rectangular-cross sectional area by Kim and Patel [35] to investigate the curved duct ...... 33- Britchford, K.M., Manners, A.P., McGuirk, J.J. and Stevens, S.J.,.
ABSTRACT The study presents in details, experimental and numerical studies on emulsion (oil-in-water) flow in rectangular cross-sectional area S-shaped diffusers. The experimental setup was designed and constructed in the fluid mechanics laboratory of the faculty of engineering, Menoufia University to obtain the experimental data since the measurements have been performed on twelve S-shaped diffusers. Different parameters including area ratio, curvature ratio, turning angle (45◦/45◦, 60◦/60◦, and 90◦/90◦), flow path (45◦/45◦, 60◦/30◦, and 30◦/60◦), inflow Reynolds number, holdup (0.03, 0.06, 0.10, 0.15, and 0.25) and emulsion stability have been considered. The static pressure distributions along the outer and inner walls of the S-diffuser including upstream and downstream tangents were measured. Based on these measurements, the energy-loss coefficient of all models could be extracted. The diffusers performance has been plotted versus inflow Reynolds number at different geometrical and inflow parameters. The studies were carried out using two types of oil-in-water (o/w) emulsions; stable o/w emulsion by using an emulsifier named by Sodium Dodecyl Sulfate (SDS) and unstable o/w emulsion without any additives at different holdup values. The experimental data for different S-diffuser configurations have been used for assessing credibility of the numerical code using ANSYS R-15.0 software Fluid Flow Fluent (FFF) - 3D with different solution methods. Computations with different turbulence closures have been carried out for prediction of the performance of S-shaped diffusers at different inflow and geometrical conditions in case of water as well as emulsion flows. With emulsion flow, besides the standard k-ε model, the mixture model was used I
as a solution multi-phase model using fine grid to obtain a more accurate flow prediction. The continuous phase (water) has been simulated using standard k-ε model by solving Reynolds-Averaged Navier-Stokes equations (RANS), while the dispersed phase (oil) has been simulated using mixture multi-phase model by solving oil liquid particle equations using 4th order Runge-Kutta method. Comparisons between present CFD code predictions and available experimental results from literature as well as the present experimental data showed good matching and better agreement. The results showed that the S-shaped diffuser energy-loss coefficient is strongly affected by the geometrical and inflow parameters. Increasing area ratio, curvature ratio, and inflow Reynolds number leads to improving diffuser performance. Whereas, decreasing the emulsion holdup (Φ) leads to decreasing diffuser performance. The turning angle plays an important role in improving the S-shaped diffuser performance. S-shaped diffuser energy-loss coefficient of water flow is lower than that of emulsion flow. Also the S-shaped diffuser energy-loss coefficient of stable o/w emulsion flow is higher than that of unstable, one. A general new correlation of energy-loss coefficient including geometrical and flow parameters for the validated studied cases of S-diffusers is extracted from the measurements.
II
ACKNOWLEDGMENTS Praise to ALLAH, LORD of the world, by whose grace this work has been completed. I would like to express my deepest appreciation and gratitude to my supervisors Prof. Kamal Abd El-Aziz Ibrahim, Prof. Wageeh Ahmed El-Askery and Dr. Ismail Mohamed Sakr. They played a key role in all aspects of my research. Their valuable comments, meaningful suggestions, encouragement, patience, understanding and support enabled me to complete such a formidable task. I would like to express my greatest gratitude and thanks to Prof. Dr. Kamal Abd El-Aziz Ibrahim for his priceless opinion and his abundant experimental and computational experience. I am deeply indebted to Prof. Wageeh Ahmed El-Askery who spent his time and energy to accomplish the current work. Special thanks to Dr. Samy Mohammad El-Behery who sacrificed his time to supply valuable opinions. Acknowledgements are also given to all staff members of mechanical power engineering department, for their encouragement and help during the progress of this work. Special thanks to my family; my daughters Noran, Rawan and Roaa and my sons Ahmed and Abd El-Haleem. Also, my beloved wife who never stops being a wonderful wife and making great efforts and contributions to our family. Her unconditional love always supports me to conquer any difficulties in our life.
Hamdy Abd El-Hameed Omara Assoc. Lect. of Mech. Power Eng. Arish Higher Institute for Engineering Higher Education Ministry, Egypt 2016 III
The dissertation is dedicated to the soul of My brother Abd El-Haleem A. Omara "My God's Mercy be upon him"
The devoted man who gave his lifetime to his big and small family by virtue of his properties precious and priceless works, and he left us a great model in giving and giving.
Hamdy A. Omara
IV
CONTENTS ABSTRACT........................................................................................... i AKNOWLEDGMENT........................................................................... iii CONTENTS............................................................................................ v LIST OF FIGURES…………………………………………………... x LIST OF TABLES……………………………………….………….... xvii NOMENCLATURE…………………………………………….......
xix
CHAPTER 1: INTRODUCTION 1.1
Motivation……………………………………………………………………………… 1
1.2
Diffuser Types and Applications………………………………….......
1.3
Diffuser Performance Parameters……………………………………………
1
1.3.1 Static Pressure Recovery Coefficient …………………………………… 1.3.2 Ideal Static Pressure Recovery Coefficient……………………........… 4 1.3.3 Diffuser Efficiency………………………………………………………… 1.3.4 Diffuser Loss Coefficient ……………………………………………..… 1.3.5 Inflow Reynolds Number………………………………………………… 1.4
Diffuser Flow Regimes………………………………………………...
7
1.5
Secondary Flow in Curved Diffuser……………………………...…
8
1.6
Design of S-Shaped Diffuser………………………………………….
11
1.7
Oil-in-Water Two-Phase Flow through S-shaped Diffuser……….
12
CHAPTER 2: LITERATURE SURVEY AND OBJECTIVES 2.1
Introduction …………………………………………………................
2.2
Literature Survey ……………………………………………………….. 2.2.1 Single Phase Flow…………………………………………………..
V
17 17
2.2.2 Two-Phase / Emulsion Flow…………………………………………..... 2.3 Conclusions of the Literature Survey…………………………….
30
2.4 Objectives of the Present Study……………………………………
32
CHAPTER 3: EXPERIMENTAL SETUP AND MEASUREMENTS 3.1 Introduction…………………………………………………………… 34 3.2 Experimental Apparatus ……………………………………..……
34
3.2.1 Emulsion -Supply Line…………………………………………………. 3.2.2 Fabrication of S-Shaped Diffusers Tested Models……………… 3.2.3 The Specifications of the Hydraulic Circuit Parts………………. 3.3 The S-Diffuser Influencing Parameters (Parametric Study) …
43
3.3.1 S-Diffuser Area Ratio………………………………………….. 3.3.2 S-Diffuser Curvature Ratio…………………………………………….. 3.3.3 S-Diffuser Turn Angle………………………………………….. 3.3.4 S-Diffuser Flow Path………………………………………….. 3.3.5 S-Diffuser Inflow Reynolds Number………………………… 3.3.6 Holdup (Φ) and Stability………………………………………... 3.4 Measurement Locations……………………………………………..
45
3.5 Measured Data………………………………………………………..
50
3.5.1 Measurements of Holdup……………………………………… 3.5.2 Measurements of Wall Static Pressure………………………... 3.5.3 Measurements of Energy Loss Coefficient…………………… 3.5.4 Measurements of Flow Rate………………………………….. 3.6 Measuring Instruments……………………………………………... 3.6.1 Pressure Gauges…………………………………………………
VI
52
3.6.2 Orifice-Meter……………………………………….…………… 3.7 Test Section Conditions………………………………………………… 53 3.8 Preliminary Measurements: Flow through Pipes………………….
53
3.9 The Experimental Procedure ……………………………..................... 59
CHAPTER 4: PHYSICAL AND MATHEMATICAL MODELS 4.1 Introduction………………………………………………………….
61
4.2 Physical model……………………………………………………….
61
4.3 Formulation Methods of Two-Phase Flow………………………… 62 4.4 Mathematical Model………………………………………………...
65
4.4.1 Mixture Two-phase Flow Model……………………………….. 4.4.2 Fluid Flow Modeling…………………………………………… 4.4.3 The Euler – Euler Model………………………………………. 4.4.4 The VOF Model……………………………………………….. 4.4.5 Turbulence Modeling…………………………………………... 4.4.5.1 Standard k-ε Model, (STD k -ε )………………………… 4.4.5.2 Standard k -ω Model, (STD k -ω )………………………... 4.4.5.3 Shear-Stress Transport k -ω Model (SST k -ω )………….. 4.4.6 Mathematical Model in a General Form……………………..
CHAPTER 5: NUMERICAL METHODOLOGY 5.1 Introduction.........................................................................................
76
5.2 Stages of Computation……………………………………………………
77
5.2.1 Model geometry and grid generation………………………………….. 5.2.2 Setup of the Solver………………………………………………………….. 5.2.3 Boundary Conditions……………………………………………
VII
5.3 Solution and Turbulence Models……………………………..
79
5.4 Convergence Criterion…………………………………………
79
5.5 Calculation and Results Displaying…………………………..
80
5.6 Numerical Schemes………………………………………….....
80
5.7 Discretization Process………………………………………….
81
5.8 Pressure Interpolation Schemes………………………………
83
5.9 Pressure-Velocity Coupling……………………………………
84
5.10 Near-Wall Treatment for Wall-Bounded Turbulent Flows….. 84 5.11 Numerical Solution Procedure…………………………………
87
CHAPTER 6: CFD CODE VALIDATION 6.1 Introduction................................................................................... 90 6.2 Validation of an S-Shaped diffuser Carrying Water …….…
90
6.2.1 Physical Model and the Grid Generation………………………………. 6.2.2 Turbulence Models and Mesh Resolution…………………
6.3 Validation of Emulsion Flow in Sudden Expansion ……..….
95
6.3.1 Reynolds Number Effect………………………………… 6.3.2 Emulsion Holdup Effect…………………………..……. 6.4 Validation of a 90° curved diffuser …..……………………....
99
6.4.1 Water Flow……………………………………………… 6.4.2 Emulsion Flow…………………………………………… 6.5 Summary and Conclusion of CFD Code Validation…………
104
CHAPTER 7: RESULTS AND DISCUSSIONS 7.1 Introduction................................................................................... 105
VIII
7.2 Solution Models and Resolution Effect……………………….
108
7.2.1 Turbulence Models……………………………………….. 7.2.2 Multiphase Models………………………………………. 7.2.3 Resolution Effect………………………………………… 7.3 Experimental and Numerical Results………………………
114
7.3.1 Wall Static Pressure coefficient Distributions………….. 7.3.2 Diffuser Performance…………………………………... 7.3.3 Loss Coefficient Correlation…………………………… 7.4 Diffuser Performance ………………………………………
129
7.5 Numerical Results……………………………………………
148
CHAPTER (8): CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE WORK 8.1 CONCLUSIONS……………………………………………………
169
8.2 RECOMMENDATIONS FOR FUTURE WORK…………….
171
REFERENCES………………………………………………………………………
173
APPENDIX A: CALIBRATION OF EMULSION FLOW METER..
182
APPENDIX B: THE EXPERIMENTAL UNCERTAINTY……………
188
APPENDIX C: ANALYTICAL STUDY OF S-SHAPED DIFFUSER
194
APPENDIX D: DESIGN DETAILS OF S-SHAPED DIFFUSERS….
197
APPENDIX E: PUBLISHED PAPERS………………………………
209
ARABIC SUMMARY…………………………………………
ح:أ
IX
LIST OF FIGURES Figure
Subject
Page
1.1 Basic diffuser geometries [1, 2]
2
1.2 Curved walled Diffusers [3]
3
1.3 Inflected walled Diffuser [4]
3
1.4 S-shaped curved-walled diffuser, present work
5
1.5 Centrifugal Pump (right) and Similar Passage between Two Impellers (left) [5]
5
1.6 Diffuser flow regimes with diffuser geometric parameters
9
1.7 Diffuser flow regimes [10]
10
1.8 Some diffusers and their geometric parameters as in Dixon [11] 10 3.1.a Schematic view of the experimental apparatus (Not to scale
36
3.1.b Physical test model of S-diffuser
37
3.2 Photograph of the test rig
38
3.3 Photograph of the S-diffusers 1, 2 and 3
39
3.4 Pipe friction loss coefficient against Reynolds number at water flow
56
3.5 Reynolds number effect on the friction factor at different holdup
56
3.6 Reynolds number effect on the friction factor at emulsion flow compared with [64]
57
3.7 Holdup effect (a) and stability effect (b) on the friction factor at 58 emulsion flow 4.1 Schematic of S-shaped diffuser test physical model (Not to scale)
64
5.1 A staggered mesh grid (a) and the mesh for 3D S-shaped 78 diffuser (b)
X
5.2
The near-wall regions of turbulent flows (ANSYS Fluent [73])
86
5.3
Flowchart of the used code program
88
5.4
Flowchart of the present numerical and experimental studies
89
6.1
The schematic coordinates system for 3D S-diffuser tested physical model [30] (not to scale)
92
6.2
The computational mesh for 3D S-shaped diffuser of Ref. [30]
93
6.3
Turbulence modeling effect on the pressure coefficient distributions of Whitelaw and Yu Ref. [30] with water flow on outer-wall (a) and inner-wall (b)
94
6.4
Grid-size effect on the pressure coefficient of water flow measured in [30]
95
6.5
The schematic diagram and the pressure profile for the sudden expansion of Ref. [58]
96
6.6
The computational grid used for the sudden expansion of Ref. [58]
97
6.7
Pressure drop for 0.2144 unstable (o/w) emulsion flowing with different velocities through a sudden expansion of Ref. [58]
97
6.8
Pressure drop for 0.3886 unstable (o/w) emulsion flowing with different velocities through a sudden expansion of Ref. [58]
98
6.9
Pressure drop for 0.6035 unstable (o/w) emulsion flowing with different velocities through a sudden expansion of Ref. [58]
98
6.10
Pressure drop for 0.3886 unstable (o/w) emulsion flowing through 99 a sudden expansion with steady/unsteady of Ref. [58]
6.11
Pressure drop for unstable (o/w) emulsions flowing through a sudden expansion at different holdup of Ref. [58]
100
6.12
The schematic diagram for a 90° curved diffuser of Ref. [71], not to scale
101
6.13
The computational grid for a 90° curved diffuser of Ref. [71]
101
6.14
Pressure recovery coefficient distributions on the outer-wall (a) and inner-wall (b) for water flow in 90° curved-diffuser of Ref. [71]
102
XI
6.15 Multi-phase modeling effect on the pressure coefficient distributions on the outer-wall (a) and inner-wall (b) for emulsion flow in 90° curved-diffuser of Ref. [71]
103
7.1 Turbulence modeling effect on the pressure coefficient distributions for model (7) carrying water
109
7.2.a Multi-phase modeling effect on the pressure coefficient distributions for model (7) carrying 0.25 stable emulsions
110
7.2.b Turbulence modeling effect on the Mixture scheme as a solution for predicting the pressure coefficient distributions of model (7) carrying 0.25 stable emulsions
111
7.3.a Computational grid
112
7.3.b Outer-wall pressure coefficient distributions with water flow
113
7.3.c Outer-wall pressure coefficient distributions with 0.25 stable emulsion flow
113
7.4 Effect of Reynolds number on pressure coefficient of model (7) carrying water
116
7.5 Effect of Reynolds number on pressure coefficient of model (7) carrying 0.25 stable o/w emulsion
117
7.6 Effect of holdup (volume fraction) on pressure coefficient of model (7) carrying water and 0.03:0.25 stable o/w emulsion
118
7.7 Effect of stability on pressure coefficient of model (7) carrying 0.03, 0.06, 0.25 stable / unstable (o/w) emulsion
119
7.8 Effect of diffuser area ratio on the diffuser pressure coefficient for water flow (M 1: AR=3, M 2: AR=2 and M 3: AR=1.5)
121
7.9 Effect of diffuser area ratio on the diffuser pressure coefficient at Φ =0.25
122
7.10 Effect of diffuser curvature ratio on the diffuser pressure coefficient for water flow (M 5: CR=12.5, M 4: CR=7.5 and M 6: CR=5)
123
7.11 Effect of diffuser curvature ratio on diffuser pressure coefficient at Φ =0.15
124
7.12 Effect of diffuser turn angle on the diffuser pressure
125
XII
coefficient for water flow (M 7: 60°/60°, M 9: 45°/45° and M 8: 90°/90°) 7.13 Effect of diffuser turn angle on the diffuser pressure coefficient at Φ =0.15
126
7.14 Effect of diffuser flow path on the diffuser pressure coefficient for water flow (M 10: 45°/45°, M 12: 30°/60° and M 11: 60°/30°)
127
7.15 Effect of diffuser flow path on the diffuser pressure coefficient at Φ =0.15
128
Φ = 0.03 stable (o/w) emulsion
130
7.16.(c), (d) Φ = 0.10 stable (o/w) emulsion
131
7.16.(a), (b)
7.16.(e), (f) Effect of Reynolds number on the energy-loss coefficient for 132 all models carrying water and stable (o/w) emulsion 7.17.(a), (b) (a) Models 1 and 9, (b) Models 2 and 4
133
7.17.(c), (d) (c) Model 3, (d) Models 5 and 10
134
7.17.(e), (f) (e) Model 6, (f) Model 7
135
7.17.(g), (h) (g) Model 8, (h) Model 11
136
7.17.(i) (i) Model 12: Effect of the emulsion holdup (volume fraction, Φ) on the diffuser energy-loss coefficient for all models carrying water and stable (o/w) emulsion
137
7.18
(a) Model 3: Φ = 0.03, 0.15, 0.25 and (b) Model 3: Φ = 0.00, 0.06, 0.10
138
7.18
(c) Model 7: Φ = 0.03, 0.15, 0.25 and (d) Model 7: Φ = 0.00, 139 0.06, 0.10
7.18
(e) Model 11: Φ = 0.03, 0.15, 0.25 and Model 11: Φ = 0.00, 0.06, 0.10: Effect of emulsion stability on S-diffuser performance for all models
140
7.19
Effect of S-diffuser area ratio on the diffuser energy-loss coefficient
141
7.20
Effect of S-diffuser curvature ratio on the diffuser energyloss coefficient
142
XIII
7.21 Effect of S-diffuser turn angle on the diffuser energy-loss coefficient 143 7.22 Effect of S-diffuser flow path on the diffuser energy-loss coefficient
144
7.23 Calculated and measured loss coefficients
147
7.24 Local cross-section plane of S-shaped diffuser
148
7.25 Position effect on computational axial velocity contours of water flow at different sections along the S-Diffuser (Model-7)
149
7.26 Position effect on computational turbulent shear stress contours of water flow at different sections along the S-Diffuser (Model-7)
150
7.27 (a) Inlet Plane (0°) and (b) Mid-first bend plane (30°)
151
7.27 (c) Inflexion Plane (60°) and (d) Mid-second bend Plane (60°/30°)
152
7.27 (e) Exit Plane (60°/60°): Reynolds number effect on computational 153 axial velocity contours of water flow at different sections along the S-Diffuser (Model 7) 7.28 (a) Inlet Plane (0°)
153
7.28
154
(b) Mid-first bend plane and (30°), (c) Inflexion Plane (60°)
7.28 (d) Mid-second bend Plane (60°/30°) and (e) Exit Plane (60°/60°)
155
7.29 Holdup effect on exit computational axial velocity contour of Model-7 at Re=24000
157
7.30 Holdup effect on the five computational axial velocity vectors for Model 7
158
7.31 Holdup effect on exit computational turbulent shear stress contour of Model-7 at Re=24000
159
7.32 Stability effect on exit axial velocity contours of Model-7 at Re=24000
160
7.33 Stability effect on exit computational turbulent stress contours of Model-7
161
7.34 Area ratio effect on computational axial velocity contour at exit plane at Re=24000
162
7.35 Area ratio effect on computational turbulent shear stress contour at exit plane at Re=24000
163
XIV
7.36
Curvature ratio effect on computational axial velocity contour at exit plane
165
7.37
Curvature ratio effect on computational turbulent stress contour at exit plane
166
7.38
Turn angle effect on computational axial velocity contours at the exit plane
167
7.39
Turn angle effect on computational turbulent stress contour at the exit plane
168
A.1
Configuration of the Orifice-meter
182
A.2
(a) Relative density of (o/w) emulsions versus holdup
184
A.2
(b) Relative viscosity of (o/w) emulsions versus holdup
185
A.3
Measured discharge coefficients Cd versus holdup
186
A.4
Measured discharge coefficients Cd versus Reynolds
187
number D.1
D.2
Geometries of the all different S-shaped diffuser tested
197:
Models with r (θ)
201
Geometries of the all different S-shaped diffuser tested
202:
Models with x (θ) and y (θ)
206
XV
LIST OF TABLES Table
Subject
Page
1.1 Different regimes for two-phase flows as reported in [15] 3.1 Geometrical parameters of the S-shaped diffusers models, for all models W=20 mm, B=80 mm and all Dims. in mm
14 42
3.2 General specifications of the used centrifugal pump as manufacturer's data 43 3.3 Tap locations of upstream and downstream tangent ducts of all models
46
3.4 Tap locations of S-diffuser models (1), (2), (3), (4) and (9)
47
3.5 Tap locations of S-diffuser models (5) and (10)
48
3.6 Tap locations of S-diffuser model (6)
48
3.7 Tap locations of S-diffuser model (7)
49
3.8 Tap locations of S-diffuser models (11) and (12)
49
3.9 Tap locations of S-diffuser models (11) and (12)
50
3.10 Specifications of the S-shaped diffuser tested runs
54
4.1 General governing equations forms of the two-phase flow
73
4.2 The general form of the turbulence models equations
74
4.3 Model constants, Damping functions, production and extra source terms
75
7.1 Summary of the investigated conditions in the present study
106
7.2 Characteristic data used in the present study
106
7.3 Different effects used in the present study
107
A.1 Orifice plate dimensions in mm.
182
B.1 Uncertainty of the energy-loss coefficients of the S-shaped diffuser models with water flow
191
B.2 Uncertainty of the energy loss coefficients for curved diffusers with unstable and stable o/w emulsions.
192
B.3 Uncertainty of the oil concentration by volume (holdup, Φ) for the tested emulsions trough the curved diffusers
193
XVI
NOMENCLATURE Symbols A
Cross-sectional area, m2
AR
S-diffuser area ratio =Aexit / Ainlet
B
Height of the S-shaped diffuser, m
CC
Concave wall
CV
Convex wall
Cd
Discharge coefficient of the orifice flow meter
Cp
Static pressure recovery coefficient
C p (I )
Ideal static pressure coefficient
Cp
i
Static pressure recovery coefficient on the inner wall
Cpo
Static pressure recovery coefficient on the outer wall
CR
S-diffuser curvature-Ratio = RC / W
DH
Hydraulic diameter, m
Kd
S-shaped diffuser energy loss coefficient
M…
Model number
P
Pressure, N /m2
Pexit
S-shaped diffuser exit Pressure, N /m2
Pinlet
S- diffuser inlet pressure, N /m2
Q
Fluid flow rate, m3 / s
Rc
Diffuser centre-line arc radius, m
Re
Reynolds number
U ref
u
Mean-streamwise velocity at the reference location (diffuser entrance), m/s Axial velocity inside the diffuser parallel to diffuser centerline, m/s
XVII
VE
Emulsion volume, m3
Vo
Oil volume, m3
Vw
Water volume, m3
V inlet
S-shaped diffuser inlet flow velocity, V inlet = U ref , m/s
V exit
S- diffuser exit flow velocity, m/s
W
Curved diffuser inlet width, m
W exit
Curved diffuser exit width, m Distance along diffuser, measured
X
from the reference location, m
Greek symbols
Holdup (Ratio of oil volume to emulsion volume)
Dynamic viscosity, kg / m.s
Kinematic viscosity, m2 / s
E / w
Emulsion/water viscosity.
E / w m
Emulsion/water density. Mercury density, kg / m3
w
Water density, kg / m3
o
Oil density, kg / m3
E
Mixture density, kg / m3
1 2
First bend angle / second bend angle, degrees
Subscripts 1
Diffuser first bend
2
Diffuser second bend
D
Diffuser condition.
E
Emulsion
XVIII
exit
Exit diffuser conditions
i
Inner wall (CC wall + CV wall)
inlet
Inlet diffuser conditions
I
Ideal
o
Outer wall (CV wall + CC wall)
O
Oil
p
Pressure
ref
Reference value
W
Water
x
Location value
List of abbreviations CFD
Computational Fluid Dynamics
EXP
Experimental data
FP
S-shaped diffuser flow path
o/w
(oil-in-water)
ow
Outer wall
iw
Inner wall
St.
Stable o/w emulsion
SDS
Sodium Dodeycl Sulfate
TA
S-shaped diffuser turn angle
Unst.
Unstable o/w emulsion
w/o
(water-in-oil)
WP
Wetted perimeter, m
List of chemical symbols CCL 4
Carbon Tetra Chloride
CH3 (CH2 )11 OSO3Na
Sodium Dodeycl Sulfate
XIX
Chapter (1)
Introduction
CHAPTER (1) INTRODUTION 1.1 Motivation The definition of a diffuser is a duct along which the mean static pressure of a flowing fluid increases as a result of decreasing kinetic energy of the flow without any external energy input. All turbo machines and many flow systems incorporate a diffuser such as the duct between the compressor and burner of a gas turbine engine, the duct following the impeller of a centrifugal compressor, and the vaned diffusers of centrifugal pump passages. Also diffuser minimizes total pressure loss in the aircraft engines by connecting the engine to the air intake S-diffusing duct. The S-shaped curved diffuser flow constitutes the most complex type of flows because the flow is diffusable, rotational and three dimensional.
1.2 Diffusers Types and Applications Diffusers can be classified according to many aspects for example, according to wall curvature to straight-walled and curved-walled diffusers. Also it can be classified according to flow speed to supersonic and subsonic flow diffusers. Turbomachinery flows are, in general, subsonic and the diffuser can be represented as a channel diverging in the direction of flow. Another classification due to cross-sectional area is conical, twodimensional and annular diffuser. Four basic types of diffuser are shown in Fig. 1.1 from which other special combinations such as the S-shaped diffuser can be made. A review of simple primary flows in straight, conical and annular diffusers is documented by Cockrell and Markland [1] and ESDU [2].
1
Chapter (1)
Introduction
(a) Straight Walled 2-D Diffuser.
(b) Straight Walled Conical Diffuser.
(c) Straight Walled Annular Diffuser.
(d) Curved Walled Annular Diffuser.
Fig. 1.1 Basic diffuser geometries [1, 2]
The static pressure recovery coefficient can be increased by 30% if a straight-walled diffuser is substituted instead of a curved-walled diffuser, see Figs. 1.2 and 1.3.
2
Chapter (1)
Introduction
(a) Bell-Shaped Diffuser.
(b) Trumpet-Shaped Diffuser.
Fig. 1.2 Curved Walled Diffusers [3]
Fig. 1.3 Inflected Walled Diffuser [4]
3
Chapter (1)
Introduction
The diffuser may have different performance when a curved-walled diffuser is associated with diffuser center-line curvature such as in S-shaped curved diffuser. The centrifugal pump has passages between its vanes look like S-diffusing ducts with diffusable, rotational and three dimensional flow, so the variations of these passages geometrical parameters will strongly affect the performance of the centrifugal pump, see Figs. 1.4 and 1.5.
1.3 Diffuser Performance Parameters Diffuser performance parameters are that indicating the performance for a given diffuser under different flow conditions. The followings parameters are some of these parameters. 1.3.1 Static Pressure Recovery Coefficient, (Cp): The static pressure recovery coefficient can be defined as in Miller [6] as, the ratio between the static pressure difference between any position (x) and reference position (ref) to the reference dynamic pressure ( ρE g × inlet dynamic head = 0.5ρE U2ref ).
P -P Cp = x ref2 0.5ρ E U ref
(1.1)
1.3.2. Ideal Static Pressure Recovery Coefficient, (Cp (I)): The ideal static pressure recovery coefficient is deduced from Bernoulli equation for ideal flow [6] by applying the continuity equation 1.2, to equation 1.1.
ρE Αinlet Vinlet = ρE Αexit Vexit
C p (I )
2 2 V inlet V exit
2 V inlet
2 U ref
(1.2) 2
U ref /AR 2
U ref
2
1
1 AR 2
(1.3)
Where ( AR ) is the area ratio of the S-shaped diffuser, AR = Aexit / Ainlet Combining eqns (1.1) and (1.3), the difference of diffuser pressure recovery coefficient ( ΔCp ) can be written as;
4
Chapter (1)
Introduction
Fig. 1.4 S-shaped Curved-walled Diffuser, Present work
View of the Passage Between Two Impellers
Operating or System Pressure
Intake or Drain
Fig. 1.5 Centrifugal Pump (right) and Similar Passage between Two Impellers (left) [5] 5
Chapter (1)
Introduction
ΔCp = Cp(exit) - Cp(inlet) = = Cp (I) -
ρ E g h loss 0.5 ρ E U ref 2
Pexit - Pinlet 0.5ρ E U 2ref = Cp (I) -
Po inlet - Po exit
(1.4)
0.5 ρ E U ref 2
Where Po inlet , Po exit are the total pressures at diffuser inlet and diffuser exit, respectively and hloss is the head loss. 1.3.3. Diffuser Efficiency (Effectiveness), ( ηd ): Diffuser efficiency is the ratio of actual static pressure rise coefficient to the ideal static pressure rise coefficient for a given diffuser [6]. Historically, diffuser performance is interrelated with that of turbo machinery, leading to the use of efficiency, ηd as a performance parameter since in practice factors such as wall friction, turbulence generation in the mainstream and flow separation in the most severe case, together mean that the ideal recovery will never be achieved. ΔCp ηd actual static pressure rise ideal static pressure rise
C p (I )
C p (I )
ρE g h loss 0.5 ρE U 2ref
C p (I )
1
h loss 2
U ref (1 AR 2 ) 2g
(1.5)
1.3.4. Diffuser Loss Coefficient, ( K d ): Diffuser loss coefficient is defined as the ratio of the loss in total pressure across a given diffuser to the reference dynamic pressure at the diffuser inlet [6].
Kd
h loss ρE g h loss Poinlet Poexit Cp (I) ΔCp 1 AR -2 + Cp(inlet) Cp(exit) 2 2 2 0.5 ρ U 0.5 ρ U Uref E ref E ref 2g
(1.6)
1.3.5. Inflow Reynolds Number, (Re): Inflow Reynolds number is defined as the ratio of the inertia force to the viscous force at the diffuser inlet. Reynolds number affects diffuser flows in two ways: Firstly, for a given diffuser geometry, variation in Reynolds number will affect the inlet boundary layer
6
Chapter (1)
Introduction
thickness and hence the diffuser loss. Secondly it will affect the growth rate of the boundary layer within the diffuser [7]. In the present study, the inflow Reynolds number is defined as:
Re =
ρE ×Uref ×DH μE
(1.7)
Where, DH is the hydraulic diameter that defined as:
DH 4 cross sectional area / wetted perimeter
1.4 Diffuser Flow Regimes The diffuser flow regimes are typically indicated in Fig. 1.6, see refs. [8] and [9]. This figure indicates four different flow regimes subjected to diffuser geometry which are "attached flow", "large transitory stall", "fully developed stall" and "jet flow". (I) In the attached flow regime, velocity and pressure distributions are steady. Through the diffuser, local axial velocity maintains positive values, i.e. no occurrence of reverse flow. (II) In the large transitory stall regime, the flow is unstable that gross fluctuations of the whole flow pattern are observed. Reversed flow regions are continually formed in and washed out of the diffuser, and large pressure fluctuations occur. (III) In the fully developed stall, the flow separates preferentially from one side wall or the other, and along this wall a stable reversed flow region is established. So that the main flow body passes along the other wall, and flow is relatively steady. (IV) In the jet flow regime, the flow separates very close to the diffuser corner so that large reversed flow regions are formed on both diffuser walls. Different diffuser flow regimes as in Fox et al. [10] are also shown in Fig. 1.7. The primary fluid mechanics problem of the diffusion process is caused by the tendency of the boundary layers to separate from the diffuser walls if the rate of diffusion is too rapid (short diffuser length). The result of too rapid diffusion is always large losses in stagnation
7
Chapter (1)
Introduction
pressure. On the other hand, if the rate of diffusion is too low, the fluid is exposed to an excessive length of wall and fluid friction losses become predominant. Clearly, there must be an optimum rate of diffusion between these two extremes for which the losses are minimized. Test results from many sources indicate that an included total divergence angle (2α) of about 2α = 7 degrees gives the optimum recovery (minimum losses) for both two-dimensional and conical diffusers as shown in Dixon [11]. In the rectangular cross-sectional area S-shaped curved diffusers the angle between the two curved diverged walls is equivalent to that of an equivalent straight walled 2-D diffuser which has the same area ratio and the same length, as shown in Fig. 1.8.b.
1.5 Secondary Flow in Curved Diffuser S-shaped diffuser flow is characterized by secondary flow effects caused by centrifugal forces acting normally to the flow direction. The formation of the secondary flows inside the S-shaped diffuser decreases the pressure recovery coefficients and therefore, increases the energy-loss coefficient according to Eq. (1.6). It is important to obtain a steady and symmetric flow downstream of the S-shaped diffuser (uniformity of S-diffuser flow). In an S-shaped curved diffuser, due to the presence of centerline curvature, fluid near the flow axis is acted upon by a larger centrifugal force than the fluid near the walls. This centrifugal pressure difference (transverse pressure gradient) forces the faster moving fluid to move outwards pushing the fluid in the boundary layer at the outer-wall around the sides towards the inner-wall; thus a significant secondary flow (normal to the primary flow direction) is produced. The high velocity fluid is shifted towards the outer wall due to the generation of secondary motion, which caused due to the combined effect of the centrifugal action of the centerline curvature and adverse pressure gradient, [12]. The main flow field in an S-shaped diffuser can be dominant by a pair of counter-rotating streamwise vortices formed by the cross-stream pressure gradient in the S-curved passage which balances the centrifugal force resulting from the turning of the fluid.
