NUMERICAL SIMULATION OF HEAT TRANSFER IN BLOOD VESSELS
SALAHUDDIN AZRAIE
A thesis submitted in fulfillment of the
requirements for the award of the degree of
Master of Engineering (Mechanical)
Faculty of Mechanical Engineering
Universiti Teknologi Malaysia
MARCH 2010
iii
ABSTRACT
There is inadequate work done on tapered and stenois blood vessel under
thermotheraphy, experimental studies being impossible to be conducted since the
system is unlike mechanical system which could be modeled in the lab. In this
study, the blood flow and heat transfer in an artery under various heating conditions
have been numerically simulated. Three different cases of blood vessels are
considered: straight vessel, converging tapered vessel and stenois vessel. Constant
heat flux was applied at several intensities in each case. It was shown that the flow
simulated at the vena contracta is analogous to the flow in a converging-diverging
nozzle. However, the dimensionless center velocity at outlet region was found to be
higher than that at the inlet region which could be due to the slow recovery of
pressure after the stenois region. The local Nusselt number fluctuated at both the
downstream and upstream which could have been caused by recirculation as the
flow entered and exit the stenois region. Results in general have shown that blood
flow in biological systems may well behave like the mechanical systems.
iv
ABSTRAK
Kajian terhadap pembuluh darah meruncing dan pembuluh ‘stenois’ dalam
keadaan terapi panas adalah kurang. Ujikaji adalah mustahil dilakukan kerana tidak
seperti sistem mekanikal, model sistem untuk ujikaji tidak boleh dilakukan di
dalam makmal. Dalam kajian ini, pengaliran darah dan pemindahan haba di dalam
saluran arteri di bawah pelbagai keadaan pemanasan telah disimulasikan secara
berangka. Tiga kes pembuluh darah telah dilihat: pembuluh lurus, pembuluh
meruncing dan pembuluh ‘stenois’. Pelbagai kekuatan fluks haba kenaan telah
dikaji dalam setiap kes. Hasil simulasi menunjukkan bahawa aliran pada vena
kontrakta adalah seperti aliran di dalam muncung tumpu capah. Halaju pusat tak
berdimensi pada kawasan alur keluar didapati lebih tinggi daripada yang terdapat
pada kawasan alur masuk. Ini mungkin disebabkan oleh pemulihan perlahan
tekanan selepas melewati kawasan stenois. Angka Nusselt tempatan turun-naik di
hulu dan hilir kawasan stenois yang mungkin disebabkan oleh edaran semula
apabila aliran memasuki dan keluar dari kawasan tersebut. Keputusan secara
umumnya menunjukkan bahawa aliran darah dalam sistem biologi berkelakuan
sama seperti aliran sistem mekanikal.
v
TABLE OF CONTENTS
CHAPTER
TITLE
TABLE OF CONTENTS
PAGE
v
LIST OF TABLES
viii
LIST OF FIGURES
ix
LIST OF SYMBOLS
xii
LIST OF APPENDICES
xiv
CHAPTER 1
INTRODUCTION
1.1
Introduction
1
1.2
Literature Review
3
1.3
Objectives
9
1.4
Scopes of Study
10
vi
CHAPTER 2
GOVERNING EQUATIONS
2.1
Continuity and Momentum Equations
11
2.2
Energy Equations
13
2.3
Related Equations
14
2.4
Assumptions
16
CHAPTER 3
METHODOLOGY
3.1
Computational Fluid Dynamics
18
3.2
Pre-processing: GAMBIT
19
3.3
Solver: FLUENT
19
3.4
Post-processing: FLUENT
21
3.5
Pre-processing: Domain of Computational
22
3.6
Solver: Parameters
26
3.7
Solver: Method of Solution
28
3.8
Expected Outcome
29
CHAPTER 4
MODEL VALIDATION
4.1
Model Validation Methodology
30
4.2
Grid Dependency Test
32
4.3
Model Validation with Numerical Results Comparison 34
vii
CHAPTER 5
RESULTS AND DISCUSSIONS
5.1
Simulation of the Normal Blood Flow in an Artery
(Straight Tube)
5.2
Simulations of Abnormal Blood Flow in an Artery under
Heating Conditions (Straight Tube)
5.3
37
39
Simulations of Blood Flow in an Artery
(Tapered and Stenois Tubes)
5.3.1
Simulations of Blood Flow in an Artery
(Tapered Tube)
5.2.2
47
47
Simulations of Blood Flow in an Artery
(Stenois Tube)
56
CHAPTER 6
CONCLUSION
6.1
Conclusion
68
REFERENCES
70
APPENDICES A-M
73
viii
LIST OF TABLES
TABLE NO
TITLE
PAGE
3.6.1
Blood and water properties
26
3.6.2
Variation for normal blood flow in an artery
26
3.6.3
Variation for abnormal blood flow in an artery under
heating condition
3.6.4
4.1.1
27
Variation for abnormal blood flow under tapered and
stenois channel
28
Summary of meshing size for grid dependency test
31
ix
LIST OF FIGURES
FIGURE NO TITLE
PAGE
1.2.1
Blood flow categories
8
1.2.2
History of blood flow studies in arteries
8
1.2.3
History of blood flow studies in veins and capillaries
9
3.3.1
Segregation solution method
3.3.2
Coupled solution method
21
3.5.1
Normal blood flow in an artery
22
3.5.2
Abnormal blood flow in an artery under heating condition 23
3.5.3
Normal blood flow in a tapered artery
3.5.4
Abnormal blood flow in a tapered artery under heating
20
23
condition
24
3.5.5
Normal blood flow in a stenois artery
25
3.5.6
Abnormal blood flow in a stenois artery under heating
condition
4.2.1
25
Percentage difference changing from one mesh size to
another
33
4.2.2
Number of iteration versus number of cells
33
4.3.1
Local Nusselt number for simultaneously developing
region in a circular duct
4.3.2
35
Comparison of local Nusselt number with correlation
results
36
x
5.1.1
Axial velocity of blood flow (q = 0W/m2, Um = 0.05m/s)
at various radii
5.1.2
Skin friction coefficient of blood flow
(q = 0w/m2, Um = 0.05m/s) at various radii
5.2.1
41
Axial velocity of blood flow (r = 5mm, Um = 0.05m/s)
under various heating conditions
5.2.5
40
Local Nusselt number of blood flow
(q = 4W/m2, Um = 0.05m/s) at various radii
5.2.4
39
Skin friction coefficient of blood flow
(q = 4W/m2, Um = 0.05m/s) at various radii
5.2.3
38
Axial velocity of blood flow (q = 4W/m2, Um = 0.0 m/s)
at various radii
5.2.2
38
42
Skin friction coefficient of blood flow
(r = 5mm, Um = 0.05m/s) under various heating conditions 43
5.2.6
Local Nusselt number of blood flow
(r = 5mm, Um = 0.05m/s) under various heating conditions 44
5.2.7
Plot of correlation of local Nusselt number for blood and
water flow
5.2.8
45
Plot of simulation and correlation of local Nusselt number
for blood flow
45
5.2.9
Zoom- in of Figure 5.2.6
46
5.3.1.1
Axial velocity of blood flow (q = 0W/m2, Um = 0.05m/s)
under various taper angles
5.3.1.2
Skin friction coefficient of blood flow
(q = 0W/m2, Um = 0.05m/s) under various taper angles
5.3.1.3
49
Skin friction coefficient of blood flow
(q = 4W/m2, Um = 0.05m/s) under various taper angles
5.3.1.5
48
Axial velocity of blood flows (q = 4W/m2, Um = 0.05m/s)
under various taper angles
5.3.1.4
47
50
Local Nusselt numbers of blood flow
(q = 4W/m2, Um = 0.05m/s) under various taper angles
51
xi
5.3.16
Axial velocity of blood flow (Um = 0.05m/s, θ = 0.50)
under various heating conditions
5.3.1.7
52
Skin friction coefficient of blood flow
(Um = 0.05m/s, θ = 0.50) under various heating conditions 53
5.3.1.8
Local Nusselt number of blood flow
(Um = 0.05m/s, θ = 0.50) under various heating conditions 54
5.3.2.1
Axial velocity of blood flow (q = 0W/m2, Um = 0.0 m/s)
with various stenois heights
5.3.2.2
Center axial velocity of blood flow
(q = 0W/m2, Um = 0.05m/s) with various stenois heights
5.3.2.3
62
Axial velocity of blood flow (a / r = 0.2, Um = 0.05m/s)
under various heating conditions
5.3.2.9
61
Local Nusselt number of blood flow
(q = 4W/m2, Um = 0.05m/s) with various stenois heights
5.3.2.8
60
Skin friction coefficient of blood flow
(q = 4W/m2, Um = 0.05m/s) with various stenois heights
5.3.2.7
59
Center axial velocity of blood flow
(q = 4W/m2, Um = 0.05m/s) with various stenois heights
5.3.2.6
58
Axial velocity of blood flow (q = 4W/m2, Um = 0.05m/s)
with various stenois heights
5.3.2.5
57
Skin friction coefficient of blood flow
(q = 0W/m2, Um = 0.05m/s) with various stenois heights
5.3.2.4
56
63
Center axial velocity of blood flow
(a / r = 0.2, Um = 0.05m/s) under various heating conditions 64
5.3.2.10
Skin friction coefficient of blood flow
(a / r = 0.2, Um = 0.05m/s) under various heating conditions 65
5.3.2.11
Local Nusselt number of blood flow
(a / r = 0.8, Um = 0.05m/s) under various heating conditions 66
xii
LIST OF SYMBOLS
cp
Specific heat
d
Diameter
F
External body forces or any user-defined sources
h
Enthalpy (ideal gases / incompressible flows)
h
Heat transfer coefficient
I
Unit tensor
J
j
Diffusion flux of species j
k eff
Effective conductivity
p
Static pressure
q
Constant heat flux
r
Radial coordinate / radius
Sh
Heat of chemical reaction and any other user-defined heat sources
Sm
Mass added to the continuous phase from the dispersed second
phase or any user-defined sources
t
Time
Tm
Fluid bulk temperature
Tw
Wall temperature
Uc
Axial velocity at center of tube
Um
Fluid average inlet velocity
x
Axial coordinate
ρ
Fluid density
xiii
ρg
Gravitational body force
τ
Tensor stress
µ
Molecular viscosity / fluid dynamics viscosity
ν
Fluid velocity
νr
Radial velocity
νx
Axial velocity
∇ν
T
Effect of volume dilation
Non-dimensional parameter
Nu
Nusselt number
Pr
Prandlt number
Re
Reynolds number
xiv
LIST OF APPENDICES
APPENDIX
A
B
TITLE
Blood properties set up in Fluent
Axial velocity profile for 0.5 tapered tube
77
Axial velocity profile for stenois height of 1.5mm
(q = 0W/m2, Um = 0.05m/s)
J
76
Axial velocity profile for stenois height of 1mm
(q = 0W/m2, Um = 0.05m/s)
I
76
Temperature contour for 0.50 tapered tube
(q = 6W/m2, Um = 0.05m/s)
H
75
Temperature contour for 0.50 tapered tube
(q = 5W/m2, Um = 0.05m/s)
G
75
Temperature contour for 0.50 tapered tube
(q = 4W/m2, Um = 0.05m/s)
F
74
Axial velocity profile for 1.50 tapered tube
(q = 4W/m2, Um = 0.05m/s)
E
74
Axial velocity profile for 1.00 tapered tube
(q = 4W/m2, Um = 0.05m/s)
D
73
0
(q = 4W/m2, Um = 0.05m/s)
C
PAGE
77
Axial velocity profile for stenois height of 2mm
(q = 0W/m2, Um = 0.05m/s)
78
xv
K
Temperature contour for stenois height of 1mm
(q = 4W/m2, Um = 0.05m/s)
L
Temperature contour for stenois height of 1mm
(q = 5W/m2, Um = 0.05m/s)
M
78
79
Temperature contour for stenois height of 1mm
(q = 6W/m2, Um = 0.05m/s)
79
xvi
CHAPTER I
INTRODUCTION
1.1
Introduction
The understanding of the function of the circulation system under both
normal and abnormal conditions, designing cardiovascular artificial devices as well
as diagnosing and treating disease is not complete without the full understanding of
blood flow. The first attempt to formulate the governing equation of motion of
blood in an artery was made by Euler in 1775. Before the age of computer, the
analytical solutions were done with much considerable simplifications. Following
the introduction of computers, blood flow equation can be treated with fewer
simplification and enable the theoretical results to be close to experimental ones
(Hunter, 1972).
Since the earlier attempt by Euler until recently development, many models
of blood and blood flow have been introduced by researchers. Earlier researchers
tend to assume blood as a Newtonian fluid which is flowing in a rigid tube wall in
order to make the calculation easier. Later on, non-Newtonian fluid assumption was
applied to study blood flow in tube with less than 1mm diameter. Lattice
xvii
Boltzmann approach was introduced around year 2009 to cater to non-Newtonian
complexity of blood flow (Ashrafizaadeh and Bakshaei, 2009). Currently, the study
of blood flow in a straight tube involves complex simulation by introducing
bifurcation and stenosis effects. Heat transfer and magnetic field effect on blood
flow have also been introduced to bring more understanding about blood rheology.
Higher heart failure mortality in most developed countries (Espino et al,
2006) makes the study of blood flow significantly important. Certain diseases lead
to heart failure due to the changing of blood flow pattern in blood vessels. From a
microscopic point of view, interaction between blood particles (red blood cells,
white blood cells and platelets) and bifurcation walls can lead to changing in
velocity patterns especially when there are irregularities in the said regions such as
fat or stenosis.
Blood contains many enzymes and hormones, and transports oxygen and
carbon dioxide from the lung to the cells of the tissues. Blood circulation also plays
an important role in thermoregulation and mass transport. The quantitative
prediction becomes more significant in human thermal comfort, drug delivery and
noninvasive measurements research. Earlier heat transfer analysis introduced noninvasive measurement of glucose in blood by means of measuring heat generation
from glucose itself. Another significant importance of heat transfer analysis in
blood vessels to surrounding walls or tissues is to understand more on drug
penetration in tumor affected tissues.
Thermal therapy is one of the widely used methods in treating tumor.
Various method of heating is introduced in medical field to enhance necrosis rate of
tumor such as ultrasound, laser irradiation, and laser ablation. Hyperthermia
therapy method is used to enhance the effect of radiotherapy and chemotherapy
(Yuan, 2009). These methods require no surgery to perform. Other method that
requires small surgery is radiofrequency tumor ablation (Santos et al, 2008). All
abovementioned methods are aim to kill tumor affected tissue. However, heating
does change behavior of blood flow and heating effectiveness relies on blood
xviii
vessels geometry. Therefore, further understanding of heating effects on blood flow
in various blood vessels geometry is indeed important to improve tumor treatment
under thermal therapy.
1.2
Literature Review
J.P. Hunter is probably the first to formulate and solved numerically the
governing equations of blood flow in a nonlinear viscoelastic artery in 1972 in his
Masters’ thesis (Hunter, 1972). He assumed that the blood was Newtonian,
