Dipartimento di Ingegneria Civile e Industriale
C
ORSO DI
L
AUREA
M
AGISTRALE IN
I
NGEGNERIA
M
ECCANICA
T
ESI DIL
AUREAASME Verification and Pressure
Sensitivity optimization of the
Cryostat of the 1.3GHz cavity
Cryostat
Candidato:
Andrea Palagi
Relatori:
Prof. Ing. Marco Beghini
Prof. Ing. Leonardo Bertini
Ing. Allan Rowe
After the most important, the most meaninful and yet the most sweetly dif-ficult 5 years of my life I find myself writing those pages on which I have pondered for a long time. Several emotions are flowing in to my mind in this moment, as every time a beautiful journey comes to an end. And it is at this time that you look back and want to thank the people who crossed your path and helped you to achieve the finish line.
First of all I want to thank my italian advisor, Marco Beghini, not only for his availability to help me but also for the meeting we had almost 6 years ago which directed me in the choice of the course of study.
As weel as my italian advisor I want to thank my american advisor, Al-lan Rowe, who gave me the opportunity to join the Fermilab staff, and all the other people who I have worked with in my american experience.
About my american experience a special thanks goes to Donato, a very passionate mechanical engineer, who helped me with some topics and in-spired me to do my best in this thesis.
My time spent in America gave me the opportunity to stay closer to one of the wisest people I have ever known, Matteo, who I want to thank not only for the loyal competition which let us cross this finish line so early but also for the moment spent together and the lessons he gave me. If it had not been for you probably I would have ended up crazy from this six months.
A special thanks goes to the people I shared the most of my last 5 years with, Alessio, Jacopo, Filippo, Lorenzo, Max and Marco which made our uni-versity career easier and more enjoyable. Thanks to you every day passed in Pisa have not been so tough.
I want to show gratitude for my friends, Federico, Tommaso, Federico, Tommaso, Luc, Simone, Matteo, Rebecca and especially Lorenzo who wel-comed me in a difficult phase of my path. It’s thank to you if I am now more loyal and a better person and now I know that everywhere I’ll go you will be always covering my shoulders.
I want to praise my parents Fiorenzo and Antonella for supporting me in everyday of my life and for teaching me to not take anything for granted in life and always struggling to achieve the best.
Last but not least I want to thank Stefano, called Ste Galera, the best brother a person could have, even if I learned that in the last years and not in my teens.
The project regards the ASME verification and the Pressure Sensitivity op-timization of the cryostat of the 1.3GHz cavity made by the Fermi National Accelerator Laboratory and to be used in the LCLS II project.
The first part of the project involves the ASM E code and the verification of the Helium Vessel with respect to the safety of the assembly. Div. 1 and Div. 2 prescription have been used when applicable.
In the second part we calculated the pressure sensitivity (df /dp) of the Cryostat in the State of the Art.
The last stage of the project is concerned to the pressure sensitivity char-acterization of the cavity and the possible assembly modification in order to reduce the df /dp.
List of Figures 3
List of Tables 7
1 Introduction 9
1.1 Superconducting Radio Frequency . . . 9
1.2 SRF Cavity . . . 10
1.3 Pressure Sensitivity . . . 11
2 State of the Art 14 2.1 The 1.3GHz Cavity . . . 14
2.2 The Helium Vessel . . . 16
2.2.1 The tube . . . 16
2.2.2 The End plates . . . 18
3 Verification according to the ASME 20 3.1 Introduction and Summary . . . 20
3.1.1 Non-Code elements . . . 21
3.2 Geometry . . . 21
3.2.1 Welds . . . 21
3.3 Loadings . . . 24
3.4 Stress Analysis Approach . . . 27
3.4.1 Applying U − 2 (g) . . . 27 3.5 Division 1 . . . 28 3.5.1 T i Cylindrical Shells . . . 28 3.5.2 Penetrations . . . 29 3.5.3 T i Bellows . . . 32 3.6 Division 2 . . . 33
3.6.1 Finite Element Model . . . 34
3.6.2 Stress Analysis Results . . . 36
4 Pressure Sensitivity Analysis 43 4.1 CAD Model . . . 43
4.1.1 The cavity . . . 43
4.1.2 The Helium vessel . . . 47
4.2 Coupled Evaluation . . . 48
4.2.1 Influence of the Tuner Stiffness . . . 51
5 Simplified Evaluation 53
5.1 Application to 1.3GHz cavity . . . 55
5.1.1 Fixed Model . . . 56
5.1.2 Axial Free Main Coupler End . . . 58
5.1.3 Axial Free Field Probe End . . . 61
5.1.4 Radial Free Main Coupler End . . . 64
5.1.5 Radial Free Field Probe End . . . 67
5.2 The characteristic equation . . . 69
5.3 Validation of the results . . . 72
6 Assembly modification 76 6.1 Force applied . . . 76
6.2 Assembly stiffness . . . 77
6.3 Helium Vessel optimization . . . 81
6.4 Other possible modification . . . 85
6.4.1 Fixed Model . . . 87
6.4.2 Axial Free Main Coupler End . . . 87
6.4.3 Axial Free Field Probe End . . . 88
6.4.4 The new characteristic equation . . . 88
6.4.5 Results . . . 89
7 Conclusion 93 A Material Properties 96 B Load Cases Results 99 C Verification of Ansys Results 106 C.1 Hoop Stress in the T i Cylinder . . . 106
C.2 Buckling of Spherical Cell . . . 107
C.3 Buckling of T i Cylinder . . . 108
D Calculations of the Titanium Bellows 111
1.1 A simplified diagram of an SRF cavity in a helium bath with
RF coupling and a passing particle beam . . . 12
2.1 Cross section view of the 1.3GHz cavity which allows to see the internal shape . . . 15
2.2 Zoom of the section of the cavity in the zone where we could see the ring . . . 15
2.3 View of the particulars in the Helium vessel. . . 16
2.4 Section of the Bellows to look at the shape. . . 17
2.5 View of the End plates to see the differences in the two Tran-sition Rings. . . 19
3.1 Differences between the Dressed cavity model and the compo-nents analyzed for the Pressure vessel verification. . . 22
3.2 Welds numbere as in Table 3.1. . . 24
3.3 Volumes for Pressure /Vacuum . . . 25
3.4 Definitions of the parameters X and Y for the calculation of the reinforcement. . . 30
3.5 Parameters to determine the Available Area and the Requested Area for the reinforcement. . . 31
3.6 The Finite Element Model. . . 34
3.7 Mesh Details. . . 35
3.8 Stress Classification Lines. . . 37
3.9 Lowest Buckling mode of the N b cavity (Pcr = 96.7M P a). . . . 39
3.10 Buckling of the conical heads. . . 40
4.1 Differences in the end part between the real model of the cav-ity and the simplified used for the FEM analysis. . . 44
4.2 Detailed view of the stiffening ring in the real cavity and in the simplified model. . . 45
4.3 Detail of the simplified model in the connection zone between the cavity and the end disk flange. You can observe that the contact has been replaced by an union between the parts and that the sharp edges has been avoided and substituted by cham-fers where possible. . . 46
4.4 Shape of the invernal volume of the cavity which is necessary for the RadioFrequency analysis. . . 46
4.5 Half model of the Helium vessel to show the differences with the real model presented in the previous chapter. . . 47 4.6 Final CAD model of the dressed cavity assembly after all the
simplifications made. . . 48 4.7 Result of the Electro Magnetic analysis to find the resonant
frequency f0 . . . 49 4.8 Axial displacement of the dressed cavity assembly when a
pres-sure of 1bar is applied in the zones where is located the Helium bath . . . 49 4.9 Result of the Electro Magnetic analysis to find the resonant
frequency f1 . . . 50 4.10 Graph putting in evidence the influence of the tuner stiffness
in the analysis of the pressure sensitivity . . . 52 5.1 Sketch of a cavity connected with a Helium Vessel by two
inter-faces where the directional displacement are probed to study the characterization. . . 54 5.2 Geometric function of the pressure sensitivity in the simple
case as reference . . . 55 5.3 A simplified diagram of an SRF cavity in a helium bath with
