1. i
MODELLING OF SURFACE ROUGHNESS EFFECT ON TURBULENT
FLOW OF YIELD POWER LAW FLUID DYNAMICS IN CONCENTRIC
ANNULUS USING ANSYS CFX
‘IBADURRAHMAN AZHARUDDIN
PETROLEUM ENGINEERING
UNIVERSITI TEKNOLOGI PETRONAS
MAY 2016
2. ii
CERTIFICATION OF APPROVAL
MODELLING OF SURFACE ROUGHNESS EFFECT ON TURBULENT
FLOW OF YIELD POWER LAW FLUID DYNAMICS IN CONCENTRIC
ANNULUS USING ANSYS CFX
by
‘Ibadurrahman bin Mohd Azharuddin
16427
Dissertation submitted in partial fulfillment of
the requirements for the
Bachelor of Engineering (Hons.)
(Petroleum Engineering)
MAY 2016
Approved by,
__________________________
(Mr. Titus Ntow Ofei)
UNIVERSITI TEKNOLOGI PETRONAS
TRONOH, PERAK
MAY 2016
3. iii
CERTIFICATION OF ORIGINALITY
This is to certify that I am responsible for the work submitted in this project, that the
original work is my own except as specified in the references and acknowledgements,
and that the original work contained herein have not been undertaken or done by
unspecified sources or persons.
___________________________
(‘IBADURRAHMAN BIN MOHD AZHARUDDIN)
4. iv
ABSTRACT
The frictional pressure losses in the pipe while the transportation operation is
performed significantly have an effect towards many other factor that led to decrease
in performance and simultaneously effect the economical value for the project. This
research is going to discuss about the effect of surface roughness towards the
turbulent flow of yield power law in eccentric annulus by using ANSYS CFX
modelling software with varying the type of pipe roughness and fluid concurrently
with various velocity. Results show that the pressure drop is increasing with pipe
roughness and the Reynolds number. The friction factor obtained from the simulation
by using fanning friction equation is compared with the Haalands friction factor
correlation. The data obtained were then be translate into Data fit software code to
come out with a new correlation with a less error in calculating the friction factor.
5. v
ACKNOWLEDGEMENT
First and foremost, highest praises and gratitude belongs to the Most Gracious and Most
Merciful, Allah Almighty. We should certainly appreciate the health, the blessings,
guidance along the ways and opportunity bestowed to us.
This work is especially dedicated to my parent, dearest family for persistently keeping
me in their prayers and continuously supporting this journey. Forever, we will always
remain indebted to their offerings and sacrifice.
The author also would like to express a gratitude towards Mr.Titus Ntow Ofei for his
guidance, monitoring, support and provision of surplus learning materials and
accessibility to all availing resources. Throughout the project, he has been more of a
coach in guiding my way. Credit also goes to Petroleum Engineering Department and
lecturers for years of knowledge and lessons.
Last but not least, Universiti Teknologi PETRONAS (UTP), as the platform where our
paths crossed with all these exceptional people. Credits are extended to all staff and
friends who contributed to this work in ways they may never know.
6. vi
TABLE OF CONTENTS
ABSTRACT..............................................................................................................iv
Table of Contents......................................................................................................vi
List of Appendixes....................................................................................................xi
Chapter 1 INTRODUCTION.....................................................................................1
1.1 Chapter Overview ...........................................................................................1
1.2 Background .....................................................................................................1
1.3 Problem Statement ..........................................................................................2
1.4 Objective .........................................................................................................3
1.5 Scope of study.................................................................................................3
Chapter 2 LITERATURE REVIEW..........................................................................4
2.1 Introduction..........................................................................................................4
2.2 Turbulent flow......................................................................................................5
2.3 Viscosity...............................................................................................................6
2.4 Non-Newtonian Fluids.........................................................................................7
2.5 Newtonian Fluids .................................................................................................9
2.6 Analysis of Annular Flow..................................................................................11
2.7 Yield Power Law................................................................................................12
2.8 CFD Modeling................................................................................................15
2.9 Fanning Friction Factor..................................................................................16
2.10 Haaland Friction Factor Equation ................................................................17
2.11 Buckingham Pi Theorem..............................................................................17
7. vii
Chapter 3 METHODOLOGY..................................................................................24
3.1Flow Phase ..........................................................................................................24
3.2 Methodology Flow.............................................................................................25
3.3 Method to determine the type of flow regime....................................................25
3.3.1 Herschel and Buckley in annulus ................................................................25
3.4 Equation Used....................................................................................................26
3.5 Buckingham Pie Theorem..............................................................................27
3.5.1 Roughness, ...............................................................................................27
3.5.2 Consistency index, ..................................................................................28
3.6 Preliminary Results............................................................................................29
3.6.1 Model Validation.........................................................................................31
3.7 Design of experiment.........................................................................................34
3.7.1 Pipe Geometry and Fluids Properties..........................................................34
3.7.2 Pipe Roughness Value.................................................................................35
Chapter 4 RESULTS................................................................................................36
4.1 Fluid 1 (YPL 1).............................................................................................36
4.2 Fluid 2 (YPL 3).............................................................................................38
4.3 Fluid 3 (YPL 5).............................................................................................40
4.4 Friction Factor Correlation............................................................................43
4.4.1 Fluid 1..........................................................................................................43
4.4.2 Fluid 2..........................................................................................................45
4.4.3 Fluid 3..........................................................................................................46
4.5 Buckingham Pie Theorem.............................................................................49
4.5.1 Equation by Datafit Solver..........................................................................49
8. viii
4.5.2 Summarization for new derive equation from Buckingham Pi Theorem
approach. ..............................................................................................................50
4.5.3 Friction Factor Correlation..........................................................................50
4.6 Gantt Chart and Key Milestones...................................................................52
4.7 Discussion .....................................................................................................53
4.7.1 Validation ....................................................................................................53
4.7.2 Fluids Analysis ......................................................................................53
4.7.3 Reynolds number.........................................................................................53
4.7.4 Shear stress, .............................................................................................53
4.7.5 Consistency Index, K ..................................................................................53
5.2.4 Flow behaviour index, n..............................................................................53
4.8 Friction Factor Analysis................................................................................54
4.9 Haalands Correlation.....................................................................................54
4.10 Buckingham Pi Theorem Analysis................................................................54
4.11 Pressure Gradient ..........................................................................................54
Chapter 5 CONCLUSION .......................................................................................55
Conclusion ...............................................................................................................55
REFERENCES.........................................................................................................56
1. Appendices....................................................................................................57
9. ix
LIST OF FIGURES
Figure 2-1Turbulent flow............................................................................................. 6
Figure 2-2-Non-Newtonian Fluid ................................................................................ 7
Figure 2-3 Shear Stress Equation viscosity................ Error! Bookmark not defined.
