U
NIVERSITÀ DI
P
ISA
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
CONTENTS 1
Contents
1 Introduction 2
2 State of the Art and Verification according to the ASME Code 3 3 Pressure Sensitivity analysis and characterization 6
4 Assembly Modification 9
4.1 Helium Vessel Modification . . . 9 4.2 Cavity Modification . . . 11
1 INTRODUCTION 2
1
Introduction
This work is related to the verification in accordance to the ASM E Boiler and Pressure Vessel code of the Cryostat (the assembly made by the Cavity and its Helium Vessel) and the possible optimization of the assembly to reduce the pressure sensitivity.
First of all we have to spend some word about the cavity. A
Radio-Frequency (RF) cavity is a metallic chamber that contains an electromag-netic field. Its purpose is to accelerate charged particles. The RF cavity is molded to a specific size and shape so that electromagnetic waves become resonant and build up inside 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.
A large variety of RF cavities are utilized in particle accelerators. In the past two decades there has been a growing number of accelerator facilities for which superconducting cavities were used. Superconducting refers 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 superconductors (e.g. Niobium). These type of metals have a very low resistance below a certain temperature (called criti-cal temperature) which depends on the material.
Figure 1: A simplified diagram of an SRF cavity in a helium bath with RF coupling and a passing particle beam
A simplified diagram of the key elements of an SRF cavity is shown in pic-ture 1. The cavity is immersed in a saturated liquid Helium bath. Pumping removes helium vapor boil-off and controls the bath temperature. The bath is 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 defor-mations and micro-oscillations of the cavity walls.
2 STATE OF THE ART AND VERIFICATION ACCORDING TO THE ASME CODE3 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 operation 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 assembly there will be a money saving for the Fermilab and for the project in which the cryo-stat is involved. The goal of this work is to modify the assembly to achieve a reduction in the pressure sensitivity.
2
State of the Art and Verification according
to the ASME Code
The first step of the work was to verify the Cryostat (The Dressed cavity assembly) in accordance to the ASM E Boiler and Pressure Vessel Code.
The assembly is made by the 1.3GHz cavity and the Helium Vessel which surround it. The cavity is a niobium SRF cavity made by an elliptical nine-cell assembly. The nine-nine-cell cavity is presented in figure 2 where we can see that it is formed by the cells and by two tubes that form the ends. 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. Between the cells there are some rings to stiffen the assembly to a point. The stiffening ring in the model have a thickness of 3mm and a width of 20.63mm. The outside radius of the ring is 56.5mm
Figure 2: Cross section view of the 1.3GHz cavity which allows to see the internal shape
The Helium vessel is the part responsible for encasing the niobium SRF bare cavity. The vessel is made of T itaniumgrade2. It has two Helium fill ports at the bottom and in the center of the vessel there is the two-phase
2 STATE OF THE ART AND VERIFICATION ACCORDING TO THE ASME CODE4 helium return line. 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.
The Vessel contains also a Bellows made by T itanium too which is needed to allows the Tuner to change the length of the assembly to restore the res-onant frequency. The Tuner is not involved in this work because it is yet to be designed.
To connect the Vessel to the cavity there are the End plates, made by three parts because of the different materials of the cavity and the vessel: The End disk flange (made by N iobium as the cavity); the Endcap Disk (made by T i − 45N b, it is the central part of the plates) and the Transition Ring (made by T itanium as the Vessel). 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. In figure 3 we can see the components just explained.
Figure 3: View of the different parts composing the vessel
Dressed SRF cavities, as defined in that document, are designed and fab-ricated following the ASM E Boiler and Pressure Vessel Code (The Code). The 1.3GHz dressed cavity as a Helium pressure vessel has materials 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 exam-ple by de-rating the allowable stress. The analyses were done using ANSYS Workbench 14.5 and Mathcad version 14.
The Boiler and Pressure Vessel Code, Section V III is divided in Division 1 (by hand calculation) and Division 2 (F EA analyses). Where Div. 1 formu-las or procedures are available, they are applied to this analisys. For those
2 STATE OF THE ART AND VERIFICATION ACCORDING TO THE ASME CODE5 cases where no rules are available, the provisions of Div. 1, U − 2(g) are
in-voked. This paragraph of the code allows alternative analyses to be used in absence of Code guidance.
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.
