1. No category

M I X I N G O F V I S C O U S N E W T O N I A NA N D
NON- NEWTONIAN FLUIDS
V. V. Chavan
u,.",of#I""]'i,1"
i,""1;"J,,
""rs
and
R.A. Mashelkar
C h e m i c aE
l r gr i n e e rni g D i v i s i o n
N a t i o n a lC h e m i c a lL a b o r a t o r y
P u n e 4 1 10 0 8 ,l n d i a
1
Conceptsand Criteria
210
Li Small ScaleScgregalion
1 . 2 L a r g eS c a l eS e g r e g a t i o(nR a n d o m )
1.3 Large ScaleSegregation(Systematic)
l 4 Laminar I\4ixing
1 . 5 D i s t r i b u l i v eM i x i n g
Meausof Mixirrg
220
-). Batch Mixing
221
3.1 Flow Pat{ornsand Velocity Distrjbution
3.2 Florv Patternsand yelocity Distribution in viscous NewtorrianFluids
3.3 Florv Pattc.rs and velocity Distribution in Non-NewtonianFruids
3'4 DischargeRates,circuration capacitiesancrcirculation Times
3.5 PowerConsumption
3.6 Mixing Tin.res
4 . ContinuousMixing
23t
4 . 1 R e s i d e n cT
eime Distribution
4.2 Micromixing
Concluding Remarks
245
Nomenclattrre
References
1. Concepts and Criteria
By 'mixing'we understandspatialintermirrglingof clifferentconstituents.
This can be brought about by severarmechaniims:(i) by mechanicalry
Mixing of Vistous Newtonian and Non'Newtonian
Fluids 2ll
breakingandseparatingtheconstituentsinsmallerelementsandthen
r e c o m b i n i n g t h e m , ( i i ) b y d e f o r m i n g t h e c o n s t i t u e n t s u stoi nthe
g s hother
earortensile'
constituent
(iii) by transieriingthe elementof one
stresses,
and (iv) by moleculardjffusion'
by forcedor natural
"onutJtion,
or more of thesemechanisms
In any industrialmixing operation.one
upon the physical propertiesof
may prevail at the sametlme, depending
conditions. It is of practical
the materialsinvolveJ and tie operating
'mixing'
by manipulatingthe con-a^1lil1s
:"- lli:
interest to propagate
is to acqutre
can be usedto our advantage'The purpos€
thesemechanisms
the word
mixture' For any quantitalive investigation
a 'homogeneous'
o:11"^o''-,It o,oog.i"ous' or 'homogeneity'shouldbe properly
i' 'of the same kind" If tbis
The dictionarymeaningof homogentou*
In the technical
be homogeneous'
were a requirement'''ooiti^tu'e "in
which
we generallyimply "compo.t:dof.jp.urts"
world by 'homogeneous'
'parts'
there
defined'
is well
these
are all of the samet ioA". If"the scaieof
This scalewas.termedby
rvill be no ambiguityin the term homogeneity'
words' the scaleof
(1953)as'scale of scrutiny'' ln other
Danckrverts
s c r u t i n y i s t h e m i n i m u m s i z e c o n s i d e r e d f o r a n y a nthat
a l y sthe
i s 'mixture
Tlreanalysisof
regionson this scaieshouldleadoneto conclude
segregated
i s h e t e r o g e n e o u s ( n o t o | t h e s a m e k i n d'mixture'
) . T h e s c a l to
e o be
f s cused'
r u t i n For
yverymuch
is
dependsupon the iu'po'" for which the
things'a
of toilet soap (rvhich is' amongstmany
example,in the
"","
of scrutiny dependsupon the
mixture of a pigment and soap) the scale
to the-prod.uctused' For
resolvingpower of the eye' It may correspond
one tablet' For a detergent
the drugsin tablettorm'it is the weight of
of commachine'In.Jhe.case
.-*hi.h
mix it is the u'eightput into rhe washing
of
grain
inclividual
the
pounde.Jrubber, it is thc t.olunreo,'.l.
a d d i t i r cc a n b c a c t i v c '
scaleat which the homoThus, the scaleof scrutinyconsidersthe
doesnot give any quantitative
geneityshouldbe scrutinized' tlo-t?utl' it
,tromog"n.iry'. statisticalnleasures'suchas experimentally
criteriontor
52 are definedas
estintatedstandardieviaiioT S or variance
s':
-
l-
.lVs-
,L \ i': ,l a ,
- C,,)2
(1.1)
coefficientof variancc C" defined
Flere the mean C,,,can be estimated' The
AS
I
.a
,:l
I
L r:
S
x 100
--7;-
(1.2)
9tn
Howevei:' these measures suffer
can be used as a measure of mixability'
referred to any generalstandard referfrom the disadvantageof not being
e n c e . T h e m o r e e l a b o r a t e a p p r o a c h e s u s e c l t o c l e s c r i b e t h e . dconveegree'or
'goodness' of mixing are diviied into three parts for .11-^.t1Y"t
(b) large-sca1esegregation(random)'
nience: (a) smatl-scut" **g'"gu'ion'
and (c) large-sca1esegregation(systeniattcl'
212 V. V. Chavanond R. A. Moshelkar
l.l
Small-Scale
Segregation
In this case,Danckwerts(1953)proposed
two quantitiesto descrjbea
mixture'Theseare(i) scaleor'r.gr.gutio'n,
and 1ii)intensityof segregation.
1.1.1 Scarr or SrcRecerloN
Scareof segregation
simprymeansthe dinrension
the
size of the crumpso[ an unmixed component that characterizes
in the mixture. For a
mixture of two conrponentswith the
n.,.un
(vorunreor
weightfractions)
C^and, *.a1,.)lthe,point,values
"on.rntration
of
the
concentration
wi' be ci and'1 - cii the!t.'point' hereis
the scareof scrutiny.If the concentrationsmeasuredat tlvo points which
are at a distancer apart are
Ce and C,, a coefficient
of correlationcan be definedas
(1.3)
At r : 0' R(r): r' within a segregated
clunrp,fi(r) wirrrecruce
fiom 1 to
0' wrrcnone of the poinrs is outsidcthe crump,
,urr.r"i;;';;;"
concertratio. is c,,, ft0')rvit bezero. This,of course,
is the idealcase.For small_
scalesegregation
it is assu'redthat beyondthe segregaled
zoneconcentrationsdo not fluctuatevery much aroundthe
mean.In other words, after
a finite value of r, thereare minor fluctuations
,h" ,puo
u to r) in R(r) around zero and as/->co, 1in"oropu.iroo.rio
R(r)-rb. Fo, large-scale
segregation,
howe'er,R(r) rviil fluctuatebetween*l-l
and --r. A typicar
correlogramfor small_scale
segregation
is shownin Fig. i.
Fig.I
A correlogrant
for small-scale
segregation
Dancklerts defined a scale of segregation
on a linear scale ,''.9')and
a v o l u m e s c a l e( Z ' ) a s f o l l o w s : J':
'fa
I;
R(r) dr
v' : 2; I
Jo
rzR1r1
ctr
I
t
I
(1.4)
(1,5)
Contributionto the area under the curve
decreases
rapidlyfor r I f
.1
I
I
Mix,ing of l/iscousNen'tonianand Non'NewtonianFluids 213
a finite f. The
and consequently the limit of tntegral can be taken at
the size of the
as
sense
-ugnltua. of S; a,,d V' will vary in the same
in a random
define
to
which by irselt is not always possible
;ffi;,
unambiguous
a
s
p . o " . r r . F o r t h i s r e a s o n i t i s c o n v e n i e n tt o u s e t h e m
is not
significance
r.urur", of scale of segregation' even though their
m
easure
t
o
e a s i l y v i s u a l i z e d .D a n c k w e r t s h a s a l s o d i s c u s s e dt h e m e t h o d s
or
S' and I/', which mainly constitute measurements of the 'point-values'
more
being
the contents in the line sample or in a volume sample (volume
than a point and much smaller than the total bulk volume)'
1.1.2 INrtuslrY op Sncnpca'rtox
The intensity of segregationcan be defined as,
*2
'|-- ---:c^(l-c^)
(1.6)
a s1;31st'lof two
where o2 is the expectedpopulation variance. consider
into two zones with
miscible liquids. InitiallS;, when they' are separated
when the conpoint concentrations either I or 0, 1is ecluirlto 1' Finally'
C,,,,l becomeszero'
ientration becomesuniform throughout at a value of
of 1 for misI gradually reducesfrom 1 to 0. The fractional decay
ii,,
c i b l el i q u i d si s g i v e nb Y :
._6Elo+{,1,:,
_+(H):
(1.7)
the lrrolecular
It can be further argued that for rapid reduction in /, either
c
o
n
s
t
a
nt flattening'
diffusibility should be very high or there must be
c
o
n
c
e
n
t r a t i o ng r a d i e L o n g a t i o n - osr u b d i v i s i o no f t h e c l u m p s s o t h a t l t i g h
u
s
ed to obtain
e n t s a r e a l r v a i ' sp r e s e n ta t t h e b o u n d a r i e s , T l - r em e t h o d s
as o2 can be
t h e s c a l eo f s e g r e g a l L o ne n a b l e o n e t o c a l c u l a t e 1 , a s l o n g
s
u
ggestedby
as
estinraled rvith sufhcient confidence- Alternatively,
nleasuresuch
Danqkrverts, an in[initely rapid reaction can be used fbr
ments.
for
Zrvietering (1959)extended the concepts proposed by Danckrverts
c o n t i n u o u sm i x i n g . T h e s e a r e d i s c u s s e di n S e c ' 4 '
1.2 Large-ScaleSegregation(Random)
viscous
w h e n m i x i n g t | o i m m i s c i b l e l i q u i d s o r d i s p e r s i n gp i g m e n t s i n
l
a
r
g
es c a l e .
i
s
o
n
a
s
e
g
r
e
g
a
t
i
o
n
t
h
e
m a t e r i a l s o r m i x i n g p o r v d e r so r p a s t e s ,
a
v
e
r
y slorv
b
e
c
o
m
e
s
d
i
f
l
u
s
i
o
n
This is quite common when molecular
F
u
nction
0
.
o
r
I
e
i
t
h
e
r
a
r
e
p r o c e s s .T h e p o i n t - v a l u e so f c o n c e n t r a t i o n
c
a
s ea r e
i
n
t
h
i
s
m
i
x
i
n
g
o
f
h 1 r ; n o n nf l u c t u a t e sw i t h i n t 1 " T h e m e a s u r e s
m
i
x
i
n
g
'
p
o
w
d
e
r
provided mainly by the work done in the area of
a s s l t l n ea
s i n c e a l I t l i e s ea p p r o a c h e sa s s u n e c o r n p l e t e r a n d o r t l n e s sw, e
considered'
brriomial distribution when a two-colnponent system is
214
V. V. Chavanand R. A. Mashelkar
Tireprobabilitydistributionin thiscaseis givenby:
Pt(C) :
(1.)".",'
- C,,)*-*'
(1.8)
w h e r e x i s t h e ' u m b e r o f p a r t i c l e si r r a s a r ' p l e a n d n " i s t h e n u m b e r o f
particles of the. component whose concentration or volume-fraction &
we denote a.s c. C,,, is the population average of c, which is
known in
nany cases.P6(C) denotes the probability that x" particlcs (of
the kind
w h o s ec o n c e n t r a t i o ni s c ) a r e i n a s a m p r eo f x p a r t i c l e s a n d
/x\
(" ./
:
x!
ilG
x.r t
( l' e)
The population mean is Cn,and the population variance o2
is (C_(I _ C,,)I x),
when lactois such as size distrib'tion of the particles
uno"'it" density
differences are considered the variance assunes different
form (Lacey,
1 9 5 7 ;v a l l e n t i n , 1 9 6 7 ) .B e f o r e t h e c o n s t i t u e n t sa r e r a n c l o m l y
distributed,
i.e., when there are two sepafate zones of the two constiiuents
the
population yariance can be obtained as C,,,(l _ C,,,).Various
methods have
been describedin literature where the experimentally obtained
variance s2
i s c o m p a r e dw i t h o l a n d , o f r , t og i v e a n e m p i r i c a l m i x i n g
index. A typical
f o r m c a n b e s e e ni n t h e i n d e x p r o p o s e db y L a c e y ( 1 9 5 7 ) .
M ' : l - q : 0 6 -I6 ;
(r . 1 0 )
T o e a s e t i r e p r o c e s so f e v a l u a t i o n , t h e b i n o m i a l d i s t r i b u t i o n m a y
be
a p p r o x i m a t e db y n o r m a l d i s t r i b u t i o n u n d e r s o m ec o n d i t i o n s( H a l d , 1 9 5 2 ) ,
T h e m i x t u r e u n d e r e v a l u a t i o n i s p o s t u l a t e d t o b e r a n d o m ; r v h e t h e rt h e
s a m p l es i z e a n d t h e n u m b e r o f s a m p l e sa r e a p p r o p r i a t ew i t h t h e h y p o t h e s i s
of randomness can be verifiet by conrparing the estirnated rnean
and
variance rvith c,,, and 12 using /-test or x2-test (see lv{ohr (1959) for
urore
d e t a i l s ) .T h e c o e { i c i e n t o f v a r i a n c c f o r t h e p o p u l a t i o n i s
c,:r',|i:
ll_=,,,
L,
ry iq;-
(l'11)
This is the parameterwhichcharacterizes
the intensityof segregatrorr.
For a fixedsamplesize(from the scaleof scrutiny)anclreasonablylarge
numberof samples
(to coverthe bulk), the testoflrandomness
or the population coefficient
of variancedetermines
the degreeof mixing. converselS.,
for a fixedvariance,Buslik(1973)suggests
the sanplesizeas the mixing
index.If the weightof the particlesis constantthen the weightof the
sampleis directlyproportionalto the numberof particles.
on thebasisof
the weightof thesampleBuslik(1973)suggesrs
a homogeneity
index.This
indexseems
to be usefulespecially
for immiscibleIiquids,wherethereere
no disintegrated
particles,Hon'ever,determination
of Buslik,sindex ir a
practicalcaseis likely to be a laboriousprocedure.For the mixins of
lvlixing of YiscousNewtonian antl Non-Newtonian Fluicls 275
€xperimerlts for
liquid or paste, a suitable approach is to carry out
scaleof scrutiny
diifcrent sample sizeswithin a iange predeterminedby the
method for
this
using
While
randomness'
tests
for
*nA,5"n p.rior. the
doesnct
one
that
the
fact
by
created
is
ambiguity
liquids and pastessome
v
a l u ef o r
f
i
c
t
i
t
i
o
u
s
a
a
s
s
u
t
n
e
t
o
b
e
w
i
l
l
b
e
s
t
way
,.olty huu. iarticles. The
c
l
o
s
e
l
y is one
m
c
r
e
x
,
v
a
l
u
e
o
f
h
i
g
h
e
r
t
h
e
x , b e a r i n gi n m i n d t h a t t h e
have a
wili
coeflicient
expected
the
that
looking *ithin the sample and
iower value.
for intensi.iy
Although thc methods describedabove provide a meastlle
segregation.
o
f
t
h
e
s
c
a
l
e
cf segregitio', thev do not give any idea aborit
w
h
i
chconsidels
(
1
9
7
3
)
In tliis ,.rp.rt the 'rethod iescribedby Kristensen
the
between
the variance within the sample, aloug with the variance
s a n r p l e ,r v i l l b e u s e f u l.
mixing of
The above ideas have been quantitatively utiiized in the
and Hali and
p a s t e sb y N { i c h a e l sa n d P u z i n a n k a s ( 1 9 5 4 ) , E a r l e ( 1 9 5 9 )
. o r i n s t a n c e ,I { a l l a n d G o d f | e } ' t a k e x a s
G o c l f . , . e(y1 9 6 5 ) ,a m o n g o t h e r - s F
, l r e f r e q u e n t L yf i n d s
t l c : q i r a r eo f i h e g r i c ld i n i e n s i o * f o r a n a r e a s a r n p l eO
i n l i t e r a t u r e w h i l e u s i L i gs t a t i s t i c a lc o n c e p t sf o r t h e i n i e r p r e t a conil-rsion
b e h e l p f L r li n
t i o n o f i l r i x i n g d a t a a n d t h e f o r e g o i n g d i s c u s s i o ns h o u l d
a v o i d i n gs t r c hc o n f u s i o u '
1.3
Large-scaleScgregation(Systemrtic)
o b s e r v e cal n d
1 n t h i s c a s e , t h e s c a l e o f s c g r e g a t i o nc a n b e v i s u a l l y
due to shear or
i n e a s u r e d .I n v e f y v i s c o u s l i q u i c l s r n i x i n g o c c t l r s e i t h e r
s t r e a m l i n ed i v i s i o n
t en r : i l ed e f o r m a t i o n s( l a m i n a r r n i x i n g ) ; o f h r e o a l l soef t h e
m
i
x
i
n
g
)
'
a n t l r e c l i s t r i b u t i o n( d i s t r i b u t i v e
is nreasttretl'
l i t h e c o n c e n t r a t r o tal t a n y p o i n t i n t h e n l i r t u r e ( o I n l i r c i )
e
t
p
p
e
a' fT h e t a p r - ' r
$
'
i
l
l
2
(
b
)
o
r
2
(
a
)
F
i
g
.
i
n
s
h
o
w
n
a plot siuiilar to the one
a
l
t
h
o
r
r
g
l
rp r e - s e t r ti 's
r
v
h
i
c
h
C
i
f
f
L
r
s
i
o
n
,
i n I j i g . , r - ( b )i s d u e 1 o m o l e c u l a r
i
n this clse,
c
l
e
f
i
n
e
d
i
:
a
s
i
l
y
b
e
c
a
n
s
e
g
r
e
g
a
t
i
o
n
n e g l i g l b t ys i i - , , i 1T1h. e s c a l eo f
a l r v a y sb e
n
o
t
r
v
i
l
l
i
t
e
t
c
.
S
i
n
c
e
i
'
.