Figure 2-5-Newtonian Fluid....................................................................................... 10
Figure 2-6-Slot-model representation of a concentric annulus.................................. 11
Figure 2-7-Farshad’s average absolute surface roughness chart for commonly used
pipe wall surfaces (Farshad and Rieke 2005) ............................................................ 15
Figure 3-1Project Milestone....................................................................................... 24
Figure 3-2-Show an optimum mesh sizes for the pipe model.................................... 30
Figure 3-3-Shows the Number of element versus the pressure gradient ................... 30
Figure 3-4-Pressure gradient versus velocity............................................................. 31
Figure 3-5-Pressure gradient versus velocity for Fluid 1........................................... 37
Figure 3-6-2D response surface for Fluid 1............................................................... 37
Figure 3-7-Pressure gradient versus velocity for Fluid 2........................................... 39
Figure 3-8-2D response surface for Fluid 2............................................................... 39
Figure 3-9-Pressure gradient versus velocity for Fluid 3........................................... 41
Figure 3-10-2D response surface for Fluid 3............................................................. 42
Figure 3-11-Friction factor VS Reynolds number for fluid 1.................................... 43
Figure 3-12-Mean Percentage Error between Haaland and Simulation+ Darcy
Equation ..................................................................................................................... 44
Figure 3-13-Friction factor vs Reynolds number for fluid 2...................................... 45
Figure 3-14- Mean Percentage Error between Haaland and Simulation+ Darcy
Equation ..................................................................................................................... 46
Figure 3-15-Friction factor VS Reynolds number for fluid 3.................................... 46
Figure 3-16- Mean Percentage Error between Haaland and Simulation+ Darcy
Equation ..................................................................................................................... 47
Figure 3-17-Friction factor for fluid 1,2 and 3........................................................... 48
Figure 3-18-Comparison for friction factor .............................................................. 50
10. x
LIST OF TABLES
Table 2-1Composition of the test fluids..................................................................... 14
Table 2-2- Variable and dimension list...................................................................... 19
Table 2-3Modelling Approach................................................................................... 20
Table 2-4-Dimensional Analysis and Similarity........................................................ 23
Table 3-1-Shows the error of accuracy between simulation and actual result........... 32
Table 3-2-Error comparison between Haalands Equation and Validation data......... 33
Table 3-3-Geometry of the pipe................................................................................. 34
Table 3-4-Fluid 1 (YPL1) properties ......................................................................... 34
Table 3-5-Fluid 2(YPL3) properties .......................................................................... 35
Table 3-6-Fluid 3 (YPL5) properties ......................................................................... 35
Table 3-7-Pipe roughness value................................................................................. 35
Table 4-1-Flow regime result for Fluid 1................................................................... 36
Table 4-2-Flow regime result for Fluid 2................................................................... 38
Table 4-3-Flow regime result for Fluid 2................................................................... 40
Table 4-4Constant Value from Data Fit Software ..................................................... 49
Table 4-5-Mass percentage error between Haaland equation, simulation+darcy and
equation by DataFit.................................................................................................... 51
11. xi
LIST OF APPENDIXES
Appendix 5.1Response surface for Fluid 1................................................................ 57
Appendix 5.3 -local sensitivity analysis for Fluid 2 .................................................. 57
Appendix 5.4 3D response surface for Fluid 2 .......................................................... 58
Appendix 5.5 local sensitivity analysis for Fluid 2.................................................... 58
Appendix 5.6 3D response surface for Fluid 3 .......................................................... 59
Appendix 5.7 Local sensitivity analysis for Fluid 3 .................................................. 59
12. xii
List of Equations
Equation 2.1-Viscosity................................................................................................. 6
Equation 2.2 Shear Stress Equation ............................................................................. 9
Equation 2.3 Frictional pressure drop........................................................................ 11
Equation 2.4-Reynolds number for annular flow....................................................... 11
Equation 2.5-Laminar flow........................................................................................ 12
Equation 2.6 Turbulent flow for annular flow ........................................................... 12
Equation 2.7-Fictional pressure drop for YPL........................................................... 12
Equation 2.8 Reynolds number for YPL.................................................................... 13
Equation 2.9 Turbulent flow for YPL........................................................................ 13
Equation 2.10-Reynolds number General formula .................................................... 17
Equation 3.1-Equivalent Reynold’s number.............................................................. 25
Equation 3.2-Critical equivalent Reynold’s number.................................................. 26
Equation 3.3-Darcy pressure loss............................................................................... 26
Equation 3.4-Haaland friction factor for rough conduits........................................... 27
Equation 3.5- Continuity............................................................................................ 27
Equation 3.6-Momentum ........................................................................................... 27
Equation 3.7-Mean percentage Error......................................................................... 27
Equation 3.8 Yield power law equation..................................................................... 32
Equation 4.2 New Correlation by Buckingham Pi Theorem ..................................... 50
13. 1
CHAPTER 1
INTRODUCTION
1.1 Chapter Overview
This research is going to discuss about the effect of surface roughness towards the
turbulent flow of yield power law in eccentric annulus by using ANSYS CFX
modeling software. One of the variables for this experiment is the roughness index
for the pipe. This research is going to focus on the friction loss in the annular by
correlate the result with total energy equation considering the major losses and minor
losses. The flow regime condition will be determine by monitoring the Reynolds
number value. The velocity profile will be record which it has a relationship between
the shear stress, shear rate and the viscosity. The behavior of the fluids will be study
and categorized accordingly.
1.2 Background
The frictional pressure losses in the pipe while the transportation operation is
performed significantly have an effect towards many other factor that led to
decrease in performance and simultaneously effect the economical value for the
project. The flow in the concentric annulus is study to monitor the effect for various
type of non-Newtonian fluid and also a few selected pipe with different roughness
effect is being used.
The research need to be done in a flow regime of turbulent flow condition by
monitoring the value of the Reynold’s number shows by the computer modeling.
The value of the Reynold’s number is controlled by the shear rate of the system.
By using the concentric annulus the annular pressure losses in a fully eccentric
14. 2
annulus could be as low as 40% of the value reported by Moises and Shah (2000)
Concentric annulus having an eccentricity which is dimensionless parameter where
it is equal to zero, while for a fully eccentric annulus is equal to one. The velocity
profile and the frictional pressure losses in the annulus have high impact to the
model that be utilize.
