1
The Character of Expiratory
Flow in the Lung
by
.John M. Co 11 ins
B.5., Rensselaer Polytechnic Institute
(1980)
M.S., Massachusetts Institute of Technology
<1982)
Submitted in Partial Fulfillment
of the Requirements for the
Degree of Doctor of Philosophy
in Mechanical Engineering
at the
Massachusetts Institute of Technology
January 21, 1988
©
Massachusetts Institute of Technology, 1988
Signature redacted
Signature of Author
Certified by
Signature redacted
'{
Accepted by
Roger D. Kamm
Thesis Supervisor
Signature redacted
Ain A. Sonin
Chairman, Department Graduate Committee
~HUIE
MAR 18 1988
l.8WIBJ\rchlvri
2
The Character
Flow
of Expiratory
in the Lung
by
John Mulloy Collins
Submitted to the Department of Mechanical engineering in
January, 1988, in partial fulfillment of the requirements for
the degree of Doctor of Philosophy.
ABSTRACT
The primary objective of this study is to predict the
expiratory pressure-flow relationship in a human lung.
Experiments and analysis of the flow in a single bifurcation
provides a means to numerically synthesize the pressure-flow
character for a complete lung, as well as providing insight
to the fluid dynamic character of expiratory flow.
A novel experimental technique was developed to measure
the pressure drop across a single bifurcation within a multiEnergy considerations allowed the
generation airway model.
pressure drop to be separated into components owing to
kinetic energy fluxes and viscous energy dissipation.
Experiments were first conducted in an idealized symmetric,
planar airway spanning the physiologically important Reynolds
Sensitivity of the results
number range from 50 to 8000.
was determined for typical pulmonary conditions;
-
Non-planar geometries
L/D ratio variations
airway shape
Non-circular
Asymmetric flow conditions
Turbulent/Laminar flow regimes.
The numerical simulations of the human lung generated
excellent agreement with available physiological data.
Different morphometric models were investigated,
demonstrating the importance of the asymmetric geometry of
the lung.
Title:
Thesis Supervisor: Dr. Roger D. Kamm
Associate Professor of Mechanical Engineering
3
Table of Contents
Page
2
Abstract
LiftVurr.f
I.
Fi
Introduction
1.1
1.2
1.3
II.
5
Objective
Motivation
Approach
Background
2.1
2.2
2.3
IV.
V.
52
Subtraction Method
Downstream Boundary Conditions
Subtraction Method Analysis
Quasi-steady Test Apparatus
Experimental Scaling
Data Collection
Data Reduction
Technique Verification
Airway Models
Estimation of Correction Factors
5.1
5.2
5.3
5.4
23
Bifurcation Flow Characteristics
Fully Developed Laminar Straight Tube F low
Fully Developed Turbulent Pipe Flow
Steady Entrance Flow
Steady Flow in a Convergent 2-D Channel
Flow in a Curved Tube
Kinetic Energy Dissipation
Definition of Pressure Drop
Summary of Pressure Loss Correlations
Experimental Methods
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
16
Forced Expiration
Flow Limitation
Work To-date
III. Conceptual Framework
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
11
Axial Velocity Profile Factor
Secondary Velocity Correction Factor
Pressure Correction
Correction Factor Implementation
78
4
Table of Contents
(Continued)
Page
VI.
6.1
6.2
6.3
6.4
6.5
6.6
6.7
Planar System Results
Embedded Bifurcation Assumption
Effect of Upstream Airway Orientation
Effect of Additional Parent Tube Length
Comparison of Results
Transition to Turbulence
Effect of Airway Collapse
VII. Flow Partition Experiments
7.1
7.2
7.3
7.4
8.1
8.2
8.3
8.4
8.5
8.6
113
Background
Experimental Procedure
Data Reduction
Results
VIII.Implications for Pulmonary Flows
IX.
93
Symmetric Airway Results
Introduction
Pressure Drop Correlations
Prediction of Model Experiments
Prediction of Physiological Model
Lung Simulations
Physiological Data
Conclusions
125
Experiments
148
References
151
Figures
155
5
List of Figures
Figure 1.3-1
bifurcation,
after
Idealized dimensions of
Pedley (1971)
A typical
Figure 2.1-1
(MEFV) curve, and the effect
Bates et al (1971)
Figure 2.3-1
pressure
in
a
pulmonary
maximal expiratory flow volume
of expiratory effort, after
Comparison of
published
frictional
losses for the idealized bifurcation.
