Some Thoughts on A High Power, Radiation Cooled, Rotating Toroidal Target for Neutrino Production J.R.J. Bennett Rutherford Appleton Laboratory, Didcot, UK 1 Introduction This note considers the power dissipation by thermal radiation from a rotating toroidal ring. The work was prompted by the paper of Bruce King [1, 2], for a water-cooled rotating band muon target. Fig. 1 and 2 show the copper-nickel band, which rotates in an atmosphere of inert gas and is sprayed by water for cooling. The dissipation is 1MW. Fig. 1. Plan view of targetry setup. Preprint submitted to Elsevier Preprint 18 February 2000
Fig. 2. Target geometry. The target specifications are: - Target Dimensions 0.6 cm thick x 6 cm high x 15.7 m circumference - Material Cu-Ni alloy (e.g. Olin 715), electrical conductivity 5% of Cu, interaction length = 15 cm - Target Rotation velocity = 3 m/sec ( 2 pulses overlap) - Solenoid Magnet 20 Tesla, 15 cm bore diameter - Proton Beam σ = 1.5 x 10 mm, 16 GeV protons,10 14 ppp, 15 Hz repetition rate 2 The Radiation Cooled Target Fig. 3 shows schematically a ring, rotating at velocity V in a vacuum. Fig. 3. Schematic of the toroid (left) and of the section traversed by the beam (right) 2
The proton beam passes through a length L and heats the ring to a temperature T max. As the ring rotates it radiates heat, according to Stefan s Law of radiation, to the water cooled walls of the vacuum chamber. The toroid enters the beam at temperature T 0. The quantity of heat absorbed and the thermal capacity relation give the temperature rise. The calculations assume a continuous or dc beam at this point. Stefan s Radiation Law : dq dt =2πrLɛσg(T 4 T 4 e ) (1) Thermal Capacity : Q = πr 2 LρS(T T 0 ) (2) which gives the power as : W = Q V L (3) where : r = the radius of the target section (1 cm) L = the effective length of the target in the beam at any one time (20 cm) ɛ = the thermal emissivity (0.3) σ = Stefan s constant (5.67 x 10 12 Wcm 2 K 4 ) g = geometry factor (1) S = specific heat at 300 0 C 1 (Ta -0.14Jg 1 K 1 ) ρ = density(ta-16.7gcm 3 ) V = peripheral velocity of the toroid (cm/s) T = temperature (K) T e = the temperature of the enclosure (300 K) T 0 = the temperature of the target entering the beam (K) The solution to these equations is shown graphically in Fig. 4. The power is plotted as a function of the radius/peripheral velocity for different values of the velocity. To aid the reader, the value of the radius is indicated at specific 1 The specific heat should be taken at 2500 K and appropriate temperatures. 3
points on the curves. The maximum surface temperature is 2500 K; other values of the target parameters are shown above. Fig. 4. Graph of power dissipated in the toroid versus radius/velocity for different velocities, V. The graph indicates, for example, that a power of 1 MW can be dissipated in a toroid of 20 m radius rotating at 1 m/s or 0.4 m radius at 10 m/s. At 10 MW power dissipation the toroid could be 200 m radius rotating at 10 m/s or 40 m radius rotating at 100 m/s. The maximum power is radiated when the toroid is rotating very quickly and the entire toroid is at the maximum temperature. The power dissipation is given by W max =2πr2πRɛσg(T 4 max T 4 e )=2.622.103.R (4) Figure 5 illustrates this for Tmax = 2500K. Fig. 5. Maximum power dissipation as a function of radius, R, ofthetoroid. 4
The power dissipation of the toroid can be increased by increasing the thermal emissivity and the cross-section radius, r. An emissivity of 0.8 is possible by effectively roughening the surface. This would increase the power dissipation by a factor of 2.5. Using tungsten would allow a higher maximum operating temperature. Another option is to alter the cross-section to increase the surface area. With a band instead of a toroid, a small diameter beam could be scanned across the area to distribute the power dissipation. To maintain the emittance of the pions the proton beam would be fixed and the band moved to provide the scanning. Fig. 6 shows this schematically. By making the band thin it would not impede the passage of the pions through the band. Fig. 6. Thin band target. It would not be difficult to increase the power dissipation in the target to 100 MW or even 1000 MW by these methods. 3 Target Length The target length in the beam may be obtained by passing the beam through the arc of the ring or angling the beam relative to the plane of the ring. Fig. 7 shows the proton beam passing through the arc of the toroid and the relationship of the effective target length to the geometry. With L =20cm, r =1cm,R must be 45 cm - which is restrictive. It is better to tilt the plane of the toroid with respect to the proton beam centre line as shown in Fig. 8. The tilt angle is less than 6 o to obtain a target length of 20 cm. 5
