. III u .l‘ {-1\\£95}
. ...N.Iff..fi. tel...F..I‘i.(i«!.f
. VEPIIVLKAI sc‘o...) 1......

..

z};

‘ré‘JflI‘?
. o ‘
1d

‘ . .1‘U
.fiu .Y.

. '
. \ .
‘. F; L93 . .
.a’ \l‘ 1". ‘th‘u‘d" . 7 ' ‘
. . - Cafe...
1 ..

t" .
’H .

uS.‘ I). ._.!l.ya§. .. ‘.\$~ .
Ilh. . $1,.
,3 .3 . .

.1?“-

‘1 O. 2h.“ . . . huh

“Omkwrlfl: “13:14.1...
Earn. ($54.3...

\. 1.... n .

.‘. AI . ( n lvymb.‘

.21. aka;

L 40‘ ..... o! :f‘;
I . .1!!! _ .

11’).
u '13.,
‘21:th
.LLI§,.IF\t-,f.. .0!‘.L.«1
.n\| u‘.-.!.vf..: (E 13:... $
5. 1.304...‘
~|1\\ ’.¢..
.1 .

,_ 1‘ '1
T;.,.

5. .
$2.! .1 .

1.ij I; .Iolvl‘vgr. 3:9:
p. 3. :5lnlu rllu§réuffrtca
10 (3-.
u€o)vH&..
)tlt? a
r . 10p

. 12¢- icl I. .1-

infam‘flu’

I.
I» It). It!» .
.fil... LN! ..... at" $3152)...
.1... . Srrt:1)if‘uf.vu

I)
if 5'“! 3.3;... t!!!
II.

.E . :1. LI 3......
. In.” gargb .luflvltflrl.
viii-.13.!
I. 50.. .1 gigs?

. in.
the:

l
:9.

[rig ‘3?"
Pill i] .3"
It!
I
(I

.1511..-
I fail i"l‘l""‘~:a)
a: ll:

. Igjflii§¢g. ..
Onfl' In
. . u v lei .tll Cs!!! .
. :i D .. . «I!

. .eHiFthrrpfl

”“43; 1

I’ll

.
09b}!!- !In In... Vifll A?

Ojratbrlfio ‘x‘lltnvpu‘lzl {Ill ‘

v .J.n .ylnx.

iiil‘r‘ .

'I'-"

 

919,093 5W3

 

~ w ““1

 

 

 

miiiifiim 1 11111311 111111
flmiversity

This is to certify that the

dissertation entitled

Evaluating the Rheological Properties
of Power-Law Fluids Using
Mixer Viscometry

presented by
Maria Elena Castell Perez

has been accepted towards fulfillment
of the requirements for

PhD. Agricultural Engineering
degree in

 

 

fw .F f/éAL

Major professor

 

Date 2/3] (70

 

MS U i: an Affirmative Action/Equal Opportunity Institution 0-12771

_—_—————__

 

 

PLACE IN RETURN BOX to remove this checkout from your record.
TO AVOID FINES return on or before date due.

 

DATE DUE DATE DUE DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

MSU Is An Alfirmetlve ActionlEqual Opportunity Institution

 

 

 

EVALUATING THE WWICAL PROPERTIES OF
POWER-LAP FIDIDS USING
MIXER VISCOHETRY

By

Maria Elena Castell—Perez

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPY

IN

Agricultural Engineering

Department of Agricultural Engineering

1990

.
I

:. :wn.r
d‘-

"‘56‘c

lei...
‘

...r.e

:‘o g.

\: qr

 

um um MUSICAL PROPERTIES GP

in (a .. ,4 roan-um FLUIDS usnic
mum

 
  
 
 
   
   
 
  
 
 
  
  
   
   

 

' t
'- .‘L. prod-1,1.-
, .

_|.V
“ANA“:
1 ,7. “Zinc...“ Maria Elena Castell Perez

. v

' . ' psi-so) 3:. BY
. ,Jlil

v

‘Lf‘ntl 1L1"

. fl

} :‘V.‘72‘»3-‘ ‘
. I ".V' .
é 1"}!éilmprocedure was developed to determine the flow properties of
' ..:"?‘7 ‘

mayor-lawfluids using mixer viecometry data. The procedure is

I?” the. mixing system: (1) Model 1, representing the analogy
attic cylinders systems with negligible end effects; (2)
resenting the same analogy with the addition of a term which
#10 and effects. In the case of a flag impeller, an alter-
.. rm evaluated, i_.e. , Model 3 representing a cylinder of

11th two blades attached.

fr viscometer (Haake Rotovisko) over the same shear

#40 l/s) with standard power-law fluids (aqueous CMC

 

 

2"?

——ve
‘n.
_
....
r:..
‘ >
v.
y x
—-.
.,1
o
‘1-
., i
u.
.
P‘-
..v
. I

 

.11:
Il'

 

solutions). The best approximation of the average shear stress of stan-
dard power—law fluids was obtained with the Model 2 (concentric
cylinders analogy with end effects) for the paddle impellers. Model 3

(d - de ) gave the best results for the flag impeller.

The procedure was evaluated with an actual power-law food
product, creamy salad dressing. In this case Model 2 also gave the best
results for the paddle impellers. For the flag impeller, Model 3 yielded
higher values of the average shear stress, with Model 1 (concentric
cylinders analogy with negligible end effects) being the best model for
approximation of the average shear stress of the tested food material.

Established mixer viscometry methods, the viscosity matching and
the slope method, were evaluated for determination of average shear
rates when agitating time—independent non-Newtonian fluids. Results
indicate that the use of a constant value of the mixer proportionality
constant, k', is not valid for all ranges of fluid rheological
properties, system geometry and operating conditions. The relationship
between the average shear rate and the impeller rotational speed was
found to be different for different rheological properties and system
geometries. The effect of the fluid properties on the value of k' is not

clearly understood and needs further investigation.

Approved

 

Major Professor Date

Approved

 

Department Chairperson Date

 

 

 

 

 
   
   
 
 
  
  
   

fill ‘l09lv Inua‘tm’ a L 1 T
mic and Mu 1

t :he (<n1;se v' i i
p?

‘.89 dppreciatinq ..

shinimhi "'1,

in, for proV‘ra u :1, f» a . x'x._-

“rel Bflginearing g».~1 a. ' ~ gun

 

Ai- Nteudad n- 13-: ‘,', a’u'

 

 

on-

I

(all:

 

l

.‘0

‘U:‘\V

pus . I . .1.
-\u n: a .bl
AP» ‘ u ‘12-.
v n . ~ .. a
o . - ~ -\~

hi-
.d-

 

 

 

 

ACKNOWLEDGMENTS

I am deeply indebted to Dr. James F. Steffe for suggestion of
the research topic and his continuous support, interest, and patient
guidance throughout the course of this study.

Sincere appreciation is extended to the guidance committee
members: Dr. Ajit K. Srivastava, Agricultural Engineering Department,
Dr. Daina M. Briedis, Chemical Engineering Department, Dr. Alden M.
Booren, Food Science and Human Nutrition Department and Dr. Juan
Menjivar, Nabisco Brands, who served as an outside examiner.

I want to thank my wonderful family, Bernardo, Rosario, Caty,
Many, Jose and Lili, for their love and words of encouragement during
these special years and for understanding my necessary long absence from
home; to Dolores Peres, who always believed in me.

Special thanks goes to Dr. Fred Bakker-Arkema for his moral
support.

I would also like to thank the Romance Languages Department
at Michigan State University for providing the financial assistance.

Special thanks is extended to the entire faculty, staff and
students of the Agricultural Engineering Department, Michigan State

University, for their moral support.

 

115i 0? TABI.
EST OF FISH.
LIST OF SYP‘B
I. MODUCI‘
2. ONEL‘IIVE
3.1mm

3.1USIll-G
3.1.1

3.2 can
we
3.2.1

3.2.2

3.2.3

3'3 ”1X13:

TABLE OF CONTENTS

LIST OF TABLES ....................................................

LIST OF FIGURES ...................................................

LIST OF SYMBOLS ...................................................

1.
2.

 

IIITRODUCTION ...................................................

(NBJECTIVES .....................................................

. LITEIAIURE REVIEW ..............................................

3.1 USING MIXING T0 EVALUATE RHEOIDGICAL PROPERTIES ............
3.1.1 Applications of Mixer Viscometry .....................

3.2 CALCULATION TECHNIQUES IN MIXER VISCOMETRY - TRADITIONAL.
METHODS ....................................................
3.2.1 Analysis of Time- Independent Behavior ................

3 .1. Viscosity Matching Method ....................

Linear Shear Stress Method ...................

Slope Method .................................

Combined Slope and Linear Shear Stress Method

Methods Using a Helical Screw ................

3.2.1.5.1 Pressure Difference Method .........

3.2.1.5.2 Direct Determination of the Flow
Curve Method .......................

0)wa
Labio'n'n
i-ai-‘i-ai-a
Ln'eisioi-a

3.2.2 Yield Stress Determination ...........................

3.2.3 Analysis of Time-Dependent Behavior ..................
3.2.3.1 Rheological Model Plus Linear Shear Rate
Method .......................................
3.2.3.2 Combined Rheological and Kinetic Model Plus
Linear Shear Rate Method .....................
3.2.3.3 Subjective Assessment of Thixotropy Using a
Vane Impeller ................................

3.2.4 Elastic Fluids .......................................
3.2.4.1 Power Requirements ...........................
3.2.4.2 The Weissenberg Effect .......................

3 .3 MIXING THEORY ..............................................

3.3.1 Power Consumption in Mixing Vessels ..................

3.3.1.1 Relation Between Flow Pattern and Power
Consumption in a Cylindrical Vessel ..........
3.3.2 Dimensional Analysis for Mixing ......................

3.3.3 Laminar Mixing Region ................................

vi

30
34
35
36
41
42
50

53
53

53
57

60

' __-_ r- a. .

 

3.3.3.1 Laminar Fluid Motion in Agitated Vessels ..... 60
3. 3. 3. 2 Laminar Flow Criterion ....................... 63
3. 3. 3.2.1 Critical Reynolds Number ........... 63
3.3 - 4 Shear Rate Distribution in a Cylindrical Mixing
Vessel ............................................... 67
3.3.4.1 Theoretical Expressions for the Rate of Shear
in a Mixing Vessel ........................... 69
3.3.4.1.1 Concentric Cylinders ............... 69
3.3.4.1.2 Empirical and Theoretical
Expressions for the Impeller
Proportionality Constant, k' ....... 73
4. m: ALTERNATIVE METHOD TO EVALUATE MIXER VISCOMETRY DATA. . . 78
4.1 DETERMINATION OF FLOW CURVES ............................... 78
4.1 - 1 Shear Stress and Shear Rate Approximations ........... 79
4.1.1.1 Model Systems ................................ 79
4.1.1.2 Shear Rate Approximations .................... 80
4.1.1.3 Shear Stress Approximations .................. 86
4.1.1.3.1 Model 1 (Concentric cylinders with
negligible end effects) ............ 86
4.1.1.3.2 Model 2 (Concentric cylinders with
end effects) ....................... 88
4.1.1.3.3 Model 3 (Flag Impeller) ............ 91
5. um um mans .......................................... 93
5-1 EQUIPMENT AND MATERIALS .................................... 93
5 1.1 The Brookfield Mixer ................................. 93
~ 1. 2 Data Acquisition System .............................. 94
5-1. 3 Impellers and Cups ................................... 94
5- 1 .4 Fluids ............................................... 96
5-2 EXPERIMENTAL DESIGN AND PROCEDURE .......................... 96
S -2 .1 Design of Experiment ................................. 96
5 2. 2 Procedure ............................................ 98
5.2.2.1 Preparation of Non- Newtonian Fluids .......... 98
5.2.22.2 Determination of Rheological Properties of
Non-Newtonian Fluids ......................... 101
5.2.2. 3 Calibration of Brookfield Viscometers ........ 101
5..2 2. 4 Data Collection .............................. 102
5'3 CALCULATIONS USING TRADITIONAL MIXER VISCOMETRY METHODS. . .. 105
5 3.1 Viscosity Matching Methods ........................... 105
5 ..3 2 Slope Method ......................................... 105
5'4 CALCULATIONS USING NEW MIXER VISCOMETRY METHOD ............. 105
5.4 1 Determination of Average Shear Rate .................. 105
5.4.2 Determination of Average Shear Stress ................ 106
5 4 3 Procedure for the use of a mixer viscometer to
directly determine the rheological properties of
power-law fluids ..................................... 107
6' “301.1: mm DISCUSSION ......................................... 109

6.1 E5111“.a
11111101
6.1.1

6.1.2 E

6.1.3!

6-2 DETERYI
USING 1

6.1 ESTIMATION OF AVERAGE SHEAR RATE USING MIXERS: TRADITIONAL

ME'I'k-IODS .................................................... 109

6. 1 - 1 Matching Viscosities ................................. 109
6.1.1.1 Power Curves Method (Metzner and Otto,

1957) ........................................ 109

6.1.1.1.1 Estimation of Average Shear Rates.. 119
6.1.1.1.2 Factors Affecting Average Shear

Rates (and k’) ..................... 129
6.1.1.1.2.1 Impeller Rotational
Speed .................. 129
6.1.1.1.2.2 Fluid Properties ....... 129
6.1.1.1.2.3 Cup Diameter ........... 130
6.1.1.1.2.4 Impeller Size (Blade
Height) ................ 131
6.1.1.2 Mixer Torque Curves Method (Mackey et al.,
1987) ........................................ 132
6.1.1.2.1 Factors Affecting k' ............... 142
6.1.1.2.1.1 Impeller Rotational
Speed .................. 142
6.1.1.2.1.2 Fluid Properties ....... 142
6.1.1.2.1.3 Cup Diameter ........... 148
6.1.1.2.1.4 Impeller Size (Blade
Height) ................ 151
6.1.1.2.2 Estimation of Average Shear Rates.. 154
6. 1 - 2 Slope Method ......................................... 160
6.1.2.1 Factors Affecting k’ ......................... 164
6.1.2.1.1 System Geometry ................... 164
6.1.2.2 Estimation of Average Shear rates and
Apparent Viscosity ........................... 171
6-‘1..3 Summary of Discussion ................................ 176
6'2 DETERMINATION OF RHEOIDGICAL PROPERTIES OF POWER-LAW FLUIDS
USING THE ALTERNATIVE MIXER VISCOMETRY METHOD .............. 177
-:E.1 Determination of the Flow Behavior Index, n .......... 177
6'-:!.2 Determination of Shear Stress-Shear Rate
Relationships ........................................ 180
6.2.2.1 Average Shear Rate In the Mixing System ...... 180
6.2.2.2 Average Shear Stress In the Mixing System.... 190
6.2.2.2.1 Torque Approximations .............. 190
6.2.2.3 Flow Curves .................................. 200
6.2.2.3.1 Ideal Fluids ....................... 200
6.2.2.3.2 Food Product ....................... 208
6'3 GENERAL RECOMMENDATIONS FOR APPLICATION OF MIXER
VISCOMETRY ................................................. 227
6..3.1 Mixer System Used in This Study ...................... 227
5..3.2 New Mixer Viscometer System .......................... 230
7' 3% m concmsmns ........................................ 233
8' Wows ma FURTHER sum .................................. 238
l
3 9' ”Pumas

APPEN'DIX A — DIMENSIONLESS ANALYSIS FOR MIXING VESSELS

 

viii

1

APPENDIX 1
APPEMIX 1

'13. REFERENCE:

(NEWTONIAN FLUIDS) ............................... 239
A.l - Development of Dimensionless Functions for
Power-Law Fluids (Used in Slope Method) ........ 243

APPENDIX B - EXPERIMENTAL RESULTS ............................. 245
APPENDIX C - PROCEDURE FOR THE USE OF THE MIXER BROOKFIELD

VISCOMETER FOR NEWTONIAN FLUIDS .............. 263

271

10. REFERENCES .....................................................

A ix

 

in}!
in

L...-
5...;

fix

r‘r.
LII

I
t")
.q

'
P19

a“

F4
L4)

:32?

Va
Pa

Y
J:

TABLE

3.1

6.4

6.5
6.6

6.7

6.8
6.9

6.10
6.11

LIST OF TABLES

Products and Rheological Characteristics Considered In
The Mixing Systems (Castell-Perez and Steffe, 1989). . . .

Studies on Mixing of Elastic Fluids ....................

Modified Impeller Reynolds Numbers for Mixing of Non-
Newtonian Fluids .......................................

Empirical and Theoretical Expressions for Determination
of the Impeller Proportionality Constant, k’ ...........

Experimental Design ....................................
Regression Results of Eqn. (6.2) (Paddle Impellers). . . .
Regression Results of Eqn. (6.3) (Flag Impeller) .......

Values of k’, defined by Eqn. (3.8) (k' - ”yew/N), for
Paddle Impellers .......................................

Values of k', defined by Eqn. (3.8) (k' - ’yav/N), for
Flag Impeller ..........................................

R°Sl'-"ession Results for the fit of Eqn. (6.10) ..........

Values of k', defined by Eqn. (6.12), for Paddle
I-peuers ..............................................

Values of I", defined by Eqn. (6.12), for Flag
Impeuer ...............................................

v‘lues of k' (evaluated using the Slope Method) ........

Values of k' (evaluated using the Slope Method) as a
“fiction of impeller rotational speed ..................

Regression Results of Eqn. (6.13) (Paddle Impellers)..
vfilues of the Flow Behavior Index of Standard Power-
lfiv Fluids Using the Mixer System (Paddles) and a
cOxicentric Cylinders Viscometer (Haake Rotovisko) ......
values of the Flow Behavior Index of Standard Power-
law Fluids Using the Mixer System (Flag) and a
cancentric Cylinders Viscometer (Haake Rotovisko) ......

Values of the Fluid Consistency Coefficient of Standard

Page

44

65

74
99
114

115

127

128

140

141

165

169

170

181

182

 

(.6

ffixe

III.-

13..

(A u

‘l 1..

0.914

Ilia

Illa!

6.16

8.1

3.2

8.3

0.1

3.2

 

Power-law Fluids Using the Mixer System (Paddles) [Model
1: Negligible end effects; Eqn. (4.14)] and a Concentric
Cylinders Viscometer (Haake Rotovisko) ................. 202

Values of the Fluid Consistency Coefficient of Standard
Power-law Fluids Using the Mixer System (Flag) [Model

1: Negligible end effects; Eqn. (4.14)] and a Concentric
Cylinders Viscometer (Haake Rotovisko) ................. 203

Values of the Fluid Consistency Coefficient of Standard
Power-law Fluids Using the Mixer System (Paddles) [Model

2: Concentric cylinders with end effects; Eqn. (4.21)]

and a. Concentric Cylinders Viscometer (Haake Rotovisko). 204

Values of the Fluid Consistency Coefficient of Standard
Power-law Fluids Using the Mixer System (Flag) [Model

2: Concentric cylinders viscometer; Eqn. (4.21)] and a
Concentric Cylinders Viscometer (Haake Rotovisko) ...... 205

Values of the Fluid Consistency Coefficient of Standard
Power-law Fluids Using the Mixer System (Flag) [Model

3: d - d ; Eqn. (6.23.1)] and a Concentric Cylinders
V18COmetgr (Haake Rotovisko) ........................... 206

Vllues of the Flow Behavior Index (n) and the Fluid

C0118 istency Coefficient of Salad Dressing Using the Mixer
system (Paddles) [Model 1, Eqn. (4.14)]) and a Concentric
cylinders Viscometer (Haake Rotovisko) ................. 219

values of the Flow Behavior Index (n) and the Fluid
con8:1.stency Coefficient of Salad Dressing Using the Mixer
system (Paddles) [Model 2, Eqn. (4.21)]) and a Concentric
cylinders Viscometer (Haake Rotovisko) ................. 220

values of the Flow Behavior Index (n) and the Fluid
consistency Coefficient of Salad Dressing Using the Mixer

sYStem (Flag) and a Concentric Cylinders Viscometer (Haake
R0tuvisko) .............................................

PgI‘tinent Quantities Involved In Fluid Agitation
t'O<:esses (Newtonian Fluids) ........................... 242

Regreesion Results of P0 - a0 (Refit1 (Fr)a‘2 (Paddle
‘pellers) ............................................. 245

Regression Results of P0 - [30 (R )191 (Fr)!92 (Flag
Illpeller) ....................... {3 ...................... 246

Values of p. , p, and p. from k' - p. (d/b)fl2 + 193

(Matching Method of Power Curves) for Power-law Fluids. 247
F'vactors for Use of the Mixer Impellers with the

Brookfield Viscometer .................................. 264
Table for the fit of Eqn. (0.2) ........................ 267

xi

 

 
 
  
 
   
 
   
 
  
   
 
   
   
   
   
 
 
 
  
  
 
   
 
   
 
 
 
  
  
 

~02 Viscosity of a Newtonian Fluid (1; - 4. 84 Pa 3)
dwith'l‘he Impeller Factors.....................

~ Menus )1-3 .. . 1.
I m Props” m .1v.1. '.

'_°“" Mn. 1939). .

I'Viaeeeem .6" m .1

- in Agit. r .1». .
Maid (Ron «‘1 .3 -. 1

flow Movie; lml' 5' .,

Whey Dian-m .n '~
"’53.. _

.- QW'AY‘O‘ ‘KH‘ .7"-
ldl Number '.‘ . ,- n. .

‘chAeA-u

>‘-_ 1‘ ‘ . 31‘) Pickle. Imml.“
‘.' la: (Mode: 1‘11
. [7&1"
. . m leech". 1H! Haws-(33..
' '. 3:123 : u
"I. and cup kam- nn 1! .4.
5‘11”)“ ~‘4h:ee-a.n.‘..

.
v‘ 541'

t“
.1"

.g Wr1uui‘l 1:- ,' 1|;.'

‘ ' Mm: R1: 83231342.»)? . -.¢.
w M1. Imil.t‘ H 25‘ -A.:\;7..-"

‘*V.I~Ollvtse...u-
.

-. Met Bunion-on; t-w— given-
. .we lmllere at», fan-or”: ’
'tiz..a...... .. .

7 - Met Relau‘mga‘e ‘... y........‘_..
, “m1. Imllexe in; 1'.wa ~ ‘
V‘O‘Jv‘ ;'~O“-e.

mun...“

1’ .111 '7 j. I"

ween “served '3. . ”5,. (me-1r; ‘Ia,r.r..‘..a\.‘,3!,.

a.” Meme to ; t: new 6.6". 1.1-1 ier

269

LIST OF FIGURES

FIGURE

3.1 Typical Impellers Used In The Determination Of

Rheological Properties With A Mixer (Castell-Perez
and Steffe, 1989) ......................................

3 . 2 Mixe :r: Viscometry Techniques ............................

I-‘lo‘ilr patterns in Agitated Vessels. A) Newtonian Fluid.
B) Elastic Fluid (Rod-climbing) ........................

3.4 8) Simplified Flow Model for Power Correlation; b)

Relative Velocity Distribution Between Paddle and Liquid
(Nagata, 1975) ..........................................

3.5 Fluid Motion in Agitated Vessels (Nagata, 1975) .........
3.5 SChematic Diagram Showing The Relation Between Power
Number and Reynolds Number for a Paddle Impeller
(Nagata, 1975) ..........................................
4-1 M"€161 Systems. A) Paddle Impeller; B) Flag Impeller;
C) Flag Impeller (Model 3 ..............................
5'1 Padtile and Flag Impellers (all dimensions are in cm.). .
5'2 :zn‘ple containers and cup base (all dimensions are in
5'3 nix1ng System ...........................................
5-1‘ schematic Diagram of Experimental System ................
6'1 Power Number/Reynolds Number Relationship for Newtonian
:I‘Slids Using the Paddle Impellers (Cup diameter - 5.5
'l .....................................................
6'2 POVer Number/Reynolds Number Relationship for Newtonian
F:leuids Using the Paddle Impellers (Cup diameter - 3.5
cll) .....................................................
6'3 POver Number/Reynolds Number Relationship for Newtonian
uids Using the Paddle Impellers (Cup diameter - 2.54
ch). ....................................................
6'4 I) Predicted Po Versus Observed Po ; b) Power Correlation
For The Paddle Impellers ................................
6.5

Q) Predicted Po Versus Observed Po ; b) Power Correlation

xiii

Page

10

51

54

61

62

81

95

97

100

104

110

111

112

117

 

6.12

6.18

6.19

6.20

6.21

6.22

For The Flag Impeller ...................................

a) Shear Stress Versus Shear Rate; b) Apparent Viscosity
Versus Shear Rate (Hydroxypropyl Methylcellulose 1%) . . ..

Average Mixing Shear Rate as a Function of Rotational
Speed of the Paddle Impellers. a) System 3; b) System 4.

Average Mixing Shear Rate as a Function of Rotational
Speed of the Paddle Impellers. a) System 5; b) System

0‘

Average Mixing Shear Rate as a Function of Rotational
Speed of the Paddle Impellers. a) System 7; b) System 8.

Average Mixing Shear Rate as a Function of Rotational
Speed of the Flag Impeller. a) CMC 1.5%; b) CMC 1% ......

Average Mixing Shear Rate as a Function of Rotational
Speed of the Flag Impeller. a) CMC 2%; b) System 3 ......

Average Mixing Shear Rate as a Function of Rotational
Speed of the Paddle Impellers (d/D-O.327). a) CMC 2%;
b) cuc 1% ...............................................

Plot of Mixer Proportionality Constant as a Function of
IIIlibeller Size for the Non-Newtonian Fluids (d/D-0.327) ..

"titer Torque Versus Viscosity Times Speed for Calculating
“1X01: Coefficient, k2 (Paddle Impellers) ................

PlTe<iicted Mixer Coefficient Versus Observed Mixer
c°¢f£icient (Paddle Impellers) ..........................

H1lter Torque Versus Viscosity Times Speed for Calculating
Mixer Coefficient, k2 (Flag Impeller) ...................

m3K6): Proportionality Constant, k' , Versus Rotational

serd, N, for Paddle Impellers When Mixing a Non-Newtonian
Fbud (CMC 1%) ..........................................

Hiker Proportionality Constant, k' , Versus Rotational
speed, N, for Flag Impeller When Mixing Non-Newtonian
Fluids. a) CMC 1%; b) CMC 2% .............................

Plot of Mixer Proportionality Constant, k' , Versus
R“Ditational Speed, N, for Non-Newtonian Fluids (Paddle
Illpellers) ...............................................

1’lot of Mixer Proportionality Constant, k’, as a Function
0f Flow Behavior Index, n (Flag Impeller) ...............

Plot of Mixer Proportionality Constant, k' , as a Function
of Fluid Consistency Coefficient, m (Flag Impeller) .....

Plot of Mixer Proportionality Constant, k’ , as a Function

118

121

123

124

125

126

133

137

138

143

144

145

146

147

 

6.23

6.24

6.28

6.29

6.38

6.39

of Flow Behavior Index, n (Paddle Impeller) ............. 149

Plot of Mixer Proportionality Constant, k', as a Function
of Fluid Consistency Coefficient, m (Paddle Impeller)... 150

Plot of Mixer Proportionality Constant, k', as a Function
of System Geometry. a) Impeller Size (d/b); b) Cup Size
(d/D)

.................................................... 52
Plot of Mixer Proportionality Constant, k' as a Function
of Fluid Consistency Coefficient, m (Pa 5 ) for Different
Paddle Impellers. a) 100 rpm; b) 20 rpm .................. 153
Average Shear Rate as a Function of Impeller Rotational
Speed for a Paddle Impeller (CMC 1%) ..................... 155
Average Shear Rate as a Function of Impeller Rotational
Speed for a Paddle Impeller (CMC 2%) ..................... 156
Average Shear Rate as a Function of Impeller Rotational
Speed for a Flag Impeller (CMC 1%) ....................... 157
Average Shear Rate as a Function of Impeller Rotational
Speed for a Flag Impeller (CMC 2%) ....................... 158
Apparent Viscosity as a Function of Average Shear Rate for
a Paddle Impeller (CMC 2%) ............................... 159
Apparent Viscosity as a Function of Average Shear Rate for
a Flag Impeller (CMC 2%) ................................. 161

a
Plots of Dimensionless Functions [P/(md Nn+1)] Versus
(l-n) for the Paddle Impellers ........................... 162
s

Plots of Dimensionless Functions [P/(md Nn+l)] Versus
(l-n) for the Flag Impeller .............................. 163

a
Plots of Dimensionless Functions [P/(md Nn+1)] Versus
(l-n) for the Paddle Impellers at Different Values of N.. 166

s
Plots of Dimensionless Functions [P/(md Nn+1)] Versus
(l-n) for the Flag Impeller at Different Values of N ..... 168

Average Mixing Shear Rate as a Function of Rotational
Speed. a) Different Cups; b) Different Paddles ........... 172

Average Mixing Shear Rate as a Function of Rotational
Speed for the Flag Impeller .............................. 173

Apparent Viscosity as a Function of Average Shear Rate
Using the Paddle Impellers. a) CMC 2%; b) CMC 1% ......... 174

Apparent Viscosity as a Function of Average Shear Rate
Using the Flag Impeller .................................. 175

‘11 . - xv

6.40

6.41

6.42

6.44

6.45

6.48

6.49

Plot of the torque on the impeller shaft M versus
rotational speed of the impeller N for 2% wt% aqueous
solution of CMC (Paddle impellers) .......................

Plot of the torque on the impeller shaft M versus
rotational speed of the impeller N for 2% wt% aqueous
solution of CMC (Flag impeller) ..........................

Experimental Versus Calculated Values of k'; a) k' Defined
by Eqn. (4.5); b) k' Defined by Eqns. (4.5) and (6.13)
for small gap (Paddle Impellers) .........................

Experimental Versus Calculated Values of k'; a) k' Defined
by Eqn. (4.7) (Paddle Impellers) ........................

Experimental Versus Calculated Values of k'; a) k’ Defined
by Eqn. (4.7); b) k' Defined by Eqn. (4 7) (d-de) ......

Measured Versus Predicted Values of Torque on the Impeller
Shaft Using Eqn. (4.14) (o) and Eqn. (4.21) (x); a) CMC
1.5%; b) CMC 1% (Paddle Impeller # 1, (d/b) - 1.8) [EE -
Model 2 (end effects); NEE - Model 1 (no end effects)]..

Measured Versus Predicted Values of Torque on the Impeller
Shaft Using Eqn. (4.14) (o) and Eqn. (4.21) (x); a) CMC

2. [EE - Model 2 (end effects); NEE - Model 1 (no end
effects)] ...............................................

Measured Versus Predicted Values of Torque on the Impeller
Shaft Using Eqn. (4.14) (o) and Eqn. (4.21) (x); a) CMC
1.5.; b) CMC 1% (Paddle Impeller # 5, (d/b) - 0.36) [EE -
Model 2 (end effects); NEE - Model 1 (no end effects)]..

Measured Versus Predicted Values of Torque on the Impeller

Shaft for All Systems and Fluids (Paddle Impellers); a)

Eqn. (4.14) (Model 1 - NEE); b) Eqn. (4.21) (Model 2 -
fl) ....................................................

Measured Versus Predicted Values of Torque on the Impeller
Shaft for All Systems and Fluids (Flag Impeller) Using
Eqn. (4.14) (Model 1) (Negligible end effects) ..........

Measured Versus Predicted Values of Torque on the Impeller
Shaft for All Systems and Fluids (Flag Impeller); a) Eqn.
(4.21) (Model 2); b) Eqn. (6.23.1) (Model 3) ............

Flow Curve for 2% wt% Aqueous Solution of CMC Determined
Using the Mixer Viscometer with a Paddle Impeller ........

Comparison of the Flow Curve for a Standard Fluid
Determined Using the Concentric Cylinders Viscometer with
the Data Obtained Using the Mixer Viscometer with the
Paddle Impellers in a Large Cup [Model 2, Eqn. (4.21)]);
a) CMC 1%; b) CMC ....................................

xvi

178

179

184

186

189

194

196

198

199

201

209

 

6.53

6.61

6.62

6.63

Comparison of the Flow Curve for a Standard Fluid

Determined Using the Concentric Cylinders Viscometer with

the Data Obtained Using the Mixer Viscometer with the

Paddle Impellers in a Small Cup [Model 2, Eqn. (4.21)]);

a) CMC 1%; b) CMC 2% .................................... 210

Comparison of the Flow Curve for a Standard Fluid

Determined Using the Concentric Cylinders Viscometer with

the Data Obtained Using the Mixer Viscometer with a

Flag Impeller [Model 3, Eqn. (6.23.1)] a) CMC 1%; b) CMC

2t ...................................................... 211

Comparison of the Flow Curve for a Standard Fluid
Determined Using the Concentric Cylinders Viscometer with
the Data Obtained Using the Mixer Viscometer with a

Flag Impeller [Model 1, Eqn. (4.14)]; a) CMC 1%; b) CMC
26 ..................................................... 212
Comparison of the Flow Curve for a Standard Fluid
Determined Using the Concentric Cylinders Viscometer with
the Data Obtained Using the Mixer Viscometer with a

Flag Impeller [Model 2, Eqn. (4.21)]; a) CMC 1%; b) CMC
2t .................................................... 213
Comparison of the Flow Curve for Salad Dressing

Determined Using the Concentric Cylinders Viscometer with

the Data Obtained Using the Mixer Viscometer with the

Paddle Impellers [Model 2, Eqn. (4.21)]; a) Large Cup; b)
Small Cup ............................................. 250

Comparison of the Flow Curve for Salad Dressing

Determined Using the Concentric Cylinders Viscometer with

the Data Obtained Using the Mixer Viscometer with a

Flag Impeller [Model 3, Eqn. (6.23.1)] ............... 216

Comparison of the Flow Curve for Salad Dressing

Determined Using the Concentric Cylinders Viscometer with

the Data Obtained Using the Mixer Viscometer with a

Flag Impeller [Model 1, Eqn. (4.14) .................... 217

Comparison of the Flow Curve for Salad Dressing

Determined Using the Concentric Cylinders Viscometer with

the Data Obtained Using the Mixer Viscometer with a

Flag Impeller [Model 2, Eqn. (4.21) .................... 218

Apparent Viscosity as a Function of Average Shear Rate for
Salad Dressing. Flag Impeller (Model 1: Concentric cylinders
with negligible end effects [NEE]) ..................... 223

Apparent Viscosity as a Function of Average Shear Rate for
Salad Dressing. Flag Impeller (Model 2: Concentric cylinders
with end effects [EE]) ................................. 224

Apparent Viscosity as a Function of Average Shear Rate for
Salad Dressing. Flag Impeller (Model 3: Concentric Cylinders

xv“ “

6.64

6.65
8.1

8.5

8.6

8.7

8.8

8.9

with and effects; d - de ) ............................. 225

Apparent Viscosity as a Function of Average Shear Rate for
Salad Dressing. Flag Impeller. a) Small Cap; b) Wide Cap. 226

Flow Diagram of General Procedure for Mixer Viscometry .. 231

a) Shear Stress Versus Shear Rate; b) Apparent Viscosity
Versus Shear Rate (Hydroxypropyl Methylcellulose 1.5%)... 248

a) Shear Stress Versus Shear Rate; b) Apparent Viscosity
Versus Shear Rate (Hydroxypropyl Methylcellulose 2%) ..... 249

Average Mixing Shear Rate as a Function of Rotational
Speed of the Paddle Impellers. a) System 1; b) System 2
(Torque Curves- Matching Method) ......................... 250

Average Mixing Shear Rate as a Function of Rotational
Speed of the Paddle Impellers. a) System 9; b) System 10
(Torque Curves- Matching Method) ......................... 251

Average Mixing Shear Rate as a Function of Rotational
Speed of the Flag Impeller. a) System 2; b) System 1
(Torque Curves- Matching Method) ......................... 252

Average Mixing Shear Rate as a Function of Impeller
Rotational Speed for Two Paddle Impellers. (Torque Curves-
Matching Method) ......................................... 253

Apparent Viscosity as a Function of Average Shear Rate for
a Paddle Impeller (CMC 1%) (Torque Curves- Matching method)
254

.........................................................

Apparent Viscosity as a Function of Average Shear Rate for
a Flag Impeller (CMC 1%) (Torque Curves- Matching method)

.........................................................

s
Plots of Dimensionless Functions [P/(md Nn+1)] Versus
(1-n) for the Flag Impeller (Slope Method) ............... 256

s

Plots of Dimensionless Functions [P/(md Nn+1)] Versus

(l-n) for the Paddle Impellers (Slope Method). a) System 3;

b) System 2 .............................................. 257

s

Plots of Dimensionless Functions [P/(md Nn+1)] Versus

(l-n) for the Paddle Impellers (Slope Method). a) System 6;

b) System 4 .............................................. 258

s

Plots of Dimensionless Functions [P/(md Nn+1)] Versus

(l-n) for the Paddle Impellers (Slope Method). a) System 8;

b) System 7 .............................................. 259

s
Plots of Dimensionless Functions [P/(md Nn+1)] Versus

(l-n) for the Paddle Impellers (Slope Method). a) System 10;
5) System 9 .............................................. 260

-...-l I I

 

8.14 Flow Curves for 1% wt% Aqueous Solution of CMC Determined
Using the Mixer Viscometer With the Paddle Impellers .....

8.15 Flow Curves for 2% wt% Aqueous Solution of CMC Determined
Using the Mixer Viscometer With the Paddle Impellers ..... 262

6.1 a) Plot of C, Versus Impeller Rotational Speed; b) Plot
of 0 versus Torque Reading (Paddle Impeller, d/b-0.45)... 266

 

LIST OF SYMBOLS

instrument constant, N m/reading
model parameter, l/s
parameter, Pa/s
constant dependent on geometry
b 2
constant, N s /m

2
surface of screw channel, m

2
surface of the edge of the screw flight, m

model parameter, dimensionless
parameter, Pa/s

impeller blade height, m
constant, dimensionless
empirical coefficient

2 -
constant, kg m sn 2

constant, l/m3

constant, -n

constant, dimensionless

constant, dimensionless
3

constant, m

correction factor

integration constant

constant, s/ma

impeller diameter, m

XX

[Eqn.
[Eqn.
[Eqn.
[Eqn.

[Eqn.

[Eqn.

[Eqn.

[Eqn.
[Eqn.
[Eqn.
[Eqn.
[Eqn.
[Eqn.
[Eqn.
[Eqn.
[Eqn.
[Eqn.

[Eqn.

(3.
(3.

.33);
.57);
.61)]
.1)]

.66)]

57)]

63)]

.66);
.86)]
.39);
.2)1
.2)]
.73);
.41)]
.29);
.50);
.114);

.1)]

 

an.

D'NMH405600.

ran-lb

m

diameter of cylindrical section, m

(Model System 3) [Eqn. (4.32)]
equivalent diameter of helical screw impeller, m
equivalent diameter of flag impeller, m

(Model System 3) [Eqn. (4.32)]
outside cylinder diameter, m

tank (container) diameter, m

defromation rate tensor, l/s [Eqn. (3.103)]
vane diameter, m

empirical coefficient [Eqn. (3.86)]
internal resisting force of fluid, N [Eqn. (3.78)]
gravitational acceleration, m/s2

distance from free surface of liquid and

bottom of cylinder, m

height of inner cylinder, m

vane height, m

impeller proportionality constant, l/rev

constant, dimensionless [Eqn. (3.16)]
rate constant, l/s [Eqn. (3.59)]
constant, l/ma [Eqn. (3.13)]
constant, ms [Eqn. (4.23)]
constant, ms [Eqn. (4.35)]
constant, Pa sn [Eqn. (3.53)]
consistency coefficient at time zero, Pa sno
length of impeller blade, m [Eqn. (4 32)]
length of impeller blade, m

inner cylinder length, m [Eqn. (3.32)]

 

 

3
o

{I
o

O

3%: ‘3 x3 .3 ::

zgaxuongnapag

I:
u

‘6

Tank length, m

-2

n or kg sn /m

consistency coefficient, Pa 5
consistency coefficient of investigated fluid, Pa sn
consistency coefficient of known fluid, Pa sn
torque, N m or kg mz/s2

equilibrium torque, N m

torque resulting from the shearing in the
two and surfaces of a cylinder, N m [Eqn. (4.10)]
initial torque, N m

torque to agitate standard solution, N m

torque required to agitate investigated fluid, N m
torque resulting from the shearing in the

cylindrical wall, N m [Eqn. (4.10)]
torque required to agitate known fluid, N m

torque in screw channel, N m [Eqn. (3.49)]
torque in gap, N m [Eqn. (3.49)]

flow behavior index, dimensionless

number of impeller blades

flow behavior index, dimensionless [Eqn. (3.53)]

flow behavior index at time zero, dimensionless

flow behavior index of investigated fluid, dimensionless
flow behavior index of known fluid, dimensionless
impeller rotational speed, rev/s

first normal stress function, Pa

empirical coefficient [Eqn. (3.86)]
parameter (2 0) [Eqn. (4.31)]

2 3
power, N m/s or kg m /s

 

WW
pa.

impeller radius, m
radial distance from the axis to any section of
the impeller, m [Eqn. (3.80)]

radius of the cylindrically rotating zone, m

arithmetic mean radius, m [Eqn. (3 32)]
inner cylinder radius, m [Eqn. (3 32)]
outer cylinder radius, m [Eqn. (3 32)]
impeller (cylinder) radius, m [Eqn. (4.17)]

inner cylinder (bob) radius, m

outer cylinder (cup) radius, m

radius of inner cylinder, m

fluid rheological parameter, dimensionless

time, min

fluid characteristic time, s

vane yield stress, Pa or N /m2

maximum measured torque (vane method), N m .
impeller velocity, m/s

impeller relative velocity, m/s

instrument constant, Pa/reading [Eqn. (3.31)]
instrument constant, Pa/reading [Eqn. (3.36)]
clearance from surface to top of impeller, m

clearance from bottom of impeller, m

W1;

Qo

instrument reading [Eqn. (3.31)]

empirical coefficient

 

xxiii

empirical coefficient

empirical coefficient

empirical cofficient

empirical coefficient

empirical coefficient

empirical coefficient

empirical coefficient

shear rate, l/s

average shear rate, l/s

standard shear rate, 1/s

geometric ratio, do/de [Eqn. (3.50)]
first normal stress coefficient, Pa 52

time dependent parameter, dimensionless [Eqns. (3.57-58)]
initial value of A, dimensionless

equilibrium structural parameter, dimensionless
Newtonian viscosity, Pa s

non-Newtonian (apparent) viscosity, Pa 5
apparent viscosity at time zero, Pa 5

apparent viscosity of investigated fluid, Pa 5
apparent viscosity of known fluid, Pa s

Casson viscosity, Pa 5

apparent viscosity at equilibrium, Pa 5
limiting viscosity at zero shear rate, Pa 3
angular velocity, rad/s

fluid density, kg/m3

shear stress, Pa

approximate initial shear stress, Pa

xxiv

 

2
0 average shear stress, Pa or N/m

av
”ax approximate shear stress, Pa [Eqn. (3 63)]
0e approximate equilibrium shear stress, Pa
0e shear stress at the end surface of the cylinder, Pa
do yield stress, Pa
”on: Mizrahi-Berk yield stress, Pa
03 shear stress of standard solution, Pa
0w shear stress at the wall, Pa
0x shear stress of investigated fluid, Pa
0y shear stress of known fluid, Pa
”Yo yield stress, Pa

ayl shear stress, Pa

T stress tensor, Pa

W angular velocity of rotation, rad/s

o e b 5

El Elasticity Number

P Torque [Eqn. (3.68)]
ha Aberystwyth Number

Po Power Number

Re Mixing Reynolds Number

Rge Generalized Reynolds Number

Ren Power-law Reynolds Number [Eqn. (6.8)]
T Torque [Eqn. (3 72)]
"i Weissenberg Number

 

CHAPTER 1
INTRODUCTION

Many food materials, especially the suspension-type, present non-
Newtonian flow characteristics. To evaluate, design, and control
processing systems for the product, the engineer needs to understand the
flow properties of these materials. This work is motivated by the need
for new instrumentation and methodologies to evaluate the rheological
behavior of suspension-type food products.

