thermo-mechanical analysis of cryo-cooled electrode system
TRANSCRIPT
TVE-F 18 015
Examensarbete 15 hpJuni 2018
Thermo-mechanical analysis of cryo-cooled electrode system in COMSOL
Joel Olofsson
Teknisk- naturvetenskaplig fakultet UTH-enheten Besöksadress: Ångströmlaboratoriet Lägerhyddsvägen 1 Hus 4, Plan 0 Postadress: Box 536 751 21 Uppsala Telefon: 018 – 471 30 03 Telefax: 018 – 471 30 00 Hemsida: http://www.teknat.uu.se/student
Abstract
Thermo-mechanical analysis of cryo-cooled electrodesystem in COMSOL
Joel Olofsson
In the planned linear accelerator called Compact Linear Collider, CLIC, electrons and positrons will be accelerated to velocities near the speed of light. A limiting factor in accelerating structures are vacuum breakdowns, which is electrical discharges from a surface as a result of a large electric field being applied. In the preparatory studies for the CLIC, Uppsala University in collaboration with The European Organization for Nuclear Research, CERN, is building a DC Spark system to analyze vacuum breakdowns. This system containing large planar electrodes will be cooled down all the way down to around 4 K in order to limit the rate of wich vacuum breakdowns happen. When cooling a system like this, which consists of different components made of different materials there is the question of how the system will be affected. The objective of this project is to investigate how the cooling will affect the stability in terms of stresses and to analyze the cool down time of the system. Another goal is to make a material recommendation for a few parts based on the results. This will be done by simulating the cooling in COMSOL Multiphysics, which is a program that uses finite element analysis to solve complex problems where different branches of physics interact. The conclusion is that the system will most likely be stable as it is and there is no need to redesign it. The choice of recommended material is alumina with the reason being it should cause the least stress and the smallest gap between the electrodes when the cooling is done. There was no big difference in the cool down time between the materials. Further studies and simulations on the system is also recommended since there are many factors not taken into consideration in this study.
ISSN: 1401-5757, UPTEC F18 015Examinator: Martin SjödinÄmnesgranskare: Martin SjödinHandledare: Marek Jacewicz
Populärvetenskaplig sammanfattningI syfte att skapa en djupare förståelse av högenergifysik har den Europeiska organisationenför kärnforskning, CERN, planerat att bygga en ny partikelaccelerator i Genève. Till skill-nad från den välkända acceleratorn "The Large Hadron Collider" kommer denna nya ac-celerator "Compact Linear Collider" (CLIC) vara linjär och elektroner och positroner skaaccelereras till hastigheter nära ljusets. Uppsala universitet har tillsammans med CERNoch många andra skapat ett elektrodsystem för att analysera de elektriska urladdningarsom sker vid accelerationen och kan påverka de accelerarande partiklarna. Detta systemska kunna kylas ned till bara några grader från den absoluta nollpunkten. Anledningentill att man vill kyla ned systemet är för att man vill begränsa de elektriska urladdningarsom sker i vakuumet mellan elektroderna när ett starkt elektriskt fält läggs över dem.
Projektet undersöker hur detta elektrodsystem uppför sig under nedkylning genom attutföra simuleringar i programmet COMSOL Multiphysics. Målet var att få en tydlig bildav hur systemets delar krymper samt hur värmen sprider sig under nedkylning. Utifråndessa resultat ville man analysera om systemet kommer vara stabilt utan att allt för storapåfrestningar skapas samt välja material till några komponenter.
Resultaten från denna studie pekar på att det tänkta systemet är stabilt och ej behöveromdesignas. Materialvalet stod mellan två liknande keramiska material där aluminiu-moxid till slut blev rekomendationen. Studien har sina begränsningar och djupare studierär rekommenderat för att skapa en tydligare förståelse för hur systemet beter sig undernedkylning.