Typical four cell
Figure 3.1-1
the parent branch downstream of a
Figure 3.2-1
laminar tube
a section of
Velocity profile in a fully developed
flow, and
the tube.
the resulting
Velocity
Figure 3.3-1
turbulent pipe flow.
Figure 3.4-1
with constant L/D
Figure 3.5-1
Dimensional
secondary flow pattern
bifurcation.
profile
acting on
force balance
in fully
developed
Stepwise contracted tube pulmonary model,
ratio's of 3.5, and area ratio's of 1.2.
Representation of
laminar flow
in a 2-
convergent channel.
Figure 3.5-2
convergent channel
Computed velocity profiles in a 2-D
flow for a range of flow conditions.
Figure 3.6-1
curved tube flow.
Secondary flow pattern
in
fully developed
Generation of the four cell secondary
pattern typical of expiratory flows from modeling a
bifurcation as two curved tubes oriented back to back.
Figure 3.6-2
Figure 3.6-3
curved tube flow.
Toroidal
coordinate system used
to
flow
analyze
Prediction of the pressure loss in a
Figure 3.6-4
curved tube, normalized by the Poiseuille loss at the same
flow rate from the theory of Dean (1927).
Experimental results for the pressure loss
Figure 3.6-5
in a curved tube, normalized by the Poiseuille loss that
would occur at the same flow rate, after Berger et al (1983)
Figure 3.9-1
Pressure loss correlations for curved
tubes, entrance flow, 2-D convergent channel flow,
turbulent flows in straight and curved tubes.
and
6
Schematic representation of the
Figure 4.1-1
for symmetric flow conditions.
method
subtraction
Figure 4.1-2
bend between two
Pressure distributions owing to a 900
long straight tube, after Ito (1960)
pipe
Schematic of the quasi-steady experimental
Figure 4.4-1
pressure measurement locations.
and
test apparatus
Kinematic viscosity of water-glycerin
Figure 4.5-1
of water content, from C.R.C.
a
percentage
mixtures as
Measured kinimatic viscosity for a 21% by
Figure 4.6-1
water glycerin-water mixture as a function of temperature
using Haake viscometer, and C.R.C. value.
Typical errors between tank level
Figure 4.6-2
readings and predicted levels based on
manometer
calibration
voltages.
output
and
fits
the curve
Entrance test experimental bellmouth and
Figure 4.8-1
geometry.
tube
straight
Tank level vs time measurements for the
Figure 4.8-2
(L/D = 6.3) at kinematic viscosities of
test
entrance
short
2
/sec
cm
0.125
0.475 and
Asymptotic region of the differential
Figure 4.8-3
a small offset error on the order of
showing
signal,
pressure
errors.
digitation
the
Error between the curve fit and the actual
Figure 4.8-4
note that the magnitude of the error are
level,
tank
measured
typically around the digitation level, and not systematic.
Calculated Reynolds numbers vs time for
Figure 4.8-5
the short tube entrance test with kinematic viscosities of
0.125 and 0.475 cm 2 /sec.
Combined dimensionless pressure vs
Figure 4.8-6
Reynolds number for both entrance tests and the subtraction
of the two, and theoretical predictions.
Results for
Figure 4.8-7
length of tubing and theory.
the loss
in the additional
Cross-section of a typical bifurcation
Figure 4.9-1
showing the cast section and aluminum coupling collars.
Momentum analysis of an exiting
Figure 5.1-1
impinging on a flat plat and turning 900.
jet
7
&
Axial velocity profiles used to calculate
Figure 5.1-2
the axial correction factor fa. A) Jan, Re=200 ;B) Chang
Menon, Re= 1060; C) Chang & Menon, Re=5712; D) Chang & Menon,
Re=6000; E) Chang & Menon, Re=6811; F) Hardin & Yu, Re=2760;
G) D.B. Reynolds, Re= 2260, Re=8000 and Re=12200
Cross plot of fa and a as calculated from
Figure 5.1-3
the numerical integrals of published velocity profiles, and
approximate relation.
Figure 5.1-4 - Results of the numerical
vs tracheal Reynolds number, and curve fit.
integrals of fa
Approximate flow profile used to estimate
Figure 5.1-5
the mean flow rate and fa factor based on boundary layer
thickness.
