ORNL Induction Probe

8/9/2019 ORNL Induction Probe

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3 4 4 5 6 0 0 2 3 8 2 2

2


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This report was prepared as an account

of

work sponsored by t h s Uni ted

Stares Government. Neither the United

S t a r e s

nor the United States Atomic

Energy Commission, nor any of their elnployees, nor any of their contractors,

subcontractors, or their employees, makes any warranty, express nr implied, or

assumes any legal liability

or

responsibility for t h e accuracy, completeness or

usefulness of any inforrnation, apparatus, product or process disclosed, or

represents that i t s use would tnot infringe privately owned r jghts.

.

. _


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OR NL-TM -l t175

Contract

N O . W-71-0;1

-eug

-26

M E T A L S

AND CERAMICS

DIVISION

DESIGN

OF ' INDUCTION PROBES

FOR F E A S U K E m N T

OF

T,EVEL

OF

LIQUID

m'rAu

C .

V .

Dodd C.

C .

Cheng

C . W. N e s t o r ,

J r .

R . B .

H o c s t r n

M A Y

1973

OAK

R I D G E N A TI O N A L LA R O R 4TO R Y

O a k

R i d g e ,

Tennessee 37830

ope

r a t e d by

UNION C A R B I D E COKPORATION

f o r t h e

3 4 4 5 5

0 0 2 3 8 2 2 2


4/122


5/122

iii

CONTENT

S

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

1

.

I N T R O D U C T I O N

. . . . . . . . . . . . . . . . . . . . . . . . .

2 . THEORETICAL ANALYSIS

. . . . . . . . . . . . . . . . . . . . .

3 -

COMPUTER CALCULATIONS

FOR

11 PROBE INSIDE BISMUTH- . . . . . .

11 . MEASUREMENTS ON LIQUID LEVEL SYSTEM . . . . . . . . . . . . .

5

SUMMARY A N D CONCLUSIONS . . . . . . . . . . . . . . . . . . .

6 A CKNOWLE DGMENTS . . . . . . . . . . . . . . . . . . . . . . .

APPENDIX

A

omputer Programs f o r a L iquid Lev el Pro be

I n s i d e

C o a x i a l C o n d u c t o r s

. . . . . . . . . . . . . . . . . . . . . .

I

. I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . .

II

D e s c r i p t i o n of INNMUL . . . . . . . . . . . . . . . . . .

A .

D R I V E R

Program . . . . . . . . . . . . . . . . . . . .

B . INhMUI. Subroutine . . . . . . . . . . . . . . . . . .

1

. X J S N T

F u n c t i o n . . . . . . . . . . . . . . . . . .

2 XILNT Function

5

. GAMCAL S u b r o u t i n e . . . . . . . . . . . . . . . .

a

.

GCA1. C S u b r o u t i n e . . . . . . . . . . . . . . .

b

.

MODBES S u b r o u t i n e . . . . . . . . . . . . . .

c

.

CND

KF:

S

S

u

b

r

ou t

i

ne

. . . . . . . . . . . . . .

i . COMKB

S u b r o u t i n e . . . . . . . . . . . . .

ii . C M I S u b r o u t i n e

. . . . . . . . . . . . . .

IC11. Execut ion oP INNMUL

. . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

A

.

TELINC.Fh Program

. . . . . . . . . . . . . . . . . .

B

. D a t a D e s c r i p t i o n . . . . . . . . . . . . . . . . . . .

I V . ATTEN Program . . . . . . . . . . . . . . . . . . . . . .

C . S ample Dat a and Re su l t s . . . . . . . . . . . . . . .

Page

_

1

1

c

d

24

33

5’f

38

35

93

99

1O?

108


6/122


7/122

I

D E S I G N OF I N D U C T I O N

PROBES FOR MEASTJESPfENT

-

OF

LEVEL

OF LIQUID IIETALS

C . V. Dodd C . C . Chengl

C .

W.

N e s t o r ,

J r . 2

E.

B.

H o f s t r a 2

ABSTRACT

This r e p o r t g i v e s g e n e r a l a n a l y s es

of

e d d y - c u r r e n t p r o b e s

f o r m ea s ur i ng t h e

l e v e l

o f l i q u i d m eta l s . T h e C ~ S P f a c o i l

e n c i r c l i n g a l e v e l c ha mb er a nd t h e

case

of a co-il i n s i d e

a

l e v e l chamber h a v e b e e n solved t h e o r e t i c , d l y , a n d c o m p u t e r

programs

a r e

inclridrAd i n t h e A p p e nd i x f o r t h e l a t t e r case .

A ; a s p e c i f i c e x am p le , w e h a v e d e s l g n e d a p r o be en c l os e d i n

molybdenum (a

good

c o n d u c t o r ) t o measure t h e level

of

m o l t e n

b i s mu th

( i t

p o o r c o n d u c to r ) . Ry u s in g a c om p ut er a n a l y s i s , t h e

q e n s i t i v i t y of

t h e

p r o b e

t o

l e v e l c h an g es i s maximized

w h i l e

t h e s e n s i t i v i t y t o u n de s i r a b l e v a r i a b l e s , s uc h

as

t e m p e r a t u r e

c h a n g e s

i s

min iini z e d . E x p e r im e n ta l me a s ure me nts d e mo n s t ra t e d

t h a t

the

l e v e l c o u l d

be

m ea su re d t o w i t h i n 4 0.089 iu. o v e r a

l e v e l

rangci

f rom

o

t o 13 i n . w i t h i n a t e m p e r a t u r e range of

600"

t o 6 5 0 ° C .

The

h i g h d e g r e e of buccess a ch i ev e d i n t h e

p r o b e d e s i g n and mcasurements f o r t h i s u n f a v o r a b l e c o m b i na t io n

o f c o n du c t or s i n d i c a t e t h a t h i g h l y a c c u r a t e ed dy -c ur r en t meas-

u re me n ts c a n

be

made w i t h a l m o s t

any

c o m b i n a t i o n

of

c o n d u c t o r s .

1.

INTLWDUCTION

T h e a b i l i t y t o m ea su re

t h e

level of B m o l t e n meta l i s very i .mportan t

i n

a

number

of

i n d u s t r i a l a nd c h em i c a l p r o c e s s e s .

W e

h a v e a n a l y z e d t h e

g e n e r a l p r o b l e m

of

m e a s u r i n g t h e 1 . ev e l

of

a c o n d u c t iv e f l u i d b y a n i nd u c-

t i o n , o r e d d y - c u r r e n t p r o c e s s . T he e d d y - c u r r e n t

p r o b e

c o n s i s t s

of

a

l on g b i f i l a r

c o i l ,

which c an e i t h e r e n c i r c l e a c h a m b e r c o n t a i n i n g t h e

l i q u i d m e t a l o r be i n s i d e a t u b e mo un te d i n t h e c ha mb er c o n t a i n i n g t h e

l i q u i d l e v e l . W e o b ta i n ed i n t e g r a l s o l . u t i o n s wh ic h were

v a l i d

f o r

the

c ha mb er e i t h e r e m pt y o r f u l l , a n d , be c au s e of

t h e

p r o b e l e n g t h ,

we

a s s u m e d t h a t

t h e

r e s p o n s e

of

t h e p r o b e

t o

l e v e l s b e t we en t h e s e t wo

ex t r eme wa

s

app

r

oxlmat e l y 1 n e a

r .

showed thLs

t o

b e a n e x c e l l e n t a s s um p ti o n.

A

r e l a x a t i o n s o l u t i o n

c o u l d

L a

t

e r experimen a1

measiir e m e m t

s

lConsu l t an t f ro m t h e U n i v e r s i t y o f T e n n es s e e.

2Ma

themat i c s

D i v i s i o n .


8/122

be us ed t o c a l c u l a t e t h e p ro be r es p on s e s t o v a r i o u s l e v e l s of l i q u i d

m e t a l , b u t w as j u dg e d t o b e

too

e x p e n s iv e t o r u n f o r t h e a d d i t i o n a l

i n fo i -m a t io n g a i n e d . W hi le t h e t e c h n i q u e i s v e r y g e n e r a l a nd c a n b e

a p pl i- e d t o a l m o st an y c o n d u c t i v e f l u i d ,

w e

a n al - yz e d a s y s t e m t h a t c o n -

s i - s t e d

of

a c o i l e n c a s e d i n mo ly bd en um ( a go od c o n d u c t o r ) u se d t o me a su r e

t h e l e v e l

of

mol t en b i smuth ( a

p o o r

c o n d u c t o r ) . T he h i g h d e g r e e o f

s u c c e s s a c h i ev e d i n t h e p ro b e dt . ; i l ; i I and measurements f o r t h i s u n fa v or a bl e

c o m b i n a t i o n of c on du ct :o rs l e a d s u s t o c o n c lu d e t h a t h i g h l y a c c u r a t e e dd y -

c u r r e n t t e c h n i q u e s c a n be d e s i gn e d a n d a p p l i e d f o r l e v e l m ea su re me nt s w i t h

a l m o s t a ny c o m b i n a t i o n o f c o n d u c t o r s .

2. THEORETICAT. ANALYSCS

The g e n e r a l c o n f i g u r a t i o n t o be c o n s id e r e d i s a n a x i a l l y s ym me tr ic

d r i v i n g c o i l l o c a t e d c o n c e n t r i c a l l y w i t h a n a r b i t r a r y riurrlber o f c y l i n -

d

r i c a 1 c o n d u c to r s w i t h a r b i t r a r y t h i c k n e s s , p erm ea b

i

1 t

y ,

p e r m i t t i v i t y ,

a n d c o n d u c t i v i t y . F o r s i m p l i c i t y , we a ss um e Lh dt a l l me dia a r e l i n e a r ,

i s o t r o p i c , a n d h om og en po us , a nd t l ie d r i v i n g c u r r e n t

i s

t ime-ha rmonic w i t h

f r e q u e n c y ,

w .

Th en , t h e c u r r e n t d e n s i t y J and v e c t o r p o t e n t i a l A w i l l

have on ly az i i nu tha l componen t s i n c y l i n d r i c a l c o o rd i n a te s

:

-3

-3

.

p

J ( x )

=

J ( r , z )

^c

6'

and

+ ..)

8

A ( x ) = A ( r , z ) ^e

( 2 )

where e . i s a n a z i m u th a l u n i t v e c t o r . The v e ct -o r p o t e n t i a l a t ( r , ~ ) ,

produ ced by a d r i v i n g c o i l w i t h a c u r r e n t d e n s i t y J ( r ' , z ' ) a t ( r l , z ' ) ,

c a n b e e x p r e s s e d a s

c;

A ( r , z ) =

G ( r , z ; r ' , z ' ) J ( r ' , z '

)

d r ' d z '

D r i v i n g

Coi 1.

where G ( r , z ; r ' , z ' ) i s t h e G r e e n 's

f u n c t i o n f o r a u n i t b - f u n c t io n c u r r e n t


9/122

.+..

a t

( r ' ,

2 ' ).

fiiiic t i o n

s

n t i s i e

In a l i n e a r , i s o t ro p i c , homogeneous medium, t h e

Green's

3

where

1.1, E, 3 i - d n a r e t h e p e r m ea b i l i t y , p e r m i t t i v i t y ,

a n d

c o n d u c t i v i t y

oE the medium.

t h e p r o p e r b ou nd ar y c o n d i t i o n s .

