studies on multicomponent solids mixing and mixtures · 2017. 12. 15. ·...
TRANSCRIPT
STUDIES ON MULTICOMPONENT SOLIDS MIXING AND MIXTURES
by
JUI-RZE TOO
B. S., National Taiwan University, 1972
A MASTER'S THESIS
submitted in partial fulfillment of the
requirements for the degree
MASTER OF SCIENCE
Department of Chemical Engineering
\SAS STATE UNIVERSITY
Manhattan, Kansas
1978
Approved b
/j. _
-
-
II - •
-•
-•
m'
TA:
- :
[ds tan
NG ANT RES
ES
BBR
1. Ill - 1
[ON OF MEAN CON7A BBR:e CI - 2
•:•• er ' - 3
n-nber III - 3
C a Li or. in ContacIII - 5
D. nstics of the ' •' C
< n^ II: "
Population Mean Contact Number.
' III -
I
::C1SICV HE DEGREE 01 SS III - 9
...
[-11
•
-
I" -
(continued)
REFERENCES III - 17
APPENDICES
IV. ESTIMATION OF MIXING INDE: i CONTACT SPOT SAMPLINGOF A MULTICOMPONT
1. INTRODUCTION " - '.
2. CONTACT NUMBER AND MIXING INDEX IV - \
2. DESCRIPTION OF AN INCOMFL1 .. MIXED STATU - 5
3.1. Dirichlet-Multinomial Model - 5
3.2. Statistical Characteristics of Random Variables
V V 'Yp
3.3. Estimation of Parameters ,£?, , S 61 Z p I \ - c
4. PRECISION OF THE ESTIMATION OF MEAN CONTACT ::R . io
5. ESTIMATION OF SAMPLE VARI. ND ITS PRECISIC -I
6. COMPUTER SIMULATION - U
6.1. Simulation of a Ternary Mixture
6.2. Mixing of Solid Particles and Estima:Index IV - I
7. CONCLUDING REMARKS
TATIONS
REFERENCES
APPENDICES
Ml: '©ICES FOR MULT ICOMPONENT MIXTURES
1. INTRODUCTH 1 - '
2. STATISTICAL ANALYSIS FOR BINARY MIXTURES - 3
3. '©ICES FOR MDLTICO
3.1. "IV | intr - 8
3.2. Pooled YarUn. |- o
3.3. Determine .iriance Mixtrix - 10
(continue
3.4. Sa -.8 of an Individual Component - 17
COMPARISON OF DI1 T DEFI 'ICES
5. CONCLUD' - 20
NOI. - 22
REFERENCES - 24
APPENDICES
CONCLUS I
1. CONCLUS I.
1. RE AT ION-
REFERENCES
ACKNOWLEDGEMEv
I - 1
: R I
INTRODUCTI
Solid - jr powder blending is an operation by which
solid materials in particulate forr are scattered rar. ?r unifor
aniong each other by the movement of the particles in a mixer. '.'- ..-
tion may be broadly defined as anv process that tends to elir.ina
inhomogeneities , or to reduce existing gradients. On the other
ty be performed to achieve an acceptabl-
control rates ol t transfer, mass transfer or chemical reaction.
ziough it is one of tne oldest unit operations, it is til] one of the
most wid._ :>loyed in industries. For example, it is essential in
plastic processing, ore smelting, fer -
r production, pharmaceutical
Lon, food manuf. and catalytical t is of chemical
in spite of the practical importance of I
operation, the fundamental theories underlying .s ope-: and the
procedure for apprai . . formancc- s have not I -y
est.:
Solids mixing pr^ • indu- are mult,
in •
-ol ended, and t: .
of utmost practical mce.
Ch literature on mixing and m . -. of raulticom-
. r t i ] .
The concept of the contact number is extended to a inulti component solids
re In the completely mixed state In chapter HI. Such an approach has
been applieo to the analysis of a binary mixture (1]. Chap: | also
an estimation of the mean cont.. spot sa j the
re -tulation of mixtures at various concentrations of
the k>. ponent. The number of spot samples necessary for a specified
value of the relative standard error is derived in this chapter.
In Chapter IV, a Dirichlc .nomial model is proposed to describe
y.ture in an incompletely mixed stau . ates c I
parameters of the Dirichlet distribution are derived and the precision of
the estimated population mean contact number is assessed. This mod <nds
the beta-binomial model for ah; ture in an incomplett
state to the multicomponent case. Computer simulations are included
to demonstrate the effectiveness cf the derived model.
To characterize the homogeneity of a solids fixture, a mixing index
usual 1 . oyed as a measure of the degree of mixedness. Over th:
rent c a for the degree of mixedness have bet. I ewed and
Fan et al. ( 3 ]. unately, the general criterion for
^ompor. i? still lacking. Chap*.
presents some approaches to define the mixing indices for rail t icomponent
.
ipter cant conclusions and proposes several
pror areas of future research.
- 3
REFERENCES
1. Y. Akao, K. Shindc, N, Vagi, L. T. Fan, R. H. Wang and F. S. Lai,Estimation cf mixing index and contact numb, c spot sampling,Powder Technol 15 (1976) 207.
2. H. Shindo, T. Yoshizawa, ';. Akao, L. T. Fan and F. S. Lai, Bs1 - on
of mixing index and contact number by spot sarcplir.
an incompletely mixed state, to appear in Powder Technol., (1978).
3. L. T. Fan, S. J. Chen and C. A. Watson, Solids mixing, . Eng. Chen.,62 (7) (1970) 53.
D the char a:
example, rocessing, ore .al
. le
A fed
.on of be
cv
a
DultJ r>::e : vpes
.
.al
M .
. - - .
1 pro; . , ecler,
• - cs to be
and .
-
ii - :
Numerous pul Lications or. binary sclics mixtures ar.c Lnar;
systetos are contained ~r. a number of surveys and review am:les. including
those published by Ueidenbaum [1], Bourne _-], Far. et al . [3-6], Yalr: tin
[7-9], Yar.: [10], Williaa [11,12], Cooke et al . [13]. Since the Study on
mixing of binary panicle systems and their oharacterioa t loo is essential
to the understanding of multicomponent systems, the recent publications on
binary mixtures arid nixing, vhicb are not included in the previous survey
and review articles, are alsc reviewed in this article.
There -are two broad and equally important aspects in the stud; Ldl
.r.g. One is concerned with the characterization of a mixture and the
other wit nechanisit and rate by which the state of a mixture changes
•
Loth aspects are included in this review.
1. m:e:;t PUBLICA v solids MIXTURES ANI MIXIH
7 section rief review I the reel res
sclics particle systems.
1.1. :es
A: '-eld [1^], three crit . are I
Tnese are - cn-sar.;. I Intensity of
segreg . on (s: . c str-
and Kersey [15] utilized Suslik's concept of homog
to define the homogeneity of a\
, which has zero standard
ion for the cone :on of a cr.inor ingredient be:- samples.
. senhar t-Rothe and Peschl [16 Eroduced concepts of testing equipme
for bulk materials har. and also described tests for characterising the
flow propertiei >wders . tfickl [17] blended I
particle size t utions to achieve a target particle size distribution.
II - 3
|developed a new for det ;le size
- <
are two approaches to the determination of the nixing index based
on the contact number. One is based • coor .jn number sampling,
Chi sp ng. By these two approaches, Shindo et al.
posed th<. models of the ition cf the conta
an incomple: I state.
.
Ra: beds. The
ef: :atior. or. the hydrodynamics of gi | Flow
necessary : gence of a reg : article stream
wert :ssed. He alsc showed that the flow pattern in the . the
EunctiOl parameters.
.1] re. ree mechanisms cf segregatic r. : ::
segre. I particles, and the rise cf coarse particles
KoseV.i e: al. tdiod segregating phenomena in a ,-.:::-
. arum mixer. Steedman e: il. [23 -gated sclids segre-
a coal gasification burner. Sugimoto [24] Itudlod tolidl Mixing
frc segregation characteristic: .xrures.
urega: - lex par: tier.. a
phenomenon caused by a shifting of fines toward the bottom of the container,
has been reported for the systems with particle diame: Cht range of
:>]. Parson (26] extended thJ
particle I -. and without the presence of agglomerates.
Row a: revieve: ~k.s
on z-z se,- gas :' red beds. 7.\ey sis: reported
lOOt imi I eroble:
II - 4
Burgess et al. [28] usee zas : laicized bees to study solids mixing
ar.d segregation. They found that mixing commences a: gas velocities just
above the minimus fluidization velocity of the non-segregat ir.g component
for particle systems of equal density, but nixing does ret commence until
the gas velocity exceeds the minimus fluidization velocitj of the se.
component for systems of different density. They aisc developed a model for
segregation and r.ixing, which is capable of predicting the cc behavior
found experimentally
.
-tzgeraid et al. [29] used a fiuidized bed to stuc;. solid Dg
and calculated solids flov patterns by use of a general ; Eaentsl moot.
Pvengarajar. et al . [30] employed the single phase backflow cell mo^
simulate the solids mixing ir. a fiuidized bed coal combustor.
-y and Merrick [21] developed a mathematical model to C -
eral concentration gradients of reacrant solids ir. a 1|
to predict the effect of coal feed spacing on the com'r . a
bed boiler. Lateral-solids mixing diffusivltiei
in a 5 ft. diameter bed.
economic advantages of using a continuous process . -
accepted. However, processes involving the h. »Ceasing
particulate solids are often designed as ba: ocesses. Such systems
should be examined to determine would be advantageous to I 9n-
tinuous nixing. Williams [11 r.uous mixing of solids,
in which the advantages of continu < equipment for continuous
• are also report! :yanami et al . [32] developed a continuous i
process for fine and cohesive powders. Th< .cal cylind' it provided
with an impeller of multistage comblike blades has bet- .en effective.
-
Carstensi - ed that the otxing of a b: ;>n-
•pherical particles - rough surfaces obeys tc rial equa: as
long as the me. the :- actions are id- and
Hwang Lng of a racer
Iht 1 exper i-
i analyzed . ;ns of a diffusion ao Lt if assumed
a function of the a:
stream.
e t e d I salk
doc f grains in a notior.Ies; .rer (K.
»r. ~ pcsec a discrete steac;
- • segre; is
mod . degree ecnes'
the -ites ( ^e particle- blended by :he
par
•
tr.
Investigated the Mixing proces
bat -| 35 ] usee a p:
1•
. Bg and her .jir.g ;ls.
-ater h ed funds Lssm . Yaaagucl
[AO] developed so=e rate equations for sclics g.
II - 6
3. MDLTK EN1 MIXTl CNG
rne publications or. Bulticom] >i et c nixtures a- review
ansively in this section.
2..1 res
Gayle e: ai. »-l , used chi-square as a cr.ter. the
.:' a mult icomponent solid] mixture inst of a c
-r.g index based or. the sample variance (or sta: :ion).
Ace; • tc ther.,
, n (0. - E •
i * i oi-1
Ei
whe
mi --s,
.- uare v : : freedom,
n &
= observed number of pa Lee of any given color in the L-
sampie ,
corresponding I in the i-th sat.;. . . asec oi
average distribution of components in the mixture.
They also proposed a segregation index as
- -?
—
hs 2 2-
I - numerical index which indicates degree of segregate the
>.• obaer - *qu«r« re,
II - 7
-square : i randoc re,
- e>.: i-»qu- :e segregate ••
Be loped a formul.. variance of a
.ven p. samples from a granular material
a k- - .on as
_
q (1 - v - v )
. -J —t = 5 (3)
vht
icb samr. .
:T.e per V. btJ
III aggregate w'r.
random samples
t volume V and
« average volume cf a particle in
as
v- - ,2v2
- ... -q.
whe: q m fractions of the part1
aggregate
.
Stange [43] characterized the quality of «". :re consisting of
s of cnt sizes, denoted b ...},... He developed the
Cbl variance of one of the component g the
component? as z. us compo:
:i - b
r i - Pi
V—
~ 7. Ef « ),+ P, (If w ),+ P. I: •-.)1 : : - 2 j j 2
- etc «0
wnere
7., p etc. = « Lent proportions of cocpor. . 2, etc .
(If. w .J. = effective mean particle weight of co-.oonent i.
expression ca extending Bus 111 .:.. (3) I over
Dulticomponent system .---
Knc:: .-:. ir.i^lycec £ mixture of particles of ;i:es
mg a technique used Li the study . - , :
various types in a renewal process up t a time. Be del
for the i»yi | Lc means, variances, and covariances of t I Ol
rent types of particles as the total \
. ed asymptot. . is
r
. Vh v (i) L?- « i
3
.
.'--.':-.
- u2(i)J , .
7,= [w - 2r.,-(i)j
where
h • number fraction of p.: •• of I
\ volume of particles li h sample,
an of the !*< on of particles «.
m(i) - second moment about the Loo of
particles c:
- 9
t •. .
- s .
Sc .plied the : th« as ic
to
:.s , and to c ..ge
..nee and .ance are et
|roposed I 'a
assu:- :he popul . covariance
are - vai pi pesed as
I s ,
is asympotc I
.
• ed as a aultivariai
, 1 • .Dutior - » asympoi
uni oroug: Lxad aggregate. 1 .d be close to
tnt Sggragai . -nd
• cd or. Statistic does not
a gooc
a very 1.
Bu duced a co: jgene-'
. . . . , a
s :
.
. .•
. :
II - 10
is difir.ec" as the negative logarithn :f the sample veight.
• .50. exter this concept to e nulticomponent mixture by i
the standard deviation of an ingredient 1 in a cor ra
fixture, eqn. (^). They assumed that the components must be re.
san>u par tii-tr- size level prior tC mixing, i.e., particles of eat. .one:
have the sa:.- particle weight.
