Particle Size Distribution II - ACS Publications - American Chemical

Chapter 23

Operational Protocols for Efficient Characterization of Arrays of Deposited Fineparticles by Robotic Image Analysis Systems

Downloaded by TUFTS UNIV on June 2, 2018 | https://pubs.acs.org Publication Date: September 24, 1991 | doi: 10.1021/bk-1991-0472.ch023

Brian H. Kaye Physics Department, Laurentian University, Sudbury, Ontario P3E 2C6, Canada Efficient characterization of the size distribution of a sample of fineparticles as viewed through a microscope requires the estimate of the population frequency in each size group. To achieve comparable efficiency in each section of the population an automated system (cybernetic instrument) should initially search large areas of a field of view to establish the range of sizes to be characterized and then establish an appropriate search algorithm for each size group so that the confidence to be placed in the data can be predicted. Procedures for automating this so called "Stratified count strategy" are discussed. Primary Count Loss Characterization by

and Secondary Count Imape A n a l y s i s

Gain

in Fineparticle

T h e p r o b l e m o f c o i n c i d e n c e effects has been e x t e n s i v e l y s t u d i e d i n s t r e a m methods o f fineparticle characterization. In stream methods o f characterization coincidence effects o c c u r i n one d i m e n s i o n a l data space that i s i n t h e o n e - d i m e n s i o n a l t r a i n o f fineparticles i n the f l u i d stream present i n the m e a s u r e m e n t z o n e o f the i n s t r u m e n t . T o u n d e r s t a n d the g e n e r a t i o n o f c o i n c i d e n c e e f f e c t s i n stream m e t h o d s c o n s i d e r t h e i d e a l i z e d s k e t c h o f a n i n t e r r o g a t i o n z o n e o f stream c o u n t e r s u c h as t h e C o u l t e r C o u n t e r o r the E l z o n e c o u n t e r as s h o w n i n F i g u r e 1 ( 1 . 2 ) . T h e actual p h y s i c a l effects o f m u l t i p l e o c c u p a n c y o f t h e s e n s i t i v e z o n e v a r i e s f r o m i n s t r u m e n t to i n s t r u m e n t b u t i n g e n e r a l m u l t i p l e o c c u p a n c y r e s u l t s i n t h e u n d e r c o u n t i n g o f a s m a l l e r s i z e , t h i s e r r o r i s c a l l e d p r i m a r y c o u n t loss a n d the f a l s e r e g i s t e r i n g o f the p r e s e n c e o f a l a r g e p s e u d o fineparticle created by the s i g n a l generated b y m u l t i p l e o c c u p a n c y o f the z o n e . T h i s latter effect i s k n o w n as S e c o n d a r y C o u n t g a i n . T h e t w o effects are k n o w n as coincidence errors. E a r l y i n the d a y s o f the d e v e l o p m e n t o f stream counters s u c h as the C o u l t e r C o u n t e r i t w a s a p p r e c i a t e d that t h e o n l y safe way to a v o i d coincidence errors i n a fineparticle characterization study w a s to c o n t i n u o u s l y d i l u t e the c o n c e n t r a t i o n o f t h e s u s p e n s i o n b e i n g u s e d i n the s t r e a m c o u n t e r u n t i l t h e m e a s u r e d fineparticle size distribution f u n c t i o n w a s i n d e p e n d e n t o f t h e f u r t h e r d i l u t i o n o f the s u s p e n s i o n . 0097-6156/91/0472-O354$06.00/0 © 1991 American Chemical Society

Provder; Particle Size Distribution II ACS Symposium Series; American Chemical Society: Washington, DC, 1991.