When dealing with food materials that are suspensions (tomato
sauce, baby foods, salad dressings, etc.), conventional narrow gap
Viscometers (such as tube, cone-and-plate and concentric cylinders
Viscometers) become unsuitable for complete characterization because of
effects such as phase separation (and consequent slip at the walls), and
blocking of the measuring gap by particle aggregates. Significant errors
in measurements can also occur due to the destruction of the particles.

An alternative instrument to use in evaluating the flow
Pr°perties of suspension-type fluids is the "impeller" or "mixer" vis-
C°meter, based upon the theory of mixing non-Newtonian fluids developed
by Metzner and Otto (1957). The method has proven to be useful for
determination of rheological properties of viscous fermentation broths
(3°ngenaar et a1., 1973; Kemblowski et a1. , 1988) and suspension—type
f°°d products (Rao, 1975; Steffe and Ford, 1984).

The Metzner and Otto approach is based on the assumption of a
direct proportionality between average shear rate in the fluid and

rotational speed of the impeller, and the proportionality constant, k',

 

depending only on the geometry of the impeller. This assumption implies
that the investigated fluid has no effect on the proportionality con-
stant. The application of an average shear rate given by the above
assumption has successfully been applied in the development of power
correlations in mixing vessels. For rheological applications, however,
it could lead to significant errors in the evaluation of the rheological
behavior of shear-dependent fluids. Hence it is necessary to explore the
implications of using the linear shear rate in mixers, to quantify the
interaction of the mixer proportionality constant, k' , and system
geometry as well as fluid properties, and to develop a procedure for
evaluating fluid properties (especifically, power-law) using a mixer
viscometer.
The movement of solid surfaces (e.g., an impeller) in contact
with a fluid causes the fluid to move in some characteristic pattern
Whi-Ch results in the development of internal stresses and the applica-
tion, on the solid surfaces, of characteristic forces which must be
contonuously counterbalanced (e.g. by a drive motor) ir order to sustain
the fluid motion (Charles, 1978). The nature of the flow pattern and the
Illafinitudes of internal stresses and applied forces depend primarily on
the geometry of the system, the rate of fluid motion and the rheological
Pmperties of the fluid.
Approximate expressions for the determination of the shear
Stress and shear rate in a vessel with an impeller can be obtained using
approximate geometries (such as the concentric cylinders analogy) for
the System, since the complicated geometry of the system makes the
solution of the proper set of differential equations of motion a dif—

ficult task .

CHAPTER 2

OBJECTIVES

The objectives of this dissertation are:

. To review and evaluate established mixer Viscometry methods.

- To analyze the influence of system geometry, operating conditions and

fluid properties on mixing shear rates when agitating power-law

fluids .

- To obtain expressions for estimation of the shear stress and the

Shear rate in the mixing systems.

- To develop a new procedure for rheological characterization of power

-1aw fluids, using mixer Viscometry data.

 

CHAPTER 3

LITERATURE REVIEW

In the food and chemical industry many operations are dependent
on effective agitation and mixing of fluids; therefore, an understanding
of the phenomena involved in mixing fluids with complex rheology (time-
dependency, shear-thinning, elasticity) is increasingly important.

The chapter leads of with a review of certain aspects of the use
of niacers and mixing principles to evaluate the rheological behavior of
fhdd or semi-solid materials and a summary of the developments in the

81333- The theory of mixing is reviewed in the second part of the chap-

ter .

3.1) U§Ifl§ MIXING TO EVALUATE RHEOLOGICAL PROPERTIES

Use of mixers for rheological evaluation was initially suggested
by Metzner and Otto (1957) with the development of a method of ap-
PrOXi-mating power-law parameters for non-Newtonian pseudoplastic (shear—
thinning) fluids using an apparent viscosity, "a calculated from data
Obtained with a mixing vessel. Their results were verified by Calderbank
(1958) and Metzner and Taylor (1960), who observed that local fluid
shear rates were found to be directly proportional to impeller speed,
f°r both Newtonian and non-Newtonian fluids, and high shear rates oc-
curred close to the impeller. Saravacos and Mayer (1967) observed
similar‘behavior when studying the rheological parameters of fruit

Purees in agitated kettles. They found that the apparent viscosity

 

decreased considerably at high agitation speeds for pseudoplastic fluids
and the shear rate at a particular speed of agitation varied with the
distance from the center of rotation.

The general principle of measurement used in mixer Viscometry is
based on the determination of the torque on the shaft of the impeller as
a function of its rotational speed. Thus, a suitable value of the ap-
parent viscosity for a non-Newtonian fluid can be obtained from
viscometric measurements if a representative value of shear rate "y in
the given vessel can be predicted. It must be noted that due to the
relatively complex flow (complex velocity profiles, hence shear rates)
escablished in the mixing vessel, mixer Viscometry only results in
aI’Prfifiximate data and only average values can be determined. The mixing
V6336]. is generally a cylindrical container with numerous possible

1mPeller configurations (Figure 3.1).

3.1.1) Applications of Mixer Viscometry

Mixers for rheological studies have been used by investigators in
the food and fermentation areas (Table 3.1). In a review of the
equipment used in fermentation studies, Solomons (1971) expressed the
need for an "in-situ" measurement system and suggested the possibility
0f uSing a rotating impeller instead of the usual "cup and bob" type of
ViscOmeter to determine the rheological properties of fermentation
cultllres. An initial application of a mixer as a viscometer was con-
sidel'ed by Bongenaar et a1. (1973), who developed a reproducible method
of characterizing the rheological properties of mold suspensions using a

turb1ne impeller instead of a rotating cylinder in a rotational

 

.- lilo
mum: K m w- moo
l

x + II“ BIA - 211°

«33* .....
mm» 5% €222:

' N.. -

b—-o—-

A'DIOI N7 [132.23] A! - Mon

.23... cg m... ”’—

 

Ann. now V 1 w- mu m"- '-
sum .

ii
5;
.3:
3:5

 

 

‘Figure 3.1: Typical Impellers Used In The Determination Of Rheological

Properties With A Mixer (Castell-Perez and Steffe, 1989).

A

Table 3.1:

Mixing Systems (CastelloPerez and Steffe, 1989).

Products And Rheological Characteristics Consideredvln The

 

 

 

 

Reference Mlxlnq Syetem(e) Product“) Characteristic

5m 0‘ d- Tm'bhe Idd We 0. )«H I'll-ll

(n73)

cum It d- He. peede Apueot puree Sno—thhnho
Dud- e! d. (1"!) Tattle Peeueeeieeflee Steer rote
Em a do ‘1‘") M . Nd“ K . m m d “I
F“ . M (l...) mu: peede m2 puree Tin: among;
Kw t
m (1000) Hull-III m. rum-mum broth 5..—mm“
KW “ d- (19”) Hdled ewe- Peueeoleeuee How an»
“nu-e: et a. (nu) w ea'ee cu mdene Vleudty
L“. et ‘- ("57) Vale am oed mention m‘ m
My et oL (1987) Flo. peede Peeedeelodla Steer-01mg
"CI-eu- a he!

“museum Veee Ru med We mu m.-
Oll- e: no. (nee) Vme. cur Apple-un- nae w—
R.’ (1.75) Flee poede m tomato puree Qua-minim
"“ ‘- cod-H1004) no. . my Pause-us- 9"“ "*-
R-de a a. (1914) 1...... Pal-In emu e. we Weee
Sun-am a a. (1 sea) ene- Meet new My
5“. et «1. mm, a. m. doe-MIM’Y‘

(name)
“In“ (urn) lmpdler rum-«em anu- Vleeedty

M & Ferd (1085) PM eedde Wt puree agar-mum

e! d. (nee) fig. ”a. Sta-let m p.“ .1...de

1% et d. (lees) Hdled ea. Tame euloe Vleeedly

Vet-ed. a cum m cyl- Kedh meeeedene mu cm.-

1.71
whey 0(t d- )(1377) Flat pedde Stu-eh em Peek moo-m
W‘, 0‘ d- (1.“) Mil“ M a” Pod Mir
] M et a. (1W1) Tue-u ”m, ”m m”..- My
Qua... u a. (1m) v... m m. M m-

 

 

 

A

 

Viscometer. This method solved the problem of particle settling and
Phase separation because the impeller achieved mixing and no water layer
was developed adjacent to the impeller blades (Charles,1978). In recent
Years, many types of rheological behavior have been studied using mixers
(Table 3.1): shear-thinning, yield stress, time-dependency, elasticity,
etc.

Some investigators have used the mixer concept to determine
SUbJective rheological parameters that are a function of measuring
instrument. An example of this is the Ottawa Starch Viscometer (Voisey
et 81. , 1977) developed for measuring the properties of starch slurries
during cooking, where the starch slurry is mixed by a flat paddle rotat-
ing at: constant speed in a stationary bowl. The torque generated by
shearing the starch slurry was recorder by a transducer, and the test
time Was considerably shorter than for other instruments such as the
Brabender Amylograph. With a similar concept in mind, Steffe et a1.
(1989) developed a mixer viscometer system to evaluate the flow behavior
of corn starch slurries during gelatinization. The method requires small
samPIE volumes (13 m1.) and short test times. A commercial version of
the i‘l'lstrument is being implemented. Walker et a1. (1988) described the
latest efforts dedicated to the evaluation of pasting behavior of
Sta“ET-"hes. An instrument developed for the wheat industry, the Rapid
v18¢"~Analizer (RVA), uses a disposable cup and plastic paddle, and
gives a pasting curve resembling the Brabender curve. The method also
requires small sample sizes (25 ml.) and short test times. The instru-
“lent has been tested and is comercially available.

Adoption of mixer Viscometry techniques has been limited by the

1‘e
1atively high cost of the necessary instrumentation. The performance

0f a low cost concentric cylinder viscometer was analyzed by Griffith

and Rao (1978) . Castell-Perez et a1. (1987) developed a low cost mixer

Viscometer system, based on the Brookfield Viscometer, for determination
0f the rheological properties of power-law fluids.

Due to the growing interest in on-line instruments for the
measurement of rheological properties of fluids during processing,
Kemblowski et al. (1988) developed a measuring system consisting of a
helical screw impeller rotating in a draught tube, based on torque
Measurements, for on-line Viscometry. Tamura et a1. (1988) investigated
the applicability of the helical screw rheometer developed by Kraynik et
31- (1984) for on-line rheological measurements.

From the point of view of improved process efficiency and quality
°f the final product, applications of mixer Viscometry techniques rather
than conventional techniques have been and can be used to successfully
j*“VeStigate the flow properties of food products, with a major potential
f°r send-solid, suspension-type food materials.

3‘2) IO T C UES IN MIXER VISCOME Y - TRADITIONAL METHODS

Various techniques using a mixer viscometer for the determination
of the rheological behavior of food materials have been developed and
tested. Procedures may vary from the geometry of the mixing system to
the type of fluid and rheological behavior considered. Because many food
Ina"tel'ials, especially the suspension-type, present non-Newtonian flow
Q‘ha‘racteristics, mixer Viscometry efforts have dealt primarily with the
rheological characterization of non-Newtonian fluids. Specific calcula-

ti
on techniques, summarized in Figure 3.2, are discussed in

 

Lek

 

10

weavaCLUoH muuoaoomfi> noxwz "N m ouswam

  

.llaefl DIIIII.

 

 

 

 
 

 

 

 

 

 

 

in:
5.59.3. .- 3.53:3.

 

_

3...: 8.8m .- .5...

38: I... a gram :2: .1 t 855.

.32. 3.8 e 8.. _ A a e
.30: .1 «0 3.51.32

_A

 

S... .. stux... .. 3.385.: are.

 

$3: .1 8 .18.

I Mz... asfllasnsob
2.2. .e .- Use-8n

 

E

garners...

 

ensure. x

.181:on

 

 

2.2:. .e t .325
$3; #38 e 3c

 

 

 

 

A82. 0:3“ e .5...
38a: 898 e .131
as: .e .o .38..“

SS: .1 3 nee-B

39am e!»

 

5.2. :8 e :8;

 

 

 

 

 

 

 

 

>0

concoaoo
loser

 

Amp::m acoucoaoocTocehv
:8: 25 a .25.:

38: 8c
2.. I .6... 8.5-: 2...!
2...: as... .- eta
,

 

 

 

 

11

the following sections.

3 2.1) Aaalxsis_af_Iima;Isdsneadeat_§shaxi2r
3.2. 1 .1) Eiscosity Matching Method

Based on the earlier work of Magnusson (1952), who proposed a
Procedure for calculation of the apparent viscosity of a non-Newtonian
fluid in mixing vessels from the power curve for Newtonian fluids, the
Work of Metzner and Otto (1957) provided engineers with a method to
determine the apparent viscosity of non-Newtonian fluids using a mixing

system. Their approach is based on the theory for power consumption in

agitated vessels .

The generally accepted empirical relationship between the Power
number, Po' and the mixing Reynolds number, Re, in the laminar flow

region (Re < 10) for a Newtonian fluid is

—-A-— (3.1)

Where A is a constant dependent on the geometry of the system. The

dimensionless numbers in Eqn. (3.1) are defined as

p0 - ——2;—; (3.2)
p d N
and
2
R __LM (3.3)
e ’7

 

12

where:
d - impeller diameter, m
N - impeller rotational speed, rev/s
p - fluid density, kg/m3

7, - Newtonian viscosity, Pa 5

and the power input, P, is related to the torque M exerted on the im-

peller by
P - 21rNM (3.4)

Combining the equations, the following relationship is obtained,

5 3 ' 2 (3.5)
pd N pd N
3 2
and P-Ad N ,, (3.6)

Thus, power measurements can be used to characterize the vis-
cosity of a Newtonian fluid in a mixing system by measuring the torque
required to turn the shaft of the impeller at a certain rotational
speed. The geometry of the system and fluid density must also be known.

When mixing a non-Newtonian fluid, especially one obeying the

I)
(“oar-law model ,

a-m-‘yn (3.7)

 

13

where:
a - shear stress, Pa
’1 - shear rate, 1/5
In - consistency coefficient, Pa 5n

n - flow behavior index, dimensionless

the viscosity increases from a minimum value closer to the impeller to a
maximum value in regions far away from the impeller (Ulbrecht and
Carreau, 1985). Metzner and Otto suggested that Eqn. (3.1) could be
Valid for a non—Newtonian fluid if an apparent viscosity, 77a, evaluated

at an average shear rate given by
"y - k’N (3.8)

were used. Equation (3.8) assumes that the average shear rate around the
1mPellet is proportional to the rotational speed of the impeller, N,
being k' the impeller proportionality constant.

Once the average shear rate, lav , has been calculated, an
&Verage apparent viscosity can then be obtained (Nagata, 1975). In their
ex1)‘"‘5'3l.mental procedure, Metzner and Otto utilized two identical sets of
mixing equipment, one containing a Newtonian fluid and the other a non-
Newt'~')l'lian fluid. Using the same impeller speed and varying the viscosity
of the Newtonian fluid so that the power measured at each impeller is
the Same, the apparent viscosities should be the same in both instru-
ments. Thus, they were able to determine the apparent viscosity, "a , of

t
he non-Newtonian fluid by knowing the viscosity of the Newtonian under

1
dentical experimental conditions in the laminar region.

at ‘r

14

The procedure for data collection and analysis was:

1. Measurement of the torque required to rotate the impeller at a fixed
rotational speed (Newtonian and non-Newtonian fluids).

2. Determination of Power Number- Reynolds Number curve for Newtonian
fluids. The value of the constant A is determined from the slope of
this curve.

3. Determination of power input and Power number, Po’ [Eqn. (3.2)] at
each impeller speed N using non-Newtonian measurements in the mixing
system.

4. From the curve obtained in (2), read the corresponding Re and find

the viscosity from the expression for the Reynolds number

JN—d (3_9)

5‘ Let n - "a . When the apparent viscosity of the non-Newtonian fluid
18 obtained, the corresponding average shear rate can be determined
from the viscometric curve (0 versus ’7 data, obtained with a
conventional viscometer), based on the assumption that n - 0/7.

6‘ calculate the value of the mixer impeller proportionality constant k’
at a specific value of the rotational speed N using Eqn. (3.8) ,
1e. k' - +/N

Holland and Chapman (1966) outlined a more complete description

of the technique developed by Metzner and Otto (1957):

1 ' w1th "a obtained from Eqn. (3.9), plot "a versus impeller speed N.

ERperimentally determine the apparent viscosity of the non-Newtonian

fluid in a conventional viscometer at various shear rates.

Plot log "a versus log ’7 .

L

 

15

~ From the plot of (1), determine the value of "a at a specified N

b

5- From the plot of (3), read the value of "1 which gives the same "a of
(4).
6- Plot these values of "7 versus the values of impeller speed N.
7- Obtain the value of k’ by measuring the slope of the plot made in
step (6).
The "viscosity matching" method assumes the value of k’ a con-
Stem; which depends only on the geometry of the impeller. Even though
this assumption has been used by numerous investigators, Metzner and
Otto recommended further analysis to determine the effect of the power-
13" parameters (m and n) in the values of the proportionality constant
1"- Wood and Goff (1973) applied the matching viscosities method to
eStimate the average shear rate in a Brabender Viscograph. The values of
iav for the impeller were obtained from a plot of shear rate versus the
viscosity (or apparent viscosity) of the Newtonian and non-Newtonian
fluids- The magnitude of the shear rate where the two viscosities inter-
sect is the average shear rate, fiav (Wood and Goff, 1973).
An alternative procedure to obtain the average shear rate is by

e
quating the two expressions for the viscosities. Hence,

"lav - m #2,, (3.10)
'“'l - -" (3.11)
and av m
1
. n - 1
1,“,- [ n/m 1 (3.12)
31 This procedure was used by Rao and Cooley (1984) and Mackey et

(1987) for determination of rheological properties of fluids

16

characterized by the power-law model.
Mackey et al. (1987) utilized the matching technique to inves-
tigate the effect of some parameters on the assumed constant value of k’

using a Brookfield Viscometer. From power requirements:

"a _ ill:— - — (3.13)
Ad N N
where k2 — 27"", the mixer coefficient, is a constant dependent on
d A

the geometry of the system. To find k2, the torque required to turn the
impeller agitating a Newtonian fluid of known viscosity is measured as a
function of the rotational speed. Torque M is plotted versus 1N and the
31°Pe is equal to l/kz. Thus, the value of constant A from Eqn. (3.13)
is determined. It is important to check that Re <10 (laminar flow
assumption),

The rheological properties of the investigated power-law fluids
are det'erlllined with a rotational cylinder viscometer. Torque versus
rotational speed data are also collected with the mixer viscometer.

Using Metzner and Otto’s approach of matching viscosities,

 

n - "a - macro”1 (3.14)
Thus,
1
1‘: H “'1
k"+[um] (3.15)

17

where k' is a function of the geometry of the system, the rheological
properties of the fluid and the operational conditions. The authors
observed that the value of k' was significantly affected by the values
of the flow behavior index and the rotational speed of the impeller. (At
rotational speeds less than 1.05 rad/s (10 rpm), the value of k’ was a
strong function of n and N). Extensive work should be done to quantify
the interaction of these parameters on the proportionality constant and

the critical limits for the impeller-to-cup diameter ratio (d/D) must be

identified .

3-2-1-2) Linea; §heag Stress Method

Bongenaar et a1. (1973) developed a technique for characterizing
the rheological properties of mold suspensions using a mixer (turbine)
viscometer_ The instrument consisted of a standard 6-blade Rushton
turbine impeller connected to a rotational viscometer and the torque on
the impeller was measured as a function of the rotational speed. Data
analy513 considered that the shear rate was given by Metzner and Otto’s
(1957) ESSumption ( iav - k'N ) and that the shear stress was directly

proportional to the torque M,

aav - k"M (3.16)

where k" is a constant independent of the value of n. A constant value
of k' (- 10) was selected based on the work done by Calderbank and M00-
Young (1959) for turbine impellers. The rheological parameters of the
Pour-law fluid were determined as follows:

1. The value of the flow behavior index of the unknown suspension, nx,

18

is obtained from the slope of the log-log plot of torque,M, versus
rotational speed N.

, the

2. To calculate the value of the consistency coefficient, mx

properties,ny and my, of a calibration fluid are measured in a

conventional rotational viscometer and the following expression is

written
n
M a m x x
J; _ x _ x N k’
M —a "Ir—n n (3.17)
Y Y y N Y k' 37’
Thus,

M n

_x_ M m (3.18)
m - y
x M nx

3’ (mm

In the case of n - (valid when In — n |< 0.1), Eqn. (3.18)
Y X X y

simplifies to

_?5_ (3.19)

with "x and My as torque values for corresponding fluids measured in the
mixer system.

The authors also worked with the "a’ defined as

 

19
a
"a - [‘7‘]. . (3.20)

where 70 is an arbitrary standard shear rate. Thus, for a power-law

 

fluid
m _ "a (3.21)
.n-l
7
and
“x _ "fix (3.22)
M n
Y ay
The procedure was also applied to fluids obeying the Casson
model,

a - a - "c "1 (3.23)

with the Casson viscosity, no , obtained from the slope of a linear plot
of 00.5 versus 70.5 . The yield stress, 00 , is obtained from extrapola-
tion of the M versus N plot, assuming that k' and k" are independent of
the Casson rheological parameters. According to Charles (1978), it is
not clear if the calculated viscosity is an intrinsic property of the
fluid. However, analysis of data is based on well-proven and widely
accepted empirical correlations and the method is useful for determining
the rheological properties of viscous fermentation broths.

The mixer Viscometry technique developed by Bongenaar et al.

(1973) was applied by Roels et a1. (1974) on their investigation of

 

20

fermentation broths. They observed that the use of a mixer prevented
phase separation and settling of the particles. Their derivation for
data analysis is as follows:

From power requirement theory in the laminar region,

2 3

P - A "aN d (3.24)
with A - 64 for a turbine impeller (using Newtonian fluids). Since the

power P is related to the torque M,

g3 3 (3.25)
M - 2" naNd
From measurement of torque M as a function of rotational speed N,
the value of the apparent viscosity "a can be obtained from Eqn. (3.25)

- -—!ZE-— (3.26)

0 3
a 64 N d

Following Metzner and Otto's suggestion that fiav- k’N, with k’ a

constant and with aav - "a 7av , or taking advantage of Eqn. (3.25),
a v. Zfiflk (3.27)
5 64d

Equation (3.27) can also be written as

u - ci a (3.28)
where :
m'
Ci- an' , (3.29)

 

21

is viewed as an instrument constant.

The measured torque as a function of N and the use of Eqns. (3.8)
and (3.28) allows for determination of rheograms for the investigated
fluid in the laminar flow region. Also, the apparent viscosity can be
calculated from Eqn. (3.26). Different rheological models were analyzed
and the authors concluded that the Casson model better described the
rheology of a fermentation broth.

This technique requires further testing and results to be com-
pared with those obtained with conventional viscometry methods. It must
be emphasized that the impeller proportionality constant, k', was as-
sumed a constant for the particular turbine impeller. When the torque is
measured as a function of N, then the constant k' can be found from

- m 7 “'1 - m (k’N) “'1 (3.30)

i-"a

Kemblowski and Kristiansen (1986) adapted the method developed by

Roels et al. (1974) to design a suitable impeller-cylinder system for
continuous on-line measurements in fermentation technology. The analysis
assumed a two-cylinder system. Thus, in a rotational viscometer, the

shear stress a is given by
a _ z a (3.31)
where:

z - a constant of the instrument, Pa/reading

a — instrument reading

 

22

The torque is then determined from the measured shear stress from

the following expression: (concentric-cylinders with narrow gap)

_ M (3.32)
Zillr
m

a
where:
11 - inner cylinder length, m
r - (r1 + r,)/2
and r - arithmetic mean of the radii of inner and outer cylinders, m
r1 - inner cylinder radius, m

r2 - outer cylinder radius, m
Combining Eqns. (3.31) and (3.32),
M - a a (3.33)
where a - instrument constant, N m/reading = 2w11r; z (3.34)

According to Eqn. (3.33), the value of the constant a, obtained
from a concentric cylinder system, is used to evaluate the torque for
the mixer system on the basis of the reading from the instrument, a. The

average shear stress is then obtained from Eqn. (3.32) as

a 1rk' (3-35)
av

or aav - Z a (3.36)

 

23

where Z - a constant for a given geometry of the impeller system,
Pa/reading
with

I 'a (3.37)

The procedure to determine the value of k' for the particular
impeller is as follows:
1. For power-law fluids, the shear dependent viscosity is given by

Eqn. (3.30) and

3L": - III1(1<'N)“'1 (3.38)
ANd

or

(3.39)

where B is found from experimental curves of torque versus rotational
speed.

2. The value of the proportionality constant k' is now a function of the
geometry of the impeller and rheological properties of the fluid,

since

T1
k' - [115,] (3.40)

with A determined from Newtonian Po versus Re curves according to

Eqn. (3.1).

24

Eqns. (3.8) and (3.36) allow determination of the curve flow for
the investigated fluid. The technique proved to be sensitive and able to
detect the variation of rheological parameters, especially the consis-
tency coefficient, m, resulting from the differences in processs
parameters. Modifications of the instrument are being made for on-line
application. It is important to recognize that this method assumes a
narrow gap between the rotating unit and the cup. Hence, care must be
taken not to violate this assumption when using a system with geometri-
cal characteristics other than the ones used in this study (six—blade

turbine impeller, H/d - 2.7 and 1.8, d/D - 0.93).

3 .2 . l . 3) Slope Method

Based on the initial work of Metzner and Otto (1957), Rieger and
Novak (1973) developed a method to determine the value of the impeller
proportionality constant for the agitation of highly viscous fluids
characterized by the power-law model, in the laminar region of flow. It
is called the "slope method" in this study.

From dimensional analysis, the following relationship is obtained

__2_ _ cm) (3.41)

Using the power relationship given by Eqn. (3.1):

P __°_(‘L (3.42)
0
en
2 2_
with R _d N “2 (3.43)

25

and
C(n) - constant, f(n, geometry)

For a Newtonian fluid,

P _ __A__n_ (3.44)

Replacing 11 with "a for a power-law fluid,

P _, 3a Lk'N)“'1 _ M (3.45)
2 R

o N d p en
Comparing Eqn. (3.45) with Eqn. (3.41),

l

C(n) - A k'“‘ (3.46)

or log C - log A - (l-n) log k' (3.47)

with C given by Eqn. (3.41)

The authors suggested that Eqn. (3.8) is valid for a particular
impeller only if a plot of log [P/(mdan+1)] versus (l-n) is a straight
line [Eqn. (3.47)]. The slope of this line is equal to -log k' . If the
plot were nonlinear, Eqn. (3.8) would be invalid since k' would not
exist. Results suggested that the linear shear rate assumption was
useful for engineering calculations of power consumption with certain
mixer impellers. However, certain dependence of the value of the con-
stant k' with the flow behavior index, n, was observed. This procedure

has been tested to determine the average shear rates of food

26

materials by Rao (1975).

3.2.1.4) Qanhinsd Slgpg and Linear Shear Stress Method
Rao (1975) combined the procedure of Rieger and Novak (1973) for

determination of the impeller proportionality constant with the proce-

dure for determination of the rheological properties of power-law fluids

developed by Bongenaar et a1. (1973). This technique will be called the

"Combined Slope and Linear Shear Stress Method" in this review, and it

has been utilized with food products by several investigators (Rao and

Cooley, 1984; Steffe and Ford, 1985; Ford and Steffe, 1986; Castell-

Perez et a1., 1987).

The procedure is as follows:

1. Collect torque versus rotational speed data of several non-Newtonian
(power-law) fluids and the investigated fluid with the mixer system.

2. Determine the rheological properties of fluids with a concentric
cylinder viscometer.

3. Plot log [P/(mdan+1)] versus (l-n) (Rieger and Novak, 1973) using
torque data obtained from mixer system. Find value of proportionality
constant k' for that particular impeller at a specific value of
rotational speed.

4. Find value of flow behavior index of investigated fluid nx, from
slope of log torque versus log rotational speed data from concentric
cylinders.

5. Determine value of consistency coefficient of the investigated fluid
mx , from mixer torque data, fluid properties and k', Eqn. (3.18)
(Bongenaar et a1., 1973).

Rao and Cooley (1984) compared the mixer Viscometry techniques

27

developed by Metzner and Otto (1957) (Viscosity Matching) and by Rieger
and Novak (1973) (Slope) for complex geometry impellers and observed
that results obtained from both techniques were in good agreement. They
also observed that the ”slope method“ could lead to large errors when

finding the value of k' , since
k' - 10"“pe (3.48)

The advantage of this method is its relative simplicity as compared to
the matching of viscosities required in Metzner and Otto's. However, the
"Viscosity matching" method seems to yield more consistent values of k'

for a particular type of impeller.

3.2.1.5) WW
3.2.1.5.1) Egesgure Difference Method

Using a helical screw, Kraynik et a1. (1984) designed an instru-
ment to measure the viscosity of concentrated suspensions of coal
particles in organic solvents. This Viscometry technique differs from
all the Others in the fact that it relates the pressure difference in
the fluid to the viscosity of the fluid. The instrument is essentially a
metering screw pump operating at zero discharge. The sample can be
pressurized and the pressure difference is measured across two ports
Spaced at different heights along the outer wall by a series of pressure
transducers (Kraynik et a1., 1984).

The authors emphasize the advantages of pressure measurements
OVer t°rQue measurements in high-pressure rotational instruments where

fitti
“3 seals are required around a rotating shaft that could affect

28

force measurements. This viscometer has potential in high-pressure
Viscometry and could be applied to determination of the pressure-
dependence of viscosity. It is also suitable for chemically-reacting and
heterogeneous fluids. Applications of this technique in on-line vis-

cometry of food products has been investigated by Tamura et a1. (1988).

3.2.1.5.2) Dirge; Detemingtign of The Flow Curve Method

Kemblowski et a1. (1988) suggested that the methods of correla-
tion of Power number as a function of the Reynolds number using the
Metzner and Otto assumption may be suitable for engineering purposes,
9‘11: not precise enough for rheological applications. The authors
developed a method which enables a direct determination of the flow
curve without the need for power data obtained with Newtonian fluids.

The analysis assumes a system of two concentric cylinders to
model the impeller system (helical screw impeller rotaing in a draught
tube). The torque on the impeller shaft is the combination of the torque
resulting from the shearing in the screw channel plus the torque result-
ing from the shearing in the gap between the edge of the screw flight

and the innner surface of the draught tube (Kemblowski et a1., 1988)-
Thus

M-Ml +14, (3.49)

h
w ere M1 ._ torque resulting from shearing in screw channel, N m

M2 ‘ torque resulting from shearing in the gap, N m

29

The technique for determination of the flow curve of the inves-

tigated fluid is the following:
1. Using a helical screw impeller rotating in a draught tube as the

measuring system, determine the torque M on the impeller shaft as a

function of rotational speed N.

2. A log-log plot of M versus N should give a straight line for power

law fluids and the slope is equal to n.

3- Calculate the parameters which characterize the geometry of the

measuring system :

A1 - surface of the screw channel, 111
2
A2 - surface of the edge of the screw flight, 111

de - equivalent diameter of the helical screw impeller according

to Chavan et al. (1972), m

4. F01: a given value of rotational speed, the shear rate is determined

from Metzner and Otto's assumption that
)7 _ k'N (3.8)

With 761v - shear rate on the surface of the "equivalent" cylinder

and 41rCr1
k'- 2 (3.50)
1-36

 

The shear stress is given by

30

- 43—, (3.51)
2Hd

aav
where:
Cr1 - correction factor - f(n, ¢) (See Calderbank and Moo-Young,
1959)
H - total height of the inner cylinder, m
M - experimentally determined torque on the
impeller shaft during the shearing of the
investigated fluid, N m
p - geometric ratio, dO/de
do - outer cylinder diameter, m
Significant changes of k' with the value of n were observed.
Comparison of experimental data with those obtained with a concentric
cylinders system showed that the mixing instrument yielded reasonable
results. Again, care must be taken when applying this method to dif-

ferent impeller geometries due to the importance of the d/D ratio in the

development of the theoretical analysis.

3.2.2) e d tr 3 Dete i ation

Knowledge of the yield stress is important in handling, process-
ing and transport of fluids. The presence of a yield stress can affect
the settling of particles in concentrated suspensions (Nguyen and Boger,
1983). Also, agitation of such fluids often gives a well-mixed region
close to the impeller and a stagnant or near-stagnant fluid in the
remainder of the container if the yield stress is not exceeded (Solomon
et a1., 1981). This well-mixed region around the impeller has been

called a cavern (Witcherle and Wein, 1981) and the boundary of the

31

cavern is defined by the surface where the local shear stress equals the
fluid yield stress (Nienow and Elson, 1988).

Nguyen and Boger (1983) investigated the applicability of the use
of a vane for yield stress measurements in concentrated non-food suspen-
sions. Haimoni and Hannant (1988) used it on cement slurries.A vane
(Figure 3.1) consists of 2-8 thin blades centered around a small
cylindrical shaft. This technique has been called the "vane method".

In this technique, the fluid under investigation is placed in a
container and the vane (which is attached to the torsional spring-
driving motor system of a concentric cylinder viscometer) is fully
immersed into the sample, then rotated very slowly at a constant speed,
and the torque required to maintain this motion is recorded as a func-
tion of time.

The technique detects the yielding moment when the torque exerted
on the vane shaft reaches a maximum. The presence of such a maximum in
the torque response is a characteristic of yield stress materials which
can be explained by the concept of structural defamation and breaking
of bonds in flocculated suspensions (Nguyen and Boger, 1985a).

From a torque balance on the surface of the impeller, the yield
stress can be calculated from the measured maximum torque, Tm , and the

dimensions of the vane, by

H l
T 33L _V _ (3.52)
o - 3 +
at!) D 3
V V

Where:

To - vane yield stress, Pa

32

Tm - maximum measured torque, N m
“V - vane height, m

Dv - vane diameter, m

The method provided satisfactory yield stress measurements only
if the vane was rotated at sufficiently low speeds. At high speeds,
significant viscous resistance together with instrument inertia and
insufficient damping may introduce errors to the measured TIn and hence
to the calculated value of yield stress. Nguyen and Boger (1983) recom-
mended some operational procedures:

1. Vane should be operated at rotational speeds below 10 rpm.

2. Depth of sample and diameter of the container should be at least
twice as large as the length and diameter of the vane to minimize any
effects caused by the walls of the container.

3. The vane should be placed at approximately the center of the
container.

4. Geometric criteria (Nguyen and Boger, 1985a) for satisfactory

measurements:
Hv/Dv < 3.5 ; D/Dv > 2.0
Zl/Dv > 1.0 ; 22/Dv > 0.5
where:

D - container diameter, m
21 - clearance from surface to top of impeller, m
2, - clearance from bottom of impeller, m

Values of yield stress obtained with the vane were compared with

33

those obtained by other methods and the agreement confirmed that the

vane method is useful for measuring accurately and directly the true

yield stress of concentrated suspensions (Nguyen and Boger, 1985a). The

advantageous features of this technique are:

1. Introduction of the vane into the sample does not significantly
disturb the sample prior to measurement.

2. Wall slip effects are eliminated and particles remain unsettled.

3. It allows measurement of the yield stress under static conditions and
within the material itself.

4. Technique requires short operation time and low cost apparatus.

5. Experiments are easy to perform and of high precision.

Keentok et a1. (1985) observed that the vane diameter had negli-
gible effect on the ratio of the diameter of the fracture zone to the
diameter of the vane. Their data supports the use of the vane for yield
stress measurements if a diameter correction is applied.

Leong et a1. (1987) measured the yield stress of brown-coal water
suspensions using this technique. Yoshimura et a1. (1987) utilized this
technique for the measurement of yield stress of oil-in-water emulsions,
conducting stress-controlled rather than shear-controlled experiments.

Qiu and Rao (1988) investigated the determination of yield
stresses of food materials using a mixer viscometer with the vane
method. The authors observed that the magnitudes of do for applesauce
were higher than those obtained by extrapolation of the Herschel-Bulkey

and the Mizrahi-Berk model,

n
a -a + [K’y J (3.53)

where :

34

2
00m - Mizrahi-Berk yield stress, Pa

2

Km - constant to be determined, Pa sn

nm - flow behavior index, dimensionless

and very similar to those magnitudes of the Bingham yield stress ob-
tained by the common procedure of extrapolation of the linear portion of
the shear stress-shear rate data. Two different impellers (a star im-
peller and a vane) were used and impeller geometry seemed to affect the
values of do. The technique proved suitable for the measurement of yield
stresses of food suspensions for a specific impeller and rotational

speed.

3.2.3) Analysis of Time-Dependent fiehavior

A better understanding of the flow properties of time-dependent
fluids is essential in handling and process design. For instance, reduc-
tion in the viscosity of the material by mechanical treatments prior to
pumping may imply lower transport energy requirements and minimization
of start-up problems usually associated with occasional pipeline shut-
down (Nguyen and Boger, 1985b).

During the mixing process, an element of thixotropic fluid ex—
periences short periods of time at high shear rates close to the
1Illpeller and longer periods in the lower shear rate regions remote from
the impeller (Edwards et a1., 1976). The rheological state of the
material will depend upon this shear history and the instantaneous shear

rate, which will affect the power requirements for the impeller.

35

3.2.3.1) Rheglogigal Mgge], 3123 Linear Shea; Rate Method

Edwards et al. (1976) developed a procedure for the calculation
of power consumption with time when a thixotropic fluid is agitated from
rest using an impeller which rotates at constant speed. Even though a
simplifying assumption, the use of Metzner and Otto's linear relation-
ship ( i - k'N ), provided a simple means of predicting the power input-
time behavior for thixotropic fluids using the Newtonian data [value of
constant A from Eqn. (3.1)] and that for time-independent non-Newtonian
fluids (value of impeller proportionality constant, k').

If the thixotropic fluid is agitated for a time t in a mixing
container using an impeller at rotational speed N, this is considered to
be equivalent to shearing the fluid in a viscometer, from the same
starting condition, at the average shear rate for time t. Thus, using
the viscometer at constant shear rate, the average apparent viscosity "a
of the thixotropic fluid at time t can be measured. Experimental proce-
dure consisted of:

1. Measurement of torque as a function of time as the impeller rotates
at constant speed.

2. Obtaining equivalent na/time data in a concentric cylinder
viscometer.

3. Calculating k' from a plot of "a versus N .

Some evidence of the dependence of k' on fluid properties was
present but the authors assumed an average value of the impeller con-
stant. They concluded.that it was possible to extend the average shear
rate ”av ) concept for time-independent non-Newtonian fluids (Metzner

and Otto, 1957) to the mixing of thixotropic fluids. The procedure

36

proved satisfactory for a variety of impellers and thixotropic fluids
provided that the impeller was capable of maintaining the entire
fluid in motion.

Nienow and Elson (1988) strongly suggested that this approach

( ”yav 0: N ) should be carefully revised when using it for time-dependent
fluids, especially shear-thickening fluids due to the still unknown flow
behavior of dilatant fluids in mixed vessels. The authors conclude that
the method for determination of k' developed by Metzner and Otto (1957)
should be carefully revised and even repeated for different fluids and

mixing systems .

3.2.3.2) Com ne eo o ical And inetic Model Plus Lingr Shear Rage
Meshes

Sestak et a1. (1982) developed a procedure for calculating the
time-dependent torque necessary for mixing inelastic thixotropic fluids
by means of impellers. The relationships of the impeller (anchor
agitator) torque versus time for constant rotational speeds when mixing
a bentonite-water suspension, were measured. The deformation histories
were expressed by means of initial values of the structural parameter,
A0 , at the beginning of any mixing experiment. A stepwise change of
rotational speed was engaged and time-dependent torque values were
measured.

A time-dependent apparent viscosity given by

r) --ߣ)- (3.54)

37

was used, and an expression for the time-dependency of the impeller
torque was obtained,

M - daa(t) (3.55)

A—
2nk'
The authors also compared several models of thixotropy and con-

cluded that Cheng's model

. n
yo yl + my (3.56)

—§%— - a(1-)) - b y A (3.57)
where:

a - yield stress, Pa
yo
0 - shear stress, Pa
yl
A - time-dependent structural parameter (f(y)),

which ranges from an initial value of 1.0 for

zero shear time to an equilibrium value, Ae’

which is less than 1.0 (Tiu and Boger, 1974)

a - model parameter, l/s

b - model parameter, dimensionless

was the best for calculations of the impeller torque-time variations for
an arbitrary past deformation history of a thixotropic fluid in a mixing
process. The ability of this model to include the influence of the past
deformation history upon the instantaneous apparent viscosity of the

fluid was also proven.

38

A more complete analysis was developed by Ford and Steffe (1986),
who combined mixer viscometry techniques with a fundamental analysis CUE
thixotropy to determine the basic parameters describing the time-
dependent behavior of starch-thickened strained apricots. Tiu and Boger

(1974) obtained a model to describe the structural breakdown of a

product:
no
a - A [ ayo + KO 1 ] (3.58)
where:
no
KO - consistency coefficient at time zero, Pa S
n0 - flow behavior index at time zero, dimensionless
A - structural parameter, accounting for time-dependent
effects, dimensionless
ayo - yield stress at time zero, Pa

with the decay of the structural equation assumed as a second order rate

equation:

JA— - -k1(A - A )2 (3.59)
(it e

where:
k1 - rate constant - f(i), l/s
Ae - equilibrium structural parameter, dimensionless

Q; - change in A with respect to time, 1/s
dt

39

The relationship between apparent viscosity and time is

1 1 + alt (3.60)

where:

fie - apparent viscosity at equilibrium, Pa 5

n - apparent viscosity at time zero, Pa 3

80

t - time, 3

k1 ’7 (3.61)
n

 

ayo + K01

and the value of Ae was calculated as:

n *‘1

A _ __e.___ (3.62)
e . n
ayo + K01

The technique developed by Ford and Steffe (1986) is as follows:
1. Use Metzner and Otto's approximation: yav- k'N (k' - 4.46 for the
paddle impeller used (Steffe and Ford, 1985)).
2. Use the linear shear stress assumption: 0 - k"M (k" - 9835 for
the system used (Ford, 1984)).