RF coupling and a passing particle beam . . . 56 5.4 Result of the Electro Magnetic analysis to find the resonant
frequency f0 in the case of fixed cavity . . . 57 5.5 Axial displacement of the bare cavity with fixed interfaces when
a pressure of 1bar is applied in the zones where is located the Helium bath . . . 57 5.6 Radial displacement of the bare cavity with fixed interfaces
when a pressure of 1bar is applied in the zones where is located the Helium bath . . . 58 5.7 Result of the Electro Magnetic analysis to find the resonant
frequency f1 in the case of fixed cavity . . . 59 5.8 Result of the Electro Magnetic analysis to find the resonant
frequency f0 in the case of cavity free only at axial displace-ment to the Main Coupler End . . . 59 5.9 Axial displacement in the case of cavity free only to axial
dis-placement at the Main Coupler End . . . 60 5.10 Radial displacement in the case of cavity free only to axial
dis-placement at the Main Coupler End . . . 60 5.11 Result of the Electro Magnetic analysis to find the resonant
frequency f1 in the case of cavity free only to axial displace-ment at the Main Coupler End . . . 61 5.12 Result of the Electro Magnetic analysis to find the resonant
frequency f0 in the case of cavity free only at axial displace-ment to the Field Probe End . . . 62 5.13 Axial displacement in the case of cavity free only to axial
5.14 Radial displacement in the case of cavity free only to axial dis-placement at the Field Probe End . . . 63 5.15 Result of the Electro Magnetic analysis to find the resonant
frequency f1 in the case of cavity free only to axial displace-ment at the Field Probe End . . . 64 5.16 Result of the Electro Magnetic analysis to find the resonant
frequency f0 in the case of cavity free only at radial displace-ment to the Main Coupler End . . . 65 5.17 Axial displacement in the case of cavity free only to radial
dis-placement at the Main Coupler End . . . 65 5.18 Radial displacement in the case of cavity free only to radial
displacement at the Main Coupler End . . . 66 5.19 Result of the Electro Magnetic analysis to find the resonant
frequency f1 in the case of cavity free only at radial displace-ment to the Main Coupler End . . . 66 5.20 Result of the Electro Magnetic analysis to find the resonant
frequency f0 in the case of cavity free only at radial displace-ment to the Field Probe End . . . 67 5.21 Axial displacement in the case of cavity free only to radial
dis-placement at the Field Probe End . . . 68 5.22 Radial displacement in the case of cavity free only to radial
displacement at the Field Probe End . . . 68 5.23 Result of the Electro Magnetic analysis to find the resonant
frequency f1 in the case of cavity free only at radial displace-ment to the Field Probe End . . . 69 5.24 Graph of the pressure sensitivity versus the displacement in
accordance to the simplified evaluation made . . . 72 5.25 Graph which shows the small differencies from the FEM model
to the two characteristic equations forecast of the pressure sen-sitivity made . . . 74 6.1 Summary of the situation explained with the pressure applied
on the end Disk . . . 77 6.2 Example of the FEA model used for the Cavity . . . 78 6.3 Model of the spring system . . . 80 6.4 Graph showing the changes in the volume and in the
displace-ment forecast varying the thickness of the Helium vessel . . . 83 6.5 Graph showing variation of the displacement with the
thick-ness of the Helium vessel . . . 84 6.6 Results of the Z displacement for the 8 mm thick Helium vessel 84 6.7 Results of the Z displacement for the coupled evaluation with
the cavity fixed at the end . . . 86 6.8 Result of the Electro Magnetic analysis to find the resonant
frequency f0 in the case of fixed cavity . . . 87 6.9 Result of the axial displacement and the f1 in the case of
6.10 Result of the axial displacement and the f1 in the case of Cav-ity free in the Field Probe End . . . 89 6.11 Graph of the pressure sensitivity versus the displacement in
accordance to the simplified evaluation made . . . 90 6.12 Result of the Electro Magnetic analysis to find the resonant
frequency f1 in the case of dressed cavity . . . 91 6.13 Result of axial displacement the case of dressed cavity with
70mm radius stiffening ring . . . 91 C.1 Path for the Hoop Stress plot . . . 106 C.2 Hoop Stress in the T i cylinder along line 1 − 2 for Pressure of
0.205M P a . . . 107 C.3 Single cell - radius for spherical shell buckling calculation . . 108 C.4 ANSYS linear buckling of the T i cylindrical shell . . . 109 D.1 Picture from the Code representing an unreinforced bellows
3.1 Summary of Weld characteristics . . . 23
3.2 Load cases . . . 26
3.3 Definition of Stresses and Coefficient in the bellows analysis, following the Code, Division 1, Appendix 26. . . 32
3.4 Complying with Appendix 26 rules for Internal pressure of 2bar (30psi). . . 33
3.5 Estimated Load History of Dressed SRF Cavity . . . 41
4.1 Results of the influence of the tuner stiffness over the pressure sensitivity of the dressed cavity . . . 51
5.1 Summary of the results of the simplified evaluation . . . 71
5.2 Displacement of the ends of the cavity for different tuner stiff-ness . . . 73
5.3 Pressure sensitivity forecasts and percantage error from the FEM model . . . 74
6.1 Stiffness of the parts according to the FEA models . . . 79
6.2 Summary of the results of thickness modification of the He-lium vessel . . . 82
A.1 Material Properties . . . 96
A.2 Allowable Stresses for each Stress Category (Units in M P a) . 97 A.3 Allowable Stresses S and estabilished value before the de-rating (Units in M P a) . . . 97
B.1 Load Case 1 - Primary Stress Results . . . 99
B.2 Load Case 1 - Bending Stress Results . . . 100
B.3 Load Case 2 - Primary Stress Results . . . 100
B.4 Load Case 2 - Bending Stress Results . . . 101
B.5 Load Case 3 - Primary Stress Results . . . 101
B.6 Load Case 3 - Bending Stress Results . . . 102
B.7 Load Case 4 - Primary Stress Results . . . 102
B.8 Load Case 4 - Bending Stress Results . . . 103
B.9 Load Case 5 - Primary Stress Results . . . 103
B.10 Load Case 5 - Bending Stress Results . . . 104
B.11 Maximum Allowable Sum of Principal Stresses (for Local Fail-ure) . . . 104
B.12 Local Failure Criterion - Niobium . . . 104 B.13 Local Failure Criterion - T i − 45N b . . . 105 B.14 Local Failure Criterion - Titanium Grade 2 . . . 105
Introduction
This work represent my working experience at the Fermi National Acceler-ator LaborAcceler-atory.Fermilab is a US Department of Energy national laborAcceler-atory specializing in high-energy particle physics. I had been working for 6 months as a Guest Engineer in the Technical Division. The Technical Division de-velops, designs, fabricates or procures, and tests accelerator and detector components.
My work during this experience concerned the 1.3GHz cavity. In particu-lar I was responsible for the verification of the Helium Vessel and the cavity according the ASM E Boiler and Pressure Vessel Code and after that to de-velop some possible solutions in order to reduce the pressure sensitivity of the cavity. Before talking about the work done we have to spend few words introducing the World of the particle accelerators in order to understand the quality of the work done.
1.1
Superconducting Radio Frequency
Superconducting Radio Frequency (SRF ) is the science and technology that involves the application of electrical superconductor to radio frequency de-vices [1]. The ultra-low electrical resistivity of a superconducting material allows an RF resonator to obtain extremely high value of power stored versus power dissipated to mantain the energy, a parameter called Qualityf actor, Q. Such very high Q resonator stores energy with very low loss and nar-row bandwidth. These properties can be exploited for a variety of appli-cations, including the construction of high-performance particle accelerator structure.