23. K A Y E

Protocols for Efficient Characterization of Arrays

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In the c h a r a c t e r i z a t i o n o f f i n e p a r t i c l e systems b y i n s p e c t i o n o f deposited fineparticles on a filter, or other flat surface, coincidence effects arise b y the c h a n c e p o s i t i o n i n g o f d e p o s i t e d f i n e p a r t i c l e s c l o s e e n o u g h to e a c h o t h e r to p o s e a n i n t e r p r e t i v e p r o b l e m as to w h e t h e r a g i v e n p r o f i l e c o n s t i t u t e s a s i n g l e f i n e p a r t i c l e o r i f the p r o f i l e s h o u l d be treated i n the l o g i c o f the c h a r a c t e r i z a t i o n s t u d y as t w o separate f i n e p a r t i c l e s . Consider f o r e x a m p l e , the s t y l i z e d f i e l d s o f v i e w f o r a c o a l dust s a m p l e and a p o l y u r e t h a n e f o a m s a m p l e o f r e s p i r a b l e dusts d e s c r i b e d b y W a l k e n h o r s t Q ) and b y L a s k i n et a l (4) s h o w n i n F i g u r e 2. A l t h o u g h i m a g e a n a l y s i s o f deposited f i n e p a r t i c l e s has been c o n s i d e r e d to be the u l t i m a t e reference m e t h o d f o r m a n y fineparticle studies it is s o m e w h a t s u r p r i s i n g that there h a v e been v i r t u a l l y n o s t u d i e s o f the effect o f c o i n c i d e n c e e f f e c t s o n the a c c u r a c y o f the f i n e p a r t i c l e c h a r a c t e r i z a t i o n study i n t w o - d i m e n s i o n a l space. P i o n e e r s i n the studies o f the s i z e d i s t r i b u t i o n o f r e s p i r a b l e dust w e r e aware o f the p r o b l e m a n d r e c o m m e n d e d c o r r e c t i o n f a c t o r s f o r s u r f a c e c o v e r e d e s t i m a t e s i f the b u i l d up o f dust f r o m a l o n g term s a m p l e r was s u c h that a large f r a c t i o n o f the surface o f the f i l t e r b e c a m e c o v e r e d w i t h dust ( £ ) . W h e n c a r r y i n g out c h a r a c t e r i z a t i o n studies by s t u d y i n g d e p o s i t e d f i n e p a r t i c l e s the p i o n e e r w o r k e r s r e c o m m e n d e d , i n any attempt to k e e p c o i n c i d e n c e e r r o r s to an a c c e p t a b l e m i n i m u m , that a n a l y s t s k e e p the percentage o f the f i l t e r p a p e r c o v e r e d by dust to b e l o w 5 % area c o v e r a g e . T h e r e are t w o p r o b l e m s w i t h t h i s r e c o m m e n d a t i o n . First of a l l , most a n a l y s t s h a v e v e r y l i t t l e e x p e r i e n c e e n a b l i n g t h e m to j u d g e w h a t c o n s t i t u t e s 5 % c o v e r a g e (see d i s c u s s i o n o f the p r o b l e m i n the n e x t s e c t i o n o f t h i s c o m m u n i c a t i o n ) and s e c o n d l y , as w i l l be s h o w n i n the s e c o n d part o f t h i s c o m m u n i c a t i o n c o i n c i d e n c e e r r o r s at 5 % c o v e r a g e c a n be a s i g n i f i c a n t feature o f the c h a r a c t e r i z a t i o n s t u d y . The elimination of spatial c o i n c i d e n c e effects i s o f v i t a l i m p o r t a n c e w h e n a s s e s s i n g the e x p o s u r e of i n d u s t r i a l w o r k e r s to r e s p i r a b l e dust h a z a r d s . F o r e x a m p l e , i f the f i n e p a r t i c l e s noted b y A , B , and C i n F i g u r e 2(a) are treated as separate e n t i t i e s , t h e n the r e s p i r a b l e dust l e v e l c o u l d be m u c h h i g h e r t h a n i f t h e y are c o n s i d e r e d to be part o f the l a r g e r u n i t s adjacent to t h e m i n the field o f view. P e r h a p s one o f the reasons f o r the s c a r c i t y o f studies o f c o i n c i d e n c e effects i n i m a g e a n a l y s i s studies stems f r o m the fact that operators h a v e been o v e r l y c o n f i d e n t o f t h e i r a b i l i t y to take d e c i s i o n s w i t h respect to the i n t e g r i t y o f i n d i v i d u a l p r o f i l e s a n d h a v e r e l i e d o n e l e c t r o n i c l o g i c to separate w h a t they c o n s i d e r to be adjacent p r o f i l e s r a t h e r t h a n i n t e g r a l entities d e p o s i t e d as a w h o l e o n the f i l t e r paper. T h u s , they h a v e d e v e l o p e d e r o s i o n - d i l a t i o n l o g i c i n a u t o m a t e d i m a g e a n a l y s i s to separate w h a t are c o n s i d e r e d c o n t i g u o u s p r o f i l e s e v e n i f the j u s t i f i c a t i o n f o r s u c h p r o c e s s i n g o f the i m a g e is m i n i m a l ( £ ) . E v e n i f one c a n rely o n e r o s i o n l o g i c to separate the c o n t i g u o u s p r o f i l e s i n a f i e l d o f v i e w t h i s places c o n s i d e r a b l e e x t r a b u r d e n o n , and h e n c e i n c r e a s e s the e x p e n s e of, the l o g i c a v a i l a b l e i n the p r o c e s s i n g o f d e p o s i t e d profiles. F r o m a c o s t effectiveness p o i n t o f v i e w it w o u l d appear d e s i r a b l e to a r r a n g e the e x p e r i m e n t a l strategies so that there are no contiguous p r o f i l e s i n the f i e l d o f v i e w . T h a t is to s i m p l i f y the l o g i c p r o c e s s d e m a n d e d o f the i m a g e a n a l y s i s s y s t e m . A basic p r o b l e m a r i s i n g i n the study o f respirable dusts deposited o n the f i l t e r is a n e c o n o m i c one. A s w i l l be s h o w n i n this c o m m u n i c a t i o n i f one attempts to keep the area d e n s i t y o f dust deposited o n a f i l t e r b e l o w the l e v e l at w h i c h s i g n i f i c a n t c o i n c i d e n c e effects o c c u r then there i s a n e e d to use a sequence o f m a n y filters d u r i n g a w o r k i n g p e r i o d . T h i s is u s u a l l y



