Liquid Mixing Fundamentals

nstructional !"#ecti$es o% This Section
By the end of this section you will be able to: Identify the geometric !hysical and
dynamic "ariables of im!ortance for the analysis of mi#ing in a stirred tan$
Assess the relati"e im!ortance of those "ariables

'asic (heological Concepts
+onsider a fluid contained between two !lates se!arated by a distance y .
,ne !late is set in motion !arallel to the other with "elocity v x .

Mathematically:
i.e.:
y
v
y
v
A
F
Area

'efinitions
xy
Fluid Viscosity (centipoise, cP)
Fluid Viscosity (centipoise, cP)

Focus o% This Section

mportant ,aria"les in the 1nal&sis o% Mixing Phenomena
)he "ariables of im!ortance in the analysis of mi#ing !henomena in stirred tan$s can be classified as: geometric "ariables
!hysical "ariables
dynamic "ariables
tan$ sha!e si6es3
)an$ sha!e e.g. cylindrical3
Internal diameter T

Baffle thic$ness

'iameter D
Blade angle
Pitch p9
Blade width height3 w

+learance off the tan$ bottom measured from the mid!oint C
+learance off the tan$ bottom measured from the im!eller bottom C b9
*!acing between im!ellers S
'isc diameter disc turbines3
i7uid <rheology= e.g. newtonian non>netwonian shear>thinning etc.3 and corres!onding !arameters e.g. !ower law e#!onent3
'ynamic "iscosity , µ

Im!eller angular "elocity ω
)or7ue τ
(elationship 'et*een N . ω
and V tip )he agitation s!eed N must be e#!ressed
in re$olutions per unit time such as: re"olutions !er minute r!m3
re"olutions !er second r!s3
)he ti! s!eed v tip is not inde!endent of N
but it is related to N as follows with ω in rad@s N is in r!s D in m v tip in m@s3:
DN D
2

By the end of this section you will be able to: +alculate &e -r in stirred tan$s
'istinguish agitation regimes
+alculate the !ower dissi!ated by an im!eller from a"ailable !ower numbers

Tur"ulence and Mixing
)urbulent flows are associated with ra!id a!!arently random %luctuations of all three com!onents of the local "elocity "ector with time
)o this day turbulence is still a relati"ely !oorly understood !henomenon
Many mixing phenomena are associated with tur"ulence

t
ChE702 27
Tur"ulent Flo*

ChE702 28
sotropic Tur"ulence

Energ& Cascade in sotropic Tur"ulent Flo*

'uring this !rocess smaller and smaller eddies are generated
,ne can conce!tually introduce an eddy &eynolds umber:
,orces-isco$s
As long as &eeddy no "iscous
dissi!ation will occur and the $inetic energy will sim!ly be transferred to smaller and smaller eddies
;owe"er at &eeddyC "iscous forces
will begin to dominate
-or &eeddyDD the eddy will not

*uch a transition occurs at the olmogoro%%+s length scale e7ual to:
where ε is the !ower dissi!ated !er unit mass and ν is the $inematic "iscosity.
λk is the si6e of the smallest edd& in
the turbulent fluid
ChE702 3+
Big whorls hv! littl! whorls Tht "!!# on th!ir v!lo$ity,
%n# littl! whorls hv! l!ss!r whorls
%n# so on to vis$osity.
ewis -. &ichardson
88>EFG3
)he !oem summari6es &ichardsonHs E20 !a!er Th! S&pply o" 'n!rgy "ro( n# to %t(osph!ri$ '##i!s .
A !lay on Jonathan *wiftHs Kr!t "l!s hv! littl! "l!s &pon th!ir b$*s to bit! +!(, %n# littl! "l!s hv! l!ss!r "l!s, n# so # in"init&(.K LGG33
Energ& Cascade3 Summar&
Energ& Cascade3 Summar&
Big whirls hv! littl! whirls, Tht "!!# on th!ir v!lo$ity,

Po*er issipation
)he po*er dissipated 5or consumed6 "& the impeller. P . is one o% the most important $aria"les to describe the !erformance of an im!eller in a tan$
P is a function of all the geometric and ph&sical $aria"les of the system

