Averages from Turbidity Measurements

Downloaded by MICHIGAN STATE UNIV on September 3, 2013 | http://pubs.acs.org Publication Date: February 12, 1987 | doi: 10.1021/bk-1987-0332.ch011

Chapter 11

Averages from Turbidity Measurements L.

H.

Garcia-Rubio

Department of Chemical Engineering, College of Engineering, University of South Florida, Tampa,FL33620

The information content of turbidity measurements from polydisperse particulate systems has been analysed on the basis of Mie theory and reported approximations to Mie theory. The results from the analysis suggest two alternative interpretations of turbidity data for the small and intermediate p a r t i c l e size regimes: an interpretation in terms of moments of the particle size distribution and an equivalent interpretation in terms of the volume to surface average p a r t i c l e diameter. It is also shown that in the intermediate particle size regime, the effective average diameter that s a t i s f i e s the mean surface average extinction efficiency does not correspond to a particular average of the particle size d i s t r i b u t i o n . The value of the effective particle diameter depends on both the wavelength and the spread of the p a r t i c l e size d i s t r i b u t i o n . It is possible, however, to interpret turbidity data in terms of an effective average diameter and the volume to surface particle diameter. The results obtained explain some of the discrepancies existing in the literature regarding the interpretation of the average particle diameters obtained from scattering measurements. The theoretical and practical implications of the results are presented and discussed.

0097-6156/87/0332-0161$06.00/0 © 1987 American Chemical Society

In Particle Size Distribution; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

162

PARTICLE SIZE DISTRIBUTION


The a n a l y s i s of p a r t i c u l a t e systems using l i g h t scattering measurements i s o f g r e a t s c i e n t i f i c and i n d u s t r i a l importance and as s u c h i t has been t r e a t e d e x t e n s i v e l y i n t h e l i t e r a t u r e ( 1 - 5 ) . F o r m o n o d i s p e r s e p a r t i c u l a t e s y s t e m s , t h e t h e o r y and t h e p r a c t i c e o f light scattering have been q u i t e s u c c e s s f u l and a r e w e l l e s t a b l i s h e d . However, t h i s i s n o t t h e case f o r p o l y d i s p e r s e s y s t e m s , where t h e t u r b i d i t y a t a g i v e n w a v e l e n g t h i s r e l a t e d t o t h e p a r t i c l e s i z e d i s t r i b u t i o n ( e q u a t i o n s 1-9). M i e Theory r e l a t e s the measured t u r b i d i t y (τ), t o t h e number, s i z e and o p t i c a l constants of suspended isotropic spherical p a r t i c l e s through the following e q u a t i o n s (1-5) :

- Ν A JO

τ(λ)

2

-

D Q ( a , m ) f ( D ))dD c

(1)

1»

or

Γ

2

J D

Q(a,m)f(D)dD

0

τ(λ)

»

rC O 2

(2)

j

j D f ( D ) )dD c

P

0

a «

—

W h e r e , τ (λ) i s t h e t u r b i d i t y a t t h e w a v e l e n g t h λ, Ν t h e n u m b e r o f p a r t i c l e s / m l , C t h e c o n c e n t r a t i o n i n g/ml, D the p a r t i c l e d i a m e t e r , f(D) t h e frequency d i s t r i b u t i o n of p a r t i c l e sizes, ρ the density of t h e p a r t i c l e s , m i s t h e c o m p l e x r e f r a c t i v e i n d e x r a t i o a n d Q(a,m) i s t h e M i e o v e r a l l e x t i n c t i o n e f f i c i e n c y . The e x t i n c t i o n e f f i c i e n c y i s g i v e n by

2

Q(a,m) - ( 2 / a )

If

the p a r t i c l e s

(3)

Σ (2n+1) Re(a + b ) n= ι η η

a r e n o n - a b s o r b i n g , e q u a t i o n 3 becomes

2

Qscat - ( 2 / a )

\

y

(2n+1)[|a | n

2

+ |b | n

2

]


(4)

11.