(f.vO. = E(f>3 J - : :
= V
re H is the effective near, weight of all particles in tne fixture. Tnus
,
eqn. (i) reduces to
:
:
= vV ; -v
2:. C\ 0.01 and U = W. gives a . rent equaticr. :;r thl homogeneity of ar
redient in t .Disponent nixture as
j) - -log [p. (100 - p ) wj, j • 1,2,. (9)
is can be used fa single
component in a mixing operation or to compare thl of
powders necessary f *ng th< horaogent .a
. componer. operation.
Cook and evaluated a Kauta DX ( g of
a multicomponent tablet preblend. Four ingredients (phenobarbitone,
butobarbitone, quinalbabitone and lactose) were i :ng
operation samples were removed fro and ass.< r each ingred:
II -•
gas 1 -ornate. 11 •• of i l nt
as cal ed accorc (9). Eac
beh .e Banner during siring. The control jae is
nee a: ary one tioe one
segre. oes not meet the
jmoge:
[53] proposed a discrt model
of a i omponent homogeneous p.
* r r. The cor. .ance, -, of the .it
; - <vJ
- -re
C. (K) - number concentration cf componentJ
::. •
(C/' bec conct
n u- : tr fractioi
number cf ceils in a mixer
.
r components in a mixer.
le composition tht completely segregated state,
. was • c model aso
* GvJ L'-t0> " 1
]
2
II - 12
where
P = transition probability of the particles fr< LI i tc ze'.
i£
C (0)
Q . .(0) = :., fj-
The variar.ee reduction ratio,'- > car. be vritten as
I - ^5 (12)
o
The degree of rtixedness r.ay be defined as
M - 1 - 1 (13)
La_ and Far. [53] proposed the idea cf using the e c: a sclic ~-
as a criterion for characterizing the mixture. Ihe entropy of the sc - .
r.ixture a: cell i car. be defined as
k
5 (N) - I C. .(N)£n C (N) (1-1
j.lJ J
and • total entropy of the syster. is then
a
S(N) - ::..£. (N; L5)
1*1
the systen is ir. the segregated t the beg:
process,
£,(0) - .6)
and therefore,a
S(0) - : S .- (17>
ii-1
Th< • . jdv of a 1 cha:. .:ero in the conoletelv secre?ated
state to a constant val comole: .»ndoti> state. Hen.. . Che en::.
of a i can be sin.il;.- ned as a measure of the decree I ess
of a i c.
II - 13
I a
mean vectors fti
chat .nee a.* « st*: I
- -ies has
bee: ited. :-e degree
e at
: - .* --^ S)
-r
re
- ...
thf cow. segregated
:e,
I | » < t of th« the co- • •
-
ihe
pi1 1 (•.-.-.-•
tveral n.cnosizec .s, Soroer [55] a<
1 usm .•
,
Lzt - t .
,
3- . . . . .......
- :.ie greatest size anc e
I assumption, the f c
I
can be used to calculate
.ncc of the I Lc horoogt as
j-l
2- q!'; 7- -
:: - 14
: vi 2 '
( : y.:, = c. u - c' — - (i - q!) o_ ( I :
- -^re
q 4
c = —,
:
1•
«
q*
X. = volume concentration in a sample,
q = theoretic volume concentration of component i.
3.2. Hixing
Gayle et al . arried out nixing experiments b-
1 mixil
Ich was designed for mixing samples of coal and coke. Gra- .
.
tch were . r in surface characteristics, shape and densi: I at iif-
I . . vi r« used. During nixing, the value of ch i-scuare, as
.;. ecr.. (2), decreases with an increase ir. the number of revolu-
tion! ---:ir.g wheel and approaches a mean lower limit asymptotical]
2Tbf initial value, x » - or chi-square (before mi: Lr is equal :
number of degrees of freedom multiplied by the number of par:. Lei
foi each temple. The number of degrees of freedom is defined as :
of . of data. The segregation index of eqn. (3), v
unity to zero as mixing proceeds from co_;le:e segregat.cn to cor.
randomization, was used to establish a conventional rate equation as
dI^ e
n i
where-
dr s
I di- if segregation
num.- ons o: er,
n rate consta:
8 • consta:
II -•
La. "an (53] developed a stochastic node! ., of ic-:.-
cor .mogeneous : a s>e | >.er
inentaliy dc ^abilities of a I
homogeneous pa: km. The oodel pe: ihe concen-
and the degree c: . copone -„o-
ger. e.
et al. [54] employed a component solids
• s hav. LtntlCl except color
I used. . ance mac:
reases as rease?
.
is define -qn. (18'. reases and approaches 1 as
the ; reases. I a Milt poner. t heterogeneous z..
expe: reases up and decreas-
asyr totidily to 1 tate because of the segre: tenet-
-
-. :
review of th( re or. binary fixtures a- I es
r.ary c or. poner. ts systems . ized bee exter.
the last several years. Zhll tfl a net onsequence cf the
zed beds in cany areas cf material process
Lag rar
.
This rev:ev alsc indicates that segrep OX
demixing, which is the counterpart of mixing, than nas
recor.e the esphasis of the re .gatic
sampling of toulti M and
.ons have been of intense interest. Vet, the literature
ance, the c: ting
II - 16
the homogeneity of a heterogeneous Bulticomponent nlxture is still la
Sir.ce any knowledge of Bulticomponent sclids mixing and mixtures shouJ
reduce to that of binary mi: ng, tc establish a bread, systematic ar.c
analytic approach tc this subject is of importance and significance.
n - :
.ns
h sat.,; le aoou COT of Che i-th sample.
B
nunbc j in ceil i at time '
ec, •. .it ion of component j
corresponding expected i-th i . based on the
avt ution of conp^ I ire
of conpor,>
negative iogar. « -lop
• ion o: conpo: ire
nu.- degree of sep- jr. of .xture
-r of components in the re
a(i) • about the or f the volune dial
delta ol
D
r of passes
r of sanples
obser ar of particles of n: . to color in t I L-l
.on probability of the particles from eel.
P, i
q ooretical volune concentration of component i
ltd In vrn. (21) p-:
1 qi
Q,. a. ^
^ e r
. nee m
II - 18
7 testing statistic as - .. (6)
t required tir.e for one pass
volume of particles ir. each sar:.;
vcl mne c :" a p - 1
"
i
v average volu a particle ir. the -ixture
weight of a sample
W weight of sample necessary tc give a standard de.
-
...
ear. weight of all particles in :
volur.e concentration ir. a sar.'.cla
Greek
u total nuDoer of cells in a r.ixer
rate constant
X anour.t of a sar.ple
•-(i; lor volune) distribution of 1
as defined in eqn. (5) =
i-1X
II. n<. fraction of par-. in cell i
-covariance natrix of a xed
:te
variance-c> trix of a
o indard n of v .on of Jom samplesof constant vo
.
'0
a rand on
id chi a segregate
II - 19
ienbo.. . T. B. and Hoopes,-
powd* <-*s ant
3. and C i
-70) \ .
-al
.
>: ...
. and C. A. War so:, I & EC AJ
. V. V. . , . • . Soc . , v . .
7. ] powders ar.
-
: s and pa'; CE 9
E es ar. I is ,
cess. Eng. , 6E 1967)
10. T. Yano, Recent .s . Assoc
, 9 (1972 .ssue (:r. Japanese .
J. C •. ula:e :.
Pow 245.
. nuous nixing of solids -.
.
.
tns, end J. Bridgwsi owder | - a
c survey. Towdcr Tcchn I .5 (19
LxCures. : - .
and J. A. 1 s , Pc 16
(1977) 189.
rt - Roche and . .es
, Cner.. En, (1977) 9
II - 20
M. Kicks, Blending solids with known particle sic^ ij :
best achieve a target particle size distril -'... presented at l
International Powder i Bulk Sold £> Frocessii ference,sement, 111., May (1577).
18. .".. S. Abdel - Azim, Short com mication - new technique for dispers -
- der fcr cetem. particle size, ?ow:er Technol., It [1977
19. H. Shindo.i T. Yoshizawa, V. Akao, L. T. Far, and F. S. timationof nixing index and contact number by spot sa .. g of a nxing in
an incompletely mixed state, to appear in Powder Technol., (1978).
20. H. Shindo, T. Yoshizawa, V. Akac. L. T. Fan and F. S. Lai. Est
mixing index and contact number by coordination number s.-.-.c- ling,
summitted tc Powder Technol. fcr publication (1976).
Particle flow and mixing in vertically vibrated beds,. ier Technol. , 15 (1976) 15".
1- . J. ftoseki, I. Inoue and K. Yamaguchi , A study on segregc: -henomensin 8 hoi tal dru National Fall Meeting of the SocietyChec. Engg. Japan, 197-..
23. U. G. Steedr.an, W. C. Corder, and H. C. Meek.s , Sclids segregation and
flow in a coa] at ion burner, CE? Jan. L977)
.
24. M. Sugimoto, A consideration of solid nixing fi character-istics i c, J. Res. Assoc. Powder Technol. Jpr. ., 1- [1977 22.
25. iu • Lta, Sere considerations of powder :
and v. Lion, Proc. of Powtech. 1971: International Po- nology•nd Bull nference, Harrogate, England, H L2-14, I
p. 103,
26. D. S. 1 segregation in fine pc ng as
n of jostling during transpc:: . n, Powder Techncl., 13
(1976) 2<
1". we and A. '. :.ow, particle :. and segregation in gas
Is - a i ., Pow . ichnol., 15 (1976) 141.
J, H. Burgess, A. C. Fane and C. J. D. . Measurement and ationof the mixing ar tion of solids in gas fluidized beds, 2nd
Congress (PAChEC'77), 2 U977) 1405.
Jatip^ ;>erimental measurementssolids movement in s tu 7 0th Annual•
30. Rengarajan, i seng and C. A mathematicalmodel of a .zed coal combustor, 70th Ann I Meetin;
- •
ley and D. f the spacing be solidEh« performance of a large reactc:
.
Sato and T. Yano, C particle*,»u i Particle Soc: I -eting (197"
33. Carstensen and I , Elencir.g shaped part.-
+ 9, Povdar Tachnol .".
. [11 .695.
:-gg an- C. L. Hwang, I .. ir. simple flow systems, J. Powder 6
Bulk Sol. 1977) \
ing and L. T. Far. "... -.:.:. a motionless Sulzer(Kc EC, Process Des. Dev., 15(1976) 381.
: mode Lin >f segregatior. in a notion-Eng. Sci. , 32 (1977) 695.
I bate; r,"
- (1976) V.
and homogenizing of granular bulk material in a
. I- - . L97( L99.
j amenta! powrie: - r Tec:
. On solids I .te equations, presentee on Povd<
gy Me-. in Japan, June 1976.
61. J. B. CayIt, 0. L. Lacty, anc Mixing of solids - chi squareas a ,14 EC, 50 (1958), L279.
Dg and sc -:th special references I -s;zedgranular nattrial, Bulletin, 165, (1950) 66.
ce. Die rr.ischgute einer tufallsmiacbung aus crei und m-Ing. Tech., 35, (1963) 580.
estimation of the variance of samples wither -
random mixture of Bulti- .zed particles, Chem. Engr. 214 (1967) CE270.
'-'
. Knott, Sampling mixtures of particles, Technotnetrics , 9, (1967)
365.
:ures of I zed particles: an applica-• of renewal theo rchnome tries , 11. (19t-°> 285.
47. .. Scheaffer. • e homoge Curt, Technometrics
,
71) 31
46. D. Buslik, A proposed universal honoger.eity ar.d nd . . Powder
Technol., 7 (19 72; 111.
49
.
J. A. Hersey and P. C. Cook, Horjogenei: :" .spersed
systems, J. Pharc. Pr.arr.ac. , 26 (1974), 126.
50. J. A. Hersey, P. Cook, M. Snyth, E. A. Eishc: a-.- E. A. Clarke,
Honogeneir t nulticonponent powder nixtures, J . 7 ...
6 3 (1974) 40 6.
51. P. C. Cook and J. A. Hersey, Evaluation of a Nauta ringa rsulticomponent fixture, Powaer Techno!., 9 (1974) 257.
52. P. C. Cook and J. A. Hersey, Evaluation cf a Nauta r.ixer in
ceutical industry II: Evaluation of a nul tic opponent rl lust. J.
Pharr.. See, RS3 (1974) 49.
53. P. S. Lai and 1. T. Fan, Application of a discretestudy of nixing of nultj component solic oarticles, Inc. Eng . Cher..Process Des . Dev., 14 (1975) 403.
5-. P.. H. Wang, L. T. Fan and J. R. Toe, Multivariate Stalof solids nixing, to appear ir. Powder Technology (1978).
K. Sorrier, Statistics ineSi with unequal \ \ ^.:es, 7C:
Annual a.I.Ch.E. Meeting, New York, K • L3-] L977).
CIL* . . . 1
ilOMO-
-r g
recesses. . lable d«: . . cms
-gree | -.e nor
ge re, are based on the .nee of the n
-pies t, i.g. ,
"
rea of deg.: s based on
met of che n.
.gree ess
. eroscopic anc gee
lomogc e. T
pes o:
as a referer.ee.
etwee: rer of
.-.•_. : a key fcl
I Of • I
I r cpe:
phases.
rto cor ixctndtd to a
•
ses of a re
.e
Ill - 2
direct counting :: tl :a:: number. I he mea :cr and :rs
precisi~r. ft - iter simulation :: the :v:-:.
hexagonal pacr:i-_: arrar.ger.er.rs a: different :oncentr E the •
component are reportec. .-.. ?:_^:e~ Ln thi ire
erved 3ti tw - -sections, t tea tained can be ::£-;
into a three-dimei . tereological a-
is:.
-
N :•
'
C RAMX
tr
I is tak-. :e:
of all particles which are in contact -. :s particular particle
er der. I :*, ar)d the
:alled the I Such 5 is called
on number M | of I *. For exar
D number of the t ensior. - ic packing
•it of the hexagonal packing I -Tient is 6.
Let A. particles be key particles in a E omponent mixture
con cles : same size, A r ,.-. I . The numberu 1 -
.