356

P A R T I C L E S I Z E D I S T R I B U T I O N II

technically and f i n a n c i a l l y unacceptable with currently available equipment. T h e r e f o r e , the a n a l y s t i s often f o r c e d to m a k e s u b j e c t i v e d e c i s i o n s w i t h r e g a r d to w h a t c o n s t i t u t e s an i n d i v i d u a l p r o f i l e w h e n l o o k i n g i n t o the m i c r o s c o p e at r e l a t i v e l y dense f i e l d s o f v i e w . A s a c o n s e q u e n c e m a n y s t u d i e s o n r e s p i r a b l e dust c h a r a c t e r i s t i c s are o f doubtful value. T h e s i t u a t i o n w i t h respect to the i n t e r p r e t a t i o n a n d assessment o f d a t a a n d r e s p i r a b l e dusts r e p o r t e d i n the s c i e n t i f i c l i t e r a t u r e is f u r t h e r c o m p l i c a t e d b y the fact that s o m e t i m e s w o r k e r s i n the f i e l d a c t u a l l y r e p o r t the s i z e d i s t r i b u t i o n o f v i s i b l e s u b u n i t s present i n the dust as d i s t i n c t f r o m the s i z e d i s t r i b u t i o n o f the o b s e r v a b l e u n i t s s i n c e they f e e l that the p r i m a r y u n i t constitutes the h e a l t h h a z a r d . C o n s i d e r for example the o u t l i n e s o f l e a d f u m e generated f r o m an e l e c t r i c f u r n a c e s t u d i e d i n a recent p u b l i c a t i o n b y T o h n o a n d T a k a h a s h i ( 1 ) . T h e s e authors report a l o g i c f o r s e p a r a t i n g the p r o f i l e s o f F i g u r e 3(a) i n t o the c o n s t i t u e n t u n i t s d i s p l a y e d i n F i g u r e 3(b). T h i s p r o b a b l y g r o s s l y o v e r estimates the h e a l t h h a z a r d o f the dust. A g a i n c o n s i d e r the a l u m i n a p r o f i l e s s h o w n i n F i g u r e 4 the authors o f t h i s s t u d y report that they treated v i s i b l e s u b u n i t s as h a v i n g b e e n f o r m e d by c h a n c e d u r i n g the d e p o s i t i o n p r o c e s s o f the capture o f the a e r o s o l b y the fibres o f the f i l t e r (&)• In t h i s latter case, the p r o b l e m i s not a s i m p l e one o f s p a t i a l r a n d o m c h a n c e c o i n c i d e n c e effects; the p r o b l e m i s c o m p o u n d e d by the fact that surface forces f o r s u c h v e r y f i n e dust e n c o u r a g e the b u i l d i n g o f c a p t u r e trees w h i c h d e l i b e r a t e l y i n t e r f e r e w i t h s i n g l e f i n e p a r t i c l e d e p o s i t i o n d y n a m i c s (°J. B e c a u s e o f the d i f f i c u l t i e s a s s o c i a t e d w i t h c o i n c i d e n c e effects b o t h b y s i m p l e r a n d o m d e p o s i t i o n a n d b y e n h a n c e d g r o w t h o f p s e u d o a g g l o m e r a t e s d u r i n g the d e p o s i t i o n p r o c e s s it b e c o m e s e s s e n t i a l w h e n a s s e s s i n g the true l e v e l s o f r e s p i r a b l e dust to h a v e a p h y s i c a l u n d e r s t a n d i n g o f the f r e q u e n c y w i t h w h i c h s i m p l e c o i n c i d e n c e effects o c c u r and h o w the g r o w t h o f c a p t u r e trees d i s t o r t fineparticle characterization studies i n t w o - d i m e n s i o n a l space. It m a y w e l l be that m a n y r e s p i r a b l e dust studies h a v e b o t h e i t h e r u n d e r a n d o v e r e s t i m a t e d the health h a z a r d s o f a s p e c i f i c dust due to the a r b i t r a r y w a y i n w h i c h apparent a g g l o m e r a t e s h a v e been treated b y the l o g i c o f the characterization procedure a n d the a n t i c i p a t i o n o f the o p e r a t o r c a r r y i n g out the s t u d y . I n t h i s c o m m u n i c a t i o n , attention w i l l be f o c u s s e d o n s p a t i a l d e p o s i t i o n c o i n c i d e n c e effects. T h e p r o b l e m o f the d i s t o r t i o n o f the s i z e d i s t r i b u t i o n o f v e r y fine r e s p i r a b l e dust b y c a p t u r e trees g r o w t h w i l l be the subject o f a future c o m m u n i c a t i o n . T h e s i m u l a t i o n studies presented i n t h i s c o m m u n i c a t i o n i n d i c a t e that s p a t i a l c o i n c i d e n c e e f f e c t s i n dust c h a r a c t e r i z a t i o n s t u d i e s c a n o n l y be c o n s i d e r e d to be n e g l i g i b l e at m u c h l o w e r filter coverage than previously anticipated or used i n industrial hygiene practice. I n the f i n a l p o r t i o n o f t h i s c o m m u n i c a t i o n the i m p l i c a t i o n s o f the studies presented here f o r the future d e s i g n o f i m a g e a n a l y s i s s y s t e m s f o r s t u d y i n g h e a l t h h a z a r d dust and o t h e r p o w d e r s are e x p l o r e d and future research o u t l i n e d . Monte

