Optimizing the use of pressurized bladders for the assembly of HL-LHC MQXFB magnets

During operation, the coil windings of superconducting magnets are subjected to strong electro-magnetic (e.m.) forces. The latter arise from the combination of large electrical transport currents and the high magnetic fields produced by the magnets.

With the objective of reacting these loads, the coils of accelerator magnets are assembled inside a mechanical structure, whose purpose is twofold. First, it is conceived to act as a force-restraining structure. Second, the structure has as well the essential role of minimizing the conductor displacement or deformation during magnet powering, ensuring the design field quality and reducing the potential risk of premature quenches caused by cracking or by frictional movements in the system.

How to create an optimum mechanical support for the superconducting coils (fulfilling the previous objectives) is a topic extensively treated in the magnet community, which ultimately leads to the so-called pre-stress or pre-load concept [1–4]. Both terms are equivalent and refer to the application of an initial and controlled load to the superconducting coils, aimed at counteracting the effect of the e.m. forces. As an example, in saddle-shape cos(θ) dipole and cos(2θ) quadrupole accelerator magnets, the Lorentz forces acting in the coils have three main components [4]: (1) an azimuthal component, tending to compress the coil towards its midplane, (2) a radial component, tending to push the coil outwardly, with a maximum effect towards the midplane, and (3) a longitudinal component, concentrated over the end coils, tending to elongate the coil. To compensate for the effects of these forces, a coil pre-load can be applied during the magnet assembly phase in the form of an initial pre-compression in the different spatial directions. Thanks to such counterbalancing forces, the deformations/displacements in the system during magnet powering can be reduced. Ideally, a full magnet pre-load would be such as to ensure that the pre-load force value equals the one of the total electro-magnetic force.

The importance of the coil pre-stress increases as we move towards higher field magnets, with larger stored energies and e.m. loads. Up to the Large Hadron Collider (LHC) magnets at CERN the preferred superconductor technology for dipole and quadrupole magnets was Nb–Ti. For the high luminosity LHC (HL-LHC) upgrade project [5], the field and field gradient requirements impose to switch to Nb₃Sn (niobium–tin) technology for some of the magnets. Nb₃Sn conductors can operate at higher fields than Nb–Ti conductors and have become today the preferred technological choice for high-field accelerator magnets. As an example, for the Nb₃Sn RRP 108/127 HL-LHC wires, the minimal critical current requirement is 1280 A mm⁻² at 15 T and 4.22 K [6]. Thus, Nb₃Sn technology seeks to unlock the previous magnetic field frontiers and to allow the design of accelerator magnets operating above 10 T [7]. A critical aspect of Nb₃Sn superconductors is, however, their mechanical strain tolerance. The critical current density of Nb₃Sn cables depends on the strain applied to the superconducting phase [8–13], this last being, in addition, a brittle intermetallic compound with a noteworthy propensity to fracture. If the loads experienced by the conductor exceed a certain threshold, J_c can be permanently reduced due to the fractures occurring in the Nb₃Sn phase and/or due to the presence of residual mechanical strain in the stabilizing matrix [14–20].

Respecting the mechanical limits at which Nb₃Sn superconductors can safely operate is hence crucial, and requires a thorough determination of the stress/strain values that the conductor can withstand under loads applied in different directions. Coil peak stress limits are in consequence set at both room (during assembly) and at cryogenic temperature (after cool-down and during energization). Their exact value depends on the particular cable employed and the load direction, for instance, the limits used for MQXFA&B programs (see next section) were initially in the 150 MPa to 200 MPa range [6, 21–24]. These thresholds apply for transverse stresses applied to superconducting cables (transverse to the cable broad face). A number of recent studies carried out at CERN for MQXF cables indicate that micro-cracking may initiate in the Nb₃Sn phases of wires subjected to transverse stresses at earlier levels, in the 100 MPa to 150 MPa range at room temperature [25, 26]. Equally of concerns, are the tensile stresses that may develop under the axial component of the Lorentz load in the coils, but these are addressed by different features in the mechanical design and are not discussed in this paper.

