Ti-alloy and W-Re Target Energy Deposition Numbers

W. Liu, T. Piggott, and J. C. Sheppard

Rev. 7: August 24, 2007

Assuming a gaussian transverse spatial distribution of undulator photons incident to a spinning target annulus of Ti-alloy, what are the values for the deposited energies, energy densities, stresses, and peak and average temperature rises in the target? How do these values change with variations in the incident beam energy and size and tangential target velocity? What is the “safe” engineering limit of the incident beam intensity?

(Note: throughout the text, the numerical values are for the case of Ti-alloy target material. The same expressions are used for W-Re target material. Both the T-alloy and W-Re values are listed in the tables) We assume the following input parameters. These will be updated as improved information becomes available. It is assumed that a K=0.92, 1.15 cm period helical undulator is used in conjunction with a 150 GeV electron beam to produce the incident photon flux. The average photon energy is ; the target quantum yield is taken to be Yg = 2%; and the capture efficiency is ec = 15% (for the case of the quarter-wave transformer). Table 1 lists the nominal beam and target parameters to be discussed.

For a required positron flux overhead of f =1.5, the required number of incident photons per electron is given as :

Table 1

Parameter / Symbol / Value / Units
Target Material / Ti-alloy / W-Re
Target absorption / aT / 8 / 4.1 / %
Target density / r / / / kg/m3
Speed of sound / vs / 4140 / 5174 / m/s
Heat capacity / Cv / 0.527 / 0.134 / kJ/kg-0K
Thermal expansion coefficient / a / / / 1/0K
Modulus / E / 114 / 400 / GPa
Gruneisen coefficient / G / 829 / 2639 / kg-m2/kJ-s2
Poisson’s ratio / n / 0.31 / 0.28 / #
Target thickness / lt / 1.43 / 0.14 / cm
Target yield / Yg / 2 / 3.3 / % (e+/g)
Capture efficiency / ec / 15 / 12 / %
Number of photons per electron / / 500 / 379 / g/e-
Tangential velocity / vt / 100 / m/s
Incident beam size / si / 1.7 / mm
Average photon energy / / 10.4 / MeV
Number of e- per bunch / nb / / e-/bunch
Number of bunches per pulse / Nb / 2625 / bunches/pulse
Inter-bunch spacing / tb / 369 / ns
Pulse Length / Tp / 968 / ms
Overhead factor / f / 1.5

The total incident photon beam power is Ei:

The amount of energy deposited per bunch is given as

The total energy deposited per pulse is

And the average power deposition is

Energy Densities (to be compared with EGS4 values):

For a gaussian profile the peak incident flux per bunch is ; assuming no cascade factor over the target length, the peak absorbed energy per unit volume per bunch is estimated to be

This corresponds to a peak deposited energy density of

Averaging over the full pulse, the peak deposited energy density is

The peak temperature rise is DTP:

And the peak compressive stress in the target is DPc (from Vesloskaya and LCC-0133):

An alternate estimate of the peak compressive stress in the target is DPc:

Note that the “alternate estimate” is most people’s primary estimate. It ignores a possible factor of about 3 by not dividing by . In earlier comparisons (LCC-0133) it was also noted that the Gruneisen formulation over-estimateS the thermal stress in comparison with the LSDyna thermal simulation codes by a factor of about 3. In LCC-0133 an arbitrary scale factor of 1/p was used to bring the hand estimates into closer agreement with the modeling code results.

The numbers need to be checked with bona fide simulations: energy deposition and thermalhydraulic…..