8
Chapter (1)
Introduction
(a) Diffuser flow regimes defined in [8]
(b) Diffuser flow regimes defined in [9] Fig. 1.6 Diffuser flow regimes with diffuser geometric parameters 9
Chapter (1)
Introduction
(a) Attached flow.
(b) Large transitory stall.
(c) Fully developed stall.
(d) Jet flow. Fig. 1.7 Diffuser flow regimes [10].
(a) Conical diffuser
(b) Straight walled 2-D diffuser
Fig. 1.8 Some diffusers and their geometric parameters as in Dixon [11].
10
Chapter (1)
Introduction
When the fluid enters S-curved passage, a pressure gradient is setting to provide the necessary inward accumulation since the sum of the kinetic energy and pressure energy is the same every where, velocities are decreasing from the inside to the outside of the Scurved passage. If the flow has a non-uniform velocity, the pressure gradient is insufficient for faster moving particles and more than sufficient for the slower ones. However, near the walls where the velocity is small, this pressure gradient along the curved walls (outer and inner walls) is not balanced by the created centrifugal force consequently the fluid is forced inward towards the central of curvature. Due to continuity, the moving central part of the flow is forced outward rapidly. The result is so called "secondary flow" in the perpendicular to the mean flow. If the curvature is significant, so that the secondary circulation is sufficiently strong and the longitudinal velocity profile will be completely altered with respect to a straight passage flow, therefore the secondary motions had a great influence on the flow development of the S-shaped diffuser as given in McMillan [13].
1.6 Design of S-Shaped Diffuser In general, diffusers are made in different shapes namely axial, radial, and curved to conform to the constraints imposed by aspects of design. S-shaped curved diffuser design and analysis are complicated compared with the other diffusers. It is very hard to obtain the optimized S-shaped diffuser design because there are several configurations of Sshaped diffusers. S-shaped diffuser parameters (geometric and dynamic parameters) control the diffuser performance and the turning angle plays an important role in improving the S-shaped diffuser performance. Also, there are parameters concerning the application in which the S-shaped diffuser will be used. Some of these parameters are those concerning (oil-in-water) two-phase flow through rectangular cross-sectional area S-shaped diffusers which will be the goal of the present work. Flow development in S-shaped curved diffusers of the present study is complicated and is influenced by the different geometrical parameters like:
11
Chapter (1)
Introduction
(1) Area ratio (AR = Aexit/Ainlet) (2) Aspect ratio (AS = B/W) (3) Curvature ratio (CR = RC/W) (4) Total divergence angle (2α) defined as the equivalent angle between the two diverging walls of a straight diffuser with the same length and area ratio as [14], see Fig. 1.8.b: 2α = 2 tan-1[W (AR-1)/2 L]
(1.8)
(5) Angles of turn of the centre line (θ1 and θ2) represent the values of the symmetric turn angles and the shape of the flow path inside the S-shaped diffuser. In case of asymmetric turn angle, inflection in the curvature is not at middle of the diffuser. In the present work the cross-sectional area of the S-shaped diffusers tested models were increased by linearly varying the width from 20 mm (kept constant for all models) at inlet to 30, 40 and 60 mm at the exit, over the total centre line length of 157, 236 and 393 mm while the height was kept constant at 80 mm. The width was equally distributed normal to the centerline. Design procedure of S-shaped diffuser has to find the following: • Dimensions of the S-shaped diffusers in mms. • Choose the S-diffuser type (symmetric / asymmetric) and the value of turn angle. • Choose the enlargement rate and the value of the S-diffuser equivalent angle. • Inlet and outlet conditions for the S-shaped diffuser and material to be fabricated. S-shaped diffuser dimensions include turn angle, width, height, area ratio, aspect ratio, diffuser length, centerline arc radius and the offset.
1.7 Oil-In-Water Two-Phase Flow through S-shaped Diffuser Under standard condition there are only four states of matter or phases; that is gas, liquid, solid and plasma phases. Multiphase flow is a quite common phenomenon that occurs both 12
Chapter (1)
Introduction
in nature and in technology. There are numerous examples of multiphase flow in industrial applications, for examples paper manufacturing, food manufacturing, solid rocket motor exhaust flows, liquid fuel combustion processes, chemical processes, lubrication systems, gas and oil extraction. The simplest case of multiphase flow is two phase flow. A two-phase flow is defined as a flow in which the mechanics of two phases with the interaction between them through their interface is considered. Ishii [15] classified the two-phase flows into four groups depending on the mixtures of phases in the flow. These groups are the flow of gas-liquid, gas-solid, liquid-solid and immiscible liquid - liquid mixtures. Two-phase flow is classified according to the geometry of the interface into three classes; separated flows, transitional or mixed flows and dispersed flows, as given in Table 1.1. Many experimental and numerical studies were dealed with different types of diffusers. Single phase flow and two-phase flows through curved diffuser were investigated without dealing with (oil-in-water) emulsion flow through S-shaped diffuser. In general, S-shaped diffuser flow is complex type of flows because of flow rotating due to diffuser curvature and the flow is 3D. These make both experimental and numerical studies were performed difficulty. Also, pressure gradients resulting from streamline curvature can produce significant secondary flows. Additionally, the adverse stream-wise pressure gradient, caused by increasing crosssectional area, can lead to flow separation, so S-shaped curved diffuser is a device that operates under an adverse pressure gradient which may generate complex flow. Flow separation in S-diffuser will cause flow distortion, which can result in high cycle fatigue, generated by unsteady dynamics loads. The S-shaped diffuser energy-loss coefficient under different flow and geometrical conditions needs however more investigations to be completed. Emulsion is a mixture of two or more immiscible liquids. One or more phases named as dispersed phase are dispersed in the other continuous phase which is known as dispersion 13
Chapter (1)
Introduction
Table 1.1 Different regimes for two-phase flows as reported in [15] Class Typical geometry configuration regimes Separated Film flow Liquid film in gas flow Gas film in liquid Annular Liquid film and gas flow core Gas film and liquid core
Mixed or transitional flow
Dispersed flow
examples Film cooling Film boiling Film boiling condensers
Jet flow
Liquid jet in gas Gas jet in liquid
Atomization jet condenser
Slug or plug flow
Gas pocket in liquid
Bubbly annular flow
Gas bubbles in liquid film with gas core
Sodium boiling in forced convection Evaporators with wall nucleation
Droplet annular flow
Gas core with droplet and liquid film
Steam generator
Bubbly droplet annular flow
Gas core with droplet and liquid film with bubbles
Boiling nuclear reactor channel
Bubbly flow
Gas bubbles in liquid
Chemical reactor
Droplet flow
Liquid droplet in gas Immiscible Liquid droplet in other Liquid
Spray cooling Oil-in-water emulsion
Particulate Flow
Solid particles in gas or Transportation liquid of coal, sand or wheat
14
Chapter (1)
Introduction
medium. The size of droplets has a significant impact on the stability and rheology of emulsion as described by Sjoblom [16]. In fact, the droplet size distribution in emulsion is a key element in determining the mass and heat transfer rates through interfacial areas in chemical, petroleum, mining, food, and pharmaceutical applications. Besides, it plays a crucial role in the design and scale-up of chemical reactors, mixers and separators, [17]. Population balance approach is generally employed to model the size distribution of the droplets, bubbles or crystals, which evolve and change with the flow due to phenomena such as nucleation, growth, coalescence, and breakage. Based upon the first stated liquid the emulsion type becomes defined because that liquid is the dispersed phase and the other is the dispersion medium. The two important sets of emulsions are an oil-in-water (o/w) emulsion and water-in-oil (w/o) emulsion. Emulsions are extremely fire resistance and highly incompressible with good cooling properties. Also, the two important types of oil-in-water emulsions are a stable o/w emulsion by using an emulsifier surfactant and unstable o/w emulsion without any additives. The specified size of the pump required to move the emulsion is controlled by determination of frictional energy loss in pipes, valves, tested models and fittings as well as the behavior of flow in the pump passages.
Injecting water into the crude oil in petroleum field to extract it easily is an important application and is one of widely used for the two-phase (o/w) emulsion flow. Emulsions are also used in many other important technological applications; used in industries, such as pharmaceutical, agriculture and food industries, used in transport of oils in pipelines, used in water treatment plants and used in chemical process systems. This thesis consists of eight chapters and five appendices: The chapters are: Chapter 1
Deals with thesis introduction and defines the various terms used in the other chapters as given above. 15
Chapter (1)
Introduction
Chapter 2 Contains the literature survey from which the objectives of the present work, can be extracted. Chapter 3 Describes and explains in details of the experimental setup and the measurements procedure used to obtain the experimental results. Chapter 4 Presents the physical and mathematical models of a single and two-phase flow (water flow and emulsion flow). Chapter 5 Describes the numerical methodology used to solve the governing system of equations for a single-phase and two-phase flows with boundary conditions. Chapter 6 Introduces validations of the current proposed code with some known test cases from the literature and the present experimental data for simulating (oil –in-water) emulsion flow in S-shaped diffusers. Chapter 7 Provides the overall experimental and numerical results obtained and discussions of their implications. Chapter 8 Concludes the main new findings of this thesis and provides the conclusions as well as recommendations for further studies in future work.
The appendices are: Appendix A Includes calibrations of the flow orifice meter. Appendix B Presents the measurements uncertainties with their calculation method. Appendix C Includes analytical study of S-diffuser. Appendix D Presents design calculations and equations of the S-diffuser tested models. Appendix E Provides a list of the papers that had been published or submitted for publication during this work.
16
Chapter (2)
Literature Review and Objectives
CHAPTER 2 LITERATURE SURVEY AND OBJECTIVES 2.1 Introduction The definition of an S-shaped diffuser is a duct through which the mean static pressure of a flowing fluid increases as a result of decreasing kinetic energy of the flow without any external energy input. The effective curved surface differs from the geometric curvature of the duct due to the growth or thinning of boundary layers. Flow on a concave bend increases the boundary layer thickness and vise versa on a convex surface. The increased boundary layer thickness on a concave surface reduces the effective radius of curvature and hence the velocity is greater than if the flow was inviscid. On a convex surface, although the boundary layer thickness is reduced as the flow accelerates, the boundary layer still effectively increases the radius of curvature and hence reduces the peak velocity compared to that of an ideal flow. This variation in velocity across the passage manifests as a static pressure variation which in turn affects the local surface flow diffusion rates and an adverse pressure gradient which may generate flow separation, depending upon the diffuser geometry and flow parameters. The review presented here covers the single and two-phase flows in diffusers. Dealing with S-shaped diffusers the literature review showed that previous studies were focused only on single phase flow in S-shaped diffusers. Review of diffuser flows is first presented followed by conclusions and objectives of the present study.
2.2 Literature Survey The ability to quickly predict complicated flow phenomena without experimental testing is obviously desirable for designers. Therefore, different studies were considered
17
Chapter (2)
Literature Review and Objectives
previously. In the present work a survey has been performed scanning the period from 1911 to 2016. The survey considers the interesting researches in the field of singlephase flow as well as (o/w) emulsion flow in straight diffusers as well as curved ducts / diffusers and pipes fittings in order to shed the main green light on the objectives of the present work. 2.2.1 Single Phase Flow Sagi and Johnston [18] studied the design and performance of curved diffusers. They concluded that for single phase air flow and at the same flow Reynolds number, the pressure coefficient recovery of the curved diffuser is lower than that of straight diffuser and the curved diffuser efficiency is lower than the straight diffuser efficiency. Patel [19] studied secondary flow in the boundary layers of a 180° channel. Some preliminary measurements were made in such flows by him during of a study on the curvature effects in nominally layers. He showed that: the vortices appear on the concave wall and also much stronger longitudinal vortices on the convex wall some distance from the duct corners are formed. These latter vortices are included by the curvature-driven secondary motion. He also, concluded that these vortices had to be reduced in order to realize two dimensionality of the flow along the duct plane. The flow behaviours through S-shaped ducts have been studied by Bansod and Bradshaw [20]. They found that; the streamwise favorable pressure gradient exists after the second bend. While the secondary flow separation of the boundary layer is generated on the far side of second bend for S-duct. So and Mellor [21] investigated experimentally the convex curvature effects in turbulent boundary layers. They noticed that, turbulent boundary layers development along a convex surface in a specially designed boundary-layer tunnel increase as curvature decreases. They also, concluded that the curvature causes the flow on convex surfaces to be stabilized with a reduction in turbulent shear stress.
18
Chapter (2)
Literature Review and Objectives
So and Mellor [22] investigated experimentally the behavior of a turbulent boundary layer on a concave wall. They revealed that the boundary layer at the minima positions is found to be twice as thick as that at the maxima positions. They also, concluded that the curvature causes the flow on concave surfaces to be destabilized with an increase in turbulent shear stress and the boundary layer is retarded and thickened. Taylor et al. [23, 24] studied the developing air flow in S-shaped ducts of square and circular cross-sections. They found that; the pressure-driven secondary flows arise in the first bend of the duct and near the outer-wall of the second bend (CC part of the second bend). These secondary flows increase the energy-loss coefficient and hence decrease the S-duct performance. Sullerey [25] performed a comparison between straight diffuser and curved diffuser. This comparison was experimentally performed under the same area ratio, effective divergence angle and air flow Reynolds number. He concluded that the straight diffuser performance is higher than that of curved diffuser and with increasing the free-stream turbulence the pressure recovery coefficient for each type increases. Azad and Burhanuddin [26] measured the wall shear stress in a diffuser air flow using a hot-wire anemometer. They carried out a correction due to the proximity of wall effects on a hot-wire measurements. They also, measured boundary layer parameters and turbulence intensity. Their results showed that the sublayer next to the wall exists in all flows (i.e., with different types of pressure gradients) and the velocity in this region is linear. They also characterized this layer which extends to a value of y+ approximately is equal to 5. Azad and Kassab [27] examined turbulent air flow in Azad diffuser throughout determination of the mean pressures, mean strain rates, energy, shear stress, length scale and balances of energy and shear stress. They observed that a retarded flow near the wall developed into a thick layer as the flow approaches the exit region of the diffuser. They also characterized the growth of the core region in the last one-third of 19
Chapter (2)
Literature Review and Objectives
the diffuser and they developed a method for measuring turbulent kinetic energy dissipation for fine scale eddies by extrapolating the dissipation calculated from the integration of spectra from different lengths of the hot wire to zero. Turbulent swirling air flows in diffusers were studied by Armfield and Fletcher [28]. They compared numerical results obtained for the class of moderate swirl in a 20° diffuser and for no swirl in an 8° diffuser with experimental results. Their mathematical model solved the full steady state time-averaged Navier–Stokes equations. They examined four turbulence models (two types of algebraic Reynolds stress model and two types of k-ε model). They found that the standard k-ε model gives poor prediction of the mean flow and therefore they used a modified form of k-ε model and the two algebraic Reynolds stress models. Belaidi et al. [29] carried out an experimental investigation in a 90° degrees curved rectangular duct to study the development of flow instability in such geometries. They concluded that, the streamwise position where the instability is first generated depends on the inflow Reynolds number and the curvature ratio. They also, found that as the curvature ratio of the bend decreases, the flow instability appears at smaller radial position for lower Reynolds number. The velocity and the pressure characteristics of the water flow in an S-shaped diffusing duct with asymmetric inlet conditions have been measured by Whitelaw and Yu [30] and simulated by Whitelaw and Yu [31]. Their results showed that rapid flow distortion is noticed in the second half of the S-duct. Turbulent flow velocity characteristics in a round cross-sectional S-shaped diffusing duct were measured by Whitelaw and Yu [32]. They found that; the region of separation is shown with two thicknesses of inlet boundary layers, and is larger with the thinner inlet boundary layer.
20
Chapter (2)
Literature Review and Objectives
Studies to clarify the flow behavior in S-shaped duct were introduced by Britchford et al. [33, 34]. They indicated that the flow within the S-shaped duct is significantly influenced by the effects of streamwise pressure gradients and flow curvature. Increasing the diffusing S-shaped duct curvature leads to minimize the losses and improve the performance. Experimental study was carried out on the developing water flow in a curved duct of rectangular-cross sectional area by Kim and Patel [35] to investigate the curved duct performance. The flow was turbulent and incompressible. They showed that: the longitudinal velocity peak is shifted towards the inner wall and turbulence is suppressed over the convex wall after 45º degrees from the starting of 90º degrees bend. The opposite occurs over the concave wall. Singh and Azad [36] measured the structure of instantaneous air flow reversals in a highly turbulent axisymmetric diffuser flow using pulsed-wire anemometry. They noticed that the adverse pressure gradient is strong enough to cause appreciable instantaneous flow reversals within Azad diffuser. They also found that the increase in entry Reynolds number leads to a decrease in the size of near-wall instantaneous reversals region as well as a decrease in the magnitude of instantaneous backflow. Bruns et al. [37] conducted an experimental investigation of a three-dimensional turbulent boundary layer in an S-shaped duct. They showed that the reverse of the curvature leads to the crossover of the transverse velocity component near the side walls. Gupta et al. [38] studied the effects of aspect ratio and curvature on flow characteristics in S-shaped diffusers using CFD. They showed that; the uniformity of longitudinal velocity and the pressure coefficient decrease with increase in the diffuser turn angle. A comparative study of RANS models of a three-dimensional turbulent flow in a rectangular cross-sectional area S-shaped channel was performed by Garbaruk et al. 21
Chapter (2)
Literature Review and Objectives
[39]. They found that; high ratings of the Spalart-Allmaras model, which are based on the results of calculations of two-dimensional flow. Numerical investigation using a three-dimensional Navier-Stokes flow solver has been performed to determine the availability of the minimization of an S-shaped duct adopting available experimental data as the inlet air-flow condition of the S-shaped duct introduced by Kim et al. [40]. They showed that the predictions agree reasonably well with the available measurements except near wall regions which are caused from the inaccuracy of the turbulence model. Thus, it was considered that the method can be applied as a tool for the design modification of the S-shaped ducts. Ng et al. [41] reported a study on an S-shaped square duct at high Reynolds numbers, and observed a flow separation at the first bend. A swirl was developed at the same location, but was attenuated at the second bend due to the formation of swirl of opposite direction. Additional flow features included the formation of a streamwise vortex or a pair of counter-rotating vortices on the outer wall of the second bend. Gopaliya et al. [42] performed computational studies on analysis of performance characteristics of S-shaped diffuser with offset. They concluded that; the pressure recovery coefficient (Cp) decreases with increase in offset. Increasing Reynolds number leads to increase of Cp. The pressure drop in sudden contractions was measured for Newtonian and nonNewtonian fluid under different flow Reynolds number by Fester et al. [43]. They showed that: (1) Contraction loss coefficient (Kcon) decreases with increasing contraction ratio (Ddownstream / Dupstream = Ratio of downstream pipe diameter to upstream pipe diameter). (2) The contraction loss coefficient is found to be inversely proportional to the inflow Reynolds number. Sinha et al. [44] used the computational fluid dynamics (CFD) to improve the performance characteristics of an S-shaped rectangular diffuser by momentum
22
Chapter (2)
Literature Review and Objectives
injection. They concluded that; the delay in separation results in improved pressure recovery coefficient and the total pressure loss coefficient decreases because of thinning the boundary layer on the CC part of the second bend (at the diffuser exit) by the momentum injection. El-Askary and Nasr [45] studied numerically and experimentally the flow in a combined rectangular cross-sectional area bend-diffuser system with straight-duct spacer length between them for different diffuser angles. A modified version of TEACH-code was used in numerical simulation and high-Re k-ε turbulence model enhanced with low-Re k- ε model for near wall treatment. Their study showed the effects of Re at different diverging angles and spacer lengths on the flow through diffuser. They found that: There is an optimum diverging angle which depends on the inflow Reynolds number and produces the minimum total pressure loss and hence good performance of such complex geometry is obtained. The value of this angle increased with increasing inflow Reynolds number and with increasing the spacer length. Good agreement between numerical and experimental data was obtained. CFD analysis of performance characteristics of S-shaped diffusers with combined horizontal and vertical offsets was conducted by Gopaliya et al. [46]. They concluded that; non-uniformity increases in case of S-shaped diffuser with rectangular outlet and with semi-circular outlet. Also increase in Reynolds number leads to increase in outlet pressure recovery for both types of diffuser under study. Sinha and Majumdar [47] investigated numerically and experimentally the flow through annular curved diffusers. They concluded that; the pressure coefficient increases sharply with area ratio and the standard k-ε model is the suitable one for simulation. Abdellatif et al. [48] performed a computational study of centerline turning angle effect on the turbulent flow through a diffusing S-duct using large-eddy simulation. They concluded that: separation regions are characterized with high turbulence levels and
23
Chapter (2)
Literature Review and Objectives
occur near the convex wall of the S-duct first bend and increase by the increase of turning angle, which hardly affects the pressure coefficient. Biswas et al. [49] investigated the flow in a constant area rectangular curved duct. They concluded that; the minimum and the maximum of wall pressure occur at the inner and the outer walls respectively under influence of curvature radius and turn angle of the curved duct. Also flow at exit is purely non-uniform in nature due to the strong secondary motion. Experimental and computational studies on S-duct diffuser air flow were conducted and various combinations of submerged-vortex generators (SVG) were tested by Paul et al. [50]. They concluded that; as skewed inflow condition, the outflow of the S-duct diffuser is distorted and reduces the duct performance so using submerged-vortex generators makes the flow more uniform at diffuser exit. The effect of momentum imparting on different performance characteristics of Sshaped diffuser has been studied by Gopaliya et al. [51]. They concluded that: momentum imparting increases the exit static pressure coefficient of the S-shaped diffusers and the non-uniformity at diffuser exit decreases by placing a rotating cylinder across the S- diffuser width. 2.2.2 Two-Phase / Emulsion Flow The hydraulic resistances of emulsion (oil-in-water) flow across orifice-meter, venturimeter, sudden contraction and sudden enlargement were measured by Nasr [52]. She did an experimental study to compare pressure drop across these geometries under different Reynolds number and different oil concentration by weight. She showed that under the same flows Reynolds number; increasing the oil concentration causes increasing in the energy losses. Under the same oil concentration; increasing flow Reynolds number causes a decreasing in energy-loss coefficient.
24
Chapter (2)
Literature Review and Objectives
Pal and Rhodes [53] investigated experimentally the laminar and turbulent flow behaviours of stable emulsions in horizontal pipeline. They noticed that; up to a dispersed phase concentration of 55.14% by volume, emulsions are Newtonian. For concentration of 65.15% o/w emulsions are non-Newtonian power-law fluids. They concluded that friction factor for stable o/w emulsion follows the usual equations of single-phase Newtonian and non-Newtonian fluids with averaged properties. They also pointed out that for dispersed phase concentration > 50% stable o/w emulsions exhibit drag reduction in turbulent flow, i.e.; the experimental friction factors fall somewhat below the single – phase equation. An experimental study concerning the laminar and turbulent pipe flow behaviors of unstable and stable (w/o) and (o/w) emulsions was performed by Pal [54]. He showed that; the unstable (o/w) emulsions exhibit drag reduction depending upon the holdup of the dispersed phase. The unstable (w/o) emulsions exhibit much stronger drag reduction than the unstable o/w emulsions. The stable emulsions exhibit relatively little drag reduction. Ibrahim and El-Zawahry [55] studied the flow of emulsion (oil-in-water) through a centrifugal pump. They showed that: the centrifugal pump can pumps emulsion fluid and it becomes unable to generate the same heads and maximum discharge as that obtained in the case of water flow. They also, found that the amount of oil in the mixture and its viscosity play an important role in the head degradation process. Buhidma and Pal [56] studied the behavior of oil-in-water emulsions through a wedge meters and segmental orifice meters of various shapes and sizes. They found that a wedge meters and segmental orifice meters are feasible flow metering devices for oilin-water emulsions and also the discharge coefficient is constant over a wide range of Reynolds number.
25
Chapter (2)
Literature Review and Objectives
Nadler and Mewes [57] investigated experimentally the flow of two immiscible liquids (oil-water mixture) in a horizontal pipe. They concluded that; (1) for the flow conditions, maximum pressure drops are found in the region of phase inversion which is observed for input water fraction between 10 and 20 %, and (2) there is no significant effect of the temperature on the flow characteristics observed. Hwang and Pal [58] studied experimentally the flow of two–phase (o/w) mixtures through sudden expansions and contractions under different dispersed phase concentrations by volume. They found that; the loss coefficient for emulsions is independent of the dispersed phase concentration (emulsion holdup) and the type of emulsions (emulsion stability). The performance of pipe bends for oil/water mixture was experimentally determined by Hwang and Pal [59]. They indicated that single-phase calibration curves of discharge coefficient can be used to predict the flow rate of a homogeneous two-phase oil/water mixture. Turian et al. [60] studied the losses due to flow of concentrated non-Newtonian slurries in bends. They found that the resistance coefficients for non-Newtonian suspension flow through bends of various angles and radii of curvature decrease with increasing Reynolds number in laminar flow. However, constant asymptotic values of resistance were approached in fully developed turbulent flow. Pal and Hwang [61] performed an experimental study on the energy-loss coefficients for flow of stable (o/w) emulsions through pipe components (valve, expansion and contraction). They showed that; emulsions are Newtonian only at low concentrations of dispersed phase. Also, the frictional losses can be successfully correlated as loss coefficient versus Reynolds number. Kassab et al. [62] performed an experimental study for flow of water with surfactant additives through sudden enlargements, sudden contractions and short bends. They
26
Chapter (2)
Literature Review and Objectives
showed that the reduction in secondary losses increases by increasing of both the surfactant concentration and Reynolds number and/or decreasing pipe diameter. Langevin et al. [63] studied the crude oil emulsion properties and their application to heavy oil transportation. They found that; the dispersing heavy oil or bitumen in water is a very efficient way to reduce the viscosity of the fluid by more than 2 orders of magnitude. Thus, formation of oil-in-water emulsions can be used to reduce the viscosity in pipelines and hence to reduce the energy losses and hence the power required to transmit the oil in pipelines. Modeling friction factor and the drag reduction behavior of the unstable w/o emulsions in turbulent flow in a horizontal pipeline were studied by Ismail [64]. He showed that: the drag reduction is considered to be due to two reasons, (1) the damping of turbulence due to the presence of dispersed phase, and (2) due to breakage and coalescence processes. He also, obtained correlations to describe the function in droplet mean diameter and emulsion holdup. The correlations were found to fit in good agreement with the experimental results of Pal [54]. A study of oil-in-water emulsion flow through pipeline using image analysis technique was performed by Khalil et al. [65]. They concerned with studying the influence of various parameters on the characteristics of stable and unstable oil-in-water emulsion. They showed that: (1) Increasing emulsion holdup causes increasing in emulsion viscosity, increasing oil droplet diameter, decreasing discharge coefficient and increasing energy-losses coefficient. (2) The energy-loss coefficients of 90° bend, fully opened gate valve, sudden enlargement and sudden contraction were arranged according to the big value respectively. (3) The resistance coefficient of pipe fittings for emulsion flow is lower than that for water flow. They also, indicated that; for holdup lower than 50% the o/w emulsion flow is Newtonian flow and for holdup higher than 50% the o/w emulsion flow is non-Newtonian flow.
27
Chapter (2)
Literature Review and Objectives
Khalil et al. [66] measured the friction losses of oil-in-water emulsions flow through pipes. They concluded that: the pipe friction factor decreases as holdup increases. Hammoud et al. [67] studied the effect of oil-in-water concentration on the performance of a centrifugal pump. They showed that the centrifugal pump head and efficiency decrease by increasing the oil concentration. They also, found that the increasing in oil concentration leads to increase of the power consumption. The pressure drop of (o/w) emulsion flow through sudden contraction and expansion in a horizontal pipe was measured by Balakhrisna et al. [68]. They showed that: the abrupt change in area during oil-water flow through pipe strongly affects on the phase distribution and the energy-loss coefficient is independent on the flow patterns. They also, indicated that: Contraction and expansion coefficients are found to be lower for the oil-in-water flow in comparison to the water flow though the same test rigs. Khalil et al. [69] studied the energy losses of oil-in-water emulsions flow through pipe fittings using image processing. They found that the energy loss coefficient increases as the emulsion holdup increases and the flow rate decreases. They also, concluded that the energy loss coefficient is found to be inversely proportional to the inflow Reynolds number. The flow through sudden expansion has been numerically investigated with two-phase oil-in-water emulsions by Manmatha and Laxman [70]. They found that; the expansion loss coefficient is independent of the Reynolds number and is not significantly influenced by the type and concentration of (o/w) emulsions. Also, they indicated that; the effect of viscosity is negligible on the pressure drop and the pressure drop increases with higher mass flow rate. Experimental (o/w) emulsion flow investigation of 90°curved diffuser was conducted by El-Askary et al. [71] and Omara H. A. [72]. They concluded that: (1) the outer-wall
28
Chapter (2)
Literature Review and Objectives
pressure coefficient is higher than that of the inner-wall. (2) The 90° curved diffuser energy-loss coefficient, Kd, is found to be inversely proportional to the inflow Reynolds number, diffuser area ratio and diffuser curvature ratio. (3) The unstable o/w emulsion exhibits higher values in pressure recovery coefficients, compared with stable o/w emulsion, while the stable and unstable o/w emulsions are lower than that of water. They also, found that the increasing of the emulsion holdup leads to increase the diffuser energy-loss coefficient and to decrease the emulsion flow discharge coefficient. ANSYS R15.0 FLUENT documentation [73] recommends that: for most complex flows, the first-order scheme may need to be used to perform the first little iterations and then switch to the second-order scheme to continue the calculation to convergence. The QUICK (Quadratic Upstream Interpolation for Convection Kinetics) discretization scheme is reported to only provide marginal improvement on the second-order scheme for rotating flows. Due to its practical importance, the two-phase flows across area expansions have been exhaustively examined with numerous simulations. As an example, for bubbly flows with high void fractions, Behzadi et al. [74] modified the standard Eulerian approach in two phase flow modeling to account for interface forces in mixtures. They also proposed a modified version of k turbulence model which was able to predict bubbles interaction with the flow eddies more efficiently through sudden expansion. Ibrahim et al. [75] computationally extended the work of Behzadi et al. [74] by considering various turbulence models. It was concluded the SST (k ‒ ω) turbulence model produces the most accurate results for the two phase flow through sudden expansions. El-Askary et al. [76] also used successfully the standard k ‒ ε turbulence model as an initiation step in simulating turbulent two-phase flow due to hydrogen production process.
29
Chapter (2)
Literature Review and Objectives
Accurate predictions and measurements of oil-in-water flow characteristics in S-shaped ducts, such as flow pattern, oil holdup and pressure gradient are important in many engineering applications. However, despite of their importance, emulsion flows have not been explored to the same extent as gas-liquid or gas-solid flows. The focused literature review showed the importance of studies performed on straight, S-shaped and curved diffusers. In spite of such importance, the second type (S-shaped diffuser) has not been sufficiently taken into consideration, either experimental or numerical. However, no previous studies have been performed on emulsion flows in S-shaped diffusers.
2.3 Conclusions of the Literature Survey The previous compressive literature review on single-phase flow through S-shaped curved diffusers and (o/w) mixture two-phase flow through pipe and pipe fittings indicates that:1. The flow in S-shaped curved diffusers under different geometric and dynamic parameters is not sufficiently studied. 2. The emulsion (oil-in-water) fluid flow through S-shaped curved diffusers has not previously studied, (for the author's knowledge). 3. The pressure recovery coefficient and efficiency of the straight diffuser are higher than that of the curved diffuser. 4. The single-phase flow behavior in an S-shaped curved diffuser is influenced by inflow Reynolds number, flow swirl, diffuser curvature ratio, diffuser area ratio and diffuser turn angle. 5. The emulsion holdup (Φ) has a great effect on the o/w emulsion two-phase flow behaviors.
30
Chapter (2)
6.
Literature Review and Objectives
The concave wall has higher pressure distribution than that of the convex wall because of the concave wall has a destabilizing effect on turbulent flow, while the convex wall has a stabilizing effect.
7.
Among all RANS turbulence models, the family of k-ε turbulence model can be used to predict the flow behavior through Pipe fittings and diffusers.
8.
The multi-phase model (Eulerian-Eulerian) model is widely used for simulation of pipe fittings such as sudden expansion (in most cases one-dimensional model).
9.
The diffuser performance depends not only on the physical shape of the duct but also upon the type of boundary conditions especially the inflow conditions as solution initialization.
10. Emulsions were (oil-in-water) type up to 64% oil (volumetric percentage) while above this ratio the emulsions were (water-in-oil) type. 11. The reduction in secondary losses increases by increasing both of oil concentration and Reynolds number and/or decreasing pipe diameter. 12.
Increasing emulsion holdup causes an increase in each of emulsion viscosity, oil droplet diameter as well as causes a reduction in each of the energy loss coefficient and the orifice discharge coefficient.
13. The pipe fittings energy-loss coefficient of emulsion flow is lower than that of water flow. 14.
The centrifugal pump performance (head and efficiency against the discharge) decreases and the pump power consumption increases as emulsion holdup increases.