GARCIA-RUBIO

a

Averages from Turbidity Measurements

(5)

η ζ (α)

ψ^(αιη) - m ψ (am) ζ^(α)

η

b

m Ψ (α) η

i|^(am) - Ψ ( a m ) ψ^(α)

m Ψ (α)

ψ^(απι) - ψ (απι) ζ^(α)

n

(6)

η η


ζ

η

163

(

α

)

ψ

" η

(

α

η

)

+

1

χ

(

η

α

(7)

)

1 / 2

Ψ (α)

~ (π α/2)

Χ (α)

- -(π α/2)

η

J£a}

(8)

1 / 2 η

Ν£α}

(9)

Where, m » (η + i k ) / n and η i s the r e f r a c t i v e index o f the s u s p e n d i n g medium. J ( a ) and N(a) a r e h a l f o r d e r i n t e g r a l B e s s e l a n d Neuman f u n c t i o n s . The s o l u t i o n t o t h e i n t e g r a l e q u a t i o n s 1 o r 2-9 i s r a t h e r d i f f i c u l t , the scattering c o e f f i c i e n t s ( a + b ) h a v e t o be η η c a l c u l a t e d a t e v e r y measured w a v e l e n g t h and e v e r y p a r t i c l e d i a m e t e r f o r a g i v e n f o r m o f t h e p a r t i c l e d i a m e t e r d i s t r i b u t i o n , where t h e f o r m o f t h e d i s t r i b u t i o n i s g e n e r a l l y unknown. T h e p r o b l e m o f s o l v i n g e q u a t i o n s 1 o r 2 f o r t h e p a r t i c l e s i z e d i s t r i b u t i o n i s known as t h e " I n v e r s e S c a t t e r i n g Problem" a n d a l t h o u g h t h i s p r o b l e m h a s b e e n t r e a t e d e x t e n s i v e l y , no g e n e r a l s o l u t i o n s have been r e p o r t e d (3> 6, 7,)* A l t e r n a t i v e approaches t o t h e s o l u t i o n o f t h e i n t e g r a l e q u a t i o n s i n c l u d e t h e assumption o f t h e shape o f t h e p a r t i c l e s i z e d i s t r i b u t i o n (3, 8-12) a n d t h e e s t i m a t i o n o f " E f f e c t i v e Particle D i a m e t e r s " (3, 13-15). I f t h e form o f t h e p a r t i c l e s i z e d i s t r i b u t i o n i s known o r i t can be assumed t h e n , the main problem i n t h e s o l u t i o n o f e q u a t i o n s 1 or 2 stems from t h e complex c a l c u l a t i o n s required t o evaluate the integral 0

0

(10)

A number o f d i s t r i b u t i o n f u n c t i o n s h a v e b e e n identified e x p e r i m e n t a l l y f o r a v a r i e t y o f systems (3) and, i n p a r t i c u l a r , t h e Log-Normal d i s t r i b u t i o n i s e x t e n s i v e l y used f o r t h e c a l c u l a t i o n o f t h e i n t e g r a l a n d f o r t h e e v a l u a t i o n o f t h e moments o f t h e p a r t i c l e s i z e d i s t r i b u t i o n ( 8 - 1 2 ) . The problem w i t h t h i s approach i s t h a t , i n g e n e r a l , t h e shape o f t h e p a r t i c l e s i z e d i s t r i b u t i o n i s unknown and t h u s , t h e average p a r t i c l e d i a m e t e r s o b t a i n e d a r e c o n d i t i o n a l upon



164


the v a l i d i t y of the assumption regarding the form of the distribution. D i r e c t a p p l i c a t i o n o f the s c a t t e r i n g equations f o r monodisperse systems t o p a r t i c u l a t e systems that a r e p o l y d i s p e r s e , w i l l clearly r e s u l t i n the e s t i m a t i o n o f average o r e f f e c t i v e p a r t i c l e diameters. The i n t e r p r e t a t i o n of these average diameters i s n o t s t r a i g h t f o r w a r d . Q u o t i n g K e r k e r ( 3 ) , " t h e r e i s no s i m p l e m e t h o d o f c o m p a r i n g t h e s i z e d e t e r m i n e d by t h e e l e c t r o n m i c r o s c o p e w i t h t h e a v e r a g e s i z e o b t a i n e d from l i g h t s c a t t e r i n g . Indeed, not only w i l l d i f f e r e n t l i g h t s c a t t e r i n g methods g i v e d i f f e r e n t a v e r a g e s , b u t each particular d i s t r i b u t i o n o f s i z e s w i l l h a v e i t s own c h a r a c t e r i s t i c average." Maron, P i e r c e and U l e v i t c h (V3) e x p e r i m e n t a l l y i n v e s t i g a t e d t h i s p r o b l e m a n d r e p o r t e d good agreement between t h e p a r t i c l e diameters obtained from s p e c i f i c t u r b i d i t y measurements t a k e n a t s e v e r a l w a v e l e n g t h s , and t h e t u r b i d i t y average p a r t i c l e diameters c a l c u l a t e d from e l e c t r o n m i c r o s c o p y m e a s u r e m e n t s o f t h e s i z e d i s t r i b u t i o n . D o b b i n s a n d J i z m a g i a n ( V4), on t h e o t h e r hand, noted t h a t f o r a wide v a r i e t y o f monomodal d i s t r i b u t i o n functions w i t h r a d i i comparable t o and l a r g e r t o t h e w a v e l e n g t h , t h e s p e c i f i c t u r b i d i t y c a n be a d e q u a t e l y represented i n terms of the volume s u r f a c e mean d i a m e t e r a n d t h e mean s u r f a c e s c a t t e r i n g efficiency.