.es of conponent A. in contact vitta a key p« .3 def I
as the contact \ ributed by conponent k.% and is denoted
1(0)'e ' We S6e : " r:C
1(0) " 1'
C '> (0)
" 3 '
3. of the Contact Number
I complet .xed state, particles of ea:r
•^cted to be randor ibuted in a sample f] e cocr .on
number sampling of sice n*. , the distribution cf the
contact numbers, C. ,-. , C.,, .,..., O^ix/os • •-
Dution, i.e.
*0 L0(0 •
Cl(0)
Ill - 3a
n=6r Ci(o)-i
7Cz(o)-3
Fig. 1. Illustration of a coordination number sampl
.
size d*"(
n«!C0(0) ")) IXO)
wl(0) p -: (0)
- cwm : -"9-iv,n?°
J CO)
Wconce ~onent
,C /r T -.us,
the expected value and the variance of the contact number.
• r.ixe- -respective!
- r.v . j - :.: Cp-
- tt*S ":.), ; (p-J (3)
:-.er hand, we have a cne-pc
segregated state, etveer. different particles
boundaries are n- .«.,
Pr{Y - Chi sample particle is oi component - 1.
j - i.2 (p-1)
?r V. • the sample particle is of component 0,
(p-1)
means t e expected value and the variance act
ited state are both zero, i.e..
EfY.] =0, ; = 1,2 ?-l) (6)s 3
[YJ = 0, : = :.: -1) (7)s j
Fron eqns. (2) and (6), ve find that :
on component .-.. changes fron zero in the zc~ I -c statr
n*X. in the complete! ec state. Akac et al. [2] nave usee : i
to define the degree cf mixedness, K, for a eneous binary sys:
Cl(0) " £
s[C
I(0)3
• '- rc i - ~ re l
c *
where C. ,~. is the mean contact number of component .-. . estimated from
sarr.oles of the mixture.
C. Estimation of the Mean Contact Number by Spot Sampling
.en a spot sample of site n is drawn from Che
the concentration of particles of component A is
x, = -^ , j - 0,1,2 ;?-: (9)
where a. denotes the number cf particles of componer.'
sample. The mean contact number is, -
,
cj(0j
- n* Xj , ;- :,: -. (10)
Suppose that k spot samples of an identical n are obtained
from the completely random mixture. The contact num.:-: i by
. ;les of component A. per key par
[CJ(0)
)
l- [n*x . L - 1,2,
Ill - 6
re subscr. . >tM • *y P* r
ample
-
er cc: r.e-: »t
' - . nunbv
he por
efore.
'j (0) tot
•re
•>::S ]
i
concentration of parci tnt)
is r le r^ear., k_, can De re; e.,
cj(0) 0^ l
< 13 >
near,
cor -.umber r.
-
Cj(0)
expe- r, C.
,
n, . - . : • .
'-B^r^l
III - 7
.. . . r-1 14
or
- _err = - -1
Equations - and L5 Imply that the estimator, C., », is bi it
is consistent, since
— V: : =Er::;-0, : aS n
rrc~. eqn. L5 , we can . .sister.: '
. , as
D)=
n-1 ~;(0)
o-l
jl[n*x
11i[x )
ii
•.
:,: ---•
Pre e £>'r.o~ that l
C' Le
j (P-D L7
• ::<? unbiased cor.
E. Precision of che Population Mean Cor. .umber «.
The precision of the escitr-ator , C' . ei population
mean c .s decenninr ?ugh evalua: . the variance
tribution, uh; (see Appendix B)
Ill - 8
3)
otnpor.-
« ki expres
l
A c:i:er. usei for des^:
:.: of variation or :he ^ndard
err
D[C' ]
- -<^' 0),
(19
S(O)1
! ... ] :
j>l-'vr[c;(0)
] CO,
-e.
=.
—
:(0)'
Chit reduces to a icnovr. express i.-
1
1(0) .
-.., )
- 1
1 number of samples, k , necessar a spc
. -t . ,
I
. : iLJ c.
n R . S . E . [e
*
(Q)]
Ill - 9
3. DEGREE DF MIXEDNESS il ON THE SAMPLE VARIANCE AN! ESTIMAT1 )F
ECISION DF THE DEGREE DF Ml ~-SS
Over 30 different criteria for the :=::ee of mix have :een
reviewed anc s jrr-.arizec by Far. e: al. [1]. Here we are ccr.cer:. .
wit] the cs:ir.a:or of the population variance of an ind ent
from soot samples. ~7, which has been frequent] j .
J
....degree of mixedness. Suppose that a. denotes the number of par:. L<
conponent A. In a spot sample of size n. Since a spot sar.rle is
be coi Lxed, "-.is random variable a '?se. --
distr. . a i". the completely mixed state if ve cr.oose A_. as at
and lump all the remaining components as the other (pseudc
Thus, we have
E ra.] -nX,, : = l,2,...,(p-l 23)
and
= nX^ . j - 1,2 Cp-1-i J j
listril can be approx. _.-;:.-
butioi • [7]. Therefor .ng ap; are tq . :-.s
can be obta. roper ty of the distl I. -ince c:
normal distrJ itJ r. (see Appendix C) .
E, - [nx (l - >:.)., j - :.:
and
;
- l - x j - i , .-
.
Thus Che relative standard t S.E. . (j
D(
R.S,
;
p
- 10
Cjl ' ?---
II - 11
u. computer simulati
a. Procedure
Randcr numbers (between ar.: ] Lth a uniforo :. ire
genera:-: by SUBROUTINE RANDU [5] tc simulate a nponent nd cure
in the coi -tely mixed s:a:e. "cr example, ir. a ternary certifies r_:::-re,
any rancc er, which is less than 0.2, e :-.:s a particle pooent
if the concentration of component a. is 20Z, i.e., if X. = 0.2. Si-ilar
we may use a random n - ich is between 0.2 and 0.5, I -en: a
Le of componen • Lf the concert I component a. is '
. i.e.,
LstrJ ited n rancc - er are gener^:
i
a spot sample cf size n, and x. ( j 1,2 is determined from eqn. 1
near, contact numbers, C ,n>i
ar re men calculate -ians c
(13;. For precision, the r.: is replicated ' s.
Sp algor.- ts beei
present wo
I
(a) Spec. pot s:. :r c , n, thl I .1 coor
the number of replications, !'. the population conce:.
each conpor.' "1,2).
(b) ?re- k, to be take
(c .macors, [C! /r..]' (j - 1,2) of zh^ pc: on
j (U; m
contact number, Cj/a\i by eqn. (lb),
(d) Compute , and : .ance, V [C. #-.*), from the
. . as
c;< ) -sa tci(o)V
i(O)1 -FT J ^(0)'.- !
i(('•
(e) Estimate the relative standard error, R.S.E.[C' ] equals
in - ::
I
Cj(0)
a , ai
^pct i is
"a
:
- UJ
)1
(30)
k ,, , - ,J
kTI ::::
.
" V < 31 >
(g) c
. as
1
v. - — (33)
r based [ e.
. as
ll cuss ion
I results o: tl . exper El or li
xture - • s 2 and 3 f - I "le
j cases tt* 4 ar.
abscissa plots the number . and the or.
I=.
III -
: * * : - o— ' ——
^
• fc— —*•—=c — — <^ —
_QD
- . r-
_
Of
n*X.
y =0*5
€
: d
: a
on
Mff * * * — * »
H^ * to
- > » T- # • f * m ~8 M y >" A A. y *
0.
• I
-.
:
*
0.2
-
r
£l •
DF 3 T SAMPLES k
Fig. 2. Results of computer simulstion »rid
1.3.
-
c.
—* ^ i [ ' i : ' i j * * » * t> a " * * *-A r—
-
_n*X, - 1. 2
- 0.2
_
"
s a • 5
: a - 10
: a - 20
—:; , J %- «*£*£ » ••s& -* w 3 at -£»»- *
n*X. - 0.6
- 0.1
: - 5
n 10
: 20
:
- r r z - c;v: z50
3. tMulti::-...
. J—
(" Z —
0.8a :•
3- H \
DC . 4-
.me 3
C
A: R . S . E
?. . S . E
S . S . E
R.S.E
?.. S.E
:'._.] (theoretical value )
-
Jl. J -1.2-
r . si:ul3:e: v..|
-]
L(o)]
3)]
.
.
12 T E 553
Compari.on of theoretical relative .t.nda
with those of numerical experiments.
-0.2 a;. "0«3
- 12d
1.0-
:.9-
7-
-
M B
c
A
*
.1.
R.S.I.
. .1.
R.S.E.
,E.
R.S.E.
.£.
Cl(0)'
(theoretical value)
C 1
- 3)J
- . am)
C 1
2(0)J
D.04-
3 T S R M P L E S
. . 5 arise: -. r.eoretiral relative scv-•-.:al experixer.t?: r.*-t. r.-20.
_ "
• X - ~
-- -
j
.,1 . rig-re - is for r.: =~, a:.- Fig. ; is for a =-.
between the results of the simulation represented
Che ::.c:re:;:il values represer.rec by £:!.: lir.-.s appears t
each case. The higher t ila C , l
re. - lard error of C' ,-»» . I: :a: a. e obsex I.Z.; (0)
-
[C',r.] ( j = 1 , 2 ) are always less zr.ar. R.S.E. [o,] - r.
.
i#€ :an scare that the precision :: : e C I I
C ! ,-» , is e esi- e. r~
. I:.a::on0) j
. S . £ .
'2 ' lei ends on t . ~sj(0)
shou Ln "igs. 6 and I- (27) E.
-.depen- • populai
sar.pl-. size, n, on t
an be ob- Ln the resulcs of ~s.
6 a: I 6, respecti Fe obs<
size, n, the relat ror of C'.. as r :, .. :
-gures .: for s.- 3Sign^
spot -s zo be taker, for a sp-
Suppose ?^?-- conc<-
. :.. La .
coordinacion nu -^n co:.
- 13.
8-
—
*
--» »—
A a - 5 (the- il values'B - 10
C -
* - 5 (simulated values)> - 10L. n -
2
no. :r
sz
:
5G
.6a. .
"- relative star'
-wq\ "-
i- »<>•«
'' '*'- exrer.
and B* - 4.
i n.
- E _
O.. CM-
_.-
: o . sLO
CCO •
:":
Line A Q :heore-B D = 10C Q 20
* a : BlBulftt | es
a 10
a - -
jl*_J<
•
nc. ::
4C DM F P ^ k
Fig. 6b. Comparison of chaoratical rtlacivt standard
trror. -hos« of nunarical
experinen:s:K
' "0.3 and n*-«
1.0-
: . f-
o
-
,
ine A
5
*I
:
a
a
o
theore- values;10
5 (
10
20
c -^ >
3. :z
s = :r 5-kples •
Cor al re.
R. • -
• ''
.
1(0) L-0.2 ud -.--6.
'
]
.
0. 9-
: l-
§0.v CM
—
•
-
—.:
-_
_
o x
•
cc - -- A
H
ilnc a3
: • i zi 1 v
D =I
o = 20
= 5 (si=u laced values)n - 10n « 20
Comparison of theoracical rtlative standard arrors,
K.S.I. iosa of nuoarical axparinants
and •.
Ill -•
!
.
nal dt —>e degree o: :ness. I h are
mainly macro [ taKt account the
pic char.. . -ex
propo.-' possesses sone superior chare
- .
'
~.al ones based on I
cor. ne contact number concept to estimation
- homogeneou nvr.z :he ?c
such a
ts.
oaper »_s,
es, as described previr al
e also been pe expressions
bar. The re:.. standard error
er estimator has been sno-- r tha\
spot sa-ples. Therefor .
mea- tl estimator is sure ... ;e es:
~1(0)'
:e
.
Ill - 15
a r of particles :: component A. in a soot sample or size nJ
:cr.:=:: I I of particles :: >nent .-. - en tre keypart-ties is of component A .
C. -, estimation Df the populati in contact numbercor.pcr.er.c A.
J
C' . unbiased estimator of the population Bean contact :.-r.;er
component A.
:.:
.
m the t-th spot sa^ie, i-l,2,...,k
^v of N replications
: deviation Df variable
expected value of variable
k r of spot samples
number c: replications in spot
n pie size of a spot sa
co . ing
and inc. . cle
. E.[y] Lve stanc iriablt
variance or vari.i .
pc en concet.
sar . ... conci:
concer.trat.cn of pa: :>rr.ponencI
o spot sample
Y. random variable of the con
^ns
Subscripts
r pletely random state
• completely segregated state
: - 16
«••
mean
.sec estiaacor
Ill - n
REFERENCES
I. L. T. Fa-, S. J. ..•--.. an ist Scl. :ir.g, ar.r .
:.-.:. Ir., . :hem. ,62 L97C
1. ';'. Akao, T. la, £. Takahashi and a. Otomo, Mi: -
mber for ire of particles with uniform size, J. ?.es. ier
321 Ln Japar.es, .
2. V. 2. Smith, P. 3. Foc:e and ?. F. Busang, Pa res,
Phys. Rev., 2~ 1929 L271.
V . Akao, H. Shindo, N. 'r'agi, L. T. Far., ?. . H. tfai ......
Estimatic ntact i -
U L., 15 - 207.
5. Lentif J- Vers:
::, p. 3^ C1967 .
6. M. Fisz, Probability Theory and Ha the ltd a, Johni SOI V.', ; 1967) 347.
H. Cramer, Mathei .ods of Statistics, ?::n: = :;: . ress,
Princetoi . .J., '1966)
.
E. E. Underwood, Quant -Wesley, : s
(197C.
.
ITI - 18
N OF EQN.