homogenous and incompressible. In his model, blood was assumed to flow in a
cylindrical tube under non-turbulent flow condition with an elastic artery wall. The
artery wall showed both elastic and collagen fiber behavior. These two behaviors
lead to a non-linear time dependent stress-strain relationship. The artery wall
became stiffer at higher transmural (through the wall of an organ) pressure due to
higher Young’s modulus value. In vito testing was used to obtain experimental data
to get the relationship between transmural pressure and arterial radius of the wall.
According to Hunter, an empirical approach was better than synthetic approach
using finite elasticity theory since the synthetic approach implied the artery wall
material is homogenous, isotropic and incompressible. Furthermore, Hunter also
considered blood flow through bifurcation in order to fully describe his heart-aorta
model. The computational blood flow through bifurcation was solved using
FORTRAN software by adopting finite difference approach. Heat transfer in blood
flow was mentioned in the last chapter. The general one dimensional heat equation
was used to calculate the transfer of heat along the artery assuming no heat loss to
the surroundings. Hunter did not consider any heat transfer between bloods to the
artery wall.
xix
Khaled et al (Khaled et al, 2003) reviewed transport theories in porous
media and its application in biological tissues. Mass diffusion and different
convective flow models are used to study the transport theories. Development of
bio-heat equation was analyzed. The most appropriate bio-heat model for heat
transfer in biological tissues was found in the theory of porous media.
Tsubota et al (Tsubota et al, 2006) introduced a new approach to describe
the behavior of blood flow in 2005 called particle method. Blood was assumed to
be incompressible and Newtonian fluid and modeled as an assembly of discrete
particles. Blood was also assumed to be homogenous fluid from macroscopic point
of view. Tsubota et al argued that continuum mechanics approach such as finite
difference method (FDM), finite volume method (FVM), and finite element method
(FEM) can be used to analyze blood flow numerically. One of obvious limitation
for this approach is the simulation is its two-dimensional analysis instead of threedimensional. However, his simulation results are yet to be verified by experimental
results. Furthermore, there is no mention of heat transfer analysis in blood vessel
using this particle method.
Blood flow and heat transfer in human finger was studied by He et al in
2008 (He et al, 2008). In his blood flow modeling, he simulated the blood flow in
large vessels with the diameter larger than or equal to 1mm. One-dimensional flow
model was used for this case. Darcy model in porous medium approach was
assumed in his blood flow modeling for micro-vessels case. Finite element model
was used to model a human finger. Magnetic resonance imaging (MRI) method
was used to obtain a geometric model of a human finger and different sequences of
images were taken to simulate flow of blood and heat transfer. From his images
data, the inflow velocity in the artery was found to be 19cm/s and in the
vein3.5cm/s. Reynolds number was 50 for this case when the arterial velocity was
the reference velocity. Ying He concluded that porous media transport model
provided good result to heat transfer analysis in living tissues (in this case, human
finger) using unified energy equation.
xx
Numerical modeling of blood flow in a variable shape artery wall was
performed by Mukhopadhyay and Layek (Mukhopadhyay and Layek, 2008). In
their study, blood was assumed to be a Newtonian fluid and the flow was simulated
in the rigid artery wall under steady state situations. Three shapes of the stenois
(abnormal narrowing in a blood vessel) were used in this modeling and the shape
was symmetric as well as asymmetric about the middle cross sectional area.
Mukhopadhyay et al agreed in their research that blood could be treated as a
Newtonian fluid since the viscosity coefficient of blood approaches a constant
value under relatively large shear rates. Furthermore, the diameter of the rigid tube
used in the modeling was equivalently the same as the diameter of a large vessel in
which Newtonian fluid condition was satisfied. Hemodynamic factors in the rigid
tube like wall shear stress, pressure and velocity were analyzed. No heat transfer
analysis was performed under this rigid artery wall assumption with variable shape
of stenois.
Thermal effect on tumor was studied by Horng et al (Horng et al, 2007)
using pulsatile blood flow. Cyclic effect of heart beat was implemented by using a
sinusoidal pulsatile velocity profile. Horng et al judged from their literature that
blood flow in all arteries undergo pulsatile flow due to the heart pump nature. The
energy transport equation with Pennes’s bioheat equation was used to solve the
heat transfer between blood vessels and surrounding tumor tissues. The blood flow
was modeled with assumptions of incompressible, laminar and Newtonian fluid.
Blood vessel was modeled as a rigid artery wall in perfect cylindrical shape. Other
blood properties such as thermal conductivity; specific heat and perfusion mass
flow rate can be found in their literature. They concluded in their study by
suggesting that steady uniform velocity profile and pulsatile velocity profile for
blood flow exhibited minor differences on thermal analysis. Heat transfer analysis
to the surrounding tissues was done in their study with an assumption of inelastic
artery wall.
The study of dynamics response of heat and mass transfer in blood flow
through bifurcation arteries was done by Chakravarty and Sen (Chakravarty and
xxi
Sen, 2005). A mathematical model of the response was developed under the
assumption of Newtonian fluid. Navier-Stokes equation was used to describe
nonlinear unsteady flow phenomenon and heat conduction and convectiondiffusion equations were used to performed thermal analysis. In this study, the
artery wall was assumed to be elastic on local fluid mechanics but not on the
stresses and strains in the wall itself. The wall motion effect on velocity profile of
blood flow was found to be insignificant. Temperature profiles at apex of arterial
bifurcation also gave insignificant difference when considering wall motion
compared to the rigid wall. In his concluding remark, there was significant
reduction in heat flux magnitude when the arterial wall was considered as rigid.
Hyperthermia therapy simulation in biological tissue was done by Yuan
(Yuan, 2008). An equivalent heat transfer coefficient in a porous model was done
by numerical analysis. In this study, all arterial blood vessels were assumed to be
straight and treated as porous medium, constant heat transfer coefficient and blood
velocity were used, and thermal properties for tissues and blood were treated as
isotropic. Ultrasound was used as a source of energy input for hyperthermia
treatment. The whole domain of calculation was considered as an assembly of
repeated hexagon units. Tri-Diagonal Matrix Algorithm was employed to solve
two-equation porous model for blood and tissue and Pennes’s bio-heat transfer
equation to predict the temperature distribution. Yuan concluded in his summary
that a decrease in vessel diameter increases the equivalent heat transfer coefficient.
The equivalent heat transfer coefficient also is not dominated by blood velocity and
heating conditions. Three equivalent heat transfer coefficient were calculated for
three different vessel diameters. Yuan did not consider in his study to for
expanding or narrowing vessel diameter or non-symmetric blood vessel.
Cooling effect of thermally significant blood vessels in perfused tumor
tissue during thermal therapy was done by Shih et al (Shih et al, 2006). A single
blood vessel with rigid wall inside and throughout the perfused tumor tissue was
used as computational domain. The velocity profile in the blood vessel was
assumed to be uniform. The temperature of blood flow and tissue were governed by
xxii
the energy transport equation. During thermal simulation, ultrasound was used as
source of heat power and the heat absorption rate of blood is assumed to be onetenth of tissue. Heating power density and heating durations were varied while the
absorbed total energy density was kept constant during the simulation. Shih et al
concluded that short duration and high intensity heating method can eliminate the
cooling the cooling effect of the blood vessels 0.2mm in diameter. Cooling effect
was stronger for blood vessel larger than 0.2mm diameter and led to incomplete
necrosis even though double heating power or density was used. Shih also did not
consider in his study to for expanding or narrowing vessel diameter or nonsymmetric blood vessel.
Radiofrequency electric current is used in radiofrequency tumor ablation
procedure. An active electrode is inserted by mean of surgery and radiofrequency
energy is applied to heat the surrounding tissue. Santos et al (Santos et al, 2008)
studied the effect of variable heat transfer coefficient on tissue temperature next to
a larger vessel during radiofrequency tumor ablation. In his journal, a large artery
10mm diameter was used and an electrode was put at two different distances away
from the artery. The blood was assumed as a Newtonian fluid and the flow was
assumed to be laminar. The vessel of artery was assumed to be straight with
constant blood velocity. Both constant and variable of the convective heat transfer
coefficient were simulated. The convective heat transfer coefficient was increased
initially and rapidly up to 6000W/m2K followed by a sharp decrease to 554W/m2K
for 1mm distance configuration. The convective heat transfer coefficient was
increased initially and rapidly up to 7000W/m2K followed by a sharp decrease to
336W/m2K for 5mm distance configuration. Santos et al concluded that assumption
of constant convective heat transfer coefficient sufficient enough to produce precise
result for thermal procedure more than 2 minutes.
The preceding paragraphs described some of the past studies related to
blood flow. Figure 1.2.1 shows blood flow major categories.
xxiii
Figure 1.2.1: Blood flow categories
Figure 1.2.2 shows summarized most of the work associated with blood
flow completed.
Figure 1.2.2: History of blood flow studies in arteries
xxiv
Figure 1.2.3: History of blood flow studies in veins and capillaries
Figure 1.2.2 and Figure 1.2.3 show that most of heat transfer studies were
done with straight vessel assumption. He et al (He et al, 2008) did performed heat
transfer study by considering bifurcation vessel but He assumed that both parent
and daughter vessels were straight vessel. Apparently, there is inadequate work
done on tapered and stenois blood vessel under heating condition.
1.3
Objective
The objective of this study is to numerically simulate the blood flow and
heat transfer in an artery under heating conditions.
xxv
1.4
Scope of Study
The scope of this study will involve:
1 – Simulation of the normal blood flow in an artery.
2 – Repeating (1) under abnormal condition of heating (i.e. chemotherapy,
thermotherapy) at various degrees.
3 – Repeating (1) for a tapered channel flow and stenois (abnormal
narrowing) channel flow.
CHAPTER I
INTRODUCTION
1.1
Introduction
The understanding of the function of the circulation system under both
normal and abnormal conditions, designing cardiovascular artificial devices as well
as diagnosing and treating disease is not complete without the full understanding of
blood flow. The first attempt to formulate the governing equation of motion of
blood in an artery was made by Euler in 1775. Before the age of computer, the
analytical solutions were done with much considerable simplifications. Following
the introduction of computers, blood flow equation can be treated with fewer
simplification and enable the theoretical results to be close to experimental ones
(Hunter, 1972).
Since the earlier attempt by Euler until recently development, many models
of blood and blood flow have been introduced by researchers. Earlier researchers
tend to assume blood as a Newtonian fluid which is flowing in a rigid tube wall in
order to make the calculation easier. Later on, non-Newtonian fluid assumption was
applied to study blood flow in tube with less than 1mm diameter. Lattice
2
Boltzmann approach was introduced around year 2009 to cater to non-Newtonian
complexity of blood flow (Ashrafizaadeh and Bakshaei, 2009). Currently, the study
of blood flow in a straight tube involves complex simulation by introducing
bifurcation and stenosis effects. Heat transfer and magnetic field effect on blood
flow have also been introduced to bring more understanding about blood rheology.
Higher heart failure mortality in most developed countries (Espino et al,
2006) makes the study of blood flow significantly important. Certain diseases lead
to heart failure due to the changing of blood flow pattern in blood vessels. From a
microscopic point of view, interaction between blood particles (red blood cells,
white blood cells and platelets) and bifurcation walls can lead to changing in
velocity patterns especially when there are irregularities in the said regions such as
fat or stenosis.
Blood contains many enzymes and hormones, and transports oxygen and
carbon dioxide from the lung to the cells of the tissues. Blood circulation also plays
an important role in thermoregulation and mass transport. The quantitative
prediction becomes more significant in human thermal comfort, drug delivery and