2.4 Objectives of the Present Study The literature review showed that previous studies were focused on single phase flow (air or water) through S-shaped diffusers. Unlike the case of single flow, no work has
31
Chapter (2)
Literature Review and Objectives
been published in the field of emulsion flow in S-shaped diffusers. Consequently, this thesis aims at studying oil-in-water emulsion flow through S-shaped diffusers to investigate the effects of geometric and dynamic parameters on the S-shaped diffuser performance. Experimental and numerical results on the energy-loss coefficient of Sshaped diffusers carrying stable and unstable (o/w) emulsions are reported in this thesis. Water and stable/unstable (o/w) emulsions with different oil concentrations by volume in range of: 0.03 0.25 are the working fluids used in the present experimental and numerical studies. Experiments are carried out on twelve models of S-shaped diffusers with different area ratio, curvature ratio, turning angle and flow path at different Sdiffuser inflow Reynolds numbers of range: 9000 Re 34000 . Based on the measured and predicted wall static pressure distributions; the S-shaped diffuser loss coefficient can be extracted and discussed under different inflow and geometrical conditions. As indicated previously, none of the previous works studied the emulsion flow in S-shaped diffuser (to the researcher's knowledge) which is an essential component in many flow systems. So, the outflow quality of the S-shaped diffuser at diffuser exit plane is the concern of the present study. The energy-loss is a key issue when designing S-shaped diffuser since with the increase in computer resources of Computational Fluid Dynamics (CFD) becomes a powerful and useful tool for engineers and designers. Therefore, the goal of the present work is to provide experimental and numerical studies on water and emulsion flows through Sshaped diffusers including the effects of different geometrical and inflow parameters on the S-diffuser performance and a validated computational study that can be used for predicting the S-shaped diffuser performance and the two additional affects (computational axial velocity and turbulent shear stress contours). The following points, however, describes the main outlines of the present work to achieve this goal: 1-
Designing and fabricating twelve S-shaped diffuser tested models.
32
Chapter (2)
2-
Literature Review and Objectives
Forming and testing the two types of o/w emulsions flow (stable and unstable o/w emulsion flow) through S-shaped diffusers to show the stability effect on the diffuser performance under different conditions.
3-
Developing a mathematical model that includes the effects of different oil concentrations on S-shaped diffusers performance. The model assumptions do not consider the effects of droplet size, coalescence and breakage.
4-
Validating the computational model with the available previous experimental data from the literature and with the experimental data obtained from the current study.
5-
Performing computational and experimental studies to investigate the effects of different parameters on the performance of S-shaped diffusers and to validate the present experimental results as following: a. Effect of S-shaped diffuser area ratio, Aexit /Ainlet. b. Effect of S-shaped diffuser curvature ratio, RC / W. c. Effect of S-shaped diffuser symmetric turn angle, θ1= θ2= θ. d. Effect of S-shaped diffuser flow path (symmetric /
asymmetric turn angle), θ1 / θ2. e. Effect of ratio of oil volume to emulsion volume (holdup), Φ. f. Effect of inlet flow Reynolds number, Re EU ref DH / E . g. Effect of o/w emulsion status (stable/unstable).
6-
Performing computations to estimate different parameters that cannot be simply measured in the laboratory such as streamline contours at different locations along the diffuser length.
7-
Concluding the main findings of the present work as well as recommendations generated from the computational and experimental studies for the future work.
33
Chapter (3)
Experimental Set-Up and Measurements
CHAPTER 3 EXPERIMENTAL SETUP AND MEASUREMENTS 3.1 Introduction The experimental work in the present study is carried out to provide experimental data for extensive model validation. This will be achieved by constructing a test rig including S-shaped diffusers tested models in the fluid mechanics laboratory of the Mechanical Power Engineering Department, Faculty of Engineering Minoufiya University. The principal objective of this work is to study the effect of the major controlling parameters that have direct effects on the S-shaped diffuser performance (S-diffuser energy-loss coefficient). These parameters are: the inflow Reynolds number, emulsion holdup and emulsion status as well as the effect of S-diffuser area ratio, curvature ratio, turn angle and S-diffuser flow path. The uncertainties of measurements are determined according to Taylor [77]. The procedures to be followed during the experiments are identified; the emulsions density, viscosity and the calibration of the orifice-meter are discussed. Twelve Sshaped diffusers tested models made of steel are tested. The models increase linearly in the width from constant 0.020 m at inlet to 0.030, 0.040 and 0.060 m at the exit, over the total centre line length of 0.157 m, 0.236 m and 0.393 m while the height was kept constant at 0.080 m. Therefore, the aspect ratio was kept constant at inlet as 4.0. The width was equally distributed normal to the centerline.
3.2 Experimental Apparatus The experimental apparatus and the instruments required to perform the experimental measurements are illustrated in Fig. 3.1, while Fig. 3.2 shows a photograph of the experimental apparatus. Experimental measurements are performed to investigate the S-diffuser performance when carrying water as well as unstable / stable (o/w) emulsion flows. This setup is designed, constructed and available to perform tests under different geometrical and inflow conditions. The setup is equipped with flow
34
Chapter (3)
Experimental Set-Up and Measurements
velocity measuring instrument (orifice flow meter) and multi-tubes manometer for all static pressure measurements as well as pressure gauges to perform preliminary tests on flow through straight pipes and studying the performance of the used pump under different emulsion statuses. The description of the main components of the circuit is given in details in the following sub-sections.
3.2.1 Emulsion -Supply Line The test section is supplied with oil-in-water emulsion through the emulsion circuit (Fig. 3.2) that consists of an emulsion main tank (1) of 0.5 m3 capacity, a suction pipe (2), a suction pressure gauges (3), a delivery pipe (4), a delivery pressure gauge (5), a centrifugal pump (6), a flow control valve (7), an orifice flow meter (8), Utube manometer (9), a converter (10), an upstream tangent duct (11), S-diffuser test section (12), a downstream tangent duct (13), a multi-tube manometer (14), a mixer electric motor (15), and calibration tank (16), see Figs. 3.1 and 3.2. The emulsion is supplied to the test rig from emulsion main tank passing through the hydraulic circuit using the centrifugal pump that delivers the emulsion through the delivery pipe. The emulsion then flows through orifice-meter, converter, upstream tangent duct, S-diffuser test section, downstream tangent duct and then to the emulsion main tank again. The flow field average velocity is controlled by adjusting the flow control valve and the considered emulsion flow rate (hence, inflow Reynolds number) can be obtained. The inflow Reynolds number is determined by measuring the flow rate using the orifice flow meter, equipped with a 16.5 mm diameter sharp-edged orifice plate placed in the delivery pipe. The pressure drop across the calibrated orifice plate is measured using a mercury U-tube manometer, from which the flow rate can be determined according equation (A.1).
3.2.2 Fabrication of S-Shaped Diffusers Tested Models Twelve models of rectangular cross-sectional area S-shaped diffuser of outer and inner curved diverged side walls and top and bottom flat parallel walls. The diffusers are designed and fabricated from steel sheets of 3 mm thickness with smooth inner surfaces.
35
Chapter (3)
Experimental Set-Up and Measurements
Details of the mounted S-shaped diffuser tested model (designed in full scale)
Suction Valve
Fig. 3.1 (a) Schematic view of the experimental apparatus (Not to scale)
36
Chapter (3)
Experimental Set-Up and Measurements
Fig. 3.1 (b) Physical test model of S-diffuser Fig. 3.1 Experimental setup
37
Chapter (3)
Experimental Set-Up and Measurements
1. Main tank
2. Suction pipe
3. Suction pressure gauges
4. Delivery pipe
5. Delivery pressure gauge
6. Centrifugal pump
7. Control valve
8. Orifice-meter
9. U-tube manometer
10. Converter
11. Upstream tangent duct
12. S-diffuser model
13. Downstream duct
14. Multi-tube manometer
15. Mixer electric motor
16. Calibration tank
Fig. 3.2 Photograph of the test rig
38
Chapter (3)
Experimental Set-Up and Measurements
(a)
(b)
(c) A Photograph shows curved diverged side-walls of S-diffusers and measuring taps of models 1&9 (a), 2&4 (b) and 3 (c) Fig. 3.3 Photograph of the S-diffusers 1, 2 and 3
39
Chapter (3)
Experimental Set-Up and Measurements
Wooden templates have been built for S-shaped diffusers geometry based on the detailed drawings to assist in the construction assembly as well as to give diffuser geometry with accurate dimensions. The wooden template for each S-shaped diffuser is fixed to the bottom base plate steel sheet. Then, the side walls are shaped along the template side surfaces and clamped strongly with it. The curved side walls sheets are welded manually (i.e., permanently) with the bottom base plate sheet and the top wall is clamped down into the curved side walls, and then they are welded. Two steel flanges were welded at the inlet and the exit of the S-shaped diffusers tested models, see Figs. 3.1.b and 3.3. The test section components are the straight upstream tangent duct and the straight downstream tangent duct which have a constant height equal to the diffuser height ( B ) of 0.080 m (i.e., the distance between the parallel top and bottom walls of the curved diffuser) and a constant cross-sectional width. All S-diffusers tested models have a constant inlet cross-sectional width of 0.020 m and a constant height of 0.080 m. Every steel S-shaped diffuser model consists of two curved and diverged side walls [CV→CC (outer wall) and CC→CV (inner wall)] and the other walls, top and bottom are flat and parallels due to the constant height, see, Fig. 3.1.b. Sixty four static pressure taps of 1 mm internal diameter are drilled normal to the wall and located along the mid-height line of both outer (o) and inner (i) straight side walls of the upstream and downstream tangent ducts as well as the outer and inner curved side walls of the S-shaped diffuser for measuring the static pressure distributions on these walls. The wall static pressure taps are spaced at an equal interval of 9° degrees in models of 45°/ 45º and 90°/ 90° and 10° degrees in models of 60°/ 60°, 60°/ 30°, and 30°/ 60° as shown in Figs. 3.1, 3.3 and D.1. The pressure recovery coefficient (CP), which is defined as the actual static-pressure rise of the diffuser to the inlet dynamic pressure, is calculated from measured static pressures. The twelve models are tested for unstable and stable o/w emulsion flow conditions. A fully developed velocity profile at diffuser entrance can be established by using a long upstream tangent duct of length 75 times the diffuser inlet width (1.50 m). Also, a downstream tangent duct of length 55 times the diffuser inlet width (1.10 m)
40
Chapter (3)
Experimental Set-Up and Measurements
is used at the S-shaped diffuser exit to sufficiently develop the flow. Emulsion on leaving the downstream duct is finally discharged into the main emulsion tank. The number of the static pressure taps on both outer and inner walls of the S-shaped diffuser and its components is symmetry (i.e., 32 taps on outer walls and 32 taps on inner walls). Photographs of the flow hydraulic circuit and three S-shaped diffusers tested models are shown in Figs. 3.2 and 3.3, respectively, for clear representation. The tested models are numbered from model 1 (M1) to model 12 (M12). Models 1&9, 2&4 and 3 have area ratio ( A R Aexit / Ainlet ) of 3.0, 2.0 and 1.5 respectively, the same curvature ratio (CR Rc /W ) of 7.5 and the same turn angle of 45°/45°, these tested diffusers have constant corresponding length of 0.236 m. Models 4, 5&10 and 6 have the same area ratio of 2.0, the same turn angle of 45°/45° and curvature ratios of 7.5, 12.5 and 5.0, respectively and have lengths of 0.236 m, 0.393 m and 0.157 m, respectively. Model 7 has area ratio of 3.0, turn angle of 60°/60°, curvature ratio of 7.5 and length of 0.283 m. Model 8 has area ratio of 3.0, turn angle of 90°/90°, curvature ratio of 7.5 and length of 0.472 m. Model 11 has area ratio of 2.0, turn angle of 60°/30°, curvature ratio of 12.5 and length of 0.393 m. Model 12 has area ratio of 2.0, turn angle of 30°/60°, curvature ratio of 12.5 and length of 0.393 m. for more details, see Appendix D, Figs. D.1 and D.2 show the designs details of twelve S-shaped diffusers models and the locations of static pressure taps. Fig. 3.1.b shows a (not-to-scale) physical model of the S-shaped diffuser tested (models 1&9) geometry and Table 3.1 shows the tabulated data of the all S-shaped diffusers tested models.
3.2.3 The Specifications of the Hydraulic Circuit Parts Referring to Fig. 3.2 the main emulsion tank (1) has a rectangular cross-section. It has height of (1.0 m), length of (0.80 m) and width of (0.60 m) and it is made from 6 mm steel sheet thickness. The suction pipe installed between the tank exit and the pump entrance has a diameter of 1.0 inch (25.4 mm) and total length of 3.20 m.
41
Chapter (3)
Experimental Set-Up and Measurements
Table 3.1 Geometrical parameters of the S-shaped diffusers models, for all models: W=20 mm, B=80 mm and all dimensions in mm.
Model
θ1 / θ2
Rc
W exit
CR
AR
2α
SExit W
(1&9)
45°/45°
150
60
7.50
3.0
9.7°
11.8
(2&4)
45°/45°
150
40
7.50
2.0
5°
11.8
(3)
45°/45°
150
30
7.50
1.5
2.5°
11.8
(5&10)
45°/45°
250
40
12.50
2.0
3°
19.64
(6)
45°/45°
100
40
5.00
2.0
7.5°
7.9
(7)
60°/60°
150
60
7.50
3.0
8°
14.14
(8)
90°/90°
150
60
7.50
3.0
5°
23.60
(11)
60°/30°
250
40
12.50
2.0
3°
19.64
(12)
30°/60°
250
40
12.50
2.0
3°
19.64
Two pressure gauges (parts (3.a) and (3.b)) are installed in the suction pipe (2) with a distance of 2.80 m between each other, in order to determine the pressure drop in the straight pipe with different o/w emulsion flow conditions. A centrifugal pump (part 6) is used to circulate the water / emulsion flow in the circuit. Preliminary measurements of friction losses of o/w emulsions flow through pipes are considered. The general specifications of the used centrifugal pump and pump performance parameters as manufacturer's data are given in details in Table 3.2. The delivery pressure gauge (part 5) is installed in the delivery pipe (4) at a distance of (0.20 m) from the pump exit. It is used to measure the delivery pressure exit from the pump as shown in Fig. 3.2. An orifice-meter (part 8) is installed in the delivery pipe at a distance of (2.90 m) from the pump exit to measure the flow rate via the
42
Chapter (3)
Experimental Set-Up and Measurements
measured pressure difference using U-tube manometer (part 9). The details of the orifice-meter will be presented in subsection 3.6.2 (Measuring Devices) by using the calibration tank (16) and the mixer (15) in case of emulsion flow. A converter (part 10) is used to convert the circular cross-section of the delivery pipe to a rectangular cross-section of the upstream tangent duct (11) to make it adapted with the shape of the S-diffuser entrance. A multi-tube manometer (part 14) is used to measure the pressure along outer and inner walls of the tested S-shaped diffuser (part 12) followed by a downstream tangent duct (part 13). A flow control valve (part 7) is used to control the flow in the circuit. Table 3.2 General specifications of the used centrifugal pump as manufacturer's data Type
N.D25 160/150
Units
Power
2
hp
Speed
2900
r.p.m
Head
20:31
mw
Capacity
7 : 15
m3water / hr
3.3 The S-Diffuser Influencing Parameters (Parametric Study) In the present study, seven important parameters are considered: (1) the S-diffuser area ratio ( A R Aexit / Ainlet ), (2) the S-diffuser curvature ratio (CR Rc /W ), (3) the S-diffuser turn angle and (4) the S-diffuser flow path as well as (5) the Sdiffuser inflow Reynolds number, (6) the emulsion holdup (Φ) and (7) the emulsion status (stable / unstable).
3.3.1 S-Diffuser Area Ratio Different exit widths (W exit ) of 30, 40 and 60 mm are considered for controlling the
area
ratio
and
the
reference
location
(position
zero
point;
x y z 0mm ) is taken at the diffuser entrance on the inner-wall. Measurements are performed on three S-diffusers with different area ratios:
A R 3.0, 2.0 and 1.5 for models 1, 2 and 3, respectively (CR 7.5 ).
43
Chapter (3)
Experimental Set-Up and Measurements
These measurements are recorded for the static-pressures distributions on both the outer and inner walls of the models and upstream and downstream tangents, therefore the corresponding S-diffuser energy loss coefficient can be deduced according to the equation C.5 in the Appendix C.
3.3.2 S-Diffuser Curvature Ratio Measurements are performed also on three S-diffusers for constant area ratio of 2.0 and with different curvature ratio: CR 12.5, 7.5 and 5.0 for models 4, 5 and 6 respectively. These measurements are recorded for the static-pressures on the outer and inner walls of the S-diffuser and its components, therefore the corresponding energy loss coefficients can be computed.
3.3.3 S-Diffuser Turn Angle Measurements are also carried out at a constant curvature ratio of CR=7.5 and AR=3.0 for three S-diffusers with different turn angles: 45°/45°, 60°/60° and 90°/90°, for models 9, 7 and 8, respectively.
3.3.4 S-Diffuser Flow Path Three S-diffusers with different two turn angles but the sum of them for each model is 90° i. e., (θ1+ θ2= 90°) are tested to study the effect of the flow path on the diffuser performance.
3.3.5 S-Diffuser Inflow Reynolds Number To study the effect of flow rate (i.e., different inflow Reynolds number) on the diffuser flow field characteristics, the flow rate is controlled to give different values of Re at the inlet of the S-diffuser. Reynolds number is based on the bulk flow rate (flow averaged velocity) and the hydraulic diameter (DH) at the diffuser inlet. The diffuser flow field measurements are performed at different Reynolds numbers range of 9,000 Re 34,000, where Re EU ref D H with D H 2WB where W B E
E is the viscosity of the fluid (water or emulsion). Measurements are recorded for
44
Chapter (3)
Experimental Set-Up and Measurements
the static-pressure distributions on the outer and inner walls of the S-diffuser and its tangents to study the effect of the Reynolds number on the pressure coefficients.
3.3.6 Holdup (Φ) and stability Measurements are also conducted at holdup ( ) of 0.03, 0.06, 0.10, 0.15, and 0.25 (values of oil concentrations by volume), to study the effect of emulsion holdup on the performance of S-diffuser models at different Reynolds number and different conditions for unstable o/w emulsion flows. Measurements are conducted on all models with unstable then stable o/w emulsions at different concentrations to investigate the effect of emulsion stability on the performance of the S-shaped diffuser by using an ionic surfactant (chemicalemulsifier) namely Sodium Dodecyl Sulfate, SDS, [ CH3(CH2 )11 OSO3Na ], which is added to the oil with 1.5% based on the water weight that exists in the mixture.
3.4 Measurement Locations The details of measurement locations of the static pressure taps of the S-shaped diffusers and their tangents to the diffuser inlet and to the diffuser exit are depending on the S-diffuser geometry and are outlined as shown in Tables from 3.3 to 3.9. Various measurement locations have been termed L-1, L-2, etc. along the geometry side wall centerline of each S-diffuser and its tangents geometry. These measurements are recorded for twelve S-diffuser models, with the same straight upstream tangent duct which has constant aspect ratio (ratio of height to the width) of AS=4 and AR=1.0. Three different straight downstream tangent ducts which have constant area ratio of AR=1.0 and different aspect ratios of AS=1.333, 2 and 2.67, respectively. The first downstream duct working with any model of AR=3 and the second downstream duct working with any model of AR=2 whereas, the third downstream duct only working with model (3).
45
Chapter (3)
Experimental Set-Up and Measurements
Table 3.3 Tap locations of upstream and downstream tangent ducts of all models (a) Upstream tangent duct of all S-diffuser models Tap Location No. S (mm) S/W -1 L.1 -20 -40 -2 L.2 -80 L.3 -4 -110 -5.5 L.4 -150 -7.5 L.5 -210 -10.5 L.6 -290 -14.5 L7 -400 -20 L.8 (b) Downstream tangent duct of S-diffuser models (1), (7), (8) and (9) o. Tap Location N S (mm) S/W 12.78 L.1 255.6 276.6 13.78 L.2 315.6 L.3 15.78 355.6 17.78 L.4 415.6 20.78 L.5 495.6 24.78 L.6 595.6 29.78 L7 715.6 35.78 L.8 855.6 42.78 L.9 1015.6 50.78 L.10 1195.6 59.78 L.11 (c) Downstream tangent duct of S-diffuser models (2), (4), (5) (10), (11) and (12)
Tap Location No.
S (mm)
S/W
L.1 L.2 L.3 L.4 L.5 L.6 L7 L.8 L.9 L.10 L.11
413 433 473 513 573 653 753 873 1013 1173 1353
20.64 21.64 23.64 25.64 28.64 32.64 37.64 43.64 50.64 58.64 67.64
46
Chapter (3)
Experimental Set-Up and Measurements
(d) Downstream tangent duct of S-diffuser model (3)
Tap Location No.
S (mm)
S/W
L.1 L.2 L.3 L.4 L.5 L.6 L7 L.8 L.9 L.10 L.11
255.6 276.6 315.6 355.6 415.6 495.6 595.6 715.6 855.6 1015.6 1195.6
12.78 13.78 15.78 17.78 20.78 24.78 29.78 35.78 42.78 50.78 59.78
Table 3.4 Tap locations of S-diffuser models (1), (2), (3), (4) and (9)
S-diffuser models (1), (2), (3), (4) and (9) Tap Location S (mm) S/W θ1º/ θ2º No. L.1 L.2 L.3 L.4 L.5 L.6 L7 L.8 L.9 L.10 L.11
0.0
0.0
0 º/ 0 º
23.6
1.18
0 º/ 9 º
47.1
2.36
0 º/ 18 º
70.7
3.54
0 º/ 27 º
94.3
4.72
0 º/ 36 º
118
5.89
0 º/ 45 º
141.4
7.07
45º/9º
165
8.25
45º/18º
188.6
9.43
45 º/27 º
212.1
10.61
45 º/36 º
235.7
11.79
45 º/45 º
47
Chapter (3)
Experimental Set-Up and Measurements
Table 3.5 Tap locations of S-diffuser models (5) and (10)
S-diffuser diffuser models (5) and (10) Tap Location S (mm) S/W θ1º/ θ2º No. L.1 L.2 L.3 L.4 L.5 L.6 L7 L.8 L.9 L.10 L.11
0.0 1.97
0 39.3 78.6 118 157.1 196.4 235.7 275 314.3 353.6 393
3.93 5.89 7.86 9.82 11.79 13.75 15.72 17.68 19.64
0 º/ 0 º 0 º/ 9 º 0 º/ 18 º 0 º/ 27 º 0 º/ 36 º 0 º/ 45 º 45º/ 9º 45º/ 18º 45 º/ 27 º 45 º/ 36 º 45 º/ 45 º
Table 3.6 Tap locations of S-diffuser model (6)
S-diffuser diffuser model (6) Tap Location No.
S (mm)
S/W
θ1º/ θ2º
L.1 L.2 L.3 L.4 L.5 L.6 L7 L.8 L.9 L.10 L.11
0 15.7 31.4 47.1 63 78.6 94.3 110 125.7 141.4 157.1
0 0.79 1.57 2.36 3.15 3.93 4.72 5.5 6.29 7.07 7.86
0 º/ 0 º 0 º/ 9 º 0 º/ 18 º 0 º/ 27 º 0 º/ 36 º 0 º/ 45 º 45º/ 9º 45º/ 18º 45 º/ 27 º 45 º/ 36 º 45 º/ 45 º
48
Chapter (3)
Experimental Set-Up and Measurements
Table 3.7 Tap locations of S-diffuser model (7)
S-diffuser diffuser model (7) Tap Location No.
S (mm)
S/W
θ1º/ θ2º
L.1 L.2 L.3 L.4 L.5 L.6 L7 L.8 L.9 L.10 L.11 L.12 L.13
0 23.6 47.1 70.7 94.3 118 141.4 165 188.6 212.1 235.7 259.3 283
0 1.18 2.36 3.54 4.72 5.89 7.07 8.25 9.43 10.61 11.79 12.97 14.14
0 º/ 0 º 0 º/ 10 º 0 º/ 20 º 0 º/ 30 º 0 º/ 40 º 0 º/ 50 º 0 º/ 60 º 60º/10º 60º/20º 60º/30º 60º/40º 60º/50º 60º/60º
Table 3.8 Tap locations of S-diffuser model (8)
S-diffuser diffuser model (8) Tap Location No.
S (mm)
S/W
θ1º/ θ2º
L.1 L.2 L.3 L.4 L.5 L.6 L7 L.8 L.9 L.10 L.11
0 47.1 94.3 141.4 188.6 235.7 283 330 377.1 424.3 471.4
0 2.36 4.72 7.07 9.43 11.79 14.14 16.5 18.86 21.22 23.57
0 º/ 0 º 0 º/ 18 º 0 º/ 36 º 0 º/ 54 º 0 º/ 72 º 0 º/ 90 º
49
90º/18º 90º/36º 90º/54º 90º/72º 90º/90º
Chapter (3)
Experimental Set-Up and Measurements
Table 3.9 Tap locations of S-diffuser models (11) and (12)
S-diffuser models (11) and (12) Tap Location No. L.1 L.2 L.3 L.4 L.5 L.6 L7 L.8 L.9 L.10
S (mm)
S/W
(θ1º / θ2º)M11
(θ1º / θ2º)M12
0 43.7 87.3 131 174.6 218.3 262 305.6 349.2 393
0 2.18 4.37 6.55 8.73 10.91 13.10 15.28 17.46 19.64
0 º/ 0 º 0 º/ 10 º 0 º/ 20 º 0 º/ 30 º 0 º/ 40 º 0 º/ 50 º 0 º/ 60 º
0 º/ 0 º 0 º/ 10 º 0 º/ 20 º 0 º/ 30 º
60º/10º 60º/20º 60º/30º
30º/10º 30º/20º 30º/30º 30º/40º 30º/50º 30º/60º
3.5 Measured Data 3.5.1 Measurements of Holdup Two different sets of emulsions are prepared using tap water and the available oil to use. The used mineral oil is low viscosity colorless, tasteless and odorless highly refined paraffinic oil, supplied by CO-OP Company, Alexandria, Egypt. It has a density of 850 kg/m3 and a kinematic viscosity limits from 13×10-6 to 19×10-6 m2/s (i.e., from 13 to 19 centi Stoke) and dynamic viscosity limits from 11×10-3 to 16×103
Pa.s (i.e., from 11 to 16 centi Poise).
In the first set of emulsions, no chemical-emulsifier (surfactant) is added so that unstable emulsion is produced. The unstable emulsion will be separated into oil and water if left without agitation (mixer) for sometime. The experiments in this set begin with tap water into which a required volume of oil varies from 0.03 to 0.25 of the emulsion volume. The holdup is defined and measured as:
Holdup
V o V o V o V w VE
(3.1)
Where, Vo is the volume of oil, Vw is the volume of water, and VE is the total (emulsion) volume. In the second set of emulsions an ionic surfactant namely
50
Chapter (3)
Experimental Set-Up and Measurements
Sodium Dodecyl Sulfate (SDS) is added to the oil as stated before in section 3.3.6. In all experiments, the mixing tank (main tank) contains a fixed volume of water (240 liter water) for 0.03, 0.06, 0.10, 0.15, and 0.25 oil concentration by volume (holdup ), the amount of paraffinic oil needed to make these concentrations are 7.5,
15.5,
27.0,
42.5,
and
80.0
liter
oil,
respectively
calculated
as
V o / (1 )V w .
3.5.2 Measurements of Wall Static Pressure The static pressure distribution is measured using 1 mm diameter pressure taps locating along the mid-height between the upper and lower parallel walls (top and bottom walls) of the S-diffuser and its tangents. All distances along the S-diffuser center-line length are normalized to the S-diffuser inlet width ( W ). The static pressure tap on the inner wall at diffuser entrance is used as a reference position (x = y = z = 0.0 mm) and for the calculation of the local pressure recovery coefficient, Cp, the static pressure tap on the outer wall at diffuser entrance is used as a reference pressure (Pref) and the velocity reference is the velocity at the diffuser entrance (Uref). The pressure taps are connected to the multi-tube manometer by using rubber hoses. The manometer has a manifold and the first and last tubes in the manifold are used for filling all tubes by the manometer fluid (Carbon Tetra Chloride) at reasonable one level and to extract air from the plastic tubes and from the system. Air is extracted from the Mercury U-tube (used with the orifice meter) by using air vents at the highest points of the Mercury U-tube. For measuring the pressure difference between every tap and the reference pressure the heights of manometer fluid in all tubes are recorded. From these data the local and overall pressure recovery coefficients of the S-shaped diffusers tested models are obtained.
3.5.3 Measurements of Energy Loss Coefficient Measurement of the total pressure is needed at inlet and outlet of the S-shaped diffuser tested models to determine the diffuser performance characteristics in terms
51
Chapter (3)
Experimental Set-Up and Measurements
of the S-diffuser loss coefficient ( K d ). The S-diffuser loss coefficient is determined for all geometrical and inlet conditions according to equation (C.5) see Appendix C.
3.5.4 Measurements of Flow Rate The flow rate is measured in all experiments using a calibrated flow orifice meter as stated in details in Appendix A, in which the discharge coefficient and flow properties at different statuses are registered.
3.6 Measuring Instruments The measuring instruments used during the experiments are classified as pressure and flow rate measuring devices.
3.6.1 Pressure Gauges Vacuum pressure device gauge is used to measure the suction pressure (-ve) before the pump at a range from -1.00 bars to 0.00 bars. Delivery pressure gauge (+ve) is used to measure delivery pressure exit from the pump with range from 0.00 bars to 6.00 bars. Multi-tubes manometer is used to measure the pressure distributions along the outer and inner walls, including the upstream tangent duct, tested diffuser and downstream tangent duct. The used manometer fluid is Carbon tetra chloride (CCl4). This liquid does not mix with water as well as the emulsions and has specific gravity of 1.6. By using this liquid the static pressure at each tapping hole can be measured by the difference-head of the used measuring liquid.
3.6.2 Orifice-Meter It is a device installed in the supply duct as shown before in Fig. 3.2 used for measuring the water as well as the emulsion flow rate passing through the system, by measuring the static head difference across the orifice using Mercury U-tube manometer. The orifice meter was calibrated for both water and emulsion flow. The calibration procedure is given in Appendix A. To measure emulsion flow Reynolds number, (Re) the density and viscosity of water and stable and unstable oil-in-water
52
Chapter (3)
Experimental Set-Up and Measurements
emulsions are measured in the laboratories of faculty of sciences, Minoufiya University by using calibrated densitometer and viscometer as stated in the Appendix A.
3.7 Test Section Conditions All tests are carried out using water flow medium and o/w emulsion flow medium at different holdups. All S-diffusers with different area and curvature ratios are tested at different flow Reynolds number and with unstable and stable o/w emulsions flow. Table 3.10 gives all test conditions carried out in the present study. The runs repeated with different turn angles and different flow paths by the same way.
3.8 Preliminary Measurements: Flow through Pipes As shown in Fig. 3.2 the two pressure gauges that mounted on the suction pipe are used with Darcy equation to measure the friction losses in the pipe. The friction coefficient is obtained in case of water flow and compared with Moody chart as shown in Fig. 3.4. The inflow Reynolds number effect on the friction coefficient at different holdup is presented in Fig. 3.5. The holdup effect on the friction coefficient at emulsion flow compared with Ref. [64] is clearly shown in Fig. 3.6. Also, Fig. 3.7.a & b have been drawn to show respectively, the holdup and stability effects on the friction coefficient in case of emulsion flow. Pipe friction loss coefficient decreases as Reynolds number increases as shown in Fig. 3.4 and the friction factor approach the values of Moody chart. As Reynolds number and holdup increase the pipe friction loss coefficient decreases as shown in Fig. 3.5. From Fig. 3.6 the friction factor approaches the values measured by khalil et al. [66] at different oil concentrations.
53
Chapter (3)
Experimental Set-Up and Measurements
As noticed in Fig. 3.7, the friction factor decreases with the presence of oil and with increasing the value of oil in water. However, the stabilized emulsion has higher values of friction loss coefficient compared with the unstable (o/w) emulsion. Table 3.10 Specifications of the S-shaped diffuser tested runs (a) Water flow Run No.
Fluid
Re
AR
CR
Model
Run-1
Water
0.00
9000 R e 34000
3.0
7.50
(1)
Run-2
Water
0.00
34000
3.0, 2.0, 1.5
7.50
(1, 2, 3)
Run-3
Water
0.00
34000
2.0
12.5, 7.5, 5
(5, 4, 6)
Run-4
Water
0.00
9000 R e 34000
2.0
12.5, 7.5, 5
(5, 4, 6)
Run-5
Water
0.00
9000 R e 34000
3.0, 2.0, 1.5
7.50
(1, 2, 3)
Run-6
Water
0.00
9000 R e 34000
3.0, 2.0, 1.5
12.5, 7.5, 5
(1, 2, 3, 5, 6 )
(b) Unstable (o/w) emulsions flow with Φ = 0.03 and 0.06 Run No.
Fluid
Re
AR
CR
Model
Run-7
Unstable o/w
0.03
9000 R e 34000
3.0
7.50
(1)
Run-8
Unstable o/w
0.03
34000
3.0, 2.0, 1.5
7.50
(1, 2, 3)
Run-9
Unstable o/w
0.03
34000
2.0
12.5, 7.5, 5
(5, 4, 6)
Run-10
Unstable o/w
0.03
9000 R e 34000
2.0
12.5, 7.5, 5
(5, 4, 6)
Run-11
Unstable o/w
0.03
9000 R e 34000
3.0, 2.0, 1.5
7.50
(1, 2, 3)
Run-12
Unstable o/w
0.03
9000 R e 34000
3.0, 2.0, 1.5
12.5, 7.5, 5
(1, 2, 3, 5, 6 )
Run-13
Unstable o/w
0.06
9000 R e 34000
3.0
7.50
(1)
Run-14
Unstable o/w
0.06
34000
3.0, 2.0, 1.5
7.50
(1, 2, 3)
Run-15
Unstable o/w
0.06
34000
2.0
12.5, 7.5, 5
(5, 4, 6)
Run-16
Unstable o/w
0.06
9000 R e 34000
2.0
12.5, 7.5, 5
(5, 4, 6)
Run-17
Unstable o/w
0.06
9000 R e 34000
3.0, 2.0, 1.5
7.50
(1, 2, 3)
Run-18
Unstable o/w
0.06
9000 R e 34000
3.0, 2.0, 1.5
12.5, 7.5, 5
(1, 2, 3, 5, 6 )
54
Chapter (3)
Experimental Set-Up and Measurements
(c) Stable (o/w) emulsions flow with Φ = 0.03, 0.06, 0.10, 0.15 and 0.25 Run No.
