An Electrochemical Investigation of Mass Transfer in a Pipe and a Simplified Bifurcation Model
Nader Mahinpey
A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Department of Chernical Engineering and Applied Chemistry and the lnstitute of Biomaterials and Biomedical Engineering University of Toronto
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for their unending love. An Electrochemical Investigation of Mass Transfer in a Pipe and a Simplified Bifurcation Model
Nader Mahinpey Doctoral Thesis, 2001 Department of Chemical Engineering lnstiute of Biomaterials and Biomedical Engineering University of Toronto
A novel technique was used to fabricate nickel flow models of a straight
pipe and a Y-bifurcation. These were used to obtain integral mass transfer coefficients by the electrochemical technique with the ferri-ferrocyanide system at
Reynolds numbers ranging from 250 to 8600, at the four Schmidt numbers of
1310, 2470, 3100 and 5655. For the straight pipe, good agreement was obtained
with previously reported mass transfer correlations. As the Schmidt number
increased, the effect of transition from laminar to turbulent flow on mass transfer
was delayed to progressively higher Reynolds numbers. With the Y-bifurcation
model, possible fiow separation and the formation of a new mass transfer
boundary layer in the daughter branches significantly influence the mass transfer
behavior.
The electrochemical technique has been used to obtain transient mass
transfer coefficients for smooth pipes. In contrast to the analytical solution of the
corresponding heat transfer problem, two distinct transient periods are observed.
The first is controlled by chemical reaction kinetics at the surface, followed by the
diffusion-controlled period in agreement with the heat transfer solution. The
transfer rate for laminar fiow is then proportional to (~lt)'~,in accordance with
Higbie's penetration theory. In turbulent flow, the transient mass transfer rate, during the second transient period, is proportional to ~e''~(~lt)'~,higher than in laminar flow.
The laser photochromic tracer method provided velocity and wall shear stress values in the plane of symmetry of a UV-transparent Plexiglas bifurcation model similar to that used in the mass transfer experiments at Reynolds numbers
500, 600, and 750. A novel copper electrodeposition technique has been used to obtain time-averaged convective local mass transfer coefficients in a straight pipe and a simplified bifurcation model. Laminar flow results for the pipe are in good agreement with the analytical Leveque solution. In the bifurcation. higher mass transfer coefficients along the inner wall and lower ones along the outer wall were observed. Both coefficients converge towards the same value further downstream. Within the branches, mass transfer and wall shear stress follow similar patterns both on the inner and outer walls. lt was found that stscY3and
Cf/2 demonstrate analogous behavior.
The lower transfer phenomena. both momenturn and mass. along the outer wall of the branches are coincident with the localization of atherosclerotic lesions and arterial plaques. I would like to express my sincere gratitude to my supervisors Professor O. Trass and Professor M. Ojha, for their continued advice, support and encouragement throughout this project. Without thern, would not have corne this far. I am sincerely grateful to Professor Basmadjian for his helpful suggestions for different parts of this study, especially in the transient mass transfer subject. I would also like to thank members of my reading cornmittee Professors. D. W. Kirk and D. E. Cormack, for their various suggestions and criticisrns. The technical support provided by Franz Schuh. Paul Jowlabar, and Fred Neub is gratefully appreciated. I would especially like to thank Franz Schuh whose innovative machine shop craftsmanship has been an asset not only for me but also for most of my fellow graduate students in the Institute. I thank Richard Leask who carried out the flow experiments and provided me with the velocity and wall shear stress data. 1 am grateful to al1 the members of the Doppler-Hemodynamics group at the lnstitute of Biomaterials and Biomedical Engineering: to Professors K. W. Johnston, R.S.C. Cobbold, and C.R. Ethier for their suggestions and support, and to al1 the students in the group for their helpfulness throughout this project. My thanks also go to the faculty members and staff of the Department of Chemical Engineering for their contribution to my career development during my studies in the Department. I also wish to thank al1 my family for their support during different stages of my life. Financial assistance from the University of Toronto Fellowship program is gratefully acknowledged. Page
m.. ABSTRACT...... I 1 I
ACKNOWLEDGEMENTS...... v
NOMENCLATURE...... m....m.m...xi
LISTOF FIGURES...... xiv
m.. m.. LISTOF TABLES...... XXI II
INTRODUCTION...... m.m...... ~...... m.mm....m...... m..m...m...... 1
1 .1 Atherosclerosis and Hemodynamics ...... 2
1.2 Atherosclerosis and Mass Transfer ...... 4
1.3 Objectives ...... 7
2 LITERATUREREVIEW ...... m....m...... m...... mm...... m. 8
2.1 Atherosclerosis and the Artery Wall ...... 8
2.2. Mass Transfer between BIood and the Arterial Wall ...... 1 2
2.2.1 Cholesterol Transport ...... 12
2.2.2 Oxygen Transport ...... 14 Table of Contents vii
2.3 Macromolecule Transport between the Intima and Media ...... 16
2.4 Platelet Adhesion on the Arterial Wall ...... 19
2.5 Abdominal Aortic Geometry ...... 21
2.6 Localization of Atherosclerosis ...... A
~NTEGRATEDMASS TRANSFER IN A PIPEAND A SIMPLIFIED
3.1 Introduction ...... ,.,...... 28
3.2 Experimental Methods ...... 29
3.2.1 The Geometry of the Bifurcation Model ...... 29
3.2.2 Preparation of the Plating Electrolyte Solution ...... 30
3.2.3 Fabrication of Electrodes ...... 32
3.2.4 The Electrochemical Technique ...... 36
3.2.5 Electrical Circuit ...... 38
3.2.6 Flow Loop and Expenmental Procedure ...... 38
3.3 Results and Discussion ...... 43
3.3.1 Pipe Data ...... 43
3.3.2 Bifurcation Data ...... 47
3.4 Conclusions ...... 52
TRANSIENTMASS TRANSFER ...... 53
4.1 Introduction ...... 33
4.2 Theory ...... 56 ... Table of Contents ml1
4.3 Experimental Procedures ...... -60
4.3.1 Flow System ...... -60
4.3.2 Electrical Circuit ...... +61
4.4 Results and Discussion ...... -63
4.4.1 Preliminary Data ...... 63
4.4.2 Laminar Flow Results ...... 65
4.4.3 Turbulent Flow Results ...... 77
4.5 Summary and Conclusions ...... 89
5 FLOWVISUALIZATION STUDIES ...... mm.m.m.m.m...... m...m....mm. ..91
5.1 Introduction ...... 91
5.2 Experimental Methods ...... 92
5.2.1 Aortic Bifurcation Flow Model Construction ...... 92
5.2.2 Flow Visualization Techniques ...... 97
5.2.3 Flow Loop and Measurements System ...... 99
5.2.4 Data Analysis ...... 101
5.2.5 Error Estimation ...... 102
5.3 Results and Discussion ...... 04
5.4 Summary and Conclusions ...... 118
6 LOCALMASS TRANSFER ...... mmm....m....m..m...m..m.m...... 119
6.1 introduction ...... 1 19
6.2 Experimental Methods ...... 120 Table of Contents i x
6.2.1 Flow System and operating conditions ...... 120
6.2.2 Physical Properties of the Electrolyte Solution ...... 123
6.2.3 Cutting ...... 124
6.2.3.1 Pipe ...... 125
6.2.3.2 Bifurcation ...... 126
6.2.4 Grinding and Polishing ...... 127
6.3 Results and Discussion ...... 129
6.3.1 Sources of Error ...... 138
6.3.2 Pipe Data ...... 141
6.3.3 Bifurcation Data ...... 149
6.3.4 The links between flow and mass transfer phenornena ..... 160
6.4 Sumrnav and Conclusions ...... j64
SUMMARYAND CONCLUSIONS...... 166
7.1 Summary and conclusions ...... 166
7.2 Relevance to Atherosclerosis ...... 173
APPENDICES...... 200
A . Sample Calculations ...... 200 Table of Contents x
A.1 Local Mass Transfer Experiments ...... 200
A.2 Averaged Mass Transfer Experiments ...... 201
B . Calibration of Rotameters ...... 203
C . Electrochemical Mass Transfer Measurements ...... 206
C .1 EIectrochemical Mass Transfer Measurement ...... 206
C.2 Advantage of Electrochemical Mass Transfer Technique ....209
C.3 Problems in Electrochemical Mass Transfer Measurements 21 1
D. Volumetric Analysis ...... 213
D. 1 lodometric Titration of CuSol ...... 214
0.2 lodometric Titration of K3Fe(CN)6...... 215
E. Experimental Uncertainty ...... 217
E. 1 Uncertainties in Single-Sample Experiments ...... 217
E.2 Uncertainties in Experimental Results ...... 218
E.2.1 Systematic Errors ...... -218
E.2.2 Random Errors ...... 219 cathode area (m2) bulk concentration (molR) skin friction wall concentration (mollL) diffusion coefficient (m2/s) diameter (m)
Faraday constant heat transfer coefficient (j/rn2 s K) copper thickness (m) current (A) limiting cell current (A) mass transfer coefficient (mls or cmls) length of the cathode (m) valence change
Prandtl number (:) volumetric fIow rate (m3/s) wall heat flux (~/rn~s) Reynolds number (3- Schmidt number (3 Nomenclature xii
Sherwood number (F)
Stanton number - ( ReShSc)
time (s)
dimensionless variable ( tua2/ v SC'")
non-dimensional tirne (E)
average velocity (m/s)
free stream fiuid velocity in x direction (mls)
axial velocity (mls)
cell voltage O/)
distance along the wall (m)
dimensionless variable (X~U,~/VT,)
distance frorn the wall in the radial direction (mm)
Greek letters a thermal diffusivity of the fluid (rn2/s) rw wall shear stress (Pa)
Y wall shear rate (1/s) +m r' dimensionless variable (t'/ (x ) ) v kinematic viscosrty (m2/s)
P density (kg/m3) S.. NornencIature xlll
6, concentration boundary layer thickness (m)
61 thermal boundary layer thickness (m)
6h hydrodynamic boundary layer thickness (m)
Su bscripts b branch ss steady state
1 trunk
2 branch inlet
3 branch outlet Page
Figure 2.1 : Anatorny of the abdominal aorta and nearby organs
(b4etter.F .H.. 1983) ...... 23
Figure 2.2: Predominant sites for the localization of atherosclerotic
lesions (DeBakey et al.. 1985)...... 27
Figure 2.3: A typical transverse section through the mid-sinus region
of the intemal carotid. level C (Ku et al.. 1985)...... 27
Figure 3.1 : Dimensions of the abdominal aortic model...... 30
Figure 3.2. Photograph of the four P.V.C. pipes. smaller scale plating setup ...... 33
Figure 3.3: Photograph of the aluminum mold of the bifurcation model
and nickel particles...... 33
Figure 3.4. Photograph of the assembled smaller plating setup ...... 34
Figure 3.5: Photograph of the assembled aluminum bifurcation mold
and the plating solution...... 34
Figure 3.6. Photograph of the smaller plating setup ...... 35
Figure 3.7.Schematic diagram of the plating tank. larger scale...... 35 . . Figure 3.8. Electrical circud...... 39
Figure 3.9: Schematic diagram of electrolyte flow system with the
aortic bifurcation...... 39
Figure 3.1 0: Polarkation curves for pipe (one anode) and bifurcation:
Re= 1109. Sc=3 1O0 ...... 41
xiv List of Figures xv
Figure 3.1 1: Sherwood number vs Reynolds number for the pipe:
1 and 2 anodes. Sc=1310...... A4
Figure 3.72: Sherwood number vs Reynolds number for the pipe:
1 and 2 anodes. Sc=2470...... 44
Figure 3.1 3: Sherwood number vs Reynolds number for the pipe (one anode)
at four different Schmidt numbers. compared with the Leveque
solution and the Chilton-Colbum analogy. at Sc=5655...... 46
Figure 3.14: Sherwood number vs Reynolds number for the bifurcation
(2 anodes) at four different Schmidt numbers...... 48
Figure 3.1 5: Sherwood number vs Reynolds number for the branches at four
different Schmidt numbers...... -50
Figure 4.1 : Schematic diagram of electrolyte system ...... 62
Figure 4.2. Electric circuit ...... 62
Figure 4.3: Mass transfer coefficients vs time at different Reynolds numbers.
Sc=2470 (earlier work) ...... 64
Figure 4.4: Mass transfer coefficients vs time at different Reynolds numbers.
Sc=3 100 (earlier work) ...... 64
Figure 4.5.A. Polanzation curve (So2650.Re=473) ...... 66
Figure 4.5.8. Cell voltage vs time (Sc=2650. Re=473) ...... 66
Figure 4.6.A. Polarization cuwe (Sc=2650. Re431 4) ...... 67
Figure 4.6.8. Cell voltage vs time (Sc=2650. Re=13 14)...... 67 Figure 4.7.A. Polarkation curve (Sc=4200 . Re=488)...... 68 Figure 4.7.B. Cell voltage vs time (Sc=4200. Re=488) ...... 68 List of Fimes xvi
Figure 4.8.A: Polarization curve (Sc4200, Re=760)...... 69
Figure 4.8.8: Cell voltage vs time (Sc=4200, Re=760)...... 69
Figure 4.9.A: Polarization curve (Sc=4200, Re=899)...... 70
Figure 4.9.8: Cell voltage vs time (Sc4200, Re=899)...... 70
Figure 4.1 O.A: Polarization curve (Sc=4200, Re=1616)...... 71
Figure 4.1 0.8: Cell voltage vs time (Sc=4200, Re=1616)...... 71
Figure 4.1 1: Mass transfer coefficients vs time at different applied voltages,
compared with the Leveque solution and the theoretical
transient line (Sc=2650,Re=473)...... 73
Figure 4.12: Mass transfer coefficients vs time at different applied voltages,
cornpared with the Leveque solution and the theoretical
transient line (Sc=2650, Re=1314)...... 73
Figure 4.13: Mass transfer coefficients vs time at different applied voltages,
compared with the Leveque solution and the theoretical
transient line (Sc=4200, Re=488)...... 74
Figure 4.14: Mass transfer coefficients vs time at different applied voltages,
compared with the Leveque solution and the theoretical
transient line (Sc4200, Re=760)...... 74
Figure 4.15: Mass transfer coefficients vs time at different applied voltages,
compared with the Leveque solution and the theoretical
transient line (Sc4200, Re=899)...... 75
Figure 4.1 6: Mass transfer coefficients vs time at different applied voltages,
compared with the Leveque solution and the theoretical List of Figures xvii
transient line (Sc=4200. Re=1616)...... 75 Figure 4.17.A. Polarization curve. turbulent regirne. (Sc=2650. Re=4891) ...... 78 Figure 4.1 7.8. Cell voltage vs time. turbulent regime. (Sc=2650. Re=4891) ...... 78 Figure 4.18.A. Polarization curve. turbulent regime. (Sc=2650. Re=5842) ...... 79
Figure 4.1 8.8. Cell voltage vs tirne. turbulent regime. (Sc=2650. Re=5842)...... 79
Figure 4.1 9.A. Polarization curve. turbulent regime. (Sc4200. Re=4396)...... 80
Figure 4.1 9.8.Cell voltage vs time. turbulent regime. (Sc4200. Re=4396)...... 80
Figure 4.20.A. Polarization curve. turbulent regime. (Sc=4200. Re=6541)...... 81
Figure 4.20.8. Cell voltage vs tirne. turbulent regirne. (Sc=4200.Re=6541) ...... 81
Figure 4.21 : Mass transfer coefficients vs time (turbulent regime). cornpared with the ernpirical solution (Sc=2650. Re=4891)...... 83 Figure 4.22: Mass transfer coefficients vs time (turbulent regime).
compared with the ernpirical solution (Sc=2650.Re=5842) ...... -83
Figure 4.23: Mass transfer coefficients vs time (turbulent regime).
compared with the empirical solution (Sc4200. Re=4396)...... 84
Figure 4.24: Mass transfer coefficients vs time (turbulent regime).
compared with the empirical solution (Sc4200.Re=6541) ...... 84
Figure 4.25: Mass transfer coefficients vs time at different Reynolds
numbers (turbulent regime). Sc=4200...... 85
Figure 4.26: Mass transfer coefficients vs time at different Reynolds
numbers (turbulent reg ime). Sc=2650 ...... 85
Figure 4.27: Mass transfer coefficient values at t=0.3 s vs applied voltage; laminar and turbulent results. Sc=4200 ...... 87 List of Fiwes xviii
Figure 4.28: Mass transfer coefficient values at t=0.3 s vs applied voltage;
laminar and turbulent results, Sc=2650...... 87
Figure 4.29: Mass transfer coefficient values at t=0.3 s vs applied voltage;
larninar and turbulent results for both Sc4200 and Sc=2650 ...... 88
Figure 5.1 : Photograph of the silicon mold...... 94
Figure 5.2: Photograph of the silicon rnold and the aluminurn core ...... 94
Figure 5.3: Schematic diagram of the poiimerization stage ...... 95
Figure 5.4: The final product of the UV-transparent plexiglas bifurcation mode1.95
Figure 5.5: (a) The arrangement of components and (b) the systern diagram
of the photochromic flow visualization and measurement
system (Couch et al., 1996)...... 100
Figure 5.6: Diagram of the four zones and measurements axes along the
bifurcation mode1...... 05
Figure 5.7: Axial velocity profile 40.41 mm upstrearn of the apex in the trunk
(Re=500) ...... 105
Figure 5.8: Axial velocity profiles at different distances downstream of the
apex at the branch towards the branch outlet (Re=500)...... 106
Figure 5.9: Nonalized wall shear rates with respect to Poiseuille wall shear
rate in the trunk (21.85 s-') along the inner and outer walls
(Re=500) ...... 1 09
Figure 5.10: Nomalized wall shear rates with respect to Poiseuille wall shear
rate in the trunk (26.22 s-') along the inner and outer walls
(Re=600)...... 7 09 List of Fimes xix
Figure 5.1 1: Nomalized wall shear rates with respect to Poiseuille wall shear
rate in the trunk (32.78 S-') along the inner and outer walls
(Re-750)- ...... 110
Figure 5.12: Nomalized inner wall shear rates vs axial position at different
Reynolds nurnbers ...... 110
Figure 5.13: Normalized outer wall shear rates vs axial position at different
Reynolds numbers...... 1 11
Figure 5.14: Schematic diagram of the formation of the secondary flow due
to the cross section deformation ...... 1 13
Figure 5.1 5: Vector plot of the secondary currents in a cross section of the
transition zone (Thiriet et al., 1992)...... 14
Figure 6.1 : Schernatic diagram of electrolyte system...... 122
Figure 6.2: Photograph of the electrodes in the bifurcation experiments...... 122
Figure 6.3: Diagram of the different cuts on the bifurcation mode1...... 126
Figure 6.4: Photograph of the layers for the earlier experiments...... 130
Figure 6.5: Photograph of the layers for the earlier experiments, damage to
the copper layer...... 130
Figure 6.6: Photograph of the layers for the earlier experiments, smearing of
either epoxy or nickel layer over the copper layer...... 131
Figure 6.7: Photograph of the layen, laminar experiments, 13 cm from the
leading edge ...... 131
Figure 6.8: Photograph of the layers. larninar experiments, 5.8 cm from the
leading edge...... 132 List of Figures xx
Figure 6.9: Photograph of the layers, laminar experiments, 7.73 cm from the
leading edge...... 132
Figure 6.10: Photograph of the layers, laminar experiments, 1 cm from the
leading edge...... 133
Figure 6.1 1 : Photograph of the layers, laminar experiments, 1 cm from the
leading edge (another location)...... 133
Figure 6.12: Photograph of the layers, laminar experiments, 2.97 cm from the
leading edge, damage to the nickel layer...... 134
Figure 6.13: Photograph of the layers, laminar experiments, 8.2 cm from the
leading edge, detachment between the 1" and zndlayers ...... 134
Figure 6.14: Photograph of the layers, laminar experiments, 13 cm from the
leading edge, detachment of al1 layers from the nickel substrate..A35
Figure 6.1 5: Photograph of the layers, laminar experiments, 2.97 cm from the
leading edge, detachment between the 2"' and 3d layers...... 135
Figure 6.16: Photograph of the layers, laminar experiments, 10.1 cm from the
leading edge, smearlng of the copper layer ...... 136
Figure 6.17: Photograph of the layers, laminar experiments, 10.6 cm from the
leading edge, smearing of the nickel layer...... 136
Figure 6.18: Photograph of the layers, laminar experiments, 3.4 cm from the
leading edge, smearing of the nickel layer...... 37
Figure 6.19: Local mass transfer distribution along the pipe
(Re= 1 750, Sc=3400)...... -142 List of Fimes xxi
Figure 6.20: Local mass transfer distribution dong the pipe
(Re= 1225, Sc=3400)...... 142
Figure 6.21 : Local mass transfer distribution dong the pipe
(Re=700, Sc=3400)...... 143
Figure 6.22: Local mass transfer distribution aiong the pipe
(Re=5925, Sc=3400)...... 145
Figure 6.23: Local mass transfer distribution along the pipe
(Re=4583. Sc=3400)...... i46
Figure 6.24: Sherwood number vs distance from the leading edge in the
trunk of the bifurcation (Re=750, Sc=J400)...... 149
Figure 6.25: Sherwood number vs distance from the leading edge in the
trunk of the bifurcation (Re=1750, Sc=3400)...... 150
Figure 6.26: Sherwood nurnber vs distance from the leading edge in the
trunk of the bifurcation (Re=4345, Sc=3400)...... 150
Figure 6.27: Local mass transfer coefficients in the plane of symmetry of
the branches (Re=750,Sc=3400) ...... 52
Figure 6.28: Local mass transfer coefficients in the plane of symmetry of
the branches (Re=1750, Sc=3400)...... 1 53
Figure 6.29: Local mass transfer coefficients in the plane of symmetry of
the branches (Re=4345, Sc=3400) ...... 1 53
Figure 6.30: Distribution of deposited copper around the circumference
3.7 mm from the apex (Re=750)...... 1 57 List of Fi-gures xxii
Figure 6.31 : Distribution of deposited copper around the circumference
43.9 mm from the apex (Re=750) ...... 157
Figure 6.32: Distribution of deposited copper around the circumference
3.7 mm from the apex (Re=1750)...... 158
Figure 6.33: Distribution of deposited copper around the circumference
43.9 mm from the apex (Re=1750)...... 158
Figure 6.34: Distribution of deposited copper around the circumference
3.7 mm from the apex (Re=4345)...... 159
Figure 6.35: Distribution of deposited copper around the circumference
43.9 mm from the apex (Re=4345)...... 159
Figure 6.36: C$2 and St scZ3along the inner wall and outer wall of the
branches in the plane of symmetry of the bifurcation
(Sc=3400, Re=750)...... 163
Figure B.1. Calibration curve for flowmeter #1 ...... 204
Figure 8.2. Calibration cuwe for flowmeter #2 ...... 204
Figure 8.3. Calibration cuwe for fiowmeter #3 ...... 205
Figure 8.4. Calibration curve for flowmeter #4 ...... 205
Figure C.1. Polarization curve for mass transfer measurements...... 208 Page
Table 2.1: Aortic diameter as a function of age, gender, and body
surface area ...... -24
Table 2.2: Abdominal aorta dimensions at different sites ...... 25
Table 6.1 : Average Sherwood number for straight pipe obtained by both
copper deposition technique and ferri-ferrocyanide technique at
different Reynolds numbers (laminar regime)...... 144
Table 6.2: Average Sherwood number for straight pipe obtained by both
copper deposition technique and ferri-ferrocyanide technique at
different Reynolds numbers (turbulent regime)...... A46
Table 6.3: Average Sherwood nurnber for the bifurcation model obtained by
both copper deposition technique and ferri-ferrocyanide technique
at different Reynolds numbers...... 154 Cardiovascular disease is the single leading cause of death in the developed wortd. A significant proportion of cardiovascular disease is attributed to atherosclerosis, a gradua1 process which leads to complete or partial occlusion of arteries throughout the human vasculature. Occlusion results from the build-up of fatty plaques around the inner circumference of arterial walls which can cause a critical reduction in the cross sectional area of the blood vessels. This may lead to severe losses in the blood supply to tissues and organs. As such, plaque build-up in the carotid arteries can reduce the flow of blood to the brain and eventually lead to a stroke. Similarly, obstruction of blood Row through the coronary arteries can result in a heart attack. This process begins virtually at birth but only becomes clinically important decades later, during middle life. Factors which are known to influence the disease process include diet, lifestyle, smoking and genetic predisposition. These factors alone, however, can not acwunt for the fact that the distribution of atherosclerotic plaque within the vascular system follows a distinct pattern. The tendency for lesions to develop at specific points of branching and curvature is highly suggestive of a local fluid mechanical effect on the genesis of the disease.
1.1 Atherosclerosis and Hemodynarnics
Clinically, it is observed that atherosclerosis and other arterial lesions do not occur randomly in the systematic vasculature. Instead, certain artenes and arterial geometric configurations have a significantly higher probability for developing the disease. In particular, arterial branches, bifurcations and curved sections have an apparent predilection for the development of such lesions. For this reason, hemodynamics has long been hypothesized to affect the development of atherosclerosis. Factors such as changing flow velocity, the magnitude of wall shear stress. oscillations in wall shear stress direction, flow separation, secondary flow, turbulence and particle residence time have al1 been implicated in plaque formation and localization (Caro, IWI; Zarins, 1983; Ku,
1985). These factors may affect the attachment, permeability, function and orientation of endotheliaf cells and their microstructures, and may influence the arterial accumulation of atherogenic substances through either infiux to or efflux from the wall.
lncreasing evidence has been observed to show that wall shear stress plays an important role in the structure and function of the vascular endothelium (Dewey, 1981; Levesque, 1985; Helmlinger, 1991; Khoo, 1988). The research with respect to the effects of wall shear stress on the generation of incipient atherosclerosis is divided into two extreme lines.