noninvasive measurements research. Earlier heat transfer analysis introduced noninvasive measurement of glucose in blood by means of measuring heat generation
from glucose itself. Another significant importance of heat transfer analysis in
blood vessels to surrounding walls or tissues is to understand more on drug
penetration in tumor affected tissues.
Thermal therapy is one of the widely used methods in treating tumor.
Various method of heating is introduced in medical field to enhance necrosis rate of
tumor such as ultrasound, laser irradiation, and laser ablation. Hyperthermia
therapy method is used to enhance the effect of radiotherapy and chemotherapy
(Yuan, 2009). These methods require no surgery to perform. Other method that
requires small surgery is radiofrequency tumor ablation (Santos et al, 2008). All
abovementioned methods are aim to kill tumor affected tissue. However, heating
does change behavior of blood flow and heating effectiveness relies on blood
3
vessels geometry. Therefore, further understanding of heating effects on blood flow
in various blood vessels geometry is indeed important to improve tumor treatment
under thermal therapy.
1.2
Literature Review
J.P. Hunter is probably the first to formulate and solved numerically the
governing equations of blood flow in a nonlinear viscoelastic artery in 1972 in his
Masters’ thesis (Hunter, 1972). He assumed that the blood was Newtonian,
homogenous and incompressible. In his model, blood was assumed to flow in a
cylindrical tube under non-turbulent flow condition with an elastic artery wall. The
artery wall showed both elastic and collagen fiber behavior. These two behaviors
lead to a non-linear time dependent stress-strain relationship. The artery wall
became stiffer at higher transmural (through the wall of an organ) pressure due to
higher Young’s modulus value. In vito testing was used to obtain experimental data
to get the relationship between transmural pressure and arterial radius of the wall.
According to Hunter, an empirical approach was better than synthetic approach
using finite elasticity theory since the synthetic approach implied the artery wall
material is homogenous, isotropic and incompressible. Furthermore, Hunter also
considered blood flow through bifurcation in order to fully describe his heart-aorta
model. The computational blood flow through bifurcation was solved using
FORTRAN software by adopting finite difference approach. Heat transfer in blood
flow was mentioned in the last chapter. The general one dimensional heat equation
was used to calculate the transfer of heat along the artery assuming no heat loss to
the surroundings. Hunter did not consider any heat transfer between bloods to the
artery wall.
4
Khaled et al (Khaled et al, 2003) reviewed transport theories in porous
media and its application in biological tissues. Mass diffusion and different
convective flow models are used to study the transport theories. Development of
bio-heat equation was analyzed. The most appropriate bio-heat model for heat
transfer in biological tissues was found in the theory of porous media.
Tsubota et al (Tsubota et al, 2006) introduced a new approach to describe
the behavior of blood flow in 2005 called particle method. Blood was assumed to
be incompressible and Newtonian fluid and modeled as an assembly of discrete
particles. Blood was also assumed to be homogenous fluid from macroscopic point
of view. Tsubota et al argued that continuum mechanics approach such as finite
difference method (FDM), finite volume method (FVM), and finite element method
(FEM) can be used to analyze blood flow numerically. One of obvious limitation
for this approach is the simulation is its two-dimensional analysis instead of threedimensional. However, his simulation results are yet to be verified by experimental
results. Furthermore, there is no mention of heat transfer analysis in blood vessel
using this particle method.
Blood flow and heat transfer in human finger was studied by He et al in
2008 (He et al, 2008). In his blood flow modeling, he simulated the blood flow in
large vessels with the diameter larger than or equal to 1mm. One-dimensional flow
model was used for this case. Darcy model in porous medium approach was
assumed in his blood flow modeling for micro-vessels case. Finite element model
was used to model a human finger. Magnetic resonance imaging (MRI) method
was used to obtain a geometric model of a human finger and different sequences of
images were taken to simulate flow of blood and heat transfer. From his images
data, the inflow velocity in the artery was found to be 19cm/s and in the
vein3.5cm/s. Reynolds number was 50 for this case when the arterial velocity was
the reference velocity. Ying He concluded that porous media transport model
provided good result to heat transfer analysis in living tissues (in this case, human
finger) using unified energy equation.
5
Numerical modeling of blood flow in a variable shape artery wall was
performed by Mukhopadhyay and Layek (Mukhopadhyay and Layek, 2008). In
their study, blood was assumed to be a Newtonian fluid and the flow was simulated
in the rigid artery wall under steady state situations. Three shapes of the stenois
(abnormal narrowing in a blood vessel) were used in this modeling and the shape
was symmetric as well as asymmetric about the middle cross sectional area.
Mukhopadhyay et al agreed in their research that blood could be treated as a
Newtonian fluid since the viscosity coefficient of blood approaches a constant
value under relatively large shear rates. Furthermore, the diameter of the rigid tube
used in the modeling was equivalently the same as the diameter of a large vessel in
which Newtonian fluid condition was satisfied. Hemodynamic factors in the rigid
tube like wall shear stress, pressure and velocity were analyzed. No heat transfer
analysis was performed under this rigid artery wall assumption with variable shape
of stenois.
Thermal effect on tumor was studied by Horng et al (Horng et al, 2007)
using pulsatile blood flow. Cyclic effect of heart beat was implemented by using a
sinusoidal pulsatile velocity profile. Horng et al judged from their literature that
blood flow in all arteries undergo pulsatile flow due to the heart pump nature. The
energy transport equation with Pennes’s bioheat equation was used to solve the
heat transfer between blood vessels and surrounding tumor tissues. The blood flow
was modeled with assumptions of incompressible, laminar and Newtonian fluid.
Blood vessel was modeled as a rigid artery wall in perfect cylindrical shape. Other
blood properties such as thermal conductivity; specific heat and perfusion mass
flow rate can be found in their literature. They concluded in their study by
suggesting that steady uniform velocity profile and pulsatile velocity profile for
blood flow exhibited minor differences on thermal analysis. Heat transfer analysis
to the surrounding tissues was done in their study with an assumption of inelastic
artery wall.
The study of dynamics response of heat and mass transfer in blood flow
through bifurcation arteries was done by Chakravarty and Sen (Chakravarty and
6
Sen, 2005). A mathematical model of the response was developed under the
assumption of Newtonian fluid. Navier-Stokes equation was used to describe
nonlinear unsteady flow phenomenon and heat conduction and convectiondiffusion equations were used to performed thermal analysis. In this study, the
artery wall was assumed to be elastic on local fluid mechanics but not on the
stresses and strains in the wall itself. The wall motion effect on velocity profile of
blood flow was found to be insignificant. Temperature profiles at apex of arterial
bifurcation also gave insignificant difference when considering wall motion
compared to the rigid wall. In his concluding remark, there was significant
reduction in heat flux magnitude when the arterial wall was considered as rigid.
Hyperthermia therapy simulation in biological tissue was done by Yuan
(Yuan, 2008). An equivalent heat transfer coefficient in a porous model was done
by numerical analysis. In this study, all arterial blood vessels were assumed to be
straight and treated as porous medium, constant heat transfer coefficient and blood
velocity were used, and thermal properties for tissues and blood were treated as
isotropic. Ultrasound was used as a source of energy input for hyperthermia
treatment. The whole domain of calculation was considered as an assembly of
repeated hexagon units. Tri-Diagonal Matrix Algorithm was employed to solve
two-equation porous model for blood and tissue and Pennes’s bio-heat transfer
equation to predict the temperature distribution. Yuan concluded in his summary
that a decrease in vessel diameter increases the equivalent heat transfer coefficient.
The equivalent heat transfer coefficient also is not dominated by blood velocity and
heating conditions. Three equivalent heat transfer coefficient were calculated for
three different vessel diameters. Yuan did not consider in his study to for
expanding or narrowing vessel diameter or non-symmetric blood vessel.
Cooling effect of thermally significant blood vessels in perfused tumor
tissue during thermal therapy was done by Shih et al (Shih et al, 2006). A single
blood vessel with rigid wall inside and throughout the perfused tumor tissue was
used as computational domain. The velocity profile in the blood vessel was
assumed to be uniform. The temperature of blood flow and tissue were governed by
7
the energy transport equation. During thermal simulation, ultrasound was used as
source of heat power and the heat absorption rate of blood is assumed to be onetenth of tissue. Heating power density and heating durations were varied while the
absorbed total energy density was kept constant during the simulation. Shih et al
concluded that short duration and high intensity heating method can eliminate the
cooling the cooling effect of the blood vessels 0.2mm in diameter. Cooling effect
was stronger for blood vessel larger than 0.2mm diameter and led to incomplete
necrosis even though double heating power or density was used. Shih also did not
consider in his study to for expanding or narrowing vessel diameter or nonsymmetric blood vessel.
Radiofrequency electric current is used in radiofrequency tumor ablation
procedure. An active electrode is inserted by mean of surgery and radiofrequency
energy is applied to heat the surrounding tissue. Santos et al (Santos et al, 2008)
studied the effect of variable heat transfer coefficient on tissue temperature next to
a larger vessel during radiofrequency tumor ablation. In his journal, a large artery
10mm diameter was used and an electrode was put at two different distances away
from the artery. The blood was assumed as a Newtonian fluid and the flow was
assumed to be laminar. The vessel of artery was assumed to be straight with
constant blood velocity. Both constant and variable of the convective heat transfer
coefficient were simulated. The convective heat transfer coefficient was increased
initially and rapidly up to 6000W/m2K followed by a sharp decrease to 554W/m2K
for 1mm distance configuration. The convective heat transfer coefficient was
increased initially and rapidly up to 7000W/m2K followed by a sharp decrease to
336W/m2K for 5mm distance configuration. Santos et al concluded that assumption
of constant convective heat transfer coefficient sufficient enough to produce precise
result for thermal procedure more than 2 minutes.
The preceding paragraphs described some of the past studies related to
blood flow. Figure 1.2.1 shows blood flow major categories.
8
Figure 1.2.1: Blood flow categories
Figure 1.2.2 shows summarized most of the work associated with blood
flow completed.
Figure 1.2.2: History of blood flow studies in arteries
9
Figure 1.2.3: History of blood flow studies in veins and capillaries
Figure 1.2.2 and Figure 1.2.3 show that most of heat transfer studies were
done with straight vessel assumption. He et al (He et al, 2008) did performed heat
transfer study by considering bifurcation vessel but He assumed that both parent
and daughter vessels were straight vessel. Apparently, there is inadequate work
done on tapered and stenois blood vessel under heating condition.
1.3
Objective
The objective of this study is to numerically simulate the blood flow and
heat transfer in an artery under heating conditions.
10
1.4
Scope of Study
The scope of this study will involve:
1 – Simulation of the normal blood flow in an artery.
2 – Repeating (1) under abnormal condition of heating (i.e. chemotherapy,
thermotherapy) at various degrees.
3 – Repeating (1) for a tapered channel flow and stenois (abnormal
narrowing) channel flow.
11
CHAPTER II
GOVERNING EQUATIONS
2.1
Continuity and Momentum Equations
Conservation of mass or continuity equation in FLUENT is written as
follows:
( )
∂ρ
+ ∇ • ρν = S m
∂t
(2.1)
ρ is fluid density,
t is time,
ν is fluid velocity,
S m is the mass added to the continuous phase from the dispersed second phase or
any user-defined sources.
For 2D axis-symmetric geometries flow condition, the Eq. (2.1) can be
written as:
ρν r
∂ρ ∂
∂
+ ( ρ ν x ) + (ρ ν r ) +
= Sm
∂t ∂x
∂r
r
(2.2)
12
x is axial coordinate,
r is radial coordinate,
ν x is axial velocity,
ν r is radial velocity.
Conservation of momentum equation in FLUENT is written as follows:
( )
()
( )
∂
ρν + ∇ • ρν ν = − ∇ p + ∇ • τ + ρ g + F
∂t
(2.3)
p is static pressure,
τ is tensor stress,
ρ g is gravitational body force,
F is external body forces or any user-defined sources.
The tensor stress τ is given by:
T
2