Experimental etermination o% Po*er Consumption
)he po*er dissipated "& $arious impellers under different conditions has been experimentall& obtained by many in"estigators

It is relati"ely easy to determine the cumulati"e o$erall po*er drawn by a mi#ing system including motor dri"es seals im!ellers etc.3
It is much more di%%icult to determine the !ower dissi!ated by the impeller alone

)he total po*er dissipation in a system is gi"en by:
If one needs to $now P i(p!ll!r P totl and
all other !ower dissi!ation sources must be $nown under the dynamic conditions in which the im!eller o!erates
)his can be 7uite difficult

A number of methods ha"e been used to measure the !ower dissi!ated by im!ellers including:
electric measurements
strain gages and tor7uemeters
Example o% Strain age S&stem %or Po*er Measurement
Tachometer
Controller
Vessel

Po*er issipation
************

Po*er issipation

=
===
re%erred to as Po or )e6
)he impeller Po*er )um"er. N p
also called Po or the ewton number )e3 is a non#i(!nsionl "ariable defined as:
If (nglish units are used then:
53DN
Po*er and Po*er )um"er
)he po*er consumed "& an impeller and the Po*er )um"er are related to each other "ia the e7uation:
where ! is a function of the im!eller ty!e and the geometric and dynamic characteristic of the system
53DN N P P ρ =

)he impeller (e&nolds num"er. (e defined as:
is a !roduct of the non>dimensional analysis.
+om!are this &e with the (e&nolds num"er %or a pipe:
µ
mpeller (e&nolds )um"er
As usual a !hysical inter!retation can be associated with the im!eller &eynolds number &e. Accordingly:
,orcesisco$s
ChE702 +8
Froude )um"er
Another non>dimensional number arising from the non>dimensional analysis is the Froude num"er. Fr defined as:
g
ChE702 +9
Froude )um"er
It can be shown that the -roude number has the following !hysical inter!retation:
,orcesna&Gra-itatio
i.e.:
)wo systems are geometricall& similar if all corres!onding dimensional ratios are the same in both systems
D
Po*er Equation %or eometricall& Similar S&stems
-or geometricall& similar including same ty!e of impeller3 stirred tan$s and im!eller all geometric ratios are the same
;ence )P does not change *ith
scale "et*een tans:
( )Fr $ DN
Po*er Equation %or 'a%%led. eometricall& Similar S&stems
Nhen "a%%les are present no "orte# occurs i.e. the gra"itational forces become unim!ortant and the Po*er )um"er "ecomes independent o% Fr:
( )e 53 $
ChE702 55
T&pical Po*er Cur$e %or mpellers in 'a%%led Tans
& &0 &00 &000 &0000 &00000
$eynolds 'um%er, $e
P o e r ' u m % e r , P o
Tur%ulent
-or (e890 the flow in a baffled tan$ is laminar
)heoretical and e#!erimental e"idence shows that:
i.e.:
Po*er Cur$e3 Laminar Flo* (egime
In the laminar %lo* region the !ower dissi!ated by an im!eller is gi"en by:
where *- is a !ro!ortionality constant that de!ends on:
ty!e of im!eller

Po*er issipation in the Laminar Flo* (egime
In the laminar regime !ower dissi!ation is: inde!endent of the #!nsity of the li7uid
directly !ro!ortional to the vis$osity
strongly affected by the gittion sp!!# P ∝N 23
strongly affected by the i(p!ll!r #i(!t!r P ∝DG3

Po*er Cur$e3 Transitional Flo* (egime
-or C0D&eDC0000 the flow regime cannot be well characteri6ed as either fully laminar or fully turbulent
'e!ending on the ty!e of im!eller P

Po*er Cur$e3 Tur"ulent Flo* (egime
At high &eynolds numbers (e:90.0003 the flow in a baffled tan$ is tur"ulent
)heoretical and e#!erimental e"idence shows that P is inde!endent of &e:
i.e.:
Po*er Cur$e3 Tur"ulent Flo* (egime
In the tur"ulent %lo* region the !ower dissi!ated by an im!eller is gi"en by:
where * is a !ro!ortionality constant e7ual to P) the asym!totic "alue of P that de!ends on:
ty!e of im!eller
5353 DN N DN k P # P ρ ρ ==

