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Studies on multicomponent solids mixing and mixtures Part III. Mixing indices
P o w d e r T e c h n o l o g y , 24 (1979) 73 - 89
73
~) E l s e v i e r S e q u o i a S ~ A . , L a u s a n n e - - P r i n t e d in t h e N e t h e r l a n d s
Studies on Multicomponent Part HI. Mixing Indices
Solids Mixing and Mixtures
L. T. FAN, J. R. TOO Department of Chemical Engineering, Kansas State University, Itfanhattan, Kans. 66506 (U.S.A.) R . M . RUBISON
Department of Statistics, Kansas State University, Manhattan, Kans. 66506 (U.S.A.) and F. S. LAI U.S. G r a i n M a r k e t i n g R e s e a r c h C e n t e r , S c i e n c e & E d u c a t i o n A d m i n i s t r a t i o n , D e p a r t m e n t o f A g r i c u l t u r e , 1 5 1 5 College Avenue, Manhattan. Kans. 66502 (U.S.A.) (Received
January 15, 1979; i n r e v i s e d form April 25, 1979)
SUMMARY
The literature on the assessment of the q u a l i t y o f s o l i d s m i x t u r e s is s c a r c e , e s p e c i a U y for multicomponent mixtures. Four app r o a c h e s t o assess t h e h o m o g e n e i t y o f a m u l t i c o m p o n e n t solids m i x t u r e have been p r o p o s e d in t h i s s t u d y . T h e m i x i n g i n d i c e s p r o p o s e d a r e e s t i m a t e d o n t h e basis o f t h e (a) p s e u d o - b i n a r y m i x t u r e c o n c e p t , (b) p o o l e d v a r i a n c e o f t h e w h o l e s y s t e m , (c) d e t e r m i n a n t o f t h e s a m p l e c o v a r i a n c e matrix, or (d) a b s o l u t e d e v i a t i o n s f r o m t h e p o p u l a t i o n means.
T h e use o f v a r i o u s m i x i n g i n d i c e s f o r m u l t i c o m p o n e n t mixtures has been s h o w n t h r o u g h a temary mixture.
1. I N T R O D U C T I O N S o l i d s m i x i n g o r p o w d e r b l e n d i n g is t h e o p e r a t i o n b y w h i c h t w o o r m o r e solid materials i n p a r t i c u l a t e f o r m a r e s c a t t e r e d r a n d o m l y in a m i x e r b y t h e r a n d o m m o v e m e n t o f t h e p a r t i c l e s . T h i s o p e r a t i o n is p e r f o r m e d t o achieve an acceptable mixing quality. In other w o r d s , t h e u l t i m a t e o b j e c t o f m i x i n g is t o attain the distribution of maximum uniformity; thus, the mixing procedure and measures of homogeneity become important aspects of this subject. T h e r e a r e t w o b r o a d a s p e c t s i n t h e investig a t i o n o f s o l i d s m i x i n g . O n e is c o n c e r n e d w i t h the c h a r a c t e r i z a t i o n o f m i x t u r e s a n d the o t h e r
with the mechanism and rate by which the s t a t e o f a m i x t u r e c h a n g e s . T h e w o r k desc r i b e d in this s t u d y is c o n c e r n e d w i t h t h e former. T h e h o m o g e n e i t y o f a solids m i x t u r e is u s u a l l y c h a r a c t e r i z e d b y use o f a m i x i n g i n d e x t o r e p r e s e n t its d e g r e e o f m i x e d n e s s . A u s e f u l a n d g e n e r a l i n d e x s h o u l d b e r e l a t e d as c l o s e l y as p o s s i b l e t o t h e s p e c i f i c c h a r a c t e r i s t i c s o f the mixture, should be independent of the mixing process, and s h o u l d be easily determ i n e d . T h e c h o i c e o f a n i n d e x also d e p e n d s o n its a p p l i c a t i o n . T h e d e s i r e d r a n g e o f t h e m i x i n g i n d e x is o f t e n b e t w e e n 0 a n d 1. Statistical analysis has been a m a j o r t o o l o f solids m i x i n g i n v e s t i g a t i o n s b e c a u s e o f t h e stochastic nature of mixing processes. Over thirty different criteria for the degree of mixedness have been reviewed and summarized b y F a n et al. [ 1 ] . T h e relationships a m o n g some of the criteria have been studied and t a b u l a t e d b y F a n a n d W a n g [ 2 ] . T a b l e 1 lists these criteria and investigators who have p r o p o s e d t h e m . Most o f t h e criteria are based on the composition analysis of equi-sized particle samples of a binary mixture. However, solid particles t o be blended o f t e n consist o f p a r t i c l e s t h a t v a r y n o t o n l y i n size a n d d e n s i t y b u t also in c h e m i c a l a n d p h y s i c a l p r o p e r t i e s . Thus, a study of mixing indices for multicomp o n e n t h e t e r o g e n e o u s s o l i d p a r t i c l e s is o f practical significance. A convenient scheme for classifying particle systems should be established to facilitate the investigation o f solids mixing. Usually, particle s y s t e m s are c l a s s i f i e d as h o m o g e n e o u s o r
74
o~ o~
o-~ -~__. ~,- " ~
o
I~ II~.-~
-~ ~ I ~
-
I~
~I~ ~
I1~
~"-'~ -~ ~I~
"~1% I
'f
~e o~
°f
t~
2
~
II II
8
~2
e.,
e...
II
II
II
i '--i
L~
0 o
I
o °.
Tit
~.~I
I
"1-~ -i-~ r,,¢
0
I o ¢IS
o. I!
U
I!
II 0 Z
75 heterogeneous (or dispersed) systems. Particles of a homogeneous system have the same p h y s i c a l p r o p e r t i e s (e.g. d e n s i t y , s i z e , s h a p e , etc.) and are distinguishable through the c h a r a c t e r i s t i c s o f t r a c e r s , s u c h as c o l o r a n d radioactivity, that do not physically alter the course of the mixing process. Here materials with physical properties that do not interact with the operation of a mixer are considered as tracers. On the other hand, particles of a heterogeneous system have different physical properties. The extent of the variation in the characteristics of a mixture m~y be expressed in terms of weight, volume or number fraction of each constituent in the mixture. The three bases yield an identical result for a homogeneous system. However, sampling by counti n g t h e n u m b e r o f p a r t i c l e s is d i f f i c u l t a n d impractical for a fine particle or powder s y s t e m . M e a s u r e m e n t b y v o l u m e f r a c t i o n is not easy, because the bulk volume of solid particles consists of the true volume of p a r t i c l e s a n d t h e v o l u m e o f v o i d a g e , i.e., t h e b u l k v o l u m e o f e a c h c o n s t i t u e n t is n o t additive. Because of the practical difficulties of measuring the volume of a mixture, the mean and variance of the weight fraction are c h o s e n h e r e as u n i t s o f m e a s u r e m e n t . In the present study, some approaches to assess t h e q u a l i t y o f a m i x t u r e h a v e b e e n proposed and the approximate distribution of a h e t e r o g e n e o u s m u l t i c o m p o n e n t m i x t u r e is derived.
2. STATISTICAL ANALYSIS OF BINARY MIXTURES The expressions of some statistical quantities associated with a binary mixture are summarized below [1]. U n b i a s e d s a m p l e v a r i a n c e * is d e f i n e d as a2_
1
~
m - - 1 y=z
(Xlj--~1.) 2
* I n d e r i v a t i o n B, t h e b i a s e d v a r i a n c e e s t i m a t o r ,
1
~
(x~i_~l.)2
m i=1 is u s e d t o -~|mplify t h e d e r i v e d f o r m u l a . F o r a l a r g e ~-~mp|e s i z e , t h e e f f e c t o f b h s will b e n e g l i g i b l e .
where m = number of spot samples, x~ = f r a c t i o n o f c o m p o n e n t 1 in t h e j t h s a m p l e , a n d X l . = s a m p l e m e a n , w h i c h is ~ XzY m i=i
Note_ t h a t t h e v a r i a n c e c a l c u l a t e d f r o m c o m p o n e n t 1 is e q u a l t o t h a t c a l c u l a t e d f r o m component 2 for a binary mixture. The ten most frequently used indices are l i s t e d i n T a b l e 1. W e c a n s e e t h a t t h e s t a n d a r d deviation (or variance) of the mixture in the completely mixed state and that of the mixture in the completely segregated state are o f t e n u s e d as t h e b o u n d a r y v a l u e s o f a m i x i n g index. A mixture with a completely ordered a r r a n g e m e n t o f p a r t i c l e s is o f t e n r e g a r d e d as p e r f e c t , i.e. t h e m i x t u r e is i d e a l l y h o m o g e nous. For the ideally homogeneous mixture, t h e c o n c e n t r a t i o n i n e a c h s a m p l e is i d e n t i c a l to the theoretical concentration. Thus, the variance of the concentration becomes zero. a2 = 0
(2)
Although spot samples taken from the perfect mixture have an identical composition, such a mixture cannot be realized by means of a conventional mixing operation. We can define another final state, the so-called ' c o m p l e t e l y m i x e d ( o r r a n d o m ) s t a t e ' t h a t is d i f f e r e n t f r o m i d e a l h o m o g e n e i t y . I t is characterized by the fact that the probability of sampling a particle of one of the comp o n e n t s f r o m a n y p o i n t i n t h e m i x t u r e is identical, even though the composition may not be identical everywhere. For a homogeneous binary system, particles in a spot sample of size N are distributed according to the b i n o m i a l . d i s t r i b u t i o n , i.e.
(1) Pr[Y1 = nl] =
N!
nz ! (N--nz)!