Secondary velocity profiles used to
Figure 5.2-1
numerically evaluate the secondary velocity correction
factor, fs; A) Chang & Menon Re=1060, @station 3, B)Chang
Menon Re=1060 @station 2s, C)Jan Re=200.
&
Prediction of fa based on the boundary
Figure 5.1-6
layer approximation for an entrance prediction, at L/D=1.75
and 3.5, curved tube flow, Sr=1/7, and 2-D convergent channel
flow, a=.05.
Normalized pressure difference measured
Figure 5.3-1
between the top and side of bifurcation leg during expiration
vs local Reynolds number.
Experimental results for the reservoir
Figure 6.1-1
level vs time for the experiments, Gen's 4-2, Nu=.475; Gen's
4-2, Nu=.125; Gen's 4-1, Nu=.475; Gen's 4-1, Nu=.125.
Subtraction of the three generation
Figure 6.1-2
"upstream" airways from the four generation airway to give
the normalized static pressure loss in the test bifurcation.
.
Decomposition of the total static
Figure 6.1-3
pressure loss across a bifurcation into a kinetic energy and
2
frictional component normalized to '4P*Re
Frictional pressure loss across a single
Figure 6.1-4
bifurcation normalized to the Poiseuille loss for the
symmetric flow experiments.
Figure 6.3-1
flow experiments.
Configurations for the nonplanar symmetric
Subtraction of the three generation
Figure 6.4-1
"upstream" airways from the three generation airway with the
additional tracheal length of L/D =3.5.
8
Frictional pressure loss in a straight
Figure 6.4-2
tube, L/D =3.5, downstream of a bifurcation, normalized to
the Poiseuille loss in the equivalent length of tubing.
Comparison of curved tube, entrance theory
Figure 6.5-1
and 2-D convergent channel flow to the band of symmetric flow
results.
Photographs of the oscilloscope traces
Figure 6.6-1
from the hot wire probe, not transition to turbulence appears
to occur near Re=1000.
Schematic representation of the bronchial
Figure 6.7-1
pressure forces, after Hughes (1974)
and
structure
support
Stereoscopic measurements of bronchial
Figure 6.7-2
cross-sections at various transmural pressures, and two lung
inflations.
Normalized frictional loss in the semiFigure 6.7-3
circular test bifurcation compared to curved tube theory
(note Reynolds number based on hydraulic diameter)
Most typical asymmetric bifurcation model
Figure 7.1-3
in the middle airways, after Olson (1971)
Experimental apparatus for the flow
Figure 7.2-1
Note separate fluid circuits.
partition experiments.
Control volume selected to analyze the
Figure 7.3-1
pressure loss in a bifurcation with asymmetric flow
conditions.
Results for the static pressure loss
Figure 7.4-1
across the variable flow leg of a bifurcation, for six
different fixed flow rates in the other leg vs flow ratio,
F.
Decomposition of the static pressure loss
Figure 7.4-2
across a bifurcation into a kinetic energy component vs
tracheal Reynolds number. (fixed daughter Reynolds # of 1506)
Figure 7.4-3 Results for the excess pressure loss due to
the mixing of two streams vs F (flow ratio)
Momentum analysis of the maximum pressure
Figure 7.4-4
recovery due to the unification of two unequal flows.
Comparison of the results for the friction
Figure 8.1-1
pressure loss in an idealized bifurcation for this and
previous investigations.
Geometry of a typical asymmetric
Figure 8.2-1
bifurcation used for analysis.
9
Comparison of numerical prediction, and
Figure 8.3-1
experimental results of Hardin & YU, for the static pressure
drop across an idealized three generation pulmonary model.
Comparison of numerical prediction, and
Figure 8.3-2
experimental results of D.B Reynolds, for the static pressure
drop across three generations of rigid "Y" connectors.
Depiction of the Zavala lung model,
Figure 8.4-1
courtesy of Medi-tech, Watertown Ma.
Dichotomous branching pattern
Figure 8.4-2
approximation for the Zavala lung model.
Comparison of numerical prediction, and
Figure 8.4-3
experimental results of Isabey and Chang, for the static
pressure drop across the Zavala lung model.
Pressure drop components which comprise
Figure 8.4-4
the calculated static pressure drop across the Zavala lung
model for uniform pressure upstream boundary conditions.
