Resultados obtenidos

The beam window will have three types of stress to cope with, each of which operates on a different timescale:

• The static stress due to pressure, though this becomes insignificant by choosing a suitable shape and thickness.

• The quasi-static transient stress caused by the initial compression of the window ma- terial due to beam heating, which is partially relieved between pulses as it is cooled.

• An elastic stress wave component that arises due to the short pulsed nature of the beam and is most significant on the timescale of microseconds.

Figure 59:

(left) The schematic diagram illustrates the propagation of a stress wave through a window caused by a bunch at time t = 0. (right) Stress wave propagation along centre axis following one single 58 ns bunch at 1.3 MW beam power for 1 mm thick beam window (tc = 322 ns)

The elastic stress waves are an extra factor on top of the static and quasi-static stress and the magnitude of the stress waves will depend on the duration of the beam pulse in relation to the beam spot size. Note that, since the beam power is to be increased to 1.3 MW by reducing the beam cycle time and not by increasing the pulse intensity (the window was designed for original 3.3×1014 protons-per-pulse), current design principles would basically work with rather minor upgrades suitable for high power operation.

Due to the different timescales involved, which makes it impractical to study all phenom- ena together using finite element analysis because of time-step requirements, the quasi-static thermal stress and elastic stress waves have been examined in separate simulations.

Elastic Stress Waves due to a Single Pulse

With intense pulsed beams that

create stress waves in targets and windows, certain combinations of material geometry and bunch structure can result in a type of stress resonance occurring. This is when stress waves produced by individual bunches constructively interfere with one another. In beam windows this can give rise to larger through-thickness longitudinal stress waves (those that reflect between inner and outer surfaces) than would otherwise occur. Consider a flat window of thicknessT, as schematically illustrated in Fig. 59. Upon interaction with a short proton pulse, compressive waves will initiate along the beam trajectory through window thickness, dissipate at the both surfaces, and be reflected as tensile wave. Characteristic timetc is the

time taken for a stress wave to propagate through the window thickness and back:

tc=

2T c , c=

p

E/ρ (2)

wherecis the speed of sound in the material,E is the elastic modulus andρis the density. The stress resonance will be maximum iftc is equal to the bunch spacing (and be minimum

iftc is equal to half of bunch spacing). For 0.3 mm window,tcis 100 ns, which corresponds

to 10 MHz stress cycle.

These wave profiles are evident in the ANSYS simulations using the latest beam bunch structure at 1.3 MW operation. To simulate the dynamic effect of a pulsed proton beam on the window, an ANSYS Multiphysics model with coupled field elements has been created. To reduce the computational load, an axisymmetric model has been approximated for simu- lations. Upper side of Fig. 60 shows stress in beam direction as function of time at window center for 1.3 MW beam operation with three different window thicknesses. Constructive interference of bunch structure (8 bunches) is visible for existing 0.3 mm thickness, where

Figure 60:

(upper) Stress in beam direction as function of time (ns) at window centre for 1.3 MW beam operation, with window thickness = 0.3, 0.4, 0.5, and 0.7 mm. (lower) Stress in the beam window as a function of thickness, after the final (8th_{) bunch of a full beam spill at 1.3 MW operation. Z Stress}

represents through thickness stress and X Stress shows radial stress, respectively.

the maximum stress reaches toSZ =∼270 MPa. When the thickness of 0.3 mm was consid-

ered as optimal, original simulations had too few elements and too large time steps leading to sub-optimal thickness. Lower side of Fig. 60 gives the maximum accumulated stress as function of thickness. The plots exhibit nice agreement to each other. In order to increase the tolerance and safety margin for the bunched-beam induced stress wave resonances, the window thickness can be chosen to be at one of these troughs in the plot.

In the real accelerator operation, there will be spill-by-spill fluctuation on beam bunch timing. Meanwhile, the analysis looks at the worst case scenario where the time between bunches is kept perfectly constant so that, at certain thicknesses, the stress waves can superimpose with each bunch. This is the constructive interference we see. If there was any fluctuation in bunch timing from bunch to bunch this would only improve things by disrupting the interference of consecutive bunches.

Transient Stress and Temperature over Multiple Pulses

Due to the rela-

tively poor thermal conductivity of the titanium alloy, the window domes need to be cooled directly across the beam spot by helium gas flowing through the gap between two domes. A Computational Fluid Dynamics (CFD) FEA simulation of the window is shown in Fig. 61. Because the pulse occurs over such a short time period, the resulting temperature jump will be independent of any cooling applied and of any material thickness. Both thermody- namic equations and ANSYS simulations have predicted a temperature rise of 156◦C with each pulse. Fig. 62 shows how the window heats up and begins to oscillate about an average temperature of about 150∼200◦C (dependent on window thickness) after only four to five pulses. This analysis assumes a heat transfer coefficient of 886 W/m2K on the internal sur- face. This value has not been verified experimentally and using ANSYS CFX as a realistic estimate for laminar helium flow between two flat plates of velocity at least 100 m/s.

Figure 61:

Helium flow streamlines for 0.4 mm-thick beam window by ANSYS CFD Analysis for 1.3 MW beam. Current operating mass flow rate of 1.1 g/s is used in this analysis.

Table 11: Summary of the Stress and Cooling Analyses for 1.3 MW Operation.

Fig. 63 shows studies to evaluate how the mass flow rate affects on the cooling per- formance. Left figure shows streamline with applying a double flow rate (2.2 g/s), where the pressure drop becomes four times higher (0.2 bar), and further increases would need to consider helium system pressurization. Right plots show maximum temperature and stress as function of heat transfer coefficient. As exhibited in the plot, even doubling the flow rate results in a tiny increase on the heat transfer coefficient, and thus improvements are small.

Stress and Cooling Analysis Summary

Tab. 11 summarizes results of the stress

and cooling analyses for typical three window thicknesses, where estimated safety factors are also given.

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