During fully-developed turbulent flow conditions a contribution to the understanding
of the flow field occurring is presented by the computational investigation of flow in
a concentric annulus for non-Newtonian fluids. The objective of this research is to
analyzed the surface roughness distribution under concentric annulus and predict
annular friction losses at a various rotation of speed.
The yield power law is apply for non-Newtonian fluid in concentric annulus by
using the high technologies computational fluid modeling the ANSYS CFX
software that able to simulates the fluid flow in turbulent flow condition.
1.3 Problem Statement
Surface roughness is one of the factors that cause the production to become less
economical since more energy is needed to keep the production going. The effect of
the surface roughness on turbulent flow of yield power law for non-Newtonian fluid
in concentric annulus is able to study by utilizing the computational fluid modeling
software ANSYS CFX.
Fluid flow is highly dependent on the viscosity of fluids. At the same time for a non-
Newtonian fluid, the viscosity is determined by the flow characteristics. In
reality most fluids are non-Newtonian, where their viscosity is dependent on shear
rate or the deformation history. Vice versa to the Newtonian fluids, a non-linear
relation between shear stress and shear rate is display by non-Newtonian fluids. The
non-Newtonian fluid yield stress or viscosity is dependent on deformation history or
time.
15. 3
1.4 Objective
The main objective of this project to study the effect of the surface roughness in
turbulent flow regime condition of yield power law for non-Newtonian fluid in
concentric annulus by using the high technologies computational fluid modeling the
ANSYS CFX software that is able to simulates the fluid flow in desired flow
condition. The specific objectives of this project are:
i. Analyzed the surface roughness distribution under concentric annulus
ii. Predict annular friction losses at a various rotation of speed.
1.5 Scope of study
Scope of study in this basically is to study the effect of surface roughness on
turbulent flow. The yield power law is the suitable law to be use especially for non-
Newtonian fluid. The fluid will be tested in concentric annulus by using the
computer modeling software ANSYS CFX. The foundation of this project would be
focus on as follows:
1. Absolute viscosity of the fluid
2. True shear rate
3. Reynold’s number of the flow
4. Friction factor
5. Relative roughness
16. 4
CHAPTER 2
LITERATURE REVIEW
2.1 Introduction
Numerical calculation of friction pressure losses in both the flow regime which is
laminar and turbulent flow for the Newtonian and non-Newtonian fluids in
concentric annuli have been done frequently and a lot of studies are available on.
There was very little has been accomplished for the effect of the surface roughness
for turbulent flow regime condition of non-Newtonian fluids through concentric
annuli.(Silva & Shah)
In order to maintain primary control of the well being drilled the drilling fluids are
primarily used during drilling operations. The function includes:
1. Keeping the hole free of cuttings
2. Suspending cuttings during trips
3. Allowing settling of cuttings on surface system
4. Transfer the hydraulic energy to the formation below the bit
Maintenance of the right drilling fluid properties and also the cost and choice of
drilling fluids are one of the important parts for completion of an oil or gas. The
properties of drilling fluids that are critical to successful drilling exercise include:
1. Mud weight
2. Yield point/value
3. Gel strength
4. Plastic viscosity and thixotropy
When the drilling fluids exhibit high viscosity at lower shear rates than at higher
shear rates it indicate that it is shear thinning. The flow regime determined the
17. 5
behavior of a drilling fluid, which is directly proportional with the ability of that
fluid to perform its basic behavior. By looking at the size, shape of the flow channel
and fluid velocity, viscosity, and fluid density the flow can be determine either
laminar or turbulent. The fluid has both laminar turbulent flow when the fluid passed
through a transition region. In order to evaluate the performance of a fluid it is
crucial to aware which of the flow regimes is present in a particular situation.
Operators and Service companies major concern is to determine the rheological
behavior of drilling fluids under static and dynamic drilling conditions. Most often,
a shear stress-shear rate graphical relationship (Rheogram), characteristic of the
type of drilling fluid, is used to understand this behavior. The rheological models
are utilize to deliver help in characterizing fluid flow.
2.2 Turbulent flow
Is an opposite of laminar condition, where vice versa to the laminar the mixing
occurs in turbulent flow where are having much higher velocity.The Reynolds
Number can characterized and quantified the condition of regime either laminar or
turbulent flows. Reynolds number is inversely proportional to the viscosity and
directly proportional to velocity.
When the Reynolds number is less than 2000 is laminar flow. When the value is
more than 4000, the flow is said to be a turbulent flow while when the value lie in
between 2000 and 4000, the flow is called transition region or critical region.
18. 6
Figure 2-1Turbulent flow
2.3 Viscosity
Commonly used unit for viscosity is in centipoise (cP) where it is equivalent to 1
mPa s (millipascal second).
Equation 2.1-Viscosity
stressshear
stressshear
viscosity
19. 7
2.4 Non-Newtonian Fluids
Figure 2-2-Non-Newtonian Fluid
In real situation non-Newtonian happen to be most fluid properties. The non-
Newtonian viscosity is dependent on shear rate or the deformation history. Vice
versa to the Newtonian fluids, the non-Newtonian fluids shows that the shear rate
and shear stress relationship is non-linear having and yield stress, or viscosity that is
dependent on deformation history or time. Five types of non-Newtonian fluids:
1. Plastic
2. Thixotropic
3. Pseudo-plastic
4. Dilatant
5. Rheopectic
When measuring each of these fluid types different considerations are required. If the
shear rate increases as the viscosity of the fluid increases the fluid is shear thickening
(See Figure 2-2).
20. 8
Mixture of cornstarch and water is a typical example for shear thickening fluids.
When the shear rate increases as the viscosity decreases, the fluids are shear thinning.
Pseudo-plastics also known as shear thinning fluids, are biological processes and
ubiquitous in industrial. Ketchup, paints and blood is a common examples.
There is several factors can caused the behavior of the non-Newtonian fluids related
to the structural reorganization of the fluid molecules due to flow. It is the
alignment of the highly anisotropic chains what results in a decreased viscosity for
polymer melts and solutions. The segregation of the different phases in the flow will
causes a shear thinning behavior for colloids.
Viscosity of fluids highly affects the fluid flow. The non-Newtonian fluid viscosity is
determined by the flow characteristics. The key in determining if a fluid is
Newtonian or non-Newtonian is characterization of viscosity is, and what range of
shear rates needs to be considered for an specific application. Often many
viscometers on the market measure index viscosity where it is lack proper
characterization of shear rate and absolute or true viscosity. The development and
modeling of applications that involve fluid flow most important parameters is the
Absolute viscosity. Thus, the shear rate that is relevant to the specific process must
be carried out at a proper characterization of viscosity.