Carlo

Studies

of

Chance

Clustering in Deposited

Fineparticle Arrays

T o g a i n a p h y s i c a l u n d e r s t a n d i n g o f the i m p o r t a n c e o f c h a n c e c l u s t e r i n g i n a d e p o s i t e d array o f m o n o s i z e d fineparticles i n t w o - d i m e n s i o n a l space one c a n use M o n t e C a r l o r o u t i n e s i n w h i c h a r a n d o m n u m b e r table is c o n v e r t e d i n t o a s i m u l a t e d field o f v i e w o f f i n e p a r t i c l e s . M o n t e C a r l o is a g e n e r a l n a m e u s e d b y m a t h e m a t i c i a n s to d e s c r i b e the m a t h e m a t i c a l m o d e l l i n g o f p r o c e d u r e s w h i c h are subject to r a n d o m v a r i a t i o n s s u c h as the success (or l a c k o f it) o f a g a m b l e r p l a y i n g roulette at M o n t e C a r l o .


23. K A Y E

357



Primary count loss « 3 of volume V Secondary count gain«1 of volume 3V F i g u r e 1: W h e n m o r e than 1 f i n e p a r t i c l e o c c u p i e s the s e n s i t i v e r e g i o n o f the i n t e r r o g a t i o n z o n e o f a stream c o u n t e r c o i n c i d e n c e effects r e d u c e the a c c u r a c y o f the f i n e p a r t i c l e c h a r a c t e r i z a t i o n study.

(a) F i g u r e 2:

(b)

D e c i d i n g w h i c h items i n a field o f v i e w o f d e p o s i t e d

fineparticle

c o n s t i t u t e separate f u n c t i o n a l entities is a c r i t i c a l d e c i s i o n i n studies o f t h e levels o f r e s p i r a b l e dust i n i n d u s t r i a l areas, (a) A s a m p l e o f c o a l dust ( R e p r o d u c e d w i t h permission

from

ref. 3.

Copyright C. C. Thomas

1972.)

(b) A

sample

of

p o l y u r e t h a n e f o a m dust. ( R e p r o d u c e d w i t h p e r m i s s i o n f r o m ref. 4. C o p y r i g h t C . C . T h o m a s 1972.)

CO

0 0.5 u,m (a)

(b)

F i g u r e 3: I n s o m e o c c u p a t i o n a l h e a l t h studies observed into "obvious?"

constituent units.

(Reproduced

fineparticles

a r e resolved

w i t h p e r m i s s i o n f r o m ref. 7.

C o p y r i g h t H o s o k a w a M i k r o n I n t e r n a t i o n a l 1988.)


358



Size fim F i g u r e 4:

I n a recent

apparently contiguous

study o f

a i r b o r n e a l u m i n a dusts t h e a u t h o r s

fineparticles a n d o p e r a t i o n a l l y i n d e p e n d e n t

c h a r a c t e r i z e d by analysis o f e l e c t r o n m i c r o g r a p h s o f the c a p t u r e d

treated

units when fineparticles.

(a) S i z e d i s t r i b u t i o n by i m a g e analysis a n d (b) size d i s t r i b u t i o n by laser a e r o s o l spectrometer.

( R e p r o d u c e d w i t h p e r m i s s i o n f r o m ref. 8.

C o p y r i g h t 1988 V C H

Publishers.)