3. Find an approximate shear stress for the sample, aax’ given by

”g M (3.63)

witil as - shear stress for standard solution, Pa

40

0x - shear stress for test sample, Pa
Ms - mixer torque when agitating standard solution, N m

Mx - mixer torque when agitating test sample, N m

4. Plot 1/(na - ne) versus time for each sample.
5. From linear regression obtain values of a1 (slope) and 1/("a - ne)
(intercept). [See Eqn. (3.60)]

6. Plot values of a1 versus fiav’

7. The torque at time zero, Mo, is obtained as

Mo 9835 b + 9
Equation (3.64) is obtained from b - 1/("a - ne) . Since
' /' the value of the 510 e b become b — 78V
"3 0 7av ’ P S, —- aao - 0e

Substituting the expressions for the shear rate and the shear stress and
manipulating the data yields Eqn. (3.64).
8. Calculate the value of "so from Eqn. (3.63) using the calculated
value of M .
o
9. Plot 0 versus 7 .
ao av
10. Find the values of K0, no and ayo from linear regression using
Herschel-Bulkley model: a - 0° - n 7 n
11. Determine Ae using Eqn. (3.62).

12. Determine k using Eqn. (3.61).

1
13. Obtain complete rheological characterization of sample [Eqns. (3.59)

and (3.60)].

41

This technique mathematically describes irreversible breakdown
and it proved useful for suspension-type products because the slip at
the wall and breakdown problems involving product loading are minimized

(Ford and Steffe, 1986).

3.2.3.3) snhjeetive Aeeeeenene ef Ibinetrony Using e Vane Impeller

When studying the time-dependent rheology of highly concentrated
and flocculated suspensions of bauxite residue (red mud), Nguyen and
Boger (1985b) found that the concentric cylinder viscometer was un-
suitable since the transient data obtained were not reproducible due to
the presence of slippage at the walls. A mixing system using a vane
impeller was found to be particularly suited for following the time-
dependent transformation of the structure of red mud suspensions without
causing any significant disturbance to the material. The procedure was
as follows:

1. Agitation of the suspension either in a capillary viscometer or in a
separate container using an anchor impeller rotating at constant
speed.

2. After a determined period of mixing, the impeller is stopped to allow
for rheological measurements.

3. Using the vane method (Nguyen and Boger, 1983), the vane is slowly
iJillllersed into the sample, then rotated at a speed of 0.1 rpm,
and torque measurements are recorded.

4° At the end of the test, the suspension is remixed and the procedure
repeated until no further changes in the flow properties are

°bServed.

42

The vane method was also employed to quantify the thixotropic
recovery with resting time. In the recovery experiment, the suspension
was allowed to rest undisturbed in closed containers and the yield
stress was determined at intervals of resting time (Nguyen and Boger,
1985b).

Experimental results provided a complete description of the
thixotropic behavior of highly concentrated red mud suspensions. The
drastic reduction in the magnitude of the rheological properties with
mixing, and the subsequent slow increase in the yield stress when at
rest, may be a way to characterize irreversible thixotropic behavior.
Simple thixotropic models were formulated for correlating the experimen-
tal results. The same technique was followed to characterize the time-

dependent behavior of brown-coal suspensions (Leong et a1. , 1987).

3.2.4) Elaetic Flnide
Many fluid and semi-solid foods exhibit viscoelastic behavior,

i.e. they exhibit viscous and elastic properties simultaneously. Due to
their complex rheology, a complete understanding of the phenomena in-
volved in mixing these fluids is important in industrial operations to
ensure proper selection of process and geometrical variables (Ulbrecht,
1974).

Even though the effects of fluid elasticity on agitators are not
totally clear, elasticity is known to affect the power required for
agitation and to produce differences in the flow fields around the
mixing impeller. Generally, it is predicted that the viscoelastic nature
of a fluid tends to reverse the direction of secondary flows induced by

centrifugal force. White et a1. (1977) observed that three different

43

flow regimes can exist in the mixing tank depending upon the level of

viscoelasticity.

3.2.4.1) W

In general, mixer viscometry techniques require the measurement
of the power required to turn the impeller agitating the fluid. The
calculation of torque (power) requirements for mixing viscoelastic
fluids is also important in the design of fermentors or processing tanks
(Prud'homme and Shaqfeh, 1984). The vast majority of studies on rheology
of agitated fluids have focused only on shear viscosity. However, vis-
cosity alone is not sufficient for calculating the torque required to
mix a viscoelastic fluid. Thus, it is important to know under what
conditions power correlations for viscoelastic fluids differ sig-
nificantly from those of inelastic fluids.

The classical apparent viscosity approach introduced by Metzner
and Otto (1957) for power consumption of non-Newtonian fluids in
agitated tanks has been considered by researchers (Table 3.2) to inves-
tigate agitation requirements of viscoelastic fluids. Their findings
indicate that mere use of the "a of non-Newtonian fluids may not be
applicable in the case of viscoelastic fluids as they exhibit different
power consumptions due to their elastic nature. However, there seems to
be a controversy as how viscoelasticity affects power requirements in
agitated tanks. Other works (mainly theoretical) in mixing of viscoelas-
tic fluids are also listed in Table 3.2.

Mashelkar et al. (1975b) observed that when agitating shear-

thinning liquids even having moderate elastic properties, the power

44

66.3.. 6.666.. to ma.x.: so 66.esum ”N.. oHnae

 

:0..e5:0:00 .030:
5.353500 .28..
8.353050 .05..
5:95—0:00 .030...
.2.... .0003 8.353500 .05..

.33. Soon. 5.2.5.500 .05..

8.353200 330d
5.353050 3.0a
5:353:00 .030...
8:95.500 330m

.00... 0500:0030!

00 0 a
Amara-.5 30.00532“
9.35.0100. :0 5.05000 .0 0.00.5
8:553:00 .05..
5.553500 00.606.
.3235... .00... 95:30.03
8.553500 .25..

2:0... 83050.50. 5 5.553500 00.60...
8.553500 Use...
.3285... .656 83866.63

.6656... .8552.
5595.0!!! :0 3.05009 .0 Sue—U

83. 12.6..
65......

65......
65:3

65......

65......
6563.

65......

6.3.3

62 6.66.3235... 0

.x: .8353

6.65:6 6559.8
6.23.
65......

6.6.66... 63.8..
65......

6686 8.1..
6.33..
66.6 65.36..

6.665.... 63:62.60

.22. .6 .6 66>

.22. 6.....8» a .8»
.32. .262 a .666...
.R2. .863... a 638..
.32. 36.666 0 66.5.9636
.32. ..o .6 .636
.32. ..6 .6 62.6.2
.83.. .6 .6 852.66:
.32. .3 .6 =38...
.22. .6 .6 6.6..
.32. .3 .6 5666..
.22.
369.6260 s 8.5.6..

.32. 83.65

.32. .3 .6 6.25
372.2. .s a :65

.32. 86:6... a .6680
.32. 62665626 a 63.30
.82. 5.8.0

.22. ..6 .6 8668

.82. .0 .6 6.5.8

.82. .6 .6 6.6.68

 

 

coco Eocene

539$ aces:

00:30.3.

 

 

45

consumption is considerably less than that predicted by the classical
apparent viscosity approach. The same behavior was observed by Ranade
and Ulbrecht (1977) and Oliver et a1. (1984). They also found that
Metzner and Otto's approach is likely to fail in scaling-up on the basis
of power consumption per unit volume due to the different power require-
ments. The need for extensive experimental work to evolve design
procedures under these conditions using the dimensionless groups
connected with the elasticity of the fluid was strongly emphasized.

ConverseLy, other investigators have observed an increase on
power requirements when mixing viscoelastic fluids (Nienow et a1., 1983;
Prud'homme and Shaqfeh, 1984; Collias and Prud'homme, 1985).]flienow and
Elson (1988) , in a review of the mixing of rheologically complex non-
Newtonian fluids in mixing tanks, concluded that viscoelastic properties
of the fluid may either decrease or increase the power requirements.
They state the impossibility of predicting which because of the complex
flow patterns developed in the mixing tank which strongly depend on the
geometry of the system and impeller, the type of fluid and the scale of
operation.

Yap et al. (1979) assumed Metzner and Otto's method was adequate
to describe the viscous properties of the fluid around duaimmeller
blade only for fluids that exhibited low elasticity and for low values
of rotational speed. This method assumes that the value of k' [from Eqn.
(3.8)] is independent of the fluid and system characteristics. The
authors developed an expression for generalized power consumption for

fluids with a low degree of elasticity:

o 93 o 91 1 23 -1
Po " 2“ nb[ (Rge) ° (1)/d) ° (d/l) ° ] (3-65)

46

where:

nb - number of blades
D - diameter of vessel, m
d - diameter of impeller, m

1 - length of impeller blade, m

23

 

2
d NP (txlav )
Rge- Generalized Reynolds Number = fl ‘
o

with t1 - fluid characteristic time, s
s - fluid rheological parameter, dimensionless

n - limiting viscosity at zero shear rate, Pa 5

This model was not successful with fluids showing a high degree
of elastic behavior.

Nienow et a1. (1983) assumed that the fiav determined by the
nwthod of Metzner and Otto can be applied to parameters other than

viscosity when studying the power requirements in aerated vessels. Thus,

W1 - [ —- 1 <+ ) (3.66)

I

where A' is obtained from N1 - A' "1b correlations. Also, the ratio of

elastic to inertial forces can be similarly derived as

W
[ 2] ( Vav) (3-67)

47

where W1 - Weissenberg Number - ¢; N

$1 - first normal stress coefficient, Pa 32

The significance of this assumption was not clear from their
findings and the authors suggested further work to test their results.

Since most viscoelastic fluids have strong shear-thinning vis-
cosity, power changes may be due to changes in fluid viscosity or
elasticity. Boger (1977/1978) observed that maltose syrup-separan solu:
tions were highly viscoelastic fluids which exhibited a nearly constant
viscosity with high elasticity (normal stress levels) over a broad shear
rate range. This model fluid, called the Boger fluid, has been used to
better assess the effects of elasticity on agitated tanks (Oliver et
a1., 1984; Prud'homme and Shaqfeh, 1984).

Prud'homme and Shaqfeh (1984) developed a correlation that ex-
plicitly includes fluid elasticity which provides a basis for assessing
whether elastic effects are likely to cause significant increases in
mixing torque (or power) requirements. Thus, the total torque is given
by the torque that would prevail in mixing a Newtonian fluid times a
contribution due to elasticity as follows:

r - (1+ma1/‘) (12.71:e + 2.41 x 10'3 Res) (3.68)

[ elastic] [ Newtonian effects]

where:
l‘ - dimensionless torque

and

N1
ma - 2 2 - elastic parameter, (3.69)
2 p r w

 

48
also called the Aberystwyth Number (Thomas and Walters, 1964)

where:
r - impeller radius, m
N, - first normal stress function, Pa

0 - angular velocity of rotation, rad/s

This correlation is based on data in the laminar flow regime and does
not account for changes in the geometry of the system and fluid vis-
cosity.

Collias and Prud'homme (1985) found that elasticity substantially
increases power requirements of turbine impellers in the viscous
(laminar) regime - the torque more than tripled for the most elastic
fluid. However, the magnitude of theleffect of elasticity depends on
both the fluid properties and the size of the vessel. A procedure to
determine the additional torque to mix an elastic fluid was developed.
Secondary flow patterns are determined by the balance between inertial

and elastic forces with

 

E - 2 (3.70)

and W - R . E (3.71)

where:

E1 - elasticity Number

A dimensionless torque, T, is determined as,

49

T - “a: - £016,121) - gazes.) (3.72)
'7

for a given geometry. The Torque Number is related to the Power Number

by,
2
2« T - P R (3.73)
o e

Viscosity and elasticity data were collected in a cone-and-plate vis-

cometer. A correlation for torque as a function of Re and E was

1
obtained by determining the «av in the vessel at each Reynolds Number

using Metzner and Otto's relationship for turbine impellers:

iav - 11 N (3.74)
Finally, the torque required, at a certain Reynolds Number, to mix a
viscoelastic fluid (a mixture of corn syrup, water, glycerin and
polyacrylamide polymer (Boger fluid)) using a turbine impeller, was
determined by adding the torque required for a Newtonian fluid the

additional torque due to elasticity, given by:
3 3 2
T - 13.12 Re + 0.01167 R6 + Re (71 E - 3200 E ) (3.75)

l l

or, in terms of Weissenberg Number [Eqn. (3.71)]:

3 2 2
T - 13.12 R + 0.01167 R + (71 W R - 3200 W. R ) (3.76)
e e i e 1 e

50

Equations (3.75) and (3.76) provide quantitative results on the effects

of elasticity on mixing torque in the laminar flow regime.

3.2.4.2) WW

Another manifestation of viscoelasticity is the climbing of the
fluid up a rotating rod associated with nonlinear effects, the normal
stress, which does not occur in Newtonian fluids (Joseph et a1. , 1984).
This phenomenon is often called the Weissenberg effect. Figure 3.3
illustrates the rod-climbing phenomenon in a vessel agitated by an
impeller. When the elastic force is sufficiently high, it overcomes the
inertia and the fluid is pulled towards the impeller. Because of the
role it may play in rheological testing and processing operations, the
possibility of using the Weissenberg effect as a method of characteriz-
ing viscoelastic fluids has been investigated (Table 3.2).

Beavers et al. (1980) showed that the free surface deformations
on a viscoelastic fluid sheared between two concentric cylinders when
the Weissenberg effect occurrs, can be used to determine rheological
data about the fluid. They also investigated the effect of the impeller
diameter to vessel diameter ratio, using two concentric cylinders. When
d/D approached unity, more complex shapes of the free surface occurred
and it showed dependence on the rotational speed, N.

Eitelberg (1983) numerically analyzed the influence of the finite
length of a rotating cylinder upon the Weissenberg effect. Results
indicate that the secondary flow influences the shape of the free sur-
face and that it is affected by the ratio of the distance from the free
surface of the fluid to the end of the rotating cylinder, h, to the

radius of the outer cylinder, r2. The main result of this study is that

Sl

 

C

 

 

 

IMF“

C) 3’“

 

 

 

”F
\

Figure 3.3: Flow Patterns In Agitated Vessels. A) Newtonian Fluid.

B) Elastic Fluid (Rod-climbing).

52

secondary flow does not reach the free surface if the distance h is
considerably greater than r2.

Even though rod (or shaft) climbing is a sure indication of
viscoelasticity, the absence of the Weissenberg effect does not imply
that the fluid is inelastic. Nienow and Elson (1988) indicated that
certain geometric (shaft diameter) and operational (impeller rotational
speed) variables as well as the presence of a yield stress, may reduce
the climbing effect.

Available mixer viscometry techniques are considered unsuitable
for viscoelastic foods due to the Weissenberg effect (Rao, 1977).
However, the need to measure the rheological parameters (particularly
elasticity) at the same time as the power data is obtained while agitat-
ing the fluids suggests the potential applicability of mixers for the
evaluation of rheological properties of viscoelastic fluid foods.
Reliable techniques could be developed which consider the effects of
geometry on the Weissenberg effect as well as power requirement deter-

minat ion .

53

3.3) W

3.3.1) WWW
3.3.1.1) Relegien fieeween Elew Bettern end Power Coneumption In A

 

Cylindrieal Vegeel.

The state of flow in a cylindrical mixing vessel is complicated
and there is some turbulence near the impeller blades. For simplicity,
it is assumed that the tangential flow is predominant and the flow can
be approximated as a type of Rankine's combined vortex (Nagata, 1975).
When a low viscosity fluid is agitated in a cylindrical vessel, a
cylindrically rotating zone around the central axis of the vessel is
formed, where the fluid rotates with the same angular velocity as that
of the impeller blade, while the flow in the outer part is similar to
that of a free vortex as shown schematically in Figure 3.4a. The central
area of the impeller [abcd] is assumed to have no relation to the power
consumption and only the outer part (the tips of the impeller) [AadD]
and [BbcC] have an important effect upon the power consumption (Nagata
et a1., 1957).

When a fluid of higher viscosity is agitated, the radius of the
cylindrically rotating zone, rc , decreases and it approaches zero at
the transition from turbulent to laminar flow. Thus, the whole impeller

area [ABCD] has a relative velocity, u to the fluid and contributes

rel ’
to the power consumption. Other vortices present in the low viscosity
region such as V, , V1' , V2 and V2' , are weak compared with the forced
vortex in the central zone (Figure 3.4a).

In the range of turbulent flow, an impeller has a relative

Velocity urel - AA' - AA”. Figure 3.4b illustrates the relative velocity

54

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 3.4: a) Simplified Flow Model for Power Correlation; b) Relative
Velocity Distribution Between Paddle and Liquid (Nagata,
1975)

55

distribution in the case of a paddle impeller. The impeller power con-
sumption to maintain the flow is considered as the energy per unit of
time to overcome the resisting forces on the blades. The internal
resisting force of fluid acting on an element of area bdl may be written
as

2
dF - pCure b d1 (3.78)

l
where C is assumed to be a constant. When the impeller rotates opposite
this resisting force at a relative velocity urel , the power consumption

is expressed by

dP - w dM (3.79)
or

dP - 0 2r dF (3.80)
where M - the moment of force acting on the impeller shaft, N m
w - angular velocity - 2xN, rad/s
r - radial distance from the axis to any section of the impeller
Thus,
dP - 4«Nr dF (3.81)
Substituting Eqn. (3.78) into Eqn. (3.81),

2
dP - 41rNprure r dl (3.82)

l

56

By integration, the power consumption for the impeller is given

14 2 1
P - f0 «Nprurelr d (3 83)

Integration of Eqn. (3.83) requires an evaluation of u which

rel’
is a function of the hydrodynamics of the vessel. Thus, the fluid flow
induced by a radial type impeller (paddle) rotating in an unbaffled
vessel can be described by using the forced and free vortex theory:

u - 0 when r < rc (3.84)

rel

2
u - ZxN ( r - rc /r ) when r > rc (3.85)

rel
where rc is the radius of the forced vortex cylinder; it is a function
of the Reynolds Number and goes to zero at small values of the Re
(laminar region).

An approximate equation for the power input in agitated vessels

in the turbulent region is then obtained (Nagata et a1. , 1957):

3
p
P _ P3 - B [10 + 0.6fRea] (3.86)
a

5 3
p N d 10 + 1.6flle

where a, p, f and B are the coefficient for the empirical Eqn.(3.86).
In the range of laminar flow, the power consumption increases
with the viscosity of the fluid, and can be characterized by the

following relationship

57

A
Po - R
e

(3.87)
Following this reasoning, Nagata (1975) developed an approximate

equation for the power consumption of paddle impellers in agitated

vessels with free surface for the complete range of flow regime.

Combining Eqns.(3.86) and (3.87),

 

a
P
P P B 10 + 0.6fRe A (3.88)
o - 3 5 - 3 a + R
p N d 10 + 1.6fRe e
[ turbulent ] [ laminar ]

with B, A, f, a and p determined experimentally. Equation (3.88) can be
applied to wide ranges of Reynolds numbers and to various paddle
geometries.
The above analysis is valid only under the following conditions:
1. The agitated fluid is Newtonian.
2. The system consists of a single impeller centered in the axis of a
vertical cylindrical vessel with a flat bottom and no baffles.
3. The fluid in the cylindrically rotating zone rotates with the same
angular velocity as the impeller.

4. The value of C in Eqn.(3.78) is constant.

3.3.2) 121W:

Power consumption data have often been correlated using dimen-
sional analysis. The variables which affect fluid motion.i111nixing are
of three types (Chavan and Mashelkar, 1980):

l. geometric (linear dimensions)

58

2. fluid properties [density (p), viscosity (0)]
3. kinematic and dynamic characteristics of flow [velocity (u),
gravitational acceleration (g), power (P)]
In mixing with rotating mechanical impellers, the velocity is
defined as the linear speed of the tip of the impeller (Rushton et a1.,

1950), so that

u - de (3.89)
where, d - impeller diameter, m
N - impeller rotational speed, rev/s
Power input by the impeller, P, is used to produce the forces irl
the mass flow and also to overcome the force of gravity, g. The power
required to rotate the shaft and blades of the impellers may be expected

to be a function of many variables:
P - f( d. D. H. b. C. L. p. n. g, N) (3.90)

Dimensional analysis (Appendix A) gives the general equation relating

the physical variables most often encountered in mixing a Newtonian

fluid,

+5 -. [LN—L1“ [Jim-1’33 [2]“ [if [2]“ a“

(3.91)

The last five terms define the effects of system and impeller

geometry. Thus, for geometrically similar systems,

59

 

_L 91:15.2. ’31 d N2 ’92 <3 92)
. . - A [ J [ ] '
pN d ’7 8
fl; fiz
Po - A (Re) (Fr) (3.93)
where:
P - Power Number - —% (3.94)
o
p N d

2

Re - Impeller Reynolds Number - —d—l:—L (3.95)

2
41-N— (3.96)

F - Froude Number -
r 8

White et al. (1934a) first defined the drag coefficient group now
known as the Power Number, Po, which characterizes the flow pattern and
represents the ratio of the power dissipated per unit volume to the
increase in kinetic energy.

The impeller Reynolds Number, Re’ has significance as a ratio of
accelerate force to viscous force. The form [ d2Np/r) ] has come into
general use for characterizing mixer operations that employ rotating
agitators (Hyman, 1962). When agitaing non-Newtonian fluids, the form of
the Reynolds Number may vary, as is shown in section 3.2.1.

The Froude Number, Fr’ is theoretically required to account for
the vortex formation as a result of the influence of gravity in an
agitated system. The influence of the FI. on power consumption seems to
be important only in unbaffled vessels outside the laminar flow region
(Green, 1953). The addition of baffles has little effect on power re-

quirements in the laminar flow region (Treybal, 1956; Nagata et a1.,

60

1957; Blasinski et a1., 1970; Nagata, 1975). Also, high viscosity fluids
(above 20 Pa 3) have sufficient internal resistance to show little if
any vortex motion, i.e. , the surface of the fluid remains essentially
horizontal (Nagata, 1975; Deak et a1., 1985).

In general, the influence of the Froude Number on the mixing
power requirement is considered negligible and practically non-existent
in the laminar region of flow. Thus, the power consumption relationship

can be expressed for each flow regions

9‘1

Laminar flow or fully baffled vessel: Po - A Re (3.97)
a1 ‘12

Turbulent flow or unbaffled vessel: Po - A Re Fr (3.98)

3 . 3 . 3) Lamina; Mixing Region

3.3.3.1) Lamina: Eluig Ligtion in Agitated Vessels

In the laminar flow region, the fluid around an impeller moves
with the impeller rotation and the fluid distant from the impeller is
almost stagnant (Figure 3.5). At very low Re, there is no turbulent flow
and the secondary circulation flow is very weak, so that the momentum
transfer from the fluid near the impeller to the more remote parts of
the fluid depends mainly upon the molecular viscosity of the fluid and
therefore the amount transferred is small and the velocity of remote
fluid is low. As the Re increases, secondary circulation flow occurs and

momentum transfer increases (Nagata et a1. , 1960).

6l

 

 

(“1"

 

 

 

 

 

 

 

 

A/

’l b t A
Transitional [D l] Baffled
Reg.

4°]
urb.
Sm. D Stagn. Zone Reg.

°f m Lam. Zone
Flow

 

 

 

U Turb. Zone

   

 

L [Dz] Non-
baffled

Figure 3.5: Fluid Motion in Agitated Vessels (Nagata, 1975)

62

Nemax A\
It. Np max 6

(”dc C ----- '1 F (2)

-‘uNp -E (3)
Mpg,"

1 103:102 103 104 105 106
R.

 

 

 

 

 

 

 

Figure 3.6: Schematic Diagram Showing The Relation Between Power Number

and Reynolds Number For a Paddle Impeller (Nagata, 1975).

63

3.3.3.2) MW

Because of the similarity of the power correlation curves [Po versus Re
(Figure 3.6)] to the friction factor plot for pipeline flow, the region
where the slope is equal to -l is considered to represent a laminar
(viscous) flow region. This has been experimentally verified by numerous
researchers. Thus, for all impellers, the laminar flow regime is charac—

terized by a linear decrease in the Power Number (P0) with Reynolds

Number (Re) ,

(3.99)

 

Applicability of Eqn. (3.99) is limited by the critical value of the Re

which depends on the geometry of the mixed system alone for Newtonian

fluids .

3.3.3.2.1) gritical Reynolds Number

In the mixing system, the transition from laminar to turbulent
flow proceeds gradually and no distinct critical Re exists for the flow
in an agitated vessel as for other hydrodynamic processes such as
pipeline flow (Re - 2100) and sedimentation (Re - 1). However, ex-
perimental values of the Reynolds Number defining the limit of laminar
flow for mixing by mechanical agitators have been determined (Rushton et
a1., 1950; Green, 1953; Hirsekon and Miller, 1953; Nagata et a1. , 1957;
Pollard and Kantyka, 1969; Nagata, 1975). Results show that the region
of purely viscous (laminar) flow extends to Re numbers from ten to one
hundred (10-100) and it seems to be influenced by the geometry of the

system (Chavan and Mashelkar, 1980). Thus, it may be incorrect to define

64

the laminar flow region unless impeller and vessel type is defined.
Laminar flow can be achieved throughout the vessel by the correct design
of the system, i.e. , selecting the correct combination of impeller and
vessel geometry, and diameter and rotational speed of the impeller.

The laminar flow region is also a function of the type of fluid
being investigated. Metzner and Otto (1957) observed that the region
extends to higher Reynolds Numbers in pseudoplastic fluids than in
Newtonian fluids. For turbine impellers, a value of Re - 10 was obtained
for Newtonian fluids while laminar flow was observed until Re - 20 for
power law fluids with 0.25 < n < 0.45.

The laminar flow region can be limited quite safely by defining
Re < 10. When this criterion is satisfied, baffles are not needed and
it allows for maximum sensitivity when calculating the average shear
rate in the agitated vessel.

When agitating non-Newtonian fluids, especially pseudoplastics,
the use of a generalized (modified) Reynolds Number enables the ap-
proximate prediction of the power of the impeller at low Re (Metzner,
1956). The values of the apparent viscosity, "a’ which are functions not
only of the fluid properties but also of the conditions under which it
is flowing (Begachev et a1., 1980), are substituted into the expression
for the Reynolds Number [Eqn.(3.95)] . A variety of modified Reynolds
Numbers have been used by investigators in mixing studies (Table 3.3),

where

R - LN—E (3.100)

By analogy with Newtonian fluids, an apparent viscosity is

65

Table 3.3 Modified Impeller Reynolds Numbers for Mixing of Non-Newtonian

Fluids.

 

Reynolds Number (Re) Researchers

Impeller Type

 

Magnusson (1952)
Hiraoka et a1. (1979)

Rushton & Oldshue (1953)
Metzner & Otto (1957)
Metzner et al. (1961)
Godleski & Smith (1962)
Nienow et al. (1983)

2 Ducla et a1. (1983)

n Reher & Bohm (1970)

a Hall & Godfrey (1970)
Nagata et a1. (1971)
Prokopec (1972)

Edwards et a1. (1976)
Takahashi et a1. (1984)
Shamlou & Edwards (1985)

Su & Holland (1967)
Rieger & Novak (1974)
Bourne et a1. (1981)

 

FForesti & Liu (1959)
Metzner & Taylor (1960)
Wichterle & Wein (1981)

d_fl___g <Bourne & Buttler (1969)

Chavan et a1. (1972, 1975)
Chavan & Ulbrecht (1972,
1973)

Rieger & Novak (1973)

Begachev et a1. (1980)
Bertrand & Courdec (1985)
Sestak et a1. (1986)

 

Paddle

Turbine

Helical

Anchor

Turbine

Helical

Anchor

Table 3.3 (cont'd.)

66

 

Reynolds Number (Re)

Researchers

Impeller Type

 

2 2-n
m

2
4.32
"p

22-

Lil—“p. [H/hlntd/dw]

-2
Nzndaf n]

m L6n + 2]

ding [k'N11‘“r_aLl
m [3n + l]

2 2-n

51.8 a
m [Mn-DI”1

infill am)”
I

3
d Ne
nD

{ Mashelkar et al. (1975)

Sawinsky et al. (1976)
(Plastic fluid)

{ Foresti & Lui (1959)

P Hall & Godfrey (1970)
Blasinski & Rzyski (1976)

 

_ Nienow et al. (1983)

Calderbank & Moo-Young
(1959)

(1961)
Beckner & Smith (1966)

Pavlushenko & Gluz (1968)

Conti et al. (1981)

{ Calderbank & Moo-Young

Disc

Anchor,
Helical

Turbine

Anchor

Helical
Turbine,
Paddle,
Screw

Turbine

Paddle,

Turbine,
Anchor

Paddle,

Turbine,
Anchor

Anchor

concentric

cylinders

Disc Turbine

 

67

conveniently defined as follows:
n8 - 0H - m (#1)“ (3.101)
for a power-law fluid. The equation shows that the apparent viscosity,

"a’ can be obtained from the known deformation or shear rate, #.

?3JL40 fiflflMLJMEELDlSDEUflflEQH_DEJLfiflLflMflflEfiiaMLKflflilfl§§EL

For non-Newtonian fluids, the apparent viscosity "a varies
throughout the mixing vessel due to variations in the shear rate (5!).
For pseudoplastic fluids (power-law fluids with 0<n<1, also called
shear-thinning fluids), the "a of the fluid in the region near the
impeller is rather low and increases progressively at regions away from
the impeller. This results in high velocities and velocity gradients in
the region near the impeller, which die away rapidly as distance from
the impeller increases (Chavan and Mashelkar, 1980).

In rheological studies, a suitable value of the "a for the non-
Newtonian fluid is required. This can be obtained from viscometric
measurements if the velocity gradient, i.e., shear rate (at least in the
immediate vicinity of the impeller) in the given vessel can be
predicted. However, it is precisely the determination of the shear rate
in the agitated vessel which presents the main difficulty since it
requires the knowledge of the velocity field. The flow of a non-
Newtonian fluid in a mixing vessel has as yet not been described
analytically because of the very complex flow structure in a vessel with

a mixer (Witcherle et al., 1984). It is therefore convenient to use

68

other methods for determination of the shear rate distribution in a
mixing vessel.

Following the idea of Magnusson (1952), Metzner and Otto (1957)
proposed the use of an average shear rate (fiav) which is proportional to
the impeller speed in pseudoplastic fluids in the laminar-flow region

which is given by

fiav - k' N (3.8)

The physical concept of the 7av has been well-established as an
important design parameter in the evaluation of the impeller performance
and prediction of power consumption (Nguyen, 1983). The fiav,represents
the rate of shear which has to exist around the impeller to produce an
observed power consumption (Oldshue, 1983). It is important to recognize
that it is a simplified approximation. Even though an experimental
result, theoretical supporting evidence may be obtained from the expres-
sion for the shear rate at the bob of a concentric cylinder viscometer
for a pseudoplastic fluid in an infinite cylinder (Ro >>> Ri), (Krieger

and Maron, 1954):

1 - 4 n N (3.102)

 

where k', from Eqn. (3.8), would be equal to 4_1 . Calderbank (1958)
n

experimentally verified the linear relationship of the shear rate around
an impeller in the laminar-flow region with the impeller speed, N.
The experimental and theoretical evaluation of the impeller

proportionality constant, k', and consequently the rate of shear in the

69

mixing vessel, has been the subject of many studies. The most common
approach is the analysis of a non-Newtonian fluid flowing between two

concentric cylinders.

3.3.4.1) Theoretieal Exereeaieae Fe; The Rate 0f Shear In A Mixing
Veeeel
3.3.4.1.1) ansentris_§xlinder§

Expressions for the determination of the average shear rate in a
vessel with an impeller have been determined by considering the mixing
system as a two-cylinder system, with the impeller as a rotating
cylinder.

It is known from theoretical hydrodynamics (Bird et al. , 1960)
that the generalized Newtonian law of internal friction for an incom-

pressible fluid is

T - 2 n D (3.103)
where
T - stress tensor, Pa
0 - fluid viscosity, Pa 3
D - deformation rate tensor, l/s

In.cylindrica1 coordinates, equation (3.103) becomes (Bird et a1.,

1960),
a _ 2 3_r . a a _ r Q__ 32 + i 33; (3.104)
rr " 6r r9 0r " 8r r r 60
a _ 20[ .1 330 + .3r] . a _ a _ r 321 + E 33; (3.105)
on r as r 92 20 " 62 r 30

70

810 600 609
_ __z . _ _ r __g __r (3.106)
azz 2" az azr arz " I: 6r + 62 :I

For a rotary motion in steady-state laminar flow, the fluid moves in a
circular pattern and only the tangential velocity prevails, i.e. , we =

we (r) and (or - wz - 0. Then, the shear stress, a, may be written as

(0
a -a -qu[-1] (3.107)

to
i - a/n - r a [L] (3.108)

Using Eqn. (3.108) an expression for the shear rate at the sur—
face of a rotating cylinder in an infinite Newtonian fluid is obtained
as

:1- -41rN (3.109)

which relates the shear rate with the rotational speed of the impeller.
Pavlushenko and Gluz (1968) referred to the use of Eqn. (3.108)
to determine the average shear rate in mixing non-Newtonian fluids by
mechanical impellers. To develop the analysis, the motion of the fluid
caused by the rotation of a cylinder is again assumed as an approximate
model of the fluid flow produced by mixing with any of the usual im-
peller types and the problem of the steady motion of a non-newtonian

fluid rotating between coaxial cylinders is considered. It is also

71

assumed that the internal cylinder with a radius Ri rotates at a con-
stant angular velocity (w - 2rN) in a stationary cylinder with a radius
of Ro - (D/d)R1. End effects are considered negligible in this

analysis.

At low velocities characterizing the flow of non-Newtonian fluids
in vessels with impellers, the inertia forces and the pressure gradient
have no marked effect on the phenomenon and may be neglected as a first
approximation (Pavlushenko and Gluz, 1968). Another assumption is that
the flow of the homogeneous, incompressible fluid is planar (because of
the symmetry). Then, the following expression is obtained from the
equations of motion in terms of stress components (cylindrical
coordinates),

2
‘i; g; (r Ora) - 0 (3.110)

This equation can be rewritten as

60
_l_ 2 __rr -

Integrating Eqn.(3.lll) once leads to,

2

r a - 01 (3.112)

r0

For a power-law fluid, a - m in . Thus, the Ora component of the

stress tensor in cylindrical coordinates is,

72

 

 

 

 

 

 

 

 

 

dr r (3.113)
Using equation (3.112), this yields
(«0) n
2+n g_ _fl
C1 - mr [dr r ] (3.114)
After integration, we - C.‘.r(n'2)/n + Car (3.115)
Using the following boundary conditions:
(i) we - ZNNRi at r - Ri (3.116)
(ii) we - 0 at r - R0
the expression for the angular velocity becomes
[1 "R 2/1'1 ‘
' .2
_ r
w - 2er (3.117)
r 1 _ P 52 2/n
I. :- R1 .1
and the expression for the shear rate is
2/n
. FR ‘ .
_e
w 1 - _r .
. _ d_. [.2 ] _ 5L _
7 r dr r r dr ZflN 1 - Ro-Z/n
. .131- 1
-g 21m 1 [JJ‘Z/n (3.118)
R
o

73

 

 

 

At r - R1, the expression simplifies to
i _ - 4111 (D/d)2/n (3.119)
n 1 _ (D/d)2/n
Considering y - k'N, then
k, _ - g; (1)/<1)”n (3.120)
n 1 _ (D/d)2/n

Equation (3.119) determines the average shear rate in a vessel with a
mixer as a function of the rheological properties of the fluid, the
mixing conditions, and the geometrical characteristics of the system and
should give more accurate results for mixing of non-Newtonian
(pseudoplastic) fluids than the use of the equation developed for

Newtonian fluids [Eqn.(3.109)].

3.3.4.1.2) Em ir ca And eoretical Ex ressions For The Im eller
Proportionality Constant, k'

The suggestion of Metzner and Otto (1957) of a constant value of
the impeller constant, k', which is a unique function of the geometry of
the system has been questioned and other expressions for the constant
have been determined (both theoretically and empirically) for the mixing
of non-Newtonian (mostly pseudoplastic) fluids.

Expressions for the impeller proportionality constant, k', are
summarized.in Table 3.4. Looking carefully at the expressions, it seems
that the value of k' can be a function of the geometry of the system
(impeller shape and size, vessel size), the rheology of the fluid

(values of shear-thinning index (n) and consistency coefficient (m) or

74

Table 3.4. Empirical and Theoretical Expressions for Determination of
the Impeller Proportionality Constant, k'.

 

kl

Researcher

Impeller Type

 

fi/N - constant

C (l-n)
C - f(Hg/d)

slope of

a
log IP/(mN“*1d>1
versus (l-n)

ls ___l___ 2
v7 1- (d/D)

.1.

(4)1'“ <d/D)2<1/d)

13.5 N + 4.43

EMPIRICAL EXPRESSIONS

Metzner and Otto (1957)

Used by: Roger & Bohm (1970),
Bongenaar et al. (1973), Roels
et a1. (1974), Nagata (1975).

{ Beckner & Smith (1966)

Rieger & Novak (1973)

Used by: Rao (1975),

Rao & Cooley (1984),

Steffe & Ford (1986),
Castell-Perez et al. (1987)

{ Sestak et al. (1976)

{ Hiraoka et al. (1979)

{ Yap et al. (1979)

{.Nguyen (1983)

a (H,/D)p(s/D)7(w/D)6 { Takahashi et a1. (1984)

Turbine, paddle

Helical, turbine,
Turbine, paddle

Anchor

Anchor, helical
Flag

Flag, star
Pitched paddle
Flag

Anchor

Paddle

Helical ribbon
(viscoelasticity)

Anchor

Helical ribbon

 

75

Table 3.4 (cont'd.)

 

 

 

k' Researcher Impeller Type
exp[4.2 (d/D) - 0.5] '{ Deak et al. (1985) Ribbon
3a - 144 (c/D) {:Shamlou & Edwards (1985) Helical ribbon
z;g_ 1/(n-l) { Kemblowski & Six-blade turbine
3
Amd Kristiansen (1986)
(l/N) an 1/(“'1) {,Mackey et al. (1987) Flag
Nm
THEORETICAL xxrnsssrous
n/(n-l)
F3L_] B Calderbank & Moo-Young Anchor
3“+1 (1959, 1961) (conc. cylinders)
.91.
___JL______ { Bourne (1965) Sphere
1 -(d/n)3/n
__fi;g___ { Mizushira et al. (1966) Cylinders
0.5 (D-d)
2/n .
£1 (D/d) { Pavlushenko & Gluz (1968) Conc. cylinders
n 1 - (D/d)2/n

Bourne & Butler Helical ribbon

4x [ n,(l - (d/D)1
(1969) (Conc. cyl.)

z/n JI/(l'n)
1- (d/D)2

 

76

 

 

 

 

 

k' Researcher Impeller Type
5_ [d/(d-D)] { Mitsuishi 8 Hirai (1969) Helical ribbon
J2 (parallel plates)
n- 1 2
0.9(n-1)
£4”) (D/d) ' 0‘75 n {Schilo (1969) Anchor
n { (D/d) - 0.75]
£9 { Prokopec (1972) Screw type
“2
1/(1-n) 2/n n/(l-n)

_ag__ _n_L_LDLee1____;_ll_ Chavan & Ulbrecht Helical

2 4' crew
I D H (1972) (conc. cyl.)

1/(n-1)
["w (ww + 2)] Paddle
« NN { Hiraoka et al. (1979) ( Numerical)
[w + 2 ]
w N
4« [n (1-" (“)1’1 { Sinevic et al. (1986) Coaxial cyl.
2/n { Nguyen & Boger (1987) Vane
4x Cr1
Kemblowski et a1. (1988) Helical Screw
2

[1 ' (l/fi) ] (Conc. cyl.)

 

77

viscosity) and the operating conditions (impeller rotational speed). The
geometry of the system also plays an important role, especially the
ratio of diameters, d/D. This is particularly true for the theoretical
expressions. In addition, the various expressions presented in Table 2.2
indicate the importance of the type of impeller to be used in the mixing
process. This should be expected since the shape of the impeller affects
the shear field in the agitated vessel.

Hall and Godfrey (1970) and Nagata (1975) confirmed that impeller
pitch, and.number of blades have no significant influence on the values
of the impeller constant but it is greatly influenced by impeller size.
The effect of the impeller height has been considered negligible by some
investigators [White and Brenner (1934), Nagata (1975)].

The value of the shear-thinning index, n, also seems important in
the determination of k'. This should be expected since the shear rate at
a given distance from the impeller has been found to be slightly greater
for a larger impeller (smaller D/d ratio) (Metzner and Taylor, 1960) and

the more shear-thinning the fluid, the greater the difference.

78

CHAPTER 4

mnmmmrmmonmsvmmmmwsommmmm

This chapter consists of the presentation of the theoretical
aspects for the development of a procedure for the rheological charac-
terization of non-Newtonian (power-law) fluids with a mixer viscometer.
The experimental verification of the developed procedure is presented in

Chapter 6.

4 - 1) 03113511153191: 0: mu QURVES

Traditional mixer viscometry techniques present two main disad-
vantages. First, the need for calibration with Newtonian fluids (Po vs
Re curves) and secondly, the use of a simplifying assumption for the
aVerage shear rate ”av - k'N), with k' a function of impeller geometry
only - Taking into account these disadvantages, the application of these
tecIt‘lniques in rheology has been questioned (Kemblowski et al., 1988;
Nienow and Elson, 1988) and the need for new procedures to evaluate the
rheological behavior of power-law fluids has been suggested.

The main purpose of this chapter is to develop a new procedure for
q 1 t7ect determination of the flow curve (shear stress-shear rate
relationships) of power-law fluids using a low-cost mixer viscometer.

The existing literature presents a similar attempt by Kemblowski et
a1 ‘ (1988) for a system of complex geometry (a helical screw impeller

Qitating in a draught tube) for application to fermentation broths using
SQInlisticated instrumentation. Other attempts [Bongenaar et a1. , 1973;

Rao . 1975; Metz et al. (1979); Kemblowski et a1. (1986)] require

79

calibration with Newtonian fluids using power correlation methods.
Expressions for the shear rate and shear stress of power law fluids have
been developed for a disc spindle (Williams, 1979) but no attempts have

been made for geometries such as the paddle or the flag impeller.