The first milestone for Nb₃Sn applications in High Energy Physics shall be achieved with the installation of the niobium–tin quadrupole magnets, MQXF, built for the high-luminosity upgrade of the LHC (HL-LHC) at CERN [5].

The mechanical assembly of these magnets is based on the so-called Bladder and Key (B&K) process pioneered at Lawrence Berkeley National Laboratory in the early 2000 s [27, 28], where the superconducting coils are assembled inside a support structure relying on aluminum shrinking cylinders (see figure 1). The desired mechanical pre-load to the coils is applied through a system of water-pressurized bladders and interference keys. When water is pumped through the bladders, the provoked expansion results into the pre-tensioning of the aluminum shrinking cylinder. The balance between the shrinking cylinder azimuthal tension and the coil azimuthal compression is then kept via the insertion of interference keys. The 'bladdering and keying' operations are usually performed in steps, gradually increasing the key size until the desired coil pre-load at ambient temperature is obtained. Once the magnet loading is completed, an outer stainless-steel shell is welded around the magnet structure to form the helium containment vessel. The design and implementation of the stainless-steel shell has been iterated to limit the mechanical coupling between the outer shell and the magnet structure [29]. During cool-down, the pre-stress level is expected to further increase up to the target value at cryogenic temperature thanks to the selection of Aluminum for the shrinking cylinder, which presents a larger thermal contraction than the rest of the magnet components. The amount of pre-stress increase, from warm to cold, is related to the aperture of the magnet, the diameter of the shell and its thickness. The decoupled outer stainless steel shell is expected to not play any role.

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To better illustrate the principle behind the shrinking cylinder-based support structure, the computed (using a 2D finite element model) strain, stress and force values for MQXF magnet after room temperature pre-load and at cryogenic temperature are summarized in table 1. The first column provides the calculated peak azimuthal stress in the inner layer pole turn conductor, $\sigma_{\theta \; \mathrm {pole} \; \mathrm {turn} \; IL}$. The two next columns indicate the expected local stress values on the coil winding pole, $\sigma_{\theta \; \mathrm {pole}}$, and on the Al-cylinder, $\sigma_{\theta \; \mathrm {cyl.}}$, at the position of the mechanical instrumentation placed in the magnet (see section 2). The following two columns provide the average stress values on the complete superconducting coil area (Avg. $\sigma_{\theta \; \mathrm {coil}}$, all coil conductor blocks) and on the Al-cylinder surface (Avg. $\sigma_{\theta \; \mathrm {cyl.}}$). The last three columns summarize the azimuthal force balance in the system ($F_{\theta \; \mathrm {pole}}$, $F_{\theta \; \mathrm {cyl.}}$) and the force ratio against the total electro-magnetic force ($F_{\mathrm{pole}}/F_{\mathrm{e.m}}$). As it can be seen, the Al-cylinder provides the azimuthal force pre-loading the coils. This force is doubled when the magnet is cooled-down to cryogenic temperature due to the differential thermal contraction between the external cylinder and the rest of the magnet components. The same effect is seen for the average coil and Al-cylinder stresses at 1.9 K: the average coil azimuthal pre-stress at cryogenic temperature is twice the one at room temperature. However, it is important to notice that when looking at the local winding pole stress values (at the position of the mechanical instrumentation), the increase of pre-load is in the order of 30%. The explanation to this behavior can be found in the fact that the values reported from the model (or measured by the mechanical instrumentation) in the winding pole correspond to a local strain/stress field. The detailed magnet geometry, in combination with non-linear frictional and bending effects, create a non-uniform stress distribution across the mechanical structure and superconducting coils. As a result, the local winding pole value provides a good estimation of the peak azimuthal stress seen in the pole turn conductor of the inner layer. However, it does not give a complete insight into the average pre-load in the magnet coils.

Table 1. Computed stress and force values using the reference MQXF 2D magnet finite element model.