. repeated b- Man cone- nber
coopc
C -- (13)k3
expec .lue of C..... In I is
E -i-i '
3) - J
r
. i
:e [6]
•
to eqn. (A-l) yields
—*0
:- 1)
Ill - 19
APPENDIX B THE der] . DF EQN. (18)
Equation 16), repeat elov, expresses laser.
of- ent A., C.,..,J ;(0)
C.j(0) n - :
,>»*;:, >•
L
L(
Thus, the variance of C. /n . in the compl random scare isj(0)
V [C[C = s—x V [ . £, ( n*x«x . ) . ]r
J( ' k2(n - : -y.:
riml
— ;—;"-, *nx 1.
r i >
n -i) r r u J
2
w 7733— »*!»k(n - 1; X
Qj
Sir.
: 2i
- 1) (r. - 2) (n - 3)
in - 1) (n -
3
- n ( -
3XC
III - 20
- -
3
up,-
a - :
(B-3)
we have
-
2 _ -
J '
-
(n - 2)
3
3X X
j
Su: rion c . es
W:
- t)
-I
*o
fr - "2)_-_ (i
X," Xo -
n - 21
(n - 1)— h Xj
1
9. 2
k.n.xQ
Cn - 1)
(r. -' 2)
(n - 1)
I - I ) "
2(2n - 3)
(n - 1)*0*
n -Q* X
kn(n - 1) :•:
- (n - 2)(Xr -7
: I -2(2n - 3) 7L X .] (18)2 :
21
- •
and (26)
.
e average number o: Lea of component the sample
among k. spot samples ai
*, a.) -
;" Fl L-l
Uaj
)i
3
:1 i [(a ) .
- a.- 2
Denoting . . . we have
k-1 k-3
k U . e :•:
:
-
--. 2 r[i/2 -
•-.for .
:<
\for u. <
Che value < the esq
. as
• - 1
J
- 1
.))-
Vis, J=
2(k - .
III - 23
2'k - n: --
Since
k - :-
we have
:;;
= e : s
.
2
k - iJ
E[i
Ic - JJ
- - X.)J
V[ i
2 k - 1J
(k - :
[n\\ (] - .'".
) ]
(k - .
Cl.
I ON
. spot sar.pl .
. c o d i Ledex
The . ^ *
.onver.I
^osed .se to
r.d gee
-
act
•2X arc • :.ecre:
e : a 1
.
: o r es •
of a
:
spoc I count the conta,. . a
.on co:. E -s es:
n
se, because
are neglected
to
. on-
•
•
s are
:ed p<
.-sed. Cc support :r.e effectiveness
- 2
2. CON! . IUMBER AND MIXING INDEX
'..'-en a particle is taker, randomly from a mixture, the r. . :f
particles which are in contact with tr.is partic_lar particle is c
the total coordination number, o*, arc this particlular particle is cal.
the sar.ple particle [1], Let A particles be key particles ir. a malt i-ccr.pcr.en:
mixture containing (p+1) types of particles of the sane sice, den
A„, A,, ..., A . The number of oarticles of coiDor.cr.: A in contact1 p
a key canticle is defined as cne contact number zct zed by
and is denoted by C .,-»
«
: (0)
In the complete Lxed state, particles of each component are e>:~ez.
to be randomly distributed in a sample from coordination number - Lng
of sice n*. Thus, the values of the contact n. or sir-.
numbers, C.,,^, C, /*»\« •••, C /AN » have the followingIKU; l(,(J; p\U)
C0(0)
" C0(0)'
Cl(0)
" Cl(0)
Cp(0)
" CpC0))
iiU -C0(0 -
Cl(0) -
c?(0)
.:
C0(0)-
Cl(0) •
'•• Cp(0)'
n»! g ,Cj(0) fn'~
c A XJ .
3iocJ(0)-
n* (1)
cJ(0)l
J
j-0
is the population conc«- -cles of component . and
c./qv is alue of the contact number C .- for the
s, the expected value and the variance of C . in the coraplet. Lxed
ate are, respe
-
.ir._
:•
"
The expected value and the variance of C ,-. in the cor egregated
o are be :aries bt••• pes
I are . .
Eg[Cj(0)
l- o, . : P
:y ' J W ?(5)
kAO e: al. posed a asec on 1 . :t
C . C . - . t. Here, ve exte: -
re.
E::3)1
(6)
1-1 irtcJt0)
]i -*o
- u
Under the assumption that the completely nixer: state exists in es
spot sa-ple, an unbiased estimator of the contact a - , -
. in as [2]
k
.\nx;x
I :
]
i
jCOJ -- kx„
denotes tr.e concentration . : A particles in - Ot
sampli size a, x. the sanpie mean concentra
:
key c at,
anc E r of s pet samples, [n I' , C' , is referJ/
an contact number contributed by component .' pcz sa-ples.
: . . ten specified in solids mixing, we n Laci ...
: = iJ
ci<o.'n^
..pec tc
ance, . ] , c re:. Btandsi
respec _],
Er)c:
(0)] ..1 j - :. . (9)
•
V Cj(0)
]
k uin-lU •>*
(10)
ind
-
- 2 2n-
3.
::ec s
:
ncer.:\ of
re ir. a re, ar. value
; 1 e
I
deal ;ns or -.ogeneity ar.
^n has beer, inert I as
an tur I .He:
we
Let z_ der.cte onpo-J J
r.e: . r the assumption t tt '-.~e ioc are -
Iterlbu ate
ger.-- I beta D (see, e.g.. -
:er.s. isP
v?
yp
J
- '"J" \l -. ,
J' 1 J
j-o J
p
:- l
j-1 - p2-1
F
:(: : -
-
L2? ? P
By.! r(ej :;: Cy.+e,)]j-0 ]
; = : j-0 - J
y y ,..., y n and I y = n- ? - j-o j
vhere Y. is a random variable de r of A partid - -
are cent air. t a spot ss the observed va :.-•_• rar.aom
variable, Y , z. che va . e randoc va ible, Z., 0. » C
-: c ... c distribution, and r . the game '
Statistical Characteristics of Random », Y_, Y„1 - P
The general..-.-, fact rial inomen: is defined as ;4]
- 2 ?
- v - v P]
wht
j
- v iv-i) ...--:), - lJ
Thus, che generalized f.. il moc aial
model, defined in eqn. (12), can be c. -e« Appendix
I\ -
p
i - - -; - -. ..
The expec
:
.-.e variance. . are o: eqr.
. respectively, as
(0,0 ,0)
a -
i-0 ^
-
.1 - E
lO.C 2,C 0)+ U
(0,0 0)
- -~
(0 : ,0)
n(n-l)e. (1+6.)'
—
l —'--(—l)p p p \ p /
L«C i-0 i-0
-
IV - s
where
• I 6
i=C -
• covariance between rancor variables Y. and '-'., C , Y ] , isK j K.
Cov[Y.,Y ]
J k
nm-J- k
•
k
l+: C B )i-0 i L- - 1-C i = i
p-
1-0L : k
? ?
wi-C - i-
k
L+i
J. 3 Estimation oi p.. ,
1 p
Estimates o: Lei di:
can be 0' ' follm
ng relationships are obtaine g each senple
mean, y , to its i do, eqn. (16):J
-
p
J1
J - -.P (20)
• re
P
i-0 a.
i w« have
-5 -
~ _-
.ISJ J
-
«2.J=2
, —(1+1
ins. 2), -e ob:
- er of spot . -•
- 10
j 1 , 2 , . . . , p 24
I e ?ara=e:er, " , car. be obtained from eqns. (21] ar.d '.2- as
y, i
i-l i
25
-. PRECIS] )F THE ESTIMATION Or MEAN :::;TACT NUMBER
v e :on8ider the precision cf the estimator, C .fl>, eqr.. [6 .
- mixture Ln an incompletely mixed state. This rretis-
from the Lstribution giver, by eqr. (12).
The concentration of A. particles, >: , in a spot sa:j
expressed as the ratio of the numbei .. par:. Les, - . CO thl
sample sice, n, i.e.,
1 -
eqns. (16) and (17) spect.
i"j'-i«i
P
- 11
ap.«:
J
.
h
• •qr.s. CB)i (27) and (28) Che cxpe » 4 /ft\]i and thej(0)
ill be derived as [see Appen- :
-
o.j(0)
J
and
VlCj(C
•
* :.-:
--(30)
and (30), ve e Stand*] -S.Z
fC ._ . . -ace as
0)'
- 12
"-C
:;(0)
:
- . - - - -.7: i-] •-_--:
1
_
+ (n-2)x(i+r)(e +eQ
) - n+r+i - l+i.
By setting ? = 1, this reduces r.o a known expressicr. for -^ binax re,
., [3]
r (t*+
p c r •' r ' = . ~ - _ • - -1(0)- n(n-l : r+2 --;-.-.
I*2n - .-, - - :- - -
-(32
Equation . samples, Che _ve
standard error to be within a specif. . I.e.,
k - (s
[2 - -
n(n-l)(x+2o :
72nr+3)en 9 - n-2)i l+i - * (n+H H (33)
•' rrespor. .. to tht con
. s . (30) t
;ns. (10) and (11 , r*tp«< (
(33) becc;
i ( Lx.,
•'
j-1
ft< R.S.E. (Cj(())
)Z
Accor . eqn. .
•. . 1 . ressed as
p n*
*'j: lnTi" • "
-
5. E5 OF SAMPL PRECIS:
Various de: ns of the degree edness based on the
sample variance have be -ed and s in et al.
Here we are concerned u .e estimator cf the population variance
of an individual component, r , which has been frequently employee
degree of mixedness. Suppose spot samples of
are take random from a ~I • incomplete /.ed
•.
- C the rancc are d is-I
buted according e Dirichlet-multinocial c. on. Sample
mea .. ecr.s. (16) and (17) , are
repeated here for convenience.
.1 - nX., - 1,2 (16)
i
] "°]
L-fj). J - :.: (">
:ore, the expected value and variance of sample variance C poiMBt
are [6]
E[S:; - (36)
- 14
and
v[s2
] - f-_ - ^ '
J - --I
for it > 1
where -.. denotes the 4th c^::^l moment about E[Y ], [see A D],J
- E (Y - E[Y ])*~
"
j J
Lat Ive Bta lard error ol . . R.S.E. .is
.s.e. ts: ) \-
ElS"!
(39)_ -
6. ' TER SIMU:
6.1 Simulation of a T<
Fl . ion o: nary
the inco;
As ienote umbers of particles from compone
. 'espec spot sample oi • n - V'-
Y and Y h.. lomial J.ist: •> parame:
.- li tach case, art
rr\\ * r areJ) 2(0)
cor ve standard er: ed
Lu« based or.
eaced
-
:ers are spec:
t spo:
•
• populatJ conce of each componen:
.
0,
. .- 1)
(
e
. k
: : re i
(2) " (eneractd as
•j:e:s
The :he D. s
. es
an
.... , %
d«-. s as
IV - 16
zo - 1 " z
i" :
.
(3) A set of multinomial random variables , each vitfa -ters
n, Z Z Z is then generated [12,13].
Steps (2) and (3) are repeated until k realizations of t
random variables (Y , Y ) have been selected.
(5) The concentration of each component is calculated ace.
eqn. (26), for each spot sample. The mean contact(
,
ft.
/
'2(0)and C
n- , are then estimated by means of eqn.
(6) Steps (2; to (5) are repeats tines, and the Bean, ?'n v,
and the variance, V[C! ,..], are computer : rs
,
respectively, as
?J(0)
=
ni}%m }m* :
= l » 2 < 40)
_
standard . R.S.E.j C'
ftJ , - - .ited as
[
1(0)J
^j(O) ' :,:
(8) The average number of pa: s of compon. he
variance, c among k spot eamples, are calculated, respectively
as
• 3)
: - - J - —
Th«. :e of b. ,'
. are evalu..
s, respe. .^s
% 2 - -
Jni- 1
J
"A ; - :
.
(10) -ased on the s.. ce,
. . :ed as
;•
-
Results r u:er Simulation and Discission
: uter : t ion .res are
snc- I. i through 5. Figure 1 LllustraCCf th( --
:ase of T " ' -. X . 2 ar "0.4. It car. be
see :coc agreeoe
expected value, e n Figs. 2 and \ abscissa I
d. :
- 17a
<j> o
* : Simulated values of C' _.
6 : Simulated values of C^O)
- ^ r> r.
. c-
I I
H[c2(0)
]= ^TT n * x
:
lj
xc
•
_
•* » * E » * I ! * * * ? >t * * * 1 i « * -»^*-
-!C',J -1(0)
J• :
" Ai
V n * A'
10 20NO. " E
. 1 . Sbtaultt of ( k-x Simula-.
n* - .
- - ... -
- 17b
-
:. 7-
.5--
.3-
.ine A:
3
- 2R.S.E. [o, J (theoretical val
31.S.E.|
a.s.B.(ci(0)
]
D:
•
I.S.E.[CJ(0)
]
:'] (s
v . I.S.E.
' .cic; (0)i
L "•»•«• i«5o,i
4C
NO. ] :z :* ''
Z _ES k
>nof : tlCAl
relative standard errors
. those of nuoerical
experiments: n* - 4A n - 2D
t - 6, X, " 3 - 2 and *2 ' °- 4
- I
: .0-1
D.9-
r. - -
• .5--
DCO.I-
;-
).2-
n i -
.
Line A,
B
C
D
;•
R.S.E. [a.] (theoretical value)
R.S.E. [al]
R.S.E.(CUD; ]
R.S.E.[6j(0)
]
R.S.E. [cj]
R.S.E. [0*]
*: R.S.E.[C£(0)
]
A: R.S.E. [C2
'
(Q)]
10 2C •
NO. : r SPOT ES k
Coa?arl»oo i*»!
ve standardse of Leal U
- 6.