PI,
(1--P1)":
(3)
w h e r e P1 is t h e w e i g h t f r a c t i o n o f c o m p o n e n t 1 i n t h e p o p u l a t i o n a n d Y1 is t h e r a n d o m variable that denotes the number of particles of component 1 in a spot sample. According to this probability density function, therefore, t h e v a r i a n c e o f t h e m i x t u r e is
76
o2 = V a t [ x 1 ]
-
(4)
PIP2
N which is the origin~ formula of Lacey [ 3 ] . In practice a mixing process rarely yields a completely mLxed mixture, either because the process is incomplete or because segregation occurs by differences in the physical properties of the components. This implies that the probability distributions of the c o m p o n e n t s during mixing are u n k n o w n . T h e probability o f sampling a particle o f a given c o m p o n e n t is p r o p o r t i o n a l t o t h e v o l u m e fraction of that component. Under the h y p o t h e s i s o f m i x t u r e s in t h e c o m p l e t e l y r a n d o m s t a t e , p a r t i c l e s in a s a m p l e c a n b e assumed to be a p p r o x i m a t e l y distributed according to the binomial distribution with p a r a m e t e r s N* a n d qz • w h e r e N* is t h e a v e r a g e n u m b e r o f p a r t i c l e s in a s a m p l e o f c o n s t a n t w e i g h t , W, a n d ql is t h e v o l u m e f r a c t i o n o f c o m p o n e n t 1. T h e v a r i a n c e o f t h e w e i g h t f r a c t i o n o f c o m p o n e n t 1 o r 2 is d e r i v e d as (see A p p e n d i ~ A ) 02_
PzP2
p2
W
PIP2
wlw2 (PlW2
(5)
+ P2Wl)
o~ = P-- ( 1 - - P 2 ) 2 P 2 + P ~ ( 1 P---P2) (7) P2 P2 w h e r e Pl a n d P2 d e n o t e t h e d e n s i t i e s o f c o m p o n e n t s 1 a n d 2, r e s p e c t i v e l y , a n d p t h e mean density. If the components have the s a m e p a r t i c l e d e n s i t y (Pl = Pl = P ) , (8)
0 2 = PIP2
3. MIXING
INDICES
FOR
MULTICOMPONENT
MIXTURES T h e p u r p o s e o f t h i s s e c t i o n is t o d e v e l o p mixing indices for a multicomponent mixture. We assume t h r o u g h o u t this section t h a t s a m p l e s are t a k e n f r o m a k n o w n v o l u m e o f a multicomponent mixture and the proportion ( b y w e i g h t ) o f e a c h c o m p o n e n t is r e c o r d e d . From the s~mple proportions, an indication o f t h e d e g r e e o f m i x e d n e s s will b e c a l c u l a t e d . Methods of sampling, determination of sample size, a n d t h e n u m b e r o f s a m p l e s d r a w n will d e p e n d o n t h e d e g r e e o f p r e c i s i o n requix.ed, the cost and practicality of sampling, and possibly physical constraints on the sampling process. 3.1
"Pseudo" binary
mixture
w h e r e W = w e i g h t o f a s a m p l e , w~, w2 = m e a n p a r t i c l e w e i g h t s o f c o m p o n e n t s I a n d 2, r e s p e c t i v e l y , a n d Px, P~ = p a r t i c l e d e n s i t i e s o f c o m p o n e n t s 1 a n d 2, r e s p e c t i v e l y . B e c a u s e , in p r a c t i c e , s o l i d p a r t i c l e s t o be m i x e d are u s u a l l y m e a s u r e d in t h e s a m e d i m e n s i o n s , this distribution seems to be a g o o d a p p r o x imation for most mixtures. Notice that eqn. (5) r e d u c e s t o e q n . (4) f o r a h o m o g e n c o u s system. Another approximate expression f o r t h e v a r i a n c e o f a m i x t u r e in t h e c o m pletely r a n d o m state was derived b y Stange [4]. Under the assumption that the probability o f s a m p l i n g a p a r t i c l e o f t h e given c o m p o n e n t is p r o p o r t i o n a l t o its v o l u m e f r a c t i o n , t h e v a r i a n c e f o r a m i x t u r e in t h e c o m p l e t e l y s e g r e g a t e d s t a t e is d e r i v e d h e r e as (see A p p e n d i x B)
Suppose that a multicomponent mixture c o n s i s t s o f c o m p o n e n t s 1, 2, 3 . . . . , k. I n describing the homogeneity of the key c o m p o n e n t , s a y , c o m p o n e n t 1, t h e m i x t u r e is c o n s i d e r e d t o c o n s i s t o f c o m p o n e n t 1 a n d another (pseudo) component x, which conrains the remaining ( k - - 1 ) components. C o m p o n e n t x is p r e s e n t in t h e m i x t u r e in a proportion of (1--/'1)- The particle weight a n d density o f c o m p o n e n t x are given, respect i v e l y , as
Uo2 = P-P---( 1 - - P ~ ) 2 P 1 + P 2 ( I -
T h u s , c o m p o n e n t x m a y b e t r e a t e d as c o m p o n e n t 2 o f a b i n a r y m i x t u r e ~- d e s c r i b e d in S e c t i o n 2. F o r i n s t a n c e , in t h e p h a r m a c e u -
Pl
Or
P---Pz) Pl
(6)
w~
=
P~ --+ 1112
~
=
(1 --t"1) /'3 + . . . + *-03
P~ Wk
(1 - - P , )
Pz +e3 +...+e~ P2
P3
(9)
(10)
Pk
77
tical industry, usually only one active ingredient and several diluents (lactose, starch, glucose, etc.) are in a dosage; therefore, the homogeneity of this active ingredient (key c o m p o n e n t ) is t h e m a i n c o n c e r n . T h e m i x i n g index for the key component may be determ i n e d as d e s c r i b e d i n t h e p r e c e d i n g s e c t i o n [5]. The present approach based on the concept of a pseudo-binary mixture can be applied to each component in a multicomponent mixture to define an index of mixedness for each c o m p o n e n t . S o m e t i m e s t h i s m e t h o d is m o ~ e useful than a tingle-valued index, because its application may indicate that a different d e g r e e o f m i x i n g is r e q u i r e d f o r e a c h c o m p o n e n t , i.e., s o m e c o m p o n e n t s m a y b e required to be mixed to a much greater extent than others [6]. 3.2 Pooled variance The unbiased sample variance of the concentration in a multicomponent mixture is d e f i n e d as f o l l o w s : a2 -
m --
i=~=l
(xii --x'i.)2Pi
(11)
where h = number of components, x o = fraction of component i in the/th spot sample, a n d ~i. = s a m p l e m e a n o f c o m p o n e n t i, =
m
$=1
When h = 2, eqn. (11) can be reduced to eqn. (1), which expresses the variance of a binary mixture. Similarly, we may express the variances for mu/ticomponent mixtures in the completely mixed and segregated states, respectively, as a~ =
E
--
i=1
Pi
P~(I--
--
P,)
homogeneous particles system, eqns. (12) and (13) can be reduced, respectively, to
02
- -
1
k
N
i=1
~] P ~ ( 1 - - P i )
(14)
h
= ~ Pf(1--P,)
o o~
(15)
i=1
Thus, we have a o2 = N a 2
(16)
Accordingly, the relationships between t h e s e m i x i n g i n d i c e s s h o w n in T a b l e 1 a r e i d e n t i c a l t o t h o s e f o r b i n a r y m i x t u r e s [ 2 ] . In Figs. I and 2 the values of the nine mixing indices are plotted against one of them, namely, Ms, for the samples of sizes 20 and 200 (no. of particles), respectively. Then M I a n d M 6 a r e l i n e a r l y r e l a t e d t o Ms. M I , M 4 , a n d M5 a p p e a r t o b e m o r e d e p e n d e n t u p o n the sample size than the others. However, only M7 is s e n s i t i v e t o t h e v a r i a t i o n in t h e t , x t e n t of mixedness in the range from the completely segregated state to the completely mixed state. 3.3 Determinant of the sample covariance matrix M u l t i v a r i a t e a n a l y s i s is a n e f f e c t i v e t o o l f o r solving multicomponent solids mixing problems [7]. Instead of being concerned with the concentration of a single component, sampling o f a k - c o m p o n e n t m i x t u r e is p r i m a r i l y c o n c e r n e d w i t h r a n d o m s a m p l e s o f (h - - 1) v e c t o r valued random response variables. Notice that o n l y (k - - 1) o f t h e k c o n c e n t r a t i o n s n e e d s t o
0 8
(12)
Pi
and
Now, we are ready to define mixing indices for multicomponent mixtures as similar to t h o s e f o r b i n a r y mixt-~tres i n T a b l e 1 . F o r t h e
00
0Z
0q HIXING
08 INDEX
"
0
Hi
Fig. 1 Comparison of mixing indices based on the 2nd approach (pooled variance) for sample size of 20.
78 1.0
T h e e s t i m a t e d v a r i a n c e s o f x~ a n d c o v a r i a n c e o f x i a n d x~. a r e , r e s p e c t i v e l y ,
~o.
MQ
i = I, 2 ..... ( k - - i )
SS~i
-
m -- 1"
(19)
and
~:o.q
x~]
Cov[xl.
= s~Q
0.2 _
SPxix~ m--1
o.';z
o.o
'
o1.
'
0%
"
ale
'
i, £ - - - - 1 , 2 . . . . , ( k - - l '
"
(i ~ ~
*'a
MIXING INOEX M,
Fig. 2. Comparison o f mixing indices based o n the 2nd approach (pooled variance) for sample size of 200. be considered. When s3mples are obtained from a (k -- 1)-van:ate population instead of a univariate population, the data can be r e p r e s e n t e d as
Thus, the sample variance-covariance t r i x is
SI2
Sample no.
i
...
(k--l)
1
Xll
X21
" - -
Xil
- - -
/(k--1)1
2
X12
X22
. . .
Xi2
. . .
X(k_l)
]
Xl I
XZ]
. . .
Xij
. . .
X(Iz--1)]
I,"~
Ilm
X2m
...
Xim
...
X(k_l)rn
-
"~1 ( k - - l )
-
.
S2(k--l)
$2(k--2)
.
ma-
S=
k S1 (k--l)
Component 1 2 ...
322
(20)
2
S(k--1)(k--1)
(21) For a homogeneous system, particles in a completely random mixture are expected to be randomly distributed in a spot sample of size N taken from the mixture. Let us de~ne a r a n d o m v e c t o r [ Y1, Y2 . . . . . Yk ] f o r a m i x t u r e o f k c o m p o n e n t s , w h e r e 1/] = n j , 0 < n j < NAT.T h i s i m p l i e s t h a t n j p a r t i c l e s o f c o m ponent] fall into the spot s3mple. Then the joint probability density function of random v a r i a b l e s , Yz, Y2 . . . . . Irk, is [ 9 ]
P r [ Y 1 = n l , Yz = n 2 . . . . . Irk = n k ]
N'. nz!
The sums of squares and cross products are d e f i n e d , r e s p e c t i v e l y , as [ 7 ]
n~,! . - . n k !
N! h
SS=,= ~
(xu--zi.)
x~----
i I,~i 712
x u )2
(17)
where nl +n2 +... +nk
and
P,
~ (xu--z~.)(x~--~~ i=I
----
(22)
i=Z
Zm j=l
SP~=
p~-i
i=i
H nil
2
i=1 =
', H
~.
, i=1
x i i x gJ
--
--
m
=
Xtl
X
8)
= N
+ P z + . . . + P,~ = I
Pi = weight fraction of the ith component in the population (= number fraction of the/th component in the population for a homogeneous system). Note that the multinomial d i s t r i b u t i o n is a g e n e r ~ i i T ~ t i o n o f t h e b i n o m i a l d i s t r i b u t i o n . A c c o r d i n g t o f~i=~ p r o b a b i l i t y
79 d e n s i t y f u n c t i o n , e q n . ( 2 2 ) , t h e variane~_s a n d c o v a r i a n c e s o f t h e r a n d o m v a r i a b l e s , Y1, Y2 . . . . Yk, a r e g i v e n , r e s p e c t i v e l y , b y [ 8 , 9 ]
•
var[Y,l = NP,(1 --P,)
(23)
Y/, Y~ ] = --NPiP~
(24)
Coy[
or equivalently,
P~(I --P~)
,
i = 1, 2 .....
(k -- 1)
(25)
N C ° v [ x i " x ~ ] = C ° v I - ~ " NYQI'
P~P~
--
i, ~ = 1 , 2
....
,(k--l)
(i¢
£)
(26)
N " Therefore, t h e v ar i an ce- co v ar i an cem a t r i x for t h e m i x t u r e in t h e c o m p l e t e l y m i x e d s t a t e can
--]
b e e x p r e s s e d as
~_
( l - - PI ) Pe
--P1 P( k -1 )
--PI P2 P2(1 - - P 2 )
I
1
(27)
N °
L--P1P(k --1)
--P2 P(k --1 )
P ck - i ) ( 1 - - / } ( k - 1 ~)_1
In the completely segregated state, each component this state, we have [8, 9] Var[x~l = P i ( 1 - - P i ) ,
is s e p a r a t e d a c c o r d i n g t o p a r t i c l e t y p e s . I n
i = 1, 2 . . . . .
(k - - 1)
(28)
and Cov[xi, x~] = --PiPe
i, £ = 1, 2 . . . . .