21. 9
2.5 Newtonian Fluids
Equation 2.2 Shear Stress Equation
.
stressShearViscositystressShear
The flow behavior of fluids between shear stress [mPa] and shear rate [1/s] which
shows a simple linear relation is described by Sir Isaac Newton (1642 - 1726).
Newtonian fluids are named after him. The proportionality constant η is the viscosity
[mPa-s] of the fluid where this relationship is known as Newton's Law of Viscosity.
Water, organic solvents, and honey is one of the example for the non-Newtonian
fluid where the temperature is the one affect the viscosity value. Referring to
equation 2.2 a plot of shear stress versus viscosity shows that an increasing shear
rates with a linear increase in stress, where the viscosity of the fluid indicates the
slope. The viscosity of Newtonian fluids remained constant even if the fluid is forced
to flow through the pipe faster, which shows that the viscosity is independent of the
rate of shear.
22. 10
Figure 2-3-Newtonian Fluid
Bingham plastics are an exception to those rules where its fluids require a minimum
stress to be applied before they flow, that stress is called yield stress. These, are
strictly non-Newtonian before it flow but when they start to flow, they will behave as
Newtonian fluids where shear rate and shear stress is linearly.
Normally the Newtonian fluids are comprised of tiny isotropic (symmetric in
properties and shape) molecules that is not oriented by flow. However, behavior
with large anisotropic molecules is also possible to have for Newtonian fluids. For
example, constant viscosity regardless of shear rate might be showed by low
concentration protein or polymer solutions. Newtonian behavior at low shear rates
with a plateau where it is called the zero shear viscosity regions is also possible for
some samples.
23. 11
2.6 Analysis of Annular Flow
The equivalent narrow slot (Fig. 2-5) represented the concentric annulus to simplify
the mathematical model. The lower plate is stationary and with an average velocity
(Vp) the top plate is moving.(Crespo, Ahmed, Saasen, Enfis, & Amani)
Equation 2.3 Frictional pressure drop
z
DD
vf
P
io
2
2
Reynolds number:
Equation 2.4-Reynolds number for annular flow
K
n
n
vDD
Re n
n
nn
io
4
13
8 1
2
y
Region II y – y
y
Figure 2-4-Slot-model representation of a concentric annulus
24. 12
where Do is the annulus OD, and Di is the ID.
Equation 2.5-Laminar flow
Re
f
16
(approximately)
for Re ≤ 3,250 – 1,150n.
Equation 2.6 Turbulent flow for annular flow
21
2
-1
750
2
1
40
Reflog
041
.
n
.
n
.
n
.
f
for Re ≥ 4,150 – 1,150n.
2.7 Yield Power Law
The YPL (Herschel-Buckley) rheology model indicated the flow behavior of drilling
fluids well over wide cases for the shear rates. As a result, the YPL model is often
preferable when accurate hydraulic predictions are required.
Pipe flow
Equation 2.7-Fictional pressure drop for YPL
z
D
vf
P
i
2
2
25. 13
Equation 2.8 Reynolds number for YPL
m
ey k
v
2
YPL
8
Re
Where,
e
e
D
v8
w
y
x
Equation 2.9 Turbulent flow for YPL
YPLRe
f
16
Where,
for Re ≥ 4.150 – 1.150n.
1-n
12
n-2
'
'
(12)
V
'
n
K
dd
Re
'
26. 14
Table 2-1Composition of the test fluids and rheological properties of the test fluids
measured by use of Anton Paar MCR 301 rheometer (reference pressure 5
atmospheric)
27. 15
Figure 2-5-Farshad’s average absolute surface roughness chart for commonly used
pipe wall surfaces (Farshad and Rieke 2005)
2.8 CFD Modeling
The Computational fluid dynamics software specially design for modeling fluid
flow and other related physical behavior is provided by the computational fluid
dynamics FLUENT CFD and ANSYS CFX software ANSYS.
ANSYS CFX and ANSYS Fluent are the primary ANSYS products in the fluids
areas. With these solutions, the simulation with a wide range of phenomena
combustion, hydrodynamics, heat transfer, mixtures of liquids/solids/gas, reacting
flows, particle dispersions, aerodynamics, and much more phenomena that is easily
and instantly solved is the steady-states and transient flow.
28. 16
By considering the engineering situation that involving the particulate mixtures and
flow of fluids the Computational Fluid Dynamics (CFD) is a great tool that is
frequently used. Complex geometries can be analyzed and modeled to an
unprecedented level. Calculation of the frictional pressure losses without entailing
significant costs through experimental could be applied.(Silva & Shah)
The transition from laminar to turbulent flow will be earlier when the
eccentricity and rotary speed is increased. Results showed that the pressure loss
is reduced with an eccentric pipe. The increase in pressure losses is observed in
the free drill-string rotation mode as increasing the rotary speed of the drill-
string. The more dynamic motion of the drill-string will give a distinct different
of pressure loss. The parameters such, flow rate, diameter ratio, eccentricity,
fluid properties, and as drill-string rotary speed is correlate with the type of flow
in the annulus whether it is turbulent, transitional or laminar. The diameter ratio
or effect of eccentricity (the ratio of the diameter of the inner pipe to that of the
outer pipe) are frequently left out to be consider in most transition-criterion
model in literature.(Erge et al.)
The inertial effects, additional shear affects and additional shear to the fluid
where both of the shear-thinning property are affected by rotating the drill-string.
Most field measurements showed an increase in frictional pressure losses as the
drill-string is being rotated.
Rotating the drill-string in laminar flow of yield-power-law (YPL) fluids through
concentric annuli can decrease the pressure losses because of the shear-thinning
ability of YPL fluids, especially in laboratory experiments.
2.9 Fanning Friction Factor
The relative roughness and Reynolds number is found to be the function for the
friction factor. The behavior of the fluid flow that is done in either smooth or rough
pipes is not identical for different type of flow regime which is between turbulent
29. 17
and laminar flow Nikuradse (1933). For laminar flow (Re < 2100), the friction factor
was independent of the surface roughness and it varied linearly with the inverse of
Reynolds number The friction factor is independent with surface roughness for
laminar flow and it varied linearly with the inverse of Reynolds number.
Generalized Reynolds number for the API standard in the function of shear stress at
the wall
Equation 2.10-Reynolds number General formula
w
V
2
GRe
2.10 Haaland Friction Factor Equation
Herschel-Buckley fluid is tested after a standard run with water has been done for the
both eccentric and fully concentric annuli. The run were done with a minimum
velocity 0.35 m/s with Reynolds number 10629 which fall in turbulent categories.