First, Fry (1969, 1973), based on a series of in vivo experiments, clairned that high wall shear stress is likely a cause for development of atherosclerosis.
They indicated that for wall shear stress levels below a certain critical value (400 dyneslcm2), the endothelial cellular elements remained histologically normal. In this subcritical wall shear stress, however, variations in wall shear stress might still influence the rate of transport of certain blood eiements e.g. alburnin and various lipoproteins. At the critical wall shear stress and above, endothelial cells began to deform and swell slightly. Also, Fry noted that the cells had significantly altered the staining properties which were indicative of chernical changes.
Besides, in this range the protein transport rate to the arterial wall of cellular elements increased. As the wall shear stress increased further, invasion and deposition of lipid material, adherence of cellular element, and deposition of fibrin occurred. Shear stresses of smaller magnitude were also found to increase the pemeability of the endothelium.
Second, Caro et al. (1969, 1971), in contrast, reported that atherosclerotic plaques preferentially fonn in regions of low wall shear stress and not high wall shear stress. They suggested that low wall shear stress retarded the transport of atherogenic substances away from the artery wall, decreased the diffusional efflux of accumuiating material from arterial wall to blood and resulted in an increased accumulation of lipids. In addition, low shear stress may interfere with Chapter 1 Introduction 4
endothelial surface turn-over rate of substances essential both to artery wall
metabolism and to the maintenance of optimal endothelial metabolic function
(Robertson, 1968). Friedman et al. (1981, 1987) perforrned pulsatile velocity
. measurements at specific sites along the walls of casts of the aortic bifurcation
and cornmon iliac arteries of humans. Wall shear stress was derived frorn the
velocity data. Their results showed that intima1 thickness correlated negatively
with mean, maximum and pulse (max-min) shear rate.
It is also believed that increased particle residence time in low shear areas
would help in the progression of the disease by inhibiting the transport of lipids
from the artery wall to the blood stream. Quantitative particle-tracking studies
have demonstrated increased particle residence time in areas of the carotid
bifurcation prone to plaque formation (Talukder, 1983). This increase in particle
residence time could thus prolong exposure of certain portions of the vessel wall
to atherogenic particles and account for focal plaque formation. Time dependent
lipid particle-vesse1 wall interactions would be facilitated and promote plaque
formation. In addition, blood-borne cellular elements that may play a role in
atherogenesis would have an increased probability of deposition and adhesion
into the vessel wall in such regions.
1.2 Atherosclerosis and Mass Transfer
Mass Transfer may represent a common link in the chain of processes by
which different atherogenic plasma substances, such as growth factors, low
density lipoproteins (LDL). fibrinogen, viruses, chemotactic agents, immune Chapter 1 Introduction 5 complexes can act to produce the variety of intimal histologic changes that we recognize as the "atherosclerotic lesions" (ROSS,1974; Smith, 1979; Hoff, 1975).
Compared with studies of the flow patterns at sites of blood flow disturbances along the arterial tree, little attention has been given to the parallel problern of associated abnormalities in arterial wall mass transfer. The relative lack of progress persists even though it has long been recognized that the role of specific hemodynamic variables in the initiation and development of atherosclerosis must be assessed by correlating flow field and mass transfer measurements with the distribution of intimal lesions in the human arteries. The reason must have been rooted in the difficulty of the subject of performing mass transfer studies. Non-invasive in vivo studies as limited by the available experimental technique do not provide suffkient spatial resolution to describe the complex flow patterns at locations of clinical interest. On the other hand. numerical solutions of heat and mass transfer problems in complex fiow fields have been constrained by cornputer abilities and numerical techniques until recently. The difficulty arose due to the thinness of mass (heat) transfer boundary layers in studying high Schmidt (Prandtl) number transport phenornena. There is a predictable relation between 6, (61) and 8h :
Sd&= SC)-'^ (w
St/8h= (pi)-'" (Iw2) where &, 6t and Sh are concentration, thermal and hydrodynamic boundary layer thickness. The reported values for the diffusion coefficient of LDL in blood have a range of 5.0 x 10" - 2.0 x cm2/s (Caro, 1971; Back, 1975). Given the Chapter 1 Introduction 6
kinematic viscosity of 0.03 cm2/s. a range of 1.5 x Io5 - 6.0 x lo5 can be
obtained for the Schmidt number. It is then found that:
SdSh4.015 (1-3)
Now, considering laminar, steady state flow with a parabolic velocity profile, Zih
can be calculated as:
Sh(x)=5.O (vxlu)'" (14)
Given a main stream velocity of 30 cmls corresponding to the velocity of the
blood in large arteries (i.e. Re4000 for a vesse1 of d=2 cm), 30 cm downstream
of the origin of the aorta (in the abdominal segment), 6h has a thickness of 0.87
cm. lnserting the hydrodynamic boundary layer, Sb, in equation (1-3), the
concentration boundary layer can be calculated to have a value of about 130 pm.
Due to a very high Schmidt number associated with the transport of blood-borne
substances, very fine grid spacing is needed to capture the concentration profile
near the surface (across the concentration boundary layer) and this in tum will
require very powerful computer facility and refined numerical techniques.
The transport of materials from blood to the arterial wzll and vice versa
may take place through mass diffusion and bulk convection. For the arterial
intima, the total blood-wall transport involves three steps: (1) a diffusion and/or convection mass transfer in the arterial lumen; (2) an uptake process at the blood wall interface; this may be associated with diffusion through the membrane, the solubiity of the species in blood or reactions at the biood/wall interface; and (3) the transport of material within the arterial wall. 1.3 Objectives
Knowledge of the mass transfer and flow behavior in the aortic bifurcation is crucial to the understanding of atherosclerosis lesion initiation and progression as well as to improvements in diagnostics methods based on the interpretation of mass transfer and fiuid flow phenornena. The overall objectives involved in carrying out this task were fivefold:
1. To construct a simplified model of the aortic bifurcation, a region prone to
atherosclerosis, based on basic geometric parameters obtained from the
Iiterature.
2. To measure integral mass transfer in a pipe and the fabricated bifurcation
model using an electrochemical technique.
3. To perform a hemodynamic assessrnent using the laser photochromic tracer
technique to charactenze the velocity and wall shear stress fields.
4. To develop a new technique to enable measurement of time-averaged local
mass transfer coefficients.
5. To suggest a possible link between the transport processes and
atherosclerosis. It should be noted that in this chapter, various Iiterature regarding the effects of mass transfer and fluid mechanics to atherosclerosis is reviewed. The fundamental mass transfer and fluid Row literature is reviewed, as required in the relevant chapters.
2.1 Atherosclerosis and the Artery Wall
Atherosclerosis involves medium- and large-sized arteries such as the abdominal, carotid, coronary, iliac, femoral, and renal arteries. The vesse1 walls of large arteries are cornposed of three layers: the intima, media and adventitia. The intima, the innermost layer, is very thin (only a few microns), and the bulk of the artery wall consists of the media and the adventitia. The inner layer of the intima consists of the endothelium, which is in direct contact with the flowing blood in the lumen. It is through this layer that the various species in the blood rnust pass in order to progress further into the wall. A thin connective tissue layer separates the endothelium from a basernent elastic membrane, the intemal elastic lamina (IEL). The IEL forms the inner boundary of the next layer, media.
The media is composed primarily of smooth muscle cells laid down in an extracellular matrix of collagen and elastin. During the initiation of atherosclerosis, it is the first two layers, the intima and media, which are of most interest. The outermost layer is the adventitia, separated from the media by the external elastic lamina (EEL). The adventitia contains fibroelastic tissue with no smooth muscle cells. It is within this layer that the vasa vasorurn are typically found, although in larger vessels the vasa vasorurn can reach into the media layer. The vasa vasorum are blood vessels that supply the adventitia with nutrients, while the intima and media are supplied from the lumen.
At birth, the subendotheiial space is so small that the endotheliurn appears affixed to the media. As the individual grows, physiologie adaptation to blood flow and wall tension leads to the growth of the intima between the endothelium and media. This adaptive intimal thickening continues over tirne and may be circumferentially diffuse or may become eccentric, depending on the local vascular environment. The distribution of eccentric intimal thickening and the distribution of advanced atherosclerosis plaques has been noted to be very Chapter 2 Literiiture Review 10 similar in many human artery studies (Stary, 1987). This thickening of the intima is due to the migration and proliferation of vascular smooth muscle cells, ernanating from the vesse1 media (Ross, 1986).
Two blood-borne components that are believed to be crucial to the atherosclerotic pathway are monocytes and low-density lipoproteins. The monocyte is a type of white blood cell central to the body's immune response system. Low density lipoprotein (LDL) is part of a family of plasma-borve lipoprotein molecules which function in the transport of lipid. It might be of interest to note that LDLs have a diameter in the range of 4-20 nm and a molecular weight of 3x1 o6 Da (Walton, 1975).
One of the most accepted hypothesis about the initiation and progression of atherosclerosis is called the "response to injuryn hypothesis (Ross, 1993). lnjury to the endothelium invokes at least two significant effects: first, an increased permeability to LDL and hence migration and trapping of LDL into the subendothelial intima, and second, attraction for blood-borne monocytes to migrate to the endothelial surface. This second effect may occur due to the formation of specific surface attractant molecules on the endothelium or through some other mechanism. LDL particles which have become trapped within the intima undergo oxidative modification and subsequently release chemoattractants (Berliner, 1995), which promote migration of the adhered monocytes through the endothelium into the arterial wall. Here, monocytes differentiate into macrophages, take up the modified LDL into their cytoplasm, and collect just undemeath the endothelial surface, forming a visible raised lesion. This lesion is known as the fatty streak and is the sign of the early atherosclerosis process. The modified macrophages, which dorninate the composition of the fatty streak, are known as foam cells, so named because of their lipid-saturated cytoplasm. This early lesion of atherosclerosis contains small amounts of SMCs but tends to occur at areas where increased migration and proliferation of SMCs from the media have led to eccentric intima1 thickening. As the process continues to progress, further SMC proliferation, monocyte recruitment, and LDL accumulation leads to the development of sites of extracellular lipid within the plaque, and the formation of a fibrous cap composed of SMCs, collagen, and elastin undemeath the endothelium. Eventually, this advanced fibrous cap rnay rupture due to unusual mechanical stress (Richardson
1989). At this point, the highly thrombogenic plaque contents and sunounding subendothelial tissue become exposed to platelets flowing in the blood. Platelet- induced activation of a clotting cascade leads to vesse1 thrornbosis and occlusion. These events lead to the formation of the complicated plaque which becomes symptomatic and clinically overt.
As far as basic atherosderosis research has corne, there are still fundamental parts of the process which remains unclear. The forernost of these is the event that triggers the inducing endothelial injury. While some hypotheses have been put forth regarding rnechanisms by which endothelial injury and dysfunction occur (Ross, 1993), it remains a controversial subject. By what mechanism do monocytes migrate from the bulk flow of blood toward the arterial wall surface? Is there an enhanced accumulation of LDL particles in the near-wall vicinity of lesion-prone areas before migration into the subendothelial space; if so, what causes this migration from bulk flow toward the wall? What influences the localization of eccentric intima1 thickening within the human vasculature, and by extension, what influences the localization of atherosclerosis?
2.2 Mass Transfer between Blood and the Arterial Wall
2.2.1 Cholesterol Transport
Based on observations that atherosclerosis develops preferentially in regions of low artenal wall shear stress, such as the descending and abdominal aorta, Caro, Fitz-Gerald and Schroter (1971 ) postulated that the distribution of early atheroma was most likely due to shear dependent mass transport phenornena. Their concept, which relates mass transfer to atherosclerosis in both hurnan and animals, was initially developed through examination of naturally occurring atherosclerosis with emphasis on transport within the lumen. Later,
Caro and Nerem performed experiments which indicated that choiesterol uptake was not due to a mass transport limitation in a straight artery (1973, 1974). This result suggested that the uptake process was a chemically controlled event.
In the work of Caro, Fitz-Gerald and Schroter (1971), a theoretical analysis of steady flow over a flat plate boundary layer was used to estimate the diffusion boundary layer thickness as a proportion of the hydrodynamic boundary layer. They argued that if diffusion resistance is small in the wall, as compared to that of the boundary layer, then cholesterol synthesized in the intima accumulates. Consequently, the diffusion boundary layer was considered to be the rate-lirniting step for the total transport process. To verify this hypothesis, Chapter 2 Literature Review 13 they used the nitrogen dioxide-wet litmus reaction to study the spatial distribution of wall flux and shear in thoracic aortic casts and mode1 branches in steady flow.
Although the predicted shear dependence of wall flux was observed, the variation of wall flux with distance along the arterial segment and the magnitude of the flux itself were different.
Similady, Caro and Nerem (1973) and Caro (1974) applied boundary layer theory to analyze their data on the flux of labeled cholesterol to the wall of the dog common carotid artery in vitro. In the experiment, a steady fiow was maintained through the arterial segment. In their analysis, the shear along the intima was assurned to be spatially uniform, and the tracer concentration at the intima1 interface was further assurned to be unifon and constant throughout the experiment. As a result, the spatial variation pattern of wall flux and the magnitude of the flux itself differed from the experiments.
Physiologic hemodynamics is much more complicated than that found in the fiow over a fiat plate. The three-dimensional geometry of arterial bifurcations and pulsatile cardiac wavefom will alter the hydrodynamic fiow field. Studies have iliustrated the complexity of the flow field found in arterial models under steady fiow (Ferguson et al., 1974, Karino et al., 1983) and the differences produced by pulsatile flow (Ku,1983). It is difficult, as pointed out by Fi&-Gerald and co-workers (1971), to predict the pulsatile hemodynamics at an artenal branch based on flow over a fiat plate. Since the aforementioned studies were camed out on simple geometries without fIow separation, the results are inconclusive. Further research in this area is necessary. Chapter 2 Literaiure Review 14
2.2.2 Oxygen Transport
The transport of oxygen from the blood through the diffusion boundary layer and
across the arterial wall has been analyzed. The vessel wall was assumed to be
permeable since arterial wall oxygen consumption rates can be detemined. The
controlling resistance was observed to be the convective transport to the blood
vessel boundary layer.
It has been postulated that an impaired supply of oxygen (hypoxia) from
the luminal blood, as may occur with blood flow disturbance, is one factor in the
atherogenic process (Wissler, 1976). Conditions of hypoxia were observed by
Dixon (1961) to increase the intracellular deposition of visible droplets of liquid
lipids, apparently due to the inability of the cells to emulsrfy and disperse the fat
they carried. Kjeldsen, Wanstrup and Astrup (1968) observed a significantly
higher degree of visible aortic atherosclerosis and aortic content of cholesterol
and triglycerids in rabbits fed on high cholesterol diet when their oxygen supply
was reduced. Zemplenyi (1968) also observed the lack of oxygen at locations where atherosclerosis occurred, and Robertson (1968) concluded that local
hypoxia at the cell level may play an important role in accelerating human
atherogenesis by the initiation of a chain reaction of a series of self-sustaining
metabolic abnormalities. Furthenore, oxygen was selected to study the transport phenornena of substances with the arterial wall because of the
availability of data on its diffusion coefficient and consurnption rates.
Friedman and Ehrlich (1975) numerically simulated luminal transport of oxygen in the vicinity of a two dimensional bifurcating arterial model. The blood flow was steady and the Reynolds number was 100. Two cases were considered: (1) the oxygen concentration along the wall was taken to be zero and the oxygen flux to the wall was then calculated; (2) the oxygen flux to the wall was specified, then the oxygen concentration along the wall was calculated.
Although their results showed wide spatial variations in interfacial mass transport, they did not find regions of impaired mural oxygen supply, possibly because blood flow separation did not occur at the Reynolds number they used. Axial diffusion of species was also neglected.
Back (1975) calculated the effects of a spatially varying shear rate on the oxygen transfer coefficient along axisymmetric arteries. He considered cases of flow acceleration, flow separation, flow reattachment, and 'mild" flow deceleration. The results indicated an increased interfacial oxygen flux for acceleration and stagnation regions, and a large decrease in flux for "mild" flow deceleration as compared to straight arteries. The analysis was limited to a boundary layer approach for both the fluid mechanics and mass transport. The axial species diffusion and the axial curvature of the wall surface were also neglected.
Back et al. (1 977) later calculated oxygen transport with pulsatik flow in a rigid arterial wall with axisymmetric irregularities. The pulsatile flow field was calculated on the basis of the reported time dependent coronary flow. It was reported that oxygen transport to the wall was dependent upon the various flow regions and, consequently, there were spatial variations in the oxygen transfer coefficient. In addition, there were considerable variations over the cardiac cycle. A reduction in oxygen transport to the arterial wall occurred at the incipient
separation location on the downstream side of a plaque, where the resistance to oxygen transport on the lumen side was believed to be at least an order of
magnitude greater than the inner vascular wall resistance. Therefore, it was concluded that the availability of oxygen for cellular respiration was essentially boundary layer controlled.
Schneideman et al. (1982) simulated oxygen transfer in a fully developed, pulsating, laminar flow in both rigid and distensible tubes as part of a study of arterial wall hypoxia and atherosclerosis. The computational mode1 was based on dimensions and Rows in the human thoracic aorta. The pulsatile velocity fields employed were based on those of Womersley with a modification to account for the radial convection relative to the moving wall. The oxygen concentration within the artenal wall was assumed to be constant. It was concluded that Womersley- type pulsatility, superposed on an axial Poiseuille fiow, had a negligible effect on oxygen transport to both ngid and distençible walls.
2.3 Macromolecule Transport between the Intima and Media
Another group of researchers has focused its studies on the transport of macromolecules through the arterial wall itself. Chronically elevated levels of plasma lipoproteins, particularly low density lipoproteins (LDL) and very low density lipoproteins (VLDL), have long been associated, in numerous epidemiologic studies, with an increased incidence of athersclerosis (Ross,
1986). Therefore, the study of the transport and accumulation of LDL and related Chapter 2 Literature Review 17 macromolecules in the arterial wall has attracted the attention of many investigators. However, the transport phenornena of large molecules between blood and the artenal wall were previously only poorly understood. Until recently, vesicular transport or fused open vesicle channels were generally thought to be the most likely pathway for large molecules to enter the vessel wall tissue. More recent studies showed that open vesicle channels were extremely rare in arterial capillaries and were absent in large vessels (Bundgaard et al., 1982). This finding was based on the fact that macrornolecules in the range of roughly 4-20 nm are too large to pass through normal intercellular junctions. Ross and coworkers (1976) postulated a "response to injuty" hypothesis and attempted to relate endothelial cell turnover and endothelial permeability to plasma LDL transport, smooth muscle cell proliferation and the receptor-mediated metabolism of the endothelial cells. In their hypothesis, a strong emphasis was initially placed on local endothelial denuding injury as a causal rnechanism. It was later rnodified after extensive experiments conclusively showed t hat overt endothelial denudation was seldom seen in early lesion formation (Ross, 1986).
Weinbaum et al. (1985) proposed a steady state endothelial leaky junction model to explain the variations in endothelial permeability to macrornolecules that have been observed in the major arteries of pigs. dogs and baboons. The model included convective transport in the intercellular clefts and media, and the diffusion behavior in the subendothelial features, but it neglected vas0 vasorum, chernical kinetics, and the diffusional resistance of the transient open junctions. In addition, it treated the vessel wall as an isotropic medium. The Chapter 2 Literature Review 18 prediction suggested that up to 50400% of the experirnentally measured local enhancernents in pemeability were due to transient openings and pooriy formed intercellular junctions around the small dying or regenerating cells dunng the tumover process.
Weinbaum and coworkers (1988) extended their previous study and developed two new models. In these models, the transient opening of a junction. which surrounds the endothelial cells undergoing cell tumover, was assumed to occur gradually from a normal spacing to a gap that was sufficiently wide to allow the passage of rnacromolecules larger than LDL. Following the maintenance of such a wide gap for a certain period of time in the cell cycle, the spacing gradually retumed to its normal size and lipid accumulation occurred. The roles of the tracer size and time factor in detemining the dimension of macromolecular leaks were considered. Nevertheless, many assumptions used in these models were not readily justifiable. The effect of convection on the transport in the wall was also ornitted.
Guided by the above theoretical models, Lin et al. (1988) camed out an experimental study on rat thoracic aortas in which Evans-albumin (EBA) was visualized by fluorescence microsccpy as the macromolecular tracer and hernatoxylin staining was used to identtfy endothelia cells in mitosis. It was shown that a close correlation existed between the leakage and endothelial cell mitosis. Later, they perfomed another experiment on the rat thoracic aortas to examine the influence of rnolecular size on the correlation between junctional leakage and cell mitosis (Lin, 1989). The leakage duration in the mitotic phase was found to Vary inversely with the tracer molecular size.
2.4 Platelet Adhesion on the Arterial Wall
Blood platelet adhesion on artenal walls has also been investigated. The studies have been focused on the interaction between platelets and the subendotheliai surface. Intact vascular endothelium nomally prevents platelet adherence because of the nonthrombogenic character of its surface and its capacity to fom antithrombotic substances such as prostacyclin and hepan3
(Baurngartner, 1973, Ross, 1986). Gaps between endothelial cells and endothelial defects are rapidly plugged by blood platelets. These platelets become a platform on which other platelets accumulate to forrn mural platelet thrombi in endothelial defects (Baumgartner, 1973). Therefore, adherence of blood platelets to the subendothelium of the injured vessel wall has been considered during the past two decades to be a fundamental event in the formation of a hemostatic plug or thrombus. Some of the platelet-connective tissue interaction may also be important in the development of atherosclerosis. such as in hypercholesterolemia and homocyastinuria after injury by intra-artenal catheters, and at perianastomotic sites after bypass surgery (Ross, 1986).
Both theoretical and experimental studies have been camed out on the transport of platelets to the de-endothelialized vessel wall, which is considered impeneable. The process of platelet adherence can be considered to occur in three steps. First, platelets are convected in streamlines parallel ta the surface. Second, platelets diffuse perpendicular to these streamlines. Finally, platelets
arrive at the surface where the rate of adhesion is controlled by platelet-surface
reactivity (Friedman, 1971). Previous results showed that platelet diffusion and adhesion processes were strongly influenced by the presence of other blood cells (especially red cells). blood flow rate, and properties of the vessel wall.
Baumgartner (1973) studied the effect of blood velocity on platelet adhesion to the subendothelial surface of anesthetized rabbit aortas. It was reported that both in vivo and in vitro platelet adhesions to subendothelium increased with blood flow velocity. This finding is consistent with the clinical observation that platelets play a greater role in arterial than in venous thrombosis. It was also mentioned that the presence of blood cells other than platelets played an important role in platelet adhesion to the subendothelium.
Turitto and Baurngartner studied the effect of blood shear rate on the interaction between platelets and subendothelium in fiowing rabbit blood (1975,
1977, and 1979). Baumgartner's perfusion technique was used. The rate of plateiet adhesion was found to depend on both the platelet-surface interaction and the platelet arrival rate at the subendothelial surface through the blood.
Under flow conditions corresponding to vessel wall shear rates less than 800
sec" (as in veins and large arteries), both their theoretical and expenmental data indicated that the adhesion rate was predorninantly diffusion controlled and detennined by the physical factors which control the arrival rate of platelets to the vicinity of the surface. As the shear rate increased from 1000 sec-' up to 2600 sec-' their experimental data predicted a gradua1 transition from diffusion to intemediate reaction control in which platelet surface kinetics were dominant.
The effects of red blood cells on platelet movement have been studied by several investigators. Basically, there are three explanations for the enhanced platelet transport in the presence of red blood cells. Based on a theoretical analysis and blood oxygenation experiments, Keller (1971) suggested that the red blood cells rotated the shear field to cause local turbulence. The mixing effect resulted in higher platelet transport and adhesion. The second explanation was reported by Goldsmith (1972) based on the observation of tracer red blood cells in concentrated ghost suspensions undergoing an erratic radial displacement.
The radial displacement was caused by continuous collision of passing cells and would induce displacement for platelets and plasma as well. The transport rate would increase proportionally with the shear rate and with the volume of red cells. The latest explanation was proposed by Aarts et al. (1988) as a "skimming layef effect based on their in vitro experiments with Baumgartnets perfusion technique. Elevated shear rate and hematocrit would shift the platelet concentration maxima toward to the vesse1 wall and result in enhanced platelet adherence to the subendothelium.
2.5 Abdominal Aortic Geometry
The abdominal aorta is a complex organ that has several branches. The most important branches are: celiac trunk, supenor mesenteric artery, right and left renal arteries and the infenor mesenteric artery. In addition, the aorta in this region is not a straight tube. In relation to the cuwature of the spine. the aorta is curved anteroposterioriy. Below the inferior mesenteric artery , abdominal aorta bifurcates into the right and left common iliac. There are other bifurcations in each of the common iliac arteries called iliac bifurcations that divide the branches into extemal and intemal iliacs (Figure 2.1).