4-1.1) WM

The movement of solid surfaces (e.g. , an impeller) in contact with
a fluid causes the fluid to move in some characteristic pattern which
results in the development of internal stresses and the application, on
the solid surfaces, of characteristic forces which must be continuously
CourIterbalanced (e.g. by a drive motor) in order to sustain the fluid
motion (Charles, 1978). The nature of the flow pattern and the mag-
hi t\ICIes of internal stresses and applied forces depend primarily on the

gec“nettry of the system, the rate of fluid motion and the rheological

p
roperties of the fluid.
Approximate expressions for the determination of the shear stress

a
nd shear rate in a vessel with an impeller can be obtained using ap-

PT:
Qutimate geometries for the system, since the complicated geometry of

th
Q system makes the solution of the proper set of differential equa-

t1
9118 of motion a difficult task.

(a.
7 1 ~ 1.1) Model Syetems
It

Consider the measuring system shown in Figures 4.1A and 4.1B.
Qt)
1“sists of an impeller (paddle or flag) of height b and diameter d,
r
Qtating in a cylindrical cup, with the impeller replaced by a cylinder

w
ith the same dimensions of the impeller, b and d. The following assump-

t.
1bus were made for the model system:

80

the fluid is purely viscous and obeys the power-law model, a - m

1)
<1)“.

11) the fluid flow produced by stirring with any impeller type is
approximated to the steady motion of fluid caused by the rotation
of a cylinder [i.e. , the impeller is replaced by a cylinder whose
dimensions are equal to those of the impeller (d and b)].

111) the shearing due to the immersed section of the impeller shaft is
negligible.

1") the resistance to flow caused by the top and bottom of the impeller

is negligible.
The first of these assumptions can be readily checked. Assumptions

( ii) and (iii) imply that the cylindrical model is sufficient to
deseribe the viscometric flow induced by a rotating impeller and that

t
he sheared surface of rotation is cylindrical. Even though this is not

3
o ’ this approximation is reasonable for the purposes of this study.

e adequacy of assumption (iv) will be tested in this investigation.

Also, since the complicated geometry of the impeller system does

no
t allow for determination of the shear rate and shear stress at a

f 1
Red point of the system, average values should be used instead.

4
‘1~1.2)§hsat_fie§s_Annmrims£isae

The expression for the shear rate at the surface of a rotat-

i
“3 cylinder in an infinite Newtonian fluid is

(4.1)

7b - 4xN
with the

w
hich relates the shear rate at the bob of the cylinder, fib,

81

Paddle

 

 

 

 

 

 

 

 

Fiepare 4.1: Model Systems. A) Paddle Impeller; B) Flag Impeller;

C) Flag Impeller (Model 3)

82

rotational speed (rev/s) of the impeller, N.
Krieger and Maron (1954) worked on the shear rate problem for non-
Newtonian fluids sheared within the (narrow) gap of a Couette viscometer
by solving the corresponding equations of motion for a cylindrical

system. The resultant shear rate equation for a fluid obeying the power-

law is,

1b " n 2/n 2/n (4'2'1)

’yb - (4.2.2)

W1 th {2 — 21rN - angular velocity, rad/s

w
I

outer cylinder (cup) radius, m

innner cylinder (bob) radius, m

UP
I

n - power-law index, dimensionless

N - cylinder rotational speed, rev/s

Equations (4.2.1) and (4.2.2) show the dependence of the shear
rate at the surface of the rotating cylinder on the flow behavior index
and the system geometry.

It is possible to approximate the complex situation of an impeller

r°tating in a tank by assuming that all fluid elements are exposed to an

83

”average shear rate” during the mixing process. Pavlushenko and Gluz
(1968) considered approximating the shear rate in a mixing system with
Eqn. (4.2.1) and (4.2.2) [See section (3.3.4.1.1) for the development]

and the shear rate at the impeller (cylinder) surface is written as,

 

 

2/n
(D/d)
7b '21?— 2/n (4.3.1)
(D/d) - l
c>:zr
. 4m (1)/<02”n (a 3 2)
1 :- _ . .
b n (D/d)2/n _ 1
‘Vuflnlealx:

D - cup diameter, 111

d - impeller diameter, m

The previous equations assume a dependence of the shear rate at
the surface of the impeller on fluid properties, impeller speed (rpm)
and, system geometry, in a form identical to that of a cylindrical im-
pe 11er. Thus, from Eqns. (4.3), an expression for the average shear rate
in the real measuring system (a vessel with a mixer) is expected to

present a similar form (but somewhat different due to the differences in

geometry) to that of Eqn. (4.3.1) and (4.3.2) as follows:

(1)/c1)“1 a
2, - p [b/d] 3 N (4.4)
av 1 (D/d)a2 _ 1

 

84

where 61 is a constant and, al, a2 and as are parameters dependent on
the power-law index, n.

Equation (4.4) differs from Equations (4.2.1), (4.2.2), (4.3.1)
and (4.3.2) in the addition of the term which takes into account the
effect of impeller variation (i.e., impeller height, b). Equation (4.4)

may also be written in the familiar form of the linear dependence of the

average shear rate in a mixer on rotational speed, proposed by Metzner

and Otto (1957),

78v - k'N (3.8)

with

(1)/d)“1 a
k' - 9. a [b/d] 3 (4.5)
(D/d) 2 - 1

 

Where k' is the impeller proportionality constant, dependent on the
system geometry (cup and impeller) and the rheological behavior of the
fluid. The average shear rate of an impeller (paddle or flag) can be
E”"‘I>ected to be a function of these parameters.

Also, a direct relationship between the average shear rate and

the geometric dimensionless numbers [(D/d) and (b/d)] can be expressed

as f0110ws:

38v - a. [ (1)/d)“1 <b/d>“3 1N (4.6)

Where

 

85
k' - fl; 1 (1)/d)“1 <b/d)°3 1 (4.7)

with ,81 a constant and, a1 and as parameters dependent on the power-law
index, n. Eqn. (4.6) differs from Eqn. (4.4) in the simplification of
the (D/d) term with the consequent elimination of a2 . Thus, Eqn. (4.6)
makes the (D/d) term a more significant parameter since the terms under

brackets in Eqn. (4.4) is close to one.
In the case of a flag impeller (d/b- constant in this

investigation), Eqn. (4.6) can be rewritten as follows:
rw-prwmflu (mm
and
k' - a. [D/dla‘ (4.9)

With 51 and (:1 similar to Eqn. (4.6).

Equations (4.4) to (4.9) allow for determination of the average
shear rate in the mixing system as a function of impeller and cup
geometry and fluid properties for each value of the impeller rotational
Speed, N.

In summary, this section presents a series of equations which

allow estimation of the average shear rate, 1 , when agitating a

av

poweIf-law fluid with an impeller (paddle or flag). The development of

the eQuations for the average shear stress will be presented in the next

section.

 

86

4.1.1.3) Wales:
4. 1.1.3.1) Hegel 1 (geneenezie eylineera aaalogy with negligible end
effects).

Consider a stationary cylindrical cup of large radius, with
laminar flow at the surface of the rotating inner cylinder of diameter
d - The fluid exerts a tangential force on the outer cylinder while the
inner cylinder exerts such a force on the fluid in contact with it. This

force is transmitted through the fluid from one layer to the next. At
any point in the fluid, the tangential force divided by the surface area
on which it acts is defined as the shear stress, a.
The torque (proportional to the drag offered by the fluid when the
cylinder is rotated) on the shaft resulting from the rotation of the
inner cylinder may be regarded as a sum of two parts: Mw , resulting

from the shearing in the cylindrical wall, and Me , resulting from the

shearing in the two end surfaces. Thus,

M-M + 2M (4.10)
w e

or

M - (21rb)ar + 2(2xr20) (4.11)

with

a - shear stress from fluid, Pa
b - inner cylinder height, 111

r - any radius (in fluid) from axis, m

 

87

Taking into account assumption (iv), the second term of the right-hand

side of Eqn. (4.11) is eliminated and Eqn. (4.11) becomes
2
M - 21rbr a (4.12)
At the innner cylinder wall (r-R),

2
M - 21rbR aw (4.13)

W1 th aw - shear stress on the wall of the inner cylinder, Pa

Let aw - aav (since the shear stress cannot be determined at a

fixed point). With R - d/2 , Eqn. (4.13) becomes

2
M - be (d/2) aav (4.14)
‘VIIEEre M - torque on the shaft resulting from the rotation of the inner
cylinder of diameter d and height b, (N m), i.e., the
experimentally determined torque on the impeller shaft during
the shearing of the fluid.
0av - average shear stress on the surface of the inner cylinder,
Pa

The average shear stress, aav , can then be determined from

Eqn. (4.14) as follows,

aav - J“,— (4.15)
«bd

88

Equation (4.15) represents the average shear stress in the real

measuring system (impeller rotating in a cylindrical cup).

4 - 1.1.3.2) liege]. 2 (geneengrie eylineere agalegy with egg effects)

The validity of assumption (iv) is questionable when the diameter
of the outer cylinder is small and, therefore, the effect of the end
S urfaces of the system (top and bottom) become significant. The
theoretical development for an approximate expression for the average
Shear stress takes advantage of the analysis used by Nguyen and Boger
( 1983) for the vane shear approach as a starting point and is as fol-
lows.

The total torque measured is composed of one component due to
shearing on the cylindrical wall and another to shearing at the two end

surfaces. The diameter of the cylindrical connecting rod is small in
comparison with the paddle diameter and so the measured torque is due to

the paddle surface only. The torque balance is given by

M-M + 2M (4.16)
w e

R

M - (21I'Rb)aw R + 2 [211' Jae(r) r2 dr] (4.17)
0

Where
M - measured torque, N m
R - impeller (cylinder) radius, m
b - impeller (cylinder) height, 111

ae(r) - shear stress at the end surface (a function of radial

89

position r), Pa

aw! shear stress at the cylindrical wall, Pa

The main assumption of this approach is that 0e is uniformly
distributed over both end surfaces and that 0e is equal to aw at the

cylindrical wall. Thus, 0e - a“.7 , and integration of Eqn. (4.17) yields

R
M - [(211’)R2b]aw + 41r Jaw r2 dr (4.18)
O
2 3 R
M - [(2101: b]aw + 411' (r /3) lo aw (4.19)

Equation (4.19) can be rewritten in terms of the impeller diameter, d.

T'tmus,
2 3
M - [(1r/2)d 13].;W + 41r -% aw (4.20)
Finally,
3 b 1
_ .114— _. + __
M 2 [a 3] aw (4.21)

Assuming that the average shear stress is equal to the shear stress at
the cylindrical wall (aw - aav ), the dependence of the average shear

stress in the real measuring system can be written as follows:

M

a -

av —K_ (4'22)

90

:3 b + l
with K - "’ "' , (4.23)

an impeller parameter dependent on the impeller dimensions only.
Equations (4.22) and (4.23) indicate that the oav can be calculated from
the measured torque, M, and impeller dimensions (d and b).

Even though the assumption of uniformly distributed 0e is valid
only for a extremelly small cylinder (d + 0), Nguyen and Boger (1985)
demonstrated that the error involved in using Eqn. (4.22) is not sig-
nificant when compared to an assumed relationship for the shear stress

at the end surfaces, i.e.,
ae(r) - (r/R)paw, when 0 s r s R (4.24)

where p - parameter ( z 0 ). The two boundary conditions to be satisfied

are:

0e - 0 at r - 0 (4.25)

a - a at r - R (4.26)
e w

Integrating Eqn. (4.17), after the introduction of Eqn. (4.24) yields

R

M - (2MR2b) aw + 2 [2x I [r/R]pr2 awdr] (4.27)
o
R p + 2

2
M - (2MB b) a + 4nd I -—£————' 0 dr] (4.28)
w w Rp w
o

91

4‘0 p + 3 R
2
M- (21rR b) a +——‘1 —R—— (4.29)
w (p+3) RP 0
41m 3
(4.30)

2
M-(21rR b) aw+-(p—+§)R

Rewriting Eqn. (4.30) in terms of the impeller diameter, d,

 

3 b 1
1rd +
M - 2 [ d 5:; J aw (4.31)

(4.31) accounts for the

The second term in the parentheses in Eqn.
(1.

error involved when using impellers (cylinders) of finite diameter,

If p - 0, Eqn. (4.31) yields Eqn. (4.21) (uniform shear stress distribu-

tion case).

4.1.1.3.3)nssL¢.LLLE12LImllsr_l

Because of the more complicated geometry of a flag impeller
another model can be considered as shown in Figure 4.1C. The reasons for
this alternative model are due to the particular geometry of the im-
Peller, which consists of a central cylinder with two blades attached.
Thus, the impeller is now replaced by an inner cylinder with an equiv-
alent diameter de’ where de is equal to half the length of each

1mPellet blade plus the diameter of the cylindrical section, dC . Thus,

cle - 2 (82) + (1c (4.32)

92

with 2 - blade length, 111

Following the same analysis for the System Model 1 [concentric
cylinders analogy with negligible end effects; Section (4.1.1.3.1)], the
corresponding expression for the average shear stress in the cylindrical

surface is given by

2M

 

av (4.33)

2
arbd
e

with de- equivalent diameter of ideal system model, m [Eqn. (4.32)]

b - cylinder height, 111

In a similar manner, the expression for the average shear stress
for the System Model 2 [Concentric cylinders analogy with end effects;

Section (4.1.1.3.2)], is as follows:

M
aav - Z (4-34)
3
1rd b + l
w - —-—€L- — ——
1th K3 2 de 3 (4.35)

In summary, a series of equations for approximation of the average
shear stress, aav , when agitating a power-law fluid with an impeller
(Paddle or flag) have been determined based on an analogy with the

concentric cylinders case.

93

CHAPTER 5

HATERIAIS ARDHETBODS

This chapter is divided into four main sections. The first sec-
tion describes the equipment and the materials used for the data
collection. In the second section, the procedure and the experimental
design are presented. The third and fourth sections present the proce—

dure for calculations using traditional and the new mixer viscometry

methods , respectively .

5.1) EQQIPMENT AND MAIQIALS

The Brookfield RVTD and HBTD mixer Viscometers were monitored by
a data acquisition system and torque responses were collected every two
seconds. Six impellers (five paddles and one flag) and three sample cups
were utilized in this investigation. Four Newtonian fluids and three

Non-Newtonian fluids were selected as the main fluids.

5.1.1) The Brookiield Mixer
The Digital Brookfield Viscometer (Brookfield Engineering

Laboratories, Inc., Stoughton, M.A.), is a rotational viscometer that
measures the torque required to rotate an immersed element (the spindle)
in a fluid. The spindle is driven by a synchronous motor through a
calibrated spring and the deflection of the spring is indicated by a
digital display. For a given viscosity, the viscous drag, or resistance

to flow (indicated by the degree to which the spring winds up). is

94

proportional to the spindle's speed of rotation and is related to the
spindle's size and shape (geometry).

Two digital viscometers, the Brookfield RVTDV-I and the
Brookfield HBTDV—I were used in this investigation. The calibration
spring torque were 7187 dyne cms and 57496 dyne cms (full scale),
respectively. Eight rotational speeds (0.5 to 100 rpm) were selected.
These Viscometers are guaranteed to be accurate within 1% of the range
in use (when utilizing the display reading) and have a sensitivity and
reproducibility of 0.2%. Digital Viscometers include a O-lOmV (or O-lV)

output signal for continuous data collection.

5.1.2) Data Acquisition System

The output signal from the Brookfield Viscometer was sent to a
Data Acquisition System (Dianachart PC-Acquisitor Model PCA-14,
Dianachart Inc., Rockaway, NJ). The acquisitor (A/D board with 48
channels) is connected to a IBM PC by using a parallel printer cable.
The viscometer torque voltage was measured and stored onto a floppy
disk. Software provides continuous on-screen display of measurements and
a series of data handling alternatives. Published accuracy is
10.02% of range (0.3pV-10V). Measurements were stored at two second
intervals. When the experiment was completed, the disk data was read

into a spreadsheet using LOTUS 1-2-3 and printed in tabular form.

5 - 1 . 3) Impellers and Cups

The impellers (Figure 5.1) were constructed from standard
BrOokfield spindle shafts (300 series stainless steel) shortened to the

required length. The paddle impeller blades were ground from a solid

95

 

.AEo cm one acoumcoemc dumv muouuoaam noun was o~oemm H—.n ouamum

may

 

3 H

«Au

 

 

 

 

Nofié

«may

 
 
 

 

 

and

:3 H 8.: o.
ivunu

 

.L r

96

piece of stainless steel that had been drilled to allow a press fit on
to the shaft. The flag impeller was supplied by Brookfield Engineering
Laboratories as a component of a commercial version of the instrument
described by Steffe et a1. (1989).

The fluid containers (Figure 5.2) consisted of cylindrical
cups, with flat bottoms, made with acrylic tubing (United States Plastic
Corp. , Lima, OH) cut to desired lenghts. The base of the cups and the
standard base were constructed from a plastic sheet extruded from cel-
lulose acetate butyrate plastic (United States Plastic Corp., Lima, OH).
These materials guarantee high optical clarity, high impact resistance
as well as light weight. Cups were constructed with fluid jackets for

temperature control.

5.1.4) Eluidg

Two types of fluids were used in this study, Newtonian and non-
Newtonian fluids. The Newtonian fluids consisted of four Brookfield
Viscosity Standards: non-toxic silicone fluids calibrated at 25°C (77°F)
with viscosities of 100 cp, 993 cp, 4840 cp and 12200 cp (0.1-12.0 Pa
5). The non-Newtonian fluids consisted of 1.0%, 1.5% and 2.0% (dry
basis) concentrations of aqueous solutions of Hydroxypropyl

Methylcellulose Premium (Methocel, Dow Chemical Co. , Midland, MI).

5. 2) EEDEDIMENIAL DESIGN AND PROCEDURE

5. 2.1) Design of Experiment

Data were collected to assess the variability in torque

measurements (needed for further data analysis) introduced by the

97

.AEo CH cum mcoamcoawp Haov swam use can muocumucoo canaom "~.m ouawwm

 

 

 

 

 

 

:3 L

 

 

nh;N

 

 

 

 

 

 

 

omom

azo

22>

mEm

32>
ace

if

ii

‘J-c

'Lr

Ll4
I 4.

le

2L.

II!

PC

tE

51

31

LI'
n

98

impeller and cup changes. Six impellers and three cups were used in a
total of 13 treatment combinations. Tests were conducted in duplicate
and the order was determined by randomization. Table 5.1 shows the
different impeller/cup combinations investigated in this study. Impeller
diameter, d,(Figure 5.3) was kept constant due to problems during
manufacture of the paddles. The impeller blade height b, was varied to
investigate the effect of impeller size. Three sample cups of different
diameter, D, were used with a range of d/D ratio from 0.3 to 0.7.

To not vary a large number of geometric constants at once, the
length of the sample cups was maintained at L - 1.5D. The fluid level,
Ii, in the cups was kept at H - 1.2D. This distance was selected to keep
all impellers sufficiently immersed in the fluid to avoid surface waves,
especially for the bigger impellers. Preliminary tests showed that the
position of the impeller, that is, the distance between the bottom of
the impeller and.the bottom of the cup, c, had no significant effect on
torque readings when placed close to the top, in the middle, or close to
the bottom. For practical considerations, impeller depth (c) was set at
c - 0.5d since it made possible the immersion of the impellers under
sufficient volume of fluid. The effect of the distance from the surface
of the fluid to the top of the impeller's blade was assumed negligible
(as in Nagata (1975). The effect of different impeller shape was inves-

tigated by using a flag impeller (Figure 5.1).

5.2.2) m
5 . 2 . 2 . 1) Dreparation of Non-Newtonian Fluids

Aqueous solutions of Hydroxypropyl Methylcellulose were prepared

by heating distilled water to 70°C (158°F) and slowly pouring the

Table i

 

 

99

Table 5.1: Experimental Design.

 

GEOMETRICAL DIMENSIONS DIMENSIONLESS VARIABLES
SYSTEM IQQI

 

n d b d/D d/b

 

Paddle Impellers

l 5.55 1.8 1.0 0.327 1.8
2 5.55 1.8 1.8 0.327 1.0
3 5.55 1.8 3.0 0.327 0.6
4 5.55 1.8 4.0 0.327 0.45
5 5.55 1.8 5.0 0.327 0.36
6 3.50 l 8 1.0 0.515 1.8
7 3.50 1 8 1.8 0.515 1.0
8 3.50 1 8 3.0 0.515 0.6
9 2.54 .8 1.0 0.709 1.8
10 2.54 1.8 1.8 0.709 1.0
Flag Impellers
1 5.55 1.5 3.0 0.273 0.5
2 3.50 1.5 3.0 0.429 0.5
3 2.54 1.5 3.0 0.591 0.5

 

where: D - cup diameter
d - impeller diameter
b - impeller blade height

100

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 5.3: Mixing System

 

 

101

percent weight of sample into the water. Mixing was carried out with a
Corning PC-351 Hot-Plate Stirrer. Solutions were cooled down and allowed

to rest for a period of 24-48 hours to eliminate air bubbles.

5.2.2.2) Determinatign of gheglggigal Droperties of Non-Newtonian Fluids

Rheological behavior of the materials was determined with a
Haake RV-l2 concentric cylinder viscometer with M-500 head and the MVFI
sensor (d/D-0.90). The viscometer is interfaced to a Hewlett-Packard 85
computer and a 3457 data acquisition system. The samples were previously
agitated for a period of 10 minutes to check for thixotropic behavior.
Torque was monitored as a function of time and reached an equilibrium
value after the completion of the test for all samples. Triplicate
replications of torque versus rotational speed data were collected for
every sample at 1-120 rpm (0.105-12.57 rad/s). The values of the flow
behavior index, n, and the consistency coefficient, m (Table 5.2), were
obtained from shear stress-shear rate data, with shear rate evaluated
using the method developed by Krieger (1968). The fluids showed power-
law behavior and no elastic characteristics, such as rod climbing. The
experiments were carried out at a constant temperature of 25°C i 1°C

(77°C) .

5.2.2.3) leibzatign of Drogkfield Viscometers

The viscosity of a Newtonian standard (Brookfield Viscosity
Standards) was determined with the RVTDV-I and the HBTDV—I Brookfield
Viscometers and cylindrical spindle # 7 (0.32 cm diameter, 5.37 cm

height) to ensure proper instrument performance and high accuracy.

102

5.5.2.4) Data collection

Figure 5u4 illustrates the overall experimental system. Once
loaded into the cup, the temperature of the fluid was controlled with a
constant-temperature water bath connected to the cup jacket with stan-
dard tubing and fittings. Temperature of the sample was allowed to
equilibrate up to 25°C (77°F). To ensure proper alignment of the im-
peller and cup system, a standard base enabled proper placement of the
cups. A guard leg was initially utilized to determine the proper align-
ment of the impellers but removed before data collection. A selected
impeller was immersed in the ution to a fixed mark with care to avoid
excessive entrainment of air bubbles. The torque reading at a selected
rotational speed was measured after steady state was reached (constant
readings). Readings were collected during one minute at the specific
value of rpm. For any given run, the rotational speed varied (a step-
wise increase) and the range of rotational speed was the operating range
of the Viscometers (0.5, 1.0, 2.5, 5, 10, 20, 50, and 100 rpm). The
tests were done in duplicate and the reproducibility of results was very
high.

Neither surface waves nor vortex formation occurred in any of the
impeller/cup/fluid combinations which satisfied the laminar flow condi-
tion (Re<10). After completion of the test (data collected with impeller
rotating at 100 rpm), the impeller was removed and the next impeller was
tested.

The above procedure was repeated for all systems and fluids.

Results will be presented and discussed in the following chapter.

Table

103

Table 5.2: Rheological Properties of Sample Fluids

 

newtonian Fluids
(Brookfield Standards)

 

3

Fluid n (Pa 3) p (kg/m )
Standard 1 0.093 i 0.001 928.570
Standard 2 0.923 i 0.001 930.000
Standard 3 4.840 i 0.001 966.829
Standard 4 12.200 i 0.001 969.421

 

anrNewtonian.F1uids

 

1 n 1 s

Fluid m (Pa 8 ) n p (kg/m )
CMC 1% 6.492 t 0.03 0.504 i 0.01 974.800
CMC 1.5% 28.417 i 0.02 0.374 t 0.007 1025.870
CMC 2% 59.275 i 0.02 0.352 t 0.003 1144.280

 

1
Means of three replications for 0-2 rev/s for Haake Viscometer data.

 

BOB

Brookfield

 

104

firmnnuflunampd

 

 

 

L.J

T- 25' C

 

 

L—i Both

 

 

 

 

 

 

 

 

 

 

 

 

Wot-r

 

 

 

 

 

 

T

Figure 5.4: Schematic Diagram of Experimental System

S

 

Doto Acqulsltor

 

 

 

 

 

 

 

 

 

 

 

 

 

 

19

V l
r4-

105
5.3) CA TIONS USING T ER VISCO Y METHO S

Three commonly used mixer viscometry methods were used for deter-
mination of the impeller proportionality constant, k' , (thus, average
shear rates) in the mixing system. The procedures are outlined in

Chapter 3.

5.3.1) W

The procedure for the two matching methods: [Metzner and Otto,
1957; and Mackey et a1., 1987] is outlined in Section (3.2.1.1). The
main equations used in these methods are Eqns. (3.1), (3.8) and (3.15).
Expressions for the Po versus Re relationships were developed for each
impeller/cup combination for both the Newtonian and the non-Newtonian
(power-laMO fluids. Also, values of k' were determined as a function of

fluid properties and system geometry.

5.3.2) filgpe Method

The procedure is outlined in Section (3.2.1.3). The main equa-
tions are Eqns. (3.41), (3.45) and (3.47). Values of the impeller
proportionality constant, k', were determined as a function of system

geometry .

5.4) A CU ONS US NG NEW R V COMETRY METHODS

5.4.1) Detetmination of Average Shear Rate

The procedure for determination of the average shear rate in the

mixing systems is as follows:

106

1) Using the values of k' obtained with traditional mixer viscometry

2)

methods, a model was found by fitting the data using stepwise
regression. Thus, the values of 31, a1, 02 and as from Eqns. (4.31)
through (4.35) were obtained for each impeller/cup combination and
fluid under study and expressions for the average shear rate were
obtained.

The values of k' obtained with traditional methods are plotted versus
the values of k' calculated using the equations above. The equation
which gives the better agreement is considered the best equation for

approximation of the average shear rate in the mixing system.

5.4.2) Dttgtmingtigg 9f Axgtage Shea; Stress
1) To check the applicability of the shear stress equations [Eqns.

(4.6), (4.13) and (4.24)], the values of torque calculated from the
corresponding equations are compared with the experimentally measured
values of torque using the impeller (mixing) system. Hence, the
torque equations for the paddle impellers are,

M - 2nb (d/2)2 a (4 5)

8V

for the concentric cylinders analogy with negligible end effects.
When end effects are considered, the equation for torque is the

following:

M - —£4— [ —g— + ——l— ] (4.12)

f(

5.4.:

107

For the flag impellers, the above equations apply in addition to
Model System 3 (d - de ),

M - 21rb (de/2)2 a (4.36)

av

where Gav - m (iav )n , for a fluid of known rheological properties,

n and m. Thus, using the values of n and m and the expression for

lav , the values of an aav are obtained for use in the torque
equations.

2) Plot calculated torque versus experimental torque. The equation that
best represents the shear stress relationship in the mixing system
will then be the one which gives better estimates of the experimental
torque values.

3) Develop flow curves (average shear stress- average shear rate curves)

for a set of geometric parameters (D, d and b).

5.4.3) o dur o e use 0 a ixer viscomete to directl determine

e eo o ic ro erties f ower-law fluids

On the basis of the considerations presented in Chapter 4, the
following procedure is proposed for the determination of the flow curve
(shear stress-shear rate relationship) of a power-law fluid:

1) Measure the torque on the impeller shaft, M, as a function of
rotational speed, N, using the mixer viscometer.

2) Determine the value of the flow behavior index, n, from the slope of
the log-log plot of M vs. N.

3) Select the appropriate equations for the average shear rate and shear

4)

5)

6)

7)

108

stress in the mixing system by following the steps outlined in
Sections (5.4.1) and (5.4.2).
For a given value of rotational speed, measured values of torque and

known system dimensions, determine the average shear stress, a

9

av

for the investigated fluid using the appropriate equation.

For a given value of rotational speed, known system dimensions and

the value of the flow behavior index of the fluid, n, determine the
average shear rate, Iav’ using the appropriate equation.

Repeat steps 4) and 5) for the complete range of rotational speeds

of the viscometer.

Evaluate the flow curves (rheograms) by plotting the ”av versus the

fiav , for the investigated fluid in log-log coordinates. The

intercept of the log-log plot is the fluid consistency coefficient,

m, in Pa 3“.

1’.‘

0V)“;

109

CHAPTER 6

RESULTS AND DISCUSSION

The results of the experimental investigation are presented and
analyzed in this section. In the first part, results obtained widn
traditional mixer viscometry methods for estimating the average shear
rate are presented and discussed. In the second part, results from the
proposed method are described and its suitability determined for
rheological.characterization of power-law fluids. A procedure for using
the Mixer Brookfield Viscometer with Newtonian fluids is presented in

Appendix C.

6.1) ESIIMAIIOM OE AVEBAQE SHEAR BATE USING MIXERS: TRADITIONAL METHODS

6.1.1) Matching Viscosities
6.1.1.1) Power Curves Methgd (Metzner and Otto, 1957)

Newtonian mixing curves for the paddle impellers in terms of
mixing power number (Po) versus Reynolds number (Re) are shown in
Figures 6.1 to 6.3. Each figure represents power data for mixing the
Newtonian fluids in the same selected sample cup. Data points are means
of two replications. These plots indicate that in the viscous regime the
power characteristics are in agreement with the relationship indicated

by Metzner and Otto (1957) for the laminar region of flow,

P -— (6.1)

i.e, they follow a straight line with a slope of -1. Regression analysis

110

 

1 .OE+07

   
 

I’UUUU‘I I I I'U'U'l U—l UI'I'Ur r U U'UUIII I I I'U'U']

1 .OE+00
d/D III 0.327

 

1 .0E+05

1 .0E+04

1 000.0

P0 = P/[ptNJatds]

 

I 'UIUII I U rUr'I' U U'U'I' I’ TUUUIU

0.0100 0.1000 ' :.0000 10.0000
Re = parNtda/‘n

0.0001 0.001 0

Figure 6.1: Power Number/Reynolds Number Relationship for Newtonian

Fluids Using The Paddle Impellers (Cup diameter - 5.5 cm).

111

 

 

1.0E+o7 I I I UUII'U‘I I I I IIIIU! I I I FUIU'I I I I IIIIII I I I VIII."
I.OE+06
. - d/D - 0.515
1-—! a
“-0 1.0E+05 * -. _
* .
fl
2:
5: 1.0904
‘\~
0- 1000.0
ll
0
0- 100.0
10.0
o d/b - 0.0
o «wmu-10
x cub-IL.
1.0 I I 'I’IUI'r I I III... I U
0.000! 0.0010

 

' I Trflr I I truitvr r I I U'III" 1
0.0100 0.1000 1.0000 10.0000

Re = ptNtdz/‘n

Figure 6.2: Power Number/Reynolds Number Relationship for Newtonian

Fluids Using the Paddle Impellers (Cup diameter - 3.5 cm).

112

 

 

1.x+o7 I IIIIII' r I IIIIIII r I I’IUI'U‘ I rIIIIItr I I IIIIIIr
1351-00
d/D - 0.700
1—-1
“-0 1.0E+05
‘0’
LEE LOE+04 1
‘\\ x x
‘1- 1000.0
11
0C: 100.0
-
10.0
o d/b - 1.0
I 0 at all: - 1.0
. I fiT "I'00r r 0 0.70"" r firI'U'U' fifi'U'Uli ' Tt""‘r fi
0.0001 0.001 0 0.01 00 0.1000 1 .0000 10.0000

Re = ptNtd2/n

Figure 6.3: Power Number/Reynolds Number Relationship for Newtonian

Fluids Using the Paddle impellers (Cup diameter - 2.54 cm).

113

on the data yielded slope values ranging from -0.98 to -1.03. Similar
results were obtained with the flag impeller. The effect of the Froude
Number, Fr , on power consumption was investigated and no significant
effect was observed (Appendix B, Tables B1 and B2) for all systems.

To account for the effect of geometry, a generalized Po versus Re

relationship was preferable and Eqn. (6.1) was transformed to

P0 - a0 Rea1 (d/b)°2 (d/0)°‘3 (6.2)
for the paddle impellers, and

Po - p, Refll (d/D) fl? (6 3)

for the flag impeller. Equations (6.2) and (6.3) are nonlinear. To

simplify the regression analysis, the following transformation was made,
log Po - log ao + allog Re + azlog (d/b) + a3log (d/D) (6.4)

log PO - log fio + fillog R8 + fizlog (d/D) (6.5)

The results of the regression are presented in Table 6.1 for the paddle
impellers and Table 6.2 for the flag impeller. The resultant power
prediction equations are,

Paddle impellers: (R2-0.990)

0 083 0 223 ,1 058 1
Po - 415.524 [ Re ° (d/b) ' (d/D) ' ] (6.6)

m

Ix: (V
(D

114

Table 6.1: Regression Results of Eqn. (6.2) (Paddle Impellers)

‘Linear Multiple Regression.Analysis

 

 

 

Regression Estimated Regression Estimated Standard t

Coeffitient Coeffitignt Error

log a0 415.524 -- --
a. -0.983 0.005 -174.50
a, -0.223 0.006 -6.52
as 1.058 0.006 17.40

 

Analysis of variance

 

 

 

 

 

 

 

Sum of Degrees of Error Mean F
Squates Freedom Squares
Regression 516.414 3 172.138 --
Residual 5.226 310 0.017 10211.5
Iotal 521.630 314
2
R - 0.990

a - 0.05

Ta

Te

115

Table 6.2: Regression Results of Eqn. (6.3) (Flag Impeller)

Linear Multiple Regression.Analysis

 

 

 

 

 

 

 

 

 

 

Regression Estimated Regression Estimated Standard t*
Coefficient Coeffitient Error
log 50 28.469 -- --
51 -0.972 0.015 -63.23
52 0.105 0.0156 0.72
Analysis of variance
Sum of Degrees of Error Mean F*
Squares Freedom Squares
Regression 146.532 2 73.266 --
Residual 3.188 87 0.036 1999.2
Total 149.710 90
R - 0.980

Test of hypothesis for 62 : C1: 82 - O

For a level of significance of a - 0.05, t(0.975,87) - 2.00

*
Since t - 0.72 < t(0.975,87), we accept C1 and conclude that 82 = 0

116

2
Flag impeller: (R -0.980)

0.972 _1
PO - 25.912 [ Re 1 (6.7)

Figures 6.4 and 6.5 illustrate the validity of Eqns. (6.6) and
(6.7) for the paddle and flag impellers, respectively. Figure 6.4a is
the plot of predicted power numbers [Eqn. (6.6)] versus the "observed"
power numbers (calculated with the measured values of torque) with the
paddle. The correlation coefficient (0.990) shows the good agreement
‘between.observed and predicted values. The slope of Figure 6.4a (0.906)
also indicates that Eqn. (6.6) predicts power numbers close to those
observed. The closer the slope and the regression coefficient to the
value of one, the better the model. Figure 6.4b presents the Po [Eqn.
(6.6)] versus Re curves for Newtonian fluids using paddle impellers.

Figure 6.5a is the plot of predicted [Eqn. (6.7)] versus observed
power numbers when using a flag impeller. The correlation coefficient
(0.980) and the slope (0.900) indicate the ability of Eqn. (6.7) to
'predict power numbers. Figure 6.5b presents the Po versus Re curves for
Newtonian fluids using the flag impeller. A t-student test was performed
to verify whether the ratio of diameters (d/D) has any significance on
the PC for the flag impeller (Table 6.2). Results indicate that the
effect of the geometric term is negligible. It is evident that Eqns.
(6.6) and (6.7) are useful to indicate the effect of controllable mixing
variables: cup diameter, impeller size and impeller shape.

Non-Newtonian mixing data (i.e. power numbers) were based on the
correlation developed for Newtonian fluids. Power numbers for the non-

Newtonian (power-law) fluids were calculated and, as Metzner and Otto

117

1.01E-l-06

 

1* ' 1 ' F7 ' I '
J R3 = 0.990 Paddle Impellers ‘
1.206+00-l *

.1

0.009054 "

0. 4451-05 -

Po (observed)

3.229054 , + I
up

 

0.00 e a
If 0 I U f t 0'
0.00 3.22905 0.44:»: 0.0090: 1.2112300 1.0115400

Po (predicted)

 

1.0:«1-07 Paddle Impellers

Po
ii
i

       

0.0001 0.0010 0.0100 0.1000 1.0000 10.0000100.00

Re“(d/b)‘”’(d/D)“"

Figure 6.4: a) Predicted Po Versus Observed PO; b) Power Correlation

For The Paddle Impellers.

118

     
   
  
  
  
   
   

1am I r I r fi I f r I
-R‘ a 0.980 Flog Impeller
1.20:1-00-4

J

0.00am -1

1
lflOE+08~

Po (observed)

300E+054

 

I' '7 r

 

l f r
£00£+08 Odllfldll L20£+GI L30£+00

Po (predicted)

h
1300 JLDO£+00

L03+07

Flog Impeller
1.0:1-00 a
’e

1am Q.

1 “+04 ”.

Po
I

1000.0
100.0 ’
10.0

b
L0

r I 'r'r' I revue

0.0001 0.0010 0.01100 011000 1.0000 10.00

Rem

Figure 6.5: a) Predicted Po Versus Observed Po; b) Power Correlation

For The Flag Impeller.

119

(1957) proposed, the corresponding Reynolds number was determined from
the plots of Figures 6.4 and 6.5. Thus, The values of an "average"
apparent viscosity, "a’ were obtained from

2
Re EEC—1— (6.8)

n ”8

6.1.1.1.1) Estimation of Averagg Shear Rates

Figure 6.6 shows the plot of shear stress or apparent viscosity,
"a , calculated as 0/1 using Haake data, versus shear rate independ-
ently measured in a concentric cylinders viscometer (Haake Rotovisko)
for a non-Newtonian fluid (CMC 1%, n-0.504, 111-6.4195 Pa s“). The plots
for the other non-Newtonian fluids are presented in Appendix B (Figures
B1 and B2). From the known shear stress (or "a ) versus shear rate

relationship, a relationship between the average shear rate, ’1 , and

av
rotational speed, N, can be established. The values of "a obtained from
Eqn. (6.8) were used to determine the corresponding lav from plots such
as Figure 6.6. The fiav thus determined are shown as a function of im-

peller speed and geometry in Figures 6.7 through 6.9 for the paddle
impellers (Figures B3 and B4 present the results for the other systems).
These plots indicate that the shear rates vary with the geometry of the
system as well with the properties of the fluid being agitated. Figure
6.7 shows the relationship between the average shear rate and the rota-
tional speed of two impellers in the same container. The impeller in
Figure 6.7a is 1 cm smaller (impeller blade height, b) than the impeller
in Figure 6.7.b. Results indicate slighlty higher shear rates at

specific values of rotational speed for the smaller impeller (Figure

6.7a).

a [P0]

1,. [PO S]

85.

120

 

68$}-

51J3-

3411‘

17H01

 

(10

' ‘T
CMC 1%

Tc 25 C

 

 

 

 

 

1
52

T— r
78 104

7"[1/8]

 

Figure 6.6: a) Shear Stress Versus Shear Rate; b) Apparent Viscosity

Versus Shear Rate (Hydroxypropyl Methylcellulose 1%).

121

30

 

 

 

 

 

 

. r—" . I r I . 1 . r . r’ . r r
1 2 "'33:? System {I 3 ,
n- .
24.1 A ”-0.352 (d/b-O.6. d/D-0.327) _
J 1
75' a
1 _
\ 1 ‘2
5:. 1 x .
t 12~ .1
'?N x
e
6" X 0 -I
" e
a . J
a
0 1 7T’ F’ T 1 ' l r' I r* r* tr r
30 . .. . , . . , . , - , . f. , .
1 z "'33:: System i 4 4
n- .
24.1 ‘ "-0.352 (d/b'OI45. d/D-00327) _,
"a: Q ‘
\ 1s ,, -
v- ) '1
5.;
0
B 1 -
‘?~ 2‘4 x
1 o ,
e
6-l _.
l 1 1 b.
O ’ F7 r r I

, . . . - .
0.0 0.2 0.51 016 ofa 110 112 151 1.0 1.3

N [rev/s]

Figure 6.7: Average Mixing Shear Rate as a Function of Rotational Speed

of the Paddle Impellers. a) System 3; b) System 4

122

Similar results are presented in Figures 6.8 and 6.9. Figure 6.8b
shows the values of the average shear rate for a small impeller in a
medium cup. Figure 6.9b shows the results for a bigger impeller (three
times) in the same cup. Figure 6.10 presents the results for the flag
impeller (Figures 6.11 and B5 show results for other conditions). The
experimental data indicate that the results are dependent on the flow
properties as well as on the geometry of the mixing system. In com-
parison with the paddle impellers, the flag impeller usually yields
lower average shear rate values. This indicates that results are depend-
ent on the shape of the impeller.