The equation by Haaland(1983) is utilized for rough conduits condition. The
roughness, , is assuming to be 0.024mm to obtained a derivation of best match by
the ‘Datafit’.( Kelessidis et al.)
2.11 Buckingham Pi Theorem
n = the number of independent variables relevant to the problem
j’ = the number of independent dimensions found in the n variables
j = the reduction possible in the number of variables necessary to be considered
simultaneously
k = the number of independent ᴨ terms that can be identified to describe the
problem, k = n - j
30. 18
k = f( 1, 2, …i)
Theorem
= Consistency index
𝜌 𝑓 = density
V = velocity
𝐷ℎ = diameter
Γ= foam quality
μ 𝑓 = viscosity
ΔP=Pressure drop
= roughness
n = flow behavior index
ω= RPM
ʄ 𝑓: {P 𝜌 𝑓, V, 𝐷ℎ , 𝛍 𝑓 , k, n , ω, , Γ, }
Dependent independent
31. 19
Table 2-2- Variable and dimension list
Variables Dimensions
ΔP ML-1
T-2
∆𝑃
𝐿
ML-2
T-2
K MT n-2
L-1
n -
Γ -
ω T-1
V LT-1
𝐷ℎ L
𝜌 𝑓 ML-3
𝛍 𝑓 ML-1
T-1
L
n=11
j’ = 3 (M, L, T).
Common individual variables: 𝜌 𝑓, V, and 𝐷ℎ
k = n – j = 11 – 3 = 8 independent ᴨ terms.
32. 20
Table 2-3Modelling Approach
ΔP
∆𝐏
𝐋
K n Γ ω V 𝑫 𝒉 𝝆 𝒇 𝛍 𝒇
L -1 -2 -1 0 0 0 1 1 -3 -1 1
T -2 -2 n-2 0 0 -1 -1 0 0 -1 0
M 1 1 1 0 0 0 0 0 1 1 0
ᴨ terms
a. Fluid Viscosity,
ᴨ1 = ρa
Vb
Dc
(ML-3
)a
( LT-1
)b
Lc
( ML-1
T-1
)
[
𝑎
𝑏
𝑐
] (
−3 1 1
0 −1 0
1 0 0
) = [
−1
−1
1
]
-3a + b + c =-1, a=1
-b=-1, b=1
a=1, c=-1
-3a+1+c=-1,
We therefore have
ᴨ1 =
ρV D
𝜇
[Reynolds number]
34. 22
-3a + b + c =-1, a=1
-b=-2, b=2,
a=1, c=0
-3a+1+c=-1,
ᴨ3 =
𝜌𝑉2
𝛥𝑃
or ᴨ4 =
𝛥𝑃
𝜌𝑉2
[Cavitation or Pressure coefficient]
d. Pressure Drop per unit length,
𝛥𝑃
𝐿
[
𝑎
𝑏
𝑐
] (
−3 1 1
0 −1 0
1 0 0
) = [
−2
−2
1
]
ᴨ4 = a
Vb
Dc
(
𝛥𝑃
𝐿
)
(ML-3
)a
( LT-1
)b
Lc
( ML-2
T-2
)
-3a + b + c =-2, a=1
-b=-2, b=2,
a=1, c=-1
-3a+1+c=-1,
ᴨ4 =
𝐿
𝛥𝑃
𝜌𝑉2
𝐷
or ᴨ5 =
𝛥𝑃
𝐿
𝐷
𝜌𝑉2
[Friction factor]
35. 23
e. Consistency index,
[
𝑎
𝑏
𝑐
] (
−3 1 1
0 −1 0
1 0 0
) = [𝑛
−1
−2
1
]
ᴨ5 = a
Vb
Dc
𝑘
(ML-3
)a
( LT-1
)b
Lc
(MT n-2
L-1
)
-3a + b + c =-2, a=1
-b=n-2, b=-n+2,
a=1, c=n
-3a+(-n+2)+c=-1,
ᴨ5 =
𝜌𝐷 𝑛 𝑉−𝑛+2
𝑘
[Unknown]
Table 2-4-Dimensional Analysis and Similarity
Parameter Definition Qualitative ratio
of effects
Importance
Reynolds
number
Always
Roughness
ratio
Turbulent,rough
walls
RE
UL
Inertia
Viscosity
L
Wall roughness
Body length
36. 24
CHAPTER 3
METHODOLOGY
Basically, the research methodology of this project is simulation study by using
ANSYS CFX software. The foundation of this project would be focus on as follows:
1. Absolute viscosity of the fluid
2. True shear rate
3. Reynolds number of the flow
4. Friction factor
5. Relative roughness
The equipment that will be used is the ANSYS CFX software to perform the
modelling. This equipment is use as the foundation to determine the effect of the
surface roughness, the friction losses and the velocity of the fluid. On top of that, the
work, Gantt chart and project planning variable for the project was shown Figure 3-1
as below.
3.1Flow Phase
As mentioned above, this report main objective is to study the effect of the surface
roughness on turbulent flow of yield power law for non-Newtonian fluid in
concentric annulus by using computer modeling software ANSYS CFX. The key
milestone of project is divided into four (4) phases as follow:
Figure 3-1Project Milestone
Literature Review
Preparation of the ANSYS CFX model
Analysis of the result
Conclusion
37. 25
3.2 Methodology Flow
3.3 Method to determine the type of flow regime
3.3.1 Herschel and Buckley in annulus
Equation 3.1-Equivalent Reynold’s number
ReCaRe(eq)
Where,
n
o
o
n
2
1
2
212 R-R
Q
R-R
12n2
k
1n
1
-1Ca
21 R-R
Q2
Re
Define Problem
Statement, Objective
and Scope of Study
Critical analysis of
literature from various
sources
CFX modelling and
simulation
Model Formulation
Validation and
Parametric Study
Implementation
and Grid
Independent
Study
Documentation
39. 27
Equation 3.4-Haaland friction factor for rough conduits
e
1.1
H
10
R
6.9
3.7D
log6.3
1
f
Equation 3.5- Continuity
0v.
t
Equation 3.6-Momentum
v
Dt
Dv
P
Equation 3.7-Mean percentage Error
n
1t
%100
MPE
t
tt
a
fa
n
In the Haaland equation there is no need to iterate the Darcy friction factor. The
accuracy of the Darcy friction factor solved from this equation is claimed to be
within about ±2 %, if the Reynolds number is greater than 3000. ( Fox et al.)
