Citation
Time-dependent enhanced heat transfer in oscillating pipe flow

Material Information

Title:
Time-dependent enhanced heat transfer in oscillating pipe flow
Creator:
Zhang, Guo-Jie, 1941-
Publication Date:
Language:
English
Physical Description:
xiii, 162 leaves : ill. ; 28 cm.

Subjects

Subjects / Keywords:
Boundary conditions ( jstor )
Heat ( jstor )
Heat flux ( jstor )
Heat transfer ( jstor )
Pipe flow ( jstor )
Pumping ( jstor )
Subroutines ( jstor )
Temperature distribution ( jstor )
Temperature gradients ( jstor )
Velocity ( jstor )
Heat -- Transmission ( lcsh )
Laminar flow ( lcsh )
Pipe -- Fluid dynamics ( lcsh )
Viscous flow ( lcsh )
Genre:
bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

Notes

Thesis:
Thesis (Ph. D.)--University of Florida, 1988.
Bibliography:
Includes bibliographical references.
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by Guo-Jie Zhang.

Record Information

Source Institution:
University of Florida
Holding Location:
University of Florida
Rights Management:
Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Resource Identifier:
024469817 ( ALEPH )
AFK9835 ( NOTIS )
19897631 ( OCLC )
AA00004834_00001 ( sobekcm )

Downloads

This item has the following downloads:


Full Text











TIME-DEPENDENT ENHANCED HEAT TRANSFER
IN OSCILLATING PIPE FLOW











By

GUO-JIE ZHANG


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY






UNIVERSITY OF FLORIDA


1988




TIME-DEPENDENT ENHANCED HEAT TRANSFER
IN OSCILLATING PIPE FLOW
By
GUO-JIE ZHANG
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1988


To my beloved motherland


ACKNOWLEDGEMENTS
The author wishes to express his deep gratitude to
the chairman of his committee, Dr. Ulrich H. Kurzweg, for
the valuable assistance and advice in guiding this research
work to its completion and for thoroughly reviewing the
entire manuscript leading to the realization of this
dissertation. Also he wishes to express his appreciation to
Drs. E. Rune Lindgren, Lawrence E. Malvern, Arun K. Varma
and David W. Mikolaitis for the many helpful discussions in
the formulation of the problem and constructive suggestions
for overcoming many difficulties in the solution process.
Thanks are expressed here also to Dean Eugene R. Chenette,
to the department chairman, Dr. Martin A. Eisenberg, and to
Dr. Charles E. Taylor for their support in allowing the
author to pursue his educational goals within this lovely
country. Part of the work presented here was funded by a
grant from the National Science Foundation, under contract
number CBT-8611254. This support is gratefully
acknowledged.
iii


TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS iii
LIST OF FIGURES vi
LIST OF TABLES ix
KEY TO SYMBOLS X
ABSTRACT xii
CHAPTERS
I INTRODUCTION 1
II FORMULATION OF THE PROBLEM 18
Governing Equations 18
Boundary Conditions 24
Model 1 24
Model 2 28
Model 3 30
Initial Conditions 33
Calculation of Tidal Displacement 35
Effective Heat Flux 37
III NUMERICAL TECHNIQUES EMPLOYED 38
Transformation 40
Crank-Nicolson Method for Momentum Equation . 46
ADI Method for Axisymmetric Heat Equations. . 48
Convergence Criteria 57
Grid Generation 59
IV NUMERICAL RESULTS AND DISCUSSION 62
Part 1. Oscillatory Pipe Flow Features .... 64
Velocity Profiles 64
Lagrangian Displacements 69
Tidal Displacements 72
Phase Lags 79
Part 2. Enhanced Heat Transfer Investigation 85
Periodic Temperature Build-up
in Thermal Pumping Process 87
IV


Temperature Distribution in Model 1 95
Temperature Distribution in Model 2 100
Temperature Distribution in Model 3 108
Heat Flux versus Tidal Displacement
(Model 2) Ill
Influence of Thermodynamic Properties .... 115
Heat Flux versus Tidal Displacement
(Model 1) 119
Influence of Wall Thickness 124
Influence of Pipe Diameter 126
Variation of Axial Temperature Gradient
In Model 3 130
Comparison of Enhanced Oscillatory Heat
Transfer and Heat Conduction 133
Enhanced Feat Flux as a Function of
Wormersley Number 135
Tuning Curves 137
V CONCLUDING REMARKS 143
APPENDIX: ETP COMPUTER CODE 149
REFERENCES 157
BIOGRAPHICAL SKETCH 161
V


LIST OF FIGURES
Figure Page
1-1 F (a) Curve 7
2-1 Thermal Pumping Device 19
2-2 Model 1 Fixed End Temperature Model 25
2-3 Model 2 Periodic Heat and Cold Sources
on Insulated Wall 29
2-4 Model 3 Pipe with Extended Conducting Sections 31
3-1 Grid System Used in the Numerical Simulations. 39
3-2 Coordinate Transformation 42
4-1 1-D Velocity Profiles in Oscillating Flow for
Wormersley Number a = 1, 10, 100, and 1000 . 65
4-2 Velocity Profiles (a = 5, after Uchida) .... 66
4-3 Magnified View of Velocity Profile near Wall . 67
4-4 Lagrangian Displacement for a =0.1, 1.0 ... 70
4-5 Lagrangian Displacement at a = 10 71
4-6 Dimensionless Cross-Section Averaged
Displacement versus Time (a = 0.1 1.0) . 73
4-7 Dimensionless Cross-Section Averaged
Displacement versus Time (a = 2 50) 73
4-8 Relationship Between Dimensionless Tidal
Displacement AX and Wormersley Number a ... 77
4-9 Relationship Between Tidal Displacement Ax and
Exciting Pressure Gradient in Water 78
4-10 Phase Variation Along Radius for Different
Wormersley Numbers 84
vi


4-11 Temperature Build-up Process in Oscillating
Flow (Model 1, a = 1, Ax = 2cm) 89
4-12 Temperature Build-up Process in Oscillating
Flow (Model 1, a = 1, Ax = 5cm) 90
4-13 Temperature Build-up Process in Oscillating
Flow (Model 1, a = 1, Ax = 10cm) 91
4-14 Temperature Build-up Process in Steady Flow
(Model 2, Uave = 1.5 cm/sec) 93
4-15 Build-up Time versus Tidal Displacement
(Model 2, a = 1) 94
4-16 Temperature Distribution in Oscillating Pipe
Flow (Model 1, a = 1, Ax = 1cm) 96
4-17 Temperature Distribution in Oscillating Pipe
Flow (Model 1, a = 1, Ax = 2cm) 97
4-18 Temperature Distribution in Oscillating Pipe
Flow (Model 1, a = 1, Ax = 5cm) 98
4-19 Temperature Distribution in Oscillating Pipe
Flow (Model 2, a = 1, Ax = 1cm) 101
4-20 Temperature Distribution in Oscillating Pipe
Flow (Model 2, a = 1, Ax = 5cm) 102
4-21 Temperature Distribution in Oscillating Pipe
Flow (Model 2, a = 1, Ax = 10cm) 103
4-22 Temperature Distribution in Oscillating Pipe
Flow (Model 2, a = 1, Ax = 20cm) 104
4-23 Temperature Distribution in Oscillating Pipe
Flow (Model 1, a = 1, Ax = 30cm) 105
4-24 Temperature Distribution in Steady Flow
(Model 2, Uave = 0.5 7.5 cm/sec) 106
4-25 Temperature Distribution in Oscillating Pipe
Flow (Model 3, a = 1, AX = 10cm) 109
4-26 Magnified View of Temperature in the Central
Pipe Section (Model 3, a = 1, Ax = 10cm) . 110
4-27 Heat Flux in Oscillating Flow and Steady Flow
(Model 2, a = 1, Water as Working Fluid). . 112
vii


117
4-28 Influence of Thermodynamic Properties of H2O
on the Enhanced Heat Flux (Model 2, a = 1,
Ax = 10 cm)
4-29 Heat Flux versus Tidal Displacement
(Model 1, a = 1, Pr = 7.03) 120
4-30 Heat Flux versus Tidal Displacement
(Model 1, a = 3, Pr = 7.03) 121
4-31 Influence of Wall Thickness on Axial Heat Flux
(Model 1, Water-Glass, a = 1, Ax = 5cm).... 126
4-32 Influence of Pipe Diameter on Heat Flux for
Fixed Frequency (Model 3, Water-glass, a = 3,
Ax = 10cm) 128
4-33 Typical Iso-Temperature Contour in Oscillating
Pipe Flow (Model 3, Water-Glass, a = 3,
Ax = 10cm) 129
4-34 Variation of Temperature T^ and T2 versus Ax
(Model 3, Water-Glass, a = 3) 131
4-35 Comparison of Enhanced Heat Transfer and Heat
Conduction in Oscillating Pipe Flow
(Model 3, Water-Glass, a = 3) 134
4-36 Variation of Axial Heat Flux versus Wormersley
Number (Model 3, HoO-Glass, Hg-Steel,
Ax = 10cm) 136
4-37 Computed Tuning Curves (Model 3, H20-Steel and
Hg-Steel, ax = 10cm) 138
4-38 Tuning Curve versus Wormersley Number
(after Kurzweg) 140
4-39 Ratio of Heat Flux due to Conduction to
Enhanced Heat Flux versus Wormersley Number
(Model 3, H20-Steel, Hg-Steel, Ax = 10cm) . 141
viii


LIST OF TABLES
Tables
Pages
4-1 Dimensionless Tidal Displacement at Different
Wormersley Numbers 72
4-2 Phase Lags Along Radius (Working Medium: H2O,
AX = 10cm, a 0.1 20) 80
4-3 Comparison of Phase Lags with Different
Working Mediums (Ax = 10cm, a = 1, 5) 83
4-4 Comparison of Enhanced Heat Flux Using
Numerical Velocity with Heat Flux Using
Analytic Velocity (Model 3, H20-Glass,
Rl = 0.1cm, R2 = 0.15cm, Pr = 7.03, a = 3) . 86
4-5 Enhanced Axial Heat Flux via Tidal Displacement. 113
4-6 Enhanced Axial Heat Flux in Steady Flow .... 114
4-7 The Influence of Properties of Water on the
Enhanced Axial Heat Flux 116
4-8 Variation of the Axial Temperature Gradient
versus Wormersley Numbers (Water-Glass,
Ax = 10 cm) 132
ix


KEY TO SYMBOLS
X
r
t
£
*?
T
L
R1
r2
w
S
P
P
c
P
K
k
Ke
Pr
x coordinate
radial coordinate
time
coordinate (x) in transformed plane
coordinate (r) in transformed plane
transformed time
pipe length
pipe inner radius
pipe outer radius
oscillating frequency
boundary layer thickness
pressure
density
specific heat
Kinematic viscosity
dynamic viscosity
thermal diffusivity
thermal conductivity
coefficient of enhanced heat diffusivity
Prandtl number
i/p 13p/3x| a measure of the maximum axial
pressure gradient (cm/sec2)
x


a
Wormersley number a = Ju>/v
T
Temperature
7
7 = dT/dx time-averaged axial temperature
gradient
r
f = r/Ri dimensionless radial distance
g
radial temperature distribution function
u
velocity
u0
representative velocity
f
velocity shape function
X
Lagrangian displacement
DX
dimensionless tidal displacement
AX
dimensional tidal displacement
Qtotal
time averaged total enhanced axial heat flow over
pipe cross-section

axial heat flux
Subscript
f
fluid
w
wall
h
hot
c
cold
th
thermal
eq
equivalent
adj
adjacent
XX


Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
TIME-DEPENDENT ENHANCED HEAT TRANSFER
IN OSCILLATING PIPE FLOW
By
Guo-jie Zhang
April 1988
Chairman: Ulrich H. Kurzweg
Major Department: Aerospace Engineering, Mechanics,
and Engineering Science
The problem of time-dependent enhanced heat transfer
in an incompressible viscous laminar fluid subjected to
sinusoidal oscillations in circular pipes which are
connected to a hot reservoir at one end and a cold reservoir
at the other end has been examined numerically in detail.
Three models were designed for the investigation of such an
enhanced thermal pumping process and a computer code (ETP)
was developed to implement all the numerical calculations.
To increase the understanding of the mechanism of
thermal pumping, the periodic velocity profiles and
Lagrangian displacements as well as tidal displacements at
various Wormersley numbers (from a = 0.1 to 1000) were
studied. Some transient problems of enhanced axial heat
transfer in oscillating pipe flow such as the periodic final
temperature build-up process in oscillating pipe flow were
also examined. The time-dependent temperature distribution
xii


in the different models was numerically studied in detail.
The enhanced axial heat flux magnitude versus different
tidal displacements with water and mercury as the working
fluids bounded
by
pipe walls
of different
material
were
observed
and
the
quadratic
coefficients
found.
The
influence
of
the
variation
of water properties on
the
enhanced axial heat flux was numerically examined and the
results show that the enhanced axial heat flux can vary
about 150 percent even within the temperature range from 0C
to 100C. The effects of wall thickness and pipe diameter
in enhanced thermal pumping were also studied and the
optimum wall thickness was found to be about 20 percent of
the pipe radius in the water-glass combination. The tuning
effect in the water-steel and the mercury-steel cases was
examined and the results show good agreement with analytic
predictions. A comparison of the enhanced axial heat flux
with the axial heat flow due to heat conduction at various
tidal displacement and Wormersley numbers shows that the
latter is quite small and negligible provided the tuning
condition is satisfied.
This study has shown that the enhanced thermal pump
is indeed a very effective tool for those problems where
large amounts of heat must be transported without an
accompanying convective mass exchange. The investigation
also indicates that turbulent flow in the reservoirs is
preferable to laminar conditions and should receive more
attention in future studies.
xiii


CHAPTER I
INTRODUCTION
Enhanced heat transport in a viscous laminar fluid
subjected to sinusoidal oscillations in a very long pipe
which connects a hot fluid reservoir at one end and a cold
fluid reservoir at the other end (Fig. 2-1) has been
recognized and studied recently by Kurzweg [15, 16, 17, 22],
The results obtained show that with this oscillatory pipe
flow the heat transferred axially from the hot end to the
cold end can be orders of magnitude larger than that
obtained by pure molecular conduction in the absence of
oscillations. In addition, the more important thing of
interest is that this heat transfer process involves no net
convective mass transport. Major assumptions made in the
above cited studies on enhanced heat diffusion are that a
constant time-averaged non-zero axial temperature gradient
is always present in the oscillating flow and that the axial
molecular conduction along the wall and in the oscillating
fluid is negligible.
Discovery of this enhanced heat transport phenomenon
was made possible by earlier studies on axial dispersion of
contaminants within steady laminar flows through capillary
tubes by Taylor in 1953 [32], and Aris in 1956 [4]. These
1


2
earlier studies show that when a small quantity of a
contaminant is introduced into a circular pipe, the
dispersion of the resultant contaminant cloud is greatly
enhanced by the flow of the fluid. Bowden's 1965 results
show that similar dispersion effects occur in oscillatory
flow [8], This enhanced axial dispersion of contaminants in
the presence of laminar oscillatory flow within capillary
tubes was studied in 1975 by Chatwin who suggested that the
assumption of constant time-averaged axial contaminant
gradient can be made [10]. Recently, Bohn et al. extended
this work to the study of gas component transfer in binary
gas mixtures when these are confined to single tubes and a
sinusoidal pressure variation is applied [7]. Further
studies in 1983 by Watson [38] show that the effective
diffusion of contaminants is proportional to the square of
the tidal displacement. This has been experimentally
verified by Joshi et al. [13] and by Jaeger [12], both in
1983, and most recently by Kurzweg and Jaeger [19], in 1987.
All these results show that the contaminant would spread
axially in both steady and oscillatory laminar pipe flow at
rates as much as five orders of magnitude higher than in the
absence of fluid motion.
The first significant research work extending these
enhanced axially diffusion studies to the heat transfer
problem in oscillatory flow within very slender pipes or
flat plate channels is due to Kurzweg [15, 16, 22]. In


3
early 1983 Kurzweg suggested that a similar dispersion
process should occur in the heat transfer area because of
the similarity in both the governing diffusion and heat
transfer equations [16], and the first preliminary theory
was formulated in 1985 [15, 16], in which, referring to
Chatwin's idea, a time averaged constant axial temperature
gradient assumption was used. The instantaneous temperature
distribution was taken to be of the form [15, 22].
T = 7[x + RX g(r)elwt] (1-1)
where 7 = 3T/3x is the time-averaged axial temperature
gradient, is the tube radius, x is the axial distance
along the capillaries (with x = L/2 and x = -L/2 denoting
the ends) L is the pipe length under consideration, w is
the oscillating frequency of fluid, g(f) is a radial
temperature distribution, and f = r/Ri is the dimensionless
radial distance. The theoretical analysis shows that under
certain conditions such enhanced axial heat diffusivity can
indeed be significantly larger than the axial molecular
conduction [15, 16] and this has been verified by some
experimental measurements by Kurzweg and Zhao [22].
In order to better understand the physical mechanisms
of this interesting and potentially useful heat transfer
technique, we shall examine in greater detail the thermal
pumping model shown in Fig. 2-1. It is assumed that a
bundle of very thin and long tubes is connected to a


4
reservoir which supplies unlimited hot liquid at one end to
a second reservoir which supplies unlimited cold liquid at
the other end. The liquid in the pipes is oscillating
axially with an amplitude such that none of the liquid which
is originally in the middle portion of pipes ever runs into
either reservoir. That is, there is no net convective mass
exchange between two reservoirs. The largest axial fluid
dimensionless displacement (when cross-section averaged) is
referred to as the nondimensional tidal displacement and is
denoted by "AX" (it should not be confused with the
dimensional tidal displacement "ax" frequently used in the
present study) At time t = 0, the fluid within the pipes
is set into axial oscillations at angular frequency w and
tidal displacement Ax. After a short transient, this
oscillatory motion will lead to very large axial heat flows
which can be readily made to exceed those possible with heat
pipes.
Before exploring the mechanisms of this enhanced heat
transport, it is necessary to introduce some new concepts
which are commonly used in the study of this type of
oscillatory motion.
As is well known [36], for high frequency viscous
laminar axial oscillations within fluid flow along rigid
pipes, the non-slip boundary condition creates a very thin
Stokes' viscous boundary layer of thickness
S = J2u/o)
(1-2)


5
where u is the fluid kinematic viscosity. For room
temperature water at a frequency of 10 Hz, this viscous
boundary layer is approximately 1.7*102 cm. The
corresponding thermal boundary layer thickness is about
fith = t/fPr (1-3)
where Pr is the Prandtl number. Note that both 5 and 5th
decrease in thickness as the oscillating frequency w
increases.
In the theoretical analysis, it is always assumed
that a fully developed velocity profile of the oscillating
flow exists within the pipes. At high frequency w, this
flow consists essentially of a slug flow over most of the
fluid core bounded by a thin boundary layer of width 8 as
discussed by Uchida [36], Neglecting end effects, the fully
developed oscillating laminar velocity profile in pipes, due
to a periodic axial pressure gradient, is found to be [19]
U(T ,t) = U0f(r)elwt (1-4)
where UQ is a representative velocity, f = r/R^ is again the
dimensionless radial distance, f(f) the velocity shape
function, and u> the angular velocity of the oscillatory
flow. The explicit form of f(f) is
1 -
f(r)
Jn(7-ir) I
j0(/^y J
(1-5)


6
where A = Ri213p/5x|/pi/U0 is the nondimensional pressure
gradient maximum acting along the capillaries, a = R\J~Jv is
the Wormersley number measuring the ratio of inertia to
viscous forces, v is the fluid kinematic viscosity of the
carrier liguid, and p is the fluid density. This velocity
profile will reduce to the familiar Poiseuille parabolic
shape as the angular frequency w becomes small, while at
moderate frequency, f(r) has the shape demonstrated by
Uchida [36].
Another new term commonly used when dealing with
oscillating flow is the cross-stream averaged dimensionless
tidal displacement AX which can be mathematically defined as
AX =
4U0
I
Jff (r)dr
(1-6)
and on integration, yields [18]
AX
I a p/ax I
1/2 p
1 +
2
a
F(a)
(1-7)
where the complex function F(a) has the form
F()
. Jn'(Z^)
1 Jo(/=!£)
with the prime denoting differentiation. Using the
definition of Kelvin function: JQ(/-ia) = ber a + i bei a


Fig 1-1
F (a) Curve [18]


8
[1], the complex function F(a) (Fig. 1-1) can be further
written as:
F(a) = Fr (a) + i Fi(a)
ber 'a
+ i
bei'a ]
ber a
+ i
bei a J
(1-8)
This dimensionless tidal displacement is related to the
maximum of the periodic pressure gradient via [18]
AX
I 3p/ax|
1/2
1
J2
1-
a
a 1
(1-9)
a 1
Apparently, for any fixed tidal displacement Ax and
oscillating frequency w, the axial pressure gradient
required is proportional to the inverse square of the
Wormersley number when a is small; however, it is
independent of a when a is very large. This also implies
that the exciting axial pressure gradient |ap/ax| is
approximately proportional to the fluid kinematic viscosity
v and inversely proportional to the square of the pipe
radius when Wormersley number is very small (it happens
only at low , small and large v, for example, oil),
while it is almost independent of the fluid kinematic
viscosity and the pipe radius when the Wormersley is very
large (it happens only at very large R^, high w, small v,


9
for instance, a liquid metal). Note, if the tidal
displacement AX is fixed, while allowing the oscillating
frequency to change, the axial pressure gradient |<3p/dx| can
become very large when the oscillating frequency becomes
large. This is mainly due to inertial effects and not so
much due to viscous drag forces which dominate the
oscillatory flow at small Wormersley number.
With the definition of the above quantities, we are
now in a position to explore the details of the enhanced
axial heat transfer in oscillatory flow within pipes. It is
assumed that a constant temperature gradient exists along
the pipe in the axial direction and that a very large time-
dependent radial temperature gradient variation is
superimposed. When the fluid moves towards the cooler side
(we term this the positive stroke), the hotter fluid within
the pipe core which is initially brought into the pipe from
the hot reservoir produces a large radial heat flow via
conduction to the cooler portions of the fluid within the
Stokes' boundary layer and to the cooler pipe wall; while
during the negative (or reverse) stroke, i.e., when the
fluid moves towards the hotter side, the higher temperature
in the boundary layer and the pipe wall conducts the heat
back into the cooler fluid core. This coupled radial heat
conduction with an axial convective transport leads to an
enhanced axial heat flux along the entire length of the
pipes.


10
Further, from the system point of view, the heated
fluid near the cold reservoir will eventually be ejected
into the cold reservoir and mixed there with the lower
temperature liquid. Contrarily, near the hot reservoir side
the fluid within the pipes which has been cooled during each
positive stroke is pushed out into the hot reservoir and
mixes with the higher temperature liquid. This process of
thermal pumping is what leads to a time-averaged heat flow
from the hot reservoir into the cold reservoir. It differs
essentially from the working principle of a normal pump.
For a normal pump one can draw the analogy with transport of
a one-way vehicle which transports passengers as well as the
carrier from one point to another. For the thermal pump one
can draw the analogy with a two-way busline which
periodically loads and unloads the passengers (heat) from
the hot reservoir to the cold reservoir, heat can be
continually transferred, and the carrier, in the time-
averaged sense, does not move. This property is
particularly important for those systems where a large
amount of heat transfer is needed while the working fluid is
required to remain in the system (as in nuclear reactors).
Note that the axial heat conduction, in general, is assumed
to be very small in this thermal pumping process compared
with the enhanced axial heat flow [16].
Apparently, the heat transport rate in thermal
pumping is governed by both the thermal properties of the


11
working medium and pipe wall and the characteristics of the
oscillatory motion of the fluid. The enhanced axial heat
flow does increase with increasing oscillating frequency as
this thins out the boundary layer and leads to an
accompanying increase in the radial temperature gradient.
This observation holds only as long as the thermal
properties of the liquid and the wall are compatible. If
the molecular conduction of the fluid in the radial
direction is very small, then even high frequency
oscillatory motion will not produce a large increase in the
rate of enhanced axial heat flow. This is because such a
system fails to supply the "passengers" enough time to be
loaded onto the "bus" and to be unloaded from the "bus".
Obviously, the system just wastes energy. On the other
hand, if the molecular conduction of the fluid in the radial
direction is very large, but the frequency of the
oscillatory motion is low, once again one can not expect
that there will be an efficient enhanced axial heat transfer
between two reservoirs because the "bus" is now moving too
slowly.
The above observation can be confirmed by an analysis
of the performance of a water-glass combination (i.e., water
is the working fluid medium within a glass tube) and of a
mercury-steel combination. For the former, it is necessary
to employ rather small diameter tubes and low frequency
oscillations with large tidal displacement, for it has


12
relatively poor heat diffusive properties as compared with
the mercury-steel case. This small tube diameter-lower
oscillation frequency set-up is necessary in order to ensure
that there is sufficient time to transfer the excessive heat
content of the bulk water core to the tube wall during the
positive stroke of each period and to permit the transfer of
excess heat from wall to the cooler core fluid during the
negative stroke. Otherwise, the water would just carry a
portion of its heat content back and forth in the pipe and
the condition for achieving optimum enhanced axial heat
transfer could not be met. For the mercury-steel case, one
can chose relatively large pipe diameters and higher
oscillation frequencies with smaller tidal displacement
because of the higher thermal conductivity. This assures
that only very short times are needed to exchange the heat
between the core of the fluid and the wall. It should be
pointed out that the suggestion of using smaller tidal
displacement is purely due to the mechanical considerations
and that one always tries to keep Ax as large as possible in
order to produce large axial heat flows.
From the above discussion, it can be concluded that
the process of enhanced heat transfer via oscillatory
pumping requires a precise tuning of parameters governing
the enhanced heat transport. Indeed there is an expected
"tuning effect" as discussed in references [16, 20, 21].


13
The tuning effect is a very important concept in the
study of the presently considered heat transfer process. It
shows that there will be an optimum combination between
thermal properties of the working medium and wall and the
characteristics of the oscillatory motion. The qualitative
aspects of the tuning effect have been observed earlier for
both the case of a flat channel and that of the cylindrical
pipe [15, 16]. From Fig. 4-38, one can see that an optimum
for axial heat transfer occurs only at or near the tuning
point which depends on the oscillating frequency and the
thermodynamic properties of the fluid and wall. As has been
pointed out by Kurzweg [20], in order to obtain the optimum
enhanced heat transfer one has to carefully select suitable
values for the pipe size, the material of pipe and the
working medium as well as the manner of oscillatory motion.
The nondimensionalized enhanced heat diffusivity is
defined as
Ke
where Ke = /ypc is the coefficient of enhanced heat
diffusivity, is the axial heat flux, 7 is the time-
averaged axial temperature gradient, p is the density of the
fluid, and c is the specific heat. One can show that this
nondimensional enhanced heat diffusivity is a function of
both the Wormersley number and the Prandtl number [21] and
hence that the dimensional axial heat diffusivity Ke is a


14
function of the tube radius the oscillating frequency w,
the kinematic viscosity u, and the square of the
dimensionless tidal displacement AX. This can be explained
from the fact that the radial heat flow is proportional to
the product of the representative radial temperature
gradient 7AX/6th an<3 the surface area per unit depth of ttAX
available for cross-stream heat transport.
The use of large tidal displacement is always
beneficial in the enhanced axial heat transfer within
oscillating pipe flow. However, in order to avoid the
direct convective net mass exchange between the two
reservoirs, the tidal displacement must be limited to less
than about one half of the pipe length.
As has already been predicted by theoretical studies
and will be confirmed by the present numerical simulations,
the axial heat transfer will be further enhanced if the
rigid surface (part of the rigid wall with finite thickness)
has a non-zero thermal diffusivity and hence heat storage
capability.
Note that the existing considerations are restricted
to laminar flow. Turbulent flow conditions can occur in
oscillating pipe flow at higher values of wAx2/V [26, 27]
and apparently would destroy the assumptions of the current
analytic model of the thermal pumping process. Fortunately,
the condition for optimum enhanced heat transfer in such
oscillating pipe flow obtained at the tuning point requires


15
very slender pipes, such that the Reynolds number is usually
small enough so that the oscillating motion falls within the
laminar range [22].
The theoretical aspects of the oscillatory enhanced
axial heat transfer process have been developed much further
than its experimental and numerical counterparts. The
theoretical predictions are quite limited and consider only
cases under certain simplifying assumptions [8], Numerical
work is necessary in order to not only to examine the
correctness of the theoretical analysis but also to further
the development of advanced enhanced thermal pumping
devices. Numerical studies are not only fast, economical
and accurate, but they also offer a handy way to access
complex geometries which can not be handled analytically.
It is the purpose of this study to extend the
analytic work on thermal pumping by a detailed numerical
study. We intend first to examine some transient problems
of axial heat transfer in oscillatory pipe flow, such as the
development of the velocity profile at various Wormersley
numbers in contrast with those of reference [36], where only
several special cases with intermediate Wormersley number a
were discussed, and to examine numerically the relationship
between the tidal displacement and the required
corresponding strength of the periodic pressure gradient as
a function of Wormersley number a and of tidal displacement
Ax.


16
Next, we examine the build-up process of the
temperature distribution in a pipe which connects a hot
reservoir at one end to a cold reservoir at the other end
and see whether there actually exists a constant time-
averaged temperature gradient along the pipe axis when the
final periodic state is eventually reached. Note that a
time-averaged linear temperature distribution along the
axial direction is an essential assumption in the existing
theoretical studies.
The third part of this investigation which forms the
main effort, is a computer-aided numerical simulation of the
thermal pumping technique, including an investigation of the
variation of the enhanced axial heat flux versus the tidal
displacement, the variation of enhanced axial heat flux
versus different Wormersley numbers, and a study of the
variation of heat flux versus different Prandtl numbers. It
also includes a study of the influence of wall thickness and
pipe diameter as well as the change of the fluid properties
on such an enhanced axial heat flux and an examination of
the tuning effect in the conducting wall case. Further, we
compare axial heat transfer in oscillating flow with that in
the steady flow, and compare quantitatively this enhanced
heat flux with that induced by the pure axial molecular
conduction under various Wormersley numbers and different
fluid-solid wall combinations. Finally, the author examines
the phase lag phenomenon in oscillatory flow in order to


17
further reveal the mechanism whereby this new heat transfer
technique functions.
The computational work presented here was completed
on the Vax-11/750 electronic computer in the Department of
Aerospace Engineering, Mechanics, and Engineering Science,
and on Vax-11/780 in the Center for Instructional and
Research Computing Activities, University of Florida,
Gainesville, Florida. The numerical approach used, for
solving the presently considered heat transfer problem,
employed a second-order Crank-Nicolson scheme and an
Alternating Direction Implicit method (ADI) with Thomas
algorithm. A Fortran computer code named ETP (Enhanced
Thermal Pumping) was developed to implement all the
calculations.


CHAPTER II
FORMULATION OF THE PROBLEM
A single tube with inner radius R^, outer radius R2
and length L (such that L >> R^ and R2) connects a large
reservoir of hot fluid at temperature T = Tjj at one end, to
another large reservoir of the same fluid at cold
temperature T = Tc at the other end (Figs. 2-1 and 2-2) .
The tidal displacement Ax and the oscillating frequency u>
are adjusted to assure that the turbulent flow does not
occur. The tube is oriented in such a manner so that the
effect of gravity on the oscillatory motion of the fluid in
the pipe is negligible. Water and mercury are employed as
the working mediums. They are understood to be Newtonian
and incompressible fluids. The variations of thermal
properties of the fluid with temperature during the heat
transport process are assumed to be negligible. With these
assumptions the problem of enhanced time-dependent heat
transfer induced by a simple harmonic oscillatory laminar
fluid motion in a very long circular pipe can then readily
be formulated.
Governing Equations
The use of a very slender capillary circular tube
with constant cross-sectional area and neglecting end
18


HOT fLUID IN HOTTER FLUID
(PRIMARY LOOP) OUT
Fig 2-1 Thermal Pumping Device
(after Kurzweg, U.S. Patent, 4,590,993 May 1986)


20
effects insures a laminar axisymmetrical one-dimensional
time-dependent motion. It is convenient to employ
cylindrical coordinates for this problem and we denote the
coordinate in the axial direction of the tube by x, and the
radial direction by r. The axial velocity can be taken to
be independent of x and, in order to satisfy the
requirements of continuity the other velocity components
must vanish. We shall further assume that the pressure
gradient induced by moving the piston (Fig. 2-1) is harmonic
and has the form [29]
1 5p
~ A cos t t2-1)
where A = l/p|ap/dx| is a constant which measures the
maximum pressure gradient existing along the x-axis.
Clearly this equation implies that we are now dealing with a
time-dependent sinusoidal pressure gradient which is
constant over the pipe cross-section at any instant and the
pressure varies linearly along the x-axis. The simplified
Navier-Stokes equation for this problem is [36]
3U / d r 3U -i
= Acos(wt) + 7"
0 < r < Ri
(2-2)
where U(r,t) is the time and radially dependent axial
velocity component.


21
The corresponding temperature T(x,r,t) of the fluid
within the pipe is governed by the heat conduction equation
[15]
3 T
at
-u
3T
ax
+ K, f
i a aT
r ar ar ^
32T ^
0 < r < R2
(2-3)
where is the inner radius of the tube and *cf the fluid
thermal diffusivity which is related the thermal
conductivity kf by
kf
Here p is the fluid density, and c is the specific heat.
Note that the viscous heating term has been neglected in
equation (2-3) since it is very small for most experimental
conditions; this is justified provided one does not deal
with very high Prandtl number fluids such as oils. The
temperature in the wall can be determined from the solution
of
3 T r 1 3 3T a2T 1
3t *w[ r gr (r 3 r) + gx2 J
Rl < r < R2
(2-4)
where kw is the thermal diffusivity of the conducting wall
in Ri < r < R2 and is defined by


22
K w
kw
P wcw
where kw, pw, and cw, are the wall conductivity, density,
and the specific heat, respectively. By introducing the
following non-dimensional terms, equation (2-2), (2-3) and
(2-4) can be treated more easily.
t* =
t
1/w
r
*
x
x
S
U*
u
A/ w
where 6 = J2v/u> is again the fluid viscous boundary layer
thickness. The dimensionless governing system of equations
can then be written as
air
at
= cos(t ) +
r a2u*
ar
TT2 +
air
ar
0 < r < RX
(2-5)
di
atJ
= cu
aT7
ax
r a2T*
2Pr
ar
Trz~ +
aT7
a 2t*
ar
Tt +
ax
^r
o < r < R!
and
(2-6)
a t
at
TT~ =
2S
a 2t*
r*2
+
aT
ar
a2T*
ax
Rl < r < R2
(2-7)


23
where
and
M Cf
Pr =
K
P cw
S =
*W
C
-A
^7
This nondimensionalization has some advantages in the
computing process to be carried out below. The
dimensionless velocity and its distribution over the cross-
section found from the momentum equation (2-5) are expected
to be universal for any Wormersley number a = R-^Jup/\x and
any associated quantities, such as the tidal displacement
and Lagrangian displacement. Its final periodic form is of
the form given by (1-4). The dimensionless temperature in
the pipe is only related to the Prantdl number and the
dimensionless velocity, while that in the wall is related to
the ratio of wall heat diffusivity to the kinematic
viscosity, as seen from equation (2-6) and (2-7).
The governing dimensionless equations (2-5), (2-6),
and (2-7) are a set of second-order parabolic type of
partial differential equations expressed in cylindrical
coordinates. To solve this set of simultaneous equations, a
corresponding set of boundary conditions and initial
conditions are required. Since the velocity is assumed to
be a function of r and t only, just two boundary conditions


24
are needed for solving the momentum equation, while for the
temperature T(x,r,t) the heat conduction equations require
two boundary conditions in both the r and x directions as
well as compatibility conditions along the interface between
the fluid and the solid wall. It should be pointed out that
the initial conditions are less important than the boundary
conditions if one seeks only a final periodic state.
Boundary Conditions
The boundary conditions for the velocity are the
usual ones for viscous flow, namely, that the velocity is
zero at the inner surface of the wall (r* = R^*) Also by
symmetry, the normal derivative of velocity at the axis is
zero. That is,
U*(R!*, t*) = 0 (2-8)
and
3U*(0, t*)
The boundary conditions for the temperature depend on the
particular model investigated.
Model 1
In this model, it is assumed that the temperature at
each end of the pipe is equal to that of the connecting
fluid filling the reservoirs when the fluid moves into the
capillary tube (i.e., during one half of each cycle). The


Fig 2-2
Model 1
L=20cm
L
INSULATING WALL
Fixed End Temperature Model
R^=0.1cm, R2=0-15cm


26
end boundary temperature at x = 0 and x = L will take the
values of temperature of the adjacent pipe fluid elements.
With this model, we tried to simulate the real experimental
observations where it can be clearly seen that a fluid jet
exits the pipe during the outstroke, while the elements of
the fluid enter the tube in a funnel pattern during the
instroke. This observation shows that there is enough time
during each oscillating cycle to allow the fluid elements
within the exiting jet to fully mix with the fluid particles
which originally are in the reservoir before the next
instroke, provided the oscillating frequency is not too
high. The end temperature boundary conditions are here
taken as
(when fluid enters pipe)
T*(0, r*, t*)
(when fluid leaves pipe)
(2-10)
(when fluid enters pipe)
(when fluid leaves pipe)
(2-11)
where T*c, T*h are the nondimensional temperatures of cold
and hot reservoirs, respectively. While T*acjj is the


27
temperature of the fluid elements which are adjacent to the
corresponding ends at a particular instant.
The thermal boundary conditions along the outer
surface r* = R2 of the pipe wall are taken to satisfy the
insulating wall condition, and along the axis of the tube
the temperature is assumed to meet the symmetric boundary
requirement, i.e., the radial derivatives of temperature
along axis are equal to zero. We thus have
d T
d r
r* = R2*
= 0
(2-12)
and
d T
dr"
r* = 0
= 0
(2-13)
The compatibility conditions along the interface
between the fluid and the solid wall are that the radial
heat flux and temperature are constant across the interface.
That is
3T
kf
dr
d T
fluid
- kw dr*
wall
at r = R]_
(2-14)
and
rji ^
ip ^
fluid
wall
at r = Rp
(2-15)
Since numerically the same nodes are chosen along the
interface, condition (2-15) will be automatically satisfied.


28
Model 2
Here it is assumed that a heat source rim of width 2b
is directly mounted on the interface between the solid wall
and the fluid at x = L/2, while two small cold rim sources
of width b each are mounted at x = 0 and x = L (see Fig. 2-
3) The wall thickness is assumed to be zero. This model
is intended to simulate the enhanced heat transfer process
of oscillatory fluid in an infinitely long pipe which is
heated and cooled by the alternative evenly distributed heat
and cold sources along the outer surface of a very thin wall
which possesses infinite heat conductivity. Apparently,
this geometry can be well simulated by the present model
with periodic boundary conditions. Mathematically the
boundary conditions are
T*(x*, Rl*, t*) = T*h
T* (x*, Rl*, t*) = T*c
or
d T
dr
7c = 0
or
X2 <
x* >
X3 *
(2-14)
0 < :
X* <
XI*
(2-15)
X4* <
X* <
L*
XI* <
X* <
X2*
(2-16)
X3 <
X* <
X4 *
X6* are nondimensional
coordinates of points which correspond to X = 0, b, L/2-b,
L/2+b, L-b, and L, respectively in Fig. 2-3. The periodic
end boundary conditions are given as


to
vo
Model 2 Periodic Heat And Cold Sources
on Insulated Wall (Except Sources Area)
L=20cm, Ri=0.1cm, b=lcm
Fig 2-3


30
T*(0, r*, t*) = T*(L*, r*, t*)
(2-17)
and
3T* i dT* i
^ L* = 0 3X* I x* = L*
(2-18)
Model 3
In model 1, the boundary conditions for the heat
equation at X = 0 and L are somewhat artificial. The
temperature distribution in the vicinity of pipe ends may
not precisely match the constant temperature end conditions.
This is particularly true in the large tidal displacement
and/or very high oscillating frequency cases since a
temperature drop occurs at the hot end and a rise at the
cold end as fluid particles exit the pipe. This leads to a
discontinuity of temperature which will lead to unexpected
dispersion errors if an even order numerical method is used
[3, 24, 30], To improve the end boundary conditions
presented in model 1, the following extended model has been
considered. It is assumed that an extension pipe which is
some 5 times the length of the central pipe is attached to
the original pipe used in model 1. The heat and cold source
are assumed to be located on the outer surface of the wall
in the extended sections of the pipe as well as at both ends
of the tube as shown in Fig. 2-4. This model is used to


Model 3 Pipe with Conducting Pieces Model
L=20cm, Ri=0.1cm
Fig 2-4


32
investigate the situation where the heat contained in the
jet is exchanged by pure heat conduction with the
surrounding fluid elements in each reservoir without any
convective mixing. In this case, the boundary conditions
along the outer surface of the wall are
T* = T*h 0 < x* < XI* (2-19)
T* = T*c X2* < X* < X3* (2-20)
and
d T*
= 0 XI* < x* < X2* (2-21)
where XI*, X2*, and X3* are nondimensional coordinates of
points which correspond to X = 5L, 6L, and 11L,
respectively, in Fig. 2-4. At both ends, we have
T*(0, r*, t*) = T*h (2-22)
and
T*(L*, r*, t*) = T*c (2-23)
As in the other models, the symmetry condition along the
axis of pipe reguires that
d T*
dr*
=0
0
(2-24)


33
and the radial flux and temperature continuity condition
along the interface are
d T*
kf dr*
fluid
dT*
kw
wall
and
(2-25)
ip *
fluid
wall
(2-26)
Here again kf and kw are the same as described in the
previous models.
Initial Conditions
The initial condition chosen depends on the problem
under consideration. If one's goal is to investigate the
periodic quasi-steady solution only, the initial conditions
chosen should be as close to the quasi-steady state as
possible so that a rapid convergence rate at low CPU time
cost is achievable. If one intends to study the transient
process, then various initial conditions should be supplied
according to purpose of the investigating cases selected.
For the velocity initial condition we choose for all our
studies
U*(x*, r*, 0) = 0
(2-27)
For the temperature, if the purpose of investigation is to
examine the build-up process of the periodic quasi-steady


34
state in the thermal field, the initial condition should be
taken as
T*(x*, r*, 0) = 0
(2-28)
However, for the other cases, in order to gain faster
convergence, the initial temperature can be assumed to have
a linear distribution in the axial direction, and thus have
the form
T*(x*, r*, 0)
T*h +
(T*c T*h)xx*
LL*
(2-29)
where LL* is the dimensionless length within which the axial
linear temperature distribution is assumed to hold, and xx*
is the dimensionless distance measured from the "origin"
which is chosen only for the purpose of this linear
temperature initialization. Both the LL* and the origin
selected depend on the model considered. For Model 1 and
Model 3, LL* is equal to L*, and the origin is taken at the
left end (Model 1), or at the left intersection between the
central pipe and the left extension pipe (Model 3). However,
for Model 2, equation (2-29) is valid only for the right
half portion of the pipe as LL* is taken to be L*/2, and the
origin is chosen at the middle section of the pipe. The
initial temperature of the left half pipe can then be found
from the plane symmetric condition about the origin cross
section.


35
Calculation of Tidal Displacement
An important quantity encountered in the study of the
enhanced heat transfer process in oscillatory pipe flow is
the tidal displacement, which is usually required to be
smaller than one half of the total pipe length in order to
avoid any convective mass exchange occurring between the two
reservoirs. It has already been defined in the introduction
chapter (1-6) and in the present computation the dimensional
tidal displacement takes the form
(2-30)
where is the inner radius of the pipe and x(r,7r/w) is the
Lagrangian displacement of the fluid elements located along
a radius within the capillary tube at t = w/w. It is
assumed these elements are initially lined up at axial
position x = L/2 (Model 1 and Model 2) or x = 5.5L (Model
3) half way between the tube ends. Numerically, the
dimensional Lagrangian displacement at time t is computed
via the equation
0
U(r,t)dt
(2-31)
It is obvious that the dimensional Lagrangian displacement
x(r,t) is a function of both time t and the radial position
r. Note that, since the existence of phase lags between the


36
stimulating pressure gradient and the displacements vary for
different Wormersley numbers, in actual calculations, the
tidal displacement is equal to the sum of the absolute
maximum cross-section averaged Lagrangian displacement and
the absolute minimum cross-section averaged Lagrangian
displacement within a period. With the same non-
dimensionalizaton procedure used in the previous section for
the governing equations, the dimensionless Lagrangian and
tidal displacement can be written as
R*
(2-32)
0
The nondimensional Lagrangian displacement within the period
can then be computed by
* / 4- \
x (r ,t )
(2-33)
where
*
x
(A/u^)
and
Ax
AX
*
(A/c2)
It is pointed out that this dimensionless tidal displacement
Ax* differs from AX defined by equation (1-7) by a constant
1/2.


37
Effective Heat Flux
The time averaged total effective axial heat flow
within the pipe has the form
(2-34)
where c is the specific heat, p is the density and again, w
is the oscillatory frequency. The per unit area effective
heat flow (termed heat flux) can then be written as
Equation (2-34) follows from the fact that pcUT is the
convective heat flux. The dimensionless total effective
heat flow can be written as
(2-36)
and the dimensionless heat flux by
*
Q*
v total
(2-37)


CHAPTER III
NUMERICAL TECHNIQUES EMPLOYED
Equations (2-5) (2-6) and (2-7) which were derived
in Chapter II can not generally be solved analytically in
any simple manner. Therefore, it is necessary to seek a
numerical solution approach to the problem.
As is well known, in order to numerically solve a set
of simultaneous governing differential equations more
accurately and efficiently, an optimized grid system plays a
very important role. For the present purpose the grid net
works were generated in the following way: within the tube,
a non-uniform grid (see Fig. 3-1) is clustered along the
radial direction in the vicinity of the inner surface of the
pipe where the larger gradients of velocity and temperature
are expected to be present, while along the x-axis the grid
is distributed uniformly except for those calculations
dealing with model 3.
The solution process was carried out in the
computational plane rather than directly in the physical
plane. Thus, a transformation which converts the governing
equations as well as the boundary condition plays an
essential role.
38


39
1). For Model 1
2). for Model 3
Fig 3-1 Grid Systems Used in the Numerical Simulations


40
The second-order implicit unconditionally stable
Crank-Nicolson scheme and the ADI method [2, 31] are
employed to break up the transformed equations into finite
difference form. A computer code designated ETP (for
Enhanced Thermal Pumping) has been developed for obtaining
the desired results. Details of this procedure will be
described in this chapter.
Transformation
The numerical solution of a system of partial
differential equations can be greatly simplified by a well-
constructed grid. It is also true that an improper choice
of grid point locations can lead to an apparent instability
or lack of convergence. Early work using finite difference
methods was restricted to some simple problems which can be
numerically solved in the physical domain. As experience
was gained, general mappings were used to transform the
physical plane into the computational domain as well as the
governing differential equations [31]. Such a grid
transformation technique brought to the numerical simulation
numerous advantages, and the computational work was no
longer restricted to a few simple geometries. A body-fitted
non-uniform grid in the physical domain could be used, which
retains the uniform spaced grid system features in the
computational domain [33, 34, 35]. However, several
requirements must be placed on the transformation: first,
the mapping must be one to one, second, the grid lines must


41
be smooth and have small skewness in the physical domain,
third, grid node point spacing should be small where large
numerical errors are expected (i.e., large solution gradient
regions) in the physical domain (Fig. 3-2).
In present study, the grid system in the physical
space is numerically generated by solving an algebraic
equation which clusters the grid lines in the region where
large gradients of the physical quantities are expected so
as to gain higher resolution of these physical quantities.
Because of the simple pipe geometry, the requirements of
grid smoothness and orthogonality are not a serious problem.
For the present purpose, the transformation is a simple one
which transforms a non-uniform grid network in the physical
plane onto a uniform grid system in the computational plane.
x* = x*(?)
r* = r*(r,) (3-1)
t* = t*(r)
The inverse transformation can be found as
= (x*)
n = r, (r*) (3-2)
r = r (t*)
where x*, r* are the dimensionless coordinates and t* the
dimensionless time in the physical domain, £, q are the
transformed coordinates and r is the transformed time in the


w
Fig. 3-2 Coordinates Transformation


43
computational domain. With this transformation, the
transformed governing equations have the form:
au
a2u*
atr
ar f(T) + a^"aV + h{v) dr,
(3-3)
aT
aT3
a2T*
aT
a r
= P
as + P2_a?2_ + P3 dr,
a2T*
+ P4^
(3-4)
and
3T
a r
aT
a2T*
aT
wi^r+ + w3~
a2 t*
+ w4^~
(3-5)
where
f(r)
a(?)
b(r?)
and
PI =
P2 =
P3 =
P4 =
at*
-cos
a r
(t*)
1
2
1
at* i
ar (ar*/ar?)2
at*r i
1 a 2r
2
dr [ r*(ar*/dr,) (ar*/ar?)J at?2
at*
r cu*
1 a2x* ^
a r
[ dx*/d£ 2Pr(ax*/a^);j as2 J
at*
i
5 r
2Pr (ax*/a£2)
at*
1 r 1
1 a2r* ^
5 r
2Pr [r* (dr*/dr,)
(ar */dr,)J dr,' J
at*
1
dr 2Pr
(ar*/ar?)2
'I
(3-6)
(3-7)
(3-8)
(3-9)
(3-10)
(3-11)
(3-12)


44
W1
W2
W3
W4
at* i (
. -i
d t
2Prw(
.(ax*/aOJ ai J
at* l
i
a r
2Prw
(ax*/ao2
at* i
1 1
a r
2Prw
[r* (dr*/dr,) (dr*/dr,)3
at*
1
1
a r
2Prw
(ar */dr,)'
d2r*>
dr,-
(3-13)
(3-14)
(3-15)
(3-16)
As mentioned in chapter II, equations (3-3), (3-4) and (3-5)
are a set of second-order parabolic partial difference
equations in cylindrical coordinates. In addition, since
the oscillating flow is considered as incompressible in the
present studies, the momentum equation (3-3) can then be
independently solved at each time step. As a result, the
time-dependent update velocity found can then be substituted
into the heat conduction equation (3-4) as a known quantity
at the same time step level. Eventually, the coupled heat
conduction equations (3-4) and (3-5) are solved
simultaneously to obtain the temperature distribution both
in the fluid and in the wall at any time. To best solve
this set of equations in terms of accuracy and efficiency,
the proper choice of numerical technique and grid net-work
is dictated by an understanding of the physical aspects of
the problem.


45
The same transformation should be also applied on all
the boundary conditions proposed in the three different
models. For the temperature compatibility conditions along
the interface between the fluid and the wall (2-25) one has
the following transformed forms:
dT* 1
dr] (dr*/dr¡) kf
fluid
dT* 1
dr/ (dr*/di])
kw
wall
(3-17)
or
dT
dr]
= K;
flow
dTJ
d r)
wall
where
kw dr*/dr) |pipe
Ka = *¡
kf 3r >Hwall
(3-18)
(3-19)
To make the subsequent form of the corresponding
finite difference governing equations less cumbersome, the
superscript will be dropped from the variables T*, r*,
t*, and x*, and in addition, U* will be replaced by V.
In the process of deriving the finite difference
governing system equations, the second-order central
differences in the domain and forward or backward
differences along the boundaries or interface of the fluid
and wall have been employed at each nodal point.