Regression analysis was used to determine the relationship between
lav and N. Results are presented in Table 6.3 for the paddle impellers
and Table 6.4 for the flag impeller. It may be seen that :yav increases
linearly with N for the paddle and flag impellers. Then, the average
shear rate has the same form as the expression assumed by Metzner and

Otto (1957) ,

lav - k' N (3.8)
However, when agitating the less viscous fluid (n=0.504, m=6.4915 Pa sn)
with a paddle impeller (Table 6.3), a better fit of the data is given by
the model lav - a + k'N (systems 3, 6, and 7) (It should be emphasized
that this is only a mathematical expression with no necessary physical
meaning). Table 6.4 shows that experimental data for the flag impeller
always follow the relationship given by Eqn. (3.8) . It seems that the
values of k' are not as highly dependent on fluid properties and system

geometry as for the paddle impellers (Table 6.3), where significant

123

 

 

 

 

 

30 ' T ' I T l r—‘I T’ T"T l . 1 tr r
x ”-3.3: System {I 5
0 1'1- .
24.1 A "-0.352 (d/b-O.36, d/D-00327) -
J
"a?
\ 18" x —I
.1 , . J
.71 12- _ , ..
O
6" x . —1
. ; 1 l
a
0 ”T I ' r' r I r r"T’ T7 ' T7 ' r’ I I
60 ' F r I ' I ' I ' I ”I I ' I T* 1 r
" "'g'gg: System f 6 ,
0 1'1- .
48-1 A "-0.352 (d/b'1.8. 0/0-0.515) x _,
.1
"To"
\ 36* x 2 ..
L ..
3 24- ..
.?~ ‘ I
’ '1
12-4 " 4
2
2 b .1
0 fi r r ' T r I r

 

T” ' r— r r r r I— '
0.0 0.2 0.4 0.6 0.8 1.0. 1.2 1.4 1.6 1.8

N [rev/s]

Figure 6.8: Average Mixing Shear Rate as a Function of Rotational Speed

of the Paddle Impellers. a) System 5; b) System 6

124

 

 

 

 

 

 

35 ' [ I r fi‘ r r 1 u 1 v f y 1 f 1
J x "-3.33: System 1? 7
O n- .
28.1 A "-0.352 (d/b-1.0, d/D-0.515) _
l ,, .
F3? :21. .
\\7 6
r- J 4
LII—J
I 14- 5 J
0% 4 X ‘
X
7- i .-
4 ° .
a
0 r* r r*rut r’ r* I tr r ' I r"r r’ r r’
50 ' I ' I I r ' I ' I ' I T’ I
. x "'33:: System I 8 ' ,
0 w- .
J .1
71" 0
\ 3 ‘ ‘
CZ: . e .
1: 204 x ..
d ‘ ‘ 1
104 O x ° ‘
e
ul 8 . 1
° b
0 e r s , . . e , . , .

0.0 0.2 0.4 016 013'110'112 1.4 1.5 1.0
N [rev/s]

Figure 6.9: Average Mixing Shear Rate as a Function of Rotational Speed

of the Paddle Impellers. a) System 7; b) System 8

125

 

 

 

 

 

 

20 a; 3/0'0573' . ' - ‘ ' I - ' . I
4 " - n-o.374. -2a.41
o d/D-O.429 m 6
16-4 a Ive-0.591 ‘
4
7 12.4
\ 3 "
E. 1 .
B 8
oh ‘4 9 d
d . 1
H g 1
< ‘ 4
a
O ’f’ I I ' I ' I r I . I r’ I rWI r
20 ' I ' I ' I ' I ' I ' I ' ' '
x d/O-O.273 ' r
J n-0.504. I .4 1
o d/D-O.429 m 6 9 5 J
164 A Iva-0.59: *
O
'7." " '
\ 12? ‘ ..
5.. ‘
I ‘ d
.5 3'4 a
l g . .
44 d
4 i
q
o D
f I ' l f I f I ' l ' l r l ' r ' ’
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

N [rev/s]

Figure 6.10: Average Mixing Shear Rate as a Function of Rotational Speed

of the Flag Impeller. a) CMC 1.5%; b) CMC 1%

126

 

 

 

 

 

20 T”I 4* I . I ' r r r‘ir’ I ' I
I x d/D-O.273 “0,352, m-59.275
o d/0-0.429 ‘
161 a Iva-0.591 ‘
J J
"a? 12 -
\ l 3 J
z. I ” ‘
I 8-4 "
‘?~
. I
4-4 ‘
4 Q 3 ‘
A a
0 v r ' r ' F ' I r I T I f r r F Y
2G I I V I ' T r ' I r I Y I T t
‘ x III-0.504 d /D-O.591 .
o n-0.374
16-« I "0.352 ‘
F—I ‘ ‘
m
\ 124 2 ‘
2. ‘ ‘
I 8-4 ‘
‘5
4 l ‘
‘4 x J
.. 1'
I g b4
o r I r I r r I f F f

 

' r‘ r r I r
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

N [rev/s]

Figure 6.11: Average Mixing Shear Rate as a Function of Rotational Speed

of the Flag Impeller. a) CMC 2%; b) System 3

127

Table 6.3: Values of k', defined by Eqn. (3.8) (k' - fiav/N ), for
Paddle Impellers.

 

' 2
FLUID SYSTEM 33 R

 

 

1 33.527 0.994
2 15.890 0.986
3 9.535 0.938*
4 10.855 0.979
CMC 1% 5 11.740 0.941
6 32.309 0.925**
7 16.154 0.899***
8 23.647 0.996
9 25.372 0.995
10 20.424 0.993
1 24.375 0.996
2 13.568 0.988
3 10.305 0.977
4 9.051 0.964
CMC 1.5% 5 8.712 0.983
6 22.564 0.981
7 12.784 0.981
8 13.859 0.998
9 21.062 0.998
10 19.511 0.989
1 17.019 0.986
2 10.776 0.976
3 9.948 0.995
4 8.834 0.991
CMC 2% 5 7.395 0.978
6 20.999 0.979
7 12.215 0.998
8 9.417 0.981
9 24.199 0.989
10 13.937 0.998

 

2
* i - 3.5072 + 6.6217N, R -0.973
av

2
** fiav - 13.695 + 20.9313N, R -0.966

2
*** fiav - 8.3535 + 9.2137N, R - 0.999

128

Table 6.4: Values of k', defined by Eqn. (3.8) (k' - yav /N) for Flag

 

 

 

 

Impeller.

2

FLUID SYSTEM, k' R
1 8.602 0.969
CMC 1% 2 9.389 0.997
3 7.704 0.974
1 7.535 0.985
CMC 1.5% 2 7.639 0.997
3 6.952 0.994
1 6.411 0.993
CMC 2% 2 6.919 0.992
3 7.087 1.000

 

129

differences are obtained for each fluid.

6.1.1.1.2) actor ec in ver e ear Rates and k'
6.1.1.1.2.l) Impeller Rotational fineeg

The relationship between the average shear rate and the impeller
speed has been shown in Tables 6.3 and 6.4 for the paddles and flag
impeller, respectively. The slope from the plots gives a value of k',
independent of the value of N. However, if individual values of k' were
obtained at specific values of N, a relationship between k' and N could

be established.

6.1.1.1.2.2) Fluid Properties

Previous research has shown a possible dependence of k'
(therefore, average shear rate) on the flow behavior index, n, widn
pseudoplastic fluids. However, this dependence has been found to be
highly dependent on system geometry and impeller shape (Calderbank and
Moo-Young, 1961; Beckner and Smith, 1966). Their results indicate a
possible decrease in the value of k' with an increase in the value of n
but results were not conclusive and this needs further study.

It is interesting to find that, in this investigation, the less
shear-thinning fluid (i.e. , larger value of n) generated higher values
of k' (higher average shear rates) than that produced.by the highly
shear-thinning fluid (lower n value) when agitating the fluid with the
paddle impellers (Figures 6.7 to 6.9). A similar behavior was observed
by Sinevic et a1. (1986) when agitating power-law fluids in a concentric
cylinders system. It is reasonable to suggest that k3 is not only a

function of the flow behavior index, n, but it also varies with the

130

value of the fluid consistency coefficient, m. In this investigation,
the less shear-thinning fluid (CMC 1%) was also the less viscous fluid.
Also, a decrease in the gap between impeller and cup (d/D) reduces the
effect of the properties of the fluid. With an impeller to cup diameter
ratio of 0.709 or greater, k' was not significantly affected by n (Table
6.3, Systems #9 and #10).

Shear rates in a cylindrical container agitated by a flag impeller
are not as significantly affected by the rheological properties of the
fluid as in the case of the paddle impellers (Figures 6.10 and 6.11),
with a slight variation at high rotational speeds for the less viscous

fluid.

6.1.1.1.2.3) Cup Diameter

The diameter of the cylindrical cup where the fhxhiis being
agitated does not have a significant effect on the value of k' when
using paddle impellers. Figures 6.7a and 6.9b show results for the same
impeller (b-3 cm; d/b-0.6) rotating in different cups. Impeller-to-cup
ratios significantly less than 1.0 (d/D-0.327; Figure 6.7a), yielded
lower shear rates values (at high rotational speeds) than those deter-
mined when using a larger impeller-to-cup diameter ratio, i.e. a smaller
gap (d/D-0.515) (Figure 6.9b). The same behavior was observed with all
the paddle impellers. However, Table 6.3 suggests that d/D may not have
a significant effect on the values of k'. This was confirmed by results
using multiple regression analysis.

As was expected, in the case of the flag impeller, the fluid

container has no significant effect (Figures 6.10a to 6.11). However, it

131

is interesting to note that the more viscous fluid (also the more shear-
thinning fluid), (Figure 6.lla) generated higher values of :Yav when
increasing the d/D ratio (smaller gap), a trend not exactly followed by
the less viscous fluids. It is suspected that the effect of the
geometric term (d/D) is highly dependent on the type of fluid being

agitated.

6.1.1.1.2.4) W

Values of the average shear rate determined for the five paddle
impellers are shown in Figure 6.12 for two non-Newtonian fluids (CMC 1%
and CMC 2%). The difference in values becomes important at high mixing
speeds, with the smaller impeller (d/b-l.8) generating the higher
average shear rates, especially when agitating a low viscosity fluid
(Figure 6.12b). This is due to the fact that a small impeller produces
low flow and thus, higher shear rates. The difference is smaller when
agitating a high viscosity fluid (Figure 6.12a). In both cases, the
difference in average shear rates generated by the different impellers
is small at low mixing speeds. It is interesting to note that the k'
values determined with the less viscous fluid (CMC 1%) show some incon-
sistency, i.e., they do not correlate with the height of the impeller
blade. This phenomenon was not observed with the more viscous fluids.
Again, the type of fluid being agitated seems to be an important factor
in the determination of the proportionality constant, k' (Figure 6.13).
Thus, the correlation between k' and the height of the impeller blade
was determined for each fluid. The best fit was obtained with the fol-

lowing model ,

132

k' - fl. (cl/m”? + p. (6.9)

with 31 , 62 and B; as regression constants. Results from regression are
shown in Table B3 (Appendix B).

In summary, results obtained with the method developed by Metzner
and Otto (1957) indicate that the value of the mixer proportionality
constant, k', is not only a function of the geometry of the system as
the authors suggested but is is also highly dependent on the properties

of the fluid under consideration.

6.1.1.2) Mixer Torque Qurves Merhgg (Mackey et al., 1987)

Following the procedure of matching Newtonian and non-Newtonian
data employed by Mackey et al. (1987), the mixer torque, M, versus nN
plots for the Newtonian fluids are shown in Figure 6.14 for all treat-
ments with the paddle impellers. As it was expected, linear results were
obtained. The slope of the lines, l/k2 (rev/m3), with k2 the mixer
coefficient, seemed to be a function of the geometric variables d/b and
d/D. Thus, the following model for the mixer coefficient, k2, was

proposed
k2 - a0 (d/b) 0. (d/D)°2 (6.10)

After linearization of Eqn. (6.10), multiple linear regression analysis
(Table 6.5) indicates an excellent fit of the data. The prediction
equation for the mixer coefficient, k2, is given by

0 684 .0 408
k, - 6121.262 (d/b) ' (d/D) ' (6.11)

133

 

 

 

 

 

 

35 I I . I I I I I II T I I III I I r
. " d/b-1.8 d/D-0.32
o d/b-1.0
284 o d/b-0.6 n-o'352 x "
4 A d/b-0.45 ,
,_., v d/bi-O.36
In .. _
\ 21
I: ‘ I
B 14- " c -
OF 4 O ' 4
7-4 x x 6 «
O
s .
' a
0 I T’ I II I I I I’III I II‘T— II’I I II II
SC) I I I I I I I I”“II’I I I I I I I I
. " djb-LB d/D-0.327 I q
o d b-1.0
48-4 o d/b-0.5 "'0'504 -
, A d/b-0.45 .
I-I v (Vb-0.36
In. . ..
\ 35
:2 4 <
t 244 " ° 1
oh
. o I ‘
X ' a
12- . ~
X
1 ' ' b1
0 ”I ‘I I I I r I FI’F’ I I I I II I II I
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

N [rev/s]

Figure 6.12: Average Mixing Shear Rate as a Function of Rotational Speed

of the Paddle Impellers (d/D-0.327). a) CMC 2% b) CMC 1%

134

35 e,.

 

k'=3.(d/b)fl.+p.

29-4

w [1/rev]

 

 

 

 

Figure 6.13: Plot of Mixer Proportionality Constant as a Function of

Impeller Size for the Non-Newtonian Fluids (d/D-0.327)

Table 6.!

 

Regressi
Coeffici

log (10
“I
“2

N-

\

Regressic
Residual

\
Total
\

0.0.0

135

Table 6.5: Regression results for the fit of Eqn. (6.10)

Linear Multiple Regression Analysis

 

 

 

 

 

 

 

 

 

 

 

 

Regression Estimated Regression Estimated Standard t*
Coefficient Coefficient Error
log a0 6121.262 -- --
a, 0.684 0.066 16.75
a2 -0.403 0.066 -5.51
Analysis of variance
*
Sum of Degrees of Error Mean F
Squares Freedom Squares
Regression 0.222 2 0.111 --
Residual 0.005 7 0.028 143.51
Total 0.227 10
2
R - 0.980

a - 0.05

136

2
R - 0.980

Figure 6.15 presents the predicted values of k2 using Eqn. (6.11)
‘versus the values of k2 determined from the slopes of the lines for the
paddle impellers (Figure 6.14). The adequacy of the prediction equatiorI
is indicated by the regression coefficient and the slope of 0.990.

Figure 6.16 presents the plots of torque (M) versus viscosity
times rotational speed (nN) to determine the values of the mixer coeffi-
cient, kg, for the flag impeller rotating in the different cups. It is
clear that the value of k2 is also a function of the geometry of the
system for the flag impeller.

The value of the mixer proportionality constant, k' , can then be

calculated from the following equation [Section 3.2.1.1, Eqn. (3.15)],

“’1 (6.12)

with k2 a function of system geometry (Tables 6.6 and 6.7). Finally, the

average shear rates are determined from
1 - k'N (3.8)
The effect of the parameters in Eqn. (6.12) on the value of k'

and, consequently, on the estimation of average mixing shear rates will

be discussed in the following section.

137

 

 

  
 

 

 

00m“: . . . r .7 , . IMMKO . 1 . . , . j .
J —-¢Hrlmd wN-Ll . J -" J mm-Lo
I’D-om: axe-um

0.00404 “ ‘-’°"°-7°' .1 0.00404 -- a/o-cm ..
o—w '1 i .4
E
2: dance a
H
U «I 1
3
8' 0.00204 /,- _
o x ,
.- 1 ''''''' .

OIm‘O" """"" _

mane

 

 

 

 

  
    

o m f I t v t
-—»4hka§§} ' a ‘ .
1-«»dflh‘aul "'“’ .
manna -
H d ‘
‘3
z: manna ‘ _
H 'o
o 4 d
3- 0.00204 -
L
O
I- i
0.00124 _

 

 

 

 

 

fiI ff I I I I
0 5 10 15 20 25
nI-N [Pctrev] nvN [Potrev]

 

Figure 6.14: Mixer Torque Versus Viscosity Times Speed for Calculating

Mixer Coefficient, k, (Paddle Impellers)

2E+04

IE+044

9200-

6300-4

kz (slope)

3400-4

500

R3 = 0.98

1i

138

T fl I r I

0 Paddle Impellers

 

 

500

251-04

‘1

r
3400

r r r

6:500 9230
k: (predicted)

 

15:04 7 29-04

 

lE+04~

9200-4

62100--l

k, [rev/m5]

3400-

500

'_

r
Paddle Impellers

U I I l U I

 

 

r r #

C3 157 25
(”WWW/mm

Figure 6.15: Predicted Mixer Coefficient Versus Observed Mixer

Coefficient (Paddle Impellers)

 

2.5

Torque [NIm I

139

 

0.00164

d

cl

 

0.001211

d

0.00084

 

if

 

 

I I . I I I I I I I I I I 'T
0 d/D-0.273; Its-17443 .
V d/D-0.429. Eta-15843 ..
+ d,/D=-0.591. Its-12488 ..
l
l.
,/ ’ .

fiIIfIr-Irrrr.,..,..,rr,..,..
3 6 9 12 15 18 21 24 27 30

ntN [Pot-rev]

Mixer Coefficient, k2 (Flag Impeller)

Figure 6.16: Mixer Torque Versus Viscosity Times Speed for Calculating

140

Table 6.6: Values of k', defined by Eqn. (6.12), for Paddle Impellers

 

 

 

 

 

kl
FLUID SYSTEM 5 rung, 20 rngi 100 rpm
1 48.594 15.176 8.206
2 36.983 14.329 8.685
3 32.451 12.772 7.823
4 26.583 11.296 6.808
CMC 1% 5 23.737 10.191 6.277
6 61.948 23.409 14.274
7 50.159 19.975 12.503
8 37.851 16.298 10.130
9 39.847 15.562 10.882
10 38.515 16.101 10.8494
1 26.094 12.872 8.325
2 27.485 13.557 8.647
3 23.803 11.939 7.636
4 21.240 10.528 6.139
CMC 1.5% 5 19.219 9.525 5.742
6 25.729 13.895 9.972
7 26.456 13.807 9.354
8 15.568 8.993 6.559
9 21.019 11.468 8.692
10 16.806 9.699 7.301
1 21.400 11.537 8.024
2 22.656 11.701 7.891
3 20.392 10.602 7.099
4 18.347 9.551 6.529
CMC 2% 5 16.347 8.683 5.665
6 22.572 12.537 9.079
7 34.282 16.127 9.605
8 18.385 10.136 7.159
9 10.903 8.914 6.099
10 18.267 10.825 8.361

 

141

Table 6.7: Values of k', defined by Eqn. (6.12), for Flag Impeller

 

 

 

kl
UID 00 m
1 39.852 11.246 6.402
CMC 1% 2 38.589 14.038 8.458
3 47.601 16.707 9.811
1 22.475 10.756 7.111
CMC 1.5% 2 19.045 9.846 6.625
3 21.788 11.445 7.833

 

1 18.135 9.558 6.801
CMC 2% 2 18.184 9.727 6.866
3 19.105 9.914 7.408

 

142

6.1.1.2.1) Factors Affecting k'
6.1.1.2.1.1) 1m elle otati na

Equation (6.12) predicts a dependence of the proportionality
constant, k',<n1the impeller rotational speed, N. Figure 6.17 shows
that as N increases, the value for the proportionality constant of the
paddle impellers decreases for any impeller and cup geometry. At values
below 10 rpm (0.1667 rps), significant changes in the value of k' are
observed. At higher rotational speeds, the proportionality constant
might reach a constant value, independent of the system geometry.

Similar results were obtained with the flag impeller (Figure
6.18). Again, the proportionality constant,ld, is signifhunufly af-
fected by impeller speeds less than 10 rpm (0.1667 rps). fUnis behavior'

was also observed by Mackey et al. (1987) for a flag impeller.

6.1.1.2.1.2) Fluid Properties
The value of k' is also dependent on the rheological properties of
the agitated fluid (Figure 6.19). When agitating a non-Newtonian fluid
with a paddle impeller, as n increases the value of k' increases and the
effect of n on k' diminishes significantly at high impellers speeds
(from 10 to 100 rpm). Also, geometry effects become less important at
that range of rotational speeds. Similar behavior was observed with the
flag impeller.
Figures 6.20 and 6.21 illustrate the variations in.tflua‘value of
k' when agitating fluids with different rheological properties with a
flag impeller at high rotational speeds (100 rpm) in different cups.

Figure 6.20 presents the changes on the value of k' with the value of n,

143

 

 

 

 

 

._ 55C) 2 l " F’ I' I’ I I 4 I II’ T I’ I I I I
E + 4' d/D'0.327. d/b-1.8 "-0.50‘
> a o d/D-0.327. d/b-0.45 ‘
Z 440- e d/D-0.327. d/b-0.36 .1
75. T 0 d/D-QSIS. d/b-1.8 4
2 I d 0-0.70 . d -1,
8 4
‘g’ 220-" -
.2
E I“ j
5 110-b- i
E ‘ :i 1
'i 3 0

O I I I I I I I YLI I I I I F I Wt'fi'
_ 550 ’ I I I r v I I I I I I I I r I I I
E £ 4' d/D-O.327. d/b-I.O "-0.504
) o d/D-O.327. d/b-O.6
z 440.. e d/O-O.515. d/b-1.0 -
E.- . o d/D-O.515. cub-0.5 ,
3 I d Oil-0.709. d b-1.0
g 330-» / / "
u
z~ ‘ <
5 220-1: .
E ..
8'
a: 1 10“ cl
: .3 .
'5 I g .

0 . I I I I I—I I I I I I F fifih—

 

 

0.0 0.2 0.4 0.5 0.8 1.0 1.2 1.4 1.5 1.8
N [rev/s]

Figure 6.17: Mixer Proportionality Constant, k', Versus Rotational
Speed, N, for Paddle Impellers When Mixing a Non-Newtonian

Fluid (CMC 1%)

144

 

 

 

 

 

 

 

— 210 I . ‘ T I ' ' F I
E II d/D - 0.273
> o d/D - 0,429 Flag Impeller. n30.504
:‘ 169- o d/D - 0.591 1
as: 4
1 J
S 128 ..
o i .
§ 874 , _
i 46- ° -
0
2 3 .
'5 I
5 F I’ I1 I I I IFl I a
E 200 . Id/D I0273I I I I I I
- " - FI - .
> 131 o d/D .. 0.429 09 Impeller. n 0 352 ..
t: 161- e d/D - 0.591 ..
1424 q
g... .
103-JP d
64-. a
444' .-
3 25+ I. .
O
s . ' . TL -—-I—‘l-
0.0 0.2 0.4 015 0.3 110 1:2 1:4 1TB 1.3 2.0
N [rev/s]

Figure 6.18: Mixer Proportionality Constant, k', Versus Rotational
Speed, N , for Flag Impeller When Mixing Non-Newtonian

Fluids. a) CMC 1%; b) CMC 2%

145

 

 

 

 

 

" 350 j I 1 I f r I Y I F 1 v 1 r 1 Y
E .1 d/b-1.0.d/D-O.327 + “0,504
> O n20.374
2288-1 I n-0.352
E J:

g 2‘15-I

U

3 4

E 144-I

O

"E

3‘

a: 72

3' «8g

3 . . .

0 r , . r . I . I , I . r . I . I o,
~210 ' I l I I I’ I V I i I ' I t I r
E .x d/b-1.o.a/o-o.515 x ”0,504
> O n-0.374
3168'“ o n-o.:552 -
45: 4
g 125-ID
o
0
=2; x
3
§ 84
r
a X
C
d: ‘42-I8
E J 8 )O‘

5. i 3 ‘

o . , . - .

 

 

 

r r I r 1' I r

IF F I I I
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

N [rev/s]

Figure 6.19: Plot of Mixer Proportionality Constant, k' , Versus

Rotational Speed, N, for Non-Newtonian Fluids (Paddle

Impellers)

146

 

 

 

 

~12 ' I
g I d/D-0.273
{ I A d/D-O.429 rpm-100
:2, 0 d D-O.591
. 10~ /
.x ,.-
1% J
'2 A
2 8-1 . .
° 0
(J J I
i? ll ‘ ‘
2 64 '.
.2
't J
o
a.
s .. ..
b
O «I
.5 I
I!

2 ' I

0-35 0.43 0.51

n. Flow Behavior

Index

Figure 6.20: Plot of Mixer Proportionality Constant, k', as a Function

of Flow Behavior Index, n (Flag Impeller)

147

 

 

 

 

12 k .

'3' - d/0-0.273 ‘

i . d/0-0.429 “Pm' ‘ 0° «
.2. o d/D-O.591

. 10 _
3

1g . .
.2 I.

g 8-1 , -
0 I u :1
5 I ‘

g 6-1 -
.2

I: . ,
8.

s u .
3

.5 ‘ I
a

2 . i f
6.00 33.00 60.00

m, Consistency Coefficient [Potsn]

Figure 6.21: Plot of Mixer Proportionality Constant, k' , as a Function

of Fluid Consistency Coefficient, 01 (Flag Impeller)

148

the flow behavior index. The lower the n, the lower the value of k'.
However, no conclusions can be made by analyzing only the effect of the
flow behavior index. Figure 6.21 shows that the more viscous fluid

(m- 59.275 Pa 3“) yields the lower values of k' (consequently, average
shear rates). Figures 6.22 and 6.23 show similar plots for a paddle
impeller (d/b-l.8) at 20 and 100 rpm (0.333 and 1.6667 rev/s). The
effect of the agitated fluid on the values of k' was the same as for the
flag impeller. No final conclusions can be drawn up to this point since
variations occur when the impellers operate in different geometric
systems and this needs further investigation. The effect of system

geometry will be discussed in the following section.

6.1.1.2.1.3) Cup Diameter

Results indicate that the value of the proportionality constant,
k', for the paddle impellers, is a function of the size of the cup
(Figures 6.22 and 6.23). The bigger cup (i.e., big impeller-to-cup-
diameter ratio, d/D-0.327) generates lower values of k' than the
produced in the smaller cups when rotating at high speeds. At lower
speeds (20 rpm) differences in the values of k' are only significant
when agitating a low viscosity fluid (CMC 1%) (Figure 6.22).
Figure 6.24b presents the changes in k' for a paddle impeller (d/b-l.0)
rotating in three different cups at 2.5, 10 and 100 rpm. It may be seen
that the diameter of the fluid container is an important parameter to
consider for the design of a mixing system for mixer viscometry tech-
niques when working at low rotational speeds. At higher speeds, the
influence of cup diameter becomes almost negligible when agitating

highly viscous fluids (II-0.352, Ill-59.275 Pa s“).

149

 

 

 

 

 

 

 

_ 20 . I .
{I 0 dz’9’0-327 rpm-100, 0,’b=1.8 ,
> I 0/0-0.515
3‘ 17. v c/D-0.709 -
‘5 1 ‘
‘g 14- '-
5 .
2?
TE: 1 1 - v -
g .. ' .
g a ‘ C -1
g .
.5

5 I o a
_ 30 . I .
E d C d/D-O.327 rpm-20' d/b-‘I 8 ‘
> I d/0-0.515
T” 25- v d/D-0.709 -
If I
3
g 20- -
a ‘ ‘
3 .. I ..
5 15 .
a . I
g. .
a: 10 ‘
g .
'5 b

5 , I .

0.35 0.43 0.51

n, Flow Behavior Index

Figure 6.22: Plot of Mixer Proportionality Constant, k', as a Function

of Flow Behavior Index, n (Paddle Impeller); a) 100 rpm;
b) 20 rpm

150

26

 

. duo-0.323 m-100 drb-1.8
- d/D-O.515 rp ' /

22-1 v d / D-0.709 '1

18-1
4

141-1.

.1

104' I -

I W

 

fixer Propenlenellty Constant. k' [I/rev]
A i L

 

30

 

O d/D-O.327 ("-20 d 1-1.8
- d/D-0.515 rp ' /

25 v Iva-0.700 -

 

We! Proportionality con-com. III DIM]
6
III
I

 

 

10-1 v
5 b
r r . l
6.00 33.00 60.00

rn, Consistency Coefficient [Po n"]

Figure 6.23: Plot of Mixer Proportionality Constant, k', as a Function
of Fluid Consistency Coefficient, m (Paddle Impeller);
a) 100 rpm; b) 20 rpm

151

Results presented in Figures 6.20 and 6.21 indicate that cup
diameter becomes important only when agitating a low viscosity fluid,
with changes in the values of k' ranging from 9 to 6.5 (l/rev), as
compared to the values of k' of 7 to 7.5 (l/rev) for a high viscosity
fluid. It is interesting to find that the value of k' is higher for the
:more shearwthinning fluid (n-O.352) when the flag impeller rotates in a
bigger cup (d/D-O.273) than the value of the less shear-thinning fluid
(n-O.504) (Figure 6.20). The trend is reversed for smaller impeller to
cup diameter ratios. These results confitm the need for a better under-

standing of the influence of fluid properties on k'.

6.1.1.2.1.4) Impeller Size (Blade Height)

Figure 6.24a shows the changes of the values of k' as a function
of impeller size (d/b). It is clear that k' is significantly affected by
the height of the impeller's blade at low impeller speeds (2.5 rpm),
especially when agitating the low viscosity fluid (n-O.504). As N in-
creases, the effect of (d/b) becomes less important. The values of k'
for the different paddle impellers rotating in a big cup are shown in
Figure 6.25. Results agree with the previous observation that k' is not
significantly dependent on impeller size when rotating at high speeds,
independently of the rheology of the agitated fluid. At a rotational
speed of 20 rpm, k' values varied from 15.2 to 10.2 (l/rev) when agitat-
ing a low viscosity fluid as compared to k' values from 11.5 to 8.7
(l/rev) for a higher viscosity fluid when using big and small paddles.

It may also be noted that the bigger impeller generated the lower

k' values (i.e., shear rates). The same behavior was observed when using

152

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

__ 20C) . , . T , . , Iv
g — ”=0-504 d/D-0.515 ,
> ---- n-O.374
"_"" 160~ --- n-O.352 -
.K
g .
g 12CL- -
U
‘3 .
‘g‘ 80- -
.2
‘t' rpm-2.5
o T
8
i 40‘ 7; """"""""" ..-_ """" .2 3;: ‘
E 1° rt" .................. ....... .
5 100 :22: """ a
o a a . , . , . , .
DJ) 0.4 CLB 1.2 1.6 2n0
ti/b
__ 125 . r I . r r, .
é d/b-1.0 —- n-O.504 ..
} .... n-0.374
1" 100- --- n-o.352 -
J
2: c1
g -75_, _
u rpm-2.5 .
'§
'2 50- -
£2 ,_..._...-...-.... .
i ‘ w- _;:l;:. ‘
5 25d ~¢KR~u ‘
, w { -..........._

.5 4 100 M ‘ d
a b
0 r r7 r I . , r. I r

0J3 CL2 0:4 0.6 CLB 1.0

Figure 6.24: Plot of Mixer Proportionality Constant, k', as a Function

of System Geometry. a) Impeller Size (d/b); b) Cup Size
(d/D)

153

14

fii

1 -
(Vb-1.8 _ ,3
d/b-1.0 rpm 100, d/D 0.327
d/b-O.5 1
d/b-O.45 +
d/b-0.36

 

d

12%

i
10--1

IO‘I.

 

Hirer Proportionality Constont, k' [l/rev]
a:
d on
$4.:

 

 

 

 

 

 

6 g 21
4 e . . - a
__ 25 . I T
E . 0 d/b'i-B rpm=20. d/D=0.327 J
> I d/b-1.0
‘7‘ 21 4 v d/b-0.6 "
8
.: 4 o d/b-0.45
§ 1. d/b-0.36
g 17- -
8
E
g 14 : -
r v
8 . v
E 10 a:
3 §
§
6 , - , b
6.00 33.00 60.00

m, Consistency Coefficient [Po 3"]

Figure 6.25: Plot of Mixer Proportionality Constant, k' , as a Function

of Fluid Consistency Coefficient, m (Pa s“) for Different

Paddle Impellers. a) 100 rpm; b) 20 rpm.

154

the Power Curves Method.

6.1.1.2.2) Estimation 9: Average Shggz gates

Using the values of the mixer proportionality constant k' deter-
mined from Eqn. (6.12), the mixing average shear rates were calculated
from Eqn. (3.8). Shear rates were also calculated using a constant value
of k' at selected rotational speeds of 10 and 50 rpm in order to deterw
mine the significance of using a constant value of k' when using mixer
viscometry techniques. Results are presented in Figures 6.26 and 6.27
for the paddle impellers (Figure 86 presents other results). It is clear
that the choice of a constant value of the proportionality constant at
10 rpm generates considerably higher average shear rates than when using
the values of k' at 50 rpm and Eqn. (6.12). For the less viscous fluid”
the larger impeller-to-cup diameter ratio (d/D-0.327) (Figure 6.26b)
produces higher shear rates than the smaller gap (Figure 6.26a). Also,
Eqn. (6.12) seems to produce unstable results at low values of rota—
tional speeds.

Similar behavior was observed when agitating a high viscosity
fluid (Figure 6.27b). However, interesting results are observed in the
small gap situation (Figure 6.27a). In this case, the value of 1" does
not significantly lead to considerable differences in the determined
shear rates. In all cases, Eqn. (6.12) presents unstability at low
rotational speeds.

For the flag impeller, the significant differences in shear rates
when using Eqn. (6.12) and a constant value of k' (at 10 rpm) are shown
in Figures 6.28 and 6.29. Variations are bigger when agitating a less

viscous fluid (Figure 6.28) than for the high viscosity fluid (Figure

155

 

 

 

 

 

 

50 r r. 1 1 W T r I I {fir I F r fi r '
— k 83-) ) n-o.504, d/0-0.709
.... ' rpm
40- __ kn (10 rpm) d/b-1.8
H
m
\
C.
I
-z~
:«v a
O‘Rr I ff f 1' r I f r r r r [ r r 1“—
55 ' n ' I ' * r r I ' r r r V r '
— : Egg.) ) n-0.504, d/0-0.32
.... ' rpm
44.- __ k. (‘0 rpm) d/b-1.8
7: ‘
\ 33' ’o"’
.2. -
s 22- x"
0% ””
a "’ " ”.flo-‘MT:
114 "” .00.
. "' b
O_F1=:,p:*”:flfi; r* I

 

 

rhr r— r r r*’* r *T Ti '
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

N [rev/s]

Figure 6.26: Average Shear Rate as a Function of Impeller Rotational

Speed for a Paddle Impeller (CMC 1%); a) Small Cap;
b) Large Cap I

25

156

 

 

 

 

I If I I I l I I r I T [ r I I I I
J - 1* ("-1 n-0.352, d/D-O.709
w (50 rpm)
'3?
\
.2.
B
-z~
0-? I I r r r I V r I r f I I r r I f
30 II I. Y, I ‘I I r ' ‘I r' I I I I r'
- 1; Egg-i ) «0.352. d/0-0.32
m. ' rpm
24... -- k' (10 rpm) d/b-1'B x”
H
m
\
:2.
I
-a-

 

 

 

 

Figure 6.27: Average Shear Rate as a Function of Impeller Rotational

for a Paddle Impeller (CMC 2%); a) Small Cap; b) Large
Cap I

157

45 .

 

_ I- (are)I I I ' T F I I I . I ' I '
k. (50 mm) n-o.so4. d/D-0.273
35" -- k' (10 rpm)

 

(I) l”
4 ”.
\ 27 ’I”
hi J ”’
’I
’I
a
6 18.1 ”’
O ”’
J ””’ O.
’ OOOOOO
” wi/
94 ”” ..........
”’
‘ I’ .......
04 0:0... 0000000000

l

 

I' I'— I Y I I I r I I' r I I T— I— T

 

Y I r I Y r f l V

r ‘ v I T —v— I V
J — k' («1.) n-0.504, d/D-0.591
«- k' (50 rpm)
44-1 -- k' (10 rpm)

in [1/81

 

’ M

’ a...

04'”.

 

f

I 1* v r r Y I r
0.0 0.2 0.4 0:6 ofa 1:0 13 1:4 1?

N [rev/s]

Figure 6.28: Average Shear Rate as a Function of Impeller Rotational

Speed for a Flag Impeller (CMC 1%); a) Large Cap;
b) Small Cap

 

[11/81

‘7.

[1l/s]

1..

Figure 6.29: Average Shear Rate as a Function of Impeller Rotational

158

 

45 —T kl. (;q )I l' I V I I I 1 I I I r I
. --.. k. (50 rpm) n-0.352, d/D-0.273
36-1 -- k' (10 rpm)
1
27..
184 ,,-""”

 

 

 

 

04 r r T I T T T— r T r U j Tf r fi . I
45 T {(1 )‘I— 1 I v F v I f I v T v ‘
— ' '4- -o. 2. d -o.591
4 .... k‘ (50 rpm) n 35 /D
36- -- k' (10 rpm)
J I
27- ,.””
18- I”’

 

 

 

Speed for a Flag Impeller (CMC 2%); a) Large Cap;
b) Small Cap

 

159

 

 

 

 

 

 

 

I. Y I I I I r
x u («.3 "0,352, d/D-O.709
o k' (50 rpm)
272 o k' (10 rpm) (Vb-1'8 ‘
9 204 ‘
o J
0.
H
o 136‘J .
? +0
68-4 S ‘
J 0‘. .
‘3 no
0 ' F r I 3‘ l ' r a
325 r . I f l ' I r f 1
x k (al.) n-0.352. d/D-O.327
o k' (50 rpm)
260 o k' (10 rpm) d/b-1'8 -
H d
1' 195 ‘
O
m ‘ ‘
H
. 1:50Ja ‘
F 49
65-1 3‘ “
4 a. x a
0 U tax o a b
r f ' r ‘ F ' f—d
0.0 6.0 12.0 18.0 24.0 30.0

7~¢[1/s]

Figure 6.30: Apparent Viscosity as a Function of Average Shear Rate for

A Paddle Impeller (CMC 2%); a) Small Cap; b) Large Cap

160

6.29).

Figure 6.30 shows the apparent viscosity-shear rate relationships
when agitating the high viscosity fluid. The use of a constant k' value
overestimates the values of "a while the use of Eqn. (6.12) predicts
lower values of "a for the shear-thinning fluid. Figure 6.31 shows the
results for a flag impeller (Other results are shown in Figures B7 and
B8).

In summary, results indicate that the proportionality constant k'
evaluated by the Mixer Torque Curves Method is significantly affected by
impeller rotational speed, fluid properties and mixing system. The
conventional mixer viscometry method of using a constant value of k’ ,
depending only on impeller geometry, may lead to significant errors when

measuring properties of shear-thinning fluids at low values of N.

6.1.2) Slope Method

The Slope Method consists of the construction of plots of log
[P/(mdsMn£l)] versus (l-n) [Eqn. (3.47)]. The value of the impeller
proportionality constant, k', is obtained from the slope of the straight
line (k' - 10$10133 ). Figure 6.32 shows a typical plot of log
[P/(mdaNn+1)] versus (l-n) for the paddle impellers. Figure 6.32a is the
plot for a paddle impeller with blades 3 cm high. Figure 6.32b is the
plot for a paddle impeller with blade heigth equal to 1 cm. Also shown
in Figure 6.32 is the line obtained by linear regression analysis (R2 =
0.980 and 0.983, respectively) of the experimental data. Similar results
were obtained with the flag impeller (Figure 6.33) rotating in a cup of
diameter (D) equal to 3.5 cm (d/D-O.591) (Results for the other

impeller/cup combinations are presented in Figures B9 through 813).

n. [Pats]

n. [Pats]

161

 

 

 

 

 

 

 

370 . 1 ' I I I
x w («.3 ”-0352, d/D-0.273
o k' (50 rpm)
296 o k' (10 rpm) "
222 '
148 '
0
d h ‘
x D a x o a Ja_J
0 fi I I ' I f I
350 r r I F I ‘— I
x k' (04-) n-O.352. d/D-0.591 .
o k' (50 rpm)
280 n k' (10 rpm) "
210 q
140 d
O
'1
70-4 1‘ ‘
J .0 ‘
O x " I a x n D
f r T I I
0.0 6.0 12.0 18.0 24.0

Figure 6.31: Apparent Viscosity Versus Average Shear Rate for a Flag

Impeller (CMC 2%); a) Large Cap; b) Small Cap

162

 

 

 

 

1.0E+04 ‘ v I ' T I l
k‘ = 9.88 (1/rev)
R3 = 0.980
_ 1000.01
32 j
n'
2 100.01"\\'\
,E, E
\\ +
Q- '1
10.0
1 0 — d/D-0.327. d/b-0.36 a ,
. I F 1* r l r [ ~1
LOE+04 . gr . I r I
k' a 16.35 (1/rev)
R2 ='- 0.983
_ 1000.0
1
2
no
'2 100.0
.5.
\\
O.

10.0

 

1.0 a . f . r r . f . 1
0.0 0.1 0.3 0.4 0.5 0.7

(1-n)

3
Figure 6.32: Plots of Dimensionless Functions [P/(md Nn+1)] Versus (l-n)

For The Paddle Impellers; a) Impeller 5 (b-S cm);
b) Impeller l (b-l cm)

163

 

LOE+04- , . r . 1 ,
k' = 16.04 (1/rev)
R3 = 0.965

1000.0

10.0

 

1.0 ' -'- g/D-(Jr.591
0.0 0.1

r

r I I v I v
0.3 0.4 0.6 0 7
(1 -n)

. d

a
Figure 6.33: Plots of Dimensionless Functions [P/(md Nn+1)] Versus (l-n)

For The Flag Impeller (Small Cap)

164

The scatter in the values of the function [P/(mdaNn+1 )] observed
in Figures 6.32 and 6.33 is not uncommon, and it can be found for im-
pellers studied by others researchers (Rieger and Novak, 1973; Rao and
Cooley, 1984). Table 6.8 presents the values of the proportionality
constant k' (evaluated using the Slope method) for the different
geometries when agitating the fluids with the paddles and flag impeller.
Considerable differences in the values of k' are observed for paddle
impellers of different size (Systems 1 to 5). The effect of the

geometric variables will be discussed in the following section.

6.1.2.1) Factors Affecting k'
6.1.2.1.1) System Geometry

The geometry of the system is an important factor in the evalua-
tion of the proportionality constant (Table 6.8). As in the matching
method (Power Curves Method), the bigger impeller produces the lower
values of k' , and an increase in the values of k' is observed in the
smaller cups (small gap). For the flag impeller, the smaller gap
produced smaller values of k'. Thus, impeller shape seems to be a factor
since different results are obtained for the two impeller types.
Deviations from linearity in the plots of Eqn. (3.47) (Figures 6.32 and
6.33) were not observed for the impellers investigated in this study.
This suggests that the value of the impeller proportionality constant is
not a function of the properties of the fluid when using this mixer
viscometry method.