Figure 4: Volumes for Pressure /Vacuum
Different Load Cases have been considered and they are listed below, where figure 4 explain the pressure volumes concerned. Below there is a brief description of the load cases:
• Load Case 1 : Warm pressurization (P1 = 0.205M P a).
• Load Case 2 : Cold operation, maximum pressure, no thermal contrac-tion (P1 = 0.41M 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 (P1 = 0.41M P a).
• Load Case 5 : Insulating and beam vacuum upset, helium volume evacuated (P2 = P3 = 0.1M P a).
Division 1 was used to demonstrate the satisfaction of the requirements for the vessel minimal thickness both for internal pressure and external pressure (Buckling), to demonstrate that reinforcements are not required for every penetration and finally to demonstrate the safety and the durability of the Titanium Bellows.
Division 2 procedures, and thus the Finite Element model, was invoked to show the sefaty of the Vessel against Collapse Pressure, Ratcheting, Local Failure, Buckling and Fatigue. Detailed information and calculations are available in the main file and are not shown in this summary due to its briefness.
3 PRESSURE SENSITIVITY ANALYSIS AND CHARACTERIZATION 6
3
Pressure Sensitivity analysis and
character-ization
The evaluation of df /dp involves a series of electromagnetic and structural analyses that can be performed with multiphysics software such as COM-SOL Multyphysics. The pressure sensitivity characterization is named Cou-pled 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)
To do this analysis a simplified CAD model was designed. The guidelines for the realization of the model were the replacement of complex geome-tries with simple geomegeome-tries where possible, the removal of details with lit-tle significance for the analysis and boolean operations of union between the welded pieces of the same material. Then, thanks to the symmetry, we were able to perform only half of the model with COMSOL reducing the time for the carrying out of the analysis. The Tuner was considered to be of infinitive stiffnes and was replaced by a fixed constraint in the Z direction.
The first step of the approach was to calculate the basic resonant fre-quency (f0) which was found to be 1.300706GHz. Then we aplied the pressure
on the model and we found the deformed shape of the cavity, crucial in order to discover the new shape of the vacuum volume inside the cavidy and thus the new resonant frequency. The third step is the mesh update and after that passage we did another Electro Magnetic analysis to find the new
res-onant frequency which was f1 = 1.300723GHz. The pressure sensitivity was
thus given by: df dp = f1− f0 p = 1.300723 · 109Hz − 1.300706 · 109Hz 103mbar = 17Hz/mbar (2)
After that we have investigated the influence of the Tuner stiffness in the calculation of the df /dp. To examine this influence we simply did several
3 PRESSURE SENSITIVITY ANALYSIS AND CHARACTERIZATION 7 coupled evaluation with the fixed constraint simulating the Tuner stiffness replaced by a spring constraint which value was updated every analysis with the stiffness we simulated. The outcomes are shown in the table 1. The reference resonant frequency is the one of the undeformed cavity so it is the same of the previous analysis that is f0 = 1.300706GHz.
Table 1: Results of the influence of the tuner stiffness over the pressure sensitivity of the dressed cavity
Stiffness[kN/mm] f1[GHz] df /dp[Hz/mbar] 0 1.300 868 162 1 1.300 831 125 2.5 1.300 800 94 5 1.300 776 70 10 1.300 755 49 20 1.300 741 35 40 1.300 732 26 80 1.300 728 22
It has been said previously that what really matters in the difference in the resonant frequency of the cavity is its deformed shape. So we can find a methodology that relies on evaluating the variation of the resonant fre-quency of a cavity by observing only the displacement at designed regions of the cavity. This approach was discovered by and Italian engineer from the Pisa University, Donato Passarelli, and has been tested only on supercon-ducting spoke frequency so we should verify if it is suitable also for our SRF cavity. The proposed method permits a reduced computational burden and a systematic approach to achieve a minimum value of pressure sensitivity in a complex system of dressed cavity.
To perform this methodology one first have to analyze which are the ar-eas of interface between the cavity and its Helium Vessel. The definition and identification of such interfaces (DOF) is of great importance for this methodology. The electromagnetic behaviour of the RF cavity is probed in relation with the displacement at such interfaces. For quasi-static consider-ation and small perturbconsider-ations, the interpolating equconsider-ation can be considered a line and one needs N simulations to define entirely the equation, where N = 1 + DOF . The steps to follow are the seguent:
• Fix all interface locations (xi= 0), apply pressure (p) (e.g. p = 1atm) and
evaluate the df /dp as in a coupled evaluation. This will be the reference pressure sensitivity for the next evaluations.