46
Crank-Nicolson Method for Momentum Equation
The second-order accurate Crank-Nicolson method is
quite well known and widely used. It is an unconditional
stable, implicit scheme for solving the parabolic types of
partial difference equations. When the Crank-Nicolson
method is applied to equation (3-3), the finite difference
algorithm at a typical node k in the radial direction and at
the time step n assumes the simpler form:
(Bkvk-i
+ Dkvk + Akvk+Jn+1 ck
k = l, 2,
, km
(3-20)
where
- -Hfn+l+ fn) (Bkvk-i+
E,_V,_- A,_V,_ . jn (3-21)
Jk k k k+1
Ak "
(2a + b )
(3-22)
^ (2 a + b )
(3-23)
Dk = 1 + ak
(3-24)
Ek = 1 ak
1
(3-25)


47
The boundary conditions along the interface (i.e.,
the inner surface of the wall, k = kmid) and the axis (k =
1) then become
V
kmid
dV
dr)
(3-26)
The initial conditions of the velocity at all nodal
points is taken to be zero.
n
k = 1, 2, 3, kmid
(3-27)
Equation (3-20) associated with the boundary conditions (3-
26) can then be written in the matrix form. This yields a
set of linear system algebraic equations which can be solved
in terms of the nodal values of velocity in the capillary
tube by using either an iterative method or a Thomas
algorithm at each time step. The explicit form is


48
*
n
' '
n+1
D, A,
V.
C.
1 1
1
1
D A
C
2 2 2
2
2
B D A_
V,
C0
3 3 3
3
-
3
Bkd-lDkd-lAkd-l
vkd-i
Vkd-1
Bkd Dkd
VW J
VKd J
(3-28)
where
B
1
kd = kmid-1
ADI Method for Axisvmmetric Heat Equations
The governing PDEs (3-3), (3-4) and (3-5) are all of
the second-order parabolic type. Thus it might be suggested
that the Crank-Nicolson scheme used in solving the momentum
equation (3-3) can also be applied to the axisymmetric heat
equations (3-4) and (3-5) and one can then take advantage
of the tridiagonal matrix form while using this
unconditionally stable technique. However, when attempting
to use such a formulation, one immediately finds that the
resulting system of linear algebraic equations is no longer
of the tridiagonal type (3-20) but rather a non-tridiagonal
matrix system requiring substantial CPU time to solve. This
difficulty can be avoided by applying the unconditionally
stable Alternating Direction Implicit method (ADI),


49
developed by Peaceman and Rachford and Douglas in 1955.
According to this scheme, the entire solution process at
each time step is "split" into two portions, i.e., the first
half of solution processes for k^*1 column (radial
direction) while the other half processes for the row
(axial direction).
With the ADI scheme, second-order central differences
are used to approximate the values of derivatives at each
nodal point in equations (3-4), (3-5). The finite
difference algorithm for those equations during the first
half of each time step for the jth column are then
.BPjj,kTjj,k-l + DPjj,kTjj,k +
AP. T
jj,k jj,k+l
n+1/2 = PXn. .
ID,*
k = 1, 2 ....kmid
(3-29)
and
BW. ,T. . +
. 33,k-l
DW.
jj,kTjj,k
+ AW. . T. . ,,
11/k 3D,k+l
n+1/2
WX
n
jj/k
k = kmid+1,...kmax
(3-30)
where the subscript jj is used to emphasize the specific
column currently to be computed. It can be seen that the
set of difference equations now is in the tridiagonal form


50
since the right-hand-side terms in the equations (3-29), (3-
30) contain only known values from the previous results and
the boundary conditions. These values can be computed by
PXjj,k (~CPjj,kTjj-l/k + EPjj,kTjj,k FPjj,kTjj+l,k)
(3-31)
and
(
-CW. . T . + EW. . T. . FW
ll/k DD-l,k DD,k D},k
. T . 1 n
(3-32)
The computational algorithm is implemented column by
column and the unknown value (Tj,^)11 can then be solved by
either an iterative or a direct method. In order to do
this, equation (3-29) needs to be assembled into the
following matrix form.
r '
PDX PA1 *
n
' T1
n+1/2
' '
PX1
PS2 pd2 pa2
T2
PX2
pb3 pd3 pa3
T3
=
PX3
PBkd-lPDkd-lPAkd-l
Tkd-1
PXkd-l
PBkd PDkd
j
T
l kd J
j
PXkd
j = 1, 2, 3 jmax
(3-33)


51
Similarly, equation (3-30) can be assembled into the matrix
form as seen in (3-34).
WDkd+l WAkd+l
n
' '
T
xkd+l
n+1/2
WXkd+l
WBkd+2 Wkd+2 WAkd+2
T
kd+2
WXkd+2
WB, WD. WA.
km km km
T
xkm
WXkm-l
* WB. WD.
kmax kmax
j
Tkmax
j
WXkmax
j = 1, 2, 3 jmax
(3-34)
It should be pointed out that the coefficients and/or
the right hand side terms in the row marked with symbol "*"
in those matrixes needs to be properly modified as well as
the limit of either subscript j or subscript k according to
the different boundary conditions in the various models
considered. For example, along the axis, the symmetry
condition requires that dT/dr = 0 and this could be
accomplished numerically by equating the temperature Tq and
T2 and by employing a new combining coefficient of PA^* =
PAi + PBi rather than the original PA^ in the first matrix
for evaluating the temperature distribution within the pipe.
In addition, if the temperature distributions along certain
parts of the outer surface of the wall are given, then the


52
limit of subscript k will be ended with k = kmax-1 rather
than k = kmax shown in equation (3-34). The same arguments
are also applicable for another subscript j in the axial
direction.
By using a similar procedures for the second half of
each time step, the finite difference algorithm for the kth
row then becomes
(CPj,kkTj-l,kk + GPj,kkTj,kk + FPj,kkTj+l,kk)
n+1 = pYn+l/2
j ,kk
j = 1, 2,
, 3 max
(3-35)
H,1:-!,!*+ Gwj,kkTj,kk+ Fwj,kkTj+i,kk)n+1 m"%2.
j = 1, 2,
j max
where the right hand terms can be computed by
(3-36)
PY1?+l/2= CP. T. + HP T AP T 1 n+1/2
^ j,kk [ ,kkij,kk-l 3,kk1j,kk ,kkij,kk+lj
(3-37)
wy"!k2= (-cBj,kkTj,kk-i + Hwj,kkTi,kk- Awj,kkTj,kk+i)n+1/2
(3-38)


53
Similarly, to emphasize that the current calculation is in
the kth row, we use the symbol kk rather than k in above
formula and the matrix form of equation (3-35) for the k^h
row can then be written as
PG2
PF2

*
n+1/2
r *
T2
' '
PY2
PC3
PG3
PF3
T3
PY3
PC4
pf
4 4
T4
PY4
PC. PG. PF.
3m-1 jm-l jm-l
Tjm-1
PXjm-l
*
i.
PC. PG.
jm jm J
k
T .
1 D J
k
PX.
1 3
n+1/2
k = 2, 3
, kmid
(3-39)
and the matrix form of equation (3-36) becomes
WG2
wf2
*
n+1/2
' 1
T2
N+l
wy2
wc3
WG3
WF3
T3
WY3
wc,
4
WG. WF .
4 4
T4
WY
4
WC. WG.
jm-l jm-1
WF.
jm-l
Tjm-1
WX.
jm-l
*
b
WC.
jm
DG.
3 J
k
T.
1 3 J
k
WX.
1 3
k = kmid+1, kmid+2 km
N+1/2
k
(3-40)


54
As indicated above, the symbol shows that the
coefficients in that row as well as the right-hand-side
terms need to be properly modified corresponding to the
different boundary conditions. The matrix terms in
equations (3-29), (3-30), (3-31) and (3-32), (3-35), (3-36),
(3-37) and (3-38) can be computed by
APj,k = "i ^P3 + P4>^
D
BPj,k P3 P4>^
J f k
CPj,k P1 P2>j,k
EPj,k = 1 2*P2>j,k
PP j ,k= "iP1 + p2).
GPj,k 1 + 2*P2)j,k
DPj,k = 1 + 2*P4>j,k
and
AW. .
D,k
= W3 + P4) .
j ,k
BW = ( W3 W4) .
j ,k
CMj,k < W1 W2>j,k
HPj,k t1 2*P4>j,k
(3-41)
- (1 2*P2>j,k
- -<4"wi + w2>j,k
= (1 + 2*W2). k
DWj f k + 2*M4>j,k
HMi,k i,k
(3-42)


55
In order to rewrite the equation in a simple form we define
the following linear operator :
LY [P]j>k = [BP, DP, AP]j,k
LY [W]j,k = [BW' DW' AW]j,k
LX [P3j/k = [CP, GP, FP]j/k
LX [W]j ^ k = [CW, GW, FW]j,k
(3-43)
and column vectors
{TY)j,k
{Tj,k-l' Tj,k' Tj,k+1 }
{TX}j,k
T.
{Tj-i,k' xj,k' *j+i,k
T.
(3-44)
Equations (3-29) which are in the radial direction can be
simply written as
n n+1/2 n
LY [P] (TY) = PX
j,k j, k j, k
(3-45)
n n+1/2
LY [W] (TY)
j ,k j ,k
n
(3-46)
j = 1, 2,
k = 1, 2,


56
Equations (3-33) and (3-34) which are in the axial
direction, then assume the following simple forms:
n+1/2
n+1
n+1/2
LX [P]
(TX)
= PY
(3-47)
j /k
j ,k
j ,k
n+1/2
n+l
n+1/2
LX [W]
(TX)
= WY
(3-48)
j ,k
j / k
j/k
j = 1, 2,
k = 1, 2,
where the right side terms can be estimated by
PX = (-CP, EP, -FP) (TX)
j,k j,k j,k
WX = (-CW, EW, -FW) (TX)
j,k j,k j, k
PY = (-BP, HP, -AP) (TY)
j,k j k j k
(3-49)
WY = (-BW, HW, -AW) (TY)
j r k j k j k


57
It is seen, as the result of the ADI "splitting
procedure which has been employed in the algorithm
associated with different boundary conditions for the
various models, that only a tridiagonal system of linear
algebraic equations needs to be solved (i.e., during step 1,
the coupled tridiagonal matrix (3-33), (3-34) are solved for
each j*-*1 column of the grid points, while during step 2, the
coupled tridiagonal matrix (3-39) and (3-40) are then solved
for each k*-h row of grid points) .
Once the periodic steady solution has been obtained,
we can calculate both the tidal displacement and the heat
flux at different locations within the pipe. The numerical
technique used here to integrate equations (2-38), (2-39)
and (2-41) for evaluating the tidal displacement and the
heat flux at each specified location can be obtained either
by the Trapezoidal rule (with end correction) or Simpson's
rule. Both numerical approaches are essentially fourth-
order methods.
Convergence Criteria
As is known, once the calculation work has started,
the time-matching process will be in the loop forever unless
a criterion can be derived that indicates when the goal of
the current computing work has been reached and further
solution-matching processes do not produce significant
increases in accuracy. Such a criterion depends on the
purpose of the calculation. If one's goal is to study the


58
transient process, one can indicate a time limit to pause or
quit the current computing work.
However, if one seeks a final periodic state
solution, one has to develop a criterion to test if the
solution can be considered acceptable.
It seems that the temperature is a good measure of
the accuracy of the overall solution process so that the
most efficient way is to apply the convergence test on the
temperature rather than on the velocity. For the present
purpose, two convergence criteria were alternatively used.
The choice of the criterion depends on what was the main
goal in the particular study case. If the main effort is to
observe the distribution of the quasi-steady temperature at
any phase angle t within a period, the testing is done by
comparing the temperature residual, i.e., by inspecting the
averaged temperature difference of each nodal point at thee
same t between adjacent periods. This can be written as
Resl
j s (T11 T22 )? k ]1/2
(JMAX)(KMAX)
<
j = 1, 2, JMAX
k = 1, 2, KMAX
(3-50)
where Til. is the value of the temperature of node (j,k)
at wt in the current period, T22. is the value of the
D I K


59
temperature of node (j,k) at the same t in the previous
period, and ej is the convergence parameter.
If the goal of the investigation is to examine the
effective heat transfer, the convergence criterion is
established by computing the residual
Res2
JSEC
* e2
(3-51)
j = 1,2, JSEC
where JSEC is the number of cross-sections where the axial
heat flux was examined, 62 is the convergence parameter, and
1, 2 are the cross-section averaged heat flux in the
current period and previous period, respectively. The
summation is carried out over all sections (JSEC) where the
heat flux is computed.
The value of the convergence parameter can be
determined by a balance of the acceptable solution accuracy
and the cost of CPU time. In the current study both el and
e2 were taken between 0.001-0.01 as the residual of the
temperature or the heat fluxes for acceptable value.
Grid Generation
As is well known, the solution accuracy and efficiency
in a large degree depends on the grid system used in the
numerical calculation. A good grid is characterized by
small skewness, high smoothness and capability of high


60
resolution in the large gradient regions in the physical
plane. It has been shown that rapid changes of grid size
and highly skewed grids can result in undesirable errors
[25]. The success of a numerical simulation of a complex
thermal fluid dynamics problem does depend strongly upon the
grid system used in the computation.
In the present study, the grid mesh was generated with
the emphasis on the high resolution capability of its simple
geometric boundaries. The technique used here is one of the
algebraic schemes which cluster the grid lines near the
region desired [34], namely,
AS. = AS (1 + e) k_1 (3-52)
where aSq is the minimum specified grid spacing next to the
wall or to some inner interfaces within the tube. The
parameter e is determined by a Newton-Raphson iteration
process so that the sum of the above increments matches the
known arc length between k = 1 and k = kmax. The grid
networks used in Model 1 and Model 2 are uniformly
distributed along the axial direction in both the fluid and
the wall; however, in Model 3, most grid points are
clustered near the central portion of the pipe so as to gain
higher resolution in the region of interest, while along the
radial direction a non-uniform grid was employed in all
three models. The minimum specified grid spacing AS0 used


61
depends on the boundary layer thickness 6, namely, the
kinematic viscosity of working fluid and the oscillating
frequency. In the present study, it was found that a good
choice of this value is AS0 = 0.055 for laminar flow cases
with a total of about 15-20 nodal points distributed along
the radius.


CHAPTER IV
NUMERICAL RESULTS AND DISCUSSION
The problem of time-dependent enhanced heat
conduction subjected to sinusoidal oscillations can now be
solved numerically for the boundary conditions appropriate
to a long capillary tube according to the various models
described in previous chapters. The computational tasks
fall into two catagories: the first part is a numerical
study concerned with the characteristics of the oscillatory
pipe flow, which includes an investigation of the velocity
profiles, the Lagrangian and the tidal displacement
trajectories. The second part of this study includes a
thorough investigation of the final periodic state
temperature build-up process in oscillating pipe flow, the
periodic temperature distribution in the pipe and wall, and
a study of the relationship between the enhanced axial heat
flux and the tidal displacement. It also includes an
investigation of the tuning effect and a comparison of the
enhanced axial heat transfer with the corresponding pure
molecular axial heat conduction as well as the investigation
of the influences of the pipe radius and the wall thickness
on the enhanced axial heat transfer.
62


63
Properties of oscillating laminar pipe flow have been
analytically discussed by S. Uchida [36] for several
different Wormersley numbers, and these results offer a
useful reference for comparing with the present numerical
studies. However, in the area of thermal fields associated
with enhanced thermal pumping there is little information
available for comparison, except for some recent results of
Kaviany [13].
The present computational work was carried out on the
Vax-11/750 computer in the Department of Aerospace
Engineering, Mechanics, and Engineering Science, and on the
Vax-11/780 in the Center for Instructional and Research
Computing Activities, University of Florida, Gainesville,
Florida. Note that it is very time-consuming to build-up a
final periodic state, for example, if the grid size is
101x22 and the time steps are 2000 per period, it takes 7.5
minutes (CPU) with the VAX-11/780 machine or almost 1 hour
(CPU) with the VAX-11/750 machine to run only one period.
It usually takes 20-30 periods to reach the final periodic
state solution.
The process of selecting model size (i.e., total
nodal points) is a synthetic balance among the storage
requirement, the solution accuracy, and the cost of CPU time
for finding an acceptable solution. Such a selection allows
each case to be solved with a minimum of expense in
computing operation thereby making it possible to do the


64
large number of runs needed to obtain enough data points to
plot curves of the desired parameters.
The results discussed in this chapter are presented
for the purpose of illustrating the effects to be
encountered in working with the enhanced thermal pump.
These results should be effective in gaining insight into
some interesting features of this enhanced heat transfer
process. Since the analytic approximations [15, 16, 18, 22]
are effective in describing the flow and the heat transfer
aspects and to give considerable insight into the problem,
they are often used to compare with the numerical results
and should give good agreement when applicable.
Part 1. Oscillatory Pipe Flow Features
In order to better understand the mechanism of the
enhanced thermal pump, it is necessary to examine the
mechanical features of the oscillating pipe flow. As
mentioned in previous chapters, the present interest in the
enhanced thermal pumping is confined to an investigation of
the central part of the slender pipe, thus the flow field
can be well approached by a 1-D time-dependent laminar model
which neglects the ends effects.
Velocity Profiles
Fig. 4-1 shows the numerically computed time-
dependent velocity profiles at different phase angles of the
exciting pressure when Wormersley numbers are equal to 1,
10, 100, and 1000, respectively. It can be clearly seen


65
a -1000
* *
*
:
2

U
c
fa
9
-f
r-rn
-r
n
# y M 9 t ise ue ie e jm see
Fig 4-1 l-D Velocity Profiles in Oscillating Flow
for Wormersley Number a = 1, 10, 100 and 1000


66
that the velocity profile at a = 1 presents a quasi
parabolic shape at any instant within a period and is in
phase with the stimulating pressure gradient. However, at
higher frequency, for example, a = 100 and 1000, the
velocity profiles can be clearly separated into two regions:
in the vicinity of the wall the flow shows a typical thin
boundary layer, while in the region far away from the wall
the fluid moves as if it were frictionless slug flow. In
fact, within this core region the velocity distribution is
independent of the distance from the wall [29]. It can be
also seen that the phases of the velocity profiles at higher
frequencies cases are shifted about k/2 with respect to the
Fig 4-2 Velocity Profile(q=5 ) [29]


67
(Rl-r)/ (Ri~r)/
(Rl~r)/
Fig 4-3 Magnified View of Velocity Profile Near Wall
(Wormersley Number a=10, H2O, 6=0.014cm)


68
stimulating pressure gradient. At the intermediate
frequency case (a = 10 of Fig. 4-1), the slug flow boundary
is not so evident, but one can still note a boundary layer
near the wall. The same pattern of the velocity profiles
associated with moderate Wormersley numbers can also be
found in the reference [36] where the velocity profile at
Wormersley number a less than 10 was presented (Fig. 4-2) .
In order to better see the variation of the time-dependent
velocity profile within the boundary layer, a set of closer
view of the velocity profile with respect to Wormersley
number at a = 10 during phase intervals wt = 30 is plotted
in Fig. 4-3.
It should be emphasized that the solutions shown here
are under the assumption that the secondary velocity in the
radial direction is negligible compared to the axial
velocity component, namely, the non-linear inertial terms
are not considered in the governing equation (2-1). This is
a reasonable approximation for moderate oscillatory
frequency w except near the ends of the pipe. However, at
high Wormersley numbers, care should also be taken to avoid
violating another assumption, namely, that of
incompressibility and the concomitant condition that the
oscillating phase does not change between the tube ends.
High Wormersley number can be obtained either by increasing
the oscillating frequency w or by increasing the pipe
diameter for a given fluid. An oscillating flow can be


69
considered incompressible if Axw/2 < 0.05C, where C is the
speed of sound in the fluid. For mercury, C = 1360 m/sec,
one requires a = 27.1, when Ax = 20 cm and = 0.1 cm. To
avoid a appreciable phase difference between the tube ends,
one requires that L/C 2tt/w. Both the restrictions are
met in the examples to be considered below.
The Lagrangian Displacements
An alternative interesting representation to the
oscillatory velocity field are the Lagrangian displacements
of the fluid elements at different radii within the pipe.
They have been plotted in Figs. 4-4 and 4-5 at time
intervals of wt = 30 for Wormersley number a = 0.1, 1.0 and
10. It is noted that since both pipe diameter and the
working fluid were fixed in this test, Figs. 4-4 and 4-5
represent the relationship between the Lagrangian
displacement and the oscillating frequency. The
trajectories plotted in Figs. 4-4 and 4-5 have been
normalized by A/w2, where A = l/p|dp/dx| is the amplitude of
the sinusoidal pressure gradient as defined in equation (2-
1) and w is the angular velocity. Suffice it here to point
out that for the lower frequency case (for example a = 0.1
and 1.0) the Lagrangian displacement trajectory shows a
foreseeable parabolic pattern at any moment. Nevertheless,
the essential distinction between low frequency oscillatory
pipe flow and steady Hagen-Poiseuille flow is that in the
former, the Lagrangian displacement trajectories as well as


70
Fig 4-4
Lagrangian Displacement for a = 0.1 and 1.0


71
Fig 4-5
Lagrangian Displacement at a = 10


72
the velocity profiles are periodic so that the fluid
particles do not translate axially upon time averaging,
while in the later case they will.
For intermediate Wormersley number (a = 10), the
trajectories of the Lagrangian displacement departs
considerably from the standard parabolic shape. This
phenomenon can be even more clearly seen in the a = 100
case. Evidently, the higher the oscillating frequency, the
thinner the boundary layer (6 = J2v/u>) .
Tidal Displacements
Figs. 4-6 and 4-7 demonstrate time variation of the
cross-section averaged dimensionless Lagrangian displacement
at Wormersley numbers in the range from a = 0.1 to 50. The
tidal displacement can be obtained by summing the absolute
maxima and the absolute minima of these curves. The
corresponding non-dimensional tidal displacements with
respect to Wormersley number from a = 0.1 to a = 12 are
listed in table 4-1 below
Table 4-1 Dimensionless Tidal Displacement
at Different Wormersley Numbers
a
o

H
1.0
M

O
3.0
4.0
5.0
\ AX
0.00246
0.14295
0.67303
1.20213
1.41160
1.49041
a
6.0
7.0
o

CO
10.0
12.0
; ax
1.56294
1.61879
1.66046
1.71900
1.76698


73
Fig 4-6 Dimensionless Cross-section Averaged
Displacement Versus Time
q = 0.1 1.0
Fig 4-7 Dimensionless Cross-section Averaged
Displacement Versus Time
q = 2 50


74
It is noticed that for very small Wormersley number
cases (a < 0.5), the cross-section averaged displacement
varies like a sinusoidal function with respect to time (the
ordinate), and as Wormersley number increases, the cross-
section averaged displacements are no longer symmetric about
the ordinate but rather favor positive values of *. If the
Wormersley number is further increased, eventually, the
cross-section averaged displacement almost entirely lies on
the right half plane. A similar feature can also be seen in
Figs. 4-4 and 4-5. In fact, this just again shows the
existence of the phase shift in the oscillating pipe flow.
When the Wormersley number is small the cross-section
averaged displacements as well as the Lagrangian
displacements show a sinusoidal variation with respect to
time and this has ?r/2 phase lag with respect to the
stimulating pressure gradient which is assumed to be a
cosine function of wt with zero initial phase angle.
However, for higher Wormersley number, the phase lags
increase to almost n with respect to the phase of the
exciting pressure gradient. Note that the Lagrangian
displacements in Figs. 4-4 and 4-5 were computed by lining
up all the fluid particles on the plane x = 0 at wt = 0,
however, by using non-zero velocity at wt = 0 as shown in
Fig. 4-1. This implies that if the phase of the exciting
pressure gradient is taken as the base of the measurement,
it is generally not possible to assure the same phase to the


75
velocity and Lagrangian displacement as well as the tidal
displacement. For example, at phase of the stimulating
pressure gradient wt = 0 the corresponding velocity phase
may be 7r/2 and the Lagrangian displacement phase may be n.
In fact, this is just as true for very high Wormersley
numbers; however, the phase difference with respect to the
exciting pressure gradient phase is less than the value
shown above for low Wormersley number. One can well see
that if the phase lags of the Lagrangian displacement or the
tidal displacement is given, one can certainly re-draw the
diagram shown in Figs. 4-6, 4-7 by lining up the phase with
itself, and then an exact symmetric pattern of the tidal
displacement curve similar to that in the very small
Wormersley number case can be obtained. Unfortunately, the
phase lags are a function of the Wormersley number and they
are not known in advance.
In order to verify the numerical method used in this
study and to check the ETP code developed, a comparison of
the computed dimensionless tidal displacement to the one
using the analytic equations (1-7) and (1-8) given by
Kurzweg [18] for Wormersley number varying from 0.1 to 100
has been calculated and plotted in Fig. 4-8. The solid line
shows the analytic solution obtained by use of equations (1-
7) and (1-8), while the dashed line shows the results with
the ETP code developed in this study. The agreement is
quite good, particularly when the Wormersley is less than


76
3.0. However, at high Wormersley number the numerical
solution shows a very slight deviation from the analytic
solution. This deviation is believed due to an inaccurate
numerical integration over the cross-section using
relatively large time steps (our time steps per period in
the calculation were between 1000-2500). A comparison with
using 104 time steps per period for Wormersley number a = 10
was studied and shows some improvement. However, using such
small time steps in the present investigation is beyond the
capacity of the current VAX computer facility used. The
numerical error becomes particularly serious as the
oscillating frequency becomes large where the extremely thin
boundary layer requires more grid nodal points to resolve
the flow variables in the vicinity of the wall.
Fig. 4-8 shows that as the Wormersley number gets
large, the dimensionless tidal displacement tends to the
limit of 2.0, which agrees with the limit of 1.0 in the
analytical solution given by Kurzweg [18] for the reason
that the normalization parameter used in [18] is twice as
large as that in the present numerical simulation.
Fig. 4-9 shows the required stimulating axial
pressure gradient used in the present study for a pipe
radius = 0.1 cm and water (u = 0.01 cm2/sec) taken as the
working medium versus the dimensional tidal displacement Ax
in cm for various Wormersley numbers (namely, oscillating
frequency). It is evident from these results that for fixed


Dimensionless Tidal Displacement AX
77
id"' 1U 101 102
Wormersley Number a
Fig 4-8 Relationship Between Dimensionless Tidal
Displacement AX and Wormersley Number a


(Uio)
Relationship Between Tidal Displacement ax
and Exciting Pressure Gradient in Water
03
Fig 4-9


79
tidal displacement, the required axial pressure gradient in
the large Wormersley number case is orders of magnitude
higher than that in the small Wormersley number case. This
may eventually put some constraint on the use of very high
frequency in the enhanced heat transfer technique.
Fortunately, to meet the tuning condition, the required
Wormersley number in this case is of order 1. As already
discussed in the introduction, in order to gain the benefit
of axial heat transfer, the use of large tidal displacement
is always preferred. However, such an increase must be
limited by the requirement of no convective net mass
transfer occurring between two reservoirs and may also be
constrained by the ability of the device to withstand the
increase in the exciting pressure gradient.
Phase Lags
We are now in the position to study the phase lags of
the Lagrangian displacement in the oscillatory pipe flow.
Fig. 4-10 shows the phase variations (in degrees) along
radius for Wormersley number varying from a = 0.1 to 4.
Some numerical results are also shown in Tables 4-2 and 4-3.
All of the data shown in Fig. 4-10 and the tables have the
phase angle measured relative to the exciting pressure
gradient. Two features can be seen in Fig. 4-10; first, in
the core portion, the phase lags are almost equal to n/2
when the Wormersley number is small, while the lags are
almost n when the Wormersley number is large, and second,


80
the phase lags vary along radius, especially in the boundary
layer. It is such phase lags that allow the existing
temperature gradient in the very thin boundary layer of the
oscillating pipe flow to act as region of temporary heat
storage. It absorbs heat when the temperature of the core
Table 4-2 Phase Lags Along Radius
(Working Medium: H2O, Ax = 10 cm)
Nodal
point
K
a = 0.1
a = 1.
a = 2.
?=r/Rl
phase
r¡= r/Rl
phase
r?=r/Rl
phase
1
0.0000
90.63
0.0000
102.06
0.0000
134.28
2
0.1667
90.63
0.0366
102.06
0.1095
134.10
3
0.3333
90.63
0.0957
102.06
0.2164
133.38
4
0.5000
90.63
0.1913
101.88
0.2836
132.66
5
0.5833
90.63
0.2593
101.70
0.3620
131.76
6
0.6667
90.63
0.3458
101.52
0.4535
130.50
7
0.7500
90.63
0.4559
101.16
0.5603
128.70
8
0.8333
90.63
0.5958
100.62
0.6849
126.00
9
0.9167
90.63
0.7737
99.72
0.8303
122.58
10
1.0000
0.00
1.0000
0.00
1.0000
0.00


81
Table 4-2 continued
Nodal
point
K
a = 3
a = 4
a = 7
7=r/Rl
phase
r?=r/Rl
phase
r?=r/Rl
phase
1
0.0000
170.83
0.0000
179.82
0.0000
181.62
2
0.0917
170.47
0.1242
179.28
0.0970
181.62
3
0.1818
169.75
0.2370
178.02
0.1892
181.62
4
0.2703
168.31
0.3395
176.04
0.2769
181.44
5
0.3573
166.51
0.4324
173.70
0.3604
180.72
6
0.4427
164.00
0.5168
171.00
0.4397
179.28
7
0.5267
161.12
0.5935
168.12
0.5152
177.12
8
0.6091
157.88
0.6630
165.24
0.5869
174.06
9
0.6901
154.64
0.7262
162.18
0.6552
170.28
10
0.7696
150.69
0.7835
159.12
0.7201
165.78
11
0.8478
146.73
0.8356
156.06
0.7818
160.74
12
0.9246
142.42
0.8828
153.18
0.8406
155.16
13
1.0000
0.00
0.9257
150.30
0.8964
149.04
14
0.9646
147.60
0.9495
142.56
15
1.0000
0.00
1.0000
0.00


82
Table 4-2 continued
Nodal
Point
K
a = 10
H
II
a
a = 20
7 =r/Rl
phase
7=r/Rl
phase
7=r/Rl
phase
1
0.0000
180.54
0.0000
179.46
0.0000
179.10
2
0.0884
180.72
0.0834
179.10
0.1635
180.54
3
0.1740
181.08
0.1647
180.18
0.3032
185.93
4
0.2566
181.80
0.2440
184.50
0.4227
182.34
5
0.3364
182.52
0.3215
190.25
0.5247
186.65
6
0.4135
182.70
0.3970
192.41
0.6120
184.14
7
0.4880
181.80
0.4709
189.53
0.6865
184.14
8
0.5599
179.64
0.5428
183.06
0.7502
181.98
9
0.6295
175.68
0.6130
174.07
0.8046
178.38
10
0.6966
170.46
0.6815
164.00
0.8512
173.71
11
0.7616
163.80
0.7484
153.20
0.8909
167.95
12
0.8243
156.06
0.8136
141.70
0.9249
161.12
13
0.8849
147.24
0.8773
130.19
0.9540
153.57
14
0.9434
137.52
0.9394
118.32
0.9788
146.01
15
1.0000
0.00
1.0000
0.00
1.0000
0.00


83
Table 4-3 Comparison of Phase Lags With Different
Working Mediums (Ax = 10)
Nodal
Point
a = 1.
a = 5.
K
r) =r/Rl
Phase
Lags
t) =r/Rl
Phase
Lags
Water
Mercury
Water
Mercury
1
0.0000
102.06
102.14
0.0000
181.62
181.62
2
0.0161
102.06
102.14
0.0714
181.44
181.26
3
0.0366
102.06
102.14
0.1429
180.90
180.54
4
0.0626
102.06
102.14
0.2143
179.64
179.46
5
0.0957
102.06
102.14
0.2857
178.02
178.02
6
0.1378
101.88
102.14
0.3571
176.04
175.86
7
0.1912
101.88
102.14
0.4286
173.34
173.37
8
0.2593
101.70
102.14
0.5000
170.28
170.11
9
0.3458
101.52
101.78
0.5714
166.50
166.51
10
0.4559
101.16
101.42
0.6429
162.18
162.20
11
0.5957
100.62
101.06
0.7143
157.50
157.52
12
0.7737
99.72
99.98
0.7857
152.10
152.13
13
0.0000
0.00
0.00
0.8571
146.16
146.37
14
0.9286
139.86
139.90
15
1.0000
0.00
0.00


Dimensionless Radius
Fig 4-10 Phase Variation Along Radius
for Different Wormersley Numbers


85
slug flow is higher than that of the boundary layer, and it
releases heat as the core temperature is relatively lower.
This large temperature gradient enhanced by the existing
velocity phase lags allows a large amount heat to be
conductively transferred radially within a very short time
and subsequently to be transferred axially by a convective
coupling.
Part 2. The Enhanced Heat Transfer Investigation
We have examined some mechanical characteristics in
oscillating pipe flow, and compared the computed solution of
the velocity field with Uchida's solution. The results for
the velocity profiles are in good agreement. Nevertheless,
before a detailed examination of the thermal field, it is
first necessary to test the current developed ETP code when
applying the temperature equations (2-6),(2-7). It is seen
that the energy equations strongly depend on the velocity
distribution and its build-up process, so that one can use
analytic periodic velocity state (Eqs. 1-4 and 1-5, with no
build-up process), and the computed velocity (with build-up
process) to verify the correctness of the resulting thermal
variables. The enhanced heat flux is a function of both
velocity and temperature (Eqs. 2-34 and 2-35) and was chosen
for a comparison of the analytic and numerical results of
the problem. Part A of table 4-4 shows the results of the
computed enhanced axial heat flux as well as the axial
conduction heat flux when using the analytic velocity


86
Table 4-4 The Comparison of Enhanced Heat Flux Using
Numerical Velocity with Heat Flux Using Analytical Velocity
(Model 3, Water-Glass, Pr = 7.03, a = 3)
A. Heat Flux Using Analytic Velocity
AX
(cm)
(w/cm2 0 K)
4>f
(w/cm2 K)
w
(w/cm2 K)
PA
(w/cm4 K)
0.9839
0.0230
0.0149
0.0811
0.0238
2.9519
0.1855
0.0132
0.0724
0.0213
4.9194
0.3864
0.0098
0.0541
0.0159
6.8868
0.5810
0.0075
0.0415
0.0122
7.8847
0.6775
0.0067
0.0369
0.0109
9.8390
0.8638
0.0055
0.0302
0.0089
B. Heat Flux Using Computed Velocity
1.0033
0.0234
0.0149
0.0811
0.0232
3.0101
0.1870
0.0133
0.0728
0.0206
5.0164
0.3899
0.0097
0.0533
0.0155
7.0226
0.5790
0.0076
0.0420
0.0117
8.0402
0.6746
0.0068
0.0374
0.0104
10.0328
0.8597
0.0055
0.0302
0.0085
* Model 3, Water-Glass
* 4> enhanced axial heat flux
* f axial heat flux by conduction in fluid
* 4>\j axial heat flux by conduction in pipe wall
* p = /AX2


Full Text
UNIVERSITY OF FLORIDA
3 1262 08556 7963


TIME-DEPENDENT ENHANCED HEAT TRANSFER
IN OSCILLATING PIPE FLOW
By
GUO-JIE ZHANG
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1988

To my beloved motherland

ACKNOWLEDGEMENTS
The author wishes to express his deep gratitude to
the chairman of his committee, Dr. Ulrich H. Kurzweg, for
the valuable assistance and advice in guiding this research
work to its completion and for thoroughly reviewing the
entire manuscript leading to the realization of this
dissertation. Also he wishes to express his appreciation to
Drs. E. Rune Lindgren, Lawrence E. Malvern, Arun K. Varma
and David W. Mikolaitis for the many helpful discussions in
the formulation of the problem and constructive suggestions
for overcoming many difficulties in the solution process.
Thanks are expressed here also to Dean Eugene R. Chenette,
to the department chairman, Dr. Martin A. Eisenberg, and to
Dr. Charles E. Taylor for their support in allowing the
author to pursue his educational goals within this lovely
country. Part of the work presented here was funded by a
grant from the National Science Foundation, under contract
number CBT-8611254. This support is gratefully
acknowledged.
iii

TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS iii
LIST OF FIGURES vi
LIST OF TABLES ix
KEY TO SYMBOLS X
ABSTRACT xii
CHAPTERS
I INTRODUCTION 1
II FORMULATION OF THE PROBLEM 18
Governing Equations 18
Boundary Conditions 24
Model 1 24
Model 2 28
Model 3 30
Initial Conditions 33
Calculation of Tidal Displacement 35
Effective Heat Flux 37
III NUMERICAL TECHNIQUES EMPLOYED 38
Transformation 40
Crank-Nicolson Method for Momentum Equation . . 46
ADI Method for Axisymmetric Heat Equations. . . 48
Convergence Criteria 57
Grid Generation 59
IV NUMERICAL RESULTS AND DISCUSSION 62
Part 1. Oscillatory Pipe Flow Features .... 64
Velocity Profiles 64
Lagrangian Displacements 69
Tidal Displacements 72
Phase Lags 79
Part 2. Enhanced Heat Transfer Investigation . 85
Periodic Temperature Build-up
in Thermal Pumping Process 87
IV

Temperature Distribution in Model 1 95
Temperature Distribution in Model 2 100
Temperature Distribution in Model 3 108
Heat Flux versus Tidal Displacement
(Model 2) Ill
Influence of Thermodynamic Properties .... 115
Heat Flux versus Tidal Displacement
(Model 1) 119
Influence of Wall Thickness 124
Influence of Pipe Diameter 126
Variation of Axial Temperature Gradient
In Model 3 130
Comparison of Enhanced Oscillatory Heat
Transfer and Heat Conduction 133
Enhanced Feat Flux as a Function of
Wormersley Number 135
Tuning Curves 137
V CONCLUDING REMARKS 143
APPENDIX: ETP COMPUTER CODE 149
REFERENCES 157
BIOGRAPHICAL SKETCH 161
V

LIST OF FIGURES
Figure Page
1-1 F (a) Curve 7
2-1 Thermal Pumping Device 19
2-2 Model 1 Fixed End Temperature Model 25
2-3 Model 2 Periodic Heat and Cold Sources
on Insulated Wall 29
2-4 Model 3 Pipe with Extended Conducting Sections 31
3-1 Grid System Used in the Numerical Simulations. . 39
3-2 Coordinate Transformation 42
4-1 1-D Velocity Profiles in Oscillating Flow for
Wormersley Number a = 1, 10, 100, and 1000 . . 65
4-2 Velocity Profiles (a = 5, after Uchida) .... 66
4-3 Magnified View of Velocity Profile near Wall . . 67
4-4 Lagrangian Displacement for a =0.1, 1.0.... 70
4-5 Lagrangian Displacement at a = 10 71
4-6 Dimensionless Cross-Section Averaged
Displacement versus Time (a = 0.1 - 1.0) . . . 73
4-7 Dimensionless Cross-Section Averaged
Displacement versus Time (a = 2 - 50) 73
4-8 Relationship Between Dimensionless Tidal
Displacement AX and Wormersley Number a ... 77
4-9 Relationship Between Tidal Displacement Ax and
Exciting Pressure Gradient in Water 78
4-10 Phase Variation Along Radius for Different
Wormersley Numbers 84
vi

4-11 Temperature Build-up Process in Oscillating
Flow (Model 1, a = 1, Ax = 2cm) 89
4-12 Temperature Build-up Process in Oscillating
Flow (Model 1, a = 1, Ax = 5cm) 90
4-13 Temperature Build-up Process in Oscillating
Flow (Model 1, a = 1, Ax = 10cm) 91
4-14 Temperature Build-up Process in Steady Flow
(Model 2, Uave = 1.5 cm/sec) 93
4-15 Build-up Time versus Tidal Displacement
(Model 2, a = 1) 94
4-16 Temperature Distribution in Oscillating Pipe
Flow (Model 1, a = 1, Ax = 1cm) 96
4-17 Temperature Distribution in Oscillating Pipe
Flow (Model 1, a = 1, Ax = 2cm) 97
4-18 Temperature Distribution in Oscillating Pipe
Flow (Model 1, a = 1, Ax = 5cm) 98
4-19 Temperature Distribution in Oscillating Pipe
Flow (Model 2, a = 1, Ax = 1cm) 101
4-20 Temperature Distribution in Oscillating Pipe
Flow (Model 2, a = 1, Ax = 5cm) 102
4-21 Temperature Distribution in Oscillating Pipe
Flow (Model 2, a = 1, Ax = 10cm) 103
4-22 Temperature Distribution in Oscillating Pipe
Flow (Model 2, a = 1, Ax = 20cm) 104
4-23 Temperature Distribution in Oscillating Pipe
Flow (Model 1, a = 1, Ax = 30cm) 105
4-24 Temperature Distribution in Steady Flow
(Model 2, Uave = 0.5 - 7.5 cm/sec) 106
4-25 Temperature Distribution in Oscillating Pipe
Flow (Model 3, a = 1, AX = 10cm) 109
4-26 Magnified View of Temperature in the Central
Pipe Section (Model 3, a = 1, Ax = 10cm) . . . 110
4-27 Heat Flux in Oscillating Flow and Steady Flow
(Model 2, a = 1, Water as Working Fluid). . . . 112
vii

117
4-28 Influence of Thermodynamic Properties of H2O
on the Enhanced Heat Flux (Model 2, a = 1,
Ax = 10 cm)
4-29 Heat Flux versus Tidal Displacement
(Model 1, a = 1, Pr = 7.03) 120
4-30 Heat Flux versus Tidal Displacement
(Model 1, a = 3, Pr = 7.03) 121
4-31 Influence of Wall Thickness on Axial Heat Flux
(Model 1, Water-Glass, a = 1, Ax = 5cm).... 126
4-32 Influence of Pipe Diameter on Heat Flux for
Fixed Frequency (Model 3, Water-glass, a = 3,
Ax = 10cm) 128
4-33 Typical Iso-Temperature Contour in Oscillating
Pipe Flow (Model 3, Water-Glass, a = 3,
Ax = 10cm) 129
4-34 Variation of Temperature T^ and T2 versus Ax
(Model 3, Water-Glass, a = 3) 131
4-35 Comparison of Enhanced Heat Transfer and Heat
Conduction in Oscillating Pipe Flow
(Model 3, Water-Glass, a = 3) 134
4-36 Variation of Axial Heat Flux versus Wormersley
Number (Model 3, HoO-Glass, Hg-Steel,
Ax = 10cm) 136
4-37 Computed Tuning Curves (Model 3, H20-Steel and
Hg-Steel, ax = 10cm) 138
4-38 Tuning Curve versus Wormersley Number
(after Kurzweg) 140
4-39 Ratio of Heat Flux due to Conduction to
Enhanced Heat Flux versus Wormersley Number
(Model 3, H20-Steel, Hg-Steel, Ax = 10cm) . . 141
viii

LIST OF TABLES
Tables
Pages
4-1 Dimensionless Tidal Displacement at Different
Wormersley Numbers 72
4-2 Phase Lags Along Radius (Working Medium: H2O,
AX = 10cm, a - 0.1 - 20) 80
4-3 Comparison of Phase Lags with Different
Working Mediums (Ax = 10cm, a - 1, 5) 83
4-4 Comparison of Enhanced Heat Flux Using
Numerical Velocity with Heat Flux Using
Analytic Velocity (Model 3, H20-Glass,
Rl = 0.1cm, R2 = 0.15cm, Pr = 7.03, a = 3) . . 86
4-5 Enhanced Axial Heat Flux via Tidal Displacement. 113
4-6 Enhanced Axial Heat Flux in Steady Flow .... 114
4-7 The Influence of Properties of Water on the
Enhanced Axial Heat Flux 116
4-8 Variation of the Axial Temperature Gradient
versus Wormersley Numbers (Water-Glass,
Ax = 10 cm) 132
ix

KEY TO SYMBOLS
X
r
t
€
r?
r
L
R1
r2
w
S
P
P
c
u
P
K
k
Ke
Pr
x coordinate
radial coordinate
time
coordinate (x) in transformed plane
coordinate (r) in transformed plane
transformed time
pipe length
pipe inner radius
pipe outer radius
oscillating frequency
boundary layer thickness
pressure
density
specific heat
Kinematic viscosity
dynamic viscosity
thermal diffusivity
thermal conductivity
coefficient of enhanced heat diffusivity
Prandtl number
i/p 13p/3x| a measure of the maximum axial
pressure gradient (cm/sec2)
x

a
Wormersley number a = Ju>/v
T
Temperature
7
7 = dT/3 x time-averaged axial temperature
gradient
r
f = r/Ri dimensionless radial distance
g
radial temperature distribution function
u
velocity
u0
representative velocity
f
velocity shape function
X
Lagrangian displacement
DX
dimensionless tidal displacement
AX
dimensional tidal displacement
Qtotal
time averaged total enhanced axial heat flow over
pipe cross-section

axial heat flux
Subscript
f
fluid
w
wall
h
hot
c
cold
th
thermal
eq
equivalent
adj
adjacent
XX

Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
TIME-DEPENDENT ENHANCED HEAT TRANSFER
IN OSCILLATING PIPE FLOW
By
Guo-jie Zhang
April 1988
Chairman: Ulrich H. Kurzweg
Major Department: Aerospace Engineering, Mechanics,
and Engineering Science
The problem of time-dependent enhanced heat transfer
in an incompressible viscous laminar fluid subjected to
sinusoidal oscillations in circular pipes which are
connected to a hot reservoir at one end and a cold reservoir
at the other end has been examined numerically in detail.
Three models were designed for the investigation of such an
enhanced thermal pumping process and a computer code (ETP)
was developed to implement all the numerical calculations.
To increase the understanding of the mechanism of
thermal pumping, the periodic velocity profiles and
Lagrangian displacements as well as tidal displacements at
various Wormersley numbers (from a = 0.1 to 1000) were
studied. Some transient problems of enhanced axial heat
transfer in oscillating pipe flow such as the periodic final
temperature build-up process in oscillating pipe flow were
also examined. The time-dependent temperature distribution
xii

in the different models was numerically studied in detail.
The enhanced axial heat flux magnitude versus different
tidal displacements with water and mercury as the working
fluids bounded
by
pipe walls
of different
material
were
observed
and
the
quadratic
coefficients
found.
The
influence
of
the
variation
of water properties on
the
enhanced axial heat flux was numerically examined and the
results show that the enhanced axial heat flux can vary
about 150 percent even within the temperature range from 0°C
to 100°C. The effects of wall thickness and pipe diameter
in enhanced thermal pumping were also studied and the
optimum wall thickness was found to be about 20 percent of
the pipe radius in the water-glass combination. The tuning
effect in the water-steel and the mercury-steel cases was
examined and the results show good agreement with analytic
predictions. A comparison of the enhanced axial heat flux
with the axial heat flow due to heat conduction at various
tidal displacement and Wormersley numbers shows that the
latter is quite small and negligible provided the tuning
condition is satisfied.
This study has shown that the enhanced thermal pump
is indeed a very effective tool for those problems where
large amounts of heat must be transported without an
accompanying convective mass exchange. The investigation
also indicates that turbulent flow in the reservoirs is
preferable to laminar conditions and should receive more
attention in future studies.
xiii

CHAPTER I
INTRODUCTION
Enhanced heat transport in a viscous laminar fluid
subjected to sinusoidal oscillations in a very long pipe
which connects a hot fluid reservoir at one end and a cold
fluid reservoir at the other end (Fig. 2-1) has been
recognized and studied recently by Kurzweg [15, 16, 17, 22],
The results obtained show that with this oscillatory pipe
flow the heat transferred axially from the hot end to the
cold end can be orders of magnitude larger than that
obtained by pure molecular conduction in the absence of
oscillations. In addition, the more important thing of
interest is that this heat transfer process involves no net
convective mass transport. Major assumptions made in the
above cited studies on enhanced heat diffusion are that a
constant time-averaged non-zero axial temperature gradient
is always present in the oscillating flow and that the axial
molecular conduction along the wall and in the oscillating
fluid is negligible.
Discovery of this enhanced heat transport phenomenon
was made possible by earlier studies on axial dispersion of
contaminants within steady laminar flows through capillary
tubes by Taylor in 1953 [32], and Aris in 1956 [4]. These
1

2
earlier studies show that when a small quantity of a
contaminant is introduced into a circular pipe, the
dispersion of the resultant contaminant cloud is greatly
enhanced by the flow of the fluid. Bowden's 1965 results
show that similar dispersion effects occur in oscillatory
flow [8], This enhanced axial dispersion of contaminants in
the presence of laminar oscillatory flow within capillary
tubes was studied in 1975 by Chatwin who suggested that the
assumption of constant time-averaged axial contaminant
gradient can be made [10]. Recently, Bohn et al. extended
this work to the study of gas component transfer in binary
gas mixtures when these are confined to single tubes and a
sinusoidal pressure variation is applied [7]. Further
studies in 1983 by Watson [38] show that the effective
diffusion of contaminants is proportional to the square of
the tidal displacement. This has been experimentally
verified by Joshi et al. [13] and by Jaeger [12], both in
1983, and most recently by Kurzweg and Jaeger [19], in 1987.
All these results show that the contaminant would spread
axially in both steady and oscillatory laminar pipe flow at
rates as much as five orders of magnitude higher than in the
absence of fluid motion.
The first significant research work extending these
enhanced axially diffusion studies to the heat transfer
problem in oscillatory flow within very slender pipes or
flat plate channels is due to Kurzweg [15, 16, 22]. In

3
early 1983 Kurzweg suggested that a similar dispersion
process should occur in the heat transfer area because of
the similarity in both the governing diffusion and heat
transfer equations [16], and the first preliminary theory
was formulated in 1985 [15, 16], in which, referring to
Chatwin's idea, a time averaged constant axial temperature
gradient assumption was used. The instantaneous temperature
distribution was taken to be of the form [15, 22].
T = 7[x + RX g (f ) elwt ] (1-1)
where 7 = 3T/3x is the time-averaged axial temperature
gradient, is the tube radius, x is the axial distance
along the capillaries (with x = L/2 and x = -L/2 denoting
the ends) , L is the pipe length under consideration, w is
the oscillating frequency of fluid, g(f) is a radial
temperature distribution, and f = r/Ri is the dimensionless
radial distance. The theoretical analysis shows that under
certain conditions such enhanced axial heat diffusivity can
indeed be significantly larger than the axial molecular
conduction [15, 16] and this has been verified by some
experimental measurements by Kurzweg and Zhao [22].
In order to better understand the physical mechanisms
of this interesting and potentially useful heat transfer
technique, we shall examine in greater detail the thermal
pumping model shown in Fig. 2-1. It is assumed that a
bundle of very thin and long tubes is connected to a

4
reservoir which supplies unlimited hot liquid at one end to
a second reservoir which supplies unlimited cold liquid at
the other end. The liquid in the pipes is oscillating
axially with an amplitude such that none of the liquid which
is originally in the middle portion of pipes ever runs into
either reservoir. That is, there is no net convective mass
exchange between two reservoirs. The largest axial fluid
dimensionless displacement (when cross-section averaged) is
referred to as the nondimensional tidal displacement and is
denoted by "AX" (it should not be confused with the
dimensional tidal displacement "ax" frequently used in the
present study) . At time t = 0, the fluid within the pipes
is set into axial oscillations at angular frequency w and
tidal displacement Ax. After a short transient, this
oscillatory motion will lead to very large axial heat flows
which can be readily made to exceed those possible with heat
pipes.
Before exploring the mechanisms of this enhanced heat
transport, it is necessary to introduce some new concepts
which are commonly used in the study of this type of
oscillatory motion.
As is well known [36], for high frequency viscous
laminar axial oscillations within fluid flow along rigid
pipes, the non-slip boundary condition creates a very thin
Stokes' viscous boundary layer of thickness
S = J2u/ (1-2)

5
where u is the fluid kinematic viscosity. For room
temperature water at a frequency of 10 Hz, this viscous
boundary layer is approximately 1.7*10“2 cm. The
corresponding thermal boundary layer thickness is about
fith = t/fPr (1-3)
where Pr is the Prandtl number. Note that both 5 and 5th
decrease in thickness as the oscillating frequency w
increases.
In the theoretical analysis, it is always assumed
that a fully developed velocity profile of the oscillating
flow exists within the pipes. At high frequency w, this
flow consists essentially of a slug flow over most of the
fluid core bounded by a thin boundary layer of width 8 as
discussed by Uchida [36], Neglecting end effects, the fully
developed oscillating laminar velocity profile in pipes, due
to a periodic axial pressure gradient, is found to be [19]
u(r ,t) = u0f(r)elwt (1-4)
where UQ is a representative velocity, f = r/R^ is again the
dimensionless radial distance, f(^) the velocity shape
function, and u> the angular velocity of the oscillatory
flow. The explicit form of f(f) is
1 -
f(r)
Jn(7-i«r) I
Jo(y-ia) J
(1-5)

6
where A = Ri213p/5x|/pi/U0 is the nondimensional pressure
gradient maximum acting along the capillaries, a = Ris
the Wormersley number measuring the ratio of inertia to
viscous forces, v is the fluid kinematic viscosity of the
carrier liguid, and p is the fluid density. This velocity
profile will reduce to the familiar Poiseuille parabolic
shape as the angular frequency w becomes small, while at
moderate frequency, f(r) has the shape demonstrated by
Uchida [36].
Another new term commonly used when dealing with
oscillating flow is the cross-stream averaged dimensionless
tidal displacement AX which can be mathematically defined as
AX =
4U0
I
Jff (r)dr
(1-6)
and on integration, yields [18]
AX
I a p/ax I
1/2 p
1 +
2
a
F(Q)
(1-7)
where the complex function F(a) has the form
F(a)
. Jn'(Z^)
1 Jo(/=!£)
with the prime denoting differentiation. Using the
definition of Kelvin function: J0(/-ia) = ber a + i bei a

0.8
a = a Veo/ v
Fig 1-1
F (a) Curve [18]

8
[1], the complex function F(a) (Fig. 1-1) can be further
written as:
F(a) = Fr (a) + i Fi(a)
ber 'a
+ i
bei'a ]
ber a
+ i
bei a J
(1-8)
This dimensionless tidal displacement is related to the
maximum of the periodic pressure gradient via [18]
AX
I 3p/ax|
1/2
1
J2
1-
a
a « 1
(1-9)
a » 1
Apparently, for any fixed tidal displacement Ax and
oscillating frequency w, the axial pressure gradient
required is proportional to the inverse square of the
Wormersley number when a is small; however, it is
independent of a when a is very large. This also implies
that the exciting axial pressure gradient |ap/3x| is
approximately proportional to the fluid kinematic viscosity
v and inversely proportional to the square of the pipe
radius when Wormersley number is very small (it happens
only at low «, small , and large v, for example, oil),
while it is almost independent of the fluid kinematic
viscosity and the pipe radius when the Wormersley is very
large (it happens only at very large R^, high w, small v,

9
for instance, a liquid metal). Note, if the tidal
displacement AX is fixed, while allowing the oscillating
frequency to change, the axial pressure gradient |<3p/3x| can
become very large when the oscillating frequency becomes
large. This is mainly due to inertial effects and not so
much due to viscous drag forces which dominate the
oscillatory flow at small Wormersley number.
With the definition of the above quantities, we are
now in a position to explore the details of the enhanced
axial heat transfer in oscillatory flow within pipes. It is
assumed that a constant temperature gradient exists along
the pipe in the axial direction and that a very large time-
dependent radial temperature gradient variation is
superimposed. When the fluid moves towards the cooler side
(we term this the positive stroke), the hotter fluid within
the pipe core which is initially brought into the pipe from
the hot reservoir produces a large radial heat flow via
conduction to the cooler portions of the fluid within the
Stokes' boundary layer and to the cooler pipe wall; while
during the negative (or reverse) stroke, i.e., when the
fluid moves towards the hotter side, the higher temperature
in the boundary layer and the pipe wall conducts the heat
back into the cooler fluid core. This coupled radial heat
conduction with an axial convective transport leads to an
enhanced axial heat flux along the entire length of the
pipes.