,
,
/
'
"
,
.
n s l t i o i t i b e t h r s i z eo f t h c -c l u s t e r s
i
ndex"
a
s
a
n
u
s
c
d
a
n
d
d
e
f
i
n
c
d
c o n s t a n t ,a r e p r e s e n t a t i v cv a l u e c a n b e
1
9
5
9
)
(
M
o
h
r
,
d
e
f
i
n
e
d
a
s
T h e i n t e n s i t y o f s e g r c g a t i o ni s u o r v
,:5r1
(r,r2)
variance is
A l t h o L r g ht h e p o l p u l a t i o n m e a n C , , ,i s k n o w n , t h e p o p u l a t i o n
intensity of
not knolvn, and thus there is no equivalent of expected
b
e
d
e
p e n c l e no
tn
w
i
l
l
i
n
t
e
n
s
i
t
v
T
h
e
m
i
x
t
u
r
c
)
.
s e g r . e g a t i o(na s i n a r a n d o n t
is
It
scrutiny.
of
scale
the
by
be
cletertri'ed
uuir.,pi"size, rvhich should
a
c
o
r
r
e
ct
l
b
r
(
r
r
,
t
l
r
a
n
b
e
t
n
o
r
e
s
h
o
u
l
d
o b v i o L r st h a t t h e s a n r p l ev o l u m e
",r)3
m
i
x
t
u
r
e
'
t
l
t
e
o
f
s
e
g
r
e
g
a
t
i
o
n
r e p r e s c n t a t i o no f t h e i n t e n s i t y o f
t h e ' l i r t l ' ri i t a r
F o r a b e t t e r u n d e r s t a n c i i n go f n l i x i n g i n t h i s c a t e g . t r y
,ilisrributive
sontedetail,
in
exantined
be
mixing' should
mixing, arid tire
216 V, Y. Chavanand R, A, Ii:[ashe]i<ar
J
cm
rl
I
I
I
I
I
- -t --
l*-i-
/cl
[e-.
it
/\
t\
ts*
(b)
Fig.2
Largc scale scgregation (s]'stematic) in the absence of
( F i - l : .2 a ) a n d j n t h e p r e s e n c e o f ( F i g . 2 b j m o l e c u h r
diliusion
1,4 LaminarMixing
The averagestriationthickness(scaleof segregation)
can be computed
from the ratio of the interfaciirlsurfacearea betwcenthe components
and
the total yolumeof the system(N,Iohr,1959).
2V
"A;
( 1 .l 3 )
This expression
is a resultof picturingthe materialas beingdeformcdby
mixingprocess
into roughlyplaneparallelsheets.
The anaiysisofSpencer
and Wiley (1951)lbr undirectiorLal
sliear(sirnpieshear.flr.iwbetrveentivo
Mixing of Viscousitieu'tonian ond Non-Newtonian Fluids 217
infinite parallelplates)leadsto the followingequationfor the changesin
the interfacialareas.
(*")':
(#,"
, - r(#,x r) cos& 1 C O S& 2 f
t ) c o s 2a 1 ( 1 . 1 4 )
where 11 and A1o are the final and i n i t i a l i n t erfacial &raos. !/1 is the
veiocity; the gradient of rvhich is in
the direction x2, cos d1 ond cos w2 ' ..:4,, :,'l'ttt/Z' :122 - 2'-t: Z :':2'
are the direction cosines of the
Att,t)?.i:u
H
H
blr
normal to the original surface with
-=fl
respect to x1 and #2, respectively'. t . ' r'
8
rg
t
The above equation points clearly lt"-.::.-:.7....:.:-./.a:,.:a.7:-,-,,7a7.-?:-717
towards important factors in the
problem,
Iaminar
for
mixing
instance, the importance of the
- . -,,,':-''' /', "2,212 t-:"^-:1"-:
orientation of the original surface
betweenthe trvo species.It is obvious
that cos o1 should be finite for the
1n:
deformations to change the interfacial area or to reduce the striation
t h i c k n e s s( s e eF i g . 3 ) .
Fig. 3 Influenceof orientationin laminar
Bergen (1959) describes a twos h e a rm i x i n g
dimensionalpicture of the ohanges
i n s t r i a t i o n t h i c k n e s sf o r a c o u e t t ef l o w b e t w e e nt l o c o n c e n t r i c c y l i n d e r s
( s e eF i g . 4 ) . I f o n e d e s c r i b e st h e f f o r v i n c y l i n d r i c a l c o o r d i n a t e st h e n t h e
:
'41-CfL1:a'
-f>"
'.rY " ) - -
' * F - ' - xr
\
\''
41.1..:\i
\
',,
/\t. I
Fig. 4
Laminar shear mixing in con.:entricc5'hndelststems
--
1
218 V. V. Chavanand R. A' Mashelkar
shearrate can be written as
i,: -rdtv'l) : f,(,OoO,)
and if the inner cylinder is rotating rvith an angular velocity J?, tlren
,,:ryi
, * 12 -{*L.t
\rCi _ ni/
At any radialposition,rvehave
, \'
: I * cot2p :
{'-l : cosec2g
\,'" /
r^.
d0
j":
-;'
when,
Furtlrermore,
lr,
l .'
rs
r
, + (,f,)'
(r.r5)
(r'16)
dr
F o r a s t e a d yf l o w ( s i r l c ei < i c e sn o t c h a n g e w i t h 0 o r t ) i n a n i n t e r v a l o f
time At, one obtains
r,
l R1, )rz ( 1. 1 7 )
' #t rr' 1
1R'/R2)21
;
:
o r c o n s i d e r i n gt l t a t A U i
revolutiotts
rs
2 r N , o w e o b t a i n i n t e r n l s o f the nuniber of
lt ,4' "l R
- , ru);'(, r, - (t tR ' i R 2 ) 2 1
( r .l 8 )
;:
T h u s t h e m i n i m u t n c h a n g e so c c u r a t r : R r a n d m a x i n r u m a t r : R 2 '
S i m p l e c a l c r - r l a t i ocna n b e d o n e t o s h o l v t h a t t h e t i m e r e q u i r e d t o o b t a i n
t h e m a x i r n u n rs t r i a t i o n t h i c k n e s st o b e m u c h l e s st h a n t h e s c a l eo f s c | u 1 i n r ' .
T h e a b o v e r e l a t i o n sh o l c lf o r t t N e r v t o n i a nf l u i d , a n d i t i s e a s y t o s h o r v
t h a t f t r r a n o n - N e i v t o n i e np o w e r l a r v { l u i d ,
/-:
/so
_
ii(r1R^,,)t"' _
(Rr//tr)2)
,,
(l.Le)
*T!\r
s lore
I t i s s e e nt h a t r v i t h i n c r e a s e dp s e i i d o - p l a s t i c i t y3 6 t u . m i x i r r g b e c o n t e n
dillicult.
S o n r ec a l c u l a t i { l n so l t h e c h a n g e s i n t h e s t r i a t i o n t h i c k n e s s h a v e b e e n
m a d e b y s h r e n k , e t a l . ( 1 9 6 3 , 1 9 6 9 ) .T h e y h a v e c l i s c u s s e ds p e c i f i c a l l yt h e
f l o w t h r o u g h a u a n n u l a r c h a n n e lr v i t h r o t a t i n g w a l l s u n d e r t h e c o n t l i t i o n s
o f a n a x i a l p r e s s u r cf k r w . l n a n o t h e r w o r k , f l o w i n t h e r e r t a t i r r gt L r b ci s
c o n s i d e r e d .O n t h e b a s i s o [ s i m p l e c a l c u l a t i o n s f o r t h e l l o r v o f v i s c o t t s
Newtonian liquids they obtained the distribution of striations aI a
particular axial lelgth in r-0'plane. Their analysis considcrsthe
t l e n d i n g o f t l o \ ' i s c o u sN e r v t o n i a nl i q u i d s a t l o r v R e y n o l d s n u m b e r s a n d
s h o r v sa g o o d a g r e e n t e nfto r t h e n t i x i n g o f b l a c k ( c a r b o n b l a c k ) a n d w h r l e
( T i O 2 ) p i g m e n t e dp o l y s t y r e n e .M o h r , e t a l ' ( 1 9 5 7 )h a v e e x t e n d e dt h e b a s i c
o f I a m i n a r s h e a rm i x i n g t o o b t a i n t h e d i s t r i b u t i o n o f s t r i a t i o n s
"on..pi,
on the
f o r a , i e x t r u i l e r .T h e y e x p l a i n t h e e f f e c t o f g e o m e l r i c a l v a r i a b l e s
bv u't
r
e
c
e
i
v
e
d
o
f
s
h
e
a
r
r ' n r t x i m u ms t r i a t i o n t h i c l i n e s s .T h e t o t a t a l n o u n t
and
rate
shear
the
product
of
eleinent was obtained as the suin of the
Mixing of Viscous Newtonian and Non-Newtonian Fluids
219
residence
time for eachpart in the flow path, The hydrodynamicsused
was approximateand wasvalid only for Newtonianliquids.Recentstudies
in hydrodynamicsof extrusionfor Newtonian(McKelvey,1959)and nonNervtonianliquids (Bigg and Middleman, 1974)aremore closeto reality.
However,the laminar shear mixing conceptshave not yet been used to
study and interpret the mixing phenomena.Furthermore, only nondirectionalflows have been considered.The direct extensionfor more
complicatedhydrodynamicsituationsis likely to be tedious.Chavan,et al.
(1975a)have usedtheseconceptsto explainthe rnixingtime (definedlater)
resultsfor a helical screw mixer in a draught tube. N{ohr, et al. have
proposedan equation for laminar shear mixing of two liquids rvith
differentviscosity.The changesin the striationthicknessare described
by
an equation
'r:3Y2
( r .20)
MtPt
r"o
w h e r e M l i s t h e n e t a m o u n t o f s h e a r i n gs t r a i n i n t h e m a j o r c o m p o n e n t ,I r 1
is the viscosity of the major component and p,2is the viscosity of the minor
component. Note that the changes in the striation thicknessrvithin the
m i n o r c o m p o n e n t( s a m e v i s c o s i t yp r . t2h r o u g h o u t ) i n t e r m s o l t h e s h e a r i n g
s t r a i n i n t h e r n a j o r c o m p o n e n ta r e p r e d i c t e d .F u r t h e r m o r e ,n o n . d i r e c t i o n a l
s t r a t i f i e df l o r v o f t w o i n t m i s c i b l el i q u i d s i s c o n s i d e r e dT. h e r e i s a p p a r e n t l y
a n e e d f o r t h e s o l u t i o n o f f l u i d d S r n a r n i cp r o b l c m s i n v o l v i n g t h e f l o w o f
t r v o o r m o r e i m m i s c i b l ep h a s e s .
M u r a k a m i e t a l . ( 1 9 7 2 ) s h o w e d e x p e r i m e n t a l l y( o n : ! c o u e a n d p l a t e
v i s c o m e t e r )t h a t o n e c a n d i s c a r d c o r n p l e t e l yM o h r ' s e q u a t i o r . r1 ' o rb l e n d i n g . T h e y o b s e r v e d t h a t t h e s t r i a t i o n t h i c k n e s so f t h c m i n o r c o i l l p o n e n t
( v i s c o s i t yp 2 ) d o e s n o t c h a n g en t u c h w i t h t i r e r , i s c o s i t 5 ' r a t i o( 0 . 6 . < -p 2 l p t 1
< 3 i . T h i s i s v a l i d o n l S , 1 y 1 . t " tnh e r r r i n o rc o m p o l t e r r ti s p l a c e d p e r p e n d i c u l a r t o t h e d i r e c t i o n o f f l o w a n d w h e n i t i s s h e a r e df o r a l i r u i t e c tl i n t e . I n
p r i n c i p l e , o n c e t h e i n i t i a l p o s i t i o n o f t h e m i n o r c o n t p o n e n t i s r : l r a n g e di t
w i l l t r y t o o r i e n t i t s e l f a n d d e f o r m s o t h a t e q ' . r a l i t yo I s t r e s s e sa u d v e l o c i t i e s i s s a t i s f i e da t t h e i n t e r f a c e s S o f a r c h a n g e si n s t r i a t i o n so n l y d u e t o s t r e s s d e f o r m a t i o n h a v c b e e n
d i s c u s s e dI.n f l o w i n g r n e d i a , t e n s i l ed e f o r n r a t i o n sa l s o p r e v a i l . T h e s e a r e
brought about by changesin the velocities in the direction of florv due tcr
g r a d i e n t ss u c h a s ! ,
"
('X
*
I
uIz
anA lt!
d.f
M o h r ( 1 9 5 9 ) p e r f o r n r e c la n a n a l l , s i s
t
f o r t h e c a l c u l a t i o no f c h a n g e si n i n t e r f a c i a la r e a u n d e r s u c hc i r c u r n s t a n c e s :
COS &l
(*,)'
,Yt
,
'
COS rv2
X,
,
'
COS &3
Xt
(r . 2 1 )
w h e r e X r , X z a n d f 3 a r e r e l a t i v e c h a n g e si n d i r e c t i o n . r r , . r 2a n d . x 3 . l - l o
d e t a i l e dc a l c t r l a t i o n so n a n a c t u a l l l o r v p r o b l e n h a v e b e e n d o n e s o f a r
trnd they are clearly desirable.
220 V. V. Chavan and R. A. Mashelkar
1.5
Distributive Mixing
When shear and tensile deformations are small, mixing occurs due to
t h e d i v i s i o n o f s t r e a m l i n e s a n d r e d i s t r i b u t i o n . S h e a r e r( 1 9 7 3 ) a n a l y s e d
the changesin the striation thickness in the plane perpendicular to the
flow for (a) an assemblyof rotating blades,(b) a stackedarray of helically
flighted ducts, (c) an assembly of planetary rollers. I-iere, the author is
p*U^t ty justitied iri assuming a plug flow, since the changesin the striaiion thicknessin the plane perpendrcularto the flow (r-0' plane) are uttder
consideration, whereas the main shear gradient exists in other plane
(.r-z plane). The tensile deformaticns can be neglected, if the rate of
uog" of velocities in the direction of florv is negligible. Thus the
.distributive mixing' can be used. The method used is as
"t
approach of
fotiows: knorving the initial striation thickness and following the flow
(while keeping proper count of the number of subdivisiols and the resulti n g r e d i s t i l U L r t i o no) n e i s a b l e t o o b t a i n a s i m p l e r e l a t i o n f o r t h e f i n a l
s t r i a t i o n t h i c k n e s sa n d t h e n u m b e r o f t i m e s t h e m a t e r i a l h a s b e e n s u b d i v i d e d a n d r e d i s t r i b u t e d .T h e h y d r o d y p a m i c sa s s u t n e si m p o r t a n c c i r t a n
i n d i r e c t w a y , i . e . , i n o b t a i n i n g t h e a v e r a g er e s i d e n c et i m e .
S p e n c e ra l d W i l e y ( 1 9 5 6 )h a v e p r o p o s e da m e t h o d w h e r e c o m p r e s s i o n
a n d d i s t r i b u t i o n c a o b e c o n s i d e r e dt o g e t h e r . T h i s i s u s e f u l e s p e c i a l l yi n
s t u d y i n gt h e b l e n d i n go n a r o l l e r m i x e r .
T h e r e c e n tr v o r k o f B i g g ( 1 9 7 5 )o n m i x i n g i n p o l y m e r f l o r v s y s t e m si s o f
c o n s i c l e r a b l ei n t e r e s t . H i s s t u d y s h o w s t h a t r e s i d e n c et i m e c l i s t r i b u t i o n
c u r v e s( s e e s e c .4 . 1 . 3 ) g i v e i n f o r m a t i o n o n t h e d i s t r i b u t i o n o l ' m a t e r i a l
along the primary'floiv direction, Ilolvever, they do not provide informat i o n o n m i x i n g i n t h e t r a n s v e r s ef l o w d i r e c t i o n f o r l a m i n a r 1 1 o ws y s t e m s .
For rnost pol,rrmerprocessingapplications, mixing ilr tfansversedirectiot.t
i s v e r y i m p o r t a n t , B i g g c o r r s i d e r sa t r u n i b e r o f c a s e s ,s u c h a s m i x i n g i n
s c r e r ve x t r u d e r s ,m i x i n g i n r o t a t i n g c y l i n d e r sa n d s t a ' r i ct t t i x e t s ,a n d s h o v ' ' s
t h a t t h e s t r i a t i o n t l t i c k n e s sa n d s t r a i n a r e c l o s e l yr e l a t e d .F r o m t h e s h a p e s
o f t h e c u r v e sr e l a t i n g s t r i a t i o n t h i c k n e s sw i t h s t r a i n i t i s c l e a r t h a t m i x e r
design is an impcrtant factor in eliiciently utilizing strain to optintize
nixing.
2.
Means of Mixing
t ixing, proper conditions will have to be created and
F o r e f f i . s i e nm
t
h e t m r c h a n i s m s d e s c r i b e di n s e c . I c o u l d p r e v a i l . T h e
s
o
maintained
m
i
x e r i s o b v i o u s l y d e p e n d e n t u p o n t h e s p e c i f i cc a s eu n d e r
s e l e c t i o no f a
c o n s i c l e r a t i o nA
. f r e q u e n t l y u s e d s e l e c t i o ng u i d e i s g i v e n b y H o a n d
Krvong (1973). lt is evident from this that the choice of agitator is largely
g o u . r n . i b y t h e l i q u i d v i s c o s i t ya n d t o a l e s s e re x t e n t b y t h e m i x i n g t a n k
i i z e . T h e g u i c l ei s s o l n e v r h alti m i t e d i n a p p l i c a t i o ns i n c es o m e v a r i a t i o n i n
t h e t l e s i g na n d o p e r a t i i ' r gv a r i a b l e sc a n s h i f t t h e s e l e c t i o np r o c e d u r ec o n s l derably. In adrlitron, ot course. probleuis of construction, endurarice of
Mixing of YiscousNeu'tonian and Non-N(utonian Ffuids 121
equipment, etc,, come in; a useful discussionof these aspects is given by
Uhl and Gray (1966). It is also conceivable that a multitude of modifications to the impeller design are possible, addition of each of rvhich
would mean that we may have to study an infinite number of individual
problems. Fortunately, this is not the case and all the mixers can be
b r o a d l y s u b d i v i d e d o n t h e b a s i s o f t h e i r s h a p e a n d a c t i o n ; a u s e f u li n f o r m a t i o n c a n b e f o u n d i n U h l a n d G r a y ( 1 9 6 6 ) .T h e s t a t i c m i x e r s w h i c h
a r e u s e d i n p i p e s h a v e b e e n d e s c r i b e db y C h e n ( 1 9 i 3 ) .