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Contents1 Introduction 5
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.1.1 Particle accelerators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.1.2 DC Spark system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.1.3 Vacuum breakdowns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.1.4 COMSOL Multiphysics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Theory 72.1 Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.1 Thermal conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.2 Specific heat capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Thermal expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2.1 Linear expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2.2 Thermal stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Cryocooler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.4 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3 Method 103.1 Defining materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 Applying boundary and initial conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.2.1 Form assembly and identity pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2.2 Boundary conditions for the heat transfer module . . . . . . . . . . . . . . . . . . 123.2.3 Boundary conditions for the solid mechanics module . . . . . . . . . . . . . . . . . 13
3.3 Multiphysics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.4 Simulating and post-processing results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.4.1 Cooling problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.4.2 Thermal contraction problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4 Results 154.1 Cool down results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.2 Contraction results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.2.1 Electrodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.2.2 Spacer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.2.3 Cryocooler bottom part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.2.4 Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.2.5 Bolts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.3 Gap calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.3.1 Inter-electrode gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.3.2 Electrode-spacer gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.3.3 Cryocooler-upper plate gap and lower electrode-lower plate gap . . . . . . . . . . . 234.3.4 Plate-Bolt gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
5 Discussion 25
6 Conclusions 26
Appendices 28
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1 Introduction
1.1 Background
1.1.1 Particle accelerators
During the 20th century the science of accelerating particles went from simple atommodels, to the ability to control the splitting of an atom, to the huge particle acceleratorswe have today to study the high energy physics of our universe. The largest and mostpowerful accelerator is currently the Large Hadron Collider (LHC) built by The EuropeanOrganization for Nuclear Research (CERN) where protons are accelerated to velocitiesvery close to the speed of light [1]. After studying the collision of proton beams in theLHC, the next step is to collide electron beams at similar energy levels. However electronsand positrons loses high amounts of energy when circulating, like in the LHC, thereforthe next generation of accelerators at CERN is the planned Compact Linear Collider(CLIC) where electrons and positrons will be accelerated linearly in a radio frequency(RF) structure consisting of two accelerators facing each other. The CLIC aims to havean accelerating gradient 20 times the LHC, thus becoming less space consuming andhopefully cheaper [2].
1.1.2 DC Spark system
In the preparatory studies for the CLIC, CERN has built a direct current (DC) systemcontaining large planar electrodes within a vacuum chamber to perform tests on. Byapplying a high voltage over the electrodes a large capacitance between them is createdand when the direction of the current changes periodically it can be used to acceleratethe electrons. In collaboration with CERN, Uppsala University is building a versionof this system which can be cooled down to cryogenic temperatures and operated in awide range of temperatures all the way down to around 4K. This system consists of twocopper electrodes which are separated by a spacer to a distance between 0 - 150 µm, buttypically around 20 µm. The upper electrode is connected to a cryocooler. The electrodesand a small part of the cryocooler are surrounded by a setup of plates and bolts whichkeeps the electrodes pressured against the spacer at all temperatures. The whole setupis built within a vacuum chamber. The whole system and a zoom in on the electrodesetup can be seen in figure 1 and 2. The objective is to have this structure operate asideally as possible, and that requires deep knowledge of how how it will react to cryogenictemperatures.
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Figure 1: DC Spark System showing the full system of the elctrode setup within a vacuum chamber.
Figure 2: Zoom in on the electrode setup. Showing the (1) electrodes, (2) spacer, (3) bottom part ofthe cryocooler, (4) plates and (5) bolts.
1.1.3 Vacuum breakdowns
Vacuum breakdowns are discharges that can happen when an electric field is very large.Breakdowns can affect the acceleration and direction of the particle and make the per-formance of the CLIC less than ideal. Breakdowns are normally detected through anincrease in current or through an increase in vacuum pressure [3]. The cooling downof the system aims to minimize the breakdown rate to increase the performance of thefuture accelerator.
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1.1.4 COMSOL Multiphysics
To study the effects of temperature change in the system the program COMSOL Multi-physics will be used. It is a software that can be used to perform finite element analysison complex setups where different branches of physics interact.
1.2 Objective
The purpose of the project is to investigate how the heat transfer inside the system willaffect the system as a whole in terms of mechanical stresses due to volumetric contractionwhen being cooled down to 4 K. The project can be separated into two different problemswhich will be called the cooling problem and the thermal contraction problem.
The cooling problem aims to simulate cool down time for the electrodes. Previouslythere have been similar simulations done on the cooling of only the electrodes and spacer.This project aims to complement those results with simulations on the whole setup. Theexpected result is that the cool down time will be longer due to the cooling power alsobeing distributed to the plates and bolts.
The end goal of the thermal contraction problem is most importantly to evaluate if themechanical design is stable in terms of thermal stresses and possibly suggest improve-ments. Another goal is to have clear results on how the distance between the electrodes,called the inter-electrode gap distance depends on temperature and material of the spacer.The spacer and the plates will be made of either alumina (Al2O3) or aluminum nitride(AlN) and by analyzing the results a suggestion for the most optimal material should bemade.
2 Theory
2.1 Heat Transfer
2.1.1 Thermal conduction
The transfer of thermal energy through a small element can be describes by Fouriers lawof heat conduction[4]:
q̇cond = −k(T )AdT
dx, (1)
where A is the cross-sectional surface area normal to dT/dx, which is the temperaturegradient in the direction of conduction and k is the temperature dependent thermalconductivity for the material. The thermal conductivity (unit W/(m·K)) is an importantproperty for this project that describes how well a material conducts heats. Thermalconductivity can vary a lot depending on the temperature.