10
Further, from the system point of view, the heated
fluid near the cold reservoir will eventually be ejected
into the cold reservoir and mixed there with the lower
temperature liquid. Contrarily, near the hot reservoir side
the fluid within the pipes which has been cooled during each
positive stroke is pushed out into the hot reservoir and
mixes with the higher temperature liquid. This process of
thermal pumping is what leads to a time-averaged heat flow
from the hot reservoir into the cold reservoir. It differs
essentially from the working principle of a normal pump.
For a normal pump one can draw the analogy with transport of
a one-way vehicle which transports passengers as well as the
carrier from one point to another. For the thermal pump one
can draw the analogy with a two-way busline which
periodically loads and unloads the passengers (heat) from
the hot reservoir to the cold reservoir, heat can be
continually transferred, and the carrier, in the time-
averaged sense, does not move. This property is
particularly important for those systems where a large
amount of heat transfer is needed while the working fluid is
required to remain in the system (as in nuclear reactors).
Note that the axial heat conduction, in general, is assumed
to be very small in this thermal pumping process compared
with the enhanced axial heat flow [16].
Apparently, the heat transport rate in thermal
pumping is governed by both the thermal properties of the

11
working medium and pipe wall and the characteristics of the
oscillatory motion of the fluid. The enhanced axial heat
flow does increase with increasing oscillating frequency as
this thins out the boundary layer and leads to an
accompanying increase in the radial temperature gradient.
This observation holds only as long as the thermal
properties of the liquid and the wall are compatible. If
the molecular conduction of the fluid in the radial
direction is very small, then even high frequency
oscillatory motion will not produce a large increase in the
rate of enhanced axial heat flow. This is because such a
system fails to supply the "passengers" enough time to be
loaded onto the "bus" and to be unloaded from the "bus".
Obviously, the system just wastes energy. On the other
hand, if the molecular conduction of the fluid in the radial
direction is very large, but the frequency of the
oscillatory motion is low, once again one can not expect
that there will be an efficient enhanced axial heat transfer
between two reservoirs because the "bus" is now moving too
slowly.
The above observation can be confirmed by an analysis
of the performance of a water-glass combination (i.e., water
is the working fluid medium within a glass tube) and of a
mercury-steel combination. For the former, it is necessary
to employ rather small diameter tubes and low frequency
oscillations with large tidal displacement, for it has

12
relatively poor heat diffusive properties as compared with
the mercury-steel case. This small tube diameter-lower
oscillation frequency set-up is necessary in order to ensure
that there is sufficient time to transfer the excessive heat
content of the bulk water core to the tube wall during the
positive stroke of each period and to permit the transfer of
excess heat from wall to the cooler core fluid during the
negative stroke. Otherwise, the water would just carry a
portion of its heat content back and forth in the pipe and
the condition for achieving optimum enhanced axial heat
transfer could not be met. For the mercury-steel case, one
can chose relatively large pipe diameters and higher
oscillation frequencies with smaller tidal displacement
because of the higher thermal conductivity. This assures
that only very short times are needed to exchange the heat
between the core of the fluid and the wall. It should be
pointed out that the suggestion of using smaller tidal
displacement is purely due to the mechanical considerations
and that one always tries to keep Ax as large as possible in
order to produce large axial heat flows.
From the above discussion, it can be concluded that
the process of enhanced heat transfer via oscillatory
pumping requires a precise tuning of parameters governing
the enhanced heat transport. Indeed there is an expected
"tuning effect" as discussed in references [16, 20, 21].

13
The tuning effect is a very important concept in the
study of the presently considered heat transfer process. It
shows that there will be an optimum combination between
thermal properties of the working medium and wall and the
characteristics of the oscillatory motion. The qualitative
aspects of the tuning effect have been observed earlier for
both the case of a flat channel and that of the cylindrical
pipe [15, 16]. From Fig. 4-38, one can see that an optimum
for axial heat transfer occurs only at or near the tuning
point which depends on the oscillating frequency and the
thermodynamic properties of the fluid and wall. As has been
pointed out by Kurzweg [20], in order to obtain the optimum
enhanced heat transfer one has to carefully select suitable
values for the pipe size, the material of pipe and the
working medium as well as the manner of oscillatory motion.
The nondimensionalized enhanced heat diffusivity is
defined as
Ke
where Ke = /ypc is the coefficient of enhanced heat
diffusivity, is the axial heat flux, 7 is the time-
averaged axial temperature gradient, p is the density of the
fluid, and c is the specific heat. One can show that this
nondimensional enhanced heat diffusivity is a function of
both the Wormersley number and the Prandtl number [21] and
hence that the dimensional axial heat diffusivity Ke is a

14
function of the tube radius , the oscillating frequency w,
the kinematic viscosity u, and the square of the
dimensionless tidal displacement AX. This can be explained
from the fact that the radial heat flow is proportional to
the product of the representative radial temperature
gradient 7AX/6th an<3 the surface area per unit depth of ttAX
available for cross-stream heat transport.
The use of large tidal displacement is always
beneficial in the enhanced axial heat transfer within
oscillating pipe flow. However, in order to avoid the
direct convective net mass exchange between the two
reservoirs, the tidal displacement must be limited to less
than about one half of the pipe length.
As has already been predicted by theoretical studies
and will be confirmed by the present numerical simulations,
the axial heat transfer will be further enhanced if the
rigid surface (part of the rigid wall with finite thickness)
has a non-zero thermal diffusivity and hence heat storage
capability.
Note that the existing considerations are restricted
to laminar flow. Turbulent flow conditions can occur in
oscillating pipe flow at higher values of wAx2/V [26, 27]
and apparently would destroy the assumptions of the current
analytic model of the thermal pumping process. Fortunately,
the condition for optimum enhanced heat transfer in such
oscillating pipe flow obtained at the tuning point requires

15
very slender pipes, such that the Reynolds number is usually
small enough so that the oscillating motion falls within the
laminar range [22].
The theoretical aspects of the oscillatory enhanced
axial heat transfer process have been developed much further
than its experimental and numerical counterparts. The
theoretical predictions are quite limited and consider only
cases under certain simplifying assumptions [8], Numerical
work is necessary in order to not only to examine the
correctness of the theoretical analysis but also to further
the development of advanced enhanced thermal pumping
devices. Numerical studies are not only fast, economical
and accurate, but they also offer a handy way to access
complex geometries which can not be handled analytically.
It is the purpose of this study to extend the
analytic work on thermal pumping by a detailed numerical
study. We intend first to examine some transient problems
of axial heat transfer in oscillatory pipe flow, such as the
development of the velocity profile at various Wormersley
numbers in contrast with those of reference [36], where only
several special cases with intermediate Wormersley number a
were discussed, and to examine numerically the relationship
between the tidal displacement and the required
corresponding strength of the periodic pressure gradient as
a function of Wormersley number a and of tidal displacement
Ax.

16
Next, we examine the build-up process of the
temperature distribution in a pipe which connects a hot
reservoir at one end to a cold reservoir at the other end
and see whether there actually exists a constant time-
averaged temperature gradient along the pipe axis when the
final periodic state is eventually reached. Note that a
time-averaged linear temperature distribution along the
axial direction is an essential assumption in the existing
theoretical studies.
The third part of this investigation which forms the
main effort, is a computer-aided numerical simulation of the
thermal pumping technique, including an investigation of the
variation of the enhanced axial heat flux versus the tidal
displacement, the variation of enhanced axial heat flux
versus different Wormersley numbers, and a study of the
variation of heat flux versus different Prandtl numbers. It
also includes a study of the influence of wall thickness and
pipe diameter as well as the change of the fluid properties
on such an enhanced axial heat flux and an examination of
the tuning effect in the conducting wall case. Further, we
compare axial heat transfer in oscillating flow with that in
the steady flow, and compare quantitatively this enhanced
heat flux with that induced by the pure axial molecular
conduction under various Wormersley numbers and different
fluid-solid wall combinations. Finally, the author examines
the phase lag phenomenon in oscillatory flow in order to

17
further reveal the mechanism whereby this new heat transfer
technique functions.
The computational work presented here was completed
on the Vax-11/750 electronic computer in the Department of
Aerospace Engineering, Mechanics, and Engineering Science,
and on Vax-11/780 in the Center for Instructional and
Research Computing Activities, University of Florida,
Gainesville, Florida. The numerical approach used, for
solving the presently considered heat transfer problem,
employed a second-order Crank-Nicolson scheme and an
Alternating Direction Implicit method (ADI) with Thomas
algorithm. A Fortran computer code named ETP (Enhanced
Thermal Pumping) was developed to implement all the
calculations.

CHAPTER II
FORMULATION OF THE PROBLEM
A single tube with inner radius R^, outer radius R2
and length L (such that L >> R^ and R2) connects a large
reservoir of hot fluid at temperature T = Tjj at one end, to
another large reservoir of the same fluid at cold
temperature T = Tc at the other end (Figs. 2-1 and 2-2) .
The tidal displacement Ax and the oscillating frequency u>
are adjusted to assure that the turbulent flow does not
occur. The tube is oriented in such a manner so that the
effect of gravity on the oscillatory motion of the fluid in
the pipe is negligible. Water and mercury are employed as
the working mediums. They are understood to be Newtonian
and incompressible fluids. The variations of thermal
properties of the fluid with temperature during the heat
transport process are assumed to be negligible. With these
assumptions the problem of enhanced time-dependent heat
transfer induced by a simple harmonic oscillatory laminar
fluid motion in a very long circular pipe can then readily
be formulated.
Governing Equations
The use of a very slender capillary circular tube
with constant cross-sectional area and neglecting end
18

HOT fLUID IN HOTTER FLUID
(PRIMARY LOOP) OUT
Fig 2-1 Thermal Pumping Device
(after Kurzweg, U.S. Patent, 4,590,993 May 1986)

20
effects insures a laminar axisymmetrical one-dimensional
time-dependent motion. It is convenient to employ
cylindrical coordinates for this problem and we denote the
coordinate in the axial direction of the tube by x, and the
radial direction by r. The axial velocity can be taken to
be independent of x and, in order to satisfy the
requirements of continuity the other velocity components
must vanish. We shall further assume that the pressure
gradient induced by moving the piston (Fig. 2-1) is harmonic
and has the form [29]
1 5p
• — ~ * A cos “t t2-1)
where A = l/p|ap/dx| is a constant which measures the
maximum pressure gradient existing along the x-axis.
Clearly this equation implies that we are now dealing with a
time-dependent sinusoidal pressure gradient which is
constant over the pipe cross-section at any instant and the
pressure varies linearly along the x-axis. The simplified
Navier-Stokes equation for this problem is [36]
3U i/ d r 3U
— = Acos(wt) + — ¿7-
0 < r < Ri
(2-2)
where U(r,t) is the time and radially dependent axial
velocity component.

21
The corresponding temperature T(x,r,t) of the fluid
within the pipe is governed by the heat conduction equation
[15]
d T
at
-u
3T
ax
+ K f
i a aT
r ar ar ^
a2T ^
0 < r < R!
(2-3)
where is the inner radius of the tube and *cf the fluid
thermal diffusivity which is related the thermal
conductivity kf by
kf
Here p is the fluid density, and c is the specific heat.
Note that the viscous heating term has been neglected in
equation (2-3) since it is very small for most experimental
conditions; this is justified provided one does not deal
with very high Prandtl number fluids such as oils. The
temperature in the wall can be determined from the solution
of
3T r 1 3 aT d2T 1
dt - *W[ r gr (r g r) + gx2 J
Rl < r < R2
(2-4)
where kw is the thermal diffusivity of the conducting wall
in Ri < r < R2 and is defined by

22
K w
kw
P wcw
where kw, pw, and cw, are the wall conductivity, density,
and the specific heat, respectively. By introducing the
following non-dimensional terms, equation (2-2), (2-3) and
(2-4) can be treated more easily.
t* =
t
1/w
r
*
x
x
S
U*
u
A/ w
where 6 = J2v/u> is again the fluid viscous boundary layer
thickness. The dimensionless governing system of equations
can then be written as
air
dV
= cos(t ) +
r a2u*
ar
TT2 +
air
ar
0 < r < RX
(2-5)
di
at’
= cu
aT7
ax’
r a2T*
2Pr
ar
Trz~ +
aT7
a 2t*
ar
it +
ax
^r
0 < r < RX
and
(2-6)
a t
at
ir~ =
2S
a 2t*
r*2
+
aT
ar
a2T*
ax
Rl < r < R2
(2-7)

23
where
and
M Cf
Pr =
K
P cw
S =
*W
C
-A
^7
This nondimensionalization has some advantages in the
computing process to be carried out below. The
dimensionless velocity and its distribution over the cross-
section found from the momentum equation (2-5) are expected
to be universal for any Wormersley number a = R^Jup/n and
any associated quantities, such as the tidal displacement
and Lagrangian displacement. Its final periodic form is of
the form given by (1-4). The dimensionless temperature in
the pipe is only related to the Prantdl number and the
dimensionless velocity, while that in the wall is related to
the ratio of wall heat diffusivity to the kinematic
viscosity, as seen from equation (2-6) and (2-7).
The governing dimensionless equations (2-5), (2-6),
and (2-7) are a set of second-order parabolic type of
partial differential equations expressed in cylindrical
coordinates. To solve this set of simultaneous equations, a
corresponding set of boundary conditions and initial
conditions are required. Since the velocity is assumed to
be a function of r and t only, just two boundary conditions

24
are needed for solving the momentum equation, while for the
temperature T(x,r,t) the heat conduction equations require
two boundary conditions in both the r and x directions as
well as compatibility conditions along the interface between
the fluid and the solid wall. It should be pointed out that
the initial conditions are less important than the boundary
conditions if one seeks only a final periodic state.
Boundary Conditions
The boundary conditions for the velocity are the
usual ones for viscous flow, namely, that the velocity is
zero at the inner surface of the wall (r* = R^*) . Also by
symmetry, the normal derivative of velocity at the axis is
zero. That is,
U*(R!*, t*) = 0 (2-8)
and
3U*(0, t*)
The boundary conditions for the temperature depend on the
particular model investigated.
Model 1
In this model, it is assumed that the temperature at
each end of the pipe is equal to that of the connecting
fluid filling the reservoirs when the fluid moves into the
capillary tube (i.e., during one half of each cycle). The

Fig 2-2
Model 1
L=20cm
L
INSULATING WALL•
Fixed End Temperature Model
R^=0.1cm, R2=0-15cm

26
end boundary temperature at x = 0 and x = L will take the
values of temperature of the adjacent pipe fluid elements.
With this model, we tried to simulate the real experimental
observations where it can be clearly seen that a fluid jet
exits the pipe during the outstroke, while the elements of
the fluid enter the tube in a funnel pattern during the
instroke. This observation shows that there is enough time
during each oscillating cycle to allow the fluid elements
within the exiting jet to fully mix with the fluid particles
which originally are in the reservoir before the next
instroke, provided the oscillating frequency is not too
high. The end temperature boundary conditions are here
taken as
(when fluid enters pipe)
T*(0, r*, t*)
(when fluid leaves pipe)
(2-10)
(when fluid enters pipe)
(when fluid leaves pipe)
(2-11)
where T*c, T*h are the nondimensional temperatures of cold
and hot reservoirs, respectively. While T*acjj is the

27
temperature of the fluid elements which are adjacent to the
corresponding ends at a particular instant.
The thermal boundary conditions along the outer
surface r* = R2 of the pipe wall are taken to satisfy the
insulating wall condition, and along the axis of the tube
the temperature is assumed to meet the symmetric boundary
requirement, i.e., the radial derivatives of temperature
along axis are equal to zero. We thus have
d T
d r
r* = R2*
= 0
(2-12)
and
d T
dr"
r* = 0
= 0
(2-13)
The compatibility conditions along the interface
between the fluid and the solid wall are that the radial
heat flux and temperature are constant across the interface.
That is
dT
kf
dr
d T
fluid
- kw dr*
wall
at r = R]_
(2-14)
and
rji ^
ip ^
fluid
wall
at r = Rp
(2-15)
Since numerically the same nodes are chosen along the
interface, condition (2-15) will be automatically satisfied.

28
Model 2
Here it is assumed that a heat source rim of width 2b
is directly mounted on the interface between the solid wall
and the fluid at x = L/2, while two small cold rim sources
of width b each are mounted at x = 0 and x = L (see Fig. 2-
3) . The wall thickness is assumed to be zero. This model
is intended to simulate the enhanced heat transfer process
of oscillatory fluid in an infinitely long pipe which is
heated and cooled by the alternative evenly distributed heat
and cold sources along the outer surface of a very thin wall
which possesses infinite heat conductivity. Apparently,
this geometry can be well simulated by the present model
with periodic boundary conditions. Mathematically the
boundary conditions are
T*(x*, Rl*, t*) = T*h
T* (x*, Rl*, t*) = T*c
or
d T
dr
■*— = 0
or
X2* <
x* >
X3 *
(2-14)
0 < :
X* <
XI*
(2-15)
X4* <
X* <
L*
XI* <
X* <
X2*
(2-16)
X3 * <
X* <
X4 *
X6* are nondimensional
coordinates of points which correspond to X = 0, b, L/2-b,
L/2+b, L-b, and L, respectively in Fig. 2-3. The periodic
end boundary conditions are given as

to
vo
Model 2 — Periodic Heat And Cold Sources
on Insulated Wall (Except Sources Area)
L=20cm, Ri=0.1cm, b=lcm
Fig 2-3

30
T*(0, r*, t*) = T*(L*, r*, t*)
(2-17)
and
3T* i dT* i
^ I x* = 0 3X* I x* = L*
(2-18)
Model 3
In model 1, the boundary conditions for the heat
equation at X = 0 and L are somewhat artificial. The
temperature distribution in the vicinity of pipe ends may
not precisely match the constant temperature end conditions.
This is particularly true in the large tidal displacement
and/or very high oscillating frequency cases since a
temperature drop occurs at the hot end and a rise at the
cold end as fluid particles exit the pipe. This leads to a
discontinuity of temperature which will lead to unexpected
dispersion errors if an even order numerical method is used
[3, 24, 30], To improve the end boundary conditions
presented in model 1, the following extended model has been
considered. It is assumed that an extension pipe which is
some 5 times the length of the central pipe is attached to
the original pipe used in model 1. The heat and cold source
are assumed to be located on the outer surface of the wall
in the extended sections of the pipe as well as at both ends
of the tube as shown in Fig. 2-4. This model is used to

Model 3 — Pipe with Conducting Pieces Model
L=20cm, Ri=0.1cm
Fig 2-4

32
investigate the situation where the heat contained in the
jet is exchanged by pure heat conduction with the
surrounding fluid elements in each reservoir without any
convective mixing. In this case, the boundary conditions
along the outer surface of the wall are
T* = T*h ,
0 < X* < XI* (2-19)
T* = T*c
X2* < X* < X3* (2-20)
and
3 T*
"IT* " 0
XI* < X* < X2* (2-21)
where XI*, X2*, and X3* are nondimensional coordinates of
points which correspond to X = 5L, 6L, and 11L,
respectively, in Fig. 2-4.
At both ends, we have
T*(0, r*, t*) = T*h
(2-22)
and
T*(L*, r*, t*) = T*c
(2-23)
As in the other models, the symmetry condition along the
axis of pipe reguires that
3 T*
3r* r*=0~ 0
(2-24)
0
(2-24)

33
and the radial flux and temperature continuity condition
along the interface are
d T*
kf dr*
fluid
dT*
kw
wall
and
(2-25)
ip *
fluid
wall
(2-26)
Here again kf and kw are the same as described in the
previous models.
Initial Conditions
The initial condition chosen depends on the problem
under consideration. If one's goal is to investigate the
periodic quasi-steady solution only, the initial conditions
chosen should be as close to the quasi-steady state as
possible so that a rapid convergence rate at low CPU time
cost is achievable. If one intends to study the transient
process, then various initial conditions should be supplied
according to purpose of the investigating cases selected.
For the velocity initial condition we choose for all our
studies
U*(x*, r*, 0) = 0
(2-27)
For the temperature, if the purpose of investigation is to
examine the build-up process of the periodic quasi-steady

34
state in the thermal field, the initial condition should be
taken as
T*(x*, r*, 0) = 0
(2-28)
However, for the other cases, in order to gain faster
convergence, the initial temperature can be assumed to have
a linear distribution in the axial direction, and thus have
the form
T*(x*, r*, 0)
T*h +
(T*c - T*h)xx*
LL*
(2-29)
where LL* is the dimensionless length within which the axial
linear temperature distribution is assumed to hold, and xx*
is the dimensionless distance measured from the "origin"
which is chosen only for the purpose of this linear
temperature initialization. Both the LL* and the origin
selected depend on the model considered. For Model 1 and
Model 3, LL* is equal to L*, and the origin is taken at the
left end (Model 1), or at the left intersection between the
central pipe and the left extension pipe (Model 3). However,
for Model 2, equation (2-29) is valid only for the right
half portion of the pipe as LL* is taken to be L*/2, and the
origin is chosen at the middle section of the pipe. The
initial temperature of the left half pipe can then be found
from the plane symmetric condition about the origin cross¬
section.

35
Calculation of Tidal Displacement
An important quantity encountered in the study of the
enhanced heat transfer process in oscillatory pipe flow is
the tidal displacement, which is usually required to be
smaller than one half of the total pipe length in order to
avoid any convective mass exchange occurring between the two
reservoirs. It has already been defined in the introduction
chapter (1-6) and in the present computation the dimensional
tidal displacement takes the form
(2-30)
where is the inner radius of the pipe and x(r,7r/w) is the
Lagrangian displacement of the fluid elements located along
a radius within the capillary tube at t = 7r/w. It is
assumed these elements are initially lined up at axial
position x = L/2 (Model 1 and Model 2) or x = 5.5L (Model
3) , half way between the tube ends. Numerically, the
dimensional Lagrangian displacement at time t is computed
via the equation
0
U(r,t)dt
(2-31)
It is obvious that the dimensional Lagrangian displacement
x(r,t) is a function of both time t and the radial position
r. Note that, since the existence of phase lags between the

36
stimulating pressure gradient and the displacements vary for
different Wormersley numbers, in actual calculations, the
tidal displacement is equal to the sum of the absolute
maximum cross-section averaged Lagrangian displacement and
the absolute minimum cross-section averaged Lagrangian
displacement within a period. With the same non-
dimensionalizaton procedure used in the previous section for
the governing equations, the dimensionless Lagrangian and
tidal displacement can be written as
R*
(2-32)
0
The nondimensional Lagrangian displacement within the period
can then be computed by
* / 4- * \
x (r ,t )
(2-33)
where
*
x
(A/u>2)
and
Ax
AX
*
(A/c2)
It is pointed out that this dimensionless tidal displacement
Ax* differs from AX defined by equation (1-7) by a constant
1/2.

37
Effective Heat Flux
The time averaged total effective axial heat flow
within the pipe has the form
(2-34)
where c is the specific heat, p is the density and again, w
is the oscillatory frequency. The per unit area effective
heat flow (termed heat flux) can then be written as
Equation (2-34) follows from the fact that pcUT is the
convective heat flux. The dimensionless total effective
heat flow can be written as
(2-36)
and the dimensionless heat flux by
*
Q*
v total
(2-37)

CHAPTER III
NUMERICAL TECHNIQUES EMPLOYED
Equations (2-5) , (2-6) and (2-7) which were derived
in Chapter II can not generally be solved analytically in
any simple manner. Therefore, it is necessary to seek a
numerical solution approach to the problem.
As is well known, in order to numerically solve a set
of simultaneous governing differential equations more
accurately and efficiently, an optimized grid system plays a
very important role. For the present purpose the grid net¬
works were generated in the following way: within the tube,
a non-uniform grid (see Fig. 3-1) is clustered along the
radial direction in the vicinity of the inner surface of the
pipe where the larger gradients of velocity and temperature
are expected to be present, while along the x-axis the grid
is distributed uniformly except for those calculations
dealing with model 3.
The solution process was carried out in the
computational plane rather than directly in the physical
plane. Thus, a transformation which converts the governing
equations as well as the boundary condition plays an
essential role.
38

39
1). For Model 1
2). for Model 3
Fig 3-1 Grid Systems Used in the Numerical Simulations

40
The second-order implicit unconditionally stable
Crank-Nicolson scheme and the ADI method [2, 31] are
employed to break up the transformed equations into finite
difference form. A computer code designated ETP (for
Enhanced Thermal Pumping) has been developed for obtaining
the desired results. Details of this procedure will be
described in this chapter.
Transformation
The numerical solution of a system of partial
differential equations can be greatly simplified by a well-
constructed grid. It is also true that an improper choice
of grid point locations can lead to an apparent instability
or lack of convergence. Early work using finite difference
methods was restricted to some simple problems which can be
numerically solved in the physical domain. As experience
was gained, general mappings were used to transform the
physical plane into the computational domain as well as the
governing differential equations [31]. Such a grid
transformation technique brought to the numerical simulation
numerous advantages, and the computational work was no
longer restricted to a few simple geometries. A body-fitted
non-uniform grid in the physical domain could be used, which
retains the uniform spaced grid system features in the
computational domain [33, 34, 35]. However, several
requirements must be placed on the transformation: first,
the mapping must be one to one, second, the grid lines must

41
be smooth and have small skewness in the physical domain,
third, grid node point spacing should be small where large
numerical errors are expected (i.e., large solution gradient
regions) in the physical domain (Fig. 3-2).
In present study, the grid system in the physical
space is numerically generated by solving an algebraic
equation which clusters the grid lines in the region where
large gradients of the physical quantities are expected so
as to gain higher resolution of these physical quantities.
Because of the simple pipe geometry, the requirements of
grid smoothness and orthogonality are not a serious problem.
For the present purpose, the transformation is a simple one
which transforms a non-uniform grid network in the physical
plane onto a uniform grid system in the computational plane.
x* = x*(0
r* = r*(r,) (3-1)
t* = t* (r )
The inverse transformation can be found as
£ = £ (x*)
n = r, (r*) (3-2)
r = r (t*)
where x*, r* are the dimensionless coordinates and t* the
dimensionless time in the physical domain, £, q are the
transformed coordinates and r is the transformed time in the

w
Fig. 3-2 Coordinates Transformation

43
computational domain. With this transformation, the
transformed governing equations have the form:
au
a2u*
atr
ar " f(T) + a<*>"aV + h{v) dr,
(3-3)
aT
aT3
a2T*
aT’
a r
= Pí¬
as + P2_a?2_ + P3 dr,
a2T*
+ P4^“
(3-4)
and
aT’
a r
aT’
a2T*
aT
wi^r+ + w3~
a2 t*
+ w4^~
(3-5)
where
f(r)
a(»?)
b(r?)
and
PI =
P2 =
P3 =
P4 =
at*
-—cos
a r
(t*)
1
2
1
at* i
ar (ar*/ar?)2
at*r i
1 a 2r
2
dr [ r*(ar*/dr,) (ar*/ar?)J at?2
at*
r cu*
1 a2x* ^
a r
[ dx*/d£ “ 2Pr(ax*/a^);j as2 J
at*
i
5 r
2Pr(ax*/a?2)
at*
1 r 1
1 a2r* ^
5 r
2Pr [r* (dr*/dr,)
(ar */dr,)J dr,2 J
at*
1
ar 2Pr
(ar*/ar?)2
'I
(3-6)
(3-7)
(3-8)
(3-9)
(3-10)
(3-11)
(3-12)

44
W1
W2
W3
W4
at* i f
. -i
d t
2Prw(
(dx*/d{)J dr,¿ J
at* l
l
a r
2Prw
(ax*/ao2
at* i
f 1 1
a r
2Prw
[r*(ar*/ar?) (ar */dr,)3
at*
1
i
a r
2Prw
(ar */dr,)'¿
a2r*>
dr,-
(3-13)
(3-14)
(3-15)
(3-16)
As mentioned in chapter II, equations (3-3), (3-4) and (3-5)
are a set of second-order parabolic partial difference
equations in cylindrical coordinates. In addition, since
the oscillating flow is considered as incompressible in the
present studies, the momentum equation (3-3) can then be
independently solved at each time step. As a result, the
time-dependent update velocity found can then be substituted
into the heat conduction equation (3-4) as a known quantity
at the same time step level. Eventually, the coupled heat
conduction equations (3-4) and (3-5) are solved
simultaneously to obtain the temperature distribution both
in the fluid and in the wall at any time. To best solve
this set of equations in terms of accuracy and efficiency,
the proper choice of numerical technique and grid net-work
is dictated by an understanding of the physical aspects of
the problem.

45
The same transformation should be also applied on all
the boundary conditions proposed in the three different
models. For the temperature compatibility conditions along
the interface between the fluid and the wall (2-25) one has
the following transformed forms:
dT* 1
dr] (dr*/dr¡) kf
fluid
dT* 1
dr/ (dr*/dr])
kw
wall
(3-17)
or
dT
dr]
= K;
flow
dTJ
dr)
wall
where
kw dr*/dr) |pipe
Ka = *—¡—
kf 3r /Hwall
(3-18)
(3-19)
To make the subsequent form of the corresponding
finite difference governing equations less cumbersome, the
superscript " * " will be dropped from the variables T*, r*,
t*, and x*, and in addition, U* will be replaced by V.
In the process of deriving the finite difference
governing system equations, the second-order central
differences in the domain and forward or backward
differences along the boundaries or interface of the fluid
and wall have been employed at each nodal point.

46
Crank-Nicolson Method for Momentum Equation
The second-order accurate Crank-Nicolson method is
quite well known and widely used. It is an unconditional
stable, implicit scheme for solving the parabolic types of
partial difference equations. When the Crank-Nicolson
method is applied to equation (3-3), the finite difference
algorithm at a typical node k in the radial direction and at
the time step n assumes the simpler form:
(Bkvk-i
+ Dkvk + Akvk+Jn+1 - ck
k = l, 2,
, km
(3-20)
where
- -Hfn+l+ fn) - (Bkvk-i+
E,_V,_- A,_V,_. , jn (3-21)
Jk k k k+1
Ak =
(2a + b )
(3-22)
^ (2 a + b )
(3-23)
Dk = 1 + ak
(3-24)
Ek = 1 - ak
1
(3-25)

47
The boundary conditions along the interface (i.e.,
the inner surface of the wall, k = kmid) and the axis (k =
1) then become
V
kmid
dV
dr)
(3-26)
The initial conditions of the velocity at all nodal
points is taken to be zero.
n
k = 1, 2, 3, kmid
(3-27)
Equation (3-20) associated with the boundary conditions (3-
26) can then be written in the matrix form. This yields a
set of linear system algebraic equations which can be solved
in terms of the nodal values of velocity in the capillary
tube by using either an iterative method or a Thomas
algorithm at each time step. The explicit form is

48
*
n
' '
n+1
' '
D, A,
V.
C.
1 1
1
1
D„ A„
2 2 2
2
2
B„ D„ A_
V,
C0
3 3 3
3
-
3
Bkd-lDkd-lAkd-l
vkd-i
vkd-i
Bkd Dkd
VW J
VKd J
(3-28)
where
B
1
kd = kmid-1
ADI Method for Axisvmmetric Heat Equations
The governing PDEs (3-3), (3-4) and (3-5) are all of
the second-order parabolic type. Thus it might be suggested
that the Crank-Nicolson scheme used in solving the momentum
equation (3-3) can also be applied to the axisymmetric heat
equations (3-4) and (3-5) , and one can then take advantage
of the tridiagonal matrix form while using this
unconditionally stable technique. However, when attempting
to use such a formulation, one immediately finds that the
resulting system of linear algebraic equations is no longer
of the tridiagonal type (3-20) but rather a non-tridiagonal
matrix system requiring substantial CPU time to solve. This
difficulty can be avoided by applying the unconditionally
stable Alternating Direction Implicit method (ADI),

49
developed by Peaceman and Rachford and Douglas in 1955.
According to this scheme, the entire solution process at
each time step is "split" into two portions, i.e., the first
half of solution processes for k^*1 column (radial
direction) , while the other half processes for the row
(axial direction).
With the ADI scheme, second-order central differences
are used to approximate the values of derivatives at each
nodal point in equations (3-4), (3-5). The finite
difference algorithm for those equations during the first
half of each time step for the jth column are then
BPjj,kTjj,k-l + DPjj,kTjj,k +
AP T
jj,kAjj,k+l
n+1/2 = PX?. „
k = 1, 2 , ....kmid
(3-29)
and
BW. . .T. . . , +
. ID j k DD/k-1
DW.
jj,kTjj,k
+ AW. . . T. . . ,,
DD/k DD rk+1
n+1/2
WX
n
jj/k
k = kmid+1,...kmax
(3-30)
where the subscript jj is used to emphasize the specific
column currently to be computed. It can be seen that the
set of difference equations now is in the tridiagonal form

50
since the right-hand-side terms in the equations (3-29), (3-
30) contain only known values from the previous results and
the boundary conditions. These values can be computed by
PXjj,k (~CPjj,kTjj-l/k + EPjj,kTjj,k FPjj,kTjj+l,k)
(3-31)
and
wx«,k
â– (
-CW. . . T . . , . + EW. . . T. . . - FW
ll/k n-l,k 11/k 11,k
í. . T . . 1 n
(3-32)
The computational algorithm is implemented column by
column and the unknown value (Tj,^)11 can then be solved by
either an iterative or a direct method. In order to do
this, equation (3-29) needs to be assembled into the
following matrix form.
r '
PD^ PA1 *
n
' T1
n+1/2
PX1
PS2 pd2 pa2
T2
PX2
pb3 pd3 pa3
T3
=
PX3
PBkd-lPDkd-lPAkd-l
Tkd-1
PXkd-l
PBkd PDkd
j
T
l kd J
j
PXkd
j = 1, 2, 3 jmax
(3-33)

51
Similarly, equation (3-30) can be assembled into the matrix
form as seen in (3-34).
WDkd+l WAkd+l
n
' '
T
xkd+l
n+1/2
WXkd+l
WBkd+2 W°kd+2 WAkd+2
T
kd+2
WXkd+2
WB, WD. WA.
km km km
T
xkm
WXkm-l
* WB. WD.
kmax kmax
j
Tkmax
j
WXkmax
j = 1, 2, 3 jmax
(3-34)
It should be pointed out that the coefficients and/or
the right hand side terms in the row marked with symbol "*"
in those matrixes needs to be properly modified as well as
the limit of either subscript j or subscript k according to
the different boundary conditions in the various models
considered. For example, along the axis, the symmetry
condition requires that 8T/dr = 0 and this could be
accomplished numerically by equating the temperature Tq and
T2 and by employing a new combining coefficient of PA^* =
PAi + PBi rather than the original PA^ in the first matrix
for evaluating the temperature distribution within the pipe.
In addition, if the temperature distributions along certain
parts of the outer surface of the wall are given, then the

52
limit of subscript k will be ended with k = kmax-1 rather
than k = kmax shown in equation (3-34). The same arguments
are also applicable for another subscript j in the axial
direction.
By using a similar procedures for the second half of
each time step, the finite difference algorithm for the kth
row then becomes
(CPj,kkTj-l,kk + GPj,kkTj,kk + FPj,kkTj+l,kk)
n+1 = pYn+l/2
j ,kk
j = 1, 2,
, 3 max
(3-35)
H.kkTj-lAk + GWj,kkTj,kk + FWj,kkTj + l,kk)n+1 - mi%2.
j = 1, 2,
j max
where the right hand terms can be computed by
(3-36)
PY1?+l/2= Í—CP. T. + HP T - AP T 1 n+1/2
D,kk [ ,kkij,kk-l ],kk1j,kk ,kkxj,kk+lj
(3-37)
wy"!ík2= (-cBj,kkTj,kk-i + HHj,kkTj,kk- Awj,kkTj,kk+i)n+1/2
(3-38)

53
Similarly, to emphasize that the current calculation is in
the kth row, we use the symbol kk rather than k in above
formula and the matrix form of equation (3-35) for the k^h
row can then be written as
PG2
PF2
•
*
n+1/2
r *
T2
' '
PY2
PC3
PG3
PF3
T3
PY3
PC4
PG* PF„
4 4
T4
PY4
PC. PG. PF.
])m-l jm-l jm-l
Tjm-1
PXjm-l
*
i.
PC. PG.
im jm J
k
T .
1 D» J
k
PX.
1 Dm
n+1/2
k = 2, 3
and the matrix form of equation (3-36) becomes
, kmid
(3-39)
WG2
wf2
*
n+1/2
' 1
T2
N+l
wy2
wc3
WG3
WF3
T3
WY3
wc,
4
WG. WF
4 4
T4
WY
4
WC.
Ijm-l
WG.
;jm-l
WF.
jm-l
Tjm-1
WX.
jm-l
*
b
WC.
Dm
DG.
3® J
k
T.
1 Dm J
k
WX.
1 Dm
N+1/2
k = kmid+1, kmid+2
, km
(3-40)

54
As indicated above, the symbol " * " shows that the
coefficients in that row as well as the right-hand-side
terms need to be properly modified corresponding to the
different boundary conditions. The matrix terms in
equations (3-29), (3-30), (3-31) and (3-32), (3-35), (3-36),
(3-37) and (3-38) can be computed by
APj,k = “< ^P3 + P4>^
D /k
BPj,k = P3 - P4>^
J > k
CPj,k " (— P1 - P2>j,k
EPj,k = Í1 " 2*P2>j,k
PP j ,k= "i—P1 + p2).
GPj,k " Í1 + 2*P2)j,k
DPj,k " Í1 + 2*P4>j,k
and
AW. .
= - W3 + P4) .
j
BW = W3 - W4) .
j /k
CMj,k - <— W1 - W2>j,k
HPj,k " t1 - 2*P4>j,k
(3-41)
- (1 - 2*P2>j,k
- -<4"wi + w2>j,k
= (1 + 2*W2). k
DWj f k - + 2*M4>j,k
HMi,k - i,k
(3-42)

55
In order to rewrite the equation in a simple form we define
the following linear operator :
LY [P]j>k = [BP, DP, AP]j,k
LY [W]j,k = [BW' DW' AW]j,k
LX [P3j/k = [CP, GP, FP]j/k
LX [W]j ^ k = [CW, GW, FW]jik
(3-43)
and column vectors
{TY)j,k
{Tj,k-l' Tj,k' Tj,k+1 }
{TX}j,k
T.
{Tj-i,k' xj,k' *j+i,k
T.
(3-44)
Equations (3-29) which are in the radial direction can be
simply written as
n n+1/2 n
LY [P] (TY) = PX
j,k j, k j, k
(3-45)
n n+1/2
LY [W] (TY)
j ,k j ,k
n
(3-46)
j = 1, 2,
k = 1, 2,

56
Equations (3-33) and (3-34) , which are in the axial
direction, then assume the following simple forms:
n+1/2
n+1
n+1/2
LX [P]
(TX)
= PY
(3-47)
j /k
j ,k
j ,k
n+1/2
n+l
n+1/2
LX [W]
(TX)
= WY
(3-48)
j ,k
j ,k
j/k
j = 1, 2,
k = 1, 2,
where the right side terms can be estimated by
PX = (-CP, EP, -FP) (TX)
j,k j,k j,k
WX = (-CW, EW, -FW) (TX)
j,k j,k j, k
PY = (-BP, HP, -AP) (TY)
j,k j , k j , k
(3-49)
WY = (-BW, HW, -AW) (TY)
j r k j , k j , k

57
It is seen, as the result of the ADI ''splitting”
procedure which has been employed in the algorithm
associated with different boundary conditions for the
various models, that only a tridiagonal system of linear
algebraic equations needs to be solved (i.e., during step 1,
the coupled tridiagonal matrix (3-33), (3-34) are solved for
each column of the grid points, while during step 2, the
coupled tridiagonal matrix (3-39) and (3-40) are then solved
for each k*-h row of grid points) .
Once the periodic steady solution has been obtained,
we can calculate both the tidal displacement and the heat
flux at different locations within the pipe. The numerical
technique used here to integrate equations (2-38), (2-39)
and (2-41) for evaluating the tidal displacement and the
heat flux at each specified location can be obtained either
by the Trapezoidal rule (with end correction) or Simpson's
rule. Both numerical approaches are essentially fourth-
order methods.
Convergence Criteria
As is known, once the calculation work has started,
the time-matching process will be in the loop forever unless
a criterion can be derived that indicates when the goal of
the current computing work has been reached and further
solution-matching processes do not produce significant
increases in accuracy. Such a criterion depends on the
purpose of the calculation. If one's goal is to study the

58
transient process, one can indicate a time limit to pause or
quit the current computing work.
However, if one seeks a final periodic state
solution, one has to develop a criterion to test if the
solution can be considered acceptable.
It seems that the temperature is a good measure of
the accuracy of the overall solution process so that the
most efficient way is to apply the convergence test on the
temperature rather than on the velocity. For the present
purpose, two convergence criteria were alternatively used.
The choice of the criterion depends on what was the main
goal in the particular study case. If the main effort is to
observe the distribution of the quasi-steady temperature at
any phase angle «t within a period, the testing is done by
comparing the temperature residual, i.e., by inspecting the
averaged temperature difference of each nodal point at thee
same «t between adjacent periods. This can be written as
Resl
j S (Til - T22 )? k ]1/2
(JMAX)(KMAX)
<
j = 1, 2, JMAX
k = 1, 2, KMAX
(3-50)
where Til. . is the value of the temperature of node (j,k)
at wt in the current period, T22. . is the value of the
D I K

59
temperature of node (j,k) at the same «t in the previous
period, and ej is the convergence parameter.
If the goal of the investigation is to examine the
effective heat transfer, the convergence criterion is
established by computing the residual
Res2
JSEC
* e2
(3-51)
j = 1,2, JSEC
where JSEC is the number of cross-sections where the axial
heat flux was examined, 62 is the convergence parameter, and
4> 1, 2 are the cross-section averaged heat flux in the
current period and previous period, respectively. The
summation is carried out over all sections (JSEC) where the
heat flux is computed.
The value of the convergence parameter can be
determined by a balance of the acceptable solution accuracy
and the cost of CPU time. In the current study both el and
e2 were taken between 0.001-0.01 as the residual of the
temperature or the heat fluxes for acceptable value.
Grid Generation
As is well known, the solution accuracy and efficiency
in a large degree depends on the grid system used in the
numerical calculation. A good grid is characterized by
small skewness, high smoothness and capability of high

60
resolution in the large gradient regions in the physical
plane. It has been shown that rapid changes of grid size
and highly skewed grids can result in undesirable errors
[25]. The success of a numerical simulation of a complex
thermal fluid dynamics problem does depend strongly upon the
grid system used in the computation.
In the present study, the grid mesh was generated with
the emphasis on the high resolution capability of its simple
geometric boundaries. The technique used here is one of the
algebraic schemes which cluster the grid lines near the
region desired [34], namely,
AS. = AS (1 + e) k_1 (3-52)
where aSq is the minimum specified grid spacing next to the
wall or to some inner interfaces within the tube. The
parameter e is determined by a Newton-Raphson iteration
process so that the sum of the above increments matches the
known arc length between k = 1 and k = kmax. The grid
networks used in Model 1 and Model 2 are uniformly
distributed along the axial direction in both the fluid and
the wall; however, in Model 3, most grid points are
clustered near the central portion of the pipe so as to gain
higher resolution in the region of interest, while along the
radial direction a non-uniform grid was employed in all
three models. The minimum specified grid spacing AS0 used

61
depends on the boundary layer thickness 6, namely, the
kinematic viscosity of working fluid and the oscillating
frequency. In the present study, it was found that a good
choice of this value is AS0 = 0.055 for laminar flow cases
with a total of about 15-20 nodal points distributed along
the radius.