• e standi
.. or'
e the
are :hese
are I reteJ)
a giv ore, we can s: i precision
,is superior to chat o: -srir.j
t popu . . .on
s
,
c n • 10, 20 and 40. an
s e •- ipprec lab ]
1st, n, on chi- ro^ •
oner. :s A, a-
of
,. as . (31). 7nese M are also us«_ .ng
BpoC sar.ples CO be :
be dc s. Suppose
and 0. * , re I spot
-.- '-rs. r.K
, :s •>.
ei .ex to be accurate wit relative re
and 5b ibout 31 spc are
nc in b< genera. np] Cha value
a spc on.
i of •
I was used to
p« re. A v 30 x 30) arr 900 loca:
- 18a
: .:
-
n ~ J
<b
•o.5H
DCO.4-
_ . -
. i-
.0-
* n = 10 (simulated values)
n = 20
n = 40
NO. : ES ksc
Comparison standardil
. . x, " D»2 and
-•-
<b
- . :-
. . .
-
> . n - 10 (simulated values)
O : n -
A : n - 40
z:
NC. :r SPOT S— Z _ES k
Fi^. 4b. Comparison oretical relative standard
er • ] with those of numerical
e: nts: t 6, t»* • 4, X- " 3*2 and
IV -"--
: .
:
D.9-
n -_
S°- w<U
' .5-
. 4-
-
Line A : n = ID (theoretical value)
fi : n = 20
C : n 40
* : n = 10 (simulated values)
<$> : n = 20
A : n . 40
20
3. OF : 5 k
Comparison ftl relative standarderror, 2.8.1 3)^' hose of numericalexperi-aenta: t - i , n * - ^ - 0.2 u> >0.4
- IM
W.7-1
_ - •- -
Line A : D - 10 (theoretical value)
3 : n 20
C : n 43
* : n 10 (simulated values)
O : n - 20
A n • 40
NO. ES k
Fig. 5b. Coooari3on of thaoretica! relative standard
. . th those of numerical
ex-»eri- ' 4, j(. 0.2 and
- If
was usee to represent the spatial distrit Jtion : a ternary r._:-::_re.
assignee a value, '0', 'i' or '2' in eacr. location t: represer.' ponent
a, , A, or .-.,-, respectively. I LtialJ the - itions in the art re
arrar.ged as shown in Fig. 6. Ihi ^-~a i-S a S-- - -- r e~r eser.tat.
of a ternary mixture in the : tely segregated state.
Mixi rocesses which gave rise tc various mixture patterns -ere
simulated by rai y interchangii :r:icles at different locations.
-: a fixed i interchanges, sampling was car:.
:ara-e:rr, r, '-as estimai 23). The . - -^re
ilated ace to eqn. (6). Xhe res-Its are Cabulat Ln Table 1.
Figure 7 snows :. - cr.art for the computer simuJ ~ee
component mixj tssessment the resultii
Lgure 8 gives the flow for generate
The Inter-
s hown .
7. CC IKS
ex t eqn
.
cut e .
for the d
.
Lon of (Y , .....1 P
ticompom -ture in an -d sea: .- :es
compi
This mcdel contains .irame: ir, Chat defines uniquely che mixing
mate v La pararae:
and che precision of che estimated population mean concacc number is assessed
equations detived In this work can be reduced to simp,
iry mixtures. •xpresiion .*e standard error can also
- 19«
)-
3 J 3 . : l i
3 : : Q 1111: liii
3 3 : w 1111C c c 1111
c : 3 1111y c
-
- : 1111: : : c c Q : 1111
1 3 3 J : ; 1111c . c 1111
3 : 3 c :
-- till
3 J 3 3 3 1111c p
L 1111: :
-- c
/* 1111c c : c 111]
3 3 c c c : c : 1111- m
3 - 3 1111C c 3 . 1111
: c c : c : : i l l i
) . : 3 : : 3 1111c : . c c c 1111: : : : c : c 1111C 3 3 3 j 3 G D c III]: : C C :
~ ft
-' i i i
: :
-
•J } 3 U 1111: 3 c 1 1 1 J
3 C : c : c w 1111 1 ' 1 \ \1 *. i i 1
C ) 3 .
«w 1111 11111
3 C c-
c : 1111 ill*'i i i i i
3 C : - : m G-
c z iiii iiiii
12 2 2222222112 2 2 2 2^2222122222222221222222222212 2:22122::12222222^:.12 2; . I 2 2 2 2 :
12 2 2 2 2 2 2 2 2 2
122222 2 222212 2 2 2 2 2 2 2 2 2
12 2 2 2. \ . 122\2 22:22:222\2222c 12222122222 2 22221 2 2 1 2 I \ : 2 2 2
122222...::12 2..... 2
12 2 2 2 2.. I I 2
12 2 2 2 2 2 2 2 1.I 2 2 2 2 2 2 .... 2
1 22 2 22 2 22 c212 2 2 2^: 2 2 c 2
12 2 2 2 2 2 2 2 2 2
1222222222212 2 2 2 2 2... 2
122222222221 : : : : 2 2 2 c c :
1 . 2 t : 2 2 2 2
12222222222\2222222222
6. A simulated ternary nixture (tvo-dinensiona? I
in the complete -<i state.
- 19b
CUED
. . to represent a tc
. e .
= L
. .. i pa i - liosen i
I
J J +
rancior. coo:
as the sampling points
..ndom sampling at the k random
I
I
o mix.
-
Z)
c «an.
. . art for aim >)m .
IS.
-
•
•.
by otr. nunfccr i
,le
.
1 m T -
:es
i
Compute the overage nunOer C
- ranee of each component aroonp,
1 i
Cc
-n[y
. (n -
r « -i J
2- y. (n -
. 8. ' ng random sampling p, ling
Opt
IV - 19d
1 2 2 c ~ c 2 c 1 Q L 1 2 2 1 2 2 2 l £ ^ ^ - c •-
? 1 2 c c c 2 2 2 I
-1 3
>
1 u 15 -\
u l <- 4 1 Z 1
3 1 2 2 J 1 c 3 1 i 1 i3
1j
1 o 2 ) i . 2 2 .
-
2 .
2/
j c 2 cn c 1 i 1
1
1)
I
->
2 C 1 G J i 4 C < I 2 2 ^rt
2 c w i i 1 Li 1 2 2 y 1 1 1 3 2 2 2 2 1 2 y 2
2 2 1 L 1 2 2 c c r-
u 1 1 1 1 4_ 2 2 I i 2 2 C 1 c C 2 2 c 2
3C 1 2^
1 I i c 2 2 1 I 2 2 u 1 1 2 1 2 1 2 2"*
^ 2 2oc 2 L . 2 1 : 3 3 1 1 1 1 1 1 2 1 1 1 2
i
2 2 1
| 2 2 j 2
2 2 2 1 2 C 2 1 2 1 2 i 1 2 r< 1 2 2 < 1 2
pC 1 2 1
c-
: 2 1 1 1 u 1 1 1 G 2 u 2 2 2 a!%
a 1 .
-
^ 3 3 1 > 3 1 2 j 1 2 p 2 2 i 2 1 1 1 1 2 1 1 i c i £ ^ 2
ii J 2 2 w 2 : 3 1 1 1
mV 1 2 c 1
r 2 ^ C w 2 1 2 c-
C
a 2 1 2 1 C ri 1 1 1
*
I 2 1 2 1 j 2i
2 J ._ 1 ^
2 J 1 1 1 1 u 1 1 2 2 2 K 11
i 1 *_ £ j*
{. 1 2 1 1
1 1 3 2 2 c 2 I 1 2 1 1 1 1 1 L i 1 1 1 2 2 2 2 «- c 1 2 :
] 2 ] 2•
j 3 1 3 1 1 ^ 1 1 i_ L11 2 C
1
1 2 —i
i 2 1 2•
^ :
2 3 3 ^ 2 c 3 1 L 2 2 2 1rM I 1 c
-> i
1n
i ^ ^ <rL < c
1 I 1 2 I w C 1'0 -}
^ 1 1 2 C 1 C n1 1 1
ft
u 2 2 Z 2 iy 2 2
2 3 1 y 2 1 c 2 Q 1 i 2 2 C c 2 2 C c c M 1 t _ t ^ 1
-
_
n1 1 2 c 2 C j •j 1 1 1 1 2 1 1 1 2 1 2 1 1 P
. 1 ^ 2
1'
a J 2 1 1 2 3 2 1 w 1i
X<^
u 2 11
2 - J 2 _N - -
Q
i 1>
C 3 l u 1rw
11 1 1
i
x 1 1 1 1 1
-
2 ^ I •. 2 1 1
1 2 1 C 2 lr
2 2 1 1 C 2 C 1 2 2 3 1*%
*. c ^ 1 J 1 2 t1
1
1
1 ) G 2 3 l j G I 2 2 1 1 u 2 1 1 1 2 1 1 c 1 2-•
C 2
1 2o
2 1 Cr*^ 2 C C u 1 2 1 1 1 1 1 1 2 B 1 2 C L . 1 1 4
u 2 2 3 C 1 C 2 2 1 i G 1 1 1 2 1 2 2~
2 2 _?1 2 2 2
2 3 3 2 1 c->
^ 2 I 1 2 2 c i 1 2 C 1 2 1 c yi
1 i c t i
1 3 2 1 C 2/>o c 1 1 1 I 1
1
1 1 1 C i 2 2 c 1 2 1 2 2 2 2
3 J j J 3->
3 2 3 2 2 2 y G I i 1 2 u^ " 1
2 „ 2 G
1 1 1 2 Q 2 c G)
*. 1 C 1 2 G 1 1 c 1 C 2 C-
- I 2 2 1 : 2 :
•I bOO random
..an^es.
- 20
be o he complete
.
utive
standa: Dr of -.oer estimator has beer, shown cc be
smaller char. .ir.ee I spot samples. Therefore,
-.dex based on the mean er estimator
based on the ce estimator.
-
references
1. Y. Akac, 7. Noca , S. Cakahash i ar.c A. Ctc-.c, Mixing - cdi
r ::r mixture :: .--.ides with linifon s.:e, J. Res. Assoc.
Powder Techno1., 8 (1971. 321 in Japanese).
2. L. T. Far., J. R. Tco, F. 5. Lai ^r. z ....solids mixing and s I. Estimation of
c or. la zi c lot geneous mixture, submitted to PowderCechnol. for publication L97
3. H. Shindo, T. Yoshizawa , .. Akao, L. T. Fan and F. S. Lai, Is .on
g ir.dex and contact number by spot sa _\ of a mixture ::.
an incompletely mixed state, to appear ir. ?;v;e: Technol., 78).
::. L. Johnson and S. Rett, 1. crj ns in :s: Continuous..-. '..lie;/ a Sons, Inc., S. Y.,
?.
3. 1. 7. Fan, J. A. Gaston, SoliInc. Eng. Cher . . 62 (19 7T.
6. . Mood, F. A. Sraybill, • C. 3oes , Introductl E E
I Statists: . ::cGraw Hill, N
.
229.
:.::.'., J. L. ] c . s . s • ter
s , Jo:. ons , .. In. , 19<
8. len, (is* ! . ,1968,p. 269.
9. J. 1 .-.e compoui---•--. . eions anong . rtions, 3io:
(1962) 6:.
10. 1. T. '; and C. S. cedures I
iriaces with non-im s , :
.
1
P. R. Tadikarsalla and, An a; echod for £<
mna and other variates, J. S- . Conput. Si: .... 3 (19
!. Has and J. B. Peacock, Sea al Discribu: . John •
. . 1970.
360 Sci. Suoroucme Package (360 ..r»ion1967).
-
.
c . ,.» number o: -
C .ft
. r&: .:nber of component
popu: -onca^ ber of1(0)
C'.-. unbiased e ir. contact r.^rier of
t Y
r of sect Bl
•2 X
ft]
a ipc
'
• nle
I abtr Ol .es sl:
. . cle
standar c . e. Y
Bti ronent
V
pc - :oncer- les of cor.pone:
sa- -oner.:
cor.ee: - ?r.er. i rot
sanple....
^ a spot Mi
n„ le
IV -
:
-
j
Subscripts
rar.ccr. variable for Che lecal ::r.:er.::i c - >nei
z local concentration of cob : A:
f( . %zrz- a function
para-eter of the Dirichlet distrj a for componentA.
j
generalized factorial moment
DC Bent
en : about z~r
estirnatcr of sar.ple variance c: onent A
-~~
Df o, for al plications
a parameter =
i-o:
co-
s cocp. segrega: catt
s abov .sol
ei
-
unbi.i or
-
OF El
assuae chat cha local cor. ited a
th< --c di.-
C. , . . - p~P
P
- -^— (l - : .- TT z (a-.
j-0
2 - c. . t2,.. .
. . : ven the local concentrations, z. , z^,...,z , is1 . p
P
! j-0j-0 J
s vec I- _ and Z is
re
"; i
t,J ^-0 J
*J
n z, •— (1 - I z) 77 z
p 1-0 J ? ... J J
j-C j-0 J
(A-3)
rancor I ?r
•:d as
Pr [Y = yj
-
= PrfY = v
• Z z
.
- = -
i • • • |
? ' ?
p-1
- •
p
r< : b
1-
-r
- = 3
.
-
Sir. . |
(u-a - . . -a)- •-
i y 4' P ?
-
CA-5)
:egral In e: -- as
1 -
1-3
p
C 1 - - z . - z )P 4.1 ? ?
c:
1-1 J
p-
1
p ? o p
r( > V
Proceeding iterative: . nally ob:
f.' p
-
•
- :
p
... ....
r- . :-. •;. .
: y ~ .
0'
-
• = :
..-.ere: - - /.
r
en as
P P
3J
- . - . Jt
p
V - .))
j-0 - j-G
- 27
APPENDIX B. DERIVATION 3F 1:
7r.~ factorial ts of tr e Dirid I c-nultinomial discril
o- tained as follows:
— - —are
K' ;: .
n n -v.
yl
= ;
i'/
-. . . .-y
o o
i-2, i - :
r( : e~- -1
p
-
fi i - P p j i
j-0 J J -
p' r 1
i - : z. v lz . . . dz-d z
,
P - -
(B-l)
According written as
V -
-( E 5 ):"'-.
" --— n
.:
j-0 J
C
'
P P 2 :
j-0 J p
(y.-V : '"' Cy-p p
Since che factorial moments of che
( : t.)