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2.1.2 Specific heat capacity
Specific heat capacity, Cp with SI-unit J/kg·K, is a material property that describeshow much energy is needed to heat up one kilogram of the specific material one degreeKelvin. This property is generally used as a constant at room temperatures, but atcryogenic temperatures this property can vary a lot.
2.2 Thermal expansion
2.2.1 Linear expansion
The coefficient of thermal expansion (CTE) describes how much a material expands orcontracts per degree Kelvin. In the linear case, i.e. at room temperature the CTE isdefined as in equation 2.
α =1
L
dL
dT, (2)
where L is the length of the object, and dL/dT is the change of length per change intemperature. A more clear way of formulating how much an object will expand or contractis by integrating equation 2 with respect to temperature, leading to:
∆L
L=
∫ Tstart
Tfinal
α(T )dT, (3)
where ∆L is the length change, L is the original length and α(T) is the temperaturedependent CTE. This expression of ∆L/L gives an exact value of how much the objectcontracts per degree Kelvin and will be used in this study.The coefficient of thermal expansion can vary a lot depending on temperature and cannotbe overlooked when designing the electrode system.
2.2.2 Thermal stress
Thermal stresses can occur when a material is exposed to a high temperature gradient orthermal expansion/contraction [5]. Different materials connected to each other are verylikely to cause stress on each other when being cooled down due to difference in CTE. Inthe electrode system there are a few key sections that should be analyzed where thermalstress might occur due to difference in contraction. Firstly at the connection betweenthe bolts and the ceramic plate but also between the plate and the copper electrode.Too much stress on a section could lead to deformation and malfunction and the systemshould therefor be designed to completely erase thermal stresses while remaining stable.
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2.3 Cryocooler
The cryocooler is the system that is used to cool down the electrode setup to cryogenictemperatures. The cryocooler used in this setup is a PT415 from cryomech[8]. The PT415 is a cryocooler of the type "Two-Stage Pulse Tube Cryocooler" which has two stageswhere cooling takes place simultaneously. One upper part which cools down to 45 K witha power of 40 W and a lower part which cools to 4.2 K with a power of 1.5 W.
2.4 Materials
The materials of interest in this study is copper, steel, alumina (Al2O3) and aluminiumnitride (AlN). The electrodes and bottom part of the cryocooler that are connected to theelectrodes will be made of copper. The spacer and plates will be made of either aluminaor aluminium nitride, where this study aims to analyse pros and cons of each material.The bolts keeping the setup together will be made of steel. The material propertiesthat needs to be defined for the simulations to work is thermal conduction, specific heatcapacity, thermal expansion, density, Young’s modulus and Poission’s ratio. In reality,all relevant properties of the materials are more or less temperature dependent howeverin this study the density, Young’s modulus and Poisson’s ratio will be held at constantvalues for time consuming reasons. Those values are taken from the COMSOL materiallibrary and can be found in table 1. For the thermal expansion, thermal conductivityand specific heat capacity of the materials, graphs will be interpolated with data fromtables 2, 3 and 4.
Table 1: Relevant non-temperature dependent material properties of Copper, Steel and Alumina. Takenfrom the COMSOL material library.
Property Copper Stainless steel 4340 AluminaDensity 8960 [kg/m3] 7850 [kg/m3] 3900[kg/m3]Young’s modulus 110e9 [Pa] 205e9 [Pa] 300e9[Pa]Poisson’s ratio 0.35 0.28 0.222
Due to lack of data, the properties for aluminium nitride will be taken directly fromthe COMSOL material library as "AlN [solid,c-axis]". Every property in this model istemperature dependent and graphs can be seen in appendice A, except for Poisson’s ratiowhich is set constant to 0.24[6].
Table 2: Values of ∆L/L for the materials of interest at different temperatures and the coefficient ofthermal expansion at room temperature.
Temperature 4 K[%]
40 K[%]
77 K[%]
100 K[%]
150 K[%]
200 K[%]
250 K[%]
α at 293 K[10−6K−1]
Copper[4] -0.324 -0.322 -0.302 -0.282 -0.221 -0.148 -0.070 16.7Steel[4] -0.296 -0.296 -0.281 -0.261 -0.206 -0.139 -0.066 15.1
Temperature 100 K[%]
150 K[%]
200 K[%]
250 K[%]
293 K[%]
400 K[%]
- -
Alumina[7] -0.078 -0.060 -0.040 -0.020 0 0.078 - -
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Table 3: Values for thermal conductivity, k [W/(m·K)], for the materials of interest at different tem-peratures.