In tur"ulent regime !ower dissi!ation is: inde!endent of vis$osity
directly !ro!ortional to the #!nsity of the li7uid
"ery strongly affected by the gittion sp!!# P ∝N G3

Sensiti$it& o% Po*er issipation
In the turbulent regime P is $er& sensiti$e to N and D
(#am!les: a 90; increase in agitation speed. N
increases the !ower dissi!ated by <<;
a 20; increase in N increases P by 7<;
a 90; increase in impeller diameter. increases the !ower dissi!ated by =9;
a 20; increase in D increases P by 9>?;

Sensiti$it& o% Po*er issipation

Po*er )um"er Cur$es %or ,arious mpellers

ChE702 66
Po*er )um"er Cur$es %or >AB Pitched-'lade Tur"ines 5>-'lades6 and 4E-< mpeller
A,ter ". M'ers an/ . . !i&ens* Persona& Comm$nication

Equation %or Po*er )um"er Cur$es
An e7uation for Power umber as a function of &e has been !ro!osed:
where A B and + are coefficients that de!end on the ty!e of im!eller.
e1000
e

Impeller Type * C
+9 1.5 0.3
50 +.0 1.0
Tur"ulent Po*er )um"er
Most lo* $iscosit& s&stems and industrial stirred tan$s o!erate in the tur"ulent regime where P is constant
A sim!le and meaningful way to com!are the po*er per%ormance o% $arious agitators is to com!are their turbulent Po*er )um"ers. )PT

)urbulent Power umbers ha"e been obtained experimentall& for many im!ellers
)y!ically P) is measured for a
<standard= configuration of the agitation system HT D/T ?@G CD3

Impeller Type 'PT
ate 5.5
'mit$ Turbine (%n"ave-Blade Turbine)

Impeller Type * PT
&.5
ChE702 7+
E%%ect o% D/T (atio on Po*er )um"er %or isc Tur"ines
0+ 0+- 0+. 0+.- 0+/ !T
0

ChE702 75
E%%ect o% D/T (atio on Po*er )um"er %or Pitched-'lade Tur"ines
Armenante et a&.* Ind. Eng. Chem. Res.* 1999.
0+ 0+- 0+. 0+.- 0+/ 0+/- 0+- 0+-- !T
0
' e

ChE702 76
E%%ect o% mpeller Clearance on Po*er )um"er %or isc Tur"ines
0 & . / - b1 !
' e
!T7&.)526 3T71 !T7&.2046 3T71 !T7&.)526 3T72 !T7&.2046 3T72 #eressi%n urve
Armenante an/ Chang* Ind. Eng. Chem. Res.* 1998.

ChE702 77

−−−=

−−=

%or isc Tur"ines
0 0.05 0.1 0.15 0.2 0.25
C b /T
%or Flat-'lade Tur"ines
0 0.05 0.1 0.15 0.2 0.25
C b /T
ChE702 80
E%%ect o% mpeller Clearance on Po*er )um"er %or Pitched-'lade Tur"ines
Armenante et a&.* Ind. Eng. Chem. Res.* 1999.
0 & . / - b1 !
' e

%or Pitched-'lade Tur"ines
0 0.05 0.1 0.15 0.2 0.25
C b /T
%or 4E-< mpellers
0 0.05 0.1 0.15 0.2 0.25
C b /T

Po*er issipation in Multiple mpeller S&stems
If the H/T ratio is larger than .2>.F multiple impellers are ty!ically used
)he Power umber and the !ower drawn by two im!ellers mounted on the same shaft and s!aced by a distance S is not usually twice that of the indi"idual im!eller
-or large S P double ≈ 2 ×P single

T T
0 0+ 0+/ 0+ 0+2 & &+ b1 !
0
0 0+- & &+- '!
0+& 0+ 0+. 0+/ !5T
0
(a) 0 0+- & &+- +-
= ,
'! &.00 1 1.))) 1.5 1.00
!T =0+/
3T =&
air entrainment
air entrainment
!T =0+/
3T =&
(a) 0 0+- & &+- +-
= ,
! &.10 &.))) &.5 &.00 1
Armenante an/ Chang* Ind. Eng. Chem. Res.* 1998. (%)
0 0+- & &+- b1 !
= ,
' ! &.0 1 1.5
(%)
(a)
0