(k - - 1) a n d i ~ ~
(29)
Therefore, t h e v a r i a n c e - e o v a r i a n c em a t r i x f o r a m u l t i c o m p o n e n t m i x t u r e in t h e c o m p l e t e l y
s e g r e g a t e d s t a t e c a n b e e x p r e s s e d as ~/}: P1) Es
=
--/}1/}2 P 2 ( 1 - - P2)
"
[--P1P,~-I,
--P1Pc~-I) "--P2P(a-1)
]
"
P2P,~-I,
(30)
~ % - 1 , ( 1 - - Pc~-l,)_J
T h e s a m p l e v a r i a n c e - c o v a r i a n c e m a t r i x , S, c h a r a c t e r i z e s t h e d e g r e e o f t h e d i s p e r s i o n o f c o n c e n t r a t i o n o f e a c h c o m p o n e n t . T h u s , t h e d e t e r m i n a n t o f t h i s m a t r i x , ISI, r e f l e c t s t h e e x t e n t o f t h e v a r i a t i o n in t h i s s y s t e m ( m i x t u r e ) s i m i l a r t o t h e v a r i a n c e in t h e u n i v a r i a t e a n a l y s i s . T h e r e f o r e , r e p l a c i n g a 2, o rz a n d a~ b y IS[, IZrl, a n d IZs[, r e s p e c t i v e l y , w e c a n a l s o d e f i n e t h e m i x i n g i n d e x f o r a multicomponent mixture. Thus, we can extend the mixing index
a2 --uz
(31)
80 proposed by Lacey [10] for a binary mixture to a multicemponent mixture by defining
M3-
ISI - - l ~ r I 1~I--1~1
(32)
O t h e r e x a m p l e s are g i v e n in T a b l e 1. N o t e that the expressions for muiticomponent m i x t u r e s are r e d u c e d t o t h o s e f o r b i n a r y mixtures. A c c o r d i n g to eqns. (27) and (30), we have E s =N~ r
Var[xi]
-
W Pi
1 --
P~
i = 2, 2, ..., (k - - 1 )
(33)
(35)
and
and iEs I = N ~ - ~ I ] E r I
s a m p l e o f c o n s t a n t w e i g h t , W, a n d qi is t h e v o l u m e fraction o f t h e ith c o m p o n e n t . The variances and covariances of weight f r a c t i o n s o f c o m p o n e n t s in a m u l t i c o m p o n e n t ~ mixture, having this probability density f u n c t i o n , are d e r i v e d , r e s p e c t i v e l y , as (see A p p e n d i x C)
p2
(34)
The relationships among the mixing indices t a b u l a t e d in T a b l e 1 are e v a l u a t e d in T a b l e 2. I n Figs. 3 a n d 4 t h e v a l u e s o f t h e n i n e m i x i n g inc:ices are p l o t t e d a g a i n s t t h a t o f M s b a s e d on the determinant of the covariance matrix a p p r o a c h f o r t h e s a m p l e s o f sizes 2 0 a n d 2 0 0 , respectively. A mixing index should be sensitive t o c h a n g e s in t h e q u a l i t y o f t h e m i x t u r e in o r d e r t o b e u s e f u l in f o l l o w i n g t h e prc~e_~= o f m i x i n g . A s h t o n a n d V a l e n t i n [ 1 1 ] pointed out that Lacey's index, eqn. (32), w a s v e r y sensitive t o c h a n g e s n e a r t h e c o m pletely segregated state and relatively insensitive d u r i n g t h e p r o c e s s o f m i x i n g n e a r t h e completely mixed state. Their more complic a t e d i n d e x , MT, w a s t h u s d e v i s e d t o e x h i b i t greater sensitivity. Mixing indices defined b y the d e t e r m i n a n t of the sample variance-covariance matrix for multicomponent mixtures can be generalized as in T a b l e 1, w h e r e ~ r a n d e s a r e t h e variance-covariance matrices of weight proportion in t h e c o m p l e t e l y m i x e d a n d c o m p l e t e l y segregated states, respectively. T h e developm e n t o f ~ r a n d e s p r e s e n t e d so f a r w a s c o n cerned only with homogeneous systems. For the heterogeneous system, we assume that the p r o b a b i l i t y o f picking a particle o f a given c o m p o n e n t is p r o p o r t i o n a l t o t h e v o l u m e fraction of that component and the probability o f f i n d i n g a p a r t i c l e o f o n e c o m p o n e n t is i d e n t i c a l e v e r y w h e r e in a c o m p l e t e l y r a n d o m mixture. Thus, particles t a k e n f r o m a s p o t s a m p l e are d i s t r i b u t e d a p p r o x i m a t e l y a c c o r d ing to the m u l t i n o m i a l distribution with t h e p a r a m e t e r s , _N* a n d qi (i = 1, 2 . . . . . k), where _N* is t h e a v e r a g e n u m b e r o f p a r t i c l e s in a s p o t
Cov[xi,
x~]
-
- -
W
P/P~,
PlP~
i, £ = 1, 2 . . . . , (k - - 1 ) a n d i ~
£
(36)
This establishes the variance-covariance m a t r i x f o r a m u l t i c o m p o n e n t m i x t u r e in t h e c o m p l e t e l y m i x e d state, ~ r . Similarly, the variances and covariances of the weight f r a c t i o n s o f t h e c o m p o n e n t s in a m u l t i c o m p o n e n t m i x t u r e in t h e c o m p l e t e l y s e g r e g a t e d s t a t e are given, r e s p e c t i v e l y , b y (see A p p e n d i x D) P
var[ ,]
(1-
=
Pi
i = 1, 2 . . . . . (k - - 1)
(37)
and
i, £ --- 1, 2 . . . . . (k - - 1) a n d i ~ £
(38)
This establishes the v a r i a n c e - c o v a r i a n c e m a t r i x f o r a m u l t i c o m p o n e n t m i x t u r e in t h e completely segregated state, Es. Notice that the variances and covariances for a heterogen e o u s s y s t e m are r e d u c e d t o t h o s e f o r a homogeneous system by setting P]L
=P2
=
• - -=Pk
and W 1
--~ W 2
-~-- . . .
W k .
3.4 Absolute deviations from the population me~iT~
The mixing index may be defined by using the absolute deviation from the population
81 A I--4
I II
T
°~
I
i.--I
'-'
I
,-~m
I
~T 7
I
I
.r--I
_= t ~.
!
L~
+
I
v-I
~l
'' I ~
I
I,--41. ~
T--,
I
!
I
i-.,I
v-i
,,-..I
1 0 I
I
I I
.,-t.
I i--.I
I I--I
~L
I 1-1
I o~ ~ilV-.
I I i.-I
!
I
I i..-I
~T
! i.-i
I
82
l I
-li
#
I
I c~
I
I~.
i~ l
l
I
T--I
I
v.-I
T-I
i
co
I
I i
J
I
I
~l ~ ~l ?
I I
r-.I
T-I T-'I
i- l
I
!
I
?
•
I
c~
I I
I
lr-I c~
E.-,
~f
0
I
I
83
I
I v~ !
!
!
i
m
v~
I
r
I
I !
!
m
!
!
I
I
I
I
I
v~
v~
1-1
•
!I~ !~1~1~
i
v ~
I
I I
!
I c~ C~IL~ e.t
!
!
I
0 I C~
c~
~G
#1~
84 z.o
.
,
.
sample mean. Throughout the whole system, t h e c o n c e n t r a t i o n o f t h e p e r f e c t m i x t u r e is exactly equal to the theoretical value, and therefore the mixing index of the perfect m i x t u r e is 1 a c c o r d i n g t o e q n . ( 3 9 ) . I f a m i x t u r e is i n t h e c o m p l e t e l y s e g r e g a t e d s t a t e , i t s m i x i n g i n d e x is 0 . T h e r e f o r e , t h e m i x i n g index, eqn. (39), always takes a value between 0 and 1 for a mixture in an arbitrary stale. This definition can also be extended to a multicomponent system as
M
z: °'8
a
~1
~ x
°"° -
/e
.
.
.
,
.
:
:
.
MIXING INDEX Nr
i~1
Fig. 3. Comparison o f mixing indices based on the
3rd a p p r o a c h (the d e t e r m i n a n t o f t h e covaxiance matrix} for sample size o f 20. mean of each individual component instead of t h e v a r i a n c e w h i c h is c o n v e n t i o n a l l y u s e d t o define the mixing index. :For instance, the following expression can be used as a mixing index for a binary mixture: rn
[Xl1 --Pz Mzz
l
(39)
= I --
Pz The denominator o f t h i s e q u a t i o n is t h e total sum of the absolute differences between the sample concentrations and the population concentration among the spot samples from a mixture in the completely segregated state (see Appendix E). In case the population mean is n o t s p e c i f i e d , i t c a n b e r e p l a c e d b y i t s 1.0
0.0
rn
o
.
.
.
~
.
.
.
.
~.l:t
.
M2
x ~o.
i
o.2
o.
13
MIXING INDEX M,
Fig. 4. Comparison of mixing indices based on the 3rd a p p r o a c h ( t h e d e t e r m i n a n t o f t h e c o v a r i a n c e matrix) for sample size of 200.
M l l
=
1
j=l
-/e
i=1
p
LPi
Pi
/
(40)
4. C O M P A R I S O N O F D I F F E R E N T D E F I N I T I O N S OF MIXING INDICES Computational effort was minimized by choosing a ternary particle system to demonstrate the use of the proposed mixing indices for multicomponent mixtures. The values of the degree of mixedness evaluated according to different definitions were compared. An actual mixture was generated by a drum mixer which was constructed from a lucite tube with a diameter of 7 cm and a length of 38 cm :nd fitted with two lucite end flanges, each with a diameter of 12.7 cm. Particles of three different types were used. Their physical properties are summarized in Table 3. Prior to mixing trials, two thin semicircular partitions were placed between the two ends of the mixer, normal to the mixer axis. They divided the mixer into three compartments of equal length. 200 g of component 1, 150 g of component 2 and 150 g of component 3 were loaded in the middle, right and left compartments, respectively. Then the bed was leveled, the partitions were removed, the cover was put in place, and the mi~er was rotated. For this experiment, the speed of the mixer was set at 30 r.p.m., and sampling was carried out at an interval of 10 minutes after the onset of rotation. With sample divider, the mixture was divided into ten sections. Each sample from each section contained 20 grams of particles. Mi~ing indices were calculated from the experimental
85 TABLE 3 P h y s i c a l c h a r a c t e r i s t i c s o f particles
Component
Material
Color
Particle diameter in. ( c m )
D e n s i t y , Pi ( g / c m 3)
Average w e i g h t , Wi (g)
1 2 3
steel lucite
silver red
lucite
white
0.!875 (0.4763) 0.1819 (0.4620) 0.1618 (0.4110)
7.60 1.18 1.18
3.438 0.487 0.343
(3) Determinant of the sample covariance matrix T h e m a t r i c e s , S, ~ r a n d ~ s , w e r e c o n structed and their determinants were then evaluated. According to the definitions in T a b l e 1, t h e i n d i c e ~ w e r e d e t e r m i n e d . T h e r e s u l t s a r e a l s o s h o w n i n T a b l e 4. (4) Absolute deviations from the population means We first calculated the sum of the absolute deviations from the population means for a mixture in the completely segregated state. Then, the sum of the absolute deviations between sample concentrations and the population means was determined. According to eqn. (40), the mixing index was computed as 0 . 6 9 9 f o r t h i s m i x t u r e .
d a t a , w h i c h a r e s u m m a r i z e d i n T a b l e 4. T h e p r o c e d u r e s a r e b r i e f l y d e s c r i b e d as f o l l o w s : (1) Pseudo-binary mixture P a r t i c l e s o f t y p e 1 w e r e a s s u m e d as t h e k e y component, and particles of types 2 and 3 w e r e j o i n t l y c o n s i d e r e d as t h e p s e u d o ~ c o m ponent. The particle weight and density of this pseudo~component were calculated from eqns. (9) and (10), respectively. The mixing indices were calculated according to ten of the more frequently used criteria given in T a b l e 1, a n d t h e r e s u l t a n t v a l u e s o f i n d i c e s a r e l i s t e d i n T a b l e 4. (2) Pooled variance The pooled variance of weight fractions of the components in a multicomponent mixture was calculated from eqn. (11). The variance f o r t h e m u ! t i c o m p o n e n t m i x t u r e in t h e completely mixed state, 02 , and that for the mixture in the completely segregated state, o2, w e r e c a l c u l a t e d a c c o r d i n g t o eqns. (12) and (13), respectively. Mixing indices were c o m p u t e d b y s u b s t i t u t i n g t h e v a r i a n c e s , or2 a n d a ~ , i n t h e e x p r e s s i o n s g i v e n i n T a b l e 1. Results are also shown in Table 4.
5. C O N C L U D I N G R E M A R K S
A s s e s s m e n t o f t h e q u a l i t y o f a m i x t u r e is one of the most important aspects in the investigation of solids mixing. The concept of homogeneity of dispersed (or heteroge-
TABLE 4 A ternary mixture and resultant values of mixing indices of this mixture according to approaches I, 2 and 3
Component Sample
1
2
3
1 2 3 4 5 6 7 8 9 10
0.345 0.450 0.410 0.390 0.475 0.565 0.520 0.610 0.575 0.355
0.450 0.075 0.150 0.275 0.275 0.225 0.295 0_250 0.325 0.585
0.205 0.475 0.440 0.335 0.250 0.210 0.185 0.140 0.100 0.060
Experimental data expressed in weight fraction.
Mixing index
Approach 1
Approach 2
Approach 3
M1 M2 M3 M4 M5 M6 M7 MS M9 /TflO
0.775 0.949 0.964 0.541 0.477 0.225 0.842 0.882 0.051 1.848
0.742 0.934 0.957 0.605 0.532 0.258 0.880 0.880 0.066 1.653
0.937 0.996 0.996 0.343 0.329 0.063 0.958 0.958 0.040 2.915
Note: approach 1 --pseudo-binary mixture; approach 2pooled variance; approach 3 -- determinant of eovarianee matrix.
86
n e o u s ) p a r t i c l e s y s t e m s is u s e f u l f o r t h e examination of the mixing and dispersion process and for the elucidation of problems t h a t are o f p r a c t i c a l s i g n i f i c a n c e . Although much effort has been expended o n this s u b j e c t , it still l a c k s g e n e r a l l y a c c e p t e d i n d i c e s t o assess t h e h o m o g e n e i t y o f a h e t e r o geneous multicomponent mixture. Four app r o a c h e s , w h i c h h a v e b e e n p r o p o s e d in this s t u d y , are o f universal a p p l i c a b i l i t y . T h e approximate joint distribution of random v e c t o r [Y1, Y2. . . . . Yk] f o r a m u l t i c o m p o n e n t m i x t u r e in t h e c o m p l e t e l y r a n d o m s t a t e has b e e n d e r i v e d . T h e u s e o f v a r i o u s mixing indices for multicomponent mixtures has b e e n s h o w n t h r o u g h a t e r n a r y m i x t u r e over a wide range of physical properties. The proposed expressions of mixing indices for m u l t i c o m p o n e n t mixtures reduce t o t h o s e f o r a b i n a r y m i x t u r e . Ar, y m i x i n g i n d e x f o r a h o m o g e n e o u s s y s t e m is a s p e c i a l case o f t h a t f o r a h e t e r o g e n e o u s s y s t e m . The choice of a mixing index depends on t h e a p p l i c a t i o n o f m i x i n g p r o c e s s . T h e relationships among the ten most frequently used i n d i c e s b a s e d o n t h e s e c o n d a n d t h i r d app r o a c h e s are evaluated. T h e m i x i n g i n d e x defined by using the absoIute deviations from the population means may have a smaller r e l a t i v e s t a n d a r d e r r o r . C l e a r l y , f u r t h e r inv e s t i g a t i o n is n e c e s s a r y i n t o t h e r e l a t i o n s h i p b e t w e e n the mixing indices based on different a p p r o a c h e s . F u t u r e s t u d i e s will c o n s i d e r t h e relative p e r f o r m a n c e o f t h e m i x i n g i n d i c e s d i s c u s s e d in this w o r k .