Temperature 4 K 10 K 20 K 40 K 77 K 100 K 150 K 200 K 295 KCopper[4] 630 1540 2430 1470 544 461 418 407 397Steel[4] 0.27 0.90 2.2 4.7 7.9 9.2 11 13 15
Temperature 0 K 100 K 150 K 200 K 250 K 273.15 K 300 K 350 K -Alumina[7] 0 133 77 55 43.4 39.7 36 30.7 -
Table 4: Values for specific heat, Cp [J/(kg·K)], for the materials of interest at different temperatures.
Temperature 4 K 10 K 20 K 30 K 50 K 77 K 100 K 150 K 200 K 300 KCopper[4] 0.09 0.88 7 27 97 192 252 323 356 386Steel (310)[4] 2 5.2 17 10 100 200 250 350 400 480
Temperature 10 K 30 K 60 K 80 K 100 K 200 K 300 K 400 K - -Alumina[7] 0.08 2.5 25 63 125 523 750 920 - -
3 MethodThe simulations for the cooling problem and the thermal contraction problem will differin that the cooling problem needs to simulate the whole setup at the same time, while thethermal contraction problem simulations will be done individually on each part, with thereason being lack of computational power. Setting up the simulations in COMSOL followsthe process of defining physics and geometries, defining materials, applying boundaryand initial conditions, applying a mesh and finally running the simulation and processthe results. In figure 3 a picture of the electrode system imported into COMSOL can beseen.
Figure 3: Full setup imported into the COMSOL software.
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3.1 Defining materials
The materials needed are copper (Cu), steel, alumina (Al2O3) and aluminum nitride(AlN). As previously stated, aluminum nitride with associated functions will be takendirectly from the COMSOL material library (see appendix for temperature dependentproperty graphs of AlN). For the other materials, the functions for the three propertieswhere temperature dependence is wanted are created by typing in the data from section2.4 in the COMSOL interpolation function node and then choosing type of interpolationand extrapolation. The interpolations are done with either linear, piecewise cubic or cubicspline and extrapolations with either constant, linear or nearest function, depending onthe property and whats realistic. An example of this can be seen in figure 4 where thefunction for thermal expansion, equation 3, is produced as a function of temperaturefor copper with data taken from table 2. Graphs over all interpolations of the materialproperties used in the simulations can be seen in the appendix.
Figure 4: Shows the COMSOL workspace for when the temperature dependent function for ∆L/Lof copper is produced. Interpolation is piecewise cubic and extrapolation is linear since the thermalexpansion is known to be more or less linear above room temperatures.
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3.2 Applying boundary and initial conditions
Figure 5: The COMSOL model builder showing all the nodes and boundary conditions.
In figure 5 the model builder in COMSOL can be seen after all the boundary and initialsconditions have been applied. In this section all conditions applied will be described.
3.2.1 Form assembly and identity pairs
In order for heat to flow freely between the boundary of two parts one need to form anassembly of the geometry and create identity pairs between every surface where differentparts are in contact. Identity pair is a definition of two surfaces in contact with eachother that allows boundary and initial conditions to be applied to both surfaces at thesame time.
3.2.2 Boundary conditions for the heat transfer module
• Solid is an automatic BC that defines the object as a solid and informs the programto use relevant values from the materials node for its solution.• Initial Values defines the initial temperature for the object. In all simulations it isset to 293.15 K (20◦C).• Thermal Insulation defines the objects boundaries to be free of heat flux.• Temperature defines a specific surface to be held at a constant temperature which inour case will be the temperature that the whole object aims to be cooled down to.
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• Heat Flux defines the thermal flux of a boundary. It is set to be the same as thecooling power of the cryocooler.• Continuity is applied on all surfaces that are in contact with another surface. Thecondition allows heat to flow freely over the boundaries of each part.
3.2.3 Boundary conditions for the solid mechanics module
• Linear Elastic Material simply defines what kind of material the object is made ofand makes COMSOL use relevant material properties from the defined material in thematerials node when calculating the solution.• Heat Flux defines the thermal flux of a boundary. It is set to be the same as thecooling power of the cryocooler.• Free allows the boundaries to not but held fixed at a specific place, but allows theobjects surfaces to move around in accordance to the physics.• Initial Values sets the displacement field and structural velocity field to zero to notimpact the result.• Rigid Motion Suppression is a condition that will stop the object to self equilibrateand keep the objects surfaces from changing its direction during the simulation.
3.3 Multiphysics
Thermal Expansion is the only node where different physics interact. Here thermalstrain is choosen so that COMSOL uses the materials ∆L/L values and not the CTE forwhen calculating thermal expansion. A strain reference temperature Tref is set to293.15 K.