(a) 0 0+- & &+- +-

+
= ,
'! &.00 1 1.5
!T =0+/
3T =&

(a) 0 0+- & &+- +-

+
= ,
! &.))) &.5

ChE702 96
Po*er Cur$es %or mpellers in 'a%%led and n"a%%led Tans
& &0 &00 &000 &0000 &00000
$eynolds 'um%er, $e
P o e r ' u m % e r , P o
*a##led Tan7
ChE702 97
Po*er Cur$es %or mpellers in 'a%%led and n"a%%led Tans
P "s. &e !lots for "a%%led s&stems
show that )P reaches an as&mptotic
$alue at high (e&nolds )um"er
P "s. &e !lots for un"a%%led
s&stems show that )P eeps
decreasing *ith (e e"en at high &eynolds umbers

Po*er and Torque
)he po*er dra*n by an im!eller P and the torque τ re7uired by the same im!eller rotating at N are related to each other by the following e7uation:
&emar$: the same !ower dissi!ation can be achie"ed using a higher tor7ue and smaller agitation s!eed or "ice "ersa
τ π τ ω N P 2==

Po*er issipation and !perating Cost o% Mixing
)he !ower dissi!ated by the im!eller P is ust the energy consumed by the im!eller !er unit time ty!ically as electric energy
;ence the operating cost of the mi#ing o!eration are !ro!ortional to P :
P ∝CostO)erating
Torque and Capital Cost
)he capital cost of a mi#ing o!eration is significantly dominated by the cost of the gear bo#
)he cost of the gear bo# is directly related to the its tor7ue rating ty!ically through an !ower law:
8.0CostCa)ita& τ ∝
mportant Mixing !perating and Scale-up Parameters
)raditionally mi#ing !rocesses ha"e been scaled up and o!erated by maintaining constant one the following !arameters: Po*er per unit liquid $olume in the
tan$ P/V , or !er unit li7uid mass P/ρV

Po*er per nit ,olume
)he !ower dissi!ated by the im!eller !er unit li7uid "olume in the tan$:
is one o% the most important mixing parameters used in scale u! of mi#ing !rocesses
)he units for P/1 are N@ $N@mG or h!@000 gal
( P
Po*er per nit Mass
)he po*er dissipated "& the impeller per unit liquid mass in the tan$ ε :
is an alternati"e to the use of P/1 since the
only difference is the !resence of ρ 3 ε is also widely used for scale>u!
)he units for ε are m2 @sG
(
( ) " #

( )
( )
( )

( )
( )

Scale-up 'ased on Constant Po*er per nit ,olume 5P/V!
( )
( )
Tip Speed and Torque per nit ,olume
-or geometrically similar systems for which D ∝ T 3 in fully turbulent regimes or for the same system at different agitation s!eeds i% the torque per unit $olume τ /1 is $e!t constant then:
3
53
3
53
*im!lifying:

/V constant !ro"ided that the geometry of the systems is similar and the flow is fully turbulent
( ) 2
2
222
3
53
tip

T&pical Tip Speed and P/V %or ,arious Mixing Equipment
9:uipment tip ms (ts)
, :;m )
0.2<0.6 1<3
0.2<+ 0.6<12
20<+0 100<200

1dditional Po*er Sources in Stirred Tans
In the "ast maority of cases mechanical po*er input in stirred tan$s is !ro"ided by impellers

Mechanical po*er can be su!!lied to stirred tan$s "ia three !rimary different sources i.e.: mechanical agitation e.g. im!ellers3
!ower deli"ered by the expansion o% a compressed gas e.g. gas dis!ersers diffusers3


Po*er nput "& as Sparging
)he mechanical po*er input contribution of a gas s!arged inside a li7uid is:
" g )P *gasgasexpanding ρ =
Po*er nput o% a Liquid Det
)he mechanical po*er input contribution of a liquid #et inected inside a li7uid is:
2
2

