2 Feb 2001Hg Jet SimulationsSteve Kahn Page 1 Simulations of Liquid Hg Jet Steve Kahn 2 February 2001 This includes work done by various people: J. Gallardo, S. Kahn, K. McDonald, R. B. Palmer, R. Samulyak, P. Thieberger, R. Weggel

2 Feb 2001Hg Jet SimulationsSteve Kahn Page 2 List of Forces, Pressures, Distortions, Deflections Induced azimuthal Eddy current. Radial forces: J Eddy  B z –Hydrostatic Pressure Axial force –Contribution from Hydrostatic Pressure –Contribution from dB z /dz Transverse forces and deflections Shear forces Transverse elliptical distortion

2 Feb 2001Hg Jet SimulationsSteve Kahn Page 3 Targeting Schematic 67 mr 100 mr beam Hg Jet Begin overlap End full overlap End overlap -60 Nozzle Begin full overlap Obviously Not to Scale Proton beam Hg Jet

2 Feb 2001Hg Jet SimulationsSteve Kahn Page 4 Magnet System Configuration Below is the coil configuration from Bob Weggel (21-Dec-00) Only coils in yellow region are used in analysis, others are far enough from target that they can be ignored.

2 Feb 2001Hg Jet SimulationsSteve Kahn Page 5 Field Calculations In this model a pole made of Vanadium Permador steel is placed in the 20 T field to act as a nozzle. –This steel has M s =2.4 T. (Figure shown on next transparency). –The nozzle is present to insure that the Hg jet enters the field intact. Opera-2D is used for field calculations. –This is a 2D finite element field solver that solves the cylindrical symmetric problem. –Only the hole in the nozzle breaks cylindrical symmetry. This only has effects in the vicinity of its aperture. The beam path is inclined with respect to the magnet axis at 67 mrad. –Justification for minimum perturbation of the beam path will be mentioned later.

2 Feb 2001Hg Jet SimulationsSteve Kahn Page 6 2D Axial Symmetric Model 1006 Steel Vanadium Permador Steel M s =2.4 T

2 Feb 2001Hg Jet SimulationsSteve Kahn Page 7 Magnetic Field in Local Coordinates Figures show B z and B y in the local coordinate system of the Hg jet. –Local system is inclined 67 mrad to solenoid axis. –Each figure shows 5 places where the the trajectory intersects axis: At 0 cm,  10 cm,  20 cm Z=0 cm is 120 cm from pole face. Z=0 cm is the far end of the target. Pole face

2 Feb 2001Hg Jet SimulationsSteve Kahn Page 8 Effect of Iron Pole on Field Graphs show a comparison of local B z and B y along Hg trajectory. Pole is made of Vanadium Permador steel which has M s =2.4 T. Pole Surface

2 Feb 2001Hg Jet SimulationsSteve Kahn Page 9 Field Derivatives Forces are proportional to field derivatives. Figure shows dB y /dz and dB z /dz along path. Spike indicates edge of pole Jaggedness of curves due to finite element nature. –dB/dz is 2 nd derivative of potential. Each element has quadratic variation of potential. dB/dz is constant in each element. Pole face

2 Feb 2001Hg Jet SimulationsSteve Kahn Page 10 Force Densities Axial force density –averaged over radius Axial hydrostatic pressure from radial force –averaged over radius Numerical evaluation uses –r 0 = 0.5 cm –v z = 20 m/sec –  = 1  10 6 ohm -1 m -1 (Hg) Formulae from R. Palmer’s note

2 Feb 2001Hg Jet SimulationsSteve Kahn Page 11 Additional Shear (or is it Torque) ByBy F F Integrating over top half:

2 Feb 2001Hg Jet SimulationsSteve Kahn Page 12 Shear Forces

2 Feb 2001Hg Jet SimulationsSteve Kahn Page 13 Angular Deflection from Transverse Force The vertical force density is This gives The resultant deflection per unit length is –This is plotted in figure Formulae from R. Palmer’s Note

2 Feb 2001Hg Jet SimulationsSteve Kahn Page 14 Integrated Angular Deflection The figure shows d  /ds and  x integrated over the path length. The Hg jet will be deflected ~0.1 mrad over the trajectory. The spacial deflection is less than 0.15 mm over the 1.5 meter path. Adjusting the field for the changing path has not been done. This probably is not important.

2 Feb 2001Hg Jet SimulationsSteve Kahn Page 15 Deceleration of Hg Jet Green curve in figure is the total axial deceleration of the Hg jet entering the magnets:  V –V(z) = V 0 –  V Blue curve shows the contribution from axial force,. Red curve shows the contribution from the hydrostatic pressure resulting from the radial force,.

2 Feb 2001Hg Jet SimulationsSteve Kahn Page 16 FronTier Code Solve Magnetohydrodynamics Problem: Boundary Conditions: Simplifications: Ignore  sec transient effects. We are interested in eddy currents which are msec effects. J as unknown variable:

2 Feb 2001Hg Jet SimulationsSteve Kahn Page 17 Hg Jet Entering and Leaving 20 T Solenoid