3.4 Simulating and post-processing results
3.4.1 Cooling problem
The boundaries of the heat transfer module and identity pairs are needed to perform thesimulations for the cooling problem. The cryocooler cools the setup in two stages. Inreality these stages are active simultaneously, but to make simulations simpler the twostages will be simulated one at a time, first from room temperature to 45 K and then from45 K down to 4 K. One point probe where temperature can be evaluated are created insideeach electrode and a time dependent study is simulated. The point probes are evaluatedand the time it takes for each point to be cooled down to desired temperature can bemeasured. An uncertainty of 0.5 K is used, meaning that when the lower electrode iswithin 0.5 K of desired temperature the setup is considered to be cooled. When thisstudy is done it is repeated with only non temperature dependent properties defined tosee how much difference it makes to take the temperature dependence into consideration.
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3.4.2 Thermal contraction problem
First of all the relevant measurements for analyzing gap distances of each object aremeasured in COMSOL. To run this simulation all boundary conditions and multi physicsdescribed above are needed, however no identity pairs are necessary since simulationsare done individually on each part. After running a stationary study and doing a pointevaluation of the three dimensional displacement of two points on the object one can applyPythagoras theorem and extrapolate how much the distance between the two points hascontracted or expanded, see equation 4.
∆ =√
(u(1)− u(2))2 + (v(1)− v(2))2 + (w(1)− w(2))2, (4)
where u, v and w represents the displacement in x, y and z-direction and the numberswithin the paranthesis represents which point. This method is applied to the parts of eachobject that are of interest for analyzing gaps and stresses in the setup. Simulations on thethe thermal expansion/contraction will be done with three different end temperatures.First to 333.15 K, which will be used to compare results with the project of F. Elmgrenand S. Warma [9], where they have done theoretical calculations and experimental gapmeasurements on a similar setup heated to 333.15 K. Second temperature is 77 K whichis the temperature of liquid nitrogen, a fluid commonly used in for cooling in cryogenicsystems. Last temperature is 4 K which is approximately the final temperature of thesystem. When all the relevant contractions have been simulated the new gap distanceswill be calculated and analyzed if they are large enough to not cause stress to the system.
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4 Results
4.1 Cool down results
COMSOL failed to perform the simulation when cooling to 4 K (for unknown reasons),instead simulations for cooling to 12 K were done. Simulations on the cool down timewhen all material properties were non temperature dependent were also done. Non tem-perature dependent properties are taken from the COMSOL material library for copperand alumina and from [6] for aluminum nitride. Figure 6 shows the simulation after 0.6 hwhen being cooled down to 45 K. Table 5 shows the cool down time simulated for the twostages depending on material of the spacer and plates. Table 6 shows the results whensimulating the same thing but with materials without temperature dependent properties,meaning they are set to constants at room temperature.
Figure 6: Simulation results for the cooling to 45 K when the spacer and plates are made of alumina
Table 5: Cool down time for the electrodes depending on spacer and plate materials.
Spacer & plate material Time 293.15 K→ 45 K Time, 45 K → 12 KAlumina 0.56 h 0.007 hAluminum nitride 0.53 h 0.0067 h
Table 6: Cool down time when simulating with non temperature dependent properties.
Spacer & plate material Time 293.15 K→ 45 K Time 45 K → 12 KAlumina 2.64 h 1.81 hAluminum nitride 1.5 h 2.2 h
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These results are compared to table 7, which is the results of S. Johannsdotter [10], wheresimulations on cooling down a smaller system consisting of only the electrodes and spacerwere done. Note that these simulations had material properties of constant values.
Table 7: Results of S. Johannsdotter. Shows the simulated cool down time for the different intervalsbut also for different cryocoolers. Note that none of these cryocoolers is the one used in this project butdo have similar characteristics.
4.2 Contraction results
4.2.1 Electrodes
(a) (b)
Figure 7: Upper electrode (a) before and (b) after cool down to 4 K. Displacement in (b) amplified bya factor ≈ 25.
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Measurements and materialsThe two electrodes are identical, meaning simulations on one electrode are sufficient.An electrode before and after cool down to 4 K can be seen in figure 7. The spacer isplaced around the smaller inner part of the electrode and will be compressed from bothdirections against the larger cylinder parts of both electrodes. The bottom part of thecryocooler will be placed around and above the smaller top cylinder of the electrode. Themeasurements that are needed to analyze stresses between these parts are the diameterof the smaller cylinders at the top and bottom, which are designed to be of the same size,and the height of the smaller bottom cylinder which together with the spacer will decidethe inter-electrode gap and the capacitance. In table 8 the relevant measurements of theelectrode can be seen.