Fluid
Re
AR
CR
Model
Run-19
Stable o/w
0.03
9000 R e 34000
3.0
7.50
(1)
Run-20
Stable o/w
0.03
34000
3.0, 2.0, 1.5
7.50
(1, 2, 3)
Run-21
Stable o/w
0.03
34000
2.0
12.5, 7.5, 5
(5, 4, 6)
Run-22
Stable o/w
0.03
9000 R e 34000
2.0
12.5, 7.5, 5
(5, 4, 6)
Run-23
Stable o/w
0.03
9000 R e 34000
3.0, 2.0, 1.5
7.50
(1, 2, 3)
Run-24
Stable o/w
0.03
9000 R e 34000
3.0, 2.0, 1.5
12.5, 7.5, 5
(1, 2, 3, 5, 6 )
Run-25
Stable o/w
0.06
9000 R e 34000
3.0
7.50
(1)
Run-26
Stable o/w
0.06
34000
3.0, 2.0, 1.5
7.50
(1, 2, 3)
Run-27
Stable o/w
0.06
34000
2.0
12.5, 7.5, 5
(5, 4, 6)
Run-28
Stable o/w
0.06
9000 R e 34000
2.0
12.5, 7.5, 5
(5, 4, 6)
Run-29
Stable o/w
0.06
9000 R e 34000
3.0, 2.0, 1.5
7.50
(1, 2, 3)
Run-30
Stable o/w
0.06
9000 R e 34000
3.0, 2.0, 1.5
12.5, 7.5, 5
(1, 2, 3, 5, 6 )
Run-31
Stable o/w
0.10
9000 R e 29000
3.0
7.50
(1)
Run-32
Stable o/w
0.10
29000
3.0, 2.0, 1.5
7.50
(1, 2, 3)
Run-33
Stable o/w
0.10
29000
2.0
12.5, 7.5, 5
(5, 4, 6)
Run-34
Stable o/w
0.10
9000 R e 29000
2.0
12.5, 7.5, 5
(5, 4, 6)
Run-35
Stable o/w
0.10
9000 R e 29000
3.0, 2.0, 1.5
7.50
(1, 2, 3)
Run-36
Stable o/w
0.10
9000 R e 29000
3.0, 2.0, 1.5
12.5, 7.5, 5
(1, 2, 3, 5, 6 )
Run-37
Stable o/w
0.15
9000 R e 24000
3.0
7.50
(1)
Run-38
Stable o/w
0.15
24000
3.0, 2.0, 1.5
7.50
(1, 2, 3)
Run-39
Stable o/w
0.15
24000
2.0
12.5, 7.5, 5
(5, 4, 6)
Run-40
Stable o/w
0.15
9000 R e 24000
2.0
12.5, 7.5, 5
(5, 4, 6)
Run-41
Stable o/w
0.15
9000 R e 24000
3.0, 2.0, 1.5
7.50
(1, 2, 3)
Run-42
Stable o/w
0.15
9000 R e 24000
3.0, 2.0, 1.5
12.5, 7.5, 5
(1, 2, 3, 5, 6 )
Run-43
Stable o/w
0.25
9000 R e 24000
3.0
7.50
(1)
Run-44
Stable o/w
24000
3.0, 2.0, 1.5
7.50
(1, 2, 3)
Run-45
Stable o/w
0.25 0.25
24000
2.0
12.5, 7.5, 5
(5, 4, 6)
Run-46
Stable o/w
0.25
9000 R e 24000
2.0
12.5, 7.5, 5
(5, 4, 6)
Run-47
Stable o/w
0.25
9000 R e 24000
3.0, 2.0, 1.5
7.50
(1, 2, 3)
Run-48
Stable o/w
0.25
9000 R e 24000
3.0, 2.0, 1.5
12.5, 7.5, 5
(1, 2, 3, 5, 6 )
55
Chapter (3)
Experimental Set-Up and Measurements
0.04
0.036
f
0.032
0.028
0.024
Water Moody Chart (/D=0.0065) C.I 0.02 30000
40000
50000
60000
70000
Re Fig. 3.4 Pipe friction loss coefficient against Reynolds number at water flow 0.04 Re, Effect 31300 39000 45000 53000 60000 66000 70000
0.036
f
0.032
0.028
0.024
0.02
0
0.05
0.1
0.15
Holdup,
0.2
0.25
Fig. 3.5 Reynolds number effect on the friction factor at different holdup
56
Chapter (3)
Experimental Set-Up and Measurements
0.04 Re, Effect at Stable o/w emulsion present study Khalil et al. [66]
0.036
f
0.032
0.028
0.024
0.02 30000
40000
50000
60000
70000
Re (a) 0.15 Stable (o/w) emulsion 0.04 Re, Effect at 0.10and 0.25 Stable o/w emulsion 0.10 0.10 0.25 0.25
0.036
of of of of
present study Khalil et al. [66] present study Khalil et al.) [66]
f
0.032
0.028
0.024
0.02 30000
40000
50000
60000
70000
Re (b) 0.10 and 0.25 Stable (o/w) emulsion Fig. 3.6 Reynolds number effect on the friction factor at emulsion flow compared with [66]
57
Chapter (3)
Experimental Set-Up and Measurements
0.04 Holdup
Effect Water
0.03 0.06 0.10 0.15 0.25
0.036
f
0.032
0.028
0.024
0.02 30000
40000
50000
60000
70000
Re (a) Holdup effect at stable (o/w) emulsion flow 0.04 Stability Effect Water 0.03 Stable 0.03 Unstable 0.06 Stable 0.06 Unstable
0.036
f
0.032
0.028
0.024
0.02 30000
40000
50000
60000
70000
Re (b) Stability effect at Φ = 0.03 and 0.06 Fig. 3.7 Holdup effect (a) and stability effect (b) on the friction factor at emulsion flow
58
Chapter (3)
Experimental Set-Up and Measurements
3.9 The Experimental Procedure In the present experimental study it is required to study the effects of flow conditions on the S-shaped diffusers (emulsion flow Reynolds number, emulsion holdup and emulsion stability). In addition, the static pressure along both Sdiffuser and diffuser tangent ducts up and down side walls are measured. The experimental procedure starts with the preparation of the experimental apparatus and measuring instrumentation, then o/w emulsion preparation and running the experiments. The o/w emulsion from the main tank has to be suctioned and delivered by passing through the test rig using a centrifugal pump. The (oil-in-water) emulsion enters through the upstream tangent duct to the test section (twelve S-shaped tested diffuser models) with measured flow Reynolds number at the diffuser entrance, then finally downstream tangent duct discharging into the main tank again. To facilitate the testing requirements, the following procedure is followed: 1-
The main tank is filled with a certain amount of water (240 liter).
2-
A certain amount of oil is added to the water. The concentration is then calculated to obtain a certain oil concentration by volume (holdup)
3-
Mixing propellers are then driven in order to obtain a homogeneous emulsion, i.e., obtaining mixture with constant holdup in all parts.
4-
Pump valves are opened, inlet control valve to the main experimental block is fully closed, pump is started and the control valve is adjusted to provide the required flow rate by opening it gradually until maximum flow Reynolds number is reached
5-
The air is extracted from the each branch part in the system, using the air vents mounted on the highest point of the tapping holes.
6-
The air is extracted from the plastic tubes and the U-tubes, using the air vents at the highest points of the U-tubes. Thus, the experimental apparatus and measuring devices are ready to carry out the required experiments.
59
Chapter (3)
7-
Experimental Set-Up and Measurements
The apparatus is completely cleaned after using each used holdup and the preparation of the apparatus and measuring devices are made as stated before for the new tested holdup with all diffusers.
8-
With unstable o/w emulsion (no-surfactant), emulsion is circulated to the experimental facility to conduct tests of diffuser performance characteristics. For each run, measurements of the static pressures along upstream duct, S-shaped diffuser and downstream duct at a constant Reynolds number are obtained for calculating the local and overall pressure recovery coefficients. The corresponding S-shaped diffusers energy loss coefficients are recorded.
9-
With holdup 0.00 (pure water), 0.03, 0.06, 0.10, 0.15 and 0.25 respectively, stable (with-surfactant) o/w emulsions are circulated to the experimental facility to obtain the pressure recovery and energy loss coefficients for all curved diffusers models under previous holdup and different Reynolds number as stated before.
10-
The measurements of the suction and delivery pressures and emulsion flow rate of the used centrifugal pump are conducted using pressure gauges and orifice-meter respectively at the previous different holdup values.
60
Chapter (4)
Physical and Mathematical models
CHAPTER 4 PHYSICAL AND MATHEMATICAL MODELS 4.1. Introduction In this chapter, the physical and mathematical models of single and two-phase flows are described. For the last three decades, Computational fluid dynamics (CFD) have been utilized as the most powerful design and simulation tool. CFD provides a reasonable numerical solution to the equations that govern fluid motion. These solutions are related to the physics of the considered problem by governing equations and models. These governing equations represent the mathematical statement of the physical laws associated with the problem considered. So robust understanding of these equations and models is essential and prerequisite. The main commercially available CFD code packages on the market are: FLUENT, CFX, STAR CD, FLOW 3D, PHOENIX. FLUENT version 6.3 and ANSYS FLUENT (includes ANSYS CFD-Post) version R 15.0. The latest CFD code was used in the present study, for simulation of the water / (oil-in-water) emulsion flows
through S-shaped diffusers to study the effects of flow and geometrical parameters on the performance of the S-diffuser. This simulation is applied using three different turbulence models are included;
standard k (STD k ), standard k (STD k ) and shear stress transport k (SST k ) as well as three schemes of the two-phase flow are used:
Volume of fluid, Mixture model, and Eulerian scheme. This chapter presents the governing equations of the selected solution scheme and turbulence models extensively.
4.2 Physical model Numerical strategies for effective computational simulations of two-phase flows contribute to the solution of industrial problems in nowadays engineering to great
61
Chapter (4)
Physical and Mathematical models
extent. To successfully simulate an o/w emulsion two phase flow, it is inevitable to first have working tools for the separated simulation of two-phase flow. A schematic of the physical model of the present problem is shown in Fig. 4.1. The studied models consist of two bends with different angles, curvature radius from 100 to 250 mm and different area ratios. The dimensions of all tested models in the present study are clearly given in the previous chapter. The physical domain of the S-shaped diffuser is composed of four sections as shown in Fig.4.1. The first section is an entrance flat straight rectangular section to diffuser inlet (upstream tangent duct), then two curved diverged side-wall sections (the two bends of S-diffuser), and flat straight rectangular section at diffuser outlet (downstream tangent duct) for flow recovery. To represent the emulsion flow behavior in the physical domain, two phase mixture of oil and water is considered. The phases are assumed to share space in proportion to their existence probabilities such that their volume fractions sums to unity in the flow field. Computational fluid dynamics (CFD) technique is used to predict the behavior of the flow in S-shaped curved side walls diffusers using different models. For a two phase fluid, the incompressible Navier–Stokes equations supplemented by a suitable turbulence model are appropriate for modeling the flow in an S-diffuser. The fundamental transport equations for mass and momentum conservation in 3-D Cartesian co-ordinate systems are solved in this study. Assuming steady and incompressible turbulent fluid flow, Reynolds averaged continuity and Navier– Stokes equations without body forces can be expressed as the main governing equations of flow in a three-dimensional domain.
4.3 Formulation Methods of Two-Phase Flow The different phases are treated mathematically as interpenetrating continua. Since the volume of a phase cannot be occupied by the other phases, the concept of phase volume fraction is introduced. These volume fractions are assumed to be continuous functions of space and time and their sum is equal to one. Conservation equations for each phase are introduced as a set of equations, which have similar
62
Chapter (4)
Physical and Mathematical models
structure for all phases. These equations are closed by providing constitutive relations that are obtained from empirical information's by application of kinetic theory. In general, two formulation methods are possible in describing two-phase flows. These formulations actually represent fundamental ways of thinking about such flows. The first is the particle-source-in-cell method where the dispersed phase is treated from a Lagrangian point of view in which the individual particles are tracked. The continuous phase is seen from an Eulerian point of view with the effect of the dispersed phase entering through source terms in the conservation equations. This method, also known as the EulerianLagrangian formulation, is physically intuitive but is not computationally practical for other than very dilute dispersed phases. The second method is the two-fluid model in which each phase is seen from an Eulerian point of view. The degree to which this description is accurate depends on how dispersed and dense each phase. Although it is not as accurate as the first method, the two-fluid model is the only alternative for practical computations when the dispersed phase is dense.
63
Chapter (4)
Physical and Mathematical models
Fig. 4.1 Schematic of S-shaped diffuser test physical model (Not to scale)
64
Chapter (4)
Physical and Mathematical models
In the present study, CFD simulations are carried out using ANSYS R-15.0 with software Fluid Flow Fluent (FFF) which applies 3D-RANS (Reynolds Averaged Navier-Stokes equations) method. The current CFD code (ANSYS R 15.0) provides us by three main models currently used in two-phase flows; Euler (Eulerian) model, mixture model and volume of fluid (VOF) model. Theoretically, the two-phase flow problem can be solved by any one of the three models. However, neither the Euler nor the volume of fluid models can be used due to the huge number of oil droplets which consumes extra -ordinary long computing time, so the mixture two-phase model is adopted in the current study, [73].
4.4 Mathematical Model 4.4.1 Mixture Two-phase Flow Model A first approximation to model emulsion flow behavior is to consider it as twophase Newtonian fluid. The first step in mathematical modeling is to decide which two-phase flow model can be used. The choice of the model depends on the type of problem of interest. The present problem can be considered as a mixed flow case in a relatively short path. There are four main models that can be used for two-phase flow simulation; Euler model, mixture model, volume of fluid model and Lagrange model. Theoretically, the two-phase flow problem can be solved by any one of the four models. Previous publications in literature have considered the emulsion as a pseudo-homogeneous fluid with suitably averaged properties as the dispersed droplets of emulsion are small and well dispersed this is for a range of holdup of 0.50 [70]. Consequently the mixture model is the chosen model in the present study. The mixture two-phase model is designed for two or more phases (fluid or particulate) and solves the mixture momentum equations and prescribes relative velocities to describe the dispersed phases. As in the Eulerian model, the phases are treated as interpenetrating continua. The effects like aggregation, breakage and coalescence phenomena will not be analyzed in the present study.
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The mixture mathematical model is based on solver proposed by [73]. The mixture two-phase model utilizes one set of momentum equations for both phases. Mixture model is referred to as: a- mixture model is used to model droplets of secondary phase that dispersed in continuous fluid phase (primary phase). b- it allows for mixing and separation of phases. c- it solves continuity and momentum equations for two phases as mixture. d- multiple species and homogeneous reactions in two phases (mixture).
Model Assumptions For mixture model simplicity, the following considerations are adopted: 1- The flow model is considered as a three dimensional oil/water mixture twophase flow, steady, and turbulent flow. 2- The flow direction is horizontal; so the gravitational acceleration is ignored. 3- The fluids in both phases (continuous, primary, phase and dispersed, secondary, phase) are Newtonian and incompressible. 4- The physical properties remain constant. 5- The surface tension forces are neglected, therefore, the pressures of both phases are equal at any cross section, [70]. 6- The flow is assumed to be isothermal; so energy equation is not needed. 7- Slip-shear lift and drag force are neglected. 8- The pressure is assumed to common to both phases. 9- The effects like aggregation, breakage, distribution size and coalescence phenomena will not be analyzed.
4.4.2 Fluid flow modeling In two-phase flows, the governing equations in both phases have to be considered according to the nature of single-phase Newtonian flow. By assuming three- dimensional, isothermal, incompressible flows, the governing equations of two- phase flows may be written in the following formulation:
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(A) Continuity equation of the mixture Model
. ρm v m 0
(4.1)
Where v m is the mixture mass-averaged velocity, [73] and ρm is the mixture density, the mixture velocity and density can be evaluated from;
k 1 n
v m
k
ρm
ρk Vk
(4.2)
n
ρm k ρk
(4.3)
k 1
With Φk is the volume fraction of phase k .
(B) Momentum equation of the mixture Model T ρm v m v m P μeff,m v m v m F
.
. kρk vdr,k vdr,k
.
n
k 1
(4.4)
Where n is the number of phases, F is a body force,
μeff,m is
the effective
turbulent viscosity of the mixture and v dr,k is the drift velocity of secondary phase k, which can be written as:
μeff,m μ m μt,m
(4.5)
vdr,k vk vm
(4.6) n
Where, μ m is the mixture measured viscosity ( μ m k μ k ) and k 1
the mixture turbulent viscosity that needs a suitable turbulence model.
67
μ t, m is
Chapter (4)
Physical and Mathematical models
(C) Volume Fraction Equation of the Secondary Phase From the continuity equation of secondary phase P, the volume fraction ( p ) equation can be obtained from:
. p ρp vm . p ρp vdr, p
(4.7)
The mixture model is used to simulate the two-liquids. The dispersion phase water liquid is simulated based on Reynolds Averaged Navier-Stokes equations (RANS) while dispersed phase oil droplet tracking procedure is used for the oil liquid. To provide a reasonable solution for engineering objectives some simplifying assumptions are made as stated before.
4.4.3 The Euler – Euler Model The Eulerian model is the most complex of the multiphase models [73]. It solves a set of (n) momentum and continuity equations for each phase. Coupling is achieved through the pressure and interphase exchange coefficients. Momentum exchange between the phases is also dependent upon the type of mixture being modeled.
4.4.4 The VOF Model The VOF model is a surface-tracking technique applied to a fixed Eulerian mesh [73]. It is designed for two or more immiscible fluids where the position of the interface between the fluids is of interest. In the VOF model, a single set of momentum equations is shared by the fluids, and the volume fraction of each of the fluids in each computational cell is tracked throughout the domain.
4.4.5 Turbulence Modeling Turbulent flows are characterized by fluctuating velocity fields. These fluctuations mix transported quantities such as momentum, energy, and species concentration,
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and cause the transported quantities to fluctuate as well. Since these fluctuations can be of small scale and high frequency, they are too computationally expensive to simulate directly in practical engineering calculations. Instead, the instantaneous (exact) governing equations can be time-averaged, ensembleaveraged, or otherwise manipulated to remove the small scales, resulting in a modified set of equations that are computationally less expensive to solve. However, the modified equations contain additional unknown variables, and turbulence models are needed to determine these variables in terms of known quantities. A turbulence model is a computational procedure to close the system of the mean flow equations. The closure equation can be employed in combination with the time-averaged continuity and momentum. The most popular nonalgebraic turbulence models are two-equations eddy viscosity models. The turbulence is simulated via choosing of different available turbulence models in ANSYS R.15 Fluid Flow Fluent (FFF) [73]. However, because of the previous and present successful results of the slandered k model, it will be widely chosen here as it is the simplest turbulence model. Before its choosing, different tests on different turbulence models are performed and the results are compared with different measurements to verify the ability of the standard k . The three chosen turbulence models are: standard k (STD k ), shear stress transport k (SST k ) and standard k (STD k ) models.
4.4.5.1 Standard k - ε Model, (STD k - ε ) The standard k model is a semi-empirical model based on model transport equations for the turbulence kinetic energy ( k ) and its dissipation rate ( ). The turbulent velocity scale is
k and k / is chosen to be the time scale. In the
derivation of the (STD k ) model, it was assumed that the flow is fully turbulent, and effects of molecular viscosity are negligible. The standard k model is therefore valid only for fully turbulent flows [73].
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Physical and Mathematical models
. ρm v m k . (μ m
t,m )k G k, m ρm ε k
(4.8)
. ρm v m . (μ m t,m ) (C1G k,m C2ρm) k
(4.9)
The turbulent (or eddy) viscosity (μt), is computed by combining k and ε as: 2 μt, m ρm Cμ kε
(4.10)
Where; C1ε, C2ε, Cμ, σk and σε are the model constants that have the following default values [73]. Cε1 = 1.44, Cε2 = 1.92, Cμ = 0.09, σk = 1.0 and σε = 1.3. These default values have been determined from experiments for fundamental turbulent shear flows including homogeneous shear flows and decaying isotropic grid turbulence. They have been found to work fairly well for a wide range of wall-bounded and free shear flows. The term G k ,m representing the production of turbulence kinetic energy, can be easily modeled from the exact equation for the transport of k , this term may be defined as:
G k ,m mi mj
mj x i
(4.11)
To evaluate G k ,m in a manner consistent with the Boussinesq hypothesis,
Gk ,m eff ,m S 2
(4.12)
Where S is the modulus of the mean rate-of-strain tensor, defined as:
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Chapter (4)
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S 2S ij S ij
(4.13)
S ij 1 mj mi 2 x i x j
(4.14)
4.4.5.2 Standard k - Model, (STD k - ) The standard k model in ANSYS Fluent is based on the k-ω model by Wilcox [78]
and [79] which incorporates modifications for low-Reynolds number effects and shear flow spreading. One of the weak points of the Wilcox model is the sensitivity of the solutions to values for k and
outside the shear layer (free stream sensitivity). While the
new formulation implemented in ANSYS Fluent has reduced this dependency, it can still have a significant effect on the solution, especially for free shear flows [78]. The standard k model is an empirical model based on model transport equations for the turbulence kinetic energy ( k ) and the specific dissipation rate ( ), which can also be thought of as the ratio of ε to k [78]. As the k model has been modified over the years, production terms have been added to both the k and
equations, which have
improved the accuracy of the model for predicting free shear flows.
The turbulence kinetic energy, k and the specific dissipation rate are obtained from the following transport equations:
. ρm v m k . (μ m
. ρm v m . (μ m
t,m )k G k, m 0.09 ρ k k
(4.15)
t,m ) (G k, m 0.09 ρk) k
(4.16)
Where; σk and σω are the turbulent Prandtl numbers for k and ω, respectively, and the mixture turbulent viscosity ( μt,m ) is computed by combining k and as follows:
μt,m ρm k
(4.17)
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The term G k ,m is given by equation (4.12) and represents the production of turbulence kinetic energy due to mean velocity gradients. G ,m represents the production of .
4.4.5.3 Shear-Stress Transport k - Model (SST k - ) The shear-stress transport k model was developed by Menter [80] and [81] to effectively blend the robust and accurate formulation of the k model in the nearwall region with the free stream independence of the k model in the far field. To achieve this, the k model is converted into a k model. The SST k model is similar to the standard k model, but includes the following refinements: The standard k model and the transformed k model are both multiplied by a blending function and both models are added together. The blending function is designed to be one in the near-wall region, which activates the standard k model and zero away from the surface, which activates the transformed k model. The SST k model incorporates a damped cross-diffusion derivative term in the equation. The definition of the turbulent viscosity is modified to account for the transport of the turbulent shear stress. The modeling constants are different. These features make the SST k model more accurate and reliable for a wider class of flows (for example, adverse pressure gradient flows and curved surfaces) than the STD k model. The SST k model has a similar form to the standard k model:
t,m )k G k, m 0.09 ρ k k
(4.18)
t,m ) (G k, m 0.09 ρk) D k
(4.19)
. ρm v m k . (μ m
. ρm v m . (μ m
Dω represents the cross-diffusion term, which can be evaluated as:
D
1.363 1 . k . x j x j
(4.20)
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Chapter (4)
Physical and Mathematical models
The solution for turbulence models is related to the computational time required for the solution of the turbulence model equation along with the other governing equation until reading convergence. Generally, two equation models ( k ) model clearly require more computational effort than the one equation models (Spalart-Allmaras) model since an additional transport equation is solved. The standard k model requires more computational effort than the standard k model due to the behavior of the enhanced model equations. Furthermore, the SST
k model tends to take more time due to the extra terms and functions in the governing equations. However, efficient programming can reduce the CPU time per iteration significantly. Convergence is another aspect influencing the solution behavior. The ability of the CFD code to obtain a converged solution depends on the suitable choice of the turbulence model for the defined problem. Considering this issue, the standard k model is known to be slightly over-diffusive in certain situations.
4.4.6 Mathematical Model in a General Form The previous widely discussed schemes of the mixture two-phase flow and the turbulence models are introduced the general form equations with its different source terms of each mixture two-phase flow and turbulence models as shown below in Tables from 4.1 to 4.3. As well as, these tables are provided with the model constants, damping functions and extra source terms. Table 4.1 General governing equations forms of the two-phase flow
Equation
Continuity
1
0
x- momentum
um
S
eff
0 u v w ( eff ) ( eff ) ( eff ) x x y x z x n
k=1
m g . ∑ k ρ k (u k u m )(u k u m )
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Chapter (4)
Physical and Mathematical models
Table 4.1 Continued
y-momentum
z-momentum
vm
wm
u v w ( eff ) ( eff ) ( eff ) x y y y z y
eff
eff
n
k=1
m g . ∑ k ρ k (v k v m )(v k v m ) u v w ( eff ) ( eff ) ( eff ) x z y z z z n
k=1
m g . ∑ k ρ k (w k w m )(w k w m )
2 2 2 x (u) y (v) z (w) x 2 y 2 z 2 + S Where is the general variable to be solved, is the variable diffusivity and represents the source term, the values of , and each corresponding equation. Also,
eff ,m , t,m
viscosities of the mixture respectively that given by;
S are given in Table 4.2 for
are the effective and turbulent
μeff,m μ m μt,m
Table 4.2 The general form of the turbulence models equations Turbulence Model Standard k-ε model
Standard k- ω model
Shear-Stress Transport k-ω model
Equation
Turbulence kinetic energy
k
μ + μt
Turbulence dissipation rate
ε
μ + μt
Turbulence kinetic energy
k
μ + μt
Gk 0.09 ρ k
Turbulence dissipation rate
ω
μ + μt
G 0.09 ρ 2
Turbulence kinetic energy
k
μ + μt
G 0.09 ρ k k
ω
μ + μt
Turbulence dissipation rate
S
G k
k
k
k
74
S
C1
k
G C 2
2
k
G 0.09 ρ 2 D (4.20)
Chapter (4)
Physical and Mathematical models
Table 4.3 Model constants, Damping functions, production and extra source terms The values of Constants
The selected Models Standard
Standard
Shear-Stress Transport
k-ε model
k- ω model
k-ω model
Cμ
0.09
0.09
0.09
k
1.0
1.0
1.0
E
1.3
1.22
1.2
Cε1
1.44
0.0
0.0
Cε2
1.92
0.0
0.0
Gk
Eq. (4.12)
Eq. (4.12)
Eq. (4.12)
Gω
0.0
ω Gk k
ω Gk k
Γk
μ + μt
μ + μt
μ + μt
Γε
μ + μt
0.0
0.0
Γω
0.0
μ + μt
μ + μt
Sk
G k
Gk 0.09 ρ k
G 0.09 ρ k k
Sε
C1 G k 2 C 2 k
0.0
0.0
Sω
0.0
G 0.09 ρ
k
k
k
75
2
G 0.09 ρ 2 D (4.20)
Chapter (5)
Numerical Methodology
CHAPTER 5 NUMERICAL METHODOLOGY 5.1 Introduction In this chapter, the numerical method used to solve the governing system of equations for single-phase and two-phase segregated flows with appropriate boundary conditions is presented. The procedure is an implicit iterative method based on the finite volume discretisation technique and uses a solution algorithm specially developed for two-phase flow which is based loosely on the SIMPLE algorithm. Special consideration is given to coupling between the phase momentum equations due to drag and to the imposition of continuity constraints and their effect on the stability of the solution algorithm. Afterwards, the appropriate boundary conditions required for the numerical procedure are implemented. The purpose of this chapter is threefold. First, is to describe the computational domain including the boundary conditions. Second, to introduce the numerical techniques adopted to solve the corresponding governing equations for singlephase flow and oil-in-water emulsion two-phase flow through S-shaped diffusers. Third, the solution procedure is described. The present numerical study is coded as ANSYS R-15.0 Fluid Flow Fluent (FFF) program which applies 3D-RANS (Reynolds Averaged Navier-Stokes equations) method. The proposed simulation aims to predict water / emulsion flow field without the presence of heat and mass transfer. Some cases are chosen from the literature and from present experimental study in order to perform a validation of the proposed model as stated in the next chapter. Generally, the finite volume discretization technique is used and iterative method is adopted to solve the nonlinear system of discretized equations. For pressure-velocity coupling, SIMPLE algorithm is considered.
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5.2 Stages of Computation Any CFD study can be divided into three main stages: 1. Pre-process: problem formulation where the region of fluid to be analyzed (the computational domain) is defined and divided up into a number of discrete elements (mesh). The properties of fluid acting on the domain, including external constraints or boundary conditions to implement realistic situations are set. 2. Solution: the solution of the CFD problem is made out whereby the governing equations are solved iteratively to compute the flow parameters of the fluid. Convergence is important to produce an accurate solution of the partial differential equations. 3. Post-process: visualization, analysis and processing of the results from the solver. Commercial packages often provide post-processing facilities that enable the creation of vector plots or contour plots to display the trends of velocity, pressure, kinetic energy and other properties of the flow. A key component of post-processing is being able to visualize complex flows.
5.2.1 Model geometry and grid generation Figure 5.1 represents a staggered grid and the mesh of S-diffuser structured grid which is created and used for computation. The regular hexahedral grid elements are generated by the commercial software, the ANSYS R15.0 FLUENT (Geometry and Mesh). According to this complex geometry, the grid is obtained using the MAP scheme for the 3-D mesh which is used to generate the mesh in 3D model adaptation refining the grid in both axial and radial directions as shown in Fig. 5.1.a. The refined grid is along the wall region and equally spaced along both spanwise and axial directions. The software package Fluid Flow Fluent of ANSYS R.15.0 (Setup) is adopted in this study that employs a finite-volume method with second order upwind scheme, and SIMPLE for pressure – velocity coupling. With water flow and emulsion flow, the computations of S-shaped diffuser with its two tangents (upstream and downstream ducts) are carried out for different grid sizes (mesh grids of coarse, medium and fine grid). The suitable resolution will be chosen based on the comparisons with the available measurements.
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dz into paper
(a) A staggered mesh grid
(b) The mesh for 3D S-shaped diffuser Fig. 5.1 A staggered mesh grid (a) and the mesh for 3D S-shaped diffuser (b)
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Chapter (5)
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5.2.2 Setup of the Solver The solution setup starts from Fluid Flow Fluent setup (FFF setup) with the general solver type pressure-based and the general solver time Steady. The solver setup is continued by selecting the material, the phases, the turbulence model and the multiphase model. The solver setup is completed by selecting boundary conditions and reference values. Finally, the choice of the solution method, the solution controls, the monitors, the solution initialization, the calculation activities, the number of iterations and the data file quantities which will be followed by the calculation.
5.2.3 Boundary Conditions The boundary conditions may be known before starting the numerical solution. It has long been recognized that diffuser performance depends not only on the physical shape of the duct but also upon the type of inflow conditions. For example, in the present study the inlet conditions are based on velocity inlet boundary condition and the exit conditions which are based on pressure outlet. The tests are conducted at different Reynolds numbers with a maximum value of 34,000 in case of water flow based on the diffuser inflow average velocity of 1.07 m/sec measured at the inlet plane (diffuser entrance) and inlet hydraulic diameter of the Sshaped diffuser as 32 mm.
5.3 Solution and Turbulence Models Preliminary investigations are done using different turbulence models and multiphase technique models to validate the CFD code with the experimental data obtained from the literature. The flow field is predicted using three turbulence models namely, standard k–ω (STD K-ω), shear stress transport k–ω (SST K-ω) and standard k–ε (STD k-ε) models. The current CFD code (ANSYS R 15.0) provides us by three main schemes currently used in two-phase flows; Euler (Eulerian) scheme, mixture scheme and volume of fluid (VOF) scheme.
5.4 Convergence Criterion The use of an iterative solution method necessitates the definition of a convergence and stopping criteria to terminate the iteration process. The measure of convergence
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Chapter (5)
Numerical Methodology
is a norm on the change in the solution vector between successive iterations. Also, the iterative algorithm is terminated after a fixed number of iterations if the convergence has not been achieved. These criteria are used to prevent slowly convergent or divergent problems from wasting computation time. Convergence in the present study is defined to be obtained after all residuals for continuity equation, all velocity components (u, v, w) for phase 1 and 2, volume fraction of phase 2, turbulence kinetic energy, turbulence dissipation rate and the stresses are less than 1x10-6.
5.5 Calculation and Results Displaying After the ANSYS finishes the solution by solving up to the monitor check convergence absolute criteria, from, ANSYS R15.0 FLUENT and ANSYS CFD-Post 3D Viewer one can obtain and display the results. CFD-Post is a flexible, state-of-the-art post-processor. It is designed to enable easy visualization and quantitative analysis of the results of CFD simulations. In CFD-Post, the 3D Viewer is accessible by clicking the 3D Viewer tab at the bottom of the panel on the right side of the interface. After loading a results file into CFD-Post, a visual representation of the geometry of the S-shaped diffuser can be seen in the 3D Viewer. Various other objects that can be viewed in the 3D Viewer, such as vectors, contours and streamlines can be also created.