Total Mechanical Po*er nput
)he total mechanical po*er input to a li7uid in a stirred "essel is:
In the !resence of a sparged gas:
+et li,-id gasexpanding impeller mechanical #otal P P P P ++=
impeller -ngassed impeller gassed P P <<

By the end of this section you will be able to: 'istinguish the flow !atterns generated
by different im!ellers under different o!erating conditions
+alculate the im!eller discharge flow from a"ailable flow numbers

mpeller Pumping 1ction
Both radial and a#ial im!ellers e#ert a pumping action within the tan$

,ortices enerated "& mpeller 'lades
Both radial and a#ial im!ellers !roduce strong $ortices behind them

&a/e
:isc

,ortices enerated "& a isc Tur"ine
A balanced "orte# !air de"elo!s behind a &ushton turbine blade con"eying away turbulent energy
*ource: John *mith Mi#ing OO

,ortices enerated "& a Pitched 'lade Tur"ine
)he single line "ortices from !itched blade or hydrofoil im!ellers are less intense that those generated by the flat blade of a &ushton turbine
*ource: John *mith Mi#ing OO

Flo* Pattern %or 1xial mpellers in 'a%%led Tans
1xial impellers tend to !um! do*n*ard or up*ard de!ending on the direction of rotation
o*n*ard pumping impellers !roduce an a#ial or angled3 main flow that: im!inges on the tan$ bottom first
mo"es u!wards near the tan$ wall
con"erges radially inwards and then
returns to the im!eller to feed it

Flo* Pattern %or (adial mpellers in 'a%%led Tans
(adial impellers !um! the li7uid radiall& forming a radial et
If C/T is sufficiently high as the li7uid et im!inges on the tan$ wall it s!lits u!wards and downwards

ChE702 129

Flo* Pattern %or (adial mpellers in 'a%%led Tans
% C/T is lo* the li7uid et im!inging on the tan$ wall onl& %orms an up*ard %lo* that first mo"es "ertically near the wall then con"erges radially inwards and returns to the im!eller to feed it <single- eight= flow !attern3

ChE702 131

mpeller Clearance and Flo* Pattern Change *ith isc Tur"ines
-or dis tur"ines a flow transition %rom dou"le-eight to single- eight regimes occurs when the C/T ratio dro!s below a s!ecific "alue: -or C/T :0/2 → <dou"le-eight= flow
!attern
-or 0.DC/T D0.2 → either flow !attern can e#ist

mpeller Clearance and Flo* Pattern Change *ith Flat-'lade Tur"ines
Also for %lat-"lade tur"ines the flow !attern changes from <double>eight= to <single>eight= regimes as C/T "aries. -or C/T :0/2A → <dou"le-eight= flow
!attern
-or 0.20DC/T D0.2F → either flow !attern can e#ist

A8iti an/ Armenante* .I1IN2 1(II * 1999.

Experimental ,elocit& Measurement
Local $elocit& measurements inside a stirred tan$ are generally di%%icult
)echni7ues include:
ixing Vessel
mpeller ischarge Flo*
)he pumping action of an im!eller results in a discharge %lo* rate out of the im!eller region Qo&t balanced by an incoming %lo* to*ard the im!eller inflow rate? Qin3. *ince mass is conser"ed it must be that:
where Q is the discharge %lo* rate
))) o-t in ==

mpeller ischarge Flo*
)he im!eller discharge flow rate Q, can be obtained by summing u! the out%lo* contri"utions from all the surfaces of the cylinder en"elo!ing the im!eller:
radial o-t $ace-pper
Flo* )um"er 5or Pumping )um"er6 N &
In order to ma$e the im!eller discharge flow rate non>dimensional one can define the Flo* )um"er. or Pumping )um"er. N &:
3DN
) N ) ≡
ChE702 1+2
Tur"ulent Flo* )um"ers
)he -low umber is to the discharge flow rate what the Power umber is to !ower



(elationship 'et*een Po*er and Flo*
In a number of industrial cases it may be ad"antageous to use im!ellers that !roduce significant circulation within the tan$ but consume little !ower.
)o determine the o!timal im!eller design and o!eration the following ratio:
should be ma#imi6ed. P )