3.5 Buckingham Pie Theorem
3.5.1 Roughness,
[
𝑎
𝑏
𝑐
] (
−3 1 1
0 −1 0
1 0 0
) = [
1
0
0
]
ᴨ1 = a
Vb
Dc
(ML-3
)a
( LT-1
)b
Lc
( L )
41. 29
3.6 Preliminary Results
In order to get the valid result for the experiment, the pipeline model has to be in an
optimum mesh sizes. This value can be determine by monitoring the value of the
given pressure gradient. The optimum mesh sizes is achieved when the pressure
gradient give a constant reading. The manipulated variable in order to determine the
optimum mesh sizes is the number of element where it is controlled by the number
of angular and radial of the pipeline model while the length of the pipe will be
constant since it is believe to give very small value to the number of element.
The number of element is choose randomly and be change after monitoring the
pressure gradient reading behavior. In this simulation, the number of element choose
is 100000, 200000, 358800, 368000, 400000, 425000, 450000 and 485000. Pressure
gradient give reading of 420 Pa/m when the number of element is 485000. The
pressure is decreased to 390 pa/m when the number of element is decreased to
100000. However the pressure gradient reading is constant when the number of
element is between 320000 and 485000 . Thus, the optimum mesh sizes is achieved
when the number of element is 385000 with the angular value 35, radial 55, and the
length is 100.
42. 30
Figure 3-2-Show an optimum mesh sizes for the pipe model
Figure 3-3-Shows the Number of element versus the pressure gradient
530.00
531.00
532.00
533.00
534.00
535.00
536.00
0 100000 200000 300000 400000 500000 600000
Pressuregradient,Pa/m
Number of Element
Element size VS Pressure Gradient
Optimum Number Of Division
Radial = 35
Angular = 55
Length = 100
43. 31
3.6.1 Model Validation
Figure 3-4-Pressure gradient versus velocity
The validation of the method is done by referring the finding from the previous paper
which is written by Kelessidis et.al, 2010 with the title ‘Experimental study and
predictions of pressure losses of fluids modelled as Herschel–Buckley in concentric
and eccentric annuli in laminar, transitional and turbulent flows’. The validation
value that is taken from the paper can be refer in Table 3-1.
0
500
1000
1500
2000
2500
3000
3500
4000
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Pressuregradient,Pa/m
velocity, m/s
Pressure gradient vs Velocity
Simulation Experiment
Mean percentage Error, 4.9%
44. 32
Table 3-1-Shows the error of accuracy between simulation and actual result
Actual Simulation
Error
[%]Velocity
[m/s] Pressure Gradient [Pa/m] Velocity [m/s]
Pressure
Gradient
[Pa/m]
0.06 1450 0.06 1540 6.2
0.14 1800 0.14 1860 3.3
0.28 2300 0.28 2430 5.7
0.42 2660 0.42 2800 5.3
0.495 2820 0.495 2955.85 4.8
0.57 3030 0.57 3234.25 6.7
0.63 3180 0.63 3284.3 3.3
0.71 3320 0.71 3415 2.9
0.84 3600 0.84 3770 4.7
The graph in Figure 3-4 is showing the result for pressure loss versus the velocity
with rotational equal to 0 for the fluid that having the following
Equation 3.8 Yield power law equation
54.0
143.059.1 w
45. 33
Table 3-2-Error comparison between Haalands Equation and Validation data
Fanning Friction factor
(Validation)
Friction factor by Haaland
Error, %
MPE,
%
2.01388 1.3413 33.4
53.4
0.4591 0.0971 78.8
0.1466 0.0400 72.7
0.0753 0.0277 63.3
0.0575 0.0245 57.4
0.0466 0.0222 52.3
0.0400 0.0206 48.5
0.0329 0.0191 42.0
0.0255 0.01735 32.0
Referring to Table 3-2, the Haaland equation is not suitable to be apply when the run
is assume to be in a smooth pipe. The error is observed to be more than 30% for all
the run with the minimum and maximum Reynolds number is 12 and 966. The fluid
flow regime is laminar since the Reynolds number value is less than 2000.
The properties of the fluid can be analyst from the Equation 3.8. The required
properties of the fluid is then be initialized in the ANSYS simulation setup. The
validation need to be done at the first place before any further experiment is
conducted. This is to ensure the result obtain in the future were valid and reliable to
be refer and analyst.
The validation result is done by using 500 iteration and RMS function with value
0.000001. The optimum mesh sizes are utilized for the geometry. The value of the
optimum mesh sizes is taken from the grid independent study that has been done
before the validation of experiment was conducted.
46. 34
3.7 Design of experiment
The experiment is going to be done by manipulate the type of yield power law (YPL)
fluid, velocity speed, and the relative roughness of the pipe. There is three type of
YPL fluid, seven type of relative roughness, and 15 value of the velocity speed. The
type of fluid that is selected from the Table 1 in literature review section. The
experiment will be run for 343 times. Below are the tables for the design of
experiment.