Pnor to fabrication of bifurcation models, an investigation was performed in the literature regarding abdominal aortic geometry. The papers within the medical database named MED-LINE were reviewed in two periods, from 1985-
1989 and from 1990-1994. This investigation showed that nomal aortic diameter varies according to age, gender, and body surface area (BSA). Table 2.1 presents some of these results. In summary, in evaluating aortic diameter, age, gender, and BSA are important factors to consider. With advancing age, aortic diameter increases at al! levels. Aortic diarneter in men is greater than in women matched for age and size. BSA has a positive correlation with aortic diameter, greatest in patients less than 50 years of age. Table 2.2 summarizes abdominal aorta dimensions at different sites. Figure 2.1: Anatomy of the abdominal aorta and nearby oqans (Netter, F.H., 1983). Table 2.1: Aortic diameter as a function of age, gender, and body surface area.
Extended thoracic aortic diarneter in miIlimeters based on age and B.S.A 1 - B.SA. (d) Aae (MI (Body Surface Area) 55 65 75 8 5
Expected celiac aortic diameter in rnillimeters based on age and B.S.A. B.S.A. (ml) Aae (vr) (Body Surface Area) 55 65 75 8 5 ( Male ) 1.3 19.232 20.332 21A32 22.532 1.9 22.376 23.476 24.576 25.676 2.5 25.520 26.620 27.720 28.820
(Femalc 13 17.212 18.3 12 19.412 20.5 12 1.9 20.356 2 1.456 22.556 23.656 2.5 23.500 24.600 25.700 26.800 Expected renal aortic diameter in rnillimeters based on age and B.S.A. B.s.A.(~') ~e fvr) (Body Surface Arta) 55 65 75 85
(Female ) 1.3 1.9 2.5 20.120 20.920 2 1.720 22.520 Expccted infiarena1 aortic diametcr based on age and BS.A B. SA.(^') Aee (MI (Body Suface Area) 55 65 75 85 (Male) 13 15.889 16.589 17.289 17.989 1.9 18.067 18.767 19.467 20.167 2.5 20.245 20.945 2 1.645 22.345 Table 2.2: Abdominal aorta dimensions at different sites.
II Abdominal Aorta diameter (Suprarenal) 1 24.lk3.2 (mm)1 D. Horejs II Abdominal Aorta diameter (Renal) 1 D. Horejs - ~bdominaiAorta diameter (Infrarenal) D. Horejs Celiac trunk diameter J .E.Moore Celiac trunk branching angle Celiac trunk distance from bifurcation 145.8t20 (mm) Superior Mesenteric Artery diameter (SaMaAm) S.M.A. branching angle S.M.A. distance from bifurcation t'=-Y128.5I17 (mm) Right renal artery diameter Right renal artery branching angle 161.4k17 (deg) 1 Right renal artery distance from bifurcation Left renal arterv diameter Left renal artery branching angle Left renal artew distance from bifurcation 109.8I17 (mm) Inferior Mesenteric Artery diameter (I.M.A.) I.M.A. branching angle 1mM.A. distance from bifurcation 1 Aortic bifurcation angle C.B.Bargeron 11.712.0 (mm) D. Horeis Left comrnon iliac ID. Horeis II Iliac bifurcation angle r External iliac 1 O.M. Pedersen II - - IIInternai iliae I 2.6 Localization of Atherosclerosis
The lesions of atherosclerosis are distributed irregularly throughout the vasculature. Some vessels are characteristically spared, whereas other sites within the artenal tree commonly contain lesions. DeBakey (1985) descnbed 5 major categones of arterial plaque distributions. The coronary arteries, the major branches of the aortic arch, and the abdominal aorta and its visceral and major lower extremity branches are sites particularly susceptible to the atherosclerosis process (Figure 2.2).
In a11 these sites, bifurcations such as the aortic bifurcation, carotid bifurcation and the left coronary bifurcation are susceptible to the formation of atheroscelrotic lesions. Lesions distribute mainly along the outer walls of the bifurcation, whereas the walls of the flow divider and the inner walls further downstream are less affected. Figure 2.3 shows the lesion thickness on a histological section in a carotid bifurcation obtained frorn a cadaver. The circumferential locations are designated in polar coordinates with O0 corresponding to the inner wall or Row divider, 180" corresponding to the outer wall opposite the flow divider, and 90" and 270° corresponding to the side wall locations. The eccentric localkation of plaque along the outer wall (180")of the carotid sinus and the minimal thickening along the inner wall of the sinus (O0) can be observed. Figure 2.2: Predominant sites for the localization of atherosclerotic lesions (DeBakey et al., 1985).
Interna1 Carotid
Cornmon Carotid
Figure 2.3: A typical transverse section through the mid-sinus region of the interna1 carotid, level C (Ku et al., 1985). 3 INTEGRATEDMASS TRANSFER IN A PIPEAND A
SIMPLIFIEDBIFURCATION MODEL
3.1 Introduction
lncreasing evidence has been observed to show that fluid mechanical factors, or hernodynamics, play an important role in vascular disease.
Convective mass transfer of blood-borne material to the arterial wall has been implicated in the localization of arterial plaques or stenoses. Atherosclerosis leading to the formation of plaques or stenoses develops at specific sites within the arterial vasculature, especially along the wall across from the flow divider or vesse1 bifilrcation, or the outer wall. This preferential localization on the outer wall of bifurcations iç the prirnary reason for implicating mechanical factors such as shear stresses or mass transfer in the disease process. Chapter 3 Integrated Mas Transfer in a 29 Pipe and a Simplified Bifurcation Model Sites of preferential thickening of the arterial wall where focal
atherosclerosis develops are exposed to low shear stress (Zarins et al., 1983; Ku
et al., 1985; Friedman et al., 1987; Moore et al., 1994). Transient flow separation
around peak flow of the cardiac cycle can develop at these sites leading to
oscillating wall shear stress direction (Ku et al.. 1985; Moore et al., 1994).
Because of the high Schmidt number associated with protein molecules, the role
of convective mass transfer in the localization of atherosclerosis t~asnot been
widely investigated either numerically or experimentally. For low density
lipoproteins (LDL), given a diffusivity of 5.0 x 10~- 2.0 x 10" cm21s (Caro et al.,
1971; Back, 1975), the Schmidt number in blood is in the range of 1.5 x 10' - 6.0
x 105.
In this part, the average mass transfer coefficients were determined in a
simplified abdominal aortic bifurcation rnodel for different Schmidt and Reynolds
nurnbers. Atherosclerosis resulting in bypass surgery or limb amputation is a
common occurrence at this site. Mass transfer coefficients were also obtained at
the same experimental conditions in a straight pipe for cornparison.
3.2 Experimental Methods
3.2.1 The Geometry of the Bifurcation Model
Section (2.4) of Chapter 2 outlines the geometry of the abdominal aorta. In this initial study, branches and curvature were not modeled. A simplified Y-
bifurcation was constnicted with a constant diameter for the main trunk and tapered branches as shown in Figure 3.1. Branches with different inlet and outlet Chapter 3 Integrated Mass Transfer in a 30 Pirre and a Sim~lifiedBifurcation Model diameters were constructed to simulate a to-scale, realistic model of cornmon
iliac arteries. The constant diameter of the parent vesse1 was used to calculate
the Reynolds number.
Figure 3.1: Dimensions of the abdominal aortic model
3.2.2 Preparation of the Plating Electrolyte Solution
The first few plating trials resulted in large cracks along the length of nickel
plated samples. Investigation revealed that the solution had been left uncovered for a long penod of time and solution had probably been oxidized and the desired
plating solution, consisting of Ni(NH3HS03)2,H3B03 and surfactant, contained Chapter 3 Integraiai Mass Transfer in O 3 1 Pipe and a Simplified Bifurcation Mode1 some impurities. Special processing was made to remove soluble chemicals.
Since the original nickel hydroxide had been made from the precipitation of nickel sulphate with caustic soda, sulphate ions were the major soluble impurities.
When sulphate ions were removed, it could be safely assumed that irnpurities were virtually removed.
Regeneration of electrolyte solution can be divided into five steps:
1) NaOH + ~i~'- Ni(OH)? -1
2) Separation of Ni(0Hh from soluble chemicals
3) Washing the soluble chemicals from Ni(OH)*
5) Adding H3BO3and surfactant to above solution
In order to get Nickel Hydroxide, 3 M caustic soda was added. After sufficient
time the liquid above the sedimentation was discarded. Inasmuch as soluble
chemicals could be found in the lower layer, distilled water was added and
uniformly mixed. Sedimentation was allowed for eight hours, and water with soluble chemicals was removed by siphoning. The same procedure was
repeated many times until addition of Ban'urn Chloride into a sample of supernatant liquid did not produce precipitation. In the next stage, under continuous mixing, sulfamic acid [(NH4)HS03] was added to the tank. Nickel
hydroxide was dissolved in sulfamic acid to form about 4 M soluble nickel Chapter 3 Integrated Mass Transfer in a 32 Pipe and a Simplified Bifurcation Mo& sulfarnate [Ni(NH3HSO3)4.Borie acid then was added to the solution with final concentration of 1 M to act as a buffer so that a constant pH could be maintained at an approximate value of five. Surfactant was added last. Its function was to prevent hydrogen bubbles from attaching to the cathode surface.
3.2.3 Fabrication of Electrodes
Since there were a few cathodes and anodes to be plated, a smaller plating container was fabncated in addition to the larger plating tank to speed up the plating step. The smaller scale set-up consisted of a 20 Mer container and four 20 centimeter lengths P.V.C. pipes. Small holes were made along the P.V.C. pipes to provide contact between anodes and solutions. S-round nickel particles were then placed inside the P.V.C. pipes (Figures 3.2 to 3.6). Four baskets of nickel rounds were used in the larger plating bath as the anode and a filter was also used to remove particles that could affect the plating quality (Figure 3.7).
Fine air bubbles formed by an air sparger produced turbulent mixing that aided in removing hydrogen bubbles attached to the cathode surface and also to increase the rate of mass transfer. The electrolyte solution in both plating baths consisted of 4 M nickel sulfamate Ni(Nt+ S03)2and 1M boric acid which acted as a buffer to maintain a pH of 5. A small amount of surfactant was added to minimize attachment of hydrogen bubbles to the cathode surface. In the next stage, positive moldç of the lumen of the main tnink and branches were machined from aluminum pipes. In addition to the bifurcation mold, aluminum molds of the straight pipes were also machined out of aluminum. These were then placed in Chapter 3 Integntteed Mass Tramfer in a 33 Pipe and a Sintplified Bifurcation MO&
Figure 3.2: Photograph of the four P.V.C. pipes, smaller scale plating setup.
Figure 3.3: Photograph of the aluminum mold of the bifurcation model and nickel particles. apter 3 IntegruieU Mus Transfer in a Pipe and a Simvlified Bifirrcation Mo&1
Figure 3.4: Photograph of the assembled smaller plating setup.
Figure 3.5: Photograph of the assembleci aluminum bifurcation mold and the plating solution. Chapter 3 Integruted Mars Transfer in a Rve d a Simplified Bifurcation Mode1
Figure 3.6: Photograph of aie srnaller plating setup. 1Air Sparger
Figure 3.7: Schematic diagram of the plating tank, larger scale. Chapter 3 Integrated Mass Transfer in a 36 Phe and a Simplified Bifurcation Mode1 nickel-plating tanks to achieve a suitable wall thickness (2-3 mm). Prior to the plating step, al1 aluminum molds were cleaned first by distilled water and then by acetone to remove any possible dirt or oils left on the surface of the molds.
Initially, a voltage of 0.6 V was applied across the anode and cathode to generate a low curent of 0.1 A in order to ensure a unifom layer of nickel deposition. After 24 hours, the voltage was increased to 1 V. At this stage, an air sparger in the larger tank and a small plastic mixer in the smaller container were used to produce turbulent mixing. Afier obtaining desira ble thickness, the samples were immersed in 6 M sodium hydroxide solution to dissolve the aluminum core. Due to the composition of alurninum pipe, a layer of dark chemicals was deposited on the inside and outside of the nickel models. A mixture of peroxide and HCI cleaned the dark layers in about 60 seconds. The concentration of HCI and peroxide were 1 M and 0.1%, respectively. Finally, a bifurcation (cathode), a 15 cm long pipe (cathode in the pipe experiments) and a
30 cm long pipe (anode) were fabricated. lntemal diameters of the pipes and also the tnink part of the bifurcation were al1 1.86 cm.
3.2.4 The Electrochemical Technique
The electrochemical technique was used to measure mass transfer coefficients between nickel surfaces and the fem-ferrocyanide solution. This method is based on a diffusion-controlled reaction at the electrode surface. As discussed in detail by Selman et al. (1978), when an electric potential is applied between two electrodes in an aqueous solution of an electrolyte, an ionic Chapter 3 Integrated Mius Trunsfer in a 37 Pipe and a Simplified Bifurcation Mode1 reduction occurs at the cathode and an oxidation at the anode. As a result a cuvent which is proportional to the number of ions reacting at the electrodes per unit time, Rows through the circuit.
At steady state, ions that are being converted at the electrode have to be supplied from the bulk of the liquid. This can occur by a diffusion process under the influence of the concentration gradient and by migration of the ions in the electric field. The rate of the electrochemical reaction increases with the potential difference between the electrode and solution, and at a sufficiently high potential difference, rnass transfer towards the electrodes becomes the rate detemining step in the electrochemical reaction. By using an extra large anode, the flux of reacting ions is influenced primarily by the cathode geometry. In this study, the following reaction occurs at the cathode
F~(CN)~%+ e- + F~(cN)~~ (3-1) and the reverse reaction at the anode. Thus, the composition of the electrolyte solution does not change.
As discussed earîier, ions may migrate due to the potential gradient. To suppress this effect or make it negligible compared to diffusion and convection, a high concentration of sodium hydroxide (inert electrolyte) is used. This also enables the Schmidt number to be adjusted since diffusivity and viscosity of the solution depend on both temperature and NaOH concentration.
If the bulk concentration of the reacting species at the cathode, Cb, is known, an average mass transfer coefficient k can be calculated from Chapter 3 Integrated Moss Transfer in a 38 Pipe and a Sirn~lifledBifurcation Mode2
where llimis the measured limiting current, n is valency change in the reaction, F is Faraday's constant and A is the surface area of the cathode.
3.2.5 Electrical Circuit
The essential features of the electrical circuit are shown in Figure 3.8. A 12 volt car battery was used as a power source. Three potentiometers with resistance 10, 8 and 8 ohms were used to regulate the voltage applied to the electrodes. The current through the circuit was obtained by measuring the potential drop across a standard one-ohm resistor with a data acquisition system by which the analog signals were converted into digital signals for cornputer processing.
3.2.6 Flow Loop and Experimental Procedure
An overall schematic diagram of the experimental flow system is given in
Figure 3.9 The experimental set-up consisted of two 200 litre polyethylene tanks on top of each other, one 20 litre polyethylene container, two rotameters, a 1/8
HP centrifuga1 pump, a glass mil and two bal1 valves.
The electrolyte was circulated through the closed-loop flow system. The Ruid was pumped from the lower tank to the upper tank çontinuously. The level of the Chapter 3 Integruted 1tfass Trunsfer iit a 39 Pipe and a Simplified Bifimatiori Mode1
Figure 3.8: Electrical circuit.
Anode Cathode Anode 250 un t5m 25crn 30 cm I I- W-H ---C
Figure 3.9: Schematic diagram of electrolyte flow system with the aortic
bifurcation. Chapter 3 Integra?ed Mass Transfer in a 40 Pipe and a Simplified Bifurcation Model electrolyte was kept constant by overfiowing the fluid through a PVC pipe to the lower tank and hence a constant head flow system was provided to generate the given flow rates. A 2.5 m long PVC pipe in front of the cathode was used to ensure fully developed flow conditions at al1 Reynolds numbers. A special connector was designed and fabricated for the bifurcation model to collect the solution flowing through the branches. This connector was machined out of a
P.V.C. rod 53 mm in diameter and 80 mm in length. Two % " pipe nipples were screwed to one side of the connector while the other side had an opening with the same size as the extemal diameter of the pipe sealed with rubber O-rings.
The branches of the bifurcation rnodel were connected to the pipe nipples by means of 1/2 '' flexible Vinylon braided tubings tightened by hose clamps. The temperature of the solution was controlled by a glass coi1 heat exchanger inserted in the lower tank. Pnor to each set of experiments, electrodes were nnsed with dilute sulfuric acid and then by acetone and, if necessary, followed by cathodic activation in 1 M sulfuric acid.
The solution was deoxygenated before each run by nitrogen bubbles. The concentrations of potassium ferricyanide and sodium hydroxide were checked by volumetric analysis (Kolthoff and Belcher, 1957) prior to each nin. To obtain
Schmidt numbers of 5655 and 3100, the runs were camed out at temperatures
17 and 28 degrees Celsius, respectively. The concentrations of K3Fe(CN), ,
K4Fe(CN), and NaOH were 1.43 x 10", 1.9 x 1O-* and 3 M, respectively. For the lower Schmidt numbers, 2470 and 1310, a concentration of 1M sodium hydroxide was used with K3Fe(CN)sand K4Fe(CN)6 at 3.125 x lo3 and 4.167 x 10j M, Chapter 3 Integruted Muss Transfer in a 41 Pipe and a SimpliffedBifurcation Mode2 respectively. All physical properties. together with Schmidt numbers, were calculated based on data provided by Bourne et. al. (1985).
Figure 3.10: Polarization curves for pipe (one anode) and bifurcation: Re=l109, Sc=3100.
For the case of the bifurcation which is a larger cathode, the ohmic potential drop in the column of fluid facing the cathode may be greater than the width of the diffusion controlled plateau that would be established if the ohmic potential was negligible. Consequently, it will not be possible to obtain diffusion controlled conditions over the whole surface because the potential difference at the end of cathode closer to the anode will have exceeded the hydrogen overvoltage before the ion concentration on the other end has falien to zero. Under these conditions a limiting current cannot be obtained. Also, since a special connecter was Chapter 3 Integrated Mass Transfer in a 42 Pipe and a Sîmvlified Bifurcation Mode1 needed to collect the solution flowing through two branches, and the length of this connecter was much longer than the regular connectors used with the straight pipe as the cathode, the ohmic potential drop was accentuated. In this case, placement of another anode upstream of the cathode reduced the resistance since the two anodes were in parallei. As a result, acceptable limiting currents were obtained in the entire range of Reynolds numbers. Figure 3.10 illustrates the polarization curves for the bifurcation with ho anodes and the straight pipe with one anode at Re=1109, and Sc=3100. Chapter 3 Integrated Mass Transfer in a 43 Pipe and a SimpIified Bifurcation Model 3.3 Results and Discussion
3.3.1 Pipe Data
For the straight pipe test-section (cathode), the use of one or two anodes
led to noticeable differences of the Sherwood number only for laminar flow.
Figures 3.1 1 and 3.12 show the results for Sc=1310 and Sc=2470, respectively.
With two anodes versus one downstrearn anode, ferricyanide ions produced by
the upstream anode lead to an increase in the near-wall concentration of the ions
at the cathode, and in turn tu a higher flux or Sherwood number. This effect is
significant at low Reynolds numbers under laminar flow, but for transitional or
turbulent flow, the excess ions in the near-wall region are rapidly removed and
diluted by the intense mixing and large wall velocity gradients. Also for laminar
Row, the effect of the upstream anode on the Sherwood nurnber was lower at
higher Schmidt numbers because of the thinner concentration boundary layer.
Figure 3.13 shows the effect of Reynolds and Schmidt numbers on the
Shennrood number for the straight pipe. All mns were conducted with one
downstream anode. Three distinct zones are observed at al1 Schmidt numbers: a
laminar zone, a transitional zone for Re=2100 to 3000, and a fully turbulent zone
for Re>3000. It is also obsewed that, as the Schmidt number was increased, transition to turbulent mass transfer occured at progressively higher Reynolds
numbers; just beyond 2000 for Sc=1310 and at about 2700 for Sc=5655. The
momentum boundary layer is fully developed by the long entrance length
provided by the PVC tube. However, the developing mass transfer boundary layer is thinner at higher Schmidt numbers. Thus, the effect of any turbulence in Chapter 3 Integraied Mass Transfer in a 44 Pipe and a Simplifted Bifurcation Model
la0 1- 1OWO M
Figure 3.1 1 : Shewood number vs Reynolds number for the pipe: 1 and 2 anodes, Sc=1310.
taa
Figure 3.12: Shemood number vs Reynolds number for the pipe: 1 and 2 anodes, Sc=2470. Chapter 3 Integrated Mus Transfer in a 45 Pipe and a Simplified Bifurcation Model the near-wall region will not be felt until turbulence reaches sufficiently close to the wall, and that happens at progressively higher Reynolds numbers. That is, increasing the Reynolds number reduced the thickness of the viscous sublayer thereby increasing the influence of turbulence on the relatively thin concentration boundary layer.
The laminar data for the pipe are correlated with a standard deviation of 3 percent by equation (3-3). It should be noted that since there were no experimental data obtained for different (dll) values, the 113 exponent was obtained from the Leveque solution (Leveque, 1928).
The Leveque solution, equation (34, is shown in Figure 3.13 for the highest
Schmidt number only.
The data are slightly lower over the full range of Reynolds numbers for laminar flow. Agreement at the other Schmidt numbers is similar as the Sc-exponents are virtually identical.
Turbulent data for Re>3300 are represented, with a standard deviation of
3%, by
Sh = 0.014~e~."~c~.~~ (3-5)
For fully developed turbulent flow, Pickett et al. (1974) reported that fully developed mass transfer conditions occur at about twelve diameters from the entrance and can be represented by the Chilton-Colbum analogy (1934) Chapter 3 Integrated Muss Transfer in a 46 Pipe and a Simplified Bifurcation Model
Sh = 0.023 R~O.~ (3-6)
The data are higher than the Chilton-Colbum analogy result, shown in Figure L 3.1 3, again, for the highest Schmidt number only, since --8 in our experiments, d which results in developing mass transfer conditions. Also, the appropriate
Reynolds nurnbers range for the Chilton-Colbum analogy is greater than 10,000 while Our experimental correlation has been obtained frorn Reynolds numbers up to 8,000. Lin et al. (1953) in their correlation, suggested the exponents to be 718 and 113. respectively. Our data are also in good agreement with the smooth surface results of Zhao and Trass (1997),given as
Sh = 0.01 3 R~O.~SCO-~~
Figure 3.13: Shennrood number vs Reynolds number for the pipe (one anode) at four different Schmidt numben, compared with the Leveque solution and the Chilton- Colburn analogy, at Sc-5655. Chapter 3 In tegrated Mass Transfer in a 47 Pipe and a Simplified Bifurcation Model 3.3.2 Bifurcation Data
For the bifurcation, the effects of Reynolds and Schmidt numbers on the
Sherwood number are presented in Figure 3.14. When flow impacts the apex of the bifurcation, a new concentration boundary layer develops along the inner walls of both branches that would induce a higher rate of mass transfer on this side of the branches. Four distinct regions are observed, most noticeable at the lowest Sc of 1310. The Sherwood number shows a linear dependence on Re up to about Re=1500. Between Re=1500 and 2100, the rate of increase of mass transfer was reduced. Along the inner wall, we expect the impinging effect.
However, along the outer wall, it appears that fiow separation rnay have been initiated at about Re=1500 or, at least, the velocity gradients at the wall would have become less steep. Further, the rate of reduction of mass transfer appears to be Schmidt number dependent, with the largest reduction seen at the iowest
Sc. Perhaps, as only modest flow separation occurs, the influence of separation may be observed more readily at a lower Sc due to the thicker concentration boundary layer.
As expected, at Re>PIOO, transition to turbulence is triggered and, thus, the Sheiwood number shows a stronger dependence on the Reynolds numbers.
While a distinct rise is observed beyond Re = 2100, the data trend in the transition zone is not noticeably different from that for the fully turbulent regime.
Transition occurs over a broader range of flow conditions since the Reynolds number in the branches is 0.83 of that in the trunk. Also, there is a cornplex Chapter 3 Integrated Mass Transfer in a 48 Pipe and a Simpîified Bifurcation Model
100
Figure 3.14: Sherwood number vs Reynolds number for the bifurcation (2 anodes) at four different Schmidt numbers.
interaction between transition, incipient turbulence and probable flow separation at the outer walls of the branches.
The laminar flow data for Re4500 gave the following correlation, with a standard deviation of 4 percent,
(3-8) where d and L are the host artery's diameter and length, which are the same as d and L for the pipe. For turbulent flow at Re>3300, with the same standard deviation,
S h=O.Oiï ~e~*~~ SC^.^^ Chapter 3 Integrated Mass Tramfer in a 49 Pipe and a Simplified Bifurcation Mode1 In general, the impinging effect along the inner wall of the branches increases the
rate of mass transfer in the laminar zone (Re up to 1500). This may explain the slightly higher slope of the data points compared to the data in the laminar
regime of the straight pipe (Figures 3.11 to 3.13), as also reflected in the respective Re-exponents of 0.43 and 0.38. Further, the transition zone is broader for the bifurcation or, in other words, transition from laminar to turbulent flow, as well as mass transfer behaviour occurs more smoothly than in the straight pipe.
To estimate the mass transfer coefficients in a daughter branch, the current in the branch was taken as one half the difference between the current in the bifurcation and that in the straight pipe. This was possible because the dimensions of the pdmary part of the bifurcation or main vessel matched those of the straight pipe, and since the branches were identical. As the connecter from the branches to the main flow was made as symmetncal as possible, it may reasonably be expected that the flow splits equally between the two branches and, hence,
For the branches, db=l 1.2 mm, the average diameter of the daughter vessels.