Different mixers act by prodLicing different flow patterns. A detailed
k n o r v l e d g eo f t h e h y d r o d y n a r n i c si n a m i x i n g . v e s s e lw i l l b e m o s t u s e l u l
i u n o t o n l y a n a l y s i n gt h e m i x e r p e r f o r m a n c e b u t a l s o i n s c a l i n gu p . A n
a p r i o r i k n o w l e d g eo f h y d r o d y n a m i c sa p p e a r st o b e a l m o s t i m p o s s i b l ei n
v i e w o [ t h e p r e s e n c eo f a c o m p l e x t i m e - d e p e n d e n tt h r e e - d i m e n s i o n a l
l e a s u r e m e n tosf v e l o c i t y
l a m i n a r o r t u r b u l e n t f l o w . D e t a i l e d e x p e r i m e n t am
d i s t r i b u t i o n i n t h e s t i r r e d v e s s e li s a n o b v i o u s a n s w e r ;b u t i n v i e w o f t h e
n u m b e r o f v a r i a b l e si n v o l v e d ,t h i s b e c o m e st o o e x p e n s i v ea n d t i m e - c o n s u m i n g . [ r o r t u n a t e l y ,a n a l t e r n a t i v ec a n b e f o u n d i n w h i c h s o r n eg r o s si n t e g r a l
q u a n l i t i e sc a n b e m o r e e a s i l y m e a s u r e d T
. h e s eg i v e u s e f u li n f o r m a t i o n c o n c e r n i n g t h e h y d l o d y n a n r i c si n t h e m i x i n g v e s s e l .I n w h a t f o l l o w s u , e s h a l l
d i s c u s si n s o n r ed e t a i l t h e d e f i n i t i o n s m
, e a s u r e m e ntle c h n i q u e sa n d u t i l i t y
o f t h e s eq u a n t i t i e s . T h e i r r e l a t i o n t o a c t u a l h y d r o d y n a m i c s v , , i l l b e d i s c u s s e di n a s u b s e q u e n ts e c t i o n . W e s h a l l c o n s i d e r t h c p r o c e s so f b a t c h
m i x i n g a n d c o n t i n u o u sr L r i x i n gs e p a r a t e l y .
3.
Batch Mixing
I n b a l c h r n i x i n g e x p e r i m e n t su s u a l l y t h e n r e a s u r e m e n tosf t h e f o l l o v l i n g
are made:
l.
2.
3.
4.
F l o w p a t t e r n s a n d v e l o c i t Sd' i s t r i b u t i o n
O v e r a l l d i s c h a r g er a t c s . c i r c u l a t i o n c a p a c i t i e sa n d c i r c u l a t i o n t i m e s
P o r v e rc o n s u n r p t i o r r
Mixing times
I n w h a t f o l l o w s , r v e s h a l l d e s c r i b et h e n r e a s u r e m e ntte c h n i q u e su s e df o r
e a c h p u r p o s e r a t h e r b r i e f J ya n d t h e n e l a b o r a t eo n t h e a v a i l a b l e i n f o r r u a tion obtained with viscous Newtonian and non-Newtonian fluids. It
s h o u l d b e e m p h a s i z e dh e r e t h a t a g r e a t d e a l o f i n f o r m a t i o n d e a l i n g w i t h
t h e a b o v e a s p e c t se x i s t s i n l i t e r a t u r e p a r l i c u l a r l y i n r e l a t i o n t o t h e t u r u u lent region florv of low-viscosity Newtonian fluids, but due to lack of
space this rvill not be reviewed here; only pertinent referenceswill be
pointed out.
3,1 Flow Patterns anil Velocity Distribution
T h e k n o w l e d g eo f f l o w p a t t e r n s a n d v e l o c i t y d i s t r i b u t i o n i s h e l p f u l i n
222 V. V. Chavanand R. A. Mashelkar
understanding the mixer performance and in building up realistic physical
m o d e l s . F u r t h e r m o r e ,i t h e l p s i n l i n k i n g i t u p w i t h t h e o t h e r i n t e g r a l
q u a n t i t i e s s u c h a s c i r c u l a t i o n c a p a c i t i e s ,c i r c u l a t i o n t i m e s , p o w e r c o n sumption, mixing time, etc. A gross observation on flow patterns
(obtained, for example, by injecting a dye tracer) gives an idea about the
o v e r a l I m o t i o n i n t h e v e s s e l ,p o i n t s o u t t h e d e a d z o n e s ,e t c . T h e d e t a i l e d
o b s e r v a t i o n so n v e l o c i t y d i s t r i b u t i o n , h o w e v e r , a r e t h e m o r e i m p o r t a n t
o n e s ,s i n c et h e y h e l p i n t h e e s t i m a t i o n o f v e l o c i t i e s ,v e l o c i t y g r a d i e n t s ,
t o t a l s t r a i n s ,e t c . , i n d i f f e r e n t p a r t s o [ t h e v e s s e l .
3.1.1 MsasuRnlrsNr TecHrqreues
There are several measurement techniques which can be used,.In the
follolving, we shall review very briefly some of them; in each case pointing out their linitations in rclation to the high viscosityNewtonian and
non-Nswtonian iiquids u,itir rvhich we are concernedhere.
(l) Particle tc<.hnique: In this case a neutrally buoyant particle of,
s a y , p o l y c t h y l e n eo r p o l 5 ' s t y r e n e
i s s u s p c n d e di n t h e l i q u i d i n t h e
v e s s e la n d m o v e m e n to f t h e f l u i d i s t r a c k e d b y a c a m e r a . B y u s i n g
a p p r o p r i a t em e a s u r e m e n p
t r o c e d u r e( s e e ,e . g . , K e l k a r , e t o l . , l g 7 3 ;
P e t e r sa n d s m i t h , 1 9 6 7 ) a t h r e e - d i m e n s i o n a 'le l o c i t y d i s t r i b u t i o n
can be obtained. The particle migration eflects (which are particuiarll, complex in viscoelastic non-Ne.,vtonian fluids) should be
taken care of; absence of such effects could be, for instnnce,
e n s u r e db y I o o k i n g f o r c i o s e d c i r c u l a t i o n l o o p s . A s t r e a k p h o t o g r a p h y m e t h o d ( e . g . S e y e ra n d M c t z n e r , 1 9 6 9 )i s e s s e n t i a l l ys i m i l a r
a n d a s u s p e n s i o no f r n i c r . b u b b l e s o f a i r i s w h a t i s u s e d i n t h i s
l e c h n i q u e . S o m e t i n ' r e sa c l o u d o f l i n e p a r t i c l e s o f a l u m i r r i u m o r
n . p h t h a l e' e ( o r o t h e r m a t e r i a l c a p a b l c o f r e f l e c t i n gl i g h t ) i s u s e d .
A thin flat hea'of
b r i g h t c o l l i m a t e dl i g h t i s p a s s e di n t o t h c
t r a n s p a r e n tl i q u i d a n d a t i v o - d i n r e n s i o n a vl e l o c i t y d i s t r i b u t i o n i s
obtained.
(it) Pitot tube, This classical technique should be used rvith great
c a r e w i t h v i s c o e l a s t i cn o n - N e r v t o n i a nf l u i d s . A n y s t a g n a t i o n p o i n t
v e l o c i t y o r p r e s s u r em e a s u r e n r e nct a n g i v e s p u r i o u sl e s u l t s ( s e ce , g .
Metzner and Astarita, 1967) due to the abnormal kinernatic
conditions w'hich the viscoelastic fluids develop under conditions
of rapidli' ciranging deformarions.
(.iii) Hot-t+'ireane,nometer: The principle of this technique is again
w e l l - e s t a b l i s h e db, u t r v h e n u s e d w i t h v i s c o e l a s t i cn o n - N e r v t o n i a n
f l r " r i d st h i s c a n g i v e s o m e p r o b l e m s . I n d e e d , t h e s t u d i e s o f J a m e s
and Aco:ta (1970) showed tha,t the heat transfer rate from a
c , v l i n d c ri m m e r s e di n a v i s c o e l a s t i cf l u i d b e c o m e si n d e p e n d e n to I
the freestream velccityafter a certlin critical velocity is reachecl.
u l t n r e n a n d l ) e n n ( 1 9 7 0 )h a v e a n a l y s e d t h i s p r o b l e n a n d s h o * , n
t h i s t . b e c o n s i s t e n tr . v i t h t h e p r e s e n c e. f a s h e a r w a v e v e l o c i t y ,
I
l
I
{
I
Mixing of ViscousNewtonian and Non-Newto:tianFluids 221
w i t h w h i c h t h e i n f o r m a t i o n i s t r a n s p o r t e di n e l a s t i c f l u i d s .
(iv\ Laser-Dopler method: This method, which involves the measurement of Dopler shift of a thin laser beam is perhaps the best to
use since the flow field is left undisturbed rhus avoiding any
anomalons effects. In recent years, there has been an increasing
trend to usethis nethod in the study of velocity distributjon in
n o n - N e w t o n i a n f l u i d s 1 e . g . ,R u d d , t 9 7 l ) .
3,2
FIon Patterns and Veloci(y Distribu(ion in
V i s c o u sN e w t o n i a nF l u i d s
The study of florv patterns for various types of inrpellers has been
e x t e n s i v e l yr e p o r t e d ( G r a y , 1 9 6 6 ) . A t l o r v R e y n o l d s n u m b e r s , a t u r b i n e
a g i t a t e d v e s s e le s s e n t i a l l yp r o d u c e sa t a n g e n t i a lm o t i o n . A t h i g h R e y n o l d s
n u n r b e r s ,t h e s e c o n d a r yc i r c u l a t i o n b e g i n s d u e t o t h e c e n t r i f u g a l f o r c e s .
T h e f l u i d i s t h r o w n a w a y r a d i a l l l , a n d b y r e a s o n so f c o n t i n u i t y , i t i s
broughtin axially,
I n t h e c a s eo f p r o p e l l c r sa n d p i r c h e db l a d e t u r b i n e s ,t i r c r ei s p r i m a r i l y
a n a x i a l r n o t i o n a l o n g r v i t h t h e t a n g e n t i a lc i r . c u l a t i o n A
. t h i u h e rR c 1 , n o k 1 s
n u m b e r s ,h o w e v e r ,r a d i a l n r o t i o n b e g i n sa n d s t a g n a n tp o c k e t sa r e f o r n r e c l
i n t h c i r o t t o m c o r n e r o f t h e v e s s e (l f o r d o w , n r v a r dp r . r n r p i nogr t h e i m p el J e r ) .
o f t e n b a f f l e s a r e u s e d r v i t h t h e s e i m p e l l e r s t o s u p i ) r e s sl l i e v o r t i c e s ( t o
r e d u c ef h e t a n g e n t i a l m o t i o n ) a n d t o p r o m o t e t h e a x i a l a n d r a d i a l n ; o t i o n .
I n t h e c a s eo f p a d d l e s a n d a n c h o r s t h e p r i m a r y m o t i o n i s t a n g e n t i a l .
T h c r a d i a l a n d a x i a l r , ' r o t i o nd e v e l o p sa t h i g h e r r o t . a t i o n ; r is p e e c l sT. J r e
R e 1 ' n o l d sn u r n b e r sa t v " h i c h a p p r e c i a b l er a d i a l a n d a x i a l n r o t i o n d ev e i o p s
a r e r n u c h h i g h e r f o r a n c h o r s t h a n t h o s c f o r p a d d l e s a n c l t u r b r n e s ;t h e
close clearance betrveen the impeller ;rnd the vesscl is apparently
r e s p o n s i b l ef o r t h i s . T h e s e c o n d a r ym o t i o n i n t h e c a s eo f a n c h o r s( i n t l r c
vcrtical plane, i.e., in the plane perpendicular to the irnpellerplaire) is
s h o r v ni n F i g . 5 ( B e c k n e r a n d s m i t h , 1 9 6 6 ) . A s t h e R e i ' n o r d s u u m b e r i s
/)2\t^
\
i n c r e a s e d( " - : : ! > l 0 ) a \ o r r e x i s d c v e l o p e db e l r i ' d t h e b l . r d e .r \ r h i g h c r
\r'
I
R e y n o l d sn u m b e r sa t w i n v o r t e x i s f o r m c d ( ' l ' 1 l e , I 0 ( , ) . 1 ' l r ed e c . e a sier r
\p
I
the clearance causes these vortices to be formed at lorver Reynolds
numbers. A detailed investigation of these vortices and also of the l.ertical
secondarymotion has been performed by peters and Smith 0969).
I n t h e c a s e o f h e i i c a l i m p e l l e r s ( r i b b o n , s c r e w s ,o r c o r n L . i n e dr i b b o n screw) the axial florv is the primary flolv. The situation is thus srmrlar to
that in the case of propeller or pitched blade turbine rrlixer-s;although the
flow patterns are somewhat more complex. Nagata et al. (1957)first
d e s c r i b e dt h e p r i m a r y c i r c u l a t i o n i n a r i b b o n a g i t a t e dv e s s e l F
. ig. 6 shows
a typicaI patt€rn for a ribbon purlping upwarcl. Thc liquitl near rhc
i n - r p e l l efrl o w s u p w a r d s i r v h i l s ts i r n u l t a n e o u s t ryo t a t i n g ) , i n w a r d s a l o n q t h e
lree surf'ace,and clolvnwardsnear the shaft and railiallr, outrvardsnear the
224 V. I'. Chavanand R. A. Mashelkar
SecondarY*
circulation
m-
ffirr
Rc.-5ff
li(''2
(b)
!-ig 5. Secondarycirculationin an ancbor agitatedtank: (a) overall
circulation,(b) flow near the blade (after Becknerand Smith'
t966)
( 1 9 7 0 )m e n t i o n t h a t t h e d i a m e t e ro f t h e c o r e o f
base. Os'hima and Yuge
the liquid coming downlvards was
about 1.5 tintes greater than the shaft
diamcter;this factor must obviously be
depending on geonletry. At higher
Re1'nolds nunrbers (R" > 20) secondaly
tloq's rvere noted by' these authors. For
r i b b o n sp u m p i n g u p w a r d si t l v a so b s e r v e d
( B o u r n ea n d B u t l e r , 1 9 6 9 )t h a t t h e l i q u i d
from the bottom travels back to the
bottom through a short path without
g o i n g t o t h e s u r f a c e .F o r r i b b o n p u m p i n g
dorvnwar<Jsthe primary flow is exactly
opposite and a secondary florv exists at
the surface. When 1he screw impeliers
are used along with a draught tube a
primary circulation occurs inside the
draught tube (like the one near the ribbon
impeller). The fluid flows downrvards in
the draught tube with a simultaneous
tangentialmotion (for a screw pumping
Fig. 6 PrimarYcirculation in a
downwards) and travels uprvards in the
nbbon agitated vessel
annular space more or less parallel to
( N a g a t ae t a l . , 1 9 5 7 )
Fluids 225
Mixing of Viscous Newtonianand Non^Newtonian
velocitiesand goesback to the
-secondarv
the vesselwall with very small angular
flows have been made as
;";e. No ou"iuutions on
;;;r;;;
exist ior a considerablerange of
yet. The primary flow wus found to
- 2.90)'The flows will somewhatdepend
Reynolds numbers 1up to n"
the draught tube' The screw
ih" gap betweJnthe impellei and
;;;
to the propellers'A.s.inthecase
without the draughttube behavessimilar
fashion' This helps in
an
of propellers'screwsare used in
"eccentric
the
increasing primary flow'
a"tipitig ine ,ecoodarymotion and in
numbers)a vortexis formedon the
At high speeds1or at higherFroude
very much upon the geometry'For
liquid surface.Its o""u""i"t depends
(Uhl and Gray' 1966)' For a ribbon
turbines it occurs at Re- 300
downwardsit occursmuch earlier
pumpingupwardsor a screwpumping
(r't-0.01). Baffiesare generallyused
than that fbr an uo"loii*p"it",
of turbine and propeller this will
to damp these vorttes' tn the case
betweenthe impeller and the liquid
obviouslydependupon the distance
surface.
3.3FlorvPatternsandYelocityDistributioninNon.NewtonianF.|uir|s
T h e r h e o l o g i c a l c o m p l e x i t i e s a s s o c i a t e d w i t h t h e ldistribution'
iquidbeingagitated
patternsand velocity
can very significantlyinflutnc" the flow
will' of course'govern the net
The nature of the rneotogitutcomplexity
The limitation of spacedoes-notenable
influenceon the hy<1rod1'namics'
of rheoiogy;hou'ever'suchinformation
us to give u."oun, ofihe science
"n
textbooks(Middleman' 1968;
may be readily found in many standard
1974)'
Sketland,1967;Astaritaand Marrucci'
is most easy,to .understand
viscosity
dependent
of shear
The influence
(shear
pseudoplastic
For instan"",ont would expectthat for
and analyse.
impeller
near
of the fluid in the
thinning)fluids,tftt "ff"ttttt viscosity
nlovesaway
progr:essively.u.t:n"
increase
and
,.gi"" ,'troulAbe rather low
iesult in ti19f. velocitiesand
from the impeller'This wilt ton"qutn'ly
t"gioo' whichwill die awayrapidly
velocitygradientsin the nearimpgller
studies(e,g.,Metznerand
frorn the impeiler.The early fhotogrJpnrc
singularlyimportant.resultu'hich
Taylor, 1960)did tonn'- ttris' fne
able to obtain on the basisof their
lvletznerand Taylor (1960) lvere
was that the averageshearrate in the
velocitydistributionmeasurement
RpM. we shallseelater(sec'3.5)that
vesseiwas linearlypr"p"rii"rr"i to
power
importantrole in the predictionof
this resultplaysan'extremely
fluids' It should be emphasizedthat
requirement,,n non-Newtonian
M e t z n e r a n o r a y t o r o u , , , u . o n e a r s o l i d i f i c a t i oincreased
n o [ a d i l arapidly
t a n t f l uwith
idinthe
core
size of
region of the i*p-fl-'' fhe
.this
as the ratio of
-io efiectbecameparticularly important
rotationalspeed;the
The angular
increased.
diu*"t". was
the tank diameter ,n" agitator
madeon
(1969)
Smith
of Petersand
velocitydistriuution,o.urul-"nts
by
observed
trends
agreementrvith the
anchor agitators are also in
Metzneranoraytorinthattheyobservedconsiderablyflattervelocity
226
V, V. Chavanand R. A. Mashelkar
profllesfor a shearthinning fluid in comparisonto a Newtonian
fluid.
Fluid elasticitycan very significantlyinfluencethe flow patiernsaround
agitators. In fact, a fundamental approach to the understanding
of
velocitydistributionaroundagitatorsmay be to studythe rotational
ffows
aroundsimplebodiessuchas spheresand discs(Kelkaret al., 1972).