	Max. $\sigma_{\theta \; \mathrm {pole} \; \mathrm {turn} \; IL}$	$\sigma_{\theta \; \mathrm {pole}}$	$\sigma_{\theta \; \mathrm {cyl.}}$	Avg. $\sigma_{\theta \; \mathrm {coil}}$	Avg. $\sigma_{\theta \; \mathrm {cyl.}}$	$F_{\theta \; \mathrm {pole}}$	$F_{\theta \; \mathrm {cyl.}}$	${F_{pole}}/{F_{e.m}}$
Step	(MPa)	(MPa)	(MPa)	(MPa)	(MPa)	(kN m−1)	(kN m−1)	%
After RT pre-load	−86	−80	53	−34	49	1530	1540	40
At 1.9 K	−113	−118	125	−73	120	3310	3220	87

The B&K assembly technique was developed as an alternative to the existing collaring strategy, which calls for heavy tooling such as a collaring press [30, 31]. In collared coil structures, as it is the case of the Nb–Ti main quadrupole and dipole magnets for the LHC, structural collars (assembled in packs) are mounted and locked by means of keys around the superconducting coils. The pre-load is applied by a geometrical interference between the collar cavity and the coils. Here, the assembly is performed under a precisely aligned hydraulic press that allows to close the cavity and enables the key insertion. To achieve this goal, one must apply a collaring force that exceeds the target coil pre-stress value. Once the keys are inserted and the collaring force removed, the collared coil assembly relaxes resulting in a significant coil pre-stress reduction with respect to the peak. The collaring technique was successfully applied to all industrial productions of Nb–Ti accelerator magnets up-to date, as once the upfront investment on the tooling has been made, it is easy to control and leads to reproducible results. As mentioned earlier, a relevant example of application of the collaring technique can be found in the case of the Nb–Ti LHC main dipole magnets. For these magnets, during the collaring process, the coils were compressed up to 115 MPa–135 MPa, depending on the method used to apply the load in the press (the maximum allowed peak stress was set to avoid the damage of the conductor insulation). Upon key insertion and force removal, an average compression of about 70 MPa for the inner layer and 75 MPa for the outer layer were obtained. After magnet cool-down to cryogenic temperature, the pre-stress in the coil decreased by between 25 MPa to 40 MPa as a result of the higher thermal contraction coefficient of the coils with respect to the outer structure [32].

When dealing with new high-field Nb₃Sn magnets, where larger electro-magnetic forces are present and where controlling peak stress during assembly becomes critical [24], the B&K technique does offer some advantages; as for instance, the possibility to have a gain of coil pre-compression at cryogenic temperature. Nevertheless, the technique has not yet been implemented in an industrial environment.

At the moment of writing this manuscript, the B&K technique has been applied to the assembly of single aperture Nb₃Sn dipole and quadrupole magnet models and demonstrators. A non-exhaustive list of the main projects that rely on the technique follows next:

• For superconducting dipole magnet models and demonstrators: SM, LR, SD, RD, HD, SMC, RMC, FRESCA2, E-RMC and RMM [33–45].
• Regarding quadrupole magnet models and demonstrators: SQ, TQS, LQ, HQ and MQXF [46–50].

In addition, 'inflatable bladders' have been as well commonly used in assemblies of large structures, as for instance, in the cold mass preparation of the ATLAS barrel toroid magnets [51].

In this manuscript we will present a comprehensive experimental and numerical modelling campaign, aimed at optimizing the use of pressurized bladders for high field magnets, such as the HL-LHC MQXFB quadrupole magnets at CERN Results on short model magnets and long prototypes have shown that the bladder inflation step may lead to a certain coil stress overshoot (reaching a maximum of ${\approx}$40 MPa in full-length MQXFB quadrupole magnet prototypes, which corresponds to 50% of the measurable warm pre-load target), similarly to the case of the collaring technique described above. The overshoot comes from the fact that in order to insert the interference keys, the opening of an additional geometrical clearance is required. Such an extra displacement is needed to deal with geometrical imperfections, manufacturing tolerances and to reduce the frictional resistance during key insertion. This requirement becomes more important for long magnets, where geometrical deviations due to assembly tolerances can be more severe than for short models.