CHAPTER IV
NUMERICAL RESULTS AND DISCUSSION
The problem of time-dependent enhanced heat
conduction subjected to sinusoidal oscillations can now be
solved numerically for the boundary conditions appropriate
to a long capillary tube according to the various models
described in previous chapters. The computational tasks
fall into two catagories: the first part is a numerical
study concerned with the characteristics of the oscillatory
pipe flow, which includes an investigation of the velocity
profiles, the Lagrangian and the tidal displacement
trajectories. The second part of this study includes a
thorough investigation of the final periodic state
temperature build-up process in oscillating pipe flow, the
periodic temperature distribution in the pipe and wall, and
a study of the relationship between the enhanced axial heat
flux and the tidal displacement. It also includes an
investigation of the tuning effect and a comparison of the
enhanced axial heat transfer with the corresponding pure
molecular axial heat conduction as well as the investigation
of the influences of the pipe radius and the wall thickness
on the enhanced axial heat transfer.
62

63
Properties of oscillating laminar pipe flow have been
analytically discussed by S. Uchida [36] for several
different Wormersley numbers, and these results offer a
useful reference for comparing with the present numerical
studies. However, in the area of thermal fields associated
with enhanced thermal pumping there is little information
available for comparison, except for some recent results of
Kaviany [13].
The present computational work was carried out on the
Vax-11/750 computer in the Department of Aerospace
Engineering, Mechanics, and Engineering Science, and on the
Vax-11/780 in the Center for Instructional and Research
Computing Activities, University of Florida, Gainesville,
Florida. Note that it is very time-consuming to build-up a
final periodic state, for example, if the grid size is
101x22 and the time steps are 2000 per period, it takes 7.5
minutes (CPU) with the VAX-11/780 machine or almost 1 hour
(CPU) with the VAX-11/750 machine to run only one period.
It usually takes 20-30 periods to reach the final periodic
state solution.
The process of selecting model size (i.e., total
nodal points) is a synthetic balance among the storage
requirement, the solution accuracy, and the cost of CPU time
for finding an acceptable solution. Such a selection allows
each case to be solved with a minimum of expense in
computing operation thereby making it possible to do the

64
large number of runs needed to obtain enough data points to
plot curves of the desired parameters.
The results discussed in this chapter are presented
for the purpose of illustrating the effects to be
encountered in working with the enhanced thermal pump.
These results should be effective in gaining insight into
some interesting features of this enhanced heat transfer
process. Since the analytic approximations [15, 16, 18, 22]
are effective in describing the flow and the heat transfer
aspects and to give considerable insight into the problem,
they are often used to compare with the numerical results
and should give good agreement when applicable.
Part 1. Oscillatory Pipe Flow Features
In order to better understand the mechanism of the
enhanced thermal pump, it is necessary to examine the
mechanical features of the oscillating pipe flow. As
mentioned in previous chapters, the present interest in the
enhanced thermal pumping is confined to an investigation of
the central part of the slender pipe, thus the flow field
can be well approached by a 1-D time-dependent laminar model
which neglects the ends effects.
Velocity Profiles
Fig. 4-1 shows the numerically computed time-
dependent velocity profiles at different phase angles of the
exciting pressure when Wormersley numbers are equal to 1,
10, 100, and 1000, respectively. It can be clearly seen

65
a - |
a » 10
p-s
?
J
:
—
=
i
a
-¡
f
L
E
= 4
i
i
-f-* * 't'
}
)
i
â– 1
7
_
3
E
;
3
F
3
i
!
i
3
-
,
a -1000
* * *"*•
m
—
U
c
fa
_
:
2
*-r-
"f
H
# y M 9» i¿« ise ue íie iw 33e Jee
Fig 4-1 l-D Velocity Profiles in Oscillating Flow
for Wormersley Number a = 1, 10, 100 and 1000

66
that the velocity profile at a = 1 presents a quasi¬
parabolic shape at any instant within a period and is in
phase with the stimulating pressure gradient. However, at
higher frequency, for example, a = 100 and 1000, the
velocity profiles can be clearly separated into two regions:
in the vicinity of the wall the flow shows a typical thin
boundary layer, while in the region far away from the wall
the fluid moves as if it were frictionless slug flow. In
fact, within this core region the velocity distribution is
independent of the distance from the wall [29]. It can be
also seen that the phases of the velocity profiles at higher
frequencies cases are shifted about k/2 with respect to the
Fig 4-2 Velocity Profile(q=5 ) [29]

67
(Rl-r)/í (Ri~r)/í
(Rl~r)/í
Fig 4-3 Magnified View of Velocity Profile Near Wall
(Wormersley Number a=10, H2O, 6=0.014cm)

68
stimulating pressure gradient. At the intermediate
frequency case (a = 10 of Fig. 4-1), the slug flow boundary
is not so evident, but one can still note a boundary layer
near the wall. The same pattern of the velocity profiles
associated with moderate Wormersley numbers can also be
found in the reference [36] where the velocity profile at
Wormersley number a less than 10 was presented (Fig. 4-2) .
In order to better see the variation of the time-dependent
velocity profile within the boundary layer, a set of closer
view of the velocity profile with respect to Wormersley
number at a = 10 during phase intervals wt = 30° is plotted
in Fig. 4-3.
It should be emphasized that the solutions shown here
are under the assumption that the secondary velocity in the
radial direction is negligible compared to the axial
velocity component, namely, the non-linear inertial terms
are not considered in the governing equation (2-1). This is
a reasonable approximation for moderate oscillatory
frequency w except near the ends of the pipe. However, at
high Wormersley numbers, care should also be taken to avoid
violating another assumption, namely, that of
incompressibility and the concomitant condition that the
oscillating phase does not change between the tube ends.
High Wormersley number can be obtained either by increasing
the oscillating frequency w or by increasing the pipe
diameter for a given fluid. An oscillating flow can be

69
considered incompressible if Axw/2 < 0.05C, where C is the
speed of sound in the fluid. For mercury, C = 1360 m/sec,
one requires a = 27.1, when Ax = 20 cm and = 0.1 cm. To
avoid a appreciable phase difference between the tube ends,
one requires that L/C « 2tt/w. Both the restrictions are
met in the examples to be considered below.
The Lagrangian Displacements
An alternative interesting representation to the
oscillatory velocity field are the Lagrangian displacements
of the fluid elements at different radii within the pipe.
They have been plotted in Figs. 4-4 and 4-5 at time
intervals of wt = 30° for Wormersley number a = 0.1, 1.0 and
10. It is noted that since both pipe diameter and the
working fluid were fixed in this test, Figs. 4-4 and 4-5
represent the relationship between the Lagrangian
displacement and the oscillating frequency. The
trajectories plotted in Figs. 4-4 and 4-5 have been
normalized by A/w2, where A = l/p|dp/dx| is the amplitude of
the sinusoidal pressure gradient as defined in equation (2-
1) and w is the angular velocity. Suffice it here to point
out that for the lower frequency case (for example a = 0.1
and 1.0) the Lagrangian displacement trajectory shows a
foreseeable parabolic pattern at any moment. Nevertheless,
the essential distinction between low frequency oscillatory
pipe flow and steady Hagen-Poiseuille flow is that in the
former, the Lagrangian displacement trajectories as well as

70
Fig 4-4
Lagrangian Displacement for a = 0.1 and 1.0

71
Fig 4-5
Lagrangian Displacement at a = 10

72
the velocity profiles are periodic so that the fluid
particles do not translate axially upon time averaging,
while in the later case they will.
For intermediate Wormersley number (a = 10), the
trajectories of the Lagrangian displacement departs
considerably from the standard parabolic shape. This
phenomenon can be even more clearly seen in the a = 100
case. Evidently, the higher the oscillating frequency, the
thinner the boundary layer (6 = J2u/u).
Tidal Displacements
Figs. 4-6 and 4-7 demonstrate time variation of the
cross-section averaged dimensionless Lagrangian displacement
at Wormersley numbers in the range from a = 0.1 to 50. The
tidal displacement can be obtained by summing the absolute
maxima and the absolute minima of these curves. The
corresponding non-dimensional tidal displacements with
respect to Wormersley number from a = 0.1 to a = 12 are
listed in table 4-1 below
Table 4-1 Dimensionless Tidal Displacement
at Different Wormersley Numbers
a
o
•
H
1.0
to
•
o
3.0
4.0
5.0
; ax
0.00246
0.14295
0.67303
1.20213
1.41160
1.49041
a
6.0
7.0
o
•
CO
10.0
12.0
| AX
1.56294
1.61879
1.66046
1.71900
1.76698

73
Fig 4-6 Dimensionless Cross-section Averaged
Displacement Versus Time
a = 0.1 - 1.0
Fig 4-7 Dimensionless Cross-section Averaged
Displacement Versus Time
q = 2 - 50

74
It is noticed that for very small Wormersley number
cases (a < 0.5), the cross-section averaged displacement
varies like a sinusoidal function with respect to time (the
ordinate), and as Wormersley number increases, the cross-
section averaged displacements are no longer symmetric about
the ordinate but rather favor positive values of *. If the
Wormersley number is further increased, eventually, the
cross-section averaged displacement almost entirely lies on
the right half plane. A similar feature can also be seen in
Figs. 4-4 and 4-5. In fact, this just again shows the
existence of the phase shift in the oscillating pipe flow.
When the Wormersley number is small the cross-section
averaged displacements as well as the Lagrangian
displacements show a sinusoidal variation with respect to
time and this has ?r/2 phase lag with respect to the
stimulating pressure gradient which is assumed to be a
cosine function of wt with zero initial phase angle.
However, for higher Wormersley number, the phase lags
increase to almost n with respect to the phase of the
exciting pressure gradient. Note that the Lagrangian
displacements in Figs. 4-4 and 4-5 were computed by lining
up all the fluid particles on the plane x = 0 at wt = 0,
however, by using non-zero velocity at wt = 0° as shown in
Fig. 4-1. This implies that if the phase of the exciting
pressure gradient is taken as the base of the measurement,
it is generally not possible to assure the same phase to the

75
velocity and Lagrangian displacement as well as the tidal
displacement. For example, at phase of the stimulating
pressure gradient wt = 0° , the corresponding velocity phase
may be 7r/2 and the Lagrangian displacement phase may be n.
In fact, this is just as true for very high Wormersley
numbers; however, the phase difference with respect to the
exciting pressure gradient phase is less than the value
shown above for low Wormersley number. One can well see
that if the phase lags of the Lagrangian displacement or the
tidal displacement is given, one can certainly re-draw the
diagram shown in Figs. 4-6, 4-7 by lining up the phase with
itself, and then an exact symmetric pattern of the tidal
displacement curve similar to that in the very small
Wormersley number case can be obtained. Unfortunately, the
phase lags are a function of the Wormersley number and they
are not known in advance.
In order to verify the numerical method used in this
study and to check the ETP code developed, a comparison of
the computed dimensionless tidal displacement to the one
using the analytic equations (1-7) and (1-8) given by
Kurzweg [18] for Wormersley number varying from 0.1 to 100
has been calculated and plotted in Fig. 4-8. The solid line
shows the analytic solution obtained by use of equations (1-
7) and (1-8), while the dashed line shows the results with
the ETP code developed in this study. The agreement is
quite good, particularly when the Wormersley is less than

76
3.0. However, at high Wormersley number the numerical
solution shows a very slight deviation from the analytic
solution. This deviation is believed due to an inaccurate
numerical integration over the cross-section using
relatively large time steps (our time steps per period in
the calculation were between 1000-2500). A comparison with
using 104 time steps per period for Wormersley number a = 10
was studied and shows some improvement. However, using such
small time steps in the present investigation is beyond the
capacity of the current VAX computer facility used. The
numerical error becomes particularly serious as the
oscillating frequency becomes large where the extremely thin
boundary layer requires more grid nodal points to resolve
the flow variables in the vicinity of the wall.
Fig. 4-8 shows that as the Wormersley number gets
large, the dimensionless tidal displacement tends to the
limit of 2.0, which agrees with the limit of 1.0 in the
analytical solution given by Kurzweg [18] for the reason
that the normalization parameter used in [18] is twice as
large as that in the present numerical simulation.
Fig. 4-9 shows the required stimulating axial
pressure gradient used in the present study for a pipe
radius = 0.1 cm and water (u = 0.01 cm2/sec) taken as the
working medium versus the dimensional tidal displacement Ax
in cm for various Wormersley numbers (namely, oscillating
frequency). It is evident from these results that for fixed

Dimensionless Tidal Displacement AX
77
10"' 1U° 101 102
Wormersley Number a
Fig 4-8 Relationship Between Dimensionless Tidal
Displacement ¿X and Wormersley Number a

(Uio)
Relationship Between Tidal Displacement Ax
and Exciting Pressure Gradient in Water
03
Fig 4-9

79
tidal displacement, the required axial pressure gradient in
the large Wormersley number case is orders of magnitude
higher than that in the small Wormersley number case. This
may eventually put some constraint on the use of very high
frequency in the enhanced heat transfer technique.
Fortunately, to meet the tuning condition, the required
Wormersley number in this case is of order 1. As already
discussed in the introduction, in order to gain the benefit
of axial heat transfer, the use of large tidal displacement
is always preferred. However, such an increase must be
limited by the requirement of no convective net mass
transfer occurring between two reservoirs and may also be
constrained by the ability of the device to withstand the
increase in the exciting pressure gradient.
Phase Lags
We are now in the position to study the phase lags of
the Lagrangian displacement in the oscillatory pipe flow.
Fig. 4-10 shows the phase variations (in degrees) along
radius for Wormersley number varying from a = 0.1 to 4.
Some numerical results are also shown in Tables 4-2 and 4-3.
All of the data shown in Fig. 4-10 and the tables have the
phase angle measured relative to the exciting pressure
gradient. Two features can be seen in Fig. 4-10; first, in
the core portion, the phase lags are almost equal to n/2
when the Wormersley number is small, while the lags are
almost n when the Wormersley number is large, and second,

80
the phase lags vary along radius, especially in the boundary
layer. It is such phase lags that allow the existing
temperature gradient in the very thin boundary layer of the
oscillating pipe flow to act as region of temporary heat
storage. It absorbs heat when the temperature of the core
Table 4-2 Phase Lags Along Radius
(Working Medium: H2O, Ax = 10 cm)
Nodal
point
K
a = 0.1
a = 1.
a = 2.
»?=r/Rl
phase
r¡= r/Rl
phase
»?=r/Rl
phase
1
0.0000
90.63
0.0000
102.06
0.0000
134.28
2
0.1667
90.63
0.0366
102.06
0.1095
134.10
3
0.3333
90.63
0.0957
102.06
0.2164
133.38
4
0.5000
90.63
0.1913
101.88
0.2836
132.66
5
0.5833
90.63
0.2593
101.70
0.3620
131.76
6
0.6667
90.63
0.3458
101.52
0.4535
130.50
7
0.7500
90.63
0.4559
101.16
0.5603
128.70
8
0.8333
90.63
0.5958
100.62
0.6849
126.00
9
0.9167
90.63
0.7737
99.72
0.8303
122.58
10
1.0000
0.00
1.0000
0.00
1.0000
0.00

81
Table 4-2 — continued
Nodal
point
K
a = 3
a = 4
a = 7
7=r/Rl
phase
r?=r/Rl
phase
r?=r/Rl
phase
1
0.0000
170.83
0.0000
179.82
0.0000
181.62
2
0.0917
170.47
0.1242
179.28
0.0970
181.62
3
0.1818
169.75
0.2370
178.02
0.1892
181.62
4
0.2703
168.31
0.3395
176.04
0.2769
181.44
5
0.3573
166.51
0.4324
173.70
0.3604
180.72
6
0.4427
164.00
0.5168
171.00
0.4397
179.28
7
0.5267
161.12
0.5935
168.12
0.5152
177.12
8
0.6091
157.88
0.6630
165.24
0.5869
174.06
9
0.6901
154.64
0.7262
162.18
0.6552
170.28
10
0.7696
150.69
0.7835
159.12
0.7201
165.78
11
0.8478
146.73
0.8356
156.06
0.7818
160.74
12
0.9246
142.42
0.8828
153.18
0.8406
155.16
13
1.0000
0.00
0.9257
150.30
0.8964
149.04
14
0.9646
147.60
0.9495
142.56
15
1.0000
0.00
1.0000
0.00

82
Table 4-2 — continued
Nodal
Point
K
a = 10
H
II
a
a = 20
7 =r/Rl
phase
»?=r/Rl
phase
»7=r/Rl
phase
1
0.0000
180.54
0.0000
179.46
0.0000
179.10
2
0.0884
180.72
0.0834
179.10
0.1635
180.54
3
0.1740
181.08
0.1647
180.18
0.3032
185.93
4
0.2566
181.80
0.2440
184.50
0.4227
182.34
5
0.3364
182.52
0.3215
190.25
0.5247
186.65
6
0.4135
182.70
0.3970
192.41
0.6120
184.14
7
0.4880
181.80
0.4709
189.53
0.6865
184.14
8
0.5599
179.64
0.5428
183.06
0.7502
181.98
9
0.6295
175.68
0.6130
174.07
0.8046
178.38
10
0.6966
170.46
0.6815
164.00
0.8512
173.71
11
0.7616
163.80
0.7484
153.20
0.8909
167.95
12
0.8243
156.06
0.8136
141.70
0.9249
161.12
13
0.8849
147.24
0.8773
130.19
0.9540
153.57
14
0.9434
137.52
0.9394
118.32
0.9788
146.01
15
1.0000
0.00
1.0000
0.00
1.0000
0.00

83
Table 4-3 Comparison of Phase Lags With Different
Working Mediums (Ax = 10)
Nodal
Point
a = 1.
a = 5.
K
r) =r/Rl
Phase
Lags
t) =r/Rl
Phase
Lags
Water
Mercury
Water
Mercury
1
0.0000
102.06
102.14
0.0000
181.62
181.62
2
0.0161
102.06
102.14
0.0714
181.44
181.26
3
0.0366
102.06
102.14
0.1429
180.90
180.54
4
0.0626
102.06
102.14
0.2143
179.64
179.46
5
0.0957
102.06
102.14
0.2857
178.02
178.02
6
0.1378
101.88
102.14
0.3571
176.04
175.86
7
0.1912
101.88
102.14
0.4286
173.34
173.37
8
0.2593
101.70
102.14
0.5000
170.28
170.11
9
0.3458
101.52
101.78
0.5714
166.50
166.51
10
0.4559
101.16
101.42
0.6429
162.18
162.20
11
0.5957
100.62
101.06
0.7143
157.50
157.52
12
0.7737
99.72
99.98
0.7857
152.10
152.13
13
0.0000
0.00
0.00
0.8571
146.16
146.37
14
0.9286
139.86
139.90
15
1.0000
0.00
0.00

Dimensionless Radius
Fig 4-10 Phase Variation Along Radius
for Different Wormersley Numbers

85
slug flow is higher than that of the boundary layer, and it
releases heat as the core temperature is relatively lower.
This large temperature gradient enhanced by the existing
velocity phase lags allows a large amount heat to be
conductively transferred radially within a very short time
and subsequently to be transferred axially by a convective
coupling.
Part 2. The Enhanced Heat Transfer Investigation
We have examined some mechanical characteristics in
oscillating pipe flow, and compared the computed solution of
the velocity field with Uchida's solution. The results for
the velocity profiles are in good agreement. Nevertheless,
before a detailed examination of the thermal field, it is
first necessary to test the current developed ETP code when
applying the temperature equations (2-6),(2-7). It is seen
that the energy equations strongly depend on the velocity
distribution and its build-up process, so that one can use
analytic periodic velocity state (Eqs. 1-4 and 1-5, with no
build-up process), and the computed velocity (with build-up
process) to verify the correctness of the resulting thermal
variables. The enhanced heat flux is a function of both
velocity and temperature (Eqs. 2-34 and 2-35) and was chosen
for a comparison of the analytic and numerical results of
the problem. Part A of table 4-4 shows the results of the
computed enhanced axial heat flux as well as the axial
conduction heat flux when using the analytic velocity

86
Table 4-4 The Comparison of Enhanced Heat Flux Using
Numerical Velocity with Heat Flux Using Analytical Velocity
(Model 3, Water-Glass, Pr = 7.03, a = 3)
A. Heat Flux Using Analytic Velocity
AX
(cm)
(w/cm20 K)
4>f
(w/cm2° K)
w
(w/cm2 ° K)
PA
(w/cm4° K)
0.9839
0.0230
0.0149
0.0811
0.0238
2.9519
0.1855
0.0132
0.0724
0.0213
4.9194
0.3864
0.0098
0.0541
0.0159
6.8868
0.5810
0.0075
0.0415
0.0122
7.8847
0.6775
0.0067
0.0369
0.0109
9.8390
0.8638
0.0055
0.0302
0.0089
B. Heat Flux Using Computed Velocity
1.0033
0.0234
0.0149
0.0811
0.0232
3.0101
0.1870
0.0133
0.0728
0.0206
5.0164
0.3899
0.0097
0.0533
0.0155
7.0226
0.5790
0.0076
0.0420
0.0117
8.0402
0.6746
0.0068
0.0374
0.0104
10.0328
0.8597
0.0055
0.0302
0.0085
* Model 3, Water-Glass
* 4> enhanced axial heat flux
* f axial heat flux by conduction in fluid
* 4>\j axial heat flux by conduction in pipe wall
* p = /AX2

87
(Eq. 1-4) for a = 1, Pr = 7.03, pipe radius = 0.1 cm and
glass walls of thickness AR = R2- Ri = 0.05 cm (which is
almost equal to the boundary layer thickness 5 = 0.047 cm
for this case) with Model 3. There were 22 points
distributed along the radius (13 points in fluid), and the
smallest distance next to the wall is equal to 0.165, while
along the axial direction 101 point were used (60 points in
the central pipe section). Thirty period runs were
considered in order to insure that the final periodic state
was achieved. An exception was the Ax = 1 cm case where
only 23 period runs were computed. Under the exact same
conditions described above, a calculation using the
numerical velocity with build-up process was tested. Part B
of table 4-4 gives the axial heat flux results using the
corresponding numerical generated velocity profile. A
comparison of results shown in Table 4-4 is quite good. One
sees that the disagreement for most terms is less than
0.001. Note for Ax of 10 cm, the ratio /f is about 15 and
gets larger as Ax is increased further.
Periodic Temperature Build-up in Thermal Pumping Process
The purpose of this facet of the present study was to
examine the temperature field build-up pattern and to
determine how long before the final periodic temperature
state can be reached. The author also intends to verify
whether or not there indeed exists a time averaged linear
axial temperature distribution within the thermal pump at

88
larger time as assumed in existing analytic solutions [15,
16, 22].
Model 1 was employed to study the temperature build¬
up history for the case where the initial velocity was zero
everywhere inside the pipe as well as at the boundary. The
initial temperature condition was assumed to be identical
with the cold end temperature at all grid points except
those at x = 0 where the dimensionless temperature T = =
1.0
Figs. 4-11, 4-12 and 4-13 show the computed
temperature build-up history at Wormersley number a = 1 at x
= L/4, and L/2 (where L = 20 cm is the pipe length), and
either on the axis r = 0 or at the wall r = R^. The tidal
displacements were chosen as ax = L/10, L/4, and L/2,
respectively. In these figures, the ordinate represents the
dimensionless temperature, while the abscissa represents the
oscillatory period runs. These temperature histories show a
sinusoidal variation. Three characteristics can be
summarized: 1) when the tidal displacement is large, the
amplitude of the temperature variation is also large, and
less adjusting time is needed; 2) the amplitude of the
sinusoidal variation of the temperature along the axis is
larger than that near the wall (see Figs. 4-4, 4-5) ; 3)
after a short transient process, the final oscillatory state
is reached. The time-averaged temperatures of this
oscillatory state at a given x-position have the same value

89
Fig 4-11
Temperature Build-up Process In Oscillating
Flow (Model 1, q = 1, AX = 2cm)

90
i.eeer- t
0 11 22 33 4< 55
. Period
Fig 4-12 Temperature Build-up Process In Oscillating
Flow (Model 1, a = 1, ax = 5cm)

91
Fig 4-13 Temperature Build-up Process In Oscillating
Flow (Model 1, a = 1, ax = 10cm)

92
for all tidal displacements. It implies that the final time-
averaged temperature depends on its x-position only. For
example, the final periodic dimensionless time-averaged
temperature is about 0.5 at the location x = L/2, while it
is about 0.75 at x = L/4 since this latter position is much
closer to the hot end. From this observation, one may well
imagine that the periodic temperature state along the
longitudinal direction is close to the linear function form
assumed in existing analytic studies [17, 22], This feature
will be further demonstrated later.
Figs. 4-11, 4-12, 4-13 also show the time needed for
a build-up to the final periodic state of the thermal field.
An unpublished conservative analytical estimation of the
build-up time by Kurzweg, based on the assumption that the
oscillating flow satisfies the tuning condition is
tx = L2/ [uAX2 A (a , Pr) ] (4-1)
where A(a,Pr) = keh/wAx2 is a function of thermal
conductivity and Wormersley number [15, 16], At the tuning
point, this value of A is about 0.02. If the tidal
displacement ax = L/2 = 10 cm and the angular frequency w =
1 (for water with = 0.1 cm), the adjustment time t^ can
be estimated from equation (4-1) to be equal to 200 sec., or
32 periods. The computed results show that the actual
adjusting time is only 20 periods.

93
t
Time t (in unit of equivalent periods)
Fig 4-14 Temperature Build-Up Process In Steady Flow
(Model 2, Ueq = 1.5 cm/sec)

Period
94
Tidal Displacement ¿x (cm)
Fig 4-15
Build-Up Time Versus Tidal Displacement
(Model 2, q = 1)
Pressure amplitude A = 1/p |dp/dx|

95
Fig. 4-15 shows the build-up time versus the tidal
displacement and applied pressure gradient corresponding to
various tidal displacements with Model 2. It can be seen
that when the tidal displacement is very small, the build¬
up time tends towards infinity, so that the so-called final
periodic state no longer exists. The final periodic
temperature state will thus be established only by heat
conduction and has a complementary error function erfc(x,t)
= 1 - erf(x,t) temperature build-up history [1, 11]. As the
tidal displacement gets larger, the build-up time needed
decreases, and when Ax is greater than about L/4 (5 cm) the
build-up time is almost constant.
In order to compare with the steady Hagen-Poiseuille
flow, a transient temperature history of such a flow with
Model 2 is also shown in Fig. 4-14. Both curves represent
the temperature computed along the axis (r = 0). The upper
temperature curve is at x = L/4, while the lower one is at x
= L/2. It is clearly seen that here there is no oscillatory
pattern at all. The final temperature state is here related
to the classical Graetz problem dealt with in great detail
in the heat transfer literature [5, 11].
Temperature Distribution in Model 1
The final periodic temperature distributions for the
three different models with either conducting walls or
insulating walls have been obtained. Fig. 4-16, 4-17 and 4-
18 show the computed periodic final temperature distribution

96
Fig 4-16 Temperature Distribution in Oscillating
Flow (Model 1, a = 1, ax = 1cm)
0
90
120
270
360
Pipe

97
96
186
270
Fig 4-17 Temperature Distribution in Oscillating Pipe
Flow (Model 1, a = 1, ax = 2cm)

Fig
98
1.000
0.500
0.500
1.000
0.500
180
270
360
4-18 Temperature Distribution in Oscillating Pipe
Flow (Model 1, q = 1, Ax = 5cm)

99
T(r,x,®) in oscillating pipe flow with Model 1. The marked
numbers 1, 2, 3 represent the temperature along the axis (r
= 0) , r = Ri/2, and at r = R]_ (wall), respectively. For
these computations, the Wormersley number was taken as a = 1
and the tidal displacement was chosen as Ax = 1 cm, 2 cm and
5 cm, respectively. Neglecting the somewhat erratic
behavior near the pipe ends, the time-dependent temperature
shows a clear linear distribution pattern along longitudinal
direction especially for the relatively small tidal
displacement case, of say, Ax = 1 cm. If one focuses on one
cross-section (x = constant) , one can clearly see the
temperature near the boundary and that at the axis are
alternately higher and lower relative to each other (Fig. 4-
18) within a period. This implies that a very large radial
conduction heat exchange occurs between the core of the flow
and the boundary layer. Note that the radial temperature
gradient becomes very large during parts of an oscillating
cycle.
The poor behavior of the temperature distribution
noted near both ends of the pipe in Fig. 4-18 at tidal
displacement of ax = 5 cm is caused by numerical dispersion
which is a common shortcoming of the second order numerical
method employed [24, 28, 30, 37]. It may be improved by
either using an odd order numerical method such as the
first-order upwind method, flux splitting technique [24] or
by employing alternative models which are able to avoid any

100
discontinuity of the physical properties at both ends, such
as Model 3.
Temperature Distribution in Model 2
In order to explore some alternative possible devices
which apply the enhanced thermal pumping technique, Model 2
(Fig. 2-3) was developed and numerically examined. The
temperature distributions in Model 2 have been studied and
results are recorded in Figs. 4-19, 4-20, 4-21, 4-22 and 4-
23 for some special cases. The working fluid used is again
water and similarly, the Wormersley number was taken as a =
1. The tidal displacements used were Ax = 1 cm, 5 cm, 10
cm, 2 0 cm, and 3 0 cm, respectively. The marked numbers 1,
2, 3 in those figures represent the temperatures at r = R]_
(wall), Ri/2 and r=0 (i.e., along the axis). It can be seen
that the temperatures in the core of the flow and that in
the boundary layer in these figures vary periodically in the
radial direction during the oscillations. The results also
show a linear axial temperature variation between the
heating and cooling sources when the tidal displacement is
small (Ax = 1 cm, Fig. 4-19) . As the tidal displacement
becomes larger (ax = 5 cm, Fig. 4-20) , the linear axial
temperature variation becomes weaker. The difference of
temperature in the core of the flow and that in the boundary
layer is evident and shows a radially alternating pattern
within an oscillation period just as seen in Model 1.
However, as the tidal displacements are further increased,

101
Fig 4-19 Temperature Distribution in Oscillating Pipe
Flow (Model 2, q = 1, Ax = 1cm)

102
Fig 4-20 Temperature Distribution in Oscillating Pipe
Flow (Model 2, o = 1, ax = 5cm)

103
Fig 4-21 Temperature Distribution in Oscillating Pipe
Flow (Model 2, a = 1, AX = 10cm)

104
Fig 4-22 Temperature Distribution in Oscillating Pipe
Flow (Model 2, a = 1, Ax = 20cm)

105
1.00rT
1 at Ri
2 at Rl/2
3 along Axis
90
180
2?0
Fig 4-23 Temperature Distribution in Oscillating Pipe
Flow (Model 1, q = 1, ax = 30cm)

106
i.ee
Ueq=7.5cm/sec
2
e.se
e.ee
Fig 4-24 Temperature Distribution in Steady Flow
(Model 2, Uave = 0.5-7.5 cm/sec)

107
as shown in Fig. 4-21, the radial temperature gradient
almost no longer exists except directly in the vicinity of
the heat source and sink area. It is obvious that the
cross-section averaged temperature in the pipe tends to
become constant and equal to the mean value of the hot and
cold source region. This phenomenon can be seen even more
clearly as the tidal displacement is further increased
(Figs. 4-22, and 4-23). One can conclude from such a
temperature distribution that the increase of axial heat
transfer ability with increasing tidal displacement is most
likely to be limited. At the same time, this axial heat
transfer capability is expected to increase by increasing
either the individual source size or the density of the
alternatively distributed hot and cold sources.
This unexpected temperature distribution in Model 2
at large tidal displacements led us to further examine the
thermal field of steady pipe flow for the boundary
conditions used in Model 2. The computed temperature
distribution with an equivalent steady averaged velocity
from 0.5 cm/sec to 7.5 cm/sec, corresponding to tidal
displacement from 1 cm to 15 cm at an oscillating frequency
of w = 1/sec are shown in Fig. 4-24. The equivalent
velocity is established from the relation Ueq = AXw/2. As
could have been anticipated, large radial temperature
gradients in this flow (similar that in the oscillating flow
with large tidal displacement), occur only at the heat

108
source and sink areas, while the radial temperature gradient
in the rest of the pipe is very small. The constant
temperature difference at the right-hand-side differs from
that on the left hand side of the heat source. This
temperature difference becomes progressively smaller as the
eguivalent velocity is increased (i.e., the fluid heating
becomes less) .
Temperature Distribution in Model 3
Model 3 was developed to examine the performance of
the thermal pump without convective heat exchange at the
ends (in the chambers) . That is, the heat is added and
withdrawn by conduction through the pipe end section (see
Fig. 2-4). The computed temperature distribution in this
model is shown in Fig. 4-25, and a magnified view of the
temperature in the central section is shown in Fig. 4-26.
The Wormersley number used is again a = 1 and the tidal
displacement ax = 10 cm. Two features are observed. First,
in contrast to the temperature computed in Model 1, the
temperature obtained here shows a very smooth pattern at the
central section ends (i.e., at x = 5L and 6L in Fig. 2-4).
Second, the linear axial temperature distribution with time-
dependent radial temperature gradient alternating in sign in
the central portion of the pipe (namely, 5L < x < 6L, in
Fig. 2-4) is retained even when the tidal displacement is as
large as ax = 10 cm (Fig. 4-26).

109
i.eee
0.500
0.000
1.000
0.500
6.000 i i I_1 1 ii 'i I ' 1 i i ! i i : i I i i : i I i ' i i i ' i i i I i )
Central
Section(L)
1111111 1111111
-Heated Section(5L)-
along Axis
at Rl/2
at Rx
1 1 ! ' ' ' ' ' ' | 1
i*Cooled Section(5L)Jf
90
-i ! I I I I I l i I I
Fig 4-25 Temperature Distribution in Oscillating Pipe
Flow (Model 3, a = 1, Ax = 10cm, L=20cm)

i.eeer

Ill
Heat Flux Versus Tidal Displacement (Model 2)
As mentioned in the previous section, a limit on the
axial heat transfer in oscillating flow with Model 2 exists,
and the heat axially transferred in the steady Hagen-
Poiseuille flow with the same model could be expected as the
upper limit of the corresponding oscillating flow.
This has been verified by a numerical calculation
with water as the working medium and the results are
recorded in Table 4-5, and 4-6, and also plotted in Fig. 4-
27. The Wormersley number used in the current test is a =
1, and the tidal displacement for the oscillating flow
varied from 1 cm to 30 cm. The corresponding equivalent
average velocity varied from 0.5 cm/sec to 15 cm/sec for the
steady flow case. In Fig. 4-27 the ordinate represents the
heat flux released from the hot source, the abscissa is the
tidal displacement or the corresponding equivalent velocity
for steady flow. The solid curve shows the heat flux
computed for the oscillating flow, while the dashed curve
shows the axial heat flux in steady flow. It can be seen
that the slope of the heat flux curve in the oscillating
pipe flow case is smaller than that in steady pipe flow in
general and particularly when the tidal displacement is
small (i.e., Ax < 3 cm). The flux increases rapidly as the
tidal displacement gets larger (ax = 4-6 cm). Still further
increasing the tidal displacement does not lead to a further
large increase of the axial heat flux. In fact the slope

Enhanced Heat Flux (Watt/cm^0K)
2.0
Fig 4-27 Heat Flux in Oscillating Flow and Steady Flow
(Model 2, a = 1, Water Working Fluid, Insulated Wall)
112

113
begins to decrease after the tidal displacement becomes
larger than about Ax = 8 cm. Eventually this slope tends to
zero as Ax increases still further. For the steady flow,
Table 4-5 Enhanced Heat Flux via Tidal Displacement
(Model 2, oscillating flow Pr = 7.03, a = 3)
AX
(cm)
Periods to
final
oscillation
state *
Max. Pressure gradient
A=l/p|dp/5x| (cm/s2)
heat flux
(w/cm2 0 K)
1.000
75
4.86
0.03
1.962
15
8.372
0.136
2.943
7
12.588
0.345
3.924
6
16.744
0.683
4.905
5
20.930
1.051
6.867
5
29.300
1.377
7.848
5
33.488
1.420
9.843
5
42.000
1.484
15.233
5
65.000
1.545
19.920
5
85.000
1.522
30.100
5
126.000
1.530
* build-up time can be estimated by equation (4-1)

114
Table
4-6 Enhanced Heat
Flux in Steady
Flow
AX
Max.
Pressure gradient
Ueq=0.5wAx
heat flux
(cm)
A=1/p|ap/ax|
(cm/s2)
(cm/s)
(w/cm2 ° K)
0.500
2.43
0.25
0.462
1.000
4.86
0.50
1.354
2.500
10.50
1.25
1.567
3.000
12.56
1.50
1.631
5.000
21.00
2.50
1.775
8.000
33.50
4.00
1.664
10.000
42.00
5.00
1.786
15.000
65.00
7.50
1.779
the slope of the heat flux curve is quite large and almost
constant at all equivalent velocities less than 0.5 cm/sec,
but it rapidly decreases when the equivalent velocity
becomes larger. Similar to the oscillating flow case, the
heat flux as the velocity increases reaches a limit. This
limit is about 20 percent greater than that for the
oscillatory flow.
The reason of the slope variation in the oscillating
flow can be explained as follows: At small tidal

115
displacement, Ke is quite small, and the molecular
conduction is dominant, so that only a very small quantity
of heat will be transferred axially. However, as the tidal
displacement increases, Ke = AwAx2 becomes considerably
larger and the enhanced heat transfer mechanism now
dominates axial conduction. The limit on axial heat flux
observed at large Ax is due to a bottleneck which develops
at the heat supply and removal sections due to insufficient
conduction heat transfer.
In the steady flow, the fluid particles are always
fresh ones with no residual heat in them so the radial
temperature gradient is relatively larger than that in the
oscillating pipe flow case. The larger radial temperature
gradient increases the heat supply and removal capability so
as to ensure an axial flux limit in excess of that possible
for oscillating pipe flow in this model.
Influence of The Thermodynamic Properties (Model 2)
It is well known that the thermodynamic properties of
the working fluid, especially the viscosity, are functions
of the local temperature. Some thermodynamic properties of
water for the temperature range of 0°C to 200° C are listed
in Table 4-7. One can see that even over such a small
temperature range the thermodynamic properties of water vary
considerably. For instance, at T = 0°C, the kinematic
viscosity v = 1.788 10-6 M2/Sec, and the corresponding
Prandtl Number is Pr = 13.6, while at temperature T = 100°,

116
Table 4-7 The Influence of Properties of
Water on the Enhanced Axial Heat Flux
o
O t-a
Pr
P
g/cm3
CP
ws/g° K
v 10“2
cm2/sec
K
w/cm° K
AX
cm
4>
at a=l
w/cm2° K
200
0.937
0.8668
4.505
0.160
0.665
10.031
1.025
160
1.099
0.9097
4.417
0.173
0.680
11.499
1.249
100
1.740
0.9606
4.216
0.294
0.680
10.029
1.500
80
2.22
0.9741
4.196
0.364
0.668
10.030
1.525
60
3.02
0.9855
4.184
0.478
0.651
10.030
1.706
40
4.34
0.9946
4.178
0.658
0.628
10.041
1.780
20
7.02
1.0000
4.182
1.006
0.597
9.374
1.476
0
13.60
1.0020
4.218
1.788
0.552
10.030
0.747
* Properties are in the saturated state [11].
n = 0.294 10-6 M2/sec and Pr = 1.74. It varies almost an
order of magnitude over this range.
The present study did not intend to investigate the
influence of property changes in full detail; however, the
performance of axial heat flux versus water properties at
temperature from 0°C to 200°C has been examined.
Though Model 2 may not be an ideal device (at large
Ax) for the application of enhanced thermal pumping, it can,
however, be a good model for demonstrating the influence of

Pr = i///c
Fig 4-28 Influence Of Thermodynamic Properties of H2O on the
Enhanced Heat Flux (Model 2, a=l, Ax= 10 cm)
Time (sec)

118
the variation of the fluid properties on the heat transfer
near the source area in oscillating pipe flow. We used the
tidal displacement of Ax = 10 cm, which according to Fig. 4-
27, lies in the maximum axial heat flux range. The enhanced
heat flux of the final periodic state is shown in Fig. 4-28
by the solid curve as a function of the Prandtl number for
water at different temperatures and also in the last column
of Table 4-7. The dashed curve shown in Fig. 4-28 is the
time needed to build-up to the final oscillation state. The
Wormersley number used in the calculations was a = 1.
According to the definition of Wormersley number
given by equation (1-10), for a fixed a and pipe radius ,
the angular velocity w is directly proportional to the
kinematic viscosity u. This also implies that the fluid
boundary layer thickness is constant (Eg. 1-2). So, the
solid curve
in Fig.
4-28,
in fact, mainly
reflects a
relationship
between
the
enhanced heat flux and the
oscillating
frequency
W .
It is found that
a maximum
enhanced heat flux can be obtained if the water of
approximate temperature T = 4 0° C is employed. The
corresponding angular velocity at this temperature level is
w = 0.658 radian/sec. Physically, the existence of a peak
value in Fig. 4-28, may be explained as follows: At higher
temperature, say, T = 100° C, the kinematic viscosity u is
quite small and hence the angular velocity w is too slow to
transport large amounts of heat axially.
As the

119
temperature decreases, v gets larger, and so does the
angular velocity u. Thus more heat can be expected to be
transported and this eventually reaches the limit mentioned
in the last section. It seems that a similar flattening
pattern in the heat flux should also appear if the
temperature is further decreased and hence the frequency w
further increased. However, as temperature decreases
further the thermal conductivity declines (Table 4-7) so
that it weakens the heat supply and removal capability at
the sources and hence makes the axial heat flux curve
decline from its maximum.
Heat Flux Versus Tidal Displacement (Model 1)
As indicated by Kurzweg in earlier analytic studies
[16, 19, 20], the enhanced axial heat transfer versus the
oscillating flow is directly proportional to the square of
the tidal displacement. A series of numerical tests with
Model 1 were run to check this prediction and to study the
influence of various fluid properties and boundary
conditions on this Ax2 effect.
The computations were separated into two groups: in
the first group, the Wormersley number was chosen very close
to the tuning point, i.e., a = 1. Water was chosen as the
working fluid, and both insulating and conducting wall
conditions (glass and steel) were considered. The
computational results are shown in Fig. 4-29. In the second
group of numerical studies, the Wormersley number was chosen

Enhanced Heat Flux (Watt/cm¿0K)
120
Fig 4-29 Heat Flux versus Tidal Displacement
(Model 1, a = 1, Pr = 7.03)

121
1 2 3 4 5 6 7 8 9 10
Tidal Displacement ax (cm)
Fig 4-30 Heat Flux versus Tidal Displacement
(Model 1, a = 3, Pr = 7.03)

as a = 3, which deviates from the tuning point. Again water
was chosen as the working medium. Similarly, both
insulating wall and conducting (glass only) wall cases were
investigated. The results are shown in Fig. 4-30. The
solid curve (Fig. 4-29) shows the enhanced axial heat flux
in oscillating pipe flow versus tidal displacement for the
insulating wall case and the dashed curve represents the
enhanced axial heat flux obtained with a steel wall. The "
. . . " curve shows the enhanced axial heat flux
computed for a glass wall. These three curves clearly
confirm the quadratic behavior,
= k^-Ax2
where k is a constant. For this specific case, it is found
that
k^ = 0.0056 for the insulating wall case,
k^ = 0.0086 for the steel wall case,
k¿ = 0.0098 for the glass wall case.
The computational results for the Wormersley number a
= 3 case are shown in Fig. 4-30. There the solid curve
represents the enhanced axial heat flux in oscillating pipe
flow with insulating wall, the dashed curve shows the glass
wall results. Similarly, the coefficient k^ in the
quadratic formula is found to have the value
k¿ = 0.0233 for the insulating wall case,
k^ = 0.0317 for the water-glass combination.

123
It is obvious that the existence of conducting walls
tends to increase heat storage capacity and hence enhance
the radial and axial heat transfer process. Thus one can
expect a larger heat flux for conducting walls than for
insulating walls at the same tidal displacement and
Wormersley number. It can be seen from Fig. 4-29 and Fig. 4-
30 that higher Wormersley number (a = 3) will generally lead
to a larger axial heat flux. In these specific cases higher
Wormersley number simply implies higher oscillating
frequency w. However, it does not mean that the tuning point
shifts to a Wormersley number equal to 3. Because here the
enhanced axial heat flux is a dimensional quantity, if this
quantity is divided by the oscillating frequency, one can
clearly see that this new quotient is larger in the a = 1
case.
If one compares the computational results obtained by
using Model 1 in this section with the previously obtained
results for Model 2 as shown in Fig. 4-27, one may at first
glance feel somewhat puzzled at discrepancies between the
results. Note, however, that there is an essential
distinction between Model 1 and Model 2. In Model 1, the
fluid elements have infinite heat exchange capability at
both ends, in fact, all the new fluid elements with constant
temperature (i.e., the reservoir's temperature) enter the
pipe at both ends during every oscillating period. This
implies that direct convective heat exchange occurs at these

124
ends during each oscillation so as to match the fast
enhanced axial heat transfer possible within the pipe. On
the other hand, for the Model 2 case, there is no such
infinite heat exchange capability at the heat source and the
sink sections. In fact, the heat being supplied and removed
is purely by molecular heat conduction at the interface
between of the thermal sources and the fluid boundary layer.
This conductive heat supply and removal mode is not
compatible with a very fast enhanced axial heat transfer
within the connecting pipe flow and actually prevents the
thermal process from working at peak efficiency. This
comparison suggests to us that one should consider a strong
and preferably turbulent convective motion in both end
reservoirs as a means to increase the efficiency of the
enhanced thermal pumping technique.
The Influence of Wall Thickness
As mentioned above, the conducting wall will, in
general, make the enhanced axial heat transfer process more
efficient since the existence of a conducting wall creates
an additional heat storage capability near the fluid-solid
interface. This additional heat storage supplements the
storage capacity of the boundary layer and hence strengthens
the enhanced heat exchange process of the whole system.
The largest radial distance which the heat can be
transferred into the wall is the wall thermal penetrating
thickness 6W. This thickness is a function of oscillating

125
frequency as well as the thermal properties of the solid
wall. As mentioned in the introduction chapter (Fig. 2-1),
the future engineering applications of the thermal pumping
technique probably will require a bundle of capillary tubes
connecting the cold and hot fluid reservoirs. In view of
space limitations and economics, one wants to minimize the
capillary wall thickness used in the thermal pump yet still
be able to make full use of the wall heat storage capacity.
Using Model 1, a numerical investigation of enhanced
axial heat flux in oscillating pipe flow in a water-glass
combination for various wall thicknesses has been performed
and the computed results recorded in Fig. 4-31. In this
study, the tidal displacement was taken as Ax = 5 cm ( =
L/4), the Wormersley number a — 1, and inner pipe radius
= 0.1 cm. Water at 20° C was used as the working medium.
If we define the effective heat flux as the enhanced axial
heat flux computed not only over the cross-section of the
fluid in the pipe but also including the solid wall area, we
can see from Fig. 4-31 that as wall thickness increases, the
effective heat flux curve first increases with increasing
wall thickness R2 - Ri but soon starts to decline.
Apparently there is an optimum value of the wall cladding
thickness. For this specific case, it has been found that
this optimum wall thickness is about 10 percent of the pipe
diameter. It also implies that the penetrating thickness 6W
into the wall in this case is about 0.02 cm, which is 20

Enhanced Heat Flux (Watt/cm2°K)
126
0.0 0.1 0.2 0.3 0.4
Wall Thickness Ar = R2 ” **i (cm)
Fig 4-31 Influence of Wall Thickness on Axial Heat flux
(Model 1, Water-Glass, a = 1, Ax = 5cm)

127
percent of the pipe inner radius. The decline of the
effective heat flux curve is due to the "unpenetrated" outer
part of the wall which plays a useless role in the enhanced
heat axial transfer process.
The Influence of Pipe Diameter
Another very important aspect of the heat transfer
technique under consideration is the performance of heat
flux versus pipe diameter. A numerical simulation with
Model 3 has been carried out to examine the pipe diameter
effects. For the test the Wormersley number was taken as a
= 3, the tidal displacement Ax = 10 cm, and the fixed wall
thickness equal to 0.05 cm. Once again 20°C water was taken
as the working fluid and the wall material was glass.
From the definition of Wormersley number (1-10),
namely, a = , it is clear that the oscillating
frequency is inversely proportional to the second power of
pipe inner radius for fixed Wormersley number and directly
proportional to the kinematic viscosity of the working
fluid. On the other hand, the enhanced axial heat flux is
an increasing monotonic function of the oscillating
frequency. This implies that an increase in inner pipe
diameter will surely lead to a decline of oscillating
frequency and hence the enhanced axial heat flux, if both
Wormersley number and the kinematic viscosity are held
constant. The numerical results are shown in Fig. 4-32.

Enhanced Heat Flux (Watt/cm¿°K)
128
Pipe diameter (cm)
Fig 4-32 Influence of Pipe Diameter on Heat flux for Fixed
Frequency (Model 3, Water-glass, q = 3, ¿x = 10cm)

o
'e lapow)
(uios *0=za 'uioi • 0=^H 'UIO0I = XV 'e =
A\o-[.£ adid But^biitoso ut
ano^uoo aanq-Baaduiai-osi iBoxdAi,
ee-fr
o
621

130
It will be noted that the $ versus curve does not
show a decrease in enhanced axial heat flux as the radius
becomes very small. It is expected from the early analytic
work that there exists an upper limit of the enhanced axial
heat flux as the inner pipe diameter becomes extremely
small. This is because that when the pipe cross-section
area is further decreased such that the central slug flow
volume which transports heat back and forth axially becomes
very small, the heat transfer mechanism of the thermal
pumping technique is destroyed since large radial
temperature gradients are no longer possible.
Fig. 4-33 shows iso-temperature contours for a pipe
diameter equal to 0.5 cm. The iso-temperature contours show
a periodic temperature pattern within an oscillation period.
It is noted that the contours at phase angles wt = 0° and
180° or 90° and 270° are not precisely anti-symmetric.
Variation of Axial Temperature Gradient in Model 3
In order to overcome the numerical instabilities
observed near the pipe ends in Model 1 and to examine the
effect of no convective heat exchange at the pipe ends,
Model 3 which is shown in Fig. 2-4, was developed and
studied.
The first numerical study carried out with this later
model was to examine the variation of the time-averaged
axial temperature gradient within the central pipe section
as the tidal displacement was increased. This time-averaged

Temperature
131
Tidal Displacement ax (cm)
Fig 4-34 Variation of Temperature Tx and T2 versus ax
(Model 3, Water-Glass, a = 3)

132
axial temperature gradient variation can be calculated by
measuring the time-averaged temperatures at the joint points
(x = 5L, and 6L, see Fig. 2-4) . The computed results are
shown in Fig. 4-34. In the calculation, the Wormersley
number was chosen as a = 1 and the working medium is water
within a glass pipe. In the figure, T2 is the time-averaged
temperature computed at the left joint (x = 5L) which is
close to the hot source, while T^, at right joint, is close
to the cold source. It is clearly seen that T2 decreases,
while Ti increases as the tidal displacements increase. It
seems that both and T2 tend to the dimensionless mean
value (i.e., 0.5) if the tidal displacement is greatly
increased. This implies that the time-averaged axial
temperature gradient in the central section will decrease
and that the enhanced axial heat flux will be weakened
Table 4-8 Variation of the Axial Temperature Gradient
versus Wormersley Number (Water-Glass, Ax = 10 cm)
(Tf, T2...• are dimensionless Temperatures)
a
At Periods
t2
Tl
AT=T2~T]_
1
29
0.622
0.387
0.235
4
39
0.705
0.295
0.411
10
38
0.788
0.213
0.575
20
38
0.853
0.149
0.704

133
although still increasing with increasing Ax. However, the
numerical results show that such a weakening of the axial
temperature gradient can be improved at higher oscillating
frequency (Table 4-8). This is because at higher w the pipe
flow develops a much thinner boundary layer in the vicinity
of the wall and hence a larger radial temperature gradient
which increases the capability of heat supply and removal
from the extended sections. In fact, the numerical results
also show that only very narrow areas of the extended
section are involved in supplying and removing heat when the
oscillating frequency is large (Fig. 4-25).
Comparison of Enhanced Oscillatory Heat Transfer
and Heat Conduction
Fig. 4-35 shows the enhanced axial heat flux computed
in the central pipe section in Model 3 and the axial
molecular conduction heat flux both in the fluid and in the
wall versus tidal displacement. The Wormersley number in
the test case is chosen as a = 3, the working medium is
water, and the solid wall material is glass. Curve
represents the variation of enhanced axial heat flux versus
the tidal displacement. This is seen to be smaller than the
heat flux due to pure molecular conduction either in the
wall (curve ^w) or in the fluid (curve 4>f) when the tidal
displacement is very small (Ax less than 1 cm in this case),
however, 4> increases dramatically when the tidal
displacement becomes larger. In contrast, the pure axial
molecular conduction declines slightly due a weakening of

Heat Flux (Watt/cm2• K)
134
Fig 4-35
Comparison of Enhanced Heat Transfer and Heat
Conduction in Oscillating Pipe Flow
(Model 3, Water-Glass, a = 3)

135
the axial temperature gradient in the central pipe section
at large Ax as discussed in the last section.
The ratio of the enhanced heat flux to the square of
the tidal displacement is also shown in the same figure
(curve p) . It indicates a declining value of p as the tidal
displacement increases. This indicates that the 2nd power
relationship between the enhanced axial heat flux and
tidal displacement Ax has also been weakened (i.e., lower
k^) if large tidal displacements are used.
Enhanced Heat Flux as a Function of Wormerslev Number
To study the influence of oscillating frequency on
the enhanced axial heat transfer, a series of numerical
tests were performed with Model 3 for various combinations
of water-steel and mercury-steel cases. Fig. 4-36 shows the
enhanced axial heat flux and the heat flux by pure molecular
heat conduction both in the fluid and in the wall as a
function of Wormersley number (i.e., a function of
oscillating frequency w, for fixed pipe radius and working
fluid).
The solid curves show the computed results in the
water-glass wall case, while the dashed curves represent the
numerical solution obtained with the mercury-steel wall
combination. The tidal displacement ax used in these
calculations is fixed and equals 10 cm. The subscripts have
the same meaning as those used in Fig. 4-35.