Figure 8.4-5
Zavala lung model.
Calculated Resistance matrix for the
A comparison of the flow distributions for
Figure 8.4-6
the Zavala lung model.
Asymmetric branching pattern for the
Figure 8.4-7
according to the Horsfeild et al (1971) lung Model 2.
lung
Comparison of numerical prediction, and
Figure 8.4-8
experimental results of Reynolds and Lee, for the static
pressure drop across a pulmonary cast below the right
intermediate bronchus.
Comparison of the overall pressure-flow
Figure 8.5-1
characteristics for the Weibel and Horsfield lung models.
Pressure drop components which comprise
Figure 8.5-2
the calculated static pressure drop across the Horsfield lung
Percentage of "long" and "short" paths
Figure 8.5-3
terminating at a given number of bifurcations into the lung.
Static pressure vs generation into the
Figure 8.5-4
lung for both the Weibel and Horsfield lung models for
several flow rates.
Normalized viscous pressure drop vs
Figure 8.5-5
generation into the lung for both the Weibel and Horsfield
lung models for several flow rates.
10
The variation of normalized viscous
Figure 8.5-6
pressure drop vs generation into the lung for Horsfield
models for several flow rates.
lung
Normalized viscous pressure drop vs
Figure 8.5-7
average airway diameter for the Horsfield lung model at
several flow rates.
Average and variation of the diameter and
Figure 8.5-8
L/D ratio vs generations into the lung for both the Weibel
and HOrsfiels lung models.
High and low Reynolds numbers Ratio
Figure 8.5-9
factors vs generation for the horsfield et al lung model.
Comparison of the effect of flow types;
Figure 8.5-10
inspiration, expiration and Poiseuille flows.
Comparison of the numerical prediction for
Figure 8.6-1
pulmonary resistance based on the horsfield lung model and
the experiements of Blide et al, and Vincent et al.
Comparison of the numerical prediction for
Figure 8.6-2
pulmonary resistance based on the horsfield lung model and
the experiements of Hyatt and Wilcox.
Comparison of the numerical prediction for
Figure 8.6-3
pulmonary resistance based on the horsfield lung model and
the experiements of Ferris et al.
Page
1.
11
Introduction
1.1 Objective
The primary objective of this work
is
to obtain a
relationship between pressure drop and flow rate for
expiratory flow from a human lung.
relationship
Specifically, a
is sought which describes the pressure drop
a single bifurcation which can be considered
inside the airway network.
insight to
across
to be imbedded
A related objective is to gain
the nature and character of expiratory pulmonary
fluid dynamics.
The results of this study are intended
it possible
to make
to synthesize the flow resistance of an entire airway system
based on predictions for each of the individual bifurcations
within the system.
Accurate predictions require that the
relationship be sufficiently general
to allow for
ideal" characteristics of a real human
lung.
the "non-
Pulmonary
parameters are known to change with position in the
with the evolving state of
lung
experimentally
inflation during expiration.
combinations of these parameters
The number of potential
which are encountered
lung and
in the
lung make
investigate each one.
it
impractical to
A dimensional
analysis of
pulmonary flows to describe steady flow across a bifurcation
(Jaffrin and Kessic,
1972)
dimensionless groupings as
identified
the relevant
the Reynolds number
and the geometric scaling parameters.
in each branch,
This study will attempt
only to establish a relationship for a typical bifurcation
Page 12
within the
parent or
of that
lung over
the of range Reynolds numbers in the
tracheal branch from 100 to
10,000.
The sensitivity
relationship to the expected range of "non-ideal"
geometry and deviations from symmetry will
also be examined.
1.2 MOTIVATION
The major motivation to
study the expiratory pressure-flow
relationship of a bifurcation is the desire to improve the
expiratory flow volume
diagnostic utility of the maximal
(MEFV) pulmonary function test.
predict
al,
Computer models exist to
(Elad et
the results of the MEFV as a diagnostic aid
1987,
Lambert et al,
1982), but
these suffer from the lack
of accurate pulmonary pressure-flow relationships.
The few correlations for expiratory flow resistance found
in the literature (Reynolds,
Reynolds,
about
1980,
1982, Hardin and Yu 1980)
three.
1979,
agree only to a factor of
While this variability could result from real
intra-study differences
surprisingly
Reynolds and Lee,
in the system geometry,
the range is
large especially in view of the statistical
averaging that would result from multiple-generation networks.