-or a fi#ed im!eller geometry it is:
i.e.:
( ) ( ) ( )253

,ariation o% Flo* and Po*er issipation

Im!ellers with blades oriented !arallel to the shaft !roduce radial %lo* and ha"e high po*er dissipation rates ltho&gh th!ir pu#ping action is signi%icant
As a conse7uence their % @P ratios is
low

Im!ellers with blades forming a small3 angle with the !lane of rotation !roduce axial %lo* and ha"e relati$el& lo* po*er dissipation rates ltho&gh th!ir pu#ping action is also signi%icant
As a conse7uence their % @P ratios will be high
1xial impellers generate less tur"ulence and shear

!ptimi@ation Strategies to Maximi@e Pumping E%%icienc&
)o maximi@e pumping e%%icienc& i.e. ma#imi6e the Q/P ratio3:
choose im!ellers with high % @P ratios
if ca!ital cost must be minimi6ed select im!ellers with the same v tip ?π'3 but

If a s!ecific flow rate Q must be achie"ed then by rearranging it is:
)o lower P at constant Q one can lower N while increasing '. )his a!!roach decreases the o!erating cost ∝ P 3.
3 +
If a s!ecific !ower in!ut P must be maintained then by rearranging it is:
+
the flow is fully turbulent.
+hanging the D/T ratio usually has little influence on P and % !ro"ided

t $ir$ is a measure of how long it ta$es
)
ChE702 157
Circulation Time
)he circulation time t $ir$ is directly related to how long it ta$es: a small neutrally buoyant tracer !article
to pass consecuti$el& through the same region e.g. the im!eller region3
a tracer to !roduce t*o consecuti$e concentration peas in the region where the detector is

By the end of this section you will be able to: 'escribe the conce!ts of blend time and
degree of uniformity and how the can be determined in the lab
+alculate the blend time for any desired degree of uniformity in a mi#ing tan$

'lend Time 5Mixing Time6
If a miscible tracer is added to a homogenous li7uid in an agitated tan$ the local concentration measured with a detector3 %luctuates *ith time
)he am!litude of the concentration fluctuations will decrease with time

ChE702 161
'lend Time

'lend Time Facts
Blend time and the achie"ement of a homogeneous state can be critical in some o!erations e.g. fast chemical reactions3
In any real mi#ing tan$ "lend time is ne$er @ero
4omogeneous phases do not mix instantaneousl&I

Experimental etermination o% 'lend Time
'etection of tracer can be accom!lished with a "ariety of techni7ues including: acid>base indicators e.g. !; meters3
ion>s!ecific electrodes
electric conducti"ity meters
light adsor!tion meters
Experimental etermination o% 'lend Time
A tracer is ty!ically added to the tan$ ty!ically at the surface3

(ensor
;racer
C 2inl
C 2inl
C 2inl
Equations %or the etermination o% 'lend Time
;ere two a!!roaches@e7uations for the determination of the blend time will be !resented:
1pproach 9: -asano Ba$$er and Penney/s a!!roach
1pproach 2: 5ren"ille/s a!!roach
1pproach 9
'lend Time and )on-ni%ormit&
)he le"el of non>uniformity or unmi#edeness3 ' is defined as:
where C o and C 2inl are the initial and
final tracer concentrations in the li7uid
Before the tracer addition t ?03 CC o
( )
− −
=

0 & . / - 5 2 &0
&
)

0 & . / - 5 2 &0
&
)

0 & . / - 5 2 &0
&
)
0 & . / - 5 2 &0
&
< ( t ) 7 ( F i n a l - ( t ) ) 5 F i n a l - %
) p(t )
Mixing (ate Constant. k

Mixing (ate Constant. k
)he greater the k "alue is: the %aster the oscillations *ill die out
the %aster "lending will be
the shorter the mixing time will be