Heat Flux ¡f> (Watt/cm^0 K)
136
Wormersley a
Fig 4-36
Variation of Axial Heat Flux Versus Wormersley
Number (Model 3, ¿x = 10cm)

137
It can be seen from Fig. 4-36 that both the enhanced
axial heat flux and axial heat flux due to pure molecular
conduction in either the wall (<¿w) or in the fluid ( f)
increase as the oscillating frequency gets larger. It is
assumed that this is partially due to the recovery of the
axial temperature gradient when the oscillating frequency
increases as discussed in the previous section. It can been
also seen that the enhanced axial heat flux is larger than
the axial heat flow by pure molecular conduction in either
case. The difference between the enhanced axial heat flux
and the axial heat flux due to pure molecular conduction in
the water-steel wall case is smaller than that in the
mercury-steel case. This implies that the water-steel
combination does not take optimum advantage of the enhanced
thermal pumping process. The axial heat flux due to pure
molecular heat conduction in the fluid, in general is very
much smaller than the enhanced axial heat flux and hence is
indeed negligible as assumed in the theoretical analysis of
this problem.
Note that if the dimensional enhanced heat flux is
divided by the corresponding oscillating frequency, we will
obtain the tuning curves which will be discussed below.
Tuning Curves
In Fig. 4-37, the left curve shows the variation of
the ratio of the enhanced axial heat flux to the
oscillating frequency u> in the water-glass wall case, while

/u (Watt* sec/cm^
138
Fig 4-37 Computed Tuning Curves
(Model 3, H20-Steel and Hg-steel, ax = 10cm)

139
the right curve represents the mercury-glass wall case as
marked in the diagram. It can be seen that both curves show
the existence of the so-called tuning effect. The optimum
value of Ri for the water-glass case (Pr = 7.03, « = 1/sec)
is found equal to 0.1 cm, and the corresponding Wormersley
number is about 1.
Fig. 4-38 shows some analytical results obtained in
reference [20, 21], There the optimum value of a for a
Prandtl number Pr = 10 is close to 1. The slight difference
in tuning peak is due to the use of a flat plate channel
geometry and different thermal wall boundary conditions in
the analytic investigation.
It is to be emphasized that the correct tuning
condition under various combinations of working fluid and
solid wall material is a crucial point in the design of
practical devices using the enhanced thermal pumping
technique. This is because the enhanced axial heat flux
which is usually orders of magnitude higher than the axial
heat flow produced by pure molecular conduction occurs only
near the tuning point and hence at a fixed radius for a
fixed value of w. Once the tuning point has been found, one
can select the appropriate pipe diameter to optimize the
heat transfer process.
We will demonstrate the above points in another way.
Fig. 4-39 shows the ratio of the axial heat flux by pure
molecular heat conduction in either the oscillating fluid or

Kg /wAX
140
a — Rju/i/
Fig 4-38 Tuning Curve versus Wormersley Number
(after Kurzweg, [16] )

141
Wormersley Number a = Rju>/u
Fig 4-39 Ratio of Heat Conduction to Enhanced Heat Flux
Versus Wormersley Number
(Model 3, H20-Steel, Hg-Steel, ax = 10cm)

142
in the solid wall to the enhanced axial heat flux for both
the water-glass and mercury-steel combinations. These
curves look like up-side-down tuning curves. They clearly
show that only at or near the tuning points the axial heat
flux generated by the pure molecular heat conduction either
in fluid or the conducting wall is quite small compared to
the enhanced heat flux and hence negligible (about 5 10-3
for the water-steel wall case, and about 5 10-2 for the
mercury-steel wall case). However, if one departs from the
tuning point this ratio grows quickly and eventually may be
greater than 1. The tube radii corresponding to the tuning
points (a = 1 for water-steel, a = 15 for mercury-steel) at
w = 25 rad/sec, correspond to = 0.02 cm for water-steel
and Ri = 0.1 cm for mercury-steel.

CHAPTER V
CONCLUDING REMARKS
We have attempted to present a systematic numerical
investigation of enhanced axial heat transfer in oscillating
pipe flow for both insulating wall and conducting wall
cases. As has been shown in Fig. 1-1, the pipes which
connect both reservoirs are the most important elements in
such enhanced thermal pumping devices and the performance of
the oscillating fluid within the pipes will determine to a
large extent if such devices are practical for a given
application such as cooling of micro-circuits or the removal
of heat from nuclear reactors without associated convective
mass exchange.
The governing differential equations derived in
chapter II are based on the assumption that viscous
dissipative terms in the energy equation are negligible
compared to the convective term UT, and that the final
oscillating flow can be well approximated by a 1-D laminar
oscillating flow with constant thermodynamic properties.
The numerical calculations presented in this
dissertation obtain solutions by breaking up the governing
equations into their finite difference form and solving the
momentum equation with the Crank-Nicolson technique, and the
143

144
energy equations with a time dependent Alternating Direction
Implicit method (ADI). A computer code named ETP (Enhanced
Thermal Pumping) was developed to implement the
calculations. It is documented in the appendix.
The physical mechanism for the enhanced thermal
pumping technique in oscillating pipe flow is an interchange
of heat between the core slug flow and the boundary layer
and the bounding conducting wall. In fact, the process acts
very much as an enhanced molecular diffusion process in
which the tidal displacement Ax plays a role similar to the
phonon mean free path in the molecular conduction and the
existence of large radial temperature gradients, orders of
magnitude higher than those existing axially, allow very
large radial conductive fluxes. Since the macroscopic
distance ax is orders of magnitude larger than the molecular
mean free path, it is not surprising that axial heat flows
orders of magnitude larger than those possible by conduction
in the absence of oscillations become possible.
This enhanced thermal pumping technique is especially
suited for those problems where it is desirable to transport
large amounts of heat without an accompanying convective
mass exchange. The removal of heat from radioactive fluids
would appear to be ideally suited for a heat transfer device
based on this enhanced heat exchange technique.
Three Models of different configuration were examined
numerically in detail. Model 1 was employed to simulate

145
constant pipe end temperature conditions as widely used in
theoretical studies. The performance of oscillating pipe
flow with this model has been examined numerically and the
computations well justify the existing analytical
approximations used. It is now clearer that the tuning
condition and a large maintained axial temperature gradient
between pipe ends is crucial for an optimal functioning
thermal pump. It was found that the enhanced axial heat
flux achievable with this model is indeed very large and can
exceed by orders of magnitude the heat flux possible by pure
heat conduction between the pipe ends. The enhanced axial
heat flux for this geometry is proportional to the second
power of the tidal displacement for both the conducting wall
and insulating wall case and with either water or mercury as
the working medium (Figs. 4-29, and 4-30). The
proportionality coefficients between and Ax2 have been
found for the above mentioned test cases. Unfortunately,
with the present second order numerical method, the
numerical solutions were distorted primarily by dispersion
error at both pipe ends where a discontinuity of the
physical properties occurs as Wormersley Number and/or tidal
displacement become large.
Model 2 was designed to examine the properties of an
alternative thermal pump design of potential use in micro-
circuit cooling. Such a configuration was successful in
transferring large quantities of heat between the iso-

146
thermal side wall sections and the fluid when small tidal
displacements were used. However, a limit on the increase
of axial heat flux with increase in tidal displacement
exists in both the oscillating and steady flow cases in this
model and the quadratic relationship no longer holds when
the tidal displacement becomes large because of an
incompatibility of large time-dependent axial heat transfer
in the thermal pumping process and the capability of the
thermal sources to supply and remove sufficient heat by
thermal conduction. Nevertheless, it made us pay more
attention to a study of the heat exchange problem at the end
points of the pipes where they connect to fluid reservoirs.
Since an effective enhanced thermal pump produced by
oscillating pipe flow requires rapid heat supply and removal
at both pipe ends any configuration unable to do this will
not allow one to take full advantage of the process.
To investigate the influence of the heat supply and
removal at the pipe ends, Model 3 was constructed by adding
very long conductive extensions at both ends of the central
pipe section. This model simulates the situation of
oscillating pipe flow for which no convective heat exchange
occurs in the end reservoirs but large areas for conduction
heat transfer are made available. Numerical results show
that the effective enhanced axial heat transfer is greatly
weakened over that achievable by good convective mixing at
the pipe ends. The reason is again that, even with an

147
enlarged heat source size (pipe extension) it is still not
able to match the very fast enhanced axial heat transfer
possible in the connecting pipe. It is found that constant
end temperature (at x = 5L and 6L) can not be maintained and
the quadratic relationship between the enhanced heat flux
and the tidal displacement will no longer hold (Fig. 4-35).
However, even with this model, the numerical results show
the tuning point concept exists. For water, this tuned
value occurs at about Wormersley number a = 1. It is also
shown that, under tuned conditions, the enhanced axial heat
transfer is orders of magnitude larger than that possible by
pure molecular conduction. However, any deviation from the
tuning condition will greatly decrease the effectiveness of
the enhanced thermal pumping.
A study on the influence of wall thickness and pipe
diameter on enhanced heat transfer was performed. The
numerical predictions show that for water, small pipe
diameter will be beneficial and the optimum wall thickness
should be about 20 percent of the pipe inner radius for the
case considered (i.e., for inner pipe radius of = 0.1cm
the best outer pipe radius should be chosen as R2 = 0.12cm).
The influence of the variation of thermal and viscous
properties of working fluid on the axial heat flux has also
been studied. The numerical solutions show that even in the
relatively narrow temperature range from 0°C to 100°C, the
axial heat flux varies more than 150 percent.

148
Extensions of the numerical studies considered here
should include: 1) developing a 3-D or at least an
axisymmetric 2-D model which is able to best approximate the
entire system including both reservoirs so that one may more
accurately study the effect of heat exchange within the
reservoirs; 2) the role of turbulence in such oscillating
flows at high frequency w and tidal displacement Ax should
receive special attention as well as the inertia forces that
are known to become large at higher oscillating frequency.
In fact, a turbulent modelling is necessary especially if
the study includes heat source and heat sink reservoirs; 3)
a consideration of the variation of the thermodynamic
properties of working fluid with temperature and hence
spatial position should also be included in any extension of
this research on enhanced heat transfer by oscillations of
viscous fluids in pipes.

APPENDIX
ETP COMPUTER CODE
An axisymmetric code with designation ETP has been
developed for computing the time-dependent Enhanced Thermal
Pumping problem with either oscillating flow (pipe or flat
plate) or steady flow. The grid is generated with a
clustering in those regions where high velocity and
temperature gradients are expected. Both Model 1 and Model
3 geometry are programed in the current code. Another
separate code (MIXPIP) is also available for the Model 2
geometry which is not discussed in this appendix.
Description of Input Variables
The following statements are used in the ETP code:
READ (5,10) NMAX,KMAX,JMAX,ICLUS,IPIPE,IPRES
READ (5,10) KMID,IREAD,JPER,ICH
READ (5,20) DS,RMUK,ROU,CP,RKS,CKS
READ (5,20) ALFA,PR,RL,T01,TO2,AMP
READ (5,20) RAD, RADI,RLEND,UEPS,TEPS
READ (5,10) (IH(I),1=1,ICH)
10 FORMAT (8110)
20 FORMAT (8F10.5)
where:
NMAX total time steps within one period
KMAX total nodal points on radius
JMAX total nodal points on axis
149

150
ICLUS = 0
= 1
IPIPE = 0
= 1
IPRES = 0
= 1
KMID
IREAD = 0
=-2
= 2
= 1
= 5
JPER
ICH = 1 .
= 4 .
DS
RMUK ....
ROU
CP
RKS
CKS
ALFA
PR
grid with clustering
uniform Grid
Pipe geometry
2-D Flat plate geometry
sinusoidal pressure gradient
constant pressure gradient
node index at interface along radius
KMID = KMAX insulating wall
< KMAX conducting wall
start a new job
start flow part only
start heat part only
(based on existing flow variables)
restart both flow & heat field calculation
restart flow field calculation only
number of period in the current run
Model 1
Model 3
distance of grid next to wall
kinematic Viscosity of fluid
density of fluid
specific heat of fluid
thermal diffusivity of solid wall
heat conductivity of wall
Wormersley number
Prandtl number

151
RL Length L of pipe or flat plate
T01 Temperature in the hot reservoir
T02 Temperature in the cold reservoir
AMP The amplitude of axial pressure gradient
AMP = -(1./R0U)*dp/DX
RAD inner radius of pipe
RADI outer radius of pipe
RLEND each extension portion length
UEPS the allowance velocity residual
TEPS the allowance temperature residual
IH(I) node index of separating segments in Model 3
Note:
to run and restart a job with this code, one of the
procedures has to be followed:
1) IREAD = 0 start both flow & heat field
= 1 restart both flow & heat field
or
2) IREAD =-2 start flow field only
=2 start heat & restart flow field
= 1 restart both flow & heat fields
A Samóle Inout
Data and
Batch Command File
The
following
sample
input
was
used
in the
computation
with Model
3 (water-
-glass)
1001
15
101
0
0
0
15
0
2
4
0.40000
0.01006
0.99820
4.18181
0.
20260
0.73000
10.00000
7.02000 20.00000
1.00000
0.
00000
29270.600
0.10000
0.15000 100.00000
0.00500
0.
00500
1
21
81
101

152
The following Command file was used to start the run
of a new case:
$ ASSIGN SAMPLO.INP FOR005
$ ASSIGN SAMPLO.DAT FOR006
$ ASSIGN FILE7.DAT FOR007
$ ASSIGN FILE8.DAT FOR008
$ ASSIGN FILE11.DAT FOROll
$ R EPT
To restart the previous run, the following command
file was used:
$
$
$
$
$
$
$
$
$
$
$
$
RENAME FILE7.DAT FILE9.DAT
RENAME FILE8.DAT FILE10.DAT
RENAME FILE11.DAT FILE12.DAT
ASSIGN SAMPL1.INP FOR005
ASSIGN SAMPL1.DAT FOR006
ASSIGN FILE7.DAT FOR007
ASSIGN FILE8.DAT FOR008
ASSIGN FILE9.DAT FOR009
ASSIGN FILE10.DAT FOROIO
ASSIGN FILE11.DAT FOROll
ASSIGN FILE12.DAT FORO12
R EPT
Note, the name of the data files could be changed according
user's taste.
Function of Subroutine in ETP Code
In the ETP code, the Program MAIN controls the main
loop in the computation, including the input and output data
control, restart control. Also it sets up the initial
condition according to the purpose of investigation. The
ETP code includes 22 subroutines. The function of each
subroutine is listed below:
Subroutine COMMENT
This subroutine gives the description of the input
and output variables listed in previous section.

153
Subroutine GRID
This subroutine is used to generate either a uniform
or a clustered grid system in the physical domain for Model
1 or Model 3; it calls the subroutine CLUS to cluster the
grid lines in the vicinity of wall and the central part of
pipe (Model 3). It also calls subroutine METPIP, METWAL to
compute the derivatives terms in the transformation.
Subroutine CLUS ÍICLUS.K2^
This subroutine is called by GRID to compute the
clustering grid distribution with equation (3-52).
Subroutine METPIP
This subroutine is called to compute the derivatives
of the coordinate transformation in the inner pipe area.
Such as 3x/3(, dx/dr¡ used in equations from (3-6) to
(3-16) as well as in the boundary condition.
Subroutine METWAL
Similarly, subroutine METWAL is called with the same
equations to compute the derivatives in transformation in
the solid wall, if a conducting wall is assumed.
Subroutine FLOW (N)
This subroutine is called to compute the velocity in
the flow field at each time step with the Crank-Nicolson
Method. Equation (3-20) or the matrix form (3-28) will be
solved in this subroutine.
Subroutine TIDAL
This subroutine is called to Compute the Lagrangian

154
displacement (2-33), tidal displacement(2-32) and the phase
lags which are relative to the exciting pressure gradient
phase.
Subroutine COFLOW
This subroutine is called to compute the coefficients
in the momentum equation at each time step, i.e., computing
equations (3-7) and (3-8).
Subroutine RHSFLO £Nj
The subroutine RHSFLO (N) is called to compute the
update right hand side terms in the momentum equation at
each time step, i.e., equations from (3-21) to (3-25).
Subroutine TRIP
A subroutine TRID (borrowed from [2]) is to solve the
tridiagonal system algebra equations, it is widely used to
compute the update velocity and temperature.
Subroutine EPSIL
Function EPSIL (borrowed from code "GRIDGEN") to find
the e value used in equation (3-52) with Newton-Raphson
root-finding technique.
Subroutine ADISU3
This subroutine is called using ADI method to compute
the temperature distribution in oscillating pipe flow in
Model 3. The insulating wall boundary condition is assumed
in the central pipe (5L < x < 6L) with fixed temperature at
the far ends of both extension pipe. Equations from (3-29),
(3-35), or the matrix from (3-33), (3-39) are solved.

155
Subroutine COTPIP
This subroutine is called by ADISU3 to calculate the
coefficients in the energy equation of pipe flow (3-41) at
each time step, which are coupled by the update velocity.
Subroutine RHSINT (N.II.LINE)
This subroutine is called by ADISU3 to compute the
update right hand side terms in the energy equation of pipe
flow with the insulating wall case, namely, solving (3-31),
(3-37).
Subroutine FLUXT fNNl
Subroutine FLUXT (NN) is called to compute the
dimensionless enhanced axial heat flux as well as the heat
flux by pure molecular conduction either in the wall or the
pipe fluid, namely, equations (2-34) to (2-37).
Subroutine OUT (NTID^
The subroutine OUT (NTID) is called to compute and
output the dimensional heat flux, tidal displacement, and
the average temperature (dimensionless).
Subroutine ADIC03
Subroutine ADIC03 is similar to ADISU3, but for the
conducting wall case.
Subroutine COTWAL
This subroutine is called to calculate the
coefficients in the wall conduction equation at each time
step by using equation (3-42).

156
Subroutine RHSCOT (N.II.LINE)
Subroutine RHSCOT is similar to RHSINT, but for the
conducting wall case.
Subroutine RHSWAL CN.II.LINE)
Subroutine RHSWAL (N,II,LINE) is called to compute
the right-hand-side terms in the wall conduction equation
with (3-32) and (3-38).
Subroutine ADISU1 (in
Subroutine ADISU1 is similar to ADISU3, but for the
case of Model 1.
Subroutine ADIC01 fN)
Subroutine ADIC01 is similar to ADISU1, but for the
conducting wall.

REFERENCES
1. Abramowitz, M., and Stegun, I. A.(editors), "Handbook of
Mathematics Functions with Formulas, Graphys, and Mathe¬
matical Tables", Dover Publications, Inc., New York
(1972) .
2. Ames, W. F. , "Numerical Method For Partial Differential
Equations," Academic Press, INC.
3. Anderson, D. A., Tannehill, J. C., and Pletcher, R. H.,
"Computational Fluid Mechanics and Heat Transfer,"
McGraw-Hill, New York (1984) .
4. Aris, R., "The Dispersion of Solute in Pulsating Flow
Through a Tube," Proceedings of Royal Society, London,
A 259, pp.370-376 (1960).
5. Arpaci V., and Larsen P., "Convective Heat Transfer,"
Prentice Hall, Englewood Cliffs, NJ (1984).
6. Baldwin B. S., and Lomax, H., "Thin Layer Approximation
and Algebraic Model for Separated Turbulent Flow," AIAA
16th Aerospace Sciences Meeting (Jan., 1978) pp.78-257
7. Bohn, D. J., Miyasaka, K., Marchak, B. E., Thompson, W.
K., Froese, A. B., and Bryan, A. C., "Ventilation by
High-Frequency Oscillation," Journal of Appl. Phys.,
Vol. 48, pp.710-716 (1980).
8. Bowden, K. F., "Horizontal Mixing in the Sea Due to
Shearing Current," Journal of Fluid Mech., Vol. 21,
p.84 (1965).
9. Chapman, A. J., "Heat Transfer," Macmillan Publishing
Co., New York (1984).
10. Chatwin, P. C., "On the Longitudinal Dispersion of
Passive Contaminant in Oscillatory Flow in Tubes".
Journal of Fluid Mech., Vol. 71, part 3, pp.513-527
(1975) .
11. Eckert, E. R. G., "Analysis of Heat and Mass Transfer,"
McGraw-Hill, New York (1972).
157

158
12. Jaeger, M. J., and Kurzweg, U. H., "Determination of the
Longitudinal Dispersion Coefficient in Flows Subjected
to High-Frequency Oscillations," Phys. Fluid, Vol. 26
No. 6, pp.1380-1382 (1983).
13. Joshi, C. H., Kamm, R. D., Drazen, J. M., and Slutsley,
A. S., "An Experimental Study of Gas Exchange in Laminar
Oscillating Flows," Journal of Fluid Mech., Vol. 133,
pp.245-254 (1983).
14. Kaviany, M., "Some Aspects of Heat Diffusion in Fluids
by Oscillation," International Journal Heat Mass
Transfer, Vol. 29, pp.2002-2006 (1986).
15. Kurzweg, U. H., "Enhanced Heat Conduction in Fluids
Subjected to Sinusoidal Oscillations," Journal of Heat
Transfer, Vol. 107, pp.459-462 (1985).
16. Kurzweg, U. H., "Enhanced Heat Conduction in Oscillating
Viscous Flows Within Parallel-Plate Channels," Journal
of Fluid Mech., Vol. 156, pp.291-300 (1985).
17. Kurzweg, U. H., "Temporal and Spatial Distribution of
Heat Flux in Oscillating Flow Subjected to an Axial
Temperature Gradient," International Journal Heat Mass
Transfer, Vol. 29, No. 12, pp.1969-1977 (1986).
18. Kurzweg, U. H., "Enhanced Diffusional Separation in
Liquids by Sinusoidal Oscillations," to appear in
Separation Sciences and Technology, in press.
19. Kurzweg, U. H., and Jaeger, M. J., "Diffusional
Separation of Gases by Sinusoidal Oscillations," Phys.
Fluids, Vol. 30, pp.1023-1026 (1987).
20. Kurzweg, U. H., and Jaeger, M. J., "Tuning Effect in Gas
Dispersion under Oscillatory Conditions," Phys. Fluids
Vol. 29, pp.1324-1326 (1986).
21. Kurzweg, U. H. and Lindgren, E. R. "Enhanced Heat
Conduction By Oscillatory Motion of Fluids in Conduits,"
A Research Proposal to the Fluid Dynamics and Heat
Transfer Program, The National Science Foundation,
under Contract Number CBT-8611254 (1986).
Kurzweg, U. H., and Zhao, L. D., "Heat Transfer by High-
Frequency Oscillations: A New Hydrodynamic Technique
for Achieving Large Effective Thermal Conductivities,"
Phys. Fluids, Vol. 27, pp.2624-2627 (1984).
22.

159
23. Li, C. P., "A Finite Difference Method for Solving
Unsteady Viscous Flow Problems," AIAA Journal, Vol. 23,
No.5 (1985).
24. MacCormack, R. W., "Current Status of Numerical
Solutions of the Navier-Stokes Equations," AIAA 23rd
Aerospace Sciences Meeting, Reno, Nevada, AIAA-85-0032
Jan. (1985)
25. Mastin, C. W., "Error Induced by Coordinate System,"
in J. F. Thomson (editor), Numerical Grid Generation,
North-Holland, New York, pp.31-40 (1982).
26. Merkli, P., and Thomann, H., "Transition to Turbulence
in Oscillating Pipe Flow," Journal of Fluid Mech., Vol.
68, pp.567-575 (1975).
27. Ommi, M., Iguchi, M., and Urahata, I., "Flow Patterns
and Frictional Losses in An Oscillating Pipe Flow,"
Bulletin of the JSME, Vol. 25, No. 202. April (1982).
28. Pulliam, T. H., "Efficient Solution Methods for the
Navier-Stokes Equations" Lecture Notes for the Von
Karman Institute (Jan., 1986).
29. Schlichting, H., "Boundary Layer Theory" McGraw-Hill,
New York (1972).
30. Shah, V. L., "COMMIX-1B: A Three-Dimensional Transient
Single-Phase Computer Program for Thermal Hydraulic
Analysis of Single and Multicomponent Systems," Vol. 1:
Equations and Numerics, Available from Superintendent of
Documents U. S. Government Printing Office, P. 0. Box
37082, Washington, D.C.20013-7982, Sep., (1985).
31. Shih, T. M., "Numerical Heat Transfer," Hemisphere
Publishing Corp., Washington (1984).
32. Taylor, F. R. S., "Dispersion of Soluble Matter in
Solvent Flowing Slowly Through a Tube," Proc. Roy. Soc.,
pp.186-203 (1953).
33. Thompson, J. F., "Grid Generation Technique in
Computational Fluid Dynamics", AIAA Journal, Vol. 22
Nov.11, pp.1506-1519, (1984).
34. Thompson, J. F., Thames, F. C. and Mastin, C. W.,
"Automatic Numerical Generation of Body-Fitted
Curvilinear Coordinate System for Field Containing Any
Number of Arbitrary Two-Dimensional Bodies," Journal of
Computational Physics., Vol. 15, pp.299-319 (1974).

160
35. Thompson, J. F., Warsi, Z. U. A., and Mastín, C. W. ,
"Boundary-Fitted Coordinate Systems for Numerical
Solution of Partial Differential Equations A Review",
Journal of Computational Physics., Vol 47., pp.1-108
(1982).
36. Uchida, S., "The Pulsating Viscous Flow Superposed on
the Steady Laminar Motion of Incompressible Fluid in a
Circular Pipe," ZAMP, Vol. VII, pp.402-421 (1956).
37. Warming, R. F., and Beam, R. M., "On the Construction
and Application of Implicit Factored Schemes for
Conservation Laws," SIAM-AMS Proceedings, Vol 11,
pp.85-129 (1978).
38. Watson, E. J., "Diffusion in the Oscillation Pipe Flow,"
Journal of Fluid Mech., Vol. 133, pp.233-244 (1983).

BIOGRAPHICAL SKETCH
Guo-Jie Zhang was born on June 26, 1941, in Beijing,
China, and spent his childhood and youth in that beautiful
capital city. After graduating from Beijing 21th High
School in 1960 he entered the University of Science and
Technology of China, Beijing, China. After 5 years of
college education in the Department of Mechanics, he
graduated in July, 1965. Since then he, as an aircraft
structure strength analysis engineer, has been engaged at
the Aircraft Research and Development Institute, Nanchang
Manufacturing Aircraft Company, China. until the spring of
1981. During that period he participated in the research on
and development of several aircraft with his main
contributions being in the area of the studies with finite
element methods. In May, 1981, he was selected to study
abroad by the Ministry of Aviation Industry of China and
after one year's English language training at Nanjing
Aeronautical Institute, he was sent to study in the
Department of Aerospace Engineering, Mechanics, and
Engineering Science, University of Florida as a visiting
scholar, and later, in May, 1983, transferred to graduate
student status. He received his Master of Engineering
degree in April, 1986, and expects to receive the degree of
161

162
Doctor of Philosophy in the spring of 1988. He is married
to Chun-hua Shi and has two children, Yan Zhang and Wei
Zhang. Mr. Guo-jie Zhang is a member of the Aeronautics
Society of China.

I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
¿gy/h 'c¿) ?/ Ulrich H. Kurzweg, Cnairman
Professor of Aerospace Engineering,
Mechanics, and Engineering Science
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
P.
- A
1
i/'-\
E. Rune Lindgren 0
Professor of Aerospace Engineering,
Mechanics, and Engineering Science
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
o yvL*jLjL*~^
Lawrence E. Malvern
Professor of Aerospace Engineering,
Mechanics, and Engineering Science

I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
Í
David W. Mikolaitis
Assistant Professor of Aerospace
Engineering, Mechanics, and
Engineering Science
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
Arun K. Varma
Professor of Mathematics
This dissertation was submitted to the Graduate Faculty of
the College of Engineering and to the Graduate School and
was accepted as partial fulfillment of the requirements for
the degree of Doctor of Philosophy.
April 1988
I'lcjLpjf' CL •
Dean, College of Engineering
Dean, Graduate School

UNIVERSITY OF FLORIDA
3 1262 08556 7963



xml version 1.0 encoding UTF-8
REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd
INGEST IEID E8TT80CM2_6VEVX9 INGEST_TIME 2011-11-03T19:32:30Z PACKAGE AA00004834_00001
AGREEMENT_INFO ACCOUNT UF PROJECT UFDC
FILES



PAGE 1

7,0('(3(1'(17 (1+$1&(' +($7 75$16)(5 ,1 26&,//$7,1* 3,3( )/2: %\ *82-,( =+$1* $ ',66(57$7,21 35(6(17(' 72 7+( *5$'8$7( 6&+22/ 2) 7+( 81,9(56,7< 2) )/25,'$ ,1 3$57,$/ )8/),//0(17 2) 7+( 5(48,5(0(176 )25 7+( '(*5(( 2) '2&725 2) 3+,/2623+< 81,9(56,7< 2) )/25,'$

PAGE 2

7R P\ EHORYHG PRWKHUODQG

PAGE 3

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

PAGE 4

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

PAGE 5

7HPSHUDWXUH 'LVWULEXWLRQ LQ 0RGHO 7HPSHUDWXUH 'LVWULEXWLRQ LQ 0RGHO 7HPSHUDWXUH 'LVWULEXWLRQ LQ 0RGHO +HDW )OX[ YHUVXV 7LGDO 'LVSODFHPHQW 0RGHO f ,OO ,QIOXHQFH RI 7KHUPRG\QDPLF 3URSHUWLHV +HDW )OX[ YHUVXV 7LGDO 'LVSODFHPHQW 0RGHO f ,QIOXHQFH RI :DOO 7KLFNQHVV ,QIOXHQFH RI 3LSH 'LDPHWHU 9DULDWLRQ RI $[LDO 7HPSHUDWXUH *UDGLHQW ,Q 0RGHO &RPSDULVRQ RI (QKDQFHG 2VFLOODWRU\ +HDW 7UDQVIHU DQG +HDW &RQGXFWLRQ (QKDQFHG )HDW )OX[ DV D )XQFWLRQ RI :RUPHUVOH\ 1XPEHU 7XQLQJ &XUYHV 9 &21&/8',1* 5(0$5.6 $33(1',; (73 &20387(5 &2'( 5()(5(1&(6 %,2*5$3+,&$/ 6.(7&+ 9

PAGE 6

/,67 2) ),*85(6 )LJXUH 3DJH ) Df &XUYH 7KHUPDO 3XPSLQJ 'HYLFH 0RGHO )L[HG (QG 7HPSHUDWXUH 0RGHO 0RGHO 3HULRGLF +HDW DQG &ROG 6RXUFHV RQ ,QVXODWHG :DOO 0RGHO 3LSH ZLWK ([WHQGHG &RQGXFWLQJ 6HFWLRQV *ULG 6\VWHP 8VHG LQ WKH 1XPHULFDO 6LPXODWLRQV &RRUGLQDWH 7UDQVIRUPDWLRQ 9HORFLW\ 3URILOHV LQ 2VFLOODWLQJ )ORZ IRU :RUPHUVOH\ 1XPEHU D DQG 9HORFLW\ 3URILOHV D DIWHU 8FKLGDf 0DJQLILHG 9LHZ RI 9HORFLW\ 3URILOH QHDU :DOO /DJUDQJLDQ 'LVSODFHPHQW IRU D /DJUDQJLDQ 'LVSODFHPHQW DW D 'LPHQVLRQOHVV &URVV6HFWLRQ $YHUDJHG 'LVSODFHPHQW YHUVXV 7LPH D f 'LPHQVLRQOHVV &URVV6HFWLRQ $YHUDJHG 'LVSODFHPHQW YHUVXV 7LPH D f 5HODWLRQVKLS %HWZHHQ 'LPHQVLRQOHVV 7LGDO 'LVSODFHPHQW $; DQG :RUPHUVOH\ 1XPEHU D 5HODWLRQVKLS %HWZHHQ 7LGDO 'LVSODFHPHQW $[ DQG ([FLWLQJ 3UHVVXUH *UDGLHQW LQ :DWHU 3KDVH 9DULDWLRQ $ORQJ 5DGLXV IRU 'LIIHUHQW :RUPHUVOH\ 1XPEHUV YL

PAGE 7

7HPSHUDWXUH %XLOGXS 3URFHVV LQ 2VFLOODWLQJ )ORZ 0RGHO D $[ FPf 7HPSHUDWXUH %XLOGXS 3URFHVV LQ 2VFLOODWLQJ )ORZ 0RGHO D $[ FPf 7HPSHUDWXUH %XLOGXS 3URFHVV LQ 2VFLOODWLQJ )ORZ 0RGHO D $[ FPf 7HPSHUDWXUH %XLOGXS 3URFHVV LQ 6WHDG\ )ORZ 0RGHO 8DYH FPVHFf %XLOGXS 7LPH YHUVXV 7LGDO 'LVSODFHPHQW 0RGHO D f 7HPSHUDWXUH 'LVWULEXWLRQ LQ 2VFLOODWLQJ 3LSH )ORZ 0RGHO D $[ FPf 7HPSHUDWXUH 'LVWULEXWLRQ LQ 2VFLOODWLQJ 3LSH )ORZ 0RGHO D $[ FPf 7HPSHUDWXUH 'LVWULEXWLRQ LQ 2VFLOODWLQJ 3LSH )ORZ 0RGHO D $[ FPf 7HPSHUDWXUH 'LVWULEXWLRQ LQ 2VFLOODWLQJ 3LSH )ORZ 0RGHO D $[ FPf 7HPSHUDWXUH 'LVWULEXWLRQ LQ 2VFLOODWLQJ 3LSH )ORZ 0RGHO D $[ FPf 7HPSHUDWXUH 'LVWULEXWLRQ LQ 2VFLOODWLQJ 3LSH )ORZ 0RGHO D $[ FPf 7HPSHUDWXUH 'LVWULEXWLRQ LQ 2VFLOODWLQJ 3LSH )ORZ 0RGHO D $[ FPf 7HPSHUDWXUH 'LVWULEXWLRQ LQ 2VFLOODWLQJ 3LSH )ORZ 0RGHO D $[ FPf 7HPSHUDWXUH 'LVWULEXWLRQ LQ 6WHDG\ )ORZ 0RGHO 8DYH FPVHFf 7HPSHUDWXUH 'LVWULEXWLRQ LQ 2VFLOODWLQJ 3LSH )ORZ 0RGHO D $; FPf 0DJQLILHG 9LHZ RI 7HPSHUDWXUH LQ WKH &HQWUDO 3LSH 6HFWLRQ 0RGHO D $[ FPf +HDW )OX[ LQ 2VFLOODWLQJ )ORZ DQG 6WHDG\ )ORZ 0RGHO D :DWHU DV :RUNLQJ )OXLGf YLL

PAGE 8

,QIOXHQFH RI 7KHUPRG\QDPLF 3URSHUWLHV RI +2 RQ WKH (QKDQFHG +HDW )OX[ 0RGHO D $[ FPf +HDW )OX[ YHUVXV 7LGDO 'LVSODFHPHQW 0RGHO D 3U f +HDW )OX[ YHUVXV 7LGDO 'LVSODFHPHQW 0RGHO D 3U f ,QIOXHQFH RI :DOO 7KLFNQHVV RQ $[LDO +HDW )OX[ 0RGHO :DWHU*ODVV D $[ FPf ,QIOXHQFH RI 3LSH 'LDPHWHU RQ +HDW )OX[ IRU )L[HG )UHTXHQF\ 0RGHO :DWHUJODVV D $[ FPf 7\SLFDO ,VR7HPSHUDWXUH &RQWRXU LQ 2VFLOODWLQJ 3LSH )ORZ 0RGHO :DWHU*ODVV D $[ FPf 9DULDWLRQ RI 7HPSHUDWXUH 7A DQG 7 YHUVXV $[ 0RGHO :DWHU*ODVV D f &RPSDULVRQ RI (QKDQFHG +HDW 7UDQVIHU DQG +HDW &RQGXFWLRQ LQ 2VFLOODWLQJ 3LSH )ORZ 0RGHO :DWHU*ODVV D f 9DULDWLRQ RI $[LDO +HDW )OX[ YHUVXV :RUPHUVOH\ 1XPEHU 0RGHO +R2*ODVV +J6WHHO $[ FPf &RPSXWHG 7XQLQJ &XUYHV 0RGHO +6WHHO DQG +J6WHHO D[ FPf 7XQLQJ &XUYH YHUVXV :RUPHUVOH\ 1XPEHU DIWHU .XU]ZHJf 5DWLR RI +HDW )OX[ GXH WR &RQGXFWLRQ WR (QKDQFHG +HDW )OX[ YHUVXV :RUPHUVOH\ 1XPEHU 0RGHO +6WHHO +J6WHHO $[ FPf YLLL

PAGE 9

/,67 2) 7$%/(6 7DEOHV 3DJHV 'LPHQVLRQOHVV 7LGDO 'LVSODFHPHQW DW 'LIIHUHQW :RUPHUVOH\ 1XPEHUV 3KDVH /DJV $ORQJ 5DGLXV :RUNLQJ 0HGLXP +2 $; FP D f &RPSDULVRQ RI 3KDVH /DJV ZLWK 'LIIHUHQW :RUNLQJ 0HGLXPV $[ FP D f &RPSDULVRQ RI (QKDQFHG +HDW )OX[ 8VLQJ 1XPHULFDO 9HORFLW\ ZLWK +HDW )OX[ 8VLQJ $QDO\WLF 9HORFLW\ 0RGHO +*ODVV 5O FP 5 FP 3U D f (QKDQFHG $[LDO +HDW )OX[ YLD 7LGDO 'LVSODFHPHQW (QKDQFHG $[LDO +HDW )OX[ LQ 6WHDG\ )ORZ 7KH ,QIOXHQFH RI 3URSHUWLHV RI :DWHU RQ WKH (QKDQFHG $[LDO +HDW )OX[ 9DULDWLRQ RI WKH $[LDO 7HPSHUDWXUH *UDGLHQW YHUVXV :RUPHUVOH\ 1XPEHUV :DWHU*ODVV $[ FPf L[

PAGE 10

.(< 72 6<0%2/6 ; U W e r" 7 / 5 U Z 6 3 3 F 3 N .H 3U [ FRRUGLQDWH UDGLDO FRRUGLQDWH WLPH FRRUGLQDWH [f LQ WUDQVIRUPHG SODQH FRRUGLQDWH Uf LQ WUDQVIRUPHG SODQH WUDQVIRUPHG WLPH SLSH OHQJWK SLSH LQQHU UDGLXV SLSH RXWHU UDGLXV RVFLOODWLQJ IUHTXHQF\ ERXQGDU\ OD\HU WKLFNQHVV SUHVVXUH GHQVLW\ VSHFLILF KHDW .LQHPDWLF YLVFRVLW\ G\QDPLF YLVFRVLW\ WKHUPDO GLIIXVLYLW\ WKHUPDO FRQGXFWLYLW\ FRHIILFLHQW RI HQKDQFHG KHDW GLIIXVLYLW\ 3UDQGWO QXPEHU LS S[_ D PHDVXUH RI WKH PD[LPXP D[LDO SUHVVXUH JUDGLHQW FPVHFf [

PAGE 11

D :RUPHUVOH\ QXPEHU D -X!Y 7 7HPSHUDWXUH G7G[ WLPHDYHUDJHG D[LDO WHPSHUDWXUH JUDGLHQW U I U5L GLPHQVLRQOHVV UDGLDO GLVWDQFH J UDGLDO WHPSHUDWXUH GLVWULEXWLRQ IXQFWLRQ X YHORFLW\ X UHSUHVHQWDWLYH YHORFLW\ I YHORFLW\ VKDSH IXQFWLRQ ; /DJUDQJLDQ GLVSODFHPHQW '; GLPHQVLRQOHVV WLGDO GLVSODFHPHQW $; GLPHQVLRQDO WLGDO GLVSODFHPHQW 4WRWDO WLPH DYHUDJHG WRWDO HQKDQFHG D[LDO KHDW IORZ RYHU SLSH FURVVVHFWLRQ W! D[LDO KHDW IOX[ 6XEVFULSW I IOXLG Z ZDOO K KRW F FROG WK WKHUPDO HT HTXLYDOHQW DGM DGMDFHQW ;;

PAGE 12

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f ZDV GHYHORSHG WR LPSOHPHQW DOO WKH QXPHULFDO FDOFXODWLRQV 7R LQFUHDVH WKH XQGHUVWDQGLQJ RI WKH PHFKDQLVP RI WKHUPDO SXPSLQJ WKH SHULRGLF YHORFLW\ SURILOHV DQG /DJUDQJLDQ GLVSODFHPHQWV DV ZHOO DV WLGDO GLVSODFHPHQWV DW YDULRXV :RUPHUVOH\ QXPEHUV IURP D WR f ZHUH VWXGLHG 6RPH WUDQVLHQW SUREOHPV RI HQKDQFHG D[LDO KHDW WUDQVIHU LQ RVFLOODWLQJ SLSH IORZ VXFK DV WKH SHULRGLF ILQDO WHPSHUDWXUH EXLOGXS SURFHVV LQ RVFLOODWLQJ SLSH IORZ ZHUH DOVR H[DPLQHG 7KH WLPHGHSHQGHQW WHPSHUDWXUH GLVWULEXWLRQ [LL

PAGE 13

LQ WKH GLIIHUHQW PRGHOV ZDV QXPHULFDOO\ VWXGLHG LQ GHWDLO 7KH HQKDQFHG D[LDO KHDW IOX[ PDJQLWXGH YHUVXV GLIIHUHQW WLGDO GLVSODFHPHQWV ZLWK ZDWHU DQG PHUFXU\ DV WKH ZRUNLQJ IOXLGV ERXQGHG E\ SLSH ZDOOV RI GLIIHUHQW PDWHULDO ZHUH REVHUYHG DQG WKH TXDGUDWLF FRHIILFLHQWV IRXQG 7KH LQIOXHQFH RI WKH YDULDWLRQ RI ZDWHU SURSHUWLHV RQ WKH HQKDQFHG D[LDO KHDW IOX[ ZDV QXPHULFDOO\ H[DPLQHG DQG WKH UHVXOWV VKRZ WKDW WKH HQKDQFHG D[LDO KHDW IOX[ FDQ YDU\ DERXW SHUFHQW HYHQ ZLWKLQ WKH WHPSHUDWXUH UDQJH IURP r& WR r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

PAGE 14

&+$37(5 ,1752'8&7,21 (QKDQFHG KHDW WUDQVSRUW LQ D YLVFRXV ODPLQDU IOXLG VXEMHFWHG WR VLQXVRLGDO RVFLOODWLRQV LQ D YHU\ ORQJ SLSH ZKLFK FRQQHFWV D KRW IOXLG UHVHUYRLU DW RQH HQG DQG D FROG IOXLG UHVHUYRLU DW WKH RWKHU HQG )LJ f KDV EHHQ UHFRJQL]HG DQG VWXGLHG UHFHQWO\ E\ .XU]ZHJ > @ 7KH UHVXOWV REWDLQHG VKRZ WKDW ZLWK WKLV RVFLOODWRU\ SLSH IORZ WKH KHDW WUDQVIHUUHG D[LDOO\ IURP WKH KRW HQG WR WKH FROG HQG FDQ EH RUGHUV RI PDJQLWXGH ODUJHU WKDQ WKDW REWDLQHG E\ SXUH PROHFXODU FRQGXFWLRQ LQ WKH DEVHQFH RI RVFLOODWLRQV ,Q DGGLWLRQ WKH PRUH LPSRUWDQW WKLQJ RI LQWHUHVW LV WKDW WKLV KHDW WUDQVIHU SURFHVV LQYROYHV QR QHW FRQYHFWLYH PDVV WUDQVSRUW 0DMRU DVVXPSWLRQV PDGH LQ WKH DERYH FLWHG VWXGLHV RQ HQKDQFHG KHDW GLIIXVLRQ DUH WKDW D FRQVWDQW WLPHDYHUDJHG QRQ]HUR D[LDO WHPSHUDWXUH JUDGLHQW LV DOZD\V SUHVHQW LQ WKH RVFLOODWLQJ IORZ DQG WKDW WKH D[LDO PROHFXODU FRQGXFWLRQ DORQJ WKH ZDOO DQG LQ WKH RVFLOODWLQJ IOXLG LV QHJOLJLEOH 'LVFRYHU\ RI WKLV HQKDQFHG KHDW WUDQVSRUW SKHQRPHQRQ ZDV PDGH SRVVLEOH E\ HDUOLHU VWXGLHV RQ D[LDO GLVSHUVLRQ RI FRQWDPLQDQWV ZLWKLQ VWHDG\ ODPLQDU IORZV WKURXJK FDSLOODU\ WXEHV E\ 7D\ORU LQ >@ DQG $ULV LQ >@ 7KHVH

PAGE 15

HDUOLHU VWXGLHV VKRZ WKDW ZKHQ D VPDOO TXDQWLW\ RI D FRQWDPLQDQW LV LQWURGXFHG LQWR D FLUFXODU SLSH WKH GLVSHUVLRQ RI WKH UHVXOWDQW FRQWDPLQDQW FORXG LV JUHDWO\ HQKDQFHG E\ WKH IORZ RI WKH IOXLG %RZGHQn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

PAGE 16

HDUO\ .XU]ZHJ VXJJHVWHG WKDW D VLPLODU GLVSHUVLRQ SURFHVV VKRXOG RFFXU LQ WKH KHDW WUDQVIHU DUHD EHFDXVH RI WKH VLPLODULW\ LQ ERWK WKH JRYHUQLQJ GLIIXVLRQ DQG KHDW WUDQVIHU HTXDWLRQV >@ DQG WKH ILUVW SUHOLPLQDU\ WKHRU\ ZDV IRUPXODWHG LQ > @ LQ ZKLFK UHIHUULQJ WR &KDWZLQnV LGHD D WLPH DYHUDJHG FRQVWDQW D[LDO WHPSHUDWXUH JUDGLHQW DVVXPSWLRQ ZDV XVHG 7KH LQVWDQWDQHRXV WHPSHUDWXUH GLVWULEXWLRQ ZDV WDNHQ WR EH RI WKH IRUP > @ 7 >[ 5; JUfHOZW@ f ZKHUH 7[ LV WKH WLPHDYHUDJHG D[LDO WHPSHUDWXUH JUDGLHQW LV WKH WXEH UDGLXV [ LV WKH D[LDO GLVWDQFH DORQJ WKH FDSLOODULHV ZLWK [ / DQG [ / GHQRWLQJ WKH HQGVf / LV WKH SLSH OHQJWK XQGHU FRQVLGHUDWLRQ Z LV WKH RVFLOODWLQJ IUHTXHQF\ RI IOXLG JIf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

PAGE 17

UHVHUYRLU ZKLFK VXSSOLHV XQOLPLWHG KRW OLTXLG DW RQH HQG WR D VHFRQG UHVHUYRLU ZKLFK VXSSOLHV XQOLPLWHG FROG OLTXLG DW WKH RWKHU HQG 7KH OLTXLG LQ WKH SLSHV LV RVFLOODWLQJ D[LDOO\ ZLWK DQ DPSOLWXGH VXFK WKDW QRQH RI WKH OLTXLG ZKLFK LV RULJLQDOO\ LQ WKH PLGGOH SRUWLRQ RI SLSHV HYHU UXQV LQWR HLWKHU UHVHUYRLU 7KDW LV WKHUH LV QR QHW FRQYHFWLYH PDVV H[FKDQJH EHWZHHQ WZR UHVHUYRLUV 7KH ODUJHVW D[LDO IOXLG GLPHQVLRQOHVV GLVSODFHPHQW ZKHQ FURVVVHFWLRQ DYHUDJHGf LV UHIHUUHG WR DV WKH QRQGLPHQVLRQDO WLGDO GLVSODFHPHQW DQG LV GHQRWHG E\ $; LW VKRXOG QRW EH FRQIXVHG ZLWK WKH GLPHQVLRQDO WLGDO GLVSODFHPHQW D[ IUHTXHQWO\ XVHG LQ WKH SUHVHQW VWXG\f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n YLVFRXV ERXQGDU\ OD\HU RI WKLFNQHVV 6 -XRf f

PAGE 18

ZKHUH X LV WKH IOXLG NLQHPDWLF YLVFRVLW\ )RU URRP WHPSHUDWXUH ZDWHU DW D IUHTXHQF\ RI +] WKLV YLVFRXV ERXQGDU\ OD\HU LV DSSUR[LPDWHO\ rf FP 7KH FRUUHVSRQGLQJ WKHUPDO ERXQGDU\ OD\HU WKLFNQHVV LV DERXW ILWK WI3U f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f 8IUfHOZW f ZKHUH 84 LV D UHSUHVHQWDWLYH YHORFLW\ I U5A LV DJDLQ WKH GLPHQVLRQOHVV UDGLDO GLVWDQFH IIf WKH YHORFLW\ VKDSH IXQFWLRQ DQG X! WKH DQJXODU YHORFLW\ RI WKH RVFLOODWRU\ IORZ 7KH H[SOLFLW IRUP RI IIf LV IUf -QLmUf MA\ f

PAGE 19

ZKHUH $ 5LS[_SL8 LV WKH QRQGLPHQVLRQDO SUHVVXUH JUDGLHQW PD[LPXP DFWLQJ DORQJ WKH FDSLOODULHV D 5?-a-Y LV WKH :RUPHUVOH\ QXPEHU PHDVXULQJ WKH UDWLR RI LQHUWLD WR YLVFRXV IRUFHV Y LV WKH IOXLG NLQHPDWLF YLVFRVLW\ RI WKH FDUULHU OLJXLG DQG S LV WKH IOXLG GHQVLW\ 7KLV YHORFLW\ SURILOH ZLOO UHGXFH WR WKH IDPLOLDU 3RLVHXLOOH SDUDEROLF VKDSH DV WKH DQJXODU IUHTXHQF\ Z EHFRPHV VPDOO ZKLOH DW PRGHUDWH IUHTXHQF\ IUf KDV WKH VKDSH GHPRQVWUDWHG E\ 8FKLGD >@ $QRWKHU QHZ WHUP FRPPRQO\ XVHG ZKHQ GHDOLQJ ZLWK RVFLOODWLQJ IORZ LV WKH FURVVVWUHDP DYHUDJHG GLPHQVLRQOHVV WLGDO GLVSODFHPHQW $; ZKLFK FDQ EH PDWKHPDWLFDOO\ GHILQHG DV $; 8 -II UfGU f DQG RQ LQWHJUDWLRQ \LHOGV >@ $; D SD[ S D )Df f ZKHUH WKH FRPSOH[ IXQFWLRQ )Df KDV WKH IRUP )mf -Qn=Af -R ef ZLWK WKH SULPH GHQRWLQJ GLIIHUHQWLDWLRQ 8VLQJ WKH GHILQLWLRQ RI .HOYLQ IXQFWLRQ -4LDf EHU D L EHL D

PAGE 20

)LJ ) Df &XUYH >@

PAGE 21

>@ WKH FRPSOH[ IXQFWLRQ )Df )LJ f FDQ EH IXUWKHU ZULWWHQ DV )Df )U Df L )LDf EHU nD L EHLnD @ EHU D L EHL D f 7KLV GLPHQVLRQOHVV WLGDO GLVSODFHPHQW LV UHODWHG WR WKH PD[LPXP RI WKH SHULRGLF SUHVVXUH JUDGLHQW YLD >@ $; SD[_ D D m f D } $SSDUHQWO\ IRU DQ\ IL[HG WLGDO GLVSODFHPHQW $[ DQG RVFLOODWLQJ IUHTXHQF\ Z WKH D[LDO SUHVVXUH JUDGLHQW UHTXLUHG LV SURSRUWLRQDO WR WKH LQYHUVH VTXDUH RI WKH :RUPHUVOH\ QXPEHU ZKHQ D LV VPDOO KRZHYHU LW LV LQGHSHQGHQW RI D ZKHQ D LV YHU\ ODUJH 7KLV DOVR LPSOLHV WKDW WKH H[FLWLQJ D[LDO SUHVVXUH JUDGLHQW _DSD[_ LV DSSUR[LPDWHO\ SURSRUWLRQDO WR WKH IOXLG NLQHPDWLF YLVFRVLW\ Y DQG LQYHUVHO\ SURSRUWLRQDO WR WKH VTXDUH RI WKH SLSH UDGLXV ZKHQ :RUPHUVOH\ QXPEHU LV YHU\ VPDOO LW KDSSHQV RQO\ DW ORZ m VPDOO DQG ODUJH Y IRU H[DPSOH RLOf ZKLOH LW LV DOPRVW LQGHSHQGHQW RI WKH IOXLG NLQHPDWLF YLVFRVLW\ DQG WKH SLSH UDGLXV ZKHQ WKH :RUPHUVOH\ LV YHU\ ODUJH LW KDSSHQV RQO\ DW YHU\ ODUJH 5A KLJK Z VPDOO Y

