EFFICIENCIES OF AXIALLY ORDERED HGMS FILTERS IN A MULTIWIRE MODEL

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HAL Id: jpa-00223635

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Submitted on 1 Jan 1984

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EFFICIENCIES OF AXIALLY ORDERED HGMS FILTERS IN A MULTIWIRE MODEL

H. Greiner, H. Hoffmann

To cite this version:

H. Greiner, H. Hoffmann. EFFICIENCIES OF AXIALLY ORDERED HGMS FILTERS IN A MULTIWIRE MODEL. Journal de Physique Colloques, 1984, 45 (C1), pp.C1-789-C1-792.

�10.1051/jphyscol:19841161�. �jpa-00223635�

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84 Regensburg, F . R. G.

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Nou,s d o n n o n s u n e f o r m u l e q u i p e r m e t d e c a l c u l e r d f u n e m a n i a r e e f f i c a c e 1s champ e t l a f o r c e m a g n 6 t i q u e d a n s un r 6 s e a u ( i n ) f i n i d e f i l s f e r r o m a g n d t i q u e s p a r a l l e l e s s a t u r 6 s d a n s u n champ p e r p e n d i c u l a i r e . Comme a p p l i c a t i o n n o u s c a l c u l o n s 1 ' 8 f f i c a c i t 8 d t u n f i l t r e HGMS o r d o n n d d u t y p e a x i a l e t m o n t r o n s q u t e l l e d e p e n d d 6 c i s i v e m e n t d e l a g e o m d t r i e du f i l t r e .

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We g i v e a n u m e r i c a l l y e f f i c i e n t f o r m u l a f o r t h e mag- n e t i c f i e l d a n d f o r c e a r i s i n g i n a ( i n ) f i n i t e l a t t i c e o f f e r r o - m a g n e t i c w i r e s s a t u r a t e d i n a p e r p e n d i c u l a r l y a p p l i e d f i e l d . We t h e n c a l c u l a t e e f f i c i e n c i e s o f a x i a l l y o r d e r e d HGMS f i l t e r s a n d show t h a t t h e y d e p e n d c r u c i a l l y o n t h e f i l t e r g e o m e t r y .

F o r s e v e r a l y e a r s t h e r e h a s b e e n some i n t e r e s t i n t h e u s e o f p e r i o d - c a l l y o r d e r e d l a t t i c e s c o n s i s t i n g o f p a r a l l e l a r r a n g e d t h i n f e r r o m a g - n e t i c w i r e s a s f i l t e r m a t r i c e s i n H i g h s r a d i e n t c a g n e t i c S e p a r a t i o n d e v i c e s / I / . D e p e n d i n g o n t h e g e o m e t r y e m p l o y e d , o n e d i s t T n g u i s h e s b e t w e e n t h e a x i a l , l o n g i t u d i n a l a n d t r a n s v e r s e c o n f i g u r a t i o n / I / . To d e v e l o p a t h e o r y o f s u c h f i l t e r s o n e h a s t o c a l c u l a t e t h e m a g n e t i c f i e l d a n d f o r c e a r i s i n g i n s u c h a w i r e m a t r i x i n a p e r p e n d i c u l a r l y a p p l i e d f i e l d s t r o n g e n o u g h t o s a t u r a t e t h e m a t r i x . T h e s o l u t i o n s s o f a r p r o p o s e d e x p r e s s e d t h e m a g n e t i c f i e l d b y l a t t i c e s u m s w i t h p o o r c o n v e r g e n c e p r o p e r t i e s w h i c h c o u l d n o t u r ~ a m b i g u o u s l y b e i n t e r p r e t e d b e c a u s e o f t h e i r c o n d i t i o n a l c o n v e r g e n c e / 2 , 3 , 4 / .

I n o u r c o n t r i b u t i o n we show t h a t t h e f i e l d s a r e g i v e n b y t h e W e i e r - s t r a s s p - f u n c t i o n a n d i t s f i r s t d e r i v a t i v e b e l o n g i n g t o t h e l a t t i c e d e f i n i n g t h e m a t r i x . A s t h e s e f u n c t i o n s c a n b e e v a l u a t e d by r a p i d l y c o n v e r g i n g t h e t a s e r i e s /5/, we w e r e a b l e t o p e r f o r m t h e e x t e n s i v e n u m e r i c a l c a l c u l a t i o n s n e c e s s a r y t o c o m p u t e c a p t u r e e f f i c i e n c i e s o f a x i a l l y o r d e r e d HGMS f i l t e r s . Our a n a l y s i s s h o w s t h a t t h e c a p t u r e e f f i c i e n c y f o r a w e a k l y m a g n e t i c p a r t i c l e i n t h e f i l t e r d e p e n d s s t r o n g l y o n t h e f i l t e r g e o m e t r y g i v e n by t h e l a t t i c e t y p e a n d t h e f i l t e r p a c k i n g f r a c t i o n . F o r f u r t h e r d e t a i l s we r e f e r t o t w o f o r t h - c o m i n g p u b l i c a t i o n s / ? O,? I / .