Figure 3.15 shows the dependence of Sh on Re for the four Schmidt numbers.
The trend in the data is sirnilar to that seen for the bifurcation, except that the four regions are more distinct since the influence of the potential separation region is restricted to the daughter branches. In particular, the data for Sc=1310 show a decrease in the Sherwood number with the Reynolds nurnber over the Chapter 3 Integrated Mnss Transfer in a 50 Pipe and a Simplified Bifurcation Mode1 Re range frorn 1400 to 2000, likely because the separation region grows as the
Reynolds number is increased.
Figure 3.15: Sherwood number vs Reynolds number for the branches at four different Schmidt numbers.
In each branch, a new concentration boundary layer developed along the
inner wall, the wall adjacent to the apex. In addition. because of the curvature, the flow streamlines are no longer parallel to the vessel wall. The velocity field distal to the apex is 3dimensional with an overall increase in wall shear stress due to the spiraling of high inertia fluid around the circumference of the branch.
Further, the wall shear stress (rw) increases because the branch diameter was roughly 60% that of the parent vessel and since T,,,,=Q/? for Poiseuille flow. Chapter 3 In tegruted MQSSTrunsfer in a 51 Pipe and a Simplified Bifurcation Model These three effects help explain the dependence of the Shenvood number on the
Reynolds number.
After the gradua1 transition, Re>3000, the data follow the pattern observed in Figure 3.14 for the whole bifurcation mode1 since the flow is now fully developed turbulent flow. Chapter 3 Iniegrated Mass Transfer in a 52 Pipe and a Simplified Bifurcation Model 3.4 Conclusions
A novel technique for the fabrication of nickel flow models was used to produce sections of a straight pipe and a Y-bifurcation. The integral rnass transfer coefficients in these test sections were determined by the electrochemical technique. Mass transfer coefficients in the pipe were in agreement with published results for both laminar and turbulent Rows. With increasing Schmidt nurnber, the Reynolds number at which the effect of turbulence on mass transfer is observed, iç delayed because of the decreasing mass transfer boundary layer thickness relative to the viscous sublayer. Further, for laminar flow, an upstream and a downstream anode led to higher mass transfer at the cathode wmpared to a single downstream anode because of the additional reacting ions in the near-wall region produced by the upstream anode.
The mass transfer patterns in the bifurcation were different because of the potential onset of flow separation along the outer wall and formation of a new boundary layer along the inner wall of the branches. Also, transition from laminar to turbulent flow as well as the corresponding mass transfer behavior take place over a broader range of Reynolds numbers because of a difference in the trunk and branch Reynolds numbers. 4.1 Introduction
In heat or mass transfer to and from flowing systems one distinguishes between the entry region of a conduit and the fully developed region which occurs further downstream. In the fomer. temperature and concentration changes are confined to a thin boundary layer near the wall; in the latter, these variations penetrate into the core of the fluid.
Heat and mass transfer coefficients in the entry region are well established for steady laminar flow. They were first studied by Leveque (1928) Cha~ter4 Transient Mass Transfer 54 and can be cast in the dirnensionless form for constant temperature or concentration boundary conditions as:
Sh = 1.615 (Re Sc (dl~))'" (4-1
Nu = 1.615 (Re Pr (dl~))'~ (4-2) For the transient period which precedes attainment of the steady state, no experimental data and little theoretical work can be found in the literature, although the importance of transient, or more generally unsteady behaviour is undeniable in many areas of transport phenornena. For example, in a pulse- plating operation, the current is tumed off periodically for short intervals to allow the concentration of metal ions to increase at the electrode surface. This procedure is useful in producing an improved deposit morphology. There are many other industrial applications where a knowledge of transient behaviour is essential for the design of control systems.
The occurrence of unsteady mass transfer is also seen in biological systems and has been discussed by Basmadjian et al. (1997) in the context of blood coagulation. "The First Hundred Seconds" and "The Breathing Reactof are two important cases of unsteady physiological mass transfer. The former, which is of particular interest here. represents the changes immediately after the coagulation cascade is triggered by a reactive event at the blood/wall interface.
The concentration boundary layer undergoes a rapid growth and, since the fate of the coagulation process may be determined by the wall concentration changes
(transients), it would be of importance to have some insights into this interval.
Furthemore, transients may give nse to large overshoots and lag times in the Chavter 4 Transient Mus Transfer 55 next phase of the coagulation cascade called "cornmon pathway" and, hence, both the steady and unsteady states need to be scnitinized (Baldwin et al.,
1994).
In The Breathing Reactof, flow pulsatility will culminate in an unsteady process. The mass transfer coefficient and the concentration boundary layer undergo periodic variations as the flow rate increases, diminishes and reverses itself during the course of a blood pulse cycle.
Soliman and Chambre (1967) treated the tirnedependent Leveque problem. Basmadjian (1990) in some parts of a detailed review, discussed unsteady state mass transfer in physiological systems. These two references are of particular reievance to the present work in which transient heat and mass transfer in the entry region of a circular pipe are examined both expenmentally and theoretically. Chapter 4 Transient Mass Transfer 56
4.2 Theory
As previously mentioned, the time-dependent boundary layer growth in the
Leveque region has been analyzed by Soliman and Chambre (1967). They generalized the steady state Leveque problem to include time dependence for the surface heat flux due to a time-step in the surface temperature and for the case of the surface temperature due to a tirne-step in the wall heat flux. Their solution for the heat flux when the Prandtl number, Pr, is very large was given in the following expanded form in powers of a dimensionless parameter t':
where
As r* O, this equation reduces to
Equations (4-3) and (4-5) are identical to within 10% of q,, (Soliman and
Chambre, 1967).
Now, one may use the analogy between heat and mass transfer to establish
in which Cb and C, are the concentrations of the reacting ion in the bulk of the solution and at the wall, respectively.
In a mass transfer controlled regime it follows that Chapter 4 Transient Mass Transfer 57
To evaluate the transient mass transfer coefficients, k, in this equation, we
introduce the steady state value, k,, obtained from the Leveque solution for the
local position x:
k d Ud2 In Sh = = 1 -076(Re Sc (dix))'" = 1.O76 (4-8) D (-)Dx
Inserting the wall shear stress, t,, from the Poiseuille solution for pipe flow
into equation (4-8) gives:
and, hence, the steady state mass transfer coefficients can be expressed as:
Replacing k, in equation (4-7) finally gives
This simplified version of equation (4-3) indicates that the transient mass transfer
coefficients for the laminar regime are. in contrast to the steady state values,
independent of velocity and distance. Its heat transfer counterpart is Chapter 4 Transient Mass Tramfer 58 where h and a are the heat transfer coefficient and thermal diffusivity,
It should be noted that the transient mass transfer coefficient k in equation (4-12) is identical to that derived by Higbie (1935) in his "penetration theov for mass transfer from bubbles rising through a stagnant liquid.
The reason is that both of these otherwise dissimilar processes are transient and diffusion controlled.
An expression for film thickness under transient conditions can be derived directly from equation (4-1 2):
Until local steady state is reached, 6 depends on diffusivity and time only, and is independent of both velocity and position.
The transient solution can be equated with the integrated Leveque solution over the length L to obtain the time required to reach steady state.
where the right hand side represents the mean integral mass transfer coefficient over the conduit length L. Solving equation (4-1 6) for time t. we obtain: Chapter 4 Transient Mass Transfer 59
However, since the transient solution is only valid to within 10% of the steady state and the approach is subsequently slower and more asymptotic, equation
(4-17) will slightly underestimate the time required to reach steady state.
It shoutd be noted that the transient period depends on length L. This reinforces the fact that the thin concentration or temperature boundary layers near the entrance region are built up rapidly, with the later, thicker ones taking a longer time to reach the steady state concentration or temperature profiles. Cha~ter4 Transient Mass Tramfer 60
4.3 Experimental Procedures
The electrochemical technique was used to measure mass transfer coefficients between nickel surfaces and the ferri-ferrocyanide solution. A more detailed descn'ption can be found elsewhere (Selman and Tobias, 1978).
4.3.1 Flow Systern
The experimental set-up shown in Figure 4.1 foned a closed loop frorn the 3 HP centrifugai pump through PVC pipe with an intemal diameter of 38 mm, to the nickel test section, two rotameters and a 200 litre polyethylene tank. An inverter was used to adjust the RPM of the pump motor and, hence, to control the flow rate. In addition to the inverter, a bypass loop was employed to control the flow rate. A 3.5m long PVC pipe before the cathode was used to ensure fully developed fiow conditions at ail Reynolds numbers. The temperature of the solution was controlled by a coi1 heat exchanger inserted in the tank.
The solution was deoxygenated before each run by nitrogen bubbles. The concentrations of potassium femcyanide and sodium hydroxide were checked by volumetric analysis (Kolthoff et al., 1957) prior to each m. To obtain Schmidt numbers of 4200 and 2650, the runs were carried out at temperatures of 18 and
28*C,respectively. The concentrations of K3Fe(CN)6,K4Fe(CN)6 and NaOH were
4 x lo4, 6 x IO-* and 2.5M. respectively. Al1 physical properties, together with
Schmidt nurnbers, were calculated based on data provided by Boume et al.
(1985). Chaoter 4 Transient Mass Trunsfer 61
4.3.2 Electrical Circuit
An 8-channel, 12-bit, analog input DAS-8 board was used to measure the voltage across the electrodes as well as the current in the cell. A 0.05 ohm current shunt was placed in series with the cell in order to measure the cell current. The voltage and current signals were isolated by using two separate signal conditioning modules. The electrode voltage signal was input to a -5V to
5V analog inpuVoutput module, which relayed an isolated signal to the DAS8 board. On the other hand, the cell current signal was input to a -100mV to 100mV analog input, -5V to 5V analog output, which relayed an amplified isolated signal to the DAS-8 board. The voltage and current data were sampled at 10 Hz for transient response experiments, and at 1/30 Hz for obtaining polarkation curves.
An overall schernatic diagram of the electrical circuit is given in Figure 4.2.
As there was a lag time between the time the data acquisition system started and the time the power supply was switched on, the first non-zero, positive current was assigned to 0.1 seconds, the srnallest time reading, to synchronise data obtained from al1 runs. Chapter 4 Tramieat Muss Transfer 62
Figure 4.1 : Schematic diagram of electroiyte system.
Figure 4.2: Electric circuit. Chapter 4 Transient Muss Transfer 63
4.4 Results and Discussion
4.4.1 Preliminary Data
Two sets of experiments have been carried out. The main objective of the first set was to obtain steady- state values of mass transfer coefficients in a srnooth, straight pipe, 18.6 mm in diameter and 150 mm in length, and a Y- bifurcation model resembling a human aortic bifurcation.
Results of that earlier work revealed two distinct regions during the transient period as shown in Figures 4.3 and 4.4. where the mass transfer coefficient k is plotted versus time. The first region with a lower slope is followed by a second one which then gradually approaches the steady state values. The point(s) at which the dope changes will hereafter be called the hump. Similar results were obtained at four different Schmidt numbers for both laminar and turbulent fiows.
Soliman and Chambre (1967) analyzed the corresponding transient heat transfer case. As expected, a general monotonic reduction of the transfer rate is predicted with no hump and no inflection point, as was clearly observed for electrochemical mass transfer in Figures 4.3 and 4.4. Since heat and mass transfer are substantially govemed by the same mechanism. a difference in the boundary conditions may justify this discrepancy. Soliman et al. imposed their boundary conditions at time t=O. However, in our mass transfer experiments, while the cell voltage was imposed at t=O, the canesponding wall boundary condition of zero ferricyanide concentration would bo established only after a finite, albeit brief period during which the reacüon at the cathode is the rate Cha~ter4 Transient Mass Transfer 64
Figure 4.3: Mass transfer coefficients vs time at different Reynolds numbers, Sc=2470 (earlier work).
Figure4.4: Mass transfer coefficients vs time at different Reynolds nurnbers, Sc=U00 (earlier work). Chapter 4 Transient Mass Transfer 65 determining step. Towards the end of this period, i.e. when the wall concentration approaches zero, the slope of the experimental k vs t plot increases and approaches or coincides with the theoretical solution for diffusion-controlled conditions. It should be noted that, as a prerequisite for the electrochemical technique, the rate of reaction must be higher than the rate of mass transfer to enable the cathode surface to consume al1 ferricyanide ions irnmediately upon reaching the surface. This should provide a wall concentration equal to zero, for voltages anywhere on the plateau of the polarization curve.
To test the above-mentioned hypothesis, a new set of experiments with a higher frequency of data collection was carried out. However, since the straight pipes and the bifurcation had been cut for local mass transfer rneasurements. an available cathode, 38 mm in diameter and 305 mm in length, and a correspondingly larger anode were used to perform the new experiments. Details regarding the electrodes and their fabrication methods are given elsewhere
(Zhao et al., 1997).
4.4.2 Laminar Flow Results
In these experiments, the polarization curve was checked at each
Reynolds number (Figures 4.5.A to 4.10.A) and the voltage across the electrodes as well as the current were recorded simultaneously with a frequency of 10 Hz, ten times higher than previously used. Based on each polarization curve, different voltages along the plateau were applied and the cunents recorded. The graphs of cell voltage vs time showed that the voltage across the cathode and Chapter 4 Transimt ikfass Transfer 66 anode reaches the final value in less than 1/10 of a second. for al1 experiments
(Figures 4.5.8 to 4.10.8). Consequently, cell voltage lag time is not a cause of the hump.
4.5.8: Cell voltage vs time (Sc=2650, Re=473). Cha~ter4 Transient Mass Transfer 67
4.6.A: Polarization curve (Sc=2650, Re=1314).
4.6.8: Cell voltage vs time (Sci2650, Rer1314). Chapter 4 Transient Muss Transfer 68
4.7.A: Polarization curve (Sc=4200, Re=488).
4.7.8: Cell voltage vs time (Sc=4200, Rez488). Chapter 4 Transient Mass Transfer 69
1MO - 1
m.0 I - 4
53.0 .- 1
70.0 .
a0 ., I I s 50.0 .- x i* 40.0 4 I I 30.0 - a -* 1 O* l 20.0 4 IO. I I
10.0 .. I - *io I
0.0 I I 0.0 W)110 1000.0 tm.0 m.0 zwwl.0 Wmv)
4.8.A: Polarization curve (Sc-4200,Re=760).
4.8.6: Cell voltage vs tirne (Sc4200, Re=760). Chapter 4 Transient Mass Transfer 70
4.9.A: Polarkation curve (Sc=4200, Re=899).
4.9.B: Cell voltage vs tirne (Sc=4200, Re=899). Chapter 4 Transient .'Mass Transfer 71
4.1 O.A: Polarization curve (Sc=4200, Re4616).
4.1 O.B: Cell voltage vs time (Sc=4200, Re=1616). Cha~ter4 Transient Mass Tramfer 72
For laminar flow, examples of mass transfer coefficients at different
Reynolds and Schmidt numbers are shown in Figures 4.11 to 4.16. In these
Figures, the line of transient behaviour obtained from the theory, equation (4-12), and the steady-state Leveque solution (1928) are depicted. The time when the hump is observed and that required to reach steady state, from equation (4-17), are also shown. In the initial, reaction rate-controlled region, since the concentration at the wall is not zero, the curves represent the ratio k(Cb-C,,,)/Cb rather than k; thus, the values are always lower than the transient theory line.
However, after the hump, when the concentration at the wall reaches zero, the current becomes the limiting current and the ordinate correctly represents the mass transfer coefficients.
It was observed that the first transient values were higher at higher applied voltages. Also, the development of the hump was delayed as the voltage across the electrodes decreased, consistent with the decreased values of k(Cb-C,,,)/Cb;
in other words, the length of time during which the surface reaction rate govems the process becomes longer as the rate is decreased. The dependence of the level of the first transient and the resultant location of the hump on voltage
refiects the dependence of the rate of the surface reaction on cell voltage.
Based on these observations, the following is proposed. When the voltage is applied to the electrodes, a mass transfer process is initiated with the formation of the concentration boundary layer. Recalling equation (4-E), one would expect, for laminar flow during the transient period, the boundary layer Cha~ter4 Transieni Mass Transfer 73 thickness to grow at a rate of tIR, independent of velocity and distance x along the cathode.
Figure 4.1 1: Mass transfer coefficients vs time at different applied cell voltages, compared with the Leveque solution and the theoretical transient line (Sc=2650, Re=473).
Figure 4.12: Mass transfer coefficients vs time at different applied cell voltages, compared with the Leveque solution and the theoretical transient line (Sc=2650, Re4314). Chapter 4 Transient AIMassTrans fer 74
Figure 4.13: Mass transfer coefficients vs time at different applied cell voltages, compared with the Leveque solution and the theoretical transient line (Sc-4200, Re=488).
Figure 4.14: Mass transfer coefficients vs üme at different applied cell voltages, compared with the Leveque solution and the theoretical transient line (Scr4200, Re=760). Chapter 4 Transient Mass Transfer 75
Figure 4.15: Mass transfer coefficients vs time at different applied cell voltages, compared with the Leveque solution and the theoretical transient line (Sc=4200, Re=899).
o. 1
Figure 4.1 6: Mass transfer coefficients vs time at different applied cell voltages, compared with the Leveque solution and the theoretical transient line (Sc=4200, Re=1616). Cha~ter4 Transient Mass Tramfer 76
Since this layer is initially very thin, the rate of mass transfer would be very high and the electrochemical process will be reaction rate limited. The initial femcyanide concentration at the cathode surface is the same as in the bulk Cb. It then falls rapidly but remains above zero at the cathode surface until the hump.
Thus. the initial measured curent (before the hump) is largely proportional to the rate of the reaction consuming the ferricyanide ions. It is not constant but is decreasing slowly due to the diminishing wall concentration. The slope of the line represents the rate of change of the reaction rate with respect to time.
Simultaneously, the concentration boundary layer is growing and increasing the resistance to the diffusion of ferricyanide ions towards the cathode surface. This results in a decrease in the rate of mass transfer, even though (Cb-
&) is increasing. These two phenomena continue until the two rates are equalized and the surface concentration drops to zero. From this point on. the rate of reaction is higher than the rate of mass transfer and the process is mass transfer controlled. After the hump, transient mass transfer is observed until the curve approaches the steady state solution. In the second transient region, mass transfer behaviour matches that for heat transfer with a constant wall temperature.
As shown in Figures 4.11 to 4.16, as the voltage across the electrodes is increased, the rate of reaction increases as well, with the k/v value at any one time remaining substantially constant, suggesting that the reaction rate varies linearly with cell voltage over the polarization plateau. Hence, the line for the reaction rate-controlled data at higher voltages intersects earlier with the line for Chapter 4 Transient Mass Trunsfer 77 the mass transfer controlled transient data resulting in the earlier observation of the hump.
A dimensional analysis was also performed to obtain empirical transient equations. Three nondimensional groups were introduced: Re, Sc and t', the latter representing non-dimensional time and given by
The results are wrrelated with a standard deviation of 3 percent. In the laminar regime
Sh = 0.46~~~-~~~~~.~~t(4-1 9)
for al1 points from the hump to the point(s) at which deviation towards steady state values is observed.
It is apparent that k is almost proportional to DI^)'' and that the impact of velocity or Reynolds number on k is negligible. Diffusivity and tirne are the main factors infiuencing transient mass transfer coefficients in the laminar regime, just as predicted by equation (4-1 2).
4.4.3 Turbulent Flow Results
Similarly, for each Reynolds number, polarkation curves were obtained and both cell voltage and current were recorded simultaneously. (Figures 4-1 7 to
4-20). Chapter 4 Transient Mass Transfer 78
Figure 4.17.A: Polarization cuwe, turbulent regime, (Sc=2650, Re=4891).
Figure 4.17.8: Cell voltage vs time, turbulent regime, (Sc=2650, Re=4891). Cha~ter4 Transient Mass Transfer 79
Figure 4.1 8.A: Polarization curve, turbulent regime, (Sc=2650, Re=5842).
0.0 * O 1 2 3 4 5 6 7 8 9 1 O t (sec}
Figure 4.1 8.8: Cell voltage vs time, turbulent regime, (Sc=2650, Re=5842). Chapter 4 Transient Mass Transfer 80
Figure 4.19.A: Polarization cunre, turbulent regime, (Sc=4200, Re=4396).
Figure 4.19.8: Cell voltage vs tirne, turbulent regime, (Sc=4200, Re4396). Chapter 4 Transieni Mass Transfer 81
Figure 4.20.A: Polarization curve, turbulent regime, (Sc=4200, Ren6541).
Figure 4.20.8: Cell voltage vs time, turbulent regime, (Sc=4200, Re=6541). Cha~ter4 Transient Mass Transfer 82
Some exarnples of data obtained for turbulent flow are shown in Figures 4.21 to
4.24. The same phenornena are observed as in laminar flow, with the main
difference being that, for the same cell voltage, the hump occurs later. The
reaction kinetics-controlled (first transient) values are comparable to those
observed in laminar flow. Once diffusional transfer is dominant, the rates for both
transient and steady state are, as expected, higher than their counterparts in
larninar fiow. Thus, it will take longer for kinetics to catch up with the higher rate
of diffusional transfer and the humps occur later. While the first transient period is
longer, the second transient, the interval consistent with heat transfer results, is
much shorter, i.e. the steady state condition is reached earlier. For laminar flow, this intewal is 50-100 seconds or longer; for turbulent flow, it is 20-30 seconds,
with shorter values for higher Reynolds and lower Schmidt numbers. The latter
conditions give higher steady state k-values which are reached eariier by the transient.
To obtain a correlation for the turbulent flow regirne, the three above-
mentioned non-dimensional groups were used again. The experimental results
up to a Reynolds nurnber of 7000 were correlated with a standard deviation of
3% by the expression
Sh = 0.09~~~-~~~~~.~'t*-O-" (4-2 1)
As in the laminar case, the range of data included in the correlation is from the hump to the point at which deviation towards steady state values is observed.
Equation (4-22) is also shown in Figures 4.21 to 4.24. Chapter 4 Transient Mass Transfer 83
o. 1 1 10 t (ri Figure 4.21: Mass transfer coefficients vs time (turbulent regime), cornpared with the empirical correlation (Sc=2650, Re=4891).
0.1 1 10 100 (4
Figure 4.22: Mass transfer coefficients vç time (turbulent regime), cornpared with the empirical correlation (Sc=2650, Re=5842). Chapter 4 Transieni Mass Transfer 84
Figure 4.23: Mass transfer coefficients vs tirne (turbulent regime), compared with the empirical correlation (Sc=4200, Re=4396).
Figure 4.24: Mass transfer coefficients vs time (turbulent regime), compared with the empirical correlation (Sc=4200, Re=6541). Chapter 4 Transient Mass Transfer 85
Figure 4.25: Mass transfer coefficients vs time at different Reynolds numbers (turbulent regime), Sc-4200.
Figure 4.26: Mass transfer coefficients vs time at different Reynolds numben (turbulent regirne), Sc=2650. Chapter 4 Transient Mass Transfer 86
Unlike larninar transient mass transfer, the Reynolds number (or velocity) does play an important role in turbulent transient mass transfer. It is also apparent that, as for laminar transient conditions, the rnass transfer coefficient k depends on (~lt)'", i.e. the correlation lines are paraliel to each other. Additional turbulent data are given in Figures 4.25 (Sc=4200) and 4.26 (Sc=2650) for different Reynolds numbers. All have been used for the correlation, equation (4-
21). Both confimi the previous observations, viz. the long first. kinetically- controlled transient, the short diffusion-controlled transient, a modest Re effect on the second transient and the stronger Re effect observed previously
(Mahinpey et al., 2000; Zhao et al., 1997) for the steady-state data.
Figures 4.27 and 4.28 show the observed k-values, Le. k(Cb-&)/Cb, versus cell voltage at t=0.3 s as an example in the first transient period for al1 data O btained at Sc=4200 and Sc=2650, respectively. Ap parently the kinetically controlled transient is substantially unaffected by flow conditions, or by the rate of diffusional transfer, and depends mainly on cell voltage. The small upward bias of the turbulent data at Sc=4200 simply suggests that turbulent eddy motion more readily replenishes the ferricyanide ion concentration at the wall, thus maintaining the &value slightly higher than under laminar flow conditions.
Figure 4.29 shows the observed k values versus cell voltage at t=0.3 s at both
Sc=4200 and Sc=2650. Possibly, the lO0C difference in operating temperature between these two sets of nins is the cause for higher reaction rate constant values at Sc=2650 compared to those at Sc=4200. Chapter 4 Transient Mass Transfer 87
Figure 4.27: Mass transfer coefficient values at t=0.3 s vs applied voltage; laminar and turbulent results, Sc=4200.
Figure 4-28: Mass transfer coefficient values at t=0.3 s vs applied voltage; laminar and turbulent results, Sc=2650. Chapter 4 Transient Mass Transfer 88
- o.,,,
Figure 4.29: Mass transfer coefficient values at t=0.3 s vs applied voltage; laminar and turbulent results for both Sc=4200 and Sc=2650. Chapter 4 Transieni Mass Tramfer 89
4.5 Summary and Conclusions
Transient mass transfer coefficients have been measured in smooth pipes using the electrochemical technique. Two distinct transient regions were obsewed in al1 experiments, different from the single transient corresponding to heat transfer. In the first transient region, the process is controlled by the surface reaction rate. During this time the wall concentration of the reactant drops to zero and the boundary layer for diffusion-controlled mass transfer is being established. In the second region, diffusion of the reacting ions through the boundary layer governs the process. In the first region, the rate of change of the mass transfer coefficient with time is slow; as the rate speeds up in the second region, a "hump" is observed where the slope changes.