5.6 Numerical Schemes The governing integral equations describing the fluid flow are solved in CFD by using a control-volume-based technique, which is accomplished as:
• The computational grid is used to divide the fluid domain into discrete control volumes.
• The governing equations are then integrated on the individual control volumes to construct algebraic equations for the discrete dependent variables (“unknowns'') such as velocity, pressure, turbulence intensity, etc.
• The discretized equations are then linearised and the resultant linear equation system is solved to yield updated values of the dependent variables.
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FLUENT provides a choice of two numerical methods [73]: the segregated solver and the coupled solver. Both numerical methods employ a finite volume discretization process, but the approach used to linearise and solve the discretized equations is either implicit or explicit. The coupled solver solves the governing flow equations simultaneously and governing equations for additional scalars sequentially. In the segregated solution method each discrete governing equation is linearised implicitly with respect to that equation's dependent variable. Implicit formulation of the coupled set of governing equations imply that the unknown value for a given variable in each cell is computed using a relation that includes both existing and unknown values from neighboring cells, resulting in more than one equation which must be solved using a point implicit (Gauss-Seidel) linear equation solver in conjunction with an algebraic multi grid (AMG) method to give the unknown quantities. Explicit formulation of the coupled set of governing equations imply that the unknown value for a given variable in each cell is computed using a relation that includes only existing values, resulting in only one equation which can be solved at a time to give the unknown quantities. In this case, a linear equation solver is not needed. The solution is updated using a multi-stage (Runge-Kutta) solver instead.
5.7 Discretization Process In order to solve a fluid flow numerically, the computational domain, including the surfaces and boundaries have to be discretized, i.e. the continuously varying quantities are approximated in terms of values at a finite number of points. This can be carried out using either one of three different methods: Finite-Difference Method (FDM), Finite-Volume Method (FVM) or Finite-Element Method (FEM). However, recent CFD packages, including FLUENT, tend to apply FVM more because it can be used on either a structured or unstructured mesh, rigorously enforces conservation, is directly relatable to physical quantities (mass flux, etc.), and is easier to program in terms of CFD code development. Two types of discretization processes exist within CFD: spatial, and temporal. The first type is discussed in more detail, while temporal discretization will be overlooked as it is
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Chapter (5)
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only applicable to time-dependent formulations or unsteady state systems by splitting the time in the continuous flow into discrete time steps.
5.7.1 Discretization Equation The governing equations in CFD consist of partial differential equations, which need to be discretized so they can be solved by the solver. In the FVM method, the governing equations of fluid flow are integrated and solved for each control volume iteratively based on the conservation laws. The discretization process results in a set of algebraic equations that resolve the variables using an integration method at a specified finite number of points within the control volumes. The flow inside the whole domain is then obtained through integration on the control volumes. The equations solved by FLUENT apply to the array of control volumes, i.e. the computational grid or mesh, generated by the spatial discretization. By default, the discrete variable values (e.g. pressure, velocities and turbulence) are calculated and stored at the cell centers by FLUENT [73]. This is accomplished using an upwind scheme whereby the face value is derived from quantities in the cell upstream, or upwind relative to the direction of the mean velocity. Four upwind advection schemes are available in FLUENT: firstorder upwind, second-order upwind, power law, and QUICK. In the first order upwind advection scheme, quantities at cell faces are determined by assuming that for any variable, the cell-centre values represent the cell-average value. Hence when first-order upwind scheme is selected, the face value of a variable is set equal to the cell-centre value of in the upstream cell. In the second order upwind advection scheme, quantities at cell faces are computed using a multidimensional linear reconstruction approach, whereby higher-order accuracy is achieved at cell faces through a Taylor series expansion of the cellcentered solution about the cell centroid. Hence the face value of a variable is computed by averaging that from the two cells adjacent to the face using a gradient method. The power-law discretization scheme interpolates the face value of a variable, using the exact solution to a one-dimensional convectiondiffusion equation.
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QUICK (Quadratic Upstream Interpolation for Convection Kinetics) discretization scheme are based on a weighted average of second-order-upwind and central interpolations of the variable. FLUENT documentation recommends that the firstorder discretization generally yields better convergence but less accurate results than the second-order scheme. First order upwind discretization is applicable without any significant loss of accuracy when the flow is aligned with a quadrilateral or hexahedral grid, so that numerical diffusion will be naturally low. Also that for most complex flows, the first-order scheme may need to be used to perform the first little iteration and then switch to the second-order scheme to continue the calculation to convergence. The power law is reported to generally yield the same accuracy as the first-order scheme. The QUICK discretization scheme is reported to compute higher-order value of the convected variable at a face more accurately on structured grids aligned with the flow direction, but to only provide marginal improvement on the second-order scheme for rotating flows solved on quadrilateral or hexahedral meshes. Because of the S-diffuser curvature and the getting more accurate results, QUICK is selected in the present work as adopted upwind advection scheme.
5.8 Pressure Interpolation Schemes Similarly, the face values of pressure from the cell values are computed using a pressure interpolation scheme. The ones available in FLUENT include: standard, linear, second order, body-force-weighted scheme and PRESTO (Pressure Staggering Option) and a brief description of all five schemes is given below. The standard scheme interpolates the pressure values at the faces using momentum equation coefficients and is only applicable when the pressure variation between cell centres is smooth. The linear scheme computes the face pressure as the average of the pressure values in the adjacent cells. The second-order scheme reconstructs the face pressure in the manner used for second-order accurate convection terms. The body force- weighted scheme computes the face pressure by assuming that the normal gradient of the difference between pressure and body forces is constant. The PRESTO scheme uses the discrete continuity balance for a “staggered” control
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Chapter (5)
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volume about the face. Fluent documentation recommends the use of the PRESTO scheme for high-Rayleigh number natural convection, high-speed rotating flows, and flows in curved domains. In the present work the PRESTO scheme is chosen as adopted pressure interpolation scheme due to the flow in S-diffuser is 3D and curved.
5.9 Pressure-Velocity Coupling The momentum equation is normally solved with a guessed pressure field, and therefore the resulting face flux, does not satisfy the continuity equation. This is rectified by applying a pressure-velocity coupling algorithm, which adds a correction to the face flux so that the corrected face flux satisfies the continuity equation. Three pressure-velocity coupling algorithms are provided in FLUENT [73]: SIMPLE (Semi- Implicit Method for Pressure Linked Equations), SIMPLEC (SIMPLE-Consistent) and PISO (The Pressure-Implicit with Splitting of Operators). The SIMPLE or SIMPLEC algorithm is recommended for steady-state calculations while PISO is recommended for transient calculations, or for cases with highly skewed meshes. The SIMPLE algorithm substitutes the flux correction equations into the discrete continuity equation to obtain a discrete equation for the pressure correction in the cell. The SIMPLEC algorithm is a variant of the SIMPLE algorithm, offering a different correction expression for the face flux correction. The use of this modified correction equation has been shown to accelerate convergence in problems where pressure velocity coupling is the main deterrent to obtaining a solution. The PISO pressure-velocity coupling scheme, is based on the higher degree of the approximate relation between the corrections for pressure and velocity. Due to the flow is assumed steady the SIMPLE algorithm is chosen as adopted pressure-velocity coupling algorithm in the present work.
5.10 Near-Wall Treatment for Wall-Bounded Turbulent Flows The k-ε model was designed for turbulent core flows and can therefore be predicted to be inaccurate in the near wall region, which is a crucial region for the
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Chapter (5)
Numerical Methodology
Successful predictions of wall-bounded turbulent flows. Special wall modeling procedures therefore need to be implemented to make these models suitable for wall-bounded flows. The boundary condition at a stationary wall is the no-slip. In order to satisfy this, the mean velocity at the wall has to be zero, thereby creating a steep velocity gradient (from zero at the walls to the mean flow velocity at the core) very close to the wall. This gives rise to the so-called “nearwall region” which can be largely subdivided into three layers (as in Fig. 5.2): The viscous sublayer (the innermost layer) where the flow is almost laminar. The buffer region (the region between the viscous sublayer and the fully turbulent layer) where both the effects of molecular viscosity and that of turbulence are equally important. The fully-turbulent layer (the outer layer) where turbulence plays a major role. In order to resolve the velocity gradient and better predict the flow behavior in the near-wall region, a higher mesh density and special wall modeling procedures are therefore required. In order to resolve the laminar sub layer, the near-wall mesh must be very fine, typically y+ ≈ 1. This imposes too large computational requirement. The enhanced wall treatment is a near-wall modeling method that combines a twolayer model with enhanced wall function to overcome this problem. Hence turbulence dissipation energy and the turbulent viscosity in the near-wall cells are completely resolved from the viscosity-affected near-wall region all the way to the viscous sub-layer. The two-layer approach divides the whole domain into a viscosity-affected region and a fully-turbulent region according to a wall distance-based on, turbulent Reynolds number, ( Re y
m y = ρm u μm ).
For the fully-turbulent region, where Rey 200, the k-ε model is adopted, while for the viscosity affected region, where Rey < 200, a one equation model is employed. The enhanced wall functions formulate the law-of-the wall as a single wall law for the entire wall region (laminar sub-layer, buffer region, and fullyturbulent outer region) by blending the linear (laminar) and the logarithmic
85
Chapter (5)
Numerical Methodology
(turbulent) laws-of-the wall. This approach allows the fully turbulent law to be easily modified and extended to take into account other effects such as pressure gradients or variable properties. This approach also guarantees the correct asymptotic behavior for large and small values of y+ and reasonable representation of velocity profiles in the cases where y+ falls inside the wall buffer region (3 < y+ < 10). The y+ is a non-dimensional parameter defined by this equation: + y y = ρm u μm
(5.1)
Where:
y is the normal distance from wall. uτ is the friction velocity ( u w / ρm ) and w is the wall shear stress.
Fig. 5.2 The near-wall regions of turbulent flows (ANSYS Fluent [73])
86
Chapter (5)
Numerical Methodology
5.11 Numerical Solution Procedure The ANSYS R15.0 CFD code is used in the present study to simulate emulsion flow in S-shaped curved diffusers and to predict the flow behavior in such geometries. The flowchart of the present used code program is shown in Fig. 5.3 to summarize the method; in which the continuous phase flow is obtained using the SIMPLE approach. A line-by-line iteration using Tri-Diagonal Matrix Algorithm (TDMA) is incorporated to solve the equations for each variable. The procedure is repeated until the arriving convergence and the maximum error of all variables between two successive coupled iterations is less than 0.005. By changing the values of the properties of the fluid, the geometry and the operating parameters, the correlations between flow dynamics and affecting parameters can be quantitatively analyzed. Figure 5.4 presents a simple flowchart for detailing all the computational and experimental studies performed in the present work.
87
Chapter (5)
Numerical Methodology
Start ANSYS Create the Geometry Create the Grid Setup the Solution Initialize the Solution Run Calculations Solve the Governing Equations Solve pressure Correction Equation Update pressure and Mass Flow rate
Solve the Turbulence Models
No
No
Solution Converged Yes
Yes Yes
Stop
Fig. 5.3 Flowchart of the used code program
88
Chapter (5)
Numerical Methodology
Fig. 5.4 Flowchart of the present numerical and experimental studies
89
Chapter (6)
CFD Code Validation
CHAPTER 6 CFD CODE VALIDATION 6.1 Introduction In this chapter, it is aimed to introduce validations of the current code using some known test cases for simulating water / emulsion flows in different geometries. These test cases are: water flow in S-shaped diffusing ducts, (o/w) emulsion flow in sudden expansion and (o/w) emulsion fluid flow in curved diffusers that will be discussed in this chapter sections. The present numerical study is coded as ANSYS R-15.0 Fluid Flow Fluent (FFF) program which applies 3D-RANS (Reynolds Averaged Navier-Stokes equations) method. The proposed code aims to predict water / emulsion flow field without the presence of heat transfer. Some cases are chosen from the literature and from present study in order to perform a validation of the proposed model as stated below. First, the code is validated with the experimental data of Whitelaw and Yu [30] in case of single phase flow (water flow in S-shaped diffusing duct). Second, the code is validated with the experimental data of Hwang and Pal [58] in case of oil / water mixture two phase flow in sudden expansion and contraction). Third, the code is validated with the experimental data of El-Askary et al. [71] in cases of water flow and emulsion flow in curved diffuser. After a through validation of the proposed solution and turbulence models presented in chapter 4 are tested for emulsion flow in different geometries of the present work.
6.2 Validation of an S-Shaped diffuser Carrying Water [30] 6.2.1 Physical Model and the Grid Generation The schematic drawing dimensions are shown in Fig. 6.1 represents the computational domain of solution that deals with water flow through a rectangular cross-section area S-shaped diffuser.
90
Chapter (6)
CFD Code Validation
The regular hexahedral grid elements are generated by the commercial software, the ANSYS R15.0 FLUENT (Geometry and Mesh). According to this complex geometry, the grid is obtained using the MAP scheme for the 3-D mesh which is used to generate the mesh in 3D model adaptation for refining the grid in both axial and radial directions as shown in Fig. 6.2. The refined grid is along the wall region and equally spaced along both radial and axial positions. The solution setup is adopted in this study that employs a finite-volume method with QUICK discretization upwind scheme, and SIMPLE algorithm for pressure-velocity coupling algorithm. With water flow and emulsion flow the computations of S-shaped diffuser with its two tangents (upstream and downstream ducts) are carried out for different grid sizes (coarse, medium and fine).
6.2.2 Turbulence Models and Mesh Resolution In the previous studies which similar to our study, CFD simulations were carried out using software Fluid Flow Fluent (FFF) which applies 3D-RANS (Reynolds Averaged Navier-Stokes equations) method. Preliminary investigations have used a range of turbulence models and multi-phase technique models to validate the CFD code with the experimental data obtained from the literature. The experimental water flow measurements in S-shaped diffusing duct given by Whitelaw and Yu [30] is predicted in the present study using three turbulence models namely, standard k (STD k ), shear stress transport k (SST k ) and standard k (STD k ) models. The comparisons show that the
pressure coefficient profile is successfully predicted by using the latest turbulence model (STD k ), see Fig. 6.3. Therefore, the STD k model can be used as a solution turbulence model for the validation code. In case of single phase flow, water flow fields of Whitelaw and Yu [30] are predicted using three grid types (coarse, medium, fine). These predictions are compared with the experimental measurements of [30].
91
Chapter (6)
CFD Code Validation
The comparisons show that the pressure coefficient profile is successfully predicted using the medium grid (1000000 cells), see Fig. 6.4.
Fig. 6.1 The schematic coordinates system for 3D S-diffuser tested physical model [30] (not to scale)
92
Chapter (6)
CFD Code Validation
Fine
medium
Coarse
Fig. 6.2 The computational mesh for 3D S-shaped diffuser of Ref. [30]
93
Chapter (6)
CFD Code Validation
0.4 0.3
CP
o
0.2 0.1 0 Water Flow at Re=40,000 CFD (SST K-) CFD (STD K-)
-0.1
CFD (STD K-)
Ref. [30]
-0.2
-2
0
2
X/W
4
6
8
(a) Outer-Wall
0.4
CP
i
0.3
0.2
0.1 Water Flow at Re=40,000 CFD (SST K-)
0
CFD (STD K-) CFD (STD K-)
Ref. [30]
-0.1
-2
0
2
Xi/W
4
6
8
(b) Inner-Wall Fig. 6.3 Turbulence modeling effect on the pressure coefficient distributions of Whitelaw and Yu Ref. [30] with water flow on outer-wall (a) and inner-wall (b)
94
Chapter (6)
CFD Code Validation
0.5 0.4
C P
0.3 0.2 0.1 Water Flow at Re=40,000
0
OW-CFD (1500,000) OW-CFD (1000,000) OW-CFD (500,000) IW-CFD (1500,000) IW-CFD (1000,000) IW-CFD (500,000) OW-Re [30] IW-Re [30]
-0.1 -0.2
-2
0
2
X/W
4
6
8
Fig. 6.4 Grid-number effect on the pressure coefficient of water flow Measured in [30] on outer wall (OW) and inner wall (IW)
6.3 Validation of Emulsion Flow in Sudden Expansion [58] 6.3.1 Reynolds Number Effect on Validation of Pressure Drop The model is further validated for the experimental data of Hwang and Pal [58] in case of oil/water mixture two phase flow in a sudden expansion with straight pipes having inner diameters of 2.037cm and 4.124cm. The schematic drawing dimensions are shown in Fig. 6.5, while Fig. 6.6 represents the domain and the
95
Chapter (6)
CFD Code Validation
computational grid which is created and meshed by the ANSYS R15.0. It is recognized that, not only the performance depends on the physical shape of the geometry but also upon the type of inflow conditions. The present study validates the experiments of Hwang and Pal [58] and also, tests them at different inflow Reynolds number (different emulsion inflow velocities) and different holdup using a suitable multi-phase model (Euler - Euler technique model). The oil used in the Ref. [58] is Bayol-35 (Esso Petroleum, Canada), which is a refined white mineral oil with a density of 780 kg/m3 and a viscosity of 0.00272 Pa-s at 25°C. Density and viscosity of water are taken as 998.2 kg/m3 and 0.001003 Pa-s, respectively. The comparisons between the present predicted pressure drop profiles of twophase flow of oil-in-water emulsions through the same sudden expansion at dispersed phase holdups of 0.2144, 0.3886 and 0.6035 with different velocities at inlet, and the experimental measurements of [58] are shown in Figs. 6.7, 6.8 and 6.9, respectively. In general, it can be seen from these previous figures that there are acceptable agreements between the predicted values of present CFD code and the experimental measurements of Hwang and Pal [58]. Further, Fig. 6.10 shows that the flow is better predicted by steady solution than unsteady solution. So, it is recommended that the steady state is adopted when running the computations by the solver of ANSYS R-15.0.
Fig. 6.5 The schematic diagram and the pressure profile
for the sudden expansion of Ref. [58]
96
Chapter (6)
CFD Code Validation
Fig. 6.6 The computational grid used for the sudden expansion of Ref. [58]
4
P (KPa)
2 0 -2 -4 -6
Unstable 0.2144 o/w Emulsion 2-Phase Flow at Different Velocities
-8
CFD (V=3.6 m/s) Ref. [58] (V=3.6 m/s) CFD (V=5.0 m/s) Ref. [58] (V=5.0 m/s) CFD (V=6.8 m/s) Ref. [58] (V=6.8 m/s)
-10 -12
-20
-10
0
10
20
30
X/D1 Fig. 6.7 Pressure drop for 0.2144 unstable (o/w) emulsion flowing with different velocities through a sudden expansion of Ref. [58]
97
Chapter (6)
CFD Code Validation
4
P (KPa)
2 0 -2 -4 -6
Unstable 0.3886 o/w Emulsion 2-Phase Flow at Different Velocities
-8
CFD (V=3.2 m/s) Ref. [58] (V=3.2 m/s) CFD (V=5.0 m/s) Ref. [58] (V=5.0 m/s) CFD (V=6.6 m/s) Ref. [58] (V=6.6 m/s)
-10 -12
-20
-10
0
X/D1
10
20
30
Fig. 6.8 Pressure drop for 0.3886 unstable (o/w) emulsion flowing with different velocities through a sudden expansion of Ref. [58]
2
P (KPa)
0 -2 -4 -6 Unstable 0.6035 o/w Emulsion 2-Phase Flow at Different Velocities
-8
CFD (V=4.0 m/s) Ref. [58] (V=4.4 m/s) CFD (V=4.8 m/s) Ref. [58] (V=4.8 m/s) CFD (V=6.0 m/s) Ref. [58] (V=6.0 m/s)
-10 -12
-20
-10
0
X/D1
10
20
30
Fig. 6.9 Pressure drop for 0.6035 unstable (o/w) emulsion flowing with different velocities through a sudden expansion of Ref. [58]
98
Chapter (6)
CFD Code Validation
4
P (KPa)
2 0 -2 -4 -6
Unstable 0.3886 o/w Emulsion 2-Phase Flow at V=3.2 m/s CFD (Eulerian Unsteady) CFD (Eulerian Steady) Ref. [58]
-8 -10 -12
-20
-10
0
10
20
30
X/D1 Fig. 6.10 Pressure drop for 0.3886 unstable (o/w) emulsion flowing through a sudden expansion with steady/unsteady of Ref. [58]
6.3.2 Emulsion Holdup Effect Emulsion holdup effect is presented when validating of the experimental measurements of Hwang and Pal [58] as shown in Fig. 6.11. At the same Reynolds number and the same geometry it is noticed that: the pressure drop of two-phase flow of oil-in-water emulsions across a sudden expansion is found to be inversely proportional to the volume fractions, Φ, which increases the losses and hence, decreases the performance.
6.4 Validation of a 90° curved diffuser [71] 6.4.1 Water Flow The present study code is further validated with the experimental data of El-Askary et al. [71] in case of water flow in a 90° curved diffuser that has inlet width of 20 mm, exit width of 40 mm and curvature radius of 150 mm. The schematic drawing dimensions are shown in Fig. 6.12 while Fig. 6.13 represents the domain of solution which is created and meshed by the ANSYS R15.0 FLUENT. The pressure recovery
99
Chapter (6)
CFD Code Validation
coefficients on the outer and inner walls are predicted by using the same previous turbulence models as shown in Fig. 6.14. From the figure it is clear that the standard k (STD k ) can adopt and shows a good agreement with the experimental data.
6.4.2 Emulsion Flow The code is also validated with the experimental data of El-Askary et al. [71] in case of two phase flow (o/w emulsion fluid flow in curved diffuser) since it is predicted by using the three multi-phase models as discussed previously. The comparisons show that the pressure coefficient profile is successfully predicted using the mixture scheme, see Fig. 6.15. Therefore, it is obvious that the mixture model is the right choice that can be used as a solution multiphase model for simulating the o/w emulsion flow. The mixture model is used here based on RANS with standard k model. Thus, from modeling different cases in this chapter it is recommended in the present numerical study that the mixture model approach with STD k should be applied for oil-in-water emulsion flow simulation in all S-shaped diffusers.
4
P (KPa)
2 0 -2 -4 -6
Unstable o/w Emulsion Holdup Effect Pressure Drop at V=5.0 m/s
-8
CFD (=0.2144) Ref. [58] (=0.2144) CFD (=0.3886) Ref. [58] (=0.3886) CFD (=0.6035) Ref. [58] (=0.6035)
-10 -12
-20
-10
0
X/D1
10
20
30
Fig. 6.11 Pressure drop for unstable (o/w) emulsions flowing through a sudden expansion at different holdup of Ref. [58]
100
Chapter (6)
CFD Code Validation
Fig. 6.12 The schematic diagram for a 90° curved diffuser of Ref. [71], not to scale
Fig. 6.13 The mesh grid for a 90° curved diffuser of Ref. [71]
101
Chapter (6)
CFD Code Validation
0.8 Water Flow at Re=29,000 CFD (SST K-) CFD (STD K-) CFD (STD K-) Ref. [71]
0.6
0.2
Cp
o
0.4
0 -0.2 -0.4 -0.6
-10
0
10
20
30
40
50
60
Xo/W (a) Outer-Wall
0.8 Water Flow at Re=29,000 CFD (SST K-)
0.6
CFD (STD K-) CFD (STD K-)
0.4
Ref. [71]
Cp
i
0.2 0 -0.2 -0.4 -0.6 -0.8
-10
0
10
20
30
40
50
60
Xi/W (b) Inner-Wall Fig. 6.14 Pressure recovery coefficient distributions on the outer-wall (a) and inner-wall (b) for water flow in 90° curved-diffuser of Ref. [71]
102
Chapter (6)
CFD Code Validation
0.8 Stable 0.10 o/w Emulsion 2-Phase Flow at Re=29,000
0.6
CFD (Eulerian) CFD (Mixture) CFD (VOF)
0.4
Ref. [71]
Cp
o
0.2 0
-0.2 -0.4 -0.6 -0.8 -10
0
10
20
30
40
50
60
Xo/W (a) Outer-Wall
0.8 Stable 0.10 o/w Emulsion 2-Phase Flow at Re=29,000
0.6
CFD (Eulerian) CFD (Mixture) CFD (VOF)
0.4
Ref. [71]
Cp
i
0.2 0 -0.2 -0.4 -0.6 -0.8 -10
0
10
20
30
40
50
60
Xi/W (b) Inner-Wall Fig. 6.15 Multi-phase modeling effect on the pressure coefficient distributions on the outer-wall (a) and inner-wall (b) for emulsion flow in 90° curved-diffuser of Ref. [71]
103
Chapter (6)
CFD Code Validation
6.5 Summary and Conclusions of CFD Model Validation Test cases are here selected from the literature and attempts to simulate the measurements are performed. The present CFD simulation is carried out using ANSYS R-15.0 to validate the experimental data of Refs. [30, 71] in case of water flow and of Refs. [58, 71] in case of emulsion flow. Once the case is simulated the results are compared with the published data in order to assess the accuracy of the present CFD code and the chased model. The validations are performed for hydrodynamics and no heat transfer in water and o/w emulsion flows is allowed. The hydrodynamics validation shows that the turbulence modulation models, the mesh, the dispersed phase holdup and the Reynolds number strongly affect the pressure coefficient distribution and hence the performance. Furthermore, the standard k-ε model is recommended in the computations of water as well as o/w emulsion flows. For simulations in case of (o/w) emulsion flow, validations with [58] and [71] are obtained with good agreement. As a suitable multi-phase technique model, the mixture model is recommended in the computation of o/w emulsion flows and the flow is better predicted by steady solution than unsteady solution. Based on these verifications, extended computations are encouraged to be performed to simulate the present measurements in case of water / emulsion flows in S-shaped diffusers.
104
Chapter (7)
Results and Discussion
CHAPTER 7 RESULTS AND DISCUSSIONS 7.1 Introduction This chapter introduces the experimental and numerical results obtained during the present study. The results are presented first for water and then for unstable/stable (o/w) emulsions flow. Different parameters of diffuser geometry and flow conditions are studied and discussed experimentally and numerically for deep understanding of oil-in-water emulsion flow behavior through S-shaped diffusers. The geometric parameters of S-shaped diffusers examined in the present study are four parameters including area ratios: 1.5, 2.0, and 3.0, curvature ratios: 5.0, 7.5, and 12.5, turning angles: 45◦/45◦, 60◦/60◦, 90◦/90◦ and flow paths: 45◦/45◦, 60◦/30◦, and 30◦/60◦. While the flow parameters investigated here are inflow Reynolds numbers ( (9000 Re 34000) , oil concentration (holdup):0.03, 0.06, 0.10, 0.15, and 0.25 (by volume) and emulsion status (unstable/stable). However, the geometrical parameters of all twelve models of the S-shaped diffusers are given in Table 7.1. This thesis focuses on the S-shaped diffuser performance. At different geometrical and inflow conditions, the local wall static pressure recovery coefficient (Cp) along middle height of both outer and inner walls of each S-shaped diffuser and its upstream and downstream tangents is obtained experimentally and numerically for both water and emulsion flows. Special attention is paid for obtain the S-shaped diffuser performance through the S-diffuser energyloss coefficients (resistance coefficients, Kd,) graphically from the corresponding pressure recovery coefficients according to equation (C.5 in Appendix C). As well as, this study presents numerically the variations of the axial velocity and the turbulent shear stress at five sections (S-diffuser inlet, mid-first bend, inflection, mid-second bend and S-diffuser exit) for each S-diffuser under different diffuser-flow parameters. To determine the suitable turbulence model, the multiphase model and the median resolution results are obtained by varying one of entered characteristic data and the others are fixed, according to Table 7.2. Experimental and numerical results obtained
105
Chapter (7)
Results and Discussion
from the present study by varying one of seven entered effects data and the others are fixed according to Tables 7.1 and 7.3. Table 7.1: Summary of the investigated conditions in the present study Area
Curvature
Turning
Flow
Flow
Emulsion
Reynolds
Axial
Turbulent
Ratio
Ratio
Angle
Path
Type
Holdup
Number
Velocity
Shear Stress
(AR)
(CR)
(TA)
(FP)
(Status)
(Φ)
(Re)
(U/Uref)
(u′v′/ U2ref)
Effects on Cp & Kd 1.5
5.0
45°/45°
45°/45°
Water
Affected by same Effects
0.00
15000
2.0
7.5
60°/60°
30°/60°
Unstable
0.03
18000
3.0
12.5
90°/90°
60°/30°
Stable
0.06
21000
30°/60°
0.10
24000
60°/30°
0.15
29000
0.25
34000
diffuser
diffuser
inlet
inlet
Mid-1st
Mid-1st
bend
bend
Inflection
Inflection
Mid-2nd bend
Mid-2nd bend
diffuser
diffuser
Exit
Exit
Numerical Only
Experimental and Numerical
Table 7.2: Characteristic data used in the present study Data
Fixed used data Variable used data
Water Flow
Emulsion Flow
Grid number
1080000
1080000
500000
1500000
Turbulence model
STD k
STD k
SST k
STD k
Mixture
VOF
Eulerian
Stable
Water
Unstable
Flow Type
Multiphase model Emulsion Status Emulsion Holdup
0.00
0.25
0.00 0.25
Reynolds Number
34000
24000
9000 Re 34000
7
7
M 1, 2, 3,………., 12
Model Number
106
Stable
Chapter (7)
Results and Discussion
Table 7.3: Different effects used in the present study
45°/45°
Unstable
Stable
60°/60°,
60°/60°,
90°/90°,
90°/90°,
Unstable
Stable
45°/45°
45°/45°
45°/45°,
45°/45°,
60°/30°,
60°/30°,
Unstable
Stable
30°/60°
30°/60°
Stable
M 1, 2, 3……12
Emulsion Holdup Effect Reynolds Number Effect Model Number Effect
5.0, 7.5,
2.0, 3.0
12.5
60°/30°,
30°/60°,
45°/45°,
60°/30°,
60°/60°,
45°/45°,
Unstable
Stable
90°/90° 30°/60°, 1.5,
5.0, 7.5,
2.0, 3.0
12.5
60°/30°,
30°/60°,
45°/45°,
60°/30°,
60°/60°,
45°/45°,
90°/90° 30°/60°, 1.5,
5.0, 7.5,
2.0, 3.0
12.5
60°/30°,
30°/60°,
45°/45°,
60°/30°,
60°/60°,
45°/45°,
90°/90°
107
Number
Unstable
30°/60°, 1.5,
Model
Stable
90°/90°
M 1, 2, 3……12
Unstable
M 1, 2, 3……12
9000≤Re≤34000
45°/45°,
Stable
9000≤Re≤34000
60°/60°,
Unstable
9000≤Re≤34000
60°/30°,
9000≤Re≤34000
12.5
30°/60°,
45°/45°,
12
0.00≤Φ≤0.25
2.0, 3.0
60°/30°,
10, 11,
0.00≤Φ≤0.25
5.0, 7.5,
7, 8, 9
0.00≤Φ≤0.25
1.5,
4, 5, 6
0.00≤Φ≤0.25
Emulsion Status Effect
30°/60°,
1, 2, 3
M 1, 2, 3……12
Number
Reynolds 9000≤Re≤34000
45°/45°
9000≤Re≤34000
Holdup
Emulsion
Status
Emulsion
Flow Path
Angle
Stable
9000≤Re≤34000
12.5
Turning
Ratio
Curvature
7.5
Unstable
9000≤Re≤34000
2
12.5
45°/45°
0.00≤Φ≤0.25
3
5.0, 7.5,
45°/45°
0.00≤Φ≤0.25
2
7.5
0.00≤Φ≤0.25
Effect
Effect
2.0, 3.0
0.00≤Φ≤0.25
1.5,
Effect Effect
Flow Path
Angle
Turning
Ratio
Curvature
Area Ratio
Area Ratio
Summary of the studied effects on S-diffuser model performance
Chapter (7)
Results and Discussion
7.2 Solution Models and Resolution Effect Preliminary investigations are done using a range of turbulence models, multiphase models and grids to examine the recommended solution models and resolution that stated before in section 6.5 to validate the CFD model with the present experimental data obtained for all tested S-diffuser models in cases of water as well as emulsion flows.
7.2.1 Turbulence Models The water flow field is predicted using the previous stated turbulence models (standard k (STD k ), shear stress transport k (SST k ) and standard k (STD k ) models). The comparisons with the present measurements show that
the pressure coefficient profile can be successfully predicted using the last turbulence model (STD k ), see Fig. 7.1.
7.2.2 Multiphase Models The case of o/w emulsion flow field is predicted using three multiphase models namely, Volume of Fluid (VOF), Eulerian and Mixture models. Predictions of these models are compared with the present experimental data. The comparisons with the present measurements show that the pressure coefficient profile can be predicted using the Mixture model, see Fig. 7.2. Thus, in the present study, the Mixture model approach is applied to oil-in-water emulsion flow simulation in all S-shaped diffusers.
7.2.3 Resolution Effect Any CFD solution heavily depends on the size and fitness of meshing; therefore, care must be taken in selecting the grid types (coarse, medium or fine). With water flow and emulsion flow the computations of S-diffuser with its two tangents (upstream and downstream) are carried out for different grid sizes (mesh cells of 500000, 1080000 and 1500000 cells). The pressure coefficient is monitored for each grid with experimental measurements. The results indicate that the medium grid (1080000) is sensible, economical and can be used as it produces suitable data in view of the comparisons with the experimental results of the pressure coefficient, see Fig. 7.3, so it is adopted as default resolution.