ACKNOWLEDGEMENT T h e authors wish to a c k n o w l e d g e the financial s u p p o r t o f the National Science Foundation (Grant ENG 73-04008A02).
N*
a v e r a g e n u m b e r o f p a r t i c l e s in a sample of constant weight, W no. o f particles o f c o m p o n e n t ] in a s p o t s a m p l e weight proportion of component j in t h e p o p u l a t i o n volume fraction of component j sample variance-covariance matrix s u m o f c r o s s p r o d u c t s o f xi a n d xQ s u m o f squares o f xi variance of weight fraction of component i covariance of weight fraction between components i and total weight of the mixture total n e t v o l u m e o f solid particles in t h e m i x t u r e variance o f the r a n d o m variable
n1
q, S
SP~ix ~ SS~ i 8ii
T V
var[Y~]
Y,. weight of a spot sample mean particle weight of component/ a r a n d o m variable d e n o t i n g the weight fraction of component i in a s a m p l e weight fraction of component i in t h e j t h s p o t s a m p l e sample mean of component i a r a n d o m variable d e n o t i n g the n u m b e r o f particles o f c o m p o n e n t ] in a s p o t s a m p l e mean density density of component i variance-covariance matrix of a m i x t u r e in t h e c o m p l e t e l y m i x e d state variance-covariance matrix of a m i x t u r e in t h e c o m p l e t e l y segregated state sample variance of weight fraction v a r i a n c e o f a m i x t u r e in t h e completely segregated state variance of a perfect mixture v a r i a n c e o f a m i x t u r e in t h e completely mixed state
W
wi xi
x~
rl p pl
X,s Er2
a~
LIST OF SYMBOLS .~
N k-1
Cov[Y~, Yj-] c o v a r i a n e e b e t w e e n t h e r a n d o m variables Y / a n d Y1 k no. of components M mixing index m no. of spot samples N n o . o f p a r t i c l e s in a s p o t s~rnple
REFERENCES 1 L . T . F a n , S . J. C h e n a n d C. A . W a t s o n , S o l i d s m i x i n g , I n d . Eng. C h e m . , 6 2 ( 7 ) ( 1 9 7 0 ) 5 3 . 2 L. T . F a n a n d R . H . W a n g , O n m i x i n g i n d i c e s , Powder
Te~h--ol., 1 1 (1975) 27.
87 3 P. M . C. L a c e y , D e v e l o p m e n t i n t h e t h e o r y o f particle mixing, J. Appl. Chem., 4 (1954) 257. 4 K. Stange, Die Mischgute einer Zufalismischung als G r u n d l a g e z u r B e u r t e i l u n g y o n M i s c h v e r suc-hen, C h e m . I n g . T e c h . , 2 6 ( 1 9 5 4 ) 3 3 1 . 5 H.G. Kristensen, Statistical properties of random and non-random mixtures of dry solids. Part I. A general expression for the variance of the composition of samples, Powder Technol., 7 (19731 249. 6 C. S c h o f i e l d , T h e d e i ' m i t i o n a n d a s s e s s m e n t o f mixture quality in mixtures of particulate solids, Powder Technol., 15 (19761 169. 7 C. Y . K r a m e r , A F i r s t C o u r s e i n M e t h o d s o f Multivariate Analysis, Virginia Polytechnic Institute, Blacksburg, Virginia, 1972. 8 R.H. Wang, L. T. Fan and J. R. Too, Multivariate statistical analysis of solids mixing, Powder Technol., 21 (19781 171. 9 M. Fisz, Probability Theory and Mathematical Statistics, Wiley, New York, 1967, p. 347. 1 0 P. M . C. L a c e y , D e v e l o p m e n t i n t h e t h e o r y o f particle mixing, J. Appl. Chem., 4 (19541 257. 1 1 M. D . A s h t o n a n d F . H . H . V a l e n t i n , T h e m i x i n g of powders and particles in industrial mixers, rrrans. Inst. Chem. Eng., 44 (1966) T166. 1 2 S. R . Miles, H o m o g e n e i t y o f s e e d l o t s , I n t . S e e d Test. Assoc. Proc., 27 (1962) 407. 1 3 P. M . C. L a c e y , T h e m i x i n g o f s o l i d p a r t i c l e s , Trans. Inst. Chem. Eng., 21 (1943) 53. 1 4 S. S. W e i d e n b a u m a n d C. F . B o n i l l a , A f u n d a m e n tal study of the mixing of particulate solids, Chem. Eng. Prog., 51 (1955) 27. 15 J.P. Beaudry, Blender efficiency, Chem. Eng., 55 (1948) 112. 1 6 T . X a n o , I. K a n i s e a n d K . T a n a k a , M i x i n g o f powders by the V-type mixer, Kagaku Kogaku, 20 (1956) 20. 1 7 M . H . W e s t m a c o t t a n d P. A . L i n e h a n , M e a s u r e ment of uniformity in seed bulks, Int. Seed Test. A s s o c . P r o c . , 2 5 ( 1 9 6 0 j 1_5 1 . 18 G. Herdan, Small Particles Statistics, Butterworths, London, 2nd edn., 1960.
PIT
V -
P2T
+ --
Pl
(A.1)
P2
The mean particle density, p, can be defined as
T
p--
V
OlP2
=
(A.2)
PxP2 + PzPl
When sampling, the probability of taking a p a r t i c l e is p r o p o r t i o n a l t o t h e v o l u m e f r a c t i o n o f t h a t c o m p o n e n t . P a r t i c l e s in t h e c o m p l e t e l y mixed state are assumed to be distributed according to the binomial distribution with p a r a m e t e r s N* a n d q l , w h e r e 24* is t h e a v e r a g e n u m b e r o f p a r t i c l e s in a s a m p l e o f c o n s t a n t w e i g h t , W, a n d q l is t h e v o l u m e f r a c t i o n o f c o m p o n e n t 1. The variance of weight fraction of comp o n e n t 1 c a n b e e x p r e s s e d as
a~ = Var[xl ]
~z -
=
We
-Vat[Y1 ]
~2 --'N*ql(1--ql)
W2
= w2 W2
w . o___p1, o__p~ w
Pl
P2
Thus, a 2 = 1"11"2
p2
W APPENDIX
A
Derivation
of eqn.
,olp 2
w x -w 2
(PI~2
-b P 2 W l )
(5)
(5)
Consider a mixture of two components, 1 a n d 2, w h i c h axe p r e s e n t i n t h e w e i g h t p r o p o r t i o n , / ' 1 a n d / ' 2 (/'1 + / ' 2 = 1 ) . S u p p o s e t h a t components 1 and 2 have particle densities P l a n d P2, r e s p e c t i v e l y , a n d also s u p p o s e t h a t the weight and volume of the mixture are T a n d V, r e s p e c t i v e l y . N o t e t h a t t h e v o l u m e , V , is t h e n e t v o l - m e o f s o l i d s , a n d n o t t h e t o t a l a p p a r e n t vol, , m e o f t h e m i x t u r e m a s s i n c l u d i n g the space between the particles. Therefore, V can be expressed as
APPENDIX
B
Derivation
of eqns. (6) and (7)
B e f o r e m i x i n g s t a r t s , a m i x t u r e is a s s u m e d to be in the completely segregated state. A m o n g t h e m s p o t s.tnaples t a k e n , m l s ~ m p l e s c o n t a i n o n l y p a r t i c l e s o f c o m p o n e n t 1, a n d the remaining (m --ml) contain only particles o f c o m p o n e n t 2. A s s n m e t h a t t h e p r o b a b i l i t y o f t a k i n g a p a r t i c l e o f a g i v e n c o m p o n e n t is proportional to its volume fraction, i.e.,
88 rn I = r n q 1
(B.1)
The sample variance can be obtained eqn. (1), in which Xl
APPENDIX
=P1
rn
i=1
1 ~, _
rft
i=1
1
[~,
(x~
-
-
p~)2
( x 1 , __ p 1 ) 2 1 = , , = 1
+
Li=l rn - - m l
(Xli
Z
--
I)1)2 l xt, = O]
i=I
1
--
-
[(1--P1)2rnx
1 and
2 are
C
Derivation of eqns. (35) and (36) Consider a multicomponent mixture having k components for which we defined the following for i = 1, 2, . . . , k: P t = density of component i W--~ = p a r t i c l e w e i g h t o f c o m p o n e n t t P~ = w e i g h t p r o p o r t i o n o f c o m p o n e n t i qi = volume fraction of component i Under the assumption as described in Subsection 3.3, particles taken from a spot sample are distributed approximately according to the multinomial distribution with p a r a m e t e r s AT* a n d qi- T h e r e f o r e , t h e v a r i a n c e o f w e i g h t f r a c t i o n o f c o m p o n e n t i is [ 9 ]
1 ~_, (xli__~i.)z
_
of components
from
Thus,
o2°
the variances identical.
+p2(m--ml)]
(B.2)
Var[x~]= Vat [ - ~ ]
m ~2
Since m I
w ~ V a t [ Y~]
= mql 0 2
-- - - / V * q ~ ( 1 W
=m
P
-- qi)
VlPl
ID 1
Vp
Thus,
=m~P~
(B.3)
Pl
V a r [ x d =
we have
--
--
i = 1, 2 .... , (k--
a 2 = --I[(1--pi)2m P---PI +Pi2(l-m
Pl
P P1)m l PI
(6)
C o v [ x i , x~] = Coy t ~ Similarly, we may also derive the variance of weight fraction for component 2 as o 2
= 192 p - - - ( 1 - P z ) 2 P 2 + P z ( 1 - ~ 2 p z
)
(7)
-
wz w2
C o y l Y . YA
O 2
- vv~ ( --N'q~qD Since p--
.OLP2
PlP2 +P2Pl
(35)
The covariance of weight ~-actions between the ith and ]th component can be expressed as
[91
- P--(I--pI)2pI + p 2 ( I - - ~ P1
i)
O 2
'
89
Thus, w
Cov[x. x./]
=
W
p 2
--
P-:Pi
i , j = l , 2 , ..., ( k - - 1 )
P,P,
Thus,
and i~-]
(36)
D
APPENDIX
i, I~ = 1 ,
Derivation of eqns. (37) and (38) For a multicomponent mixture in the completely segregated state, the derivation of the variance of weight fraction for each component is similar to that for a binary mixture (see Appendix B). Thus, we have Var[x/]
C o v [ x i , x~] = P i P , f 1
= P-~-(1--P/)2P/+p2(1--p~-.P/),
i=1,2,...,
Pi
(37)
"
Under the assumption Appendix A, we have
as described
in
mi = mqi
(D.1)
and
APPENDIX
Cov[xi, x ~ ] = -
1
m 2:
D2
m
andi¢
(38)
£
E o
= mql
(D.2)
where rni denotes the number of samples containing only particles of component i, a n d m ~ t h a t o f c o m p o n e n t ~. T h e c o v a r i a n c e o f weight fractions between components i and is e x p r e s s e d a s
1
2,...,(k--I)
P,P]
SUm of the absolute differences between the s a m p l e c o n c e n t r a t i o n s a n d its p o p u l a t i o n concentration For a binary mixture in the completely segregated state, we assume that m 1 samples contain only particles of component 1 among the m spot samples to be taken. Under the same assumption imposed in Appendix B, we have In 1
m ~ = rnq~
PiP
Therefore, the sum of the absolute differences between the sample concentrations and i t s p o p u l a t i o n c o n c e n t r a t i o n is
~.