Table 8: Relevant measurements of the electrode
Height of the lower part of the electrode, h 10 mmDiameter of smaller cylinders, d 62 mm
Simulation resultsIn table 9 the simulation results for when simulating with the defined temperature depen-dent properties can be seen. In table 10 simulations with constant properties are done tofor comparison to previous table.
Table 9: Temperature dependent properties change in h and d for relevant temperatures.
Temperature ∆helectrode [mm] ∆delectrode [mm]333.15 K +0.00668 +0.041477 K -0.030 -0.1874 K -0.032 -0.201
Table 10: Change in h and d when simulating with non temperature dependent properties.
Temperature ∆helectrode [mm] ∆delectrode [mm]77 K -0.0367 -0.2284 K -0.0491 -0.305
When comparing how much impact the temperature dependent properties has on thesimulations, we see that for the simulation to 77 K the simulation with temperaturedependent properties for copper are 82.2 % of the simulations with constant properties.At 4 K this number is 65.9 %.
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4.2.2 Spacer
Figure 8: Spacer.
Measurements and materialsThe spacer will be made in either alumina or aluminum nitride and can be seen in figure8. The measurements relevant for analyzing stresses and the inter-electrode gap are theinner diameter and the height of the spacer which is measured to be:
Table 11: Relevant measurements of the spacer.
Inner diameter of the spacer, dspac 62.2 mmHeight, hspac 20.02 mm
Simulation resultsThe results of the spacer simulations can be seen in table 12 where the change in distancefor the relevant measurements and materials can be seen.
Table 12: Total change of the measured distance for the different materials and temperatures.
Temperature ∆dspac, Al2O3 [mm] ∆dspac, AlN [mm] ∆hspac, Al2O3 [mm] ∆hspac, AlN [mm]333.15 K +0.0146 +0.00642 +0.00495 +0.0020677 K -0.0532 -0.0168 -0.0171 -0.005394 K -0.0662 -0.0168 -0.0213 -0.00539
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4.2.3 Cryocooler bottom part
Figure 9: Bottom part of the cryocooler.
Measurements and materialsThe bottom part of the cryocooler will be made of copper and all the inner and outerdiameters of the bottom disks, which will be in contact with the electrodes and plate, isrelevant for analyzing gaps and stresses in the setup and can be seen in table 13. Pictureof the cryocooler bottom part can be seen in figure 9.
Table 13: Relevant measurements of cryocooler copper part.
Diameter of small bottom plate, d1 cooler 52 mmOuter diameter of large bottom plate, d2 cooler 68 mmInner diameter of large bottom plate, d3 cooler 62 mm
Simulation resultsIn table 14 the change in temperature for the three noted diameters can be seen dependingon temperature.
Table 14: Temperature dependent change in d1, d2 and d3 for relevant temperatures.
Temperature ∆d1 cooler [mm] ∆d2 cooler [mm] ∆d3 cooler [mm]333.15 K +0.0348 +0.0455 +0.041477 K -0.157 -0.205 -0.1874 K -0.168 -0.220 -0.201
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4.2.4 Plates
(a) (b)
Figure 10: Upper plate (a) and lower plate (b).
Measurements and materialsThe plates will be made in either alumina or aluminum nitride and can be seen in figure10. The measurements relevant for analyzing stresses are the larger inner diameters andthe diameter of the bolt holes and can be seen in table 15. The difference between theplates is the inner diameters, since the upper one will be in contact with the cryocoolerand the lower is directly in contact to the lower electrode.
Table 15: Relevant measurements of the plates.
Larger inner diameter of the upper plate, dU 34.1 mmLarger inner diameter of the lower plate, dL 31.1 mmDiameter of bolt holes, bplate 6 mm
Simulation resultsSimulations on the plates could not be done due to lack of computational power to handlethe small mesh size required for the small bolt holes.
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4.2.5 Bolts
Figure 11: One of the five steel bolts in the setup.
Measurements and materialsThe bolts are made of steel and the only relevant measurement is the diameter of thebolt to make sure it will fit into the bolt holes of the plate without causing stress to thesystem. The bolt is seen in figure 11 and the relevant measurement in table 16
Table 16: Relevant measurements of the bolts
Diameter of a bolt, dbolt 5 mm
Simulation results The change in diameter for the bolt at different temperatures com-pared to the initial temperature can be seen in table 17
Table 17: Change in diameter of the bolt for different temperatures.
Temperature ∆dbolt [mm]333,15 K +0.0030477 K -0.014054 K -0.0148
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4.3 Gap calculations
4.3.1 Inter-electrode gap
The inter-electrode gap will depend on the height of spacer and the height of the smallerpart of the electrodes that lies within the spacer. At room temperature, 293.15 K thespacer is 20.02 mm, and the electrode part is 10 mm which results in the gap being 20µm. New gap distance g can therefor be calculated by equation 5 and results can be seenin table 18.
ginterelectrode = hspac + ∆hspac − 2 ∗ (helectrode + ∆helectrode). (5)
Table 18: Table over the inter-electrode gap distance for the different temperatures and spacer materialsalumina and aluminum nitride.