23. K A Y E


359


T h u s the s i m u l a t i o n o f the a r r i v a l o f p e s t i c i d e droplets o n a l e a f w o u l d be m o d e l l e d by a M o n t e C a r l o r o u t i n e ( 1 Q ) . In v i e w o f the fact that m a n y early studies o f d e p o s i t e d f i n e p a r t i c l e s r e c o m m e n d that one c a n s a f e l y use 5 % b y area f i e l d s o f v i e w to c a r r y out s i z e d i s t r i b u t i o n studies the f i r s t study c a r r i e d o u t w a s to s i m u l a t e the c l u s t e r i n g o c c u r r i n g b y c h a n c e i n a d e p o s i t e d m o n o s i z e d a e r o s o l at a 5 % c o v e r a g e e x p e c t a t i o n . The random n u m b e r t a b l e u s e d i n the e x p e r i m e n t s reported i n t h i s study is s h o w n i n Figure 5 (12). (Note - many personal computers have random number g e n e r a t o r p r o g r a m s h o w e v e r , the g e n e r a t i o n o f t r u l y r a n d o m n u m b e r s i s a d i f f i c u l t task. F o r this reason w e used the table o f F i g u r e 5 w h i c h is t a k e n f r o m a n e x t e n s i v e study a n d t a b u l a t i o n o f r a n d o m n u m b e r tables ( & ) . E a c h o f the d i g i t s i n the table o f figure 5 o c c u r w i t h a frequency o f 0 . 1 . T h e r e f o r e , to s i m u l a t e the appearance o f a f i n e p a r t i c l e f i e l d o f v i e w c o v e r e d b y m o n o s i z e d p r o f i l e s u p to 5 % o f the f i e l d one needs an u n b i a s e d m e t h o d f o r c o n v e r t i n g h a l f o f any g i v e n d i g i t i n t o s i m u l a t e d f i n e p a r t i c l e s . A n o b j e c t i v e routine f o r c h a n g i n g a d i g i t into a s i m u l a t e d f i n e p a r t i c l e at a 5 % l e v e l o f area c o v e r a g e is to take a d i g i t as r e p r e s e n t i n g a f i n e p a r t i c l e i f a d i g i t i n another r a n d o m n u m b e r table is o d d , s i n c e o n average 5 0 % o f the d i g i t s i n the other reference table w i l l be o d d . T h u s e v e r y 9 i n the table o f F i g u r e 5 c o u l d b e c o m e a dust p a r t i c l e i f w h e n w e encounter a 9 i n the table a d i g i t i n a c o m p a n i o n r a n d o m n u m b e r table is o d d . O n e c o u l d also use the c r i t e r i o n that the d i g i t i n the other table h a d to be e v e n w h e n a g a i n h a l f o f the d i g i t s i n the first table w o u l d be c o n v e r t e d i n t o s i m u l a t e d fineparticles. In F i g u r e 6 s i x s i m u l a t e d f i e l d s o f v i e w at 5 % c o v e r a g e are shown. T h e n o t a t i o n i n the top left h a n d c o r n e r i n d i c a t e s the r u l e u s e d to generate the s i m u l a t e d f i e l d o f v i e w . T h u s , i n the f i e l d o f v i e w o f F i g u r e 6(a) z e r o was c o n v e r t e d to a s i m u l a t e d d e p o s i t e d f i n e p a r t i c l e i