Support for this hypothesis was gained from measurements of reaction rate versus ceIl voltage (on the polarization curve plateau). A linear relationship was observed. The location of the hump, at which control is changed. moved toward later times as cell voltage was decreased.
In the mass transfer-controlled transient region, the transfer rate is proportional to (~lt)'", in accordance with the Higbie penetration theory and independent of velocity and distance. By analogy, the data also confin the heat transfer results.
The same phenornena were observed in transient turbulent flow. While the
kinetically controlled transient is comparable to its laminar counterpart, the transfer rate during the diffusion-controlled transient is higher than that in laminar flow. This results from the influence of velocity or Reynolds number on the Chapter 4 Transient Mass Transfer 90 transfer rate which varies with ~e'"(~/t)'".Those observations and the higher steady state transfer in turbulent fiow result in a longer kinetics-controlled first transient and a shorter mass transfer-controiled second transient period compared to those in laminar flow. 5.1 Introduction
In the past 20 years fluid mechanics has begun to be appreciated as a key factor, not only in understanding the etiology of the disease but also in serving as a regulator and modifier of cellular biology in both normal and diseased arteries.
There has been increasing amount of evidence that local hemodynamics is indeed an important factor in the atherosclerotic disease process and there is an important relationship between the disease processes and the characteristics of blood flow in the arteries. Atherosclerotic plaques are typically found at branch points and bifurcations (Stary, 1987). There has been speculation that the characteristics of flow in such regions may participate in the development of atherosclerosis. As mentioned earlier, the abdominal aorta is a site where Chapter 5 Flow Visualization Studies 92 atherosclerosis and the aneurysms can occur. The basic goal of this chapter is to determine hemodynamics of a simplified aortic bifurcation model using laser photochromic technique.
5.2 Experimental Methods
5.2.1 Aortic Bifurcation Flow Model Construction
The fabrication procedure for the bifurcation model was a rnulti-step molding technique which was developed for the construction of physical models of arterial geometries.
The first step consisted of making an aluminum core with the same dimensions as the mass transfer nickel model. In the second step, Sylgard 184 silicone elastomer kit (Dow Coming, Midland, MI) was used to generate a silicone mold. The Sylgard consisted of two components: the elastomer and curing agent. These two cornponents, respectively, were mixed together in the ratio of approximately 9:1 by mass and the mixture was then degassed. The machined aluminum core was placed inside an acrylic box and the mixture was poured up to middle of the box. After heating for 3 hours at 55-60°C,the silicone material cured around the cast. Subsequently, the mold release grease (Apiezon) was nibbed on the surface of cured silicone to facilitate detachment of the next cured silicone layer from the first half. The same process was repeated for the upper half to obtain the completed silicone rnold (Figure 5.1).
The third step involved fornation of a cast from the silicone mold. A low melting point metal (Cerrolow 117, Cern Metal Products, Bellefonte, PA), whose Cha~ter5 FIo w Ksualization Studies 93 chief component is bismuth, was used in the mold to produce the cast. Most molten metals when solidified in molds shrink and pull away from the molds, failing to reproduce fine mold detail. Because Cerro Alloys expand and push into mold detail when they solidify, they are excellent materials for duplication and reproduction processes. This characteristic of expansion andfor non-shrinkage, combined with low melting temperature (47'~)and ease of handling, are the major reasons for their extensive use.
Before pouring the low melting point metal into the silicon mold, a three piece alurninum core was machined and screwed together to provide support for the subsequent low melting point metal cast. This three-piece aluminum core resembled a smaller size of the original aluminum core. The aluminum core was then placed inside the silicon mold and screwed to the acrylic box and two ends of the mold were closed (Figure 5.2). The low melting point metal was poured into the annulus of the open end within an oven at approxirnately 55-60°C.Once filled, the silicone mold was slowly cooled in an ice water bath to promote metal crystallization from one end. This gradua1 cooling procedure further rninimized the formation of the small pits on the surface of the metal cast during the metal crystallization process. Using a soldering iron and extra metal alloy, a few remaining srnaIl pits were repaired. Eventually, a fine file, followed by a fine sand paper, was used to polish the cast and remove the remaining imperfections. Chaoter 5 Flow k"tsualrtatr*onStudies
Figure 5.1 : Photograph of the silicon mold.
Figure 5.2: Photograph of the silicon mold and aie aluminum core. Methyl Methacrylate Monomer UV LlGHT C I +++++++++ I
De-inhibited Methyl Methacrylate f
a- Fans De-inhibited Methyl Meaiacrytate
Figure 5.3: Schematic diagram of the polimerization stage.
Figure 5.4: The final product of the UV-transparent Plexiglas bifurcation
model. Chapter 5 Flow Visualization Studies 96
The fourth step involved polyrnerization of the methyl methacrylate
(Polysciences Inc.), an organic monomer, around the alloy cast. Methyl rnethacrylate has proved to be suitable for this purpose, as it is UV transparent and easy to mold. For use with laser photochromic flow analysis, it is necessary that the flow model be transparent to ultraviolet (UV) light. Since methyl methacrylate monomer contained inhibiting agents, a calumn half filled with De- hibit-100 macroreticular ion exchange resin was used to remove these agents. A resin column was made using a burette with approximately 40 mL of resin packed inside. Originally, the column was regenerated by passing methanol equal to 2.5 times the resin volume and ailowing it to flow through the resin column in one hour. The inhibited monomer then passed through the column at a rate of about four bed volumes per hour (16OmUhr). Subsequently, a catalyst,
2,2'-Azobisisobutyronitrile (Polysciences Inc.), was added in the amount of 20 mg per 100 mL of monomer in order to increase the rate of curing. The de-inhibited monomers were then added layer by layer (1-2 cm depth) to the transparent box containing the bifurcation model. This stage was performed in layers so as to minimize runaway reactions within the methyl methacrylate solution, which has been observed to cause defonations in curing polymer. Each additional layer was added before the underlying layer had fully cured so as to minirnize the occurrence of visible interfaces between layers. A UV light was placed below the box when the level of solution was below the suspended alloy cast. Once the level of polymer had reached the cast, the light was placed above the box so as to avoid a UV light shadow of the model which impeded polymerization within the Chapter 5 Flow Visualization Studies 97 shadow. Also, two fans were used to control the temperature of the solution due to heat evolved because of the polymenzation. The length of the final curing process was approximately three weeks (Figure 5.3).
The ffih and last step involved rnelting out the alloy at approximately 50°C in an oven. After the alloy was melted out, a residue of the alloy remained along the inner surface of the transparent rnold. The inner surface was then cleaned and polished with a knot of cotton cloth. Finally, a high pressure jet of polishing solution through the mold removed any further remaining alloy deposits. Figure
5.4 shows the final product of the UV-transparent Plexiglas bifurcation model.
5.2.2 Flow Visualization Techniques
Due to the difficulties of performing in vivo measurements, in vitro studies have played a major role in characterizing arterial velocity and shear stress fields. However, the determination of the wall shear stress for arterial blood flow in vitro conditions presents severe technical limitation. For example, methods such as laser Doppler (Ku et al., 1985; Shu and Hwang, 1991) and hot wire(Rittgers et al., 1978) anemometries provide low spatial resolution, because the velocity measurements are only made at a single radial location. As well, low temporal resolution makes it difficult to accurately estimate the instantaneous wall shear stress under non-periodic or complex Row conditions. In other words, at best, these techniques provide only an estimation of the average shear stress.
Flush mounted hot film techniques have proved to be effective in estimating wall shear stress, although they provided limited information regarding its direction Chanter 5 Flow Ksualization Studies 98
(Nandy and Tarbell, 1986). There are other techniques of flow visualization to
measure velocity profile such as dye injection (Clark, 1980), hydrogen bubbles
(Ahmed et al., 1984) and particle imaging velocimeter (Karino et al., 1977;
Ad rian, 1989). Attempts made with these flow visualization techniques usually
included invasive methods by introducing particles into the flow stream.
A unique flow visualization method called, the laser p hotochromic tracer
method, developed at the University of Toronto by Popovich and Hummel (1967),
enables detemination of instantaneous velocity profiles without disturbing the flow. In early studies, only a limited amount of quantitative measurements could
be obtained. Later improvements implemented by Ojha et al. (1988,1989) have
significantly reduced these shortwmings. Modification of the optics has resulted
in sharper and thinner traces, which has improved the overall accuracy in the velocity measurernents. In particular, because of its high spatial and temporal
resolution and its ability to provide simultaneously the velocity profile at multiple
sites in a Row channel, the technique enables a relatively large and complex flow field to be exarnined in a detailed manner in both fiuid mechanic and biofluid mechanic fields.
The ongoing development of flow measurement techniques has been continued to improve our understanding of fundamental and cornplex flows (Park et al., 1994; Couch et al., 1996). Chapter 5 Flow VîsualizationStudies 99
5.2.3 Flow Loop and Measurement System
The experimental apparatus and methodology have been described in detail elsewhere (Couch et al., 1996), and will be only briefly outlined here. All flow measurements in this study were performed by Richard Leask.
The setup of the experimental apparatus used for flow measurements is depicted in Figure 5.5. The test fluid contained 50 ppm of 1,3,3,-trimethyl-6-nitro-indoline-
2-spiro-2-2-benzopyrane (TNSB, an organic photochromic dye) dissolved in deodorized kerosene (Shell-sol 715). This chernical has a short UV absorption spectrum and is soluble in organic liquids. The aforementioned concentration provides sharp traces with suitable penetration depth. The dynamic viscosity and density of the solution at 20'~are 1.43 cP and 0.76 g cm3, respectively. W hen exposed to focused ultraviolet light from the laser, the indicator undergoes a reversible photochemical reaction, and tums opaque in less than 1 ps. The measurement system consisted of a pulsed nitrogen laser (VSL 337ND-10Hz, 3 mJ, h=337 nm. Laser Science Inc., Newton, MA), a convex lens (f = 380 mm), an electronic strobe, a high-resolution charge-coupled device (CCD) camera with a digital interface (Kodak MegaPlus 1400, Eastman Kodak, San Diego, CA), a macro lens, a digital frame grabber (Dipix XPG 1000, Ottawa, Ontario), a rnotorized stage and controller, and a programmable wavefom purnp (UHDC,
London. Ontario). A wmputer (PC compatible) was used to control and synchronize the individual components of the system, and to record and analyse the experimental data. Chapter 5 Flow Visualization Studies 1O0
Nitrbytli I~XI and optics .-\ --S. *
Figure 5.5: (a) The arrangement of cornponents and (b) the systern diagram of the photochromic flow visualization and measurement system (Couch et al., 1996). Chapter 5 Fhw Pïsual&ztion Studies 101
To ensure fully developed entrance conditions the trunk part of the acrylic model was extended 1 meter by a 18.6 mm diameter glass tube.
5.2.4 Data Analysis
The methodology relies on formation of "photochromic traces", obtained by firing a single laser beam at multiple locations across the test section. This was accomplished by mounting the test section on a traversing stage, and moving it with respect to the laser beam so that the entire region of interest in the test
section was sampled. In more detail, for each position of the test section, the following steps were taken.
An initiai image ("background imagen) of the model was acquired with a CCD
camera and image-processing board, and stored in a temporary buffer on the
imaging board.
An individual dye line was formed at an angle to the host tube axis using a UV
laser. One image of this line was acquired O ms after firing the laser. The
background image was subtracted from this image to isolate the dye line.
The line was extracted by image processing software and stored as the
"undisplaced trace" at that location.
Five more "background images" and individual dye lines were acquired in
tandem. lmaging of these dye lines was delayed by 10 ms after firing the
laser. After subtracting the respective background from each dye image, an
average image containing a single dye trace was obtained by pixel-wise Chapter 5 Flow Vr'sualization Studies 102
averaging of intensities. The resulting average trace was extracted and
recorded as the "displaced tracen.
The model was then moved to the next grid position (0.5 I O.Olmm displacement) and the entire process was repeated. After traversing the measurement region of the test section, the above procedure was repeated using a different angle between the host tube and the laser beam.
Wall shear stress was calculated from a set of digitized points. The digitized trace was fit to a third order polynomial using 50 near-wall digitized points on the trace, and the slope of the fmed polynomial evaluated at the wall was then calculated.
5.2.5 Error Estimation
There are several sources of error that affect the velocity measurernents obtained with the photochromic system and the grid reconstruction technique.
Details can be found elsewhere (Couch et al., 1996). A summary of the sources of error will be outlined briefly here as follows:
One source of the experimental error involved the alignment of the model with respect to the bearn. This error is minimized by the fact that the path of the beam is held fixed while the model is translated within the rneasurement plane during an experiment. In addition, fiducial marks on the surface of the test section provide a target for aligning the entry and exit of the beam. It is estimated that al1 traces lie within 15 pm of the measurement plane. Given 50 Fm as the dimension of the trace in this direction, alignment emr is negligible. Positioning the model Chapter 5 Flow Visualization Studies 103 with respect to the beam axis is achieved with the use of precision stepper motors and translation stages. Although the resolution of the stage is 2.5 Fm, the absolute reproducibility of the system over the travelling range is less than f10
Fm in each coordinate direction.
Another source of error was optical distortion resulting from the mismatch of refractive indices between the test fluid (1.43) and acrylic test section (1.49). In a previous evaluation of the photochromic method, the effects of optical distortion were determined to cause an error of less than +1% of the centerline velocity
(Ojha et al.. 1988).
Timing error is the other source of error. All timing is generated by three
16-bit digital counters. The flash delay error is determined by the accuracy of the counters (< 1 ps), and the variation in the triggering of the laser (< 1 ps) and electronic flash (< 1 ps). This results in an overall error of less than 0.2% in the flash delay of 2 ms used for the velocity measurements.
Overall,.the contributions of different sources of error depend on the scale and the complexity of the flow model. It is approximated that the overall eror in estirnating the mean axial velocity is about f1 %. As mentioned before, wall shear stress has been calculated based on the slope of the fitted polynomial to the digitized traces. The standard deviations refiect how well the third order polynomial fit to the velocity profile. As a result, an average of 540% uncertainty is attached to the wall shear rates data. The higher extreme belongs to the wall shear rates corresponding to the apex and hip area, due to additional difficulties in fMing the wall slope. Chapter 5 Fiow Visualization Studies 104
5.3 Results and Discussion
The laser photochromic tracer method was utilized to obtain hemodynamics in a Y-bifurcation mode1 along the plane of symmetry. The velocity field and wall shear stress patterns were characterized for steady flow conditions with a fully developed inlet flow at three Reynolds numbers; 500, 600. and 750. A limitation in pump output did not allow a Reynolds number higher than 750. Examples of the axial velociiy profiles at different locations in the plane of symmetry are shown for Re=500 in Figures 5.7 and 5.8. Figures 5.9 to 5.11 show the normalized wall shear rate along the outer and inner wall with respect to Poiseuille wall shear rate in the trunk for three different Reynolds numbers in the plane of symmetry. It should be noted that the Reynolds numbers are calculated based on the inlet diameter (18.6 mm). As the existing fluid possesses a constant viscosity at a given temperature, the Newtonian assumption is valid and thus the wall shear stress patterns follow exactly the same patterns as wall shear rates.
The abscissa in Figures 5.9 to 5.13 represents the distance dong the length of the outer wall of both the tnink and branch. By convention, zero is perpendicular to the apex in the plane of symmetry. The positive numbers manifest the axial distance from the apex towards the branch outlet. Negative numbers represent the distance from the apex towards the leading edge (tnink inlet) along the axis of the trunk. Due to field of view constraints, two runs were carried out; 1: from apex, parallel to the branch axis, to 40 mm downstrearn and Chapter 5 Flow Visualization Studies 1O5
2: from apex to 60 mm upstream parallel to host axis. Figure 5.6 presents the
measurement axes along the bifurcation model.
Figure 5.6: Diagram of the four zones and measurements axes along the bifurcation model.
Figure 5.7: Axial velocity profile 40.41 mm upstream of the apex in the trunk (RerJOO). Figure 5.8: Axial velocity profiles at different distances downstream of the apex at the branch towards the branch outlet (Re=500).
These results show no evidence of flow separation up to a Reynolds number of 750. It is also evident that the general behavior of the results at al1
Reynolds numbers is similar. lnner wall shear rate starts with a magnitude about
6.5 times of the trunk's and exponentially declines to 2.5 tirnes of the value in the trunk by less than 2 inlet branch diameters. In contrast, the wall shear rates ai the outer wall of the branch start about 1.2 times that of the inlet and increase linearly towards the values of the outer wall further downstream.
Due to symmetry of the model, flow was assumed to split equally between the two branches. However, since wall shear stress is directly proportional to the flow rate and inversely proportional to the radius cubed (T,,,,=W?), a global Chapter 5 Flow fisudization Studies f 07
increase in the wall shear stresses in the branches is expected compared to
those in the trunk.
The flow field may be categorized by four different sections as follows.
First, the flow field in the inlet region, trunk, is a Poiseuilie-like fiow with a
parabolic velocity profile. Figure 5.7 shows the velocity profile at Re=500, 40.41
mm upstream of the apex, where Poiseuille flow governs the flow field. Due to the limitations of the laser beam penetrations. it was not possible to capture the velocity profiles for the whole diameter in the trunk. About Il mm from the wall was the farthest distance that velocity profiles could be characterized.
The second zone. which is the junction region of the inlet channel to the branches, is called the transition zone. This region starts from about 30 mm upstream of the apex, where the cross section of the trunk changes from circular to elliptical, and ends where the trunk bifurcates into two smaller tapered branches.
The third zone is initiated from the bifurcation apex to two inlet branch diameters downstream. The shear rate on the inner wall starts about six times greater than the values at the outer wall at the beginning of this zone and declines exponentially.
The fourth zone encompassing the remainder of the branch, resumes a
Poiseuille-like fiow, where the outer and inner wall shear rates converge towards the same values. The axial velocity profiles for six different locations of the branch in the plane of symmetry at Re=500 are shown in Figure 5.8. The high velocity fluid is seen to move to the inner wall due to centrifuga1 action. As the Cha~ter5 Flow Visualization Studies 1O8 fluid travels further along the branch, the peak axial velocity shifts back towards the center of the pipe. Also, because of gradual reduction of the branch diameter, the axial peak velocity increases as fluid moves towards the branch outlet. The rnonotonic reduction of the branch diameter can be seen in Figure 5.8, as a reduction in velocity velocity profile width.
At the branch inlet (third zone), flow impacts the apex of the bifurcation and fons a new boundary layer along the inner wall of the branches that induces a higher rate of momentum transfer on this side of the branches. In addition, because of the change in the direction of flow due to the curvature, the high inertia fluid is skewed towards the inner wall by centrifugal forces and hence flow streamlines are no longer parallel to the vesse1 wall. The velocity field distal to the apex is 3-dimensional with an overall increase in wall shear stress due to the spiraling of high inertia fluid around the circumference of the branch. Figures
5.9 to 5.1 1 show a difference in the magnitude of the shear rates along the inner and outer walls at the apex for the different Reynolds numbers.
However, it should be noted that secondary flow caused by the centrifugal force is expected to be strongly dependent on the bifurcation angle and radius of curvature. The relatively small bifurcation angle and hence large radius of cuwature in the model studied here would cause a gradual change of the flow from the trunk into the branches. This would reduce the intensity of secondary flow compared to the impinging effect at the branch inlet. Chapter 5 Flow Visua,rization Studies 109
Figure 5.9: Normalized wall shear rates with respect to Poiseuille wall shear ratte in the tmnk (21.85 s") along the inner and outer walls (Re=500).
5.10: Normalized wall shear rates with respect to Poiseuille wall shear rate in the trunk (26.22 sg') dong the inner and outer walls (Re=600b Cha~ter5 Flow Visualization Studies 110
Figure 5.11: Normalized wall shear rates with respect to Poiseuille wall shear rate in the trunk (32.78 s") along the inner and outer walls (Re=750).
Figure 5.12: Normalized inner wall shear rates vs axial position at different Reynolds numbers. Chapter 5 Flow Visualization Studies 11 1
Figure 5.13: Normalized outer wall shear rates vs axial position at different Reynolds numbers.
lmmediately after the apex, however. the branches straighten out resulting in a redevelopment of Poiseuille-like flow creating a convergence of the inner and outer wall shear stresses. The difference between the outer and inner wall shear rate drops from 6 times ai the apex to 1.3 times, 40 mm (about three branch diameten) from the apex. It is expected that outer and inner wall shear rates reach the same values at the latter parts of the branches. However, as it can be clearîy seen from the velocity profiles in the branches (Figure 5.8), flow accelerates continually due to the gradua1 reduction of the cross section. This results in an increase of the wall shear rates. In other words, a pseudo Poiseuille flow govems the wall shear rate behavior in the latter part of the branches. Chapter 5 Flow Visualization Studies 112
Another interesting flow feature can be seen in Figures 5.9 to 5.1 1, upstream of the apex which is consistent for al1 Reynolds numbers in this study.
Normalized wall shear rates begin to rise about 30 mm upstream of the apex, reaching their maximum value about 10 mm further and drop again quickly to the
Poiseuille value 10 mm upstream of the apex.
One possible reason for the presence of such a trend in the transition zone might stem from the geometry of the model in this zone. The flow model used in these experiments was designed to simulate, in a simplified way, the transition zone adjacent to the flow divider in the bifurcation. Consequently, its inlet cross section was circular, which then gradually defoned downstream into an elliptical cross section of progressively greater eccentdcity. Evidently the converging walls of the tube act locally to produce flow sources, while the diverging walls generate sinks. The resulting secondary motions due to a net convective curent, leaving the wnverging wall and directed towards the diverging wall in each quadrant of the flow section (Figure 5.1 4).
At the latter part of the transition zone slight invagination in the cross section, owing to the converging top and bottom walls and diverging lateral walls, may change the velocity profile. Thiriet et al. (1992) studied numerically steady flow in a 3-D symmetric model of the human aortic bifurcation with a branch angle of 70 degrees. Their geometry in the transition region simulates the principal feature of the transition zone in the model of the present study. They did not calculate the wall shear rate; however, they showed the axial velocity isocontours and vector plots and referred to the presence of secondary flow with Chanter 5 Flow Vr'sualization Studies 113 source regions close to the converging upper and lower walls toward the wre region, directly and along the walls (Figure 5.15). They rnentioned that the sink region seemed to belong to the centerplûne, moving from the tube axis to the lateral wall further downstrearn. Figure 5.13 shows a nse in wall shear rate 30 mm proximal to the apex due to a decrease in the cross sectional area and also, as previously mentioned, change of the cross section geometry. The inlet cross section is circular however as the cross-section tapers; the converging walls are furrowed more and more deeply toward the flow divider with a peak 20 mm prior to the apex, probably because of the transverse currents generated by the combined effect of diverging and converging walls. Hereafter the cross sectional area diverges again towards the branches resulting in a decrease in the wall shear rate up to the bifurcation area where the flow splits into the branches and zone three is introduced.
Place of the wall shear stress measurements 7 Plane of symmetry
Figure 5.14: Schematic diagram of the formation of the secondary fiow due
to the cross section deformation. Chapter 5 Flow Ksualization Studies 114
The formation of secondary flow originating from a change in the cross section was also observed by Snyder et al. (1983). Hot-wire anemometry was used in a geometric shape of major bronchial airways to obtain axial and transverse velocity components for two different Reynolds numbers of 380 and
1540. They observed the development of transverse currents different from the secondary motions induced by the curvature prior to the bifurcation.
Figure 5.15: Vector plot of the secondary currents in a cross section of the
transition zone (Thiriet et al., 1992).
Transverse pressure gradient are present at the branch inlet. This phenornenon has been seen in previous studies. Wall shear stress results for flow in the 90" curved vesse1 with a curvature ratio of 1/6 have shown such a trend (Park. 1998). Bovendeerd et al. (1987) and Smith (1976) have suggested Cha~ter5 Flow Visualization Studies 115 that this phenornenon is driven by a transverse pressure gradient in the vessel cross-section which generates secondary flow towards the inner wall, resulting in a local increase and decrease in the wall shear stress at the inner and outer walls, respectively. A pressure gradient that drives flow towards the inner wall, is also clearly present at the inlet and appears to extend upstream. The upstream pressure variation in a 90" vessel measured by Ito (1960) and theorized by Smith
(1976),is the driving force behind the skewed velocity profile and local variations in wall shear stresses.
Figures 5.12 and 5.13 show the normalized inner and outer wall shear rates with respect to Poiseuille values at given Re in trunk vs axial position, respectively. Given 540% uncertainty attached to the wall shear rates data, the relative values almg the nonalized inner and outer walls in the whole rnodel except the second zone (latter part of the trunk) do not change significantly over the range of Reynolds nurnbers studied. For the same flow model, the physical and transport properties are the same for all three runs. At a given position, the momentum transfer rate is thus solely dependent on the fluid velocity. Therefore, the normalized data for different fluid velocities should fall into the sarne trend.
However, the influence of the Reynolds number in the second zone, which is govemed by the presence of a secondary flow due to the deformation of the cross section, can be observed.