Further skepticism is based on the seemingly unrealistic
approached by the correlations at both high and
limits
low Reynolds
numbers.
A second, but equally compelling motivation to
the expiratory pressure-flow relationships
is the profound
of fundamental understanding of expiratory flows,
normal breathing.
While several
investigate
lack
even during
models of inspiratory
Page 13
resistance exist
(Pedley et al
Jaffrin and Kessic
1970a and b,
Pedley et al
1971,
no equivalent extensive study of
1972),
Similarities
expiratory flow resistance has been conducted.
are expected between inspiration and expiration,
differences between the two flows,
but
preclude
as discussed below,
a direct comparison.
1.3 Approach
The approach used in this study is primarily experimental.
treatments of the problem are limited by the
Analytical
complexity associated with a fully three dimensional flow
extending from low
('10)
to moderately high
Some progress can be made, however,
numbers.
simple analogous flows.
technique,
in developing
was developed to provide the pressure difference
This new method avoids the restrictive
assumption made by most previous
investigators that the form
loss in a single bifurcation is the same for
the airway as a whole.
errors
and to aid
experimental approach, called the subtraction
across a bifurcation.
for
in the use of
insight.
A novel
for the
Reynolds
These are included to serve as both
theoretical support for the results,
physical
('10,000)
It also avoids the substantial
involved in the direct measurement of
across a single generation due
the pressure drop
to the considerable variation
pressure over the cross-section.
experiments,
that found
A series of
"companion"
including flow visualization and hot
anomometery were used to establish
wire
transition from laminar
to
in
Page 14
turbulent flow.
Experiments were conducted on a network of bifurcations
modeled after an "idealized bifurcation" presented by Pedley
(1977).
The network from which casts were made was designed
and fabricated in two pieces on a numerically controlled
(Univ. of
milling machine by Drs J. Hammersley and D.E. Olson.
Figure 1.3-1 presents the dimensions of the idealized
Mich.)
bronchial bifurcation,
which can be further
characterized as
having;
w An ratio of the total daughter tube area to parent tube
area of about 1.2, based on a mean diameter ratio of .75
v A variable ratio of the radius of curvature of the tube
center-line to the tube radius, with a mean of about
The tube curvature is greatest in the daughter
1/7.
tube at the bifurcation, and gradually straightens once
the branching angle is reached.
a A changing cross-sectional shape which is circular
the parent and daughter tubes, blended with
progressively elliptical to dumbell shapes in the
junction of the bifurcation.
a
in
A branching angle, with a sharp flow divider, which
varies from '640 in the large airways to 1000 in the
smaller airways with a average value of about 700.
m Tube length/diameter ratio of 3.5 between bifurcations.
x Smooth walls as a result of a thin mucas layer.
The idealized bifurcation was chosen rather
from a real
lung to maximize experimental control,
minimize the number of unknowns.
understanding
and
to
than a cast
Effort was directed first
to
the most simple planar symmetric flow conditions
then extended to determine the sensitivity of
typical
and to
the results
variations in upstream boundary conditions, flow
Page 15
asymmetries and changes
in bifurcation geometry.
Applicability of the results found
in the constructed
idealized models is investigated by comparing predicted and
measured values of the pressure-flow relationships in rigid
human
lung casts.
Page 16
2.
Background
2.1 Forced Expiration
Despite the present lack of detailed understanding,
long been recognized as a valuable tool
forced expiration has
in assessing pulmonary function.
Hutchinson
the technique as indicator of pulmonary vital
in 1958,
recognized
capacity and
the
It was not until
pulmonary dysfunction over a century ago.
insightful work of Hyatt et al,
(1846)
however,
that forced
expiration results were described in the more useful form of
maximum expiratory flow vs.
tests are currently used
lung
inflation.
Forced expiration
(Hyatt) to indicate the advent and
progression of such important pulmonary disorders as:
a Emphysema
v Bronchitis
v Cystic Fibrosis
a Asthma
The utility of the MEFV function test
is further enhanced
because it can be generated from a relatively simple clinical
evaluation
(Hyatt 1983).
the flow vs.
The MEFV curves are generated from
time traces obtained from a forced vital capacity,
maneuver.
FVC,
To obtain the FVC traces,
total
lung capacity
a subject inhales maximally
( to
) and then exhales into a flow meter as
forcefully and completely
as possible to residual
volume.