PAGE 22

IRU LQVWDQFH D OLTXLG PHWDOf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f WKH KRWWHU IOXLG ZLWKLQ WKH SLSH FRUH ZKLFK LV LQLWLDOO\ EURXJKW LQWR WKH SLSH IURP WKH KRW UHVHUYRLU SURGXFHV D ODUJH UDGLDO KHDW IORZ YLD FRQGXFWLRQ WR WKH FRROHU SRUWLRQV RI WKH IOXLG ZLWKLQ WKH 6WRNHVn ERXQGDU\ OD\HU DQG WR WKH FRROHU SLSH ZDOO ZKLOH GXULQJ WKH QHJDWLYH RU UHYHUVHf VWURNH LH ZKHQ WKH IOXLG PRYHV WRZDUGV WKH KRWWHU VLGH WKH KLJKHU WHPSHUDWXUH LQ WKH ERXQGDU\ OD\HU DQG WKH SLSH ZDOO FRQGXFWV WKH KHDW EDFN LQWR WKH FRROHU IOXLG FRUH 7KLV FRXSOHG UDGLDO KHDW FRQGXFWLRQ ZLWK DQ D[LDO FRQYHFWLYH WUDQVSRUW OHDGV WR DQ HQKDQFHG D[LDO KHDW IOX[ DORQJ WKH HQWLUH OHQJWK RI WKH SLSHV

PAGE 23

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f IURP WKH KRW UHVHUYRLU WR WKH FROG UHVHUYRLU KHDW FDQ EH FRQWLQXDOO\ WUDQVIHUUHG DQG WKH FDUULHU LQ WKH WLPH DYHUDJHG VHQVH GRHV QRW PRYH 7KLV SURSHUW\ LV SDUWLFXODUO\ LPSRUWDQW IRU WKRVH V\VWHPV ZKHUH D ODUJH DPRXQW RI KHDW WUDQVIHU LV QHHGHG ZKLOH WKH ZRUNLQJ IOXLG LV UHTXLUHG WR UHPDLQ LQ WKH V\VWHP DV LQ QXFOHDU UHDFWRUVf 1RWH WKDW WKH D[LDO KHDW FRQGXFWLRQ LQ JHQHUDO LV DVVXPHG WR EH YHU\ VPDOO LQ WKLV WKHUPDO SXPSLQJ SURFHVV FRPSDUHG ZLWK WKH HQKDQFHG D[LDO KHDW IORZ >@ $SSDUHQWO\ WKH KHDW WUDQVSRUW UDWH LQ WKHUPDO SXPSLQJ LV JRYHUQHG E\ ERWK WKH WKHUPDO SURSHUWLHV RI WKH

PAGE 24

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f DQG RI D PHUFXU\VWHHO FRPELQDWLRQ )RU WKH IRUPHU LW LV QHFHVVDU\ WR HPSOR\ UDWKHU VPDOO GLDPHWHU WXEHV DQG ORZ IUHTXHQF\ RVFLOODWLRQV ZLWK ODUJH WLGDO GLVSODFHPHQW IRU LW KDV

PAGE 25

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

PAGE 26

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

PAGE 27

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f KDV D QRQ]HUR WKHUPDO GLIIXVLYLW\ DQG KHQFH KHDW VWRUDJH FDSDELOLW\ 1RWH WKDW WKH H[LVWLQJ FRQVLGHUDWLRQV DUH UHVWULFWHG WR ODPLQDU IORZ 7XUEXOHQW IORZ FRQGLWLRQV FDQ RFFXU LQ RVFLOODWLQJ SLSH IORZ DW KLJKHU YDOXHV RI Z$[9 > @ DQG DSSDUHQWO\ ZRXOG GHVWUR\ WKH DVVXPSWLRQV RI WKH FXUUHQW DQDO\WLF PRGHO RI WKH WKHUPDO SXPSLQJ SURFHVV )RUWXQDWHO\ WKH FRQGLWLRQ IRU RSWLPXP HQKDQFHG KHDW WUDQVIHU LQ VXFK RVFLOODWLQJ SLSH IORZ REWDLQHG DW WKH WXQLQJ SRLQW UHTXLUHV

PAGE 28

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

PAGE 29

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

PAGE 30

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f ZLWK 7KRPDV DOJRULWKP $ )RUWUDQ FRPSXWHU FRGH QDPHG (73 (QKDQFHG 7KHUPDO 3XPSLQJf ZDV GHYHORSHG WR LPSOHPHQW DOO WKH FDOFXODWLRQV

PAGE 31

&+$37(5 ,, )2508/$7,21 2) 7+( 352%/(0 $ VLQJOH WXEH ZLWK LQQHU UDGLXV 5A RXWHU UDGLXV 5 DQG OHQJWK / VXFK WKDW / !! 5A DQG 5f FRQQHFWV D ODUJH UHVHUYRLU RI KRW IOXLG DW WHPSHUDWXUH 7 7MM DW RQH HQG WR DQRWKHU ODUJH UHVHUYRLU RI WKH VDPH IOXLG DW FROG WHPSHUDWXUH 7 7F DW WKH RWKHU HQG )LJV DQG f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

PAGE 32

+27 I/8,' ,1 +277(5 )/8,' 35,0$5< /223f 287 )LJ 7KHUPDO 3XPSLQJ 'HYLFH DIWHU .XU]ZHJ 86 3DWHQW 0D\ f

PAGE 33

HIIHFWV LQVXUHV D ODPLQDU D[LV\PPHWULFDO RQHGLPHQVLRQDO WLPHGHSHQGHQW PRWLRQ ,W LV FRQYHQLHQW WR HPSOR\ F\OLQGULFDO FRRUGLQDWHV IRU WKLV SUREOHP DQG ZH GHQRWH WKH FRRUGLQDWH LQ WKH D[LDO GLUHFWLRQ RI WKH WXEH E\ [ DQG WKH UDGLDO GLUHFWLRQ E\ U 7KH D[LDO YHORFLW\ FDQ EH WDNHQ WR EH LQGHSHQGHQW RI [ DQG LQ RUGHU WR VDWLVI\ WKH UHTXLUHPHQWV RI FRQWLQXLW\ WKH RWKHU YHORFLW\ FRPSRQHQWV PXVW YDQLVK :H VKDOO IXUWKHU DVVXPH WKDW WKH SUHVVXUH JUDGLHQW LQGXFHG E\ PRYLQJ WKH SLVWRQ )LJ f LV KDUPRQLF DQG KDV WKH IRUP >@ S f f§ a ‘ $ FRV fW Wf ZKHUH $ OS_DSG[_ LV D FRQVWDQW ZKLFK PHDVXUHV WKH PD[LPXP SUHVVXUH JUDGLHQW H[LVWLQJ DORQJ WKH [D[LV &OHDUO\ WKLV HTXDWLRQ LPSOLHV WKDW ZH DUH QRZ GHDOLQJ ZLWK D WLPHGHSHQGHQW VLQXVRLGDO SUHVVXUH JUDGLHQW ZKLFK LV FRQVWDQW RYHU WKH SLSH FURVVVHFWLRQ DW DQ\ LQVWDQW DQG WKH SUHVVXUH YDULHV OLQHDUO\ DORQJ WKH [D[LV 7KH VLPSOLILHG 1DYLHU6WRNHV HTXDWLRQ IRU WKLV SUREOHP LV >@ 8  G U 8 L f§ $FRVZWf f§  U 5L f ZKHUH 8UWf LV WKH WLPH DQG UDGLDOO\ GHSHQGHQW D[LDO YHORFLW\ FRPSRQHQW

PAGE 34

7KH FRUUHVSRQGLQJ WHPSHUDWXUH 7[UWf RI WKH IOXLG ZLWKLQ WKH SLSH LV JRYHUQHG E\ WKH KHDW FRQGXFWLRQ HTXDWLRQ >@ 7 DW X 7 D[ I L D D7 U DU DU A 7 A U 5 f ZKHUH LV WKH LQQHU UDGLXV RI WKH WXEH DQG rFI WKH IOXLG WKHUPDO GLIIXVLYLW\ ZKLFK LV UHODWHG WKH WKHUPDO FRQGXFWLYLW\ NI E\ NI +HUH S LV WKH IOXLG GHQVLW\ DQG F LV WKH VSHFLILF KHDW 1RWH WKDW WKH YLVFRXV KHDWLQJ WHUP KDV EHHQ QHJOHFWHG LQ HTXDWLRQ f VLQFH LW LV YHU\ VPDOO IRU PRVW H[SHULPHQWDO FRQGLWLRQV WKLV LV MXVWLILHG SURYLGHG RQH GRHV QRW GHDO ZLWK YHU\ KLJK 3UDQGWO QXPEHU IOXLGV VXFK DV RLOV 7KH WHPSHUDWXUH LQ WKH ZDOO FDQ EH GHWHUPLQHG IURP WKH VROXWLRQ RI 7 U 7 D7 W rZ> U JU U Uf J[ 5O U 5 f ZKHUH NZ LV WKH WKHUPDO GLIIXVLYLW\ RI WKH FRQGXFWLQJ ZDOO LQ 5L U 5 DQG LV GHILQHG E\

PAGE 35

. Z NZ 3 ZFZ ZKHUH NZ SZ DQG FZ DUH WKH ZDOO FRQGXFWLYLW\ GHQVLW\ DQG WKH VSHFLILF KHDW UHVSHFWLYHO\ %\ LQWURGXFLQJ WKH IROORZLQJ QRQGLPHQVLRQDO WHUPV HTXDWLRQ f f DQG f FDQ EH WUHDWHG PRUH HDVLO\ Wr W Z U r [ [ 6 8r X $ Z ZKHUH -YX! LV DJDLQ WKH IOXLG YLVFRXV ERXQGDU\ OD\HU WKLFNQHVV 7KH GLPHQVLRQOHVV JRYHUQLQJ V\VWHP RI HTXDWLRQV FDQ WKHQ EH ZULWWHQ DV DLU DWf FRVW f U DXr DU 77 DLU DU U 5; f GL DWFX D7 D[f U D7r 3U DU 7U]a D7 D Wr DU 7W D[ AU R U 5 DQG f D W DW 77a 6 D Wr Ur D7 DU D7r D[ 5O U 5 f

PAGE 36

ZKHUH DQG 0 &I 3U 3 FZ 6 r: & $ A 7KLV QRQGLPHQVLRQDOL]DWLRQ KDV VRPH DGYDQWDJHV LQ WKH FRPSXWLQJ SURFHVV WR EH FDUULHG RXW EHORZ 7KH GLPHQVLRQOHVV YHORFLW\ DQG LWV GLVWULEXWLRQ RYHU WKH FURVV VHFWLRQ IRXQG IURP WKH PRPHQWXP HTXDWLRQ f DUH H[SHFWHG WR EH XQLYHUVDO IRU DQ\ :RUPHUVOH\ QXPEHU D 5A-XS?[ DQG DQ\ DVVRFLDWHG TXDQWLWLHV VXFK DV WKH WLGDO GLVSODFHPHQW DQG /DJUDQJLDQ GLVSODFHPHQW ,WV ILQDO SHULRGLF IRUP LV RI WKH IRUP JLYHQ E\ f 7KH GLPHQVLRQOHVV WHPSHUDWXUH LQ WKH SLSH LV RQO\ UHODWHG WR WKH 3UDQWGO QXPEHU DQG WKH GLPHQVLRQOHVV YHORFLW\ ZKLOH WKDW LQ WKH ZDOO LV UHODWHG WR WKH UDWLR RI ZDOO KHDW GLIIXVLYLW\ WR WKH NLQHPDWLF YLVFRVLW\ DV VHHQ IURP HTXDWLRQ f DQG f 7KH JRYHUQLQJ GLPHQVLRQOHVV HTXDWLRQV f f DQG f DUH D VHW RI VHFRQGRUGHU SDUDEROLF W\SH RI SDUWLDO GLIIHUHQWLDO HTXDWLRQV H[SUHVVHG LQ F\OLQGULFDO FRRUGLQDWHV 7R VROYH WKLV VHW RI VLPXOWDQHRXV HTXDWLRQV D FRUUHVSRQGLQJ VHW RI ERXQGDU\ FRQGLWLRQV DQG LQLWLDO FRQGLWLRQV DUH UHTXLUHG 6LQFH WKH YHORFLW\ LV DVVXPHG WR EH D IXQFWLRQ RI U DQG W RQO\ MXVW WZR ERXQGDU\ FRQGLWLRQV

PAGE 37

DUH QHHGHG IRU VROYLQJ WKH PRPHQWXP HTXDWLRQ ZKLOH IRU WKH WHPSHUDWXUH 7[UWf WKH KHDW FRQGXFWLRQ HTXDWLRQV UHTXLUH WZR ERXQGDU\ FRQGLWLRQV LQ ERWK WKH U DQG [ GLUHFWLRQV DV ZHOO DV FRPSDWLELOLW\ FRQGLWLRQV DORQJ WKH LQWHUIDFH EHWZHHQ WKH IOXLG DQG WKH VROLG ZDOO ,W VKRXOG EH SRLQWHG RXW WKDW WKH LQLWLDO FRQGLWLRQV DUH OHVV LPSRUWDQW WKDQ WKH ERXQGDU\ FRQGLWLRQV LI RQH VHHNV RQO\ D ILQDO SHULRGLF VWDWH %RXQGDU\ &RQGLWLRQV 7KH ERXQGDU\ FRQGLWLRQV IRU WKH YHORFLW\ DUH WKH XVXDO RQHV IRU YLVFRXV IORZ QDPHO\ WKDW WKH YHORFLW\ LV ]HUR DW WKH LQQHU VXUIDFH RI WKH ZDOO Ur 5Arf $OVR E\ V\PPHWU\ WKH QRUPDO GHULYDWLYH RI YHORFLW\ DW WKH D[LV LV ]HUR 7KDW LV 8r5r Wrf f DQG 8r Wrf 7KH ERXQGDU\ FRQGLWLRQV IRU WKH WHPSHUDWXUH GHSHQG RQ WKH SDUWLFXODU PRGHO LQYHVWLJDWHG 0RGHO ,Q WKLV PRGHO LW LV DVVXPHG WKDW WKH WHPSHUDWXUH DW HDFK HQG RI WKH SLSH LV HTXDO WR WKDW RI WKH FRQQHFWLQJ IOXLG ILOOLQJ WKH UHVHUYRLUV ZKHQ WKH IOXLG PRYHV LQWR WKH FDSLOODU\ WXEH LH GXULQJ RQH KDOI RI HDFK F\FOHf 7KH

PAGE 38

)LJ 0RGHO / FP / ,168/$7,1* :$//f )L[HG (QG 7HPSHUDWXUH 0RGHO 5A FP 5 FP

PAGE 39

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f 7r Ur Wrf ZKHQ IOXLG OHDYHV SLSHf f ZKHQ IOXLG HQWHUV SLSHf ZKHQ IOXLG OHDYHV SLSHf f ZKHUH 7rF 7rK DUH WKH QRQGLPHQVLRQDO WHPSHUDWXUHV RI FROG DQG KRW UHVHUYRLUV UHVSHFWLYHO\ :KLOH 7rDFMM LV WKH

PAGE 40

WHPSHUDWXUH RI WKH IOXLG HOHPHQWV ZKLFK DUH DGMDFHQW WR WKH FRUUHVSRQGLQJ HQGV DW D SDUWLFXODU LQVWDQW 7KH WKHUPDO ERXQGDU\ FRQGLWLRQV DORQJ WKH RXWHU VXUIDFH Ur 5 RI WKH SLSH ZDOO DUH WDNHQ WR VDWLVI\ WKH LQVXODWLQJ ZDOO FRQGLWLRQ DQG DORQJ WKH D[LV RI WKH WXEH WKH WHPSHUDWXUH LV DVVXPHG WR PHHW WKH V\PPHWULF ERXQGDU\ UHTXLUHPHQW LH WKH UDGLDO GHULYDWLYHV RI WHPSHUDWXUH DORQJ D[LV DUH HTXDO WR ]HUR :H WKXV KDYH G 7 G U Ur 5r f DQG G 7 GU Ur f 7KH FRPSDWLELOLW\ FRQGLWLRQV DORQJ WKH LQWHUIDFH EHWZHHQ WKH IOXLG DQG WKH VROLG ZDOO DUH WKDW WKH UDGLDO KHDW IOX[ DQG WHPSHUDWXUH DUH FRQVWDQW DFURVV WKH LQWHUIDFH 7KDW LV 7 NI GU G 7 IOXLG NZ GUr ZDOO DW U 5@B f DQG UML A LS A IOXLG ZDOO DW U 5S f 6LQFH QXPHULFDOO\ WKH VDPH QRGHV DUH FKRVHQ DORQJ WKH LQWHUIDFH FRQGLWLRQ f ZLOO EH DXWRPDWLFDOO\ VDWLVILHG

PAGE 41

0RGHO +HUH LW LV DVVXPHG WKDW D KHDW VRXUFH ULP RI ZLGWK E LV GLUHFWO\ PRXQWHG RQ WKH LQWHUIDFH EHWZHHQ WKH VROLG ZDOO DQG WKH IOXLG DW [ / ZKLOH WZR VPDOO FROG ULP VRXUFHV RI ZLGWK E HDFK DUH PRXQWHG DW [ DQG [ / VHH )LJ f 7KH ZDOO WKLFNQHVV LV DVVXPHG WR EH ]HUR 7KLV PRGHO LV LQWHQGHG WR VLPXODWH WKH HQKDQFHG KHDW WUDQVIHU SURFHVV RI RVFLOODWRU\ IOXLG LQ DQ LQILQLWHO\ ORQJ SLSH ZKLFK LV KHDWHG DQG FRROHG E\ WKH DOWHUQDWLYH HYHQO\ GLVWULEXWHG KHDW DQG FROG VRXUFHV DORQJ WKH RXWHU VXUIDFH RI D YHU\ WKLQ ZDOO ZKLFK SRVVHVVHV LQILQLWH KHDW FRQGXFWLYLW\ $SSDUHQWO\ WKLV JHRPHWU\ FDQ EH ZHOO VLPXODWHG E\ WKH SUHVHQW PRGHO ZLWK SHULRGLF ERXQGDU\ FRQGLWLRQV 0DWKHPDWLFDOO\ WKH ERXQGDU\ FRQGLWLRQV DUH 7r[r 5Or Wrf 7rK 7r [r 5Or Wrf 7rF RU G 7 GU F RU ; r [r ; r f ;r ;,r f ;r ;r /r ;,r ;r ;r f ; r ;r ; r ;r DUH QRQGLPHQVLRQDO FRRUGLQDWHV RI SRLQWV ZKLFK FRUUHVSRQG WR ; E /E /E /E DQG / UHVSHFWLYHO\ LQ )LJ 7KH SHULRGLF HQG ERXQGDU\ FRQGLWLRQV DUH JLYHQ DV

PAGE 42

WR YR 0RGHO f§ 3HULRGLF +HDW $QG &ROG 6RXUFHV RQ ,QVXODWHG :DOO ([FHSW 6RXUFHV $UHDf / FP 5L FP E OFP )LJ

PAGE 43

7r Ur Wrf 7r/r Ur Wrf f DQG 7r L G7r L A /r ;r [r /r f 0RGHO ,Q PRGHO WKH ERXQGDU\ FRQGLWLRQV IRU WKH KHDW HTXDWLRQ DW ; DQG / DUH VRPHZKDW DUWLILFLDO 7KH WHPSHUDWXUH GLVWULEXWLRQ LQ WKH YLFLQLW\ RI SLSH HQGV PD\ QRW SUHFLVHO\ PDWFK WKH FRQVWDQW WHPSHUDWXUH HQG FRQGLWLRQV 7KLV LV SDUWLFXODUO\ WUXH LQ WKH ODUJH WLGDO GLVSODFHPHQW DQGRU YHU\ KLJK RVFLOODWLQJ IUHTXHQF\ FDVHV VLQFH D WHPSHUDWXUH GURS RFFXUV DW WKH KRW HQG DQG D ULVH DW WKH FROG HQG DV IOXLG SDUWLFOHV H[LW WKH SLSH 7KLV OHDGV WR D GLVFRQWLQXLW\ RI WHPSHUDWXUH ZKLFK ZLOO OHDG WR XQH[SHFWHG GLVSHUVLRQ HUURUV LI DQ HYHQ RUGHU QXPHULFDO PHWKRG LV XVHG > @ 7R LPSURYH WKH HQG ERXQGDU\ FRQGLWLRQV SUHVHQWHG LQ PRGHO WKH IROORZLQJ H[WHQGHG PRGHO KDV EHHQ FRQVLGHUHG ,W LV DVVXPHG WKDW DQ H[WHQVLRQ SLSH ZKLFK LV VRPH WLPHV WKH OHQJWK RI WKH FHQWUDO SLSH LV DWWDFKHG WR WKH RULJLQDO SLSH XVHG LQ PRGHO 7KH KHDW DQG FROG VRXUFH DUH DVVXPHG WR EH ORFDWHG RQ WKH RXWHU VXUIDFH RI WKH ZDOO LQ WKH H[WHQGHG VHFWLRQV RI WKH SLSH DV ZHOO DV DW ERWK HQGV RI WKH WXEH DV VKRZQ LQ )LJ 7KLV PRGHO LV XVHG WR

PAGE 44

0RGHO f§ 3LSH ZLWK &RQGXFWLQJ 3LHFHV 0RGHO / FP 5L FP )LJ

PAGE 45

LQYHVWLJDWH WKH VLWXDWLRQ ZKHUH WKH KHDW FRQWDLQHG LQ WKH MHW LV H[FKDQJHG E\ SXUH KHDW FRQGXFWLRQ ZLWK WKH VXUURXQGLQJ IOXLG HOHPHQWV LQ HDFK UHVHUYRLU ZLWKRXW DQ\ FRQYHFWLYH PL[LQJ ,Q WKLV FDVH WKH ERXQGDU\ FRQGLWLRQV DORQJ WKH RXWHU VXUIDFH RI WKH ZDOO DUH 7r 7rK [r ;,r f 7r 7rF ;r ;r ;r f DQG G 7r ;,r [r ;r f ZKHUH ;,r ;r DQG ;r DUH QRQGLPHQVLRQDO FRRUGLQDWHV RI SRLQWV ZKLFK FRUUHVSRQG WR ; / / DQG / UHVSHFWLYHO\ LQ )LJ $W ERWK HQGV ZH KDYH 7r Ur Wrf 7rK f DQG 7r/r Ur Wrf 7rF f $V LQ WKH RWKHU PRGHOV WKH V\PPHWU\ FRQGLWLRQ DORQJ WKH D[LV RI SLSH UHJXLUHV WKDW G 7r GUr f

PAGE 46

DQG WKH UDGLDO IOX[ DQG WHPSHUDWXUH FRQWLQXLW\ FRQGLWLRQ DORQJ WKH LQWHUIDFH DUH G 7r NI GUr IOXLG G7r NZ ZDOO DQG f LS r IOXLG ZDOO f +HUH DJDLQ NI DQG NZ DUH WKH VDPH DV GHVFULEHG LQ WKH SUHYLRXV PRGHOV ,QLWLDO &RQGLWLRQV 7KH LQLWLDO FRQGLWLRQ FKRVHQ GHSHQGV RQ WKH SUREOHP XQGHU FRQVLGHUDWLRQ ,I RQHnV JRDO LV WR LQYHVWLJDWH WKH SHULRGLF TXDVLVWHDG\ VROXWLRQ RQO\ WKH LQLWLDO FRQGLWLRQV FKRVHQ VKRXOG EH DV FORVH WR WKH TXDVLVWHDG\ VWDWH DV SRVVLEOH VR WKDW D UDSLG FRQYHUJHQFH UDWH DW ORZ &38 WLPH FRVW LV DFKLHYDEOH ,I RQH LQWHQGV WR VWXG\ WKH WUDQVLHQW SURFHVV WKHQ YDULRXV LQLWLDO FRQGLWLRQV VKRXOG EH VXSSOLHG DFFRUGLQJ WR SXUSRVH RI WKH LQYHVWLJDWLQJ FDVHV VHOHFWHG )RU WKH YHORFLW\ LQLWLDO FRQGLWLRQ ZH FKRRVH IRU DOO RXU VWXGLHV 8r[r Ur f f )RU WKH WHPSHUDWXUH LI WKH SXUSRVH RI LQYHVWLJDWLRQ LV WR H[DPLQH WKH EXLOGXS SURFHVV RI WKH SHULRGLF TXDVLVWHDG\

PAGE 47

VWDWH LQ WKH WKHUPDO ILHOG WKH LQLWLDO FRQGLWLRQ VKRXOG EH WDNHQ DV 7r[r Ur f f +RZHYHU IRU WKH RWKHU FDVHV LQ RUGHU WR JDLQ IDVWHU FRQYHUJHQFH WKH LQLWLDO WHPSHUDWXUH FDQ EH DVVXPHG WR KDYH D OLQHDU GLVWULEXWLRQ LQ WKH D[LDO GLUHFWLRQ DQG WKXV KDYH WKH IRUP 7r[r Ur f 7rK 7rF 7rKf[[r //r f ZKHUH //r LV WKH GLPHQVLRQOHVV OHQJWK ZLWKLQ ZKLFK WKH D[LDO OLQHDU WHPSHUDWXUH GLVWULEXWLRQ LV DVVXPHG WR KROG DQG [[r LV WKH GLPHQVLRQOHVV GLVWDQFH PHDVXUHG IURP WKH RULJLQ ZKLFK LV FKRVHQ RQO\ IRU WKH SXUSRVH RI WKLV OLQHDU WHPSHUDWXUH LQLWLDOL]DWLRQ %RWK WKH //r DQG WKH RULJLQ VHOHFWHG GHSHQG RQ WKH PRGHO FRQVLGHUHG )RU 0RGHO DQG 0RGHO //r LV HTXDO WR /r DQG WKH RULJLQ LV WDNHQ DW WKH OHIW HQG 0RGHO f RU DW WKH OHIW LQWHUVHFWLRQ EHWZHHQ WKH FHQWUDO SLSH DQG WKH OHIW H[WHQVLRQ SLSH 0RGHO f +RZHYHU IRU 0RGHO HTXDWLRQ f LV YDOLG RQO\ IRU WKH ULJKW KDOI SRUWLRQ RI WKH SLSH DV //r LV WDNHQ WR EH /r DQG WKH RULJLQ LV FKRVHQ DW WKH PLGGOH VHFWLRQ RI WKH SLSH 7KH LQLWLDO WHPSHUDWXUH RI WKH OHIW KDOI SLSH FDQ WKHQ EH IRXQG IURP WKH SODQH V\PPHWULF FRQGLWLRQ DERXW WKH RULJLQ FURVVn VHFWLRQ

PAGE 48

&DOFXODWLRQ RI 7LGDO 'LVSODFHPHQW $Q LPSRUWDQW TXDQWLW\ HQFRXQWHUHG LQ WKH VWXG\ RI WKH HQKDQFHG KHDW WUDQVIHU SURFHVV LQ RVFLOODWRU\ SLSH IORZ LV WKH WLGDO GLVSODFHPHQW ZKLFK LV XVXDOO\ UHTXLUHG WR EH VPDOOHU WKDQ RQH KDOI RI WKH WRWDO SLSH OHQJWK LQ RUGHU WR DYRLG DQ\ FRQYHFWLYH PDVV H[FKDQJH RFFXUULQJ EHWZHHQ WKH WZR UHVHUYRLUV ,W KDV DOUHDG\ EHHQ GHILQHG LQ WKH LQWURGXFWLRQ FKDSWHU f DQG LQ WKH SUHVHQW FRPSXWDWLRQ WKH GLPHQVLRQDO WLGDO GLVSODFHPHQW WDNHV WKH IRUP f ZKHUH LV WKH LQQHU UDGLXV RI WKH SLSH DQG [UUZf LV WKH /DJUDQJLDQ GLVSODFHPHQW RI WKH IOXLG HOHPHQWV ORFDWHG DORQJ D UDGLXV ZLWKLQ WKH FDSLOODU\ WXEH DW W ZZ ,W LV DVVXPHG WKHVH HOHPHQWV DUH LQLWLDOO\ OLQHG XS DW D[LDO SRVLWLRQ [ / 0RGHO DQG 0RGHO f RU [ / 0RGHO f KDOI ZD\ EHWZHHQ WKH WXEH HQGV 1XPHULFDOO\ WKH GLPHQVLRQDO /DJUDQJLDQ GLVSODFHPHQW DW WLPH W LV FRPSXWHG YLD WKH HTXDWLRQ 8UWfGW f ,W LV REYLRXV WKDW WKH GLPHQVLRQDO /DJUDQJLDQ GLVSODFHPHQW [UWf LV D IXQFWLRQ RI ERWK WLPH W DQG WKH UDGLDO SRVLWLRQ U 1RWH WKDW VLQFH WKH H[LVWHQFH RI SKDVH ODJV EHWZHHQ WKH

PAGE 49

VWLPXODWLQJ SUHVVXUH JUDGLHQW DQG WKH GLVSODFHPHQWV YDU\ IRU GLIIHUHQW :RUPHUVOH\ QXPEHUV LQ DFWXDO FDOFXODWLRQV WKH WLGDO GLVSODFHPHQW LV HTXDO WR WKH VXP RI WKH DEVROXWH PD[LPXP FURVVVHFWLRQ DYHUDJHG /DJUDQJLDQ GLVSODFHPHQW DQG WKH DEVROXWH PLQLPXP FURVVVHFWLRQ DYHUDJHG /DJUDQJLDQ GLVSODFHPHQW ZLWKLQ D SHULRG :LWK WKH VDPH QRQ GLPHQVLRQDOL]DWRQ SURFHGXUH XVHG LQ WKH SUHYLRXV VHFWLRQ IRU WKH JRYHUQLQJ HTXDWLRQV WKH GLPHQVLRQOHVV /DJUDQJLDQ DQG WLGDO GLVSODFHPHQW FDQ EH ZULWWHQ DV 5r f 7KH QRQGLPHQVLRQDO /DJUDQJLDQ GLVSODFHPHQW ZLWKLQ WKH SHULRG FDQ WKHQ EH FRPSXWHG E\ r r ? [ U W f f ZKHUH r [ $XAf DQG $[ $; r $Ff ,W LV SRLQWHG RXW WKDW WKLV GLPHQVLRQOHVV WLGDO GLVSODFHPHQW $[r GLIIHUV IURP $; GHILQHG E\ HTXDWLRQ f E\ D FRQVWDQW

PAGE 50

(IIHFWLYH +HDW )OX[ 7KH WLPH DYHUDJHG WRWDO HIIHFWLYH D[LDO KHDW IORZ ZLWKLQ WKH SLSH KDV WKH IRUP f ZKHUH F LV WKH VSHFLILF KHDW S LV WKH GHQVLW\ DQG DJDLQ Z LV WKH RVFLOODWRU\ IUHTXHQF\ 7KH SHU XQLW DUHD HIIHFWLYH KHDW IORZ WHUPHG KHDW IOX[f FDQ WKHQ EH ZULWWHQ DV (TXDWLRQ f IROORZV IURP WKH IDFW WKDW SF87 LV WKH FRQYHFWLYH KHDW IOX[ 7KH GLPHQVLRQOHVV WRWDO HIIHFWLYH KHDW IORZ FDQ EH ZULWWHQ DV f DQG WKH GLPHQVLRQOHVV KHDW IOX[ E\ r 4r Y WRWDO f

PAGE 51

&+$37(5 ,,, 180(5,&$/ 7(&+1,48(6 (03/2<(' (TXDWLRQV f f DQG f ZKLFK ZHUH GHULYHG LQ &KDSWHU ,, FDQ QRW JHQHUDOO\ EH VROYHG DQDO\WLFDOO\ LQ DQ\ VLPSOH PDQQHU 7KHUHIRUH LW LV QHFHVVDU\ WR VHHN D QXPHULFDO VROXWLRQ DSSURDFK WR WKH SUREOHP $V LV ZHOO NQRZQ LQ RUGHU WR QXPHULFDOO\ VROYH D VHW RI VLPXOWDQHRXV JRYHUQLQJ GLIIHUHQWLDO HTXDWLRQV PRUH DFFXUDWHO\ DQG HIILFLHQWO\ DQ RSWLPL]HG JULG V\VWHP SOD\V D YHU\ LPSRUWDQW UROH )RU WKH SUHVHQW SXUSRVH WKH JULG QHWn ZRUNV ZHUH JHQHUDWHG LQ WKH IROORZLQJ ZD\ ZLWKLQ WKH WXEH D QRQXQLIRUP JULG VHH )LJ f LV FOXVWHUHG DORQJ WKH UDGLDO GLUHFWLRQ LQ WKH YLFLQLW\ RI WKH LQQHU VXUIDFH RI WKH SLSH ZKHUH WKH ODUJHU JUDGLHQWV RI YHORFLW\ DQG WHPSHUDWXUH DUH H[SHFWHG WR EH SUHVHQW ZKLOH DORQJ WKH [D[LV WKH JULG LV GLVWULEXWHG XQLIRUPO\ H[FHSW IRU WKRVH FDOFXODWLRQV GHDOLQJ ZLWK PRGHO 7KH VROXWLRQ SURFHVV ZDV FDUULHG RXW LQ WKH FRPSXWDWLRQDO SODQH UDWKHU WKDQ GLUHFWO\ LQ WKH SK\VLFDO SODQH 7KXV D WUDQVIRUPDWLRQ ZKLFK FRQYHUWV WKH JRYHUQLQJ HTXDWLRQV DV ZHOO DV WKH ERXQGDU\ FRQGLWLRQ SOD\V DQ HVVHQWLDO UROH

PAGE 52

f )RU 0RGHO f IRU 0RGHO )LJ *ULG 6\VWHPV 8VHG LQ WKH 1XPHULFDO 6LPXODWLRQV

PAGE 53

7KH VHFRQGRUGHU LPSOLFLW XQFRQGLWLRQDOO\ VWDEOH &UDQN1LFROVRQ VFKHPH DQG WKH $', PHWKRG > @ DUH HPSOR\HG WR EUHDN XS WKH WUDQVIRUPHG HTXDWLRQV LQWR ILQLWH GLIIHUHQFH IRUP $ FRPSXWHU FRGH GHVLJQDWHG (73 IRU (QKDQFHG 7KHUPDO 3XPSLQJf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

PAGE 54

EH VPRRWK DQG KDYH VPDOO VNHZQHVV LQ WKH SK\VLFDO GRPDLQ WKLUG JULG QRGH SRLQW VSDFLQJ VKRXOG EH VPDOO ZKHUH ODUJH QXPHULFDO HUURUV DUH H[SHFWHG LH ODUJH VROXWLRQ JUDGLHQW UHJLRQVf LQ WKH SK\VLFDO GRPDLQ )LJ f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r [r"f Ur UrUf f Wr WrUf 7KH LQYHUVH WUDQVIRUPDWLRQ FDQ EH IRXQG DV ‹ ‹ [rf Q U Urf f U U Wrf ZKHUH [r Ur DUH WKH GLPHQVLRQOHVV FRRUGLQDWHV DQG Wr WKH GLPHQVLRQOHVV WLPH LQ WKH SK\VLFDO GRPDLQ e T DUH WKH WUDQVIRUPHG FRRUGLQDWHV DQG U LV WKH WUDQVIRUPHG WLPH LQ WKH

PAGE 55

Z )LJ &RRUGLQDWHV 7UDQVIRUPDWLRQ

PAGE 56

FRPSXWDWLRQDO GRPDLQ :LWK WKLV WUDQVIRUPDWLRQ WKH WUDQVIRUPHG JRYHUQLQJ HTXDWLRQV KDYH WKH IRUP DX DXr DWU DU I7f DAD9 K^Yf GU f D7 D7 D7r D7f D U 3n DV 3BD"B 3 GU D7r 3Af f DQG 7f D U D7f D7r D7f ZLAU Za D Wr ZAa f ZKHUH IUf D}"f EU"f DQG 3, 3 3 3 DWr f§FRV D U Wrf DWr L DU DUrDU"f DWrU L D U GU > UrDUrGUf DUrDU"fDW" DWr U FXr D[r A D U > G[rGe f 3UD[rDAfM DV DWr L U 3U D[rDef DWr U DUr A U 3U >Ur GUrGUf DU rGUfGUn DWr GU 3U DUrDU"f n, f f f f f f f

PAGE 57

: : : : DWr L L G W 3UZ D[rD2DL DWr O L D U 3UZ D[rDR DWr L  D U 3UZ >Ur GUrGUf GUrGUf DWr D U 3UZ DU rGUfn GUr! GU f f f f $V PHQWLRQHG LQ FKDSWHU ,, HTXDWLRQV f f DQG f DUH D VHW RI VHFRQGRUGHU SDUDEROLF SDUWLDO GLIIHUHQFH HTXDWLRQV LQ F\OLQGULFDO FRRUGLQDWHV ,Q DGGLWLRQ VLQFH WKH RVFLOODWLQJ IORZ LV FRQVLGHUHG DV LQFRPSUHVVLEOH LQ WKH SUHVHQW VWXGLHV WKH PRPHQWXP HTXDWLRQ f FDQ WKHQ EH LQGHSHQGHQWO\ VROYHG DW HDFK WLPH VWHS $V D UHVXOW WKH WLPHGHSHQGHQW XSGDWH YHORFLW\ IRXQG FDQ WKHQ EH VXEVWLWXWHG LQWR WKH KHDW FRQGXFWLRQ HTXDWLRQ f DV D NQRZQ TXDQWLW\ DW WKH VDPH WLPH VWHS OHYHO (YHQWXDOO\ WKH FRXSOHG KHDW FRQGXFWLRQ HTXDWLRQV f DQG f DUH VROYHG VLPXOWDQHRXVO\ WR REWDLQ WKH WHPSHUDWXUH GLVWULEXWLRQ ERWK LQ WKH IOXLG DQG LQ WKH ZDOO DW DQ\ WLPH 7R EHVW VROYH WKLV VHW RI HTXDWLRQV LQ WHUPV RI DFFXUDF\ DQG HIILFLHQF\ WKH SURSHU FKRLFH RI QXPHULFDO WHFKQLTXH DQG JULG QHWZRUN LV GLFWDWHG E\ DQ XQGHUVWDQGLQJ RI WKH SK\VLFDO DVSHFWV RI WKH SUREOHP

PAGE 58

7KH VDPH WUDQVIRUPDWLRQ VKRXOG EH DOVR DSSOLHG RQ DOO WKH ERXQGDU\ FRQGLWLRQV SURSRVHG LQ WKH WKUHH GLIIHUHQW PRGHOV )RU WKH WHPSHUDWXUH FRPSDWLELOLW\ FRQGLWLRQV DORQJ WKH LQWHUIDFH EHWZHHQ WKH IOXLG DQG WKH ZDOO f RQH KDV WKH IROORZLQJ WUDQVIRUPHG IRUPV G7r GU@ GUrGUcf NI IOXLG G7r GU GUrGL@f NZ ZDOO f RU G7 GU@ IORZ G7G Uf ZDOO ZKHUH NZ GUrGUf _SLSH .D rf§cf§ NI U !+ZDOO f f 7R PDNH WKH VXEVHTXHQW IRUP RI WKH FRUUHVSRQGLQJ ILQLWH GLIIHUHQFH JRYHUQLQJ HTXDWLRQV OHVV FXPEHUVRPH WKH VXSHUVFULSW r ZLOO EH GURSSHG IURP WKH YDULDEOHV 7r Ur Wr DQG [r DQG LQ DGGLWLRQ 8r ZLOO EH UHSODFHG E\ 9 ,Q WKH SURFHVV RI GHULYLQJ WKH ILQLWH GLIIHUHQFH JRYHUQLQJ V\VWHP HTXDWLRQV WKH VHFRQGRUGHU FHQWUDO GLIIHUHQFHV LQ WKH GRPDLQ DQG IRUZDUG RU EDFNZDUG GLIIHUHQFHV DORQJ WKH ERXQGDULHV RU LQWHUIDFH RI WKH IOXLG DQG ZDOO KDYH EHHQ HPSOR\HG DW HDFK QRGDO SRLQW

PAGE 59

&UDQN1LFROVRQ 0HWKRG IRU 0RPHQWXP (TXDWLRQ 7KH VHFRQGRUGHU DFFXUDWH &UDQN1LFROVRQ PHWKRG LV TXLWH ZHOO NQRZQ DQG ZLGHO\ XVHG ,W LV DQ XQFRQGLWLRQDO VWDEOH LPSOLFLW VFKHPH IRU VROYLQJ WKH SDUDEROLF W\SHV RI SDUWLDO GLIIHUHQFH HTXDWLRQV :KHQ WKH &UDQN1LFROVRQ PHWKRG LV DSSOLHG WR HTXDWLRQ f WKH ILQLWH GLIIHUHQFH DOJRULWKP DW D W\SLFDO QRGH N LQ WKH UDGLDO GLUHFWLRQ DQG DW WKH WLPH VWHS Q DVVXPHV WKH VLPSOHU IRUP %NYNL 'NYN $NYN-Q FN N O NP f ZKHUH +IQO IQf %NYNL (B9B $B9B MQ f -N N N N $N D E f f A D E f f 'N DN f (N DN f

PAGE 60

7KH ERXQGDU\ FRQGLWLRQV DORQJ WKH LQWHUIDFH LH WKH LQQHU VXUIDFH RI WKH ZDOO N NPLGf DQG WKH D[LV N f WKHQ EHFRPH 9 NPLG G9 GUf f 7KH LQLWLDO FRQGLWLRQV RI WKH YHORFLW\ DW DOO QRGDO SRLQWV LV WDNHQ WR EH ]HUR Q N NPLG f (TXDWLRQ f DVVRFLDWHG ZLWK WKH ERXQGDU\ FRQGLWLRQV f FDQ WKHQ EH ZULWWHQ LQ WKH PDWUL[ IRUP 7KLV \LHOGV D VHW RI OLQHDU V\VWHP DOJHEUDLF HTXDWLRQV ZKLFK FDQ EH VROYHG LQ WHUPV RI WKH QRGDO YDOXHV RI YHORFLW\ LQ WKH FDSLOODU\ WXEH E\ XVLQJ HLWKHU DQ LWHUDWLYH PHWKRG RU D 7KRPDV DOJRULWKP DW HDFK WLPH VWHS 7KH H[SOLFLW IRUP LV

PAGE 61

r Q n n Q $ 9 & 'f $f &f %f 'f $B 9 & %NGO'NGO$NGO YNGL 9NG %NG 'NG 9: 9.G f ZKHUH % NG NPLG $', 0HWKRG IRU $[LVYPPHWULF +HDW (TXDWLRQV 7KH JRYHUQLQJ 3'(V f f DQG f DUH DOO RI WKH VHFRQGRUGHU SDUDEROLF W\SH 7KXV LW PLJKW EH VXJJHVWHG WKDW WKH &UDQN1LFROVRQ VFKHPH XVHG LQ VROYLQJ WKH PRPHQWXP HTXDWLRQ f FDQ DOVR EH DSSOLHG WR WKH D[LV\PPHWULF KHDW HTXDWLRQV f DQG f DQG RQH FDQ WKHQ WDNH DGYDQWDJH RI WKH WULGLDJRQDO PDWUL[ IRUP ZKLOH XVLQJ WKLV XQFRQGLWLRQDOO\ VWDEOH WHFKQLTXH +RZHYHU ZKHQ DWWHPSWLQJ WR XVH VXFK D IRUPXODWLRQ RQH LPPHGLDWHO\ ILQGV WKDW WKH UHVXOWLQJ V\VWHP RI OLQHDU DOJHEUDLF HTXDWLRQV LV QR ORQJHU RI WKH WULGLDJRQDO W\SH f EXW UDWKHU D QRQWULGLDJRQDO PDWUL[ V\VWHP UHTXLULQJ VXEVWDQWLDO &38 WLPH WR VROYH 7KLV GLIILFXOW\ FDQ EH DYRLGHG E\ DSSO\LQJ WKH XQFRQGLWLRQDOO\ VWDEOH $OWHUQDWLQJ 'LUHFWLRQ ,PSOLFLW PHWKRG $',f

PAGE 62

GHYHORSHG E\ 3HDFHPDQ DQG 5DFKIRUG DQG 'RXJODV LQ $FFRUGLQJ WR WKLV VFKHPH WKH HQWLUH VROXWLRQ SURFHVV DW HDFK WLPH VWHS LV VSOLW LQWR WZR SRUWLRQV LH WKH ILUVW KDOI RI VROXWLRQ SURFHVVHV IRU NAr FROXPQ UDGLDO GLUHFWLRQf ZKLOH WKH RWKHU KDOI SURFHVVHV IRU WKH URZ D[LDO GLUHFWLRQf :LWK WKH $', VFKHPH VHFRQGRUGHU FHQWUDO GLIIHUHQFHV DUH XVHG WR DSSUR[LPDWH WKH YDOXHV RI GHULYDWLYHV DW HDFK QRGDO SRLQW LQ HTXDWLRQV f f 7KH ILQLWH GLIIHUHQFH DOJRULWKP IRU WKRVH HTXDWLRQV GXULQJ WKH ILUVW KDOI RI HDFK WLPH VWHS IRU WKH MWK FROXPQ DUH WKHQ %3MMN7MMNO '3MMN7MMN $3 7 MMN MMNO Q 3;Q ,'r N NPLG f DQG %: 7 NO ': MMN7MMN $: 7 N 'NO Q :; Q MMN N NPLGNPD[ f ZKHUH WKH VXEVFULSW MM LV XVHG WR HPSKDVL]H WKH VSHFLILF FROXPQ FXUUHQWO\ WR EH FRPSXWHG ,W FDQ EH VHHQ WKDW WKH VHW RI GLIIHUHQFH HTXDWLRQV QRZ LV LQ WKH WULGLDJRQDO IRUP

PAGE 63

VLQFH WKH ULJKWKDQGVLGH WHUPV LQ WKH HTXDWLRQV f f FRQWDLQ RQO\ NQRZQ YDOXHV IURP WKH SUHYLRXV UHVXOWV DQG WKH ERXQGDU\ FRQGLWLRQV 7KHVH YDOXHV FDQ EH FRPSXWHG E\ 3;MMN a&3MMN7MMON (3MMN7MMN )3MMN7MMONf f DQG ‘ &: 7 (: 7 ): OON ''ON ''N '`N 7 Q f 7KH FRPSXWDWLRQDO DOJRULWKP LV LPSOHPHQWHG FROXPQ E\ FROXPQ DQG WKH XQNQRZQ YDOXH 7MAf FDQ WKHQ EH VROYHG E\ HLWKHU DQ LWHUDWLYH RU D GLUHFW PHWKRG ,Q RUGHU WR GR WKLV HTXDWLRQ f QHHGV WR EH DVVHPEOHG LQWR WKH IROORZLQJ PDWUL[ IRUP U n 3'; 3$ r Q n 7 Q n n 3; 36 SG SD 7 3; SE SG SD 7 3; 3%NGO3'NGO3$NGO 7NG 3;NGO 3%NG 3'NG M 7 O NG M 3;NG M MPD[ f

PAGE 64

6LPLODUO\ HTXDWLRQ f FDQ EH DVVHPEOHG LQWR WKH PDWUL[ IRUP DV VHHQ LQ f :'NGO :$NGO Q n n 7 [NGO Q :;NGO :%NG :rNG :$NG 7 NG :;NG :% :' :$ NP NP NP 7 [NP :;NPO r :% :' NPD[ NPD[ M 7NPD[ M :;NPD[ M MPD[ f ,W VKRXOG EH SRLQWHG RXW WKDW WKH FRHIILFLHQWV DQGRU WKH ULJKW KDQG VLGH WHUPV LQ WKH URZ PDUNHG ZLWK V\PERO r LQ WKRVH PDWUL[HV QHHGV WR EH SURSHUO\ PRGLILHG DV ZHOO DV WKH OLPLW RI HLWKHU VXEVFULSW M RU VXEVFULSW N DFFRUGLQJ WR WKH GLIIHUHQW ERXQGDU\ FRQGLWLRQV LQ WKH YDULRXV PRGHOV FRQVLGHUHG )RU H[DPSOH DORQJ WKH D[LV WKH V\PPHWU\ FRQGLWLRQ UHTXLUHV WKDW G7GU DQG WKLV FRXOG EH DFFRPSOLVKHG QXPHULFDOO\ E\ HTXDWLQJ WKH WHPSHUDWXUH 7T DQG 7 DQG E\ HPSOR\LQJ D QHZ FRPELQLQJ FRHIILFLHQW RI 3$Ar 3$L 3%L UDWKHU WKDQ WKH RULJLQDO 3$A LQ WKH ILUVW PDWUL[ IRU HYDOXDWLQJ WKH WHPSHUDWXUH GLVWULEXWLRQ ZLWKLQ WKH SLSH ,Q DGGLWLRQ LI WKH WHPSHUDWXUH GLVWULEXWLRQV DORQJ FHUWDLQ SDUWV RI WKH RXWHU VXUIDFH RI WKH ZDOO DUH JLYHQ WKHQ WKH

PAGE 65

OLPLW RI VXEVFULSW N ZLOO EH HQGHG ZLWK N NPD[ UDWKHU WKDQ N NPD[ VKRZQ LQ HTXDWLRQ f 7KH VDPH DUJXPHQWV DUH DOVR DSSOLFDEOH IRU DQRWKHU VXEVFULSW M LQ WKH D[LDO GLUHFWLRQ %\ XVLQJ D VLPLODU SURFHGXUHV IRU WKH VHFRQG KDOI RI HDFK WLPH VWHS WKH ILQLWH GLIIHUHQFH DOJRULWKP IRU WKH NWK URZ WKHQ EHFRPHV &3MNN7MONN *3MNN7MNN )3MNN7MONNf Q S NNLMNNO NNMNN NNLMNNOM f Z\N F%MNN7MNNL +ZMNN7LNN $ZMNN7MNNLfQ f

PAGE 66

6LPLODUO\ WR HPSKDVL]H WKDW WKH FXUUHQW FDOFXODWLRQ LV LQ WKH NWK URZ ZH XVH WKH V\PERO NN UDWKHU WKDQ N LQ DERYH IRUPXOD DQG WKH PDWUL[ IRUP RI HTXDWLRQ f IRU WKH NAK URZ FDQ WKHQ EH ZULWWHQ DV 3* 3) f r Q U r 7 n n 3< 3& 3* 3) 7 3< 3& SIf 7 3< 3& 3* 3) P MPO MPO 7MP 3;MPO r L 3& 3* MP MP N 7 '} N 3; } Q N NPLG f DQG WKH PDWUL[ IRUP RI HTXDWLRQ f EHFRPHV :* ZI r Q n 7 1O Z\ ZF :* :) 7 :< ZF :* :) 7 :< :& :* MPO MP :) MPO 7MP :; MPO r E :& MP '* p N 7 p N :; p N NPLG NPLG NP 1 N f

PAGE 67

$V LQGLFDWHG DERYH WKH V\PERO r VKRZV WKDW WKH FRHIILFLHQWV LQ WKDW URZ DV ZHOO DV WKH ULJKWKDQGVLGH WHUPV QHHG WR EH SURSHUO\ PRGLILHG FRUUHVSRQGLQJ WR WKH GLIIHUHQW ERXQGDU\ FRQGLWLRQV 7KH PDWUL[ WHUPV LQ HTXDWLRQV f f f DQG f f f f DQG f FDQ EH FRPSXWHG E\ $3MN L A3 3!A %3MN 3 3!A I N &3MN f§ 3 3!MN (3MN  r3!MN 33 M N Lf§3 Sf *3MN  r3fMN '3MN  r3!MN DQG $: 'N : 3f M N %: f§ : :f M N &0MN f§ : :!MN +3MN W r3!MN f r3!MN ZL Z!MN r:f N ':M I N r0!MN +0LN O rZ!LN f

PAGE 68

,Q RUGHU WR UHZULWH WKH HTXDWLRQ LQ D VLPSOH IRUP ZH GHILQH WKH IROORZLQJ OLQHDU RSHUDWRU /< >3@M!N >%3 '3 $3@MN /< >:@MN >%:n ':n $:@MN /; >3MN >&3 *3 )3@MN /; >:@M A N >&: *: ):@MN f DQG FROXPQ YHFWRUV ^73@ 7:@ 7
PAGE 69

(TXDWLRQV f DQG f ZKLFK DUH LQ WKH D[LDO GLUHFWLRQ WKHQ DVVXPH WKH IROORZLQJ VLPSOH IRUPV Q Q Q /; >3@ 7;f 3< f M N M N M N Q QO Q /; >:@ 7;f :< f M N M N MN M N ZKHUH WKH ULJKW VLGH WHUPV FDQ EH HVWLPDWHG E\ 3; &3 (3 )3f 7;f MN MN MN :; &: (: ):f 7;f MN MN M N 3< %3 +3 $3f 7
PAGE 70

,W LV VHHQ DV WKH UHVXOW RI WKH $', VSOLWWLQJf SURFHGXUH ZKLFK KDV EHHQ HPSOR\HG LQ WKH DOJRULWKP DVVRFLDWHG ZLWK GLIIHUHQW ERXQGDU\ FRQGLWLRQV IRU WKH YDULRXV PRGHOV WKDW RQO\ D WULGLDJRQDO V\VWHP RI OLQHDU DOJHEUDLF HTXDWLRQV QHHGV WR EH VROYHG LH GXULQJ VWHS WKH FRXSOHG WULGLDJRQDO PDWUL[ f f DUH VROYHG IRU HDFK Mrr FROXPQ RI WKH JULG SRLQWV ZKLOH GXULQJ VWHS WKH FRXSOHG WULGLDJRQDO PDWUL[ f DQG f DUH WKHQ VROYHG IRU HDFK NrK URZ RI JULG SRLQWVf 2QFH WKH SHULRGLF VWHDG\ VROXWLRQ KDV EHHQ REWDLQHG ZH FDQ FDOFXODWH ERWK WKH WLGDO GLVSODFHPHQW DQG WKH KHDW IOX[ DW GLIIHUHQW ORFDWLRQV ZLWKLQ WKH SLSH 7KH QXPHULFDO WHFKQLTXH XVHG KHUH WR LQWHJUDWH HTXDWLRQV f f DQG f IRU HYDOXDWLQJ WKH WLGDO GLVSODFHPHQW DQG WKH KHDW IOX[ DW HDFK VSHFLILHG ORFDWLRQ FDQ EH REWDLQHG HLWKHU E\ WKH 7UDSH]RLGDO UXOH ZLWK HQG FRUUHFWLRQf RU 6LPSVRQnV UXOH %RWK QXPHULFDO DSSURDFKHV DUH HVVHQWLDOO\ IRXUWK RUGHU PHWKRGV &RQYHUJHQFH &ULWHULD $V LV NQRZQ RQFH WKH FDOFXODWLRQ ZRUN KDV VWDUWHG WKH WLPHPDWFKLQJ SURFHVV ZLOO EH LQ WKH ORRS IRUHYHU XQOHVV D FULWHULRQ FDQ EH GHULYHG WKDW LQGLFDWHV ZKHQ WKH JRDO RI WKH FXUUHQW FRPSXWLQJ ZRUN KDV EHHQ UHDFKHG DQG IXUWKHU VROXWLRQPDWFKLQJ SURFHVVHV GR QRW SURGXFH VLJQLILFDQW LQFUHDVHV LQ DFFXUDF\ 6XFK D FULWHULRQ GHSHQGV RQ WKH SXUSRVH RI WKH FDOFXODWLRQ ,I RQHnV JRDO LV WR VWXG\ WKH

PAGE 71

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mW ZLWKLQ D SHULRG WKH WHVWLQJ LV GRQH E\ FRPSDULQJ WKH WHPSHUDWXUH UHVLGXDO LH E\ LQVSHFWLQJ WKH DYHUDJHG WHPSHUDWXUH GLIIHUHQFH RI HDFK QRGDO SRLQW DW WKHH VDPH mW EHWZHHQ DGMDFHQW SHULRGV 7KLV FDQ EH ZULWWHQ DV 5HVO M V 7 7 f" N @ -0$;f.0$;f M -0$; N .0$; f ZKHUH 7LO LV WKH YDOXH RI WKH WHPSHUDWXUH RI QRGH MNf DW ZW LQ WKH FXUUHQW SHULRG 7 LV WKH YDOXH RI WKH .