Temperature Spacer materialAlumina [µm] Aluminum nitride [µm]
333,15 K 11.58 8.70293.15 K 20.0 20.077 K 63.26 75.014 K 63.50 79.41
The results can be compared to the results of F. Elmgren and S. Warma[9], which canbe seen in figure 12. In their project copper electrodes and an alumina spacer of similarsize were heated from 22◦C to 60◦C. The plots show different measurements of how thegap distance changes with temperature.
(a) (b)
Figure 12: Results of experimental and theoretical gap distance for F. Elmgren and S. Warma.
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4.3.2 Electrode-spacer gap
The gap between the inner diameter of the spacer and the diameter of the electrodepart that is inside the spacer is calculated by subtracting the change in diameter forthe different temperatures and materials and calculataing the difference, see equation 6.Results can be seen in table 19.
gelectrode−spacer = (dspac + ∆dspac)− (delectrode + ∆delectrode). (6)
Table 19: Table over the electrode-spacer gap distance for the different temperatures and spacer mate-rials.
Spacer material Alumina [µm] Aluminum nitride [µm]333,15 K 173.1 165.0293,15 K 200 20077 K 334.0 370.54 K 334.7 384.1
4.3.3 Cryocooler-upper plate gap and lower electrode-lower plate gap
The simulations needed to get numerical values for these gaps could not be done dueto the plate simulations needing more computational power. However, it is the sameprinciple as the electrode-spacer gap where the inner material is made of copper and theouter material is either alumina or aluminum nitride. The difference in diameters at roomtemperature is also 200 µm.
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4.3.4 Plate-Bolt gap
The failure to simulate contraction for the plate makes this gap harder to analyze. Whatcan be done instead is to compare the material data for thermal contraction of steel,alumina and aluminum nitride and see that steel are expected to contract much morethan both alumina and aluminum nitride, see figure 13.
Figure 13: Percentual contraction vs temperature for the three materials of interest.
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5 DiscussionThe choice of material does not impact the cool down time more than a few minutes whereboth will take a little more than half an hour from room temperature to 12 K. Whencomparing to the case of properties not being temperature dependent and the previousstudy, it differs the most for the cooling in the interval with the lowest temperatures.The cool down time for this interval 45 K to 12 K is very low and this can be traced tothe thermal conductivity being high and the specific heat being very low in this interval.This is common for many materials. Therefor it should not take long time at all to coolthe setup another 8 degrees down to 4 K.
Alumina will contract and expand more than aluminum nitride. It will also result ina smaller inter-electrode gap. Comparing my results with those of F. Elmgren and S.Warma it is observed that my simulations for heating the setup to 60◦C reduce the inter-electrode gap with approximately 42 % while both their experimental and theoreticalgap distance reduces around 10-20 %. This should be caused mostly by the propertiesin the simulations not being accurate, but it is also hard to know exactly what kind ofcopper and alumina they used. Their theoretical graph uses a different CTE value atroom temperature compared to this study. Further studies are recommended.
By observing the calculated gap distances it is noted that all cases except the bolt gapand inter-electrode gap involves the inner part being made of copper and the outer partof the ceramic material, and that copper always will contract the most of these materi-als according to the available data. From those observations the setup is evaluated tobe designed such that there is no chance of the gap distances changing in a way thatwould cause stresses due to the inner parts ending up being larger than the original gapdistance. The bolt gaps can in a similar way be evaluated to not cause any stresses byobserving figure 13, where steel should contract the most. Stresses should only occurbetween parts that are directly in contact, with no gap, such as the the top and bottomsurface of the spacer to the electrodes. These stresses can be minimized by minimizingthe relative change in contraction between the materials.
There are a lot of things that make the simulations less than ideal. As mentioned allsix materials properties that COMSOL uses for its calculations are temperature depen-dent in reality, but in the simulations half of them are constant. The three propertieswhere temperature dependence has been applied is however the properties that is mostsensitive to temperature change. Another big thing is the lack of data for lower temper-atures. When the materials were defined the values outside the data points dependedon what type of extrapolation were done. As can be seen in the appendice and section3.1, some were set to constant and others to linear depending on what property it wasand what felt the most realistic. This will impact the simulations to differ from realitywhere some contractions could turn out to be both smaller and larger than simulated.Another thing to note is that the properties for steel are not completely accurate as someare taken from the source as "Stainless steel 304" and some as "Stainless steel 4130". Asfor the heat transfer, small heat losses due to radiation are likely to happen but have notbeen accounted for in the simulations. COMSOL also has it’s own limitations in that
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the finite element analysis it performs is dependent on a mesh which in turn dependson computational power. The COMSOL solver has not been looked at deep enough tomake a comment of how this might have impacted the result. Another important aspectin understanding the behavior of the system will be to study whether the current that isapplied will affect the heat transfer and make any notable impact.