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Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19841161

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JOURNAL DE PHYSIQUE

F i g . 1 L a t t i c e w i t h p r i m i t i v e v e c t o r s F i g . 2 P r a c t i c a l R e a l - 2W1 a n d 2W3 d e f i n e d by t h e w i r e s i n a i z a t i o n o f a n o r d e r e d p l a n e p e r p e n d i c u l a r t o t h e m , ^HGMS f i l t e r

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fi a n d C i s a n a r b i t r a r y c o m p l e x number. Complex n u m h e r s are i n t e r - p r e t e d a s t w o d i m e n s i o n a l v e c t o r s a n d v i c e v e r s a . E x p r e s s i o n (1) p o s s - e s s e s t h e p e r i o d s o f t h e l a t t i c e a n d s a t i s f i e s t h e r e l e v a n t m a g n e t o - s t a t i c e q u a t i o n s a n d b o u n d a r y c o n d i t i o n s a s shown i n / l o / ,

T h e m a g n e t i c f i e l d i n a f i n i t e l a t t i c e o f e.9. r e c t a n g u l a r o r e l l i - p t i c a l s h a p e c e n t e r e d a t t h e o r i g i n o f c o o r d i n a t e s i s g i v e n t o a v e r y g o o d d e g r e e o f a p p r o x i m a t i o n by e x p r e s s i o n (1) i f t h e v a l u e

i s a t t r i b u t e d t o t h e c o n s t a n t C. T h i s c o n s t a n t c a n b e r e a d i l y e v a l - u a t e d f o r a g i v e n f i l t e r a n d r e p r e s e n t s d e m a g n e t i z i n g e f f e c t s d u e t o t h e F i n i t e f i l t e r g e o m e t r y .

To c a l c u l a t e t h e m a g n e t i c f o r c e o n a w e a k l y m a g n e t i c p a r t i c l e o f v o l u m e V a n d r e l a t i v e s u s c e p t i b i l i t y ; r ( i n t h e f i l t e r we h a v e t o s u b - s t i t u t e e x p r e s s i o n (1) i n t h e u s u a l f o r m u l a f o r t h e m a g n e t i c f o r c e . A f t e r some a l g e b r a o n e o b t a i n s f o r t h e m a g n e t i c f o r c e

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F i l t e r e f f i c i e n c i e s i n a M u l t i w i r e Model

H a v i n g o b t a i n e d t h e m a g n e t i c f o r c e o n a s m a l l w e a k l y m a g n e t i c p a r t i c l e a n d a s s u m i n g t h e ' v a l i d i t y o f S t o k e s t l a w f o r t h e h y d r o d y n a m i c f o r c e , i t s e q u a t i o n s o f m o t i o n a r e r e a d i l y f o r m u l a t e d , A s u s u a l t h e i n e r t i a l a n d g r a v i t a t i o n a l f o r c e $ a r e n e g l e c t e d . A s i n t h e a x i a l c o n f i g u r a t i o n t h e f l u i d s t r e a m s p a r a l l e l t o t h e w i r e s i t i s e a s y t o c a l c u l a t e t h e l a m i n a r v e l d c i t y p r o f i l e f r o m t h e N a v i e r S t o k e s e q u a t i o n by a s s u m i n g p e r i o d i c b o u n d a r y c o n d i t i o n s , T h e f o r m o f t h e e q u a t i o n o f m o t i o n re- v e a l s t h a t t h e p a r t i c l e t r a j e c t o r i e s a n d h e n c e t h e f i l t e r e f f i c i e n c y

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a b o u t t h e same i n t h e l a m i n a r a n d p o t e n t i a l f l o w m o d e l f o r e q u a l f l o w r a t e s . F i g . 3 s h o w s t h e i n f l u e n c e o f t h e l a t t i c e t y p e o n t h e f i l t e r e f f i c i e n c y . I t a p p e a r s t h a t a q u a d r a t i c l a t t i c e w i t h d i a g o n a l l y a p p - l i e d f i e l d y i e l d s t h e o p t i m a l c o n f i g u r a t i o n . F i g . 5 d i s p l a y s t h e i n - f l u e n c e o f t h e f i l t e r p a c k i n g f r a c t i o n o n f i l t e r e f f i c i e n c y . F i g . 6 a n d 7 c o m p a r e t h e p r e d i c t i o n s o f t h e o n e - w i r e - m o d e l s a n d o u r m u l t i - w i r e m o d e l w i t h m e a s u r e d f i l t e r e f f i c i e n c i e s .

F i g . 5 E f f i c i e n c y v s . p a c k i n g F i g . 6 E f f i c i e n c y o f a q u a d r a t i c f r a c t i o n f o r f i x e d v a l u e s of t * l a t t i c e o f p a c k i n g f r a c t i o n 8.7%

Q u a d r a t i c l a t t i c e t y p e , f i e l d x x x e x p . V a l u e s r e f ./6/

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Packing fraction 1 % 1

Fig.7 Saturation of efficiency with packing fraction for t = 200

exp. values from /7/

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theoret. values from /7/

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theoret. values this work Conclusion

Our multiwire model of axially ordered HGMS filters clearly shows the influence of the filter geometry on the filter efficiency. These efPects cannot be described by one-wire-models. The predictions of our model are in good agreement with measured values.

References

/I/ R.R.8irss and M.R.Parker in Progress in Filtratign and /2/ I.Eisenstein, IEEE Trans.flagn.

/3/ I.Eisenstein, IEEE /4/ W.H.Simons,

/5/ A.Erdelyi, editor Higher Transcendental Functions

",

^vo1.2,

chapter XIII, 1953

/6/ S.Uchiyama, S.Kondo, M.Takayasu, IEEE Trans.flagn. 1 2 (1 976) 895 /7/ R.R.Birss, R.Gerber, M.R.Parker IEEE Trans.Nagn.12 1978 892 /8/ R.R.Birss, R.Gerber, M.R.Parker IEEE Trans.Nagn.14 1978 389 /9/ R.R.Birss, R.Gerber, M.R.Parker IEEE Trans.Magn.14 1978 1165

publication

I I

/lo/

H.Greiner, H.HofFmann, 3.Magnet.Mag.Mat.

,

accepted for /11/ H.Greiner, H.Hoffmann, J.Magnet.flag.Mat.

,

accepted for

publication

*work supported by Oeutsche Forschungsgerneinschaft