2 ρ

D

w h e r e t h e mean s u r f a c e

3 2

scattering

0

i s given

by

»00

f CO Q(a,m) - J

efficiency

D

2

Q(a,m) f ( d ) dD/ J

D

2

f ( D ) dD

Since t h e r a t i o Q(a,m)/ D was s h o w n , n u m e r i c a l l y , t o be i n d e p e n d e n t o f t h e shape o f t h e d i s t r i b u t i o n , D o b i n s and J i z m a g i a n c o m p u t e d t a b l e s o f t h e mean s u r f a c e s c a t t e r i n g e f f i c i e n c y as a f u n c t i o n o f t h e v o l u m e s u r f a c e mean d i a m e t e r f o r t h e e v a l u a t i o n o f D d i r e c t l y from t r a n s m i s s i o n measurements. More r e c e n t l y , Bagchi and V o i d , a n a l y s e d l a t e x p a r t i c l e s c o n s i d e r a b l y l a r g e r than t h e wavelength of the incident radiation where t h e extinction c o e f f i c i e n t i s approximately independent of the p a r t i c l e s i z e . T h e i r r e s u l t s suggest that, i n t h e p a r t i c l e range i n v e s t i g a t e d , t h e s p e c i f i c t u r b i d i t y c a n be d i r e c t l y r e l a t e d to the f i r s t three moments o f t h e p a r t i c l e s i z e d i s t r i b u t i o n a n d t o t h e v o l u m e t o surface average p a r t i c l e diameter. U n f o r t u n a t e l y , the polydispersities of the l a t i c e s investigated were r e l a t i v e l y s m a l l p r e v e n t i n g any d e f i n i t i v e s t a t e m e n t as t o t h e average that best correlates the data. 3 2

3 2

Ideally, i t i s desirable t o solve equations 1 or 2 f o r the complete s i z e d i s t r i b u t i o n whenever p o l y d i s p e r s e s y s t e m s a r e b e i n g a n a l y s e d . However, from t h e c o m p u t a t i o n a l point of view, t h e d i r e c t application of the equations f o r monodisperse systems i s more


11.

165


GARCIA-RUBIO

appealing. I t i s therefore necessary t o establish both, the physical meaning o f the " e f f e c t i v e " average diameters as well as t h e i n f o r m a t i o n c o n t e n t o f t h e t u r b i d i t y d a t a i n t e r m s o f t h e moments o f the p a r t i c l e s i z e d i s t r i b u t i o n . Problem

F o r m u l a t i o n And A n a l y s i s


By f i r s t c o n s i d e r i n g t h e R a y l e i g h r e g i m e a n d e x p a n d i n g t h e c o m p l e x scattering c o e f f i c i e n t s i n power s e r i e s o f a , i t c a n be shown t h a t t h e e x t i n c t i o n c a n b e a p p r o x i m a t e d b y ( 1 , 3t 1 7 )

Q(a,m) = V

a + V a

3

1

1

+ I\ a* + V a

5

(12)

+

or Q(a,m) = r D + Γ x

D

3

3

u

+ I\ D

+ Γ

D

5

5

(13)

+

Where t h e c o e f f i c i e n t s Γ and r's are i m p l i c i t functions of the o p t i c a l constants and thus o f the wavelength ( s e e Appendix I ) . For monodisperse systems, replacement o f e q u a t i o n 13 i n t o e q u a t i o n 2, y i e l d s a n a p p r o x i m a t i o n t o t h e t u r b i d i t y i n t e r m s o f powers o f t h e p a r t i c l e d i a m e t e r

[ T

τ(λ) =

x

+ Γ

3

D

2

+ I\ D

3

+ Γ

1

5

D* +

]