PAGE 72

WHPSHUDWXUH RI QRGH MNf DW WKH VDPH mW LQ WKH SUHYLRXV SHULRG DQG HM LV WKH FRQYHUJHQFH SDUDPHWHU ,I WKH JRDO RI WKH LQYHVWLJDWLRQ LV WR H[DPLQH WKH HIIHFWLYH KHDW WUDQVIHU WKH FRQYHUJHQFH FULWHULRQ LV HVWDEOLVKHG E\ FRPSXWLQJ WKH UHVLGXDO 5HV -6(& r H f M -6(& ZKHUH -6(& LV WKH QXPEHU RI FURVVVHFWLRQV ZKHUH WKH D[LDO KHDW IOX[ ZDV H[DPLQHG LV WKH FRQYHUJHQFH SDUDPHWHU DQG W! M! DUH WKH FURVVVHFWLRQ DYHUDJHG KHDW IOX[ LQ WKH FXUUHQW SHULRG DQG SUHYLRXV SHULRG UHVSHFWLYHO\ 7KH VXPPDWLRQ LV FDUULHG RXW RYHU DOO VHFWLRQV -6(&f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

PAGE 73

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f NB f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

PAGE 74

GHSHQGV RQ WKH ERXQGDU\ OD\HU WKLFNQHVV QDPHO\ WKH NLQHPDWLF YLVFRVLW\ RI ZRUNLQJ IOXLG DQG WKH RVFLOODWLQJ IUHTXHQF\ ,Q WKH SUHVHQW VWXG\ LW ZDV IRXQG WKDW D JRRG FKRLFH RI WKLV YDOXH LV $6 IRU ODPLQDU IORZ FDVHV ZLWK D WRWDO RI DERXW QRGDO SRLQWV GLVWULEXWHG DORQJ WKH UDGLXV

PAGE 75

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

PAGE 76

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f ZLWK WKH 9$; PDFKLQH RU DOPRVW KRXU &38f ZLWK WKH 9$; PDFKLQH WR UXQ RQO\ RQH SHULRG ,W XVXDOO\ WDNHV SHULRGV WR UHDFK WKH ILQDO SHULRGLF VWDWH VROXWLRQ 7KH SURFHVV RI VHOHFWLQJ PRGHO VL]H LH WRWDO QRGDO SRLQWVf LV D V\QWKHWLF EDODQFH DPRQJ WKH VWRUDJH UHTXLUHPHQW WKH VROXWLRQ DFFXUDF\ DQG WKH FRVW RI &38 WLPH IRU ILQGLQJ DQ DFFHSWDEOH VROXWLRQ 6XFK D VHOHFWLRQ DOORZV HDFK FDVH WR EH VROYHG ZLWK D PLQLPXP RI H[SHQVH LQ FRPSXWLQJ RSHUDWLRQ WKHUHE\ PDNLQJ LW SRVVLEOH WR GR WKH

PAGE 77

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

PAGE 78

D r r r f§ 8 F ID I UUQ }U Q \ 0 } Wm LVH XH LH }H MP VHH )LJ O' 9HORFLW\ 3URILOHV LQ 2VFLOODWLQJ )ORZ IRU :RUPHUVOH\ 1XPEHU D DQG

PAGE 79

WKDW WKH YHORFLW\ SURILOH DW D SUHVHQWV D TXDVLn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f >@

PAGE 80

5OUf 5LaUf 5OaUf )LJ 0DJQLILHG 9LHZ RI 9HORFLW\ 3URILOH 1HDU :DOO :RUPHUVOH\ 1XPEHU D +2 FPf

PAGE 81

VWLPXODWLQJ SUHVVXUH JUDGLHQW $W WKH LQWHUPHGLDWH IUHTXHQF\ FDVH D RI )LJ f WKH VOXJ IORZ ERXQGDU\ LV QRW VR HYLGHQW EXW RQH FDQ VWLOO QRWH D ERXQGDU\ OD\HU QHDU WKH ZDOO 7KH VDPH SDWWHUQ RI WKH YHORFLW\ SURILOHV DVVRFLDWHG ZLWK PRGHUDWH :RUPHUVOH\ QXPEHUV FDQ DOVR EH IRXQG LQ WKH UHIHUHQFH >@ ZKHUH WKH YHORFLW\ SURILOH DW :RUPHUVOH\ QXPEHU D OHVV WKDQ ZDV SUHVHQWHG )LJ f ,Q RUGHU WR EHWWHU VHH WKH YDULDWLRQ RI WKH WLPHGHSHQGHQW YHORFLW\ SURILOH ZLWKLQ WKH ERXQGDU\ OD\HU D VHW RI FORVHU YLHZ RI WKH YHORFLW\ SURILOH ZLWK UHVSHFW WR :RUPHUVOH\ QXPEHU DW D GXULQJ SKDVH LQWHUYDOV ZW r LV SORWWHG LQ )LJ ,W VKRXOG EH HPSKDVL]HG WKDW WKH VROXWLRQV VKRZQ KHUH DUH XQGHU WKH DVVXPSWLRQ WKDW WKH VHFRQGDU\ YHORFLW\ LQ WKH UDGLDO GLUHFWLRQ LV QHJOLJLEOH FRPSDUHG WR WKH D[LDO YHORFLW\ FRPSRQHQW QDPHO\ WKH QRQOLQHDU LQHUWLDO WHUPV DUH QRW FRQVLGHUHG LQ WKH JRYHUQLQJ HTXDWLRQ f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

PAGE 82

FRQVLGHUHG LQFRPSUHVVLEOH LI $[Z & ZKHUH & LV WKH VSHHG RI VRXQG LQ WKH IOXLG )RU PHUFXU\ & PVHF RQH UHTXLUHV D ZKHQ $[ FP DQG FP 7R DYRLG D DSSUHFLDEOH SKDVH GLIIHUHQFH EHWZHHQ WKH WXEH HQGV RQH UHTXLUHV WKDW /& m WWZ %RWK WKH UHVWULFWLRQV DUH PHW LQ WKH H[DPSOHV WR EH FRQVLGHUHG EHORZ 7KH /DJUDQJLDQ 'LVSODFHPHQWV $Q DOWHUQDWLYH LQWHUHVWLQJ UHSUHVHQWDWLRQ WR WKH RVFLOODWRU\ YHORFLW\ ILHOG DUH WKH /DJUDQJLDQ GLVSODFHPHQWV RI WKH IOXLG HOHPHQWV DW GLIIHUHQW UDGLL ZLWKLQ WKH SLSH 7KH\ KDYH EHHQ SORWWHG LQ )LJV DQG DW WLPH LQWHUYDOV RI ZW r IRU :RUPHUVOH\ QXPEHU D DQG ,W LV QRWHG WKDW VLQFH ERWK SLSH GLDPHWHU DQG WKH ZRUNLQJ IOXLG ZHUH IL[HG LQ WKLV WHVW )LJV DQG UHSUHVHQW WKH UHODWLRQVKLS EHWZHHQ WKH /DJUDQJLDQ GLVSODFHPHQW DQG WKH RVFLOODWLQJ IUHTXHQF\ 7KH WUDMHFWRULHV SORWWHG LQ )LJV DQG KDYH EHHQ QRUPDOL]HG E\ $Z ZKHUH $ OS_GSG[_ LV WKH DPSOLWXGH RI WKH VLQXVRLGDO SUHVVXUH JUDGLHQW DV GHILQHG LQ HTXDWLRQ f DQG Z LV WKH DQJXODU YHORFLW\ 6XIILFH LW KHUH WR SRLQW RXW WKDW IRU WKH ORZHU IUHTXHQF\ FDVH IRU H[DPSOH D DQG f WKH /DJUDQJLDQ GLVSODFHPHQW WUDMHFWRU\ VKRZV D IRUHVHHDEOH SDUDEROLF SDWWHUQ DW DQ\ PRPHQW 1HYHUWKHOHVV WKH HVVHQWLDO GLVWLQFWLRQ EHWZHHQ ORZ IUHTXHQF\ RVFLOODWRU\ SLSH IORZ DQG VWHDG\ +DJHQ3RLVHXLOOH IORZ LV WKDW LQ WKH IRUPHU WKH /DJUDQJLDQ GLVSODFHPHQW WUDMHFWRULHV DV ZHOO DV

PAGE 83

)LJ /DJUDQJLDQ 'LVSODFHPHQW IRU D DQG

PAGE 84

)LJ /DJUDQJLDQ 'LVSODFHPHQW DW D

PAGE 85

WKH YHORFLW\ SURILOHV DUH SHULRGLF VR WKDW WKH IOXLG SDUWLFOHV GR QRW WUDQVODWH D[LDOO\ XSRQ WLPH DYHUDJLQJ ZKLOH LQ WKH ODWHU FDVH WKH\ ZLOO )RU LQWHUPHGLDWH :RUPHUVOH\ QXPEHU D f WKH WUDMHFWRULHV RI WKH /DJUDQJLDQ GLVSODFHPHQW GHSDUWV FRQVLGHUDEO\ IURP WKH VWDQGDUG SDUDEROLF VKDSH 7KLV SKHQRPHQRQ FDQ EH HYHQ PRUH FOHDUO\ VHHQ LQ WKH D FDVH (YLGHQWO\ WKH KLJKHU WKH RVFLOODWLQJ IUHTXHQF\ WKH WKLQQHU WKH ERXQGDU\ OD\HU -YX!f 7LGDO 'LVSODFHPHQWV )LJV DQG GHPRQVWUDWH WLPH YDULDWLRQ RI WKH FURVVVHFWLRQ DYHUDJHG GLPHQVLRQOHVV /DJUDQJLDQ GLVSODFHPHQW DW :RUPHUVOH\ QXPEHUV LQ WKH UDQJH IURP D WR 7KH WLGDO GLVSODFHPHQW FDQ EH REWDLQHG E\ VXPPLQJ WKH DEVROXWH PD[LPD DQG WKH DEVROXWH PLQLPD RI WKHVH FXUYHV 7KH FRUUHVSRQGLQJ QRQGLPHQVLRQDO WLGDO GLVSODFHPHQWV ZLWK UHVSHFW WR :RUPHUVOH\ QXPEHU IURP D WR D DUH OLVWHG LQ WDEOH EHORZ 7DEOH 'LPHQVLRQOHVV 7LGDO 'LVSODFHPHQW DW 'LIIHUHQW :RUPHUVOH\ 1XPEHUV D R f + 0 f 2 ? $; D R f &2 D[

PAGE 86

)LJ 'LPHQVLRQOHVV &URVVVHFWLRQ $YHUDJHG 'LVSODFHPHQW 9HUVXV 7LPH T )LJ 'LPHQVLRQOHVV &URVVVHFWLRQ $YHUDJHG 'LVSODFHPHQW 9HUVXV 7LPH T

PAGE 87

,W LV QRWLFHG WKDW IRU YHU\ VPDOO :RUPHUVOH\ QXPEHU FDVHV D f WKH FURVVVHFWLRQ DYHUDJHG GLVSODFHPHQW YDULHV OLNH D VLQXVRLGDO IXQFWLRQ ZLWK UHVSHFW WR WLPH WKH RUGLQDWHf DQG DV :RUPHUVOH\ QXPEHU LQFUHDVHV WKH FURVV VHFWLRQ DYHUDJHG GLVSODFHPHQWV DUH QR ORQJHU V\PPHWULF DERXW WKH RUGLQDWH EXW UDWKHU IDYRU SRVLWLYH YDOXHV RI r ,I WKH :RUPHUVOH\ QXPEHU LV IXUWKHU LQFUHDVHG HYHQWXDOO\ WKH FURVVVHFWLRQ DYHUDJHG GLVSODFHPHQW DOPRVW HQWLUHO\ OLHV RQ WKH ULJKW KDOI SODQH $ VLPLODU IHDWXUH FDQ DOVR EH VHHQ LQ )LJV DQG ,Q IDFW WKLV MXVW DJDLQ VKRZV WKH H[LVWHQFH RI WKH SKDVH VKLIW LQ WKH RVFLOODWLQJ SLSH IORZ :KHQ WKH :RUPHUVOH\ QXPEHU LV VPDOO WKH FURVVVHFWLRQ DYHUDJHG GLVSODFHPHQWV DV ZHOO DV WKH /DJUDQJLDQ GLVSODFHPHQWV VKRZ D VLQXVRLGDO YDULDWLRQ ZLWK UHVSHFW WR WLPH DQG WKLV KDV "U SKDVH ODJ ZLWK UHVSHFW WR WKH VWLPXODWLQJ SUHVVXUH JUDGLHQW ZKLFK LV DVVXPHG WR EH D FRVLQH IXQFWLRQ RI ZW ZLWK ]HUR LQLWLDO SKDVH DQJOH +RZHYHU IRU KLJKHU :RUPHUVOH\ QXPEHU WKH SKDVH ODJV LQFUHDVH WR DOPRVW Q ZLWK UHVSHFW WR WKH SKDVH RI WKH H[FLWLQJ SUHVVXUH JUDGLHQW 1RWH WKDW WKH /DJUDQJLDQ GLVSODFHPHQWV LQ )LJV DQG ZHUH FRPSXWHG E\ OLQLQJ XS DOO WKH IOXLG SDUWLFOHV RQ WKH SODQH [ DW ZW KRZHYHU E\ XVLQJ QRQ]HUR YHORFLW\ DW ZW r DV VKRZQ LQ )LJ 7KLV LPSOLHV WKDW LI WKH SKDVH RI WKH H[FLWLQJ SUHVVXUH JUDGLHQW LV WDNHQ DV WKH EDVH RI WKH PHDVXUHPHQW LW LV JHQHUDOO\ QRW SRVVLEOH WR DVVXUH WKH VDPH SKDVH WR WKH

PAGE 88

YHORFLW\ DQG /DJUDQJLDQ GLVSODFHPHQW DV ZHOO DV WKH WLGDO GLVSODFHPHQW )RU H[DPSOH DW SKDVH RI WKH VWLPXODWLQJ SUHVVXUH JUDGLHQW ZW r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f DQG f JLYHQ E\ .XU]ZHJ >@ IRU :RUPHUVOH\ QXPEHU YDU\LQJ IURP WR KDV EHHQ FDOFXODWHG DQG SORWWHG LQ )LJ 7KH VROLG OLQH VKRZV WKH DQDO\WLF VROXWLRQ REWDLQHG E\ XVH RI HTXDWLRQV f DQG f ZKLOH WKH GDVKHG OLQH VKRZV WKH UHVXOWV ZLWK WKH (73 FRGH GHYHORSHG LQ WKLV VWXG\ 7KH DJUHHPHQW LV TXLWH JRRG SDUWLFXODUO\ ZKHQ WKH :RUPHUVOH\ LV OHVV WKDQ

PAGE 89

+RZHYHU DW KLJK :RUPHUVOH\ QXPEHU WKH QXPHULFDO VROXWLRQ VKRZV D YHU\ VOLJKW GHYLDWLRQ IURP WKH DQDO\WLF VROXWLRQ 7KLV GHYLDWLRQ LV EHOLHYHG GXH WR DQ LQDFFXUDWH QXPHULFDO LQWHJUDWLRQ RYHU WKH FURVVVHFWLRQ XVLQJ UHODWLYHO\ ODUJH WLPH VWHSV RXU WLPH VWHSV SHU SHULRG LQ WKH FDOFXODWLRQ ZHUH EHWZHHQ f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f WDNHQ DV WKH ZRUNLQJ PHGLXP YHUVXV WKH GLPHQVLRQDO WLGDO GLVSODFHPHQW $[ LQ FP IRU YDULRXV :RUPHUVOH\ QXPEHUV QDPHO\ RVFLOODWLQJ IUHTXHQF\f ,W LV HYLGHQW IURP WKHVH UHVXOWV WKDW IRU IL[HG

PAGE 90

'LPHQVLRQOHVV 7LGDO 'LVSODFHPHQW $; LGn 8r :RUPHUVOH\ 1XPEHU D )LJ 5HODWLRQVKLS %HWZHHQ 'LPHQVLRQOHVV 7LGDO 'LVSODFHPHQW $; DQG :RUPHUVOH\ 1XPEHU D

PAGE 91

8LRf 5HODWLRQVKLS %HWZHHQ 7LGDO 'LVSODFHPHQW D[ DQG ([FLWLQJ 3UHVVXUH *UDGLHQW LQ :DWHU )LJ

PAGE 92

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f DORQJ UDGLXV IRU :RUPHUVOH\ QXPEHU YDU\LQJ IURP D WR 6RPH QXPHULFDO UHVXOWV DUH DOVR VKRZQ LQ 7DEOHV DQG $OO RI WKH GDWD VKRZQ LQ )LJ DQG WKH WDEOHV KDYH WKH SKDVH DQJOH PHDVXUHG UHODWLYH WR WKH H[FLWLQJ SUHVVXUH JUDGLHQW 7ZR IHDWXUHV FDQ EH VHHQ LQ )LJ ILUVW LQ WKH FRUH SRUWLRQ WKH SKDVH ODJV DUH DOPRVW HTXDO WR Q ZKHQ WKH :RUPHUVOH\ QXPEHU LV VPDOO ZKLOH WKH ODJV DUH DOPRVW Q ZKHQ WKH :RUPHUVOH\ QXPEHU LV ODUJH DQG VHFRQG

PAGE 93

WKH SKDVH ODJV YDU\ DORQJ UDGLXV HVSHFLDOO\ LQ WKH ERXQGDU\ OD\HU ,W LV VXFK SKDVH ODJV WKDW DOORZ WKH H[LVWLQJ WHPSHUDWXUH JUDGLHQW LQ WKH YHU\ WKLQ ERXQGDU\ OD\HU RI WKH RVFLOODWLQJ SLSH IORZ WR DFW DV UHJLRQ RI WHPSRUDU\ KHDW VWRUDJH ,W DEVRUEV KHDW ZKHQ WKH WHPSHUDWXUH RI WKH FRUH 7DEOH 3KDVH /DJV $ORQJ 5DGLXV :RUNLQJ 0HGLXP +2 $[ FPf 1RGDO SRLQW D D D }" U5O SKDVH Uc U5O SKDVH U" U5O SKDVH

PAGE 94

7DEOH f§ FRQWLQXHG 1RGDO SRLQW D D D U5O SKDVH U" U5O SKDVH U" U5O SKDVH

PAGE 95

7DEOH f§ FRQWLQXHG 1RGDO 3RLQW D + ,, D D U5O SKDVH U5O SKDVH } U5O SKDVH

PAGE 96

7DEOH &RPSDULVRQ RI 3KDVH /DJV :LWK 'LIIHUHQW :RUNLQJ 0HGLXPV $[ f 1RGDO 3RLQW D D Uf U5O 3KDVH /DJV Wf U5O 3KDVH /DJV :DWHU 0HUFXU\ :DWHU 0HUFXU\

PAGE 97

'LPHQVLRQOHVV 5DGLXV )LJ 3KDVH 9DULDWLRQ $ORQJ 5DGLXV IRU 'LIIHUHQW :RUPHUVOH\ 1XPEHUV

PAGE 98

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nV VROXWLRQ 7KH UHVXOWV IRU WKH YHORFLW\ SURILOHV DUH LQ JRRG DJUHHPHQW 1HYHUWKHOHVV EHIRUH D GHWDLOHG H[DPLQDWLRQ RI WKH WKHUPDO ILHOG LW LV ILUVW QHFHVVDU\ WR WHVW WKH FXUUHQW GHYHORSHG (73 FRGH ZKHQ DSSO\LQJ WKH WHPSHUDWXUH HTXDWLRQV ff ,W LV VHHQ WKDW WKH HQHUJ\ HTXDWLRQV VWURQJO\ GHSHQG RQ WKH YHORFLW\ GLVWULEXWLRQ DQG LWV EXLOGXS SURFHVV VR WKDW RQH FDQ XVH DQDO\WLF SHULRGLF YHORFLW\ VWDWH (TV DQG ZLWK QR EXLOGXS SURFHVVf DQG WKH FRPSXWHG YHORFLW\ ZLWK EXLOGXS SURFHVVf WR YHULI\ WKH FRUUHFWQHVV RI WKH UHVXOWLQJ WKHUPDO YDULDEOHV 7KH HQKDQFHG KHDW IOX[ LV D IXQFWLRQ RI ERWK YHORFLW\ DQG WHPSHUDWXUH (TV DQG f DQG ZDV FKRVHQ IRU D FRPSDULVRQ RI WKH DQDO\WLF DQG QXPHULFDO UHVXOWV RI WKH SUREOHP 3DUW $ RI WDEOH VKRZV WKH UHVXOWV RI WKH FRPSXWHG HQKDQFHG D[LDO KHDW IOX[ DV ZHOO DV WKH D[LDO FRQGXFWLRQ KHDW IOX[ ZKHQ XVLQJ WKH DQDO\WLF YHORFLW\

PAGE 99

7DEOH 7KH &RPSDULVRQ RI (QKDQFHG +HDW )OX[ 8VLQJ 1XPHULFDO 9HORFLW\ ZLWK +HDW )OX[ 8VLQJ $QDO\WLFDO 9HORFLW\ 0RGHO :DWHU*ODVV 3U D f $ +HDW )OX[ 8VLQJ $QDO\WLF 9HORFLW\ $; FPf ZFP .f !I ZFPr .f W!Z ZFP r .f 3$ ZFPr .f % +HDW )OX[ 8VLQJ &RPSXWHG 9HORFLW\ r 0RGHO :DWHU*ODVV r HQKDQFHG D[LDO KHDW IOX[ r W! I D[LDO KHDW IOX[ E\ FRQGXFWLRQ LQ IOXLG r !?M D[LDO KHDW IOX[ E\ FRQGXFWLRQ LQ SLSH ZDOO r S I!$;

PAGE 100

(T f IRU D 3U SLSH UDGLXV FP DQG JODVV ZDOOV RI WKLFNQHVV $5 5 5L FP ZKLFK LV DOPRVW HTXDO WR WKH ERXQGDU\ OD\HU WKLFNQHVV FP IRU WKLV FDVHf ZLWK 0RGHO 7KHUH ZHUH SRLQWV GLVWULEXWHG DORQJ WKH UDGLXV SRLQWV LQ IOXLGf DQG WKH VPDOOHVW GLVWDQFH QH[W WR WKH ZDOO LV HTXDO WR ZKLOH DORQJ WKH D[LDO GLUHFWLRQ SRLQW ZHUH XVHG SRLQWV LQ WKH FHQWUDO SLSH VHFWLRQf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

PAGE 101

ODUJHU WLPH DV DVVXPHG LQ H[LVWLQJ DQDO\WLF VROXWLRQV > @ 0RGHO ZDV HPSOR\HG WR VWXG\ WKH WHPSHUDWXUH EXLOGn XS KLVWRU\ IRU WKH FDVH ZKHUH WKH LQLWLDO YHORFLW\ ZDV ]HUR HYHU\ZKHUH LQVLGH WKH SLSH DV ZHOO DV DW WKH ERXQGDU\ 7KH LQLWLDO WHPSHUDWXUH FRQGLWLRQ ZDV DVVXPHG WR EH LGHQWLFDO ZLWK WKH FROG HQG WHPSHUDWXUH DW DOO JULG SRLQWV H[FHSW WKRVH DW [ ZKHUH WKH GLPHQVLRQOHVV WHPSHUDWXUH 7 )LJV DQG VKRZ WKH FRPSXWHG WHPSHUDWXUH EXLOGXS KLVWRU\ DW :RUPHUVOH\ QXPEHU D DW [ / DQG / ZKHUH / FP LV WKH SLSH OHQJWKf DQG HLWKHU RQ WKH D[LV U RU DW WKH ZDOO U 5A 7KH WLGDO GLVSODFHPHQWV ZHUH FKRVHQ DV D[ / / DQG / UHVSHFWLYHO\ ,Q WKHVH ILJXUHV WKH RUGLQDWH UHSUHVHQWV WKH GLPHQVLRQOHVV WHPSHUDWXUH ZKLOH WKH DEVFLVVD UHSUHVHQWV WKH RVFLOODWRU\ SHULRG UXQV 7KHVH WHPSHUDWXUH KLVWRULHV VKRZ D VLQXVRLGDO YDULDWLRQ 7KUHH FKDUDFWHULVWLFV FDQ EH VXPPDUL]HG f ZKHQ WKH WLGDO GLVSODFHPHQW LV ODUJH WKH DPSOLWXGH RI WKH WHPSHUDWXUH YDULDWLRQ LV DOVR ODUJH DQG OHVV DGMXVWLQJ WLPH LV QHHGHG f WKH DPSOLWXGH RI WKH VLQXVRLGDO YDULDWLRQ RI WKH WHPSHUDWXUH DORQJ WKH D[LV LV ODUJHU WKDQ WKDW QHDU WKH ZDOO VHH )LJV f f DIWHU D VKRUW WUDQVLHQW SURFHVV WKH ILQDO RVFLOODWRU\ VWDWH LV UHDFKHG 7KH WLPHDYHUDJHG WHPSHUDWXUHV RI WKLV RVFLOODWRU\ VWDWH DW D JLYHQ [SRVLWLRQ KDYH WKH VDPH YDOXH

PAGE 102

)LJ 7HPSHUDWXUH %XLOGXS 3URFHVV ,Q 2VFLOODWLQJ )ORZ 0RGHO T $; FPf

PAGE 103

LHHHU W 3HULRG )LJ 7HPSHUDWXUH %XLOGXS 3URFHVV ,Q 2VFLOODWLQJ )ORZ 0RGHO D D[ FPf

PAGE 104

)LJ 7HPSHUDWXUH %XLOGXS 3URFHVV ,Q 2VFLOODWLQJ )ORZ 0RGHO D D[ FPf

PAGE 105

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f @ f ZKHUH $D3Uf NHKZ$[ LV D IXQFWLRQ RI WKHUPDO FRQGXFWLYLW\ DQG :RUPHUVOH\ QXPEHU > @ $W WKH WXQLQJ SRLQW WKLV YDOXH RI $ LV DERXW ,I WKH WLGDO GLVSODFHPHQW D[ / FP DQG WKH DQJXODU IUHTXHQF\ Z IRU ZDWHU ZLWK FPf WKH DGMXVWPHQW WLPH WA FDQ EH HVWLPDWHG IURP HTXDWLRQ f WR EH HTXDO WR VHF RU SHULRGV 7KH FRPSXWHG UHVXOWV VKRZ WKDW WKH DFWXDO DGMXVWLQJ WLPH LV RQO\ SHULRGV

PAGE 106

W 7LPH W LQ XQLW RI HTXLYDOHQW SHULRGVf )LJ 7HPSHUDWXUH %XLOG8S 3URFHVV ,Q 6WHDG\ )ORZ 0RGHO 8HT FPVHFf

PAGE 107

3HULRG 7LGDO 'LVSODFHPHQW [ FPf )LJ %XLOG8S 7LPH 9HUVXV 7LGDO 'LVSODFHPHQW 0RGHO T f 3UHVVXUH DPSOLWXGH $ S _GSG[_

PAGE 108

)LJ VKRZV WKH EXLOGXS WLPH YHUVXV WKH WLGDO GLVSODFHPHQW DQG DSSOLHG SUHVVXUH JUDGLHQW FRUUHVSRQGLQJ WR YDULRXV WLGDO GLVSODFHPHQWV ZLWK 0RGHO ,W FDQ EH VHHQ WKDW ZKHQ WKH WLGDO GLVSODFHPHQW LV YHU\ VPDOO WKH EXLOGn XS WLPH WHQGV WRZDUGV LQILQLW\ VR WKDW WKH VRFDOOHG ILQDO SHULRGLF VWDWH QR ORQJHU H[LVWV 7KH ILQDO SHULRGLF WHPSHUDWXUH VWDWH ZLOO WKXV EH HVWDEOLVKHG RQO\ E\ KHDW FRQGXFWLRQ DQG KDV D FRPSOHPHQWDU\ HUURU IXQFWLRQ HUIF[Wf HUI[Wf WHPSHUDWXUH EXLOGXS KLVWRU\ > @ $V WKH WLGDO GLVSODFHPHQW JHWV ODUJHU WKH EXLOGXS WLPH QHHGHG GHFUHDVHV DQG ZKHQ $[ LV JUHDWHU WKDQ DERXW / FPf WKH EXLOGXS WLPH LV DOPRVW FRQVWDQW ,Q RUGHU WR FRPSDUH ZLWK WKH VWHDG\ +DJHQ3RLVHXLOOH IORZ D WUDQVLHQW WHPSHUDWXUH KLVWRU\ RI VXFK D IORZ ZLWK 0RGHO LV DOVR VKRZQ LQ )LJ %RWK FXUYHV UHSUHVHQW WKH WHPSHUDWXUH FRPSXWHG DORQJ WKH D[LV U f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

PAGE 109

)LJ 7HPSHUDWXUH 'LVWULEXWLRQ LQ 2VFLOODWLQJ )ORZ 0RGHO D D[ FPf 3LSH

PAGE 110

)LJ 7HPSHUDWXUH 'LVWULEXWLRQ LQ 2VFLOODWLQJ 3LSH )ORZ 0RGHO D D[ FPf

PAGE 111

)LJ 7HPSHUDWXUH 'LVWULEXWLRQ LQ 2VFLOODWLQJ 3LSH )ORZ 0RGHO D $[ FPf

PAGE 112

7U[pf LQ RVFLOODWLQJ SLSH IORZ ZLWK 0RGHO 7KH PDUNHG QXPEHUV UHSUHVHQW WKH WHPSHUDWXUH DORQJ WKH D[LV U f U 5L DQG DW U 5@B ZDOOf UHVSHFWLYHO\ )RU WKHVH FRPSXWDWLRQV WKH :RUPHUVOH\ QXPEHU ZDV WDNHQ DV D DQG WKH WLGDO GLVSODFHPHQW ZDV FKRVHQ DV $[ FP FP DQG FP UHVSHFWLYHO\ 1HJOHFWLQJ WKH VRPHZKDW HUUDWLF EHKDYLRU QHDU WKH SLSH HQGV WKH WLPHGHSHQGHQW WHPSHUDWXUH VKRZV D FOHDU OLQHDU GLVWULEXWLRQ SDWWHUQ DORQJ ORQJLWXGLQDO GLUHFWLRQ HVSHFLDOO\ IRU WKH UHODWLYHO\ VPDOO WLGDO GLVSODFHPHQW FDVH RI VD\ $[ FP ,I RQH IRFXVHV RQ RQH FURVVVHFWLRQ [ FRQVWDQWf RQH FDQ FOHDUO\ VHH WKH WHPSHUDWXUH QHDU WKH ERXQGDU\ DQG WKDW DW WKH D[LV DUH DOWHUQDWHO\ KLJKHU DQG ORZHU UHODWLYH WR HDFK RWKHU )LJ f ZLWKLQ D SHULRG 7KLV LPSOLHV WKDW D YHU\ ODUJH UDGLDO FRQGXFWLRQ KHDW H[FKDQJH RFFXUV EHWZHHQ WKH FRUH RI WKH IORZ DQG WKH ERXQGDU\ OD\HU 1RWH WKDW WKH UDGLDO WHPSHUDWXUH JUDGLHQW EHFRPHV YHU\ ODUJH GXULQJ SDUWV RI DQ RVFLOODWLQJ F\FOH 7KH SRRU EHKDYLRU RI WKH WHPSHUDWXUH GLVWULEXWLRQ QRWHG QHDU ERWK HQGV RI WKH SLSH LQ )LJ DW WLGDO GLVSODFHPHQW RI D[ FP LV FDXVHG E\ QXPHULFDO GLVSHUVLRQ ZKLFK LV D FRPPRQ VKRUWFRPLQJ RI WKH VHFRQG RUGHU QXPHULFDO PHWKRG HPSOR\HG > @ ,W PD\ EH LPSURYHG E\ HLWKHU XVLQJ DQ RGG RUGHU QXPHULFDO PHWKRG VXFK DV WKH ILUVWRUGHU XSZLQG PHWKRG IOX[ VSOLWWLQJ WHFKQLTXH >@ RU E\ HPSOR\LQJ DOWHUQDWLYH PRGHOV ZKLFK DUH DEOH WR DYRLG DQ\

PAGE 113

GLVFRQWLQXLW\ RI WKH SK\VLFDO SURSHUWLHV DW ERWK HQGV VXFK DV 0RGHO 7HPSHUDWXUH 'LVWULEXWLRQ LQ 0RGHO ,Q RUGHU WR H[SORUH VRPH DOWHUQDWLYH SRVVLEOH GHYLFHV ZKLFK DSSO\ WKH HQKDQFHG WKHUPDO SXPSLQJ WHFKQLTXH 0RGHO )LJ f ZDV GHYHORSHG DQG QXPHULFDOO\ H[DPLQHG 7KH WHPSHUDWXUH GLVWULEXWLRQV LQ 0RGHO KDYH EHHQ VWXGLHG DQG UHVXOWV DUH UHFRUGHG LQ )LJV DQG IRU VRPH VSHFLDO FDVHV 7KH ZRUNLQJ IOXLG XVHG LV DJDLQ ZDWHU DQG VLPLODUO\ WKH :RUPHUVOH\ QXPEHU ZDV WDNHQ DV D 7KH WLGDO GLVSODFHPHQWV XVHG ZHUH $[ FP FP FP FP DQG FP UHVSHFWLYHO\ 7KH PDUNHG QXPEHUV LQ WKRVH ILJXUHV UHSUHVHQW WKH WHPSHUDWXUHV DW U 5@B ZDOOf 5L DQG U LH DORQJ WKH D[LVf ,W FDQ EH VHHQ WKDW WKH WHPSHUDWXUHV LQ WKH FRUH RI WKH IORZ DQG WKDW LQ WKH ERXQGDU\ OD\HU LQ WKHVH ILJXUHV YDU\ SHULRGLFDOO\ LQ WKH UDGLDO GLUHFWLRQ GXULQJ WKH RVFLOODWLRQV 7KH UHVXOWV DOVR VKRZ D OLQHDU D[LDO WHPSHUDWXUH YDULDWLRQ EHWZHHQ WKH KHDWLQJ DQG FRROLQJ VRXUFHV ZKHQ WKH WLGDO GLVSODFHPHQW LV VPDOO $[ FP )LJ f $V WKH WLGDO GLVSODFHPHQW EHFRPHV ODUJHU D[ FP )LJ f WKH OLQHDU D[LDO WHPSHUDWXUH YDULDWLRQ EHFRPHV ZHDNHU 7KH GLIIHUHQFH RI WHPSHUDWXUH LQ WKH FRUH RI WKH IORZ DQG WKDW LQ WKH ERXQGDU\ OD\HU LV HYLGHQW DQG VKRZV D UDGLDOO\ DOWHUQDWLQJ SDWWHUQ ZLWKLQ DQ RVFLOODWLRQ SHULRG MXVW DV VHHQ LQ 0RGHO +RZHYHU DV WKH WLGDO GLVSODFHPHQWV DUH IXUWKHU LQFUHDVHG

PAGE 114

)LJ 7HPSHUDWXUH 'LVWULEXWLRQ LQ 2VFLOODWLQJ 3LSH )ORZ 0RGHO D $[ FPf

PAGE 115

)LJ 7HPSHUDWXUH 'LVWULEXWLRQ LQ 2VFLOODWLQJ 3LSH )ORZ 0RGHO R D[ FPf

PAGE 116

)LJ 7HPSHUDWXUH 'LVWULEXWLRQ LQ 2VFLOODWLQJ 3LSH )ORZ 0RGHO T $; FPf

PAGE 117

)LJ 7HPSHUDWXUH 'LVWULEXWLRQ LQ 2VFLOODWLQJ 3LSH )ORZ 0RGHO D $[ FPf

PAGE 118

U7 DW 5L DW 5O DORQJ $[LV )LJ 7HPSHUDWXUH 'LVWULEXWLRQ LQ 2VFLOODWLQJ 3LSH )ORZ 0RGHO T D[ FPf

PAGE 119

LHH 8HT FPVHF HVH HHH )LJ 7HPSHUDWXUH 'LVWULEXWLRQ LQ 6WHDG\ )ORZ 0RGHO 8DYH FPVHFf

PAGE 120

DV VKRZQ LQ )LJ WKH UDGLDO WHPSHUDWXUH JUDGLHQW DOPRVW QR ORQJHU H[LVWV H[FHSW GLUHFWO\ LQ WKH YLFLQLW\ RI WKH KHDW VRXUFH DQG VLQN DUHD ,W LV REYLRXV WKDW WKH FURVVVHFWLRQ DYHUDJHG WHPSHUDWXUH LQ WKH SLSH WHQGV WR EHFRPH FRQVWDQW DQG HTXDO WR WKH PHDQ YDOXH RI WKH KRW DQG FROG VRXUFH UHJLRQ 7KLV SKHQRPHQRQ FDQ EH VHHQ HYHQ PRUH FOHDUO\ DV WKH WLGDO GLVSODFHPHQW LV IXUWKHU LQFUHDVHG )LJV DQG f 2QH FDQ FRQFOXGH IURP VXFK D WHPSHUDWXUH GLVWULEXWLRQ WKDW WKH LQFUHDVH RI D[LDO KHDW WUDQVIHU DELOLW\ ZLWK LQFUHDVLQJ WLGDO GLVSODFHPHQW LV PRVW OLNHO\ WR EH OLPLWHG $W WKH VDPH WLPH WKLV D[LDO KHDW WUDQVIHU FDSDELOLW\ LV H[SHFWHG WR LQFUHDVH E\ LQFUHDVLQJ HLWKHU WKH LQGLYLGXDO VRXUFH VL]H RU WKH GHQVLW\ RI WKH DOWHUQDWLYHO\ GLVWULEXWHG KRW DQG FROG VRXUFHV 7KLV XQH[SHFWHG WHPSHUDWXUH GLVWULEXWLRQ LQ 0RGHO DW ODUJH WLGDO GLVSODFHPHQWV OHG XV WR IXUWKHU H[DPLQH WKH WKHUPDO ILHOG RI VWHDG\ SLSH IORZ IRU WKH ERXQGDU\ FRQGLWLRQV XVHG LQ 0RGHO 7KH FRPSXWHG WHPSHUDWXUH GLVWULEXWLRQ ZLWK DQ HTXLYDOHQW VWHDG\ DYHUDJHG YHORFLW\ IURP FPVHF WR FPVHF FRUUHVSRQGLQJ WR WLGDO GLVSODFHPHQW IURP FP WR FP DW DQ RVFLOODWLQJ IUHTXHQF\ RI Z VHF DUH VKRZQ LQ )LJ 7KH HTXLYDOHQW YHORFLW\ LV HVWDEOLVKHG IURP WKH UHODWLRQ 8HT $;Z $V FRXOG KDYH EHHQ DQWLFLSDWHG ODUJH UDGLDO WHPSHUDWXUH JUDGLHQWV LQ WKLV IORZ VLPLODU WKDW LQ WKH RVFLOODWLQJ IORZ ZLWK ODUJH WLGDO GLVSODFHPHQWf RFFXU RQO\ DW WKH KHDW

PAGE 121

VRXUFH DQG VLQN DUHDV ZKLOH WKH UDGLDO WHPSHUDWXUH JUDGLHQW LQ WKH UHVW RI WKH SLSH LV YHU\ VPDOO 7KH FRQVWDQW WHPSHUDWXUH GLIIHUHQFH DW WKH ULJKWKDQGVLGH GLIIHUV IURP WKDW RQ WKH OHIW KDQG VLGH RI WKH KHDW VRXUFH 7KLV WHPSHUDWXUH GLIIHUHQFH EHFRPHV SURJUHVVLYHO\ VPDOOHU DV WKH HJXLYDOHQW YHORFLW\ LV LQFUHDVHG LH WKH IOXLG KHDWLQJ EHFRPHV OHVVf 7HPSHUDWXUH 'LVWULEXWLRQ LQ 0RGHO 0RGHO ZDV GHYHORSHG WR H[DPLQH WKH SHUIRUPDQFH RI WKH WKHUPDO SXPS ZLWKRXW FRQYHFWLYH KHDW H[FKDQJH DW WKH HQGV LQ WKH FKDPEHUVf 7KDW LV WKH KHDW LV DGGHG DQG ZLWKGUDZQ E\ FRQGXFWLRQ WKURXJK WKH SLSH HQG VHFWLRQ VHH )LJ f 7KH FRPSXWHG WHPSHUDWXUH GLVWULEXWLRQ LQ WKLV PRGHO LV VKRZQ LQ )LJ DQG D PDJQLILHG YLHZ RI WKH WHPSHUDWXUH LQ WKH FHQWUDO VHFWLRQ LV VKRZQ LQ )LJ 7KH :RUPHUVOH\ QXPEHU XVHG LV DJDLQ D DQG WKH WLGDO GLVSODFHPHQW D[ FP 7ZR IHDWXUHV DUH REVHUYHG )LUVW LQ FRQWUDVW WR WKH WHPSHUDWXUH FRPSXWHG LQ 0RGHO WKH WHPSHUDWXUH REWDLQHG KHUH VKRZV D YHU\ VPRRWK SDWWHUQ DW WKH FHQWUDO VHFWLRQ HQGV LH DW [ / DQG / LQ )LJ f 6HFRQG WKH OLQHDU D[LDO WHPSHUDWXUH GLVWULEXWLRQ ZLWK WLPH GHSHQGHQW UDGLDO WHPSHUDWXUH JUDGLHQW DOWHUQDWLQJ LQ VLJQ LQ WKH FHQWUDO SRUWLRQ RI WKH SLSH QDPHO\ / [ / LQ )LJ f LV UHWDLQHG HYHQ ZKHQ WKH WLGDO GLVSODFHPHQW LV DV ODUJH DV D[ FP )LJ f

PAGE 122

LHHH % 8 f ,, n, n , , , , n , n L , , &HQWUDO 6HFWLRQ/f +HDWHG 6HFWLRQ/f DORQJ $[LV DW 5O DW 5[ L A&RROHG 6HFWLRQ/f-I L , , O L , )LJ 7HPSHUDWXUH 'LVWULEXWLRQ LQ 2VFLOODWLQJ 3LSH )ORZ 0RGHO D $[ FP / FPf

PAGE 123

LHHHU

PAGE 124

,OO +HDW )OX[ 9HUVXV 7LGDO 'LVSODFHPHQW 0RGHO f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f 7KH IOX[ LQFUHDVHV UDSLGO\ DV WKH WLGDO GLVSODFHPHQW JHWV ODUJHU D[ FPf 6WLOO IXUWKHU LQFUHDVLQJ WKH WLGDO GLVSODFHPHQW GRHV QRW OHDG WR D IXUWKHU ODUJH LQFUHDVH RI WKH D[LDO KHDW IOX[ ,Q IDFW WKH VORSH

PAGE 125

(QKDQFHG +HDW )OX[ W! :DWWFPA.f )LJ +HDW )OX[ LQ 2VFLOODWLQJ )ORZ DQG 6WHDG\ )ORZ 0RGHO D :DWHU :RUNLQJ )OXLG ,QVXODWHG :DOOf

PAGE 126

EHJLQV WR GHFUHDVH DIWHU WKH WLGDO GLVSODFHPHQW EHFRPHV ODUJHU WKDQ DERXW $[ FP (YHQWXDOO\ WKLV VORSH WHQGV WR ]HUR DV $[ LQFUHDVHV VWLOO IXUWKHU )RU WKH VWHDG\ IORZ 7DEOH (QKDQFHG +HDW )OX[ YLD 7LGDO 'LVSODFHPHQW 0RGHO RVFLOODWLQJ IORZ 3U D f $; FPf 3HULRGV WR ILQDO RVFLOODWLRQ VWDWH r 0D[ 3UHVVXUH JUDGLHQW $ OS_GS[_ FPVf KHDW IOX[ I! ZFP .f r EXLOGXS WLPH FDQ EH HVWLPDWHG E\ HTXDWLRQ f

PAGE 127

7DEOH (QKDQFHG +HDW )OX[ LQ 6WHDG\ )ORZ $; 0D[ 3UHVVXUH JUDGLHQW 8HT Z$[ KHDW IOX[ I! FPf $ S_DSD[_ FPVf FPVf ZFP r .f WKH VORSH RI WKH KHDW IOX[ FXUYH LV TXLWH ODUJH DQG DOPRVW FRQVWDQW DW DOO HTXLYDOHQW YHORFLWLHV OHVV WKDQ FPVHF EXW LW UDSLGO\ GHFUHDVHV ZKHQ WKH HTXLYDOHQW YHORFLW\ EHFRPHV ODUJHU 6LPLODU WR WKH RVFLOODWLQJ IORZ FDVH WKH KHDW IOX[ I! DV WKH YHORFLW\ LQFUHDVHV UHDFKHV D OLPLW 7KLV OLPLW LV DERXW SHUFHQW JUHDWHU WKDQ WKDW IRU WKH RVFLOODWRU\ IORZ 7KH UHDVRQ RI WKH VORSH YDULDWLRQ LQ WKH RVFLOODWLQJ IORZ FDQ EH H[SODLQHG DV IROORZV $W VPDOO WLGDO

PAGE 128

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f ,W LV ZHOO NQRZQ WKDW WKH WKHUPRG\QDPLF SURSHUWLHV RI WKH ZRUNLQJ IOXLG HVSHFLDOO\ WKH YLVFRVLW\ DUH IXQFWLRQV RI WKH ORFDO WHPSHUDWXUH 6RPH WKHUPRG\QDPLF SURSHUWLHV RI ZDWHU IRU WKH WHPSHUDWXUH UDQJH RI r& WR r & DUH OLVWHG LQ 7DEOH 2QH FDQ VHH WKDW HYHQ RYHU VXFK D VPDOO WHPSHUDWXUH UDQJH WKH WKHUPRG\QDPLF SURSHUWLHV RI ZDWHU YDU\ FRQVLGHUDEO\ )RU LQVWDQFH DW 7 r& WKH NLQHPDWLF YLVFRVLW\ Y 06HF DQG WKH FRUUHVSRQGLQJ 3UDQGWO 1XPEHU LV 3U ZKLOH DW WHPSHUDWXUH 7 r