The flow field in a bifurcated vessel is cornplicated and difficult to describe
(Malcolm and Roach, 1979; Walburn and Stein, 1982). Although there aie few studies in aortic bifurcation, mostly numerical, it is not possible to directly Chapter 5 Flow K,cudi~ationStudies 116
compare the present results with any measurements. Nevertheless. the flow
picture obtained from this study is in qualitative agreement with other results.
Wille (1984) studied numerically steady flow in a symmetric model of the
human aortic bifurcation, for a Reynolds number of 10. However, this single
value is far too low. His pressure isobars showed a high-pressure zone at the
apex as opposed to a low pressure gradient at the outer wall of the bifurcation.
As mentioned, these results did not show any evidence of flow separation
in the range of Reynolds numbers conducted. However, along the outer wall of the branch, separation may occur depending upon flow conditions and geometrical configurations such as: bifurcation angle, radius of curvature, branch to trunk area ratio and sharpness of the corner around the hip. Yung et al. (1989)
investigated numerically a symmetrical bifurcation with an area ratio of 2 and a
branch angle of 60 degrees. Due to the sharp comer at the outer wall, flow separation was observed at Reynolds numbers as low as 200. Walbum and
Stein (1981) conducted a numerical investigation in a symrnetrical bifurcation with a branch to trunk area ratio of 0.8 and a bifurcation angle of 70 degrees but a very smooth comer at the other wall (around the hip). They did not find separation for steady fiow at Reynolds numbers between 500 and 1500.
Although, the bifurcation angle of their numerical model was larger by 1O degrees compared to bifurcation angle of Yung et al. study, flow separation did not occur even at the higher Reynolds numbers. This discrepancy can be expiained due to the smoothness of the comer at the outer wall. Chapter 5 Flow Ksunlkatiun Studîes 117
Nazerni et al. (1989) studied numerically the influence of the angle of the bifurcation and the inlet flow conditions on the magnitude of the shear stress near the apex in a two dimensional syrnmetric bifurcation. They observed fiow separation for a bifurcation angle of 50" but not for a 15" bifurcation with a
Reynolds number of 500. They also concluded that high shear stresses are generally present along the divider wall or wedge surface, quite independent of the branching angle. Low shear stresses, including negative wall shear stresses, occur at the outer wall of bifurcation. Their non-dimensional wall shear stress drops drastically right before the apex due to the enlargement of the cross sectional area, before it again rises after the apex. This fluctuation was more severe for the wider angle of the bifurcation. Chapter 5 Flow Crsual&ztionStudies 118
5.4 Sumrnary and Conclusions
The laser photochromic tracer method has been used to study fluid phenornena under steady laminar flow in the plane of symmetry of a simplified aortic bifurcation at Reynolds numbers 500, 600, and 750. The flow field in this study may be categorized by four different zones. In the first zone, Poiseuille-like
Row govems the trunk. The second zone, initiated from the latter part of the tnink. is characterized by an increase in wall shear rates. Deformation of the cross section may create a secondary flow in this region resulting in higher shear rates. The converging upper and lower walls are sources, with Row toward the core region, directly and along the walls. The sink region seems to belong to the centerplane, moving from the tube axis to the lateral wall further downstrearn.
The third region is characterized by higher shear rates along the inner wall, mostly due to the formation of a new boundary layer, and lower shear rates along the outer wall. This region is initiated from the apex and extends to approximately two branch diameters downstrearn.
The fourth zone starts further downstream, where again Poiseuille-like flow redevelops and the outer and inner wall shear rates converge towards the same values. Flow accelerates continually along the length of the branches due to the gradua1 reduction of the cross section, implying an increase of the wall shear rates in the latter part of the branches. 6.1 Introduction
In chapter 3, the integrated mass transfer coefficients were presented for a pipe and a simplified bifurcation model using the fem-ferrocyanide electrochemical technique. This technique has also been used to determine local mass transfer coefficients by other researchers (Reiss and Hanratty, 1962,
1963). A very thin and small electrode is placed in different locations and the obtained cuvent is used to calculate the rate of transfer of ions in that particular place. There are some technical shortcomings in placing such a small electrode.
Besides, the placement of the electrode rnight upset the fluid flow to some extent and hence change the mass transfer coefficients. To overcome some of the disadvantages of Hanratty's technique, a novel electrodeposition technique, Chapter 6 Local Mass Transfer 120 here, is used to obtain time-average convective local rnass transfer coefficients in a straight pipe and a Y-bifurcation model, thus complementing results in chapter
3. This method is based on a diffusion-controlled deposition rate at the cathode surface. With an electrolyte solution of H2S04 (high concentration) and CuS04
(low concentration), copper is dissolved at the anode and deposited ont0 the cathode. Cu -CU*' + 2e (at the anode) CU'+ + 2e -Cu (at the cathode )
After a certain period of time, the thickness of copper deposition is measured to determine the local mass transfer distribution.
6.2 Experimental Methods
6.2.1 Flow System and operating conditions
The system foned a closed loop from the 3 HP centrifuga1 pump, through the test sections and the flow meters to the tank (Figure 6.1). The entrance section to the cathode had a length of 3 meters to ensure fully developed flow conditions at al1 Reynolds numbers. The test sections were arranged with the sequence of anode, cathode and anode for both pipe and bifurcation experiments. The anodes were made of copper and the cathode was made of nickel. The upstream and downstream anodes had a length of 30crn for the pipe experiments. To be consistent with the average rnass transfer experiment set up for the bifurcation experiments, the upstrearn and downstream anodes were 15 cm and 30 cm long, respectively (Figure 6.2). Solutions were deaerated with Chapter 6 Local Muss Transfer 121 nitrogen for at least 30 min prior to each nin and nitrogen was swept over the solution during the experiments. Nitrogen used for the purging the solutions was cylinder grade (Canadian Oxygen limited). During an experimental run, the system was first brought to the desired operating temperature (25%) with water flowing through a heat exchanger; polarization curves were checked at various flow rates and their midpoints used to set the operating voltages. The electrical circuit was the same as that used for the ferri-ferrocyanide electrochemical system (average mass transfer experiments). The concentration of each of the species entering the solution was checked pnor to each nin by chemical analysis
(Appendix D). The concentration of CuS04 and H2SO4 were determined by iodometric titration (Kolthoff et al., 1969) and by titration with standardized NaOH, respective1y.
To be able to perform experiments more than once for each nickel modei, a multiple layer approach was taken. After a period of tirne, when a layer of copper had been deposited, a thin layer of nickel was deposited over the wpper layer followed by the deposition of copper again under different conditions, etc.
To do this, a piece of nickel rod (6.35 mm or 0.25 inch dia.. 99.5%, Alfa AESAR
Inc., stock # 11454) had been centered in the straight pipe before being inserted in the nickel plating tank. A three piece nickel core (the same material as mentioned above) was also machined and screwed together for the deposition of nickel in the bifurcation model. This three-piece nickel are resernbled a smaller size of the bifurcation model and could easily ffi inside the model. Chapter 6 Lod Mms Trunsfer 122
A \ WATER IN
Figure 6.1 : Schematic diagram of electrolyte system.
Figure 6.2: Photograph of the electrodes in the bifurcation Experiments. Cha~ter6 Local .Mms Trunsfer 123
Experiments were run at three different Reynolds numbers of 1750. 1225 and 700 for the laminar case and two Reynolds numbers of 5925 and 4583 for the turbulent case in the straight pipe. Times of deposition were 12, 12 and 16 hours for the Reynolds numbers of 1750, 1225 and 700, respectively. For the turbulent case, lengths of experirnents were 15 and 18 houn at Reynolds numbers 5925 and 4583, respectively. Experiments also were conducted at three different Reynolds numbers of 4345, 1750 and 750 for the bifurcation model with deposition times of 11, 24, 31 hours, respectively.
The concentrations of copper sulphate were 0.008 M and 0.005 M for the laminar and turbulent nins of the pipe experiments, respectively. The concentration of copper sulphate for the bifurcation experirnents was 0.001 M.
The concentration of sulphuric acid for both pipe and bifurcation experiments was
3.1 M.
6.2.2 Physical Properties of the Electrolyte Solution
Kinematic viscosity of the solution was measured using a Cannon-Fenske
Opaque viscometer, size 50 (ASTM D 445). Kinernatic viscosity was measured as 1.38~10~(m2/s) at 25O C. Also, the density of the solution was measured as
1 188 (kg/m3).
The diffusion coefficient of cupric ion in sulphuric acid solution was calculated
DP based on the Einstein-Stokes theory. Arvia et al. (1966) found the ratio - to be T
(2.23I0.37)x 10"~ for the solution of cuprk sulphate and sulphunc acid Chapter 6 Local Mass Transfer 124 for a temperature ranging from 18 to 40°C. This ratio is reasonably constant within the whole range of viscosity and temperature investigated. Hence; difbsivity of cupric ions in the sulphuric acid was calculated as 4.05 x 10-l' (rn2/s) and Schmidt number 3400 for both pipe and bifurcation experiments.
6.2.3 Cutting
The next stage was to cut the samples at specific locations. Straight pipe and bifurcation had to be filled out with some resins, otherwise the very thin deposited layer detached easily due to mechanical forces induced from different stages of sample preparation such as: cutting, grinding, polishing. Different materials were tested to fiIl out the samples but al1 of them failed to be suitable for this purpose. Some of them did not have enough low viscosity (fiuidity) to fil1 out the details of the sample and some shrank after filling out, especially in the cutting process. Eventually, Hysol TE5246 aluminum filled resin cured with Hysol
Hd3615 hardener were found to be the most suitable materials for filling purposes. There were other hardeners. However, since Hd3615 had the lowest viscosity of ail (lo3 poise), it was chosen to harden the resin. Density of the mixture was 1.7 g/cm3. A plastic seal was machined to cover one end of the sample prior to the filler preparation. f he following procedure was used to prepare the filer:
Wam up Hysol TE5246 resin to 50' C.
Weigh the cornponents Chapter 6 Local Mass Transfer 125
Mix mechanically TE5246 resin with HD3615 hardener with the ratio of 10 to
1 for about 1O minutes.
De-air the mixture for about 10 minutes.
Wam-up the mixture 50" C to lower its viscosity.
Pour the mixture in the pipe and bifurcation.
Cure at room temperature for 36 hours.
Post-cure the samples at 70" C for about 2 hrs.
Remove the samples from the oven and let it cool to room temperature.
6.2.3.1 Pipe
n the next stage, the length of the pipe was measured precisely with a calibre Subsequently, a regular band saw with a fine blade (32 teeth per inch) was employed to cut the sample. After the resin was cured, a curve (meniscus) was formed on the surface. As a result, 2 mm of this end (corresponding to the leading edge) was cut. It should be noted that the other end was kept unchanged to have a fixed reference point when cutting was perfoned. The place where the band saw blade was introduced, was located about 1.5 mm downstream of the desirable place, because 1 mm was removed due to the cutting and 0.5 mm due to tuming the surface flat by means of a DoAll 13" Lathe. The measurements of the length of the samples by means of the calibre were repeated at this stage to precisely locate the distance of both surfaces from the leading edge. In middle of the cutting process, we noticed that liquid nitrogen could be used for two Cha~ter6 Locd Mass Transfer 126 purposes. First and more importantly, to smooth out the cut surfaces and second to release heat generated from the cutting process.
6.2.3.2 Bifurcation
Three major cuts were perfomed on the bifurcation model (Figure 6.3).
Figure 6.3: Diagram of the different cuts on the bifurcation model.
The first cut detached the tmnk from the two branches. Since the exact location of the apex was not known, with the second cut, 3.7 mm was the closest distance that wuld have been achieved from the apex. The third cut was perfomed diagonally with an angle of 17.5" with respect to the trunk's normal plane. In order to obtain the location of the apex at the second branch, the remainder was machined from the top slowiy by means of a DoAll 13" Lathe thus approaching the apex at the second branch. By this procedure, 0.4 mm was the closest distance achieved from the apex after grinding and polishing. Subsequently, the tmnk and two branches were cut at specific locations and then samples were Chapter 6 Local ?duss Tramfer 127
ground and polished. It should be noted that the leading edge was the reference
line for the trunk, while the ends of the branches were chosen as the reference
lines for the branches. When the exact location of the apex was determined, al1
lengths were converted to the distance from the apex.
6.2.4 Grinding and Polishing
The next step was to grind and polish the samples to make them prepared for the microscope reading. Grinding was performed by Silicon Carbide Grinding
Paper. Grinding papers were placed on a rotating disc and the samples were then introduced to the rotating device while water flowed ont0 the grinding paper to release the heat generated from grinding the samples. Grinding started with
grinding paper with a grit number of 240 followed by the grit numbers 320, 400,
600, 800, 1200, respectively. Unlike grit size 240, which is the roughest one
among the above group of grinding papers, grit size 1200 possesses the finest
grain size. Between each step of the grinding, sarnples were rinsed with
methanol and then cleaned ultrasonically to remove any possible dirt fomed on
the surfaces due to the grinding process.
Polishing the samples at the next stage made the sample surface even
smoother. Besides, without the polishing stage the distinction of the different
Iayers of nickel and copper was difficult due to a weak contrast. Polishing was
performed using a Oiamond Suspension in two stages. At first a suspension
polished the surfaces down to 6 microns followed by a finer diarnond suspension which polished the samples down to 1 micron. Again sarnples were cleaned ultrasonically between the stages.
After the grinding and polishing stages were finished, the lengths of the samples were measured to calculate the final distance of each surface from the leading edge or from the apex in the case of branches of the bifurcation model. Chapter 6 Locd Mass Transfer 129
6.3 Results and Discussion
As this was the first time that copper electrodeposition has been used to study mass transfer in a convective diffusion system, many prelirninary experiments were perfomed to assess optimal protocols for the experimental method. A set of experiments was carried out to find out the feasibility of the multiple layer approach. In this set, the electrolyte system was the same as the electrolyte system of the main experiments (Figure 6.1). lnstead of nickel cathode, a piece of copper with the same material and diameter of the copper anode but 150 mm in length was cut. A layer of nickel (about 15-17 p) was electrodeposited and then copper expenments were nin. Results of that earlier work revealed that although the layers were intact and quite distinguishable in some places along the circumference (Figure6.4). due to softness of the copper layers, these layers were damaged at some other places (Figure 6.5).
Preliminary experiments revealed that smearing of the epoxy or metal layers made the interface of the layers unclear (Figure 6.6). To cope with these imperfections, it was decided that after finishing the last copper deposition run, another extra nickel layer should be deposited to support the copper layers from being damaged or smeared. The observations of the samples from the final runs showed that this extra layer helped to reduce the smearing effects to some extent. Figures 6.7 to 6.18 are the images taken by a CCD camera at different locations along the circumference and length of the pipe for laminar experiments.
As it shown in Figures 6-13 to 6-15, the layers are detached in some places. This Cha~ter6 Local Mass Transfer
Figure 6.4: Photograph of the layers for the earlier experiments.
Figure 6.5: Photograph of the layers for the earlier experiments, damage to the copper layer. Chapter 6 Local Mass Transfer
Figure 6.6: Photograph of the layers for the earlier experiments, srnearing of either epoxy or nickel layer over the copper layer.
Figure 6.7: Photograph of the layen, laminar experiments, 13 cm from the leading edge. Chapter 6 Locai Mass Transfer
Figure 6.8: Photograph of the layers, laminar experirnents, 5.8 cm from the leading edge.
Figure 6.9: Photograph of the layen, laminar experiments,
7.73 cm from the leading edge. Chapter 6 Local Mass Tramfer
Figure 6.10: Photograph of the layers, laminar experiments,
1 cm from the leading edge.
Figure 6.1 1: Photograph of the layers, laminar experiments,
1 cm from the leading edge (another location). '6 Local Mass Transfer
Figure 6.12: Photograph of the layers, laminar experiments, 2.97 cm from the leading edge, damage to the nickel layer.
Figure 6.13: Photograph of the layers, laminar experiments, 8.2 cm from the leadin edge, detachment between the lstand 2"! layers. Figure 6.14: Photograph of the layers, laminar experiments, 13 cm from the leading edge, detachment of al1 layers from the nickel substrate.
Figure 6.15: Photograph of the layers, laminar experiments, 2.97 cm from the leading edge, detachment between the 2"d and 3& layers. Local Mass Tramfer
Photograph of the layers, laminar experiments, 10.1 cm from the leading edge, smearing of the copper layer
10 P H
Figure Photograph of the layers, laminar experiments, 10.6 cm from the leading edge, smearing of the nickel layer Chanter 6 Local Mass Transfer
C
Figure 6.1 8: Photograph of the layers, laminar experiments, 3.4 cm from the leading edge, smearing of the nickel layer.
detachment is due to mechanical forces induced from different stages of sample preparation, especially cutting stages. In the cutting stages, we noticed that the detachment occurs where the band saw is introduced to the surface and layers located on the other side of the surface are perfectly attached. The materials which filled the inside of the models are soft (compared to the metals) and are pushed over by the band saw blade. On some locations, al1 layers are detached from the nickel substrate (Figure 6-14) and on other locations, detachments occur between the layers (Figure 6-13). As mentioned before, the effect of srnearing was reduced by depositing the last nickel layer. However, as it is obvious in Figures (6-1 6 to 6-18), there is still smearing of the nickel over the copper layers in some places, smearing of the copper over the nickel layers in Chapter 6 LodMQSS Transfer 138 other places and occasionally even the srnearing of epoxy over both nickel and copper layers.
6.3.1 Sources of Error
The major source of the error in the local mass transfer results is due to the inaccuracies in the measurements of copper thickness. As it was shown in the Figures 6.16, 6.17 and 6.1 8, smearing of the copper, nickel and epoxy made the interface of the layers unclear. The detachment of the layers is another source of uncertainties. Attempts were made to collect the data from places with the least arnount of damage, yet scatter is observed in the different graphs.
Because of variability in the copper thickness, h,, and because of inherent and unavoidable random errors in the rneasuring procedure, the best estimate of the
"truenvalue of h, is then given by the average or mean value of h,:
n
To quantify the spread of the wpper thickness, the standard deviation of h, is calculated over the desirable surfaces.
For both the pipe and trunk of the bifurcation the thickness measurements were performed along the circumference where there were distinguishable interfaces.
A maximum of 13% variation is attached to the individual measurements. a is associated with the error in each measurement .hd, at a given surface. The best Cha~ter6 Local Mass Transfer 139 estimate of h,, which is the anthmetic mean of the copper deposition over the circumference of the surface, accounts for errors in the calculation of the mass transfer coefficients. In this case, standard deviation of the mean may be a reasonable representative of the uncertainty associated with the data. This
"standard deviation of the meann is called the standard error and is often denoted
For the laminar and turbulent runs of the pipe and tmnk of the bifurcation, depending on the uniformity of the deposited copper, 5 to 10 measurements were taken; therefore, the standard error would be less than 13%.
In the two branches of the bifurcation the scatter of the deposited copper was higher, even for small segments of the circumference and especially at the apex area. This can be partly attributed to the compfex nature of the flow in that area. The reported thickness of the copper at a given angle is indeed the average of up to 10 observations in 15' of that particular location. In other words, the measurements are performed over an arc of 10' and the middle of the arc is considered for that set of measurements. Due to the syrnmetry of the cross section, the extra error induced by this assumption is minimal in the measurements at the plane of symmetry and the plane normal to this plane. A maximum 23% uncertainty interval (apex area) is attached to the individual measurernents along the arcs in the branches. Further downstream, a maximum
17% variation was observed. As discussed before, depending on the number of measurements for each segment, the standard deviation of the mean is smaller than those of individual measurements.
Another source of error in the quantification of local mass transfer coefficients was the difisivity of cupric ion in sulphuric acid solution. The diffusion coefficient is calculated based on the Einstein-Stokes constant reported by Arvia et al. (1966). 17% uncertainty may incur as systematic error in the mass transfer calculations.
The time-dependent boundary layer growth was analyzed in chapter-4.
Based on those data, the time needed to obtain steady state mass transfer was in the order of (100-150) seconds for the low Reynolds numbers in the laminar regime and even less for the turbulent fiow. This time domain is extremely short compared to the iength of experiment and hence the contribution of unsteadiness to our local mass transfer data is quite negligible.
The other sources of error, such as uncertainties in the concentration, temperature, density, viscosity, time and dimensions of the rnodels contribute to error. However, the influence of these sources or errors are less significant compared to the uncertainties associated with thickness measurements and diffusivity of the cupnc ions in the electrolyte solution. Chapter 6 Lord .Mass Transfer 141
6.3.2 Pipe Data
For the laminar straight pipe three different runs were performed;
Re=1750, 1225. 700, respectively. The effect of distance from the leading edge
on the Sherwood number for different Reynolds nurnbers is presented in Figures
6.d 9 to 6.21. lt should be noted that the Sherwood number at each location has
been calculated based on the average thickness over the respective
L circumference. - = 8 in Our experiments, which results in developing mass d transfer conditions. The formation of the concentration boundary starts right from the leading edge and grows exponentially along the pipe resulting in an increase
in the resistance to the diffusion of the ions towards the electrode surface.
Consequently, the rate of mass transfer decreases corresponding to the
concentration boundary layer growth. Mass transfer coefficients in the entry
region for steady laminar flow were studied by Leveque (1928) and can be
presented for the local position in the dimensionless forrn for constant
concentration boundary condition as
Sh = 1.O76 (Re Sc (dlx))'" (6-4)
and are shown as solid Iines in Figures 6.19-6.21. Data points are in relative
agreement with the Leveque solution. Uncertainties in the thickness
measurernent can partly be responsible for the observed differences. Chapter 6 Local Mass Transfer 142
Figure 6.19: Local mass transfer distribution along the pipe
Figure 6.20: Local mass transfer distribution along the pipe
(Re=1225, Sc=3400). Chapter 6 Locai Mass Transfer 143
Figure 6.21: Local rnass transfer distribution along the pipe
(Re=700, Sc=3400).
The other possible reason for the slight difference of the data point with the analytical solution of Leveque may be due to the use of an upstream anode.
The influence of the upstrearn anode on Sherwood number in the average mass transfer experiments can be observed in Figures 3.1 1 and 3.12. Because of the additional near-wall ions produced by the upstream anode, a higher mass transfer coefficient is observed. Based on Our ferri-ferrocyanide mass transfer results for Sc=2470 (Figure 3.12), an increase of 3%, 8% and 10% on average can be expected for the Reynolds numbers of 1750,1225, and 700, respectively.
In the local mass transfer expenments, Sc=3400. The thinner concentration boundary layer associated with this larger Schmidt number dilutes the influence of upstream anode. Chapter 6 Local Muss Transfer 144
The following equation has been used to calculate the averaged
Shenvood number for the local mass transfer experiments:
At the same tirne, equation (3-3) has been used to compare the average data obtained by the integration for the local experiments with those obtained by the ferri-ferrocyanide experiments. 60th sets of data are tabulated in the following table. The readings of the copper thickness around the circumference were not unifom and taken at specified angles for the al1 surfaces. These readings were obtained from circumference segments with the least amount of incurred damages. A9 in equation (6-5) is the angle between two consecutive measurements and AL is the axial distance between hivo consecutive cuts.
Table 6.1: Average Shenvood number for straight pipe obtained by both copper deposition technique and ferri-ferrocyanide technique at different Reynolds numbers (laminat regime).
Average Sh based on the Average Sh based on the integration of local mass correlations obtained from transfer experiments average mass transfer (Copper deposition) experiments (Ferri-ferrocyanide technique) 1750 143.2 135.6 Cha~ter6 Local .Muss Transfer 145
The averaged local values are 6%, 12%, and 4% greater than those obtained by the correlation from ferri-ferrocyanide experiments, respectively. It should be noted that there was no exact match between the local mass transfer
(copper) and average mass transfer (ferri-ferrocyanide) experiments. Employing equation (3-3) to obtain the average Shewood number for cornparison with the local mass transfer experirnent will introduce 3% error which is the standard deviation of the fitted equation to the data points.
Two different nins were performed for the turbulent straight pipe;
Re=5925, 4583, respectively. Figures 6.22 and 6.23 illustrate the Shenvood number versus distance from the beginning of the pipe. Again, equation (6-5) has been employed to integrate the local Shenvood numbers both around the circumference and along the length of the straight pipe. Table (6.2) represents the averaged Shenvood number obtained from two different approaches.
Figure 6.22: Local mass transfer distribution dong the pipe
(Rer5925, Sc=3400). Cha~ter6 Local Mass Transfer 146
Figure 6.23: Local mass transfer distribution along the pipe (Re=4583, Sc=3400).
Table 6.2: Average Sherwood number for straight pipe obtained by both copper deposition technique and ferri-ferrocyanide technique at different Reynolds numbers (turbulent regime).
Average Sh based on the Average Sh based on the integration of local mass correlations obtained frorn transfer experiments average mass transfer (Copper deposition) experiments (Ferri-ferrocyanide technique) 459 Cha~ter6 Local Muss Transfer 147
The difference between these two sets of numbers is well below the uncertainties explained in section (6.3.1). The excellent, and to some extent unexpected, agreement between these two different approaches may anse from two possible reasons. First, al1 measurements rnay have been taken from the places with the least amount of imperfections (i.e. smearing); or second, the errors from different sources and for different places have cancelled out each other such that the overall uncertainty has become smaller than the individual sources.