τ = µ  ∇ν + ∇ν  − ∇ •ν I 

 3

(2.4)
µ is the molecular viscosity,
I is the unit tensor,
T
∇ν is the effect of volume dilation.
For 2D axis-symmetric geometries flow condition, the axial momentum
equation can be written as:
∂
(ρ ν x ) + 1 ∂ (r ρ ν x ν x ) + 1 ∂ (r ρ ν r ν x ) = − ∂p +
∂t
r ∂x
r ∂r
∂x
1 ∂   ∂ν x 2
 1 ∂   ∂ν x ∂ν r 
+
− ∇ •ν  +
 + Fx
r µ  2
r µ 
r ∂x   ∂x 3
∂x 
 r ∂r   ∂r
( )
(2.5)
For 2D axis-symmetric geometries flow condition, the radial momentum
equation can be written as:
13

∂ν 
∂
(ρ ν r ) + 1 ∂ (r ρ ν x ν r ) + 1 ∂ (r ρ ν r ν r ) = − ∂p + 1 ∂ r µ  2 ∂ν r + x 
∂t
r ∂x
r ∂r
∂r r ∂x   ∂x
∂r 
1 ∂   ∂ν r 2
+
r µ 2
− ∇ •ν
r ∂r   ∂r 3
ν
ν 2µ

∇ •ν + ρ x + Fr
 − 2 µ 2r +
3 r
r
r

( )
2
( )
(2.6)
∇ •ν is defined as:
∇ •ν =
∂ν x ∂ν r ν r
+
+ .
∂x
∂r
r
2.2
Energy Equation
(2.7)
The energy equation in FLUENT is solved in the following form:


∂
(ρ E ) + ∇ • ν (ρ E + p ) = ∇ •  k eff ∇ T − ∑ h j J j + τ eff •ν  + S h
∂t
j


(
)
(
)
(2.8)
k eff is the effective conductivity,
J j is the diffusion flux of species j ,
k eff ∇ T is energy transfer due to conduction,
∑h
j
J j is energy transfer due to species diffusion,
j
∑ (τ
eff
)
•ν is energy transfer due to viscous dissipation,
j
S h is the heat of chemical reaction and any other user-defined heat sources,
E is defined as:
E=h−
p ν2
+ ,
ρ 2
(2.9)
h is enthalpy (ideal gases) and is defined as:
h = ∑Y j h j ,
j
(2.10)
14
or
h is enthalpy (incompressible flows) and is defined as:
h = ∑Y j h j +
j
p
ρ
.
(2.11)
In Eq. (2.10) and Eq. (2.11), Y j is the mass fraction of species j and h j is
defined as follows:
T
h j = ∫ c p , j dT
(2.12)
Tref
Tref is at 298.15K.
2.3
Related Equations
 r
Axial velocity, u = U m 2 1 − 
  r0
where
u is axial velocity
U m is fluid average inlet velocity
r is radial position
r0 is tube radius