With our work, we prove that the B&K assembly procedure can be optimized to eliminate the coil over-stress during bladder operations thanks to the proper positioning of the bladders within the structure. The removal of the coil stress overshoot allows to meet the requirements on peak stress limits with contingency. Furthermore, the conclusions and lessons learnt presented here can be applied to any other high field magnet design based on the bladders and keys assembly process. Note that the stress overshoot is something that cannot be straightforwardly avoided in the collaring technique (unless relying on the so-called 'tapered key' technique described in [30]).

In section 2 of the paper, a brief introduction to the MQXF magnet design is provided. Section 3 explain the use of pressurized bladders for the assembly of MQXF magnets at CERN Finally, the optimization study and the discussion of the results are presented in sections 4 and 5 respectively.

The collaboration between CERN and LARP/AUP on MQXF started with an initial development phase, after the design activities, during which a number of so-called short model magnets have been produced to qualify, and whenever required, optimize the design and fabrication process. MQXF short models, referred to as MQXFS, feature a magnetic length of 1.2 m and share the same cross-section as that of MQXFA and MQXFB.

The way in which the MQXFA and MQXFB magnets are assembled has been established based on the short model experience [73]. In this regard, the review of the magnet assembly procedure results of paramount importance for our later development and it is explained next.

3.2.1. Coil size measurements.

3.2.2. Coil shimming.

3.2.3. Coil-pack centering.

3.2.4. Magnet pre-load.

The final pre-load of the magnet takes place after the coil-pack has been centered. The interference in all quadrants is steadily increased from 13.2 mm up to the target value, which for MQXF has been assigned based on the test results of short model magnets. On this subject, different pre-load levels were explored during the production of MQXFS model magnets. The target pre-load for the long magnets was chosen as that of the short models showing the best performance during cold test. Instead of an interference value (which can differ due to the actual geometry of the coils and magnet structure), the objective pre-load is better defined in terms of the Al-cylinder and coil strain (or stress) measured by the mechanical instrumentation at the selected locations. The explored parameter space in terms of average pole azimuthal stress (over the four coils) ranged from −30 MPa in MQXFS6 up to −100 MPa in MQXFS5 [80]. The final target values selected for MQXFA [81] correspond to MQXFS4 and are set at ($-80 \pm 8$) MPa of average azimuthal compression in the coil winding poles and ($58 \pm 6$) MPa of average azimuthal tension in the Al-cylinder (four angular measurements at each longitudinal position). As it will be shown next, these values will be modified for MQXFB magnet after our optimization study.

For the last two stages, the centering and actual pre-load, two main alternatives have been historically explored in terms of the bladder—key insertion operations: (i) the full bladder strategy, and (ii) the quadrant-by-quadrant option.

In (i) all bladders in the structure are pressurized at the same time, i.e. the bladders from each quadrant are all connected in parallel to the same pressure. Such water supply scheme creates an equal and symmetric force distribution within the magnet, which preserves the symmetry of the system. Conversely, in (ii), the bladder operation is done by pressurizing each quadrant one by one. In this scheme, the force distribution within the structure is asymmetric as illustrated in figure 2.

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The major advantage of the quadrant-by-quadrant scheme is that the pressure required to further enlarge the clearance between masters (when gradually increasing the key size) can be reduced [28]. Consequently, the risk of failure due to the burst of the bladders is significantly decreased. This strategy (quadrant-by-quadrant) has been used in the assembly of all MQXFS model magnets at CERN both for the centering and magnet pre-load. For MQXFA magnets, a mixed procedure combining the full bladder and quadrant-by-quadrant approaches is currently used: at the moment of writing this paper, the full bladder strategy is employed up to a key size of ${\approx}$13.15 mm. From thereon, the quadrant-by-quadrant procedure is used. For MQXFB magnets, the three first prototypes followed the short model experience, using exclusively the quadrant-by-quadrant method. As explained thereafter, a new procedure has been devised and will be applied from now on, also retrofitted to the last short model magnets assembled.