f the c o n v e r s i o n reference d i g i t i n a c o m p a n i o n table was o d d . I n F i g u r e 6(b) the d i g i t 1 was c o n v e r t e d to a s i m u l a t e d dust p a r t i c l e w h e n e v e r the c o n v e r s i o n reference d i g i t w a s e v e n and so o n . It is i n t e r e s t i n g to note the f l u c t u a t i o n s i n r e a l c o v e r a g e w h e n the a n t i c i p a t e d c o v e r a g e i s 5 % b y area. It c a n be seen that the actual area coverage v a r i e d f r o m a l o w o f 4 . 5 6 % to a high 6.28%. It is also i n t e r e s t i n g to note i n v i e w o f the current interest i n f r a c t a l g e o m e t r y that these 6 f i e l d s o f v i e w c o n s t i t u t e s t a t i s t i c a l l y s e l f s i m i l a r systems (2..UJ. These f i e l d s o f v i e w c a n be used to train operators i n w h a t constitutes a g i v e n l e v e l o f c o v e r a g e a n d w h a t s t a t i s t i c a l l y s e l f s i m i l a r systems look l i k e . I n i t i a l l y operators w h e n s h o w n these f i e l d s o f v i e w v a r y w i d e l y i n t h e i r estimate o f the area c o v e r a g e . In an i n f o r m a l e x p e r i m e n t w i t h p e o p l e n e w to the f i e l d o f s i z e c h a r a c t e r i z a t i o n a n s w e r s r a n g i n g f r o m 1 0 % to 1% were g i v e n . F r o m the p o i n t o f v i e w o f this c o m m u n i c a t i o n the important aspect o f the s i m u l a t e d f i e l d s o f v i e w o f F i g u r e 6 i s the l e v e l o f c l u s t e r i n g that has occurred. T h u s , clusters as h i g h as 4 u n i t s have o c c u r r e d and clusters o f 3 and 2 o c c u r r e l a t i v e l y f r e q u e n t l y . In F i g u r e 7 the s i z e d i s t r i b u t i o n o f the s i m u l a t e d c l u s t e r s r e p r e s e n t i n g s e c o n d a r y c o u n t g a i n c l u s t e r s at 5 % c o v e r a g e c a n be seen. It i s o b v i o u s that p r i m a r y count loss and s e c o n d a r y c o u n t g a i n are s i g n i f i c a n t f a c t o r s e f f e c t i n g the a c c u r a c y o f the c h a r a c t e r i z a t i o n study at 5 % c o v e r a g e . I f the p i x e l s i z e represented a s i z e w h i c h w a s c o n s i d e r e d to be the t h r e s h o l d o f the s i z e c o n s t i t u t i n g a r e s p i r a b l e h a z a r d the n u m b e r o f r e s p i r a b l e dust f i n e p a r t i c l e s w o u l d be u n d e r estimated by a factor o f 1.4 , a factor w h i c h c o u l d result i n a dangerous l e v e l o f dust b e i n g reported as a safe acceptable l e v e l . T o e x p l o r e the density o f coverage w h i c h is desirable, i f one must seek out a c o v e r a g e at w h i c h s i g n i f i c a n t s p a t i a l c o i n c i d e n c e effects o n a f i l t e r s u r f a c e are a v o i d e d , the appearance o f f i l t e r papers at an a n t i c i p a t e d