Tlic

s o l u t i o n

o f

E q .

( )

for

e a c h mcadiurn

rlirrst a l s o s a t i s f y

We s h a l l f i r s t

c o n s i d e r a ? ) - fu n c t io n c o i l c o a x i a l

w i t l i

k a k f - ?

c y l i n d r i c a l cu n d u ct o r s; k-1

of

the m i n s i d e

t h e

c o i l

arid

k ' - 1

o f

t l i c n i

o u t s i de th e c o i l , a s

shown

i n F i g . 1. Tlicl g e n e r a l s o l u t i o n o f E q . ( 1 1 )

i n

any

r e g i o n ,

t i ,

may

be

o b t a i n c d

b y

s e p a r a t i o n

of v a r i a b l e s .

S e t t i n g

a n d d i - v i d i n g E q .

1 1 )

by

R ( r ) Z z )

g i v e s :

T h e s

ul,

s

c r

i

p t

s

n 011 the

p

c r m e a b i. 1 i t

y , p e r m i t

t

i

v

j.t

y,

a nd

c

ond

i.ic t i v i : y

d e n o t e t h e

values

of

t h e s e

p a r a m e t e r s i.n t h e r e g i o n n .

We

s h a l l

choose

the

s e p a r a t i o n " c o n s ta n t " , (X'.., o be n e g a t i v e

a n d

de f i -ne

'C. V.

Dodd

and

W.

E . Deeds, J .

A p p l .

Phys ,

2 829-2838 (1968).


10/122

C X FI L-

CWG

7 2 - 3 0 7

2

F i g .

1.

Mul t ip l e Concen t r i - c Conduc to r s

i n

t h e P r e s e n c e o f a Delta

F u n c t i o n C o i l .

Then

we

c a n

w r i t e f o r Ehe z

drpendence

S o l v i n g his d i f f e r e n t i a l e q ua t io n g i v e s :

L ( Z )

A

sin L - ( z - z 0 ) i B cos i e - z 0 ) .

We c a n

d r o p

t h e s i n e t e r m

d u e t o

t h e symmetry a b o u t

z l .


11/122

5

The

r a d i a l t e r m h a s t fi e f o l l o w i n g d e pe n de n ce :

.

T h i s

d i f € e r e n t i a l e q u a t i on

has

t h e

f o l l u w i

ng

s o l u t i . o n :

where

I tY r )

and

K (Q r ) a r e

m o d i f i e d

Ressel

f u n c t i o n s

of

f i r s t o r d e r .

The c om p le te s o l u t i o n t o

t h e

Green's f u n c t i o n i n e a c h r e g i o n i s a n

1 n I n

i n t e g r a l o v e r t h e s e p a r a t i o n c o n s t a n t

0:

f o r

n

L 1,2,

. . .

k ;

l l , ? ' ,

.

. . k'. T h e unknown cons i an t s


12/122

4

ORNL-DWG 72-3046

/-

F i g .

2.

T o p

V i e w

of a Typ ic al Boundary Between

Two

Regions .

U s i n g

t he

G r e e n ' s f u n c t i o n s f r o m

E q .

( 1 0 )

i n

E q .

( 1 2 )

w e

o b t a i n :


13/122


14/122

We c a n r e v e r s e t li e o r d e r

o f

i n t e g r a t i o n of t h e i n t e g r a l s c o n t a i n i ng

: - s s e l

f u n c t i o n s an d u s e t h e F o u r i e r i n t e g r a l t he or em ,

F:quation

(16)

t h e n b e c o me s :

A

s i m i l a r o p e r a t i o n

on

E q . ( 1 1

)

g i v e s :

E q u a t i o n s (1

)

and ( 1 - ) r e p r e s e n t

t h e

r e l a t i o n s b e tw ee n t h e c o n s t a n t s

f o r

a n y

two r e g i o n s i n s i d e t h e c o i l . We

s h a l l

now so lve f o r a l l t h e

unkoowyii c o n s t a n t s i n t h e f o l l o w i n g m a n n er . S i n c e t h e i n n e r m o s t r e g i o n

h a s only one unknown cons tan t :

C I , w e

s h a l l s o l v e f o r t h e unknown co n -

s t a n t s i n t h e s e co nd r e g i o n

i n

t e r m s of i t . N ext we s h a l l s o l v e f o r

t h e

u n k n o wn c o n s ta n t s i n t h e t h i r d r e g io n i i i t e rms o f C t h e n t h e f o u rt l l ,

u n t i l we r e ac h t h e r e g i o n c o n t a i n i n g t h e c o i l , k . We s h a l l d o t h e s a m e

k h i n g

f o r t h e r e g io n s outside t h e c o i l , s t - a i t i n g w i t h t h e o u t e r m o s t a n d

w o rk i ng i nw a rd s, s o l v i n g f o r e a ch r e g i o n i n t e rms o f D', u n t i l

we

r e a c h

t l ie r e g i on k ' . We s h a l l t h e n u s e E q . ( 1 0 and E q .

( l e )

f o r t h e c o i l

r e g i o n s k a n d k'. T h i s w i l l g i v e t w o e q u a t i o n s f o r t h e t wo un kn ow ns ,

C

and

D

a nd we c a n s o lv e f o r th em.

h i s

w i l l t h e n a l l o w u s t o

- > ] r i t e

h e e x p r e s s i o n f o r t h e u nknown c o n s t a n t s i n a ny r e g i oi i .

1'

1

1

1 ' '

S o l v i n g E q s .

( 1 0

and

(16)

o r t h e u nknown c o n s t a n t s i n

a n y

r e g i o n ,

n + l ,

i n terms

o f

t h e unknown c o n s t a n t s i n r e g i o n n . w h e r e t h e c o i l i s

n o t b e twe e n

l l i ~

e g i o n s , g i v e s :


15/122


16/122

10

T h i s t r a n s f o r m a t i o n

m a t r i x

g i v e s t h e r e l a t i o i l b et wPen t h e c o n s ta n t s i n

a n y two re g i o n s n o t c o n t a i n i n g

t h e

c o i l b e t w ee n t h em . I t

i s

t h e s a me

f o r r e g i o n s i n s i d e

and

o u t s i d e t h e c o i l , w i t h t h e e x c ep t i o n t h a t n s h ou l d

b e r e p l a c p d b y n f o r r e g i on s o u t s i d e t h e c o i l t o c or re sp on d t o o ur

n o t a t i o n .

S t a r t i n g f ro m t h e i n n e rm o s t r e g i o n (n:

1)

an d g o i n g t o t h e s ec on d

g i v e s :

The

c on st ar it s i n t h e t h i r d r e g i o n ca n

b e

o b t a i n e d

b y :

a n d t h e c o n s t a n t s i n t h e f o u r t h r e g i o n b y :

The g e n e r a l e x p r e s s i o n f o r t h e n t h r e g i o n

i s

(26)

To

make o u r e x p r e s s i o n s s h o r t e r ,

we

s h a l l

define:


17/122

v ( n ) T

. . . T

-3,2 T

q1 '

-n, n

-

1

-11

- 1., 1

-

a n d when 11 := lr, w e sha1.1 d r o p t h e a r g u m e n t. T hu s w e h a v e

We h a v e

v e r y

s imi la r e q u a t i -0 1 1 s

f o r

the>

r e g i o n s o u t s i d e

t h e

c o i l

:

where w e

h a v e rriade

a s i m i l a r d e f i n i t i o n :

and we h a v c d r o p p e d t h e a rg u me n t w h e n 11' k' T l ~ u s , we c a n

noTo

w r i t e

t h e c o n s t a n t s

i n

a n y L e qi o n i n

terms

o f t h e co n s t a n t i n

t h e

innerrrlos t

region,

a ~

[ o r

t h e

ouLerrnost region,

a [2,] b y means

o f t h e

t r a n s f o i n i a t i o n m a t r i c e s , V(n) and

U ( n ) . We

s h a l l w r i t

e

E q s . ( 1 ' 0

and

(18) f o r t h e

r e g i o n s

on

e i t h e r

s i d e of

t h e c o i l ,

k

a n d

k .

h a v e r :: L ,

s o

t l i a t

7 ( r -r'

) = 1 . A l s o = 2

=

Cr a nd 13 - H

- ; io, s o

t h a t the

e q u a t i o n s becorric:

-1 -1 '

v

Y

Herr w e

II 11

k

I<

0

k

-

k '

and

ck~,(uor'

4 D K

(Der')

z

c lC ,1

(a r ' )

+

D ~ ,(i-y

r ' )

.

(36)

k 1 1 0


18/122

U si ng o u r m a t r i x n 3 t a t i o n ,

we

c a n w r i t e

2

Dk =

V , C

cl 1

-- LT D ,

'k' 16 1 '

and

We now have

two

equa t i ons and on ly two unknowns .

S o l vi n g t h e s e g i v e s

and

( 4 4 )

whe r e t h e d e n o m in a to r h a s a g a i n b ee n s i m p l i f i e d

by

u s e of t h e Wronsk ian ,

and t h e f a c t t h a t P o

=

2

.

Now

w e

c a n s u b s t i t u t e t h e v a l u es of t h e

0


19/122

c o n s ta n t s i n E y s . k 3 ) a n d (41 i ) i n t o

E q s .

(51

)

and (53) t o g i v e t h e

c o n s t a n t s i n a n y r e g i o n .

Wc'

c a n w r i t e

t h e

G r e e n ' s f u n c t i o n

f o r a n y

r eg io ti i n s i d e t h e c o i l a s

The

G r ee n 's f u n c t i o n f o r a ny r e g i on o u t s i d e t h e c o i l i s

Once

we

h a v e t h e G r e e n ' s f u a c t i o n ,

we

c a n

g e t t h e

v e c t o r p o t e n t i a l

b y u s i n g

E q . ( 3 ) .

c r o s s s e c t i o n , a s shown

in

F i g .

3 .

We

a dd ed a n o t h e r r e g i o n , t h a t c o n -

t a i n s t h e c o i l , and d e si g n at e d

i t

r e g i o n c . For a

dense1.y

a n d u n i f o r m l y

wound c o i l ,

t h e

c u r r e n t d e n s i t y

J ( r ' ,

z '

) i s

a p p r o x i m a t e l y :

The most common type of c o i l is on(: of r e c t a n g u l a r

wherc

n

i s t h e

number

of

t u r n s

p e r

u n i t a r e a , an d

I i s

t h e c u r r e n t p e r

t u r n . S u b s t i t u t i n g

E q s . ( ) + ( )

and

(4))

i n t o

E q .

(3) g i v e s t h e v e c t o r

p o t e n t i a l f o r any r e g i o n i n s i d e t he c o i l a s

C


20/122

14

F i g .