2



(2.13)
15
Skin friction coefficient, C f =
64 Re
4
(2.14)
where
Re is Reynolds number
Re =
ρUm d
µ
(2.15)
ρ is fluid density
µ is fluid dynamics viscosity
d is tube diameter
Axial temperature gradient,
2q
dT
=
dx ρ U c c p r
(2.16)
where
c p is fluid specific heat
q is constant heat flux
Temperature difference between wall and fluid bulk,
2
Tw − Tm =
11 U c r ρ c p dT
96
k
dx
(2.17)
where
Tw is wall temperature
Tm is fluid bulk temperature
hd
k
(2.18)
Hydrodynamic entrance length, Lh = 0.056 Re d
(2.19)
Thermal entrance length, Lt = 0.05 Re Pr d
(2.20)
Nusselt number, Nu =
where
h is heat transfer coefficient
16
where
Pr is Prandlt number
Pr =
cp µ
2.4:
k
(2.21)
Assumptions
Blood is assumed to be homogenous, incompressible, and a Newtonian
fluid (Hunter, 1972, He et al, 2005, Tsubota et al, 2006, Horng et al, 2007, Jafari et
al, 2008, Mukhopadhyay and Layek, 2008, Santos et al, 2008, and Yuan, 2008).
The blood can be assumed as a Newtonian fluid since temperature in normal human
body is almost constant all the time at 37 degree Celsius. According to Fahraeus
and Lindqvist, blood flow through tubes more than approximately 1mm diameter
has a constant apparent blood viscosity (Waite and Fine, 2007 and Chandran et al,
2007).
Blood flow in artery is assumed to be uniform velocity profile instead of
pulsative velocity profile in a perfect cylindrical channel. Even though the actual
flow in artery is in pulsative velocity profile, thermal analysis result shows no
significant difference if uniform velocity profile is used (Horng et al, 2007).
Arterial wall is assumed to be rigid channel instead of elastic channel. Elastic
arterial wall was used by Hunter (Hunter, 1972) and Chakravarty and Sen
(Chakravarty and Sen, 2005) while rigid arterial wall was used by Horng et al
(Horng et al, 2007), Mukhopadhyay and Layek (Mukhopadhyay and Layek, 2008),
Santos et al (Santos et al, 2008), and Yuan (Yuan, 2008). Study by Chakravarty and
Sen (Chakravarty, 2005) showed that both velocity profile and thermal analysis in
elastic arterial wall provided insignificant difference compare to the rigid one.
17
Figure 1.2.2 and Figure 1.2.3 in the previous chapter show that there are
lacks of study for blood vessel with a tapered and stenois channel under various
heating conditions. Most of the heat transfer studies were done by assuming the
blood vessel is straight. Therefore, simulation of the normal blood flow in an artery
with various shapes of channels under various heating conditions will performed in
this study.
18
CHAPTER III
METHODOLOGY
3.1
Computational Fluid Dynamics
There are many commercially available softwares to perform computational
fluid dynamics (CFD) in general. FLUENT, POLYFLOW and CFX are among the
major ones used in flow modeling. In this study, FLUENT version 6.2.26 and
GAMBIT 2.3.16 (GAMBIT) from ANSYS Incorporation (ANSYS) are used. There
are three elements in the CFD package in ANSYS, which are:
1. Pre-processing
2. Solver
3. Post-processing
19
3.2
Pre-processing: GAMBIT
Pre-processing is an input process for any flow modeling simulation in
ANSYS software. In this stage, geometry of desired part or system region is
defined. The region of interest is called computational domain. Once the
computational domain is defined, the domain is subdivided into predefine number
of smaller region. Fluid properties also will be defined in this stage. Boundary
layers will be defined on the computational domain such as wall, center line, inlet
and outlet. Overall accuracy of flow simulation is heavily relies in this stage. In
general, the fineness of the grid, the results will be more accurate.
GAMBIT is relatively old pre-processing software for ANSYS package.
ANSYS is no longer support GAMBIT once it is replaced by ANSYS Design
Modeler and ANSYS Meshing in ANSYS version 12 under ANSYS. Now,
GAMBIT is no longer considered the latest state-of-the-art pre-processing software
for engineering analysis, but it still suitable to run simulation for this study.
3.3
Solver: FLUENT
Flow calculations are performed by solver in CFD package. The governing
integrals equations for the conservation of mass and momentum, and energy
equation are solved using two numerical methods. FLUENT 6.2 version provides
two methods:
1- Segregated solver
2- Coupled solver
20
Control-volume-based technique is used in both solvers. In general, both
solvers consist of:
1- The computational domain is divided by computational grid into discrete
control volumes.
2- Algebraic equations for the discrete dependent variables are constructed by
integrating the governing equations on the individual control volumes.
3- Values of the dependent variables are updated by linearization of the
discrete equations and solution of the resultant linear equation system.
The segregation solver solves the governing equations of continuity,
momentum, and energy sequentially. The overview of the segregated solution
method is shown in Figure 3.3.1.
Update properties.
Solve momentum equations.
Solve continuity equation. Update pressure, face mass flow rate.
Solve energy, species, turbulence, and other scalar equations.
Converged?
Stop
Figure 3.3.1: Segregation solution method
The coupled solver solves the governing equations of continuity,
momentum, and energy simultaneously. The overview of the coupled solution
method is shown in Figure 3.3.2.
21
Update properties.
Solve continuity, momentum, energy, and species equations simultaneously.
Solve turbulence, and other scalar equations.
Converged?
Stop
Figure 3.3.2: Coupled solution method
3.4
Post-processing: FLUENT
Post-processing is the last element in CFD package. All interpretation of
numerical results as well as CFD image of simulation is produced in this stage.
Beside as a solver, FLUENT also can be used during post-processing stage.
22
3.5
Pre-processing: Domain of Computational
Figure 3.5.1 shows domain of computational for simulation of normal blood
flow in artery.
Figure 3.5.1: Normal blood flow in an artery
Figure 3.5.2 shows domain of computational for simulation of abnormal
blood flow in artery under heating condition.
23
Figure 3.5.2: Abnormal blood flow in an artery under heating condition
Figure 3.5.3 shows domain of computational for simulation of normal blood
flow in a tapered artery.
Figure 3.5.3: Normal blood flow in a tapered artery
24
Figure 3.5.4 shows domain of computational for simulation of abnormal blood flow
in a tapered artery under heating condition.
Figure 3.5.4: Abnormal blood flow in a tapered artery under heating condition
Figure 3.5.5 shows domain of computational for simulation of normal
blood flow in a stenois artery.
25
Figure 3.5.5: Normal blood flow in a stenois artery
Figure 3.5.6 shows domain of computational for simulation of abnormal
blood flow in a stenois artery under heating condition.
Figure 3.5.6: Abnormal blood flow in a stenois artery under heating condition
26
3.6
Solver: Parameters
The following data for blood (Horng et al, 2007, and Shih et al, 2006) and
water properties in Table 3.6.1 will be use during simulation:
Table 3.6.1: Blood and water properties
Properties
Density, ρ
Dynamics
viscosity, µ
Specific heat, c p
Thermal
conductivity, k
Unit
kg
m3
kg
ms
Ws
kg K
Water at 310K
993
Blood at 310K
1060
6.95 ×10 −4
3.00 ×10 −3
4174
3770
W
mK
0.628
0.500
4.6
22.6
Prandlt
number, Pr
The following variations will be use during simulation of normal blood flow
in artery:
Table 3.6.2: Variation for normal blood flow in an artery
Case #1
Constant length
Diameter #1
Diameter #2
Diameter #3
Vel. pattern #1
Vel. pattern #1
Vel. pattern #1
27
The following variations will be use during simulation of abnormal blood
flow in artery under heating condition:
Table 3.6.3: Variation for abnormal blood flow in an artery under heating
condition
Case #1
Constant heating
source
Case #2
Constant
diameter
Diameter #1
Diameter #2
Diameter #3
Vel. pattern #1
Vel. pattern #2
Vel. pattern #3
Heat flow pattern
Heat flow pattern
Heat flow pattern
#1
#2
#3
Heating source #1
Heating source #2
Heating source #3
Vel. pattern #1
Vel. pattern #2
Vel. pattern #3
Heat flow pattern
Heat flow pattern
Heat flow pattern
#1
#2
#3
The following variation will be use during simulation of abnormal blood
flow under tapered (Case #1-3) and stenois (Case #4-6) channel:
28
Table 3.6.4: Variation for abnormal blood flow under tapered and stenois channel
Case #1
Tapered angle #1
Tapered angle #2
Tapered angle #3
No heating
Vel. pattern #1
Vel. pattern #2
Vel. pattern #3
Case #2
Tapered angle #1
Tapered angle #2
Tapered angle #3
Vel. pattern #1
Vel. pattern #2
Vel. pattern #3
Heat flow pattern
Heat flow pattern
Heat flow pattern
#1
#2
#3
Case #3
Heating source #1
Heating source #2
Heating source #3
Constant tapered
Vel. pattern #1
Vel. pattern #2
Vel. pattern #3
angle
Heat flow pattern
Heat flow pattern
Heat flow pattern
#1
#2
#3
Case #4
Stenois height #1
Stenois height #2
Stenois height #3
No heating
Vel. pattern #1
Vel. pattern #2
Vel. pattern #3
Case #5
Stenois height #1
Stenois height #2
Stenois height #3
Constant heating
Vel. pattern #1
Vel. pattern #2
Vel. pattern #3
source
Heat flow pattern
Heat flow pattern
Heat flow pattern
#1
#2
#3
Case #6
Heating source #1
Heating source #2
Heating source #3
Constant stenois
Vel. pattern #1
Ve. pattern #2
Vel. pattern #3
height
Heat flow pattern
Heat flow pattern
Heat flow pattern
#1
#2
#3
Constant heating
source
3.7
Solver: Method of Solution
The segregated solver will be used in this study. In section 2.3, blood is
assumed to be incompressible fluid. Therefore, the segregate solution method is
sufficient to run this simulation. Under-relaxation factor for energy equation is set
29
to 1.0 since there are no temperature-dependent properties or buoyancy forces
present.
3.8
Expected Outcome
1)
Velocity and heat flow patterns when the vessels are heated at
various degrees.
2)
Comparative analysis between artery flow and mechanical systems
under similar conditions to help us to understand future trends since
mechanical systems tend to be more predictable.
30
CHAPTER IV
MODEL VALIDATION
4.1
Model Validation Methodology
Three models are constructed in order to evaluate all three cases as
mentioned in Chapter 1. Model 1, Model 2 and Model 3 are constructed for a flow
in straight tube geometry, a flow in tapered tube geometry, and a flow in stenois
tube geometry respectively.
Water is used as working fluid instead of blood for validation purposes.
Grid dependency test is performed on Model 1 by doing the simulations under
various meshing sizes. Each simulation results are then compared to correlation
results.
Grid dependency test is performed for Model 1 for two reasons. First, to
ensure that selected meshing size is sufficient for the simulations. Second, to ensure
the simulation results are converged and with agreement with theoretical results.
31
Four sizes of meshing are selected for this test. The following table shows
summary of meshing size for grid dependency test.
Table 4.1.1: Summary of meshing size for grid dependency test
Mesh Type
Element in X-
Element in Y-
Number of
direction
direction
Elements
Mesh A
50
10
500
Mesh B
100
20
2000
Mesh C
150
30
4500
Mesh D
200
40
8000
Local Nusselt numbers are calculated from simulation results for each mesh
size then compared to correlation results. The following correlation equations are
suggested Kakac and Yener (1995) for simultaneously developing region in a
circular duct.
Constant temperature difference (Langhaar velocity profile)
Nu = 4.36 +
 Re Pr 

0.104 
 x 
 d 
 Re Pr 

1 + 0.016 
 x 
 d 
0.8
(4.1)
Constant heat input (Parabolic velocity profile)
Nu = 4.36 +
 Re Pr 

0.023 

x

 d 
 Re Pr 

1 + 0.0012 

x

 d 
0.8
(4.2)
32
Constant heat input (Langhaar velocity profile)
Nu = 4.36 +
 Re Pr 

