocol for a continuousoperation compatible with the accelerator data acquisi-tion and archiving system. The analysis pipeline has alsoto be included in the main data analysis process in orderto have alarm kind surveillance of occurring events. Afirst analysis using Rott’s approach and wave approachshowed that the system is complex and that a differentmethod has to be considered if we are to understand andfind a better solution to the problem. We therefore planon starting an R&D program for a dedicated code foranalysing cryogenic thermo-acoustics in complex geome-tries. At the same-time, we plan on deploying a tuneable,damper like test setup that mimics variable resistance,inductance and capacitance thermodynamic RLC circuitin order to better characterise the impedance of a TAOloaded cryogenic system.
Acknowledgements