3 .

C o i l w i t h R e c t a n g u l a r Cross S e c t i . o n C o n c e n t r i c w i t h C y l i n -

d r i c a l C o n d u c t o r s .

R e v e r s i n g the o r d e r of i n t e g r a t i o n a n d i n t e g r a t i n g o v e r t h e d i me n si o ns

of

t h e c o i l g i.ve s

a(z-R ) -

sin

( n ) I a

r )

f

V (n)Kl(anr)]

._

..-..

... . ..

.-

1

A (n)(r,

z ) =

1

7T

2

0

(u22v11

-

u 1 2 v 2 1 )


21/122

15

where v w

have

d e f i n e d t h e f u n c t i o n r ;

and

where r and r a r e now taken

a s

t h e c o i l i n n e r and o u t e r r a d i i , and

s h o u l d n o t

b e

c on fu se d w i t h t h e o u t e r r a d i i

of

t h e f i r s t t v 7 0 r e g i o n s .

The v e c t o r p o t e n t i a l f o r a ny r e g ion o u t s i de t h e c o i l i s

1 2

T h e r e g i o n of t h e c o i l r e q u i r e s s p e c i a l t r e a t m e n t .

To

f i n d t h e v e c t o r

p o t e n t i a l a t

a

p o in t i n the c o i l r e g i o n , r , v7e must add

t h e

s o l u t i o n of

A ( > ( r , z ) f o r

a

c o i l g o in g f ro m

r

t o r t o t h e s o l u t i o n

of

2


22/122

7

a

a

a

c

h

-

n

i

-

kt

0

W

7

0

W

.

0

d

0

d

W

d

N

H

u

n

kd

0

W

n

L

d

%

0

t

rhr

H

&

L

0

u

d

’

p

r

L

a

w

d

W

0

2

W

d4

8

0

-

h

d

0

M

G

0

M

i

-

n

L

s

o

u

t

G

4

H

n

L

0

4

n

r

.

0

u(

d

d

0

L

0

d

v

0

7

W

I

d4

4

H

4

0

h

k

r

L

X

-

X


23/122

1 7

We s h a l l

u s e t h e d e f i n i t i o n s g iv en i n E q s . (50)

and

(51)

f o r

1(r2>rl)

and K ( r r ) , and we s h a l l use t he r e l a t i o n s

2 '

1

a n d

E q u a t i o n (53)

t he n

becomes


24/122

18

We

h a v e s ho wn i n a n

c a r l l e r

paper ’ t h a t t h e i n t e g r a l

o f

t h e e x p r e s si o n

i n t h e l a r g e s qu a re b r a ck e t s i s

rn

- e da

(55)

( r , r ) J ( a r )

0 2

2 2 2 1

1

0

where

1

( r ,r

)

5

2 2

1.

c1

Making t h i s s u b s t i t u t i o n g i v e s

:

) - s i n a(z-R

___

’IT

0

We

h av e now d et e r m i ne d t h e v e c t o r p o t e n t i a l f o r a n y re g i o n . Onc e

t h e v e c t o r p o t e n t i a l h a s b ee n d et e rm i ne d ,

we

c a n c a l c u l a t e a ny p h y s i c a l l y

o b s e r v a b l e e l e c t r o m a g n e t i c i n d u c t i o n p h en om en on f r om i t . The p a r t i c u l a r

p a r a m et e r s t h a t w e w is h t o c a l c u l a t e a r e t h e m u t u a l i m pe da nc e a nd t h e

s e l f - i m p e d a n c e of two i d e n t i c a l b i f i l a r c o i l s .

The

mutua l impedance

be tween two c o i l s , 1 and 2 , i s

t h e

v o l t a g e i n du c ed i n one c o i l

by

a u n i t

c u r r e n t i n t h e o t h e r :


25/122

1 9

S i n ce b o t h c o i l s a r e i d e n t i c a l and o ccup y t h e samc r e e f o n , we can

expand

E q . ( 5 7 ) and perforiii the i n t - ’ g r a t i o n o v e r t h e s ec on d c o i l t o o b t a i n

Th e c o i l i m p ed a nc e

i s

and s i n c e t h e c o i l s a r e i d e n t i c a l and i n

the

same r e g i o n , M =

I f

w e t a k e t h e s p e c i a l case

where

t h e r e

are

110 c o n d u c t o r s o u t s i d e the

c o i l

w e

h a v e

E q .

(59)

becomes

1 2

=

l

t h e u n i t m a t r i x . Th en

U22

=

1

and

U I 2

=

0

so

t h a t


26/122

20

A

c o mp u te r p ro g ra m, ENCMTJL, t o e v al u a

s i o n w i l l b e gi v en i n a

l a t e r

r e p o r t .

t h e c o i l ,

we h a v e

x

= 1,,

and

E q .

(5 9 )

r

A

c o mp u te r p ro g ra m, ENCMTJL, t o e v a l ua t - e t h e i n t e g r a l p a r t

o f

t h i s e xp re s- -

s i o n w i l l b e gi v en i n a

l a t e r

r e p o r t .

O r , if we

h av e no c o n d u c t o r s i n s i d e

t h e c o i l ,

we h a v e

x

= 1,, and E q .

(59) becomes

2

2n p

2

0

-

( '2- ' J-> (r2-r1)

%2

-

t e

t h e i n t e g r a l p a r t

o f

t h i s

O r , if we h a v e n o c o n d u c to

becomes

.pres--

i n s i d e

A compute r p rogram, Il???bIuL, t o e v a l u a t e

t h e

i n t e g r a l p a r t o f t h i s

e x F r e s s i o n i s i n t h e a pp en di x.

T h u s ,

by u s e o f t h e s e p r o g ra m s w e c a n c a l c u l a t e t h e m u tu al i n d u ct a n ce

and c o i l imrledance of c o i l s i n a l i q u i d l e v e l p r ob e,

c o n s i s t i n g of m u l t i p l e

c o nd u ct o rs i n s i d e o r o u t s i d e the c o i l s .

We

s h a l l now c o n s i d er t h e e f f e c t s

of t h e c x t e r n a l e l e c t r i c a l c i r c u i t .

The e q u i v a l en t c i r c u i t

o f

a l i q u i d l e v e l p r ob e i s show n i n F i g .

4 .


27/122

21

ORNL-DWG

72-6710

F i g

P r o b e .

Vo DRIVING VOLTAGE

Ro

Cg

Z ,

SERIES RESISTANCE IN THE DRIVING CIRCUIT

S HUNT CA P A CI T A NCE OF THE DRIVING CIRCUIT

R6

M MUTUAL IMPEDANCE BETWEEN THE DRIVER AND PICK-UP COIL

Zpu IMPEDANCE

O F

THE PICK-UP COIL

R,

C,

D.C. RESISTANCE O F THE DRIVER COIL

I M P E DA NCE

OF

THE DRIVER

C OI L

D.C. RESISTANCE OF THE P ICK-UP COIL

S HUNT CA P A CI T A NCE

OF

TH E P ICK-UP CIRCUIT

R g A M P L I F I E R I N P U T I M PE D AN C E

I LOOP CURRENT

, 4 .

Simplified C i r c u i t D ia gr am f o r a n E dd y C u r r e n t L i q u i d L e v e l

W e c a n

w r i t e

t h e f o l l o w i n g s e t of e q u a t i o n s f o r t h e v o l t a g e d r o p s a r ou n d

e a c h

of

t h e

l o o p s

i n t h e c i r c u i t :


28/122

We c an use determinants and solve f or the currelit in the f i n a l loop,

I,,,

produced by an applied voltage V

.

c;’

c:

I ,

\J

0

We

shall solve

€or

the current,

I,,

multiply it

by

t i le

resistance

R

to

dptermine the input voltage to the amplifier, and then multiply by

t h e

amplifier gain G to determine the output voltage:

9


29/122

23

R e a r r a n g i n g terms so t h a t E q . (68) c l e a r l y r e ma in s f i n i t e when t h e

c a p a c i t a n c e go es t o z e r o , we f i n d t h a t

From E q .

( 6 9 )

we c an c a l c u l a t e t h e p h as e s h i f t b etw ee n t h e v o l t a g e

d r i v i n g t h e e d d y - c u r r e n t p r o b e an d t h e a m p. li fi ed v o l t a g e r e c e i v e d b y t h e

p ha se s h i f t d e t e c t o r . S in c e t h e d r i v e r c o i l and pi ck up c o i l a r e i n t h e

same

r e g i o n a nd

a r e

i d e n t i c a l ,

w e

h a v e

where

M

w i l l b e g i ve n by e i t h e r E q . (61) o r E q .

( 6 2 ) .

T o e v a l u a t e E q .

( 6 9 ) , t h e r e i s a compute r p rogram, ATTEN,which c a l c u l a t e s t h e magn i tude

an d p h a se o f t h e o u tp u t v o l t a g e f o r v a r i o u s v a l u e s o f t h e t e r m s i n

E q .

( 6 9 )

The c i r c u i t p a r a m et e r s , Ro, C 6 , R

and C a r e v a l u e s t h a t may b e

9 ’ 7

v a r i e d ( w i th i n c e r t a i n l i m i t s ) w i t h a n e x t e r n a l p l ug -i n a t t e n u a t o r .

Th e a t t e n u a t o r , when p r o p e r l y c h o s en , h a s t h e f o l l o w i n g e f f e c t s :

(1)

The

p h a s e

s h i f t d ue t o t e mp er at ur e d r i f t s c a us i ng

v a r i a t i o n s i n t h e dc r e s i s t a n c e v a l u es of

E:

and

R

c an b e e s s e n t i a l l y e l im i n a te d .

6

7

(2) The

L-C

n et wo rk i n t h e d r i v i n g and p ic ku p c i r c u i t s

i n F ig . 4 a n d t h e m u t u a l c o u p l i n g b e t w ee n t h e c i r -

c u i t s com bin e t o a c t as a b an d- pa ss f i l t e r t h a t

r e d u ce s t h e n o i s e i n t h e i n s t r u m e n ta t i o n .

3 ) A r e d u c t i o n i n s e n s i t i v i t y o c c u rs and t h e p h a se

s h i f t s d u e t o v a r i a t i o n s i n t h e o t h e r parameters

i n c r e a s e . H ow ev er , t h e s e e f f e c t s ca n b e made

n e g l i g i b l e i f t h e a t t e n u a t o r

i s

p r o p e r l y d e s i g n e d .

The a t t e n u a t o r d e s i g n w i l l b e c o ns id e re d i n g r e a t e r d e t a i l i n t h e n e xt

c h a p t e r .