[Xl]--P1 ] =
,j=l
[Xlj--Pl]Ix,j=l
1=1
(xij--~.)(x~--x~.)
j=l
m! ~
+
m--m t ~, i X l i - - P 1 I [x,./=O ./=1
m
2~ (xij - - P D ( x ~ --P~)
= (1 --Px)mx
.i=1
= (l--Px)mql
m i
+Pl(m
--ml)
+Pxm(l
--qx)
( ~ - - P~)(x~ --Pz)[~i=~. ~j=o m
•
+
=(I--PI)m P--PI+Ptm(I-- ~
Z 1 (xo - - PD(x~s --P~) l~ij= o. x~s=*
--P,) +
--
(E.I)
m --mi--rn ~ j=l
-
= mPx -~x(I
1 --
[mi(1
(x,j --e~) (x~ --e~)]x,i: o. ,.u:o] --PD("--P~)
+ rr~(--~)(1
m
--P~)
If the s y s t e m is a h o m o g e n e o u s equation reduces to
system, this
m
~_, I x l i - - P , l
= 2raP1(1 --P1)
(E.2)
1=1
+ (m --m i -- m~)PiPd -
1
- - [mP~Pe - - miP~ - - m ~ P i ] m
Equations (E.1) and (E.2) are employed deriving eqn. (39).
in
73
~) E l s e v i e r S e q u o i a S ~ A . , L a u s a n n e - - P r i n t e d in t h e N e t h e r l a n d s
Studies on Multicomponent Part HI. Mixing Indices
Solids Mixing and Mixtures
L. T. FAN, J. R. TOO Department of Chemical Engineering, Kansas State University, Itfanhattan, Kans. 66506 (U.S.A.) R . M . RUBISON
Department of Statistics, Kansas State University, Manhattan, Kans. 66506 (U.S.A.) and F. S. LAI U.S. G r a i n M a r k e t i n g R e s e a r c h C e n t e r , S c i e n c e & E d u c a t i o n A d m i n i s t r a t i o n , D e p a r t m e n t o f A g r i c u l t u r e , 1 5 1 5 College Avenue, Manhattan. Kans. 66502 (U.S.A.) (Received
January 15, 1979; i n r e v i s e d form April 25, 1979)
SUMMARY
The literature on the assessment of the q u a l i t y o f s o l i d s m i x t u r e s is s c a r c e , e s p e c i a U y for multicomponent mixtures. Four app r o a c h e s t o assess t h e h o m o g e n e i t y o f a m u l t i c o m p o n e n t solids m i x t u r e have been p r o p o s e d in t h i s s t u d y . T h e m i x i n g i n d i c e s p r o p o s e d a r e e s t i m a t e d o n t h e basis o f t h e (a) p s e u d o - b i n a r y m i x t u r e c o n c e p t , (b) p o o l e d v a r i a n c e o f t h e w h o l e s y s t e m , (c) d e t e r m i n a n t o f t h e s a m p l e c o v a r i a n c e matrix, or (d) a b s o l u t e d e v i a t i o n s f r o m t h e p o p u l a t i o n means.
T h e use o f v a r i o u s m i x i n g i n d i c e s f o r m u l t i c o m p o n e n t mixtures has been s h o w n t h r o u g h a temary mixture.
1. I N T R O D U C T I O N S o l i d s m i x i n g o r p o w d e r b l e n d i n g is t h e o p e r a t i o n b y w h i c h t w o o r m o r e solid materials i n p a r t i c u l a t e f o r m a r e s c a t t e r e d r a n d o m l y in a m i x e r b y t h e r a n d o m m o v e m e n t o f t h e p a r t i c l e s . T h i s o p e r a t i o n is p e r f o r m e d t o achieve an acceptable mixing quality. In other w o r d s , t h e u l t i m a t e o b j e c t o f m i x i n g is t o attain the distribution of maximum uniformity; thus, the mixing procedure and measures of homogeneity become important aspects of this subject. T h e r e a r e t w o b r o a d a s p e c t s i n t h e investig a t i o n o f s o l i d s m i x i n g . O n e is c o n c e r n e d w i t h the c h a r a c t e r i z a t i o n o f m i x t u r e s a n d the o t h e r
with the mechanism and rate by which the s t a t e o f a m i x t u r e c h a n g e s . T h e w o r k desc r i b e d in this s t u d y is c o n c e r n e d w i t h t h e former. T h e h o m o g e n e i t y o f a solids m i x t u r e is u s u a l l y c h a r a c t e r i z e d b y use o f a m i x i n g i n d e x t o r e p r e s e n t its d e g r e e o f m i x e d n e s s . A u s e f u l a n d g e n e r a l i n d e x s h o u l d b e r e l a t e d as c l o s e l y as p o s s i b l e t o t h e s p e c i f i c c h a r a c t e r i s t i c s o f the mixture, should be independent of the mixing process, and s h o u l d be easily determ i n e d . T h e c h o i c e o f a n i n d e x also d e p e n d s o n its a p p l i c a t i o n . T h e d e s i r e d r a n g e o f t h e m i x i n g i n d e x is o f t e n b e t w e e n 0 a n d 1. Statistical analysis has been a m a j o r t o o l o f solids m i x i n g i n v e s t i g a t i o n s b e c a u s e o f t h e stochastic nature of mixing processes. Over thirty different criteria for the degree of mixedness have been reviewed and summarized b y F a n et al. [ 1 ] . T h e relationships a m o n g some of the criteria have been studied and t a b u l a t e d b y F a n a n d W a n g [ 2 ] . T a b l e 1 lists these criteria and investigators who have p r o p o s e d t h e m . Most o f t h e criteria are based on the composition analysis of equi-sized particle samples of a binary mixture. However, solid particles t o be blended o f t e n consist o f p a r t i c l e s t h a t v a r y n o t o n l y i n size a n d d e n s i t y b u t also in c h e m i c a l a n d p h y s i c a l p r o p e r t i e s . Thus, a study of mixing indices for multicomp o n e n t h e t e r o g e n e o u s s o l i d p a r t i c l e s is o f practical significance. A convenient scheme for classifying particle systems should be established to facilitate the investigation o f solids mixing. Usually, particle s y s t e m s are c l a s s i f i e d as h o m o g e n e o u s o r
74
o~ o~
o-~ -~__. ~,- " ~
o
I~ II~.-~
-~ ~ I ~
-
I~
~I~ ~
I1~
~"-'~ -~ ~I~
"~1% I
'f
~e o~
°f
t~
2
~
II II
8
~2
e.,
e...
II
II
II
i '--i
L~
0 o
I
o °.
Tit
~.~I
I
"1-~ -i-~ r,,¢
0
I o ¢IS
o. I!
U
I!
II 0 Z
75 heterogeneous (or dispersed) systems. Particles of a homogeneous system have the same p h y s i c a l p r o p e r t i e s (e.g. d e n s i t y , s i z e , s h a p e , etc.) and are distinguishable through the c h a r a c t e r i s t i c s o f t r a c e r s , s u c h as c o l o r a n d radioactivity, that do not physically alter the course of the mixing process. Here materials with physical properties that do not interact with the operation of a mixer are considered as tracers. On the other hand, particles of a heterogeneous system have different physical properties. The extent of the variation in the characteristics of a mixture m~y be expressed in terms of weight, volume or number fraction of each constituent in the mixture. The three bases yield an identical result for a homogeneous system. However, sampling by counti n g t h e n u m b e r o f p a r t i c l e s is d i f f i c u l t a n d impractical for a fine particle or powder s y s t e m . M e a s u r e m e n t b y v o l u m e f r a c t i o n is not easy, because the bulk volume of solid particles consists of the true volume of p a r t i c l e s a n d t h e v o l u m e o f v o i d a g e , i.e., t h e b u l k v o l u m e o f e a c h c o n s t i t u e n t is n o t additive. Because of the practical difficulties of measuring the volume of a mixture, the mean and variance of the weight fraction are c h o s e n h e r e as u n i t s o f m e a s u r e m e n t . In the present study, some approaches to assess t h e q u a l i t y o f a m i x t u r e h a v e b e e n proposed and the approximate distribution of a h e t e r o g e n e o u s m u l t i c o m p o n e n t m i x t u r e is derived.
2. STATISTICAL ANALYSIS OF BINARY MIXTURES The expressions of some statistical quantities associated with a binary mixture are summarized below [1]. U n b i a s e d s a m p l e v a r i a n c e * is d e f i n e d as a2_
1
~
m - - 1 y=z
(Xlj--~1.) 2
* I n d e r i v a t i o n B, t h e b i a s e d v a r i a n c e e s t i m a t o r ,
1
~
(x~i_~l.)2
m i=1 is u s e d t o -~|mplify t h e d e r i v e d f o r m u l a . F o r a l a r g e ~-~mp|e s i z e , t h e e f f e c t o f b h s will b e n e g l i g i b l e .
where m = number of spot samples, x~ = f r a c t i o n o f c o m p o n e n t 1 in t h e j t h s a m p l e , a n d X l . = s a m p l e m e a n , w h i c h is ~ XzY m i=i
Note_ t h a t t h e v a r i a n c e c a l c u l a t e d f r o m c o m p o n e n t 1 is e q u a l t o t h a t c a l c u l a t e d f r o m component 2 for a binary mixture. The ten most frequently used indices are l i s t e d i n T a b l e 1. W e c a n s e e t h a t t h e s t a n d a r d deviation (or variance) of the mixture in the completely mixed state and that of the mixture in the completely segregated state are o f t e n u s e d as t h e b o u n d a r y v a l u e s o f a m i x i n g index. A mixture with a completely ordered a r r a n g e m e n t o f p a r t i c l e s is o f t e n r e g a r d e d as p e r f e c t , i.e. t h e m i x t u r e is i d e a l l y h o m o g e nous. For the ideally homogeneous mixture, t h e c o n c e n t r a t i o n i n e a c h s a m p l e is i d e n t i c a l to the theoretical concentration. Thus, the variance of the concentration becomes zero. a2 = 0
(2)
Although spot samples taken from the perfect mixture have an identical composition, such a mixture cannot be realized by means of a conventional mixing operation. We can define another final state, the so-called ' c o m p l e t e l y m i x e d ( o r r a n d o m ) s t a t e ' t h a t is d i f f e r e n t f r o m i d e a l h o m o g e n e i t y . I t is characterized by the fact that the probability of sampling a particle of one of the comp o n e n t s f r o m a n y p o i n t i n t h e m i x t u r e is identical, even though the composition may not be identical everywhere. For a homogeneous binary system, particles in a spot sample of size N are distributed according to the b i n o m i a l . d i s t r i b u t i o n , i.e.
(1) Pr[Y1 = nl] =
N!
nz ! (N--nz)!