The
advantage
here is that at least the shapecomplexities
of the agitators
(which precludethe possibilityof any theoreticalstudy)are avoidedancl
the velocitydistributioncan be viewedfrom a theoreticalanele.
If one considersa third order viscoelasticfluid which "portrays
the
characteristics
of a finite shear thinning viscosityand elasiicity(mani_
festingitself in the form of finite normal stressdifferencein viscometric
flows)then for a sphereof radius R rotating at an angular velocity
f),
oneobtainsthe streamfunctionI as (waltersand savins,
lg6s;
^ (oRz
- i l + z t n .i ,' .),l1l (/ ,t- R \ t ^s,i-n, 02
n ,' -c o ,s' ( 3 . 1 )
* ( r , o'' ) : *, i l ) n '
\ e , / - - ' t ,1' ;
;)
The projcctions
of srreanrrines
on a pranecontainingthe axisof rotation
havebeenshotvnin Fis. 7.
F i g ' 7 s c c o n d a r vf r t ' * ' a r . u n d a s n h e r er o t a t i n gi n
a v i s c o e l a i t i cr i q u i r l :
(a) Nor:r'rarf.rces donrinating, (b) Centrifugal
forcesdorninaring,
(c) irirtrntaland centrifugalforcts comparable
It is eviclent that different flow situations rviil
arise f,or differe't varues
o f t h e p a r a r n e t er p R z l u 2 ( r v h i c h i s a r a t i o o f i n e r t i a l
s t r e s s e st o n o r n r a l
stresses
for a third-orderfluid).This uumber praysan important
part in
deterrnining
the florvfielclaroundspheres,
uno i, iik"ty to ituy an equa'y
rmportantpart in describing
the flow fierd around u!ituto.r, and corsequentlyin determining
the circulationand mixingtimes.
wirile analysingthe flow field around.a spheie in
detair, threecases
arisedepending
on the valueof the purur.t"i eR2pr. When ppzlszl-1,
the normal stresses
determinethe diiection of the secondaryflows in
the
entire spacearoundthe sphere.In sucha case,the fluid
moves in highry
segregated
fashionand jn fact, at pointsr : 6azlRand 0, : sin_rt/di),
q'ill
the fluid
movein closedcircresaround the axis of rotatio'
rvithout
gettingmixednith the bulk of the fluid (Fig. 7a).
When p.R2/a:
) 3, the
Mixing of VlscousNewtonian and Non-Newtonian Fluids 227
centrifugalforcesdominatethe entireflow field (Fig. 7b), andelasticitywill
haveno significanteffectin modifyingthe flow pattern. For a fluid with
L < pR2lez< 3, there will be a sphereof radius R inside which closed
loopsand consequently,
highly segregated
zones,tvill be observed.Outsidethe sphereof radiusR therervill be a centrifugalzone(Fig, 7c).
In fact, sucha flow reversalhas also beenobservedby Watersand King
(1971)in the caseof a discrotatingin viscoelastic
fluids. Giesekus
(1963)
studied the changesin flow patternsdue to the interactionof elasticity
and inertia in the caseof commercialagitatorsand found qualitatively
similar flow patterns. Feldkamp (1962) gives an account of extensive
photographicwork to illustratethesephenomena.Kelkar (1972)obtained
detailedthree-dimensional
velocitydistributionsin viscoelasticfluidsby
using the pa{ticle techniqueand confirmedthe results of the earlier
workers.
Ide and White (19'74)undertookan extensiveinvestigationof the influenceof rheologicalpropertieson flow patterns,especially
with a view to investigateits implicationsin bulk polymerisingstyreneto polystyrene.
They used spheres,discs and turbines as agitators. The observedflow
patternswerein goodagreement
rvith the earlierworkers.They alsouseda
screw-propeller
agitator.In Newtonianfluids, the actionof the screwpropeller is to push the fluid verticallyjn the tank. In the caseof viscoelastic
fluids,they found that normal stresseffectsreinforcedthe scrervpropulsion
mechanism.
Theseresultsshouldbe contrasted
with the differentbehaviour
of the screwin the draft tube systemdiscussed
later.
The studyof Ide and White (1974)is particulariyinterestingsincethey
observedflow patterns during bulk polymerizationof st1'rene.They
foundthat at low conversions,
lhe florv patternsaround the sphereand
otheragitatorsinvolveda primary axisymmetric
flow and secondary
flow
of a fluid beingdrawn in at the poles axially and expelledat
consisting
the equator.As the conversionjncreased,the secondaryflolv lleld around
a spheredividedinto two regions,one adjacentto the sphereconsisting
of closedcirculatingmotions and secondat larger distancesfrom the
sphere.Similar flow patternsrvereobservedwith other agitators.
The implicationsof suchchangesin the secondaryflolv patternson the
pelformanceof a mixing vesselshouldbe clearly appreciated,
The photographicstudy of Kelkar et al. (1972)is particularJyimportantin this
connection.Their velocitydistributionmeasurements
in viscoelastic
fluids
shorvedquite clearly that although the secondaryflorv patternscan be
changedquite significantly,the primary flow remainspracticallyunaffected. Sincethe major contributionto the shear stressesin the vicinity of
the rotating body comesfrom the variation of the primary florvvelocity
with radial distance,the power consumptionappearsto be
component
relatively unaffectedby the modifrcationsbrought in due to elasticity.
However,the secondaryflow patternsrvill have a significantefrecton
distribution,molecular-weight
residence-time
distribution (in the caseof
polymerizationreactors),mixing times, circulation times and related
228 V. V. Chavan and R. A. Mashelkar
in the
someof thesewill be discussed
kinematicallycontrolledprocesses;
sectionslater on.
appropriate
' 'Veiy
little experimentalwork has been publishedon the influenceo[
elasticiiyon flow patternswith other agitators.Petersand Smith (1969)
haveprovidedsomeexperimentaldata on the influenceof elasticityon
flow patternsaround anchor agitators.The most significantdifferences
fluids appeared
*ere iound in front of the agitaior blade.The viscoelastic
point'
stagnation
a greaterdegreeof disturbancearoundthe
to experience
indeterminate
and there was sameevidenceof buildingup of a rather
boundary layer in front of the blade. Furthermore,viscoelasticliquids
enhancedverticalcirculation'
produceda consiclerably
on the velocity distribution in vessels
elasticity
fluid
of
influence
The
studied.chavan,
stirredwith helicalagitatorshas not been extensively
wherein
observations
qualitative
some
provided
however,
et al. {1975a),
geometrlsome
for
least
at
and
elastic
liquids
highly
for
they showedthat
the angularflorv increasedrvhereasthe axial florvrvas
cal configurations,
of
damped considerably.The detailed velocity distribution.studies
dataobtaintypical
son:e
shows
8
Carreauet al. (1976)supportthis. Fig.
CMC solution (which showednegligible
ed by theseworkers.Foi a 2o//o
elasficity),the strong axial upflow and downflow can be clearly seen'
q
N
0"00
a c05
(J
qJ
:r
"(0
-)<
(u
LO
tf)
s
.a
LT
C.J
tr
-\
02
04
0.6 08
s raditts,2r t
Cimens.,;on/e-s
F'ig.8 Axial velocity distribution in a ribbon agitatedvessel
(CarreauP. J. et al.,1976)
Fluids 229
Mixing of ViscoasNewtonian and Non-Newtonian
eiastic)'the axial flow
whereasfor a \o/oPAAsolution(whichwas strongly
for
responsible
can be seento be significantlydamped' The mechanisms
the
that
is suspected
sucha phenomenonare not quite clear,however,it
of rapidlychanging
fields
in
kinematic
liquids
p.""ri^', behaviorof elastic
in
Mlre detailedstudiesare obviouslyneeded
delormationis responsible.
this area.
g,4DischargeRates,CirculationCapacitiesandCirculationTimes
an important integral hydrodynamic
From the velocitymeasurements,
q.,un.i.ycanbeeasilyobtained.Itislheoveralldischargerateorthe
passesof a fluid element
uu..ug. time elapsedb.t*."n the successive
summaryof the
tf,.ou!n the impeiler. Gray (1966)givesa comprehensive
calculationsfor the
inu.rtlgutionsof turbinesand propellers,where the
measurement'
overaliradial and axial flow are done from the velocity
3.4.1 Mle.suneueNrTECIINIQUES
Adirectmeasurementondischargeflowispossibleinthecaseof
flow is predomioantly
screwsand propellerswith draft tube where the
to circulateback to
axia1.In this mithod, insteadof allowing the liquid
cvlinderby
the draughttubeit was made to overflowinto a measuring
extendingthetubeabovetheliquidlevel,Theliquidisfedintothetube
means.A correctestimateof the axial throughputcan
by some-external
is leve]ledwith the
onty t]" obtainedif the liquid insidethe draught tube
Sometimes
rirnof the vessel,i'e', no ixtra gravitationalheadis imposed'
(1972)has
this method has been used without proper care' Chavan
rates thus rneastlred
Jir.urr"d the probableerrors. Often the discharge
is incorrect
obviously
This
alone.
impeller
to
the
to b" due
are consider.dsincethepresenceofthedraughttubeanditspositioninginthevessel
rate'
the valuesof the discharge
very nruchinfluence
completeone loop can
to
element
fluid
Th, ou.rug" time takenby the
a small pulse of an
by
ejecting
or
particle
be measuredby following a
electrolyteanrjnreasuringtt'"conductivityoftheliquidleavingthe
to calcup..ipn..V of the inrpetler,The circulaliontime' 0"' can be used
late the liquid PumPingrate,q, as
q:%
Y
(3.2\
dimensionless
rvhere z is the volume of the liquid in the tank. The
and defined
number
circulation
a
numberwhich is normallyusedis called
in termsof the inrpellerdiameteras
e,:#,
(3.3)
discharge-flow
A few comments neerl to be made about the relevance of
measurement.
230 V. V. Chavan and R. A. Mashelkar
The discharge-floiv measurements from velocity profiles are quite
accurate, but those using the experimental average-circulation time can
be questionable.It is obvious that a small parlicle follows a route depen'
d i n g u p o n i t s i n i t i a l p o s i t i o n . U n d e r s t e a d y - s t a t ec o n d i t i o n s o n e l o o p
should correspond to one circulation. Thus the circulation-time rneasure'
m e n t s s h o l r l dv e r y r n u c h d e p e n du p o n t h e i n i t i a l p o s i t i o n o f t h e p a r t i c l e
o r t h e t r a c e r , I n a n a g i t a t e d v e s s e l ,h o w e v e r , t h e r e m a y b e s o m e r a n d o m
m o t i o n ( C h a v a n , 1 9 7 2 ) . T h u s , b e c a u s eo f t h e r a n d o m n a t u r e o f t h e
i n i t i a l p o s i t i o n o f t h e p a r t i c l c b e f o r e e a c h c i r c u l a t i o n , t h e d i s c h a r g er a t e
given by equation3.1 is a statistical estimation of the overalI discharge
r a t e . I n t h i s p r o c e d u r eo f r n e a s u r e m e not f c i r c u l a t i o n t i m e o n e c a n a l s o
o b t a i n a s t a t i s t i c a ld i s t r i b u t i o n o f t h e c i r c u l a t i o n t i m e s , w h i c h a g a i n i s a n
e s t i m a t i o no f t h e v e l o c i t y d i s t r i b u t i o n i n t h e p l a n e p e r p e n d i c u l a rt o t h e
f l o w . O ' s h i m a a n d Y u g e ( 1 9 7 0 )a n d I v o n u e a n d S a t o ( 1 9 6 7 )h a v e o b t a i n e d
such informationon variousimoellers.
3.4.2 Exprnurrexrat Oesrnr',q,rroNs
G r a y ( 1 9 6 6 )h a s s u m m a r i z e dt h e i n f o r m a t i o n o n c i r c u l a t i o n t i m e s a n d
d i s c h a r g er a t e s f o r t u r b i n e a n d p r o p e l l e r a g i t a t o r s . T h i s i n f o r m a t i o n ,
h o r v e v e r p, e l t a i n s t o R e > 1 0 5 ,i . e . , f o r t u r b u l e n t c o n d i t i o n s w h e r e C ; i s
i n d e p e n d e no
t f Re. Gi-ey has compared the circulation numbers on the
b a s i so f e q u a l p o w e r c o n s u n r p t i o n .A s a n e x a m p i e , o n e c a n n o t e t h a t a
r o t a t i n g d i s c r e q u i r e d a b o u t 3 0 0 t i m e s m o r e e n e r g yt o g i v e t h e s a m e d i s charge rate as an eight-bladed turbine. The circulation numbers for
p r o p e l l e r sw e r e t y p i c a l l y i n t h e r e g i o n o f 0 . 4 t o 0 , 6 , w h e r e a s f o r t u r b i n e s
they rverein the region of 0.6 to 2. Since we are mainly concerned with
the mixing of high viscositl.liquids, such information is probably not
t o t a l l y r e l e v a n tt o u s .
T h e l i t e r a t u r e o n c i r c u l a t i o n c a p a c i t i e s( a n d t i m e s ) f o r h i g h v i s c o s i t y
l i q u i d a g i t a t i o n r v i t h h e l i c a l i m p e l l e r s h a s b e e n w e l l s u m n r a r i z e db y
C h a v a n ( 1 9 7 2 ) . T h e c i r c u l a t i o n n u m b e r v a r i e s b e t v v e e n0 . 4 t o l f o r
r i b b o n s a n d b e t w e e n 0 . 0 I t o 0 . 5 f o r s c r e l v si n a d r a f t t u b e . T h e s e a r e
i n d e p e n d e n to f R e y n o l d s n u m b e r ( f o r R e < 1 0 o r R e < 5 0 ) b u t a r e
strongly influenced by the geometrical configuration. Studies with a
l a r g e n u m b er o f p u r e l y 'v i s c o u sl i q u i d s r e v e a l e dt h a t c i r c u l a t i o n n u n r b e r s
were not at alI influenced by shear-thinning characteristics.The
o b s e r v a t i o n sl v i t h s t r o n g l y e l a s t i c l i q u i d s , h o w e v e r , s h o w e d t h a t t h e
circulation was damped as a result of elasticity and the circulation
numbers were lowered quite significantly. Chavan and Ulbrecht (1974)
have provided a correlation which adequately described the data for
p u r e l y v i s c o u s f l u i d s ; w h e r e a s C h a v a n e t a l . ( 1 9 7 5 a )h a v e d e s c r i b e da
methodto correlatethe data on elastic liquids fcrr a given geometrical
c o n f i g u r a t i o n .T l i e p i r p e r b y C h a v a n e t a l . ( 1 9 7 5 b ) i s a l s o o f i n t e r e s t .
C i r c u l a t i o n t i r n e sf o r o t h er a g i t a t o r s o p e r a t i u g u r d e r l a m i n a r c o n d i t i o i t s
i n v i s c o e l a s t i cl i q u i d s h a v e n o t b e e nm e a s u r e d a s y e t . T h e p e c u l i a r f l o w
Mixlng of ViscousNewtonian and Non-Newtonian Fluids 231
reversalsand the two-zonevelocitydistributionregionsaf ising out of the
of elasticityshouldstronglyinfluencethe circulationtimes. The
presence
enhancedvertical circulation in elastic liquids reported by Petersand
smith (1969) for anchor agitated vesselsand by Feldkamp (1962)for
propellerswould imply that the circulation numbersmay increaseon
as in the caseof helical agitaaccountof elasticityrather than decrease
tors. Detailedstudiesexploringtheseaspectswould be highly desirable.
Specificcomments need to be made concerning the influence of
circulationtimes and circulation time distribution in polymerization
reactors.An interestingstudy in this contextis mentionedby Olson and
Stout (1967)with specificreferenceto condensationpolymerizationof
bifunctlonalpolymers.The study is with a specificreferenceto a turbine
impellerwtriChproducescirculationloopsof the polymer; the viscosities
of ihe polymermass are so high that material in the circulation loops
The water of condensationis removedby diffusive
remainssegregateci.
transportto the surfaceof the polymerizingmass' The contributionof
the diffusiveprocessto the rate of reactionis approxinatedby assun.ling
that the water concentfationis proportionalto the circulationtime of the
The effectof mixingupon the productmolecularfluid elements,
segregated
*.ig[t distribution has been found and it has been showt] that the
assumedmixing pattern significantli'alters it' Indeed' if the circulation
the polymermay be expected
time is low at the end of polymerization,
to have poor uniformity. This exampleclearly illustratesthe role of
systems.
circulationloopsin batchpolymerization
3.5 PorverConsumPtion
The spaceavailablehere is too limited to cover the enormouspower
data whichhavebeenpubiishedoverthe years;nor is it the
consumption
review.Thepublishedreviews(Uhl and Gray, 1966;Wohl,
of
this
purpose
and
Beli, 1967)give a good appraisalof such infornation.
iV68; f"nny
of the power consulnption
can
makea reliableestin1ate
one
At present,
arrangements
geometrical
of
number
for
a
and
for variousimpellerdesigns
availableon
also
is
of
great
deal
information
for Nervtonianliquids.A
here a
present
We
rvill
multiphase
systems.
liquidsand
non-Newtonian
ready-made
giving
a
thau
rather
viewpointof the analysisof the data,
designmethod.
TEcHNIQUES
MEesunnuENT
The methodsof measuringpower input to a stirred vesselhave been
by Holland and chapman (1966).Thesemethodsfall into four
discussed
groups.
1. Measuringthe power suppliedto an electricmotor by measuring
current, voltage and polver factor and correctingthe copper and iron
lossesand gearbox {riction losses.
2. The reactiontorque on a motot stator is nteasuredby suspending
3.5,I
232 V. V. Chavan and R. A. Mashelkar
suspension
and the forcerequiredto hold
the entiremotor on a torsionless
it stationaryis measured.
3. The mixing vesselis nrountedon a frictionlessturntable and the
forcerequiredto hold the tablestationaryis measured,
4. The torquein the mixer shaft is obtainedby measuringthe twist
overan intentionallyweakpart of the shaft.
Method (l) is not preferredfor researchwork as it is difhcult to get
precisevaluesof the power unlessthe lossesare correctly estimated.It
seemsthat methods(3) and (4) are generallyused for simplicity and
accuracy.