To give particles energy as they move through an accelerator, physicists use cavities containing electric fields that oscillate. The changes in electric field help push the particles from one cavity to the next. These oscillations occur with the same frequency as radiowaves, which is why this form of ac-celeration is called Radio-Frequency. Electromagnetic fields are excited in the cavity by coupling in an RF source with an antenna. When the RF fre-quency fed by the antenna is the same as that of a cavity mode, the resonant fields build to high amplitudes. Charged particles passing through apertures
in the cavity are then accelerated by the electric fields and deflected by the magnetic fields.
Superconducting refers to the way in which electric current is carried through these accelerating cavities. Electric current in a cavity may create friction, unless the cavity is created using special metals called supercon-ductors. These type of metals have a very low resistance below a certain temperature (called critical temperature) which depends on the material. This means that these metals conduct electricity almost perfectly. Even in a superconductor, by the way, if electric current passing through a cavity encounters any bump or impurities the flow of electricity is interrupted and energy can be lost as heat. This is why cavities must be very clean and polished to a smooth finish.
1.2
SRF Cavity
A Radiofrequency (RF) cavity is a metallic chamber that contains an elec-tromagnetic field [2]. Its primary purpose is to accelerate charged particles. RF cavities can be structured like beads on a string, where the beads are the cavities and the string is the beam pipe of a particle accelerator, through which particles travel in a vacuum.
To prepare an RF cavity to accelerate particles, an RF power generator supplies an electromagnetic field. The RF cavity is molded to a specific size and shape so that electromagnetic waves become resonant and build up in-side the cavity. Charged particles passing through the cavity feel the overall force and direction of the resulting electromagnetic field, which transfers energy to push them forwards along the accelerator.
The field in an RF cavity is made to oscillate at a given frequency, so timing the arrival of particles is important. The ideally timed proton, with exactly the right energy, will see zero accelerating voltage when the particle accelerator is at full energy. Protons with slightly different energies arriving earlier or later will be accelerated or decelerated so that they stay close to the energy of the ideal particle. In this way the particle beam is sorted in discrete packets called "bunches".
A large variety of RF cavities are utilized in particle accelerators. His-torically they have been made of copper, a good electrical conductor, and operated near room temperature with water cooling. The water cooling is necessary to remove the heat generated by the electrical loss in the cavity. In the past two decades, though, there has been a growing number of ac-celerator facilities for which superconducting cavities were considered more suitable for the accelerator than the normal conducting copper versions. The motivation for using superconductors in RF cavities is not to achieve a net power savings. Though superconductors have very small electrical resis-tance, the little power that they do dissipate is done at so very low temper-ature, typically in a liquid helium bath at 1.6K to 4.5K . The refrigeration power to mantain the cryogenic bath at low temperature in the presence of heat from small RF power dissipation is dictated by the Carnot efficiency
and can easily be comparable to the normal conductor power dissipation of a room-temperature copper cavity. The motivation for using superconducting RF cavity are, instead, the following:
• High duty cycle or continuous wave (cw) operation. SRF cavity allow the excitation of high electromagnetic fields at high duty cycle, or even cw, in such regimes that a copper’s cavity electrical loss could melt the copper, even with robust water cooling
• Low beam impedance. The low electrical loss in an SRF cavity al-lows their geometry to have large beampipe apertures while still main-taining a high accelerating field along the beam axis. Normal conduct-ing cavities need small beam apertures to concentrate the electric field as compensation for power losses in wall currents. However, the small apertures can be deleterious to a particle beam due to their spawning of larger wakefields, which are quantified by the accelerator parameters termed beamimpedance and lossparameter .
• Nearly all RF power goes to the beam. The RF source driving the cavity need only provide the RF power that is absorbed by the particle beam being accelerated, since the RF power dissipated in the SRF cav-ity is negligible. This is in cotrast to normal-conducting cavities where the power loss can easily equal or exceed the beam power consumption. The RF power budget is important since the RF source technologies have costs that increase dramatically with increasing power.
The most common fabrication technology for a SRF cavity is to form thin walled (1 − 3mm) shell components from high purity niobium sheets by stamping. These shell components are welded then together to form cavities. A simplified diagram of the key elements of an SRF cavity is shown in pic-ture 1.1. The cavity is immersed in a saturated liquid Helium bath. Pumping removes helium vapor boil-off and controls the bath temperature. The he-lium vessel is often pumped to a pressure below hehe-lium’s superfluid lambda point to take advantage of the superfluid’s thermal properties. Because su-perfluid has very high thermal conductivity, it makes an axcellent coolant. In addition, superfluids boil only at free surfaces, preventing the formation of bubbles on the surface of the cavity, which would cause mechanical per-turbations. An antenna is needed in the setup to couple RF power to the cavity fields and, in turn, to any passing particle beam. The cold portions of the setup need to be extremely well insulated, which is best accomplished by a vacuum vessel surrounding the helium vessel and all cold components. The full SRF cavity containment system, including the vacuum vessel and many other details, is the cryomodule.
1.3
Pressure Sensitivity
As explained before a cavity is immersed in a saturated liquid Helium bath which is pumped in order to control the bath temperature. The bath is
Figure 1.1: A simplified diagram of an SRF cavity in a helium bath with RF coupling and a passing particle beam
keeped at a certain pressure and the cavity resonant frequency depends on this pressure. The pressure fluctuations in the Helium bath inevitably due to the compressibility of the fluid cause cavity detuning by elastic deforma-tions and micro-oscilladeforma-tions of the cavity walls. This detuning implies that the resonant frequency of the cavity changes because of the deformation of the Niobium core of the cavity. Any small shift from the resonant frequency of the cavity requires significant increase in power to mantain the electro-magnetic field constant. For a cavity on resonance, the electric and electro-magnetic stored energies are equal. If a small perturbation is made on the cavity wall. this will generally produce an unbalance of the electric and magnetic energies, and the resonant frequency will shift to restore the balance. The Slatter perturbation theorem describes the shift of the resonant frequency, when a small volume ∆V is removed from a cavity of volume V [3]. For these reasons, the cavity sensitivity to Helium pressure is an important pa-rameter which must be taken in consideration during the design of a dressed cavity system. The evaluation of df /dp involves a series of electromagnetic and structural analyses that can be performed with multiphysics software such as COMSOL Multyphysics. The pressure sensitivity characterization is named Coupled Evaluation and these are the several steps to follow in order to calculate the pressure sensitivity:
• Electro Magnetic analysis Eigen frequency simulation to find the resonant frequency (f0)
• Static Structural analysis Find the deformation under given pres-sure load (p)
• Moving Mesh analysis Update the mesh after deformation inducted by the applied pressure
• Electro Magnetic analysis Eigen frequency simulation to find the resonant frequency after deformation (f1)
• Evaluation At this point the pressure sensitivity can be found as df
dp =
f1− f0
p (1.1)
When the pressure is applied and the resonant frequency is changed there is a tool designed specifically to restore the eigenfrequency of the cavity to its initial value. This tool is named Tuner, and its function is to mantains the tuning of the RF cavity after cooldown of the vessel and during opera-tion of the RF cavity. The tuner is an active system to restore the cavity resonant frequency. In fact it is equipped with a stepper motor that needs energy to function. Thus this method for tuning the cavity is an energy wasteful method and if we can decrease the pressure sensitivity of the as-sembly there will be a money saving for the Fermilab and for the project in which the cryostat is involved.