PI,
(1--P1)":
(3)
w h e r e P1 is t h e w e i g h t f r a c t i o n o f c o m p o n e n t 1 i n t h e p o p u l a t i o n a n d Y1 is t h e r a n d o m variable that denotes the number of particles of component 1 in a spot sample. According to this probability density function, therefore, t h e v a r i a n c e o f t h e m i x t u r e is
76
o2 = V a t [ x 1 ]
-
(4)
PIP2
N which is the origin~ formula of Lacey [ 3 ] . In practice a mixing process rarely yields a completely mLxed mixture, either because the process is incomplete or because segregation occurs by differences in the physical properties of the components. This implies that the probability distributions of the c o m p o n e n t s during mixing are u n k n o w n . T h e probability o f sampling a particle o f a given c o m p o n e n t is p r o p o r t i o n a l t o t h e v o l u m e fraction of that component. Under the h y p o t h e s i s o f m i x t u r e s in t h e c o m p l e t e l y r a n d o m s t a t e , p a r t i c l e s in a s a m p l e c a n b e assumed to be a p p r o x i m a t e l y distributed according to the binomial distribution with p a r a m e t e r s N* a n d qz • w h e r e N* is t h e a v e r a g e n u m b e r o f p a r t i c l e s in a s a m p l e o f c o n s t a n t w e i g h t , W, a n d ql is t h e v o l u m e f r a c t i o n o f c o m p o n e n t 1. T h e v a r i a n c e o f t h e w e i g h t f r a c t i o n o f c o m p o n e n t 1 o r 2 is d e r i v e d as (see A p p e n d i ~ A ) 02_
PzP2
p2
W
PIP2
wlw2 (PlW2
(5)
+ P2Wl)
o~ = P-- ( 1 - - P 2 ) 2 P 2 + P ~ ( 1 P---P2) (7) P2 P2 w h e r e Pl a n d P2 d e n o t e t h e d e n s i t i e s o f c o m p o n e n t s 1 a n d 2, r e s p e c t i v e l y , a n d p t h e mean density. If the components have the s a m e p a r t i c l e d e n s i t y (Pl = Pl = P ) , (8)
0 2 = PIP2
3. MIXING
INDICES
FOR
MULTICOMPONENT
MIXTURES T h e p u r p o s e o f t h i s s e c t i o n is t o d e v e l o p mixing indices for a multicomponent mixture. We assume t h r o u g h o u t this section t h a t s a m p l e s are t a k e n f r o m a k n o w n v o l u m e o f a multicomponent mixture and the proportion ( b y w e i g h t ) o f e a c h c o m p o n e n t is r e c o r d e d . From the s~mple proportions, an indication o f t h e d e g r e e o f m i x e d n e s s will b e c a l c u l a t e d . Methods of sampling, determination of sample size, a n d t h e n u m b e r o f s a m p l e s d r a w n will d e p e n d o n t h e d e g r e e o f p r e c i s i o n requix.ed, the cost and practicality of sampling, and possibly physical constraints on the sampling process. 3.1
"Pseudo" binary
mixture
w h e r e W = w e i g h t o f a s a m p l e , w~, w2 = m e a n p a r t i c l e w e i g h t s o f c o m p o n e n t s I a n d 2, r e s p e c t i v e l y , a n d Px, P~ = p a r t i c l e d e n s i t i e s o f c o m p o n e n t s 1 a n d 2, r e s p e c t i v e l y . B e c a u s e , in p r a c t i c e , s o l i d p a r t i c l e s t o be m i x e d are u s u a l l y m e a s u r e d in t h e s a m e d i m e n s i o n s , this distribution seems to be a g o o d a p p r o x imation for most mixtures. Notice that eqn. (5) r e d u c e s t o e q n . (4) f o r a h o m o g e n c o u s system. Another approximate expression f o r t h e v a r i a n c e o f a m i x t u r e in t h e c o m pletely r a n d o m state was derived b y Stange [4]. Under the assumption that the probability o f s a m p l i n g a p a r t i c l e o f t h e given c o m p o n e n t is p r o p o r t i o n a l t o its v o l u m e f r a c t i o n , t h e v a r i a n c e f o r a m i x t u r e in t h e c o m p l e t e l y s e g r e g a t e d s t a t e is d e r i v e d h e r e as (see A p p e n d i x B)
Suppose that a multicomponent mixture c o n s i s t s o f c o m p o n e n t s 1, 2, 3 . . . . , k. I n describing the homogeneity of the key c o m p o n e n t , s a y , c o m p o n e n t 1, t h e m i x t u r e is c o n s i d e r e d t o c o n s i s t o f c o m p o n e n t 1 a n d another (pseudo) component x, which conrains the remaining ( k - - 1 ) components. C o m p o n e n t x is p r e s e n t in t h e m i x t u r e in a proportion of (1--/'1)- The particle weight a n d density o f c o m p o n e n t x are given, respect i v e l y , as
Uo2 = P-P---( 1 - - P ~ ) 2 P 1 + P 2 ( I -
T h u s , c o m p o n e n t x m a y b e t r e a t e d as c o m p o n e n t 2 o f a b i n a r y m i x t u r e ~- d e s c r i b e d in S e c t i o n 2. F o r i n s t a n c e , in t h e p h a r m a c e u -
Pl
Or
P---Pz) Pl
(6)
w~
=
P~ --+ 1112
~
=
(1 --t"1) /'3 + . . . + *-03
P~ Wk
(1 - - P , )
Pz +e3 +...+e~ P2
P3
(9)
(10)
Pk
77
tical industry, usually only one active ingredient and several diluents (lactose, starch, glucose, etc.) are in a dosage; therefore, the homogeneity of this active ingredient (key c o m p o n e n t ) is t h e m a i n c o n c e r n . T h e m i x i n g index for the key component may be determ i n e d as d e s c r i b e d i n t h e p r e c e d i n g s e c t i o n [5]. The present approach based on the concept of a pseudo-binary mixture can be applied to each component in a multicomponent mixture to define an index of mixedness for each c o m p o n e n t . S o m e t i m e s t h i s m e t h o d is m o ~ e useful than a tingle-valued index, because its application may indicate that a different d e g r e e o f m i x i n g is r e q u i r e d f o r e a c h c o m p o n e n t , i.e., s o m e c o m p o n e n t s m a y b e required to be mixed to a much greater extent than others [6]. 3.2 Pooled variance The unbiased sample variance of the concentration in a multicomponent mixture is d e f i n e d as f o l l o w s : a2 -
m --
i=~=l
(xii --x'i.)2Pi
(11)
where h = number of components, x o = fraction of component i in the/th spot sample, a n d ~i. = s a m p l e m e a n o f c o m p o n e n t i, =
m
$=1
When h = 2, eqn. (11) can be reduced to eqn. (1), which expresses the variance of a binary mixture. Similarly, we may express the variances for mu/ticomponent mixtures in the completely mixed and segregated states, respectively, as a~ =
E
--
i=1
Pi
P~(I--
--
P,)
homogeneous particles system, eqns. (12) and (13) can be reduced, respectively, to
02
- -
1
k
N
i=1
~] P ~ ( 1 - - P i )
(14)
h
= ~ Pf(1--P,)
o o~
(15)
i=1
Thus, we have a o2 = N a 2
(16)
Accordingly, the relationships between t h e s e m i x i n g i n d i c e s s h o w n in T a b l e 1 a r e i d e n t i c a l t o t h o s e f o r b i n a r y m i x t u r e s [ 2 ] . In Figs. I and 2 the values of the nine mixing indices are plotted against one of them, namely, Ms, for the samples of sizes 20 and 200 (no. of particles), respectively. Then M I a n d M 6 a r e l i n e a r l y r e l a t e d t o Ms. M I , M 4 , a n d M5 a p p e a r t o b e m o r e d e p e n d e n t u p o n the sample size than the others. However, only M7 is s e n s i t i v e t o t h e v a r i a t i o n in t h e t , x t e n t of mixedness in the range from the completely segregated state to the completely mixed state. 3.3 Determinant of the sample covariance matrix M u l t i v a r i a t e a n a l y s i s is a n e f f e c t i v e t o o l f o r solving multicomponent solids mixing problems [7]. Instead of being concerned with the concentration of a single component, sampling o f a k - c o m p o n e n t m i x t u r e is p r i m a r i l y c o n c e r n e d w i t h r a n d o m s a m p l e s o f (h - - 1) v e c t o r valued random response variables. Notice that o n l y (k - - 1) o f t h e k c o n c e n t r a t i o n s n e e d s t o
0 8
(12)
Pi
and
Now, we are ready to define mixing indices for multicomponent mixtures as similar to t h o s e f o r b i n a r y mixt-~tres i n T a b l e 1 . F o r t h e
00
0Z
0q HIXING
08 INDEX
"
0
Hi
Fig. 1 Comparison of mixing indices based on the 2nd approach (pooled variance) for sample size of 20.
78 1.0
T h e e s t i m a t e d v a r i a n c e s o f x~ a n d c o v a r i a n c e o f x i a n d x~. a r e , r e s p e c t i v e l y ,
~o.
MQ
i = I, 2 ..... ( k - - i )
SS~i
-
m -- 1"
(19)
and
~:o.q
x~]
Cov[xl.
= s~Q
0.2 _
SPxix~ m--1
o.';z
o.o
'
o1.
'
0%
"
ale
'
i, £ - - - - 1 , 2 . . . . , ( k - - l '
"
(i ~ ~
*'a
MIXING INOEX M,
Fig. 2. Comparison o f mixing indices based o n the 2nd approach (pooled variance) for sample size of 200. be considered. When s3mples are obtained from a (k -- 1)-van:ate population instead of a univariate population, the data can be r e p r e s e n t e d as
Thus, the sample variance-covariance t r i x is
SI2
Sample no.
i
...
(k--l)
1
Xll
X21
" - -
Xil
- - -
/(k--1)1
2
X12
X22
. . .
Xi2
. . .
X(k_l)
]
Xl I
XZ]
. . .
Xij
. . .
X(Iz--1)]
I,"~
Ilm
X2m
...
Xim
...
X(k_l)rn
-
"~1 ( k - - l )
-
.
S2(k--l)
$2(k--2)
.
ma-
S=
k S1 (k--l)
Component 1 2 ...
322
(20)
2
S(k--1)(k--1)
(21) For a homogeneous system, particles in a completely random mixture are expected to be randomly distributed in a spot sample of size N taken from the mixture. Let us de~ne a r a n d o m v e c t o r [ Y1, Y2 . . . . . Yk ] f o r a m i x t u r e o f k c o m p o n e n t s , w h e r e 1/] = n j , 0 < n j < NAT.T h i s i m p l i e s t h a t n j p a r t i c l e s o f c o m ponent] fall into the spot s3mple. Then the joint probability density function of random v a r i a b l e s , Yz, Y2 . . . . . Irk, is [ 9 ]
P r [ Y 1 = n l , Yz = n 2 . . . . . Irk = n k ]
N'. nz!
The sums of squares and cross products are d e f i n e d , r e s p e c t i v e l y , as [ 7 ]
n~,! . - . n k !
N! h
SS=,= ~
(xu--zi.)
x~----
i I,~i 712
x u )2
(17)
where nl +n2 +... +nk
and
P,
~ (xu--z~.)(x~--~~ i=I
----
(22)
i=Z
Zm j=l
SP~=
p~-i
i=i
H nil
2
i=1 =
', H
~.
, i=1
x i i x gJ
--
--
m
=
Xtl
X
8)
= N
+ P z + . . . + P,~ = I
Pi = weight fraction of the ith component in the population (= number fraction of the/th component in the population for a homogeneous system). Note that the multinomial d i s t r i b u t i o n is a g e n e r ~ i i T ~ t i o n o f t h e b i n o m i a l d i s t r i b u t i o n . A c c o r d i n g t o f~i=~ p r o b a b i l i t y
79 d e n s i t y f u n c t i o n , e q n . ( 2 2 ) , t h e variane~_s a n d c o v a r i a n c e s o f t h e r a n d o m v a r i a b l e s , Y1, Y2 . . . . Yk, a r e g i v e n , r e s p e c t i v e l y , b y [ 8 , 9 ]
•
var[Y,l = NP,(1 --P,)
(23)
Y/, Y~ ] = --NPiP~
(24)
Coy[
or equivalently,
P~(I --P~)
,
i = 1, 2 .....
(k -- 1)
(25)
N C ° v [ x i " x ~ ] = C ° v I - ~ " NYQI'
P~P~
--
i, ~ = 1 , 2
....
,(k--l)
(i¢
£)
(26)
N " Therefore, t h e v ar i an ce- co v ar i an cem a t r i x for t h e m i x t u r e in t h e c o m p l e t e l y m i x e d s t a t e can
--]
b e e x p r e s s e d as
~_
( l - - PI ) Pe
--P1 P( k -1 )
--PI P2 P2(1 - - P 2 )
I
1
(27)
N °
L--P1P(k --1)
--P2 P(k --1 )
P ck - i ) ( 1 - - / } ( k - 1 ~)_1
In the completely segregated state, each component this state, we have [8, 9] Var[x~l = P i ( 1 - - P i ) ,
is s e p a r a t e d a c c o r d i n g t o p a r t i c l e t y p e s . I n
i = 1, 2 . . . . .
(k - - 1)
(28)
and Cov[xi, x~] = --PiPe
i, £ = 1, 2 . . . . .
(k - - 1) a n d i ~ ~
(29)
Therefore, t h e v a r i a n c e - e o v a r i a n c em a t r i x f o r a m u l t i c o m p o n e n t m i x t u r e in t h e c o m p l e t e l y
s e g r e g a t e d s t a t e c a n b e e x p r e s s e d as ~/}: P1) Es
=
--/}1/}2 P 2 ( 1 - - P2)
"
[--P1P,~-I,
--P1Pc~-I) "--P2P(a-1)
]
"
P2P,~-I,
(30)
~ % - 1 , ( 1 - - Pc~-l,)_J
T h e s a m p l e v a r i a n c e - c o v a r i a n c e m a t r i x , S, c h a r a c t e r i z e s t h e d e g r e e o f t h e d i s p e r s i o n o f c o n c e n t r a t i o n o f e a c h c o m p o n e n t . T h u s , t h e d e t e r m i n a n t o f t h i s m a t r i x , ISI, r e f l e c t s t h e e x t e n t o f t h e v a r i a t i o n in t h i s s y s t e m ( m i x t u r e ) s i m i l a r t o t h e v a r i a n c e in t h e u n i v a r i a t e a n a l y s i s . T h e r e f o r e , r e p l a c i n g a 2, o rz a n d a~ b y IS[, IZrl, a n d IZs[, r e s p e c t i v e l y , w e c a n a l s o d e f i n e t h e m i x i n g i n d e x f o r a multicomponent mixture. Thus, we can extend the mixing index
a2 --uz
(31)
80 proposed by Lacey [10] for a binary mixture to a multicemponent mixture by defining
M3-
ISI - - l ~ r I 1~I--1~1
(32)
O t h e r e x a m p l e s are g i v e n in T a b l e 1. N o t e that the expressions for muiticomponent m i x t u r e s are r e d u c e d t o t h o s e f o r b i n a r y mixtures. A c c o r d i n g to eqns. (27) and (30), we have E s =N~ r
Var[xi]
-
W Pi
1 --
P~
i = 2, 2, ..., (k - - 1 )
(33)
(35)
and
and iEs I = N ~ - ~ I ] E r I
s a m p l e o f c o n s t a n t w e i g h t , W, a n d qi is t h e v o l u m e fraction o f t h e ith c o m p o n e n t . The variances and covariances of weight f r a c t i o n s o f c o m p o n e n t s in a m u l t i c o m p o n e n t ~ mixture, having this probability density f u n c t i o n , are d e r i v e d , r e s p e c t i v e l y , as (see A p p e n d i x C)
p2
(34)
The relationships among the mixing indices t a b u l a t e d in T a b l e 1 are e v a l u a t e d in T a b l e 2. I n Figs. 3 a n d 4 t h e v a l u e s o f t h e n i n e m i x i n g inc:ices are p l o t t e d a g a i n s t t h a t o f M s b a s e d on the determinant of the covariance matrix a p p r o a c h f o r t h e s a m p l e s o f sizes 2 0 a n d 2 0 0 , respectively. A mixing index should be sensitive t o c h a n g e s in t h e q u a l i t y o f t h e m i x t u r e in o r d e r t o b e u s e f u l in f o l l o w i n g t h e prc~e_~= o f m i x i n g . A s h t o n a n d V a l e n t i n [ 1 1 ] pointed out that Lacey's index, eqn. (32), w a s v e r y sensitive t o c h a n g e s n e a r t h e c o m pletely segregated state and relatively insensitive d u r i n g t h e p r o c e s s o f m i x i n g n e a r t h e completely mixed state. Their more complic a t e d i n d e x , MT, w a s t h u s d e v i s e d t o e x h i b i t greater sensitivity. Mixing indices defined b y the d e t e r m i n a n t of the sample variance-covariance matrix for multicomponent mixtures can be generalized as in T a b l e 1, w h e r e ~ r a n d e s a r e t h e variance-covariance matrices of weight proportion in t h e c o m p l e t e l y m i x e d a n d c o m p l e t e l y segregated states, respectively. T h e developm e n t o f ~ r a n d e s p r e s e n t e d so f a r w a s c o n cerned only with homogeneous systems. For the heterogeneous system, we assume that the p r o b a b i l i t y o f picking a particle o f a given c o m p o n e n t is p r o p o r t i o n a l t o t h e v o l u m e fraction of that component and the probability o f f i n d i n g a p a r t i c l e o f o n e c o m p o n e n t is i d e n t i c a l e v e r y w h e r e in a c o m p l e t e l y r a n d o m mixture. Thus, particles t a k e n f r o m a s p o t s a m p l e are d i s t r i b u t e d a p p r o x i m a t e l y a c c o r d ing to the m u l t i n o m i a l distribution with t h e p a r a m e t e r s , _N* a n d qi (i = 1, 2 . . . . . k), where _N* is t h e a v e r a g e n u m b e r o f p a r t i c l e s in a s p o t
Cov[xi,
x~]
-
- -
W
P/P~,
PlP~
i, £ = 1, 2 . . . . , (k - - 1 ) a n d i ~
£
(36)
This establishes the variance-covariance m a t r i x f o r a m u l t i c o m p o n e n t m i x t u r e in t h e c o m p l e t e l y m i x e d state, ~ r . Similarly, the variances and covariances of the weight f r a c t i o n s o f t h e c o m p o n e n t s in a m u l t i c o m p o n e n t m i x t u r e in t h e c o m p l e t e l y s e g r e g a t e d s t a t e are given, r e s p e c t i v e l y , b y (see A p p e n d i x D) P
var[ ,]
(1-
=
Pi
i = 1, 2 . . . . . (k - - 1)
(37)
and
i, £ --- 1, 2 . . . . . (k - - 1) a n d i ~ £
(38)
This establishes the v a r i a n c e - c o v a r i a n c e m a t r i x f o r a m u l t i c o m p o n e n t m i x t u r e in t h e completely segregated state, Es. Notice that the variances and covariances for a heterogen e o u s s y s t e m are r e d u c e d t o t h o s e f o r a homogeneous system by setting P]L
=P2
=
• - -=Pk
and W 1
--~ W 2
-~-- . . .