30/122

24

3 . COMPUTER CALCULATIONS

F O R

A

PRO BE I N SI D E

BISMUTH

The t y pe o f l i q u i d l e v e l p r ob e an a ly z ed i n t h i s r e p o r t

i s

shown in

F i g . 5 . I t c o n s i s t s o f a l o n g b i f i l a r c o i l i n s i d e a molybdenum c o n-

t a i n e r . Th e d ra w in g

i s

s ym ni et ri c a b o u t t h e c o i l a x i s , s o o n ly h a l f

of

t h e p r o b e i s shown ( i n c r o s s s e c t i o n ) . 'The p r o b e

i s

p l a c e d i n s i d e a

m oly bd en um c a v i t y , a nd t h e l e v e l of t h e m o l te n b is m ut h i n

t h e

c a v i t y

c a n b e m e a s u r e d .

A

c u r r e n t f l ow i ng i n t h e d r i v i n g c o i l p ro du ce s a n

e l ec t ro m a gn e t i c f i e l d t h a t

i s

m o d i f i e d b y t h e p r e s e n c e of t h e c o n d u c t o r s .

The p ic ku p c o i l d e t e c t s t h i s f i e l d . I n t h e de s ig n

of

t h i s p r o b e , w e

used t h e f o l l o w i n g p r o c e d u r e . We f i r s t maximized t h e s e n s i t i v i t y of

t h e p r ob e t o c ha ng es i n l i q u i d l e v e l , t h e n m in im iz ed t h e e f f e c t s

of

t h e

u n d e s i r a b l e v a r i a b l e s s u c h a s t e m p e ra t u r e d r i f t , and f i n a l l y maximized

t h e s e n s i t i v i t y t o e r r o r r a t i o .

ORNL -DW G

71

9737

I

L i q u i d L e v e l

P r o b e

F i g . 5 .

L i q u i d

L e v e l P ro be I n s i d e C o n d u c t o rs .


31/122


32/122

26

w i t h a w a l l t h i c k n e s s o f a p p r o xi m a t el y 1' m i l s , t h e r e i s n o c h a n ge i n

p h as e a s t h e 1 iq ui .d l e v e l

i s

v a r i - e d . A s t h e w a l l t h i- c kn e s s

i s

i n c r e a s e d

f u r t h e r , t h e c u r v e s o f p h a se w i t h a n d wi .t h ou t b i s m u th b ecome q u i t e Ear

a p a r t i n d i c a t i n g t h a t t h e r e

wil l

b e a l a r g e ph as e s h i f t a s t h e l i q u i d

l e v e l

i s

v a r i e d . T he p h a s e d i f f e r e n c e a p p r o a c h e s a maximum a t a p p r o x i -

m a t e l y 4-(m i l s and t h e n d e c r e a s e s a g a i n t o z e r o a s t h e c u r ve s c ro s s a t

14? m i l s , appro ach an o t he r maxi-mui., s ep a ra t i -on , and t he n c ro s s ag a i n .

The b e h a v i o r

o f

t h e m ag n it ud e c u r v e s o f t h e v o l t a g e i s v e r y s i m i l a r ,

w i t 1 1

t h e c u rv es c r o s s in g and approach i -ng a maximum

a n d

t h e n c r o s s i n g

agai .n. The f i - r s t maximum, w hich i s n o t shown i n F i g .

6,

i s t h e l a r g e s t

an d o c c u r s a t z e r o w a l l t h i c k n e s s f o r b o t h t h e m ag n it ud e c h an ge a nd

t h e p has e s h i f t . S in ce t h i s i s i m p r a c t i c a l i n o u r d es i gn , we e l e c t e d

t o

c o n c e n t r a t e o n

t l ie

seco nd maximum. The v a l u e

o f

w a l l t h ic k n e s s a t

which t h i s maximum occ ur s i s a l s o a f u n c t io n

o i

f r e q ue n c y . I n F i g .

r[,

W E h a v e a v e r y s i m i l a r p l o t o f m a g n i tu d e a n d p ha s e o f t i le p i c k u p coi .1

v o l t a g e a s f u n c t i o n s o f t h e c o n t a i n e r w a l l t h i c k n e s s , wi.th an d w i t h ou t

b i sm u t h, a t a f r e q u e n c y of 22 kHz . F igure 6: w h i c h was r u n a t 10 kHz,

i s

v er y s i m i l a r t o F i g .

7,

e x ce p t t h e v a l u e s of w a l l t h i c k n e s s f o r

maximum magni tude and phase change a r e s ma l l e r at-

the

h i g h e r f r e q u e n c y .

A t

20 kHz th e wa l l t h i c kn es se s f o r maximum phase and magni tude change

o cc ur a t 2.7 m i l s and 80 m i l s , r e s p e c t i v e l y , c om pare d t o jk 7 m i l s and

110

m i l s , r e s p e c t i v e l y , a t

10

kHz.

I n g e n e r a l ,

w e

a r e a b l e t o p l o t optimum w a l l t h i c k n e s s

f o r

maximum

s e n s i t i v i t y a g a i n s t f re qu en cy ,

a s

s ho wn i n F i g .

fl

Ln F ig . we have

o p t i m i z e d t h e w a l l t h i c k n e s s f o r t h e iiiaximum p h a s e s h i f t , a nd t li e

amount o f phase s h i f t we ge t a t opt imum

i s

a l s o p lo t t e d a g a i n s t t h e

f r e q u e n c y . B e ca u se t h e c u r v e s a r e r ou nd ed , i t

i s

d i f f i c u l t

t o

e x a c t l y

d e t e rm i n e t h e o ptimum w a l l t h i c k n e s s f o r

a

g i v e n f r e q u e n c y . Th e

s h a p e

o f t h e c u r v e s v a r i e s w i th c o i l s i z e , s o w e made s i m i l a r p l o t s f o r u p t i -

uium w a l l t h i c k n e s s f o r maximum s e n s i t i v i t y t o b o t h m a g n it u d e a nd p h a s e

c ha ng es v s f re q u en c y u s i n g d i f f e r e n t s i z e c o i l s .

A

compos i t e o f t h e s e

p l o t s f o r p hase s h i f t

i s

show n i n F i g . 3. F rom F ig . 'j, we conc luded

t h a t a l i q u i d l e v e l pr ob e w i t h

a

w a l l t h i c k n e s s o f 30 m i l s would have

a d eq u a te s e n s i t i v i t y . Once t h e w a l l t h i c k n e s s h a s b e en f ix e d , t h e n

w e


33/122

PY-


34/122

N

x

I

c

1

(

U

l

4

H

S

3

S

V

'

H

d

f

l

X

k

8

S

N

Y

3

1

V

M


35/122

c a n detemt:i.ne the

o p t i n i i m f r e q u e n c y

f o r

maximum s e n s i t . i v i t y a n d t h e s e n s i -

t i .vf . ty at . t h a t f r eq u en c y f o r a ny s i z e c o i l . I n F i g . 10,

WE ?

h a v e

opt i .mize t l

f o r maximum p h a s e sl-iiEt, and w e have made s i ln i I a r

pl .o ts

f o r mal:i.murn atnp1.i-

t u d e c1-iange, From t h i s pLoL we

d e c i d e d t h a t

w e

would have a d e q u a t e sen.s:i -

t i v i t y i . f the

c o i l

form

01)

was

O.72,Y

i n .

a f requency

of 16 kHz, w e

h a v e 0.1.8 r a d i a n s o r a p p r o x i m a t e l y 10 phase

s h i - f t a s t:he 1 i q u i . d I.e-vel

i.s v d r i e d f r o m f u l l . ' to e m p t y . W l i l t . t h i s i s

not: t l ie most

s e n s i t

i - ~ e onf igura t : io[ j . ,

t h e sen:;-i.ti.vii:y i s riluch

nmre

t h a n

adequigte, and tile compromises

m a d e

t h u s f a r i n s u r e a

good

i nexpens i -ve

m e c h a n i c a l . d e s i g n . W s h a l l . now t u r n o u r

at

With t h i s s i z e coi.1. o p e r a t e d

a t

d r i f t s .

2:

7c

5

0

...

\

-

..

...

O W L - DWG 71.- 7 798

0 . 2 5

R I V L H

c r i i l

31CKUP Colt

L E N G T H O F C O I L - 4 3 i n

TEMPERATIJRF - 700"C

31

RFSiS1

V I l Y=15O$km

Mu R E S I S T I V I T Y - 2 O p C L L m

1AVlTE OD = 7 A

W IRE

D l A h l E T E R z r2 - r , r-45rri Is

W A L L

T H I C K N E S S = A

A = 3 0 m i i s

0.375 0.500 0.625 0.750 0.875 1 0 0 0

LAVITE OD

f i n

1

F i g . 10.

O p t i i r n u m F requency f o r Maximum Phase S h i f t and P h a s e S h i f t

a t Maximum

P l o t t e d

A g a i n s t C o i l Form 01).


36/122

I n t h e c a l c u l a t i o n s t o m in im iz e d r i f t s , w e i n c l u d e d t h e o u t e r m ol yb -

denum con t a ine r a s show n i n F i g .

5.

We assumed t h a t b o t h t h e c o i l s h e a t h

and t l ie o u t e r c o n ta i ne l ' w er e i n f i n i t e l y l o n g ; t h e c o i l was a c t u a l l y

13.625

i n . l o n g.

The m aj or c o n t r i b u t i o n t o d r i f t s i s t h e

w i d e

r a n g e

of

o p e r a t i n g

t e m p e r a t u r e of t h e p r o b e . Th e t e m p e r a t u r e c o n t r i b u t i o n s ca n b e b r o k e n

t h e f o l l o w in g :

'There i s a c ha ng e i n t h e a c f i e l d d u e t o c h an ge s i n t h e

r e s i s t i v i t y

of

t i i t . molybdenum and b i smu th . T h i s ca us es

a

c h an g e i n t h e s e l E - im p e da n c e o f b o t i i t h e d r i v e r a nd

p i c k u p c o i l s an d t h e m u tu a l c o u p l i n g b et w ee n t he m .

T here i s a ch an ge i n t h e d c r e s i s t a n c e o f b o th c o i l s .

T hermal expans ion

oC

t h e c o i l a nd c o n du cl o r s a l s o

c a u s e s a c h a n g e i n t h e i mp e da n ce s a n d m u t u a l c o u p l i n g .

\?e s h a l l c o n s i d e r e a ch of t h e s e f a c t o r s s e p a r a t e l y .

I n F i g .

11

we h a v e p l o t t e d t h e p h a s e o f t l ie v o l t a g e o f t h e p i- ck up

c o i l a g a i n s t fr eq u en cy f o r v a r i o u s t e m p e r a t u re s . We c a n s ee t h a t t h e s e

c ur ve s i n t e r s e c t

a t

a c e r t a i n f r e q u e n c y , w h i ch m ea ns t h a t t h e te m p e r a -

t u r e c o e f f i c i e n t of tlhe phase

i s

e s s e n t i a l l y z e r o air t h a t f r e q u e n c y .