State of the Art
In this chapter we are going to analyze the cavity and its Helium vessel at the current state. This state is a modification of a previous design and is to be verified by the ASME code. We are going
The dressed SRF Cavity we take into account is called the LCLS II He-lium Vessel RF Cavity Assembly. The dressed cavity consists of the niobium nine-cell 1.3GHz cavity, with a unique serial number, and the titanium He-lium vessel, also with unique serial number. The LCLS II HeHe-lium Vessel RF Cavity Assembly consists essentially of two sub-assemblies: the niobium SRF (bare) cavity and the titanium Helium vessel weldment.
The design of the niobium nine-cell cavity is the same as the cavities used in the TESLA facility at Desy (Hamburg, Germany), which has been in operation for the past 10 years. The design of the Helium vessel is a modification of the TESLA design. The location of the titanium bellows is a modification of the TESLA design that is a result of collaboration between Fermilab and DESY. The old design had the bellow in the middle of the tube while this new design presents the bellow at the Field Probe end of the assembly.
2.1
The 1.3GHz Cavity
The niobium SRF cavity is an elliptical nine-cell assembly. The nine-cell cavity is presented in figure 2.1 where we can see that it is formed by the cells and by two tubes that form the ends. A single cell, or a dumbbell, consists of two half-cells that are welded together by an Electron Beam process at the equator of the cell. The thickness of the cells is 2.8mm. The iris diameter of the cell is 70mm while the outside diameter of the cell is 206.9mm.
The length of an half-cell depends on the position where it is: the mid half-cells have a theoretical length of 57.70mm; the long end half-cells has a length of 57mm and the short end half-cell has a theoretical length of 56mm. This means that the bare cavity assembly is not symmetric for a plane that cut its axis in the half. Later, when we would like to simplify the model to do the FEA analyses, this will be important.
Between the cells there are some rings to stiffen the assembly to a point.
Figure 2.1: Cross section view of the 1.3GHz cavity which allows to see the internal shape
Figure 2.2: Zoom of the section of the cavity in the zone where we could see the ring
Some flexibility in the length of the cavity is required to tune the cavity and optimize its resonant frequency. The stiffening ring in the model have a thickness of 3mm and a width of 20.63 mm and are to be welded with the cavity with the Electron Beam process too. The outside radius of the ring is 56.5mm.
In figure 2.2 we can see the details in the shape of the cells and the ring. The ring does not surround the cavity for 360◦ because it is made by two sec-tion cutted 5mm above the equator. The result is that the cavity is more loose in the Z direction and the tuneability is improved. The figure highlights also that the cell shape is basically two oval, a little one near the itis and a big one on the other side, combined by a line tangent to both. The small features near the largest diameter are meant to help the processo of Electron Beam welding.
2.2
The Helium Vessel
The Helium vessel is the part responsible for encasing the niobium SRF bare cavity. The vessel is made of Titanium grade 2. It has two Helium fill ports at the bottom and in the center of the vessel there is the two-phase helium return line. At the side of the vessel there are tabs which support the vessel within the HTS cryostat or cryomodule. In figure 2.3 we can see the details of the model. The vessel is long 965mm and its thickness is 5mm for the most part of its length. The great opening is 95.5mm of diameter while the other two openings, the ones for the Helium filling, are only 16mm of diameter.
2.2.1
The tube
(a) Section of the tube of the vessel. (b) Two phase return line.
(c) Helium fill ports. (d) Details of the modification made
to help the welding process.
At the end of the tube the thickness is changed because of the welding preparation process. Here the thickness is reduced to 2.5mm because the tube will be joined woth the other parts of the assembly by a Tungsten Inert Gas welding process. At the Main Coupler end the tube is attached to the end plate, while at the Field Probe end the tube is joined by the bellows, made by Titanium too, which provide the flexibility in length to the assembly. This flexibility in the vessel allows also for accomodating the change in the nine-cell cavity length due to thermal contraction at cryogenic temperature and for tuning the niobium cavity during operation.
The Titanium bellows is an unconventional one. Indeed it starts its con-volution at a radius of 116.3mm and it ends them at a diameter of 109.7mm. This can be seen in Figure 2.4 where we can infer that this bellows will have an non-integer number of convolutions (2.5) which, for analytic reasons, will be approximed to 2 in the calculations. The total bellows length is 42mm and the length of the part with the convolutions is 22mm.
Figure 2.4: Section of the Bellows to look at the shape.
A lever tuner (yet to be designed) supports the vessel at the bellows. Two control systems act on the lever tuner to change the length of the vessel, and thus change the length of the cavity. A slow-control tuner system that con-sists of a stepper motor that changes the vessel length is the first one. The stepper motor extends the length of the cavity by less than 2mm to bring it to the desired resonant frequency to counteract the combined effects of ther-mal contraction and pressurization during cool down. Once the cavity is at cryogenic temperature the slow tuner system is shut off. The second one is a fast-control tuner system consisting of two piezoelecteic actuators prevents detuning of the cavity during operation due to Lorentz Forces and noise sources (microphonics). The piezos provide an increase in bellows length (bellows expansion) of 13µm during operation. The vessel is expected to have a lifetime of 10 years.
2.2.2
The End plates
The end plates are the parts responsible for connecting the Helium vessel tube and bellows to the cavity. Each plate is composed by three parts:
• End disk flange: The End disk flange is the part that connect the subassemply to the cavity. It is made of Niobium as the cavity because it will be welded with the cavity. Its shape is also determined by the fact that the End disk flange has also the function of last support ring. • Endcap Disk: The Endcap disk is a conical part that connect the other two parts of the subassembly. It is made by an alloy: the T i − 45N b. This alloy is made from 55% of Titanium and by 45% Niobium. It is fundamental because it will be welded to the other two parts that are made by different materials.
• Transition Ring: The transition ring is made of Titanium grade 2 and is responsible to attach the assembly to the Helium vessel tube. This part will be different depending on whether it is in the Main Coupler end or in the Field Probe end. In fact in the Main Coupler End it is to be welded to the tube while in the Field Probe end it will be welded to the Titanium Bellows.
From Fig. 2.5 we can see what we were talking about. The End disk flanges and the Endcap disks have exactly the same shape both in the Field Probe end and in the Main Coupler end. The transitio ring could not be tha same in both sides. Indeed it has to interact with different part: in the Field Probe end it is welded to the Bellows while in the Main Coupler end it is welded to the tube. These parts have different diameter so the Transition ring in which will join the Bellow will be a circular crown with a smaller outside diameter.
The iris diameter of the End plates (which shapes reminds us a circular crown) is the outside diameter of the iris of the cavity that is 72mm. The outside diameter depend on the position of the End plate: th Main Coupler End plate is welded with the Helium Vessel tube so its outside radius is the radius of the tube in that spot (117.5mm). In the Field Probe end plate the outside diameter is the smaller diameter of the bellows or 213mm.
(a) Field Probe end.
(b) Main Coupler end.
Figure 2.5: View of the End plates to see the differences in the two Transition Rings.
Verification according to the
ASME
In this chapter we want to summarizes how the cavity, as a Helium vessel, follows the requirements of the FESHM Chapter 5031.6 for Dressed SRF Cavities [4]. Dressed SRF cavities, as defined in that document, are de-signed and fabricated following the ASM E Boiler and Pressure Vessel Code (The Code). The 1.3GHz dressed cavity as a Helium pressure vessel has ma-terials and complex geometry that are not conducive to complete design and fabrication following the Code. However, where the Code can not be followed, we show that the vessel is safe in accordance with FESHM Chapter 5031.6 for example by de-rating the allowable stress. The analyses were done using ANSYS Workbench 14.5 and Mathcad version 14.
3.1
Introduction and Summary
This analysis is intended to demonstrate that the LCLS II 1.3GHz SRF cav-ity conforms to the ASME Boiler and Pressure vessel Code, Section VIII, Div. 1, to the greatest extent possible. Where Div. 1 formulas or procedures are available, they are applied to this analisys. For those cases where no rules are available, the provisions of Div. 1, U − 2(g) are invoked [5]. This para-graph of the code allows alternative analyses to be used in absence of Code guidance.