3.5.2 D,cre AI'IeLvsts
Sincepowerconsurnptionwas first measuredfor rotating impellersin
usingdimesional
the late lgth century,thedatahaveoftenbeencorrelated
analysis.It may be expectedto be a functionof the follorvingvariables:
(l) Geometricalvartables: Impeller diameter(d), tank diameter(l),
liquid depth(n)
(ti) Material properties: Density (p), viscosity{pr)
(iii) Processt'ariables; Rotationalspeedof the impeller (l/), gravita(g)
tional acceleration
can simplybe writtenas,
The relationship
(3.4)
P :f(d, D,lt, p, p, N, g)
gives
analysis
Dirnensionai
dN2\",
P
(3's)
^,,ld2NP\",1
i
,ts,,,t3p:- / l?/-
r v h e r eK i s a c o n s t a n tf o r p a r t i c u l a r s e t o f g e o m e t r i c a vl a r i a b l e sa n d c o u l d
be obtainedas
*: t,(-3)"(4\"'
'\a
| \t)
(r.6)
The dimensionlessgroups (Pldsll3e), (d2NplD and (d!'lzlg) are known as
power number (Po), Reynolds number (Re) and Froude number (Fr),
respectively.
T h e a b o v ea n a l y s i si s v a l i d o n l y u n d e r t h e f o l l o w i n g c o n d i t i o n s : ( a )e i t h e r
a single liquid or two miscible liquids having similar properties are
present in the vessel, (b) the temperature changes due to energy
dissipation are so small that variation in fluid properties becauseof the
temperature changesis negligible, (c) the florv behavior of the liquid can
b e c h a r a c t e r i z e db y a s i n g l ep a r a m e t e r ,i . e . , v i s c o s i t y .T h u s t h e l i q u i d i s
Newtonian.
The values of E, e1 and e2 have been determined by a number of
different
workers for several different geometrical variables. e1 aSSUIIles
v a l u e si n t h e r a n g e o f - 1 t o 0 d e p e n d i n gu p o n t h e o p e r a t i n g r a t l g e o f
R e y n o l d sn u m b e r s . l t h a s b e e n f o u n d t o h a v e a v a l u e o f - I b e l o w a
certain critical value of Reynolds number. This critical Reyrrolds number
Mixing of ViscousNewtonian and Non'Newtonian Fluids 233
varies between l0 to 100 and it seemsto be influencedby the geometryof
the system. At very high Reynolds numbers (Re > l0a) power number
* I is
becomesindependent of Reynolds number. The region where e1 is
c o m m o n l y u n i e r s t o o d a s l a m i n a r r e g i o n . I t m a y b e p r o p e r t o d e s i g n a t ei t
as creeping flow region as the majority of the non-viscometricflow situations give rise to such relationships only when the inertia terms in the
Naviei-Stokes equations are dropped. The effect of Froude number on the
power consumption would be apparent only when somepower is consumed
in producing significant waves on the surfaceof the liqgid or in sustaining
u uort*" in liquid around the impeller shaft. It is found that for unbafred
vesselsthese effects are negligible below Reynolds nnmber of 300. This
limit is exceededfor Re > 103when the baffie system is adequateorlvhen
the impeller is suitably off-centred. Nagata and Yokoyama (1955) state
that the effect of Froude number is so negligible as to be indeterminate
except by a very accurate dynamometer. When such effectsare present, a
praciical method of analysis is available (LJhl and Gnay, 1966 and
Skeliand,1967).
F o r s t a n d a r d d e s i g n sK a n d e 1 i n e q u a t i o n 3 . 5 c a n b e e a s i l y o b t a i n e d .
H o l v e v e r ,t o o b t a i n a g e n e r a l i z e dc o r r e l a t i o n ,a s e r i e so f e x p e r i m e n t s w i t h
wide variation in geometrical variables will be required. Often, the
g e o m e t r i c a lc o r r e l a t i o n sa r e o b t a i n e d w i t h o n e o r t r v o v a r i a b l e sa n d e v e n
when these variables are not independently varied, their dependenceon
other variables is negLected.This is incorrect mathematically as well as
p h y s i c a l l y .T h e m o s t i m p o r t a n t v a r i a b l e s ,h o l v e v e r ,c a n b e o b t a i n e d l v i t h
p r o p e r p h y s i c a l r e a s o n i n go r b y c o n d u c t i n gs p e c i a le x p e r i m e n t s '
For inelastic non-Newtonian fluids, dinensional analysiscan be perf o r m e d o n l i n e s s i m i l a r t o t h o s e f o r N e r v t o n i a nf l u i d s . T h e r e s u l t sw i l l
depend upon the specificmodel chosen.Thus, for instance, \\'e obtain for
a polver-law model
and fol an Ellis model
or: ,r(q#,r)
( 3 . 7)
,,: o,(#,{#,")
(3.8)
H e r e ( d 2 N 2 - ' p l K ) a n d ( d 2 N p l p d a r e t h e c o r r e s p o n d i n gR e y n o l d sn u m b e r s '
The dimensionlessnumber (l/po/'o) was introduced by Kelkar et a1.(1972)
and called a viscositynumber. Obviously, wirh the aid of dimensional
a n a l y s i s a l o n e o n e c a n n o t g e t q u a n t i t a t i v ei n f o r r n a t i o na b o u t t h e i n v o l v e m e n t o f t h e d i m e n s i o n l e s ps a r a m e t e r so f t h e r h e o l o g i c a lm o d e l .
The first attempt to incorpcrate the paranleters which describe the
deviation of any inelastic fluid from the Newtonian characteristicsinvolve
t h e u s e o f r e s u l t so b t a i n e di n a p i p e f l o r v .T h i s a n a l o g yw a s f i r s t s u g g e s t e d
b y M e t z n e r a n d O t t o ( 1 9 5 7 ) .I t h a s b e e ns h o w n b y S k e l l a n d( 1 9 6 7 )t h a t s u c h
a R e y n o l d s n u m b e r d o e s n o t g i v e a L r n i q u ep o \ \ ' e rc u r v e f o r a w i i i e r a n g e
o f f l o w b e h a v i o u r i n d e x . M i t s u i s h i a n d H i r a i ( 1 9 6 9 )s t a t e t h a t t h i s i s n o t
234 V. V. Chavan and R, A. Mashelkar
a failure of the analogybut of the power-lawapproximation.Theysuggest
an alternativemethod involving the use of Ellis and Sutterby models.
Chavan(1972)has conclusivelyshownthat evenwith such models,pipeflow analogydoesnot offer a uniquecurve,evenin the laminar region.
In order to obtain a powercorrelationin termsof the rheologicalparameters,the knowledgeof shearrates(at leastin the immediatevicinity
In an extremelysignificantcontribution,
of the impeller)is necessary.
Metznerand Otto (1957)showedthat the averageshearrate in the vessel
can be assumedto be directly proportional to the rotational speedat
leastin the laminar region(seeMagnusson(1952)for relatedwork). This
to obtain the
shearrate can be obtainedas follows.First, it is necessary
plot of Po-Re for Newtonian fluids for the systemunder consideration.
Then the power number is calculatedfrom the power data for the nonReynolds
Newtonianliquids.For this power number the corresponding
numbercouldbe obtainedfrom the Newtonianplot. The averagevjscosity
givenby the Reynoldsnumber would givethe correspondingshearrate
(iu") from the viscometricdata. A relationshipsuchas
iu":
:
j
ll
(3.9)
k"N
has beenobtainedby variousworkersfor variousimpellersand the values
liquids by
of k" havebeen reported.For the agitationof pseudoplastic
paddles
l0
turbines,propellers
k" assumesa value betiveen and t3.
and
For dilatantliquidsMetzneret al. (1961)obtaineda linearrelation(up to
z : 1 . 5 ) .T h e v a l i d i t yo f t h e l i n e a rr e l a t i o n s h iipn e q u a t i o n3 . 1 w a s ' s u b by Metznerand Taylor (1960)(seesec.3.1). From
established
sequently
the availablejnlormationit appearsthat one can accepta linear relationship undercreepingflow conditions.Holever, thereis someuncertainty
about the extensionof this methodfor higher Reynoldsnumberand also
about whetherft" can have a universalvalue even in the lorv Reynolds
et al., l96l) showthat
numberrange.The resultson turbine (M.etzner,
Reynoldsnumber*, : {#,
the generalized
(wlrerep, : F"(i",\ obtain-
data)can givea uniqu"lu.r, up to Re : 160(iaminar
ed from viscometric
regionendsat Re : 191within f 20 percent.However,this is notobserved in everycaseand Skelland(1967)commentsthat suchextensionis
not justified on accountof insufficientevidence.The constantof proportionality,ft,, canotherwisebe a constantfor a particularpseudoplastic
similar system.In other words,k, should be
liquid and geometrically
generallya functionof geometryand rheology.The problemscan best be
overcomeby consideringa power-lawrelationship.In creeping-flow(or
laminar)regionone can obtaina relationship
Po: r(k"),-,(t$-:t)-'
(3.l 0)
From experimentaldata on power consumptionon Nervtonianand
(asdescribed
inelasticliquidsand from the florvcurve,k, can be measured
(y.") will be known to give K and
earlier).Also, the rangeof shear-rales
Mixing of ViscousNewtoniatt and Non-Newtonian Fluids 235
n, calderbank and Moo-Young (1959)and Beckner and smith (1966)
have obtained/c"in this way and relatedit to the geometryand rheology.
After knowing the operating rotational speedsand the values of ko
from literaturean approximateestimateof the shearratescan be obtain'
ed. When k" is dependenton n, a trial and error proceduremay be needed
to estimatekr. Horvever,a rough estimationcould be possiblefrom the
Thus, for exanple,for helical
previousinformationin similarsituations.
rotationalspeedsrangefrom
if
then
to
100;
from
l0
impellersk" varies
rates
of
shear
Of I to 1000sec-tmay be
range
then
a
O.i to l0 sec-l
appropriate.
Noi" that equation3.9 doesnot give an averag€shear-ratein the entire
thus obtainedcan only characvesselin all the regimes.The shear-rates
terizethe flow near the impellerand that too only in the laminar region.
at the tip of tbe impellerare
Even in the laminar region,the shear-rates
as much as seventimeshigherthan thosegivenby equation3.9.At higher
Reynoldsnumbers,the maximum shear-ratesin the vorticesbehindthe
bladehavebeenfound to be of an order of magnitudehigher(Van't Riet
and Smith,1973).
The power consumptiondata for Binghamplasticfluidscanbe correlat(seeNagataet al.,
ed by usingsimpledimensionalanalysisconsiderations
1970).
The influenceof fluid elasticity on power consumptionis negligible
under creepingflorv conditions(Kelkar et al., 1972;Chavan, 1972and
Riegerand Novak, Ig74). At higher Reynoldsnumbers,however,i1
the secondaryflowsand one obtainsa
appearsthat elasticitysuppresses
in
comparisonto a purelyviscousliquid
porver
consumption
reductionin
(Kale et al., 1973).
3.6 Mixing Times
criterion,basicallydesignedto
Mixing time is a purely experimental
gives an idea about
of agitators.It essentially
comparethe performance
A
homogenization.
of
required
degree
a
carry
out
to
required
time
the
the
for
comparing
good
basis
can
a
time
form
mixing
measured
properly
towardsthe completionof mixing.
contributionof the hydrodynamics
TecHNlQurs
3.5.1 MEnsuntnaaNr
Primarily,a smalIquantityof liquid differingfrom the bulk in either
concentrationof a traceror temperatureand otherwisehaving the same
physicalpropertiesis mixed rvith the bulk liquid in the vessel'The
changesin the tracer concentration(or temperaturein the thermal
involving
methods)can be measuredas a functionof time. Techniques
(Schlieren
indices
refractive
absorption
spectrometry,
conductivity,atomic
nlethod)or light intensity(dye additionor decolorizationusing fast
ale flequently
thiosulphate)
or iodilte-sodiurn
suchas acid-base
reactions
(Coyle
observed
be
can
visually
used.The c,olorizationor decolorization
236 V. V. Chavanant{ R. A. Mashelkqr
et al., 1970).[n visual methods, there can be subjectiveerrors and hence
these should be avoided. The instrumental methods provide point values
of dillerent quantities. The selectionof a point in the vesselfor measurements and spotting the end-point are two major difficulties. Complete
mixing on molecular level in the entire vessel is diltcult to achieve and
diffficuit to measure. The merits and demerits of various techniques of
measurementhave been discusseclby Ford et al. (1972).
3.6.2 D.qre ANelvsts
The change in the selectedintensive property (concentration, temperat u r e , r a d i o a c t i v i t y ) a s a f u n c t i o n o f t i n t e i s m e a s u r e dd u r i n g m i x i n g t i m e
experimentation. Ifl mixing time is so defined that the value of the
m e a s u r e dq u a n t i t y i s , s a y , v ii t h i n l 0 t o 2 5 p e r c e n t o f t h e f i n a l l y e x p e c t e d
a v e r a g ev a l u e , t h e n a r a t i o n a l i n t e r p r e t a t i o no f t h e d a t a c a n b e o b t a i n e d
(Hoogendoorn and Den Hartog 1967, Chavan 1972). The phenomena
such as molecular dtffusion or conduction will obviously influence the
mixing time. Horvever, it seemsthat the first 75 to 90 per cent change is
mainly controlled by hydrodynamics (such as convectionor shear or
t e n s i l ed e f o r m a t i o n o r t u r b u l e n c e ) .T h e o v e r a l l c h a n g e sd u e t o d i f f u s i o n
r v i l l b e r e l a t i v e l ys l o w i n t h i s p e r i o d . I t w i l l a s s u m ea n i m p o r t a n t r o l e a s
t h e t r a c e r i s d i s p e r s e dm o r e a n d m o r e i n t h e b u l k . [ n t h e f i n a l s t a g e so f
t h e p r o c e s si t w i l l b e t h e m a j o r c o n t r i b u t i n g f a c t o r .
E x t e n s i v ei n f o r m a t i o no n m i x i n g t i m e s h a s b e e nr e p o r t e db y G r a y ( 1 9 6 6 ) '
a n d H o a n d K w o n g ( 1 9 7 3 ) ;t h e l a t t e r a u t h o r s h a v e g i v e n a p a r t i c u l a r l y
u s e f u ls u m m a r yo f t h e p u b l i s h e dw o r k o n m i x i n g t i m e s f o r b o t h N e w ' t o n i a n
a n d v i s c o i n e l a s t i cn o n - N e w t o n i a u f l u i d s . C h a v a n ( 1 9 7 2 )h a s d e s c r i b e dt h e
e f f e c to f n o n - N e l v t o n i a nb e h a v i o r o n m i x i n g t i m e s i n v e s s e l sa g i t a t e dr v i t h
helical impellers,His findings could briefly be summarizedas follows.
T h e n - r r x i n gt i m e s a r e i n v e r s e l y p r o p o r t i o n a l t o t h e r o t a t i o n a l s p e e d f o r
v e r y l o u ' a n d v e r y h i g h R e y n o l d sn u m b e r s . I n b e t w e e n ,t h e y v a r y w i t h a
n e g a t i v e p o w e r o f R e y n o l d s n u m b e r ; t h e e x p o n e n t si n g e n e r a l b e i n g
b e t l v e e n- 0 . 3 a n d - 2 . T h e l i r n i t i n g R e y n o l d sn u m b e r , f o r w h i c h l { d , , ,i s
c o n s t a n t ,i s m u c h h i g h e r f o r t h e a x i a l f l o w i m p e l l e r s t h a n f o r t h e o t h e r
i m p e l l e r s .I n t h i s r e g i o n , t h e s h e a r t h i n n i n g p r o p e r t i e s d o n o t i n f l u e n c e
t h e d i m e n s i o n l e sm
s i x i n g t i m e ( n o t e t h e s i m i l a r c o n c l u s i o n so n c i r c u l a t i o n
t i n e s d e s c r i b e di n s e c .3 . 4 ) .
only recently certain papers have appearedwhich examine the influence
. h a v a ne t a l . ( 1 9 7 5 a ,
o f f l u i d e l a s t i c i t yo n m i x i n g t i m e s i n a g i t a t e dv e s s e l sC
1 9 7 5 b )f o u n d e x p e r i m e n t a l l yt h a t m i x i n g t i m e s i n c r e a s e ds i g n i f i c a n t l y a s
a r e s u l t o i f l u i d e l a s t i c i r yw h e n h e l i c a la g i t a t o r sw e r e u s e d .U l b r e c h t ( 1 9 7 4 )
h a s n i c e l y s u m m a r i z e dt h i s w o r k . I n d e e d , t h e o b s e r v a t i o n sm a d e i n t h e
f o r e g o i n gc o n c e r n i n gt h e d a m p i n g o f a x i a l c i r c u l a t i o n a s a r e s u l t o f f l u i d
e l a s fi c i t y d o s u g g e s ta n i n c r e a s ei n m i x i n g t i m e s , a n d t h e e x p e r i m e n t a l
l i n d i n g s o f C a r r e a u ( 1 9 7 6 )a l s o b e a r t h i s o u t . O n t h e o t h e r h a n d , a n
e n h a u c e dr r e r l i c a lc i r c u l a t i o n a s a r e s u l t o f f l u i d e l a s t i c i t y i n t h e c a s e o f
Mixing of Viscous Newtonian antl Non-Newtonian Fluids 237
mixing
anchoragitatedvessels(smith, 1970)gives considerablyshortef
No
reported.
been
have
timesi soiretimesreductionsby a factor of four
be
thus
can
generalconclusionsregarding the effect of fluid elasticity
carefully
be
to
have
i.u*n and the detailld hydrodynamic changes
examinedin eachcase,
The recentstudyby Yap et al. (t97s) is particularlyilluminatingsofar
fluids are concerned.with special
as problemsof scale-upfor viscoelastic
to ribbon ugitutort they showedthat for geometricallysimilar
reference
volume in
rnixersthe criterionto be observedis that the power per unit
the mixersbe equal for equal degreesof homogeneityof the mix. This
observationis basedon the inverserelation betweenthe power number
and the Reynoldsnumber anrl the fact that the number of impeller
is constant.Theseconditions
revolutionsfor a givendegreeof homogeneity
nonviscous(pseudoplastic)
purely
as
are satisfiedfor Newtonianas well
viscoelastic
highly
for
differences
Newtonianfluids,but thereare significant
in
liquids. As a matter of fact it is found that the nixing efficiency
of
case
in
thc
that
than
liquidsis almost2 to 5 times less
viscoelastic
liquids
equivalentNewtonianfluids.Furthermore,in highly viscoclastic
-^-}
d
r.1Y
91"
/?(Dia.)D)
\&t/
Fig. 9 Comparisonof mixer performance(Hoogendoornand
Den Hartog' 1967)
l Turbine + Baffles
1a Turbine
2 3 Inclined Blade Paddles
3 3 Inclined Blade Paddles* Draught Tube
3a I Inclined Blade Paddle + Draught Tube
4 Screw
5 Screw+ Draught Tube
6 Ribbon
7 Propeller A I Draught Tube
8 PropetlerB + Draught Tube
8c ProPeller B
9 Anchor
238 Y. V. Chavan and R. A. Mashelkar
mixing emcietcy does not appear to be very sensitive to the geometry.