T hese z e r o t e m p e ra t u r e c o e f f i c i e n t p o i n t s o c c u r a t a p p ro x i m a te l y 25

kHz f o r t h e c a s e i n wh ich t h e l i q u i d l e v e l p r ob e i s e mp ty a nd a t a b o u t

.jO kHz f o r t h e p r ob e b e i n g f u l l o f b i sm u t h. More e x a c t c a l c u l a t i o n s

w i t h a n e x t e rn a l c i r c u i t a t ta c h ed

t o

t h e p r ob e f o r an o p e r a t i n g f r e -

quency of

i ' )+.300kHz

a r e s hown i n t h e t a b l e b e l o w.

T a b l e 1 . C a l c u l a t e d P h as e S h i f t V a l u es ( i n D e g r e e s )

w i t 1 1 Bismuth and

A i r

a t S e v e r a l T e m p e ra t ur e s

M a t e r i a l


37/122

W N L - I IWG 71- 9076R

U

u

4

c

I,

1

F i g .

11.

Phase o f t h e Pickup

Coi.1 Vol t

a g e P l o t t e d

A g a i n s t

Frequency

f o r Different T e m p e r a t u r e s .

I n

t h e t e m p e r a t u r e

range

of p a r t i c u l a r i n t e r e s t ,

~ O O

o

f;:;oO(:, we have

c h o s e n our p a r a m e t e r s s o t h a t

we have

e x a c t : l y t h e same p o s i . t : i v e s l o p e

of phase

with te rr ipe ra t :u re

(4-0. $ " / ~ 0 " c )

n

b i s n ~ r t : t ia s t h e

negative

s l o p e

(-'3.06"/50")

i n a i r . T h ese

two e n d

p o i n t s r e p r e s e n t

t h e

worst t,ernpera

-.

t u r e c o e f f i c i e n t ,

and

i t

i s e x pe c te d t o

d e c r e a s e t o z e r o

f o r

t h e c a s e 01

t h e

p r o b e

b e i n g

h a l f

f u l l .

T h i s

worst

c a se

o f t h e c a l c ul . a te t l - t e m p e r a t u r e

c o e f f i c i e n t i s e q u i va l - e nt t o

I-0.0015

n . / " C e r r o r i n t h e l i q u i d

level .

measurement .

S i - m i l a r

c a l c u l a t i o n s

were made

f o r t h e ma g n i tu d e of t h e p i c k u p

c o i l

v o l t a g e , b u t t h e r e

i s no

f r e q u en c y a t w h ic h t h e

magnii-udch

r e ma in s

constant

w i t h t e n i p e r at u r c c h a n ge s .


38/122

we

n e x t c o n s i de r e d t h e v a r i a t i o n i n p ha s e d ue to c h a ng e s i n d c

r e s i s -

t a n ce of t h e c o i l . A s we d i s c us s e d e a r l i e r ,

a

v a l u e o f c a p a c i t a n c e and

r e s i s t a n c e i n t h e d r i v i n g a nd d e t e c t i n g c i r c u i t s c an b e c ho se n ti ia t

w i l l

g i v e e s s e n t i a l l y n o ch an ge i n p ha se a s t h e d c r e s i s t a n c e o f t h e c o i l i s

v a r i e d , a n d will

also

a c t as a f i l t e r t o r ed u c e t h e s ys te ii i n o i s e . How ev er,

when the R - C ne twork

i s

a d j u s t ed t o g i v e e x a c t ly z e r o te mp e ra tu re d r i f t

w i t h t h e b is m ut h p r e s e n t , i t w i l l n o t g i.v e e x a c t l y z e r o d r i f t w i t h t h e

b i s mu t h a b s e n t . T h e r e f o r e , t h e n e tw o rk

i s

a dj us ce d t o g i v e sm al l d r i f t s

of o p p o s i t e s i g n s i n t h e two c a s e s . Bu t t h e more

we

f i . l t e r t o r p du ce

s y s t e m

n o is c , t h e g r e a t e r t h e s e

s m a l l

d r i f t s become. I n a d d i t i o n , t h e

p ha se s h i f t s d ue t o v a r i a t i o n s i n t h e c a p a c i t a n c e and r e s i - s t a n c e v a l u e s

i n t i le c i - r cu i t i n c r e a s e a s

w e

r e d uc e t h e s y st e m n o i s e . T h e r ef o r e ,

w e

m us t co mp ro mi se b e tw e en t h e s y s t e m n o i s e an d d r i f t s . The f o l l o w i n g t a b l e

summar izes t he d r i f t s be tween 60i3'and 6;d) 'c , u s i n g

-*j6.:j

f o r t h e s e r i e s

r e s i s t a n c e of t h e d r i v e r c i r c u i t and t l ie s h u n t r e s i s t a n c e o f t h e p ic k u p

c i r c u i t and 5e.30 pF f o r t h e s h u n t c a p a c i t a n c e .

T a b l e 2. Summary o f P h a se S h i f t s D ue t o V a r i a t i o n s

i n Va r io u s C i r c u i t P a ra m e te r s

P a r a m e t e r V a r i e d

Va r ia t i o n

i n

P aramete r

($)

P h a s e S h i f t

( d e g r e e s

)

D r i v e r C o i l R e s i s t a n c e

P ic k up C o i l R e s i s t a n c e

D r i v e r Sh u nt C a p a c i t a n c e

P i c k u p S hu n t C a p a c i t a n c e

D r i v e r C i r c ui

t

S e r i e s

R e s i s t a n c e

P i c k u p C i r c u i t S h u n t

R e s i s t a n c e

Opera t i ng F requency

1 >

1s

1

1

1

1

1

0.004

0.001

The p h a s e s h i f t s g i v e n

i n

T a b l e 2 a r e t h e a b s o l u t e maximum v a l u e s

t h a t o c c u r a ny wh er e i n t h e t e m p e r a t u r e r a n g e w i t h o r w i t h o u t b i sm u t h .


39/122

53

T h e f i l i a l c o r i t - r i b u t i o n t o d i r f t s t h a t

w e

con s id e re d was t he rm31

e x p a n s i o n o f th e sys tem. In cl ud in g t li er rual. expans i.o i i e f f e c t s , a tempc?ra-

t u r e

v a r i . a t i o n f r o m 600"

t o

6 5 0 ~

aused

a

0.05" d i f f e r e n c e i n t h e p l ~ a s p

change b e t w e e n t h e two t e m p e r a t u r e s , which

can

b e compensated C o r w i t h

a s m a l l f r e q u e n c y c h a n g e .

Additional information o n t h e d e s i g n uf t h i s

probe

is g i v e n i n

I

othe l -

r e p o r t s .

T hr ee d i f f e r e n t

sets of

measurements

h a v e

be en performed on

[:he

l i - q u i d

l eve l .

p r o b e .

T ie f i r s t - s e t o f iiieasuz-cI~metlTs

w a s

p e rf o rm e d a t a

f r e q u e n c y

of

10

kHz

a nd a t room t em per a tu re . T he p r imary purpose o f

t hese measurement s

was

t o

t e s t

tl-ie

linear

.t:y of

t h e

p l - ~ b ~ ,

1- t c1

tile

secc2iitl

p u r p o s e W ~ ~ S

o check t h e a c c u r a c y

o f

t h e c a l c u l a t i o n s

T h e

p r o b e c o n s i s t e d

of

t h e c o i l e n cl os ed i n

a

moI,ybdenuin sheath, I,ut

w i t h -

oii't

any o u te r iiioly1)dennum c o n t a i n e r . The bi.siuu1:11 was

rcpl.aced by 12

r i n g s 01' Snc,onel ,

each machined

t o

1

_ _ 0 . 0 l i j

i n . t h i c k .

The

phase

s h i f t

was

r Q c o r d e d

t:o w L t 1 i i n O.OI." a s t h e

r i n g s

were

added, siim.lI.ating

a n

i n cr e as i -n g l i q u i d l e v e l .

A ] .east-squarc:s

f i t

was

marl


40/122

The measurements

were

made a t a f r eque ncy o f

27.'7

kHz l o r v a r i o u s

l e v e l s o f b i sm u th a nd a t v a r i o u s

t e m p e r a t u r e s b e t w e e n 550°C and 700"C.

T h e r e s u 1 . t ~a r e p l o t t e d i n F i g . 12. T h e p h a s e s h i f t i s a l i n e a r f u n c t i o n

o f l i q u i d l e v e l o v e r a r a ng e f ro m z e r o t o a bo u t 13 i n . The s l o p e o f t h e

c u rv e v a r i e s s l i g h t l y w i t h t em p e ra t ur e , with the minimum temperature

c o e f f i c i - e n t o c c u r r i n g b e t we e n 600" C and 650"

.

The t em p e r at u r e c o e f f i c i e n t o f

t h e

l e v e l r e a d in g was

0.009

n . / " C

I O

f o r t h e chamber e il ip ty and 0.00211 i n . : C f o r t h e c h am be r f u l l o f b i s mu t h.

T he h i g h e r t e m p e r a t u r e c o e f f i c i e n t w i t h t h e c ha mb er e m pt y w as p r o b a b ly

du e t o t h e f e r r o m a gn e t ic o u t e r c a s i n g .

A s e r i e s

of

more a c c u r a t p c a l c u l a t i o n s , c o n s id e r i 'i g m ore d i f f e r e n t

p a r a m e t e r s , sh owed t h a t m o re a c c u r a t e m e a s ur e m en t s c o u l d b e made a t

2)-

kHz f o r t h e t e m p e r a t u r e r a n g e o f 600"~o 650"~. he l e v e l i n t h e p r ob e

14

35

36 37

38

39 40 41

42 43 1 4

45

46

47

PH ASE

SH IFT I deq l

F i g . 12. P ha se S h i f t P l o t t e d A ga i n s t L i q u i d L e v el f o r V a r i o u s

T empera tu res Measured a t a F requency o f 27.7 kHz.


41/122

.55

was s e t a t

t h e maximum, de ter min ed by th e amount of b i sm uth

.i.n

t h e s y s t e m

a t t h e t i m e . The m an om ete r r e a d i n g i n d i c a t e d

10

i n . , b u t l a t e r m ea su re -

m en ts i n d i c a t e t h a t t h e a c t u a l l e v e l was a b ou t 13.5 i n . T ab le 3 shows

b o t h c a l c u l a t e d a nd m e as u re d v a l u e s o f m a g n i t u d e a nd p h a s e . The a c t u a l

c a l c u l . n t i o n s w er e made w i t h t h e p r o be e i t h e r f u l l o r em pt y of b i s m u th ,

and t h e 1 0 - i n . c a l c u l a t i o n s r e p r e s e n t a 1 .i ne ar i n t e r p o l a t i o n b et we e n z e r o

and

13.625

i n . o f b i s m u t h .