IV'i
(B-3)
lB-4)
•quacion (B-3) car :en as
- 28
5 °
! .:•-
I-
J
3J
"
3-1 J ~>
;
:z: 3
? ?
-
. „ J-l
- -1 - -
.1 :
? : p
, : + : -1)( I - : ...(Ij-o
- -
APPENDIX C DERIVATION OF E(
,::.isec estimator ~i the contact r.jr.ier : ent
A.i ecn. z ) , is repeated below,
a*[x 1
K-'..
(8)
The expected value E ' nN ] isj(0)
I
:.
—^r • -r- - E [x. x n ] .
L-l
-
n-l
-.ce
E( - ei-'-
a a
" i ^;V-:
*i / *
w e
EIC' Jn*
r.-l
n*X (30)
ace V[C'Q)
1 is
:j(0)l——
- ' J flx
rXQ
2 ' "51
-
" V '
{c "
. :-
• -in - r ] (c-5)
(C-6)
.:•'••-.
r.
--;-3n-:,^: -
U-•-
: ^5--.. ^— cc-i
o-l n-2= '
IJ -—
-
(C-9)
andD ft-1
E - r—* (C-10)
:' .
-
- r.- •
- ft+i b+t+U] (c-
e
2
-]
J q
(n-l)9.6.— -—;
.--_ n-3)e e.
(n-2, (n+ - 7_ - r.-l —
-
IV - 21
r-- o+T+1)] cc-]
. stitution of eqns. CC-2) and (C-12) ir.ro ~c?.. \- . \
2
ives
:.:
VfC'1
j(0)(r.-l)
_
Cn-J-- -
n t (t+1)*(t+2 [t+3
(2(3-3n-3T-2m 8n6, - (n-2)* r+1 Jn+€J J
+ T(T+l)(tH-T+l
n* (iH-iQ9
-7-7n(n-l)kX T
fc
(T+l)fc
(T+2: --
- -.
+ Cn-2)x(T+] - ---::-.---
- 32
CO eqr. . ^escer. aoa«ncs
. AS
n(n-l)( .
-l : (7^— ct--
3• - - ' (D-3)
-3)]
- - -
j ]: :
-: --
. . . : .-.ec
" !61uj -
'
(0-5)
^:n:en:s I :ar. be expresst- of
(D-6)
-1)] * E V ] (D-7)*• J
u' - E - + JE[T » -1 -
-. • -2 -3)] + 6EIY.CT.-1) (1-2)1
, «qn. (38) ,efl as
"
-
-3a;
- 23
Substituting eqns. (D-6 through (D-9) into eqn. D-1C , we final]
the ceiailec expression sf the fourth central moment of tl ilet-
nultinomial cis:r: tioti for component .-. ..
-
:>road I One
_:ie
ranges . Tr.e work
coru f ornn
So. opera.. - Lch two or
are sca::e:
1 e ? . ed
.
aspects of
assess of
a i . the Tht
: e snc
L CO the er
I tion,
nges .
(c) cause nc ^r.ce ir. tr.e .or,,
id) be ie to as . and
(e) be i and I I Lgprous.
. -e the homo^cnc . r,g index
A useful and g
-3tcd as closcl; :tie I .
-
.
be . . s applica-
ranges bt •
-
1
.
V - 2
Statistical analysis r.as beer, a major tool of solids nixing investiga-
tions because of the stochastic nature of nixing processes. Over thirty
different criteria for the degree of r.ixedness have beer, reviewed and
E .-r.arized by Far. et al. [2). Tr.e relationships among so~e cf the criteria
have been studied and tabulated by Far. and Wang [3]. Table 1 lists these
criteria and investigators who have proposed then. Most cf the criteria
are based on the composition analysis of equi—sized particle samples of a
nary mixture. However, sclid particles tc be blended often consist not
only of particles cf varying size and density but also of widely d:::rr-
chemical ar.: physical properties. Tr.us, a study cf mixing indices for
D -1 ti component heterogeneous solid particles is of practical sign:.. e.
A convenient scheme for clas.- tag particle - ^. should be es
facilitate the investigation cf sclids c. Usually, par:
systems are classified as homogeneous or heterogeneous vor dispersed)
systems. Particles of a homogeneous system have the sane- physical properties
(e.g. den;.- , size, shape, etc.) and are distinguisl the
: ics cf tracers, such as color and radioac: . ~:.ich dc r.
course of the mixing process. He: «
physic. _ .:h do not interact nil the opt: . of a
are considered as tracers. On the other hand, particl .eous
system have different physical properties.
The extent of .at ion in the c: .sties of a mixture may
be • ::.s of v.. Lght, volume or number fraction of each con:'. ::-
:e [4,5). The three bases yield an identical r> I or
a homogeneous system. However, sampl . counting the number of particles
is t and impractical for a fine particle or powder system. Measure*
lume fraction is not easy, because the bulk volume of sc
c r- I
— - z- -
.
-
— i
- . -c >
E -— E
K-
- ---
- -
3 z-
ei-
X C- -
— C- -zB. z
z— z —X ——-
V.
z
-
- —
-e B
z -
< >
e-u
*^»
-c —
- 1 —r-
c c. r rs,
t- c — —
.
— — Mc =
& — *~
.
rUJl
—
.
u |—
-—
—
Wl -Jc B1 - ^ ^>
1 ' 1 1
at-M Uil M UJI '
C < s:•^ S!
d—
,
—: un
r t
I
NO
N•
E =•- *
i 1
•
- c^< >?
I (N.
c e
t
s
- —- —- = —
— — i- - u Br - —— — u.DC - e. X!—
•
a u Iu . 4J
> X —— z -—
I-
— E"
r* z r.
CM C —B —
>> c Bo -r C- ez- I
- 3 -C
-rr-
I
- -
— —-B =_ _K
3E I
I
09
v.
»-
t/SI M
1 1-
^TMi Wl—
M
-- •
c E_; C
—-
>^-
~5 •z
.
zt E
a— ET c=— *N
—•-.' —— EX— H
E —
§ -~9 Z
t>s —— K= '.
iE C- -C
Ic
• m
> .— ^ CM U
= D — 1
—
G 1
B k 1 1 1 1—i
— = N NO CM O - C uo: - — C O B C
- E — 1 r • 1
K —- _— n <• If
r - s 2: r JE
yr
M_ -
- .
particles consists of the true volume of partnles and t ze,
i.e., the bulk vclume of each cor.s:::.er. : is net at- . Since the practi-
cal difficulties of measuring the • ix a nixer exist, the zear. ar.t
variar.ee of the weight fratticr. are chosen here as unit easure
I te literature or. the assessaer.: ::" mixture quality is g ry
scarce. This is especially true for mul ticomponer.t i es. Gayle et al.
[6] used chi-scuare as a criterion instead of conventional defil
. tng index, which utilize the sample variance ior sta:.:„r^ deviation
Granules used in their experiments were sir.ilar m surface cl I --S,
sr.ape ant density but different in eclor only. Luslik [7] propose^
numerical homogeneity index to express q.. y varying degrees
homogeneity, nar.o the negative logarithr. cf the sample w<
to obtain a standard deviation o: 1*. . Heri.ey et . jse con-
cept to ca ty of ad ingre_.
hey also assumed tha: | components z to the
same particle size level prior t I :ng.
Ifl the present at . some approaches to assess the c a i:>.-
VC been propos roxim^ cneous
derived.
2. 51 . XSIS OF BINARY MZXTU1
Thi tsaioni of some statistical quantities associate. a
are s zed below [2):
A sample variance is defined as
'-
«Z (x
li' (1 >
-1 1J
An ur.biaso- sa- ance is defined as
2
ere
' s
.
x. or. of compc: th saa:
- sarrie Dean which is - I >.
e cal :ncr.: -qual tc that cal:
:re. The popular: or
k2
]- (3)
e see: rd deviation or variance
eiy rand- ed state and I
segregated state are cfter. used as the boundary values
aoong a group of samj .
drav the co-pletcly rancor state is ce:
[10) as-: -2
-
-...-.'_ (4)-
V VI
re
. « ve:: : and 2, respective.- (P
x+ P
2- 1),
- . El of compone-
•
and 2, rttp€£td >. .
i -
s, .- standard d Lona cf 7
• of components 1
anc spective
• - C of I sar.rie.
.e is .. (4) is reduced to
- 5
P,*2 -
If each component consists of particles c: Che s.v :.;le weight) w,
(w, = w_ = w) , ecr.. (5) car. be written as
2 V ' T,
i w : :
=n
p:p2
(6)
h is the original formula of Lacey [11] where N is I ei c: par: .-
ties in a spct sample.
~er the assumptic: t the pro: el Llity of sar
the given component is proportional tc its volume fraction, t.ne variance
for a mixture in the completely segregated state is derived here as .'see
Appen- .
.
o — CI—I I P- p "(1 - — P )
- -- !-:•.
:: -
.
"1 - — P
2 ) (7)
vheri end . . denote trie densities of components 1 and . ..v,
and z thl the components have the same particle de-
(p, p, e)i
- P P (8)
can be seen that th- .ance of a mixture in the completely segre-
gated state (dependent of sair.; .ded that sampling
taken at the interface between the components and tha: variance
concentration (. ion) : d sta:
la inversely proportional to sample sise. Thl : en most free used
.ces for binar ires are the* Table
V - t
A a r r «.
geneo
tin'
(9)P
iough spc
poslton, such a : Lng
op-
state thai is d it ideal homogeneity. It i- ed by
of sf. ' .. e components from
.lure , even though the c«. I be
tdenti e. For a homogeneous binary system, particles in a t|
sample of are distributed according tO Iial difttt . • .
Pr|V V ^T ri"la-r/2
(10>
where V i<- ble that, denotes the DUiabei .cs of cor
ponent lira.' on,
th< . s
o « Var [x, ]
r 1
- Var [J)
which is eqr.. (6) .
In pr Lng pro.. •
C because the pro or b' :<y
d i
:
thai the probability distributions of the components curing . are
Down. The probability :f sampling a particle of a given component is
proportional cc the volume fracrior. cf rhar corponer.: Dnder che
• -jsis cf mixtures In Che completely randor. scare, parricles ir. a
sample car. be assumed to be apprcximarelv ciFtri: according re rhe
binomial distribution with parameters N s.r\i q^ , where K is the • . ed
b er of pamrles ir. a sample cf constant weight, W, and c. is the vcl
fraction of conponenr 1. Tne variance cf rhe weight fraction cf coupons
1 or 2 is derived as [c :...: B]
pl°2 12 f
2 1
Since, ir. practice, solid particles to be r.ixec are usual] isured
sar_ snsions, this distri seems I a good
-res. Notice that ear.. 11 reduces to eqr. . (' hocogen-
eous system.
3. [CES FOR H IICOMPONE ::TUKES
Th pose c: • section is to develop mixing li
cor ture. We SSSumt throughout this section
: a mul
(by weight .corded. • sampl OS,
an indication of the degree of nixednei • v. - 1 1 be calculated. ods of
sampling, determination of sample lise, es dr..
on the degree of pri I cost and practical,
of sampling, and possit al constraints on the sampling process.
"Pseuc
Suppose that a -poner. curt o ....
aogeneity of the key component, say, component | ne
e is con co com 1 and another (psuedo-j com-
reaair..:.. . -. • onents
.
is
pre: .. prop of (1-P-). The p.-. • and
de: are give as
-- : -v
— - -- - ... -v.
-
U-Pj)m
o p
L2 ' k
. cospc : e treated as conponent 2 of I bit re as
de^ r. sec: ?:r Instance, I Lndustry,
us u ent and several diluents (lactose, stare .
.cose, etc.) are ir. a dosage; ther<: the homogeneity '
key conpon< the ca ir. concern. The r. I.ie
be dtl .e: as described ir. the 7receding section UJ,
teh based on the concept G -re
be a; tc each component in a oulticotr.poner ' to define an
each co~ •: thod is acre useful
. cate that a
for II .oaponer.t, i.e., sone
ts nay be requ:- mil a nuch greater e> other
(1).
- c
3.2 ?c:led '.ariance
Zr.e saccle variance cf the concentration ir. a - iti ent mixture
is defined as follows:
k
9 --^r- Z ' Z (>:. .- x. )V (14
m—1 _ l ij i . i
where
k = number of cor.pcnents,
c - Dumber of spct sanples,
>: = fraction cf component i In tbe j—th spct sample
>: .
= sar.ple near, of component i
= - : x ±i
When k 2, eqn. 14 car. be reduced re eqn. (2 • - expresses : ::-
ance of i ture. S. ly, we may express the- variance! for
uulticor.ponent mixture! in the completely mixed and segre, .
respe as
c:
- : k.1 P] (1 - — V (15)
and
1-1111
. - ompon«. res
as sir. to those for binar Table 1. Pol the homogenous
Par'
tta, eqr.s. (15) and (16) can be red. . reaped . :o
(1-PJ1
:-.
-
. -e have
,-
Ac ne relat. . ps bet- :hese I -ices sh.
-
Table 1 ere iden:ical to those for jres [3]. - -
plot the value? of the nim s aga:
.ind 200
car. be a< related I
6
a??- be more depender. ers. However,
the variat. Eht e>:t. xedne^ :ne
regated state tc the c .xec state.
.e Covariance M
inalysil is an effectivt tool f. : icompone
: observing the c. ion of a
of a k-cor.por- erned
I es c: (k-1) ve .ec rancor respcr.se v | e:- . Nrtice
oncer need tc be consider^ .en sample?
are o": a (k-l.'-varia te popula* \eac of ^ riate popu-
lation, tht data can be represented as
.co-
sample T\v (k-1)
(k-l)l
(k-
V - 10a
.
".DFX KB
. 1 Comparison of mixing indices based on the 2nd approach(pooled variance) for sample 6ize of 20.
•. -
'
e- 1 1r i--
i r 1 1
/ l\
/^2.