It was suspected that the slope method may mask the effect of the
impeller rotational speed on the value of k' . Since some researchers

have sometimes used a single value of the impeller rotational speed for

165

Table 6.8: Values of k' (evaluated using the Slope Method)

Paddle Impellers

 

 

 

 

 

 

SYSTEM k' 3
d/D, d/b
1 0.327 1.80 16.354 0.983
2 0.327 1.00 12.576 0.985
3 0.327 0.60 10.407 0.981
4 0.327 0.45 10.000 0.960
5 0.327 0.36 9.880 0.980
6 0.515 1.80 16.89 0.960
7 0.515 1.00 15.99 0.960
8 0.515 0.60 10.306 0.904
9 0.709 1.80 17.730 0.972
10 0.709 1.00 9.827 0.920
Flag Impeller
2
SYSTEM k' R
1 0.273 20.070 0.980
2 0.429 15.090 0.970

3 0.591 16.04 0.965

 

166

 

 

 

 

1.0E+05 . I . r a I
d/D=0.327. d/b=0.6
"g
__ 1.0E+04 3
E o
- . r a;
”13 1000.0
g. A 100 rpm
0. + 50
100.0 0 20

n 10

o 5

I 2.5

10.0 I . r . , . r . T . aj
1.0E+05 v 1 I I ' I ‘— I r 7
d/D=0.327. d/b-1.8
,_ 1.0E+04
g .
z n
”13 1000.0 g Q‘
-
.3. A 100 rpm -.
o. + 50
100.0 0 20

o 10

o 5

I 2.5

10.0. a ,— 1 I F bj
0 0 0 1 0.3 0 4 0 6 0.7

(1-n)

3
Figure 6.34: Plots of Dimensionless Functions [P/(md Nn+1)] Versus (l-n)
For The Paddle Impellers at Different Values of N;

a) Impeller 3 (b- 3cm); b) Impeller l (b-l cm)

167

the development of plots such as Figure 6.32, this effect was inves-
tigated in this study. Figure 6.34 shows the changes in the slope of the
lines when selecting different values of N for the paddle
impellers. Similar results were obtained with the flag impeller (Figure
6.35) . Table 6.9 presents the values of the proportionality constant k'
(evaluated by using the slope method) for the different system
geometries when agitating the fluids with the paddles and flag impeller
obtained at three different values of N (5, 20 and 100 rpm). It is clear
that if only a particular value of the impeller rotational speed is used
for the evaluation of k' , the selection of N becomes a critical step in
the method since significant differences in the value of k' are observed
at the different values of N: the lower the value of N, the higher the
value of k'. Also, as N increases (100 rpm), k' becomes less dependent
on system geometry.

Regression analysis was conducted for the data for the paddle
impellers and results indicated that the diameter ratio (d/D) was not
statistically significant (Table 6.10). The following model was

proposed:

k' - a. (d/b)fl1 N’32 (6.13)
Thus,

0 41 0 347
k' - 9.365 (d/b) ' N ' (6.14)

Comparing the magnitude of k' (Table 6.8) for the flag impeller
with the one obtained for the paddle impeller with same blade height
(Impeller # 3: b - 3 cm; b/d - 0.6), the magnitude of k' is higher for

the flag impeller. Results in Table 6.9 also indicate that higher values

168

 

 

 

 

LOE+05 - . . , .
d/D= 0.273 Flog Impeller
_ 1.05-+04
' ' N
a 1"
"9 1000.0
5 “h
‘< ~
e. ' .
100.0
+ 50 rpm
I 215
10.0 , . , . . r 8|
1.00.05 , . r , r r .
d/D- 0.591 Flog Impeller
LOE+04 g
.E g m
"1: 1000.0 "' no
é '-
__‘ A 100 rpm
E: + 50
100.0 0 20
o 10
o 5
10.0. "' .15. . . . - - b
0.0 0.1 0.3 0.4 0.6 0.7

(l-n)

a
Figure 6.35: Plots of Dimensionless Functions [P/(md Nn+1)] Versus (l-n)

For The Flag Impeller at Different Values of N; a) Large

Cap; b) Small Cap

 

169

Table 6.9: Values of k' (evaluated using the Slope Method) as a Function
of Impeller Rotational Speed

Paddle Impellers

 

 

 

 

 

 

 

SYSTEM, k'
d/D d/b 5 rpm 20 rpm 100 rpm
1 0.327 1.80 30.196 13.542 8.685
2 0.327 1.00 20.705 11.719 8.154
3 0.327 0.60 19.217 9.895 7.020
4 0.327 0.45 14.348 9.732 6.469
5 0.327 0.36 14.290 9.375 6.038
6 0.515 1.80 56.959 13.651 8.526
7 0.515 1.00 34.825 13.343 7.839
8 0.515 0.60 19.405 9.932 6.807
9 0.709 1.80 28.859 16.528 12.365
10 0.709 1.00 17.312 11.151 8.405
Flag Impeller
SYSTEM k'
5 rpm 20 rpm 100 rpm
1 0.273 134.986 14.544 7.592
2 0.429 52.150 12.478 7.577

3 0.591 40.885 9.760 6.883

 

170

Table 6.10: Regression Results of Eqn. (6.13) (Paddle Impellers)

Linear Multiple Regression Analysis

 

 

 

Regression Estimated Regression Estimated Standard t

Coefficient Coefficient Error

108 30 9.365 -- --
51 0.410 0.073 5.96
82 0.347 0.073 -11.39

 

Analysis of Variance

 

 

 

 

 

 

 

 

Sum of Degrees of Error Mean F*
Saueres Freegem Squares
Regression 1.366 2 0.683 --
Residual 0.226 27 0.008 81.6
Tgtel 1.592 30
2
R - 0.900

a - 0.05

 

171

of k' are encountered for the flag impeller, especially when using low
rotational speeds. However, at higher values of N, the differences in
the magnitude of k' become negligible and the average value of k' for
the paddle impellers (k' - 8.03 i 1.78 1/rev) is higher than the value
for the flag impeller (k' - 7.35 2t 0.40 l/rev). This may be due to the
larger surface area of the paddle impellers.

There is no published data on the value of k' for paddle impellers
using this mixer viscometry method. Steffe and Ford (1985) reported k'
values of 4.64 (at 60 rpm) for a pitched flag impeller. This magnitude
is in reasonably good agreement with the average value of k' - 7.35 i
0.40 l/rev (at 100 rpm) (Table 6.9) determined in this investigation.
When using different values of N, the average value is k' - 17.067 i
2.64 l/rev as compared to the value of 13.8 obtained by Rao and Cooley
(1984) (d/D-0.8, d/b-&.5). Differences in the magnitudes of k' are due

to differences in geometry.

6.1.2.2) Estimatiop of Average Shea; Rates and Apparent Viscosity

Average shear rates were determined using Eqn. (3.8). Figures
6.36 and 6.37 present the average shear rates for the paddles and the
flag impeller, respectively. The effect of impeller and cup size on
average shear rates are presented in Figures 6.36a and 6.36b for the
paddle impellers. As it was expected, no significant effect of cup
diameter is observed.

Average apparent viscosities were calculated with Eqn. (3.13).
Figure 6.38 presents the values of "a for two non-Newtonian fluids as a
function of iav for the different paddle impellers. It may be seen that

the choice of impeller will produce different values of "a' Figure 6.39

25

172

 

 

 

 

 

'7’ Ti ' ’7 ' T I I I '*_1 I’ I I
4 x d/D-0.327 d/b‘1.0
O d/D-0.515
20"1 c: d/D-OJOS q
'7." J ‘
\\‘ 15 _
C. l ! .
B 10
‘5 .4 ‘1
l a
J
5 . l
J I ..
0 r r l I l ' I - I v* I ' I"' a
25 T f I f I I t F 1 T y f y
. x d/b-1.8 d/D-O.327 ,
o d/b-1.0
204 o d/b-O.8 ..
J a (Vb-0.45 q
'_' V d b-0.36
0) 15d / -
\ x
n
B 10 ..
oh ‘1 x O
4 g T
5-4 a q
a i J
o . , . , . b
0.0 0.2 0.4

, .
0.6

r . , . r
0.8 1.0 1.2

N [rev/s]

 

Figure 6.36: Average Mixing Shear Rate as a Function of Rotational

Speed. a) Different Cups; b) Different Paddles

 

173

 

 

 

20 ‘ I ' I fir I 1 I r I I I , I Y ‘
X d/D-O.273
. o d ”3-0.429 Flog impeller ‘
a d D-0.591
16-4 / _‘
J
X
'7' 124 -
\
3:. I .
.zi 8-4 x g
0
_‘ O
44 g _
I 3
0 T I r I

0.0 0.2 0.4 0.6. 018' 110' 1T2 1.4 1.5'

N [rev/s]

Figure 6.37: Average Mixing Shear Rate as a Function of Rotational

Speed for the Flag Impeller

 

1.8

 

174

 

 

 

 

 

4s +— r . e I . r .
J x d/b-1.8
o d/b-1.o n-0.352, d/D-O.32
364 o d/b-O.6
.4 A (Vb-0.45
2? 274 V d/b-O.36
O
0. 1
I—l
a 18-4 ‘0
F? a"
‘0‘, x
9'4 "D O x
1 V ‘ D O X
0 # I I r l r r
10 . , I r I . l .
I z 33:}: n-O.504, (VD-0,327
84 o d/b-O.6
I A d/b-O.45
I-—v - .
2’ 64 v d/b 036
C)
&. x.
. 44 °x'
P J ‘o a x
u
2-I a o x , ‘ I: o x
0 fl I I r I I r
0.0 3.2 6.4 9 5 12.3

Figure 6.38: Apparent Viscosity as a Function of Average Shear Rate

7~[1/s}

Using The Paddle Impellers. a) CMC 2%; b) CMC 1%

 

 

175

 

 

 

 

 

 

55 ' T ' I I I r
I * "3:3: "3:21;. I
0 our. .rn- .
44-4 :1 n-0.352. m-59.275 d/D-0.591 -
I I
H D
5’: 33~ ..
O
o. J a .
h—l
o 22% _
S? 4 o a J
O O
114 -
O
+ x x x 31
0 l I ' I fi fi
45 . I . I . I . r
+ * "-3'331- "2:212, I
o n- . , m- .
36-4 a «0.352. III-59.275 d/D'O-273 -
F_‘ 4 o ' -
5': 27- -I
O
o. I o .
H
. 18-4 _
F? 4 o n j
|
O D
94 a 1
.I 4
x x x x b
O ' I ' T ' I ' 1* r*’
0.0 3.0 6.0 9 0 12.0 15.0

Figure 6.39: Apparent Viscosity as a Function of Average Shear Rate

Using The Flag Impeller; a) Small Cap; b) Large Cap

176

shows results for the flag impeller. The size of the sample container
also results in different values of "a’ and the bigger the cup, the
smaller the values of "a“

6.1.3) WW

Three mixer viscometry methods were reviewed and evaluated. The
values of the mixer proportionality constant k' for determination of the
mixing average shear rate iav evaluated using the three mixer viscometry
methods have been presented in Tables 6.3 through 6.9 in the previous
sections. The assumption iav - k'N [Eqn. (3.8)] is the basis for the
three procedures. However, the original assumption of the work of
Metzner and Otto (1957) of a constant value of k' is not always valid
for variations in operating conditions (impeller rotational speed, N),
fluid rheological properties and system geometry.

It is important to note that the three methods predict variations
in k' with the geometry of the mixing system for the ranges investigated
in this study. However, the flower Curves Method shows little variation
with geometry for the flag impeller (Table 6.4). When working at high
mixing speeds [N equal to 100 rpm (0.167 rps)], the values of k' ob-
tained with the Iorgug Cuges Method and the Slope Method show excellent
agreement. Under this operating condition, the effect of the other
parameters (fluid rheology and system geometry) become less significant.
Thus, the assumption of a constant k' value is valid under these terms.

All the three mixer viscometry techniques require the determina-
tion of parameters using computer analytical techniques, i.e., small
changes in experimental data result in big changes in results of regres-
sion analysis. The W can be very tedious and deviation

from the basic assumption for the average shear rate ("yav - k'N) may

 

177

occur when using different impellers. The Mixer Torgue Curves and the
Slope methods are simpler since they require less data handling.
Results indicate that the basic assumption of traditional mixer
viscometry methods of a direct proportionality between average shear
rate in the agitated fluid and the rotational speed of the impeller
[Eqn. (3.8)], with k' depending only on the geometry of the impeller,
may be incorrect when working at low rotational speeds. Variation in
fluid rheological properties has also proven to be an important factor

for determination of the average shear rate in the mixing system.

6.2) ERMI AT OLOGICAL 0 RTIES OF POWER-LAW FLUIDS USING

THE ALTERNATIVE MIXER VISCOMETRY METHOD

This section presents the experimental verification of the
proposed procedure for determination of rheological properties of
power-law fluids using a measuring system which consists of an impeller

(paddle or flag) rotating in a cylindrical container.

6.2.1) Determination of the Flow Behavior IndexI n

The measured torque on the impeller shaft is presented as a func-
tion of the rotational speed of the impeller for each non-Newtonian
fluid, with the geometry of the impeller as a parameter. Figure 6.40
shows the results for the CMC 2% solution for the different paddle/cup
combinations. Figure 6.41 shows results for the flag impeller. It fol-
lows from these plots that for every geometry of the impeller system the

experimental points may be approximated by a straight line. All

178

 

1 .000

0.100

M [Ntm]

0.010

 

0.001 .

 

0.100

M [Ntm]

0.010

 

 

0.10 ' 1.00 ' ' ”10100' ' 300700 1000.00
N [rpm]

Figure 6.h0: Plot of the torque on the impeller shaft M versus
rotational speed of the impeller N for 2% wt% aqueous
solution of CMC (Paddle Impellers) (See Table 6.9 for

definition of systems)

179

1.000 . . n...

 

 

 

 

: , . - ....., . . N..... . W-..“
: CMC 27: Flog 3
0.1003 -.
E 3 . 1
* '1 '1
z I . .
&—-l
2
0010-: -.
1 , 1
‘ H (VD-0.591 '
' H d/D-O.429 '
0.001 . . ......r e. rm... .fi ....°.'," “pf???"
0.10 1.00 10.00 100.00 1000.00

N [rpm]

Figure 6.41: Plot of the torque on the impeller shaft M versus
rotational speed of the impeller N for 2% wt% aqueous

solution of CMC (Flag Impeller) A

180

lines have similar slope, and this slope is equal to the value of the
flow behavior index, n. Tables 6.11 and 6.12 present the values of the
rheological parameter for all the systems. It is seen that the mag-
nitudes of the flow behavior index n obtained with the impeller system
for the standard fluids are in good agreement with those using the
concentric cylinders viscometer when operating at the same range of

shear rates (0-40 l/s).

6.2.2) Determination of Shear firregs-Shear Rate Relationships

6.2.2.1) Average Shear gate 1;; The Miring System
To verify the applicability of Eqns. (4.4), (4.6) and (4.9) to

approximate the values of the mixing average shear rates, the values
of k' determined using the three investigated mixer viscometry tech-
niques [Section (6.1)] were used (as average values) for comparison with
the theoretical expressions. Initial values of the parameters of the
equations ([91, a1, a2 and as) were assigned following the concentric
cylinders analogy to find the values of the constant 191 and
the parameters a, , a2 and as that showed best agreement with the

experimental values of k'. These are the following:

131-411'

011 - a2 - 2/n and,

as - n/2, with n - power-law index.
Thus, Eqn. (4.5) can be written as follows,

2/n n/2

 

k' - 4n [ (D/d) (6.13)

(d/b)
(1)/d)”n - 1 J

 

181

Table 6.11: Values of the Flow Behavior Index of Standard Power-Law
Fluids Using the Mixer System (Paddles) and a Concentric
Cylinders Viscometer (Haake Rotovisko)

 

rElow Behavior Index- n
1 2
Mixer Congenrrig Cylinders

 

 

 

__ELQID_______§X§IEM
1 0.902 i 0.003
2 0.834 t 0.004
3 0.896 i 0.003
4 0.887 t 0.002
CMC 1% 5 0.842 t 0.002 0.829 i 0.005
6 0.808 i 0.003
7 0.807 i 0.005
8 0.857 i 0.004
9 0.869 i 0.006
10 0.857 t 0.005
n 0.856 t 0.030
avg
1 0.599 t 0.008
2 0.759 i 0.004
3 0.726 i 0.004
4 0.735 i 0.003
CMC 1.5% 5 0.718 i 0.002 0.718 i 0.005
6 0.599 t 0.007
7 0.687 i 0.005
8 0.584 i 0.006
9 0.660 i 0.005
10 0.640 i 0.005
n 0.670 i 0.060
avg
1 0.679 t 0.004
2 0.688 t 0.003
3 0.648 i 0.004
4 0.684 i 0.002
CMC 2% 5 0.676 t 0.003 0.528 i 0.003
6 0.646 i 0.001
7 0.662 i 0.002
8 0.653 t 0.003
9 0.641 i 0.001
10 0.628 i 0.001
n 0.660 i 0.020
avg

 

1
Brookfield Mixer. Shear rate range: 0-30 l/s [Eqns. (6.18) and (6.19)]

2
Haake MV-III (d/D-0.73). Shear rate range: 0-40 l/s

182

Table 6.12: Values of the Flow Behavior Index of Standard Power-Law
Fluids Using the Mixer System (Flag) and a Concentric
Cylinders Viscometer (Haake Rotovisko)

 

Flow Behevior Index. n

 

 

 

1 2
ELQID SXSIEM Mixery Concentric Cylinders
1 0.933 i 0.005
CMC 1% 2 0.805 t 0.002 0.829 i 0.005
3 0.967 t 0.007
n 0.901 i 0.085
avg
1 0.811 t 0.004
CMC 1.5% 2 0.711 i 0.003 0.718 t 0.005
3 0.708 t 0.001
n 0.743 i 0.058
avg
1 0.690 i 0.003
CMC 2% 2 0.667 i 0.002 0.528 i 0.003
3 0.658 i 0.002
n 0.672 i 0.016
avg

 

1
Brookfield Mixer. Shear rate range: 0-30 1/s [Eqn. (6.23)]

2
Haake MV-III (d/D-0.73). Shear rate range: 0-40 1/s

 

183

Equatiru1 (6.13) differs from the original equation for the shear

rate at the cylindrical bob,

 

 

 

qb .. gg (1)/<1)”n (‘1-3)
n (D/d)2/n - 1
or
ab - am (D/dlz/n (6.14)
n (D/d)2/n _ 1
with 51 - (ZEN , al - a2 - 2/n, in the value of parameter 131 , since

the division by n is not present in the equation for the paddle Ln-
pellers. As stated before, the effect of the power-law parameters on the
value of k' is not clearly understood. For an impeller, the dependence
on the value of n seems to be less significant than in the case of the
concentric cylinders. The values of a1 and 012 are identical to the
concentric cylinders analogy.

Values of k' obtained with Eqn. (6.13) are shown in Figure 6.42a.
It is interesting to note that the above expression gives reasonably
good results, except for the small gap case (d/D a 0.709). (These data
are represented by the crosses (+) in Figure 6.42a). It is clear that
Eqn. (6.13) predicts considerably higher values of k' when the impeller-
to-cup diameter ratio is small [ (d/D) 2 0.709 ]. Thus, Eqn. (6.13) is a
good approximation of the data when the diameter ratio follows within
the range of 0.327 S d/D 5 0.515. An equation to approximate the values

of k' when (d/D) was 2 0.709 was obtained (based on the concentric

184

30.00

 

 

 

 

   
  

 

J 0 Wide Glap I t I i
+ Small 1
e 25.20- GOP _
(J
1+: .
c d
2 20.404 0 T.
T: .
Q) * ‘I +
g- 15.60-1 0 ° ° ..
‘5; 4 ° +
- e :' + ‘
x 1 0.80- 3.9 Paddles -1
6.00 ' r r ‘ I a
30.00 . ,
4 1o “fideIGap
+ Small Gap
9 25.204
(I
4.;
I:
g 20.40—
e: .1
3.
x 1 5e 60"
C)
\J .1
3‘ 1 0.80- Paddles
0.00 . n.1,

 

 

6.00 ' 10130 ' 15:60 ' 20140 25120 ' 30.00

k' (calculated)

Figure 6.42: Experimental Versus Calculated Values of k'; a) k' defined
by Eqn. (4.5); b) k' defined by Eqns. (4.5) and (6.13)

for small gap (+) (Paddle Impellers)

185

cylinders analogy) by expressing the experimental values of k' by:

n/2

 

(6.15)

2.12
1.1 _ (m [ (010)

(«M»)
(1)/d)”In - 1 J

Due to the empirical nature of Eqn. (6.15) it is difficult to
physically explain the difference in the value of a1 - (2-n)/n. However,
this result indicates that the gap between the impeller and the fluid
container (cup) is a factor to be carefully taken into account when
considering the use of mixer viscometry methods.

Figure 6.42b presents results for the values of k' calculated
using Eqn. (6.13) for 0.327 s d/D 5 0.515 and Eqn. (6.15) for (d/D) i
0.709. The results are satisfactory for all the impellers and power-law
fluids investigated. The agreement between experiment (mixer viscometry
techniques) and theory [Eqn. (6.15)] was usually better than 10%.

Simplified equations were attempted by using the same procedure of

data fitting. The final equations obtained were as follows:

n/2
k1 - a, [——f§j§i§72 ] (6.16)

Simplifying, Eqn. (6.16) becomes

k' - 4n (1)/b)“/2

(6.17)
Figure 6.43 shows the values of k' calculated from Eqn. (6.17)
versus that determined from experimental data. Results show a reasonable

agreement between predicted and average values. Even though not as good

 

186

30.00

 

 

 

 

I I I I

25.20- a -
A O
«8 0 a o
1:
C, 20040-1 -1
E 0
e: o o
8. ° °
3 15.604 0 ° .-
V O
- -1 o :1
x. '6'

10.304 , .116 Paddles .

6.00 . , . , . r e F r
6.00 10.80 15.60 20.40 25.20 30.00

k' (calculated)

Figure 6.43: Experimental Versus Calculated Values of k'; a) k' defined

by Eqn. (4.7) (Paddle Impellers)

187

as Eqn. (6.13), results fall within the range of k' values obtained
using mixer viscometry techniques. The maximum error was less than 20%.
It would seem that for the purposes of engineering design the use of
Eqn. (6.17) is reasonable.

Thus, results indicate that it is possible to approximate the
value of the impeller proportionality constant, k' , for a particular
impeller (paddle) by using Eqn. (6.13) for large to medium gaps, and by
Eqn. (6.15) for small gap (d/D 2 0.709). Only the geometry of the system
and the value of the flow behavior index, n, for the investigated fluid
are needed. Equations (6.13) and (6.15) [as well as Eqn. (6.17)] avoid
the dangerous assumption of k' being a constant independent of the fluid
properties. Also, it can.be said that k' remains constant at each value

of rotational speed. In terms of the average shear rate, these equations

 

 

become,
2/n
78v - 4n ‘D/g;n ] (d/b)“/2 } N (6.18.1)
(D/d) - 1
0r
2/n
. (D/d) n/2
1 - 20 (d/b) (6.18.2)
3" [ (1)/<1)”n - 1 l

for a paddle impeller and 0.327 s d/D 5 0.515 , and

zfin
18v - {4w [ (D/d) J (d/b)“/2 } N (6.19.1)

(D/d)2/n - 1

 

188

01'

(D/d)z;In

(D/d)2/n - 1

“/2 (6.19.2)

Iav - 20 [ J (d/b)
for d/D 2 0.709 (with l 2 b 2 5 cm and 0.5 2 n 2 0.9). Also, a
simplified approximation is given by

n/2

+av - [4w (D/b) 1 N (6.20.1)

01'

13v - [2«o(0/b)“/2] (6.20.2)

for the geometric range investigated (0.327 s d/D 5 0.709).

In the case of a flag impeller (b/d-0.5), Eqn. (4.9) gave good
results for the values of k' , with 51" 411 and 011 - n/2. An expression
such as Eqn. (4.9) was preferred for the flag impeller because Eqn.
(6.13) did not yield very good results for this type of impeller. Thus,
Eqn. (4.9) becomes,

k' - 4n [ (1)/d)”2

l (6.21)
Equation (6.21) predicts the value of the proportionality con-

stant, k', for a flag impeller rotating in a power-law fluid. Figure

6.44a shows the results obtained from Eqn. (6.21) when d- 1.5cm. Figure

6.44b shows the case for Model 3 (d - de ). It seems that the assumption

189

 

 

 

 

 

 

C
o
H
c:
o
E
1:
o
a.
x
0
V
3‘
d 1
C.‘ 25001 1
0
«OJ d
c
g 22.00~ , ° .
e: . O
o
a- 18.004 0 "1
3 1 ° 0
- 0e e .
x 14500-1 "‘
. Flag .
10.00 . . . . . b

 

10.00 14100 13100 22100 26100 30.00
k' (calculated)

Figure 6.44: Experimental Versus Calculated k'; a) k' defined by

Eqn. (4.7); b) R’ defined by Eqn. (4.7) (d—de )

190

of a cylinder of equivalent diameter, de , as responsible for the
shearing of the fluid when using a flag impeller tends to overestimate
the values of k'. However, the values of k' fall within the range of k'
determined from the traditional mixer viscometry methods with a maximum
error of 15%. Again, the assumption of a constant k' value, only a
function of impeller geometry, is avoided and proved not to be tnnua for
the cases investigated in this study.

Thus, the average shear rate in a mixing system (flag impeller)
can be approximated by the following equation,

’1

- {411' [ (1)/d)“2 1} N (6.22.1)

8V

or

n/2

’1 v - 201 (DA!) ] (6.22.2)

a

6.2.2.2) Average Shear Strees in The Mixing System
6.2.2.2.1) Iergue hpprorimeriohe

To check the applicability of the shear stress equations, Um;
experimentally measured values of torque for every mixing system were

compared to those calculated using the following equations:
Model 1 (concentric cylinders analogy with negligible end effects):

M - 2th (d/2)2 aav (4.14)

191

Model 2 (concentric cylinders analogy with end effects):
a

fi L+_L
M- 2 I: d 3 :laav (4.21)

Model 3 (Flag impeller; d- de ):

2

M - 2«b (de/2) aav (6.23 l)

and
d3
n
___e L-Il

M 2 [ de 3 ] aav (6.23.2)

where

aav'- m (y'av) flwith n from mixing system or a conventional
concentric cylinders viscometer, and m from concentric cylinders vis-
cometer. The value of the average shear rate can be evaluated using the
appropriate equations [Eqns. (6.13) and (6.15) for the paddles and
(6.21) for the flag for the ranges investigated in this study].ffima
average shear rate could also be determined by traditional mixer vis-
cometry methods.

Figure 6.45 presents the values of torque for two fluids (1% and
1.5% CMC solutions) in a system consisting of a small paddle impeller
(d/b-l.8) rotating in a large cylindrical cup (d/D-0.327). It can be
seen that the Model 1 (concentric cylinders analogy with negligible end
effects) [Eqn. (4.14)] gives better prediction of the torque values than
the Model 2 (concentric cylinders analogy with end effects) [Eqn.

(4.21)]. For a more viscous fluid (2% CMC), the Model 2 gives better

results for the torque values (Figure 6.46). These results indicate that

192

 

6111: 1' .5: (Li/6- 1.11. 670-0327) ‘

   
     

 

Measured Torque [Ntm]x10'

 

 

 

 

Measured Torque [Nem]x10'

x in
f t I I I’ 0 a
0.50 0.18 0.54 0.52 0.40
Calculated Torque [Nem]x 10a

 

 

   

Figure 6.45: Measured Versus Predicted Values of Torque on the Impeller
Shaft Using Eqn. (4.14) (o) and Eqn. (4.21) (x); a) CMC
1.5%; b) CMC 1% (Paddle Impeller # l, (d/b) - 1.8) [EE -

Model 2 (end effects); NEE - Model 1 (no end effects)]

 

 

193

the type of fluid being agitated is a significant factor for considera-
tion when using mixer viscometry methods.

When using a larger paddle impeller (d/b-0.36) in the same cup
[(d/D - 0.327] (Figure 6.47), the differences between the torque \nilues
predicted by the two models becomes smaller, with the assumption of
negligible end effects (Model 1) showing better agreement between
experiment and theory than the assumption of the presence of end effects
(Model 2). Thus, when working in a large cup, the end effects can be
assumed negligible.

For the other mixing systems, Model 2 (presence of end effects)
(Figure 6.48) predicts values in close agreement with the experimentally
measured torque values. This is due to the presence of a smaller gap
between the impeller and.the wall of the cup and the use of Eqn. (4.21)
seems to account for any effect of the solid boundaries. Figure 6.48a
indicates that Eqn. (4.14) predicts lower values of torque. Figure 6.48b
indicates the applicability of Eqn. (4.21) to represent the torque on
the shaft resulting from the rotation of the paddle impeller
(approximated by a cylinder).

Based on the results presented in Figures 6.45 through 6.48, it can
be concluded that when using a mixer viscometer with a small paddle in a
large cup [(d/D) 5 0.327], assumption (iv) is valid, and the average

shear stress can be approximated by Eqn. (4.15),

a - 3L,- (4.15)

for standard power-law fluids of low to medium viscosity. For a highly

viscous standard fluid (CMC 2%), the effect of the solid boundaries

194

 

   
    
 

 

 

 

2.00 . , . , . 1 e 1 ,

., . CMC 22: (d/b=1.8, d/D=0.327)

C3

3'2 1.60-1

l—!

E .

*1

z

"" 1.20-4

0

:3

a.

3

I— 0.604

13- J

0

L.

a

c, 0.40-4

0)

2 ‘ )6 NE
0.00-J . , . , . , .°IEE.

0.100 0.40 0.80 1.20 1.60 2.00

Calculated Torque [Ntm]x 103

Figure 6.46: Measured Versus Predicted Values of Torque on the Impeller
Shaft Using Eqn. (4.14) (o) and Eqn. (4.21) (x) CMC 2% [EE

- Model 2 (end effects); NEE - Model 1 (no end effects)]

195

 

 

 

 

 

no 3.00 I I I I t I

3'; .l CMC 1.5% (d/b=0.36, d/D=0.327)

,_,

E 2.40-

I'

z

I—l

., 1.80-4

3

U’ -l

3

,__ 1.20..

‘0 .1

3’- J

3 0.60

g NE

2 0.00-l , . , E":
0.00 0 60 1.20 1.80 2.40 3.00

3

 

 

 

Measured Torque [Ntm]x10

 

 

x NEE‘

6 SE
0.00 . r . , . , . , . Al
0.00 0.16 0.32 0.46 0.64 0.60

Calculated Torque [N*m]x 103

Figure 6.47: Measured Versus Predicted Values of Torque on the Impeller
Shaft Using Eqn. (4.14) (o) and Eqn. (4.21) (x); a) CMC
1.5%; b) CMC 1% (Paddle Impeller # 5, (d/b) - 0.36) [EE -

Model 2 (end effects); NEE - Model 1 (no end effects)]

 

 

 

196

5.00 .
Paddles

r* ' ”’r ‘2’ I . I I

L_

Measured Torque [Nem]x10’

 

 

 

 

 

 

 

 

“o
3..
E
9
.2..
O
3
g
g...
g
I!
0.00 1 1 - r r . ”E‘s"
0.00 1 .00 2.00 3.00 4.00 5.130

Calculated Torque [Nem]x 103

Figure 6.48: Measured Versus Predicted Values of Torque on the Impeller
Shaft For All Systems And Fluids (Paddle Impellers); a)

Eqn. (4.14) (Model 1 - NEE); b) Eqn. (4.21) (Model 2 - EE)

 

197

becomes significant and assumption (iv) is questionable. Thus, the

average shear stress can be approximated by

-1
3

a - -1§— [ -g— + -%— ] M (6.24)

Equation (6.24) also applies to the other impeller/cup combina-
tions for the range of viscosities of the standard fluids.

The torque values calculated by equations (4.14), (4.21) and
(6.23.1) were compared to those measured experimentally with the flag
impeller. Figure 6.49 presents the measured torque values versus those

calculated using Eqn. (4.14) (Model 1):

M - 2nb (d/2)2 Gav (4.14)

It is evident that this model yields values of torque considerably
higher than the experimental values. Figure 6.50a presents tflue results

using Eqn. (4.21) (Model 2):

.1d. .0. + .1.
M - 2 [ d 3 ] aav (4.21)

and it yields considerably lOwer values oi'torque. Equation (6.23.1)

(Model 3),

2
M - 2wb (de/2) aav (6 23 1)

seems to be the one that better represents the torque in the impeller

198

 

 

3.00
I'D d
CD
.35, 2.40-
E
* d
E.
1.604
0
3 .l
6'
1- 1.204
. 'u
a: .
3
8 0.60-
0
2 .
0.00
0.00

I ' I ' l ' I

Model 1 (No end effects)

 

 

o 9 "
0
$ 1
10
e . ‘6 . ° "
66 ° Flag .
”I" °
' 0.60 t 1.20 1.60 2.40 3.00

Calculated Torque [Nurm])<103

Figure 6.49: Measured Versus Predicted Values of Torque on the Impeller

Shaft For All Systems And Fluids (Flag Impellers) Using

Eqn. (4.14) (Model 1) (Negligible end effects)

 

199

 

 

 

 

 

 

 

 

2.00 . F . , . ,
”c Model 2 (End effects)
.3 1.604
E
O
.2...
3 1.204
6
1- 0.604
3 0.404
a
0.00 f f , e
2.00 v I V fi ' l ' I
'0 . Model 3 (d I de) .
3:". 1.604 0 -
E 1.204 ° ‘
*- Geno-1 "
. . ,
00w- -
. Flag ,
0-00 r. r c r c I 1 1— .— b
0.00 0.40 0.80 1 .20 1 .60 2.00

Calculated Torque [Nalsrr1]x103

Figure 6.50: Measured Versus Predicted Values of Torque on the Impeller
Shaft For All Systems And Fluids (Flag Impeller); a) Eqn.

(4.21) (Model 2); b) Eqn. (6.23.1) (Model 3)

200

shaft (Figure 6.50b).
Thus, the average shear stress for the flag impeller may be ex-

pressed by

3 '1
«d
e _L i
a — 2 [ (18 + ] M (6.25)

 

av 3

Equation (6.25) implies that the effect of the end boundaries (top
and bottom) is important when using a mixing system with a flag im-

peller.

6.2.2.3) Elow Curves
6.2.2.3.1) Ideal Eluids

Average shear stress-average shear rate curves were obtained for
the standard power-law fluids using Eqns. (6.18) and (6.19) for evalua-
tion of the average shear rate of paddle impellers and Eqn. (6.21) for
the flag impeller. The average shear stress was determined using Eqn.
(4.21) for the paddles and Eqn. (6.23.1) for the flag impellrnt. Figure
6.51 presents a typical shear stress-shear rate plot for the CMC 2%
solution obtained using the mixer viscometer with a paddle impeller.
Similar results were obtained with the other fluids and geometries
(Figures 314 and BIS).

As indicated before, the values of the flow behavior index were
calculated as the slope of the log-log plot of torque, M, versus the
rotational speed of the impeller, N. The values of the consistency
coefficient, m, were obtained by linear regression of the power model,
a - m (y'av) fl The values are shown in Tables 6.13 through 6.17 for

av

every system geometry and Models analized in this investigation.

 

201

250.00

 

 

 

 

*' I r, T’ t r ' r r
d I4L61- d '4L32
. ° I” m 7 one 2:: .
200.00- ..
O
F; ISOeOO'l -
0L
H I
. O
b 100.00. "'
° 1
50.00.. a Ra I 0.990 .
. o n - 0.666 n .
)0 m - 26.35 Pace
0000 r fi r r r r Ti 7 r f
0.00 6.00 l 2.00 1 6.00 24.00 30.00

‘7- [1/81

Figure 6.51: Flow Curve For 2% wt% Aqueous Solution of CMC Determined

Using the Mixer Viscometer with a Paddle Impeller

202

Table 6.13: Values of the Fluid Consistency Coefficient of Standard
Power-Law Fluids Using the Mixer System (Paddles) [Model
1: negligible end effects; Eqn. (4.14)] and a Concentric
Cylinders Viscometer (Haake Rotovisko)

 

n

Consistency Coefficient, m (Pa 3 2

 

 

 

1 2
FLUID SYSTEM Mixer Congentgic Cylinders
1 2.139 i 0.003
2 2.664 t 0.004
3 2.835 t 0.003
4 3.173 i 0.005
CMC 1% 5 3.557 i 0.002 2.619 t 0.003
6 2.492 i 0.003
7 2.737 i 0.004
8 2.823 i 0.002
9 2.937 i 0.001
10 2.906 t 0.003
m 2.826 i 0.378
avg
1 18.892 i 0.004
2 12.261 i 0.007
3 13.781 i 0.005
4 14.702 i 0.004
CMC 1.5% 5 15.687 i 0.003 15.620 i 0.004
6 18.892 i 0.002
7 15.299 i 0.002
8 22.967 i 0.003
9 21.619 i 0.004
10 24.742 i 0.003
m 17.877 i 4.200
avg
1 33.209 i 0.002
2 30.656 i 0.004
3 33.379 i 0.002
4 32.980 i 0.001
CMC 2% 5 35.142 i 0.003 33.170 i 0.005
6 37.737 i 0.002
7 34.793 i 0.004
8 38.149 i 0.001
9 45.239 i 0.002
10 47.738 i 0.005
m 36.900 i 5.540
avg

 

1
Brookfield Mixer. Shear rate range: 0-30 l/s [Eqns. (6.18), (6.19)]

2
Haake MV-III (d/D-0.73). Shear rate range: 0-40 l/s

203

Table 6.14: Values of the Fluid Consistency Coefficient of Standard
Power-Law Fluids Using the Mixer System (Flag) [Model
1: negligible end effects; Eqn. (4.14)] and a Concentric
Cylinders Viscometer (Haake Rotovisko)

 

n

Qongigtency Coefficient, m (Pa 5 2
1 2
ELDIQ SXSIEM Mixer Concentric Cylinders

 

 

 

1 0.750 1 0.007
CMC 1% 2 1.288 1 0.005 2.619 1 0.003
3 1.155 1 0.004
m 1.064 1 0.280 3
avg
1 4.152 1 0.003
CMC 1.5% 2 7.100 1 0.003 15.620 1 0.004
3 8.961 1 0.002
m - 6.738 1 2.421
avg
1 12.400 1 0.004
CMC 2% 2 15.926 1 0 002 33.170 1 0.003
3 21.318 1 0.003
m 16.548 1 4.491
avg

 

1
Brookfield Mixer. Shear rate range: 0-30 1/s [Eqn. (6.21)]

2
Haake MV-III (d/D-0.73). Shear rate range: 0-40 l/s

 

204

Table 6.15: Values of the Fluid Consistency Coefficient of Standard
Power-Law Fluids Using the Mixer System (Paddles) [Model

2:

concentric cylinders with end effects; Eqn. (4.21)])

and a Concentric Cylinders Viscometer (Haake Rotovisko)

 

n

angistengy CnefficientI m (Pa 3 )

 

 

 

1 2
FLUID SYSTEM Mixggi Concentric Cylinders
1 1.526 t 0.003
2 2.046 t 0.003
3 2.643 t 0.002
4 2.759 t 0.005
CMC 1% 5 3.161 t 0.002 2.619 i 0.003
6 1.562 i 0.003
7 1.056 i 0.004
8 2.353 i 0.001
9 1.836 t 0.002
10 2.219 t 0.004
m 2.116 t 0.642
avg
1 7.699 i 0.004
2 9.362 t 0.007
3 11.484 i 0.005
4 12.784 i 0.004
CMC 1.5% 5 14.006 i 0.003 15.620 i 0.004
6 11.808 i 0.002
7 11.473 i 0.002
8 19.139 i 0.003
9 13.523 i 0.004
10 18.556 i 0.002
m 13.001 i 3.644
avg
1 22.115 i 0.002
2 24.334 i 0.004
3 26.354 i 0.001
4 28.348 i 0.001
CMC 2% 5 31.378 i 0.003 33.170 i 0.005
6 23.595 i 0.003
7 27.543 i 0.004
8 31.794 i 0.002
9 26.356 i 0.003
10 36.085 1 0.004
m 27.790 i 4.268
avg

 

1
Brookfield Mixer. Shear rate range: 0-30 l/s

2
Haake MV-III (d/D-0.73). Shear rate range: 0-40 l/s

205

Table 6.16: Values of the Fluid Consistency Coefficient of Standard
Power-Law Fluids Using the Mixer System (Flag) [Model
2: concentric cylinders with end effects; Eqn. (4.21)]
and a Concentric Cylinders Viscometer (Haake Rotovisko)

 

 

 

nc e ie t Pa sn
1 2
S xe Co centric C linders
1 0.640 i 0.007
CMC 1% 2 1.104 i 0.005 2.619 t 0.003
3 1.120 f 0.004
m 0.954 t 2.720
avg
1 4.001 i 0.003
CMC 1.5% 2 6.350 1 0.003 15.620 i 0.004
3 7.574 i 0.002
m 5.975 t 1.816
avg
1 12.210 i 0.004
CMC 2% 2 13.782 i 0.002 33.170 i 0.003
3 18.324 1 0.003
m 14.772 1 3.175
avg

 

1
Brookfield Mixer. Shear rate range: 0-30 l/s

2
Haake MV-III (d/D-0.73). Shear rate range: 0—40 1/s

 

206

Table 6.17: Values of the Fluid Consistency Coefficient of Standard
Power-Law Fluids Using the Mixer System (Flag) [Model
3: d - de ; Eqn. (6.23.1)] and a Concentric Cylinders
Viscometer (Haake Rotovisko)

 

 

 

 

enc e ient Pa sn
1 2
__ELQ.I_D__§1$_TEM—_mxer COW.
1 1.522 1 0.006
CMC 1% 2 2.648 1 0.004 2.619 1 0.003
3 2.328 1 0.004
m 2.166 1 0.580
avg
1 8.408 1 0.003
CMC 1.5% 2 14.379 1 0.003 15.620 1 0.004
3 18.417 1 0.003
m 13.735 1 5.035
avg
1 25.120 1 0.004
CMC 2% 2 32.250 1 0.002 33.170 1 0.003
3 43.280 1 0.002
m 33.550 1 9.149
avg

 

1
Brookfield Mixer. Shear rate range: 0-30 l/s [Eqn. (6.23.1)]

2
Haake MV-III (d/D-O.73). Shear rate range: 0-40 l/s

207

Tables 6.13 and 6.14 present the values of the fluid consistency
coefficient, m, obtained by replacing the mixing system with the Model 1
[negligible end effects, Eqn. (4.14)], for the paddles and flag im-
peller, respectively. Comparison with the values obtained using the
concentric cylinders viscometer show that the results with the paddle
impellers are satisfactory. However, the flag impeller gives con-
siderably lower values of the rheological parameter, m, when using Model
1 (Table 6.14). Also, the effect of the gap seems to be more important
for this impeller, since the value of the consistency coefficient ob-
tained with the small gap case (System 3, d/D - 0.591) is twice as high
as the value of the consistency coefficient in the large gap case
(System 1, d/D - 0.273) (See Table 6.14). A small gap also gives higher"
‘malues of the rheological parameter when using a paddle impeller
(Systems 9 and 10, d/D - 0.327), especially for the fluid of higher
viscosity (CMC 2%) (Table 6.13).

Tables 6.15 and 6.16 present the values of the consistency coeffi-
cient, m, calculated by replacing the mixing system with the Model 2
[presence of end effects, Eqn. (4.21)], for the paddles and flag im-
peller, respectively. No big differences with the values from Tables
6.13 and 6.14 are observed, with results from Model 2 [Eqn. (4.14)]
lower than those obtained with Model 1 [Eqn. (4.21)]. The same behavior
was observed in results for the flag impeller (Table 6.16).