(14)

2ρ

F o r p o l y d i s p e r s e s y s t e m s , r e p l a c e m e n t o f e q u a t i o n 13 i n t o e q u a t i o n 2 for every p a r t i c l e diameter, y i e l d s an approximation t o the t u r b i d i t y i n t e r m s o f r a t i o s o f moments o f t h e p a r t i c l e size d i s t r i b u t i o n w i t h o u t h a v i n g t o make a s s u m p t i o n s r e g a r d i n g t h e s h a p e of t h e d i s t r i b u t i o n :

τ(λ) »

[ I\ D 2p D

3

+ Γ

3

D

5

+ I\ D

6

+ Γ

5

D

7

+

]

(15)

3

W h e r e Dn r e p r e s e n t s t h e n t h moment o f t h e p a r t i c l e For monodisperse systems i n t h e large p a r t i c l e t h e e x t i n c t i o n e f f i c i e n c y c a n be assumed c o n s t a n t e q u a l t o 2, e q u a t i o n 2 b e c o m e s

diameter. s i z e regime, where and a p p r o x i m a t e l y

τ(λ) - 3 C I / ( p D) (16)


166


For p o l y d i s p e r s e s y s t e m s e q u a t i o n 2 becomes

- 3 C i

τ(λ)


Where D

/(p

i n the

large particle

D )

regime

(17)

3 2

i s the volume t o s u r f a c e average p a r t i c l e

3 2

size

diameter.

E q u a t i o n s 15 a n d 17 p o i n t t o s o m e o f t h e d i f f i c u l t i e s i n the i n t e r p r e t a t i o n o f t h e t u r b i d i t y d a t a i n t e r m s o f a v e r a g e s o r moments o f t h e p a r t i c l e s i z e d i s t r i b u t i o n . F r o m e q u a t i o n 15 i t i s c l e a r t h a t the i n t e r p r e t a t i o n of the diameter d e p e n d s on t h e l e v e l of approximation used to evaluate the e x t i n c t i o n efficiency. F u r t h e r m o r e , i t s u g g e s t s t h a t , i n p r i n c i p l e , a s i g n i f i c a n t number o f moments a n d t h e r e f o r e t h e s h a p e o f t h e p a r t i c l e s i z e distribution c a n be e v a l u a t e d d i r e c t l y f r o m t u r b i d i t y e x p e r i m e n t s . The only l i m i t a t i o n being the s i g n a l to noise r a t i o of the measurements. On t h e o t h e r hand, i n t h e l a r g e p a r t i c l e s i z e regime (equation 17), o n l y one a v e r a g e , t h e v o l u m e t o s u r f a c e a v e r a g e i s a v a i l a b l e . I f t h e p a r t i c l e s i z e d i s t r i b u t i o n i s k n o w n , o n l y t w o moments ( t h e s e c o n d a n d t h e t h i r d ) c a n be o b t a i n e d d i r e c t l y f r o m t u r b i d i t y experiments. The p r o b l e m o f d e c i d i n g on a p a r t i c u l a r average diameter ( o r d i a m e t e r s ) t o be u s e d i n t h e c a l c u l a t i o n of the extinction e f f i c i e n c y c a n be f o r m u l a t e d a s f i n d i n g a v e r a g e p a r t i c l e d i a m e t e r s , Dav, s u c h t h a t

(18) ο where aav subject

= ïïDav/λ

t o the f o l l o w i n g

constraints:

1.

The a v e r a g e d i a m e t e r ( s ) , a s behavior, should s a t i s f y the equations ( i e : the r e s u l t i n functional f o r m as e q u a t i o n s

r e p r e s e n t a t i v e of the p o p u l a t i o n a n a l y t i c behavior of the scattering g e q u a t i o n s s h o u l d have t h e same 1*1 a n d 1 6 ) .

2.

The e q u a t i o n s o b t a i n e d f o r t h e a v e r a g e p a r t i c l e diameter(s) should reduce to the equations developed f o r monodisperse systems ( i e : the r e s u l t i n g equations s h o u l d reduce t o e q u a t i o n s 14 a n d 16 f o r m o n o d i s p e r s e s y s t e m s ) .