)on-ni%ormit& Peas
)he absolute "alues of the height o% the oscillation peas p8t), in the 7 t cur"e will determine whether a re7uired le"el of homogeneity has been achie"ed
)he "alues of p8t) can be found from:
( ) t k et 1 t p −== ma=

p8t) determines the le$el o% non- homogeneit& non>uniformity3
,ne can ar"itraril& decide when sufficient uniformity has been achie"ed by selecting a small enough p8t) "alue e.g. 0.0F im!lying that the largest fluctuation is FS of the final 7 "alue3
-or t →∞ p8t) →0
egree o% ni%ormit&. )
It is con"enient to introduce the egree o% ni%ormit& 9 defined as:
where 9 is ust the com!lement of p for e#am!le if p?0.0F 9 ?EFS im!lying that the li7uid is EFS homogeneous3.
)hen:

)hen:
)his e7uation relates the "lend time. t ) re7uired to achie"e a desired le"el
of 9 to 9 and * . -or e#am!le the time re7uired to achie"e EES homogeneity is:
( ) k
'lend Times to 1chie$e ,arious ) +s
( ) ( )
(#am!le: t 9 for a s!ecific 9 and that
for EES t EE3:
( ) ( )

determine the mi#ing rate constant *
As usual dimensional analysis is used:
t')eim)e&&er

)hen:

=
Mixing (ate Constant

)he e7uation for */N is then:
where the !arameters and b de!end on the t&pe o% impeller used
5.0
Impeller a b
1.01 2.30
0.6+1 2.19
Chemineer E3 0.272 1.67
-or a %ixed set of geometric "ariables i.e.: same impeller
same D/T ratio
same +/T ratio
If */N is constant:
i.e.:
for same im!eller same D/T ratio same H/T ratio3 irrespecti$e o% scale
constant=N t 4

'lend Time and mpeller Speed
)he higher the agitation speed is the shorter the "lend time will be
-or geometricall& similar systems this e7uation:
does not change *ith scale 5eometrically similar small and large
"essels ha"e the same "lend time only if the agitation s!eed N is the same at "oth scales
N t 4
-rom:
it follows that i% mixing time is to remain unchanged during scale-up the agitation s!eed N must remain constant !ro"ided geometric similarit& is maintained
constant=N t 4
E%%ect o% !ther Factors on 'lend Time
)he procedure outlined "e%ore can be used to obtain the blend time t 9 for
the case in which:
the flow is tur"ulent

Correcti$e %actors can be a!!lied to t 9 to account for:
di%%erent %lo* regimes

where: " &e correcti"e factor for the effect of &e
" µ : correcti"e factor to account for the effect of "iscosity differences
" ρ correcti"e factor to account for the effect of density differences
( )
ChE702 198
t 9 &e→∞R µ T?R ρ ?03 is the
<standard= t 9 i.e. under fully turbulent
conditions with an added fluid ha"ing the same "iscosity an density as the li7uid in the tan$3 calculated as outline beforeR
t 9 &eR µ TR ρ 3 is the mi#ing time
calculated to account for the effect of &e "iscosity and density differences.

'lend Time at i%%erent (e&nolds )um"ers
" ;! is the correction factor to account for &e
effects when &e is below 0000 and the fluid is not fully turbulent. &emar$: " ;! ?
for &e0000.
,nce t 9 has been calculated for &e 0000
it is !ossible to obtain t 9 at other &eynolds

&0 &00 &000 &0000 &00000 &9?00
$e
0+&
% u
E%%ect o% ,iscosit& (atio on 'lend Time
" µ : is the correction factor to account for "iscosity effect when the "iscosity of the added fluid is greater than that of the li7uid in the tan$.
In order to calculate " µ : the "iscosity ratio:
must be determined first.
asano et a&.* Chem. Eng .* 199+

" ρ is the correction factor to account
for the effect of differences in densities between the added li7uid an the li7uid in the tan$.
If the density difference is 6ero <standard= case3 " ρ?.
In order to calculate " ρ the &ichardson
umber &i must be calculated first.