PAGE 129

7DEOH 7KH ,QIOXHQFH RI 3URSHUWLHV RI :DWHU RQ WKH (QKDQFHG $[LDO +HDW )OX[ R 2 + 3U 3 JFP &3 ZVJr Y f FPVHF ZFPr. $; FP I! DW D O ZFPr r 3URSHUWLHV DUH LQ WKH VDWXUDWHG VWDWH >@ Q 0VHF DQG 3U ,W YDULHV DOPRVW DQ RUGHU RI PDJQLWXGH RYHU WKLV UDQJH 7KH SUHVHQW VWXG\ GLG QRW LQWHQG WR LQYHVWLJDWH WKH LQIOXHQFH RI SURSHUW\ FKDQJHV LQ IXOO GHWDLO KRZHYHU WKH SHUIRUPDQFH RI D[LDO KHDW IOX[ YHUVXV ZDWHU SURSHUWLHV DW WHPSHUDWXUH IURP r& WR r& KDV EHHQ H[DPLQHG 7KRXJK 0RGHO PD\ QRW EH DQ LGHDO GHYLFH DW ODUJH $[f IRU WKH DSSOLFDWLRQ RI HQKDQFHG WKHUPDO SXPSLQJ LW FDQ KRZHYHU EH D JRRG PRGHO IRU GHPRQVWUDWLQJ WKH LQIOXHQFH RI

PAGE 130

3U LF )LJ ,QIOXHQFH 2I 7KHUPRG\QDPLF 3URSHUWLHV RI +2 RQ WKH (QKDQFHG +HDW )OX[ 0RGHO D O $[ FPf 7LPH VHFf

PAGE 131

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f IRU D IL[HG D DQG SLSH UDGLXV WKH DQJXODU YHORFLW\ Z LV GLUHFWO\ SURSRUWLRQDO WR WKH NLQHPDWLF YLVFRVLW\ X 7KLV DOVR LPSOLHV WKDW WKH IOXLG ERXQGDU\ OD\HU WKLFNQHVV LV FRQVWDQW (J f 6R WKH VROLG FXUYH LQ )LJ LQ IDFW PDLQO\ UHIOHFWV D UHODWLRQVKLS EHWZHHQ WKH HQKDQFHG KHDW IOX[ DQG WKH RVFLOODWLQJ IUHTXHQF\ : ,W LV IRXQG WKDW D PD[LPXP HQKDQFHG KHDW IOX[ FDQ EH REWDLQHG LI WKH ZDWHU RI DSSUR[LPDWH WHPSHUDWXUH 7 r & LV HPSOR\HG 7KH FRUUHVSRQGLQJ DQJXODU YHORFLW\ DW WKLV WHPSHUDWXUH OHYHO LV Z UDGLDQVHF 3K\VLFDOO\ WKH H[LVWHQFH RI D SHDN YDOXH LQ )LJ PD\ EH H[SODLQHG DV IROORZV $W KLJKHU WHPSHUDWXUH VD\ 7 r & WKH NLQHPDWLF YLVFRVLW\ X LV TXLWH VPDOO DQG KHQFH WKH DQJXODU YHORFLW\ Z LV WRR VORZ WR WUDQVSRUW ODUJH DPRXQWV RI KHDW D[LDOO\ $V WKH

PAGE 132

WHPSHUDWXUH GHFUHDVHV Y JHWV ODUJHU DQG VR GRHV WKH DQJXODU YHORFLW\ X 7KXV PRUH KHDW FDQ EH H[SHFWHG WR EH WUDQVSRUWHG DQG WKLV HYHQWXDOO\ UHDFKHV WKH OLPLW PHQWLRQHG LQ WKH ODVW VHFWLRQ ,W VHHPV WKDW D VLPLODU IODWWHQLQJ SDWWHUQ LQ WKH KHDW IOX[ VKRXOG DOVR DSSHDU LI WKH WHPSHUDWXUH LV IXUWKHU GHFUHDVHG DQG KHQFH WKH IUHTXHQF\ Z IXUWKHU LQFUHDVHG +RZHYHU DV WHPSHUDWXUH GHFUHDVHV IXUWKHU WKH WKHUPDO FRQGXFWLYLW\ GHFOLQHV 7DEOH f VR WKDW LW ZHDNHQV WKH KHDW VXSSO\ DQG UHPRYDO FDSDELOLW\ DW WKH VRXUFHV DQG KHQFH PDNHV WKH D[LDO KHDW IOX[ FXUYH GHFOLQH IURP LWV PD[LPXP +HDW )OX[ 9HUVXV 7LGDO 'LVSODFHPHQW 0RGHO f $V LQGLFDWHG E\ .XU]ZHJ LQ HDUOLHU DQDO\WLF VWXGLHV > @ WKH HQKDQFHG D[LDO KHDW WUDQVIHU YHUVXV WKH RVFLOODWLQJ IORZ LV GLUHFWO\ SURSRUWLRQDO WR WKH VTXDUH RI WKH WLGDO GLVSODFHPHQW $ VHULHV RI QXPHULFDO WHVWV ZLWK 0RGHO ZHUH UXQ WR FKHFN WKLV SUHGLFWLRQ DQG WR VWXG\ WKH LQIOXHQFH RI YDULRXV IOXLG SURSHUWLHV DQG ERXQGDU\ FRQGLWLRQV RQ WKLV $[ HIIHFW 7KH FRPSXWDWLRQV ZHUH VHSDUDWHG LQWR WZR JURXSV LQ WKH ILUVW JURXS WKH :RUPHUVOH\ QXPEHU ZDV FKRVHQ YHU\ FORVH WR WKH WXQLQJ SRLQW LH D :DWHU ZDV FKRVHQ DV WKH ZRUNLQJ IOXLG DQG ERWK LQVXODWLQJ DQG FRQGXFWLQJ ZDOO FRQGLWLRQV JODVV DQG VWHHOf ZHUH FRQVLGHUHG 7KH FRPSXWDWLRQDO UHVXOWV DUH VKRZQ LQ )LJ ,Q WKH VHFRQG JURXS RI QXPHULFDO VWXGLHV WKH :RUPHUVOH\ QXPEHU ZDV FKRVHQ

PAGE 133

(QKDQFHG +HDW )OX[ I! :DWWFP.f )LJ +HDW )OX[ YHUVXV 7LGDO 'LVSODFHPHQW 0RGHO D 3U f

PAGE 134

7LGDO 'LVSODFHPHQW D[ FPf )LJ +HDW )OX[ YHUVXV 7LGDO 'LVSODFHPHQW 0RGHO D 3U f

PAGE 135

DV D ZKLFK GHYLDWHV IURP WKH WXQLQJ SRLQW $JDLQ ZDWHU ZDV FKRVHQ DV WKH ZRUNLQJ PHGLXP 6LPLODUO\ ERWK LQVXODWLQJ ZDOO DQG FRQGXFWLQJ JODVV RQO\f ZDOO FDVHV ZHUH LQYHVWLJDWHG 7KH UHVXOWV DUH VKRZQ LQ )LJ 7KH VROLG FXUYH )LJ f VKRZV WKH HQKDQFHG D[LDO KHDW IOX[ LQ RVFLOODWLQJ SLSH IORZ YHUVXV WLGDO GLVSODFHPHQW IRU WKH LQVXODWLQJ ZDOO FDVH DQG WKH GDVKHG FXUYH UHSUHVHQWV WKH HQKDQFHG D[LDO KHDW IOX[ REWDLQHG ZLWK D VWHHO ZDOO 7KH FXUYH VKRZV WKH HQKDQFHG D[LDO KHDW IOX[ FRPSXWHG IRU D JODVV ZDOO 7KHVH WKUHH FXUYHV FOHDUO\ FRQILUP WKH TXDGUDWLF EHKDYLRU W! NA$[ ZKHUH NI! LV D FRQVWDQW )RU WKLV VSHFLILF FDVH LW LV IRXQG WKDW NA IRU WKH LQVXODWLQJ ZDOO FDVH NA IRU WKH VWHHO ZDOO FDVH N IRU WKH JODVV ZDOO FDVH 7KH FRPSXWDWLRQDO UHVXOWV IRU WKH :RUPHUVOH\ QXPEHU D FDVH DUH VKRZQ LQ )LJ 7KHUH WKH VROLG FXUYH UHSUHVHQWV WKH HQKDQFHG D[LDO KHDW IOX[ LQ RVFLOODWLQJ SLSH IORZ ZLWK LQVXODWLQJ ZDOO WKH GDVKHG FXUYH VKRZV WKH JODVV ZDOO UHVXOWV 6LPLODUO\ WKH FRHIILFLHQW NA LQ WKH TXDGUDWLF IRUPXOD LV IRXQG WR KDYH WKH YDOXH NA IRU WKH LQVXODWLQJ ZDOO FDVH NA IRU WKH ZDWHUJODVV FRPELQDWLRQ

PAGE 136

,W LV REYLRXV WKDW WKH H[LVWHQFH RI FRQGXFWLQJ ZDOOV WHQGV WR LQFUHDVH KHDW VWRUDJH FDSDFLW\ DQG KHQFH HQKDQFH WKH UDGLDO DQG D[LDO KHDW WUDQVIHU SURFHVV 7KXV RQH FDQ H[SHFW D ODUJHU KHDW IOX[ IRU FRQGXFWLQJ ZDOOV WKDQ IRU LQVXODWLQJ ZDOOV DW WKH VDPH WLGDO GLVSODFHPHQW DQG :RUPHUVOH\ QXPEHU ,W FDQ EH VHHQ IURP )LJ DQG )LJ WKDW KLJKHU :RUPHUVOH\ QXPEHU D f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nV WHPSHUDWXUHf HQWHU WKH SLSH DW ERWK HQGV GXULQJ HYHU\ RVFLOODWLQJ SHULRG 7KLV LPSOLHV WKDW GLUHFW FRQYHFWLYH KHDW H[FKDQJH RFFXUV DW WKHVH

PAGE 137

HQGV GXULQJ HDFK RVFLOODWLRQ VR DV WR PDWFK WKH IDVW HQKDQFHG D[LDO KHDW WUDQVIHU SRVVLEOH ZLWKLQ WKH SLSH 2Q WKH RWKHU KDQG IRU WKH 0RGHO FDVH WKHUH LV QR VXFK LQILQLWH KHDW H[FKDQJH FDSDELOLW\ DW WKH KHDW VRXUFH DQG WKH VLQN VHFWLRQV ,Q IDFW WKH KHDW EHLQJ VXSSOLHG DQG UHPRYHG LV SXUHO\ E\ PROHFXODU KHDW FRQGXFWLRQ DW WKH LQWHUIDFH EHWZHHQ RI WKH WKHUPDO VRXUFHV DQG WKH IOXLG ERXQGDU\ OD\HU 7KLV FRQGXFWLYH KHDW VXSSO\ DQG UHPRYDO PRGH LV QRW FRPSDWLEOH ZLWK D YHU\ IDVW HQKDQFHG D[LDO KHDW WUDQVIHU ZLWKLQ WKH FRQQHFWLQJ SLSH IORZ DQG DFWXDOO\ SUHYHQWV WKH WKHUPDO SURFHVV IURP ZRUNLQJ DW SHDN HIILFLHQF\ 7KLV FRPSDULVRQ VXJJHVWV WR XV WKDW RQH VKRXOG FRQVLGHU D VWURQJ DQG SUHIHUDEO\ WXUEXOHQW FRQYHFWLYH PRWLRQ LQ ERWK HQG UHVHUYRLUV DV D PHDQV WR LQFUHDVH WKH HIILFLHQF\ RI WKH HQKDQFHG WKHUPDO SXPSLQJ WHFKQLTXH 7KH ,QIOXHQFH RI :DOO 7KLFNQHVV $V PHQWLRQHG DERYH WKH FRQGXFWLQJ ZDOO ZLOO LQ JHQHUDO PDNH WKH HQKDQFHG D[LDO KHDW WUDQVIHU SURFHVV PRUH HIILFLHQW VLQFH WKH H[LVWHQFH RI D FRQGXFWLQJ ZDOO FUHDWHV DQ DGGLWLRQDO KHDW VWRUDJH FDSDELOLW\ QHDU WKH IOXLGVROLG LQWHUIDFH 7KLV DGGLWLRQDO KHDW VWRUDJH VXSSOHPHQWV WKH VWRUDJH FDSDFLW\ RI WKH ERXQGDU\ OD\HU DQG KHQFH VWUHQJWKHQV WKH HQKDQFHG KHDW H[FKDQJH SURFHVV RI WKH ZKROH V\VWHP 7KH ODUJHVW UDGLDO GLVWDQFH ZKLFK WKH KHDW FDQ EH WUDQVIHUUHG LQWR WKH ZDOO LV WKH ZDOO WKHUPDO SHQHWUDWLQJ WKLFNQHVV 6Z 7KLV WKLFNQHVV LV D IXQFWLRQ RI RVFLOODWLQJ

PAGE 138

IUHTXHQF\ DV ZHOO DV WKH WKHUPDO SURSHUWLHV RI WKH VROLG ZDOO $V PHQWLRQHG LQ WKH LQWURGXFWLRQ FKDSWHU )LJ f WKH IXWXUH HQJLQHHULQJ DSSOLFDWLRQV RI WKH WKHUPDO SXPSLQJ WHFKQLTXH SUREDEO\ ZLOO UHTXLUH D EXQGOH RI FDSLOODU\ WXEHV FRQQHFWLQJ WKH FROG DQG KRW IOXLG UHVHUYRLUV ,Q YLHZ RI VSDFH OLPLWDWLRQV DQG HFRQRPLFV RQH ZDQWV WR PLQLPL]H WKH FDSLOODU\ ZDOO WKLFNQHVV XVHG LQ WKH WKHUPDO SXPS \HW VWLOO EH DEOH WR PDNH IXOO XVH RI WKH ZDOO KHDW VWRUDJH FDSDFLW\ 8VLQJ 0RGHO D QXPHULFDO LQYHVWLJDWLRQ RI HQKDQFHG D[LDO KHDW IOX[ LQ RVFLOODWLQJ SLSH IORZ LQ D ZDWHUJODVV FRPELQDWLRQ IRU YDULRXV ZDOO WKLFNQHVVHV KDV EHHQ SHUIRUPHG DQG WKH FRPSXWHG UHVXOWV UHFRUGHG LQ )LJ ,Q WKLV VWXG\ WKH WLGDO GLVSODFHPHQW ZDV WDNHQ DV $[ FP /f WKH :RUPHUVOH\ QXPEHU D DQG LQQHU SLSH UDGLXV FP :DWHU DW r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

PAGE 139

(QKDQFHG +HDW )OX[ :DWWFPr.f :DOO 7KLFNQHVV $U 5 f rrL FPf )LJ ,QIOXHQFH RI :DOO 7KLFNQHVV RQ $[LDO +HDW IOX[ 0RGHO :DWHU*ODVV D $[ FPf

PAGE 140

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r& ZDWHU ZDV WDNHQ DV WKH ZRUNLQJ IOXLG DQG WKH ZDOO PDWHULDO ZDV JODVV )URP WKH GHILQLWLRQ RI :RUPHUVOH\ QXPEHU f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

PAGE 141

(QKDQFHG +HDW )OX[ I! :DWWFPr.f 3LSH GLDPHWHU FPf )LJ ,QIOXHQFH RI 3LSH 'LDPHWHU RQ +HDW IOX[ IRU )L[HG )UHTXHQF\ 0RGHO :DWHUJODVV D $[ FPf

PAGE 142

2 / )LJ 7\SLFDO ,VR7HPSHUDWXUH &RQWRXU LQ 2VFLOODWLQJ 3LSH )ORZ D[ FP 5A FP r FPf 0RGHO D

PAGE 143

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r DQG r RU r DQG r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

PAGE 144

7HPSHUDWXUH 7LGDO 'LVSODFHPHQW D[ FPf )LJ 9DULDWLRQ RI 7HPSHUDWXUH 7[ DQG 7 YHUVXV D[ 0RGHO :DWHU*ODVV D f

PAGE 145

D[LDO WHPSHUDWXUH JUDGLHQW YDULDWLRQ FDQ EH FDOFXODWHG E\ PHDVXULQJ WKH WLPHDYHUDJHG WHPSHUDWXUHV DW WKH MRLQW SRLQWV [ / DQG / VHH )LJ f 7KH FRPSXWHG UHVXOWV DUH VKRZQ LQ )LJ ,Q WKH FDOFXODWLRQ WKH :RUPHUVOH\ QXPEHU ZDV FKRVHQ DV D DQG WKH ZRUNLQJ PHGLXP LV ZDWHU ZLWKLQ D JODVV SLSH ,Q WKH ILJXUH 7 LV WKH WLPHDYHUDJHG WHPSHUDWXUH FRPSXWHG DW WKH OHIW MRLQW [ /f ZKLFK LV FORVH WR WKH KRW VRXUFH ZKLOH 7A DW ULJKW MRLQW LV FORVH WR WKH FROG VRXUFH ,W LV FOHDUO\ VHHQ WKDW 7 GHFUHDVHV ZKLOH 7L LQFUHDVHV DV WKH WLGDO GLVSODFHPHQWV LQFUHDVH ,W VHHPV WKDW ERWK DQG 7 WHQG WR WKH GLPHQVLRQOHVV PHDQ YDOXH LH f LI WKH WLGDO GLVSODFHPHQW LV JUHDWO\ LQFUHDVHG 7KLV LPSOLHV WKDW WKH WLPHDYHUDJHG D[LDO WHPSHUDWXUH JUDGLHQW LQ WKH FHQWUDO VHFWLRQ ZLOO GHFUHDVH DQG WKDW WKH HQKDQFHG D[LDO KHDW IOX[ ZLOO EH ZHDNHQHG 7DEOH 9DULDWLRQ RI WKH $[LDO 7HPSHUDWXUH *UDGLHQW YHUVXV :RUPHUVOH\ 1XPEHU :DWHU*ODVV $[ FPf 7I 7ffff DUH GLPHQVLRQOHVV 7HPSHUDWXUHVf D $W 3HULRGV W 7O $7 7a7@B

PAGE 146

DOWKRXJK VWLOO LQFUHDVLQJ ZLWK LQFUHDVLQJ $[ +RZHYHU WKH QXPHULFDO UHVXOWV VKRZ WKDW VXFK D ZHDNHQLQJ RI WKH D[LDO WHPSHUDWXUH JUDGLHQW FDQ EH LPSURYHG DW KLJKHU RVFLOODWLQJ IUHTXHQF\ 7DEOH f 7KLV LV EHFDXVH DW KLJKHU Z WKH SLSH IORZ GHYHORSV D PXFK WKLQQHU ERXQGDU\ OD\HU LQ WKH YLFLQLW\ RI WKH ZDOO DQG KHQFH D ODUJHU UDGLDO WHPSHUDWXUH JUDGLHQW ZKLFK LQFUHDVHV WKH FDSDELOLW\ RI KHDW VXSSO\ DQG UHPRYDO IURP WKH H[WHQGHG VHFWLRQV ,Q IDFW WKH QXPHULFDO UHVXOWV DOVR VKRZ WKDW RQO\ YHU\ QDUURZ DUHDV RI WKH H[WHQGHG VHFWLRQ DUH LQYROYHG LQ VXSSO\LQJ DQG UHPRYLQJ KHDW ZKHQ WKH RVFLOODWLQJ IUHTXHQF\ LV ODUJH )LJ f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f RU LQ WKH IOXLG FXUYH c!If ZKHQ WKH WLGDO GLVSODFHPHQW LV YHU\ VPDOO $[ OHVV WKDQ FP LQ WKLV FDVHf KRZHYHU LQFUHDVHV GUDPDWLFDOO\ ZKHQ WKH WLGDO GLVSODFHPHQW EHFRPHV ODUJHU ,Q FRQWUDVW WKH SXUH D[LDO PROHFXODU FRQGXFWLRQ GHFOLQHV VOLJKWO\ GXH D ZHDNHQLQJ RI

PAGE 147

+HDW )OX[ I! :DWWFPR.f )LJ &RPSDULVRQ RI (QKDQFHG +HDW 7UDQVIHU DQG +HDW &RQGXFWLRQ LQ 2VFLOODWLQJ 3LSH )ORZ 0RGHO :DWHU*ODVV D f

PAGE 148

WKH D[LDO WHPSHUDWXUH JUDGLHQW LQ WKH FHQWUDO SLSH VHFWLRQ DW ODUJH $[ DV GLVFXVVHG LQ WKH ODVW VHFWLRQ 7KH UDWLR RI WKH HQKDQFHG KHDW IOX[ WR WKH VTXDUH RI WKH WLGDO GLVSODFHPHQW LV DOVR VKRZQ LQ WKH VDPH ILJXUH FXUYH Sf ,W LQGLFDWHV D GHFOLQLQJ YDOXH RI S DV WKH WLGDO GLVSODFHPHQW LQFUHDVHV 7KLV LQGLFDWHV WKDW WKH QG SRZHU UHODWLRQVKLS EHWZHHQ WKH HQKDQFHG D[LDO KHDW IOX[ I! DQG WLGDO GLVSODFHPHQW $[ KDV DOVR EHHQ ZHDNHQHG LH ORZHU NAf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f 7KH VROLG FXUYHV VKRZ WKH FRPSXWHG UHVXOWV LQ WKH ZDWHUJODVV ZDOO FDVH ZKLOH WKH GDVKHG FXUYHV UHSUHVHQW WKH QXPHULFDO VROXWLRQ REWDLQHG ZLWK WKH PHUFXU\VWHHO ZDOO FRPELQDWLRQ 7KH WLGDO GLVSODFHPHQW D[ XVHG LQ WKHVH FDOFXODWLRQV LV IL[HG DQG HTXDOV FP 7KH VXEVFULSWV KDYH WKH VDPH PHDQLQJ DV WKRVH XVHG LQ )LJ

PAGE 149

+HDW )OX[ I! :DWWFPA .f :RUPHUVOH\ D )LJ 9DULDWLRQ RI $[LDO +HDW )OX[ 9HUVXV :RUPHUVOH\ 1XPEHU 0RGHO $[ FPf

PAGE 150

,W FDQ EH VHHQ IURP )LJ WKDW ERWK WKH HQKDQFHG D[LDO KHDW IOX[ I! DQG D[LDO KHDW IOX[ GXH WR SXUH PROHFXODU FRQGXFWLRQ LQ HLWKHU WKH ZDOO Zf RU LQ WKH IOXLG I! If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

PAGE 151

I!X :DWWr VHFFPA )LJ &RPSXWHG 7XQLQJ &XUYHV 0RGHO +6WHHO DQG +JVWHHO D[ FPf

PAGE 152

WKH ULJKW FXUYH UHSUHVHQWV WKH PHUFXU\JODVV ZDOO FDVH DV PDUNHG LQ WKH GLDJUDP ,W FDQ EH VHHQ WKDW ERWK FXUYHV VKRZ WKH H[LVWHQFH RI WKH VRFDOOHG WXQLQJ HIIHFW 7KH RSWLPXP YDOXH RI 5L IRU WKH ZDWHUJODVV FDVH 3U m VHFf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

PAGE 153

.J Z$; D f§ 5MXY )LJ 7XQLQJ &XUYH YHUVXV :RUPHUVOH\ 1XPEHU DIWHU .XU]ZHJ >@ f

PAGE 154

:RUPHUVOH\ 1XPEHU D 5MX!X )LJ 5DWLR RI +HDW &RQGXFWLRQ WR (QKDQFHG +HDW )OX[ 9HUVXV :RUPHUVOH\ 1XPEHU 0RGHO +6WHHO +J6WHHO D[ FPf

PAGE 155

LQ WKH VROLG ZDOO WR WKH HQKDQFHG D[LDO KHDW IOX[ IRU ERWK WKH ZDWHUJODVV DQG PHUFXU\VWHHO FRPELQDWLRQV 7KHVH FXUYHV ORRN OLNH XSVLGHGRZQ WXQLQJ FXUYHV 7KH\ FOHDUO\ VKRZ WKDW RQO\ DW RU QHDU WKH WXQLQJ SRLQWV WKH D[LDO KHDW IOX[ JHQHUDWHG E\ WKH SXUH PROHFXODU KHDW FRQGXFWLRQ HLWKHU LQ IOXLG RU WKH FRQGXFWLQJ ZDOO LV TXLWH VPDOO FRPSDUHG WR WKH HQKDQFHG KHDW IOX[ DQG KHQFH QHJOLJLEOH DERXW IRU WKH ZDWHUVWHHO ZDOO FDVH DQG DERXW IRU WKH PHUFXU\VWHHO ZDOO FDVHf +RZHYHU LI RQH GHSDUWV IURP WKH WXQLQJ SRLQW WKLV UDWLR JURZV TXLFNO\ DQG HYHQWXDOO\ PD\ EH JUHDWHU WKDQ 7KH WXEH UDGLL FRUUHVSRQGLQJ WR WKH WXQLQJ SRLQWV D IRU ZDWHUVWHHO D IRU PHUFXU\VWHHOf DW Z UDGVHF FRUUHVSRQG WR FP IRU ZDWHUVWHHO DQG 5L FP IRU PHUFXU\VWHHO

PAGE 156

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

PAGE 157

HQHUJ\ HTXDWLRQV ZLWK D WLPH GHSHQGHQW $OWHUQDWLQJ 'LUHFWLRQ ,PSOLFLW PHWKRG $',f $ FRPSXWHU FRGH QDPHG (73 (QKDQFHG 7KHUPDO 3XPSLQJf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

PAGE 158

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f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

PAGE 159

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

PAGE 160

HQODUJHG KHDW VRXUFH VL]H SLSH H[WHQVLRQf LW LV VWLOO QRW DEOH WR PDWFK WKH YHU\ IDVW HQKDQFHG D[LDO KHDW WUDQVIHU SRVVLEOH LQ WKH FRQQHFWLQJ SLSH ,W LV IRXQG WKDW FRQVWDQW HQG WHPSHUDWXUH DW [ / DQG /f FDQ QRW EH PDLQWDLQHG DQG WKH TXDGUDWLF UHODWLRQVKLS EHWZHHQ WKH HQKDQFHG KHDW IOX[ DQG WKH WLGDO GLVSODFHPHQW ZLOO QR ORQJHU KROG )LJ f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f 7KH LQIOXHQFH RI WKH YDULDWLRQ RI WKHUPDO DQG YLVFRXV SURSHUWLHV RI ZRUNLQJ IOXLG RQ WKH D[LDO KHDW IOX[ KDV DOVR EHHQ VWXGLHG 7KH QXPHULFDO VROXWLRQV VKRZ WKDW HYHQ LQ WKH UHODWLYHO\ QDUURZ WHPSHUDWXUH UDQJH IURP r& WR r& WKH D[LDO KHDW IOX[ YDULHV PRUH WKDQ SHUFHQW

PAGE 161

([WHQVLRQV RI WKH QXPHULFDO VWXGLHV FRQVLGHUHG KHUH VKRXOG LQFOXGH f GHYHORSLQJ D RU DW OHDVW DQ D[LV\PPHWULF PRGHO ZKLFK LV DEOH WR EHVW DSSUR[LPDWH WKH HQWLUH V\VWHP LQFOXGLQJ ERWK UHVHUYRLUV VR WKDW RQH PD\ PRUH DFFXUDWHO\ VWXG\ WKH HIIHFW RI KHDW H[FKDQJH ZLWKLQ WKH UHVHUYRLUV f WKH UROH RI WXUEXOHQFH LQ VXFK RVFLOODWLQJ IORZV DW KLJK IUHTXHQF\ Z DQG WLGDO GLVSODFHPHQW $[ VKRXOG UHFHLYH VSHFLDO DWWHQWLRQ DV ZHOO DV WKH LQHUWLD IRUFHV WKDW DUH NQRZQ WR EHFRPH ODUJH DW KLJKHU RVFLOODWLQJ IUHTXHQF\ ,Q IDFW D WXUEXOHQW PRGHOOLQJ LV QHFHVVDU\ HVSHFLDOO\ LI WKH VWXG\ LQFOXGHV KHDW VRXUFH DQG KHDW VLQN UHVHUYRLUV f D FRQVLGHUDWLRQ RI WKH YDULDWLRQ RI WKH WKHUPRG\QDPLF SURSHUWLHV RI ZRUNLQJ IOXLG ZLWK WHPSHUDWXUH DQG KHQFH VSDWLDO SRVLWLRQ VKRXOG DOVR EH LQFOXGHG LQ DQ\ H[WHQVLRQ RI WKLV UHVHDUFK RQ HQKDQFHG KHDW WUDQVIHU E\ RVFLOODWLRQV RI YLVFRXV IOXLGV LQ SLSHV

PAGE 162

$33(1',; (73 &20387(5 &2'( $Q D[LV\PPHWULF FRGH ZLWK GHVLJQDWLRQ (73 KDV EHHQ GHYHORSHG IRU FRPSXWLQJ WKH WLPHGHSHQGHQW (QKDQFHG 7KHUPDO 3XPSLQJ SUREOHP ZLWK HLWKHU RVFLOODWLQJ IORZ SLSH RU IODW SODWHf RU VWHDG\ IORZ 7KH JULG LV JHQHUDWHG ZLWK D FOXVWHULQJ LQ WKRVH UHJLRQV ZKHUH KLJK YHORFLW\ DQG WHPSHUDWXUH JUDGLHQWV DUH H[SHFWHG %RWK 0RGHO DQG 0RGHO JHRPHWU\ DUH SURJUDPHG LQ WKH FXUUHQW FRGH $QRWKHU VHSDUDWH FRGH 0,;3,3f LV DOVR DYDLODEOH IRU WKH 0RGHO JHRPHWU\ ZKLFK LV QRW GLVFXVVHG LQ WKLV DSSHQGL[ 'HVFULSWLRQ RI ,QSXW 9DULDEOHV 7KH IROORZLQJ VWDWHPHQWV DUH XVHG LQ WKH (73 FRGH 5($' f 10$;.0$;-0$;,&/86,3,3(,35(6 5($' f .0,',5($'-3(5,&+ 5($' f '6508.528&35.6&.6 5($' f $/)$355/772$03 5($' f 5$' 5$',5/(1'8(367(36 5($' f ,+,f ,&+f )250$7 f )250$7 )f ZKHUH 10$; WRWDO WLPH VWHSV ZLWKLQ RQH SHULRG .0$; WRWDO QRGDO SRLQWV RQ UDGLXV -0$; WRWDO QRGDO SRLQWV RQ D[LV

PAGE 163

,&/86 ,3,3( ,35(6 .0,' ,5($' -3(5 ,&+ '6 508. 528 &3 5.6 &.6 $/)$ 35 JULG ZLWK FOXVWHULQJ XQLIRUP *ULG 3LSH JHRPHWU\ )ODW SODWH JHRPHWU\ VLQXVRLGDO SUHVVXUH JUDGLHQW FRQVWDQW SUHVVXUH JUDGLHQW QRGH LQGH[ DW LQWHUIDFH DORQJ UDGLXV .0,' .0$; LQVXODWLQJ ZDOO .0$; FRQGXFWLQJ ZDOO VWDUW D QHZ MRE VWDUW IORZ SDUW RQO\ VWDUW KHDW SDUW RQO\ EDVHG RQ H[LVWLQJ IORZ YDULDEOHVf UHVWDUW ERWK IORZ t KHDW ILHOG FDOFXODWLRQ UHVWDUW IORZ ILHOG FDOFXODWLRQ RQO\ QXPEHU RI SHULRG LQ WKH FXUUHQW UXQ 0RGHO 0RGHO GLVWDQFH RI JULG QH[W WR ZDOO NLQHPDWLF 9LVFRVLW\ RI IOXLG GHQVLW\ RI IOXLG VSHFLILF KHDW RI IOXLG WKHUPDO GLIIXVLYLW\ RI VROLG ZDOO KHDW FRQGXFWLYLW\ RI ZDOO :RUPHUVOH\ QXPEHU 3UDQGWO QXPEHU

PAGE 164

5/ /HQJWK / RI SLSH RU IODW SODWH 7 7HPSHUDWXUH LQ WKH KRW UHVHUYRLU 7 7HPSHUDWXUH LQ WKH FROG UHVHUYRLU $03 7KH DPSOLWXGH RI D[LDO SUHVVXUH JUDGLHQW $03 58frGS'; 5$' LQQHU UDGLXV RI SLSH 5$', RXWHU UDGLXV RI SLSH 5/(1' HDFK H[WHQVLRQ SRUWLRQ OHQJWK 8(36 WKH DOORZDQFH YHORFLW\ UHVLGXDO 7(36 WKH DOORZDQFH WHPSHUDWXUH UHVLGXDO ,+,f QRGH LQGH[ RI VHSDUDWLQJ VHJPHQWV LQ 0RGHO 1RWH WR UXQ DQG UHVWDUW D MRE ZLWK WKLV FRGH RQH RI WKH SURFHGXUHV KDV WR EH IROORZHG f ,5($' VWDUW ERWK IORZ t KHDW ILHOG UHVWDUW ERWK IORZ t KHDW ILHOG RU f ,5($' VWDUW IORZ ILHOG RQO\ VWDUW KHDW t UHVWDUW IORZ ILHOG UHVWDUW ERWK IORZ t KHDW ILHOGV $ 6DPOH ,QRXW 'DWD DQG %DWFK &RPPDQG )LOH 7KH IROORZLQJ VDPSOH LQSXW ZDV XVHG LQ WKH FRPSXWDWLRQ ZLWK 0RGHO ZDWHU JODVVf

PAGE 165

7KH IROORZLQJ &RPPDQG ILOH ZDV XVHG WR VWDUW WKH UXQ RI D QHZ FDVH $66,*1 6$03/2,13 )25 $66,*1 6$03/2'$7 )25 $66,*1 ),/('$7 )25 $66,*1 ),/('$7 )25 $66,*1 ),/('$7 )252OO 5 (37 7R UHVWDUW WKH SUHYLRXV UXQ WKH IROORZLQJ FRPPDQG ILOH ZDV XVHG 5(1$0( ),/('$7 ),/('$7 5(1$0( ),/('$7 ),/('$7 5(1$0( ),/('$7 ),/('$7 $66,*1 6$03/,13 )25 $66,*1 6$03/'$7 )25 $66,*1 ),/('$7 )25 $66,*1 ),/('$7 )25 $66,*1 ),/('$7 )25 $66,*1 ),/('$7 )252,2 $66,*1 ),/('$7 )252OO $66,*1 ),/('$7 )25 5 (37 1RWH WKH QDPH RI WKH GDWD ILOHV FRXOG EH FKDQJHG DFFRUGLQJ XVHUnV WDVWH )XQFWLRQ RI 6XEURXWLQH LQ (73 &RGH ,Q WKH (73 FRGH WKH 3URJUDP 0$,1 FRQWUROV WKH PDLQ ORRS LQ WKH FRPSXWDWLRQ LQFOXGLQJ WKH LQSXW DQG RXWSXW GDWD FRQWURO UHVWDUW FRQWURO $OVR LW VHWV XS WKH LQLWLDO FRQGLWLRQ DFFRUGLQJ WR WKH SXUSRVH RI LQYHVWLJDWLRQ 7KH (73 FRGH LQFOXGHV VXEURXWLQHV 7KH IXQFWLRQ RI HDFK VXEURXWLQH LV OLVWHG EHORZ 6XEURXWLQH &200(17 7KLV VXEURXWLQH JLYHV WKH GHVFULSWLRQ RI WKH LQSXW DQG RXWSXW YDULDEOHV OLVWHG LQ SUHYLRXV VHFWLRQ

PAGE 166

6XEURXWLQH *5,' 7KLV VXEURXWLQH LV XVHG WR JHQHUDWH HLWKHU D XQLIRUP RU D FOXVWHUHG JULG V\VWHP LQ WKH SK\VLFDO GRPDLQ IRU 0RGHO RU 0RGHO LW FDOOV WKH VXEURXWLQH &/86 WR FOXVWHU WKH JULG OLQHV LQ WKH YLFLQLW\ RI ZDOO DQG WKH FHQWUDO SDUW RI SLSH 0RGHO f ,W DOVR FDOOV VXEURXWLQH 0(73,3 0(7:$/ WR FRPSXWH WKH GHULYDWLYHV WHUPV LQ WKH WUDQVIRUPDWLRQ 6XEURXWLQH &/86 ,&/86.A 7KLV VXEURXWLQH LV FDOOHG E\ *5,' WR FRPSXWH WKH FOXVWHULQJ JULG GLVWULEXWLRQ ZLWK HTXDWLRQ f 6XEURXWLQH 0(73,3 7KLV VXEURXWLQH LV FDOOHG WR FRPSXWH WKH GHULYDWLYHV RI WKH FRRUGLQDWH WUDQVIRUPDWLRQ LQ WKH LQQHU SLSH DUHD 6XFK DV [ G[GUc XVHG LQ HTXDWLRQV IURP f WR f DV ZHOO DV LQ WKH ERXQGDU\ FRQGLWLRQ 6XEURXWLQH 0(7:$/ 6LPLODUO\ VXEURXWLQH 0(7:$/ LV FDOOHG ZLWK WKH VDPH HTXDWLRQV WR FRPSXWH WKH GHULYDWLYHV LQ WUDQVIRUPDWLRQ LQ WKH VROLG ZDOO LI D FRQGXFWLQJ ZDOO LV DVVXPHG 6XEURXWLQH )/2: 1f 7KLV VXEURXWLQH LV FDOOHG WR FRPSXWH WKH YHORFLW\ LQ WKH IORZ ILHOG DW HDFK WLPH VWHS ZLWK WKH &UDQN1LFROVRQ 0HWKRG (TXDWLRQ f RU WKH PDWUL[ IRUP f ZLOO EH VROYHG LQ WKLV VXEURXWLQH 6XEURXWLQH 7,'$/ 7KLV VXEURXWLQH LV FDOOHG WR &RPSXWH WKH /DJUDQJLDQ

PAGE 167

GLVSODFHPHQW f WLGDO GLVSODFHPHQWf DQG WKH SKDVH ODJV ZKLFK DUH UHODWLYH WR WKH H[FLWLQJ SUHVVXUH JUDGLHQW SKDVH 6XEURXWLQH &2)/2: 7KLV VXEURXWLQH LV FDOOHG WR FRPSXWH WKH FRHIILFLHQWV LQ WKH PRPHQWXP HTXDWLRQ DW HDFK WLPH VWHS LH FRPSXWLQJ HTXDWLRQV f DQG f 6XEURXWLQH 5+6)/2 e1M 7KH VXEURXWLQH 5+6)/2 1f LV FDOOHG WR FRPSXWH WKH XSGDWH ULJKW KDQG VLGH WHUPV LQ WKH PRPHQWXP HTXDWLRQ DW HDFK WLPH VWHS LH HTXDWLRQV IURP f WR f 6XEURXWLQH 75,3 $ VXEURXWLQH 75,' ERUURZHG IURP >@f LV WR VROYH WKH WULGLDJRQDO V\VWHP DOJHEUD HTXDWLRQV LW LV ZLGHO\ XVHG WR FRPSXWH WKH XSGDWH YHORFLW\ DQG WHPSHUDWXUH 6XEURXWLQH (36,/ )XQFWLRQ (36,/ ERUURZHG IURP FRGH *5,'*(1f WR ILQG WKH H YDOXH XVHG LQ HTXDWLRQ f ZLWK 1HZWRQ5DSKVRQ URRWILQGLQJ WHFKQLTXH 6XEURXWLQH $',68 7KLV VXEURXWLQH LV FDOOHG XVLQJ $', PHWKRG WR FRPSXWH WKH WHPSHUDWXUH GLVWULEXWLRQ LQ RVFLOODWLQJ SLSH IORZ LQ 0RGHO 7KH LQVXODWLQJ ZDOO ERXQGDU\ FRQGLWLRQ LV DVVXPHG LQ WKH FHQWUDO SLSH / [ /f ZLWK IL[HG WHPSHUDWXUH DW WKH IDU HQGV RI ERWK H[WHQVLRQ SLSH (TXDWLRQV IURP f f RU WKH PDWUL[ IURP f f DUH VROYHG

PAGE 168

6XEURXWLQH &273,3 7KLV VXEURXWLQH LV FDOOHG E\ $',68 WR FDOFXODWH WKH FRHIILFLHQWV LQ WKH HQHUJ\ HTXDWLRQ RI SLSH IORZ f DW HDFK WLPH VWHS ZKLFK DUH FRXSOHG E\ WKH XSGDWH YHORFLW\ 6XEURXWLQH 5+6,17 1,,/,1(f 7KLV VXEURXWLQH LV FDOOHG E\ $',68 WR FRPSXWH WKH XSGDWH ULJKW KDQG VLGH WHUPV LQ WKH HQHUJ\ HTXDWLRQ RI SLSH IORZ ZLWK WKH LQVXODWLQJ ZDOO FDVH QDPHO\ VROYLQJ f f 6XEURXWLQH )/8;7 I11O 6XEURXWLQH )/8;7 11f LV FDOOHG WR FRPSXWH WKH GLPHQVLRQOHVV HQKDQFHG D[LDO KHDW IOX[ DV ZHOO DV WKH KHDW IOX[ E\ SXUH PROHFXODU FRQGXFWLRQ HLWKHU LQ WKH ZDOO RU WKH SLSH IOXLG QDPHO\ HTXDWLRQV f WR f 6XEURXWLQH 287 17,'A 7KH VXEURXWLQH 287 17,'f LV FDOOHG WR FRPSXWH DQG RXWSXW WKH GLPHQVLRQDO KHDW IOX[ WLGDO GLVSODFHPHQW DQG WKH DYHUDJH WHPSHUDWXUH GLPHQVLRQOHVVf 6XEURXWLQH $',& 6XEURXWLQH $',& LV VLPLODU WR $',68 EXW IRU WKH FRQGXFWLQJ ZDOO FDVH 6XEURXWLQH &27:$/ 7KLV VXEURXWLQH LV FDOOHG WR FDOFXODWH WKH FRHIILFLHQWV LQ WKH ZDOO FRQGXFWLRQ HTXDWLRQ DW HDFK WLPH VWHS E\ XVLQJ HTXDWLRQ f

PAGE 169

6XEURXWLQH 5+6&27 1,,/,1(f 6XEURXWLQH 5+6&27 LV VLPLODU WR 5+6,17 EXW IRU WKH FRQGXFWLQJ ZDOO FDVH 6XEURXWLQH 5+6:$/ &1,,/,1(f 6XEURXWLQH 5+6:$/ 1,,/,1(f LV FDOOHG WR FRPSXWH WKH ULJKWKDQGVLGH WHUPV LQ WKH ZDOO FRQGXFWLRQ HTXDWLRQ ZLWK f DQG f 6XEURXWLQH $',68 LQ 6XEURXWLQH $',68 LV VLPLODU WR $',68 EXW IRU WKH FDVH RI 0RGHO 6XEURXWLQH $',& I1f 6XEURXWLQH $',& LV VLPLODU WR $',68 EXW IRU WKH FRQGXFWLQJ ZDOO

PAGE 170

5()(5(1&(6 $EUDPRZLW] 0 DQG 6WHJXQ $HGLWRUVf +DQGERRN RI 0DWKHPDWLFV )XQFWLRQV ZLWK )RUPXODV *UDSK\V DQG 0DWKHn PDWLFDO 7DEOHV 'RYHU 3XEOLFDWLRQV ,QF 1HZ
PAGE 171

-DHJHU 0 DQG .XU]ZHJ 8 + 'HWHUPLQDWLRQ RI WKH /RQJLWXGLQDO 'LVSHUVLRQ &RHIILFLHQW LQ )ORZV 6XEMHFWHG WR +LJK)UHTXHQF\ 2VFLOODWLRQV 3K\V )OXLG 9RO 1R SS f -RVKL & + .DPP 5 'UD]HQ 0 DQG 6OXWVOH\ $ 6 $Q ([SHULPHQWDO 6WXG\ RI *DV ([FKDQJH LQ /DPLQDU 2VFLOODWLQJ )ORZV -RXUQDO RI )OXLG 0HFK 9RO SS f .DYLDQ\ 0 6RPH $VSHFWV RI +HDW 'LIIXVLRQ LQ )OXLGV E\ 2VFLOODWLRQ ,QWHUQDWLRQDO -RXUQDO +HDW 0DVV 7UDQVIHU 9RO SS f .XU]ZHJ 8 + (QKDQFHG +HDW &RQGXFWLRQ LQ )OXLGV 6XEMHFWHG WR 6LQXVRLGDO 2VFLOODWLRQV -RXUQDO RI +HDW 7UDQVIHU 9RO SS f .XU]ZHJ 8 + (QKDQFHG +HDW &RQGXFWLRQ LQ 2VFLOODWLQJ 9LVFRXV )ORZV :LWKLQ 3DUDOOHO3ODWH &KDQQHOV -RXUQDO RI )OXLG 0HFK 9RO SS f .XU]ZHJ 8 + 7HPSRUDO DQG 6SDWLDO 'LVWULEXWLRQ RI +HDW )OX[ LQ 2VFLOODWLQJ )ORZ 6XEMHFWHG WR DQ $[LDO 7HPSHUDWXUH *UDGLHQW ,QWHUQDWLRQDO -RXUQDO +HDW 0DVV 7UDQVIHU 9RO 1R SS f .XU]ZHJ 8 + (QKDQFHG 'LIIXVLRQDO 6HSDUDWLRQ LQ /LTXLGV E\ 6LQXVRLGDO 2VFLOODWLRQV WR DSSHDU LQ 6HSDUDWLRQ 6FLHQFHV DQG 7HFKQRORJ\ LQ SUHVV .XU]ZHJ 8 + DQG -DHJHU 0 'LIIXVLRQDO 6HSDUDWLRQ RI *DVHV E\ 6LQXVRLGDO 2VFLOODWLRQV 3K\V )OXLGV 9RO SS f .XU]ZHJ 8 + DQG -DHJHU 0 7XQLQJ (IIHFW LQ *DV 'LVSHUVLRQ XQGHU 2VFLOODWRU\ &RQGLWLRQV 3K\V )OXLGV 9RO SS f .XU]ZHJ 8 + DQG /LQGJUHQ ( 5 (QKDQFHG +HDW &RQGXFWLRQ %\ 2VFLOODWRU\ 0RWLRQ RI )OXLGV LQ &RQGXLWV $ 5HVHDUFK 3URSRVDO WR WKH )OXLG '\QDPLFV DQG +HDW 7UDQVIHU 3URJUDP 7KH 1DWLRQDO 6FLHQFH )RXQGDWLRQ XQGHU &RQWUDFW 1XPEHU &%7 f .XU]ZHJ 8 + DQG =KDR / +HDW 7UDQVIHU E\ +LJK )UHTXHQF\ 2VFLOODWLRQV $ 1HZ +\GURG\QDPLF 7HFKQLTXH IRU $FKLHYLQJ /DUJH (IIHFWLYH 7KHUPDO &RQGXFWLYLWLHV 3K\V )OXLGV 9RO SS f

PAGE 172

/L & 3 $ )LQLWH 'LIIHUHQFH 0HWKRG IRU 6ROYLQJ 8QVWHDG\ 9LVFRXV )ORZ 3UREOHPV $,$$ -RXUQDO 9RO 1R f 0DF&RUPDFN 5 : &XUUHQW 6WDWXV RI 1XPHULFDO 6ROXWLRQV RI WKH 1DYLHU6WRNHV (TXDWLRQV $,$$ UG $HURVSDFH 6FLHQFHV 0HHWLQJ 5HQR 1HYDGD $,$$ -DQ f 0DVWLQ & : (UURU ,QGXFHG E\ &RRUGLQDWH 6\VWHP LQ ) 7KRPVRQ HGLWRUf 1XPHULFDO *ULG *HQHUDWLRQ 1RUWK+ROODQG 1HZ
PAGE 173

7KRPSVRQ ) :DUVL = 8 $ DQG 0DVWQ & : %RXQGDU\)LWWHG &RRUGLQDWH 6\VWHPV IRU 1XPHULFDO 6ROXWLRQ RI 3DUWLDO 'LIIHUHQWLDO (TXDWLRQV $ 5HYLHZ -RXUQDO RI &RPSXWDWLRQDO 3K\VLFV 9RO SS f 8FKLGD 6 7KH 3XOVDWLQJ 9LVFRXV )ORZ 6XSHUSRVHG RQ WKH 6WHDG\ /DPLQDU 0RWLRQ RI ,QFRPSUHVVLEOH )OXLG LQ D &LUFXODU 3LSH =$03 9RO 9,, SS f :DUPLQJ 5 ) DQG %HDP 5 0 2Q WKH &RQVWUXFWLRQ DQG $SSOLFDWLRQ RI ,PSOLFLW )DFWRUHG 6FKHPHV IRU &RQVHUYDWLRQ /DZV 6,$0$06 3URFHHGLQJV 9RO SS f :DWVRQ ( 'LIIXVLRQ LQ WKH 2VFLOODWLRQ 3LSH )ORZ -RXUQDO RI )OXLG 0HFK 9RO SS f

PAGE 174

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nV (QJOLVK ODQJXDJH WUDLQLQJ DW 1DQMLQJ $HURQDXWLFDO ,QVWLWXWH KH ZDV VHQW WR VWXG\ LQ WKH 'HSDUWPHQW RI $HURVSDFH (QJLQHHULQJ 0HFKDQLFV DQG (QJLQHHULQJ 6FLHQFH 8QLYHUVLW\ RI )ORULGD DV D YLVLWLQJ VFKRODU DQG ODWHU LQ 0D\ WUDQVIHUUHG WR JUDGXDWH VWXGHQW VWDWXV +H UHFHLYHG KLV 0DVWHU RI (QJLQHHULQJ GHJUHH LQ $SULO DQG H[SHFWV WR UHFHLYH WKH GHJUHH RI

PAGE 175

'RFWRU RI 3KLORVRSK\ LQ WKH VSULQJ RI +H LV PDUULHG WR &KXQKXD 6KL DQG KDV WZR FKLOGUHQ
PAGE 176

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n? ( 5XQH /LQGJUHQ 3URIHVVRU RI $HURVSDFH (QJLQHHULQJ 0HFKDQLFV DQG (QJLQHHULQJ 6FLHQFH FHUWLI\ WKDW KDYH UHDG WKLV VWXG\ DQG WKDW LQ P\ RSLQLRQ LW FRQIRUPV WR DFFHSWDEOH VWDQGDUGV RI VFKRODUO\ SUHVHQWDWLRQ DQG LV IXOO\ DGHTXDWH LQ VFRSH DQG TXDOLW\ DV D GLVVHUWDWLRQ IRU WKH GHJUHH RI 'RFWRU RI 3KLORVRSK\ /DZUHQFH ( 0DOYHUQ 3URIHVVRU RI $HURVSDFH (QJLQHHULQJ 0HFKDQLFV DQG (QJLQHHULQJ 6FLHQFH

PAGE 177

, FHUWLI\ WKDW KDYH UHDG WKLV VWXG\ DQG WKDW LQ P\ RSLQLRQ LW FRQIRUPV WR DFFHSWDEOH VWDQGDUGV RI VFKRODUO\ SUHVHQWDWLRQ DQG LV IXOO\ DGHTXDWH LQ VFRSH DQG TXDOLW\ DV D GLVVHUWDWLRQ IRU WKH GHJUHH RI 'RFWRU RI 3KLORVRSK\ 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nOFM/SMIn &/ f 'HDQ &ROOHJH RI (QJLQHHULQJ 'HDQ *UDGXDWH 6FKRRO

PAGE 178

81,9(56,7< 2) )/25,'$