Undulator Length:

The number of photons per meter per electron is Ng:

For K=0.92 and lu = 1.15 cm, Ng = 1.90. Thus for ng = 500 photons/e-, the required undulator length is

Table 2:Ti-alloy and W-Re target material comparison, with overhead factor of f = 1.5

Parameter / Symbol / Value / Units
Target Material / Ti-alloy / W-Re
Target thickness / lt / 1.43 / 0.14 / cm
Target absorption / aT / 8 / 4.1 / %
Target yield / Yg / 2 / 3.3 / % (e+/g)
Capture efficiency / ec / 15 / 12 / %
Number of photons per electron / / 500 / 379 / g/e-
Undulator Length / Lu / 263 / 199 / m
Incident photon energy / Ei / 43.7 / 33.1 / kJ/pulse
Energy deposition per bunch / dEb / 1.33 / 0.52 / J/bunch
Energy deposition per pulse / DEp / 3.5 / 1.4 / kJ/pulse
Average power deposition / DPavg / 17.5 / 6.8 / kW
Peak absorbed energy per bunch / dUb/Vol / 5.2 / 20.3 / MJ/m3-bunch
Peak absorbed energy per bunch / dUb/Mass / 1.2 / 1.1 / kJ/kg-bunch
Peak absorbed energy per pulse / DUp/Mass / 53 / 49 / kJ/kg-pulse
Peak temperature rise / DTp / 101 / 362 / 0K/pulse
Peak compressive stress, G / DPcG / 82 / 1019 / MPa
Peak compressive stress, TE / DPcTE / 98 / 637 / MPa
Ultimate tensile strength / UTS / 700@5000C / 900@10000C / MPa

Comparison with EGS4:

Table 3

Parameter / Value EGS / Hand Calculation / Units
Target Material / TiAlV / W-Re / TiAlV / W-Re
Spot size (sx=sy) / 1.85 / 1.47 / 1.85 / 1.47 / Mm, rms
Peak absorbed energy per bunch / 2.75 / 27 / 2.76 / 22.4 / MeV/cm^3/g
Peak absorbed energy per bunch / 4.4 / 32.8 / 4.4 / 27.2 / MJ/m3-bunch
Peak absorbed energy per bunch / 0.97 / 1.7 / 0.98 / 1.4 / kJ/kg-bunch
Peak absorbed energy per pulse / 49 / 67 / 49 / 56 / kJ/kg-pulse
Peak temperature rise / 93 / 504 / 93 / 418 / 0K/pulse
Peak compressive stress, G / 75 / 1404 / 75 / 1167 / MPa
Peak compressive stress, TE / 90 / 886 / 90 / 736 / MPa

The overriding assumption in the hand calculation is that the peak energy deposition in the target is uniform over the length of the target. This works better in the case of Ti-alloy than for the W-Re material in comparison to the EGS4 results. The increase in energy deposition along the target length is accompanied by a compensating increase in spot size. This assumption is not as good in the case of the thinner W-Re target. For the W-Re, the EGS4 numbers are about 20% higher than for the hand calculation.

Comparison with LSDyna3D:

The expressions for the compressive stress, DPcG and DPcTE , have modified by factors of 1/p and (1-2n), respectively to better “match” the stresses calculated with LS Dyna3D. Table 4 lists the peak compressive stresses for Ti-alloy and W-Re for a given peak temperature rise as estimated by the above expressions and by LSDyna3D. For both the Ti-alloy and W-Re case, the DPcTE estimate more closely agrees with the LSDyna3D simulations for a given DTp .

Table 4

Parameter / LSDyna3D / Hand Calculation / Units
Target Material / TiAlV / W-Re / TiAlV / W-Re
Peak temperature rise / 148 / 504 / 148 / 504 / 0K/pulse
Peak compressive stress, G / 163 / 844 / 119 / 1405 / MPa
Peak compressive stress, TE / 143 / 887 / MPa

Comparison for different capture efficiencies:

Table 5:

Parameter / Units
Target Material / TiAlV / TiAlV / TiAlV / TiAlV
Incident Spot Size / 1.7 / 1.7 / 1.7 / 1.7 / mm
Capture Efficiency / 13 / 16 / 20 / 30 / %
Number of photons per electron / 577 / 469 / 375 / 250 / g/e-
Undulator Length / 303 / 246 / 197 / 131 / m
Average power deposition / 20 / 16 / 13 / 9 / kW
Peak temperature rise / 117 / 95 / 76 / 50 / 0K/pulse
Peak compressive stress, TE / 113 / 92 / 74 / 49 / MPa