For more accurate simulations it might be necessary to perform tests on the relevantmaterials in the same environment as in the planned accelerator to gather accurate dataand then do simulations where all properties are temperature dependent and radiation,friction and the electric field are take into account.
6 ConclusionsThe system is evaluated to be designed with large enough gaps to not cause stresses whenbeing cooled, no matter alumina or aluminum nitride as material choice for the spacerand plates. The choice of material does also not affect the cool down time in any majorway.
If a smaller inter-electrode gap at 4 K is desired alumina is the material to pick. Asmaller gap would not require the same voltage to get the desired capacitance. Aluminawill contract more than aluminum nitride but copper will contract the most so the rela-tive change in contraction between parts of said materials will be smaller with alumina.Therefor alumina as choice of material might be better in terms of reducing stresses dueto the parts sliding against each other. One would have to look into the friction propertiesof the materials, but if they are similar alumina might prolong the time the setup can beused before the parts are worn out.
All in all this study concludes that alumina is the better choice of material, but also thatthis study gives a very surface level understanding of the behavior of the system since alot of factors have not been accounted for. From the results no suggestions for improvingthe setup can be made. To get a more precise understanding on how the setup will behaveit is recommended to perform simulations with most importantly more accurate materialproperties, but also with radiation, friction, electricity and other factors.
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References[1] CERN, "The Large Hadron Collider", https://home.cern/topics/large-hadron-
collider. 25.05.2018.
[2] CERN, "CLIC in a nutshell". http://clic-study.web.cern.ch/content/clic-nutshell.25.05.2018.
[3] Shipman Nicolas, Manchester U. and CERN, Experimental study of DC vac-uum breakdown and application to high-gradient accelerating structures for CLIC,http://inspirehep.net/record/1381397/.
[4] J.W. Ekin. Experimental Techniques for Low Temperature Measurements: CryostatDesign, Material Properties and Superconductor Critical-Current Testing. OxfordUniversity Press, 2006.
[5] Wikipedia, "Thermal Stress". https://en.wikipedia.org/wiki/Thermal_stress.25.05.2018
[6] Accaratus, "Aluminum Nitride, AIN Ceramic Properties".http://accuratus.com/alumni.html. 25.05.2018
[7] Touloukian, Y.S.; Powell, R.W.; Ho, C.Y. and Klemena, P.G. (1971). Thermophys-ical properties of Matter-The TPRC Data Series–(Vol 2. Thermal Conductivity-Nonmetallic solids, Vol 5. Specific Heat-Nonmetallic solids, Vol 13. ThermalExpansion-Nonmetallic solids). CINDAS/Purdue University, West Lafayette.
[8] Cryomech, "PT415". http://www.cryomech.com/cryorefrigerators/pulse-tube/pt415/. 25.05.2018
[9] F. Elmgren and S. Warma, "Micrometer gap distance", 2018, Uppsala University.
[10] S. Johannsdotter, “Design of the Cryocooled DC Discharge System with Heat Trans-fer Simulations in COMSOL”, 2017. Uppsala University.
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Appendices
Figure 14: Graphs over the temperature dependent properties for aluminum nitride plotted versustemperature (K). Taken from the COMSOL material library.
(a) Thermal expansion ∆L/L (1) (b) Thermal conductivity (W/(m*K))
(c) Specific heat capacity Cp (J/(kg*K)) (d) Density (kg/m3)
(e) Young’s Modulus E (Pa)
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Figure 15: Graphs over the temperature dependent properties for copper plotted versus temperature(K). Taken from the COMSOL material library. Thermal expansion graph can be found in the definingmaterials section.
(a) Specific heat capacity Cp (J/(kg*K)) (b) Thermal conductivity (W/(m*K))
Figure 16: Graphs over the temperature dependent properties for steel plotted versus temperature(K). Taken from the COMSOL material library.
(a) Specific heat capacity Cp (J/(kg*K)) (b) Thermal conductivity (W/(m*K))
(c) Thermal expansion ∆L/L (1)
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Figure 17: Graphs over the temperature dependent properties for alumina plotted versus temperature(K). Taken from the COMSOL material library.
(a) Specific heat capacity Cp (J/(kg*K)) (b) Thermal conductivity (W/(m*K))
(c) Thermal expansion ∆L/L (1)
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