This cavity contains several features which are not supported by the Code. These are related primarily to materials, weld types, and non de-structive examination, and are addressed in details in the next subsection of this report. These are accepted as unavoidable in the context of SRF cavi-ties, and every effort is made to demonstrate through consideration of their implications in the analysis.
Advantage is taken of the increase in yield and ultimate strength which occurs in the N b and T i components at the operating temperature of 1.88K. The design pressure specified for this analysis are 2bar at 293K and 4bar at 1.88K. This analisys confirms that the MAWPs of the vessel can be safely set at these pressure. Negligible margin for increase is available at 293K
but the cold MAWP could be increased substantially above 4bar.
In addition to these fundamental operating limits, the cavity was also shown to be stable at external pressures on the T i shell of 1bar, and internal pressures on the N b cavity of 1bar; these loadings could coccur under fault conditions, when the beam and insulating vacuums have been compromised, and the Helium volume has been evacuated.
3.1.1
Non-Code elements
With regards to the Design Verification, the LCLS II 1.3GHz cavity does not comply with Div. 1 of the code in the following ways:
1. Category B joints in titanium must be either Type 1 butt welds (welded from both sides) of Type 2 butt welds (welded from one side with back-ing strip) only. Some category B (circumferential) joints are Type 3 butt welds (welded from one side with no backing strip).
2. All joints in titanium vessel must be examined by the liquid penetrant method. No liquid penetrant testing was performed on the vessel. 3. All electron beam welds in any material are required to be
ultrason-ically examined along their entire length. No ultrasonic examination was performed on the vessel.
The evaluation of Type 3 butt welds in the titanium is based on a de-rating of the allowable stress by a factor of 0.6, the factor given in Div. 1, Table U W − 12 for such welds when not radiographed. The exceptions listed above do not address Code requirements for material control, weld procedure certification, welder certification.
3.2
Geometry
This analysis is based on geometry obtained from the drawings obtained from the Fermilab Drafting department. In figure 3.1 (a) is swhown the Dressed SRF cavity, complete with magnetic shield, piping and a mock-up of the lever tuner. For the analysis, only the N b cavity, the conical T i − 45N b heads and titanium shell and bellows are modeled, as well as the flanges to which the Helium vessel is constrained. Also these components are shown in figure 3.1 (b). Different colour in this picture represent different materials. the indivitual cavity component names used in the report have been already shown in the "State of the Art" chapter.
3.2.1
Welds
This section describes the welds as a precursor to the weld stress evaluation. Welds are produced by the EBW (Electron Beam Welding) process in the N b
(a) Dressed LCLS II SRF Cavity.
(b) Cavity components included in the analysis.
Figure 3.1: Differences between the Dressed cavity model and the compo-nents analyzed for the Pressure vessel verification.
and the N b − to − T i transitions, and the TIG (Tungsten Inert Gas) process in the T i − T i welds.
All welds on the Dressed SRF cavity are designed as full penetration butt welds. All welds are performed from one side, with the exception of the T i − 45N b to T i transition welds. Those welds are performed from two sides. No backing strips are used for any welds. 3.1 summarizes the weld numbering, including the Code classification of both joint category and weld type, and the corresponding efficiency. The location of the welds as numbered in 3.1 are shown in figure 3.2.
Table 3.1: Summary of Weld characteristics Weld Materials Joined Weld pro-cess Joint Cate-gory Code Weld Type Joint Effi-ciency 1 N b − N b EBW B 3 0.6 2 N b − N b EBW B 1 0.7 3 N b − N b EBW − 3 0.6 4 N b − T i45N b EBW B 3 0.6 5 T i45N b − T i EBW B 1 0.7 6 T i − T i T IG C 7 0.6 7 T i − T i EBW B 3 0.7 8 T i − T i EBW B 3 0.6 9 N b − N b EBW B 3 0.6 10 N b − N b EBW B 3 0.6 11 N b − N b EBW B 3 0.6 12 T i − T i EBW C 7 0.6
Let us show a brief description of every welding: 1. End tube spool piece to End cap flange
2. End tube spool piece to RF half cell 3. End cap flange to RF half cell 4. End cap flange to end cap disk 5. End cap disk to transition ring
6. 1.3GHz 9 cell RF cavity (transition ring) to Bellows assembly
7. (Field Probe End) Bellows assembly to LCLS II Helium vessel assem-bly
8. Bellows convolution to weld cuffs 9. Support ring to half cell
11. Half cell fo half cell
12. (Main Coupler End) Transition ring to LCLS II Helium vessel assem-bly
Figure 3.2: Welds numbere as in Table 3.1.
3.3
Loadings
The dressed cavity is shown in figure 3.3. There are three volumes which may be pressurized or evacuated:
1. The LHe volume of the Helium vessel
2. The volume outside the cavity typically evacuated for insulation
3. The volume through which the beam passes on the inside of the N b cavity itself.
The pressure in these volumes are denoted as P 1, P 2, and P 3, respec-tively.
With regard to pressure, typical operation involves insulating vacuum, beam vacuum and a pressurized LHe volume. Atypical operation may oc-cour if the insulating or beam vacuums are spoiled, and the LHe space si-multaneously evacuated. This reverses the normal operational stress state of the device, producing an external pressure on the T i shell, and an internal pressure on the N b cavity; however this pressure is limited to a maximum differential of 1bar.
Figure 3.3: Volumes for Pressure /Vacuum
In addition to the pressure loads, the cavity also sees dead weight forces due to gravity which are reacted at the T i flanges, as well as thermal con-tractions when cooled to the operating temperature of 1.88K, and a strain-controlled extension by the blade tuner after cool down. All of these loadings are considered in the analysis.
Three of these loads - gravity, liquid head and pressure - produce both primary and secondary stresses. The remaining loads - thermal contraction and tuner extension - are displacement controlled loads which produce sec-ondary stresses only. This results in five load cases.
This load cases are shown in table 3.2, along with the temperatures at which the resulting stresses were assessed, and the stress categories that were applied. Below there is a brief description of the load cases:
• Load Case 1 : Warm pressurization.
• Load Case 2 : Cold operation, maximum pressure, no thermal contrac-tion.
Table 3.2: Load cases
Load case Loads Temperature Stress Categories 1 1. Gravity 293K Pm, Pl, Pl+ Q
2. P1 = 0.205M P a
2
1. Gravity
1.88K Pm, Pl, Pl+ Q 2. LHe liquid head
3. P1 = 0.41M P a 4. P2 = P3 = 0 3 1. Cool down to 1.88K 1.88K Q 2. Tuner extension of 1.5mm 4 1. Gravity 1.88K Pm, Pl, Pl+ Q 2. LHe liquid head
3. Cool down to 1.88K 2. Tuner extension of 1.5mm 4. P1 = 0.41M P a 5. P2 = P3 = 0 5 1. Gravity 293K Pm, Pl, Pl+ Q 2. P1 = 0 3. P2 = P3 = 0.1M P a
• Load Case 3 : Cool down and tuner extension, no primary loads. • Load Case 4 : Cold operation, full LHe inventory, masimum pressure,
primary and secondary loads.
• Load Case 5 : Insulating and beam vacuum upset, helium volume evacuated.
The most critical loading condition we can suppose is the fourth, where alle the loadings are considered, while the fifth is only the atypical operation which occours only in failing conditions of the cryostat.