Further work is decidedly warranted in this area, since most polymer
melts handled in practice show significant viscoelasticphenomena,
The information on mixing and power consumption can be suitably
combined to provide a rational basis for comparison of performance of
different agitators. A typical comparison has been shown in Fig. 9 where,
a t a p a r t i c u l a r m i x i n g t i m e , t h e p o w e r r e q u i r e d t o a c h i e v et h e d e s i r e d
No such comparison has been
degreeof mixing increaseswith (0!,,Ply.D3).
compiled for mixing of non-Newtonian fluids, and it will be clearly
desirable to do this.
It should be emphasizedthat we have consideredthe mixing of only a
single fluid. When blending of two fluids with widely different rheological
properties is considered,the situation becomesmuch more complex. Only
a start on studying this important problem has been now made (see,e.g.,
Ford and Ulbrecht, 1975).
4.
Continuous Mixing
In the foregoing, we have described some aspects of mixing related to
a d e s c r i p t i o n ' o fg r o s s p a r a m e t e r ss u c h a s t h e p o w e r c o n s u m p t i o n ,m i x i n g
time, circulationtime, etc. Such information is useful from the point of
v i e w o f e s t i m a t i n g t h e s u i t a b i l i t y o f a g i v e n m i x e r c o n f i g u r a t i o nf o r a
g i v e n p u r p o s e .I n d e e d , t h i s i n f o r m a t i o n n r a y b e e n t i r e l y a d e q u a t e w h e n
non-reactive processesare to be carried out (such as blending of two
m i s c i b l en o n - r e a c t i n gf l u i d s ) . H o w e v e r , w h e n o n e c o n s i d e r st h e u s e o f a
mixing device for thc purpose of contacting two or more chemically
r e a c t i n gc o m p o u n d s ,t h e s i t u a t i o n b c c o m e sm u c h m o r e c o m p l e x . I n d e e d ,
t h e n : i x i n g d e v i c en o w h a s t h c j o b o f p r o v i d i n g o p t i m u m t i m e - c o n r p o s i t i o n , t i m e - t e m p e r a t u r ea n d t i m e - s h e a re n v i r o n m e r . r tS. u c ha n e n v i r o n m e n t
t h e n d e p e n d su p o n t h e f l u i d r n e c h a n i c si n t h e d e v i c e ,a n d r v eh a v ee m p h a s i z e di n s e c . 3 . 3 a s t o h o w c o n p l i c a t e d t h i s p r o b l e m o f u n d e r s t a n d i n gt h e
d e t a i l e d f l u i d m e c h a n i c si s . I { o r v e v e r , o v e r t h e y e a r s , v a r i o u s m e t h o d s
have been devised to account for the modification of reactor behav.iour
c a u s e db y t h e m i x i n g a n d c o n t a c t i n g p a t t e r n s . W e s h a l l b r i e f l y r e v i e r v
t h e s en o r v . I t s h o u l d b e e m p h a s i z e dt h a t t h e b a s i c a s p e c t sr e l a t e d t o t h e
field of nou-ideal flow have been well surnmarizedin somc cxcellent
r e v i e w ss u c h a s L e v e n s p i e la n d B i s c h o f f( 1 9 6 3 ) , B i s c h o f f ( 1 9 6 6 ) , N e u m a n
( 1 9 7 4 )a n d i n t h e r e c e n t l y r e v i s e d t e x t b o o k o f L e v e n s p i e l( 1 9 7 2 ) . C o n s e quently, the fundamental deflnitions will not be reviervedin depth. What
will be considered, however, are the special problem areas concerned
w i t h t h e h i g h t y v i s c o u sl i q u i d s .
4.1
ResidenceTime Distribution
The developmeno
t f t h e c o n c e p to f r e s i d e n c et i r n e d i s t r i b r : t i o n a t l e a s t
in tlie modern chenricalengineeringera dates back to Danckrverts'classic
lvlixing of Yiscous Newtonian and Non-Nawtonian Fluids
239
paper (Danckwerts, 1953); although earlier papers could be found out'
ibi concept of residencetime distribution avoids the need to know the
exact flow pattern in the vessel,but seeks to know only the information
regarding how long the moleculesstayed in the vessel,or in other words
the residencetime distribution (RTD) of the fluid. A stimulus-response
technique can be obviously used to obtain information on RTD.
T h e t w o i d e a l c a s e so f p l u g f l o w a n d p e r f e c t m i x i n g a r e o b v i o u s b u t
for jntermediate cases, several age distribution functions have to be
d e f i n e d i n o r d e r t o d e s c r i b eq u a n t i t a t i v e l y t h e a g e d i s t r i b u t i o n o f t h e
'C
fluid. The definitions of the well-known E, F and curves follow simply
and a standard textbook (such as Levenspiel, 1972) should be referred
t o u n d e r s t a n dt h e p h y s i c a l m e a n i n g o f t h e s e . I n t h e f o l l o w i n g w e s h a l l
discussbriefly the methods by rvhich such information can be obtained,
the methods of interpretation and the manner in which this information
should be used.
4. l.l
MresuReltBNr N{ErHoDS
I t i s u n d e r s t a n d a b l et h a t i n p l u g f l o w c o n d i t i o n s R T D w i l l c o n s i s to [ a
delta function, since all the nolecules pass through with the same residence time. If mixing takes place, then there is a spread in RTD. In case
s o m em a t e r i a l s t a y s f o r l o n g ( d e a d s p a c e )o r p a s s e st h r o u g h v e r y q u i c k l y
( b y p a s s i n g ) ,w e h a v e a g r e a t e r s p r e a d .T h u s , i f w e i n j e c t a t t h e i n l e t o f
t h e m i x i n g d e v i c e a n d o b s e r v et h e t r a c e r b e h a v i o u r a t t h e o u t l e t , w e g e t
i n f o r m a t i o no n m a c r o m i x i n gw h i c h r e l a t e st o t h e d u r a t i o n o f t h e s t a y o I
v a r i o u s f r a c t i o n s i n t h e v e s s e l ,b u t n o t t o w h a t t h e y d i d i n t h e v e s s e l .
The latter part of the information is called micromixing and rvill be dealt
rvith in the next section.
B a s i c a l l y ,a s t e p c h a n g ei n c o n c e n t r a t i o n a t t h e i n l e t i s w h a t i s u s e d
b u t o t h e r s i g n a l ss u c h a s a s i n e w a v e ( K r a m e r s a n d A l b e r d a , 1 9 5 3 ) a n d
evenrandom noise (Angus and Lapidus, 1963) may be used. It is also
p o s s i b l et o u s e u n s t e a d yb e h a v i o u r r e a c t i n gs y s t e m s( L e l t i , 1 9 6 5 )o r s t e a d y
state measurementsat different flow rates or temperatures(Hoare, 196l).
H o r v e v e r ,a n i n e r t t r a c e r h a s o b v i o u s a d v a n t a g e so v e r a r e a c t i n g t r a c e r .
A n o v e l m e t h o d i s t h e o n e d e v e l o p e db y G o l d i s h e t a l . ( 1 9 6 5 )w h o u s e d a
s p e c i a l t r a c e r ( c o m p l e t e l ys o l u b l e i n t h e t e s t f l u i d ) , l v h i c h i s c o l o u r i e s s
until activated by a flash photolysis. This means that a flash of light
producesa pulse oI colored material without actually having to physically
inject a tracer into the sYStem.
We have discussedthe problems of choosing a suitable tracer for the
m i x i n g t i m e m e a s u r e m e n t sj n s e c . 3 . 6 . T h e p r o b l e m s o f t h e c h o i c e o f
tracer in the study of RTD when handling non-Newtonian ftuids are
quite similar. In a certain range, at least, the rheological properties can
be very sensitive to electrolyte concentration, pH, temperature, etc'
Consequentlya
, g r e a t c a r e u s u a l l y n e e d st o b e e x e r c i s e di n t h e c h o i c e o f
appropriate tracot.
240 V. V. Chavanand R. A' Mashelkar
of the tracer (without disturThe points concerning the rapid injection
measufement at the outlet
bing the normai flow frtiern) anO a correct
are described by Bischoff
this
are imporrant. some ploilr.r, concerning
.
(' 1 9 6 3 )a n d W h i t e ( 1 9 6 2 ) .
the tracer tun.tl! l n i f i c a n t l y
measuring
and
of
injecting
technique
Th;
experiments. The
influence the interpritatio"n of the residence-time
(
1
9
7
1 )a r e p ^ a r t i c u l a r l y
l n d T u r n e r ( 1 9 7 0 )a n d T u r n e r
p a p e r sb y L e v e n s p i e a
m
e
t
h
ods of injection'
two
ifiu*inuiing in tiis regard. They considered
the flow through
to
proportion
the first in which the iracer is added in
in which the tracer is
each point in the injection pl-ane.and.sicond
plahe' They also considered two
added uniformly acros, the injection
reading is taken
, ne in which the mixing-cup
m e t h o d so f m e a s u r e m e n t o
over the cross-sectional
and second in rvhich the average concentration
planeofthemeasurementismeasured'Thedatainterpretationwill
that has been
upon the corlbination ol injection-measurement
l"f.na
used.
4 . 1, 2
lxtsRpRlrnttoN oF Respoxsn
oncelheRTDcurveisobtained,agrossdeviationfromtheidealized
easy to detect' However' consideplug no* or completely mixed models is
r a b l e i n | o r m a t i o n c a t r b e e x t r a c t e d f r o m t h e R T D c u r v e . T h e d e(1972)'
tailsof
given by Levenspiel
unofyring the RTD curves have been
H i n r r u e l b l a u a n d B i s c h o f f ( 1 9 6 8 ) a n d o t h e r s . N a o r a n d S h i n n a r ( b1y9' 6 3 )
d e t e r m i n i n g d e a d s p a c ea n d
h a v e p r o v i d e d f a i r l y s e n s i t i v em e a n so f
pu,,inglromthellTDcurr.e.BischoffandMcCracken(1966)lial'econsi.
i' relation to actual problems'
l.oO ir. utility of thesc dillerent methods
in predicting the performThis information should idcally be helpful
i
It
I
I
I
t
I
I
I
anceofachenricalreactor'Whethertl-risispossibleorngt,vern
ynuch
c o n s i d e r e di s l i n e a r o r o n b
e
i
n
g
s
.
v
s
t
e
m
t
h
e
w
h
e
t
h
e
r
u
p
o
n
A.p.na,
p r o c e s s e si t - f o i l o w s t h a t m e r e
linear. By additivity property of linear
to depict the behaviour
be'adequate
stinrulus responseinioilnution should
ofavesselaSal.eector,aslongasthereactionrateislinearinconcent r a t i o n , H o t v e v e r , l b r n o n . l i n e a r s y s t e m s , w e r e a l l l . n e e d p o i n t - t o -isp o i n t
the exact history of each molecule
information becausea knowledge of
aspectwill be dealt rvith in
n e c e s s a r y( m i c r o m i x i n g i n f o r m a t i o n ) ' T h i s
a later section'
of the tracer data involvesthe
A further point in the interpretatton
der,elopmentofflownrodels.Thisrrsuallyinvolvesthedefuritionofaflow
modelcontainingcertainparan]etersandthencorrelatingtheseparameters
intermsoftheSyStemvariables'Hope|ully,thesecorrelationscanthen
s i t u a t i o n s( w i t h o u t a c t u a l l y h a v i n g
b e u s e dt o p r e d i c t t h e b e h a v i o ro f n e w
thetracerdataonthenewsituation)'Thisishelpiulindesignpractice.
I t i s i m p o r t a n t t o n o t e h e r e t h a t t h e p a r a r r r e t e r s s h o u l d h a v e s o m ei np ha y s i c a l
parameter
As pointed out by Bischoff (1966)' a volunte
**"i"g.
flowmodelnrayhaveavaluelargerthanthatoftheactualph-vsical
Mixing of Viscous Ne.wtonianand Non-NewtonianFtuicls
241
vorumeof the reactor,and althoughthis may not cause
a seriousprobrem
in mere curve fitting, it racksthe desirableflature of being
uut" to a"ripher the actualoperationof the reactorbeingstudied.
A numberof models(someof the important ones are
reviewedin the
ensuingdiscussion)
may be formuratedon the basisof the tracerinformation. Basically,this consistsof breakingup a rargemodel
into smailer
regions(rvhichmay or may not be on the basis oi the
feel for the fluid
mechanicsin the system),each of which may be represented
by a dispersionmodel, by-passing,dead space,etc, The puru*.t.r,
introduced
representflowsto, sizesof and extentof mixing in,
variousregionsand
do a good job in termsof curvefitting. However,
the same tracer informationcan' at times,give rise to severardifferent
flow modersor paralneters'It thus appea-rsthat, wheneverpossibre,
it is importantto base
the modelon physicarreasoning,so rhai the parameters
are physica'y
rueaningful.
we shallnorvconsiderin so'redetairthe experimentar
and tircoretical
work which has appearedin the past on the aspects
of RTD in u"rf
viscoussystems,especially,the non-Newtonjan,yri.ntr.
4.1.3 RTD r^-Cro-srr>
CoNpurrs
The simplestcaseto considcris a straightcircurartube
in rvhicha f id
florvsunderlaminarconditions.with in;easing
useof tuburarporyrneri_
zationreactors,the study of RTD under'suchconditions
assumes
importThe parabolicprofirerbr a Nervtonianflurd causcsa
wicrespreadin the
residence
tinresand tfu E curveis easilyobtainedas
n:fir, l(o<*
(4.r)
- 0, elser.vhere
G.Z)
The sa're problemcan be treateclfor a non-Nswtonian
fluid as weU.If
an ostwaald-de-waele
power-rawfluid descriptionis used, tlien thc
corresponding
agedistributionfunctionsmay be easirye,r,aruatccr.
szabo
and Naunran(1969)and cintron-corderoet al. (196g)have
examined
this
problem.Novosadand urbrecht( t966)nraticuseof
theseagedistributions
to predictthe conversionin elementaryreactionscarried
out in tubular
reactors.
In
recent paper' osborne (1975) has deveropedpurely
.a
convective
modelsfor tubularreactors.He pointi out that polym.riraiion
reactors
are often fairly short-rvith farrly rapid reactiontui have
a pronounced
velocity profile. He deveroperd
mathematicalexpressions
so that any of
the four commonly used tracer techniqucscan be used to
obtain an
empiricalprofileindex.He hasarsogivena 'retrrod to calcurate
the distributionof productsresurtingfrom a sequence
of first orderreactions.
Edrvards
and Saletan(r967)in a simirarway carcurated
the effectof nonuniformveJocity
distributionon RTD by usinga quarticprofileequation
242 V. V. Chavanand R, A" Ivtashelkar
w i t h a n u n t r i l r o w np a r a m e t e r w h i c h r ep r e s en t e d t h e r a t i o o f t h e c e n t r e l i n e
to the mean velocity.
I t s h o u l c l b e e m p h a s i z e dt h a t p o i y m e r i z a t i o n f l o w r e a c t o r s c o n b i n e
l a m i n a r f l o r v w i ( h l o w d i f f u s i v i t i e s .T h e h i g h e r v i s c o s i t y o f t h e m o r e
c o m p l e t e l yp o l y m e r i z e dm a t e r i a l n e a r t h e w a l l s e r v e st o s h a r p e nt h e v e l o '
c i t y p r o f l l e p r e d i c t e d f o r a N c w t o n i a n f l u i d . C e n t r e - l i n e v e l o c i t i e sa s
m u c h a s e i g h t t i m e s t h e a v e r a g ev e l c c i t i e s h a v e b e e n o b s e r v e d( E d w a r d s
a n d S a l e t a n, 1 9 6 7 ) . I n d c e d ,B r a s i e ( 1 9 6 8 )f i n d s t h a t t h e e x p e r i m e n t a lR T D
m e a s u r e l n e n t sa p p e a r t o c o r r e s p o n d m o r e t o t h e e l o n g a t e dv e l o c i t y p r o f i l e s ( c o r r e s p o n d i n gt o d i l a t a n c y ) t h a n t o t h e f l a t t e n e d p r o l i l e s ( c o r r e s p o n e l i n gt o p s e u d o p l a s t i c i t yi n h e r e n t1 o t h e p o l y n r e r i cs o l u t i o n ) .T h e s t r o n g
coupling of the heat effectsr',,ilhreactiou in such reactots makes it rather
u n l i k e l y t h a t t h e s i m p l i f i e d a n a l y s i s n r e n t i o n e da b o v e w i l l b e a p p l i c a b l e
t o p o l y m e r i z a t i o r lr e a c t o r s .H o w e v e r , t h e r e a r e o t h e r s i t u a l i o n sw h e r e t h e
a n a l y s i s m a y b e a p p l i c a b l e . T h i s c o n c e r n st h e t h e r i n a l p a s t e u r i z a t i o no f
l i q u i d f o o d s , r v h i c h i s u s u a ! l y c a r r i e d o u t i n t u b u l a r c o n d r - r i t sT. h e d e a t h
r a t e o I m i c r o - o r g a n i s m is. sd i r ec t l y p r o p o r t i c n a l t o t h e i r p o p u l a l i o n C e n s i t l '
h c n c et h e s 1 ' s t e r nt e s e m l - - l eas f i r s t - o r d e r r e a c t i o n . T h e l a r g e l i t e r a t u r e i n
t h e a r e a o f f o o d t e c h n o l c , g y( s e e, e . g , , C h a r m , l 9 7 l ) s h o u l d b e r e f e r r e dt o
t o u n d e r s t a n dl h e i r n p l i c a t i o n so f t h i s .