3.3.1. Origin of the Overshoot.

An important consequence of using interference keys in the B&K technique is that a certain clearance needs to be created by the bladder force in order to insert the keys. As previously introduced, this additional opening is needed in order to deal with geometrical imperfections, assembly tolerances and to reduce the frictional resistance during key insertion. As an example, to insert a key of a given size, k_s , the distance in between masters shall be $k_s + c$, where c is the required clearance.

To understand the importance of this additional opening in the 7.2 m long MQXFB assembly, let us focus on the simplest case where the full bladder method is used and no bending or non-linear effects are present. In this case, the need for reaching a clearance between masters that is larger than the final size of the interference key, translates into a larger pre-compression force seen by the coils during the bladder operation. Indeed, in this case, the displacement created by the bladders can be seen as an equivalent key of size $k_s + c$. When the bladder pressure is released, the coil pre-load relaxes from its peak to its final value. This phenomenon is similar to what happens in the case of a collared-coil assembly upon key insertion completion and release of the collaring press force.

The case of the quadrant-by-quadrant operation is more complex and requires a detailed simulation of the full process using finite element modelling techniques. In such a case, the non-symmetric loading of the structure introduces important bending and non-linear frictional effects. In addition, before having introduced the last interference key in all quadrants, a mixed state composed of different key thickness in the four quadrants appears. The simulation of the MQXF quadrant-by-quadrant assembly process using ANSYS APDL^® is shown in figure 3 images (a) and (b) and in figure 4 (top). The plots show the computed azimuthal stress distribution map in the superconducting coils (figures 3(a) and (b)) and the corresponding calculated local stress values at the position of the mechanical instrumentation (figure 4—top). In the simulation, the geometry of the coils and structural parts corresponds to the reference design one, without geometrical imperfections; the interference has been adjusted to obtain −80 MPa at the end of the magnet pre-load, and the target clearance for key insertion is 300 $\mu\mathrm{m}$. The model predicts that when the bladders in one quadrant are inflated, the two poles opposite to the active bladders experience an increase of their azimuthal compression. The predicted overshoot in pole stress is approximately 11 MPa. On the contrary, the az. compression decreases in the two poles placed next to the active bladders. In terms of the computed azimuthal peak stress in the coil blocks (figures 3(a) and (b)), the overshoot equals 17 MPa. Therefore, the peak coil load during magnet assembly happens during the inflation of the last bladders due to the combination of a non-symmetric structure deformation (bending and non-linear effects) and the clearance requirement.

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3.3.2. The 7.2 m long MQXFBP3 example.

Figure 5 shows one example from the experimental mechanical measurement performed during the assembly of the third MQXFB quadrupole prototype magnet (MQXFBP3). The plots depict the derived azimuthal stress (from the actual strain measurements, assuming a bi-axial stress state) on each winding pole (top) and on the Al-cylinder (bottom). Red rectangles point to the last four bladder-key operations. The measurement location corresponds to the middle of the magnet (MI). With MQXFBP3 assembled using the quadrant-by-quadrant strategy, a pole stress overshoot in the order of 20 MPa is seen for all signals. The final stress average can be computed from the plot as ${\approx}{-}80$ MPa. Expanding the analysis to the three MQXFBP3 magnet positions where mechanical instrumentation is available (recall figure 1), the peak azimuthal stress on the coils during bladdering is −138 MPa in the LE side, −125 MPa in the middle (MI) and −120 MPa in the RE side. In previous prototypes, the maximum az. stress measured on a coil during bladdering were −151 MPa for MQXFBP1 and −136 MPa for MQXFBP2 (notice that this last value is not shown in figure 6 due to a large coil imbalance after centering that prevents the calculation of average values).

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3.3.3. Assembly data summary.