Fig. 7. Proton beam passing through the arc of the toroid. Fig. 8. Geometry with the tilted target, θ 2r/L = 1/10 radians = 5.7 degrees. 4Pulsed Effects If the proton beam is pulsed, then some of the power dissipation relations are modified. Assume that the protons have a very short pulse length (1 ns) and a repetition rate of f (Hz). When the toroid moves slowly, the volumes illuminated by consecutive pulses of the proton beam overlap and gradually separate as the toroid rotates faster or the repetition rate falls. The pulses are just separate when the speed and frequency are given by V = Lf (5) For a target length of 20 cm and a repetition rate of 100 Hz, this speed is 20 m/s. At higher speeds the pulses are separate and the toroid is effectively larger than necessary. The parts of the toroid heated by the beam may cool more before they pass back into the beam, but the maximum temperature is still limited (by T max 2500 K for tantalum). Thus the peak power that can be dissipated is given by W = Q V L = πr2 ρls(t max T e )f (6) Putting in the numbers for the target, gives W =0.32 f MW, which is 32 MW at 100 Hz repetition rate. 6
Fig. 9. Temperature rise versus velocity for different powers. 4.1 Temperature Rise, Uniform Beam The temperature of the toroid entering the beam rises abruptly. The temperature change is, T = W πr 2 ρsv (7) and, putting in values for tantalum, T =0.136 W V (8) At 1 MW the temperature rise is 136 K with a velocity of 10 m/s. Large temperature rises are to be avoided because of the thermal stress induced in the structure. The tantalum is weak at high temperatures. Fig. 9 shows the temperature rise as a function of velocity. Large rises, towards 2000 K, are found at low velocities. At 10 MW power dissipation the velocity needs to be 70 m/s to keep the temperature rise down to 200 K. The stresses and high velocities are likely to limit the rotating toroid to 10 MW. As the toroid starts to radiate the power, a gradient is set up across the section radius, r. In equilibrium this gradient is given by δt r = P 0 4πK (9) 7
where P 0 is the power dissipated per unit length of the target, P 0 = W 2πR (10) At 1 MW, with V = 10 m/s and R = 4 m, the radial temperature gradient is under 40 K. 4.2 Temperature Rise If the beam does not heat the target uniformly over the section radius, r, but has, for example, a parabolic distribution, then the centre will instantaneously rise in temperature above that of the outer radius. The power/unit length can be expressed as a function of the radius, ρ P (ρ) = 2P [ ( ) ] 0 ρ 2 1 πr 2 r (11) In this case, with r = 1 cm, the radial temperature rise at the centre, ρ =0, is δt r = 2 T (12) π If the temperature rise is 136 K for the uniform beam case, then there will be an additional rise of 87 K to the centre of the target. In fact the situation is more complicated than this since the beam, whether uniform or not, enters the toroid at an angle and produces additional longitudinal and radial temperature gradients. The exact gradients and their stresses are being considered by Chris Densham [3]. These calculations may show that the allowed power dissipation is restricted to quite low values - perhaps 1 MW or less. There may be ways of reducing the thermal stresses by slicing the target up into short sections or discs so that the longitudinal stresses are reduced. However, calculations by King [2]of the stresses in the cupronickel rotating band target indicate that the problem is not severe at the 1 MW level. This is, at least in part, due to the higher specific heat of the cupronickel. There are other high temperature materials which have higher specific heats than tantalum which should be considered for the radiation cooled target. 8
5 Mechanics It is proposed to consider the electromagnetic levitation and guidance of the toroid and rotation by linear motors. This will mean that there will be no moving parts (except for the toroid) in the vacuum and no physical contact with the toroid. No work has been done on this aspect to date. 6 Summary A toroid or band, levitated and rotating in vacuum, and thermally radiating its power to water-cooled vacuum chamber walls could provide a simple, clean and reliable high power target for pion/muon production. It would not require beam windows between the incoming proton beam and the outgoing pion beam. Powers of tens of MW or more are capable of being dissipated. However, pulsed proton beams limit the power dissipation. Further, severe restrictions may be introduced by the thermally induced stresses. These problems are aggravated at low repetition rates. The pulsed situation requires further study. Also, levitation and rotation of the toroid have yet to be addressed. References [1]B.J. King, S.S. Moser, R.J. Weggel, N.V. Mokhov, A Cupronickel Rotating Band Pion Production Target for Muon Colliders, Proc. PAC 99 - New York City, NY, U.S.A., 29 March-2 April, 1999. [2]B.J. King, Rotating Band Pion Production Target for Muon Colliders, See these proceedings. [3]C.J. Densham, See these proceedings. 9