0.036 

x

 d 
 Re Pr 

1 + 0.011

x

 d 
0.8
(4.3)
Each meshing sizes are then compared with theoretical results for
validation. The smallest size of meshing which is in agreement with theoretical
results within 5% of error is the chosen as the optimum meshing size. The smallest
size is selected because it consumes less calculation time compare to bigger one
and still produces reasonable results.
Once Model 1 is validated, it can be used for blood flow simulations. The
blood flow simulations are performed for all cases as mentioned in Table 3.6.2 to
Table 3.6.4 and results are shown in the following sections.
4.2
Grid Dependency Test
The simulations for Model 1 are performed with fluid average inlet velocity
is taken at 0.05m/sand constant wall heat flux is taken at q = 4W/m2. The radius is
taken at r = 5mm. The following figures show results of the simulations for grid
dependency test.
33
Grid Dependency Test
8
Grid Dependency Test
7
6
Difference (%)
5
4
3
2
1
0
1000 cells to 2000 cells
2000 cells to 4500 cells
4500 celss to 8000 cells
-1
Meshing Improvement
Figure 4.2.1: Percentage difference changing from one mesh size to another
Iterations Convergence vs Number if Cell
800
700
Number of Iteration
600
500
400
300
200
100
0
0
1000
2000
3000
4000
5000
6000
7000
8000
Number of Cell
Figure 4.2.2: Number of iteration versus number of cells
Figure 4.2.1 shows that the local Nusselt number varies around 7.3% as
meshing size is changed from 1000 cells to 2000 cells. As the meshing size is
refined further from 2000 cells to 4500 cells, results virtually do not vary at all.
34
Same result shown as the meshing size is refined further more from 4500 cells to
8000 cells. Therefore, further refinement of meshing size give no better
convergence results.
Figure 4.2.2 shows number of iteration required to get convergence result
versus number of cell. Number of iteration required increases as number of cells
increase. From Figure 4.2.1 and 4.2.2, Mesh B is selected since it has the optimum
size compare to other meshing size.
4.3
Model Validation with Numerical Results Comparison
The following figure shows correlation plots of local Nusselt number
variation for simultaneously developing region in a circular duct. Flow conditions
are same as in grid dependency test and water properties are taken at 310K as
shown in Table 3.6.1.
35
Variation of Local Nusselt Number
10
Langhaar Vel. Profile (Cons. Temp. Diff.)
Parabolic Vel. Profile (Cons. Heat Input)
9
Langhaar Vel. Profile (Cons. Heat Input)
Nux
8
7
6
5
4
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
x/L
Figure 4.3.1: Local Nusselt number for simultaneously developing region in a
circular duct
The following figure shows the simulation result with a parabolic velocity
profile and a Langhaar velocity profile. Local Nusselt number for the simulation
converge at dimensionless distance higher than 0.2. Percentage of error around 3%
compares to parabolic velocity profile correlation and 1% compares to Langhaar
velocity profile correlation.
Higher percentage of error occurs at dimensionless distance less 0.2. Since
present study only concentrates on fully developed region, higher percentage of
error at the said region is ignored in this study.
36
Comparison of Local Nusselt Number
20
Simulation
Parabolic Vel. Profile
Langhaar Vel. Profile
18
16
Nux
14
12
10
8
6
4
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
x/L
Figure 4.3.2: Comparison of local Nusselt number with correlation results
37
CHAPTER V
RESULTS AND DISCUSSIONS
5.1
Simulation of the Normal Blood Flow in an Artery (Straight Tube)
The axial velocity of blood flow simulations at constant average inlet
velocity of 0.05m/s and zero wall heat flux boundary condition at various radii is
shown in Figure 5.1.1. Maximum axial velocities occur at center of the tube and
gradually become zero at the wall. At the same radius and average inlet velocity,
the axial velocity of blood is quantitatively the same as that of water. This is
because the velocity profile is only influenced by the average inlet velocity and
tube diameter as stated in Equation 2.13.
38
Axial Velocities Pattern
7
Water (r = 5mm)
Blood (r = 5mm)
6
Blood (r = 6mm)
R adial Position (mm)
Blood (r = 7mm)
5
4
3
2
1
0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
U / Um
Figure 5.1.1: Axial velocity of blood flow (q = 0W/m2, Um = 0.05m/s) at various
radii
Skin Friction Coefficient
0.10
Cf
0.08
0.06
Water (r = 5mm)
Blood (r = 5mm)
Blood (r = 6mm)
0.04
Blood (r = 7mm)
0.02
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
x/L
Figure 5.1.2: Skin friction coefficient of blood flow (q = 0w/m2, Um = 0.05m/s) at
various radii
39
Figure 5.1.2 shows the skin friction coefficient of blood flow simulations at
constant average inlet velocity of 0.05m/s and zero heat flux at various radii. The
skin friction coefficient increases as the tube radius decreases. The skin friction
coefficient remain constant at dimensionless length of tube greater than 0.1. This is
because the skin friction coefficient does not depend on length but on radius. Water
has lower skin friction coefficient compared to blood under the same conditions.
This is due to the ratio of density and dynamics viscosity for water which is almost
four times lower than blood in general.
5.2
Simulation of Abnormal Blood Flow in an Artery under Heating
Condition (Straight Tube)
Axial Velocities Pattern
7
Water (r = 5mm)
Blood (r = 5mm)
6
Blood (r = 6mm)
R adial Position (mm)
Blood (r = 7mm)
5
4
3
2
1
0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
U / Um
Figure 5.2.1: Axial velocity of blood flow (q = 4W/m2, Um = 0.0 m/s) at various
radii
40
The axial velocities of blood flow at constant average inlet velocity at
0.05m/s and constant heat flux of 4W/m2 boundary condition at various radii is
shown in Figure 5.2.1. The profiles are similar to the case without flux. Heating of
the fluid at 4W/m2 did not seem to affect the velocity profiles probably because it is
too small. This observation is expected because the velocity profiles are, again,
influenced by tube diameters and average inlet velocity only. A property that might
have been affecting the velocity is the viscosity which with the heat flux applied
did not affect the velocity profile here.
Skin Friction Coefficient
0.10
Cf
0.08
0.06
Water (r = 5mm)
Blood (r = 5mm)
Blood (r = 6mm)
0.04
Blood (r = 7mm)
0.02
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
x/L
Figure 5.2.2: Skin friction coefficient of blood flow (q = 4W/m2, Um = 0.05m/s) at
various radii
Figure 5.2.2 shows the skin friction coefficient of blood flow simulations at
constant average inlet velocity of 0.05m/s and constant heat flux of 4W/m2 under
various radii. The skin friction coefficient values are comparable with Figure 5.1.2
which is simulated under zero heat flux. With or without heat being applied, the
skin friction coefficient values do not change since it does not depend on the heat
input. This is also because the velocity profile remains the same as shown in Figure
5.2.1.
41
Variation of Local Nusselt Number
16
Water (r = 5mm)
Blood (r = 5mm)
Blood (r = 6mm)
Blood (r = 7mm)
Nux
12
8
4
0.2
0.4
0.6
0.8
1.0
x/L
Figure 5.2.3: Local Nusselt number of blood flow (q = 4W/m2, Um = 0.05m/s) at
various radii
Figure 5.2.3 shows the local Nusselt number of blood flow simulations at
constant average inlet velocity of 0.05m/s and constant heat flux of 4W/m2 at
various radii. Higher local Nusselt number occurs at dimensionless distance less
than 0.6. This is because the flow is still in the thermally developing region. The
local Nusselt number begins to converge after a dimensionless distance higher than
0.6. The local Nusselt number for blood in bigger tubes is higher compared to the
smaller tubes. Bigger tubes have larger surface area which leads to higher heat
absorption from externally applied heat flux. The local Nusselt number for water is
clearly lower compared to blood under the same conditions at dimensionless
distance less than 0.5. This is because water has a higher thermal conductivity
(kwater = 0.628W/m0C) compared to blood (kblood = 0.500W/m0C) and Nusselt
number is decreasing with increasing of thermal conductivity. Average Nusselt
number of blood flow is found to be 5.14, 5.79 and 6.83 for radius of 5, 6 and 7mm
respectively. Average Nusselt number of water flow is found to be 4.96 for 5mm
tube.
42
Axial Velocities Pattern
5
Water (q = 4w/m^2)
Blood (q = 4w/m^2)
Blood (q = 5w/m^2)
R adial P osition (mm)
4
Blood (q = 6w/m^2)
3
2
1
0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
U / Um
Figure 5.2.4: Axial velocity of blood flow (r = 5mm, Um = 0.05m/s) under various
heating conditions
Figure 5.2.4 shows the axial velocity of blood flow simulations at constant
average inlet velocity of 0.05m/s and tube radius of 5mm under various heating
conditions. Maximum axial velocity occurs at the center of the tube and gradually
becomes zero at the wall. Each heating condition yields the same velocity profiles
as expected. Heating of the fluid at these ranges of heat flux did not seem to affect
the velocity profiles probably because it is too small. The results here are identical
to those in Figures 5.2.1 and 5.1.1.
43
Skin Friction Coefficient
0.10
Cf
0.08
0.06
Water (q = 4w/m^2)
Blood (q = 4w/m^2)
Blood (q = 5w/m^2)
0.04
Blood (q = 6w/m^2)
0.02
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
x/L
Figure 5.2.5: Skin friction coefficient of blood flow (r = 5mm, Um = 0.05m/s)
under various heating conditions
Figure 5.2.5 shows the skin friction coefficient of blood flow simulations at
constant average inlet velocity of 0.05m/s and tube radius of 5mm under various
heating conditions. As the applied heat flux increases, the skin friction remains
unchanged. This is because the skin friction coefficient does not depend on heat
input (see Equation #2.14).
44
Variation of Local Nusselt Number
16
Water (q = 4 w/m^2)
Blood (q = 4 w/m^2)
Blood (q = 5 w/m^2)
Blood (q = 6 w/m^2)
Nux
12
8
4
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
x/L
Figure 5.2.6: Local Nusselt number of blood flow (r = 5mm, Um = 0.05m/s) under
various heating conditions
Figure 5.2.6 shows the local Nusselt number of blood flow simulations at
constant average inlet velocity of 0.05m/s and tube radius of 5mm under various
heating conditions. The local Nusselt number is higher in thermally developing
region which is at dimensionless distance less than 0.4. The local Nusselt number
converges to a constant value for dimensionless distance higher than 0.4. Blood
under various heating conditions show almost identical local Nusselt number.
Water produces lower local Nusselt number compared to blood under the same
conditions as clearly shown at dimensionless distance less than 0.5.
Comparison of the results for this study and those using the correlation used
by Langhaar is shown in Figure 5.2.8. Using his correlation showed slight
differences between water and blood as seen in Figure 5.2.7. Simulation results
showed that large disagreement occurs at the inlet developing region. It maybe
possible to modify Langhaar’s correlation to be used for blood but further
investigation will be required and this is beyond the scope of this study.
45
Variation of Local Nusselt Number
10
Parabolic Vel. Pro. (Cons. Heat Input Blood)
Langhaar Vel. Pro (Cons. Heat Input Blood)
9
Par. Vel. Pro. (Cons. Heat Input Water)
Langhar Vel. Pro. (Cons. Heat Input Water)
Nux
8
7
6
5
4
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
x/L
Figure 5.2.7: Plot of correlation of local Nusselt number for blood and water flow
Comparison of Local Nusselt Number
14
Simulation Blood
Parabolic Vel. Profile Blood
Langhaar Vel. Profile Blood
13
12
11
Nux
10
9
8
7
6
5
4
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
x/L
Figure 5.2.8: Plot of simulation and correlation of local Nusselt number for blood
flow
46
Variation of Local Nusselt Number
5.0
Water (q = 4 w/m^2)
Blood (q = 4 w/m^2)
Blood (q = 5 w/m^2)
4.9
Blood (q = 6 w/m^2)
Nux
4.8
4.7
4.6
4.5
0.90
0.95
x/L
Figure 5.2.9: Zoom- in of Figure 5.2.6
Figure 5.2.9 shows the zoom in region of Figure 5.2.6. Local Nusselt
number converges to a constant value as expected. Blood under higher applied heat
flux produces slightly higher local Nusselt number. This is because local Nusselt
number is directly proportional to the amount of applied heat. Water produces
slightly higher local Nusselt number compared to blood under the same imposed
conditions at fully developed region. Possible explanation for this situation is due
to water has lower skin friction coefficient compared to blood as shown in Figure
5.2.5. Average Nusselt number is found to be 5.14 for each case of applied heat
flux.
Figures 5.1.1, 5.2.1 and 5.2.4 have shown that blood profiles follow the
general characteristic of flow in a circular straight tube. Furthermore, axial velocity