N o t e t h a t t h e f o r m u l a t i o n s t a t e d by e q u a t i o n 18 i s d i f f e r e n t from t h a t o f D o b b i n s e t a l ( E q u a t i o n 1 1 ) i n t h a t we a r e s e e k i n g a s o l u t i o n t o t h e i n t e g r a l e q u a t i o n 18 i n t e r m s o f o n e o r s e v e r a l o f the moments of the p a r t i c l e s i z e d i s t r i b u t i o n and i n t h a t , s i m u l t a n e o u s l y , we a r e i n q u i r i n g a s t o t h e i n f o r m a t i o n c o n t e n t o f the s c a t t e r i n g function with regards to the p a r t i c l e size distribution.



11. GARCIA-RUBIO

167


T h e r e a r e two r e g i o n s i n t h e diameter space where t h e b e h a v i o r of t h e a v e r a g e d i a m e t e r ( s ) c a n e a s i l y b e i n v e s t i g a t e d : t h e r e g i o n where t h e diameter i s s m a l l compared t o t h e wavelength o f t h e incident r a d i a t i o n ( i e : the Rayleigh regime) a n d t h e r e g i o n where t h e d i a m e t e r i s much l a r g e r t h a n t h e w a v e l e l e n g t h ( i e : t h e l a r g e particle size regime). I n the l a r g e p a r t i c l e s i z e regime, i t i s c l e a r f r o m e q u a t i o n 17 t h a t t h e o n l y a v e r a g e o b t a i n e d f r o m t u r b i d i t y measurements i s t h e volume t o s u r f a c e average d i a m e t e r . I t i s a l s o e v i d e n t t h a t e q u a t i o n 17 h a s t h e same f u n c t i o n a l f o r m a s e q u a t i o n 16 a n d t h a t i n f a c t r e d u c e s t o e q u a t i o n 16 f o r m o n o d i s p e r s e systems. Thus s a t i s f y i n g t h e c o n s t r a i n t s imposed f o r the s e l e c t i o n o f the average diameters. Note t h a t although r a t i o s o f s u c c e s s i v e moments w i l l s a t i s f y t h e c o n s t r a i n t s imposed, o n l y D r e s u l t s d i r e c t l y from e q u a t i o n 2. 3 2

For t h e a n a l y s i s i n t h e s m a l l p a r t i c l e s i z e regime, r a t h e r than approximating t h e s c a t t e r i n g e f f i c i e n c y w i t h a power s e r i e s a s i t w a s d o n e t o o b t a i n e q u a t i o n 1 5 , e q u a t i o n 18 w i l l b e r e p l a c e d f i r s t into equation 2 t o obtain

Q(aav,m) τ

( ) λ

±

β

±

(19)

Α

2 ρ

D

3 2

Then o n t h e b a s i s o f t h e f i r s t c o n s t r a i n t , t h e e x t i n c t i o n i s expanded i n power s e r i e s o f a a v t o y i e l d

τ(λ) =

3

C

1

[τ

ϊ

2 ρ D

Dav + Γ

3

Dav

3

+ I \ Dav* + Γ

5

Dav

5

efficiency

+ ..]

(20)

3 2

n o t e t h a t by f o l l o w i n g t h i s a p p r o a c h t h e e r r o r s i n t h e a p p r o x i m a t i o n o f Q(a,m) a r e n o t i n t e g r a t e d . T h e r e a r e many a v e r a g e s t h a t when s u b s t i t u t e d i n t o e q u a t i o n 20 w i l l r e d u c e i t t o e q u a t i o n 14 f o r m o n o d i s p e r s e distributions. However, o n l y t h e D a v e r a g e y i e l d s t h e same f u n c t i o n a l f o r m a s e q u a t i o n 14 f o r p o l y d i s p e r s e a n d m o n o d i s p e r s e p a r t i c l e size d i s t r i b u t i o n s . T h e r e f o r e , i t f o l l o w s t h a t t h e volume t o surface average diameter s a t i s f i e s t h e c o n s t r a i n t s imposed a t b o t h , t h e s m a l l a n d t h e l a r g e p a r t i c l e s i z e r e g i m e s . E q u a t i o n s 15 a n d 20 provide twoequivalent i n t e r p r e t a t i o n s f o r the t u r b i d i t y data i nt h e s m a l l p a r t i c l e s i z e r e g i m e . E q u a t i o n 20 i s p a r t i c u l a r l y a t t r a c t i v e because a s i n g l e average c o u l d be used f o r t h e i n t e r p r e t a t i o n o f t u r b i d i t y d a t a i n b o t h t h e s m a l l and l a r g e p a r t i c l e s i z e regimes. However, i t r e m a i n s t o d e m o n s t r a t e i f t h i s i s a l s o t r u e f o r t h e intermediate s i z e regime ( i e : D - λ ) . To s e a r c h f o r t h e a v e r a g e d i a m e t e r ( s ) t h a t w o u l d satisfy e q u a t i o n 18 i n t h e i n t e r m e d i a t e p a r t i c l e s i z e r e g i m e , a n u m e r i c a l a p p r o a c h was t a k e n . T h e o b v i o u s a p p r o a c h i s t o c a l c u l a t e t h e r i g h t h a n d s i d e o f e q u a t i o n 18 f o r a v a r i e t y o f d i s t r i b u t i o n s a n d a 3 2