/ M&V/'
21
//yujO. /yy^ 1
1 -
~2L
'—m
II -
-
X^6
" 9\> /J
/
-
J^^-^"J><^' _
,.
~~^5 ^^^"*
.
—NIXING INDf.X Ma
1.0
Comparison of mixing indices based on the 2 nd approach(pooled variance) for sample si.-.e of 200.
- 11
..-:
E
X . .
- -k
2j
2a>:
••--
---
le near, cf the l-th component is defined as
:•: = —- x . .
= ij
The sins of squares and cross products are defined, respectively, as
; , - >*= _ ; x . .- x . )
j-1 « l<
2 1
j-1 " j=l J
•
and
"v."'
(^- :
r. r.
- x .. x - -
( : x . . ) ( : i
J-3c
j-1 « j-1
The estitaated variances of x and co . oe of x and x are, r<
[x.] - si
SS
m-11-1, -. .... [k-3 -
and
cov. .-_. ;- su
s?
All
. i
. :he sample \ jriance matrix is
. (V--
-
s -
.. •-:
Sl(k-1) .
,-.S (k-DU-D
a com: re are
expec: randc spot ;>ns
the re. Let us de: random vec:
r.er. r.s, vnere'J
. C _ r.
j
• par-
:ne spot
randc .ables.1 -
\,
" \ ]
SU pn. ln,!...iL I
rl
r2
!
i-ia
-
P - -...-7,-1
......
«.
-. ture (for 1
homogeneous system)
-I
Note that the ::_! ttnc~ial distribution is a ger.eraliza: . _ - .al
distribution. According tC this pr l ty density function, ecr.. 25),
the ir.eans , variances and ccvariances of the rai .es, V., Y«, •-, V.
,re given, respectively, by: [14,15]
i . ; w>
VarlY. ] = NP,(1-F.) ,
Cov[Y.. , T ] = -HP.P.l i. _ -
21
C27
--
or equivalent!; .
Elx,] Ehf]
= V
I
= Var[-Is
P.(l-P.)l 1
(30)
Gov , xt
] = Cov[-jj- .
P.Pr
i x
(31)
Therefore, t ..ce-covariance ma: the C<
can be expressed as
PjUpPj)
- P1P2
'P P-1)
-P P
P2(l-P
2)
- I P
?(k-l)
-
:
'(k-I
P(k-l)
(11)
}
'
segrege^ :ate, each different cospc:
separated a | pe. I ate, we have|
end
Var[xJ -
-PiPi
(33)
:ance cat • icocpone:
can be expressed as
. . .
P2(l- ...
l~Pl>
?,
i
p:*(k-i)
_i (l-p )
(35)
e variar.ce-cevariance r . £, characterises the degree of
the dispers coraponer.:. Xhue ( Chf
. the extern
riencf ir. the u: .alysis. Therefore,
tnd 0- by S I , I I , and , respectively. - car. also
Dent B -is, we can extt
ex
2- -
(36)
0-0m L
: :
°o*
:ey [16] for a I ure to a multicor
ng-
-s
are t the expressions
are reduced tc those : "ures.
15
According t: eqns. ( 22y and (25;, we have
cr.:
The relationships among the mixing es tabulated in Tatle 1 are evalua -
ir. Table 2. Figures 3 and 6 plot the values oi Inine n -ices
against that of M- for the sar.rles of sizes 20 and 20C , respectively. A
mixil Lndex should be sensitive to changes ir. tl c: a
mixture ir. order to be useful ir. following the progress of mixing.
Vaier.- L7] pointed out that Lacey'f index, ecn. (3"), was very
sen: changes near the completely segregated state and rela:
insensitive during the process of nixing near the ccr.pletel; : :e.
Their more complicated index, !'._, was thus devised tc k :er
sensitivity
.
:.ed by the determinant :e-
covariance matrix for mul t iconpone ..tures can be gent:
. where I and Z are the var iance-covariar.ee mi c: w— r —
s
proportion in t. pletely ^r.ixed and completely segregated stau .
respectively. The development c: _r„ an ^ - F rc:St
only with homogeneous systems. Co: eterogeneous case. ut
probability of selecting a particle of a given compont proportional to
volur.< :hat conponent and the probabil. I ng a particle
ne compone:.- tical everyvhi a coaplet< andoc re.
Under these assumptions, particles take: a spot sample are die: ed
approximately according to the multinomial distribute e parameters,
where N is the expected number of particles in a
—r
i
-
E
-
1
I
oc-—X—
I—-
>—r
-
r^ —
Z
- <
'
r
-II—.-1
•—
-
ri
<m— <
•JTt M|
I
-
- <.
— <i
-r
-^
--.
M
- <I
s:<
i
<
-
- <
ri
<
i
X
i?
j
— _ir
<
~s
l«
ir
i
I?
X
-
—
C
I
I
c
H
£ M <
rw v£
£
«
aKC
(9
r
ac
I?
- <
cc
- <I
H <B
pm t-*
r
cXi
HM
1
St X
k
— 'w
^
•H|g
w* — <oc
SB
1
-»<
oc
1
1
, C* I
:
1
— L<
a-n — <
1 1
1
y*
o <
I I
I
Grs — <
1rB
=
—-
T.
I
—
3: i
i
n
•-
~s
i
X
<- ^I -
N
I — <
«MX.
I
<—
r
<
- <i
"T"
i
i
Hl'
i
- <c
- <I
ri
^
-
- k
<?
-l<
N
i
iA ,
k'ttf\
z -'— tn
"C--
z
zzU
I
r-
^<
> Ii
.'^
-•5
< rs r^
—Xii
-7<
^<
5c
£ - <
c
T.
""5I
ri
IT. I
ii II w*e
- <
3 _i<
i
-'<
a:
i
<
-t -->
•
:
:
r-»<
I I
.. _ . .
V* i-JI
o(N — <
_.<
I I
[-
0.? . »l
MIX. INL 8
Fig. 3 Comparison of r.ixinp indices based on the 3rd approach( the detcrrirj.int: of the covariance matrix) for samplesize of 20.
7 - I
MI X ING INT i
1.0
B
. 4 Comparison of mixing indices based on the 3rd approach( the determinant of the covariance mat: r san-.
size of 200.
:r»e
componc:
.mces and c: -• .ons cf components in a
joporu e, ha cen; >re
re Appenc
|
•
:- - p<>. 0)
and-
Ccv . : ,| : .
,2 ' (41)
. ishes .nee m .re
• c, I . cht variances and covariances
I | components in c
the segregstec are givi . respectively, by [see Appendix D)
]- £ (l-l * l - •
- -1- p. . i - L, : -:
: . .
l]-P
jL- - .
. I - 1, : k and 1 t I (*3)
es the var iance-covar iance matrix for a .ompor. :re
te, £ • Not e that the variances and co-
•rogeneous I I reduced to those for a homogeneous
px
. o2
- . . . .pk
-...
. 17
1 . - . Sa-ple Mean of ar. Individual "omponent
Ar'-ac et al. '!£] have proposed a mixing index base,: the zz?.-
b er as
C - Cs
c - cr s
r
where C is ar. estimator of the population contact r. a -;>::ure
ar. .- .irary state, C^ arid C are near, contact nurDe:: of th< Bixturei I
Lately r.ixed state and completely segregated Bt respect J ly.
index. :.. . , eqn. --, taK.es the value of C In the completely
segregated state, a value betveen and 1 li ... in .xed state,
an. e COnplei d state. Recently, Shindo et al. :st:
Lnomial mod<_ ^escribe zhe mixture in ar. incomplete state
Che parameter, t, vhich is unique, car, be employed to
Lng index
".: "—
(45)
shown that t'r. . Ltion can be extender |
Bixing index as [20]
Kll " 771 CW
The relative standard error of the mean contact number estimator has been
sh, - be sn., n that c: | variance estimator of spot samples 120]
be defined by using the sample mean
each ind. I compon. tJ d of -!,ich is con.
used to define the i inaex. For instance, I g expression U
be used as a r index for a bina- :ure;
LI
.—
i
. (47
The de: equa: .1 sur e abs ences
ifl sar s and th< conct .on among
the- >-s froc a cota: segregated state (See
hi populi. . • can be
le near.. The co: re
DUghoul % equc.I , and
therefore, the :he pe: I accc:
• e co-.pletely segregated state,
ecr. . 47), - takta ;-- between C and 1
also be ex:
.t system as
: f : :•: - ^ p.
:,
"~ x - - -
± Ci-p ) + (1 -£ p )]
.
i
4.
Tc was choser.
to c use ( o posed ices for
es. Iht values of the degree ol ss ev. d according to
ions were compart
t was- gerv 1 was c.
- •
•
. ler.f. • • ed
• e end es, M Ittr of 1!
ere ustd. M are sumrarized
V -
8 —CC DC~ —
i 3>C /-v
_
£ C CC CC
-—-
— &
— t
r: — oc
—'—
c c
-c—
u
I
c —3
tt
-
C
I<C — ~
BI=
— rs
-
seal- Li e placed between
ends of the cixer, normal They divide xer
nree coapar- cf equal iengt: . % cocpor.
- one I nents 2 *1 - I loaded
: s , re «. bed was
le-. re removed, the cover was put in place, and the
ll :ated. 7r.e speed of the s.ixer was set at 30 r.p. exper.-
..1 cf 1C minutes after
the onset rle divider was employee tc divide
tap] teined froa eac -as
:ces were CM ErOfl the eapenrer.tal data,
Lt A. as
- s:
(1) Pseu jre
was assuDed as the key cocooner . I particles of
p«a 1 ar,; 3 wereJ
:dered as the pseudo-conponer • Ibi particle
weir .: der.^ pseudc-cor.poner.t were calculated fron eqns. (12)
. respt The dices were mg to
te:. :cre fre~ >ed criteria g. D Table 1, and the results
4.
"(2) Pooled variance
r.ce of w< Lght i ens of the ccrr.por. -
cocponer. •-
: ror. cqn . (14). The variance foi
xture in the COl xed state, , and that for the
:the completely segregated state, . -ere calculated accc-
(15) and (16), resp es were
1 and :,e expressi. '- Results
are alsc tabulated I .^4 .
r-
,_
'^ r^ C P" c- pi cc a c IP
t f cr c <~ CM o |P| ir «d ^—' 7 a a- r~ m 3 C" c- c 3i-
C C d c C C o c 3 o CMc~
<
PJ
J=w CM <j r~v tr cm cc c c vT mX <r r~ IP, C PI LT cc X . LT
C r-^ cr a \^ L^ cm cc X c vCbC- O c c c c C c o o •—
•
c_
<
»*
£-
a u~ a- *tf•—
i
r^ LT CM CM ^— X'
r^. <r £ ^~* r-^ CNI •4 a <r- r^ 9 LT ^ CM cc X c Xa • •
a C C d O C c e c c »—
'
«£
•oe.—
o_- r. <— ^T LT c r~- X c #—
2:B z E jE z 3E *- S - £—X~
_JL . i
r^ c IT C C IP C c cc IT ~? r- t—
•
X *C cpi r. <J ^3 r~ CM r . — «—
•
p* cC O c C o O a c c c
o c IP L~ cCM l/- r^ \r r»~ r^ r^. O X
• — r. r. cm CM CM PI L-
o C c C C O c o c
O o o c C lA• t—
<
• r^ O CM r—
•
r*.
•—
«
c- • • r> <7 in n
<
C o c O C o o C C o
Cc
si ,- r">• vD X ^ o—
c(-
z.
c
-m c— >~— -
X.— I—
E -" w
•_ C_ u
- tr c ari
c — It—
•z 1- u *~
w4 (C- — PI
.£ > z— M -:1 "H ^ C-7 | - CC
C »< 1 - Xc z — — - CM
i c E
i i
— E
C
•— CM r^ is: C
_z _n _= 3 -
- — cr: c c:
-
1 > t-
u u c —c- a a _ aC- c C- z e< < <; «^
- 3— .
.
z 0— -Z -
« -
-
c c
a c
x 3.— —E x
-
~
•*. c —3
-
Xv
c
— —
- 2C
sample c ance Mtr
and : , were con r.ants— ^x —
s
to the d ie
determined, e also tabu. It t*.
:ual component was calculated. The .
tal s -rences between sample concer. ns and samj
mean was -.. (,48), tht -as cr
as :e
.
KS
Che quality t one ol most | aspe:
of homo gene I -
per;- • - terogeneous) par t systems is useful for the
process an r the tlu -or. of probl'
are c : ance.
is been expended or. this subject, it still lacks
a:ce: :es to assess the homogent-:;. oj :eroger.e
re. Four approaches, vhic 7rcposec
^al a: • appr. ution
Y ] for a multiconponer '.-e
completely rar.co- state has been derived. The use of vario . :ng indices
-compc- been she a tf re over
a v rrcperties.
ropost :cs for
•nose for a ': foi BOgtOM
a spc | heterogeneous systea.
- -
The choice of a r.ixir.g index depends on the appltcat icr. c: nixing pro-
cesses. 7r,e relationships aaong the ter. aost frequently used - :es
based en the second and third approaches are alsc evaluated. The nix.
ejiz'r. sarple Bean Bay have sculler relative standard error.
In trie future, we shall study the feas:: of e: r.g
index, as defined by chi-square [6], to a universal index for a hetero-
geneous nultico^pcnent r.ixture.
-
jNS
A
C Mtimai of the population conta :>er
C near, conta: : letely randoc state
near. .gregated state
Co^ .ance between the randor .ibles T. an:
expected value of the r
r nents
Dg index
= no. of spot sanples
no. of particles in a spot sanple
expez te onsta:
weight, W,
no. of particles cf cocponer.t J ir. a spot sample
v. lior. of component J
luce fraction of component j
sacple variance-covariance isatrix
ross products of >: an:
res c f x
standard deviation of particle weight ol ••
: j
ice of
COV
. xture
total net vcluae of solid particles in the cixture
variance I ,e randoc variable
'
r
;
PJ
qJ
s
ss
;j
.
