calculations of liquid hg jet evaluated for realistic magnet system

31 Jan 2001 Magnetic Field Effects Simulations Steve Kahn

Calculations of Liquid HgJet Evaluated for Realistic Magnet

SystemSteve Kahn

31 January 2001

This summarizes work done by various people:

J. Gallardo

S. Kahn

R. B. Palmer

P. Thieberger

R. Weggel

K. McDonald


List of Forces, Pressures, Distortions, Deflections

• Induced azimuthal Eddy current.

• Radial forces: JEddyBz

– Hydrostatic Pressure• Axial force

– Contribution from Hydrostatic Pressure

– Contribution from dBz/dz

• Transverse forces and deflections• Shear forces• Transverse elliptical distortion


Magnet System Configuration

Coil Zcentral Half-Width Rinner Router J cm cm cm cm Kamps/cm2

HC1 -29.35 42.75 18.00 23.43 2.796 HC2 -26.3 45.80 23.43 34.01 2.010 HC3 -25.05 47.05 34.01 44.19 1.475 SC1 -54.40 50.30 60.45 123.97 2.808 SC2 40.10 34.20 60.45 109.98 2.809 SC3 144.75 60.45 73.53 97.03 3.011 SC4 292.25 77.05 74.95 85.33 4.278 SC5 484.65 105.35 82.00 88.40 5.323 SC6 647.45 47.45 50.52 53.41 7.736 SC7 767.55 62.65 46.90 49.48 8.431 SC8 916.60 76.40 46.12 48.46 8.703 SC9 1089.8 86.80 46.68 48.54 8.932

SC10 1281.9 95.30 45.33 46.89 9.134 SC11 1479.8 92.60 47.15 48.52 9.311 SC12 1712.4 130.0 45.09 46.27 9.447 Table 1: Coil parameters for target capture magnet system.

•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.


Targeting Schematic

67 mr

100 mr

beam

Hg Jet

Beg

in

overla

p

En

d fu

ll o

verlap

En

d

overla

p

-60No

zzle

-45 -30 Beg

in

full

overlap

0 +15

Obviously Not to Scale

Proton beam Hg Jet

Parameter Symbol Early Study 2 Current Study 2 Hg Jet Radius RHg 0.5 cm 0.5 cm

Jet Incline Angle jet 100 mrad 67 mrad

Angle Between Jet and Beam crossing 0 mrad 33 mrad

Jet Velocity Vjet 20 m/sec 30 m/sec Time Between Pulses t 20 ms

Length of Jet Ljet 30 cm 60 cm

Parameter Symbol Value Conductivity , cm 108

Density , gm/cm3 13.4 Surface Tension Tsurface, n/m 0.456


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


Saturated Pole

Iron is highly saturated.

< 1.36 everywhere


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.

Effect of Iron Pole on By

-20000

2000400060008000

10000120001400016000

-200 -100 0 100 200Path Position, cm

By,

ga

uss

No Fe

Fe

Effect of Iron on Bz

0

50000

100000

150000

200000

250000

-200 -100 0 100 200

Path Position, cm

Bz,

ga

us

s

No Fe

Fe

Field Differences from Iron Pole

-25000

-20000

-15000

-10000

-5000

0

5000

-200 -100 0 100 200

Path Position, cm

de

l B

, g

au

ss

del By

del Bz

Pole Surface


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.

Local Vertical Field

-4000

-2000

0

2000

4000

6000

8000

10000

12000

14000

16000

-200 -150 -100 -50 0 50 100 150 200

Z position, cm

0 cm

10 cm

20 cm

-10 cm

-20 cm

Local Axial Field at 0.67 mr Incline

0

50000

100000

150000

200000

250000

-200 -150 -100 -50 0 50 100 150 200

Path position, cm

0 cm

10 cm

20 cm

-10 cm

-20 cm

Pole face


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.

Field Derivatives

-2000

-1500

-1000

-500

0

500

1000

1500

2000

2500

3000

-200 -150 -100 -50 0 50 100 150 200

Path Position, cm

dBz/dz

dBy/dz

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 = 1106 ohm-1 m-1 (Hg)

22

4

dz

dBv

rf zz

dz

dBB

dz

dv

rf z

zo

p 8

2

Formulae from R. Palmer’s note

Force Densities

-4.E+07-3.E+07-2.E+07-1.E+070.E+001.E+072.E+07

-60 -10 40 90 140 190

Axial Position, cmFo

rce

Dens

ity,

nt/m

3

1000*fz(out)

fr(out)


Additional Shear (or is it Torque)

By

F

F

)sin(2

),( dz

dBvr

Brf zyz


Force on Coil Half

Force on Top Half

-4000

-3000

-2000

-1000

0

1000

2000

-60 -10 40 90 140 190

Axial Position, cm

dF

/dz,

nt/

m

dFz/dz

dFtop/dz


Angular Deflection from Transverse Force

• The vertical force density is

• This gives

• The resultant deflection per unit length is

– This is plotted in figure

)sin(ry ff

dz

dB

dz

dBrv

dz

dFzyy 4

8

dz

dB

dz

dBr

dz

dvzyy

8

2

dz

dB

dz

dB

v

r

dz

d zy

8

2

Formulae from R. Palmer’s Note

Angular Deflection

-5.0E-050.0E+005.0E-051.0E-041.5E-042.0E-042.5E-04

-60 40 140

Axial Position, cmd(

thet

a)/d

z, ra

dian

/m


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.

Integrated Deflection

00.0020.0040.0060.008

0.010.012

-60 -10 40 90 140 190

Axial Position, cm

Def

lect

ion

, cm

Integrated Angular Deflection

0.E+00

2.E-05

4.E-05

6.E-05

8.E-05

-60 40 140

Axial Position, cm

Def

lect

ion

, rad

ian

s


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>.

Average Reduction in Velocity

-0.06

-0.04

-0.02

0

0.02

0.04

-60 -10 40 90 140 190

Axial Position, cm

del V

, m/se

c

Total

Axial Force

Hydrostatic


Elliptic Distortions

• Horizontal Force:

– If there is an incline angle, By0. There can be an axial eddy current:• fx=jzBy

• jz~cos

• Radial Inward Force– Azimuthal eddy current gives radially inward force:

• fr=jBz

• Eccentricity: = y/x Fx/Fy

cosyy

zx Bdz

dBrvf

3

3

2r

dz

dBBv

dz

dF yy

H

FxBy

dz

dBBv

rf z

zr 2

3

3

1r

dz

dBBv

dz

dF zz

Rx

Fr

dzdB

B

dz

dBB

dzdB

B

zz

yy

zz 2

~0.98 at target

calculations of liquid hg jet evaluated for realistic magnet system

Documents

field calculationsin

pole surfacechart1

hg jet radiusrhg0

d finite element field

current study

dbzdztransverse forces

radial forces

mcdonaldlist of forces