168


s u f f i c i e n t l y broad range o f t h e i r parameter v a l u e s and t h e n , s e a r c h for a d i a m e t e r t h a t when s u b s t i t u t e d i n t o t h e e x p r e s s i o n f o r t h e e x t i n c t i o n e f f i c i e n c y ( i e : e q u a t i o n 3 ) , s a t i s f i e s e q u a t i o n 18. A c o m p a r i s o n b e t w e e n D a v a n d t h e moments o r r a t i o s o f moments o f t h e d i s t r i b u t i o n i n question w i l l indicate which average d i a m e t e r s a t i s f i e s e q u a t i o n 1 8 . A m a j o r d i f f i c u l t y w i t h t h e above approach stems from the diameter b e i n g a m u l t i v a l u e d f u n c t i o n o f t h e e x t i n c t i o n e f f i c i e n c y ( 1-3). I n o t h e r words, t h e r e i s not a unique v a l u e f o r t h e d i a m e t e r as a f u n c t i o n o f t h e e x t i n c t i o n efficiency. An a l t e r n a t i v e a p p r o a c h t h a t c i r c u m v e n t s t h e u n i q u e n e s s p r o b l e m i s to s u b s t i t u t e t h e moments a n d t h e r a t i o s o f moments o f t h e d i s t r i b u t i o n i n t o t h e e x t i n c t i o n e f f i c i e n c y and then s e l e c t those t h a t b e s t a p p r o x i m a t e t h e c o n d i t i o n s t a t e d by e q u a t i o n 18. T h e e f f e c t o f t h e s h a p e o f t h e d i s t r i b u t i o n c a n be t a k e n i n t o c o n s i d e r a t i o n , w i t h o u t l o s s o f g e n e r a l i t y , i f i t i s assumed t h a t t h e s h a p e o f t h e d i s t r i b u t i o n i n q u e s t i o n c a n be a p p r o x i m a t e d w i t h a w e i g h t e d sum o f g a u s s i a n d e n s i t y f u n c t i o n s ( 1 6 )

f(D)

subject

- ^

to

v ( j ) N(D, y ( j ) ,

j.

(21)

o(j))

v(j) » 1

and where p ( j ) and o ( j ) a r e t h e p a r a m e t e r s f o r t h e j t h a p p r o x i m a t i o n function.

Replacement

o f e q u a t i o n 21 i n t o

Q(aav,m) - ^

18

yields

v( j ) Q ( a a v ( j ) , m) / ^

v(j) D (j) 2

(22)

E q u a t i o n 22 i n d i c a t e s t h a t t h e s u r f a c e a v e r a g e s c a t t e r i n g e f f i c i e n c y c a n be e x p r e s s e d a s a a l i n e a r c o m b i n a t i o n o f t h e a v e r a g e e f f i c i e n c y for each of the a p p r o x i m a t i o n f u n c t i o n s . T h e r e f o r e , i t i s only necessary t o f i n d which averages s a t i s f y e q u a t i o n 8 f o r a g a u s s i a n d i s t r i b u t i o n . E x t e n s i v e c o m p u t a t i o n s w i t h a v a r i e t y o f parameter v a l u e s i n d i c a t e s t h a t Dav does n o t c o r r e s p o n d t o any p a r t i c u l a r a v e r a g e o r r a t i o o f moments o f t h e p a r t i c l e s i z e d i s t r i b u t i o n . T h i s c a n be s e e n i n f i g u r e s 1 - 4 . W h e r e t h e f r a c t i o n a l e r r o r s , b e t w e e n t h e s u r f a c e average e x t i n c t i o n e f f i c i e n c y and t h e e x t i n c t i o n c a l c u l a t e d with t r i a l averages of the p a r t i c l e s i z e d i s t r i b u t i o n ( ε « 1 Q ( a a v , m ) / Q ( a , m ) ) , a r e s h o w n a s f u n c t i o n o f t h e s i z e p a r a m e t e r a. C l e a r l y , i t appears t h a t d i f f e r e n t p a r t i c l e a v e r a g e s w i l l be m o r e e f f e c t i v e a t d i f f e r e n t s i z e parameter v a l u e s . I n g e n e r a l , averages i n c l u d i n g h i g h e r moments a p p e a r t o p e r f o r m b e t t e r o v e r a l l . The r e a s o n f o r t h e o b s e r v e d p e r f o r m a n c e c a n be b e t t e r u n d e r s t o o d w i t h