108
Chapter (7)
Results and Discussion
1
Upstream
S-Diffuser
Downstream
0.9 0.8 0.7
Cpo
0.6 0.5 0.4
Water Flow at Re=34000 CFD (SST k-) CFD (STD k-) CFD (STD k-) EXP. data
0.3 0.2 0.1 0
-20
-10
0
10
20
30
40
50
60
70
Xo/W (a) outer-wall 1
Upstream
S-Diffuser
Downstream
0.9 0.8 0.7
Cpi
0.6 0.5 0.4
Water Flow at Re=34000 CFD (SST k-) CFD (STD k-) CFD (STD k-w) EXP. data
0.3 0.2 0.1 0
-20
-10
0
10
20
30
40
50
60
70
Xi /W (b) inner-wall Fig. 7.1 Turbulence modeling effect on the pressure coefficient distributions for model (7) carrying water
109
Chapter (7)
Results and Discussion Upstream
0.5
S-Diffuser
Downstream
0.45 0.4
Cp o
0.35 0.3 0.25 0.2 0.15
0.25 Stable Emulsion Flow at Re=24000 Num. [STD k-] CFD (Eulerian Scheme) CFD (Mixture Scheme) CFD ( VOF Scheme) EXP. data
0.1 0.05 0 -0.05 -20
-10
0
10
20
30
40
50
60
70
Xo/W (i) outer-wall 0.5
Upstream
S-Diffuser
Downstream
0.45 0.4 0.35
Cp
i
0.3 0.25 0.2
0.25 Stable Emulsion Flow at Re=24000 Num. [STD k-] CFD (Eulerian Scheme) CFD (Mixture Scheme) CFD ( VOF Scheme) EXP. data
0.15 0.1 0.05 0
-20
-10
0
10
20
30
40
50
60
70
Xi/W (ii) inner-wall Fig. 7.2.a Multi-phase modeling effect on the pressure coefficient distributions for model (7) carrying 0.25 stable emulsions
110
Chapter (7)
Results and Discussion Upstream
0.5
S-Diffuser
Downstream
0.45 0.4 0.35 0.3
Cp o
0.25 0.2 0.15
0.25 Stable o/w Emulsion Flow at Re=24000 Mixture Scheme
0.1
CFD (SST K-)
0.05
CFD (STD K-)
0
CFD (STD K-) EXP. data
-0.05
-20
-10
0
10
20
30
40
50
60
70
Xo/W (i) outer-wall 0.5
Upstream
S-Diffuser
Downstream
0.4
Cp i
0.3
0.25 Stable o/w Emulsion Flow at Re=24000 Mixture Scheme
0.2
CFD (SST K-)
0.1
CFD (STD K-) CFD (STD K-) EXP. data
0
-20
-10
0
10
20
30
40
50
60
70
Xi/W (ii) inner-wall Fig. 7.2.b Turbulence modeling effect on the Mixture scheme as a solution for predicting the pressure coefficient distributions of model (7) carrying 0.25 stable emulsions
111
Chapter (7)
Results and Discussion
Fig. 7.3 (a) Computational grid
112
Chapter (7)
Results and Discussion
1
Upstream
S-Diffuser
Downstream
0.9 0.8 0.7
Cpo
0.6 0.5 0.4 Water Flow at Re=34000 CFD Using STD k- Model 500000 Cell 1080000 Cell 1500000 Cell EXP. data
0.3 0.2 0.1 0 -0.1
-20
-10
0
10
20
30
40
50
60
70
Xo/W Fig. 7.3 (b) Outer-wall pressure coefficient distributions with water flow
0.5
Upstream
S-Diffuser
Downstream
0.45 0.4 0.35
Cp
o
0.3 0.25 0.2 0.25 Stable Emulsion Flow at Re=24000 CFD Using Mixture Model
0.15 0.1
500000 Cell 1080000 Cell 1500000 Cell EXP. data
0.05 0 -0.05 -20
-10
0
10
20
30
40
50
60
70
Xo/W (c) Outer-wall pressure coefficient distributions with 0.25 stable emulsion flow Fig. 7.3 Computational grid and resolutions effect on the results for model (7)
113
Chapter (7)
Results and Discussion
However, this is not the grid used in all simulations but the resolution depends on the studied case, but keeping in consideration for all cases; the nearest wall non-dimensional distance y by increasing the mesh quality at mesh creating and check it in the solution setup of the ANSYS R15.0 program.
7.3 Experimental and Numerical Results The present study considers and discusses experimentally and numerically the pressure coefficient and the performance of the S-shaped diffusers as well as presents diffuser loss coefficient correlation, respectively as the following.
7.3.1 Wall Static Pressure coefficient Distributions In the present study, it is aimed to extract the static pressure recovery coefficient distributions, Cp, for the S-shaped curved diffusers with different area ratios: (1.5, 2, and 3), different curvature ratios ( CR Rc /W 5, 7.5, and 12.5 ), different flow paths (45◦/45◦, 60◦/30◦ and 30◦/60◦), and different turn angles (45◦/45◦, 60◦/60◦ and 90◦/90◦), see Table 7.1. The results are presented experimentally and numerically for the unstable and stable (o/w) emulsions at different holdup () values ranging from 0.03 to 0.25. Measurements are performed at different diffuser inflow Reynolds numbers range of 9000 Re 34000 where
Re
EU ref D H with D 2W B 2(20)(80) 32mm (the diffuser inlet H W B (20 80) E
hydraulic diameter). The pressure recovery coefficients are first presented to obtain the corresponding energy-loss coefficients of the S-shaped curved diffusers carrying water as well as oil-in-water emulsion with the previous stated values of holdup for stable and unstable emulsions. Wall static pressure is measured as stated before in 3.5.2 and by applying equation (C.4.a), pressure recovery coefficients are obtained. The experimental and numerical studies are conducted for the twelve S-diffuser tested models to indicate the effect of inflow Reynolds number, (34000, 24000,
114
Chapter (7)
Results and Discussion
18000 and 12000), on the static pressure recovery coefficient on outer and inner walls of diffuser model 7, as shown in Figs. 7.4 and 7.5. From the results in Figs. 7.4 and 7.5, it is clear that the static pressure recovery coefficient of the S-shaped diffuser is found to be directly proportional to the diffuser inflow Reynolds number with a noticeable increase in the adverse pressure gradient inside the S-shaped diffuser at high Reynolds number. This leads to an increase of the boundary layer thickness on the walls and hence may leads to the chance of separation in the S-shaped diffuser. However, the increase of inflow velocity of emulsion flow reduces the loss coefficient as discussed later. Therefore, the performance of S-shaped diffuser can be improved with accelerating the inflow. Generally, the higher wall pressure distribution is observed on the outer walls of all models, because of the presence of radial pressure gradient generated by the centrifugal force due to longitudinal curvature. One notices that there is a distinct linear reduction of pressure along the outer-wall and the innerwall of the upstream and downstream tangent ducts of the S-shaped diffuser. The emulsion holdup and emulsion stability / instability are studied numerically and experimentally and it is found that: the increase of holdup leads to clear decrease in the static pressure recovery coefficient of the Sshaped diffuser and the unstable (o/w) emulsion pressure coefficient is higher than that of stable, as shown in Figs. 7.6 and 7.7. The flow resistance increases with high holdup because of its high viscosity and its large size oil droplet. The unstable emulsion works as a lubricant causing decrease in the wall friction coefficient and therefore, causing increase in pressure coefficient.
115
Chapter (7)
Results and Discussion CFD, EXP, CFD, EXP, CFD, EXP,
Re=34000 Re=34000 Re=24000 Re=24000 Re=18000 Re=18000
1 0.9 0.8 0.7
Cp
o
0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1
-20
-10
0
10
20
30
40
50
60
70
Xo/W (a) outer-wall CFD, EXP, CFD, EXP, CFD, EXP,
Re=34000 Re=34000 Re=24000 Re=24000 Re=18000 Re=18000
1 0.9 0.8 0.7
Cp
i
0.6 0.5 0.4 0.3 0.2 0.1 0 -20
-10
0
10
20
30
40
50
60
70
Xi /W
(b) inner-wall Fig. 7.4 Effect of Reynolds number on pressure coefficient of model (7) carrying water
116
Chapter (7)
Results and Discussion
1 CFD, Re=24000 EXP, Re=24000 CFD, Re=18000 EXP, Re=18000 CFD, Re=12000 EXP, Re=12000
0.9 0.8 0.7
Cp
o
0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1
-20
-10
0
10
20
30
40
50
60
70
Xo/W (a) outer-wall
1 CFD, Re=24000 EXP, Re=24000 CFD, Re=18000 EXP, Re=18000 CFD, Re=12000 EXP, Re=12000
0.9 0.8 0.7
Cp
i
0.6 0.5 0.4 0.3 0.2 0.1 0
-20
-10
0
10
20
30
40
50
60
70
Xi /W (b) inner-wall Fig. 7.5 Effect of Reynolds number on pressure coefficient of model (7) carrying 0.25 stable o/w emulsions
117
Chapter (7)
Results and Discussion CFD, water CFD, 0.03 (o/w) CFD, 0.06 (o/w) CFD, 0.10 (o/w) CFD, 0.15 (o/w) CFD, 0.25 (o/w)
EXP., water EXP., 0.03 (o/w) EXP., 0.06 (o/w) EXP., 0.10 (o/w) EXP., 0.15 (o/w) EXP., 0.25 (o/w)
1 0.9 0.8 0.7
Cpo
0.6 0.5 0.4 0.3 0.2 0.1 0 -20
-10
0
10
20
30
40
50
60
70
[Xo/W] (a) outer-wall CFD, water CFD, 0.03 (o/w) CFD, 0.06 (o/w) CFD, 0.10 (o/w) CFD, 0.15 (o/w) CFD, 0.25 (o/w)
EXP., water EXP., 0.03 (o/w) EXP., 0.06 (o/w) EXP., 0.10 (o/w) EXP., 0.15 (o/w) EXP., 0.25 (o/w)
1 0.9 0.8 0.7
Cp
i
0.6 0.5 0.4 0.3 0.2 0.1 0
-20
-10
0
10
20
30
40
50
60
70
[Xi/W]
(b) inner-wall Fig. 7.6 Effect of holdup (volume fraction) on pressure coefficient of model (7) carrying water and 0.03:0.25 stable o/w emulsions
118
Chapter (7)
Results and Discussion CFD, 0.03 unst. CFD, 0.03 stable CFD, 0.06 unst. CFD, 0.06 stable CFD, 0.25 unst. CFD, 0.25 stable
EXP., 0.03 EXP., 0.03 EXP., 0.06 EXP., 0.06 EXP., 0.25 EXP., 0.25
unst. stable unst. stable unst. stable
1 0.9 0.8 0.7
Cpo
0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1
-20
-10
0
10
20
30
40
50
60
70
[Xo/W] (a) outer-wall CFD, CFD, CFD, CFD, CFD, CFD,
0.03 0.03 0.06 0.06 0.25 0.25
EXP., 0.03 unst. EXP., 0.03 stable EXP., 0.06 unst. EXP., 0.06 stable EXP., 0.25 unst. EXP., 0.25 stable
unst. stable unst. stable unst. stable
1 0.9 0.8
Cp
i
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
-20
-10
0
10
20
30
40
50
60
70
[Xi/W] (b) inner-wall Fig. 7.7 Effect of stability on pressure coefficient of model (7) carrying 0.03, 0.06, 0.25 stable / unstable (o/w) emulsions
119
Chapter (7)
Results and Discussion
Results including the effects of S-diffuser geometrical parameters, area ratio, curvature ratio, turn angle and flow path on the diffuser outer wall / inner wall pressure coefficients at Re=34000 (water flow) and Re=24000 (emulsion flow) for all tested models are introduced in Figs. from 7.8 to 7.15. However, increasing the area ratio causes enhancement of the axial pressure gradients and this can be clearly visible in Figs. 7.8 and 7.9 in which the walls pressure increases for models 1 and 2 of the higher area ratios. The jump of the inner wall pressure distribution at Xi/W=5 which is followed by a strong decrease of the pressure at Xi/W=7, and then a gradual increase may be due to the inflection of the wall curvature. However, the presence of emulsion strongly reduces the walls pressure values. The variations of curvature ratio shown in Figs. 7.10 and 7.11 strongly affect the performance of the S-diffuser because of the generation of strong secondary flows and the higher centrifugal force with small radius of curvature as clear in model (6). The turn angle affects also the pressure recovery coefficient by change of the turn angle (models 7, 8, and 9) since strength of the generated counter rotating vortices inside the curved diffuser is higher for the higher angle of turn and this leads to increasing the losses and the flow resistance inside the S-diffuser model (8), see Figs. 7.12 and 7.13 for both water and emulsion flows. The flow path plays also an important role on diffuser performance, since it is noticed clearly that Cp increases with symmetric turn angle and decreases with asymmetric turn angle (models 10, 11, and 12), i. e. as the flow path changes as 45º/45º, 30º/60º, and 60º/30º the pressure coefficient strongly decreases respectively as shown in Figs. 7.14 (water flow) and 7.15 (emulsion flow). This behavior may be due to the enhanced intensity of internal vortices due to side walls curvature and the flow resistance increases. From the results in Figs. from 7.8 to 7.15, it is noticed that: the pressure recovery coefficient of the S-diffusers with water / emulsion flow is found to be directly affected by area ratio, curvature ratio, symmetry of the S-shaped diffuser turn angle and inflow Reynolds number as well as the flow status (water or emulsion flow).
120
Chapter (7)
Results and Discussion
1 0.9 0.8 0.7
Cp
o
0.6 0.5 0.4 0.3 0.2
CFD, EXP, CFD, EXP, CFD, EXP,
0.1 0 -20
-10
0
10
20
30
40
50
M1 M1 M2 M2 M3 M3
60
70
Xo /W (a) outer-wall
1 0.9 0.8 0.7
Cp
i
0.6 0.5 0.4 0.3 CFD, EXP, CFD, EXP, CFD, EXP,
0.2 0.1 0
-20
-10
0
10
20
30
40
50
M1 M1 M2 M2 M3 M3
60
70
Xi /W (b) inner-wall Fig.7.8 Effect of diffuser area ratio on the diffuser pressure coefficient for water flow (M 1: AR=3, M 2: AR=2 and M 3: AR=1.5)
121
Chapter (7)
Results and Discussion
1 0.9 0.8 0.7
CFD, EXP, CFD, EXP, CFD, EXP,
M1 M1 M2 M2 M3 M3
50
60
Cp
o
0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1
-20
-10
0
10
20
30
40
70
Xo/W (a) outer-wall
1 CFD, M1
0.9
EXP, M1 CFD, M2
0.8
EXP, M2 CFD, M3 EXP, M3
0.7
Cp
i
0.6 0.5 0.4 0.3 0.2 0.1 0
-20
-10
0
10
20
30
40
50
60
70
Xi/W (b) inner-wall Fig.7.9 Effect of diffuser area ratio on the diffuser pressure coefficient at Φ =0.25
122
Chapter (7)
Results and Discussion
1 0.9 0.8 0.7
Cp
o
0.6 0.5 0.4 0.3 CFD, M5 EXP, M5 CFD, M4 EXP, M4 CFD, M6 EXP, M6
0.2 0.1 0 -0.1
-20
-10
0
10
20
30
40
50
60
70
Xo /W (a) outer-wall 1 0.9 0.8 0.7
Cp
i
0.6 0.5 0.4 0.3
CFD, M5 EXP, M5 CFD, M4 EXP, M4 CFD, M6 EXP, M6
0.2 0.1 0
-20
-10
0
10
20
30
40
50
60
70
Xi /W (b) inner-wall Fig. 7.10 Effect of diffuser curvature ratio on the diffuser pressure coefficient for water flow (M 5: CR=12.5, M 4: CR=7.5 and M 6: CR=5)
123
Chapter (7)
Results and Discussion
1 CFD, M5 EXP, M5 CFD, M4 EXP, M4 CFD, M6 EXP, M6
0.9 0.8 0.7
Cp
o
0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1
-20
-10
0
10
20
30
40
50
60
70
Xo /W (a) outer-wall
1 CFD, M5 EXP, M5 CFD, M4 EXP, M4 CFD, M6 EXP, M6
0.9 0.8 0.7
Cp
i
0.6 0.5 0.4 0.3 0.2 0.1 0
-20
-10
0
10
20
30
40
50
60
70
Xi /W (b) inner-wall Fig. 7.11 Effect of diffuser curvature ratio on diffuser pressure coefficient at Φ =0.15
124
Chapter (7)
Results and Discussion
1 0.9 0.8 0.7 0.6
Cp
o
0.5 0.4 0.3
CFD, M7 EXP, M7 CFD, M9 EXP, M9 CFD, M8 EXP, M8
0.2 0.1 0 -0.1
-20
-10
0
10
20
30
40
50
60
70
Xo /W (a) outer-wall
1 0.9 0.8 0.7
Cp
i
0.6 0.5 0.4
CFD, M7 EXP, M7 CFD, M9 EXP, M9 CFD, M8 EXP, M8
0.3 0.2 0.1 0
-20
-10
0
10
20
30
40
50
60
70
Xi /W (b) inner-wall Fig. 7.12 Effect of diffuser turn angle on the diffuser pressure coefficient for water flow (M 7: 60°/60°, M 9: 45°/45° and M 8: 90°/90°)
125
Chapter (7)
Results and Discussion
1 CFD, M7 EXP, M7 CFD, M9 EXP, M9 CFD, M8 EXP, M8
0.9 0.8 0.7 0.6
Cp
o
0.5 0.4 0.3 0.2 0.1 0 -0.1
-20
-10
0
10
20
30
40
50
60
70
Xo /W (a) outer-wall
1 CFD, M7 EXP, M7 CFD, M9 EXP, M9 CFD, M8 EXP, M8
0.9 0.8 0.7
Cp
i
0.6 0.5 0.4 0.3 0.2 0.1 0
-20
-10
0
10
20
30
40
50
60
70
Xi /W (b) inner-wall Fig. 7.13 Effect of diffuser turn angle on the diffuser pressure coefficient at Φ =0.15
126
Chapter (7)
Results and Discussion
1 0.9 0.8 0.7
Cp
o
0.6 0.5 0.4 0.3 CFD, M10 EXP, M10 CFD, M12 EXP, M12 CFD, M11 EXP, M11
0.2 0.1 0 -0.1
-20
-10
0
10
20
30
40
50
60
70
Xo /W (a) outer-wall
1 0.9 0.8 0.7
Cp
i
0.6 0.5 0.4 CFD, M10 EXP, M10 CFD, M12 EXP, M12 CFD, M11 EXP, M11
0.3 0.2 0.1 0
-20
-10
0
10
20
30
40
50
60
70
Xi /W (b) inner-wall Fig. 7.14 Effect of diffuser flow path on the diffuser pressure coefficient for water flow (M 10: 45°/45°, M 12: 30°/60° and M 11: 60°/30°)
127
Chapter (7)
Results and Discussion
1 CFD, M10 EXP, M10 CFD, M12 EXP, M12 CFD, M11 EXP, M11
0.9 0.8 0.7 0.6
Cp
o
0.5 0.4 0.3 0.2 0.1 0 -0.1
-20
-10
0
10
20
30
40
50
60
70
Xo /W (a) outer-wall
1 CFD, M10 EXP, M10 CFD, M12 EXP, M12 CFD, M11 EXP, M11
0.9 0.8 0.7
Cp
i
0.6 0.5 0.4 0.3 0.2 0.1 0
-20
-10
0
10
20
30
40
50
60
70
Xi /W (b) inner-wall Fig. 7.15 Effect of diffuser flow path on the diffuser pressure coefficient at Φ =0.15
128
Chapter (7)
Results and Discussion
7.4 Diffuser Performance The effect of Reynolds number on the energy-loss coefficient for all models carrying water and stable (o/w) emulsions at different holdups are presented experimentally and numerically in Fig. 7.16. For all tested S-shaped diffuser models the energy-loss coefficient is found to be inversely proportional to the diffuser inflow Reynolds number because of the enhanced turbulence generated in the S-shaped diffuser with high Reynolds number. For emulsion flow, increasing Reynolds number causes break-up for the large oil droplets and this leads to lower flow resistance. The effect of the emulsion holdup (volume fraction, Φ, ranged from 0.03 to 0.25) on the diffuser energy-loss coefficient for all models carrying water and stable (o/w) emulsion are presented experimentally and numerically in Fig. 7.17. From the results in Fig. 7.17 it is found that the energy-loss coefficient of the S-diffuser models are strongly affected by the holdup (volume fraction) by different values from model to other. The comparisons in Figs. 7.16 and 7.17 reveal that the flow in S-shaped diffuser model (3) exhibits higher values of the diffuser energy-loss coefficient than the other models, while model (7) exhibits the lowest value of the energy-loss coefficient compared with the other tested models. Therefore, the best design of all tested S-shaped diffusers is model (7) which has area ratio of 3, curvature ratio of 7.5, and symmetric center-line turn angle of (60°/60°). The effect of emulsion stability (stable / unstable) on S-diffuser performance for models 3, 7, and 11 are presented experimentally and numerically in Fig. 7.18 in which the energy-loss coefficient of the S-diffuser models is affected by the emulsion stability.
129
Chapter (7)
Results and Discussion 1.8 1.6 1.4
EXP (M 8) CFD (M 8) EXP (M 1&9) CFD (M 1&9) EXP (M 7) CFD (M 7)
EXP (M 12) CFD (M 12) EXP (M 2&4) CFD (M 2&4) EXP (M 5&10) CFD (M 5&10)
EXP (M3) CFD (M3) EXP (M11) CFD (M11) EXP (M 6) CFD (M 6)
Kd
1.2 1 0.8 0.6 0.4 0.2 0
6
9
12
15
18
21
Re10
3
24
27
30
33
36
(a) Φ = 0.0 (water) 1.8 EXP (M 12) CFD (M 12) EXP (M 2&4) CFD (M 2&4) EXP (M 5&10) CFD (M 5&10)
EXP (M 3) CFD (M 3) EXP (M 11) CFD (M 11) EXP (M 6) CFD (M 6)
1.6 1.4
EXP (M 8) CFD (M 8) EXP (M 1&9) CFD (M 1&9) EXP (M 7) CFD (M 7)
Kd
1.2 1
0.8 0.6 0.4 0.2 0
6
9
12
15
18
21
Re103
24
27
30
33
36
(b) Φ = 0.03 stable (o/w) emulsion Fig. 7.16 Effect of Reynolds number on the energy-loss coefficient for all models carrying water and stable (o/w) emulsion, continued
130
Chapter (7)
Results and Discussion
1.8 EXP (M 3) CFD (M 3) EXP (M 11) CFD (M 11) EXP (M 6) CFD (M 6)
1.6 1.4
EXP (M 12) CFD (M 12) EXP (M 2&4) CFD (M 2&4) EXP (M 5&10) CFD (M 5&10)
EXP (M 8) CFD (M 8) EXP (M 1&9) CFD (M 1&9) EXP (M 7) CFD (M 7)
Kd
1.2 1
0.8 0.6 0.4 0.2 0
6
9
12
15
18
21
Re103
24
27
30
33
36
(c) Φ = 0.06 stable (o/w) emulsion 1.8
EXP (M 3) CFD (M 3) EXP (M 11) CFD (M 11) EXP (M 6) CFD (M 6)
1.6
EXP (M 12) CFD (M 12) EXP (M 2&4) CFD (M 2&4) EXP (M 5&0) CFD (M 5&0)
EXP (M 8) CFD (M 8) EXP (M 1&9) CFD (M 1&9) EXP (M 7) CFD (M 7)
1.4
Kd
1.2 1
0.8 0.6 0.4 0.2 0
6
9
12
15
18
21
Re103
24
27
30
33
36
(d) Φ = 0.10 stable (o/w) emulsion Fig. 7.16 Effect of Reynolds number on the energy-loss coefficient for all models carrying water and stable (o/w) emulsion, continued
131
Chapter (7)
Results and Discussion
1.8 1.6 1.4
Kd
1.2 1
0.8 0.6 0.4 0.2 0
EXP (M 12) CFD (M 12) EXP (M 2&4) CFD (M 2&4) EXP (M 5&10) CFD (M 5&10)
EXP (M 3) CFD (M 3) EXP (M 11) CFD (M 11) EXP (M 6) CFD (M 6)
6
9
12
15
18
21
Re103
24
EXP (M 8) CFD (M 8) EXP (M 1&9) CFD (M 1&9) EXP (M 7) CFD (M 7)
27
30
33
36
(e) Φ = 0.15 stable (o/w) emulsion 1.8 1.6 1.4
Kd
1.2 1
0.8 0.6 0.4
EXP (M 3) CFD (M 3) EXP (M 11) CFD (M 11) EXP (M 6) EXP (M 6)
0.2 0
6
9
12
15
EXP (M 12) CFD (M 12) EXP (M 2&4) CFD (M 2&4) EXP (M 5&10) CFD (M 5&10)
18
21
Re103
24
EXP (M 8) CFD (M 8) EXP (M 1&9) CFD (M 1&9) EXP (M 7) CFD (M 7)
27
30
33
36
(f) Φ = 0.25 stable (o/w) emulsion Fig. 7.16 Effect of Reynolds number on the energy-loss coefficient for all models carrying water and stable (o/w) emulsion
132
Chapter (7)
Results and Discussion
1.8 1.6
0.15 -EXP 0.15 -CFD 0.25 -EXP 0.25 -CFD
0.06 -EXP 0.06 -CFD 0.10 -EXP 0.10 -CFD
0.00 -EXP 0.00 -CFD 0.03 -EXP 0.03 -CFD
1.4
Kd
1.2 1
0.8 0.6 0.4 0.2 0
6
9
12
15
18
21
Re103
24
27
30
33
36
(a) Models 1 and 9 1.8
0.00 -EXP 0.00 -CFD 0.03 -EXP 0.03 -CFD
1.6 1.4
0.06 -EXP 0.06 -CFD 0.10 -EXP 0.10 -CFD
Kd
1.2
0.15 -EXP 0.15 -CFD 0.25 -EXP 0.25 -CFD
1
0.8 0.6 0.4 0.2 0
6
9
12
15
18
21
Re103
(b)
24
27
30
33
36
Models 2 and 4
Fig. 7.17 Effect of the emulsion holdup (volume fraction, Φ) on the diffuser energy loss coefficient for all models carrying water and stable (o/w) emulsion, continued
133
Chapter (7)
Results and Discussion
1.8 1.6 1.4
Kd
1.2 1
0.8 0.6 0.4 0.00 -EXP 0.00 -CFD 0.03 -EXP 0.03 -CFD
0.2 0
6
9
12
15
0.15 -EXP 0.15 -CFD 0.25 -EXP 0.25 -CFD
0.06 -EXP 0.06 -CFD 0.10 -EXP 0.10 -CFD
18
21
Re103
24
27
30
33
36
(c) Model 3 1.8
0.00 -EXP 0.00 -CFD 0.03 -EXP 0.03 -CFD
1.6
0.15 -EXP 0.15 -CFD 0.25 -EXP 0.25 -CFD
0.06 -EXP 0.06 -CFD 0.10 -EXP 0.10 -CFD
1.4
Kd
1.2 1
0.8 0.6 0.4 0.2 0
6
9
12
15
18
21
Re103
24
27
30
33
36
(d) Models 5 and 10 Fig. 7.17 Effect of the emulsion holdup (volume fraction, Φ) on the diffuser energy loss coefficient for all models carrying water and stable (o/w) emulsion, continued
134
Chapter (7)
Results and Discussion
1.8
0.00 -EXP 0.00 -CFD 0.03 -EXP 0.03 -CFD
1.6
0.06 -EXP 0.06 -CFD 0.10 -EXP 0.10 -CFD
1.4
Kd
1.2
0.15 -EXP 0.15 -CFD 0.25 -EXP 0.25 -CFD
1
0.8 0.6 0.4 0.2 0
6
9
12
15
18
21
Re103
24
27
30
33
36
(e) Model 6 1.8
0.00 -EXP 0.00 -CFD 0.03 -EXP 0.03 -CFD
1.6
0.06 -EXP 0.06 -CFD 0.10 -EXP 0.10 -CFD
0.15 -EXP 0.15 -CFD 0.25 -EXP 0.25 -CFD
1.4
Kd
1.2 1
0.8 0.6 0.4 0.2 0
6
9
12
15
18
21
Re103
24
27
30
33
36
(f) Model 7 Fig. 7.17 Effect of the emulsion holdup (volume fraction, Φ) on the diffuser energy loss coefficient for all models carrying water and stable (o/w) emulsion, continued
135
Chapter (7)
Results and Discussion
1.8
0.00 -EXP 0.00 -CFD 0.03 -EXP 0.03 -CFD
1.6
0.15 -EXP 0.15 -CFD 0.25 -EXP 0.25 -CFD
0.06 -EXP 0.06 -CFD 0.10 -EXP 0.10 -CFD
1.4
Kd
1.2 1
0.8 0.6 0.4 0.2 0
6
9
12
15
18
21
Re103
24
27
30
33
36
(g) Model 8 1.8 1.6 1.4
Kd
1.2 1
0.8 0.6 0.4 0.00 -EXP 0.00 -CFD 0.03 -EXP 0.03 -CFD
0.2 0
6
9
12
15
0.15 -EXP 0.15 -CFD 0.25 -EXP 0.25 -CFD
0.06 -EXP 0.06 -CFD 0.10 -EXP 0.10 -CFD
18
21
Re103
24
27
30
33
36
(h) Model 11 Fig. 7.17 Effect of the emulsion holdup (volume fraction, Φ) on the diffuser energy loss coefficient for all models carrying water and stable (o/w) emulsion, continued
136
Chapter (7)
Results and Discussion
1.8
0.00 -EXP 0.00 -CFD 0.03 -EXP 0.03 -CFD
1.6 1.4
0.06 -EXP 0.06 -CFD 0.10 -EXP 0.10 -CFD
Kd
1.2
0.15 -EXP 0.15 -CFD 0.25 -EXP 0.25 -CFD
1
0.8 0.6 0.4 0.2 0
6
9
12
15
18
21
24
27
30
33
36
Re103 (i) Model 12 Fig. 7.17 Effect of the emulsion holdup (volume fraction, Φ) on the diffuser energy loss coefficient for all models carrying water and stable (o/w) emulsion It is noticed that the unstable (o/w) emulsion flow has lower resistance than the stable because in the unstable, the oil droplets lubricate the diffuser walls and this leads to low friction coefficient which decreases the losses. The energy-loss coefficient of the S-shaped diffusers decreases as the area ratio or the curvature ratio increases as shown in Figs. 7.19 and 7.20, respectively. The reduction of energy loss with the increase of area ratio is due to the higher pressure recovery generated in the diffuser of high area ratio. The increase of curvature ratio is due to the increase of curvature radius and hence a reduction of internal secondary flows. The effect of S-diffuser symmetric turn angle is shown in Fig. 7.21, and the loss coefficient is small affected by change of the turn angle. The diffuser loss coefficient is strongly affected by diffuser flow path since it is noticed clearly that it increases with the symmetric turn angle and decreases with the asymmetric turn angle as shown in Fig. 7.22.