3.4
Stress Analysis Approach
The goal of this analysis is to qualify the vessel to the greatest extent pos-sible in accordance with the rules of the Code, Section V III, Div. 1. This Division of the Code provides rules covering many cases; however, there are features of this cavity and its loadings for which the Division has no rules. This does not mean that the vessel cannot be qualified by Div. 1, since Div. 1 explicitly acknowledges the fact that it does not prevent formulaic proce-dures ("rules") covering all design possibilities. From U − 2(g):
This Division of Section V III does not contain rules to cover all details of design and construction. Where complete details are not given, it is intended that the Manufacturer, subject to the acceptance of the Inspector, shall provide details of design and construction which will be as safe as those provided by the rules of this Division.
Division 1 rules relate to both geometries and loads. For either, there are few rules applicable to the features of the cavity. The only components of the cavity which can be designed for internal and external pressure by the rules of Div. 1 are the Ti shells and the Ti bellows. In the Ti shell, there are two penetrations for connection of externals for which the required reinforcement can also be determined by Code rules.
The conical heads have half-apex angles exceeding 30 degrees, and no knuckles; Div. 1, Appendix 1, 1 − 5(g) states that their geometry falls under U − 2(g). The N b cavity itself resembles an expansion joint, but does not con-form to the geometries covered in Div. 1, Appendix 26. Therefore, U − 2(g) is again applied. U G − 22(h) states that "temperature gradients and differen-tial thermal contractions" are to be considered in vessel design, but provides no rules to cover the cavity. In this analysis, all thermal contraction effects are addressed under U − 2(g). The cavity is also subjected to a controlled displacement loading from blade tuner. There are no rules in Div. 1 covering such a loading, so U − 2(g) is applied.
3.4.1
Applying U − 2 (g)
U − 2(g) is satisfied in this analysis by the application of the design-by-analysis rules of the Code, Section V III, Div. 2, Part 5. These rules
pro-vide protection against plastic collapse, local failure, buckling, fatigue, and ratcheting. The specific sections of Part 5 applied here are:
• Plastic collapse - satisfied by an elastic stress analysis performed according to 5.2.2.
• Ratcheting - satisfied by an elastic stress analysis performed accord-ing to 5.5.6.1.
• Local failure - satisfied by an elastic stress analysis performed ac-cording to 5.3.2.
• Buckling - satisfied by a linear buckling analysis performed according to 5.4.1.2(a).
• Fatigue assessment - the need for a fatigue analysis is assessed ac-cording to 5.5.2.3.
In general, an elastic stress analysis begins by establishing stress clas-sification lines (SCLs) through critical sections in the structures according to the procedures of Part 5, Annex 5A, so they are chosen near the disconti-nuities and are through the thickness of the part. The stresses along these lines are then calculated (in this case, by a FEA), and "linearized" to produce statically equivalent membrane stress and bending stress components. The allowable stress for each component depends on the category of the stress. This category (or classification) depends on the location of the SCL in the structure, and the origin of the load.
Stresses near discontinuities have higher allowables to reflect their abil-ity to redistribute small amounts of plasticabil-ity into surrounding elastic ma-terial. Stresses produced solely by strain-controlled loads (e.g., thermal con-tractions and blade tuner extension) are given higher allowables regardless of their location in the structure. Allowable stresses are expressed in terms of multiples of S, which is the allowable general primary membrane stress. The values of S used in this analysis are given in the Appendix.
3.5
Division 1
In this section we want to present the calculation made to show that the ASME code, Section V III division 1 is satisfied.
3.5.1
T i Cylindrical Shells
Thickness for internal pressure The minimum thickness for the T i cylin-drical shells under internal pressure can be calculated from U G − 27(c)(1):
t = P · R
Where t is the required thickness to satisfy the Code, P is the pressure which is 0.205M P a in the warm case and 0.41M P a in the cold case, R is the inside radius of the shell (115mm), E is the efficiency of the seam weld (for Type 3 TIG weld: one sided butt weld, no radiography) that is 0.6 and S is the maximum allowable membrane stress, calculated from the material properties and iti is 79M P a for the warm case and 255M P a for the cold case. Substituting the values in the equation we find out that the minimum required thickness when warm is 0.49mm. When cold and pressurized to 0.41M P a it is 0.31mm. The actual minimum thickness of the shells is 2.5mm near the ends. Therefore, the T i cylindrical shells meet the minimum thick-ness requirements of U G − 27 for internal pressure.
Thickness for external pressure (Buckling) The minimum thickness required for the T i cylindrical shells under external pressure can be calcu-lated from U G − 28(c). This procedure uses charts found in the Code, Section II, Part D. These charts are based on the geometric and material character-istics of the vessel.
If we use the actual length L = 965mm, the real diameter D0 = 230mm and the thickness of 1.4mm we found that the parameter to be used in the charts are L/D0 = 2.2 and D0/t = 165. From the Code, Section II, Part D, Subpart 3, Figure G, the factor A is 0.0003. The allowable pressure is then:
P = 2
3 · A · Em· t D0
= 0.11M P a (3.2) Where Em is the young modulus of Titanium (107GP a) and the other pa-rameters have already been introduced. The substitution give P = 0.11M P a. This is approximately equal to the 0.105M P a maximum external vessel for which the vessel must be qualified.
The actual minimum thickness of the Ti shell is 2.5mm. This occurs near the ends, and it is unlikely that the collapse is well predicted by this thick-ness, due to its short length, and proximity to the conical head, which will tend to stiffen the region. If we assume, however, that the entire shell is this thickness, and repeat the calculations above, the allowable external pres-sure is 0.23M P a. If we assume the collapse is better predicted by the pre-dominant thickness of 5mm, then the factor A = 0.0009, and the allowable external pressure is 0.7M P a.
In any case, the required minimum thickness of 1.4mm is less than the actual minimum thickness anywhere on the T i cylindrical shell. Therefore, the T i shell satisfies the Code requirement for external pressure.
3.5.2
Penetrations
The T i cylindrical shell contains three penetrations two of which have the same diameter. The largest of these penetrations is 3.76 inches (95.5mm) in diameter. From U G − 36(c)(3):
"Openings in vessels not subject to rapid fluctuations in pressure do not require reinforcement other that inherent in the construction under the fol-lowing conditions: welded, brazed, and flued connections meeting applicable rules and with a finished opening not larger than 3.5 inches diameter - in vessel shells or heads with a required minimum thickness of 3/8 inch or less." The minimum required thickness of the shell is largest for the case of 0.205M P a pressurization (warm case). This thickness (calculated before) is 0.49mm. This is less than 9mm (3/8in). The two smaller penetrations have a diameter of 16mm (0.63in) which is smaller than 3.5in therefore no additional reinforcement is required for these penetrations. However the largest pen-etration has a diameter of 95.5mm (3.76in) so for this penpen-etration we need further calculations to see if the reinforcement is needed or not.
Figure 3.4: Definitions of the parameters X and Y for the calculation of the reinforcement.
From the definitions given by figure 3.4 we can write that X is equal to the diameter (X = 95.5mm) and Y is equal to 2.5 times the thickness so it is 12.5mm. We can now calculate the other parameters introduced in the Code and represented in figure 3.5:
• d = diameter of the nozzle = 95.5mm • t = thickness of the vessel = 5mm • tn= thickness of the nozzle = 1.65mm
• tr = minimum required thickness of the vessel = 0.49mm • trn= minimum required thickness of the nozzle = 0.25mm
Now we can calculate the parameters to see if the reinforcement is re-quired. We have to calculate the Requested Area and the Available Area both of the Vessel and of the Nozzle. The we have to evaluate if the Re-quested Area is larger than the available area. If so the reinforcement is needed. If not there is no need of reinforcement.
• Requested Area:
Ar = d · tr= 48.2mm2 • Vessel Available Area
A1 = (2 · X − d) · (t − tr) = d · (t − tr) = 433mm2 • Nozzle Available Area
A2 = 2 · Y · (tn− tnr) = 5 · t · (tn− tnr) = 35mm2
Since the Requested Area is smaller than the total Available Area the reinforcement is not needed neither for the dual phase opening.