E%%ect o% ensit& i%%erence on 'lend Time
)o account for the effect of density differences the &ichardson umber is introduced:
22 DN
" g Ri
asano et a&.* Chem. Eng .* 199+

'lend Time and eometric Similarit&
In %ull& tur"ulent geometricall& similar systems the e7uation below still holds:
)his im!lies that "lend time experiments can be conducted in small scale equipment to determine the abo"e constant and that this e7uation can be used for scale-up !ur!oses
constant=N t 4
)he procedure to calculate t 9 is then:
set the desired "alue of 9
fi# D, T, H and the im!eller ty!e
set N
calculate the <standard= blend time

Scale-up and 'lend Time
( )
( )
( ) ( )
( )

=
&ecall that:
If P/V is $e!t constant during scale>u! of geometricall& similar systems:
the "lend time increases with the linear3 scale %actor raised to the 2H< po*er
N t 4

=

'lend Time and !ther Time Scales
It is always im!ortant to ma$e sure that blend time is much shorter than the other time scales that may be im!ortant to the !rocess

'lend Time in Small Tans and Large Tans
Blend time is ty!ically short in small la"orator& tans but much longer in larger tans

An em!irical mi#ing rule of thumb states that:
)
ChE702 21+
-rom the definition of circulation time t $ir$ 1/Q3 and -low umber N Q:
it follows that:

can be obtained recalling that:
)hen:

-inally:
( ) ( )

⋅

⋅⋅==
"

N Q?0.8
If 9 ?EES α ?QL3 ⇒ t 33?F.G×t $ir$
If 9 ?EFS α ?0.F×QL?G0.F3 ⇒ t 36 ?G.F×t $ir$
'lend Time and Circulation Time

ChE702 220
)he !re"ious results confirm that the "lend time is t&picall& a multiple o% the circulation time
t 9 @t $ir$ is ty!ically in the range G> for t 36 @t $ir$ and Q>E for t 33 @t $ir$
)hese results "alidate the em!irical mi#ing rule of thumb stating that
<Blend )ime= ≈ Q × <+irculation )ime=
'lend Time and Circulation Time

1pproach 2
'lend Time Equation in Tur"ulent (egime3 1pproach 2
-or mi#ing in the tur"ulent regime &eC00003 5ren"ille EE23 found:
0.GG D D @T D 0.F0
C @T ? 0.GG
5.05.1
3?195
20.5
'lend Times to 1chie$e ,arious ) +s
( ) ( )
If 9 ?EFS is the reference degree of homogeneity:
-or e#am!le:
'lend Time Equation in Tur"ulent (egime3 1pproach 2
-or mi#ing in tur"ulent regime &eC00003 the 5ren"ille e7uation becomes:
0.GG D D @T D 0.F0
C @T ? 0.GG
( ) 5.05.1
3?1
ChE702 226

-rom:
and:
mpeller E%%icienc& in Tur"ulent (egime3 1pproach 2
-rom the !re"ious e7uation for turbulent regime it is: all impellers o% the same diameter are
equall& energ& e%%icient i.e. achie"e the same t 9 at the same !ower in!ut@mass3
shorter t ) are achie"ed with larger impellers at the same po*er inputHmass
blend time is independent of fluid !ro!erties when scaling at constant po*er

'lend Time Equation in Transitional (egime3 1pproach 2
-or mi#ing in the transitional regime C200D&eDC00003 5ren"ille EE23 found:
0.GG D D @T D 0.F0
C @T ? 0.GG
2
'lend Time Equation in Transitional (egime3 1pproach 2
-or mi#ing in transitional regime C200D&eC00003 the 5ren"ille e7uation becomes:
0.GG D D @T D 0.F0
C @T ? 0.GG
( ) 2
-rom:
and:
mpeller E%%icienc& in Transitional (egime3 1pproach 2
-rom the !re"ious e7uation for the transitional regime it is: all impellers o% the same diameter are equall&
energ& e%%icient i.e. achie"e the same t 9 at the same !ower in!ut@mass3
shorter t ) are achie"ed with larger impellers at the same po*er inputHmass
blend time is proportional to $iscosit& and in"ersely !ro!ortional to density

ChE702 233
Conclusions3 Po*er. Flo* and 'lend Times in Mixing Tans
nder turbulent conditions the po*er dissi!ated by an im!eller depends on: agitation s!eed P ∝ N <3
im!eller si6e P ∝ D6 3
ty!e of im!eller P ∝ P3
density of the fluid P ∝ =3
1xial im!ellers and radial im!ellers generate different circulation patterns
In general a#ial im!ellers generate more flow !er unit of !ower dissi!ated than radial im!ellers

Documents

Liquid Mixing Fundamentals