W k .
3.4 Absolute deviations from the population me~iT~
The mixing index may be defined by using the absolute deviation from the population
81 A I--4
I II
T
°~
I
i.--I
'-'
I
,-~m
I
~T 7
I
I
.r--I
_= t ~.
!
L~
+
I
v-I
~l
'' I ~
I
I,--41. ~
T--,
I
!
I
i-.,I
v-i
,,-..I
1 0 I
I
I I
.,-t.
I i--.I
I I--I
~L
I 1-1
I o~ ~ilV-.
I I i.-I
!
I
I i..-I
~T
! i.-i
I
82
l I
-li
#
I
I c~
I
I~.
i~ l
l
I
T--I
I
v.-I
T-I
i
co
I
I i
J
I
I
~l ~ ~l ?
I I
r-.I
T-I T-'I
i- l
I
!
I
?
•
I
c~
I I
I
lr-I c~
E.-,
~f
0
I
I
83
I
I v~ !
!
!
i
m
v~
I
r
I
I !
!
m
!
!
I
I
I
I
I
v~
v~
1-1
•
!I~ !~1~1~
i
v ~
I
I I
!
I c~ C~IL~ e.t
!
!
I
0 I C~
c~
~G
#1~
84 z.o
.
,
.
sample mean. Throughout the whole system, t h e c o n c e n t r a t i o n o f t h e p e r f e c t m i x t u r e is exactly equal to the theoretical value, and therefore the mixing index of the perfect m i x t u r e is 1 a c c o r d i n g t o e q n . ( 3 9 ) . I f a m i x t u r e is i n t h e c o m p l e t e l y s e g r e g a t e d s t a t e , i t s m i x i n g i n d e x is 0 . T h e r e f o r e , t h e m i x i n g index, eqn. (39), always takes a value between 0 and 1 for a mixture in an arbitrary stale. This definition can also be extended to a multicomponent system as
M
z: °'8
a
~1
~ x
°"° -
/e
.
.
.
,
.
:
:
.
MIXING INDEX Nr
i~1
Fig. 3. Comparison o f mixing indices based on the
3rd a p p r o a c h (the d e t e r m i n a n t o f t h e covaxiance matrix} for sample size o f 20. mean of each individual component instead of t h e v a r i a n c e w h i c h is c o n v e n t i o n a l l y u s e d t o define the mixing index. :For instance, the following expression can be used as a mixing index for a binary mixture: rn
[Xl1 --Pz Mzz
l
(39)
= I --
Pz The denominator o f t h i s e q u a t i o n is t h e total sum of the absolute differences between the sample concentrations and the population concentration among the spot samples from a mixture in the completely segregated state (see Appendix E). In case the population mean is n o t s p e c i f i e d , i t c a n b e r e p l a c e d b y i t s 1.0
0.0
rn
o
.
.
.
~
.
.
.
.
~.l:t
.
M2
x ~o.
i
o.2
o.
13
MIXING INDEX M,
Fig. 4. Comparison of mixing indices based on the 3rd a p p r o a c h ( t h e d e t e r m i n a n t o f t h e c o v a r i a n c e matrix) for sample size of 200.
M l l
=
1
j=l
-/e
i=1
p
LPi
Pi
/
(40)
4. C O M P A R I S O N O F D I F F E R E N T D E F I N I T I O N S OF MIXING INDICES Computational effort was minimized by choosing a ternary particle system to demonstrate the use of the proposed mixing indices for multicomponent mixtures. The values of the degree of mixedness evaluated according to different definitions were compared. An actual mixture was generated by a drum mixer which was constructed from a lucite tube with a diameter of 7 cm and a length of 38 cm :nd fitted with two lucite end flanges, each with a diameter of 12.7 cm. Particles of three different types were used. Their physical properties are summarized in Table 3. Prior to mixing trials, two thin semicircular partitions were placed between the two ends of the mixer, normal to the mixer axis. They divided the mixer into three compartments of equal length. 200 g of component 1, 150 g of component 2 and 150 g of component 3 were loaded in the middle, right and left compartments, respectively. Then the bed was leveled, the partitions were removed, the cover was put in place, and the mi~er was rotated. For this experiment, the speed of the mixer was set at 30 r.p.m., and sampling was carried out at an interval of 10 minutes after the onset of rotation. With sample divider, the mixture was divided into ten sections. Each sample from each section contained 20 grams of particles. Mi~ing indices were calculated from the experimental
85 TABLE 3 P h y s i c a l c h a r a c t e r i s t i c s o f particles
Component
Material
Color
Particle diameter in. ( c m )
D e n s i t y , Pi ( g / c m 3)
Average w e i g h t , Wi (g)
1 2 3
steel lucite
silver red
lucite
white
0.!875 (0.4763) 0.1819 (0.4620) 0.1618 (0.4110)
7.60 1.18 1.18
3.438 0.487 0.343
(3) Determinant of the sample covariance matrix T h e m a t r i c e s , S, ~ r a n d ~ s , w e r e c o n structed and their determinants were then evaluated. According to the definitions in T a b l e 1, t h e i n d i c e ~ w e r e d e t e r m i n e d . T h e r e s u l t s a r e a l s o s h o w n i n T a b l e 4. (4) Absolute deviations from the population means We first calculated the sum of the absolute deviations from the population means for a mixture in the completely segregated state. Then, the sum of the absolute deviations between sample concentrations and the population means was determined. According to eqn. (40), the mixing index was computed as 0 . 6 9 9 f o r t h i s m i x t u r e .
d a t a , w h i c h a r e s u m m a r i z e d i n T a b l e 4. T h e p r o c e d u r e s a r e b r i e f l y d e s c r i b e d as f o l l o w s : (1) Pseudo-binary mixture P a r t i c l e s o f t y p e 1 w e r e a s s u m e d as t h e k e y component, and particles of types 2 and 3 w e r e j o i n t l y c o n s i d e r e d as t h e p s e u d o ~ c o m ponent. The particle weight and density of this pseudo~component were calculated from eqns. (9) and (10), respectively. The mixing indices were calculated according to ten of the more frequently used criteria given in T a b l e 1, a n d t h e r e s u l t a n t v a l u e s o f i n d i c e s a r e l i s t e d i n T a b l e 4. (2) Pooled variance The pooled variance of weight fractions of the components in a multicomponent mixture was calculated from eqn. (11). The variance f o r t h e m u ! t i c o m p o n e n t m i x t u r e in t h e completely mixed state, 02 , and that for the mixture in the completely segregated state, o2, w e r e c a l c u l a t e d a c c o r d i n g t o eqns. (12) and (13), respectively. Mixing indices were c o m p u t e d b y s u b s t i t u t i n g t h e v a r i a n c e s , or2 a n d a ~ , i n t h e e x p r e s s i o n s g i v e n i n T a b l e 1. Results are also shown in Table 4.
5. C O N C L U D I N G R E M A R K S
A s s e s s m e n t o f t h e q u a l i t y o f a m i x t u r e is one of the most important aspects in the investigation of solids mixing. The concept of homogeneity of dispersed (or heteroge-
TABLE 4 A ternary mixture and resultant values of mixing indices of this mixture according to approaches I, 2 and 3
Component Sample
1
2
3
1 2 3 4 5 6 7 8 9 10
0.345 0.450 0.410 0.390 0.475 0.565 0.520 0.610 0.575 0.355
0.450 0.075 0.150 0.275 0.275 0.225 0.295 0_250 0.325 0.585
0.205 0.475 0.440 0.335 0.250 0.210 0.185 0.140 0.100 0.060
Experimental data expressed in weight fraction.
Mixing index
Approach 1
Approach 2
Approach 3
M1 M2 M3 M4 M5 M6 M7 MS M9 /TflO
0.775 0.949 0.964 0.541 0.477 0.225 0.842 0.882 0.051 1.848
0.742 0.934 0.957 0.605 0.532 0.258 0.880 0.880 0.066 1.653
0.937 0.996 0.996 0.343 0.329 0.063 0.958 0.958 0.040 2.915
Note: approach 1 --pseudo-binary mixture; approach 2pooled variance; approach 3 -- determinant of eovarianee matrix.
86
n e o u s ) p a r t i c l e s y s t e m s is u s e f u l f o r t h e examination of the mixing and dispersion process and for the elucidation of problems t h a t are o f p r a c t i c a l s i g n i f i c a n c e . Although much effort has been expended o n this s u b j e c t , it still l a c k s g e n e r a l l y a c c e p t e d i n d i c e s t o assess t h e h o m o g e n e i t y o f a h e t e r o geneous multicomponent mixture. Four app r o a c h e s , w h i c h h a v e b e e n p r o p o s e d in this s t u d y , are o f universal a p p l i c a b i l i t y . T h e approximate joint distribution of random v e c t o r [Y1, Y2. . . . . Yk] f o r a m u l t i c o m p o n e n t m i x t u r e in t h e c o m p l e t e l y r a n d o m s t a t e has b e e n d e r i v e d . T h e u s e o f v a r i o u s mixing indices for multicomponent mixtures has b e e n s h o w n t h r o u g h a t e r n a r y m i x t u r e over a wide range of physical properties. The proposed expressions of mixing indices for m u l t i c o m p o n e n t mixtures reduce t o t h o s e f o r a b i n a r y m i x t u r e . Ar, y m i x i n g i n d e x f o r a h o m o g e n e o u s s y s t e m is a s p e c i a l case o f t h a t f o r a h e t e r o g e n e o u s s y s t e m . The choice of a mixing index depends on t h e a p p l i c a t i o n o f m i x i n g p r o c e s s . T h e relationships among the ten most frequently used i n d i c e s b a s e d o n t h e s e c o n d a n d t h i r d app r o a c h e s are evaluated. T h e m i x i n g i n d e x defined by using the absoIute deviations from the population means may have a smaller r e l a t i v e s t a n d a r d e r r o r . C l e a r l y , f u r t h e r inv e s t i g a t i o n is n e c e s s a r y i n t o t h e r e l a t i o n s h i p b e t w e e n the mixing indices based on different a p p r o a c h e s . F u t u r e s t u d i e s will c o n s i d e r t h e relative p e r f o r m a n c e o f t h e m i x i n g i n d i c e s d i s c u s s e d in this w o r k .