3.7.1 Pipe Geometry and Fluids Properties
Table 3-3-Geometry of the pipe
Inner diameter 0.1397 [m]
Outer diameter 0.2032 [m]
Hydraulic length 1.5000 [m]
Table 3-4-Fluid 1 (YPL1) properties
Shear stress 0.29 [Pa]
K 0.07 [Pa.s^m]
m/n 0.55 -
Density 1200 [Kg/m^3]
Π 3.142 -
Temperature 70 [°F]
47. 35
Table 3-5-Fluid 2(YPL3) properties
Shear stress 4.09 [Pa]
K 2.44 [Pa.s^m]
m/n 0.33 -
Density 1200 [Kg/m^3]
Π 3.142 -
Temperature 75 [°F]
Table 3-6-Fluid 3 (YPL5) properties
Shear stress 1.59 [Pa]
K 0.39 [Pa.s^m]
m/n 0.51 -
Density 1200 [Kg/m^3]
Π 3.142 -
Temperature 80 [°F]
3.7.2 Pipe Roughness Value
Table 3-7-Pipe roughness value
Material Average Absolute Roughness [m]
Internally plastic-coated pipe 5.08E-06
Honed -bare carbon steel 1.25E-05
Electro polished-bare Cr13 3.00E-05
Cement lining 3.30E-05
Bare carbon steel 3.51E-05
Fiberglass lining 3.81E-05
Bare Cr13 5.33E-05
49. 37
Figure 4-1-Pressure gradient versus velocity for Fluid 1
Figure 4-2-2D response surface for Fluid 1
2900.00
3000.00
3100.00
3200.00
3300.00
3400.00
3500.00
3600.00
3700.00
3800.00
3900.00
0.00E+00 1.00E-05 2.00E-05 3.00E-05 4.00E-05 5.00E-05 6.00E-05
PRESSUREGRADIENT[PA/M]
PIPE ROUGHNESS [M]
PRESSURE GRADIENT VS PIPE ROUGHNESS
FOR VARIOUS VELOCITY
4.45 m/s
4.42 m/s
4.39 m/s
4.36 m/s
4.33 m/s
4.30 m/s
4.27 m/s
4.24 m/s
4.21 m/s
4.18 m/s
4.15 m/s
4.12 m/s
51. 39
Figure 4-3-Pressure gradient versus velocity for Fluid 2
Response Surface result
Figure 4-4-2D response surface for Fluid 2
4000.000
4200.000
4400.000
4600.000
4800.000
5000.000
5200.000
5400.000
5600.000
5800.000
6000.000
0.E+00 1.E-05 2.E-05 3.E-05 4.E-05 5.E-05 6.E-05
PRESSUREGRADIENT[PA/M]
PIPE ROUGHNESS [M]
PRESSURE GRADIENT VS PIPE ROUGHNESS FOR
VARIOUS VELOCITY 4.50
m/s
4.45
m/s
4.40
m/s
4.35
m/s
4.30
m/s
4.25
m/s
4.20
m/s
4.15
m/s
4.10
m/s
4.05
m/s
4.00
m/s
53. 41
Figure 4-5-Pressure gradient versus velocity for Fluid 3
3800.00
4000.00
4200.00
4400.00
4600.00
4800.00
5000.00
5200.00
5400.00
5600.00
0.E+00 1.E-05 2.E-05 3.E-05 4.E-05 5.E-05 6.E-05
PRESSUREGRADIENT[PA/M]
PIPE ROUGHNESS [M]
PRESSURE GRADIENT VS PIPE ROUGHNESS FOR
VARIOUS VELOCITY 4.50 m/s
4.45 m/s
4.40 m/s
4.35 m/s
4.30 m/s
4.25 m/s
4.20 m/s
4.15 m/s
4.10 m/s
4.05 m/s
4.00 m/s
3.95 m/s
3..90 m/s
3..80 m/s
54. 42
Response Surface result
Figure 4-6-2D response surface for Fluid 3
The simulation needs to be run more than a thousand times before a valid result is
achieved. As referring to the table above, the run is done for approximately 343
times and the final output for each of the experiment to measure the pressure loss of
the system simultaneously correlate the pressure drop value with surface roughness
effect.
The pressure is then be correlated with the surface roughness to observed the effect
on the turbulent flow in annuli. It is recommended to estimates the pressure losses in
concentric annuli since it gives an accurately reading, especially for ECD-critical
operations such as managed pressure, extended reach and offshore drilling. A
reduced pressure loss is calculated due to the shear thinning ability of the YPL fluids
as the drill string rotates.
55. 43
4.4 Friction Factor Correlation
4.4.1 Fluid 1
Figure 4-7-Friction factor VS Reynolds number for fluid 1
0.00470
0.00480
0.00490
0.00500
0.00510
0.00520
0.00530
4800 5000 5200 5400 5600 5800
Fanningfrictionfactor,f
Reynolds number, NRe
f VS NRe (Fluid 1)
5.33E-05 m
3.81E-05 m
3.51E-05 m
3.30E-05 m
3.00E-05 m
1.25E-05 m
5.08E-06
57. 45
4.4.2 Fluid 2
Figure 4-9-Friction factor vs Reynolds number for fluid 2
0.007400
0.007500
0.007600
0.007700
0.007800
0.007900
0.008000
0.008100
0.008200
16500.0 17500.0 18500.0 19500.0 20500.0 21500.0 22500.0 23500.0
Fanningfrictionfactor,f
Reynolds number, NRe
f VS NRe (Fluid 2)
5.33E-05 m
3.81E-05 m
3.51E-05 m
3.30E-05 m
3.00E-05 m
1.25E-05 m
5.08E-06 m
58. 46
Figure 4-10- Mean Percentage Error between Haaland and Simulation+ Darcy
Equation
4.4.3 Fluid 3
Figure 4-11-Friction factor VS Reynolds number for fluid 3
0.006000
0.006500
0.007000
0.007500
0.008000
0.008500
16000.0 17000.0 18000.0 19000.0 20000.0 21000.0 22000.0 23000.0
Fanningfrictionfactor,f
Reynolds Number, NRe
Mean Percentage Error comparison for fluid 2
Haaland Equation R1
Darcy Equation R1
Darcy Equation R2
Haaland Equation R2
Darcy Equation R3
Haaland Equation R3
0.0068000
0.0069000
0.0070000
0.0071000
0.0072000
0.0073000
0.0074000
0.0075000
5600.0 6100.0 6600.0 7100.0 7600.0
Fanningfrictionfactor,f
Reynolds number, NRe
f VS NRe (Fluid 3)
5.33E-05 m
3.81E-05 m
3.51E-05 m
3.30E-05 m
3.00E-05 m
1.25E-05 m
5.08E-08 m
Pipe Roughness 1 (5.08E-06), MPE 14.9%
Pipe Roughness 2 (3.30E-05), MPE 13.4%
Pipe Roughness 3 (3.30E-05), MPE 12.1%
60. 48
Figure 4-13-Friction factor for fluid 1,2 and 3
Referring to Figure 20, the graph is plotted in log graph for X-axis. The maximum
Reynolds number for fluid 1 is 22575 which higher than fluid 1 and 2 where the
maximum value is 5601 and 7383. The Reynolds number obtained is observe to be
turbulent flow for all fluid since the value is higher than 4000. The friction factor
decreasing with increasing of the Reynolds number. The maximum friction factor
obtained from fluid 2 is 0.0075 and the minimum friction factor obtained from
fluid 1 is 0.0047.
0.004500
0.005000
0.005500
0.006000
0.006500
0.007000
0.007500
0.008000
0.008500
4500 45000
Frictionfactor,f
Reynolds Number, NRe
Friction factor VS Reynolds number for fluid 1, 2, 3
Fluid 1
Fluid 2
Fluid 3
61. 49
4.5 Buckingham Pie Theorem
Coefficient of multiple regression, 2
R = 0.9972
4.5.1 Equation by Datafit Solver
Y =
a+b*ln(x1)+c*ln(x1)^2+d*ln(x1)^3+e*ln(x1)^4+f*ln(x1)^5+g*ln(x2)+h*ln(x2)^2+i
*ln(x2)^3+j*ln(x2)^4
Table 4-4Constant Value from Data Fit Software
99% Confidence Intervals
Variable Value
a 1.394000
b 0.307842
c 0.077300
d 0.009711
e 0.000610
f 0.000015
g -0.424466
h 0.074960
i -0.005836
j 0.000169
/D1
k
VD
2
2)(-nn
62. 50
4.5.2 Summarization for new derive equation from Buckingham Pi Theorem
approach.