The r e s i s t o r v a l u e s a r e t h o s e o f b o t h t h e d r i v e r s e r i e s and t h e p i c ku p

s h u n t r e s i s t a n c e s , an d t h e s e c a n b e v a r i e d t o nc li i- ev e a " f i n e tu n in g " i n

t h e t e m p e r a t u r e c o e f f i c i e n t o f t h e p r o b e . T he p h a s e d if fe rc -. nc :c i

i s

t h e

p h a s e a t t h e l o w er t er np pr at :u re s u b t r a c t e d f ro m t h e p h a se a t t h e u p p e r

t e m p e r a t u r e .

w ou ld h e e q u a l b u t o p p o s i t e i n s i g n w i t h t h e p r o be i u l .1 a nd e mp ty , a s

shown f o r

t he

c a l c u l a t e d v a l u e s u s i - n g

36.3 ~2.

p ha se d i f f e r e n c e was -1.0.1.2 r a t h e r t h an O.06 c a l c u la t e d f o r t h e 1 0 - i n .

l e v e l . By i n c r e a s i n g t h e Val-ue s o f t h e r e s i s t o r s i n t h e a t t e n u a t o r , we

a r e a b l e t o d e c r ea s e bo t h

t h e

c a l c u l a t e d a nd m e as ur ed p h a s e d i f f e r e n , c e s .

The b e s t v a l u e of r e s i s t o r s t o g i v e minimum d r i f t wou1.d p rob abl y b e

s l i g h t l y l a r g e r t ha n

36.3

R. T h e l e v e l was a1 .so v a r i e d u s i n g t h e 75-a

r e s i s t o r s i n . t h e a t t e n u a t o r . The t e m p er a t ur e c o e f f i c i e n t of t h e piinse

v a r i e d f r o m

+O.OO&? / C (-i-O.U352

i n , / °C) w i . t h

t h e

chamber empty t o O.O($ /

" C

w i t h t h e c ham ber f u l l .

Ey

d e c re a s in g t h e v a l u e of t h e r e s i s t o r s , i t

s h ou ld b e p o s s i b l e t o o b t a i n

a

t em p e ra t ur e c o e f f i c i e n t

of

-t-O.OOl.6

i.n./"C

wi t h t h e chamber empty , - 0 . 0016

i n . / " C

w i t h t h e c h am be r f u l l and z e r o

w i t h t h e c ha mb er h a l f f u l

1.

I n t h e i d e a l c a se , t h e p h a se s h i f k s be tw ee n 600°C and

650°C

However , t h e measured

T h e c a l c u l a t e d s l o p e a t 600°C was

0 . 8 ) + j o / i n .

compared

t o

a measured

v a l u e o f 0 . 8 6 6 " / i n .

m a g ne t ic p e r m e a b i l i t y o f t h e o u t e r c o n t a i n e r i s on ly approx ima te ly known.

P a r t o f t h i s e r r o r may b e du e t o t h e f a c t t h a t t h e

W i t l i

t h e i n f o r m a t i

on

g a in c cl i n t h e s e measurements , we

wi l l

b e a b l e

t o c a l i b r a t e t h e p r ob e u se d i n t h e

l o o p

w i t h a n o u t e r c o n t a i n e r

of

molyb-

denum, a s

shown

i n F i g .

3 .

The l o we r c a l i b r a t i o n p o i n t f o r t h e p ro b e

c a n b e o b t a i n e d b e f o r e b i s mu t h

i s

e v e r ad de d t o t h e s y s te m . To g e t t h e

u p pe r c a l i b r a t i o n p o i n t , t h e bi sm u th l e v e l m u s t b e r a i s e d o v e r t h c t o p

o f t h e p ro b e , an d t h e

phase

r e a d i n g r e c o r d e d .

I f

0.110'

i s s u b t r a c t e d


42/122

C

pi

m

U

.

A

r

4

\

D

r

t

c

W

tl

n


43/122

3‘I

f ro m t h i s p h a se re a d in g , t h i s g i v e s the v a l u e of t he pha . se

w h e n

t h e l i q c i d

l e v e l

i s

15 i n . The phase wi l l . be a l i n e a r f u n c t io n

of

t h e b i smuth l e v e l

be tween 0 i n . and

13

i n . W r i n g t h e n i n e ~ ii on .t hso f t e s t i n g o f t h e

probe,

t h e z e r o p o i n t a p p e a r e d t o move, but t h e s l o p e was e s s e n t i a l 1 ~ y o n: ;t am t

( w i t h i n

1 +$).

T h e r e f o r e , b y as s um i ng a c o n s t a n t s l o p e 2 i . t

i s

possLbIe

t o r e c a l i b r a t e t h e p ro b e b y r a i s i n g the b i smuth L eve l o v e r t l i e probe .

T h e r e

w a s

a c o n s i d e r a b l e a m o u n t

of

c o r r o s i o n i . n t h e

l.oc)j) wli.:i.ch

probi~b1.y

accoun ted f o r t l ie s h i f t i n t he z e r o l e v e l p o i n t .

A

p a s s i v e KLC phase

c a l . i . b r a t o r was

c o n s t r u c t e d t o e l i m in a t e i n s t r u -

m e n . t

v a r i a t i o n s

It

g i v e s two measurements of

t h e

i.nst:rurrtt:nt gain. ( s ~ . o p e

and

two

:i.ndependei-it: nei3 sure Inen t s of t h e z e r o p o i n t .

T h e

1.orig-tc1.rm s t a -

b i I - i t y

o f

t h e c a I ih r a t :o r i s a p p r o x i m a t e l y 0 . 0 1 ” .

A d d i t i o n al i n f o r ma t i on on t h e l i n e a r i t y

t e s t s , ‘

t h e i n i t i a l . mcasure-

6

ments

a r e

g iv e n i n

o t h e r

r e p o r t s .

in. t l ie t e s t loop , and t l ie

: f i n a l

measurements‘ i n t h e t e s t l oo p

>

.

SUMMARY ANT) CONCX1ISIONS

The problem

f o r

a c . oi l e i t h e r e n c i r c l. i n g o r enc-.>sed by

a n

a r b i t r a r y

number

of

c o n d u c t o r s h a s bee11 s o l v e d . A c c u r a t e a nd versa : i Je computer

programs have

be t ~ i

w r i t t e n w h ic h n u m e r i c a l l y

eval .uate

t h e s e

s o l . u t i o n s

G o o d a g r e e m e n t h a s

been

o b t a i n e d b e tw e e n t h e c a l c u l a t i . o n s a nd e x p e i r i . m c ? r i t s .

These programs h a v e b ee n. a p p l i e d to d e s i g n very a c c u r a t e l i q u i d l e v e l

p r o b e s even

t hough

w e

a r e l o o k i n g t h r o u g h a

good

conductor (molybdenum)

arid m e as ur in g t h e l e v e l

o f

a p o or c o n d u c t o r ( b i s m u t h ) .

‘Id. E . McNeese e t l . ,-

E n g i n e e r in . g D ev el op me nt S t u d i e s f o r Mol.ten-

S a l t B r ee d er Reactor P r o c e s s in g No. 13, ORNL-TM--3776 ( i n p r e p a r a t i o n >.

I,. E .

McNeesc It

ala,

E nginee r ing Deve lopment Studies f o r

Molten-

S a l t

Breeder R

O I W L - T M - ~ ~

( i n

p r e p a r a t i o n ) .

‘L.

E . McNeese c t

l . ,

E ng inec r ing Deve lopment S t ud i e s F o r r l _ l t c n -

S a l t B r e e d e r R e a ct o r P r o c e s s i n g

N o .

16,

O I W L - T M - I I O ? ~ ( i n

p r e p a r a t i o T -


44/122

6.

ACKNOWLEDGMZNTS

The

a u t h o r s wish

to

acknowledge

t h e

a s s i s t a n c e o f C . C . L u

in

t h e

p r e j

irninary programming, t

r a s s i s t a n r - o f

I 0 . Weeren,

I , .

D.

Chitwood,

a n d

0 . E . Conner i n p e rf o rm i ng

t h e

n e a s u r c m e n t s ,

Id.

E .

Deeds

slid

L . E .

McNeese

f o r e d i

Ling t h e i e p o r t , J a n i c e

Shannon f o r preparing the

f i n a l

manusc r ip t and Jerry R o t l i and L. E . McNeese for providing ovcrall

giii dailce


45/122

APPENDIX A


46/122

APPENDIX A

ComKuL-er.. P ro g ra m s f o r 3 Liq uid J ,evel Probe

I n s i d e

...

C o a x i a l C o n d u c t o r s

I. lNTKODUCTION

T hi s compute r p rogram

i s

u s e d t o ca l c ul a f rv t h e n o r m a l i z e d c o i l i mp ed -

a n ce of b o t h t h e d r i v e r and p i c ku p c o i l s o f an e d d y - c u r re n t p ro be , s u r -

r o un d ed by m u l t i p l e c o a x i a l c y l i n d r i c a l c o n d u c t o r s . T he p ro gr am mi ng

c o n s i s t s of a f u n d a m e n t a l s u b r o u t i n e namcd

I . N M L ,

which i s c a l l e d i n t o

e x e c u t i o n b y a m ai n D R I V E R program. INNMLTL b r a nc h e s t o e i g h t o t h e r new

s u b r o u t i n e s d u r i n g e x e c u t i o n .

Z N N N U T ,

i t s c l

f

i s

a d ap te d f ro m a n e a r l i e r

p rogram i n t h e BASIC l ang uage .

The FORTRAN-IV computer languagc i s u s e d , w i t h

REALYE

a r i t h m e t i c , o n

t h e IBF1/36G c o m p u t e r s .

l a t i o n s w i t h s u f f i c i e n t a c c ur ac y i n t h e c a s e of a " t h i n" r e g i on among

t h e c o n d uc t in g m e di a . I n c o n n e c ti o n w i t h tlij-s, yoine s p e c i a l s u b r o u t i n e s

w er e a l s o d ev el op ed f o r t h e c a l c u l a t j o n o f t h e n e c e s sa r y m od if ie d Re sse l

f u n c t i o n s , w i t h ac cu ra c y of a t l e a s t 15 s i g n i E ic a n t d ec im al d i g i t s f o r

r e a l a rg u m en t s, a nd lu s i g n i f i c a n t d i g i t s f o r c om plex a rg ii me nt s.

New work w a s d on e t o

c a r r y

o u t t h e ne ed ed c a l c u -

An o u t l i f i e of t h e p ro gr am f o l l o w s , w i t h d i s c u s s j o n an d p ro gr am l i s t i n g

o f e ac h s e p n r a t e r o u t i n e . The e i g h t s u b r o u t i n e s r e q u i r r d by I N W L a r e

named

X I I N T , X J J N T ,

GAMCLL, GCALC; MODBES,

CMDKES, COMKB,

and C M I . They

a r e ne ed ed f o r i n t e g r a t i o n s of x3 ( x ) and XI (x ) , f o r c a l c u l a t i o n oE t h e

gamma f a c to r , and f o r i i ie i r l o d i f i e d B e s s e l f u n c t i o n s .

1 1

L a s t a n i n t e r a c t i v e p r og ra m w i l l b e g i v e n f o r g e n e r a t i o n of t h r d a t a

b l o c k F i l e , which I n l i s t b e s u b m i t t e d

w i t h

t h e p r og ra m f o r e x e c u t i o n

011

t h e I B M , / ~ ~ O o m p u t e r .