ACKNOWLEDGEMENT T h e authors wish to a c k n o w l e d g e the financial s u p p o r t o f the National Science Foundation (Grant ENG 73-04008A02).
N*
a v e r a g e n u m b e r o f p a r t i c l e s in a sample of constant weight, W no. o f particles o f c o m p o n e n t ] in a s p o t s a m p l e weight proportion of component j in t h e p o p u l a t i o n volume fraction of component j sample variance-covariance matrix s u m o f c r o s s p r o d u c t s o f xi a n d xQ s u m o f squares o f xi variance of weight fraction of component i covariance of weight fraction between components i and total weight of the mixture total n e t v o l u m e o f solid particles in t h e m i x t u r e variance o f the r a n d o m variable
n1
q, S
SP~ix ~ SS~ i 8ii
T V
var[Y~]
Y,. weight of a spot sample mean particle weight of component/ a r a n d o m variable d e n o t i n g the weight fraction of component i in a s a m p l e weight fraction of component i in t h e j t h s p o t s a m p l e sample mean of component i a r a n d o m variable d e n o t i n g the n u m b e r o f particles o f c o m p o n e n t ] in a s p o t s a m p l e mean density density of component i variance-covariance matrix of a m i x t u r e in t h e c o m p l e t e l y m i x e d state variance-covariance matrix of a m i x t u r e in t h e c o m p l e t e l y segregated state sample variance of weight fraction v a r i a n c e o f a m i x t u r e in t h e completely segregated state variance of a perfect mixture v a r i a n c e o f a m i x t u r e in t h e completely mixed state
W
wi xi
x~
rl p pl
X,s Er2
a~
LIST OF SYMBOLS .~
N k-1
Cov[Y~, Yj-] c o v a r i a n e e b e t w e e n t h e r a n d o m variables Y / a n d Y1 k no. of components M mixing index m no. of spot samples N n o . o f p a r t i c l e s in a s p o t s~rnple
REFERENCES 1 L . T . F a n , S . J. C h e n a n d C. A . W a t s o n , S o l i d s m i x i n g , I n d . Eng. C h e m . , 6 2 ( 7 ) ( 1 9 7 0 ) 5 3 . 2 L. T . F a n a n d R . H . W a n g , O n m i x i n g i n d i c e s , Powder
Te~h--ol., 1 1 (1975) 27.
87 3 P. M . C. L a c e y , D e v e l o p m e n t i n t h e t h e o r y o f particle mixing, J. Appl. Chem., 4 (1954) 257. 4 K. Stange, Die Mischgute einer Zufalismischung als G r u n d l a g e z u r B e u r t e i l u n g y o n M i s c h v e r suc-hen, C h e m . I n g . T e c h . , 2 6 ( 1 9 5 4 ) 3 3 1 . 5 H.G. Kristensen, Statistical properties of random and non-random mixtures of dry solids. Part I. A general expression for the variance of the composition of samples, Powder Technol., 7 (19731 249. 6 C. S c h o f i e l d , T h e d e i ' m i t i o n a n d a s s e s s m e n t o f mixture quality in mixtures of particulate solids, Powder Technol., 15 (19761 169. 7 C. Y . K r a m e r , A F i r s t C o u r s e i n M e t h o d s o f Multivariate Analysis, Virginia Polytechnic Institute, Blacksburg, Virginia, 1972. 8 R.H. Wang, L. T. Fan and J. R. Too, Multivariate statistical analysis of solids mixing, Powder Technol., 21 (19781 171. 9 M. Fisz, Probability Theory and Mathematical Statistics, Wiley, New York, 1967, p. 347. 1 0 P. M . C. L a c e y , D e v e l o p m e n t i n t h e t h e o r y o f particle mixing, J. Appl. Chem., 4 (19541 257. 1 1 M. D . A s h t o n a n d F . H . H . V a l e n t i n , T h e m i x i n g of powders and particles in industrial mixers, rrrans. Inst. Chem. Eng., 44 (1966) T166. 1 2 S. R . Miles, H o m o g e n e i t y o f s e e d l o t s , I n t . S e e d Test. Assoc. Proc., 27 (1962) 407. 1 3 P. M . C. L a c e y , T h e m i x i n g o f s o l i d p a r t i c l e s , Trans. Inst. Chem. Eng., 21 (1943) 53. 1 4 S. S. W e i d e n b a u m a n d C. F . B o n i l l a , A f u n d a m e n tal study of the mixing of particulate solids, Chem. Eng. Prog., 51 (1955) 27. 15 J.P. Beaudry, Blender efficiency, Chem. Eng., 55 (1948) 112. 1 6 T . X a n o , I. K a n i s e a n d K . T a n a k a , M i x i n g o f powders by the V-type mixer, Kagaku Kogaku, 20 (1956) 20. 1 7 M . H . W e s t m a c o t t a n d P. A . L i n e h a n , M e a s u r e ment of uniformity in seed bulks, Int. Seed Test. A s s o c . P r o c . , 2 5 ( 1 9 6 0 j 1_5 1 . 18 G. Herdan, Small Particles Statistics, Butterworths, London, 2nd edn., 1960.
PIT
V -
P2T
+ --
Pl
(A.1)
P2
The mean particle density, p, can be defined as
T
p--
V
OlP2
=
(A.2)
PxP2 + PzPl
When sampling, the probability of taking a p a r t i c l e is p r o p o r t i o n a l t o t h e v o l u m e f r a c t i o n o f t h a t c o m p o n e n t . P a r t i c l e s in t h e c o m p l e t e l y mixed state are assumed to be distributed according to the binomial distribution with p a r a m e t e r s N* a n d q l , w h e r e 24* is t h e a v e r a g e n u m b e r o f p a r t i c l e s in a s a m p l e o f c o n s t a n t w e i g h t , W, a n d q l is t h e v o l u m e f r a c t i o n o f c o m p o n e n t 1. The variance of weight fraction of comp o n e n t 1 c a n b e e x p r e s s e d as
a~ = Var[xl ]
~z -
=
We
-Vat[Y1 ]
~2 --'N*ql(1--ql)
W2
= w2 W2
w . o___p1, o__p~ w
Pl
P2
Thus, a 2 = 1"11"2
p2
W APPENDIX
A
Derivation
of eqn.
,olp 2
w x -w 2
(PI~2
-b P 2 W l )
(5)
(5)
Consider a mixture of two components, 1 a n d 2, w h i c h axe p r e s e n t i n t h e w e i g h t p r o p o r t i o n , / ' 1 a n d / ' 2 (/'1 + / ' 2 = 1 ) . S u p p o s e t h a t components 1 and 2 have particle densities P l a n d P2, r e s p e c t i v e l y , a n d also s u p p o s e t h a t the weight and volume of the mixture are T a n d V, r e s p e c t i v e l y . N o t e t h a t t h e v o l u m e , V , is t h e n e t v o l - m e o f s o l i d s , a n d n o t t h e t o t a l a p p a r e n t vol, , m e o f t h e m i x t u r e m a s s i n c l u d i n g the space between the particles. Therefore, V can be expressed as
APPENDIX
B
Derivation
of eqns. (6) and (7)
B e f o r e m i x i n g s t a r t s , a m i x t u r e is a s s u m e d to be in the completely segregated state. A m o n g t h e m s p o t s.tnaples t a k e n , m l s ~ m p l e s c o n t a i n o n l y p a r t i c l e s o f c o m p o n e n t 1, a n d the remaining (m --ml) contain only particles o f c o m p o n e n t 2. A s s n m e t h a t t h e p r o b a b i l i t y o f t a k i n g a p a r t i c l e o f a g i v e n c o m p o n e n t is proportional to its volume fraction, i.e.,
88 rn I = r n q 1
(B.1)
The sample variance can be obtained eqn. (1), in which Xl
APPENDIX
=P1
rn
i=1
1 ~, _
rft
i=1
1
[~,
(x~
-
-
p~)2
( x 1 , __ p 1 ) 2 1 = , , = 1
+
Li=l rn - - m l
(Xli
Z
--
I)1)2 l xt, = O]
i=I
1
--
-
[(1--P1)2rnx
1 and
2 are
C
Derivation of eqns. (35) and (36) Consider a multicomponent mixture having k components for which we defined the following for i = 1, 2, . . . , k: P t = density of component i W--~ = p a r t i c l e w e i g h t o f c o m p o n e n t t P~ = w e i g h t p r o p o r t i o n o f c o m p o n e n t i qi = volume fraction of component i Under the assumption as described in Subsection 3.3, particles taken from a spot sample are distributed approximately according to the multinomial distribution with p a r a m e t e r s AT* a n d qi- T h e r e f o r e , t h e v a r i a n c e o f w e i g h t f r a c t i o n o f c o m p o n e n t i is [ 9 ]
1 ~_, (xli__~i.)z
_
of components
from
Thus,
o2°
the variances identical.
+p2(m--ml)]
(B.2)
Var[x~]= Vat [ - ~ ]
m ~2
Since m I
w ~ V a t [ Y~]
= mql 0 2
-- - - / V * q ~ ( 1 W
=m
P
-- qi)
VlPl
ID 1
Vp
Thus,
=m~P~
(B.3)
Pl
V a r [ x d =
we have
--
--
i = 1, 2 .... , (k--
a 2 = --I[(1--pi)2m P---PI +Pi2(l-m
Pl
P P1)m l PI
(6)
C o v [ x i , x~] = Coy t ~ Similarly, we may also derive the variance of weight fraction for component 2 as o 2
= 192 p - - - ( 1 - P z ) 2 P 2 + P z ( 1 - ~ 2 p z
)
(7)
-
wz w2
C o y l Y . YA
O 2
- vv~ ( --N'q~qD Since p--
.OLP2
PlP2 +P2Pl
(35)
The covariance of weight ~-actions between the ith and ]th component can be expressed as
[91
- P--(I--pI)2pI + p 2 ( I - - ~ P1
i)
O 2
'
89
Thus, w
Cov[x. x./]
=
W
p 2
--
P-:Pi
i , j = l , 2 , ..., ( k - - 1 )
P,P,
Thus,
and i~-]
(36)
D
APPENDIX
i, I~ = 1 ,
Derivation of eqns. (37) and (38) For a multicomponent mixture in the completely segregated state, the derivation of the variance of weight fraction for each component is similar to that for a binary mixture (see Appendix B). Thus, we have Var[x/]
C o v [ x i , x~] = P i P , f 1
= P-~-(1--P/)2P/+p2(1--p~-.P/),
i=1,2,...,
Pi
(37)
"
Under the assumption Appendix A, we have
as described
in
mi = mqi
(D.1)
and
APPENDIX
Cov[xi, x ~ ] = -
1
m 2:
D2
m
andi¢
(38)
£
E o
= mql
(D.2)
where rni denotes the number of samples containing only particles of component i, a n d m ~ t h a t o f c o m p o n e n t ~. T h e c o v a r i a n c e o f weight fractions between components i and is e x p r e s s e d a s
1
2,...,(k--I)
P,P]
SUm of the absolute differences between the s a m p l e c o n c e n t r a t i o n s a n d its p o p u l a t i o n concentration For a binary mixture in the completely segregated state, we assume that m 1 samples contain only particles of component 1 among the m spot samples to be taken. Under the same assumption imposed in Appendix B, we have In 1
m ~ = rnq~
PiP
Therefore, the sum of the absolute differences between the sample concentrations and i t s p o p u l a t i o n c o n c e n t r a t i o n is
~.
[Xl]--P1 ] =
,j=l
[Xlj--Pl]Ix,j=l
1=1
(xij--~.)(x~--x~.)
j=l
m! ~
+
m--m t ~, i X l i - - P 1 I [x,./=O ./=1
m
2~ (xij - - P D ( x ~ --P~)
= (1 --Px)mx
.i=1
= (l--Px)mql
m i
+Pl(m
--ml)
+Pxm(l
--qx)
( ~ - - P~)(x~ --Pz)[~i=~. ~j=o m
•
+
=(I--PI)m P--PI+Ptm(I-- ~
Z 1 (xo - - PD(x~s --P~) l~ij= o. x~s=*
--P,) +
--
(E.I)
m --mi--rn ~ j=l
-
= mPx -~x(I
1 --
[mi(1
(x,j --e~) (x~ --e~)]x,i: o. ,.u:o] --PD("--P~)
+ rr~(--~)(1
m
--P~)
If the s y s t e m is a h o m o g e n e o u s equation reduces to
system, this
m
~_, I x l i - - P , l
= 2raP1(1 --P1)
(E.2)
1=1
+ (m --m i -- m~)PiPd -
1
- - [mP~Pe - - miP~ - - m ~ P i ] m
Equations (E.1) and (E.2) are employed deriving eqn. (39).
in