Table 6.17 presents the results obtained by replacing the flag
impeller with a cylinder of dimensions as in Model 3 [d - de , Eqn.
(6.23.1)]. It can be seen that this model gives considerably higher

values of the fluid consistency coefficient than the previous models. In

 

208

comparison to the values obtained using a concentric cylinders vis-
cometer, results also show better agreement.

Rheogxams (shear stress-shear rate curves) were developed for all
fluids and systems. Figure 6.52 shows the flow curves for the 1% CMC
(Figure 6.52a) and the 2% CMC (Figure 6.52b) solutions using the mixer
viscometer with the paddle impellers in a large cup [(d/D) - 0.327]. The
model selected for approximation of the average shear stress was Model 2
(concentric cylinders analogy with end effect) since it gave more con-
sistent values of torque (Figure 6.48b). Results were compared with
those obtained with the concentric cylinders viscometer (Haake Rotovisko
with the MV-III sensor which provided a range of shear rates similar to
that obtained with the Brookfield viscometer and the mixer impellers)
and proved to be satisfactory. Figures 6.53a and 6.53b present results
for a small gap [(d/D) - 0.709]. Results indicate that the method works
better when the agitated fluid is highly viscous (CMC 2%) than when
agitating a low viscosity fluid (CMC 1%) when a small gap is present.

Figure 6.54 presents the flow curves obtained for the flag im-
peller using the Model 3 (d - de ). Results show excellent agreement
with those obtained using the concentric cylinder viscometer, with
better results for the more viscous fluid (CMC 2%). For comparison,
Figures 6.55 and 6.56 present the flow curves obtained using Model 1
[Eqn. (4.14)] and Model 2 [Eqn. (4.21)], respectively. It is clear tfluat
the Model 3 [Eqn. (6.23.1)] is still the best approximation for the

mixing system when agitating standard power-law fluids.

6.2.2.3.2) Eood Erodugt

Flow curves for an actual food product (Rancher's Choice Creamy

209

1000.00 . mmy . .

 

 

 

 

 

 

 

 

”.ny . .mmf y rnwnr . ,,
o d/b-1.8 («VD-0.327)
o d/b-0.6
e d/b-OJG
100"” + H6611. (IN-lll) ,
1'— O 3'...
&: . 60
.1 10.00 ° °
' “.4!
E: 'e' "
1.00 °
CMC 13
0.10y , ..yyqu . ....;IIFy .yn—Jlfv . ..ynllr . a
1000.001W t 1 fit: r r: u - rt
: o d/b-1.6 (am-0.827) ' ' ""5
I o «Vb-0.6 I
1 A d/b-OJO ‘
‘ + Pkuflut(flwhdl) . ’ ‘
.25? d 4»"’0 ‘
._.. 100.001 " ‘ 1
a 1 23': :
I: j e 1
d «0‘ . II
1 ~ . q
. . . one 2:: 1
10.00““ . . . vr'I'l—‘l—I-F'I'I'W—l-‘FFl'I'Ifi—l—I-I'I'IEI'I'
0.01 0.10 1.00 10.00 100.00 1000.00

 

‘7- [1/8]

Figure 6.52: Comparison of the Flow Curve for a Standard Fluid
Determined Using the Concentric Cylinders Viscometer
with the Data Obtained Using the Mixer Viscometer with
the Paddle Impellers in a Large Cup [Model 2, Eqn.

(4.21)]; a) CMC 1%; b) CMC 2%

210

1 000.00

 

 

  

o d/b-1.6 (d/0-0.708)
o d/b-l .0
+ Hooke (IN-Ill)

 

 
 

 

 

 

 

1 00.00
H
I! ,0
E. 1 '
oe
.g " oe
10.00 ,4" 0
.‘ae
C110 13
1 .00 I W fi I W r a
lOOKLIN11--r-r1w11ur-w-1Frvunqr . . ..yuf”. ...yan—** y
1 o d/b-l .6 “ID-0.700) “a?
: o d/b-1.0 1
' + Moshe (UV-ll) 1
. «o
'3' ‘ 0. a: ‘
E... 100.00-1 3116 -:
6 ' 1
b 3 . I" :
a . d
- 9 a
. ° one 2: .
10.00-4mm . , . n. b
0.101 0.10 1 .00 10.00 1 00.00 1 000.00

 

1"- [Us]

Figure 6.53: Comparison of the Flow Curve for a Standard Fluid

Determined Using the Concentric Cylinders Viscometer

with the Data Obtained Using the Mixer Viscometer with

the Paddle Impellers in a Small Cup [Model 2, Eqn.

(4.21)]; a) CMC 1%; b) CMC 2%

 

211

1000.00

 

I 00"!!! r U IIUItI'r I fititiq T U VIII-a!
d/0-0.273
d/0-0.420
d/0-0.501

100.00 Hooke (IN-Ill)

1 0.00

 

Um [P0]
'0' 9 a 0
3+
If
e“.
“x
’o
“o
’0
0+
'3
o
W

 

 

 

 

 

1.00 °
CMC 13

0.10I . . ”Hwy . ..yyy.q— . . ”Hwy . . ......,
1000.00,, . . turner , . uum . . ”may amt"
I o d/D-0.273 1
. O d/D-0.429 :
‘ A d/D-0.591 "
‘ + Hooke (UV-Ill) to. " '
I ‘ .

:3" . 1 "°
3 I 0.3 :
b 1 ‘ +5 1
-1 . fi «1
a; "
‘ 011023 '
b
10.00-[W y . yynnf . y.r"uf . rutrm
0.131 0.10 1 .00 1 0.00 l 00.00 1 000.00

7"- [1/81

Figure 6.54: Comparison of the Flow Curve for a Standard Fluid
Determined Using the Concentric Cylinders Viscometer
with the Data Obtained Using the Mixer Viscometer with
a Flag Impeller [Model 3, Eqn. (6.23.1)]; a) CMC 1%;

b) CMC 2%

 

 

212

 

 

 

 

 

1000.00 . mum, y . ......y . yywm, . r ....u,
o d/0-0.273
o d/0-0.420
A d/0-0.591
100'00 + Hooke (MV-lll) ,
7 + ”’+
a +
I: fine
4" b
E‘ 10.00 ‘4‘” be °
0 + he
b be
1.00 a
O
CMC 1%
a
0.10! y rrrva . .......r . ....yyq—. Irrrrfl“: ....yy.‘
1000.00 . . . .....y r . . .....y y.....y ......y .
o d/0-0.273
a d/DI!0.420 4.
A d/0-0.501 «O
+ Pkuflu|(Ndefl) * 1"
P; 100.00 1:.
Q +’ ‘
<3 13’s a.
0. ... "0
H ‘u.
6
b 10.00 1:.
C110 23b
1.00. . ...nrr—I—v-l-I-n-u“. .yrnflj:. .yywqu. run
0.01 0.10 1 .00 1 0.00 100.00 1 000.00

7?.[1/31

Figure 6.55: Comparison of the Flow Curve for a Standard Fluid
Determined Using the Concentric Cylinders Viscometer
with the Data Obtained Using the Mixer Viscometer with
a Flag Impeller [Model 1, Eqn. (4.14)]; a) CMC 1%;

b) CMC 2%

 

213

 

 

 

 

1000000 I I IIIIII' I I I (YTTYI I I IIIIIII I rII
o d/0-0.273
n d/0-0.420
1 00.00 A d/D-0.591
+ Hooke (av-111) , ,
E” . ‘
... 10.00 be 5°
6 0. Que
b 9 50
1.00 a 3"
0
one 17:
0.101 . 11...“, . . ...., . e. ...", . . ”3.,
1000000 I I I VUI'UIr I I I IIIIII I I I III!!! I I I Irrt
o d/D-0.273
a d/0-0.422
a d/0-0.591 . '
.— 100.00 1 "w“ ("V-m) .. ° ‘5.
e ‘ ‘30
(3 o a
0. . ° °o
1—1 . ‘6 ‘6.
6 ., 1
10.00 o
cuc 2::
1.001 . . ......y . . r . . . 1...?
0.10 1.00 1050 106.00 100"0.00

7"- [1/81

Figure 6.56: Comparison of the Flow Curve for a Standard Fluid
Determined Using the Concentric Cylinders Viscometer
with the Data Obtained Using the Mixer Viscometer with
a Flag Impeller [Model 2, Eqn. (4.21)]; a) CMC 1%;

b) CMC 2%

 

214

Dressing; Kraft, INC., Glenview, Illinois) were estimated to check the
applicability of the proposed procedure. Figure 6.57 presents the
results using the paddle impellers [Model 2, Eqn. (4.21)] in comparison
with the ones obtained with the concentric cylinders viscometer. The
agreement of the data is evident for different impeller and cup sizes.
Tables 6.18 and 6.19 present the values of the rheological parameters (n
and m) for the salad dressing, evaluated using Model 1 [Eqn. (4.14)] and
Model 2 [Eqn. (4.21)] for approximation of the average shear stress with
the paddle impellers, respectively. The proposed procedure seems to be
able to approximate the rheological behavior of a food product when
using the mixer viscometer with the paddle impellers. The approach takes
into account the variations in impeller and cup geometry.

It is interesting to note that the differences due to variation in
impeller are less than the observed in the developed curve flows for the
standard fluids (See Figures 6.53 and 6.57, for instance). The same
trend was observed with the flag impeller (Figures 6.56 and 6.58).

For the flag impeller, peculiar results were obtained. Table 6.20
shows the values of the rheological parameters of the tested fluid with
the three models investigated in this study. Figure 6.58 shows the flow
curves obtained for the food product by approximating the flag impeller
with a cylinder of diameter (18 (Model 3). Approximation of the shear
stress-shear rate data with Model 3 gives higher values of the flow
curve as compared with the concentric cylinders viscometer (Haake
Rotovisko with MV-III). Figures 6.59 and 6.60 present results using
Model 1 and 2, respectively. Results indicate that the two models show
excellent agreement with the concentric cylinders data for the food

Product as compared with Model 3 (Figure 6.58). It might be said that

215

1000.00 . . .....r—I—r-l-I-nwfi .
o d/b-1.8 MAJ-0.327)

 

 

'7 Ir

 

 

 

o d/b-0.6
A d/b-OJd
Hooks ouvuqu)
'— 1 00.00
I:
0..
h-‘ a
6 11.1313"
1 0.00
1.00fi trttr‘fi'irriI—TTT It:
1 000.00
o d/b-1.8 (d/0-0.700)
u d/b-1.0
1+ Hunk..fllvhdfl)
1 00.00
H
‘3 10¢ .
1: " I'v':%.'fl'
1 0.00
Salad Dreeehg,
b
1.00 '
0.01 0.10 1 .00 1 0.00 1 00.00 1 000.00

7- [V8]

Figure 6.57: Comparison of the Flow Curve for Salad Dressing Determined
Using the Concentric Cylinders Viscometer with the Data
Obtained Using the Mixer Viscometer with the Paddle
Impellers [Model 2, Eqn. (4.21)]; a) Large Cup; b) Small

Cup

216

 

 

 

 

1000.00 . . . . yyyaE .yynr. 1
g o d/0-0.273 ' 1'
1 a d/D-0.420 :
1 A d/D-4L591 .
, + Hooke (IN-ill) ,
100.00-:1 -
'3' 3 d 5, * .
0L 1 ‘* d* .
1.... . ‘ ‘ + 1
8 1 . w" -
I I
1 Flag Impeller I
. Salad Dressing ‘
1 .00-W u rt rm!!— u I u I m
0.01 0.10 1 .00 10.00 1 00.00 1 000.00

'7- [1/81

Figure 6.58: Comparison of the Flow Curve for Salad Dressing Determined
Using the Concentric Cylinders Viscometer with the Data
Obtained Using the Mixer Viscometer with a Flag Impeller
[Model 3, Eqn. (6.23.1)]

 

217

 

 

 

 

 

1 Flog Impeller
. Solod Dreeehq
1.00-mi. rumM—r yuuur r..."
0.131 0.1 O 1 .00 10.00 1 00.00 1 000.00

7- [1/8]

Figure 6.59: Comparison of the Flow Curve for Salad Dressing Determined
Using the Concentric Cylinders Viscometer with the Data
Obtained Using the Mixer Viscometer with a Flag Impeller

[Model 1, Eqn. (4.14)]

 

1000.00 ‘

1 00.001

- 3

<3 4

Q. ..

H I
e!

10.00-é

1 .00-

00‘31

 

+>ao,

 

 

 

218
. . .mq—fi-mwrr—q—u—u-nm . . . . y...
d/0-0.273 '
d/0-0.420
d/0-0.501
Hooks (LN-Ill)
+
...
O
6
I
oe’ 3'"
e
Flog Impeller
Salad Dressing
. . ..mq—. ..mIIF . ....n
0.1 0 1 .00 l 0.00 1 00.00 1000.00

7.. [1/8]

Figure 6.60: Comparison of the Flow Curve for Salad Dressing Determined

Using the Concentric Cylinders Viscometer with the Data

Obtained Using the Mixer Viscometer with a Flag Impeller

[Model 2, Eqn. (4.21)]

 

‘

 

 

219

Table 6.18: Values of the Flow Behavior Index (n) and the Fluid
Consistency Coefficient of Salad Dressing Using the Mixer
System (Paddles) [Model 1, Eqn. (4.14)] and a Concentric
Cylinders Viscometer (Haake Rotovisko)

 

 

 

n te e i ie Pa sn
1
SYSTEM Ming;__!1§nnn§ter
1 0.248 t 0.003 29.811 1 0.003
3 0.243 1 0.001 20.186 1 0.002
5 0.236 1 0.001 30.057 1 0.005
6 0.239 1 0.002 25.475 1 0.005
8 0.227 1 0.003 19.039 1 0.003
9 0.257 t 0.002 32.007 1 0.004
10 0.258 1 0.002 27.701 1 0.003
Average 0.244 i 0.011 26.325 1 5.029

 

2
Haake data: n - 0.399 0.005

1
m - 15.00 1 0.003 Pa s“

1
Brookfield Mixer. Shear rate range: 0-30 1/s [Eqns. (6.18) and (6.19)]

2
Haake MV-III (d/D-0.73). Shear rate range: 0-40 l/s

 

220

Table 6.19: Values of the Flow Behavior Index (n) and the Fluid
Consistency Coefficient of Salad Dressing Using the Mixer
System (Paddles) [Model 2, Eqn. (4.21)] and a Concentric
Cylinders Viscometer (Haake Rotovisko)

 

 

 

n s stenc Coe icient m Pa sn
1
SYSTEM MW:
1 0.248 1 0.002 19.314 1 0.003
3 0.236 t 0.003 15.484 1 0.004
5 0.243 t 0.001 18.023 1 0.002
6 0.239 1 0.002 18.793 1 0.005
8 0.227 1 0.003 15.867 1 0.002
9 0.257 1 0.002 18.313 1 0.003
10 0.258 1 0.002 18.999 1 0.004
Average 0.244 1 0.011 17.853 1 1.554

 

2
Haake data: n - 0.399 1 0.005 n
i

m - 15.00 0.003 Pa 3

1
Brookfield Mixer. Shear rate range: 0-30 1/s [Eqns. (6.18) and (6.19)]

2
Haake MV-III (d/D-0.73). Shear rate range: 0-40 1/s

 

221

Table 6.20: Values of the Flow Behavior Index (n) and the Fluid
Consistency Coefficient of Salad Dressing Using the Mixer
System (Flag) and a Concentric Cylinders Viscometer
(Haake Rotovisko)

 

 

 

 

 

 

n te en m Pa sn
1
__§y§1§u Mixg; Vigggneter
Model System 1
1 0.257 1 0.003 14.916 1 0.004
2 0.230 1 0.002 15.030 1 0.004
3 0.235 1 0.002 16.249 1 0.003
Average 0.240 t 0.014 15.398 1 0.739
Model System 2
1 0.257 t 0.003 12.570 1 0.005
2 0.230 1 0.002 12.907 1 0.003
3 0.235 1 0.002 12.740 1 0.003
Average 0.240 1 0.014 12.739 1 0.168
Model System 3
1 0.257 1 0.003 28.691 1 0.003
2 0.230 1 0.002 29.240 1 0.002
3 0.235 1 0.002 30.690 1 0.004
Average 0.240 1 0.014 29.873 1 0.742
2
Haake data: n - 0.399 1 0.005 n
m - 15.00 1 0.003 Pa 5

1
Brookfield Mixer. Shear rate range: 0-30 1/s [Eqn. (6.21)]

2
Haake MV-III (d/D-0.73). Shear rate range: 0-40 1/s

222

the area of the flag impeller of diameter d and blade height b con-
tributes to the torque values when agitating the food product. In the
case of a standard power-law fluid such as the CMC solutions, this is
not the case, since the assumption of the total area of the flag im-
peller yielded higher values of torque as compared with the geometry
given by Model 3 (d - de ). These results indicate that the fluid to be
investigated using mixer viscometry is of great importance when select-
ing the appropriate equations to approximate the flow curves since the
food product (salad dressing) shows more shear-thinning behavior (n=
0.244 1 0.011) than the standard CMC solutions (0.5 s n S 0.9).

In terms of comparison, plots of the apparent viscosity, "a ,
versus the average shear rate, :yav , were developed for the different
Models using the flag impeller and the results obtained with the con-
centric cylinders viscometer. Figures 6.61 to 6.63 present the results
for the three models analyzed in this investigation. The agreement of
results is clear and excellent. It may be seen that the differences
obtained with the Model 3 (d - de ) is not so drastic (Figure 6.63).
Figure 6.64 presents the results for two different gaps. Figure 6.64a
shows the results when agitating the food product with a flag impeller
and a small gap is present (d/D-0.59l). Figure 6.64b shows the results
for the wide gap case (d/D-0.273). It may be noted that the effect of
the mixing system (in this case, the impeller-to-cup diameter ratio,
d/D) is not significant.

In.summany, analytical systems were developed for the mixing
systems based on the original development for a power-law fluid agitated
in a concentric cylinders viscometer. These systems (or models) take

into account differences in impeller shape (paddle or flag) and

n. [Pms]

223

 

 

 

 

 

40.00 . I . r I I .
o d/D-O.273
. o d/D-O.429 F109 q
A d/D-O.591
32°00“ + Hooks (W—lll) Model 1 (NE) "‘
24.0046 ..
16.00~ ..
O
- +
4'
8.00- 00+ 4
d 3*» 4
‘ an“
on A *
0.00 ,— rw rfi , , .—
0.00 8.00 16.00 24.00 32.00

7.. [1/8]

40.00

Figure 6.61: Apparent Viscosity as a Function of Average Shear Rate For

Salad Dressing. Flag Impeller (Model 1: concentric

cylinders with no end effects [NEE]).

 

224

 

 

 

 

40.00 . , . , . ,
o d/0-0.273
. u d/D-O.429 ”“9
A d/D-0.591
3100* + Hooks (NV-Ill) M°d°' 2 (EB) -
:3 24.004 ..
O
E. 1 ‘
5 16.004 -
d g cl
8 OO- *4-
an, ' "
*4
a Q +
an
0.00 . , . , . °° ,‘ . j .
0.00 8.00 16.00 24.00 32.00 40.00

‘7- [1/8]

Figure 6.62: Apparent Viscosity as a Function of Average Shear Rate For
Salad Dressing. Flag Impeller (Model 2: concentric

cylinders with end effects [EE]).

 

 

n. [Pms]

225

 

 

 

 

40.00 . , . , I
o d/D-0.273
. a d/D-O.429 ”“9 4»
A d D-4L5 1
552-004 + Hick. (Msv—m) MM“ 3 (d'd°) -’
L
o
24.00-1 ..
J
16.004 i q»
d Q q»
a.00~ " 9' -.
4'
. ++‘* ‘bA 0
* On A +
0.00 F 1 T ‘ . .
' 0.00 8.00 16.00 24.00 32.00

40.00

Figure 6.63: Apparent Viscosity as a Function of Average Shear Rate For

Salad Dressing. Flag Impeller (Model 3: concentric

cylinders with end effects; (1 - de ).

n.- [Pets]

1). [Pas]

226

 

 

 

 

 

 

40.00 . , e r r I
J o Model ; Eggs) Flog q
a Model
32.00-4 A Model :5 (d-de) d/D-O.591 _
, + Hook. (av-m) (Smou GOP)
A
24.00-4 0 ..
.n .
16.00“ ‘ c-
0
II a d
8.00-1 1», ‘ ..
D + ’
.1 g “ ... .1
0.00 r r e r r r 1 r g
40.00 . , . , . , r T
, °1°331£2$ +
D 0
32.00-1 A Model 3 (4.4.) d/D-O.273 _
, + Hook. (uv—m) (W140 Gap) 1
I
24.00-J, ..
du d
16.00~ A ..
O
" 0‘
I A
8.00-1 3.,“ ..
II a ’fi . ’ +
‘ 3 + 0
0.00 e r r , 1 1 1 f g
0.00 8.00 1 6.00 24.00 32.00

7.. [1/81

Figure 6.64: Apparent Viscosity as a Function of Average Shear Rate For

Salad Dressing. Flag Impeller. a) Small Cap; b) Wide Gap.

 

227

differences in system geometry (impeller-to-cup diameter ration, d/D,
and impeller blade height, b). Thus, flow curves can be directly detemw
mined by only measuring the torque needed to rotate the impeller as a
function of the rotational speed of the impeller. The selection of the

model can be checked with data from a conventional viscometer when

available.
6.3 QENEBAL REQOMMEHQAIIQNS Egg IE5 AEELICAIION QE MIXER VISCOMETRY

This section presents the recommendations for a general procedure
Untxing system and unknown power-law fluid). Figure 6.65 shows the flow
diagram of the procedure. The procedure has been developed for use with
a simple and easy to handle data collection system. However, it can be

applied to different combinations of Viscometers and mixing systems.

6.3.1 Mixer System Used in This Study

The procedure is as follows:

1. Maintain a constant temperature of the fluid.

2. Impeller and cup dimensions (d/D and d/b) must be known.

3. Other geometric parameters should remain constant for a set of
experiments (fluid height, distance from bottom of impeller to bottom
of container).

4. Check for time-dependent effects (there should be none for power law
fluids). Agitate fluid for a period of 10 minutes to reach an
equilibrium value of torque.

5. Select range of rotational speeds. The criterium to follow is

 

228

Re 5 10. Also, check for presence of surface waves or turbulence.

The maximum allowable value of rotational speed, N, will be that
which does not produce formation of surface waves on agitated fluid.
Operating values of the rotational speed (rev/s) greater than 20 rpm
are recommended.

. Measure the torque in the shaft required to agitate the fluid, M, as
a function of impeller rotational speed, N.

. From log-log plot of M vs. N, evaluate the value of the flow behavior

index, n, of the fluid.

 

. To determine the flow curve of the fluid, a model is required.

Therefore, expressions for an average mixing shear rate, fiav , and

an average shear stress, a , were developed. Different models were

av
evaluated to determine the best model for flow curve determination.

. If working with a paddle impeller [ 0.36 s (d/b) S 1.8], use Eqn.
(6.19.1) or (6.19.2) if d/D 2 0.709 (small gap) and Eqn. (6.18.1) or
(6.18.2) if [0.327 S (d/D) 5 0.515] (wide to medium gap). For a flag

impeller [(d/b)-0.3], use Eqn. (6.21). Thus,

 

 

2/n
. (D/d) n/2
7 _ 4, (d/b) N (6.18.1)
av { [ (D/d)2/n _ 1 ] }
or
2/n
. (D/d) n/2
1 - 20 (d/b) (6.18.2)
av [ (D/d)2/n _ 1 ]

for a paddle impeller and 0.327 S d/D 5 0.515 , and

229

zfifi
73v - {4x [ (D/d’ J (d/b)n/2 } N (6.19.1)

(D/d)2/n - 1

 

or

Zfin
. (D/d) n/2
7 - 20 (d/b) (6.19.2)
8" [ (1)/<1)”n - 1 J

 

for d/D 2 0.709 (with 1 z b 2 S cm and 0.5 2 n 2 0.9).

Flag impeller (b/d—0.5),

k' - 4x [ (1)/d)“2

1 (6.21)

10. Determine the best model to approximate the average shear stress.
A way to check the applicability of the shear stress equations
[Model 1 (Eqn. (4.14); Model 2 (Eqn. (4.21) and Model 3 (Eqn.
(6.23.1) and (6.23.2)] is to compare the values of torque calculated
using these equations with the experimentally measured values of
torque. The best equation will be the one that shows better
agreement with the experimental data. Thus,

Model 1 (concentric cylinders analogy with negligible end effects):

M - 2xb (cl/2)2 aav (4.14)

Model 2 (concentric cylinders analogy with end effects):

 

11.

12.
13.

230

__NL _‘Q_+;
M 2 [ d 3 ] Gav

Model 3 (Flag impeller; d- de ):

2
M - 2nb (dc/2) aav

and

(4.21)

(6.23.1)

(6.23.2)

Develop plots of average shear stress versus average shear rate for

the fluid (flow curves). The value of the consistency coefficient,

m, can be evaluated by linear regression of the plots.

Evaluate the apparent viscosity of the fluid from "a - m (iav )

n-l

If available, compare results with data obtained with a traditional

concentric cylinders viscometer. Make sure the range of shear rates

are comparable.

6.3.2 New Mixer Viscometer System

in this study is used, the following procedure is recommended:

If an impeller other than a flag or a paddle impeller considered

. Follow previous steps (l-7).

. Evaluate k' of the particular impeller using the Slope method. The

main reason for selection of this mixer viscometry method is its

 

231

 

combination (d/D.d/b,

1

I xeeo otner yariaoles

 

" select impeller/cup <:E::>

Add no .
"noel“? FLAG Fino x wfin

. traditional
3564/) ‘ d/b-O.3 mixer VlSC. methods
ye

 

constant (c.h,
i.etcl

 

 

 

 

 

 

 

 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

check for 33 from find exorgsswon tor k‘
' -. ' d tnus for mixin
tine deoenden. 33v fr°r Eqn,(5.21) ‘“ the 1
' Eqn. (6.18) cm) 0.709 svst;m
agitate fluid
for main. at 1 yes 123?? "add .
‘selected | iiyfrom ' .' la

 

[Eqn.(6.19)

v V

calculate Gig

 

 

  

 

 

   
   

n
range of rotational
speed of viscoeeter select Model
(1. 2 or 3)

 

 

  

 

 

 

00

go to
R.‘ 10 lower value ®
yes yes

 

 

 

 

 

 

I is
appropri ate _

 

 

 

 

 

 

 

elasure torque H as
a function of N

find values of
fluid consisten
coeff..n. fro-
linear regressi

V

 

 

 

 

free log-log plot

of H vs. I. find
flow behavior 1

 

   
     

 

 

 

 

 

evaluate
* 7""1‘3‘ )fl-l
l l '
3.1.3:.“

 

 

Figure 6.65: Flow Diagram of General Procedure For Mixer Viscometry

232

simplicity. If differences in fluid properties are considered, the
Viccccicy_mccching_mcchcc should be used.

. Seek a range of variable where k' is constant (i.e., (d/D), (d/b) and
N).

. With the values of k' obtained in (2) find an expression for k' as

a function of system geometry and fluid properties. As a start, use

the ccncentric cyiigcerc cncicgy cppzoach developed in this study
[Equations (6.18.1), (6.18.2), (6.19.1), (6.19.2), or (6.21)].

 

Different expressions may be obtained when different (d/D) ranges are
considered.

. From (3), find expressions for fiav'

. Follow previous step (10) to find approximations of the aav of the
fluid under study for the new mixing system.

. Follow steps (ll-13) to develop the flow curves (shear stress-shear
rate relationship).

. Repeat procedure for different impeller/cup combination.

 

233

CHAPTER 7

SUMMARY’ARD CONCLUSIONS

1. A procedure for determination of the rheological behavior of power-

law fluids using a mixer viscometer was developed.

2. Based on the concentric cylinders system analogy, an expression for
the average shear rate in the mixing system can be determined.
Average shear rates are a direct function of impeller to cup diameter
ratios, impeller rotational speed and the value of the flow behavior

index, n.

3. The size of the gap has a significant effect on the determination
of expressions for the average shear rate in the mixing system.
Hence, two different expressions were obtained for different d/D
ranges: Eqn. (6.18) for 0.327 s d/D 5 0.515 and Eqn. (6.19) for

d/D 2 0.709 when using paddle impellers.

 

 

2/n
. (D/dl_ n/2
1 - a, (d/b) N (6.18)
av { [ (D/d) 2/n _ 1 ] }
4
. (D/d) n/2
7 - 4, (d/b) N (6.19)
a" { [(1)/d) 2/n - I] }

4. Impeller shape plays an important role in the development of

equations for development of average shear stress-average shear rate

 

234

relationships for power—law fluids. Thus, the expression for the
average shear rate for a flag impeller is different from the ones
obtained for the paddle impellers [Eqns. (6.18) and (6.19)}, and is
given by Eqn. (6.21).

- k'N - 4n [ (1)/d)”2

iav ] N (6.21)
. The analogy made with the concentric cylinders with the addition
of end effects is the best model [Eqn. (4.21)] for approximation of
the average shear stress of power-law fluids when agitated with a
paddle impeller rotating in a cylindrical cup. Results apply for all
the different impeller/cup combinations and power-law fluids tested
in this study ( 0.327 s d/D 5 0.709; 0.36 S d/b S 1.8; 0.5 S n s

0.9).

M - -49— {—9— + -%— ] aav (4.21)

. The best model for approximating the average shear stress when
agitating a standard power-law fluid with a flag impeller rotating
in a cylindrical cup is given by the assumption of the flag as a
cylindrical surface with two blades attached, with a total diameter

equal to de , with negligible end effects [From Eqn. (6.23.1)],

2
nd
e

av 2

 

. Flow curves (average shear stress versus average shear rate) were

 

235

determined for power-law fluids without the need for calibration

with Newtonian fluids.

. The applicability of the WW procedure

was experimentally verified with both ideal (standard) and actual

(food product) power-law fluids.

. Comparison of results obtained with the conccntcic cylinderc

cncicgy procedure with those obtained using a conventional
concentric cylinders viscometer show excellent agreement for the

range of impeller and cup sizes investigated in this study.

10. The ccnccncgic cylinders cnciogy procedure is capable of estimating

11.

12.

13.

the rheological behavior of a food product when using the mixer

viscometer with the paddle impellers.

The QOBCGDEELQ cylincccs cnaiogy procedure is simple and requires

little data collection (torque measurements at selected values of
impeller rotational speed) in comparison to the more tedious

approach of established mixer viscometry methods.

The developed concentric cylinder procedure has proven to yield
excellent approximation of the rheological behavior of power-law

fluids using a low-cost viscometer.

Established mixer viscometry methods have been reviewed and

evaluated for power-law (shear-thinning) fluids.

 

 

14.

15.

16.

17.

18.

236

System geometry (impeller and cup), fluid properties and operating
conditions (impeller rotational speed) were varied to investigate
the effect of these parameters on the value of the mixer
proportionality constant, k', using traditional mixer viscometry

methods.

Traditional mixer viscometry methods (viscosity matching and slope)
predict variations in the mixer proportionality constant, k', with
geometry of the mixing system. The power curve method is very
tedious and deviation from the basic assumption (48v - k'N) may
occur. The mixer torque Curve and the slope methods are simpler

with the slope method requiring less data handling.

The common assumption of fiav - k'N, with k' a constant depending on
the geometry of the impeller only, may lead to significant errors in
the values of the average shear rate when measuring properties of
shear-thinning fluids at low rotational speeds. The determined

critical rpm range was found to be N S 20 rpm.

The effect of system geometry and fluid rheological behavior becomes
almost negligible at higher rotational speeds. Thus, the assumption

of a constant k' is valid under these conditions (N > 20 rpm).

There is a relation between the average shear rate and the type
of fluid being agitated. However, the influence of the power-law
rheological parameters, m and n, on mixer viscometry methods is not

clearly understood. It seems that the more shear-thinning the fluid

9-"

 

19.

237

(lower value of n), the lower the average shear rate. However, it
must be noticed that the fluid with lower n values in this study was
also the more viscous fluid (larger value of m). Thus, further
investigation is necessary to identify the effect of the power-law

parameters in mixer viscometry methods.

The shape of the impeller has an important influence on average
shear rates as determined by established mixer viscometry methods
since different values of k' are obtained for the two impeller
shapes investigated in this study, especially at low values of

rotational speed, N.

 

( -

 

238

CHAPTER 8

SUGGESTIONS FOR.PUTURB STUDY

. To study the influence of the power-law parameters (n and m) on
the value of the impeller proportionality constant, k'; one
possibility is to vary the values of the rheological parameters

and identify separate relationships for each parameter.

. To test the developed procedure (concentric cylinders analogy) with

different types of fluids and impeller shapes.

. To test the developed procedure (concentric cylinders analogy) for a

wider range of shear rates using different viscometers.

. To review and evaluate the mixer viscometry methods when agitating
fluids that do not obey the power-law model. The presence of a yield

stress and time dependent characteristics should also be evaluated.

. To extend the results from this investigation to scale-up of mixing

tanks.

. To evaluate shear fields in mixing systems.

 

 

APPENDICES

 

 

 

A
u
m
u

239

APPENDIX.A

DIMENSIONLESS ANALYSIS FOR MIXING VESSELS (NEWTONIAN FLUIDS)

When using dimensional analysis to design an experiment, the first
step is to define the pertinent quantities as shown in Table A.1. The
number of required pi terms (dimensionless quantities) is defined from
Buckingham's theorem as the number of pertinent quantities minus the
number of dimensions. Therefore the number of pi terms required to
design the experiment is 10 minus 3, which equals 7.

By describing the motion of a fluid in a mixing vessel only in
terms of length [LJ, time [T] and mass [M], the following set of dimen-

sionless products is obtained (Rushton et al., 1950):

1
. Thus,

fi1 52 53 _
N D

«1 (D) - d p

000 fil 'fl2 3fi3 -1
nl-LTM-L T (M/L) D

L: 51 ' 353 ‘ 1 ‘ 0 51 ‘ 1
T: ~62 - 0
M: 63 - 0 n1 - d/D = D/d

By inspection, «2 - H/d
n3 - b/d

n‘ - L/d

 

240

7'5 - C/d

fl1 32 53
With u - n/p, «6 (v) - d N p V then,

0 O 0 fil '32 3 BS 2 '1
L T M - L T ( M/L ) ( L /T )

 

L: 51 ' 3fi3 ' 2 ' 0 51 ' 2

1

o o o fi1 2 s 53 2 '
L T M - L T ( M/L ) (l/T )

L3 fl1 ' 3fl3 ' 1 ' 0 51 ' 1
T: ~82 + 2 - o p, - 2
M: fis-O
2
d N

«7 - “g“ - Froude Number, Fr

51 52 53 ’1
N

"a (P) ' d p P . Thus,

1

51 '5

O 0 0 2 3 33
L T M - L T ( M/L )

2 3
( ML /T )

 

241

L1 31 ' 353 ’2 ' 0 51 ' 5
M: fls ' 1 - 0 fis - 1

5 a
N ' 5 a
P d N p

- Power Number, P

According to the Buckingham's theorem (Langhaar, 1981), the
power consumption of mixer impellers is given by an.equathn10f the

form:

:82 193 34 :35 :86 .37
P - f [ (Re) (Fr) (D/d) (H/d) (L/d) (b/d) (c/d) ]

or [Eqn. (3.91)]

 

In the current investigation, (H/d), (L/d) and (c/d) remained

constant. Therefore, Eqn. (3.91) becomes

 

 

 

242

Table A.l: Pertinent Quantities Involved In Fluid Agitation Processes
(Newtonian Fluids)

 

Pertinent Quantities

 

Number Symbol Description Units Dimensions
Independent Variables

 

l d Impeller m L
diameter
2 D Tank diameter m L
3 b Impeller blade m L -
height ‘
4 H Fluid depth m L
5 L Tank length m L
_1
6 N Impeller rev/s T
Speed
3 ,3
7 p Fluid kg/m ML
Density
,1 ,1
8 n Fluid kg/ms ML T
Viscosity
2 _2
9 g Gravitational m/s LT
Acceleration

 

Dependent Variable

2 3
10 P Power kgm/ 5 ML T

 

The fundamental factors affecting the mixing,process are the
configuration of the system, the behavior of the fluid and the process
control variables (rotational speed). The most significant variables
that can be manipulated to affect power consumption are rotational
speed, N; impeller and tank geometry, (d, D, b); and fluid properties, p
and 17 (Temperature dependency is built in p and 11. Therefore, tempera-
ture was not included as one of the pertinent quantities for the
analysis).

 

 

243

A.1 Development of Dimensionless Functions for Power-Law Fluids (Used in

Slope method)

 

When agitating non-Newtonian (power-law fluids), the viscosity n
is replaced by the flow behavior index, n, and the fluid consistency

coefficient, m (kg sn'Z/m). Thus, the power requirement is a function of

P-f(d. N. p.m. n)

 

31 32 33 34 35
d N

0 - C P p m (n is dimensionless and enters in m)

2 3 31 32 33 3 34 “,2 35
0 - (ML /T ) L T (M/L ) (MT /L)

L: 281 + fiz - 364 - 35 - 0
T: -3al - p. + (n-2)fl5 - 0

M: 6, + 34 + 35 - 0

Let fil - B4 - 0. Then, 35 - -l

 

32 ' ’3
33 ' -n-1
n _ P
1 3
m d Nn+1

Let 31 - 0 and 8‘ - 1. Then, 65 - -1

33 ' -n-2

 

244

32'2

 

 

APPENDIX.B

EXPERIMENTAL RESULTS

3"—
. '1

 

245

APPENDIX B

EXPERIMENTAL RESULTS

“1

Table 3.1: Regression Results of Po - ao(Re) (Fr)m2 (Paddle Impellers)

Linear Multiple Regression.Analysis

 

 

 

Regression Estimated Regression Estimated Standard t*

Coefficient Cosfficient Error

log a0 88.299 -- --
011 -0.976 0.023 -43.53
a2 -0.023 0.023 -1.08

 

Analysis of Variance

 

 

 

 

 

 

 

Sum of Degrees of Error Mean F
Squares Freedom Squares
Regression 48.346 2 24.173 --
Residual 0.585 312 0.022 1115.37
Totsl 481931 314
2
R = 0.988

Test of hypothesis for a2 : Cl: a2 - 0
C2: (12"0

For a level of significance of 0.05, t(0.975,w) = 1.960

*
Since t - 1.08 < t(0.975,m), we accept C1 and conclude that a2 = O

 

I
4 1

 

246
Table 3.2: Regression Results of Po - 190(Re)’81 (Fr)flz (Flag Impeller)

Linear.Mu1tiple Regression.Ana1ysis

 

 

 

 

 

 

 

Regression Estimated Regression Estimated Standard t*
Coefficienc Qccffiicicnt Error
log 60 12.567 -- --
81 -0.983 0.045 -20.07
fig -0.056 0.045 —1.87
Analysis of variance
Sum of Degrees of Error Mean F*
Squsres Freedom Squares
Regression 61.373 2 30.669 --
Residual 2.015 87 0.032 958.66
Totsl 63.388 90
2
R - 0.968

Test of hypothesis for 62 : C1: 62 - 0
Can/92,10

For a level of significance of 0.05, t(0.975,87) = 1.990

*
Since t - 1.87 < t(0.975,87), we accept C1 and conclude that 62 = 0

 

247

Table B.3: Values of 81, fiz and 8, from k' - 81 (d/b)fi2 + 53 (Matching
Method of Power Curves) For Power-Law Fluids

 

 

2
FLUID 51 52 33 R
1 5.029 2.618 10.159 0.985
2 5.898 1.764 7.734 1.000 yr-
3 4.137 1.459 7.179 0.971

 

Fluid 1: Hydroxypropyl Methylcellulose 1% (n=0.504, m=6.49 Pa sn)
Fluid 2: Hydroxypropyl Methylcellulose 1.5% (n-0.374, m=28.42 Pa sn)

Fluid 3: Hydroxypropyl Methylcellulose 2% (n-0.352, m-59.27 Pa 5“)

 

 

248

 

 

 

 

 

 

 

I I . I _ I I I I . I .0
J CMC 1.5% T: 25 C o o o I
0
144.01 0 ° "‘
'1 o d
O
F; 108.0" 0 -1
0- J o I
H
b 72.0- "‘
4 o .
36.0.1 —1
fl 4
3
o o . f . I . r - I If I .
10.0 2 l ‘ I V r ' I T
8.0-4 ‘
H 4 o ‘
a) GJJW ‘
C,
l J G
5.4
o 4.0“ O "
§ . O a
O O
2.0-1 o - I
0 ° ° 0 o o,
b
CLO I r 71% 1‘ r r” ‘7 pi r
0 30 50 90 120 150

 

Figure 3.1: a) Shear Stress Versus Shear Rate; b) Apparent Viscosity

Versus Shear Rate (Hydroxypropyl Methylcellulose 1.5%)

F‘”_‘”

 

a [P0]

’1- [PO 3].