Unlike the laminar case, there is no analflical solution for developing mass transfer conditions in turbulent fiow. However, various analogies have been suggested to estimate the rate of transfer between the flowing fluid and the solid surface. The turbulent mass transfer data rnay thus be compared with the
Chilton-Colbum analogy (1934), one of the most well known analogies between transport phenomena. It is worth mentioning that Chilton-Colbum have presented this analogy based on an empirical presentation of the effect of the Prandtl number on heat transfer to fiuids in turbulent Row. They also proposed that the same relation should hold for mass transfer, and would seem to have been the first to emphasize the three way analogy between mass transfer, heat transfer and friction in turbulent fiow. However, it should be noted that the Chilton-Colbum analogy is for fully developed mass transfer conditions. Also, this analogy has been obtained for Reynolds numbers greater than 10,000. The integrated
Sherwood number is 30% higher for Re=5925 and 25% higher for Re=4583 than the Chilton-Colbum result. Based on the conditions applicable to this analogy, the level of discrepancy between Our integrated local value and the Chilton-
Colbum analogy may be expected.
Pickett and Ong (1974) reported that fully developed mass transfer conditions in the turbulent regime occur at about twelve diameters from the entrante. Based on the experimental results, they proposed the following equation with a standard deviation of 10% for Ud s 10.1 :
Sh=0.125 ~e~ SC'^ (dlL) (6-6)
The above equation underestimates the Shewood number by as much as 13% and 5% compared to the integrated local mass transfer coefficients for Reynolds numbers of 5925 and 4583, respectively.
The slope of the data points in graphs 6.22 and 6.23 are - 0.08 and
-0.09for Reynolds numbers of 5925 and 4583, respectively. This implies that the mass transfer entry region is shorter in turbulent flow than in laminar flow.
Based on the Leveque solution, mass transfer coefficients in laminar flow are proportional to x4.? The results of Pickett and Ong (1974) indicate that
Shewood number is correlated with dimensionless electrode length to a power of - 0.2 for turbulent flow in a ceIl with a rectangular cross section, as it has been shown in equation (6-6). Beyond this short entry region, the Sherwood number rapidly approaches a constant value, wrresponding to fully developed mass transfer. Chapter 6 Local Mass Transfer 149
6.3.3 Bifurcation Data
For the bifurcation, three runs were perforined; Re=4345, 1750, 750, respectively. Local mass transfer coefficients along the trunk for deiFferent
Reynolds numbers are shown in Figures 6.24 to 6.26. These results are in close agreement with the data for the straight pipe and the same general features can be observed.
Figure 6.24: Sherwood number vs distance from the leading edge in the trunk of the bifurcation (Re=750, Sc=3400). Chapter 6 Local Msss Transfer 150
orna (cm)
Figure 6.25: Shemood number vs distance from the leading edge in the trunk of the bifurcation (Rer1750,Sc=3400).
Figure 6.26: Sherwood number vs distance from the leading edge in the trunk of the bifurcation (Re=4345,Sc=3400). Cha~ter6 Local Muss Transfer 151
The Sherwood number at different distances from the apex in the plane of symmetry and at different Reynolds numbers are shown in Figures 6.27 to 6.29.
Scattering in the data is observed, which is mainly due to the sources of uncertainties mentioned in section 6.3.1. However, mass transfer coefficient at al1 three Reynolds numbers at the inner wall are larger than those along outer wall.
As fiow impinges the apex, a new concentration boundary layer is formed. Since the thickness of the boundary layer is very thin at the wall adjacent to the apex, the resistance to the diffusion of ions towards the electrode surface is small,
resulting in a higher mass transfer in this side of the branch. This phenomenon
resembles the formation of the boundary layer at the leading edge of a pipe. The
Sherwood number at the inner wall of the apex is about twice as large as the
Shewood nurnber at the outer wall. In chapter 5, it was shown that the high
velocity is skewed towards the inner wall resulting in the higher wall shear stress.
Further downstream. flow redevelops and a high rnomentum velocity zone is
again inclined towards the center of the pipe causing inner and outer wall shear
stress to converge towards the same value. The same phenornena can be seen
here. A careful inspection reveals that inner and outer wall Sherwood numbers
converge about 6D,4D and 20 from the apex for the Reynolds numbers of 750,
1750 and 4345, respectively. Figures 6.30 to 6.35 represent the distribution of
deposited copper around the circumference 3.7 mm and 43.9 mm from the apex
at different Reynolds numbers. In these figures zero (or 360) corresponds to the
inner wall and 180 denotes the outer wall, moving from inner wall to the outer
wall clockwise. The thickness of copper is higher at the inner wall area and lower Cha~ter6 Local M~ssTransfer 152 at the opposite wall, 3.7 mm from the apex. It seerns that further downstream; i.e.
43.9 mm from the apex, the thickness of concentration boundary layers would merge towards the same value, due to redevelopment of flow. This fact can be observed in our flow studies. Based on the axial velocity profile graph (Figure
5.7), the peak velocity is tumed towards the centerline and a Poiseuille-like flow is re-established. This phenornenon is probably directly proportional to the
Reynolds number; as in figure 6.34, the thickness of deposited copper is still lower in the outer side wall area compared to the other side for Re=750. The length required to obtain the fully developed mass transfer boundary layer for turbulent fiow is shorter (about 20 in our experiments) in cornparison to the laminar flow and the outer and inner wall merge faster towards the fully developed value.
Figure 6.27: Local mass transfer coefficients in the plane of symmetry of the branches (Re=750, Sc=34DO). Cha~ter6 Local Muss Transfer 153
Figure 628: Local mass transfer coefficients in the plane of symmetry of the branches (Re=1750, Sc=3400).
Figure 6.29: Local mass transfer coefficients in the plane of symmetry of the branches (Re=4345, Scr3400). Chapter 6 Local Mas Tmnsfer 154
A comparison between the Sherwood number of the branches for al1
Reynolds numbers shows discrepancies to some extent. Aside from the uncertainties involved in the local rnass transfer experiments, it was presumed that the flow splits equally between the two branches. Although care was taken to construct the model as well as the connectors from the branches to the main flow as symmetrical as possible, flow might divide unequally between two branches; therefore, resulting in a different Reynolds number in each branch. However, since the difference between the branch mass transfer coefficients is more obvious at the apex area and diminishes further downstream, the error in measuring the thickness may be the dominant factor in creating such a d iscrepancy.
To compare these results with the average mass transfer experiments performed by fem-ferrocyanide technique, equation (6.5) was used for the tnink and branches and the average Shennrood number was calculated by this rnethod.
Data are tabulated as follows:
Table 6.3: Average Sherwood number for the bifurcation model obtained by both copper deposition technique and ferri-ferrocyanide technique at different Reynolds numbers.
Re Average Sh based on the Average Sh based on the integration of local mass correlations obtained from transfer experiments average mass transfer (Copper deposition) experiments (Ferri-ferrocyanide technique) 4345 342.9 377.5 Cha~ter6 Local Mass Transfer 155
Equations (3-8) and (3-9)were used to produce results in the third column. A 5-
10 percent difference is observed which is lower than the uncertainties involved in the individual rneasurements of the copper experiments. This might be caused by taking the average over the local values, which results in a smoothed result.
Flow separation is the other possible phenomena that might take place at the outer wall of the bifurcation. If the mornentum contained in the Ruid layers adjacent to the surface cannot overcome the force exerted by the adverse pressure gradient, the boundary layer separates from the surface. At this point the velocity gradient vanishes. Along the outer wall of the branch, separation may occur depending upon flow conditions and geometrical configurations such as: bifurcation angle, radius of curvature, branch to trunk area ratio and sharpness of the corner around hip. Our ferri-ferrocyanide results (Figure 3.14 and 3.15) suggest that flow separation may have been initiated at about Re=1500 or, at least, the velocity gradients at the wall would have become less steep.
Furthermore, although fiow separation is a flow phenornenon, the Sc dependence can be observed in those results. That is, the influence of separation may be observed more readily at the lowest Schmidt number. The effect of flow separation will be more easily felt for the lower Schmidt number due to a thicker concentration boundary layer.
As described in chapter 5, the laser photochromic tracer method was used to obtain velocity profile and wall shear stress. Re=750 was the highest Reynolds number in flow visualization experirnents. Flow study results are quite consistent with mass transfer data, as there was no sign of separation up to Re=750. Low Chapter 6 Local Mass Transfer 156
mass transfer coefficients along the outer wall at the apex area couid be a result
of flow separation. On the other hand, if there is a boundary layer separation, it
would be expected to find locally high deposited copper spots corresponding to
the reattachment point. Such a high mass transfer point(s) was not found in our
s pecimens. However, the low spatial resolution resulted from the invasive technique of thickness measurement and/or limited sample range (0.4 cm downstrearn of the apex downward) may have been insufficient to capture the re-
attachment point. The speciiic way of cutting the bifurcation mode1 did not allow
us to study upstream of the apex at the outer wall (hip area). After ail. Sc=3400 in
the local mass transfer experiments and according to the average mass transfer
results, very small reduction rate of rnass transfer in this range of Schmidt
number can be expected. Perhaps, as only modest flow separation, if any,
occurs, the influence of separation may be observed more readily at a lower Sc.
It should be noted that mass transfer techniques are an implicit way of
studying flow phenornena; Le. flow separation. To explicitly study Row, a flow
visualization technique should be implemented. The laser photochromic tracer
method has a great potential to study flow with fine resolution. A more powerful
pump or a gravity flow with the large liquid level height can provide the higher
Reynolds numbers necessary to study the structure of the possible flow
separation at the outer wall. Chapter 6 Local Mass Transfer 157
Figure 6.30: Distribution of deposited copper around the circumference 3.7 mm from the apex (Re=750).
O
Figure 6.31 : Distribution of deposited copper around the circumference 43.9 mm from the apex (Re=750). Chapter 6 Local lhss Tramfer 158
Figure 6.32: Distribution of deposited copper around the circumference 3.7 mm from the apex (Re=1750).
Figure 6.33: Distribution of deposited copper around the circumference 43.9 mm from the apex (Re4750). Chapter 6 Local Mass Transfer 159
Figure 6-34: Distribution of deposited copper around the circumference 3.7 mm from the apex (Re=4345).
Figure 6-35: Distribution of deposited copper around the circumference 43.9 mm from the apex (Re=4345). Cha~ter6 Local ,Muss Transfer 160
6.3.4 The links between flow and mass transfer phenornena
All of the three important transfer processes between solid surfaces and fluid streams depend both on molecular properties and on the motion of eddies in the turbulent stream. It might thus be expected that the three are closely related and that separate fields of mass transfer, heat transfer, and fluid friction would merge to become a single su bject. The quantitative results between the three processes are usually described in the form of "analogiesn between heat and momentum transfer or between mass and momentum transfer. This is the case because fluid fow has been studied in much more detail than either heat or mass transfer, but all three could equally be, and are sometimes, considered jointly.
In laminar flow the exact solution of the boundary layer equation by
Blasius over a flat plate has been extended by Pohlhausen to include heat transfer. His solution can equally well be applied to mass transfer when the
Prandtl number is replaced by the Schmidt nurnber which represents the ratio of the molecular transfer of momentum to that of mass. Pohlhausen's exact solution is represented by:
where St is defined by:
Sh k St, =-=- ReSc U and Cf is represented by: Chapter 6 Local Mass Transfer 161
Equation (6-7) rnay be used in the laminar regime to find a link between the rnass transfer and wall shear stress data. It should be noted that the
exponent 23 on Sc (Pr) has been shown to hold well for turbulent flow, as well.
One important issue that should be considered is that equation (6-7)has been established for flow over a Rat plate and the similar approach for the pipe.
However, at bifurcations or bends there is a Jdimensional flow field with a transverse pressure gradient resulting frorn the sewndary fiow. The above mentioned analogy, as well as the other analogies will hold true provided that skin friction, Cf, is based on shear friction and not total drag. Although both fon drag and Sh are related to Ruid mechanics, the mechanism is different.
Therefore, it is expected that a pressure drop, which may be partially due to form drag, and Sh do not exhibit analogous behavior. As a result. the contribution of form drag to the total pressure loss is the important factor that should be taken into account. Based on reported data at rough surfaces (Zhao et al., 1997), at high Reynolds numbers when the friction factor is constant, the largest part of resistance to flow is due to form drag. However, at low Reynolds numbers, fom drag is not predominant as suggested by the results of momentum transfer of rough surfaces.
Figure 6.36 represents both St, SC= and C42 versus distalice from the apex in the plane of symmetry of the bifurcation model. Left hand side of the equation (6-7), St, SC^, has been calculated from local mass transfer data of the branches for Re=750 in the bifurcation (Figure 6.27). The branch wall shear stress data (Figure 5.10), obtained by the laser photochromic tracer technique, at Re=750 has been employed in the calculation of G/2 (right hand side of the equation 6-7). To rnake the mass transfer data distinguishable from the fluid friction data, G/2 has been plotted as solid lines. As no smoothing has been applied to the wall shear stress data, some fluctuations are observed in the solid lines (Figure 6.36). A qualitative agreement is observed. As it was explained in chapter 5, wall shear stress data is extracted from velocity profiles at different locations which are unaffected by form drag. In other words, the friction factor in
Our experiments is based on shear friction and not total drag. This is an essential condition for analogies to be valid.
The mass transfer and friction data follow a similar pattern both on the inner walls with a higher transfer rate and on the outer walls with a lower one.
Subsequently, both transfer rates converge towards the same value further downstream. In section 5.2.5 of chapter 5, d0Herent sources of errorç were discussed. It was concluded that, overall, 5-10% uncertainty is attached to the wall shear rate data. The higher extrerne belongs to the wall shear rates corresponding to the apex and hip area, due to additional difficulties in fitting the wall slope. Error analysis in the local mass transfer experiments revealed that up to 23% uncertainty may be associated with the individual data points around both the apex and hip area. The higher level of uncertainties in the apex and also hip area may account for the greater discrepancy between the two transfer rates in these areas. The differences between the mass transfer and friction data are well below the maximum uncertainties further downstream. Chapter 6 Local &Zuss Trms fer 163
-Cfi2. inner wall -CfI2. outer waU A St ScZ3. mer wall 8 St Sc2f3. outer wall
O t O M 30 40 50 60 70 00 90 1O0 Olstanco form the apex (mm)
Figure 6.36: Cd2 and St sca3 along the inner wall and outer wall of the branches in the plane of symrnetry of the bifurcation (Sc=3400, Re=750). Chapter 6 Local Mass Transfer 164
6.4 Summary and Conclusions
Time-averaged convective local mass transfer coefficients have been obtained for a straight pipe and a bifurcation model by a novel electrodeposition technique. This method is based on a diffusion-controlled deposition rate at the cathode surface. The anode is made of copper and the cathode is made of nickel. The electrolyte solution consists of H2SO4(hig h concentration) and CuSO4
(low concentration). After a period of tirne, a layer of copper is deposited on the cathode surface. The thickness of this layer is proportional to the rate of mass transfer. A multiple layer approach was taken to perform more than one set of experiments on each model. In this approach, a thin layer of nickel separates the deposited copper layer under different flow conditions. The rnodels were first filled with a polymer-based resin and cut, subsequently, cross-wise at different locations using a band-saw. An optical microscope was used to measure the thickness of copper layers.
For the pipe, laminar flow was studied at Reynolds numbers 1750, 1225, and 700. Laminar results in the straight pipe were cornpared with the analflical
Leveque solution. Some data points are slightly overestimated which partly might be due the uncertainties in measuring the thickness of the copper layers. The local mass transfer values were integrated to make a cornparison with the average Shenivood number obtained previously by the fem-ferrocyanide electrochemical technique. A close agreement resulted from this cornparison.
For the turbulent straight pipe two different runs were performed; Re=5925 and 4583. The integrated local mass transfer values showed a good agreement Chapter 6 Local Mass Transfer 165 with the average Shewood number obtained previously, using the fem- ferrocyanide electrochernical technique. Turbulent results also showed a shorter mass transfer entry region than in laminar flow.
For the bifurcation, higher rnass transfer coefficients along the inner wall were observed because of the formation of a new concentration boundary layer.
The inner and outer wall mass transfer coefficients converge towards the same value further downstream. Mass transfer results in the trunk, for the laminar flow regime, are comparable with those obtained for the laminar straight pipe and hence with the Leveque analytical solution. Average Sherwood numbers, obtained by integrating the local values, are in good agreement with the
integrated mass transfer coefficients obtained previously using the fem- ferrocyanide electrochemical technique.
In an attempt to find a link between flow and mass transfer phenomena, it
was found that S~SC~and G/2 demonstrate analogous behavior in the
branches. Qualitative agreement is observed, most likely because wall shear
stress data are calculated from velocity profile. The larger discrepancy in the
apex area may result from the larger level of uncertainty in the apex area for both
the wall shear stress and local mass transfer data. 7 SUMMARYAND CONCLUSIONS
7.1 Sumrnary and Conclusions
A novel technique for the fabrication of nickel flow models has been used to
produce sections of a straight pipe and a simplified bifurcation model.
The electrochemical technique with the fem-ferrocyanide system has been
applied to obtain integral rnass transfer coefficients in these test sections at
Reynolds numbers ranging from 250 to 9000, at the four Schmidt numbers of
131 0,2470,3100 and 5655. Cha~ter7 Summarv cnd Conciusions 167
Mass transfer results in the pipe are in agreement with published results for
both laminar and turbulent flows and are correlated, respectively, by the
following equations:
S h = 0 .O 1~R~~.~~sc~.~~
The use of an upstream anode in addition to the downstream anode led to
higher mass transfer at the cathode with laminar flow because of the
additional near-wall ions produced by the upstream anode.
For the straight pipe, with increasing Schmidt number, the Reynolds number
at which the effect of turbulence on mass transfer is obsewed, is delayed
because of the decreasing mass transfer boundary layer thickness relative to
the viscous su blayer.
For the bifurcation, the following two equations represent the averaged mass
transfer results in the laminar (Re<1500), and turbulent (3300 reg imes, res pectively: Sh=O.Oi? R~O-~SCO*~~ The mass transfer patterns in the bifurcation were different because of the potential onset of flow separation along the outer wall and formation of a new boundary layer dong the inner wall of the branches. Alsol transition from laminar to turbulent flow as well as the corresponding mass transfer behavior Cha~ter7 Summary and Conclusions 168 take place over a broader range of Reynolds nurnbers because of a difference in the trunk and branch Reynolds nurnbers. The electrochemical technique with the ferri-ferrocyanide system has been used to obtain transient mass transfer coefficients for smooth pipes. Two distinct transient regions were observed in al1 experiments, different from the single transient corresponding to heat transfer. In the first transient region, the process is wntrolled by the surface reaction rate. During this time the wall concentration of the reactant drops to zero and the boundary layer for diffusion-controlled mass transfer is being established. In the second region, diffusion of the reacting ions through the boundary layer govems the process. In the first region, the rate of change of the mass transfer coefficient with time is slow; as the rate speeds up in the second region, a "humpn is observed where the slope changes. Support for this hypothesis was gained from measurements of reaction rate versus cell voltage (on the polarization curve plateau). A linear relationship was observed. The location of the hump, at which control is changed, moved toward later times as cell voltage was decreased. In the mass transfer-controlled transient region, the transfer rate for laminar flow is proportional to R DI^)'", in accordance with the Higbie penetration theory and independent of velocity and distance. By analogy, the data also confirm the heat transfer results. Chapter 7 Summaty and Conclusions 169 For laminar flow during the transient period, the boundary layer thickness grows at a rate of tqn, independent of velocity and distance. An expression for film thickness can be derived as: The time which is required to attain to the steady state in the laminar regime can be expressed as: The same phenomena were observed in transient turbulent flow. While the kinetically controlled transient is comparable to its laminar wunterpart, the transfer rate during the diffusion-controlled transient is higher than that in larninar flow. This results from the influence of velocity or Reynolds number on the transfer rate which varies with ~e'"(~/t)'~.Those observations and the higher steady state transfer in turbulent flow resuit in a longer kinetics- controlled first transient and a shorter mass transfer-controlled second transient period compared to those in laminar fiow. The laser photochromic tracer method has been used to study fluid phenomena under steady laminar flow in the plane of symmetry of a simplified aortic bifurcation at Reynolds numbers 500, 600, and 750. The flow field in this study may be categorized by four different zones: 1) In the first zone, Poiseuille-like flow govems the tnink. Chapter 7 Summarv and Conclusions 170 2) The second zone, initiated from the latter part of the trunk. is characterized by an increase in wall shear rates. Deformation of the cross section may create a secondary flow in this region resulting in higher shear rates. The converging upper and lower walls are sources, with flow toward the core region, directly and along the walls. The sink region seems to belong to the centerplane, moving from the tube axis to the lateral wall further downstream. 3) The third region is characterized by higher shear rates along the inner wall, mostly due to the formation of a new boundary layer, and lower shear rates along the outer wall. This region is initiated from the apex and extends to approximately two branch diameters downstream. 4) The fourth zone starts further downstream, where again Poiseuille-like flow redevelops and the outer and inner wall shear rates converge towards the same values. Flow accelerates continually along the length of the branches due to the gradua1 reduction of the cross section, implying an increase of the wall shear rates in the latter part of the branches. Time-average local mass transfer coefficients have been obtained by a novel electrodeposition technique in a straight pipe and a Y-bifurcation fiow model which is an idealized model of the human aotto-iliac bifurcation. This model was the same one which used previously to study average mass transfer by the ferri-ferrocyanide electrochemical technique. This novel technique is based on a diffusion-controlled deposition rate at the cathode surface. The anode is made of copper and the cathode is made of Cha~ter7 Summcm and Conclusions 171 nickel. Electrolyte solution consists of H2SO4(high concentration) and CuS04 (low concentration). After a period of time when a layer of copper is deposited on the cathode surface, the thickness of this layer is proportional to the rate of mass transfer. For the pipe, laminar flow was studied at Reynolds numbers 1750, 1225, and 700. Laminar mass transfer results for the pipe agreed well with the analytical Leveque solution. The local mass transfer values were integrated to make a comparison with the average Shenvood number obtained previously by the fem-ferrocyanide electrochemical technique. A close agreement resulted from this comparison. For the turbulent straight pipe two different runs were performed; Re=5925 and 4583. The integrated local mass transfer values showed a good agreement with the average Shenrvood number obtained previously, using the ferri-ferrocyanide electrochemical technique. Also, turbulent results showed a shorter mass transfer entry region than in larninar flow. For the bifurcation, local mass transfer coefficients have been obtained at Reynolds numbers of 4345, 1750, and 750. Local mass transfer coefficients in the tnink follow the same general patterns as those in the straight pipe. In the branches, the higher mass transfer coefficients along the inner wall are due to the formation of a new concentration boundary layer. The inner and outer wall mass transfer coefficients converge towards the same value further downstream. Chapter 7 Summaty and Conclusions 172 The convergence of the inner and outer wall mass transfer seems Reynolds number dependent. lnner and outer wall Sherwood numbers converge about 6D and 4D from the apex for the Reynolds numbers of 750 and 1750, respectively. For turbulent regime the inner and outer merge more rapidly wmpared to the larninar flow. In our experiments, after about 2D from the apex, the inner and outer wall Sherwood number rnerge towards the same value for Re4345 In the trunk, the developing mass transfer boundary layer reçults in a different pattern of the mass transfer coefficients as opposed to their flow (shear stress) wunterpart. The rate of mass transfer decreases with distance white the fully developed flow produces a constant wall shear stress in the tnink. Within the branches, rnass transfer and wall shear stress follow similar patterns both on the inner and outer walls. The Pohlhausen theoretical analysis of heat transfer in the laminar boundary layer for flow over a flat plate has been used to find a possible link between fiow and mass transfer phenornena at the branches. stscY3 and G/2 showed analogous behavior in the branches. It seems that since wall shear stress data is extracted from velocity profiles, the contribution of fom drag on skin friction is quite negligible. The lower transfer phenomena, both momentum and mass, along the outer wall of the branches are coincident with the localization of atherosclerotic lesions and arterial plaques. Cha~ter7 Sumrniwv and Conclusions 173 7.2 Relevance to Atherosclerosis In this study, the obtained mass transfer coefficients and wall shear stress along the outer wall were lower cornpared to those along the inner wall. As rnentioned before, the outer walls of bifurcations are susceptible to the atherosclerotic plaques. Lesions distribute mainly along the outer walls of the bifurcation, whereas the walls of the flow divider (apex area) and the inner walls further downstream are less affected (Zarins, 1983). It has been suggested that a threshold value may exist below which plaque deposition tends to occur (Ku, 1985). In addition. low shear stress rnay interfere with endothelial surface turn- over of substances essential both to artery wall metabolism and to the maintenance of optimal endothelial metabolic function (Robertson, 1 968). Low wall shear stress induces endothelial injury (dysfunction) increasing permeability and expression of cellular adhesion molecules on the endothelial surface. Arterial remodelling and adaptive intima1 thickening with monocyte accumulation and foam cell formation adjusts wall shear stress. In other words, the physiological response to low shear stress demands a reduction in diameter and as a consequence, thickens the intima. The outer walls also showed a Iower mass transfer coefficient. As a result of low mass transfer rate, the rnacromolecules could not easily diffuse back to the mainstrearn resulting in an increases of the particle residence time. An increase in particle residence time could thus prolong exposure of certain portion of the vesse1 wall to atherogenic particles resulting in an increase of wall Chapter 7 Summmv and Conclusions 174 concentration of macromolecules which, in tum, acwunt for focal plaque formation. The other important factor is the filtration velocity of plasma through the vessel wall. The higher the filtration velocity, the higher and faster the accumulation of the