• Fix all interfaces but one at time, apply the pressure and evaluate the directional displacement at the free interface and again the df /dp.
3 PRESSURE SENSITIVITY ANALYSIS AND CHARACTERIZATION 8 • Assuming a linear behaviour, this time one extrapolates the linear function correlating the pressure sensitivity and the displacements of each DOF , calculating the slopes (Ai) and the constant terms (qi) to
obtain:
(df /dp)i = Ai· xi+ qi (3)
• In virtue of the superposition principle, true for the assumptions done, one can compile the function:
df dp = DOF X k=1 Ai· xi+ qi (4)
The values Ai, qi are function of the structural design of the cavity. Each
geometrical change of the cavity has influence on the characteristic function and a new function must to be extracted. For the cavity in our analysis we noted that the places where the cavity is in connection with the vessel are the two end disk flanges. The Figure 5 explain the situation, we can note thet in this case DOF = 4 thus we did 5 coupled evaluation to discover the characteristic equation.
Figure 5: A simplified diagram of an SRF cavity in a helium bath with RF coupling and a passing particle beam
The results of this analysis are summarized in table 2 where you can find, for each case, the Degree of freedom implied, the displacement at the degree of freedom in µm, the base eigenfrequency f0, the new eigenfrequency
f1 after the application of the pressure, the value of the dpdf and the value of
the coefficient of the characteristic equation related to the case.
Given these results we are now able to write the final characteristic equa-tion for the 1.3 GHz cavity which is:
df
4 ASSEMBLY MODIFICATION 9 Table 2: Summary of the results of the simplified evaluation
DOF displ.[µm] f0[M Hz] f1[M Hz] df /dp[Hz/mbar] Ai
Fixed 0 1.300 706 1.300 716 10 q = 10 x1 26.5 1.300 706 1.300 725 5 19.5 0.36
x2 26.5 1.300 706 1.300 726 5 19.5 0.36
x3 −0.3 1.300 706 1.300 716 0 10 0
x4 −0.3 1.300 706 1.300 716 0 10 0
This simplified forecast well confirms the data already found from the coupled evaluations. With this simplification we can now try to find a way to correlate the pressure sensitivity to the assembly stiffness and then try to made changes in the helium vessel to modify its stiffness and see the influence in the pressure sensitivity. From the equation we can also infer that the changes in the pressure sensitivity will not be very effective because the lower limit is 10Hz/mbar even at zero displacement of the cavity, this aspect will be further investigated later.
The equation have been also compared to the results taken from the var-ious coupled evaluations made at different tuner stiffness and we saw that the equation weel forecast the pressure sensitivity with errore within the 5% so it can be considered a reasonable approximation. This leads us to use the equation to continue our analysis and to use it in order to try to mod-ify the Helium vessel of the assembly with the aim of reducing the pressure sensitivity with a saving of time in the optimization process.
4
Assembly Modification
In this section we want to introduce the effort we made to reduce the pres-sure sensitivity of the dressed cavity. The first idea is to modify the Helium Vessel and the other idea is to modify the cavity itself.
4.1
Helium Vessel Modification
After we found the characteristic equation we have to find a systematic ap-proach to optimize the thickness of the vessel in order to achieve a substan-tial reduction in the pressure sensitivity. We calculated the applied force by the pressure on the end plates and the assembly stiffness to estimate the end displacement of the cavity considering the assembly like a spring. Then we modified the thickness to see the displacement variation and the volume variation and we chose a compromize between them.
To find the applied force on the ends we noticed that the end plates can be seen as circolar cronws, and knowing the internal diameter (d = 84mm) and the outside diameter (D = 230mm) it was easy to calculate the area (A = 36000mm2). The pressure applied is 1bar thus the force was 3600N .
4 ASSEMBLY MODIFICATION 10 At this point we calculated the stiffness of the entire assembly. To do so first we calculated the stiffness of the parts involved along the Z axis with the F EM and then we looked deeply how the parts are connected and uses the series and parallel formulas to define a spring-system stiffness model of the assembly. The spring-system is made by a parallel between the cavity and a series made by the Tuner, the Helium Vessel and the Main Coupler End. The bellows and the Fiel Probe End are not considered because they are in parallel with the Tuner and the tuner is infinetly rigid. The total stiffness of the assembly was found to be Ktot = 1.77 · 105N/mm. So the displacement
was calculated as δcav = F/Ktot = 20µm and agrees very weel with the FEA
model which gave us a displacement of 19µm.