]
- 23
U weight of a spct san -
w nean particle weigh' of component j
X a rancor, variable denoting the weight fractioi ponenti in a sample
X weight fraction of component i in the 1—th spot sanzle
x. sample mean of conponent ii .
Y. a random variable denoting the number of particles of-1 component j in a spot sample
Greek
: mean density
:
.
density of component i
—
r
1
r
var iance-covariance matrix of a mixture in the complet-
mixed state
variance-covariance matrix of a mixture in the C
segregated state
2sample .-nee of weight traction
2c variance of a mixture in the comp . --gated s:.
_ance of a perfect mixture
23 variance o: a state
a pa ex .'- i: -n [19]
>n. Solids t
1970) 53.
3. in am 5)
H. C iom and nonrandoco: A ger.
composition of sacple;,, Po% .7 < IS 73) 249.
. il Engineering,,
' - . 1958, p. 2
Og of sol Ldf - c'r.: I
a*
D. Bus'. proposed univi :
Technol., 7 (1973) 111.
.'. C. Coo- . of pharmaceutical. Phannac, 26 (1974), 126.
9. J. A. . . :h, E. A. Bishop and E. A. Clarke,of multicoiaponent pow.
63 (1974) 408.
10. Die mischguicbev. von z. rsuchen, Chemie-Ing.-Tech: ... (1954) 331.
the theory o: ng, J. App.
Cht . • U954), 257.
K. S cf v.. - :h unequal par tic".
I .Ch.E ' Nov. 13-17 (1977) .
13. C. nine in I :hods of Mult i\
Polytechnic ksburg,
R
.
. . . T. Fan and J. R. Too. ' al ar.
of solid .ng, t in Po- Logy (197!-
Probabil
.
& Sons, In: . .'
. (1967) 347.
theory of p. J.
Appl, Cher... 4 Co
powdei and part
industrial ns. Inst. Chen;. L ., -•. (1966) T166.
• -
18. Y. Akao, T. Noda, S. Takahashi and A. Otomo, Mixing index :
number for mixture of particles with uniform size, J. : .-.ssoc.
Powder Te L, 5 (1971) 321 (in Japanese).
19. H. Shindo. T. Yoshizawa, Y. Akao, L. T. Fan and F. S. Lai, Estimation
of nixing index and contact number by spot sampling of a nixing ir.
an incompletely nixed state, tc appear in Powder Iechnol., (1976) .
20. Y. Akao, H. Shindo, N. Yagi, L. T. Fan, R. B. Wang and F. S. Lai,
Estimation of Mixing Index and contact number by spct sampling,
Powder Technol., 15 (1976) 207.
21. S. R. Miles, Homogeneity of seed lots, Int. Seed Test. Assoc.
27 (1962) 407.
I ,M.C. Lacey, The nixing of solid particles, Trans. Ins:. CmL943) 53.
23. £. S. Weicenbaum and C. F. Bonilla, A fundamental studyirticulate solids, Chem. Eng . Prog., 51 (1955) 1".
2C J. P. Beaudry, Blender efficiency, Cnem. Engr., 55 (1948) 11-.
23. T. Yano, I. and K. Tanaka, povdei the V-typeer, Kagaku Kogaku 2C (19 56) 20.
26. M. K. Westmacott and P. A. Linehan, Measurement c: u: :r. seedbulks, Int. Seed Test. Assoc. Proc
. , 25 (1960) 1
17. G. Herdan, Small Particles Statistics, 2nd cd..
196C.
-
7)
Befort is assur a In the complex
segregated •! ng the c I uples tal
particles c: ) cor. cs
at 2. As. a par - of a
:ocpon
The sar . ance car. be ob * eon. -r.ich
.
- I
• - - (> - P ):
";
:
- -r .
(a - P )
x "0-
t -]
: \ ' '[ ' " ci)]
n
a — P.Pl
1(A-3)
we I
;: - I
C 1;
:
-r:- _,.,
-
2= — c - p,r?. + ?. ;: - — p.)
Similarly, ve may alsc derive the variance of velgl
as
(7)
Taction ] 2
- _ -o \-\>r I *2-.
- - _ .^_ p )
Since
: ;
rlp2 2°1
the variances of components I and 2 are ider.iical
- 28
Appendix 5. Equation (11)
.der a tvo comp. 2 , which art prcse:
the : pro: - 1 . Suppose that components
1 respectively, and also suppose
. -rse of t te
that the the net volume of solids, and not the . appa:
the mixture nasi *he space be- l-s.
-essed as
I
dens:: , . an be defined as
m —V
" ?r: * ?:
:
:
a par the
volume that component. Particles in the co~:le: bead 11
assur .stributed according to the bJ --'
-
parameters K and q. , vher I expected number of pari in a
. M, q is the volume fraction of ccr.pon-
of componcr. : 1 can be expressed as
-
1
- *r • Var;.w
_ j
- 2 C
_2w
Tnus,p p W • V
::
- ^0
-I)
Consider a Qui ticocpont e having -
J the fol for 1 J
- lie:. nt -
- pa: _-nt i
P. weight proportion of coaponer. : I
- •.
-. assumption as describee froa
and c . 7. :•:.:•. : : . ..
| Lftnci I
. is [15 J
I I
X±
] - Var:-rV:
i
-:
- -P. C -- P.)
w- «
Tnus,
- ?t
(1 - TT" ty. 1 - 1, 2 (40)
-•eight fr«Cl • -on the and j-th cc
be expressed a^
-
• ] - COVl-TT" . -5-]
—1 J
-2ii
*(- U c . c . )~~ l
i ';
8 M f :— (- — ? — ?.
w2 V :
i'
1
:
Thus ,
- I
Cov[x. , x
.j = - — ?,?; .
tPj 1J
-:
i = i, k an: i * •
- 32
Equation (-
c coc; -gregated state, the
der
.
riance o: Liar
I bina- :ure (see Appe: , vi have
: - -•I .
i i
-'
- • - • • • • •'
Under the on as describe opendix A, we have
:-:
z z. c (r>-
iapl«f cor.-
zonpor.' covariance of we. -«reer.
essec as
•
Cov| . ] - - Z (>:,.- >:, )•:-. -01
i i**. .
" tj
;. <*: -.
- VD
(x.. - P^:. -
U" 1
j-1- p
i)(*ti - V
;
ij °' *l)1
.
(x
J-l- V V
'ij
.if- -: - p,)(-p,) - n»B (-p.)(i - ?.
+ (»-- p p ]1 X J, A.
- ir..Pr
- ir.rP.]
* i l i. t i'
= P! - q P - q P
V - 33
LRUS ,
....}- p p [1 - f- - -£-],
K 1 i- p. p.I i.
42
1,1-1,2 k & 1 f* I
- 34
Append f the Absolute Differences between the Staple Concentrationsa: Population Conce: >n
For a binar egregated •( .we assunc that
cs contain cr.ly p.. cs of component 1 among the t spo: samples
to be Un ^n imposed in Apj
:
Therefore, the sue of the absolute ences between the sample concen-
•.
::-r .
'
- P
-
- (1 - Px)« -
- - I
(1 - P )« -t-?i-;:;:- _ p )
- BP If- (1 - P ) (1 - 7- ? )]
aeneous syster. equa*. educe.c
- ?, !- 2mP, (1 - P,)
Eqi- and '.1-2) are employee In deriving eqr.. (47)
: - !
CHAPTER VI
CONCLUSIONS AND RECOMMENDATIONS
This chapter summarizes significant conclusions of this work and
gives several recommendations for further studies.
1. CONCLUSIONS
(1) The concept of the contact number to estimate the mixing index
is successfully extended to multicomponent mixtures. Such an
application has been employed only for the mixing of binary
mixtures 111. The multinomial model is proposed to describe a
multicomponent mixture in the completely mixed state. H ~al
experiments have been performed to verify the theoretical is-
ions of the distribution of the contact number. The mean contact
numbers and their precision from the computer simulations of
mixtures with regular packing arrangement are reported.
(2) A theoretical model, namely, the Dirichlet-multinomial model, -
derived to describe a multicomponent mixture in an incompl
mixed state. This model contains parameter that defines uniquely
based on the contact nu: ...is
parameter, I , is derived and the precision of the estimated po; -
lation mean contact number is assessed. Computer simulations a:
included to demonstrate the effi he derived modi
I andard error o: MAO contact nu
has been shown rstima.
of spot sample*. the precision index
based on the mean contact number estimator is su. at
CO based on the variance estimate
-
(3) Literature assessment of mixture cu. scarce, as
Ls topic it acks generally accepted principles, in spite of
practical significance. This ls especia ue for
component solids mixing. Several approaches to assess the homo-
geneity of a multicomponent solids r.ixture have been proposed
Chapter V. An approximate distribution of a hetero^
in the corcplet' also reported.
proposed here are estimated on the I of the
(a) pseudo binar re conct
(b) pooled variance of the whol'
(c) determinant of the sample covariance matrix, or
(d) sample mean of each component in the mixture.
(1) The non- ideal mixing behavior of cohesionless solid particles in a
• izontal drur r>een mode a continue
process [2]. The Markov chain may also be applied to Simula-
the non-ideal mixing behavior of solids particles in a dr
Lai and Fan [3] proposed a discrete steady-stat kov chain
model for the mixing process of a multicomponent homogenec
particle sy D a motionless mixer. Wang and Fan [«] also
proposed a Markov chain model for axial I :tion of bir.
solid particles. It is recommended that this model ma- • s-
••.component heterogeneous particles blended by a
motionless mixer.
(3) Other »f stochastic models, c. ne birth-death model
(5,6] and the restricted random walk model [7,8". be employed
to simulate solids and demixir.
(4) Multivariate analysis is an effective tool I ilti-
component solids nixing problems [9,10,11]. Since tl
are statistically dependent among themselves, v jle
out one from the others for treal In bulk saapling Lti-
component heterogeneous i:.i>::ure. The vai be cc
jointly. Thus, the following multivariate statistical methods
may be exploited to analyze sarpling results:
(a) multivariate analyses of variance a:
(b) multiple, partial, ar.d canonical cor hniqui
(c) discriminant and classification analys. .
(d) factor analysis and canonical ana
(5) Nonparametric statistical tests can be : data v.
different scales of measurement, such as nc .
or ratio, without knowing the distribution cf | pulation [12),
The distributions of the components during mixing are usua.
unknown, and therefore, nonparametric stat; :iould
be effective tools to anal samplingi olids
nixing. Applicability of these | on solids mixing i
mi :s demonstrated [13]. Natural .in be
omponent mixtures and r.
- u
RE1
• ng anc•
t turn:
Po- . 15 (19 76) .
ion model o: In a
Jruit) i "7.
• -i. Ap;
K . II.
5. Course in .c Process*
•. . I
6
.
as tic Pi es , Holdeiv
Ler, Hie Theory of Stochastic ProcJohn V
. 3rd ed . , 19 70.
8. Y. Cha; . id
i
zed
. 1976.
9. C. ods of Mull
La, 197
10. T. . :. Introdu to Multivariate Si :cal Anal
Jo. York, 1958.
plication in Education andBrooks/Cole Publishing Co.. 1975.
•trie'
. . and L. T. Far., Ar.
s to ' solid . Powder Technol . , L(
JOWLEDGEME
The author wishes to express hlfl sincere gratitude to Dr. I. 7. "an
for his excellent guidance, constant encouragement and creative advice
in directing this research. To Dr. R. >'.. Rubisor. goes e special debt
of gratitude for his contious interest and help during the course of
this study. Appreciation is extended to Dr. F. S. Lai and Dr. W. P.
V.'alawender for serving on the author's graduate committee and for review:
this thesis manuscript. Special gratitude is extended to r e,
'.'.. V. V.'u, and his parents for their steady encouragement.
:s research was financially supported in part by the National
Science Foundation ( Grant ENG 73-04008A02 ) , and the Agricultural Resear
Service. United State Deparment of Agriculture. This sup: ratefully
acknowledged. Computing time and secretary service provided by the
Department of Chemical Engineering of Kansas State University is also
appreciated
.
STUDIES I ITCOMPONENT SOLIDS MIXING AND MIXTURES
by
-RZE TOO
B. S., National Taiwan Vnive:
ABSTRACT OF A MASTER'S THESIS
submitted in partial fulfillment of the
requiremc: I he degree
MASTER OF SCIENCE
Department of Chemical Engineering
NSAS STAT: IRSITY
-nhattan, Kansas
1978
ABSTRACT
The concept of the contact number is extended tc e oultic< at ioli
mixture in the completely mixed state. Chapter 3 presents an es cm of
the mean contact number by measuring the concentration of particles by s:
sampling. This chapter also reports the mean contact numbers and their
precision determined from the computer simulation of mixtures with regular
packing arrangements at different concentrations of the key component. The
number of spot samples necessary for a specified value of the relative
standard error is also derived. All equations derived here are reduced to
the corresponding expressions for two-component mixtures.
A theoretical model, namely, the Dirichlet-multinomial mod- '.
. i . vec
for the distribution of (Y. ,s.'
, 7 N In a spot sample oi size r. of a12 p
multicomponent mixture ir. an incompletely mixed state, where Y. denotes the
number of particles of component A.. This model contains the parameter .7".
that defines uniquely- the mixing index based on the contact qui -timate
of this parameter ,7", is derived and the precision ol Na-
tion mean contact number is assessed. Computer simulation is included to
support the effectiveness of the derived model.
.terature on the assessment of the quality of solids r.ixtu:
very scarce. Tins is especially true for multicomponent mixtures. Four
approaches to assess the homogen. :" a mu ent solids I I have
been|
»-d in this study. The n indices ed are estimated on
the basis of the
pseudo blfU
.ed variance of th- <• svstem,
(c) de terminal ' sample covariance ma
sample mean of each component in • re.
The use of various mixing indices for multicotnpon« es has been
throtlf ternarv mixtures.