Equation 4.1 New Correlation by Buckingham Pi Theorem
f2f1f
Where,
32
1])(ln[00971101])0.0773(ln[1])0.307(ln[1.3941 .f
54
1])(ln[00001501])(ln[000610 ..
432
2])(ln[00016902])(ln[00583602])[0.07496(ln2])(ln[42446602 ...f
4.5.3 Friction Factor Correlation
Figure 4-14-Comparison for friction factor value between friction factor by Haaland,
friction factor by simulation + Darcy and friction factor by datafit solver
0.0045
0.0055
0.0065
0.0075
0.0085
0.0095
0.0105
4000.0 9000.0 14000.0 19000.0 24000.0
Frictionfactor,f
Reynolds number, NRe
Comparison of friction factor for three method
Friction (Haaland)
Friction (simulation+Darcy)
Friction Datafit
63. 51
Table 4-5-Mass percentage error between Haaland equation, simulation+darcy and equation
by DataFit
Comparison Mass percentage Error, %
Simulation VS Haaland 41.5
Datafit VS Haaland 41.6
Simulation VS Datafit 0.7
64. 52
4.6 Gantt Chart and Key Milestones
PCB4022 FINAL YEAR PROJECT 1
No. Tasks 1 2 3 4 5 6 7 8 9 10 11 12 13 14
1 Selection of Topic
2
Literature Review and Study on the ANSYS CFX
modelling software
3 Submission of Extended Proposal
4 Proposal Defence
5
Further Study on Surface Roughness, Yield power
law and turbulent flow.
6 Simulation for ANSYS CFX
7 Submission of Interim Draft Report
8 Submission of Interim Report
9 Field Work
65. 53
4.7 Discussion
4.7.1 Validation
The validation results are observe to have error with less 10 %. The maximum error
obtained is 6.7% where in most cases, a percent error or difference of less than 10%
will be acceptable while an error that give a difference of more than 10% will have
high probability that some mistake has occurred. Thus the model is assume to be
valid to be use in further experiment.
4.7.2 Fluids Analysis
From the calculation made, the Fluid 2 is observed to have the highest Reynolds
number follow with the Fluid 3 and the Fluid 1 is having the lowest Reynolds
number. This can be analyzed by comparing the properties of the fluids:
4.7.3 Reynolds number
Fluid 2> Fluid 3> Fluid 1
4.7.4 Shear stress,
Fluid 2> Fluid 3> Fluid 1
4.7.5 Consistency Index, K
Fluid 2> Fluid 3> Fluid 1
5.2.4 Flow behaviour index, n
Fluid 2< Fluid 3< Fluid 1
66. 54
4.8 Friction Factor Analysis
For all fluids the friction Factor is increasing when the pipe roughness
increase. Thus the relation between friction factor and pipe roughness is
directly proportional.
The friction factor will increase when the Reynolds number is increased.
Thus the relation between Reynolds number and friction factor is directly
proportional.
4.9 Haalands Correlation
MPE Fluid 2, 3%< MPE Fluid 3, 22% < MPE Fluid 1, 87%
The mass percentage error, MPE between the Haaland friction factor equation and
the darcy friction factor equation is decreasing when the Reynolds number is
increasing. The MPE for fluid 2 is 12% where the Reynolds number is 22000. The
analysis made that the MPE will be below 10 % when the Reynolds number is more
than 25000. Thus a conclusion can be made from the result obtained that the Haaland
equation is suitable to be apply for the fluid flow with the Reynolds number of more
than 25000. This has been tested in the validation of the model when comparing the
friction factor value between haaland correlations with the actual value.
4.10 Buckingham Pi Theorem Analysis
The value for coefficient of multiple regression obtained from Data Fit solver is
0.9972 which consider to be very precise. The mass percentage error, MPE for the
friction factor between simulation and Data Fit is 0.7 %.
4.11 Pressure Gradient
The pressure drop is increase when the shear stress, Reynolds number,
and consistency index increased. Thus, the relation is directly
proportional.
67. 55
The pressure drop is increase when the flow behavior index is decreased.
Thus, the relationship is inversely proportional.
CHAPTER 5
CONCLUSION
Conclusion
Based on the analysis made, several hypotheses can be made:
1. Reynolds number will increase when the shear stress and consistency index
increased. Thus the relation between Reynolds number, shear stress and
consistency index is directly proportional.
2. Reynolds number will increase when the flow behavior index is decreased.
Thus the relation between the Reynolds number and flow behavior index is
inversely proportional.
3. The relation between velocity and Reynolds number is directly proportional.
4. The turbulent flow is start with velocity 3.8 m/s for fluid 2 and 3 while for
fluid 1 Is 4.04 m/s which is higher than the other two fluids. This different in
velocity value for turbulent to be achieve is due to the properties of the fluid.
68. 56
REFERENCES
1. Crespo, F. E., Ahmed, R. M., Saasen, A., Enfis, M., & Amani, M. Surge-and-
Swab Pressure Predictions for Yield-Power-Law Drilling Fluids. doi:
10.2118/138938-PA
2. Erge, O., Ozbayoglu, E. M., Miska, S., Yu, M., Takach, N., Saasen, A., &
May, R. The Effects of Drillstring-Eccentricity, -Rotation, and -Buckling
Configurations on Annular Frictional Pressure Losses While Circulating
Yield-Power-Law Fluids. doi: 10.2118/167950-PA
3. Silva, M. A., & Shah, S. N. Friction Pressure Correlations of Newtonian and
Non-Newtonian Fluids through Concentric and Eccentric Annuli.
4. Fox, W. Pritchard, P. McDonald, J. Wiley and Sons, "Introduction to Fluid
Mechanics, Seventh Edition."2010/.
5. V. C. Kelessidis, P. Dalamarinis, R. Maglione, "Experimental study and
predictions of pressure losses of fluids modeled as Herschel–Bulkley in
concentric and eccentric annuli in laminar, transitional and turbulent
flows."2011/6/30/.
6. B. J. McKEON, C. J. SWANSON, M. V. ZAGAROLA, R. J. DONNELLY
AND A. J. SMITS, "Friction factors for smooth pipe flow." 27 April/4/
2004/.
7. F. F. Farshad and H.H. Rieke "Surface-Roughness Design Values for
Modern Pipes."28/4/2006/.
8. A.H. Kamel and A. S. Shaqlaih, "Frictional Pressure Losses of Fluids
Flowing in Circular Conduits."21/4/2015/.