Based on the assembly experience and on the cold test results from the first MQXFB prototypes [29], a detailed investigation campaign on the MQXFB magnet production was launched at CERN in Spring 2021. The goal of this campaign was to carry out a thorough root cause analysis of the performance limitation of the first two MQXFBP1&2 prototypes and derive possible risk mitigation measures. The analysis identified three possible root causes related to (1) cold mass assembly, (2) magnet assembly and (3) coil manufacturing. As a part of (2), it was decided to review the pre-load targets and to study as to minimize the peak stress level seen by the superconducting coils during assembly at room temperature. In order to reach this objective, the removal of the stress overshoot during bladder operations was strongly pursued. It should be noted that according to the magnet finite element computations, the coil peak stress does not occur neither during assembly at room temperature nor during energization, but at the end of cool-down in the inner layer coil pole block. This effect could be alleviated by introducing a notch in the winding pole. However, it was deemed more appropriate to avoid this notch so as: (i) to limit the displacement of the pole turn conductor corner during magnet energization (which increases in presence of the notch, as a result of the reduction in the pole rigidity), (ii) to avoid a discontinuity in the pole tip geometry towards the coil extremities (a smooth transition to a solid pole should be engineered at the coil ends), and (iii) to allow for the installation of the mechanical instrumentation used for pole strain assessment [70].

The cure to the stress overshoot was found thanks to the analysis of the force distribution in the system when bladders are pressurized and by imitation to the process followed at CERN for the loading of other magnets and demonstrators [82]. Following the original MQXF bladder positioning selected at the beginning of the project, the bladder operations exert a simultaneous load in the coils and Al-cylinders. This force distribution eventually results in the appearance of the undesired stress overshoot. Without any modification to the magnet design, if the bladder force were to be applied exclusively at the interface between iron yoke parts, one could obtain a net gain in cylinder tension without any part of this force being applied to the coils. In simpler words, the solution would consist in placing additional bladders in the iron yoke cooling holes that would open up the cylinder without exerting any force onto the coil-pack. As just stated, such a solution is inspired by the previous experience coming from the assembly of racetrack model magnets (SMC, RMC, E-RMC and RMM [39–43]) and the FRESCA2 dipole magnet [44, 45] at CERN For these magnets, additional bladders located in the iron yoke are used in combination with the coil-pack ones. By doing so, one can reach the required assembly force while keeping the bladder pressure at reasonable values, i.e. $\lt$400 bar (see next paragraphs).

The practical implementation of this modification in MQXF was investigated using the two-dimensional FE model of the magnet. The desired force distribution in the system could be achieved by introducing the additional bladders as shown in figure 7. These auxiliary bladders act exclusively on the yoke and Al-shrinking cylinders, unloading the coil-pack.

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For the operation of the bladders, it was decided to switch to the full-bladder pressurization strategy (all quadrants active at the same time, including the cooling hole bladders). The reasons behind this choice are two-fold: (i) to reduce the complexity and number of bladder cycles per magnet assembly by a factor of four, and (ii) upon completion of the centering phase, to keep a symmetric operation of all bladders so as to allow the correct coil-pack position when the new cooling hole bladders are active. The main drawback of the full operation could be, however, the need of excessive bladder pressures. Nevertheless, the performed FE simulations predict that the latter can be kept below the acceptable 400 bar level, when targeting for the reference magnet pre-load. Such a pressure value has been set based on practical experience, and it can be seen as an acceptable level that avoids the most-common failure mode of the bladders.

For the centering phase, the quadrant-by-quadrant strategy was kept (no use of new cooling hole bladders), as it has been deemed more appropriate. In this case, the main advantage comes from its asymmetric nature, which can help to correct initial deviations in the coil-pack positioning with low bladder pressures. Note that for quadrant operations, at the beginning of centering (when no interference is present), the only reaction force opposing the coil-pack displacement comes from the weight of the coil-pack and the frictional interaction with the structure.