Certain modif,cations of the slraight tube configuration can dramatic a l l y a l t e r t h e h y d r o d y n a m i c s a n d c o n s e q u e n t l y t h e r e s i d e n c et i m e
distribution. As an example, consider the coiling of a straight tube. A
c e n t r i f u g a l l yd r i v e n s e c o n d a r yf l o w i s s e t u p r v h i c h i s s u p c r i u . r p o s eodn
t h e p r i m a r y a x i a l f l o r v . T h i s h a s t h e e l T e c to f r , a r r o r v i n gd o w n t h e R T D
c o n s i d e r a b l y .A s p e c l so f ' t h i s h i l ' r , rtlr e e ns t u d i e c lt h c o r -tei c a i l y b y R u t h v c n
( 1 9 7 1 )a n d ex p er i n r e n t a l l yb y T r i v e d i a n d V a s u d ev a ( 1 9 7 , \ )f o r N e r v t o n i a n
f l u i d s . T h e n o n - N e l v t o n i a n f l u i d l l o w i n c o i l e d t r r b e si s b e i n g e x t e n s i v e l l r
s t u d i e do n l y r e c e r r t l y( s e et l t e r v o r k o f M a s h e l k a r a n d D e v a r a j a n ,1 9 7 5 )
a n i l c o n s e c l u e n t lnvo R T D s t u d i e sh a v e b e e np e r f o r n r e ral s I ' e t . f - - o i l e ct lL r b c
a s a r e a c t o r c o n t i g u r a t i o n ,o f c o u r s e ,h a s a u a d c l j t i o n a la r i v : i n t a g et h a t t h e
s t r o n g s e c o n c l a r ym o t i o n p r o d u c e s h i g h e r t r c i L t i r a r i s f c r c o e f l i c i e n t s .
F u r t h e r r n o r e ,i t n - r a yp r e v e n t t h e b u i l d u p o f h i g i r v i s c o s i t y m a t e r i a l
during polymerization at the rvalis aird consequently n;rrrorv down tlte
m o l e c u i a r w e i g h t d i s t r i b u t i o n . C o i l e d t u l . ; ea s a r e a c t o r s h o w s a g r e a t
p r o m i s e a n d h e n c e t h e a s p e c t so f R T D s h o u l d b c c e r t a i n l y l o o k e d i n t o .
S i m i l a r c o m m e n t s h o l d f o r n o n - c i r c u l a r t u b e s a s r a , e l l ,a l t h o u g h t h e
s t r e n g t ho t ' t h e s e c o n d a r yc i r c u l a t i o n i s m u c h s n t a l l e r i n t i r i s c a s e .N o n Newtonian viscoelastic materials have the peculiarity thal rectilinear
motion (as in the case of Newtonian fluids) is not possible and a fourc e l l e ds ec o n d a r y c i r c u l a t i o ns e t si n ( s e eG r e e n a n d R i v l i n , 1 9 5 6 ) .T h i s r v i l l
also narrow down RTD.
4.1.4 RTD rN Acrrareo Vr.sslls
T h e m u l t i - p a r a m e t er m o d e l s d e v e l o p e dt o d e s c i i l : eR T D i n c o n t i n t t o r - i s
Miring
of l'iscous Net+tonit'n and Non-Newtcnian
FLtids
243
stirred tank reactors envision sucl non-idealities as by-passing, stagnant
zones,piston flow, etc. The review by Olson and Stout (1967) shows that
R T D i n a w e l l - s t i r r e d r e a c t o rc o r r e s p o n d st o t h e e x p o n e n l i a l d i s t r i b u t i o n
of a completely mixed reactor lvhen the circulation rate is much greater
than about five times the throughput. Such a condition may be easily
a c h i e v e di n l o w v i s c o s i t ys ) ' s t e m sb, u t i n h i g h v i s c o s i t y s y s t e m s( s u c h a s
p o l y m e r i z a t i o n a n d f e r n r e n t a t i o nr e a c t o r ) , t h e y a r e f a r m o r e d i f f i c u l t t o
achieve.
T h e h i g h v i s c o s i t ys y s t e m so f o u r i n t e r e s tw i l l i n v a r i a b l y o p e r a t eu n d e r
l a m i n a r c o n d i t i o n s .T h e i n f o r m a t i o n o n c i r c u l a t i o n r a t e ( s e es e c . 3 . 4 ) a n d
a r i x i n g t i n r e ( s e es e c .3 . 6 ) s h o u l d b e h e l p f u l i n d e s i g n i n g t o a c h i e v eg o o d
m i x i n g . F o r e x a m p l e ,i n a c o n t i n u o u ss y s t e m ,i f v , , ek e e p t h e r a i x i n g t i m e
r a t h e r s m a l l i n c o m p a r i s o nt o t h e m e a n r e s ; d e n c ct i m e , t h e n R T D f a i r l y
close lo perfect mixing may be achieved. The studies on RTD in
c o n t i n u c u sa g i t a t e d v e s s e l sa s s u m eg r e a t i m p o r t a n c e .A l t h o u g h e n o u g hi s
k n o w n i n t h e a r ea o f l o r v - v i s c o s i t yl i q u i C s ,l i t t i e i n f o r m a t i o n i s a v a i l a b l e
f o r h i g h v i s c o s i t yl i q u i d s .
Z a l o u d i k ( 1 9 6 9 )p r o v i d e d R T D d a t a f o r a g i t a t e d v e s s e l sj n w h i c h f l a t u s e dt o a g i t a t ec o r n - s y r u p s o l u t i o n s i n t h e l a m i n a r
bladed turbines
"vere
and early transition regicn. Significant deviations from the conrpletely
nixed system were detecled and a ttvo-parameter model r',,asdevelopedto
d e s c r i b eR T D . S t o k e sa n d N a u m a n ( 1 9 7 1 )a n a l y s e dt h e d a t a o f I l l a n k s a n d
S t o k e s( 1 9 7 0 ) ,r v h i c hw e r eo b t a i n e dr v i t h p o l y s t y r e n es o l u t i o n so f v i s c o s i t i e s
i n t h e r a n g e o f . { t o 2 0 0 p o i s e .D u a l f i a t - b l a d e dt u r b i n e sa n d a c o r n b i n a t i o n
o l p i t c h e d - e n d f i a t - b l a d e dt u r b i n e s w e r e u s e d . S t o k e s a n d N a u m a n
m a d e u s e o f a s i n g l e p a r a n r e t c r t a n k s - j n ' s r -irc s r r r o c i c l .R e c e n t l y l i 4 o o Y o u n g a r r d C h a n ( 1 9 7 1 )h a v e p r o v i d c d R T D d a t a o n b o t h N e w t o n i a n
a l d n o n - N e w t o n i a n p o r v e r - l a t v1 1 u i d sa g i t a t e d t i y ' l l a t - l r l a d c d t u r t i n e s .
T h e m o d e l u s e d[ " i 1 ' N { o o - Y o u nagn d C h a n h a c ]a c o m b i n a t i o no f c o n r p l e l e l y
n r i x e c lr e g i o n sw i l h a d c a c l - s p a caen i i a p l u g l i o w r e g i o n . T h c y c o r r c l a l e r l
tirc model parameters in tertns of the Reynolds nunrber, a pseudoplasticity index and the product of the agitator speed and the ntan
r e s i d e n c et i m e . T h e r a n g e o f t h e p a r a m c t e r s c o n s i d e r e db 1 ' I \ ' l o o - Y o u n g
and Chan includes values typical ol industrial polymerizationreactors.
H o l v e v e r ,a n c h o r , h e l i c a l s c r e wa n d r i b b o n a n d p i t c h e d t u r b i n e a g i t a t o r s
are more commonly usedfor polymerization and it would be important
t o o b t a i n i n f o r m a t i o n o n t { T D i n s u c h s y s t e m s .F u r t h e r i l o r s r , t h e m o d e l
l i q u i d s t e s t e d b y M o o - Y o u n g a n d C l t a n w e r e p u r e l y v i s c o t t s .\ Y h e n a
v i s c o e l a s t i cl i q u i d i s a g i t a t e d , t h c i n t e r a c t i o n o f i n e r t i a a r , d e l a s t i c i l ym a y
p r o d u c e v e r y p e c u l i a r f l o w p a t t e r n s( s e es e c .3 . 3 ) a c d t h i s m a y s i g n i f i c a n t l y
i n f l u e n c e t h e r e s i d e n c et i m e d i s t r i b u t i o n .
It is only in recentycars tliat the importanceof mixing considerations
i r r p o l y m e r i z . a t i a nr e a c t o r si s b e i n g r e c o g n i z e d l.n s h a r p c o n t r a s t t o m i x i n g
s t u d i e si n s i m u l a t e df l u i d s , t h e r e i s a n i n c r e a s i n gt l e n d t o r v a r d se x a m t n i r t g
the role of mixing in actual polymerization leactors. The vlorl: of Cc'le
( 1 9 7 6 ) i s p a r t i c u l a r l y ' r ' a l u a b l ei n t h i s r e g a r d . C o l e s t u d i e d t h e n o n - i d e a l
244 V. V. Chawn and R. A. Mashelkor
m i x i n g p a t t e r n s i n c o n t i n u o u s s t i r r e d t a n k p o l y m e r i z a t i o n r e a c t o r sa n d
characterized these in terms of a mixing model, Anionic solution
polymerization of butadiene was then undertaken to study the influence
of mixing. correlations were then developed betrveen the operating
p a r a m e t e r s ,m i x i n g m o d e l p a r a m e t e r sa n d m o l e c u l a r w e i g h t d i s t r i b u t i o n
parameters.
I t i s c o n c e i v a b l et h a t s i m i l a r e f f o r t s c o u l d b e u n d e r t a k e n f o r t h e
c o r r e l a t i o no f o t h e r p o l y m e r c h a r a c t e r r ' s t i c ss,u c h a s c o p o l y m e r s e q u e n c e
d i s t r i b u t i o n , e t c . I t i s c l e a r t h a t s u c h s t u d i e s w o u l d b e o f i m m e n s eh e l p
t o e n g i n e e r sd u e t o t h e f o l l o w i n g r e a s o n s :
l . c h a n g e s i n p r o d u c t c h a r a c t e r i s t i c sd u e t o s c a l e - u pe f f e c t s a s n e l y
products move from laboratory scale to production scale could be
a n t i c i p a t e d a n d c o r r e c t e d f o r . T h i s s p e e d su p t h e s c a l e - u p s e q u e n c eb y
el i m i n a t i n g t r i a l a n d e r r o r c x p e r i m e n t a t i o ni n I a r g e e q u i p m e n t .T h i s w i l l
t r e c o n t ep a r t i c u l a r l y i t n p o r t a n t a s t h e n e e c lf o r p r o d u c t sr v i t h m o r e d e t a i l e d
s p e c i f i c a t i o n sf o r m o r e d e n i a n d i n ga p p l i c a t i o n sw i l ! i n c r e a s e .
2 . o p e r a t i n g c o n d i r i o n si n l a b o r a t o r y a n d p i l o t p l a n t e q u i p m e n tc o u l d
b e s el e c t e dj u d i c i o u s l y s o t h a t t h e p l a n t s c a l em i . r i n g c o n c l i t i o n sc o u l d b e
duplicated.
3. The need for design changesin production equipment may be shown
early so that the nature of the changesneededcould be inferred.
' 4 . M o r e i n t e l l i g e n t d e s i g no f f u t u r e e q u i p m e n t a t a n y s c a l e c o u l d
be
made, provided tcst results from the existing equipment are considered
r e l a t i v e t o t h e p r o d u c t p r o p e r t i e sd e s i r e d .
4.2
il{icronrixing
As remarked earlier, for non-liuear reactions the residence tiure
d i s t r i b u t i o nb y i t s c l f d o c sn o t d e f i n et h e s t a t e o f n i i x i n g s i n c e i t n r e r e l y
d e f i n e sw h a t i s t e r r n e da s r n a c r o m i x i n ga n d n o t n r i c r o n r i x i n g A
. conrpletely
s e g r e g a l e ds v s t e r r r e p r e s e n t so n e l i m i t o f m i c r o m i x i n g , t h a t i s , w h e r e
there is no rnixing at all. There is, of course, all upper limit o'
i n i c r o m i x i n g ,w h i c h c o r r e s p o n d st o t h e m a x i m u n r a m o u n t o f m o l e c u l a r
l e v e l m i x i n g p o s s i b l ea n d Z w i e t e r i n g ( 1 9 5 9 )h a s t e r m e d t h i s a s t h e s r a t eo f
m a x i m u m m i x e d n e s s .I n t h e c o m p l e t e l y s e g r e g a r e dr e a c t o r t h e ' r i x i n g
between fluid elements u,hich have spent different times in the reactor
o c c u r si n t h e e x i t s t r e a m ,w l i e r e a si n t h e m a x i m u u r m i x e d n c s sc a s em i x i n s
o c c u r si m m e d i a t e l y, v i z . a t t h e e n t r a n c e
I f c o n s i d e r a t i o n sa r e r e s t r i c t e do n l y t o t h e c o n v e r s i o no f t h e r e a c t a n t s
(and not to the detailsof the product yield) then the maximum mixedness
gives the lorvest possible conversion of reactants firr reaction orders
great€f than one and highest conversion for reaction c-,rdersless than one.
c o m p l e t e s e g r e g a t i o ng i v e s t h e h i g h e s t p o s s i b l e c o n v e r s i o n f o r o r t l e r s
greater than one and the lorvest conversion for orders less than one. ,l'he
c o n ' e r s i o n f o r a 6 1 s 1 - o r d erre a p t i o ni s e n t i r e l y d e t e r r n i n e db 1 , R T D
an<l
Mixing of ViscousNewtonian and Non-Newtonian Fluicls 245
i t i s i n d e p e n d e n to f t h e l e v e l o f m i c r o m i x i n g . N o v a d a n d T h y n ( 1 9 6 6 )
have given useful charts to depict these effects.
It is clear that we have to be able to describe the intermediate state of
segregation(or micromixing) by the use of certain models. Some efforts
have been niade in this direction. Curl (1963), for instance, has proposed
a coalescencemodel wherein he considers a model of dispersed phase
d r o p l e t sf l o w i n g t h r o u g h a n i d e a l s t i r r e d t a n k . I f t h e r e a c t i o n o c c u r s i n
the droplets and they do not coalesce, then this phase will act as a
macrofield. On the other hand, if they do coalesceand dispersethen they
a p p r o a c h t h e m i c r o f l u i d b e h a v i o u r . S p i e l m a na n d L e v e n s p i e l( 1 9 6 5 ) h a v e
shown how the conversion can be obtained from such a model by using a
Monte-Carlo procedure.
Ng and Rippin (1965) have proposed the idea of a two environnent
model, wherein a reactor r,vitharbitrary RTD is viewed as consisting of
c l u m p s o f u n m i x e d f l u i d ( e n t e r i n ge n v i r o n m e n t )a n d a l r e a d y m i x e d f l u i d
( l e a v i n ge n v i r o n m e n t ) .I n t h e u s u a l f o r m u l a t i o n o f t h e t w o e n v i r o n m e n t
m o d e l , t h e d i v i s i o n b e t w e e n t h e s e g r e g a t e da n d m a x i m u m n r i x e d n e s si s
based on the age of the fluid elenent. Typically, a fluid elters the reactor
in the segregatedenvironment and leaves in the maximum nixedness
e n v i r o n m e n t .C h e n a n d F a n ( 1 9 7 1 ) h a v e , h o l v e v e r , p r o p o s e d a r e v e r s e d
t w o e n v i r o n m e n t m o d e l a s b e i n g m o r e r e p r e s e n t a t i v eo l t h e p o l y m e r
r e a c t o r s .I t i s a s s u m e d t h a t t h e r e a c t a n t s a r e i n i t i a l l y w e l l m i x e d , b u t
b e c o m es e g r e g a t e da s t h e v i s c o s i t i e si n c r e a s ed u r i n g p o l y m e r i z a t i o n .
5.
Concluding Remarks
The present review has been confined to the mixing of viscous fluitls.
A c o l s i d e r a b l e r e s e a r c he f f o r t i s c u r r e n t l y u n d e r r v a y a l l o v e r t h e q o r l d
a n d n e n ' a v e n l l e sf o r r e s e a r c ha r e c e r t a i n l y e m e r g i n g .A l t h o u g h t h e p r e s e n t
s ost of the important areas,certain flelds have not
r e v i e r ve n c o m p a s s em
. ome of the important areas, for example,
b e e n a d e q u a t e l l ' c r ' : n v e r t e dS
are as follows:
l. Demand of unilorm polymermelt quaiity in lerms of both compos i t i o n a n d t e m p e r a t u r eh a s g i v e n r i s e t o t h e a p p l i c a t i o n o f m o t i o n l e s s m i x e r s . I n f o r m a t i o n o n t h e s em i x e r s i s g r o w i n g ( s e e ,e . g ' , S c h o t t
e t a l . , 1 9 7 5 , C h e n a n d M a c D o n a l d , 1 9 7 3 ) .A p p l i c a t i o n s o f t h e s e
m i x e r s i n o t h e r a r e a s , e . g . , g a s d i s p e r s i o nj n v i s c o u s l i q u i d s 1 s e e ,
e . g . , S m i t h , 1 9 7 8 )i s a l s o b e i r g c o n s i d e r e d .T h i s i s a n e v o l v i n g a r e a
of researchand considerablebasic and applied research inputs wili
be necessrry to take these mixers to a commercially viable stage.
2 . C u r i o u s a n o m a l i e sr e s u l t i n p o l y m e r - a d d i t i v es y s t e m s ' I t h a s b e e n
shown (White and Lee, 1974)that during Poiseuille florv the lorver
v i s c o s i t yf l u i d o f t h e t w o f l u i d s w i l l m i g r a t e t o t h e r e g i o n o f l i i g h e s t
s h e a r ( e . g . m o u l d r e l e a s ea g e n t s ) .C o n v e r s e l y ,s o l i d p a r t i u i e s i l a
246 Y. V. Chavanand R. A. Mashelkar
poiymer-particle system rvill nrigrate to the region of lowest shear
cluring Poiseuille llow accounting for resin ricir surfaces in filled
'demixing'
s y s t e n l s( K u b a t a n d S z a l a n c z i ,1 9 7 4 ) .T h e s ep r o b l e m so f
a r e q u i t e i m p o r t a n t a n d a r e r e c e i v i n gw i d e a t t e n t i o n o f t h e r e s e a r c h er s .