Figure 3.5: Parameters to determine the Available Area and the Requested Area for the reinforcement.
3.5.3
T i Bellows
The design of metallic expansion joints (e.g., bellows) is addressed by Ap-pendix 26 of the Code. The formulas permit calculation of internal and ex-ternal pressure limits. In a bellows, the pressure may be limited not only by stress, but by squirm (internal pressure), and collapse (external pressure.) The analysis shows that the bellows with an internal MAWP of 2.0bar (30psi) at room temperature or an external MAWP of 1.0bar (14.5psi) follows the rules of Appendix 26. The allowed value S is for titanium at room tempera-ture (see Appendix).
Table 3.3 defines the stresses that are examined in the bellows analysis. Table 3.4 summarizes how the calculated or actual stresses comply with the allowed stresses. The details of the Appendix 26 calculations are presented in the Appendix.
Table 3.3: Definition of Stresses and Coefficient in the bellows analysis, fol-lowing the Code, Division 1, Appendix 26.
Symbol Description Units
S1 Circumferential membrane stress in the bellows tan-gent, due to pressure P
psi S2e Circumferential membrane stress due to pressure P
for end convolutions
psi S2i Circumferential membrane stress due to pressure P
for end convolutions
psi S11 Circumferential membrane stress due to pressure P
for the collar
psi S3 Meridional membrane stress due to pressure P psi S4 Meridional bending stress due to pressure P psi
P Design pressure psi
S Allowable stress of bellows material psi Cwc Weld joint efficiency of collar to bellows (no
radiogra-phy, single butt weld)
Sc Allowable stress of collar material psi Kf Coefficiend for formed bellows
Psc Allowable internal pressure to avoid column instabil-ity
psi Psi Allowable internal pressure based on in-plane
insta-bility
psi Pa Allowable external pressure based on instability psi
Longitudinal Weld in Bellows Convolution The allowable stress S = 79M P a for the bellows convolution assumes a weld joint efficiency of 1.0. The bellows is hydro formed from a rolled tube with a longitudinal (seam) weld that is not radiographed. Let’s evaluate the weld by de-rating the allowable stress S by a factor of 0.6, which is the factor for a Type 3 weld that is not
Table 3.4: Complying with Appendix 26 rules for Internal pressure of 2bar (30psi). Calculated Stress [psi] Allowed Value [psi] Requirement Applicable paragraph S1 = 428 S = 11500 S1 < S 26 − 6.3.1 S11 = 441 Cwc· S = 6900 S11 < Cwc· S 26 − 6.3.2 S2e = 995 S = 11500 S2e < S 26 − 6.3.3(a)(1) S2i = 5545 S = 11500 S2i < S 26 − 6.3.3(a)(2) S3+ S4 = 4275 Kf · S = 34500 S3+ S4 < Kf· S 26 − 6.3.3(d) P = 30 Psc = 6.47 · 107 P ≤ Psc 26 − 6.4.1 P = 30 Psi = 198 P ≤ Psi 26 − 6.4.2 Ext. P= 14.5psi Pa = 1077 Ext. P< Pa 26 − 6.5
radiographed. The de-rated allowable stress is 79M P a · 0.6 = 47.4M P a. This is still greater than the calculated circumferential stresses of S1, S2e, and S2i in the convolutions.
Fatigue Analysis for Titanium Bellows The equations in the Code for fatigue analysis of a bellows are not valid for Titanium. The manufacturer of the titanium bellows for the helium vessel provided design calculations following the Standards of the Expansion Joint Manufacturers Association [6]. The allowable fatigue life is calculated with the equation:
Nc= c St− b
a
(3.3) where a b, and c are material and manufacturing constants. The man-ufacturer uses the same material and manufacturing constants as what EJMA uses for austenitic stainless steel. In addition, the manufacturer in-cludes a safety factor of two in their calculation of the allowable number of cycles since the titanium bellows is a custom-made project. The manufac-turer calculated an allowable number of cycles to be N C = 375600.
The slow tuner system has the capability of increasing the vessel length less than 2mm after each cool down. The bellows extension will occur 200 times over the lifetime of the vessel. This is far less than the allowable number of cycles, so the bellows is designed well within the limits of fatigue failure. Detailed Code calculations are shown in the Appendix.
3.6
Division 2
In this section we want to present the results of the analysis made to follow the Division 2 rules so we are showing the results of the F EA analysis done.
3.6.1
Finite Element Model
A 3d fnite element half model was created in NX 7.5 and then imported into the ANSYS software. Elements were 10-node tetrahedra, and 20-node hex-ahedra. Material behavior was linear elastic. The lever tuner is very rigid. Axial constraint of the helium vessel was therefore simulated by constrain-ing the outer surface of each flange in the Z (axial) direction. This constraint places the line of action at a maximum distance from the shell, producing the maximum possible moment on the welds between the T i blade tuner flanges and the shell.
Figure 3.6: The Finite Element Model.
For the cool down loading, the distance between the T i flanges was as-sumed to close by an amount equivalent to the shrinkage of a rigid stainless steel mass spanning the flanges. The constraint against gravity is simulated by fixing the flange outer surface nodes at 180 degrees in the Y (vertical) di-rection. The finite element model is shown in Figure 3.6. Figure 3.7 shows the mesh detail at various locations within the model. The complete model was used to demonstrate satisfaction of the plastic collapse, ratcheting, and local failure criteria. Subsets of the model were also used to address the linear buckling of the N b cavity and conical head.
3.6.2
Stress Analysis Results
The complete finite element model was run for the five load cases. Stress classification lines, shown in Figure 3.8, were established through the criti-cal sections of the structure. The stresses along these lines were linearized with ANSYS, and separated into membrane and bending components. The linearized stresses (expressed in terms of Von Mises equivalent stress, as required by 5.2.2.1(b)) are categorized according to the Code, Div. 2, Part 5, 5.2.2.2 into primary and secondary stresses.
The primary and secondary stresses along each SCL for each of the five load cases are given in the Appendix. Where more than one weld of a given number is present the weld with the highest stresses was assessed. The stresses from the Appendix are used to demonstrate satisfaction of two of the criteria listed in the Stress Analysis Approach of this report: Protection against plastic collapse, and protection against ratcheting. Demonstrating protection against local failure employs the complete model, but requires the extraction of different quantities.
The required minimum thicknesses of the T i shells for internal and ex-ternal pressure have been previously calculated by Div. 1 rules. Therefore, no SCLs addressing the T i shell thickness far from welds or other disconti-nuities are established here. See the Appendix for verification that the FEA produces the correct hoop stress in the T i shell.
Collapse Pressure The criterion for protection against plastic collapse is given in Div. 2, 5.2.2. The criterion is applied to load cases in which primary (load-controlled) stresses are produced. For this analysis, this is Load Case 1, Load Case 2, and Load Case 5. The following stress limits must be met (per 5.2.2.2.4(e)), where S is the maximum allowable primary membrane stress:
• Pm = primary membrane stress≤ S
• Pl = primary local membrane stress≤ 1.5 · S
• Pl+ Pb = primary local membrane + primary bending≤ 1.5 · S
In this work, the Pl classification is limited to SCL B (weld 2). All other membrane stresses extracted on the SCLs are classified under the more con-servative Pm, which is then used in place of Pl in the third formula above.
Examining the table in the Appendix where the results are listed it is found that the closest approach to the limiting stress for any case occour at SCL D (weld 4, the weld between the end disk flange and the transition ring) in Load Case 1, where the primary membrane stress plus the primary bending stress of 10.1M P a compares to an allowable of 18M P a.