11.

GARCIA-RUBIO

AveragesfromTurbidity Measurements

169

F i g . 1. A p p r o x i m a t e b e h a v i o r o f t h e f r a c t i o n a l e r r o r a s a f u n c t i o n o f t h e m e a n s i z e p a r a m e t e r αχ » π Ό /\ and t h e f r a c t i o n a l standard d e v i a t i o n σ. W h e r e Ώ i s t h e mean p a r t i c l e d i a m e t e r . ι

1



170


F i g . 2 . A p p r o x i m a t e b e h a v i o r o f t h e f r a c t i o n a l e r r o r as a f u n c t i o n o f t h e mean s i z e p a r a m e t e r α * π D /X a n d t h e f r a c t i o n a l s t a n d a r d d e v i a t i o n σ. Where D i s the s u r f a c e average p a r t i c l e diameter. 2

2

2



11.

GARCIA-RUBIO


171

0.4 J

F i g . 3. A p p r o x i m a t e b e h a v i o r o f t h e f r a c t i o n a l e r r o r a s a f u n c t i o n o f t h e mean s i z e p a r a m e t e r α * π D /X and t h e f r a c t i o n a l s t a n d a r d d e v i a t i o n σ. Where D i s t h e mean v o l u m e t o s u r f a c e average p a r t i c l e diameter. 3 2

3 2

3 2


172


r e f e r e n c e t o f i g u r e 5. A s s u m e t h a t f o r a m a t e r i a l h a v i n g a g a u s s i a n p a r t i c l e s i z e d i s t r i b u t i o n and t h e e x t i n c t i o n p r o p e r t i e s r e p r e s e n t e d i n t h e f i g u r e , m e a s u r e m e n t s a r e t o be t a k e n a t a w a v e l e n g t h λ c o r r e s p o n d i n g t o a s i z e p a r a m e t e r α - 4. N o t e t h a t , i n i t i a l l y , m o s t of the p a r t i c l e s i z e d i s t r i b u t i o n l i e s b e f o r e t h e f i r s t extinction maximum. T h i s i m p l i e s t h a t t h e w e i g h t i n g on D i n e q u a t i o n 18 w i l l increase together w i t h the diameter squared. Therefore, the a v e r a g e d i a m e t e r t h a t s a t i s f i e s e q u a t i o n 18 w i l l c o r r e l a t e b e t t e r w i t h a v e r a g e s c o n t a i n i n g h i g h e r moments o f t h e p a r t i c l e s i z e d i s t r i b u t i o n (ie: the t u r b i d i t y a v e r a g e DT). Now assume t h a t two more m e a s u r e m e n t s a r e t o be t a k e n a t w a v e l e n g t h s A / 2 a n d λ / 4 . For t h i s p u r p o s e t h e p a r t i c l e s i z e d i s t r i b u t i o n i s mapped a g a i n o n t o t h e α domain. Note t h a t the p a r t i c l e s i z e d i s t r i b u t i o n has changed r e l a t i v e t o the f i r s t measurement, the p a r t i c l e s i z e d i s t r i b u t i o n a p p e a r s t o have been s h i f t e d and b r o a d e n e d as t h e w a v e l e n g t h was reduced. C l e a r l y , the r e s u l t from changing the wavelength i s a c h a n g e i n t h e r a n g e o f v a l u e s o f Q(a,m) w e i g h t i n g t h e D i n equation 18 t h u s r e s u l t i n g n o t o n l y i n d i f f e r e n t v a l u e s o f t h e s u r f a c e average e x t i n c t i o n e f f i c i e n c y but a l s o , i n d i f f e r e n t v a l u e s of t h e a v e r a g e d i a m e t e r t h a t w i l l s a t i s f y e q u a t i o n 18. A s t h e w a v e l e n g t h i s reduced, the d i s t r i b u t i o n continues to s h i f t u n t i l the l a r g e p a r t i c l e s i z e r e g i m e i s r e a c h e d a n d t h e v a l u e o f Q(a,ra) b e c o m e s independent of the shape of the p a r t i c l e s i z e d i s t r i b u t i o n . I n c r e a s i n g t h e v a r i a n c e o f the p a r t i c l e s i z e d i s t r i b u t i o n w i l l have a s i m i l a r e f f e c t o n t h e a v e r a g e s s a t i s f y i n g e q u a t i o n 18 s i n c e t h e i n c l u s i o n o f l a r g e r and s m a l l e r s i z e s w i l l a s y m m e t r i c a l l y a f f e c t t h e w e i g h t i n g o n D . A n i n d i c a t i o n a s t o how d i f f e r e n t m o m e n t s o f t h e p a r t i c l e s i z e d i s t r i b u t i o n a f f e c t the c a l c u l a t i o n of the e x t i n c t i o n e f f i c i e n c y c a n be o b t a i n e d d i r e c t l y f r o m t h e i n t e r p r e t a t i o n of Q ( a , m ) f o r a n o m a l o u s d i f f r a c t i o n ( i e : [m - 1]

