# Presentation: 417711

## 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

## List of Forces, Pressures, Distortions, Deflections

- Induced azimuthal Eddy current.
- Radial forces:  JEddyBz
- Hydrostatic Pressure
- Axial force
- Contribution from Hydrostatic Pressure
- Contribution from dBz/dz
- Transverse forces and deflections
- Shear forces
- Transverse elliptical distortion

## Targeting Schematic

**100 mr**

**beam**

**Hg Jet**

**Begin overlap**

**End full overlap**

**End overlap**

## 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.

## 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 Ms=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.

## 2D Axial Symmetric Model

- 1006 Steel

- Vanadium Permador Steel  Ms=2.4 T

## Magnetic Field in Local Coordinates

- Figures show Bz and By 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**

## Effect of Iron Pole on Field

- Graphs show a comparison of local Bz and By along Hg trajectory.
- Pole is made of Vanadium Permador steel which has Ms=2.4 T.

- Pole Surface

## Field Derivatives

- Forces are proportional to field derivatives.
- Figure shows dBy/dz and dBz/dz along path.
- Spike indicates edge of pole
- Jaggedness of curves due to finite element nature.
- dB/dz is 2nd derivative of potential.  Each element has quadratic variation of potential.  dB/dz is constant in each element.

**Pole face**

## Force Densities

- Axial force density
- averaged over radius
- Axial hydrostatic pressure from radial force
- averaged over radius
- Numerical evaluation uses
- r0 = 0.5 cm
- vz= 20 m/sec
- = 1106 ohm-1 m-1 (Hg)

- Formulae from R. Palmer’s note

## Additional Shear (or is it Torque)

- By

- F

- F

**Integrating over top half: **

## Shear Forces

## 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

## 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.

## Deceleration of Hg Jet

- Green curve in figure is the total axial deceleration of the Hg jet entering the magnets: V
- V(z) = V0–V
- Blue curve shows the contribution from axial force, <fz >.
- Red curve shows the contribution from the hydrostatic pressure resulting from the radial force, <fp>.

## 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:**

## Hg Jet Entering and Leaving 20 T Solenoid