T l i i s

program, named T E L J N C ,

i s

s t o r e d o n t h e

PDP-10

d is k, and may be used f rom

t h e

t e l e t y p e t o prep '3i-e t h e d a t a i n t h e n e c e s -

s a r y f or ma t f o r t h e D R I V E K . T he d a t a c o u l d , o f c o u r s e , b e s e t up f o r

e x e c u t i o n o f t h e p ro g ra m b y a n y o l -h e r c o n v e n i e n t m e t h o d.


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4 .

11.

DESCRIPTION OF

INNIYIJL

A .

D R I V E R

Program

The D R I V E R

main

p ro gr am r e a d s t h e d a t a f o r

t h e

c a s e s

t o

b e e x e c u t e d

b y

the INNfWL

s u b r o u t i r i e . A f t e r r e c e i vi n g t h e d a t a a n d

making

p r e l i m i n a r y

c a l c u l a t i o n s i f n e c e s sa r y , t h e D R I V E R t h e n b r a n c h ( i s t o INNMLJT,. A f t e r t h e

c a l c u l a t i o n

of

n o r m a l i z e d

c o i l

impedance

i s

comple t ed by INWRTL and

i t s

o t h e r s u b r o u t i ne s , t h e n

Lhe

DRIVER program prepares

a

summary print

o u t ,

showing data and

r e s u l t s .

Data must be g i v e n

t o t h e

DRIVER program f o r t h e

c o i l .

di .mensions, fo r

the a i r n o r m iI . i za t i o ii f a c t o r , an d

f o r

i n f o r m a t i o n a b o u t each r:oriductor

o u t s i d e t:h e c o i l . Th e

inner

r a d i u s , t h e r e I . n t l v e p e r m e a b i l it y ,

2

t h e v a l u e

of

t h e q u a n t i t y (~~p . n i? r e n ee de d f o r

each

c o n d u c t o r .

m

and

b 3 E L '

2

m

he D R I V E R program

i s

w r i t t e n

t o

r e c e i v e t h e q u a n t i t y ,

q J , u F

, i n any

one

of

t h r e e d : i . f f e r e n t

w a y s . F i r s t ,

t l i e v a l u e

of the

q u a n t i t y may b e

g iv e n d i r e c t l y f o r e a c h c o n d u c t o r . S ec on d, t h e f a c t o r

o

may b e g i v en f o r

each c o n d u c t o r . In

t h a t :

case, t he p e r m e a b i l i t y ,

p i i s

IT. x 1.0 x

li .

R A

Then t h e o p e r a t i n g f r e qu e n c y a l s o i s g i v e n i n o r d e r t o o b t a i n 1d

for

t h e

c o i l , and F i n

meters , i s

e i t h e r g i ve n o r c a 1. c ul o te d f o r t h e C L ~ ~ J . ,

- 7

m'

T hi rd , t h e r e s i s t i v i t y ,

p (pil-cm)

may be g iv en a s d a t a

f m ach

c o n -

8

d u c t o r . The r e l a t i o n b e tw ee n

p

and

i~ i s

g i v e n b y :

q u a n t i t y w y a r

i s

c a l c u l a t e d

a s O.. - ,Ogi9 j ?

x f r e q .

x

r

x

ilREL/p. The

c o n s t a n t

i s a

p r o d u c t of c o n v er s io n f a c t o r s

--

( 2 r / - ) ( O . O 2 5 4 ) hi x I . O ) ( l O ) .

T h e re f o re i n this t h i r d case, t h e known d a t a must i n c l u d e t h e d r i v i n g

f r e q u e n c y ( H e r t z ) , a n d

F

( i n . ) , a s

w e l l a s

r e l a t i v e p e r m e a b i l i t y . The

d e t a i l s o f s e t t i n g up t h e d a t a f o r s u b mi s s io n

with

t he compute r p rogram

wi1.L

b e gi v e n i n t h e

l a s t -

s e c t i o n .

o CL 10

p .

Then t h e

2

2

m

F o l l o w i n g

i s

a l i s t o f t l ie program DRIVER.


48/122

A

0

y

i

x

L

03

r

I

4

v

D

C

L

M

P

0

U

F P

-

v

(

T

x

3

m

-

A

-

C

r

I

-

D


49/122


50/122

44

9

10

1 I

5

1a Q

185

1 9 0

1 1

2


51/122

45

11

l a ) INNMUL

T he fo rmula which

i s

c a l c u l a t e d by t h e INN?IUL s u b r o u t i n e

i s

t h e

n o r m a l i z e d c o i l i mp ed a nc e ,

of b o th t h e d r i v e r

and

p ic ku p c o i l s i n a b i f i l a r c o i l s ur r ou nd ed by

m u l t i p l e c o a x i a l c o n d u ct o rs .

T h e q u a n t i t y

i s

given by Eq. (50) .

I t

i s c a l c u l a t e d by a b r an c h t o t h e

X I I N T

f u n c t i o n

s u b p r o g r a m . S i m i l a r l y , t h e q u a n t i t y ,

i s

g i v e n by E q .

r a t i o ,

U12/U22,

o r gamma, i s c a l c u l a t e d b y

a

b r a n c h f r o m INNMUL t o t h e

GAMCAL

s u b r o u t i n e .

(56) and i s c a l c u l a t e d b y

the

X J J N T subprogram.

The


52/122

The X I I N T ,

XJJNT,

and GAMCAL s u b r o u t i n e s w i l l

b e

d i s c us s e d s e p a r a t e l y .

The i n t e g r a n d i n t h e n u m er a to r o f t h e i mp ed an ce f o r m u la i n INNMUL

-(XI,

c o n t a i n s a t:errn w i t h t h e f a c t o r ( X -C E

v a l u e s o f

CiL

l e s s

t h a n

0.5,

t h i s f a c t o r i s c a lc u l a t e d by t h e r a t i o n a l

a p p r o x i m a t i o n ,

-

l ) , where I,

=

k - A l . For

2

8

T h i s

i s

d e r i v e d a s a c o n t i n u e d f r a c t i o n a pp ro xi ma nt f ro m t h e P a d e ' t a b l e

a s s o c i a t e d w i t h t h e p ower

s e r i e s ,

-a.

I f CXL

i s

l a r g e r t h an 20, e i s d ro pp e d f ro m t h e c a l c u l a t i o n of

01+

r

--

1 ) . F o r a?, be tween

0.5

and

20,

e

i s

e v a l u a t e d b y L h e e x p o n e n t i a l

f u n c t i o n s ub pr og ra m a v a i l a b l e

i n t h e 1 ~ ~ / 3 6 3ORTRAN l i b r a r y .

-OX A

T h e q u a n t i t y (1

-

cos(ol?,)

w hi ch a p p e a r s i n o n e

t e r m

o f L h e i n t e g r a n d

2 L

2

s o b t a i n e d by t h e h a l f - a n g l e f o rm u la f ro m si n (- ) .

The o v e ra l l i n f i n i t e i n t e g r a l

i s

a p p r o x i m a t e d i n INNMUL

for

a n u p p e r

l i m j t o f 16. T h e a v e r a g e o f t h e i n t e g r a n d

i s

c a l c u l a t e d f o r

100

e q u a l l y -

s p a c e d p o i n t s f r o m

0

t o 1 . F o r e a c h o f t h e r e m a i n i n g 15 i n t e r v a l s oL

u n i t l e n g t h , t h e a v er a ge of t h e i n te g r a n d f o r 20 e q u a l l y s p ac e d p o i n t s

i s u s ed . T h i s h a s be en f ou nd t o g i v e a s u f f i c i e n t l y good a p p r o xi m at i on

t o t h e i n t e g r a l and

i s

named

17

i j I 6 i n t he p rograu i .

A t

t h e same

t i m e

d ur in g t h i s c a l c u l a t i o n i n

'LNNMTJL,

i n t e g r a t i o n

i s

p e rf o rm e d i n d e p e n d e n t l y o n t h e s e c o n d t e r m o f t h e i n t eg r a n d t o o b t a i n

a n e s t i m a t e d a i r v a l u e , named

19.

The a i r va l ue c o n t r i b u t e s t o t h e r e a l

p a r t o f t h e i n t e g r a l , 17, and

i s

b y f a r t h e s l o w e s t p a r t t o co nv er ge .

We

'Als ton

S .

Househo lde r ,

A

G l o s s a r y f o r N u m e r ic a l A n a l ys i s 2

ORNL-2704,

pp 73

-7'4


53/122

c n l c u l a t e t h e air v a l u e i n s c p a r a t e program and givc. t h ~a l u e

a s

p a r t

o f

t h e

d a t a to t h i s p r o g r a m , a s Ag.

The

a i r v a l ue d e p e n d s o t l ly on t h e

o n

t h e c o i l

dimensions

a nd not on

the

c o n d u c t o r , 2 n d

i s

( T h i s term a p p e a r s b o t h i n t h e n u m e r a t o r

and

the denorn ina to r . )

T h e r e f o r e , we c a n o b t a i n

t h e

c o m p l e t e l y c o n v e r g e d

real part

oi t h e i n t e -

g r a l

by

a d d i n g t h e d i f f e r e n c e b et we en A3 and 19. The completely c o n -

v e r g e d r e a l p a r t i s I'( t

(Arj-Iy).

The

normalized impedance is

A listing

of IplNMuL f o l l o w s .


54/122

48

c

C

c

C

C

C

C

C

c

c

C

c

C

C

c

C

C

1

t i l =

I N N E X C U L L

i < A D I U > / MLAN

C D I L t I A D I U S

r


55/122

49

c

G l = O . D O

I;Z =H%

P R I N T 3

3

F ~ K M A T C ~ X ,X ' P

1 4 A 9

'AIR V A L U E U r 5 X ~H E A L

P A t < T ' > G X , ' I M A G

P A R T ' / / / )

C

c b E G I N LMBP DN A L P H A > l l H I C H IS T H E V A K I A L ~ L E NAMED X

c

125 J 7 = 0 * O

J S = O . D O

J ' 9 = O o D O

X = t i * i i 1 * 0 * 5DO

130 W9 = X * L

G I l = )< *X

O = O f * Q 1

Q 6 = 2 * W 1

c

C

C

135

1 4 0

C

c

c

C

c

c

145

KL

C

GAMMA )= ii6 J

I

M

C G A M M A

1=F7


56/122


57/122

11.

B e l

and

2 )

X J J N T

and X I I N T

T he se t w o f u n c t i o n s u bp ro gr am s e v a l u a t e t h e e x p r e s s ions,

and

2

x I 1 ( x ) dx ,

xr-l

r e s p e c t i v e l y . These a r e equal. t o t h e va1.i.ies g i v e n f o r

J ( r , > ,

) i n E q .

( $ 6 )

and

I ( r < , r )

i n

E q . (50).

The method

of

i n t e g r a t i o n in l ~ o t i

ases ,

for- a n

u p p e r l i m i t n o t g r e a t e r t h a n

5, i s

t h e summati -on o f a p o w e r

s e r i e s

expre:;-

s i o n

u n t i l t h e

l.nst

i n c l u d e d t e r m i s I.ess t h a n t imes t h e c u r r e n t

s u m .