3 . P e c u l i a rm i x i n g p r o b l e m s a r i s e d u r i n g t h e t v r o p h a s e c o n t a c t o f a
g a s a n d a n o n - N e w t o n i a nl i q u i c l a n d t h e s u b s e q q e nht e a t a n d m a s s
t r a n s p o i t p r o c e s s e s( s e e , 0 . 3 . , M a s h e l k a r , 1 9 7 6 , A s t a r i t a a n d
M a s h e l k a r , 1 9 7 7 ) .T h i s a r e a i s a l s o i n a s t a g eo f i n f a n c y '
I n t h e p r e s e i t tr e v i e w w e h a v e a t t e m p t e dt o a n a l y s e s o m e a s p e c t so f a
topic rvhich not too long ago rvas being treated entirely as an art and is
n o w r a p i d l y a c h i e v i n ga s o u n d s c i e n t i f i cb a s i s .T h e i n t r o d u c t i o n o I w e l l d e f i n e d s t a t i s t i c a l l n e a n s f o r e s t i m a t i n g t h : g o o d n e s so f m i x i n g , t h e
l i n k i n g w i t h t h E l a i n i n a r a n d d i s t r i b u t i v e m i x i n g , i n c r e a s e de r n p l r a s i so t l
a t t e m p t s t o a n a l y s e a s u b s t r u c i u r er a t h a r t h a n o b s e r v a t i o n o f a g r o s s
p h e n o r n e n o na n d l u r t h e r m o r e t h s l i n k u p o f s u c h s u b s t r u c t i t r e t o t h e
h y d r o d y n a m i c sh a v e a i d e d c o n s i d e r a b l yi n e v o l v i n gs u c h a s c i e n t i f i cb a s i s .
T h e b l a c k b o x v i e r v p o i n ti s v a n i s h i n ga n d t h i s i s a w e l c o m es i g n .
I n t h e a r e a sj u s t c o v e r e d w e h a v e p r o v i d e d a n a l y s i s a s r v e l l a s c o n s t r u c t i v e s u g g e s t i o n fso r f u t u r e r e s e a r c hi n e a c h i n d i v i d u a l c a s e . I t s h o u l d
b e e n p h a s i z e dt h a t o u r a t t e m p t h a s b e e n t o b e r e p r e s e n t a t i v er a t h e r t h a n
e n c y c l o p a e d i cs o t h a t a r a t i o n a l a n a l y - s i os f t h e p r e s e n td a y k n o w l e d g eo f
t i r e s u b j e c tc o u l d b e m a d e . I t i s h o p e d t h a t t i r e r e v i e v r r v i i l b e u s e f u l f o i
r e s e a r c h e ras n c l d e s i g n e r sa l i k e -
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lndex
PseudoplasticitY
R e v o l u t i o n so f t h e s t i r r e r
r..c
per second
No. of revalutions(see
equatioil I ' 17)
ation
E s t i m a t e dv a r i a n c e
Linear scaleof segregation
Time
Y e l o c i t i i r t . r ' ,t' i i r e c t i u t t
Yelocit; in 6 directiun
V o l u m e s c a l eo f s e g r e g a tion
Bu[k voluntc
Co-ordilrates
R e l a t i v e c h a n g e si n d i s tance in xt, xz, x3 direction
N u m b e r o f P a r t i c l e si n
a sample
N u m b e r o f P a r t i c l e so f
c o n l p o n e n tu n d e r c o n s i deration
248 Y. Y. Chavqn and R. A. l'fashelkar
GREEK
LETTERS
c,
c o Sc 1 ,
cos c!2,
cos rf,3
o"r,uz
p
i
Tuu
0
0'
0,,
0"
p
Parameter in the Ellis Fa
po
fluid model
Direction cosines
Pr, Pz
lvlaterial parameters for
a third order fluid
Angle defined in equa'
tion Ll6
Shear rate
Average shear rate
Residence time (dimensioniess)
Angular coordinate
N{ixing time
Circulation time
Yiscosity
p
o2
"3
of
r0
,b
A
Apparent viscosity
Parameter in the Ellis
fluid model
Viscosities of comPo'
nents (l) and (2)
Density
population
Expected
variance
Expected variance bet'
ween two comPletelY
segregatedcomponents
Expected variance of a
binomial distribution
Parameter in the Ellis
fluid model
Stream function
Angular velocitY
R E F E R E N C ES
Angus, R. M. and L, Lapidus, AIC4E "r', 9' (1963),810'
Astarita, G. and G, Marrucci, Principlesof Non-NewtonianFluid Mechanics,McGraw
Hill, Berkshire(1974).
Astarita, G. and R' A, Mashelkar, Chem.Errg.(Lond')' (1977)' 100'
Beckner,J. L. and J' M. Smith, Trans.Inst, Chem.Engrs',44,(L966),224'
Bergen,J. T., in Processitrgof Therntoplasricr,Bernhardt, E. C' (Ed.), Reinhold, N.Y.
(19s9).
g,tt. Sci', 15, (1975)'684.
Bigg, D., Pols'p1.
In'J- Eng' Chem.Fundls.,13' (1974),66'
Middleman,
S.
Bigg, D. and
Ertg',11,(1963)'129'
J,
Chenr.
Bischoff.K. B. Can,
(1966)'I3.
Chettt.,58'
Bischoff,K. ts.,Ind' Etrg.
Ind. Etry-Chent.,58'(1966)'18'
McCracken,
Bischoff,K. B. and E. A.
presentedat the 67th National AIChE Meetpapef
Blanks,R. F. and R. L. Stokes,
i n g , A t l a n t a .( I 9 7 0 ) .
Blaskinski,H., B. Kochanski,and E. Rzyski, Irtt.Chent.Eng. ProcessInd., lO'(1970),
I'16.
Bourne,J.R., Chetn.Ens.,l80' (1966),202.
Bourne,J. R. and H. Butler, Trans.Ittst. Chen. Engrs.,47' (1969)'lil'
st. Louis, (1968)'
Brasie,w. c., paper presentedat the 6lrd National AlchE lr,Ieeting,
Buslik, D., PowderTechnologv,T' (1973)'l1l.
calderbank, P. H. anctM. B. Nloo-Young,Trnirs.Inst. Chent.Engrs.,37,(1959)'26.
Carreau,P. J., L Pattersonand C. Y. Yap,Cun. J. Chem.Eng',54, (1975)'135'
of Fotd Engineering,The Avi Publishing, connecticut (1971).
charm, 5., Ftrndamentals
Thesis,
University of Salford (1972).
V.,
Ph.D.
V.
Chavan,
J.
Chavan,V. V. and Ulbrecbt, Ind. Eng. Chem'(Proc. Des. Develop'>'12,(1973r'472'
Chavan.V. V., M. Arumugam and J. Ulbrecht, AIChE J ,21, (1975a),613'
-'
Chavan,V. V., D. E. Ford and IvI. Arumugam, Can.,J' Chetn.Eng ,53, (1975b)'628'
Chen,S. J., J. Soc.Cosnret.Chent',24,(1973),639.
, h e m . . U n g , 8 5( ,1 9 7 3 ) 1' 0 5 .
C h e n .S . J . a n d A . R . I \ { a c D o n a l dC
Chen. M. S. K. and L. T. Fan, Curt.J. Chem.Ens.,49, (1971)'704.
cintron-cordero, R., R. A. Mostello, and J. A. Biesenberget,can. J. Chent.Ens., 46,
( 1 9 6 8 ) 4, 3 4 .
Cole, W. M., AICLE Synp. Set ,72 (160),(1976),51.
Coyle,C. K., H. E. Hiscbland, B. J.Michel and J. Y. Oldshue,AlChEJ., 16' 11970),
903.
C u r l , R . L . , A I C \ E J , 9 , ( 1 9 6 3 )i,7 5 '
Danckwerts,P. Y-, Researc}(London). 6, (l9il), 355.
250 Y. V. Chavanand R. A. Mashelkar
Danckwerts,P. Y., Chem.Eng. Sci.,2, (1953),l.
Danckwefts,P. Y., Appl.,Scf.Res.3 (A),(1g52),27g.
I
Earle, R. L.,Trans. lnst. Chent.Engrs.,37,(1959),297.
Edwards,W. M. and D. J. Saletan,Ind. Eng. Chem.,59,(1967),37.
Feldkamp,K,, Thesis,TechnischeHochschuleBraunschweig,(1962).
Ford, D. E., R. A. Mashelkar and J. Ulbrecht,Proc. Tech.Int., f 7, (1972),803.
Ford, D, E. and J. Ulbrecht,AICUE J.,21, (1975),1230.
Giesekus,tL, Rheol,Actq,3, (1961),59.
G o fd i s h ,L . H . , J , A . K o u t s k y a n d R . J . A d l e r , C h e m .E n g . 9 c i . , 2 0 ,( i 9 6 5 ) ,1 0 1 1 .
Gray, J, B.,itt Mixing: Thzoryand Praclite, Vol. 1, Acad., N.Y. (1966).
Green,A. E. and R. S. Rivlin, Quart. A1.tpl.Math.,14, t1956),299.
Hald, A., Statistical Theor.yfor EtryineeringApplications, V/iley, N.Y. (1952).
H a l l , K . R . a n d J . C . G c d f r e y ,A I C h E . t C h ES y m p .S e r .N o . 1 0 ,( L o a d o n : I n s t n .C h e m .
Engrs.)(1955).
Himmelblau,I). M. and K. B. Bischoll,ProcessAnalysisand Sinulation: Delenninistic
Systerns,
Wiley, N.Y. (1958).
Ho, F. C. and A. Kwong, Chant.Ettg.,8$,(1973'1,95.
Hoare, M. R., Ind. Etig. Chein.,53,(1961),197.
Holland, F. A. and F. S. Chapnran, Ltquid l{i-ting cmd Processing itt Siirrel Tanks,
R e i n h o l c lN, . Y . ( 1 9 6 6 ) .
lloogendoorn,C. J. and A, P. Den IIartog, Chein.Eug. ici.,22, (1967t,1689.
Ide, Y. and J. L. White, J. Appl. Pol;tn. Sci.,l*, (19741,299'1.
Kogolcu,5, i1967), 121.
Ivonue, I. arrd K. Sato, Krr.ga./i.'r
James,D. F'.and J. Acostrt,J. Flrril ll{et:h.,42,(,1970),269.
, d J . U l l r r e c h t ,C h e n . E n g . T e c l t . , 4 6 , ( 1 9 7 1 ) , 6 9 K a l e ,D . D . , R . A . N { a s h e l k a r n
Kelkar, J. V., Ph.D. Thesis,University of Saiford,(1972).
Kelkar, J. V., R. A. Nlashtlkar and J. Ulbrecltt, Tittns. Irtst.Chetn.Etrgrs.,50,(.1972),
-1qJ.
Kelkrr, J. V., R. A. Mesh:lkar, and J. Ulbrccltt, J. Appl. Pol.t'tn.Sci.,17,(1973),3059.
l ( r a m e r s ,F I . a n d G . A l b e r d a ,C h : n t .E u g .S c i . ,2 , ( 1 9 5 1 ) 1
, 71.
Kristensen,tI. G., .4rch,P/tunr. Chein.Sci.,l, (1973),I02.
Iiubat, J. and A. Szalanczi,Folyn Elg. ,Sci.,ld, (1974),873.
L a c e y ,P . N { . C . , - f . A p p l . C l r e n t . , ' 1 , ( 1 9 5 1 ) , 2 5 7 .
Lelti, U., Ind, Rtg. Chent.Fun:l.,4,(1965),360.
Levenspiel,O., ChernicalReacti:tttEngineering,
Wiley, N.Y. (1972).
Levenspiel,O. and K. B. Bischoli',Adv. Chen. Ens.,1, (1963),95.
Levenspiel,O. and J. C. R. Turner, Clrcnt.Eng.9ci.,25, (1970),1605.
M a g n u s s o nK, . , l v a , 2 3 , ( 1 9 5 2 )8, 6 ,
Nlashelkar,R. A., Cltem.Inil. Develop.,10, (1976),17.
Mashelkar,R. A. and G. V. Devarajtn,Trans.Inst. Chen. Engrs.,54,(1976),100.
MaKelvey,J. M., PolJ,ntt Processtng,
Wiley, N.Y. ( 1962).
I V l e t z n eA
r ,. B . a n d C , A s t a r i t a ,A I C | E , f . , I 3 , ( 1 9 6 7 ) , 5 5 0 .
M e t z n e r ,A . B . a n d I l . E . O r t o , A I C | E 1 . , 3 , ( 1 9 5 7 ) , 3 "
lVletzner,
A. B. and J. S. Taylor, AlChg "/.,6, (1960),109.
l v l e t z n e fA, , 8 . , R . H . F e e h s [, { . L R o m o s ,R . B . O u r a n J J . P . I t i t h i l l . A t C i t E i . , 1 ,
(1e61),3.
'
251
Mixing of ViscousNewtonian and Non'Newtonian Fluids
(1954)'604'
Michaels,A. S. and V. Puzinankas,Chent.Eng' Prog',50,
(1968)'
N'Y'
Middleman, S. The FIow of High Polyn'rers'Wiley,
Milsuishi, N. and N. J' Hirai, J. Chent.Engg' Jap',2,(1969),117'
(Bernhardt'E' C'' (Ed')' Reinhold' N'Y''
of Thermoplastics'
Mohr, W. l). ia Processing
( Iese).
Mohr,W.D.,R.I-.SaxtonandC'H.Jepson,Ind.Eng'Chem.'49,(1957)'i855.
Moo-Young,M.BandK.W.Chan,Can'J'Chent'Ertg''49'(1971)'187'
Jap.,5' (1972),l'
Murakami, Y., K' Fujimoto and S. Votarri,.I' Chem'Engg'
Press'(1975)'
Nagata, 5., Mi'ting'. Principlesantl Applications,I{alsted
Gotoh' !' Chem' Eng' Jtrp''3'
Nagata,S., M. Nishikarva,H, Tade, H' llirabayashi and
(t970),237.
Kogaku'21' \1957)'278'
Nagata, S., T. Yanagimotoand T' Yokoyama, Kagaku
Univ'' 14' (1955)'253'
Nagata,S. and T. Yokoyama,Men' Fac' Ettgg',Kyoto
Naor, P. and R. Shinnar,lncl-Eng'Cltznt'Fund ,2,(IS6l\'278'
Macromol' Chent.,C 10, (1974)'7-;.
Newman, E' 8,, ./' Macrono]. Sci.-.Reys.
31' (i966)' 3710'
Cotttrnun''
Novad, Z. andJ. Thyn, Collect'C:ech' Cltent'
(196q' 405'
Novosad,Z' artdJ. Ulbrecht, Chen' Etts' Sci''21'
Symp' on Chem' Reaction
European
3rd
Pr'oc'
Rippin,
T
D.
W.
Ng, D. Y. C' and
(1964)'
Engg., An.rsterdam
Olson,J.H.andL.E.Stout,inMi.vittg:Theorl,atttlPraclice,Vol.2,V.W.Uhland
J. I3. Grav (Eds.),AcademicPress,N'Y'' (1967)'
(1970)'115'
O'Slrima,E' and K. Yuge' Kagnku!{ogaku,39'
(
1
9
7
5
)
,
159'
. ng. Sci',30'
O s b o r n e ,F . T ' , C h e n t E
59' (1967)'10'
C!rcn''
Eng'
Iid'
J'
Belt,
K.
and
R'
W.
Penney,
Etg" 40' (1967)'T360'
Peters,D. C. and J. M. Smith, Trurts Inst'Clent'
17' (1969)'268'
Peters.D. C. and J. NI. Smith, Cun J' Chenr'Eng''
52' (1974)'285'
Rieger,F. and V. Novak, Trans' Ittst'Chetn'Eng''
1
1 1 '{ } 2 '( 1 9 7 1 ) ' 2 1 '
S
e
r
'
N
o
'
S
v
m
p
P
r
o
g
r
'
E
n
g
.
R u d d . N { .J . , C h e m '
I
113'
(1971)'
Sci',26,
Eng'
Rutlrven,D. M., Cltetn'
Eng. Ptcg.,TI (l), (1975)'54'
schott, N. R., B. weinstein,and D. La Ilombard, Ciurn.
(1969)'126'
Seyer,F. A. and ;\' B. Metzner' '4IChE J',15,
(1973),
l09l'
28,
Shearer,C. 1., Chent.Eng. Sci ,
-f.
Al|rely,Jr., SPE Trans.,3,(1963)']92.
Shrenk,w' J., K. J. Cleerman,anc
MotlernPlostics'45' (1969)'159'
Shrenk,W. J., D. S' Chisholm,:LndT' Alfrey'
Skelland,A.H.P.,Non-NewtoniailFlorealJl{eatTt,t;rsfer'wiley,N'Y'(1967)'
6 ' ( 1 9 5 1 ) 1' 1 3 '
S p e n c e rR, ' S . a n d R . M W i t e v ,J ' C o t l o i dS c i ' '
(1965)'247'
Spielman,L' and O' Levenspiel,Chen' Eng' Sci''20'
Smith, J' M., Chem.Errg.(Lond ), (1970)'45'
Smith, J. M., Chem-Erg' (Lond'), (1978)'827'
Stokes,R.L.andE.B.Nauman,Can'I'Chent'Ettgg''49'(1971)'187"
(1969)'575'
Szabo,T. T. and E. B' Nauman, AIC|E "l'',f5'
30' (1975)'317'
Trivedi, R. N. and K. Vasudeva'Chen' Eng' Sci''
549'
(1971)'
Sci',26,
Eng'
Turner, J. C. R , Chen'
tmd Ptac!ice' Academic Press'I{'Y''
Uhl, V. W. and l. B- Gra-v(Eds'), Mixrrr6r':Thecry
(1966).
Ulbrecht, 1., Chem.Eng.,286' (i971), 347'
252 T. V. Chavan and R. A. Mashelkar
Ultmrn, J" S. and M. M. D:nn, Trans, Soc.Rheol,,14,(1970),307.
Valfentin,F. H. H., Chent.Ettg.,(1967),99.
Walters,K. and J. G. Savins,Trans.Soc. Rheol.,9,(1965),407.
Waters, N. D. and M. J. King, Q. L Mech. Appl. Maths., Zl, (1971),331.
White, E. T., Imp. Coll. Chem.Eng. Soc. J., 14, (1962),j2.
White, J. L. and B. L. Lee, Trans. Sbc.Rheol., tB, (1974), 467.
Wohl, M. H., Chettt.Ens.75, (1968),113.
Yap, C. Y., I. Patterson,and P. J. Carreau,unpublishedwork, (1978).
Zafoudik, P., Br. Chem.Eng., Ll, (1969),6i7.
Zwielerirg, T. N., Chen. Eng. Sci.,1f , (1959),1.
t