profiles of blood do not change with the increasing amount heat applied in the
simulation.
47
Figures 5.2.3 and 5.2.9 have shown that the tube radius does affect the local
Nusselt numbers when the amount of heat flux applied was varied. Bigger radius of
tube absorbs more heat compared to smaller one.
5.3
Simulation of Blood Flow in an Artery (Tapered and Stenois Tubes)
5.3.1
Simulation of Blood Flow in an Artery (Tapered Tube)
Axial Velocities Pattern
Out: 0.0 degree
D im en sio n less R ad ial P o sitio n at
T ap ered Ou tlet
Out: 0.5 degree
Out: 1.0 degree
Out: 1.5 degree
0
0.5
1
1.5
U / Um
2
2.5
3
3.5
Figure 5.3.1.1: Outlet axial velocity of blood flow (q = 0W/m2, Um = 0.05m/s)
under various taper angles
The axial velocity (at tapered outlet) of blood flow simulations under
various converging taper angles is shown in Figure 5.3.1.1. The simulations were
completed at constant average inlet velocity of 0.05m/s and zero heat flux.
48
Maximum axial velocity occurs at the tube center and gradually becomes zero at
the wall for the no slip condition. As the tapered angle increases, maximum axial
velocity at the center of the tube also increases. It is also observed that velocity
profile for bigger tapered angle assumes flat shape instead of parabolic shape. This
observation can be explained using conservation of mass principle. To conserve
mass, the inflow rate must equal the outflow rate. Since increasing of the tapered
angle reduces the cross sectional area, thus average velocity increases. This result
agrees qualitatively well with Chakravarty and Mandal (2000). It is noted that
Chakravarty and Mandal (2000) used different geometrical dimensions as
compared to present study. They found that the velocity profile increased with the
increase of the tapered angle.
Skin Friction Coefficient
0.30
Taper 0.0 degree
Taper 0.5 degree
0.25
Taper 1.0 degree
Taper 1.5 degree
Cf
0.20
0.15
0.10
0.05
0.0
0.2
0.4
0.6
0.8
1.0
x/L
Figure 5.3.1.2: Skin friction coefficient of blood flow (q = 0W/m2, Um = 0.05m/s)
under various taper angles
Figure 5.3.1.2 shows the skin friction coefficients of blood flow simulations
at constant average inlet velocity of 0.05m/s and zero heat flux under various
converging tapered angles. It is observed that the skin friction coefficient values are
the same value at the tapered inlet. This is because the tube diameters are the same
49
at the tapered inlet region. The skin friction coefficient for zero tapered angle
remains constant for the entire dimensionless length of tapered region. On the other
hand, the skin friction coefficient increases as the dimensionless distance from
tapered region inlet increases. Increments of skin friction coefficient at a particular
angle seem to occur at a constant rate. Higher converging taper angle has a higher
rate of increment. For the 0.50 angle, the gradient is constant at 0.04. For the 1.00
angle, the gradient is constant at 0.11. For the 1.50 angle, however, the gradient is
0.17 until x/L = 0.5 and then increases further with a gradient of 0.25. This
observation is expected because the skin friction is inversely proportional to the
Reynolds number as stated in Equation #2.14. Reynolds number is directly
proportional to diameter. As the tapered angle increases, the diameter decreases and
lead to decreasing in the Reynolds number.
Axial Velocities Pattern
Out: 0.0 degree
D im en sio n less R ad ial P o sitio n at
T ap ered Ou tlet
Out: 0.5 degree
Out: 1.0 degree
Out: 1.5 degree
0
0.5
1
1.5
U / Um
2
2.5
3
3.5
Figure 5.3.1.3: Axial velocity of blood flows (q = 4W/m2, Um = 0.05m/s) under
various taper angles
Figure 5.3.1.3 shows the axial velocity (at tapered outlet) of blood flow
simulations at constant average inlet velocity of 0.05m/s and constant heat flux of
4W/m2 under various converging taper angles. As the tapered angle increases,
50
maximum axial velocity at the center of the tube also increases. Figures 5.3.1.3 and
5.3.1.1 are identical since the axial velocity patterns remain the same with the
amount of heat flux applied. This is because the velocity profile is influenced by
the tube diameter and average inlet velocity only. Refer to Appendix B to D for
axial velocity profile plots from Fluent.
Skin Friction Coefficient
0.30
Taper 0.0 degree
Taper 0.5 degree
0.25
Taper 1.0 degree
Taper 1.5 degree
Cf
0.20
0.15
0.10
0.05
0.0
0.2
0.4
0.6
0.8
1.0
x/L
Figure 5.3.1.4: Skin friction coefficient of blood flow (q = 4W/m2, Um = 0.05m/s)
under various taper angles
Figure 5.3.1.4 shows the skin friction coefficient of blood flow simulations
at constant average inlet velocity of 0.05m/s and constant heat flux of 4W/m2 under
various converging tapered angles. The skin friction coefficient values are the same
as that under no heat flux condition. This observation is expected since the average
inlet velocity does not change with or without heat flux under present study. Skin
friction coefficient also is not affected by the heat flux as stated in Equation #2.14.
51
Variation of Local Nusselt Number
5.0
N ux
4.5
4.0
Taper 0.0 degree
3.5
Taper 0.5 degree
Taper 1.0 degree
Taper 1.5 degree
3.0
0.0
0.2
0.4
0.6
0.8
1.0
x/L
Figure 5.3.1.5: Local Nusselt numbers of blood flow (q = 4W/m2, Um = 0.05m/s)
under various taper angles
Figure 5.3.1.5 shows the local Nusselt number of blood flow simulations at
constant average inlet velocity of 0.05m/s and constant heat flux of 4W/m2 under
various converging taper angles. It is observed that the local Nusselt numbers are at
the same at the tapered inlet region. The local Nusselt numbers for zero tapered
angle remains constant. As the tapered angle increases, the local Nusselt number
decreases at almost a constant rate. Decrement trends of the local Nusselt number
can be explained from a geometrical point of view. As the tapered angle increases,
the surface area decreases. Therefore, less heat is able to be absorbed by blood as it
passes through and it leads to a lower local Nusselt number. Average Nusselt
number is found to be 5.14, 4.60, 4.37 and 4.04 for 0.0, 0.5, 1.0 and 1.5 taper
angles respectively.
52
Axial Velocities Pattern
5
R ad ial P o sitio n at T ap ered O u tlet
(m m )
q = 4w/m^2
q = 5w/m^2
q = 6w/m^2
4
3
2
1
0
0
0.5
1
U / Um
1.5
2
2.5
Figure 5.3.1.6: Axial velocity of blood flow (Um = 0.05m/s, θ = 0.50) under
various heating conditions
Figure 5.3.1.6 shows the axial velocity of blood flow simulations at
constant average inlet velocity of 0.05m/s and constant converging tapered angle of
0.50 under various heating conditions. Maximum axial velocity occurs at the center
of the tube and gradually becomes zero at the wall. Each heating condition yields
the same profile as expected. This is because the velocity profile does not change
with the changing of amount of applied heat unless the viscosity is greatly affected.
The heat flux applied in this study seemed insignificant enough to influence the
velocity profile.
53
Skin Friction Coefficient
0.20
q = 4w/m^2
q = 5w/m^2
q = 6w/m^2
Cf
0.15
0.10
0.05
0.0
0.2
0.4
0.6
0.8
1.0
x/L
Figure 5.3.1.7: Skin friction coefficient of blood flow (Um = 0.05m/s, θ = 0.50)
under various heating conditions
Figure 5.3.1.7 shows the skin friction coefficient of blood flow simulations
at constant average inlet velocity of 0.05m/s and constant converging tapered angle
of 0.50 under various heating conditions. Each heating condition produces identical
skin friction coefficient value. The skin friction coefficient increases as the
dimensionless distance from tapered inlet region increases. Increment of the skin
friction coefficient occurs at a same constant rate. The amount of heat flux applied
in this study is quite insignificant to influence the velocity profile change as
mentioned earlier, the skin friction coefficient value remains the same under
various heating conditions.
54
Variation of Local Nusselt Number
4.8
q = 4 w/m^2
q = 5 w/m^2
q = 6 w/m^2
Nux
4.6
4.4
4.2
4
0.0
0.2
0.4
0.6
0.8
1.0
x/L
Figure 5.3.1.8: Local Nusselt number of blood flow (Um = 0.05m/s, θ = 0.50)
under various heating conditions
Figure 5.3.1.8 shows the local Nusselt number of blood flow simulations at
constant average inlet velocity of 0.05m/s and constant converging tapered angle of
0.50 under various heating conditions. The local Nusselt number shows downward
trend at almost a constant rate. The highest local Nusselt number occurs at the inlet
of the tapered region while the lowest local Nusselt number occurs at the outlet
tapered region. Note that the local Nusselt number is higher for higher amount of
applied heat flux. This observation is expected since local Nusselt number is
directly proportional to heat transfer coefficient and heat transfer coefficient is
directly proportional to heat flux. Therefore, as amount of applied heat flux
increases, heat transfer coefficient increases, thus local Nusselt number increases as
well. It is also observed that local Nusselt number dipped close to the outlet region.
Increasing the amount of heat flux seems to move this sudden decrease region
further downstream. However, further studies into lengthening the tapered region
may show if this trend is repeated with higher heat flux. Average Nusselt number is
found to be 4.64, 4.65 and 4.66 for 4, 5 and 6W/m2 heat flux respectively. Refer to
Appendix E to G for temperature contour plots from Fluent.
55
Figures 5.3.1.1, 5.3.1.1 and 5.3.1.6 showed that blood follows the general
characteristic of flow in a converging tapered tube. Axial velocity profiles of blood
do not change regardless of the heat being applied. Profiles are identical for both
straight and converging tapered tube.
Figures 5.3.1.2, 5.3.1.4 and 5.3.1.7 showed that skin friction coefficient
does not vary with applied heat but vary with tapered angle. This is because tapered
region has different cross sectional area which leads to a changing radius and
consequently to a changing area. If tapered angle was zero, it is completely
analogous to constant radius tube cases as shown in Figures 5.1.2, 5.2.2 and 5.2.5.
Figures 5.3.1.5 and 5.3.1.8 showed that applied heat flux has a less
significant effect on local Nusselt number compared to that from the variation of
tapered angle. Rate of changes of local Nusslet number are different for each case
in Figure 5.3.1.5. This is because as dimensionless distance changes, radius also
changes due to the tapered angle. Different tapered angle produces different rate of
radius reductions. Therefore, local Nusslet number decreases at a difference rate.
On the other hand, in Figure 5.3.1.8, the rate of changes of local Nusselt number
remains the same under various heating conditions in this study. This is because the
amount of heat flux being applied is constant as dimensionless distance moves
from inlet to outlet.
56
5.3.2
Simulation of Blood Flow in an Artery (Stenois Tube)
Axial Velocities Pattern
5
Cen: a / r = 0.2
Cen: a / r = 0.3
Radial Position (mm)
4
Cen: a / r = 0.4
Out: a / r = 0.2
Out: a / r = 0.3
3
Out: a / r = 0.4
2
1
0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
U / Um
Figure 5.3.2.1: Axial velocity of blood flow (q = 0W/m2, Um = 0.0 m/s) with
various stenois heights
The axial velocity of blood flow simulations at constant average inlet
velocity of 0.05m/s and zero heat flux under various stenois heights is shown in
Figure 5.3.2.1. It is observed that the maximum axial velocity occurs at the center
of the tube. The axial velocity gradually reduces to zero at the tube wall. Velocity
shapes gradually changed from a parabolic profile to a flat profile at the center of
the stenois. Note that all stenois cases have identical velocity profiles at the inlet
region of the stenois. This observation can be explained using conservation of mass
principle. To conserve mass, the inflow rate must equal the outflow rate. Since
increases of stenois height reduces the cross sectional area, thus average velocity
increases at that region. This leads to a pressure decrease, according to the
Bernoulli equation. As the stenois height reduces, the cross sectional area increases,
thus average velocity decreases. In this region, the pressure increases. This
behavior is analogous to flow in converging-diverging nozzle. This observation
57
agrees well with Chakravarty and Mandal (2000). They also found that as the
stenois height increases, the velocity increases. It is also found that the velocity
profile at the center of the stenois is comparable to the converging tapered case as
shown in Figure 5.3.1.1.
Center Axial Velocities Pattern
4.0
3.5
Uc / Um
3.0
2.5
a / r = 0.2
a / r = 0.3
2.0
a / r = 0.4
1.5
0
0.2
0.4
x/L
0.6
0.8
1
Figure 5.3.2.2: Center axial velocity of blood flow (q = 0W/m2, Um = 0.05m/s)
with various stenois heights
Figure 5.3.2.2 shows the center axial velocity of blood flow simulations at
constant average inlet velocity of 0.05m/s and zero heat flux under various stenois
heights. It is observed that the maximum dimensionless center axial velocity occurs