I f

t h e

upper 1.iinf-t

of the i n t e g r a l

is

g r e a t . e r t h a n 7, a s y m p t o t i . c

formu.--

l a s a r e u s e d

i n

b o t h s u b r o u t i n e s . I n g en .e ra l a p p l i c a t i . o n , we assume

i .a i

t h e s e s u br o u t i n e s t h a t

b o t h

1.imits a r e n o n r i e g a t i v e and

tilie

l o w e r

l i m i t i s

less t h a n O K e q u a l t o t h e u p pe r I .i mi t

on

t h e i n t e g r a l .

r._ 1

2 1

9

An e x p l a n a t i o n f o l l o w s

f o r

the summation method ,

w h i c l t i s uscil i n t h e

c a se

t h a t t h e u pp e r

l i m i t i s l e ss

t h a n o r

e q u a l

t o

5.

For t h e p u r p o s c o f e x p l a i n i n g t h e g e n e r a l mcthod

o f i n t e g r a t i o n , l e t

f x ) b e a ny f u n c t i o n

o f

a

r e a l

v a r i a b l e w it h t h e

Taylor

s e r i e s

expanPion,

Term wise i n t e g r a t i o n y i e l d s

( 771

9W.

A . Si m pson

et GP j omputer Yrogrnins f o r Some Eddy-Curren t

Problems

- 1970, ORNL-TM-3335 p . 259

( J u n e lg'(1

) .


58/122

52

I n t r o d u c e a new sequence,

2 2

2

b-

--

a

-

..

,

and

s o

o n . I n r e l a t i.onnd i t

i .s

s e en t h a t

q

0

=I,

q

1

= = b- . a ,

q 2

t o t h i s s eq ue nc e, o u r i n t e g r a l may be w r i t t e n a s

W

T h i s i s t h e s e r i e s t o b e a p p r o x i m a t e d .

R e c ur r en c e r e l a t i o n s may b e d c r i v e d t o g e t t h e t er m s o f t h e q - se q u en c e

i n t h e f o l l o w i n g w ay. Use t h e a 1g eb ra i. c i d e n t i t y

(‘19)

n

n

-1 n - 1 a b ( b n - 2

-

a n - 2 )

bn -

a

= (cl-tb)(b

- - - a

)

-

11

T h i s g i v e s , w i t h n q

T he n d i v i s i o n by n and s i m p 1 i f i . c a t i o n g i v e s

::: (bn - a ), n qn :: ( a + b ) ( n - l ) q n-1 - a b ( n - 2 )

q

11-2-

n

S i m i l ar l y , one c a n s t a r t w i t h t h e i d e n t i t y

n

2 2 n - 2

n - 2

2 2 11-4 n - 4 )

b n - a = : ( a + b ) ( b - a ) - a b ( b - a

t o o b t a i n a s e co nd u s e f u l r e c u r r e n ce r e l a t i o n ,

(80

( E 2 )

I n t h e f u n ct i on s X J J N T a nd XIINT, t h e s e c on d r e c u r r e n c e r e l a t i o n i s used

f o r n g r e a t e r t h an

3 ,

of xJl(x) and

xT1(x) .

s i nc e a l t e r n a t e

t e rms

a r c z e r o f o r t h e power s e r i e s


59/122

53

I n

o r d er t o a p p l y

t h i s

i n t e g r a t i o n p r oc e du r e,

l ook

f i r s t a t t h e

power se r i e s" f o r

~ ] . ( x ) :

Then

m 2r

+

1

r

( r - i - l )

(86

k )

2k-l

k/2-1 )

(k/2)

(k

)

k

X

X I . x ) =

k=2

( e v e n v a l u e s

o n l y )

I n t h e p r ec e di n g e x p r e s s i o n , (k

)

d en om i na to r, t o c o i n c i d e w i t h t h e

f

k=

0

h a s b e e n m u l t i p l i e d i n t o n u m e r at o r a nd

T a y l o r s e r i e s

k

k k

x .

Then

i t i s

e a s i l y seen t h a t

t h e

c o e f f i c i e n t s f o r an even number

k

a r e :

= 15/8, w i t h

c6

= 1, c4 = y 2 ,

2

k

T he se e ve n- i nd ex ed c o e f f i c i e n t s a r e r e l a t e d by

c

o t h e r

c o e f f i c i e n t s ,

co ,

e l ,

c 7 ,

c5,

. .

a r e z e r o .

c i e n t s h o l d f o r

Jck = ( k + l ) / k .

A l l

k-t,

The same

c o e f f i -

J

'OH. J e f f r e y s a nd 3. S.

Jeffreys,

Methods

of

M a t h e m a t i c a l Phys ic s ,

p .

574,

C am b ri dg e U n i v e r s i t y P r e s s , 1950.


60/122

54

x J l ( x )

x

J

r2lrl

w i t h

t h e

e x c e pt i on t h a t t h e

terms

a r e a l t e r n a t i n g i n s i g n .

The summation m et hod , which j s d e s c r i b e d a b o v e , i s c o m b i n e d w i t h t h e

9

a s y m p t o t i c f or mu la s’ , f o r t h e c a s e t h a t t h e l o we r

l i m i t

i ; l e s s t h a n o r

e qu a l t o 5 , b u t t h e u p p er 1 i i i i i . t

i s

g r e a t p r t h a n

5.

L is t i ngs o f XI INT and X J J N T f o l l o w ,


61/122

c.

i;

f

9 0

1 C 5

c.

c


62/122

56


63/122

.

\

z

L

\

L

C

i

z


64/122

1

5r'

1 r 5

3 C 5

3

0

3 1 1

::

)r

3 1

5


65/122

59

11. B ( 3 )

GAMCAL

uL2/

he GAMCAL s u b r o u t i n e i s used by INNMLJL

t o

compute gamma r a t i o ,

UZ2, w hic h a p p e ar s i n t h e f i r s t t e rm

of

t h e i n t e g ra n d of i n t e r e s t i n

INNMUL

.

I t

i s n e c es s a ry i n t h e c a l c u l a t i o n s o f

GAMCAL t o

h a v e g i v e n t he quan-

~

F 2

named

%, f o r

e a c h c o n d u c t o r .

I n t h i s p ro d uc t i?

i s

t h e

Wlik

k m ’

m

i t y ,

mean

c o i l r a d i u s i n m e t e r s . The a n g u l a r f r e qu e n c y

of

t h e d r i v i n g c u r r e n t

i s

g iven by

w .

The o t h e r v a r i a b l e s I arid

7

a r e t h e p e r m e a b i l i t y a nd

c o n d u c t i v i t y , r e s p e c t i v e l y , of t h e c o n d u c t o r . F o r e a c h c o n d u c t o r ,

k k

where a i s t h e v a r i a b l e of i n t e g r a t i o n

i n

INNMLJL. Then,

The i n n e r r a d i u s and r e l a t i v e p e r m e a b i l i t y m u s t a l s o

be

known f o r

e a c h c o nd u ct o r f o r t h e c a l c u l a t i o n s t o o b ta in gamma. Cau t ion m u s t b e

e x e r c is e d i n t h e c a s e o f a l a r g e p e r m ea b i l i ty t h a t t h e c a l c u l a t i o n s do

n o t l e a d t o c om pu te r o v e r f l o w o r u n d e r f l o w.

The u s e of t h e z e r o - t h and f i r s t o r d e r m o d i li e d B e s s e l f u n c t i o n s

h a s b ee n s ee n i n t h e c h a p t e r d e a l in g w i th t h e o r e t i c a l a n a l y s i s .

s u b r o u t i n e s h a v e b e en w r i t t e n

t o

s e c u r e

more

s i g n i f i c a n t f i g u r e s th a n

p r e v i o u sl y a v a i l a b l e i n t h e c a l c u l a t i o n s o f

GAMCAL.

T h e s e s u b r o u t i n e s ,

names

MODBES,

CMDBES,

COMKB,

and CM I , w i l l b e d e s c r i be d i n d i v i d u a l l y .

New

A d i s t i n c t i o n

i s

made be tween a norma l condu c to r r e g io n and a “ t h in ”

c o n d u c t o r r e g i o n .

A

r e g i o n

i s

c l a s s i f i e d a s t h i n i .f


66/122

t h e o u t e r b o u nd a ry

of

r e g i o n k ,

k-1’

h e e l e m e n t s of t h e T - m a t . r i x a t

r

a r e

T

11 k , k - 1 ) = z K 0 ( z ) I 1

(a

- l r k - l 1

(89)

T 1 2 ( k , k - l ) :::

z K ~ ( z ) K

(a - l r k - l

1

(90

and

T22 k , k - 1 ) = z l O ( z ) K1(a - l r k - l )

(92

(pk/bLk- l ) ak - l rk - l K() (%-l rk- l ) ‘1 (’)

where /. = CI r

(93

k k - l *

For cross ing a normal . ,

o r

“ t h i c k “

reg ion ,

t h e s e m a t r i x e l em e n t s a r e

computed

b y GAMCAL

a c c o rd i n g t o t h c ab ov e r e l a t i o n s h i p s .

H owev er, i f t h e r e g i o n i s “ t h i n ” , a s p e c i a l p r o c e d u r e i s u s e d .

I n

t h a t ca s c, n o t e t h a t a t

t h e

i n n e r

boundai-y, r of

t h e r e g i o n k, t h e

m a t r i x e l e me n ts a r e

k’


67/122

61

To

c r v s s r e g i o n k, t h e

T-rna t r ix

i s

the p r o d u c t T

In t h e

f o l l o w i n g

e q u a t i o n s we

w i l 1

d r o p

t h e ( k i 1 , k - 1 )

f o r convenier icc .

- k + l , k :k,k-l Tki1,k-I.

T h e

matrix

eI.ernents

of the

p r o d u c t

a r c ?

t h e n

T12 = T

1

(k - I - l , k )

T

1 2( k , k - I )

- -

T l f ( k + l , k )

T

2 k , k - 1 ) ,

(100)

TZ1 =;

T,.,

( k + l , k ) T .

(k,k-1) i-

Tpp(k-i-1.,k)

T

( k , k - 1 )

,

(7.01.

)

1

1.1 . ..

21

a n d

(k-i-1,k) T12(k,k-1) 1 T, , , (k t l ,k)

T , > , , ( k , k - l )

(102)

21.

L . L- ..

T,,,,

=

T

:..d

This matrix

rnuI . t i p1 . i ca t i on y i e l d s t h e

following

complcx

e l e m e i ~ t s

of t h e

p r o d u c t m a t r i x :


68/122

n

9

N

W

s

N

W

d

H

Y

/

-

u

-

g

4

,

H

W

-

n

-

d

W

I

N

W

w

d

m

N

H

v

d

-

i

d

W

I

N

v

W

W

d

H

.

N

W

b

i

N

W

s

G

N

U

W

3

r

7

4

v

N

W

r

H

5

N

W

s

o

N

W

-

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