Thus we decided to augment the Helium Vessel thickness and to see the displacement variation to find the optimum. From the graph 6 we can see that the displacement forecast has a linear decrease until the thickness of 8mm and then ot begins a slighter decrease. From the other data in the Thesis we saw that at this thickness the Helium vessel has a 62% of volume more. With the data of our knowledge it seems that the best compromizes between an increase of the volume and a decrease of the displacement of the cavity is the 8mm thickness.
0 2 4 6 8 10 12 14 16 18 20 22 24 10 15 20 25 thickness [mm] δcav [µm ] Displacement Variation
Figure 6: Graph showing variation of the displacement with the thickness of the Helium vessel
The 8mm thickness model was further investigated with a F EA model. The sum of the total displacement of the cavity was found to be 11µm, which is below the result of the forecast we made with in this section with the simplyfing assumption. The new pressure sensitivity is 14Hz/mbar. the He-lium vessel is made of titanium which is a very expansive material and the money saving resultant from the reduction of the pressure sensitivity could be exceeded by the raising costs of the raw material necessary to build the vessel. The solution presented will undergo to a review by the management of Fermilab which will make a differential cost analysis.
4 ASSEMBLY MODIFICATION 11
4.2
Cavity Modification
We have seen in the previous section that the maximum results achievable in the attempt of reducing the pressure sensitivity is influenced by the cav-ity. In fact the transverse stiffness of the cavity is too low and makes it too deformed when the pressure is applied even with the ends fixed. The idea we had to do that without changing the shape of the cavity (fundamental to its primary purpose: accelerate charged particles) is to change the position of the stiffening ring of the cavity. This modification affect the cavity itself so the characteristic equation already found can no longer be used.
Hence the idea is to enlarge the radius of the stiffening ring. Seen from the analyses made during the characterization of the cavity that the most deformable zone is the one at 71.6mm radius from the axis of the cavity we decide to put the stiffening ring (a ring of 3 mm thickness) at 70mm from the axis so the mean radius of the thickness will be in that zone. With this new ring we started another simplified evaluation.
The characteristic equation for the model with the larger stiffening ring is: df /dp = 0.4 · (x1 + x2) + 6. We can see that the slope is almost the same
(0.4 vs. 0.36), and this is in accordance with the fact that the shape of the cavity is not changed that much, but the new straight line is the 40% below the original one. Indeed we can see that for the cavity fixed at the end the result of the df /dp is as low as 6Hz/mbar instead of 10Hz/mbar as we found in the simplified evaluation previously made.
Then we applied the actual Helium Vessel at this cavity model to see the results of the pressure sensitivity. The new result of a pressure sensitivity was found to be 11Hz/mbar, which is an improvement of the 35% if we con-sider 0 the minimum result of the df /dp and an improvement of the 54% if the maximum reachable is considered the value of 6Hz/mbar of the new im-proved cavity. The change in the diameter of the stiffening rings does not even affect the Helium vessel and its volume so the cost of the new solu-tion is cheaper than the previous improvement we have presented. In this case however the cavity tend to be more rigid. With a stiffer assembly some problems can occour.
We talked before about the Tuner. Its function is to model the length of the dressed cavity assembly to rebalance the resonant frequency. This task is undoubtedly made easier by a reduced stiffness of the cavity. If we move the stiffening rings to make the cavity more rigid, the optimal behaviour of the Tuner could be compromized. However the Tuner is not part of this work and is not considered, but when the optimization process will be carried out further by the Fermilab staff this issue should be taken into account.
LIST OF FIGURES 12
List of Figures
1 A simplified diagram of an SRF cavity in a helium bath with
RF coupling and a passing particle beam . . . 2
2 Cross section view of the 1.3GHz cavity which allows to see the
internal shape . . . 3
3 View of the different parts composing the vessel . . . 4
4 Volumes for Pressure /Vacuum . . . 5
5 A simplified diagram of an SRF cavity in a helium bath with
RF coupling and a passing particle beam . . . 8
6 Graph showing variation of the displacement with the