The right side of figure 3, with images (c) and (d), shows the azimuthal stress distribution in the superconducting coils during the last bladder operation (c) and after the insertion of the interference keys (d) for the revised procedure. The figure compares these new results to the original quadrant-by-quadrant option, shown on the left side. Contrary to the quadrant-by-quadrant case (with a computed peak azimuthal stress in the coil block, $\sigma_{y \; \mathrm {peak}}$ = −105 MPa), the use of the new cooling hole bladders results into a decrease of the coil peak stress during bladder operations ($\sigma_{y \; \mathrm {peak}}$ = −72 MPa). Since the same pressure is used for all bladders, the average compressive azimuthal stress in the coil is instead larger than in the quadrant by quadrant method (−50 MPa compared to −42 MPa during bladder pressurization). This compressive value could be adjusted by using a larger pressure in the cooling hole bladders than in the coil-pack ones. However, this modification has not been considered necessary, as the average value remains well within the tolerable limits for the conductor.

The expected azimuthal stress at the mechanical instrumentation position is shown in the bottom of figure 4. During final bladder pressurization the pole stress is actually 20 MPa lower than after the final key insertion according to the FE model.

A very instructive plot used in the assembly of B&K based magnets is the so-called transfer function graph, which plots the azimuthal stress level in the coil winding poles as a function of the azimuthal stress in the external Al-cylinder. It gives the essential information on how much azimuthal force is provided by the cylinder and which force reaches the superconducting coils. Interestingly, according to the 2D FE model, the use of the new bladder procedure translates into a modification of the magnet transfer function (see figure 8, which plots the computed transfer function for the different assembly methods). For the same Al-cylinder stress, the azimuthal stress level in the winding pole is lower when all bladders are operated at the same time. Detailed investigations of this behavior have revealed that the reduction in coil stress is a result of the global frictional interaction between the different parts of the structure. The latter modifies the local stress seen in the coil winding poles. The simulation also predicts that when the magnet is cooled-down to cryogenic temperature, the friction effects disappear and the stress state is equivalent to that of a magnet assembled with a quadrant-by-quadrant strategy.

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Prior to the application of the new procedure to the assembly of a full-length MQXFB quadrupole magnet, the method was firstly put in practice in a full-length mechanical assembly test (MQXFBMT3) and in a short quadrupole magnet model (MQXFS7e).

4.2.1. The 7.2 m long MQXFBMT3.

In details, the MQXFBMT3 mechanical assembly test was a magnet assembled with four non-conform Nb₃Sn coils, whose objective was to refine the MQXFB magnet assembly procedures and to qualify the implementation of the new bladder scheme. In addition, the magnet was heavily instrumented, including two supplemental longitudinal positions for the mechanical instrumentation. Being a so-called mechanical assembly, the magnet was disassembled few weeks after the end of the magnet loading, no cool-down to cryogenic temperature was performed.

Figure 9 shows the results of the mechanical measurements carried out in the central position of the MQXBFMT3 assembly test. As desired, when the new bladder scheme is employed, the peak winding pole azimuthal stress disappears during bladder pressurization. This behavior is obtained at the expense of a larger tension in the Al-cylinder, as shown in the bottom of figure 9. The azimuthal stress in the Al-cylinder reaches 130 MPa in MQXFMBT3, in comparison with the 80 MPa seen for MQXFBP3 (quadrant-by-quadrant, figure 5). Despite this increase, the values are well below the yield limit of the material (420 MPa at 300 K [83]). Note that sharp geometries have been removed from MQXFB Al-cylinders to avoid undesired stress concentration problems.

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The absolute peak stress in the coils now appears after the insertion of the last loading keys and not anymore during bladder operation. Focusing on the final assembled state, the experimental results confirm as well that with the new procedure, the stress imbalance between coils remains of the same order of magnitude than for previous magnet loadings (±15 MPa, see also figure 6).

In global terms, the MQXFBMT3 mechanical test campaign was successfully completed. No showstopper was found for the practical implementation of the new bladder scheme. Despite a larger displacement under pressure (bladder stroke) when compared to their iron master counterparts, the new cooling hole bladders worked reliably and confirmed the feasibility for their use.