137
Chapter (7)
Results and Discussion
1.8 1.6 1.4
Kd
1.2 1
0.8 0.6 0.4 0.2 0
6
9
12
15
0.25 St.-EXP 0.25 St.-CFD 0.25 Unst.-EXP 0.25 Unst.-CFD
0.15 St.-EXP 0.15 St.-CFD 0.15 Unst.-EXP 0.15 Unst.-CFD
0.03 St. -EXP 0.03 St. -CFD 0.03 Ust. -EXP 0.03 Ust. -CFD
18
21
Re103
24
27
30
33
36
(a) Model 3: Φ = 0.03, 0.15, 0.25 1.8 Water -EXP Water -CFD
1.6
0.06 St.-EXP 0.06 St.-CFD 0.06 Unst.-EXP 0.06 Unst.-CFD
1.4
0.10 St.-EXP 0.10 St.-CFD 0.10 Unst.-EXP 0.10 Unst.-CFD
Kd
1.2 1
0.8 0.6 0.4 0.2 0
6
9
12
15
18
21
Re103
24
27
30
33
36
(b) Model 3: Φ = 0.00, 0.06, 0.10 Fig. 7.18 Effect of emulsion stability on S-diffuser performance for sample models, continued
138
Chapter (7)
Results and Discussion
1.8 1.6 1.4
0.25 St.-EXP 0.25 St.-CFD 0.25 Unst.-EXP 0.25 Unst.-CFD
0.15 St.-EXP 0.15 St.-CFD 0.15 Unst.-EXP 0.15 Unst.-CFD
0.03 St. -EXP 0.03 St. -CFD 0.03 Ust. -EXP 0.03 Ust. -CFD
Kd
1.2 1
0.8 0.6 0.4 0.2 0
6
9
12
15
18
21
Re103
24
27
30
33
36
(c) Model 7: Φ = 0.03, 0.15, 0.25 1.8 0.06 St.-EXP 0.06 St.-CFD 0.06 Unst.-EXP 0.06 Unst.-CFD
Water -EXP Water -CFD
1.6 1.4
0.10 St.-EXP 0.10 St.-CFD 0.10 Unst.-EXP 0.10 Unst.-CFD
Kd
1.2 1
0.8 0.6 0.4 0.2 0
6
9
12
15
18
21
Re103
24
27
30
33
36
(d) Model 7: Φ = 0.00, 0.06, 0.10 Fig. 7.18 Effect of emulsion stability on S-diffuser performance for sample models, continued
139
Chapter (7)
Results and Discussion
1.8 0.03 St. -EXP 0.03 St. -CFD 0.03 Ust. -EXP 0.03 Ust. -CFD
1.6 1.4
0.15 St.-EXP 0.15 St.-CFD 0.15 Unst.-EXP 0.15 Unst.-CFD
Kd
1.2 1
0.25 St.-EXP 0.25 St.-CFD 0.25 Unst.-EXP 0.25 Unst.-CFD
0.8 0.6 0.4 0.2 0
6
9
12
15
18
21
Re103
24
27
30
33
36
(e) Model 11: Φ = 0.03, 0.15, 0.25 1.8 1.6
0.10 St.-EXP 0.10 St.-CFD
0.06 St.-EXP 0.06 St.-CFD 0.06 Unst.-EXP 0.06 Unst.-CFD
Water -EXP Water -CFD
1.4
0.10 Unst.-EXP 0.10 Unst.-CFD
Kd
1.2 1
0.8 0.6 0.4 0.2 0
6
9
12
15
18
21
Re103
24
27
30
33
36
(f) Model 11: Φ = 0.00, 0.06, 0.10 Fig. 7.18 Effect of emulsion stability on S-diffuser performance for sample models
140
Chapter (7)
Results and Discussion
1.8 1.6 1.4
Kd
1.2 1
0.8 0.6 0.4
0
6
9
12
15
EXP (M 1) CFD (M 1)
EXP (M 2) CFD (M 2)
EXP (M 3) CFD (M 3)
0.2
18
21
Re103
24
27
30
33
36
(a) 0.25 stable (o/w) emulsion 1.8
EXP (M 1) CFD (M 1)
EXP (M 2) CFD (M 2)
EXP (M 3) CFD (M 3)
1.6 1.4
Kd
1.2 1
0.8 0.6 0.4 0.2 0
6
9
12
15
18
21
Re10
3
24
27
30
33
36
(b) Water Fig. 7.19 Effect of S-diffuser area ratio on the diffuser energy-loss coefficient
141
Chapter (7)
Results and Discussion
1.8 1.6 1.4
Kd
1.2 1
0.8 0.6 0.4
EXP (M 6) CFD (M 6)
0.2 0
6
9
12
15
EXP (M 4) CFD (M 4) 18
21
Re103
24
EXP (M 5) CFD (M 5) 27
30
33
36
(a) 0.25 stable (o/w) emulsion 1.8
EXP (M 4) CFD (M 4)
EXP (M 6) CFD (M 6)
1.6
EXP (M 5) CFD (M 5)
1.4
Kd
1.2 1
0.8 0.6 0.4 0.2 0
6
9
12
15
18
21
Re103
24
27
30
33
36
(b) Water Fig. 7.20 Effect of S-diffuser curvature ratio on the diffuser energy-loss coefficient
142
Chapter (7)
Results and Discussion
1.8 1.6 1.4
Kd
1.2 1
0.8 0.6 0.4
EXP (M 7) CFD (M 7)
0.2 0
6
9
12
15
EXP (M 8) CFD (M 8) 18
21
Re103
24
EXP (M 9) CFD (M 9) 27
30
33
36
(a) 0.25 stable (o/w) emulsion 1.8
EXP (M 7) CFD (M 7)
1.6
EXP (M 9) CFD (M 9)
EXP (M 8) CFD (M 8)
1.4
Kd
1.2 1
0.8 0.6 0.4 0.2 0
6
9
12
15
18
21
Re103
24
27
30
33
36
(b) Water Fig. 7.21 Effect of S-diffuser turn angle on the diffuser energy-loss coefficient
143
Chapter (7)
Results and Discussion
1.8 1.6 1.4
Kd
1.2 1
0.8 0.6 0.4
0
EXP (M 11) CFD (M 11)
EXP (M 10) CFD (M 10)
0.2 6
9
12
15
18
21
Re103
24
EXP (M 12) CFD (M 12) 27
30
33
36
(a) 0.25 stable (o/w) emulsion 1.8
EXP (M 11) CFD (M 11)
EXP (M 10) CFD (M 10)
1.6
EXP (M 12) CFD (M 12)
1.4
Kd
1.2 1
0.8 0.6 0.4 0.2 0
6
9
12
15
18
21
Re103
24
27
30
33
36
(b) Water Fig. 7.22 Effect of S-diffuser flow path on the diffuser energy-loss coefficient
144
Chapter (7)
Results and Discussion
From the results in Figs. from 7.19 to 7.22 it is generally concluded that: energy-loss coefficient of the S-diffusers with water flow is found to be inversely proportional to area ratio, curvature ratio, symmetry of the Sshaped diffuser turn angle and inflow Reynolds number. The S-diffuser energy-loss coefficient approaches asymptotic values for high diffuser inflow Reynolds number. For the same radius of curvature, i.e. an equal main-flow length, the pressure gain from the widest diffuser (the highest area ratio) is higher than the energy lost due to intensive secondary flows produced; it means that the diffuser energy loss decreases with increasing the area ratio, see Fig. 7.19. The main reason of the curvature ratio effects is the decreasing of the centrifugal force created in the S-shaped curved diffuser with increasing the curvature ratio, which is responsible for secondary flow generation. As the curvature ratio decreases the energy loss in the S-shaped diffuser increases for all different flows, because of the generation of strong secondary flows (with small radius of curvature) superimposed on the mean flow, see Fig. 7.20. This behavior is attributed to the change of turbulence intensity which may be affected by the area ratio and the curvature ratio of the S-diffusers. The diffuser energy-loss coefficient is affected by change of the symmetric turn angle because of the different three models turn angles are symmetric and as symmetric turn angle increases energy-loss coefficient increases, since the strength of the counter rotating vortices is being higher for the higher angle of turn and this causes relatively high flow resistance through the S- diffuser, see Fig. 7.21. At the same Reynolds number, the same curvature radius and the same sum of the two bends turn angles ( 1 2 90 ) the energy-loss coefficient of the S-diffuser increases with asymmetric turn angle and decreases with symmetric turn angle (45°/45°), see Fig. 7.22.
145
Chapter (7)
Results and Discussion
This behavior tends to the created secondary flows because of the strong turn of S-shaped diffuser. This indicates the lower resistance to the flow in the case of symmetric turn angle. The presence of asymmetric turn angle increases the drag of flow to move in the S-diffuser and this enhances the energy loss of the S-diffuser.
7.3.3 Loss Coefficient Correlation The present study presents general new correlations of the S-diffuser loss coefficient for all present studied cases exploring the ranges of geometrical parameters and flow parameters. These correlations are extracted from the present experimental data. Correlations equations in water flow and stable (o/w) emulsion flow can be extracted from the curve fitting and can be written as the following: (a) Water flow
Kd
= W
1138100.119 1.162011
Re
× θ1
-0.3520896
× CR
0.70444
-0.1048264
× [θ 2 / θ 1 ] × AR
0.695365
(7.1)
(b) Emulsion flow
Kd = E
9763.069 × Φ0.348498 × θ10.03964059 × [θ2 / θ1 ]-0.069 Re0.8128904 × CR0.0882963 × AR0.8632483 (7.2)
The calculated loss coefficients according to these correlations equations are plotted against the measured loss coefficients and valid all cases studied in the present work with error ± 40 % in case of water flow and ± 30% in case of emulsion flow as shown in Fig. 7.23.
146
Chapter (7)
Results and Discussion
1.2
Calculated Loss Coefficient
+40% 1 0.8 0.6
-40%
0.4 0.2 0
0
0.2
0.4
0.6
0.8
1
1.2
Measured Loss Coefficient (a) Water flow
Calculated Loss Coefficient
2
+30%
1.6
1.2
-30% 0.8
0.4
0
0
0.4
0.8
1.2
1.6
Measured Loss Coefficient (b) Emulsion flow Fig. 7.23 Calculated and measured loss coefficients 147
2
Chapter (7)
Results and Discussion
7.5 Numerical Results For all the following numerical results the directions and the wall locations in 2-D with X axis into paper are considered as below.
Fig. 7.24 Local cross-section plane of S-shaped diffuser Computations of the streamwise mean velocity, U (axial velocity) and the turbulent shear stress ( -u v ) contours at different five computation stations (inlet plane, mid-first bend plane, inflection plane, mid-second bend plane and exit plane) along the S-shaped diffuser are simulated in local cross-sectional area shown in Fig. 7.24 and presented as shown in Figs. 7.25 and 7.26, respectively. The figures show that the maximum velocity core and maximum turbulent shear stress core are approaching the outer-wall of the S-diffuser by approaching the exit plane as clear in Figs. 7.25 and 7.26. This is due to the inflection of curvature of walls after the inflection plane. Figures 7.27 and 7.28 indicate the Reynolds number effect on the computational axial velocity and turbulent stress contours of water flow at the different five planes along the S-diffuser (model-7). It can be seen from these figures that the axial velocity and turbulent stress are strongly affected by the inflow Reynolds number due to the same reasons given in Figs. 5.25 and 26. The complexity of the flow is clearly viewed at the inflection plane.
148
Chapter (7)
Results and Discussion
[m/s]
(a) Inlet Plane
[m/s]
(b) Mid first bend Plane
[m/s]
(c) Inflection Plane
[m/s]
(d) Mid second bend Plane
[m/s]
(e) Exit Plane
Fig. 7.25 Position effect on computational axial velocity contours of water flow at different sections along the S-Diffuser (Model 7)
149
Chapter (7)
Results and Discussion
Pa
(a) Inlet Plane
Pa
(b) Mid first bend Plane
Pa
(c) Inflection Plane
Pa
(d) Mid second bend Plane
Pa
(e) Exit Plane
Fig. 7.26 Position effect on computational turbulent shear stress contours of water flow at different sections along the S-Diffuser (Model 7)
150
Chapter (7)
Results and Discussion
(i) Re=34000
(ii) Re=24000
(iii)
[m/s]
Re=18000
(a) Inlet Plane (0°)
(i) Re=34000
(ii) Re=24000
(iii) [m/s]
Re=18000
Fig. 7.27 (b) Mid-first bend plane (30°) Fig. 7.27 (b) Mid-first bend plane (30°)
151
Chapter (7)
Results and Discussion
(i) Re=34000
(ii) Re=24000
(iii)
[m/s]
Re=18000
(c) Inflexion Plane (60°)
(i) Re=34000
(ii) Re=24000
(iii)
[m/s]
Re=18000
Fig. 7.27 (d) Mid-second bend Plane (60°/30°)
152
Chapter (7)
Results and Discussion
(i) Re=34000
Re=24000
(iii)
[m/s]
Re=18000
(e) Exit Plane (60°/60°) Fig. 7.27 Reynolds number effect on computational axial velocity contours of water flow at different sections along the S-Diffuser (Model 7)
(i) Re=34000
(ii) Re=24000
(iii) Re=18000
Fig. 7.28 (a) Inlet Plane (0°) 153
Chapter (7)
Results and Discussion
(i) Re=34000
(ii) Re=24000
(iii) Re=18000
(b) Mid-first bend plane (30°)
(i) Re=34000
(ii) Re=24000
(iii) Re=18000
Fig. 7.28 (c) Inflexion Plane (60°) 154
Chapter (7)
Results and Discussion
(i) Re=34000
(ii) Re=24000
(iii) Re=18000
(d) Mid-second bend Plane (60°/30°)
(i) Re=34000
(ii) Re=34000
(iii) Re=34000
(e) Exit Plane (60°/60°) Fig. 7.28 Reynolds number effect on computational turbulent stress contours of water flow at different sections along the S-Diffuser (Model 7)
155
Chapter (7)
Results and Discussion
The holdup effect on exit computational axial velocity contour of model-7 at Re=24000 for dispersed phase concentrations of Φ = 0.0, 0.10 and 0.25 by volume is presented in Fig. 7.29 that indicates that the axial velocity is to be inversely proportional to the dispersed phase holdup. This behavior tends to the higher flow resistance associated with the higher holdup. Also, this behavior is clear in Fig. 7.30 that shows the holdup effect on axial velocity vectors at the different five planes for model-7. The holdup effect on exit turbulent shear stress contours of model-7 at Re=24000 is presented in Fig. 7.31 that shows that the turbulent shear stress increases as the holdup decreases. Stability effect on exit computational axial velocity and exit computational turbulent stress contours for model-7 at Re=24000 with dispersed phase concentration of 0.25 are computed in Figs. 7.32 and 7.33 that show the axial velocity and turbulent stress of the unstable o/w emulsion are close to that of the stable o/w emulsion. Area ratio effect on exit (at exit plane) computational axial velocity and turbulent shear stress contours for models 1, 2 and 3 at Re=34000 of water flow are presented in Fig. 7.34 and 7.35 that show increasing velocity with small cross-section area (model-3) according to the continuity equation. The turbulent shear stress increases also as area ratio decreases.
156
Chapter (7)
Results and Discussion
[m/s] (a) Water
( b ) 0 . 1 0 [m/s]
Unstable (o/w)
[m/s] (c) 0.25 Unstable (o/w) Fig. 7.29 Holdup effect on exit computational axial velocity contour of Model-7 at Re=24000
157
Chapter (7)
Results and Discussion
[m/s] (a) Water
[m/s] (b) 0.10 Unstable (o/w)
[m/s] (c) 0.25 Unstable (o/w) Fig. 7.30 Holdup effect on the five computational axial velocity vectors for Model 7
158
Chapter (7)
Results and Discussion
(a) Water
(b) 0.10 Unstable (o/w)
(c) 0.25 Unstable (o/w) Fig. 7.31 Holdup effect on exit computational turbulent shear stress contour of Model-7 at Re=24000
159
Chapter (7)
Results and Discussion
[m/s] (a) 0.25 Unstable (o/w)
[m/s] (b) 0.25 Stable (o/w)
Fig. 7.32 Stability effect on exit axial velocity contours of Model-7 at Re=24000
160
Chapter (7)
Results and Discussion
(b) 0.25 Stable (o/w)
(a) 0.25 Unstable (o/w)
Fig. 7.33 Stability effect on exit computational turbulent stress contours of Model-7
161
Chapter (7)
Results and Discussion
[m/s]
(a) AR = 3.0 (Model-1)
(b) AR = 2.0 (Model-2)
[m/s]
[m/s]
(c) AR = 1.5 (Model-3)
Fig. 7.34 Area ratio effect on computational axial velocity contour at exit plane at Re=24000
162
Chapter (7)
Results and Discussion
(a) AR = 3.0 (Model-1)
(b) AR = 2.0 (Model-2)
(c) AR = 1.5 (Model-3) Fig. 7.35 Area ratio effect on computational turbulent shear stress contour at exit plane at Re=24000
163
Chapter (7)
Results and Discussion
Figures 7.36 and 7.37 show the curvature ratio effect on computational axial velocity and turbulent stress contours for water flow at exit plane. It can be seen from these figures that the axial velocity and turbulent stress are to be inversely proportional to diffuser curvature ratio due to the centrifugal force effect results from S-diffuser curvature. Turn angle effect on computational axial velocity and turbulent shear stress contours at the exit plane for models 7, 8 and 9 with inflow Reynolds number of 34000, of water flow are presented in Figs. 7.38 and 7.39, respectively that show the highest axial velocity and turbulent shear stress are in model-7 then decrease inside model-8 and the lowest are in model-9. This behavior tends to the effect of forming counter-rotating vorticities near diffusers exits.
164
Chapter (7)
[m/s]
Results and Discussion
(a) CR = 7.5 (Model-4)
[m/s] (b) CR = 12.5 (Model-5)
[m/s]
(c) CR = 5.0 (Model-6)
Fig. 7.36 Curvature ratio effect on computational axial velocity contour at exit plane
165
Chapter (7)
Results and Discussion
(a) CR = 7.5 (Model-4)
(b) CR = 12.5 (Model-5)
(c) CR = 5.0 (Model-6) Fig. 7.37 Curvature ratio effect on computational turbulent stress contour at exit plane
166
Chapter (7)
Results and Discussion
[m/s] 60°/60° (Model-7)
[m/s] 90°/90° (Model-8)
[m/s] 45°/45° (Model-9) Fig. 7.38 Turn angle effect on computational axial velocity contours at the exit plane (water flow, Re=34000)
167
Chapter (7)
Results and Discussion
60°/60° (Model-7)
90°/90° (Model-8)
45°/45° (Model-9) Fig. 7.39 Turn angle effect on computational turbulent stress contour at the exit plane (water flow, Re=34000)
168
Chapter (8)
Conclusions and Recommendations for Future Work
CHAPTER (8) CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE WORK 8.1 CONCLUSIONS Comprehensive experimental and numerical studies were successfully performed to investigate the o/w emulsion flow development within twelve tested models of S-shaped diffusers using test rig built up in the advanced fluid mechanics laboratory of the faculty of engineering, Menoufiya University. Measurements of static pressure distributions along the inner and outer walls of the S-shaped diffusers allowed the static pressure recovery coefficient and performance of the twelve tested models, in terms of total pressure loss to be assessed. These results were used to validate the ANSYS R-15.0 Fluid Flow Fluent CFD code (FFF), which was then used to predict the performance of the S-shaped diffusers with (o/w) emulsion flow. ANSYS R-15.0 proved a quick and useful alternative code to assess the effect of geometric and dynamic parameters on the S-diffuser performance. Three turbulence models namely STD k , STD k and SST k and three multi-phase schemes namely mixture, volume of fluid (VOF) and Eulerian were examined in the present study. The standard k-ε model and the mixture approach are used in simulations. The continuous phase (water) is simulated using Reynolds-Averaged Navier-Stokes equations (RANS), while the dispersed phase (oil) is simulated using mixture multi-phase scheme. The numerical solution of S-diffuser flow was based on the SIMPLE algorithm. A numerical multi-phase model based on mixture approach was used in the computations. It predicted reasonable static pressures and pressure recovery coefficients along the S-diffuser. The predictions of the numerical
169
Chapter (8)
Conclusions and Recommendations for Future Work
model were validated with previous work data taken from the literature as well as the present experimental work. Measurements and computations on the mid-plane between the two flat parallel walls (top wall and bottom wall), of the S-shaped diffuser tested models at varying inflow Reynolds numbers, holdup values and emulsion status, shows that the S-diffuser energy-loss coefficient is lowest with water flow and increases as holdup increase or Reynolds number decrease. Also, the results on the unstable and stable (o/w) emulsion flow, shows the Sdiffuser energy-loss coefficient was increased by 13.5% when an emulsifier (SDS) was added to the emulsion to make it stable (o/w) emulsion. Dealing with the twelve tested models, shows that the S-diffuser energy-loss coefficient increases as S-diffuser area ratio decreases and when the flow path changes as 45º/45º, 30º/60º, 60º/30º, the energy-loss coefficient strongly increases respectively. It can therefore be concluded from the results that the energy-loss coefficient of S-shaped diffusers is to be inversely proportional to each of the inflow Reynolds number, S-diffuser area ratio, S-diffuser curvature ratio and emulsion holdup inverse (1/Φ). Emulsion stability decreases the S-diffuser performance and the turning angle plays an important role in improving the S-shaped diffuser performance. The following main points can be drawn from the present study; 1.
The standard k model (STD k ) is suitable for performing simulation of water / emulsion flows in S-shaped diffusers.
2.
The mixture model (Mixture Scheme) gives the best performance in predicting the o/w emulsion flow in S-diffuser compared with the other multi-phase turbulence models used in the present study.
3.
The pressure recovery coefficient of unstable o/w emulsion flow is higher than that of stable o/w emulsion flow, while water flow has the highest.
170
Chapter (8) 4.
Conclusions and Recommendations for Future Work
The energy-loss coefficient of S-shaped diffusers is found to be inversely proportional to each of the inflow Reynolds number, Sdiffuser area ratio, S-diffuser curvature ratio and emulsion holdup inverse (1/Φ).
5.
As the turn angle (TA) increases from 45º/45º to 90º/90º, the energyloss coefficient increases because of the presence of counter rotating vorticities.
6.
As the flow path (FP) changes as 45º/45º, 30º/60º, 60º/30º, the energyloss coefficient strongly increases, respectively.
7.
The best design model of all tested models of the S-diffusers is model (7) which has area ratio of 3, curvature ratio of 7.5, and symmetric center-line turn angle of (60°/60°), which exhibits the lowest value of the energy-loss coefficient.
8.
The unstable (o/w) emulsion exhibits lower values of the energy-loss coefficients, compared with stable (o/w) emulsion.
9.
For the S-diffusers energy-loss coefficient, the (o/w) emulsion flows have the worest diffuser performance compared with the case of water flow.
10. New correlations equations are extracted for all tested cases for computing the loss coefficient of S-diffusers handling water as well as emulsions flows.
8.2 RECOMMENDATIONS FOR FUTURE WORK Recommendations for future work are required since recently S-shaped diffusers are drawing more attention in the design of the economical Sshaped diffuser parameters in order to optimize the designing system. Based upon the discussions and conclusions as mentioned earlier and to further extension and improvement of the current model predictions, the following research work points are recommended:
171
Chapter (8)
1-
Conclusions and Recommendations for Future Work
Applying more advanced turbulence model such as non-linear models or Reynolds stress model provides more accurate predictions.
2-
The study would have to be supported by measurements of turbulent structure, especially in regions of diffusions.
3-
Studying the effect of the induced flow using SGV (swirl generator vortex) on the S-Shaped diffuser performance will be a considerable extension to the current work.
4-
Studying the effect of addition of a rotating upstream on a circular cross-sectional area S-Shaped diffuser performance.
5-
The presence of solid particles (sand) with different sizes in emulsion flows will shed light on the real cases.
172
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181
Appendix A
Calibration of Emulsion Flow Meter
APPENDIX A CALIBRATION OF EMULSION FLOW METER In this section, calibration of the orifice flow meter is intended to determine discharge coefficient corresponding to various dispersed phase holdup, 0.0 ≤ Φ ≤ 0.25. A sharp-edged orifice meter is interposed in the flowing fluid supply line. The inner diameter of the orifice is tapered at 45° degree and two pressure taps are formed at distances of (D and D/2) from orifice location as shown in Fig. A.1 and the dimensions of the orifice meter are shown in Table A.1.
Fig. A.1 Configuration of the Orifice-meter
Table A.1 Orifice plate dimensions in mm. Pipe diameter, D
Orifice diameter, d
Plate thickness, t
Lu
Ld
25.4
16.4
3
25.4
12.7
182
Appendix A
Calibration of Emulsion Flow Meter
The calibration is carried out using emulsion as the working fluid, calibration collecting tank, and a stop watch. The time elapsed for collecting certain volume of emulsion as well as the pressure drop across the orifice plate is measured as:
Ao A p
Q = Cd
A2p - Ao2
2gH
(A.1)
Where, H is the pressure head difference across the orifice plate in meters of fluid flowing through the system that can be expressed in terms of manomeric head ( H o ) as following:
m
H H o
E
1
(A.2)
where m and E are the densities of the manometer and flowing fluids, respectively. The densities and viscosities of different holdup o/w emulsions are measured in laboratories of faculty of sciences, Minoufiya university using densitometer and viscometer. The measuring values of emulsions density and viscosity are summarized in a worksheet table by which the relation between the density as well as viscosity against the holdup for stable and unstable (oil-inwater) emulsions are drawn as in [71]. Fig. A.2.a shows the stable and unstable emulsion holdup effect on the relative density (emulsion density to water density) and the holdup effect on the relative viscosity (emulsion viscosity to water viscosity) is clearly shown in Fig. A.2.b. As noticed, the density decreases while the viscosity increases as the holdup increases. The Reynolds number in the calibration operation is defined according to:
Re
E U P D H Q E Q E E D H E D H
183
(A.3)
Appendix A
Calibration of Emulsion Flow Meter
Where, U P is the flow velocity in the pipe, D H is the pipe hydraulic diameter which equal to the pipe diameter,
E and E is the emulsion dynamic and
kinematic viscosities, respectively. The discharge coefficients, Cd given in Eq. (A.1) is evaluated during calibration of the flow orifice meter and plotted against the Reynolds number as well as against the emulsion holdup as shown in Figs. A.3 and A.4, respectively. These figures show that the discharge coefficient decreases as the dispersed phase holdup, Ф increases. Also, the emulsion stability decreases the discharge coefficient.
E w (Stable) E w (Unstable) 1
E w
0.96
0.92
0.88
0.84
0
0.1
0.2
Holdup
0.3
0.4
Fig. A.2 (a) Holdup effect on the emulsion relative density
184
Appendix A
Calibration of Emulsion Flow Meter
E w E w
( Stable ) ( Unstable )
14 12
E w
10 8 6 4 2 0
0
0.1
0.2
0.3
0.4
Holdup () (b) Holdup effect on the emulsion relative viscosity Fig. A.2 Relative density (a) and viscosity (b) of (o/w) emulsions versus holdup
185
Appendix A
Calibration of Emulsion Flow Meter
0.61 Re=9000 Re=12000 Re=18000 Re=24000 Re=29000 Re=34000
0.6
0.59
Cd
Unstable (o/w) emulsion
0.58
0.57
0.56
0.55
0.54
0
0.05
0.1
0.15
0.2
0.25
0.3
Holdup () (a) Holdup effect on measured Cd of unstable o/w emulsion 0.61 Re=9000 Re=12000 Re=18000 Re=24000 Re=29000 Re=34000
0.6
0.59
Stable (o/w) emulsion
Cd
0.58
0.57
0.56
0.55
0.54
0
0.05
0.1
0.15
Holdup ()
0.2
0.25
(b) Holdup effect on measured Cd of stable o/w emulsion Fig. A.3 Measured discharge coefficients Cd versus holdup
186
0.3
Appendix A
Calibration of Emulsion Flow Meter
0.6 0.58 0.56
Cd
0.54 0.52 Pure Water 0.03 Unstable 0.03 Stable 0.06 Unstable 0.06 Stable 0.10 Stable 0.15 Stable 0.25 Stable
0.5 0.48 0.46 0.44
0
15000
30000
45000
60000
Re Fig. A.4 Measured discharge coefficients Cd versus Reynolds number
Therefore, the orifice-meter calibration curves show that: (1)
The emulsion density ( ρE ) decreases while the emulsion viscosity ( E ) increases as the emulsion holdup increases ( ).
(2)
The emulsion flow discharge coefficient ( Cd ) decreases as the E
emulsion holdup increases. (3)
The discharge coefficient of unstable o/w emulsion flow is higher than that of stable o/w emulsion flow ( Cd
187
Unst. E
Cd
St. E
).
Appendix B
The Experimental Uncertainty
APPENDIX B THE EXPERIMENTAL UNCERTAINTY B.1 Experimental Uncertainty of Diffuser Performance Before being documented for any analysis or design, the data obtained from experimental studies should be validated. The experimental errors may be of two types, namely fixed and random errors. Fixed error can be removed by proper calibration or correction while random error cannot be removed. The factors that introduce random error are uncertain by their nature. Uncertainty analysis should be conducted on all data collected from all measurements in order to quantify the data and validate the accuracy. As known, the flow resistance coefficient of the tested S-shaped diffuser, K d (S-diffuser energy loss coefficient) is given by:
K d hloss 22g hloss U ref
2g 2gB 2 W h loss (Q / A )2ref Q2
2
(B.1)
In which, Q is the actual volume flow rate measured by the calibrated orifice meter and hloss is the loss head through the S-shaped diffuser. The precision uncertainty of K d is due to uncertainties for all independent variables of them. The precision uncertainty
d d
can be represented as given in Taylor [77]:
2 B K B hloss K d hloss d K h K B B h d d loss loss d 2 2 d K d W Q K d Q W K Q Q K d W W d
188
1/2
2
(B.2)
Appendix B
The Experimental Uncertainty
Where, hloss / hloss , B / B , W /W and Q / Q are the precision uncertainties for independent variables hloss , B ,W and Q , respectively. The reasonable estimation of the uncertainty intervals due to random error in the measured length is taken as plus or minus half the smallest scale division (the least count) of the instrument used in conducting the measurement of the lengths. Thus, hloss , B and W are taken to be 0.5 mm. The discharge Q is measured using the calibrated orifice meter and calculated from the orifice meter equation, therefore:
Q Q
H o Q
Q H o H o H o
2
D p Q
Q D P D P D P
2
d O Q d O Q d O d O
1/2 2
(B.3)
By substituting Eq. (B.3) into Eq. (B.2), the precision uncertainties of the energyloss coefficient can be obtained; therefore we can actually estimate the performance of emulsion flow in S-shaped diffuser through the energy-loss characteristics. Table B.1 and B.2 present the range of energy loss coefficient and percentage precision uncertainty of resistance coefficient,
d d
(i. e., Ukd % ) during all
measurements, of water and emulsion flows.
B.2 Experimental Uncertainty in Emulsion Preparation In preparing emulsion in Chapter 3, the error in the oil concentration by volume (holdup ) can be obtained from equation (3.1) as:
Vo V V o w V o V w V E VE
V w 1 VE V o V E 2
Vo
V o 1 1 V o V E
189
(B.4)
Appendix B
The Experimental Uncertainty
= V o = V 2E VE V w
V w
V w 1) V w VE
(B.5)
The precision uncertainty can be represented as given in Taylor [77]:
V o V o V o V o
V w V w V w V w
Vo (1- ) Vo
(1- )
Vo Vo
2
2
-(1
2
Vw Vw
)
1/2 2
Vw Vw
1/2 2
(B.6)
1/2 2
Where, V o V o and V w V w are the precision uncertainties for independent variables V o and V w , respectively. A glass flask with a maximum volume of one liter is used to measure the volume of the water and oil as well as the least count of the used glass flask with a gradual scale is 10 cm3 . The reasonable estimate of the uncertainty intervals due to random error in the measured volume is taken as plus or minus half the smallest scale division of the instrument used in conducting the measurements of the volumes, thus,
V o and V w are taken to be 5 cm3.
For example, at emulsion holdup Ф=0.03; the amount of oil is (7.5) liter oil is added to a constant volume of pure water of (240) liter water and the computed error in the amount of oil equals 37.5 cm3 (0.0375 liter) computed as: erroroil = 0.005 × 7.5 = 0.0375 liter oil. The computed error in the volume of pure water equals 1200 cm3 (1.2 liter) computed as: errorwater = 0.005 × 240 = 1.2 literwater. Therefore, the percentage precision uncertainty of the oil concentration in the emulsion by volume (holdup Ф), (i. e., U % ) is 2.04 % according to
190
Appendix B
The Experimental Uncertainty
equation (B.6), similarly for Ф=0.06, 0.10, 0.15, and 0.25 the uncertainties can be obtained as shown in Table (B.3).
B.3 Uncertainty in the Manometer Head (Length) The error in the length is ±0.5 mm.
B.4 Uncertainty in the Time and Flow Velocity The uncertainty in the flow velocity is ±2.75% (Uu % 2.75) . Since, U E UE
S S
2
1/2 2
t
t
(B.7)
B.5 Uncertainty in the Pressure The error in the pressure is ±0.01 kPa.
Table.B.1 Uncertainty of the energy-loss coefficients of the S-shaped diffuser models with water flow S-diffuser tested model
Range of Kd
Flow type
percentage uncertainty U
kd
%
1&9
0.09 Kd 0.69
Water
2.2 Uk % 2.7
2&4
0.26 Kd 0.86
Water
2.9 Uk % 3.2
3
0.48 Kd 1.1
Water
2.8 Uk % 3.2
5&10
0.18 Kd 0.80
Water
3.6 Uk % 4.0
6
0.36 Kd 0.96
Water
2.7 Uk % 3
7
0.05 Kd 0.65
Water
2.6 Uk % 2.9
8
0.14 Kd 0.74
Water
3.0 Uk % 3.5
11
0.42 Kd 0.98
Water
3.3 Uk % 3.9
12
0.31 Kd 0.91
Water
3.1 Uk % 3.6
191
d
d
d
d
d
d
d
d
d
Appendix B
The Experimental Uncertainty
Table.B.2 Uncertainty of the energy loss coefficients for curved diffusers with unstable and stable o/w emulsions. O/W Emulsion type
percentage uncertainty
S-diffuser tested model
Range of Kd
1&9
0.27 Kd 0.89
Stable
2.55 Uk % 3.25
1&9
0.19 Kd 0.78
Unstable
2.35 Uk % 2.95
2&4
0.78 Kd 1.38
Stable
3.30 Uk % 3.71
2&4
0.68 Kd 1.23
Unstable
3.05 Uk % 3.48
3
1.44 Kd 1.98
Stable
3.75 Uk % 4.25
3
1.14 Kd 1.81
Unstable
3.45 Uk % 3.88
5&10
0.54 Kd1.21
Stable
3.91 Uk % 4.36
5&10
0.35 Kd 0.94
Unstable
3.63 Uk % 3.96
6
1.08 Kd 1.73
Stable
3.15 Uk % 3.45
6
0.98 Kd 1.48
Unstable
2.82 Uk % 3.05
7
0.16 Kd 0.78
Stable
2.82 Uk % 3.23
7
0.09 Kd 0.69
Unstable
2.65 Uk % 2.98
8
0.42 Kd 1.03
Stable
3.21 Uk % 3.71
8
0.34 Kd 0.94
Unstable
3.08 Uk % 3.55
11
1.25 Kd 1.87
Stable
3.63 Uk % 4.44
11
0.98 Kd 1.52
Unstable
3.42 Uk % 4.18
12
0.93 Kd 1.57
Stable
3.28 Uk % 3.93
12
0.78 Kd 1.39
Unstable
3.18 Uk % 3.73
192
Uk % d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
Appendix B
The Experimental Uncertainty
Table.B.3 Uncertainty of the oil concentration by volume (holdup, Φ) for the tested emulsions trough the curved diffusers.
Range of V o ( liter )
Range of V w ( liter )
The percentage uncertainty U %
1
0.03
V o 0.0 375
V w 1.5
U% 2.02
2
0.06
V o 0.0 775
V w 1.5
U% 2.18
3
0.10
V o 0.135
V w 1.5
U %
2.32
4
0.15
V o 0.213
V w 1.5
U %
2.4
5
0.25
V o 0.40
V w 1.5
Holdup
No.