Ratcheting Protection against ratcheting, the progressive distortion of a component under repeated loadings, is provided by meeting the require-ments of Div. 2, 5.5.6. Specifically, the following limit must be satisfied:
∆Sn,k ≤ Sps (3.4) where ∆Sn,k is the primary plus secondary equivalent stress range and Sps is the allowable limit on primary plus secondary stress range. The stress range ∆Sn,k must take into account stress reversals; however, there are no stress reversals in normal operation of the cavity, so for this analysis ∆Sn,k is equal to the primary plus secondary stresses given in the tables presented in the Appendix.
Examination of the tables shows that the cavity satisfies the ratchet-ing criterion; the closest approach to the allowable primary plus secondary stress range limit occurs for Load Case 4 (gravity + liquid head + 0.4M P a + blade tuner extension + cool down) in the T i bellows. For this load case, the calculated primary plus secondary stress range reaches 73% of the allowable.
Local Failure The criterion for protection against local failure is given in Div. 2, 5.3.2:
σ1 + σ2+ σ3 ≤ 4 · S (3.5) where σ1, σ2, σ3 are the principal stresses at any point in the structure, ans S is the maximum allowable primary membrane stress, multiplied by a joint factor if applicable. The criterion is assumed to be satisfied if the sum of the principal stresses calculated at every element centroid in the model meets the stress limit for the material.
The results for each material and each load case are given in the Tables in the Appendix. The closest approach to the allowable limit occurs in the iris support ring welds for Load Case 4 (cold, 0.41M P a internal pressure, tuner extension), which reaches 0.94 of the allowable. For all other materials/load cases, the principal stress sum lies below the allowable.
Buckling The buckling of the Titanium shells and bellows is addressed by Div. 1 rules in the previous section of this report. The Code, Div. 1, does not contain the necessary geometric and material information to perform a Div. 1 calculation of N b cavity collapse. Therefore, the procedures of Div. 2, part 5, 5.4 Protection against collapse from buckling are applied.
A linear elastic buckling analysis was performed with ANSYS. A design factor was applied to the predicted collapse pressure to give the maximum allowable external working pressure. This design factor, taken from 5.4.1.3(c) for spherical shells, is 16. Only the cavity was modeled. The ends are con-strained in all degrees of freedom to simulate the effect of the attachment to the conical heads and T i shells of the Helium vessel.
The predicted buckled shape is shown in Figure 3.9. The critical pressure is 96.7M P a. Applying the design factor gives this component a maximum al-lowable external working pressure of 6M P a, which is far greater than the required M AW P of 0.1M P a external. The ANSYS buckling pressure seems
large; as a check, a calculation of the collapse of a sphere of similar dimen-sions to those of a cell was done. This calculation, given in the Appendix of this report, produces a similar result.
The buckling pressure of the conical heads was calculated by the linear buckling approach used for the Nb cavity. A model of the head only was made. It was constrained against axial motion where it connects to the Ti shell, but allowed to rotate freely, and translate radially. The predicted buck-ling shape is shown in Figure 3.10. The critical buckbuck-ling pressure is 358M P a. Applying the design factor of 2.5 (from 5.4.1.3(b) for conical shells under ex-ternal pressure) gives an M AW P for exex-ternal pressure of 143M P a, which is well above the actual maximum pressure of 0.1M P a.
Figure 3.10: Buckling of the conical heads.
Fatigue Assessment The need for a fatigue analysis can be determined by applying the fatigue assessment procedures of Div. 2, Part 5, 5.5.2.3, Fatigue Analysis Screening, Method A. In this procedure, a load history is estabil-ished which determines the number of cycles of each loading experienced by the Dressed SRF Cavity. These numbers are compared against criteria which determine whether a detailed fatigue analysis is necessary.
First of all we have to introduce some parameters which will be used in this analysis:
• NδF P = expected number of full-range pressure cycles, including startup and shutdown
• NδP O= expected number of operating pressure cycles in which the range of pressure variation exceeds 20% of the design pressure for integral construction or 15% of the design pressure for non-integral construc-tion
• NδT E = effective number of changes in metal temperature difference between any two adjacent points
• NδT α = number of temperature cycles for components involving welds between materials having different coefficients of thermal expansion that cause the value of (α1 − α2) · T to exceed 0.00034
Table 3.5: Estimated Load History of Dressed SRF Cavity
Loading Designation Number of Cycles
Cool Down NδT E 100 Pressurization NδF P 200 Tuning Nδtuner 200
The load history consists of multiple cool down, pressurization, and tun-ing cycles. Estimates for the number of cycles of each load a cavity might experience are given in Table 3.5. The information of table 3.5 is used with the criterion elencated below to determine whether a fatigue analysis is nec-essary:
• Attachments and nozzles in the knuckle region of formed heads
NδF P + NδP O+ NδT E+ NδT α ≤ 350 (3.6) • All other components that do not contain a flaw
NδF P + NδP O+ NδT E+ NδT α ≤ 1000 (3.7) • Attachments and nozzles in the knuckle region of formed heads
NδF P + NδP O+ NδT E+ NδT α ≤ 60 (3.8) • All other components that do not contain a flaw
NδF P + NδP O+ NδT E+ NδT α ≤ 400 (3.9) The tuning load has no direct analog to the cycle definitions presented. Therefore, it will be assigned its own definition as a cyclic load (Nδtuner) and treated additively. For the Nb cavity, construction is integral, and there are
no attachments or nozzles in the knuckle regions of the heads. Therefore, the applicable criterion is:
NδF P + NδP O+ NδT E+ NδT α ≤ 1000 (3.10) which, substituting the values become:
100 + 200 + 200 = 500 ≤ 1000 (3.11) The criterion is satisfied, and no fatigue assessment is necessary for the N b cavity.
Pressure Sensitivity Analysis
We explained before the steps necessary to do a Pressure Sensitivity Analy-sis and so to calculate the df /dp. The df /dp analyAnaly-sis involves a series of FEM analyses and each of them requires an accurate mesh in order to find a ra-tional result. Thus it is very important to simplify the CAD model to achieve an inferior number of degrees of freedom so we do not run out of RAM and we can get to the result in a minor time.
4.1
CAD Model
The 3D models are made with the CAD designer N X7.5 which is the CAD modeler used at Fermilab. The models are built with the aim to obtain a sim-plest possible model without compromizing the analysis accuracy. With that intention we need a model without all the details which could lenghten the simulation without contributing to the final solution or could create errors in the disctretization model and singular point.
To investigate the pressure sensitivity only the cavity and the helium ves-sel are modeled. The tuner assembly has not been included in the analysis and in its place we have put a spring constraint that simulate the stiffness of the tuner, the only parameter of the tuner that matters in this analysis. The guidelines for the realization of the model are the seguent:
• replacement of complex geometries with simple geometries, where pos-sible
• removal of details with little significance for the analysis
• Boolean operation of union between the welded pieces of the same ma-terial
4.1.1
The cavity
The complete model of the cavity has been analyzed in the previous chap-ter. In a pressure sensitivity analysis the main important part is the one contained between the two endcap disks. So all the coupler ports and the
HOM ports outside from this volume are not important for the simulation and therefore will not be included in the CAD model. This can be seen in picture 4.1 where the difference in the end parts of the exact model and the simplified model is highlighted.
(a) Main Coupler End de-tailed.
(b) Main Coupler End sim-plified.
(c) Field Probe End de-tailed.
(d) Field Probe End simpli-fied.
Figure 4.1: Differences in the end part between the real model of the cavity and the simplified used for the FEM analysis.
The dumbbell assembly is mantained the same because its shape is opti-mized to achieve certain results in terms of resonant frequency. Some words should be said over the stiffening ring: in the real model those are made by two parts and are drilled. In our CAD model those holes would only lead to a finer mesh in their proximity without contributing to the solution. For this reason the real rings is replaced by ones made by one part only and unite