290.0

249

 

232.04

174uO-+

116.0-1

58.0-1

 

0.0

T

I ' I
CMC 2% T= 25 C

 

 

 

25.0

20.0~

15.0-

10.0-1

5.0-

 

 

0.0

 

BIO ' 100

Figure B.2: a) Shear Stress Versus Shear Rate; b) Apparent Viscosity

Versus Shear Rate (Hydroxypropyl Methylcellulose 2%)

 

 

250

 

100

 

 

 

 

 

 

' I ' T’ T‘ I I"T I ' I I I ' I I
i x n=0.504 System {I 1
o n=0.374
80+ A 1130.352 (d/b‘1.8, d/D=00327) _,
f— + 3
{2. 60-1 x 4
::J J
.gi 140-1 0 1
I x . 1
204 I, ..
x A
4 3 2 ,a I
O r r’ ' I ' ”Ti ' I ' I r* r r”r’ ' r r
60 r T ' l T ' I T I ' [ VT j r I r
x n=0.504 System # 2 J
o n80.374
431 1 “0.352 (d/b=1.0. d/0=0.327) q
I J
"a? -
\\‘.36-I
.2. 1 .
.gi 124-1 5 a
J x A '1
12‘1 a -I
I , .
b
0 r ‘T' f”r"r”r‘ vi I r I I r’ I I I I r
0 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8
N [rev/s]

Figure B.3: Average Mixing Shear Rate as a Function of Rotational Speed

of the Paddle Impellers. a) System 1; b) System 2 (Torque

Curves - Matching Method)

 

 

250

 

 

 

 

 

 

 

 

 

‘ I ' T ' T— T I ' 1 r 1 r T 1 l '
o n-O.374
80—1 A n=O.352 (d/b‘1.8, d/D=0.327) _
.—-. 1
{ 60" x T
::J a
a 404 o .I
ah I
I .. . I
204 . -
X 5
4 3 2 a,J
o . I . I . f . I . I I I . I I I I
60 T F ' I ‘ T ' I T I ' [ fl j r I r
.1 x "’0'504 System {I 2 .
o n=-O.374
48.1 A 0-0.352 (d/b=1.0. d/D=0.327) _I
1 1
F07 - A
\ 35
J:; 4
.{L 24-1 5 ..
‘ x A '1
I 9 .
b
O r r T T T r I ' T ' I ' T r
00 02 04 0.6 08 1O 12 14 16 18

Figure B.3: Average Mixing Shear Rate as a Function of Rotational Speed
of the Paddle Impellers. a) System 1; b) System 2 (Torque

Curves - Matching Method)

251

 

 

 

 

 

50 I I ' I ' I 1 I ' ”T ‘j I ‘* I ' I ' '
I. " "-3.33: System # 9 3
O n- .
48.4 A n-0.352 (d/b-1.8. (II/030.709) I
.— ‘ I
Q 364 a .4
::a J x I
.g 24' -1
4 o 1
12 x
X o .
I . ~
‘ 8
O ' r ”f ‘T’ r’ r4’r' rfi’* I ’ Ti’r’ T'*** T—“T
45 T T I I T V— T1 r r U T r r r 1 I
I x "'33:: System I? 10 .
0 fl- .
364 A "-0.352 (d/b-1.0. d/D-O.709) _.
I 2 4
TD... 27 -I
\ l ‘
I: I .
B 18% 3 a
‘?~
J .
A
9-I . _
x ‘ ‘
. . b

 

 

 

 

Figure 3.4: Average Mixing Shear Rate as a Function of Rotational Speed
of the Paddle Impellers. a) System 9; b) System 10 (Torque

Curves - Matching Method)

252

 

20

 

 

' I r’ I I I T— I ' r I ‘T’ r‘ I ' r
. x "-0-504 d/D-0.429
o 0-0.374
16m A III-0.352 d
X
I I
H
m o _
\ 124 .
L ‘
3 ..
.5 8‘4 ;
I x ‘
44 _
" B
4 6 4
a
0 ' r’ ' r* r I I r7 r r r ”I f r* T’ r T”
zotww.1..II..I-fI
‘ x 11-0.504 (”0.027
0 n-O.374
16+ 5 n-O.352 -
J X
F; 12- O -I
\
,. a
I—l 1
8 .I
‘?~ (I z 1
O
X
41 x o d
J g ‘ .
b
O 1 I . VT I ,r, r r—er. I .

 

 

 

I r r ' ff
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

N [1/rev]

Figure 3.5: Average Mixing Shear Rate as a Function of Rotational Speed
of the Flag Impeller. a) System 2; b) System 1 (Torque

Curves - Matching Method)

 

 

253

 

    

 

 

 

 

50 -
I I I T I T I T—I f T I r T'-
— /.,-1,I n-0.504. d/Da-O.709 J
x d/b-O.36 i
4o-I ,/ -—.
I
"a? 3.0-I ’x Q
\ I," ",
s; I a 4
B 20" ,”’ _‘I
IN /’ . l
4 ,”’ ............... 1
10-1 ........ 2 -
.I ’ ......... .I
o "' """ a
r r T r I r T r I I fil’ I r r r 1'
30 f r ‘11 I I j T I I— I' l 1 r I— I T
I — d/b-1-8 n-O.352, d/D-O.327
x d/b-O.36 .
244 ’,,’
J ,I”
184 ’a” D

1.. [1/81

 

1

rA—g_J___L-~J-_._L_L-_1__ul-- A

 

Figure 8.6: Average Shear Rate as a Function of Impeller Rotational
Speed for Two Paddle Impellers (Torque Curves - Matching

Method)

254

 

 

 

 

 

 

 

20
I. T IT I I T I
x k («.5 n.0504, d/D-O.709 j
o k' (50 rpm) ‘
161 a k' (10 rpm) d/b-1'8 4:
J” 4
5’3 124 '1-
c1 4P
.9. . ‘:
é 8* I
I' J
I
4.4 ‘x ‘
J x“ I
m D
0 T r T F T— T T r T—
15 1. l ' I ' I ' I '
x k (on) n-0.504. d/D-O.327 .
o k' (50 rpm)
124 o k' (10 rpm) d/b‘1'8 "
+ J
7; '° '
I 9-I "
o
o. J. ‘
LII—l
. 6% D -4
S . I.
3-4 q'x.x|- A
I " ° ° a
O I f r [T ' I t F t
0.0 10.0 20.0 30.0 40.0 50.0
in [1/81

Figure 3.7: Apparent Viscosity as a Function of Average Shear Rate
for a Paddle Impeller (CMC 1%) (Torque Curves - Matching

Method)

15

255

 

12%

no [PO*S]

X L‘(oq3
O k' (50 rpm)
n '3 1" (10 rpm)

I II I

I T T

«0.504, d/D-O.273 4

 

r ' l

U r F

 

X

I T I

n-o.504. d/D-O.591

I

n— _1_- J-I_._l---...l... —_J.-__l—J .aL—_4_.J___4.--_.l.._i. ---l.-- L. _ J-- -- __J

 

 

' r’
.0 10.0

I

r
20.0 30 0

'F- [1/81

 

50.0

Figure 8.8: Apparent Viscosity as a Function of Average Shear Rate

For a Flag Impeller (CMC 1%) (Torque Curves - Matching

Method)

256

 

 

 

 

 

 

l.OE-l-05 f I e I I I I I 3
d/D= 0.429 Flog Impeller 3.
l.OE+O4 E 5 j
32 § m 1'
4’, 1000.0 "' 22: 4
3 A 100 rpm
E + 50
10013 0 20 d
U 10 3 a.
o 5 1
215
10.0 I ‘ r I I l f IT I I l
l.OE+05 * I I . I . I I I .
d/D= 0.273 Flog Impeller j
LOE+O4
'5— 3 ‘ as: 3
3 o 5
a? :3: @g '1
'53 1000.0 1.8 a? o '5
..Es A 100 rpm .8 o
E + 50 . n .
100.0 0 20 ' “a
o 10 5
O 5 ‘
10.0 I " .2'5 . t I l I , T
0.0 O 1 O 3 0.4 O 6 O 7

(l-n)

3
Figure 8.9: Plots of Dimensionless Functions [P/(md Nn+1)] Versus (l-n)

For The Flag Impeller (Slope Method)

257

 

 

 

 

 

 

LOE+04 I I I I I I , I . ,
k‘ = 10.41 (1/rev)
R3 = 0.981
,__, 1000.0
3
I:
a.
‘2 100.0
E
I—l
\\
o.
10.0
— d/D-0.327. d/b-0.6
1-0 I '* r** I I I I I I ' I
1 0E+04 I I I I I I I I I
k‘ = 12.58 (1/rev)
R3 =3 0.985
1 o — cup-0.327. d/b-1.0
- l ' I ' I I I r I I ”’fi
0.0 0.1 0.3 0.4 0.6 0.7

(l-n)

3
Figure 8.10: Plots of Dimensionless Functions [P/(md Nn+1)] Versus (l-n)

For The Paddle Impellers (Slope Method)

258

 

 

 

 

 

 

 

100E+04 V I V r r I V T
k' = 26.89 (1/rev)
F?3 = 0.960
... 1000.0
3;
z
a.
1: 100.0
.5.
\\
o.
10.0
-— 410-04515. d/b-1.8
1-0 I I” I’ I’ I I I I I I
I.oe+04 I . I . I l I
k‘ = 10.00 (1/rev)
R2 =3 0.960
.. 1000.0
1
3 L
2 100.0 .
N-
S - -_ I
\\
o.
10.0
— d/D-O.327. d/b-O.45
1-0 I'* II ”1* II I I l' I I I I
0.0 0.1 0.3 0.4 0.6

(l-n)

3
Figure 8.11: Plots of Dimensionless Functions [P/(md Nn+1)] Versus (l-n)

For The Paddle Impellers (Slope Method)

 

259

 

1 .OE-I-04

. I I I
k' = 10.31 (1/rev)

 

 

 

I?3 = 0.904
I. 1000.0
3
Z:
a.
13 100.0
E.
\\
O.
10.0
— Iva-0.515. d/b-O.6
1'0 I ' I I I f I r I I
1 OE+O4 f r I r r I I a
k' = 15.99 (1/rev)
F!3 = 0.960
,_ 1000.0
3
I:
a.
1: 100.0
E
H
\\
0..
10.0

 

1 o — ago-0.515. d/b-1.0 '
- I I I— I I I r I r I
0.1 0.3 0.4 0.6 0.7

(l-n)

 

3
Figure 3.12: Plots of Dimensionless Functions [P/(md Nn+1)] Versus (l-n)

For The Paddle Impellers (Slope Method)

260

1.0E+04 I I I I I I I I I
k‘ = 9.827 (1/rev)
R3 = 0.920
1000.0
100.0
10.0
I I I I
' i
O

 

P / [mod'i-NWI’]

 

 

LOE+04 I I I. I I I I I
k‘ = 17.73 (1/rev)
R2 =- 0.972

10.0

 

— d/D-O.709. (Vb-1.8
I r V I I I I'
0.0 0.1 0.3 0.4 0.6

(1 -n)

3
Figure 8.13: Plots of Dimensionless Functions [P/(md Nn+1)] Versus (l-n)

For The Paddle Impellers (Slope Method)

261

 

 

 

 

1000.00 .--.....I I I ......I . ...I?
one 1:: - d/D-O.327
100.00
F.. o
I! an
X
E. 10.00 a- ° "
I x-
!) .x.
1.00 " °
I. x cub-0.30
. I d/b-(M
0.10 I . I ......I e. ......I I I I.-.I.? “(”3‘."I.IIII
1000.00 . I ..-...I I ...I-..I - ......II .
one 1:: - d/D-O.515
100.00
:5 I.
h—I 10.00 3'0 O
I x-
b 8.. 0
1.00 "II
"' x «Vb-0.6
x I d/b-1.0
0.10 . ...II- - fl..." .-I.I.° ‘97".‘9.
0.01 0.30 1.50 Iofoo {03.00

7.. [1/81

Figure 3.14: Flow Curve For 1% wt% Aqueous Solution of CMC Determined

Using the Mixer Viscometer with the Paddle Impellers

262

 

 

 

 

1000.00 I I ..III I I I . ..III I I I I.
CMC 23 - d/D-0.327
100.00 x. 3' °
"" XI 0
O
o- 8. O
‘-‘ X- o
I x. o
b
IOJDO x. 01
”I
x d/b-OJG
I (Vb-0.6
1.00 I WIN-II I I. .....I III... .4. .-.-..I
‘000000 I tilltvt vrvvu' v rrvvuu' r I U rrrt'
CMC 23 - d/D-O.515
" file
0
'3' 100.00 a. 0
file
£331 ”he:
'3 ‘IO
10.00 in
’hO ' '
x d/b-(M
I d/b-1.0
1.00I I . ...I-If I I . .....r ...-II?- “ébf‘fiI-Iq
0.01 O.1O 1 .00 10.00 1 00.00

7.. [1/81

Figure 8.15: Flow Curve For 2% wt% Aqueous Solution of CMC Determined

Using the Mixer Viscometer with the Paddle Impellers

263

APPENDIX.C

PROCEDURE FOR USE OF THE MIXER BROOKFIELD

VISCOMETER FOR NEWTONIAN FLUIDS

This section provides the user of Brookfield Viscometers with the
information required to obtain viscosity readings with the "mixer
inmellers" (flag and paddle type). This information consists of the
impeller "factors" and viscosity ranges for use of the impellenfimq)
combinations with a Brookfield Viscometer. These factors are equivalent
to the Factor Finder supplied with the Viscometer for other spindle

geometries.

6.1) Determination of Impeller Factors

Brookfield Viscometers use a Factor for every spindle/speed
combination the user selects. It is simply multiplied by the Viscometer
reading to evaluate viscosity (in centipoise).

Table 0.1 presents the values of the Factors for the flag and
paddle impellers for evaluation of the viscosity of Newtonian fluids.
The recommended range of viscosity for the Viscometer model is almo
shown to assure proper use of the equipment.

The procedure for data collection and analysis was:

1. The torque required to rotate the impeller, at each value of
rotational speed for several Newtonian fluids of known viscosity at

constant temperature, was measured.

264

Table C.1: Factors for Use of the Mixer Impellers with the Brookfield

 

 

 

 

 

Viscometer
Paddle Impellers
FACTOR VISCOSITY RANGE (Pa 3)
SYSTEM VISCOMETER MODEL Min - Max Min - Max
HBTD ifiyTD, HBTD RVTD

1 45.1/N 5.7/N 4.51 - 451 0.57 - 57

2 33.8/N 4.2/N 3.38 - 338 0.42 - 42

3 22.7/N 2.8/N 2.70 - 227 0.28 - 28

4 19.0/N 2.3/N 1.90 - 190 0.23 - 23

5 15.5/N 2.0/N 1.50 - 150 0.20 - 20

6 45.0/N 5.0/N 4.50 - 450 0.50 - 50

7 29.0/N 3.3/N 2.90 - 290 0.33 - 33

8 21.0/N 2.3/N 2.10 - 210 0.23 - 23

9 34.0/N 4.1/N 3.40 - 340 0.41 - 41

10 22.5/N 2.8/N 2.25 - 225 0.28 - 28

Flag Impeller

l 55.0/N 6.5/N 5.50 - 550 0.65 - 65

2 55.0/N 6.6/N 5.50 - 550 0.66 - 66

3 42.5/N 5.3/N 42.5 - 425 0.53 - 53

N in rpm

Recommended RPM - 10 - 100

3
1 Pa 5 - 10 cp

Viscosity ranges (in Pa 5):
Maximum: Factor x 100
Minimum: Factor x 10

 

265

2. Plots of Newtonian viscosity, n, versus torque, M, for each value of
rotational speed, N and each system geometry (impeller/cup
combination) were made. Figure C.1b is a typical plot for a selected
impeller/cup combination. 3. A relationship between Newtonian

viscosity, 0, and torque reading, was found to be of the form:

n - C1 M (C.l)
where:
3

C1 - constant, s/m

M - torque reading, N m

4. The proportionality constant, C1, was plotted as a function of N for
each system geometry as in Figure CIla.

5. From Eqn. (0.1) and the plot of (4), the following expression for
the Newtonian viscosity was obtained for the different systems

(Table CIZ):

B2
97 - B, (N/60) M (0.2)

with N - impeller rotational speed, rpm
3
B1 - constant, l/m

32 - constant, -n

Thus, the Newtonian viscosity, 0, can be calculated from the fixllow-

ing equation:

0 - FACTOR * Torque (C.3)

266

3.00E+05

 

 

 

 

 

 

   

 

 

 

J -- 'Y- 4301.70;[x«(-'-0.3773)] I 4
14054-0541 -
4'.
5'" n 1
g 13054-054: -
75~ 4 I J
H I
I. 12054-054 -
o C
J I I
6.00E‘I‘04’4 ‘g "i
4 “‘ q
0.00 I "‘.""“."--,-~ -- W ------- - - I 3
0.00 0.40 0.80 1 .20 1.60 2.00
N [rev/3]
14 ' I ' I ' I I
12.. f r ,0 ..I
+ 1,, .
10-4 I, I J
H J ll ’ d
g g... d/b-0.36 I.
3:. 5 '
and
P -- rpm-2.5 d
4... — rpm-5 ..
-- rpm-10 ‘
2 - rpm-20 -
-- run-50 ‘
0 I I . , I I'— m"? b
0.0000 0.0000 0.001 2

 

0.0010 0.0024 0.0030

Torque [Ntm]

Figure C.1: a) Plot of 01 Versus Impeller Rotational Speed; b) Plot

of n Versus Torque Reading (Paddle Impeller, d/b-O 45)

Table C.2: Table for the fit of Eqn.

267

Paddle Ilpellers

(C.2)

 

 

 

 

SYSTEM 81 82
1 131.6462 -0.9541 0.999
2 97.2958 -0.9827 1.000
3 65.1144 -0.9753 1.000
4 54.3087 -0.9750 1.000
5 46.0179 -0.9773 1.000
6 139.0464 -0.8527 0.990
7 92.6439 -0.9l34 0.989
8 61.7416 -0.9388 0.995
9 99.0534 -0.9771 1.000
10 65.6778 -0.9905 1.000

Flag Lupeller
1 162.9035 -0.9816 1.000
2 158.2457 -0.9755 1.000
3 124.8972 -0.9973 1.000

 

268

B2
with FACTOR - 81(N/60)

IFactors were converted to constants for use with the Brookfield

display readings as follows:

,0
[FACTOR x spring constant (Nm) x 10 ] x N - FACTOR2 (C.3)

FACTOR:

with FACTOR - ——7§—- (C 4)

Table C.2 shows average values for every system (at all values of
N).

The mixer impeller Factors for every impeller/cup combination
presented in Table C.2 allow the user for direct determination of vis-
cosity readings. Torque readings were converted to Viscometer display
readings to facilitate the procedure.

Table 0.3 shows the values of viscosity obtained with the dif-
ferent impeller/cup combinations at two selected values of N (10 and 50
rpm). It may be seen that the maximum error obtained is about 10% which
indicates that the prediction factors provide accurate estimation of the

Newtonian viscosity.

C.2) ro dure Fo D e inat o O ewton n Vi cosit

The following procedure is a useful tool for analysis of vis-
cosity data of food.products of unknown behavior. It is also a starting
point from which more advanced techniques can be explored.

The procedure for determination of Newtonian viscosity with the

Mixer Brookfield Viscometer is as follows:

 

269

Table C.3: Values of Viscosity of a Newtonian Fluid (n - 4.84 Pa 3)
Obtained with The Impeller Factors

Paddle Impellers

 

 

 

 

0 (Pa 8)
SYSTEM N (rpm) HBTD RVTD % Error
1 10 4.71 4.76 -2.7 «1.7
50 4.50 4.55 -7.0 -6.0
2 10 5.00 4.98 3.3 2.9
50 4.75 4.72 -1.9 -2.5
3 10 4.68 4.62 -3.3 -4.5
50 4.66 4.60 -3.7 -4.9
4 10 4.99 4.84 3.0 0.0
50 4.93 -- * 1.9 --
5 10 4.95 5.11 2.3 5.6
50 4.84 -- 0.0 --
6 10 5.25 4.67 8.5 -3.5
50 4.66 4.79 -3.7 -l.0
7 10 4.79 4.95 -l.0 2.3
50 4.53 4.66 -6.4 -3.7
8 10 5.29 4.99 9.3 3.1
50 4.96 -- 2.5 --
9 10 4.85 4.67 0.2 -3.5
50 4.70 4.55 -2.3 -5.9
10 10 4.95 4.93 2.3 1.9
50 4.91 4.89 1.4 1.0
Flag Impeller
1 10 4.60 4.55 -4.9 -5.9
50 4.56 4.49 «5.7 -7.2
2 10 5.11 5.21 5.6 7.6
50 4.85 5.04 0.2 4.1
3 10 4.61 4.62 -4.7 —4.5
50 4.51 4.51 -6.8 -6.8

 

Reading out of range of viscometer

270

I Determine if the fluid is Newtonian. To do so, record display

reading with impeller rotating at different values of rotational
speed. A plot of the reading values as a function of N should give a
straight line with slope equal to 1.

. Select impeller/cup combination and value of rotational speed
according to standard Brookfield procedure. Generally, a reading
display of 10 or higher assures that proper selection of equipment
has been made.

. Read digital display from Viscometer.

. Find FACTOR for selected system and N in Table 6.1.

. To obtain viscosity readings (Pa 3), multiply Brookfield reading by

the Factor at specified value of rotational speed (in rpm) from step

(2).

271

REFERENCES

Beavers, G.S., Yoo, J.Y. and Joseph, D.D. 1980. The free surface on a
liquid between cylinders rotating at different speeds. Part III. Rheol.
Acta 19, 19-31.

Beckner, J.L. and Smith, JIM. 1966. Anchor-agitated systems: power input
with Newtonian and pseudoplastic fluids. Trans. Instn. Chem. Engrs. 44,
T224-T235.

Begachev, V.I., Gurvich, A.R. and Braginski, L.N. 1980. A generalized
method of power calculation in mixing highly viscous Newtonian and non-
Newtonian media. Theor. Found. Chem. Engng. 14(1), 98-104.

Bertrand, J. and Courdec, J.P. 1988. Numerical and experimental study of
flow induced by an anchor in viscous, Newtonian and pseudoplastic
fluids. Int. Chem. Engng. 28(2), 257-270.

Bird, R.B., Stewart, W.E. and Lightfoot, E.D. 1960. Transport Phenomena.
John Wiley & Sons Inc.

Blasinski, H. and Rzyski, E. 1976. Mixing of non-Newtonian fluids with
turbine, propeller and paddle agitator. Int. Chem. Engng. 16(4), 751-
754.

Blasinski, HI, Kochanski, B. and Rzyski, E. 1970. The Froude number in
the mixing process. Int. Chem. Engng. 10(2), 176-179.

Boger, DIV. 1977/1978. A highly elastic constant-viscosity fluid. J.
Non-Newt. Fluid Mech. 3, 87-91.

Bohme, 0., V0 , R. and Warnecke, W. 1985. Die freie oberflache einer
flussigkeit uber einer rotierenden scheibe. Rheol. Acta 24, 22-33.

Bongenaar, J.J.T.M., Kossen, N.W.F., Metz, B. and Meijboom, F.W. 1973. A
method for characterizing the rheological properties of viscous fermen-
tation broths. Biotech. Bioengng. 15, 201-206.

Bourne, J.R. 1965. Brit. J. Appl. Phys. 16, 1411.

Bourne, J.R. and Butler, H. 1969. Power consumption of helical ribbon
impellers in viscous liquids. Trans. Instn. Chem. Engrs. 47, T263-T270.

Bourne, J.R., Buerli, M. and Renegass, W. 1981. Power and heat transfer
to agitated suspensions: Use of heat flow calorimetry. Chem. Engng. Sci.
36, 782-784.

Calderbank, P.H. and Moo-Young, MIB. 1959. The prediction of power
consumption in the agitation of non-Newtonian fluids. Trans. Instn.
Chem. Engrs. 37, 26-33.

 

272

Calderbank, P.H. 1958. Part I: The interfacial area in gas-liquid con-
tacting with mechanical agitation. Trans. Instn. Chem. Engrs. 36, 443-
463.

Calderbank, P.H. and Moo-Young, M.B. 1961. The power characteristics of
agitators for the mixing of Newtonian and non-Newtonian fluids. Trans.
Instn. Chem. Engrs. 39, 338-347.

Castell-Perez, MIE. and Steffe, J.F. 1989. Using Mixing to evaluate

rheological properties. Chapter 14 in Visggglagtig Ergperties of Solid,
Elgi§_§n§_§§m1_§glig_figgg§ (Rao, M.A., ed.) Elsevier Applied Science

Publishers Ltd., Barking, England.

Castell-Perez, M.E., Steffe, J.P. and Morgan, R.G. 1987. Adaptation of a
Brookfield (HBTD) viscometer for mixer viscometry studies. J. Texture
Stud. 18, 359-365.

Charles, M. 1978. Technical aspects of the rheological properties of
microbial cultures. Adv. Biochem. Engng. 8, 1-61.

Chavan, V.V. and Ulbrecth, J. 1973. Power correlation for off-centred

helecal screw impellers in highly viscous Newtonian and non-Newtonian
liquids. Trans. Instn. Chem. Engrs. 51, 349-354.

Chavan, V.V. and Ulbrecht, J. 1972. Power correlation for helical ribbon
impellers in inelastic non-Newtonian fluids. Chem. Engng. J. 3, 308-311.

Chavan, V.V. and Mashelkar, RIA. 1980. Mixing of viscous Newtonian and

non-Newtonian fluids. In Advances in Transport Ergggsses, New Delhi,
210-246.

Chavan, V.V., Arumugam, M. and Ulbrecht, J. 1975. On the influence of
liquid elasticity on mixing in a vessel agitated by a combined ribbon-
screw impeller. AIChE J. 21(3), 613-615.

Chavan, V.V., Jhaveri, A.S. and Ulbrecht, J. 1972. Power consumption for
mixing of inelastic non-Newtonian fluids by helical screw agitators.
Trans. Instn. Chem. Engrs. 50, 147-155.

Chaplin, L. 1987. Mixing of highly viscoelastic fermentation broths.
Presented at the 11th Conference on Mixing, August 1987, Henniken, NH.
In Nauman, E.B., Etchells, A.W.III and Tatterson, 0.8. 1988. Mixing: The
state-of-the-art. Chem. Engng. Prog. 84, 58-69.

Collias, D.J. and Prud'homme, R.K. 1985. The effect of fluid elasticity
on power consumption and mixing times in stirred tanks. Chem. Engng.
Sci. 40(8), 1495-1505.

Conti, RI, Sicardi, S. and Specchia, V. 1981. Effect of the stirrer
clearance on particle suspension in agitated vessels. The Chem. Engng.
J. 22, 247-249.

Crochet, M.J. and Walters, K. 1983. Numerical methods in non-Newtonian
fluid mechanics. Ann. Rev. Fluid Mech. 15, 241-260.

 

 

273

Deak, A., Havas, G. and Sawinsky, J. 1985. The power requirements for
anchor, ribbon and helical screw agitators. Int. Chem. Engng. 25(3),
558-565.

Dealy, J.M. and Vu, T.K.P. 1977/1978. The Weissenberg effect in molten
polymers. J. Non-New. Fluid Mech. 3, 127-140.

Ducla, J.M., Desplanches, H. and Chevalier, J.L. 1983. Effective vis-
cosity of non-Newtonian fluids in a mechanically stirred tank. Chem.
Engng. Comm. 21, 29-36.

Edwards, M.F., Godfrey, J.C. and Kashani, M.M. 1976. Power requirement
for the mixing of thixotropic liquids. J. Non-New. Fluid Mech. l, 309-
322.

Eitelberg, G. 1983. Weissenberg effect and its dependence upon the
experimental geometry. Rheol. Acta 22, 131-136.

Ford, E.W. 1984. Rheological analysis of starch-thickened apricots using
mixer viscometry techniques. M.S. Thesis, Department of Agricultural
Engineering, Michigan State University, East Lansing, Michigan.

Ford, EIW. and Steffe, J.F. 1986. Quantifying thixotropy in starch-
thickened, strained apricots using mixer viscometry techniques. J.
Texture Stud. 17, 71-85.

Foresti, R. Jr. and Liu, T. 1959. How to measure power requirements for
agitation of non-Newtonian liquids in the laminar region. Ind. and Eng.
Chem. 51(7), 860-864.

Godleski, S. and Smith, JIC. 1962. Power requirements and blend times in
the agitation of pseudoplastic fluids. AIChE J. 8(5), 617-620.

Green, S.J. 1953. Agitation in process design. Trans. Instn. Chem.
Engrs. 31, 327-347.

Griffith, D.L. and Rao, V.N.M. 1978. Flow characteristics of non-
Newtonian foods utilizing a low-cost rotational viscometer. J. Food
Science 43, 1876-1877.

Haimoni, A. and Hannant, D.J. 1988. Developments in the shear vane test

to measure the gel strength of oilwell cement slurry. Adv. Cement Res.
1(4), 221-229.

Hall, K.R. and Godfrey, JIC. 1970. Power consumption by helical ribbon
impellers. Trans. Instn. Chem. Engrs. 48, T201-T208.

Hiraoka, SI, Yamada, I. and Mizoguchi, K. 1979. Two dimensional model
analysis of flow behavior of highly viscous non-Newtonian fluid in
agitated vessel with paddle impeller. J. Chem. Engng. Japan 12(1), 56-
62.

Hirsekon, F.S. and Miller, SIA. 1953. Agitation of viscous solid-liquid
suspensions. Chem. Engng. Prog. 49(9), 459-466.

 

274

Hoffman, A.H. and Gottenberg, W.G. 1973. Determination of the material
functions for a simple fluid from a study of the climbing effect. Trans.
Soc. Rheol. 17(3), 465-486.

Holland, F.A. and Chapman, F.S. 1966. Liquid Mixing and Processing in
firirrgd_1ank§. Chapman and Hall, Ltd. London

Hyman, D. 1962. Mixing and agitation. Adv. Chem. Engng. 3 (Drew, Hoopes
and Vermeilen, eds.)

Joseph, D.D., Beavers, C.S., Cers, A., Dewald, C., Hoger, A. and Than,
P.T. 1984. Climbing constants for various liquids. J. Rheol. 28(4), 325-
345.

Kale, D.D., Mashelkar, R.A. and Ulbrecht, J. 1973. Drag reduction in
rotational viscoelastic boundary layer flows. Nature Phys. Sci. 242, 29-
30.

Keentok, Mo., Milthorpe, J.F. and O'Donovan, E. 1985. On the shearing

zone around rotating vanes in plastic liquids: theory and experiment. J.
Non-New. Fluid Mech. 17, 23-25.

Kemblowski, Z. and Kristiansen, B. 1986. Rheometry of fermentation
liquids. Biotech. Bioeng. 28, 1474-1483.

Kemblowski, 2., Sek, J. and Budzynski, P. 1988. The concept of a rota-
tional rheometer with helical screw impeller. Rheol. Acta 21, 82-91.

Kraynik, A.M., Aubert, J.H., Chapman, R.N. and Gyure, D.C. 1984. The
helical screw rheometer: A new concept in rotational rheometry. In
Proceedings of the Annual Technical Conference of the Society of
Plastics Engineers, 14 Fairfield Drive, PIO. Box 0403, Brookfield
Center, Connecticut 06804.

Krieger, I.M. and Maron, S.H. 1954. Direct determination of the flow
curves of non-Newtonian fluids. III. Standardized treatment of vis-
cometric data. J. Appl. Phys. 25(1), 72-75.

Laughs“. ELL. 1980. W. Robert
E. Krieger Publishing Company Inc., Huntington, New York.

Leong, Y.K., Creasy, D.E., Boger, D.V. and Nguyen, Q.D. 1987. Rheology
of brown coal-water suspensions. Rheol. Acta 26, 291-300.

Mackey, K.L., Morgan, R.G. and Steffe, J.F. 1987. Effects of shear-

thinning behavior on mixer viscometry techniques. J. Texture Stud. 18,
231-240.

Magelli, F., Fajner, D. and Pasquali, G. 1986. Backmixing in multistage

mixer columns - III. Slightly viscoelastic liquids. Chem. Engng. Sci.
41(9), 2431-2433.

Magnusson, K. 1952. Effektbehouets beroende au den effektiva vis-
kositeten vid omroring av strukturviskosa vatskor. IVA 23(2), 86-91.

 

27S

Mashelkar, R.A., Kale, D.D. and Ulbrecht, J. 1975a. Rotational flows of
non-Newtonian fluids. Part I. Turbulent flow of inelastic fluids around
discs. Trans. Instn. Chem. Engrs. 53, 143-149.

Mashelkar, R.A., Kale, D.D. and Ulbrecht, J. 1975b. Rotational flow of
non-Newtonian fluids. Part II. Torque suppression with agitators. Trans.
Instn. Chem. Engrs. 53, 150-153.

Metzner, A.B. 1956. Non-Newtonian technology: Fluid mechanics, mixing
and heat transfer. Adv. Chem. Engng. l, 77.

Metzner, A.B. and Otto, RIE. 1957. Agitation of non-Newtonian fluids.
AIChE J. 3(1), 3-10.

Metzner, A.B. and Taylor, J.S. 1960. Flow patterns in agitated vessels.
AIChE J. 6(1), 109-114.

Metzner, A.B., Feehs, R.H., Lopez Ramos, H., Otto, R.E. and Tuthill,
J.D. 1961. Agitation of viscous Newtonian and non-Newtonian fluids.
AIChE J. 7(1), 3-9.

Mitsuishi, N. and Hirai, N. 1969. Power requirements in the agitation of
non-Newtonian fluids. J. Chem. Engng. Japan 2(2), 217-224.

Mizushina, T., Ito, R. and Hiraoka, S. 1966. In Egg; and Mass Iransfier
(in Russian), Vol 4. Nauka i Tekhnika, Minsk, 287.

Nagata. S. 1975. WW3.- John Wiley & Sons.
New York.

Nagata, SI, Nishikawa, M., Tada, H. and Gotoh, S. 1971. Power consump-
tion of mixing impellers in pseudoplastic liquids. J. Chem. Engng. Japan
4(1), 72-76.

Nagata, S., Yamamoto, K., Hashimoto, K. and Naruse, Y. 1960. Studies on
the flow patterns of liquids in a cylindrical mixing vessel, over a wide
range of Reynolds number. Mem. Fac. Engng. Kyoto Univ. 22(1), 68-85.

Nagata, S., Yokogama, T. and Yaragimoto, M. 1957. A study on the mixing
of high-viscosity liquid. Chem. Engng. Japan 21, 278-286.

Nguyen, Q.D. and Boger, D.V. 1983. Yield stress measurement for con-
centrated suspensions. J. Rheol. 27(4), 321-349.

Nguyen, Q.D. and Boger, D.V. 1985a. Direct yield stress measurement with
the vane method. J. Rheol. 29(3), 335-347.

Nguyen, Q.D. and Boger, D.V. 1985b. Thixotropic behavior of concentrated
bauxite residue suspensions. Rheol. Acta 24, 227-237.

Nguyen, Q.D. and Boger, D.V. 1987. Characterization of yield stress of
fluids with concentric cylinder viscometers. Rheol. Acta 26, 508-515.

 

276

Nienow, AIW. and Elson, T.P. 1988. Aspects of mixing in rheologically
complex fluids. Chem. Engng. Res. Des. 66, 5-15.

Nienow, A.W., Wisdom, D.J., Solomon, J., Machon, V. and Vlcek, J. 1983.
The effect of rheological complexities on power consumption in an
aerated, agitated vessel. Chem. Engng. Comm. 19, 273-293.

Oldshue, J.Y. 1983. Elgig_uixing_1§§hnglggy. McGraw-Hill Co., New York.

Oliver, D.R., Nienow, A.W., Mitson, RIJ. and Terry, K. 1984. Power
consumption in the mixing of boger fluids. Chem. Engng. Res. Des. 62,
123-127.

Pavlushenko, 1.5. and Gluz, M.D. 1968. Shear rate gradient in a vessel
with stirrer. Theor. Found. Chem. Engng. 2(1), 105-109.

Pollard, J. and Kantyka, TIA. 1969. Heat transfer to agitated non-
Newtonian fluids. Trans. Instn. Chem. Engrs. 47, T21-T27.

ProkOpec, L. 1972. Agitated polymerization reactor for highly viscous
media. Instn. Chem. Engng. 12(4), 712-719.

Prud'homme, RIK. and Shaqfeh, E. 1984. Effect of elasticity on mixing
torque requirements for Rushton turbine impellers. AIChE J. 30(3), 485-
486.

Qiu, 0.0. and Rao, M.A. 1988. Role of pulp content and particle size in
yield stress of apple sauce. J. Food Sci. 53, 1165-1170.

Ranade, V.R. and Ulbrecht, J. 1977. Gas dispersion in agitated viscous
inelastic and viscoelastic liquids. Proc. 2nd. European Conference on
Mixing, Cambridge, F683-F100.

Rao, M.A. 1975. Measurement of flow properties of food suspensions with
a mixer. J. Texture Stud. 6, 533-539.

Rao, M.A. 1977. Rheology of liquid foods - A review. J. Texture Stud. 8,
135-168.

 

Rao, M.A. and Cooley, H.J. 1984. Determination of effective shear rates

in rotational viscometers with complex geometries. J. Texture Stud. 15,
327-335.

Reher, E. and Bohm, R. 1970. Chem. Tech. 22, 134. Cited by Rieger, F.,
Novak, Z. and Havelkova, ). 1988. The influence of the geometrical shape

on the power requirement of ribbon impellers. Int. Chem. Engng. 28(2),
376-383.

Rieger, F. and Novak, V. 1973. Power consumption of agitators in highly
viscous non-Newtonian liquids. Trans. Instn. Chem. Engrs. 51, 105-111.

Rieger, F. and Novak, V. 1974. Power consumption scale-up in agitating
non-Newtonian fluids. Chem. Engng. Sci. 29, 2229-2234.

277

Roels, J.A., Van den Berg, J. and Voncken, R.M. 1974. The rheology of
mycelial broths. Biotech. Bioengng. 16, 181-208.

Rushton, J.H. and Oldshue, J.Y. 1953. Mixing - Present theory and prac-
tice. Part I and II. Chem. Engng. Prog. 49(4/5), 161-168, 267-275.

Rushton, J.H., Costich, E.W. and Everett, HIJ. 1950. Power characteris-
tics of mixing impellers. Chem. Engng. Prog. 46(8), 395-404.

Sadowska, M., Naczk, M., Sikorski, ZIE. and Ziminska, H. 1982. Effect of
fish protein preparations on rheological properties of meat sausages. J.
Texture Stud. 13, 371-379.

Saravacos, 0.0. and Moyer, J.C. 1967. Heating rates of fruit products in
an agitated kettle. Food Technol. 21(3), 372-376.

Sawinsky, J., Havas, G. and Deak, A. 1976. Power requirement of anchor
and helical ribbon impellers for the case of agitating Newtonian and
pseudoplastic liquids. Chem. Engng. Sci. 31, 507-509.

Schilo, D. 1969. Power requirement of tangential mixers in the mixing of
non-Newtonian liquids. Chem.-Ing. tech. 5, 253.

Sestak, J., Houska, M. and Zitny, R1 1982. Mixing of thixotropic fluids.
J. Rheol. 26(5), 459-475.

Sestak, J., Zitny, R. and Houska, M. 1986. Anchor-agitated systems:

power input correlation for pseudoplastic and thixotropic fluids in
equilibrium. AIChE J. 32(1), 155-158.

Shamlou, P.A. and Edwards, MIF. 1985. Power consumption of helical
ribbon mixers in viscous Newtonian and non-Newtonian fluids. Chem.
Engng. Sci. 40(9), 1773-1781.

Sinevic, V., Kuboi, R. and Nienow, A.W. 1986. Power numbers, Taylor
numbers and Taylor vortices in viscous Newtonian and non-Newtonian
fluids. Chem. Engng. Sci. 41(11), 2915-2923.

Solomon, J., Elson, T.P. and Nienow, A.W. 1981. Cavern sizes in agitated
fluids with a yield stress. Chem. Engng. Comm. 11, 143-164.

Solomons, G.L. 1971. Fermentation equipment. Adv. Appl. Micro. 14, 231-
247.

Steffe, J.F. and Ford, E.W. 1985. Rheological techniques to evaluate the
shelf-stability of starch-thickened, strained apricots. J. Texture Stud.
16, 179-192.

Steffe, J.F., Castell-Perez, M.F., Rose, K. and Zabik, MIE. 1989. Rapid
testing method for characterizing the rheological behavior of gelatiniz-
ing corn starch slurries. Cereal Chemistry 66(1), 65-68.

 

 

278

Takahashi, K., Yokota, T. and Konno, H. 1984. Power consumption of
helical ribbon agitators in highly viscous pseudoplastic liquids. J.
Chem. Engng. Japan 17(6), 657-659.

Tamura, M.S., Shoemaker, C.F. and Henderson, J.M. 1988. Helical screw
rheometer for on-line viscosity measurements. Paper # 519, presented at
IFT Meeting, New Orleans, LA.

Thomas, R.H. and Walters, K. 1964. The motion of an elastico-viscuos
liquid due to a sphere rotating about its diameter. Quart. J. Mech. and
Appl. Math. 17, 39-42.

Tiu, C. and Boger, D.V. 1974. Complete rheological characterization of
time-dependent food products. J. Texture Stud. 5, 329-338.

Treybal, R.E. 1956. Mechanically aided liquid extraction. Adv. Chem.
Engng. 1, 229-329.

Ulbrecht, J. 1974. Mixing of viscoelastic fluids by mechanical agita-
tion. The Chem. Engr. June, 347-367.

Ulbrecht, J.J. andeau, P. 1985. Mixing of viscous non-Newtonian

fluids. Chapter 4 in Mining gr Liguigs by Mecnnnignl Agitation. V01 1.

(Ulbrecht, J.J. and Patterson, C.K., eds.). Gordon and Breach Science
Publishers, New York.

Vocadlo, J.J. and Charles, M.E. 1971. Measurement of yield stress of
fluid-like viscoplastic substances. The Can. J. Chem. Engng. 49(10),
576-582.

Voisey, P.W., Paton, D. and Timbers, GIE. 1977. The Ottawa starch
viscometer- A new instrument for research and quality control applica-
tions. Cereal Chem. 54(3), 534-557.

Walker, 0.8., Ross, A.S., Wrigley, C.W. and McMaster, G.J. 1988.
Accelerated starch-paste characterization with the Rapid Visco-Analyzer.
Cereal Foods World 33(6), 491-494.

Whalen, P.J., Anantheswaran, R.G. and Shahani, K.M. 1987. Power consump-
tion of a whey-corn mash as compared to a full corn mash during
liquefaction. Biotec. Letters 9(7), 523-528.

White, A.M. and Brenner, E. 1934. Studies in agitation. V. The correla-
tion of power data. AIChE J. 30, 585-597.

White, A.M., Brenner, E., Phillips, CIA. and Morrison, M.SI 1934a.
Studies in agitation. IV: Power measurements. AIChE J. 30, 570-584.

White, J.L., Chankraiphon,S. and Ide, Y. 1977. Rheological behavior and
flow patterns around agitators in polymer solutions. Trans. Soc. Rheol.
21(1), 1-18.

Wichterle, K. and Wein, 0. 1981. Threshold of mixing of non-Newtonian
liquids. Int. Chem. Engng. 21(1), 116-120.

279

Wichterle, K., Kadlec, M., Zak, L. and Mitschka, P. 1984. Shear rates on
turbine impeller blades. Chem. Engng. Commun. 26, 25-32.

Williams, R.W. 1979. Determination of viscometric data from the
Brookfield R.V.T. viscometer. Rheol. Acta 18(3), 345-359.

Wood, FIW. and Goff, T.C. 1973. The determination of the effective shear
rate in the Brabender Viscograph and in other systems of complex
geometry. Die Starke 25, 89-91.

Yagi, H. and Yoshida, F. 1975. Gas absorption by Newtonian and non-

Newtonian fluids in sparged agitated vessels. Ind. Engng. Chem. Process
Des. Dev. 14(4), 488-493.

Yap, C.Y., Patterson, W.I. and Carreau, P.J. 1979. Part III. Non-
Newtonian fluids. AIChE J. 25(3), 516-521.

Yoshimura, A.S., Prud'homme, R.K., Princen, H.M. and Kiss, AID. 1987. A
comparison of techniques for measuring yield stresses. J. Rheol. 31(8),
699-710.