at the center of the stenois. As the stenois height increases, maximum
dimensionless center axial velocity increases. This observation is expected since
the flow simulated here at the vena contracta is analogous to the flow in a
converging-diverging nozzle. It is also observed that the dimensionless center axial
velocity at outlet region is higher compared to the inlet region. This is because the
pressure is not fully recovered after the stenois due to possible of backflow. This
observation agrees well with Mukhopadhyay and Layek (2008), their axial velocity
profile being similar to that in Figure 5.3.2.2.
58
Skin Friction Coefficient
a / r = 0.2
0.8
a / r = 0.3
a / r = 0.4
Cf
0.6
0.4
0.2
0.0
0
0.2
0.4
0.6
0.8
1
x/L
Figure 5.3.2.3: Skin friction coefficient of blood flow (q = 0W/m2, Um = 0.05m/s)
with various stenois heights
Figure 5.3.2.3 shows the skin friction coefficient of blood flow simulations
at constant average inlet velocity of 0.05m/s and zero heat flux under various
stenois heights. It is observed that the skin friction coefficient is higher for higher
stenois height at dimensionless distance less than 0.6. For dimensionless distance
more than 0.6, the highest stenois height does not produce a higher skin friction
coefficient. This is due to the skin friction coefficient decreasing with increasing
product of average velocity and tube diameter. Changes of skin friction coefficient
are due to the stenois increasing as the stenois height increases. The skin friction
coefficient rate of change at both inlet and outlet region of stenois are different
since the rate of change of axial velocities are different as well, as shown in Figure
5.3.2.2. This is because the skin friction affects the Reynolds number and the
Reynolds number influences the fluid velocity. Refer to Appendix H to J for axial
velocity profile plots from Fluent.
59
Axial Velocities Pattern
5
Cen: a / r = 0.2
Cen: a / r = 0.3
Radial Position (mm)
4
Cen: a / r = 0.4
Out: a / r = 0.2
Out: a / r = 0.3
3
Out: a / r = 0.4
2
1
0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
U / Um
Figure 5.3.2.4: Axial velocity of blood flow (q = 4W/m2, Um = 0.05m/s) with
various stenois heights
Figure 5.3.2.4 shows the axial velocity of blood flow simulations at
constant average inlet velocity of 0.05m/s and constant heat flux of 4w/m2 under
various stenois heights. It is observed that the maximum axial velocity occurs at the
center of the tube. The axial velocity gradually reduces to zero at the tube wall.
Velocity shapes gradually changed from a parabolic profile to a flat profile at the
center of the stenois. All stenois cases have identical velocity profiles at the inlet
region of stenois. It is noted that the velocity profiles are identical to Figure 5.3.2.1
for the case of zero heating. This is because only the tube diameter and the velocity
affect the velocity profile.
60
Center Axial Velocities Pattern
4.0
3.5
Uc / Um
3.0
2.5
a / r = 0.2
a / r = 0.3
2.0
a / r = 0.4
1.5
0
0.2
0.4
x/L
0.6
0.8
1
Figure 5.3.2.5: Center axial velocity of blood flow (q = 4W/m2, Um = 0.05m/s)
with various stenois heights
Figure 5.3.2.5 shows the center axial velocity of blood flow simulations at
constant average inlet velocity of 0.05m/s and constant heat flux of 4W/m2 under
various stenois heights. The center axial velocity values are similar to that in Figure
5.3.2.2 which is simulated under zero heat flux condition.
61
Skin Friction Coefficient
a / r = 0.2
0.8
a / r = 0.3
a / r = 0.4
Cf
0.6
0.4
0.2
0.0
0
0.2
0.4
0.6
0.8
1
x/L
Figure 5.3.2.6: Skin friction coefficient of blood flow (q = 4W/m2, Um = 0.05m/s)
with various stenois heights
Figure 5.3.2.6 shows the skin friction coefficients of blood flow simulations
at constant average inlet velocity of 0.05m/s and constant heat flux of 4w/m2 under
various stenois heights. The skin friction coefficient values are similar to those in
Figure 5.3.2.3 which is simulated under zero heat flux condition.
62
Variations of Local Nusselt Numbers
a / r = 0.2
5.0
a / r = 0.3
a / r = 0.4
4.5
Nux
4.0
3.5
3.0
2.5
2.0
0
0.2
0.4
0.6
0.8
1
x/L
Figure 5.3.2.7: Local Nusselt number of blood flow (q = 4W/m2, Um = 0.05m/s)
with various stenois heights
Figure 5.3.2.7 shows the local Nusselt number of blood flow simulations at
constant average inlet velocity of 0.05m/s and constant heat flux of 4W/m2 under
various stenois heights. The local Nusselt number decreases as the stenois height
increases. The local Nusselt number is at minimum at the center of the stenois
height. Higher stenois height has lower local Nusselt number in general. This is
because higher stenois height has a smaller radius and the smaller radius leads to a
smaller surface area. Therefore, less heat is able to be absorbed and it leads to a
lower local Nusselt number. It is also observed that local Nusselt number at the
inlet is slightly higher compared to the outlet. This could be due to the cooling
effect since the outlet region of stenois height has higher axial velocity compared to
the inlet region (Figure 5.3.2.5). For a dimensionless distance less than 0.5, the
local Nusselt number is comparable to the tapered case as shown in Figure 5.3.1.5.
Average Nusselt number is found to be 4.12, 3.80 and 3.49 for stenois height ratio
of 0.2, 0.3 and 0.4 respectively.
63
Axial Velocities Pattern
5
In: q = 4w/m^2
In: q = 5w/m^2
In: q = 6w/m^2
Radial Position (mm)
4
Cen: q = 4w/m^2
Cen: q = 5w/m^2
Cen: q = 6w/m^2
Out: q = 4w/m^2
3
Out: q = 5w/m^2
Out: q = 6w/m^2
2
1
0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
U / Um
Figure 5.3.2.8: Axial velocity of blood flow (a / r = 0.2, Um = 0.05m/s) under
various heating conditions
Figure 5.3.2.8 shows the axial velocity of blood flow simulations at
constant average inlet velocity of 0.05m/s and constant stenois height of 1mm
under various heating conditions. It is observed that maximum axial velocity occurs
at the center of the tube. The axial velocity gradually reduces to zero at the tube
wall. Velocity shapes gradually changed from a parabolic profile to a flat profile at
the center of the stenois. It is noted that all heating conditions cases have identical
velocity profiles as in the tapered case discussed previously.
64
Center Axial Velocities Pattern
2.75
q = 4 w/m^2
q = 5 w/m^2
q = 6 w/m^2
Uc / Um
2.50
2.25
2.00
1.75
0
0.2
0.4
x/L
0.6
0.8
1
Figure 5.3.2.9: Center axial velocity of blood flow (a / r = 0.2, Um = 0.05m/s)
under various heating conditions
Figure 5.3.2.9 shows the center axial velocity of blood flow simulations at
constant average inlet velocity of 0.05m/s and constant stenois height of 1mm
under various heating conditions. It is observed that maximum dimensionless
center axial velocity occurs at the center of the stenois. It is noted that all heating
conditions cases have identical velocity profiles.
65
Skin Friction Coefficient
q = 4w/m^2
0.25
q = 5w/m^2
q = 6w/m^2
0.20
Cf
0.15
0.10
0.05
0.00
0
0.2
0.4
0.6
0.8
1
x/L
Figure 5.3.2.10: Skin friction coefficient of blood flow (a / r = 0.2, Um = 0.05m/s)
under various heating conditions
Figure 5.3.2.10 shows the skin friction coefficient of blood flow simulations
at constant average inlet velocity of 0.05m/s and constant stenois height of 1mm
under various heating conditions. It is observed that maximum skin friction
coefficient occurs at the center of the stenois. It is noted that all heating conditions
cases have identical skin friction coefficient values. The skin friction coefficient
values remain the same regardless of the amount of heat flux applied.
66
Variation of Local Nusselt Numbers
4.8
q = 4 w/m^2
4.6
q = 5 w/m^2
q = 6 w/m^2
4.4
Nux
4.2
4.0
3.8
3.6
3.4
0
0.2
0.4
0.6
0.8
1
x/L
Figure 5.3.2.11: Local Nusselt number of blood flow (a / r = 0.8, Um = 0.05m/s)
under various heating conditions
Figure 5.3.2.11 shows the local Nusselt number of blood flow simulations
at constant average inlet velocity of 0.05m/s and constant stenois height of 1mm
under various heating conditions. It is observed that minimum local Nusselt
number occurs at the center of the stenois height. It is noted that the local Nusselt
number variations with the variation in heat flux are small at both the inlet and
outlet of the stenois regions. It is observed that the local Nusselt number fluctuated
at the downstream of stenois as well as upstream. This was observed in the tapered
case (Figure 5.3.1.8). The phenomenon could be due to the recirculation caused by
the sudden decrease in flow region at the speed studied. The increase in fluctuations
after the stenois could be contributed by the slow decrease in velocity after the
stenois (Figure 5.3.2.9). Average Nusselt number is found to be 4.12, 4.15 and 4.17
for 4, 5 and 6W/m2 heat flux respectively. Refer to Appendix K to M for
temperature contour plots from Fluent.
Figures 5.3.2.1, 5.3.2.4 and 5.3.2.8 showed that blood follows general
characteristic of flow in stenois tapered tube. Axial velocity profiles of blood do
67
not change with the amount of heat being applied under the stenois condition. The
observations are identical for straight tube, converging tapered tube and stenois
tube.
Figures 5.3.2.2, 5.3.2.5 and 5.3.2.9 showed that maximum center axial
velocities occur at the center of stenois. Under the same stenois condition, center
axial velocities do not change with the amount of heat being applied.
Figures 5.3.2.3, 5.3.2.6 and 5.3.2.10 showed that skin friction coefficient
does not vary with applied heat but vary with stenois height. If stenois height was
zero, it is completely analogous to the constant radius tube cases as shown in
Figures 5.1.2, 5.2.2 and 5.2.5.
68
CHAPTER VI
CONCLUSION
6.1
Conclusion
Simulation of blood flow in a straight tube under various radius and heating
conditions were completed. Both blood and water flows exhibit identical axial
velocity patterns for flow in a straight circular tube. The velocity, skin friction
coefficient and local Nusselt number were found to be unaffected by the heating
conditions, the latter changed insignificantly with variations in tube radius.
Blood flow simulations in converging tapered tube were performed under
various tapered angles and heating conditions. Axial velocities under zero heat flux
agreed well with past results of Chakravarty and Mandal (2000). Axial velocity
patterns did not change under various heating conditions but did so under various
tapered angles. Skin friction coefficients did not vary under various heating
conditions but did under various tapered angles. Local Nusselt numbers decreased
as dimensionless distance from tapered region increased.
Blood flow simulations in stenois tube were performed under various
stenois heights and heating conditions. Center axial velocity pattern under zero heat
69
flux agreed well with past work of Mukhopadhyay and Layek (2008). Axial
velocity patterns did not change under various heating conditions but did under
various stenois heights. Skin friction coefficients did not vary under various heating
conditions but did under various stenois heights. Local Nusselt numbers varied
with stenois heights with small variations under various heating conditions.
The results in general have shown that blood flow in biological systems
may well behave like the mechanical systems. Since experimental studies on blood
flow in biological system are near impossible to be conducted under desirable
conditions, their behavior can possibly be predicted with the current findings and
future extensions of the current work. It has been demonstrated that heat transfer in
blood flow within tapered cross-section is affected by various heating conditions.
This could be further investigated in terms of the various parts of the biological
system if and when responding to thermal treatment.
70
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73
APPENDICES
Appendix-A: Blood properties set up in Fluent
74
Appendix-B: Axial velocity profile for 0.50 tapered tube (q = 4W/m2, Um =
0.05m/s)
Appendix-C: Axial velocity profile for 1.00 tapered tube (q = 4W/m2, Um =
0.05m/s)
75
Appendix-D: Axial velocity profile for 1.50 tapered tube (q = 4W/m2, Um =
0.05m/s)
Appendix-E: Temperature contour for 0.50 tapered tube (q = 4W/m2, Um =
0.05m/s)
76
Appendix-F: Temperature contour for 0.50 tapered tube (q = 5W/m2, Um =
0.05m/s)
Appendix-G: Temperature contour for 0.50 tapered tube (q = 6W/m2, Um =
0.05m/s)
77
Appendix-H: Axial velocity profile for stenois height of 1mm (q = 0W/m2, Um =
0.05m/s)
Appendix-I: Axial velocity profile for stenois height of 1.5mm (q = 0W/m2, Um =
0.05m/s)
78
Appendix-J: Axial velocity profile for stenois height of 2mm (q = 0W/m2, Um =
0.05 m/s)
Appendix-K: Temperature contour for stenois height of 1mm (q = 4W/m2, Um =
0.05 m/s)
79
Appendix-L: Temperature contour for stenois height of 1mm (q = 5W/m2, Um =
0.05 m/s)
Appendix-M: Temperature contour for stenois height of 1mm (q = 6W/m2, Um =
0.05 m/s)