particles at the blood-endothelium boundary. Probably the endothelial LDL permeability is concentration dependent and increases as the wall concentration elevates (Karino, 1990). The other factor that might alter the permeability of the vessel wall is the blood pressure. The lower the near-wall velocty (or wall shear rate) and the higher the blood pressure, the higher the permeability of the wall. In conclusion, although wall shear stress can affect the morphology of endothelial cells, thus possibly affecting their biological function, mass transfer phenornena at the blood-vesse1 wall boundary almost certainly play an equally important role in the localized pathogenesis and development of atherosclerosis in human circulation. The current work is, to our knowledge. the first application of the electrochemical technique for mass transfer rneasurements in a situation of relevance to hemodynamics. There is obvious room for much further effort and refinement. As already mentioned, in addition to the average mass transfer coefficients, local measurements are essential for a better understanding of both flow and mass transfer phenomena. This is particulariy important for some issues such as potential flow separation from the outer walls of the bifurcation branches. Some such measurements have been made with a new copper plating technique and results are presented in chapter 6. Flow visualization results (chapter 5) for selected laminar flow conditions shed further light on the phenomena in a 175 simplified bifurcation model. A combination of these three tools would provide a very powerful means to experimentally study any transport phenornena with even complicated geometry. However, as it was the very first step in exploiting such an electrodeposition in the technique to study local mass transfer, the need for further improvements in the technique are evident. One of the drawbacks in this technique was the invasive technique of cutting the model to capture the thickness of deposited copper. Here, two different methods are proposed which may eliminate such a shortcoming. Ultrasonic pulse methods of nondestructive testing (N DT) are CU rrently being used in industry to measure the thickness of solid layers. One of the most widely used ultrasonic NDT methods for this purpose is the "pulse-echo" method, which involves the use of transducers which act first as emitters of short ultrasonic pulses and then as receivers to detect echoes from interfaces. A fairly accurate thickness measurement can be achieved based on measurement of time difference between the echoes. Ultrasonic instruments with resolutions of a few microns are currently available on the market (Ensminger, 1988). Using a light source may be the other potential means to measure the thickness. The reason that an object is red is because the reddish spectrum of the light is reflected while the other spectrums are absorbed. Two color laser beams such as: red and blue or red and green, can be used. The amount of the absorption of the blue or green which in tum is proportional to the thickness of copper, can be the basis for the measurement of the Chapter 8 Recommendatiuns 177 deposited copper layer. One other advantage of this approach is that since the trançparency of the copper layer may govern the degree of accuracy of this technique, the layers are preferred to be as thin as possible. Hence, the time of deposition will be decreased dramatically compared to those implemented in this study. Pulsed flow requires adaptation of the technique to transient conditions. Only after these conditions are well understood, can one properly interpret data obtained under pulsatile flow conditions, for average as well as local values. Chapter 4 provided some insights as to how the electrochemical system can be adapted for unsteady fiow. Employing a proper pump which is capable of producing a physiological pulsatile flow, will provide more realistic mass transfer results. Although the electrochemical technique with the ferri-ferrocyanide has been largely used in different mass transfer experiments, the kinetics of the reaction at the surface is not clearly understood. The reaction rate issues were brought up in the transient mass transfer section of this study. It seems that there is much to be discovered about the first transient region dunng which the surface reaction rate is wntrolling the process. Further experiments can be conducted which cover a wider range of femcyanide concentration, cell voltage and solution temperature. This information will provide the necessary means to explore reaction order and dependence of the reaction on the voltage and temperature. Chapter 8 Recommendations 178 Once the features of the surface reaction are untangled, a model can be established to explain the mechanisms involved in the first transient period of the mass transfer experiments. The geometry used here is obviously a great simplification of the actual aortic geometry. It is possible to build a more realistic model, with appropriate curvature, further branching and undulating walls and to make measurements. Flow phenornena have been studied both experimentally and with the computerized fiuid dynamics (CFD) approaches in such complex geometries. Also, mass transfer measurements may well be done at some point. The Schmidt number range in blood is some 100 times higher than the range covered here. Some work with glycerin added to the system has been initiated. It can increase the Schmidt number by one order of magnitude but poses its own experimental problems. The inclusion of glycerin and sodium hydroxide to the electrolyte solution of fem-ferrocyanide provides an unfortunate combination which results in deterioration of ferricyanide ions. The rate of deterioration apparently is directly proportional to both the sodium hydroxide concentration and the temperature. The obtained information can be employed to perform the experiments efficiently in order to study mass transfer in a very high Schmidt number. The laser photochromic tracer method has a great potential to study flow with fine resolution. A more powerful pump or a gravity flow with a large liquid level height can provide the higher Reynolds numbers necessary to study the structure of the possible flow separation at the outer wall of the aortic bifurcation model. It should be noted that the average physiological Reynolds number in the human abdominal aorta is 640 (Milnor, 1989), which is basically covered by the flow visualization experiments. However, a reduced cross sectional area due to stenosis or other cardiovascular disorders will increase the Reynolds number and possibly cause local turbulence. The Reynolds number range in mass transfer experiments is extended to turbulent fiows also in order to study the fundamentals of transition frorn laminar to turbulent flows. 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Circulation Research, 53, 502-514. Zemplenyi, T., (1968), Enzyme Biochemistry of the artenal wall as related to atherosclerosis. Chapter 16 Lioyd-luke, London. Re ferences 199 Zhao, W., and Trass, O., (1997), Electrochemical mass transfer measurements in rough surface pipe flow: geometrically similar V-shaped grooves. /nt. J. Heat Mass Transfer; 40,2785-2797. Appendix A: Sample Calculations; Local and Averaged Mass Transfer Experiments A.1 Local Mass Transfer Experiments (Copper Deposition) Temperature = 298 (K) Density of the solution = p, = 1188 (kg/m3) Kinematic viscosity = v = 1.38~1o4 (m2/s) Diffusivity = 4.049 x 10'" (m2/s) Concentration of cupric sulphate = 8 (mole/m3) = 0.008 (kmol/m3) Time of deposition = t = 43200 (s) Thickness of deposited copper = h,= 7.2 x 10.~(m) Density of the copper metal = p, = 8960 (kg/m3) lnside diameter of the pipe = d = 1.86 x IO-^ (m) A.2 Averaged Mass Transfer Experiments (ferri-ferrocyanide experiments) As mentioned previously, difïùsivity of fenicyanide in the electrolyte solution. and hence Schmidt number are adjusted by changing the concentration of sodium hydroxide and temperature. Two different concentrations of sodium hydroxide at two temperatures provided four different Schmidt numbers for the steady state averaged mass transfer experiments, and one range of sodium hydroxide at two temperatures provided two ranges of Schmidt number for the transient mass transfer coefficients. Here, only a sample calculation for the pipe in steady state experiments will be demonstrated. All physical properties, together with Schmidt numbers, were calculated based on data provided by Bourne et ai. (1985). Temperature = 301OK Density of the solution = p, = 1 120 (kglm3) Kinematic viscosity = v = 1.45 x 1o4 (m2/s) Diffusivity = 4.71 x 10"~(m2/s) Concentration of femcyanide = 14.3 (mol/m3) Concentration of ferrocyanide = 19 (mollm3) Concentration of sodium hydroxide = 3000 (mol/m3) Limiting current = 0.0359 (A) Surface area = 8.77 x 10' 3 rn 2 Sc=31O0 Flow rate = [(0.39 x 59) + 3.42 ] x 1od x =2.77 10-5 (rn3lS) (8.04- 1) Q 2*77 O-' Velocity = -- ' = O. 102 (m/s) TT Tr -d2 -x(0.0186)~ 4 4 Appendix B: Calibration of Rotameters Electrolyte flow rates were measured by three rotameters. Figures 8.1-8.4 are the calibration curves for the volume flow rates against rotameter readings, determined experimentally by weighing the amount of water fiowing through the meter over a period of time. The least-squares curves are fitted through the points shown. The equations of these calibration curves are: Flowmeter #1: Q (mlls) = 6.12 x (% Flow) + 11.32 Flowmeter #2: Q (rnlls) = 1.48 x (% Flow) Flowmeter #3: Q (mlls) = 0.39 x (% Flow) + 3.42 Flowmeter #4: Q (mlls) = 52.56~(% Flow) + 24.39 Amendix B Calibration of Rotameters 204 O 10 20 30 40 50 60 70 80 90 100 K Flow Ftgure 6.1. Callbration curve for flowmeter #î O 10 2a 30 40 50 60 70 80 90 100 X Flow Flgure B9. Calibration cunre for flowmeter #2 Amendix B Calibrarion of Rotam eters 205 60 K Flow Flgure 63.Callbntion curve for iiowmeter #3 O 10 20 30 40 5C) 60 70 80 90 % Flow Figure 8.4. Calibration curve for fiowmeter #4 Appendix C: Electrochemical Mass Transfer Measurernents C.1 Electrochemical Mass Transfer Measurement Mass transfer in electrochemical system involves transport of ions from the bulk of solution to the electrode surface in three steps. For instance, the ionic flux, N, for specie il consists of ionic migration under the electric field, transport by convection due to the bulk velocity, diffusion due to the concentration gradient. By the addition of a highly conductive inert electrolyte, ionic migration can be suppressed. The component of velocity perpendicular to the electrode is zero at the wall. The convection is eliminated there. Thus, transport is principally by diffusion near the wall. The effective diffusion includes both molecular and eddy diffusion. Their contributions to mass transfer are included in mass transfer coefficients. When a voltage is applied to electrodes, an electrochemical reaction occurs at each electrode with three steps. These are (1) Diffusion of the reactants to the electrode. (2) Reaction at the electrode surface, (3) Diffusion of reaction products from the electrode surface to the bulk, The first one is controlled by flow conditions, geometry, concentration of reacting species and physical properties. The second step is controlled by the temperature, geometry, surface conditions as well as the applied voltage. The third step is strongly dependent on the concentration of products at the surface. In electrochernical mass transfer measurements, it is preferable to have cathode Appendix C Electrochemicc! Wïss Tramfer Meusuremen ts 207 control. In the cathode control condition. as any of the three can be the rate controlling step, the experimental conditions are adjusted to warrant diffusion control. The anode is made larger; the concentration of ferrocyanide is higher than that of femcyanide to guarantee cathodically wntrolled conditions. Of the three steps taking place at the cathode, the third step will not be rate controlling at any stage due to low ferro-ferricyanide concentration. Therefore, the reaction rate at the cathode surface should be faster than the diffusion rate in order to achieve diffusion control. This can be accomplished by applying a correct voltage across electrodes. This principle can be easily illustrated by the relationship between current and voltage across electrodes. As shown in Figure (C-1), at low applied voltages across electrodes, the surface reaction rate is slow, and the current through electrodes is small. As the voltage is increased, the current increases to a limiting value from which point further increase in voltage will have no effect on the magnitude of current. This region is called the polarized region. The current is called limiting current. In the polarized region, diffusion rate of ferricyanide to the electrode surface controls the magnitude of current and the ions reaching the cathode surface react instantaneously. Then the reactant concentration at the surface is essentially zero. Further increase in the voltage will have no effect on the current up to a certain voltage beyond which the cuvent starts to increase at a very steep rate again. This is due to side reactions such as hydrogen and oxygen evolution which will commence when their correct over-potentials have been reached. Apoendix C Electrochemicaf Mass Transfer Measuremen ts 208 The magnitude of current at the polarized region is a measurernent of the diffusion rate through the concentration boundary layer to the electrode surface. Based on Faraday's iaw, the average mass flux to the cathode is given by where A is the electrode surface area. Mass transfer coefficient can be expressed by the following equation. N=K(C, -C,) (c-2) Hence, the mass transfer coefficient, Kt cm be easily calculated if values of Ili,, Cband C, can be measured. Figure CA: Polarization curve for mass transfer measurements Amendix C Electrochemiciïi ~tirrssTransfer Measurements 209 Among the three, lm and Cb are easily measurable. C, can not be measured to any degree of accuracy in conditions when reaction rate is controlling. However, in the polarized region under diffusion control, the value of C, is virtually zero. This makes calculation of mass transfer coefficients possible. After studying vanous possible electrolytes, Lin et al. (1953) and Ranz (1958) concluded that a mixture of potassium ferro-ferricyanide in the bulk presence of sodium hydroxide is the best and most suitable electrolyte for mass transfer studies. This view has not been changed over tirne. Although other electrolytes have been proposed and used, none has been used as extensively as the above. The following reactions will occur in this system. F~(CN)~>+ e -F~(cN)~' (at the cathode) F~(cN)~~-- F~(cN)~~- + e (at the anode) As the voltage between the electrodes is increased, the concentration G, of ferricyanide ions at the cathode surface falls and ultimately reaches zero. Under this condition, the electrolytic current is govemed by the diffusion of the ferricyanide ions through the concentration boundary layer that is formed adjacent to the cathode and is constant over a wide range of voltage as previously described. C.2 Advantage of Electrochemical Mass Transfer Technique The phenomenon of limiting current at a high stirring rate was recognized by physical chemist Nernst in 1904. A link between experiment and theory was established for mass transfer at a rotating disk by Levich (1962). Since Wagner Amendix C Eiectrochemicd 1'Mriss Transfer Measurements 310 (1949) used the electrochernical technique in 1949 to study free convective mass transfer, and Lin et al. (1951) used it for convective rnass transfer in the turbulent pipe flow, the method has been well established as an excellent technique for mass transport studies at liquid-solid interfaces. This technique has been widely used for varieties of problerns (Eisenberg et al., 1951; Fortuna et al., 1972; Gabe at al., 1987;Goldstein et al., 1990; Hubbard et al., 1966; Kim et al.. 1991; Law et al., 1981; Wranglen et al., 1962; Nikov et al., 1987; Selman et al., 1971). The popularity of electrochemical mass transfer measurement can be attributed to the advantages it holds over other common methods of measuring mass transfer rates, such as the dissolution of a coating on the wall of the duct. These advantages are: Precision: the dissolution rnethod involves the measurement of very small changes in the coating thickness. The current, however, is a direct measurement of the mass transfer rate with a suitable arnmeter. Speed: the cell current can be measured in a few seconds, while the dissolution method requires a new coating for each measurement, and several minutes of transfer for a sufficient thickness change. Constant surface conditions: the dissolution of a coated surface causes changes in shape and roughness which rnay affect the transfer rates, while the surface in the electrochemical method undergoes no such change. The last factor is, of course, the overriding reason for the choice of this technique in the present study. Moreover, as pointed out by Grassmann (1979). further advantages of electrochemical mass transfer technique are: Ap~endixC Efectrochemical Mass TransferMeasurements 211 (A) Local measurements are possible. Reiss and Hanratty (1962,1963) employed circular cathodes with diameters between 0.13 mm and 6 mm. such dimensions are possible since, contrary to thermal isolation, electrical isolation needs only very small distances. Electrolysis cells using a reference electrode in addition to the cathode and anode have also been employed in order to control the potential of the working electrode (cathode) with respect to the reference electrode (Sydberger and Lotz. 1982; Rizk et al.. 1996). (2) The possibility of instantaneous rneasurements allows investigation of the pattern of turbulence or mass transfer of unsteady and fiuctuating processes. However, care has to be taken since the capacity of the electric double layer, which seems to be 10-70 microfaraday 1 (cm)2 may cause some delay. The time resolution may be reduced to about one micro second. (3) It is nearly impossible to measure heat transfer coefficients of liquids with high Prandtl and Reynolds numbers. The heat generated by dissipation would be prohibitive. Here the electrolytic method fills the gap. C.3 Problems in Electrochemical Mass Transfer Measurements In using this experirnental technique, there are three problems: inability to obtain diffusion control. side reactions and conditions of electrode surface. According to Lin et al. (1951 ) and Ranz (1958). the ferro-ferricyanide system has the highest reaction rate and there is no contamination of surfaces from main reaction as al1 ions involved are highly soluble. Although there is a small amount of deposition occurring from side reactions, it is not a severe problem. One major Amendix C Electrochemicd Bfûss Transfer Measurements 212 problem that Dawson (1958) had, was the rapid electrode deactivation. In order to overcome this problem, he had to use a laborious cleaning procedure which involved acid, alkaline and cathodic cleaning of the electrode before each run. Even after this cleaning, there was a problem of reproducibility of the results. Dawson's problerns anse from the industrial grade nickel for his surface fabrication. Here, a new fabrication method was used to make nickel models and very good polarization curves were obtained for al1 the runs. The deactivation of the surfaces was found to be not severe in this study due to the pure nickel. Appendix D: Volumetric Analysis Volurnetric analysis is a quantitative method of anaiysis in which the ionic species being determined is completely reacted with an accurately measured volume of a reagent of exactly known concentration called the standard solution. There are two ways of obtaining a solution of exactly known concentration. If the chemical compound is available in a state of high purity, the solution may be prepared by dissolving an accurately weighed quantity of primary standard substance in an accurately measured volume of solution. Otherwise, a solution of approximately known concentration must be standardized against a primary standard substance by titration. Primary standard substance must possess certain characteristics: 1) should be easiiy obtained and preserved in a state of high purity; irnpurities should not exceed 0.01 -0.02 % 2) should have a high molecular weight 3) reaction with solutions being standardized must be stoichiometric 4) error in titration end point determination must be negligible, and a suitable indicator for end point detemination must be available. Every volumetric analysis involves a titration; the process by which a standard solution (or titrant) is delivered from a burette into a solution of the substance being analyzed until the amount of standard solution added is exactly equivalent to the arnount of substance being determined. This point of the Appendix D V'haetric Analvsis 214 titration is temied the equivalence point and is detected by some physical change which occurs in solution at or near the equivalence point. The point at which the changes manifest themselves is called the end point of the titration. The most common visual signal involves a color change but there are other signals, for example, formation of a precipitate which occurs in some reactions. Sometimes the species present in either the titrant or the solution to be titrated is responsible for the physical changes; otherwise a carefully selected auxillary solution. known as an indicator signals the end of a titration. Reactions comprising volumetric analyses fall into one or two categories. The first of these consists of the ion combination reactions, including proton transfer reactions e.g. the titration of a sample of hydrochloric acid with sodium hydroxide. The second category consists of oxidation-reduction reaction in which the chernical change is achieved through transfer of electrons. e.g. the titration of a sample of ferricyanide solution with sodium thiosulphate. D.l lodometric Tittation of CuS04 cu2+reacts with iodide ions according to the following reaction equation: 2 cu2++ 51- 2Cui + la- The reaction is essentially an oxidation-reduction reaction in which cu2+ is reduced by iodide ions to Cu* and it is the basis for the analysis of copper. Cul is precipitated while 13- remains in solution. Sodium thiosulphate which reacts quantitatively with the triiodide ion is used as the titrant solution. Appendix D Valum et ric An al vsis 2 15 A dilute solution of starch is used as the indicator since starch gives a dark blue wlor with 13-. Because the indicator decomposes in solutions which contain a high concentration of iodine it is not added until most the iodine has been titrated as indicated by a light yellow color of the solution. Near the end of the titration when the solution turned pale yellow, potassium thiocyanate was added to improve the end point and then Na2S2S3titrant was added until the blue color produced by the indicator just disappeared. The procedure for titration was the following. To a certain volume (i.e. 20 ml) of the solution sample, 10 ml of 1N potassium iodide (KI) solution was added. After mixing the solution, the liberated iodine (brown in color) was titrated with the sodium thiosulphate, immediately. W hen the brown color had changed to a pale yellow, 3ml of starch indicator solution was added, which produced a black-blue color. The titration was contiued until the blue color had faded somewhat, but had not altogether disappeared. At this stage, potassium thiocyanate (KSCN), was added and mixed thoroughly. Again the titration continued until the bluish-gray color just changed to white, with a pinkish tinge. Three titrations were usually perfomed, and the average titre was used in calculations. D.2 lodometric titration of K3Fe(CN)6 The procedure for titration of ferricyanide was the f~llowing.A sample of the solution was acidified with concentrated hydrochloric acid, and then 1M potassium iodide (KI) was added to the sarnples. Liberated iodine was titrated with standard sodium thiosulphate to a light yellow color (from an original reddish brown color) before starch solution was added. Reactions occur according to the following equations: 13- 2 F~(CN)~>+ 3 I- 2 F~(CN)~&+ 13- 13- + 2 3 1- + ~405- In strong acid solution, the first reaction proceeds from left to right but is reversed in neutral solution. After initially adding thiosulphate, a few drops of 1% starch solution was added and titration continued until the solution turned colorless, indicating the end-point had been reached. Appendix E: Experimental Uncertainty The purpose of this section is to discuss the major sources of error in the mass transfer experimental results. The evaluation of experimental uncertainties must be partially based on the experimenter's judgement. Hence such an evaluation is necessarily somewhat subjective. Nevertheless, an attempt has been made to set limits which are compatible with the observed degree of agreement among redundant measurements and the degree of reproducibility among repeated measurements. Replicated data are not available for most runs. Measurements of limiting current were usually taken only once. Only a few trials used repeated measurements to assure nickel surface activity. The deviation in the limiting current obtained from repeated measurernents was lower than 1% in most cases. which probably shows a reasonably good reproducibility. However, this small drift may not be a realistic representation of errors associated with the mass transfer experiments, because it does not agree with scatters observed in the grap hical results. Uncertainties associated with the results should be estimated using a reliable statistical method. E.l Uncertainties in SingleSample Experiments In many engineering experiments, it is costly, and sometimes impractical, to estimate al1 of the uncertainties of observations by repetition. For single- sample experiments, the uncertainty distribution is defined as the distribution of errors that the experimenter believes would be found if the variables were &mendix E Experim enta1 Un certaintv 218 sampled a great many times (Kline et al., 1953). If the experimental result, R, is a function of n independent variables (equation E-1) whose uncertainties are known, the uncertainty for R can be calculated using equation (E-2). R = R (vi, VZ, ...... vn ) In these equations, wl, wz, ...... w, are the uncertainties for independent variables (vl, v2, ...... vn) and w~ is the uncertainty interval for the result (R). E.2 Uncertainties in Experirnental Results The experirnental data of single quantities for mass transfer are divided into four subgroups. The first one is the geometric parameters such as L and D; the second includes various physical quantities such as temperature, concentration, resistance and voltage drop across the resistor; the third group includes Row rates; the last one includes the physical properties of density, viscosity, and diffusivity. There are two main sources of errors: systematic errors, random errors. E.2.1 Systematic Errors Systematic errors refers to an error which is present for every measurements of a given quantity; it may be caused by a bias on the part of the experimenter, a mis-calibrated measuring instrument, a sample of measurements selected which do not fairly represent the whole population, etc. Systematic errors refiect the accuracy of the experirnent. In Our mass transfer experiments, systematic errors likely originate from three different sources: (a) Errors in calibration of an instrument. e.g. calibration of rotameters, standard concentration of sodium thiosulphate and hydrochloric acid for chemical analysis. (b) lnability of an instrument to reflect the true values of the variables being measured, e.g. partial deactivation of nickel surfaces. (c) lnability to duplicate the exactly same physical property and same operating conditions when the measurements are performed in two different experiments. All the equipment was calibrated prior to use. This included the following: thermometers, rotameters, resistors, voltmeters, ammeters, multimeters and scales. In different stages of experirnents, they were checked again and no drifting was found. In conclusion, al1 attempts were made to eliminate the possible sources of systematic errors. E.Z.2 Random Errors The other source of error is random error. This refers to the spread in the values of a physical quantity from one measurernent of the quantity to the next, caused by random and indeterminate fluctuations. This type of error can also be caused by the limitations of the measuring equipment, which is sometimes referred as reading error. Random errors affect the precision of the experirnent. Aspendix E Exmvhental Uncertain& 220 Possible random errors are: Scatter in the instrument calibration data. Reading errors of an indicating instrument; e.g. a Ructuating temperature. fluctuating voltage and misjudging the neutral point in chemical analysis. Errors associated with fluctuating physical operating conditions such as the flow rate and temperature variations. Errors in geometric conditions. Random errors associated with the instrument calibrations are defined by the standard deviations of the data from their curve fitting equations. The experimental uncertainty estimates are summarized in the following tables. As shown, some quantities are directly measured variables and others are calculated variables. The estimation of the calculated variables is based on the method of Kline and McClintock (1953). Variable Value or ranqe Maximum Probable Error A bsolute Rela rive Geornetric Parameters Length Length Diameter Diameter Amendix E Eiperimeniai Uncertain tv 22 1 Phvsical Ouan tities Temperature Sodium thiosulphate concentration Sodium thiosulphate volume Ferricyanide volume Ferricyanide concentration Sodium thiosulphate concentration (LW Sodium thiosulphate volume Ferricyanide volume Fenicyanide concentration HCI NaOH NaOH Appendix E Experim enta1 Un certaintv 222 Resistor Resistor Voltage Rotameters Rotameter 1 15-100 divs 0.5 divs 0.5-3'10 Rate of 0.5-3% Rotameteri Rotameter 2 15-100 divs 0.5 divs 0.5-3 % Rate of 0.5-3 '10 Rotameter2 Rotameter 3 15-100 divs 0.5 divs 0.5-3 % Rate of 0.5-3% Rotameter3 Rotameter 4 15-100 divs 0.5 divs 0.5-3 % Rate of 0.5-3% Rotameter4 Physical Properties Density Viscosity Diffusivity The uncertainties of the reduced variables are calcuiated by writing the variables as functions of the measured independent variables and applying equation (E-2). Reynolds number uncertainty Maximum= 3.1% Schmidt number uncertainty Maximum= 4.1 % Sherwood number uncertainty Maximum= 4.3%