U % 2.66
B.6 Uncertainty in the Emulsion density The uncertainty in the emulsion density is ± 3.26 % (U % 3.26) since, E
E E
m E E m E E m E m E
1 V E VE
mE
m E
2
2
V E V E E E V E V E
V mE V E E m V 2 V E E E 2
m E m E
2
V
VE
1/2 2
E
193
1/2 2
1/2
2
(B.8)
Appendix C
Analytical Study of S-shaped Diffuser
APPENDIX C ANALYTICAL STUDY OF S-SHAPED DIFFUSER Emulsions can generally be treated as pseudo-homogeneous fluids with averaged properties as the dispersed droplets of emulsions are small and are well distributed. Consequently, one can apply the single-phase flow equation to correlate the pressure loss data for emulsion flow through S-diffusers. It should be noted that emulsions are Newtonian only at low to moderate concentrations of dispersed phase (oil concentration up to 46.9%), see Pal and Hwang [59]. Bernoulli's equation is valid along any streamline in any frictionless flow, and it can be modified to include the energy losses due to the presence of S-shaped diffuser wall friction and the superimposed vortices produced by curvature or recirculation. Thus, for the S-shaped diffuser of curved side walls, the energy equation applied between the inlet and exit of the S-diffuser is:
2
hinlet
2
V inlet
hexit +
2g
V exit 2g
hloss hinlet hexit
hloss
+hloss
V
2 inlet
(C.1)
2 V inlet /AR 2
2g
2 2 Pexit V inlet 1 -1 / AR E g 2g
(C.2)
Pinlet
194
(C.3)
Appendix C
Analytical Study of S-shaped Diffuser
hexit are
Where hinlet and while
the pressure heads at inlet and exit, respectively,
V inlet and V exit are the time-averaged mean velocities at inlet and exit,
respectively and
E is the mixture (emulsion) density.
The local static pressure recovery coefficient ( C p ) is used in describing Sshaped diffuser performance and can be defined as:
Cp
( Px Pref ) 0.5 E U ref 2
(C.4.a)
The overall pressure recovery coefficient of the S-diffuser is defined as:
Cp
d
( PExit Pref ) 0.5 E U ref 2
(C.4.b)
Where, Px is the local wall measured static pressure and Pref is the reference pressure and
U ref
is the reference mean velocity. C p
d
is the overall static
pressure recovery coefficient of the S- diffuser and PExit is the average static pressure at the diffuser exit. The C p represents the ratio between the local static pressure difference between any location ( x ) and reference pressure tap location ( ref location) to the reference dynamic pressure. The reference pressure tap location is taken at the Sdiffuser entrance on the outer-wall (Pref = P1,
test section),
while the distance
reference is taken at the S-diffuser entrance at the inner-wall (S1, test section = 0.0 mm) as shown in Fig. 3.1.b.
195
Appendix C
Analytical Study of S-shaped Diffuser
Also, S-shaped diffuser performance can be represented in terms of the diffuser loss coefficient ( K d ). From energy equation using equations (C.3 and C.4) the S-diffuser loss coefficient can then be determined from applying the following equation:
Kd
1 (1 ( C C ) p (exit ) p (inlet ) AR 2
(C.5)
From which the loss head through S-shaped diffuser can be computed as:
hloss
= Kd
2 V inlet
(C.6)
2g
Where C p (inlet ) and C p (exit ) are the pressure coefficients at inlet and exit of the S-shaped diffuser, respectively.
196
Appendix D
Design Details of S-Shaped Diffusers
APPENDIX D DESIGN DETAILS OF S-SHAPED DIFFUSERS D.1 Geometry of S-Shaped Diffusers (r,θ) According to Eq. (D.1)
197
Appendix D
Design Details of S-Shaped Diffusers
198
Appendix D
Design Details of S-Shaped Diffusers
199
Appendix D
Design Details of S-Shaped Diffusers
200
Appendix D
Design Details of S-Shaped Diffusers
Fig. D.1 Geometries of the all different S-shaped diffuser tested Models with r (θ)
201
Appendix D
Design Details of S-Shaped Diffusers
D.2 Geometry of S-Shaped Diffusers (x,y) According to Eq. (D.2)
202
Appendix D
Design Details of S-Shaped Diffusers
203
Appendix D
Design Details of S-Shaped Diffusers
204
Appendix D
Design Details of S-Shaped Diffusers
205
Appendix D
Design Details of S-Shaped Diffusers
Fig. D.2 Geometries of the all different S-shaped diffuser tested Models with x (θ) and y (θ)
206
Design Details of S-Shaped Diffusers
Appendix D
D.3 S-Shaped Diffuser General Equation of r (θ) and z (θ) into paper
θ1
θ2
°
°
°
= 30 , 45 , 60 , 90
°
Total
= 30 , 45 , 60 , 90 , θ 2 °
Total
, θ1 = 0° : θ1
°
W = Winlet +
°
°
R o = R C
R i = R C
= 0° : θ 2
(Wexit Winlet ) 90°
R C = 100, 150, and 250
mm
W 2
W 2
Equation ( D.1)
207
Total
Total
Design Details of S-Shaped Diffusers
Appendix D
D.4 S-Shaped Diffuser General Equation of x (θ), y (θ) and z (θ) into paper °
°
°
°
θ1 = 0° : θ1 Total
°
°
°
°
θ 2 = θ 2 Total : 0°
θ1 Total
= 30 , 45 , 60 , 90 ,
θ 2 Total
= 30 , 45 , 60 , 90 ,
For the first bend
:
Jouter J inner 0and Iouter Iinner 1
For the second bend : Iouter
= Iinner 0and Jouter J inner 1
Outer Wall Polyline
Wθ 1
2
x o = Iouter ( R C -
) Sin θ 1
(2 R C Sin θ 2Total ) + Jouter Wθ 2 ) Sin (θ 2Total - θ 2 )) -(( R C + 2 W yo = Iouter ( R C + W ) - (( R C - θ 1 ) Cos θ1 ) 2 2 W ( R C + 2 ) - (2 R C Cos θ 2Total ) + Jouter Wθ 2 ) Cos (θ 2Total - θ 2 )) +(( R C + 2 Inner Wall Polyline
x i Iinner
Wθ 1 ( R + ) Sin θ 1 C 2
( 2 R Sin θ 2 ) C Total + Jinner Wθ 2 )Sin (θ -(( R C 2Total - θ 2 )) 2 Wθ W 1 yi Iinner ( R C + ) - (( R C + ) Cos θ 1 ) 2 2 W (( R C + 2 ) - ( 2 R C Cos θ 2Total )) + J inner Wθ 2 ) Cos (θ 2Total - θ 2 )) + (( R C 2 Equatio n ( D.2)
208
Appendix E
Analytical Study Of S-Diffuser
APPENDIX E PUBLISHED PAPERS
This appendix presents a list of the papers that were published or under review during the present work.
1-
K. A. Ibrahim, W. A. El-Askary, I. M. Sakr and Hamdy A. Omara., "Experimental and Numerical Studies on Water Flow through Rectangular Cross-Sectional Area S-Shaped Diffusers", Engineering Research Journal. Faculty of Engineering, Minoufiya University, Vol. 38, No. 4, PP: 309-325, (2015).
2-
K. A. Ibrahim, W. A. El-Askary, I. M. Sakr and Hamdy A. Omara., "Experimental and Numerical Studies on Emulsion (OilIn-Water) Flow in S-Shaped Diffusers", 17th Int. Conference on Applied Mechanics
and Mechanical
Engineering,
Military
Technical College, Kobry El-Kobbah Cairo, Egypt, (AMME 17), Part I: Scientific Sessions, Session No. 3, MP-1 (14), (April 19-21, 2016). 3-
K. A. Ibrahim, W. A. El-Askary, I. M. Sakr and Hamdy A. Omara., “Evaluation of S-Shaped Diffuser Performance with Emulsion (Oil in Water) Flow", International Journal of Fluid Mechanics Research, (under review).
209
ملخص رسالة الدكتوراه المقدمة من المهندس /حمدى عبدالحميد على عماره بقسم هندسة القوى الميكانيكية بكلية الهندسة -جامعة المنوفية في موضوع
دراسات عمهيه و عذديه عهي سريان مستحهة (زيث في ماء) فى وواشر عهي شكم حرف S
اىسشٝاُاىَسرؽيةٕ٘ سشٝاُ ٍضٝطٍِاشْ ٍِِٞاىس٘ائوٗ،اؼذٍَْٖاٝر٘صعفٜشنوقطشاخ طغٞشجفٜظَٞغأّؽاءاألرخشٍِٗ.اىَؼشٗ أُاىسائواىَرشرد ٝسَٚتاىط٘ساىذارخيٜأٗاىط٘س اىَرقطغفٜؼِٞأُاىسائواىَشردٝؼش تاسٌاىط٘ساىخاسظٜأٗاىط٘ساىَسرَشّٗ.ظشاًىَاذرَرغ تٔ اىَسرؽيثاخ ٍِ رخاطٞح ذقيٞو اإلؼرناك أشْاء سشٝاّٖا ف ٚرخط٘ط ّقيٖا ىزا ٝرٌ اسرغاله اىَ٘ائغ اىَقييحىإلؼرناكفّٚقواىَ٘ائغاىصقٞيحٍصواىثرشٗهاىخاًترنيفحأقو ٗمرطثٞقٌٍٖٝرٌإسرخذاًذذفق ٍسرؽية (صٝد فٍ ٚاء ) السرخشاض اىْفظ اىخاً ٍِ تاطِ األسع ػِ طشٝق ؼقِ اىْفظ تاىَاء. ٗذسرخذًاىَسرؽيثاخأٝضا ًفٜاىؼذٝذٍِاىظْاػاخاألرخشٍ،ٙصواألدٗٝحٗاىضساػحٗاىظْاػاخ اىغزائٞحٗ.اىْ٘اششاىرٚػيٚشنوؼش Sضشٗسٝحفٜذذفقاىسائوؼٞس ٝرثاطأاىسائوٗٝرغٞش إذعإٔفٜٗقدٗاؼذٗذسرخذًٕزٓاىْ٘اششفٜاىؼذٝذٍِاىرطثٞقاخٍْٖاػيٚسثٞواىَصاهذظٌَٞ ٍَشاخٍضخحاىطشداىَشمض .ٛ ٗ قذ أٗضؽد ٍشاظؼح األتؽاز اىساتقح فٍ ٚعاه اىسشٝاُ رخاله ّاشش ػي ٚشنو ؼش Sأُ اىرشمٞض مئػيٚاىسشٝاُأؼادٙاىط٘س(ٍاءإٔٗ٘اء)ٗال٘ٝظذأٙتؽسؼرٚذاسٝخٔ(ػيٚؼذ ػيٌ اىثاؼس) ػِ سشٝاُ ٍسرؽية فّ ٚاشش Sىزا ماُ اىٖذ ٍِ اىثؽس اىؽاى ٕ٘ ٚدساسح ذأشٞش ٍ Sؼَيٞا ً ٗػذدٝا ً ف ٚؼاىر ٚسشٝاُ اىَاء
اىؼ٘اٍو اىَخريفح ػي ٚأداء ّاشش ػي ٚشنو ؼش ٗاىَسرؽيةٗىرؽقٞقٕزااىٖذ ماّدٕزٓاألػَاه : أ
ذممٌذظممٌَٗٞذظممْٞغاشْممٚػشممشَّ٘رظ ما ًٍخريف ما ًٍممِاىْ٘اشممشػيممٚشممنوؼممش ٗSتْسممةٍسمماؼح ذسمممممممممممماّٗٗ,[)1.5)،(2.0)،)3.0(]ٙسممممممممممممةإّؽْمممممممممممماء](ٗ[)5.0)،(7.5)،(12.5صٗاٝمممممممممممما إّؽْاء](ٗ [)45º/45º)،(60º/60º)،(90º/90º(،(60º/30º،( 30º/60ºرىلفٍٜؼَودْٝاٍٞنا اىَ٘ائغاىَرقذٍحت قسٌْٕذسحاىق٘ٙاىَٞناّٞنٞحتنيٞحاىْٖذسحتشثِٞاىنمً٘ظاٍؼمحاىَْ٘فٞمح،ؼٞمسذَمد ػَيٞحاىرظْٞغٍِأى٘اغاىؽذٝذاىظيةمَاذٌذظٌَٗٞذظْٞغشالشحَّارضٍخريفحٍمِاألّفما اىخيفٞمح اىَسمممرقَٞحتَسممماؼاخٍقطمممغشاترمممحذسممماٍٗ)1.5)،(2.0)،)3.0(]ٙمممشجٍمممٍِسممماؼحٍقطمممغٍمممذرخو اىْاشش](ٗ,[)80×20 mmط٘هشاتمدٝسماٍٗ55ٙمشجٍمِػمشعٍمذرخواىْاشمش(ّٗ,(Wفمقٗاؼمذ فقظأٍاٍٍٚسرقٌٞتَساؼحٍقطمغذسماٍٗٙسماؼحٍقطمغٍمذرخواىْاشمشٗ،طم٘هٝسماٍٗٗ75Wٙؽم٘ه تط٘هٝساٗ40Wٙؼٞسٍذرخواىَؽ٘هٍشتغاىشنوتط٘هضيغٍٗ26 mmخشظٍٔسرطٞيٜاىشنو تَساؼحٍقطغذساٍٗٙساؼحٍقطغٍذرخواىْاشش](ٗ،[)80×20 mmذٌأٝضا ًذصثٞداإلسذفاعفٚممو ٍِاىْ٘اششٗاألّفا تقَٞحذساٗ .80 mmٙ
ذْاٗىداىذساسحاىرعشٝثٞحتاىَؼَورخظائضسشٝاٍُائغٍسرؽية(صٝدفٍٜاء)رخالهاىْ٘اششػيٚ شنو ؼش Sؼٞس ذٌ ذ٘صٝغ اىضغظ ػي ٚط٘ه اىعذساُ اىخاسظٞح ٗ اىذارخيٞح ىيْ٘اشش ٗاىَْثغ ٗاىَظة ٗ .ذسرْذ ٍؼاٍالخ فقذاُ اىطاقح ىنو َّ٘رض ػي ٚقٞاساخ ٍفظيح ىر٘صٝؼاخ اىضغظ ػيٚ ط٘هظذساُاىْاششتَافٜرىلاىَْثغٗاىَظةٗقذذٌذقٌٞٞأداءٕزٓاىْ٘اششذؽدذأشٞشػذجػ٘اٍو ٍخريفحٕٗ ٚاىقَٞح اىَرَاشيح ىضاٗٝرٚاإلّؽْاء األٗىٗ ٚاىصاّٞح ىيْاشش ، (60º/60º) ، (90º/90º(] S ) ، [)45º/45ºاىقَٞحاىَخريفحىضاٗٝرٚاإلّؽْاءىيْاششٍٗعَ٘ػَٖا ( 09ºذأشٞشٍساساىسشٝاُ دارخو اىْاشش ](ّ ،) [)45º/45º)، (60º/30º) ، (30º/60ºسثحاإلّؽْاء(ّسثحّظفقطشاىرق٘طىَؽ٘س اىْاششاىَْؽْٚإىٚػشعٍذرخواىْاشش) ّ ،سثحاىَساؼح(ّسثحٍساؼحٍقطغاىْاششػْذاىَخشض إىٍ ٚساؼحٍقطغاىْاششػْذاىَذرخو)ّ ،سثحاىضٝدفٚاىخيٞظ(ّسثح ؼعٌاىضٝدإىٚؼعٌاىخيٞظ) ٗسقٌسْ٘ٝىذصىيسشٝاُ(ّسثح ق٘جاىقظ٘ساىزاذٚإىٚق٘جاىيضٗظح)،تاإلضافحإىٚاإلسرقشاسٝحٗػذً اإلسرقشاسٝح(ؼاىحاإلذضاُٗؼاىحػذًاإلذضاُ)ىيخيٞظاىَسرخذًٗ،ماّداىْسةاىؽعَٞحىرشمٞضاىضٝد فٚاىخيٞظٕ .’%25’% 15’% 10’% 6’%3’%0:ٚ
مَاذٌإظشاءدساسحػذدٝحذؼرَمذػيمٚمم٘دػمذدٝ (ANSYS R 15.0- FLUENT)ٙعٞمذإسرنشما سي٘كسشٝإُزااىَائغاىَسرؽيةرخالهاىْ٘اششاىرٚػيٚشنوؼش ٗSذسرخذًاىَْ٘رضاىشٝاضٚ اىَسرْثظٗذؼشعاىْرائطاىعذٝمذجػيمٍٚؼاٍموفقمذاُاىطاقمحأشْماءسمشٝاُاىَسمرؽيةاىغٞمشٍسمرقشٗ اىَسرقشرخالهٕزٓاىْ٘اششٍٗقاسّرٖاتاىْرائطاىَؼَيٞحٗ.اشرَيد اىذساسح ػيٍ ٚقاسّح أداء شالشح َّارض ىؽساب اإلضطشابفٚاىسشٝإُٗمٗ(STD K-ω), (SST K-ω), (STD k-ε) ٚممزىلٍقاسّمح أداء شالشحطش ؼوػذدٙىيسشٝاُشْائٚاىط٘سٍٕٗ(Mixture), (Eulerian), (VOF) ٚغاىْرائط ب
اىرعشٝثٞحاىَْش٘سجٟرخشِٗٝاىخاطمحتاىسمشٝاُاإلضمطشاتٚرخمالهاىْ٘اشمشاىَْؽْٞمحتاإلضمافح إىممٚاىقٞاسمماخاىؽاىٞممحٗ.قممذأرٖممشخٕممزٓاىَقاسّمماخأُطشٝقممحاىخيممٞظ(ٍ)Mixtureممغَّمم٘رض اإلضطشاب ) (STD k-εذؼطٚأفضوّرائطػذدٝحىيرْثؤتسي٘كاىسشٝاُىزىلذٌاسرخذاٍٖافٚ اىذساسحاىؽاىٞح.مَاقاٍداىذساسحتاىقٞاطاىؼذدٙىيسشػحاىَؽ٘سٝحٗاإلظٖماداخاإلضمطشاتٞح (مْر٘ساخاىسشػحٗاإلظٖاد) .
ٍِاىْرائطاىَؼَيٞحذٌإسرخالصٍؼادىحذعشٝثٞحظذٝذجذؼثشػٍِؼاٍوفقذاىطاقحىيْاششذؼثٞشاً ٍرقاستا ً ٍغ ّظٞشٓ اىَقاط ٗرىل ىنو َّارض اىْ٘اشش اىر ٚذٌ دساسرٖا ف ٚرو ؼاالخ اىسشٝاُ اىَخريفحاىَذسٗسح. ٗذرنُ٘اىشساىحٍِشَاّٞحأت٘ابتاإلضافحإىٚقائَحتاىَشاظغٗرخَسحٍالؼقَٗٝنِذيخٞظمٖا ماىراى -:ٜ
انثاب األول: ٗٝشَواىَقذٍحاىخاطمحتاىثؽمسٗاىرمٜذرضمَِإَٔٞمحٗإسمرخذاٍاخمموٍمِاىْاشمشاىمزٙػيمٚ شنوؼش ٗ Sاىَائغاىَسرؽية(ٍاء+صٝمد)فمٍٚعماالخاىؽٞماجٗممزىلٝؽرم٘ٙػيمٚأّم٘اع ٗأشناهاىْ٘اششاىَْؽْٞح . انثاب انثاوي:
ٗٝشَوٍشاظؼمحشماٍيحىحتؽمازاىسماتقحاىرمٜذمٌّشمشٕافمٍٜعماهقشٝمةٍمٍِعماهاىثؽمس.شمٌ ْٝرٖٕٜزااىثابتإٌٔاإلسرْراظاخٍِاألتؽازاىساتقحٗمزىلاىٖذ ٍِاىثؽساىعاس .ٙ
انثاب انثانج: ٗٝؼشعششؼا ًذفظٞيٞا ًىيعٖاصاىَؼَيٜاىَسرخذًٗٗطفا ًىيْ٘اشمشاىَخريفمحاىَْؽْٞمحػيمٚشمنو ؼمش ٗ Sذؽذٝممذٍ٘اقمغاىقٞمماطاىَخريفمحتاىرفظممٞوٍٗنّ٘ماخاألظٖممضجاىَسمرخذٍحفممٜػَيٞممح اىقٞاط.مزىلٝقذًذؽذٝمذاىؼ٘اٍمواىَخريفمحىيْاشمشٗاىرمٚذمرؽنٌفمٚأدائمٔٝٗ.ؽرمٕ٘ٙمزااىثماب أٝضا ًػيٚدساسحذأشٞشّسثحذشمٞمضاىضٝمدفمٚاىخيمٞظػيمٍٚؼاٍمواإلؼرنماكألّاتٞمةاىسمشٝاُ. ٗقذأٗضؽداىْرائطاىَؼَيٞحاىرٜذٌاىؽظ٘هػيٖٞافٕٚزااىثابأٍُسمرؽيةاىضٝمدفمٚاىَماء ٝقيوٍٍِؼاٍواإلؼرناكألّاتٞةاىسمشٝاٍُقاسّمحٍمغاىَماءٗتضٝمادجّسمثحاىضٝمدفمٚاىَماءفمئُ اإلّخفاعفٍٚؼاٍمواإلؼرنماكٝضٝمذمَماأٍُؼاٍمواإلؼرنماكىيَسمرؽيةاىغٞمشٍسمرقشأقموٍمِ ّظٞشٓاىَسرقش.شٌْٝرٖٕٜزااىثابتخط٘اخإظشاءاىرعماسباىَؼَيٞمحاىَسمرخذٍحفمٜاىقٞاسماخ اىؽاىٞح . خ
انثاب انراتع: ٗٝقذًٗطفاىَْ٘رضاىشٝاضٚىيؽواىؼذدٙؼٞسٝؼشعاىَؼادالخاىؽامَحىيسشٝإُٜٗػثاسج ػمممٍِؼمممادالخاإلسمممرَشاسٝحٗمَٞمممحاىرؽمممشكىيَسمممرؽيةاىَضمممطشبٍمممغاسمممرخذاًشالشمممحَّمممارض إضطشابٍخريفحىؽسماباىيضٗظمحاإلضمطشاتٞحٕٗم(STD K-ω), (SST K-ω), (STD k-ε) ٚ
ٗرىلػِطشٝقؼسابطاقحاىؽشمحاإلضطشاتٞحٍٗؼذهاىرشردىٖاٗ.مزىلٝؼمشعشالشمحطمش ؼوػذدٙىيسشٝاُشْائٚاىط٘سٕٗ.(Mixture), (Eulerian), (VOF)ٚ انثاب انخامس: ٗٝؼشعٗطفا ًىيذساسحاىؼذدٝحاىرٚذؼرَذػيٚم٘دؼموػمذدٙىمذْٝاٍٞنااىَ٘ائمغٝعٞمذإسرنشما سي٘كاىسمشٝاُفمٚاىَسماساخاىَخريفمحٗرىملتئسمرخذاًٝٗ.ANSYS R 15.0شمرَوػيمٍٚشاؼمو اىؽواىؼذدٍٙغذ٘ضٞػاىشنواىْٖذسٚىيَْ٘رضاىَسمرخذًٗذقسمَٔٞاىمٚػمذدٍْاسمةٍمِاىخالٝما اىؼذدٝممحٗذؽذٝممذرممشٗ اىسممشٝاُػْممذاىمذرخ٘هٗمممزىلػْممذاىخممشٗض.مَماذممٌذ٘ضممٞػطممش ستممظ اىضغظتاىسشػحٗٗضغٍخطمظىيثشّماٍطاىَسمرخذًفمٚاىؽمواىؼمذدٙػمِطشٝمقاىؽاسمةاالىمٚ ٗٗضغٍخطظىَشاؼواىؼَواىثؽصٚاىؽاى .ٚ
انثاب انسادس: ٗٝشرَوػيٍٚقاسّحتِٞذْثؤاخاىَْ٘رضاىشٝاضٚاىسماتقشمشؼٍٔمغّرمائطذعشٝثٞمحسماتقحٗرىمل ىيرأمذٍِطؽحٗظ٘دجاىن٘داىؼذدٙاىَقرشغاسرؼَاىٔٗ.رىلفٚروَّارضإضطشابٍخريفمحٗ طش ؼوػذدٍٙخريفحىيسشٝاُشْائٚاىط٘سٗ.قذأٗضؽداىَقاسّاخأُأفضوطشٝقحٍسمرخذٍح فٚاىؽوٕٚطشٝقحاىرؽيٞواىؼذدٙتئسرخذاًطشٝقحاىخيٞظٍغَّ٘رضاإلضمطشاب) (STD k-ε مَاأٗضؽداىَقاسّاخذ٘افقا ًظٞذاًتِٞاىْرائطاىْظشٝحاىؽاىٞحٗاىْرائطاىَؼَيٞحاىساتقح .
انثاب انساتع: ٗٝشرَوػيٚػشعٗذؽيٞوذفظٞيٜىيْرائطاىَؼَيٞحٗاىؼذدٝحاىرٜذٌاىؽظ٘هػيٖٞاىنواىْ٘اشش اىَْؽْٞحاىرٚػيٚشنوؼمش ٕٗ Sمٚػثماسجػمِّرمائطذمأشٞشّسمثحاإلّؽْماءّٗرمائطذمأشٞشّسمثح اىَساؼحّٗرائطذأشٞشّسثحذشمٞضاىضٝدفٚاىخيٞظّٗرائطذأشٞشسقٌسْ٘ٝىذصىيسشٝاُػيٍٚؼاٍمو ذ٘صٝغاىضغظاإلسراذٞنٍٜٗؼاٍواىفقذىنوّاششٍِاىْ٘اششاىَْؽْٞحاىَسرخذٍحفمٕٜمزااىثؽمس. ٗأٝضا ًذٌدساسحذأشٞشاإلسرقشاسٝحٗػذًاإلسرقشاسٝحىيخيٞظاىَسرخذًػيٍٚؼاٍوذ٘صٝمغاىضمغظ اإلسراذٞنٍٗٚؼاٍواىفقذىنوّاششٍِاىْ٘اششاىَْؽْٞحاىَسرخذٍحفٕٜزااىثؽس.مَاقاٍد
ز
اىذساسحت٘ضغٍؼادى حًذعشٝث ٞحًٍسرخيظ حًٍِاىذساسحاىَؼَيٞحذمشتظممواىؼ٘اٍمواىَمؤششجػيمٚ أداءاىْاشش(اىزٙػيٚشنوؼش ٍ(Sغٍؼاٍوفقذاىطاقحىيْاششٗرىيلىنوّاششٍمِاىْ٘اشمش اىَْؽْٞحاىَسرخذٍحفٕٜزااىثؽسفٚؼاىرٚسشٝاُاىَاءٗاىَسرؽيةٗ.قمذػثمشخٕمزٓاىَؼادىمح ػٍِؼاٍوفقذاىطاقحىيْاششذؼثٞشاًٍرقاستا ًٍغّظٞشٓاىَقاط . ٗأرخٞشاًذٌاىقٞاطاىؼذدٙىيسشػحاىَؽ٘سٝحٗاإلظٖاداخاإلضطشاتٞحىنواىَْارضٍؽمواىذساسمح فٚط٘سجمْر٘ساخ( .)Turbulent shear stress and axial velocity contours
انثاب انثامه: ٗٝؼممشعٕمممزااىثممابرخالطمممحاىؼَممواىثؽصممم،ٚؼٞممسٝؽرممم٘ٛػيممٜإٔمممٌاإلسممرْراظاخاىرمممٜذمممٌ اسرخالطٖإٍِزٓاىذساسحاىَؼَيٞحّٗ٘ظضإَٖٔافٜاٟذ :ٜ
اىذساسحاىؼذدٝحذسرطٞغاُذرْثأتسمي٘كاىسمشٝاُدارخمواىْ٘اشمشاىَْؽْٞمحاىرمٚػيمٚشمنو ؼش ٗ Sذؼطّٚرائطذرفقٍغّرائطاىذساسمحاىَؼَيٞمحَّٗم٘رضاالضمطشاب(STD k-
)εمافٚىيرْثؤتسي٘كسشٝاُاىَائغاىَسرؽية(صٝدفٍٚاء)رخالهاىْ٘اششاىَخريفح.
اىؽممواىؼممذدٙىيسممشٝاُشْممائٚاىطمم٘س(سممشٝاُاىَممائغاىَسممرؽية)تاسممرخذاًَّمم٘رضاىخيممٞظ ٝؼطّٚرائطّظشٝحأقشبىيْرائطاىرعشٝثٞحٍِغٞشٕاتاسرخذاًَّ٘رضؼوػذدٙأرخش.
ىعَٞمممغَّمممارضاإلرخرثممماسىيْ٘اشمممشاىَْؽْٞمممحاىَسمممرؼَيحفمممٚؼاىمممحسمممشٝاُاىَممماءأٗسمممشٝاُ اىَسرؽيثاخاىَسرقشجٗغٞشاىَسرقشجٝنٍُ٘ؼاٍوذ٘صٝغاىضغظاإلسمراذٞنٚػيمٚاىعمذاس اىخاسظٜىيْاششأمثشٍِرىلػيٚاىعذاساىذارخيٜتسثةاىرأشٞشاىَشمضٗٙذق٘طاىْاشش.
ٍؼاٍالخذ٘صٝغاىضغظاإلسراذٞنٚىيَسرؽيثاخذنُ٘أقوٍِذيلاىرٜىيَٞمآٗ،فمٚؼاىمح اىَسرؽيثاخاىَسرقشجذنُ٘ذيلاىَؼاٍالخأقوٍْٖمافمٚؼاىمحاىَسمرؽيثاخغٞمشاىَسمرقشج ؼٞساإلسرقشاسٝحىيخيٞظاىَسرخذًذؼَوػيٚصٝادجفقذاىطاقح .
ٍؼاٍوذ٘صٝغاىضغظاإلسراذٞنٚػيٚاىعمذساُاىعاّثٞمحىيْاشمشاىَْؽْمٚاىمزٙػيمٚشمنو ؼش ٝ Sرْاسةطشدٝاًٍغموٍِّسثحاإلّؽْاءّ،سثحاىَساؼحٗ،سقٌسْ٘ٝىذصىيسمشٝاُ، ٗػنسٞاًٍغموٍِّسثحذشمٞضاىضٝدفٚاىخيٞظتاىؽعٌٍٗقذاسصاٗٝحاالّؽْاء.
أداءاىْاششراخاىَساساىَرَاشوىضاٗٝرٚاالّؽْاءأفضوٍِّظٞشٓاىغٞشٍرَاشو.
َّ٘رضاىْاششسقٌ7أفضواىَْارضفٚاالداءالٍّٔرَاشواىَسماسٗىمٔصاٗٝمحاّؽْماءغٞمش مثٞشجّسثٞاّٗسثحذق٘طٍر٘سطحمَاأُىّٔسثحٍساؼحأػي.ٚ
أٝضاًاىَؼادىمحاىرعشٝثٞمحاىعذٝمذج()new correlationاىرمٚذمٌاسرخالطمٖاٍمِاىقٞاسماخ اىؽاىٞحذؼثشػِقٌٍٞؼاٍوفقذاىطاقحىيْاششفٚموؼاالخاىسشٝاُاىَخريفحاىرٚذٌ ض
دساسرٖاذؼثٞشاًٍرقاستا ًٍغٍاْٝارشٕاٍِاىقٌٞاىَقاسحىَؼاٍوفقذاىطاقح.
ْٝثغٜاىر٘ضٞػأُاىْرائطٗاالسرْراظاخاىرٜقذٍرٖإزٓاىذساسحأُذنٍُ٘فٞمذجألٗىكمل اىزِٝٝؼَيُ٘فٜاىظْاػاخاىنَٞٞائٞحٗاىثرشٗىٞح.
شٌْٝرٖٕٜزااىثاباألرخٞشتطشغأفناسٍٗقرشؼاخٍسرقثيَٝٔٞنِإظشائٖافٍٜعاهسشٝاٍُائغ ٍسرؽيةرخالهاىْ٘اششاىَْؽْٞحاىرٚػيٚشنوؼش . S
قائمة انمراجع: ٗذشممَواىَشاظممغاىَسممرخذٍحفممٕٚممزااىثؽممسٗػممذدٕاٍ81شظؼمماذممٌّشممشٕافممٚاىفرممشجٍممِػمماً ()ً1011إى .)ً2016(ٚ
انمالحق :
مهحق أ: ٗٝشمممرَوػيمممٍْٚؽْمممٚاىَؼممماٝشجىعٖممماصقٞممماطٍؼمممذهاىسمممشٝاُ(اىقمممشصرٗاىفٕ٘مممحاىضمممٞقح) ىيَسرؽيثاخفٚؼاىحاىرشمٞضاخاىَخريفحىيضٝدفٚاىَاءٗذأشٞشدسظحذشمٞضاىضٝدػيٍٚؼاٍمو اىرظش . مهحق ب: ٗٝشرَوػيٚؼساتاخاألرخطاءاىَؼَيٞحفٜػَيٞحاىقٞاطىيَرغٞشاخاىَقاسحٗاىرٚماّدذرشاٗغ ت .4.44%ٗ5.35%ِٞ
مهحق ت: ٗٝؽر٘ٙػيٚؼساتاخ ذؽيٞيٞحىيْاششاىَْؽْٗٚؼساباىَؼاٍالخاىَخريفحاىرمٚذمرؽنٌفمٚأدائمٔ (ٍؼاٍوذ٘صٝغاىضغظاإلسراذٞنٍٗٚؼاٍوفقذاىطاقح). مهحق ث: ٗٝشرَوػيٚاىرظََٞاخٗاىؽساتاخاىرفظٞيٞحىنوَّ٘رضٍِاىَْارضاىرٚذٌدساسرٖا. مهحق ج: ٗٝشرَوػيٚاالتؽازاىَْش٘سجٍِاىؼَواىثؽصٚاىؽاى .ٚ
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