The
Page 17
total
is the expired
rate verses time.
The FVC curves
plotting as flow vs.
expired
the flow
and re-
lung volume to obtain the Maximal
regions;
(Hyatt, 1983),
one which
independent.
is effort
is effort
region occurs
during
lasting until
about 80% of vital capacity.
the
initial rapid
consists of
dependent and
The effort dependent
another which
increase
in flow,
The effort
occurs with the
independent region, or flow limited region,
decay
record
integrated
are
see Figure 2.1-1
The MEFV curve,
to
is used
(MEFV) forced expiration curve.
Expiratory Flow Volume
two characteristic
device
volume)
Typically a Spirometer
lung vital capacity.
an attached recording
with
lung volume minus residual
expired volume (total
in flow rate below approximately 75% of vital capacity.
Curves corresponding to various degrees of effort are also
shown
clearly demonstrating
in Figure 2.1-1,
the effect of
effort.
The utility of the MEFV curves in detecting pulmonary
dysfunction
is primarily a result of the flow
of expiration.
limited portion
During flow limitation for a given gas,
the
flow is only a function of the mechanical properties of the
lung and the geometry of the
lung.
2.2 Flow Limitation
The flow
limiting
nature of
described by analogy to
expiratory
flow has been
the flow limitation which occurs in
thin walled collapsible tubes.
Shapiro
(1977a),
of the physiological and medical aspects of flow
in a summary
in collapsible
Page 18
tubes,
this way,
describes the effect
compliant)
"
The
(extremely
airways are easily collapsed by an in-wardly acting
transmural pressure;
area makes for
the resulting decreased cross-sectional
increased air speeds;
this
in turn produces an
increased pressure drop associated with the Bernoulli dynamic
pressure and with frictional
Efforts to model
resistance"
forced expiration have proven extremely
difficult because of the coupled effects of the airway
structure and fluid dynamics.
Additionally,
mechanical properties of the airways and
behavior in a bifurcation is
information on the
the detailed flow
lacking.
To understand how these interactions and
the effect of
friction within the lung affects flow limitation,
to examine the equations for
A one dimensional model
to predict
walled,
it
is useful
the thin-walled compliant tubes.
has been developed by Shapiro
the structural/fluid dynamic
(1977b)
interactions in a thin
massless straight compliant tube with varying wall
properties and rest area.
The interactions are shown to depend
on the following:
" The nature of the vessel structure in the form of
a " tube law " which describes the vessel area as a
function of transmural pressure ( internal-external
pressure ) and vessel stiffness, Kp.
" The external pressure acting on the vessel,
both the end pressure boundary conditions.
and
" Frictional dissipation within the flow.
" Changes in rest area along
the vessel.
The nature of the interactions are summarized
in the
Page 19
differential equation below;
2.2.1
=
2)
-
Ao
U
local
dx +
+
+
gC2
Dh
is the mean flow speed, S
where U
tube wave speed, P
is the
dp'
dKp
dx
dx
to C,
is the ratio of U
to the the tube stiffness,
g
is the fluid density, g
acceleration,
Z
is elevation,
factor which can depend on the
geometry, Dh
the
is the
p'
local rest area,
gravitaional
dx
+ Kp-
p'
is the static pressure,
transmural pressure normalized
Ao
1
C2
2S2f
d(P+gg2)
-dAo
dU
(1-
Kp,
is
is some friction
f
local Reynolds number and
is the hydraulic diameter defined as
4*area/perimeter, and x is the distance down the tube.
Flow
limitation occurs when the axial
velocity of
When flow
fluid matches the local wave speed of the tube.
or "choking"
limitation,
occurs,
not be
the flow will
This
increased by reductions in downstream pressure.
phenomena is exactally the same as that which
trans-sonic compressible gas flow and
or
the
is found
in
in the flow over a weir
"waterfall".
Much work remains before equation 2.2.1
with more confidence to expiratory flow.
estimates can now be applied to each of
equation.
alone,
This work will concentrate
can be applied
At best,
only
the terms in the
on the frictional losses
and not attempt to describe the remaining terms.