MQXFS7e short model is the fifth assembly of the MQXFS7 magnet. With the objective of further verifying the finite element model predictions, the magnet was fully unloaded from its previous configuration (MQXFS7d) and then re-assembled twice. The first assembly (MQXFS7e1) was performed using the original quadrant-by-quadrant procedure. Upon completion of the magnet pre-load, it was unloaded immediately. In the second assembly (MQXFS7e2), the magnet was pre-loaded using the newly optimized method with additional bladders in the yoke cooling holes. Note that both assemblies (MQXFS7e1 and MQXFS7e2), reproduced the magnet pre-load target of MQXFS7d, previously cold tested. To verify the mechanical behavior at cryogenic temperature of MQXFS7e2, the magnet was cold tested in the same conditions than MQXFS7d.

The goal of this full campaign was three-fold: (i) to verify that the new loading procedure had no detrimental effect on magnet performance, (ii) to further study the impact of the procedure in the magnet transfer function, and (iii) to confirm that as predicted by the model, the magnet mechanical behavior at cryogenic temperature is independent of the assembly sequence, even if the winding pole azimuthal stress value is different at warm.

The responses to (i) and (iii) were found during the MQXFS7e cold test phase [84]. The magnet, assembled with the new procedure, reached the same current level as its previous version pre-loaded using the quadrant-by-quadrant bladder scheme (MQXFS7d). In addition, the mechanical measurements performed at cryogenic temperature confirmed that the mechanical behavior at cold is identical for both strategies. This last statement is supported by the two plots of figure 10, which show the stress variation during a current ramp for the winding pole and Al-cylinder. Solid lines are used for the azimuthal stress direction, dotted lines for the longitudinal one. The tests show the same response during cold magnet powering, despite a measurable difference in the transfer function at warm (not shown here and explained in next paragraph).

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The answer to (ii) is graphically represented in figure 11. The top plot shows the comparison on the pole azimuthal stress achieved by both loading strategies, as a function of the interference key size. The bottom graph shows the same information regarding the cylinder azimuthal stress. As predicted by the model, the same cylinder stress results into a lower local stress in the winding pole in the case of the new bladder strategy. The pole stress difference for a cylinder azimuthal tension of 40 MPa (13.7 mm key in this magnet) is 14 MPa. The value is in very good agreement with the expected difference based on the FE model, 15 MPa (see figure 8). As already mentioned, this difference disappears at cryogenic temperature.

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All together, the obtained experimental results confirm the FE model outcome and provide the necessary confidence for the application of the new procedure in the assembly of full-length MQXFB prototypes and series magnets.

Having validated the new pre-loading procedure both in the short magnet model and in the full-length MQXFBMT3 test, the technique was then applied for the assembly of a real full-length prototype magnet.

Thanks to the introduction of the auxiliary bladder system, more stringent pre-load targets and stress limits could be established. The new values seek to minimize as much as possible the risk for conductor degradation due to excessive conductor strain. They also take into consideration the change in the magnet transfer function during loading and the expected final state at cold. The exact values are listed hereinafter:

• The average azimuthal Al-cylinder stress in the magnet (average of all the sensors located in the three longitudinal positions) shall be: $+58\,\pm\,6$ MPa.
• The average azimuthal stress in the coil winding poles (average of all the poles, including the three longitudinal positions) shall be: $-70\,\pm \,10$ MPa. This corresponds to a reduction of 10 MPa with respect to the original target. However, the pre-load at cold would be the same.
• The peak azimuthal stress in any of the coil winding poles (at any longitudinal position) must be always within −100 MPa.

The results from the magnet assembly are shown in the summary plot of figure 6. All the targets and stress limits have been fulfilled. The magnet was assembled with a maximum absolute peak stress value (winding pole) of −74 MPa. Comparing to previous prototypes, a peak stress reduction in the order of 70 MPa has been achieved.