360


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05023 30376 61815 22056 38223

42445 60217 36878 66879 10899

81479 12916 72093 48908 13677

06582 71034 63479 44909 14360

57832 41493 15847 12820 74016

19864 00810 81812 75666 06527

09655 40746 72644 70382 80281

46091 22328 85935 86600 53788

89898 98099 82053 21025 70695

27000 91285 44887 72892 17812

12588 54345 26995 36748 43757

72677 34034 74237 42544 07029

50662 37310 88921 00765 78410

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51209 63399 92664 04582 09606

61781 98036 90519 16897 33152

22706 17711 13201 05507 81105

94834 15957 54268 00115 79698

24773 26772 92950 95513 22892

95921 40023 80560 47712 80406

10413 86215 28286 92943 81722 72870 67899 21900 13822 81956

85039 41583 86454 18132 08991

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27026 73009 72880 26865 03425

13873 52091 60952 42058 39700

35350 86401 61152 97353 53545

94513 64081 02839 43889 16848

38339 10484 30079 41507 26169

56642 74682 45524 57139 00705

53730 44393 46409 42113 17743

56710 69167 12928 85637 58687

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32666 99982 87109 89488 12041

16744 75423 48129 81433 81800

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89506 84147 53494 63646 60474

63437 82941 04187 64467 34712

81051 47088 79955 48089 90626

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59752 62126 81231 15504 68616

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93433 15839 40379 02513 28219

82926 82679 16182 03315 98681

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52025 09677 54408 55861 72830

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98960 18795 15805 11393 85666

40993 78102 99528 09048 03020

84206 30630 34002 55210 27931

76311 16175 69239 71579 01660

91982 77998 77643 95191 44950

F i g u r e 5:

R a n d o m n u m b e r table used i n s i m u l a t i o n e x p e r i m e n t s .

(Reproduced

w i t h p e r m i s s i o n f r o m ref. 12. C o p y r i g h t F r e e Press P u b l i s h e r s . )


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