The effect of frictional
dissipation on flow
can be seen from equation 2.2.1
to be unique.
always acts to remove energy from the flow,
limitation
Friction which
drives the flow
Page 20
toward a "choked"
speed
condition,
to meet the local
flows, S<1,
increasing/decreasing
For
tube wave speed.
the decrease in
transmural
friction reduces the tube area,
flow toward the tube wave speed.
which,
the flow
sub-critical
pressure caused by
in turn,
accelerates the
the
In super-critical flows,
decrease in transmural pressure results in
the opposite effect,
a decrease in the flow speed toward the wave speed.
It
is
expected then that friction will play an important role in
determining not only the required expiratory effort but also
the
location of the flow limiting
or "choked"
point.
2.3 Work To-date
Relatively few studies have examined the frictional
dissipation in an airway network during expiratory flow.
Pedley (1977)
has made the the observation
everyone doing a model
that
"it
is as if
experiment on inspiration ran out of
time or energy when it came time to repeating the measurements
for expiration".
The few studies which are available consist mainly of
experimental
model studies in airway networks which are either
idealized planar geometric models or pulmonary cast models.
Scherer
(1972) developed a theoretical model
expiratory flow through a single bifurcation.
required by Scherer's analysis,
a two dimensional
to describe
The assumptions
such as restricting
form limits the utility of the model.
The comparison of available experimental studies
1982,
the flow to
Hardin and Yu
1980, Reynolds 1980,
(Reynolds
Reynolds and Lee 1979)
Page 21
shows high variability
Figure 2.3-1
results.
results for
in both the form and magnitude of
inferred
graphically presents the
pressure loss in a standardized
the frictional
bifurcation based on the correlations published for the results
The standardized bifurcation
of the 4 different model studies.
chosen as the basis of comparison is a typical
bifurcation as put forth by Pedley
depicted in Figure 1.3-1.
pulmonary
and graphically
(1977),
the pressure loss are
The results for
normalized by the corresponding fully developed Poiseuille flow
pressure loss through an equivalent
length tube.
The magnitude of the results do not agree to better than
The nature of these results have
within a factor of about 3.
some puzzling attributes which appear to be inconsistent with
First, at
intuition.
to
3.5 higher
low Reynolds number the results are 1.5
than would be expected
of fully developed Poiseuille flow.
of results of Hardin (1982),
on asymptotic
based
Second,
limits
with the exception
little similarity exists with the
form of the pressure losses found in flows which are expected
These similarities
that a Re'.5 dependence is
discussed
in more detail
Wilson 1982).
maximal
framework section.
in the conceptual
pressure flow relationship found by D.B.
incorporated
into expiration models
Elad et al
flow on viscosity,
and will be
likely to occur,
The inadequacy of these relationships
(1985)
appears to be underestimated
is also seen when
Reynolds
(Elad et
found that
especially at
tend to indicate
al
(1980)
the
is
1985, Lambert
the dependence of
low lung volumes,
in comparison with physiological
&
to be generally similar.
Page 22
Staats et al
(1980),
et al
and Wood and Bryan (1969).
(1963),
The Lambert
&
data from experiments conducted by Schilder
Wilson model was able to produce a better viscosity dependence
agreement with experiments, but required
that the pressure
flow losses at
losses be 3.5 times the corresponding laminar
very low Reynolds numbers.
A possible reason for
method used
these discrepancies may lie in the
in previous studies for extracting
flow relations from global
experiments,
measurements.
In all previous
the form of the pressure drop across a single
bifurcation was assumed to be that found for
whole.
local pressure-
the model
as
Curve fitting was then used to determine the
coefficients for a single bifurcation.
for the relationship,
which
the entire network, may mask
a single bifurcation.
Imposition of this form
is actually a weighted average over
the true form of the relation for
Page 23
3.
Conceptual
Framework
3.1 Bifurcation Flow Characteristics
The complexity of the expiratory flow within a bifurcation
is currently beyond the capabilities of theoretical analysis.
The form of
the Navier-Stokes equation which describes the flow
cannot be solved analytically,
Current computational capabilities are
typified by the work of Wille
months of processor
(1984),
which consumed about 2
time to solve the similar
steady
of only
inspiratory flow problem at a Reynolds number
The remaining alternative to understand
the results of model
limits of
the practical
full three dimensional flow are beyond
current computers.
treatments of the
and numerical
experiments
10.
is to use
the flow
in bifurcations to develop an
understanding of the nature and character of the flow.