2

2

2

9 ( n + k ) + 24(n - k ) + 16

( 2nk + 4nk ( n - k - 3 ) ) 2

+ ^

2

Where: (n

2

2

2

2

2

+ k ) + n - k - 2

Pi

ρ

_ 6nk

ρ

=

2

R l

2

2

2

3

2

2

2

2

2

(η - k ) " + 22(η - k ) - -Ι60(η - k ) - 200(η - k )

2

400 - 4 n k

+

2

2

2

2

2

[24(η - k ) + 4 n k Ζ

2

3

2

2

3

2

+ 39]

2

2

2nk[2(n - k ) - 12(n - k ) + 205(n ~ k ) Ζ

2

2

+ 8n k (n

Ζ

-

2

2

2

2

- k Ζ

2

+ 9) - 198]

2

2

(η + k ) + 4 (η - k ) + 4

Replacement of the value α » πϋ/λ i n t o equation 17, y i e l d s equation 17a. Where the i t h c o e f f i c i e n t i s now given by

Γ

Α

- Γ·

(π/λ)

1


178


APPENDIX I I

T h i s a p p e n d i x r e p o r t s t h e w e i g h t s f o r t h e moments o f t h e p a r t i c l e s i z e d i s t r i b u t i o n o b t a i n e d f r o m an e i g h t o r d e r T a y l o r S e r i e s a p p r o x i m a t i o n t o t h e s c a t t e r i n g e f f i c i e n c y f o r t h e anomalous d i f f r a c t i o n case


K

l

A)

- -B ( A t -

x

0

2

K

2

= (B/2!)(A t

K

3

= (-B/3!)(A t

- 2A t

0

-

x

3

K„ * ( B A ! ) ( A e

5

= (-B/5!)(A t

K

6

« (B/6!)(A t

K

7

- (-B/7!)(A t

H

x

6

- 6Α^

0

2

*

3

- 10A t

3

2

5

A,)

+ 4A t

2

- 5k t

5

0

- 3A t - 6A t

x

K

2

2

3

- MA t

o

2

- 3Ait

0

A )

+

A J

+ 10A t

2

+ 20A t

3

3

- 15A f 2

3

+ 5A„t + 15A„t

A ) 5

2

- 6A t

-

- 21A t

2

-

- 56A t

3

-

5

A.)

A t

8

- 7A t

=* ( - B / 8 ! ) ( A t

5

- 28A t

6

+ 35A,t"

2

3

+ 35A t H

5

7

8

- 8A t

7

t

28A t

- 21A t

A )

0

2

+ 56A t

2

+ 8A t

6

where

6

x

•

6

K

7

0

7

-

3

5

+ 70A t H

H

A ) 8

t * 2-nk/\

0

Β » 4cos(6)/Y

2

Ύ = 2π(η - η ) / λ 0

The A c o e f f i c i e n t s

0

c a n be g e n e r a l i z e d f o r e v e n

for

η even

An = ( c o s ( 2 B ) - n ) ( c o s 6 ) y

for

η odd

An « (cosB s i n ( 2 8 ) - η s i n B )

R E C E I V E D November 12,

and odd i n d e c e s

n

Ύ

Π

1986


5

Averages from Turbidity Measurements

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