Similarities can then be drawn with the flow characteristics
in similar,
but
less complicated flows.
complicated flows are better understood,
Generally,
and
less
the proper
analogies taken together can form a conceptual
which to
the
framework
in
interpret expiratory flows.
Flow visualization and hot-wire velocity mapping studies
performed
Hugh-Jones
1983,
in airway models and
1959, Schroter and
Chang and Menon 1985,
in single bifurcations
(West and
Sudlow 1969, Patra and Afify
Jan 1986
) have lead to a number
of consistent observations on expiratory flows;
is that a four cell secondary flow
of the junction for tracheal
downstream
pattern occurs
a The most notable
Page 24
This flow
Reynolds numbers from 50 to several thousand.
pattern is shown schematically in Figure 3.1-1.
a The secondary flow pattern is developed either within
the junction or just downstream of it.
a Flow patterns are qualitatively similar for the range of
inlet conditions tested; fully developed parabolic inlet
flows, blunt inlet flows, and flows with secondary flows
similar to those found in the outlet.
a The magnitude of the secondary flow velocity is between
approximately 1/5 and 1/2 of that in the bulk axial flow.
n Axial velocity profiles just downstream of the junction
contain a dip which is rapidly transformed into a peak
in the plane of the bifurcation, but remains flat in the
plane normal to the bifurcation.
m Distances greater than about one diameter downstream of
and are
are "blunt",
the junction, the axial velocities
thin boundary layer.
characterized by a relatively
a For the case of a long straight parent branch, the
secondary flows generated in the bifurcation are
convected downstream through the equivalent of many
branch lengths.
Inspection of these flow characteristics, and the geometry
of a bifurcation lead to the consideration of several flow
types as possible analogies to steady expiratory flow.
The
following are the flow problems for which possible analogies
apply,
and are well understood based on experimental
theoretical
and
investigations.
a Fully developed steady laminar flow
in a straight
a Fully developed steady turbulent flow
tube.
in a straight
a Steady entry flow in a straight tube.
a Steady flow in a convergent 2-D channel.
s Fully developed steady flow
a Kinetic energy dissipation.
tube.
in a curved tube.
Page 25
3.2 Fully Developed Laminar Straight Tube Flow
tube flow was one of the earliest
Fully developed straight
analogies drawn to describe pulmonary flows
(Rohrer 1915).
The analogy stems from geometric similarities given that
airways,
in the most elementary form, can be considered a
series of straight tubes.
The analogy neglects the effects of
curvature on the secondary flows and that each tube section is
typically too short for a fully.developed flow to develop.
Schlichting
(1979)
gives the entrance
length required
to
establish a fully developed Poiseuille profile to be;
Le=0.03*Re*Dia
3.2.1
A fully developed flow will therefore not evolve within a
length of 3.5*Dia.
bifurcation which has a typical
1977) until
the Reynolds number drops below about
Reynolds numbers significantly below 100,
be fully developed.
length of the tube will
dissipation in a fully developed straight
serves as a
though,
limit for
low asymptotic
(Pedley
100.
For
an increasing
The frictional
tube flow therefore
the flow
in a bifurcation.
Derivation of the fully developed pipe flow equations can
be found
in most
text books
on fluid mechanics.
The problem
represents the Navier-Stokes equation in its most simple form;
3.2.2
-
=
dx
where u
is the axial
velocity,
-
du 2
dP
dr
r
2
is the radial position,
x the
Page 26
axial position,
p the pressure,
and p is the fluid viscosity.
Solving 3.2.2 for the pressure drop across a tube of
length L gives;
Del P = 32pLU/D2
3.2.3
where U
is the bulk averaged velocity defined as the flow rate
divided by area,
and D is the tube diameter.
Equation 3.2.3
can be written in another form which introduces a
characteristic pressure P*, and
3.2.4
the Reynolds number
Del P =32 L/D P* Re
where p* is given by;
y
*
3.2.5
P
;Nu 2
2
=
=
gD2
D2
where Nu is the kinematic viscosity.
A scaling argument which provides more
insight to the
nature of the pressure drop can be used to arrive at the same
result to within a constant factor.
While this is not
necessary due to the susceptibility of the problem to
analysis,
it helps to establish a comparison for
rigorous
other more
complicated effects.
From a force balance on a section of
shown in Figure 3.2-1,
the pressure
tube with a length L,
force acting on the tube
cross-sectional area, can be seen to be balanced by the wall
shear stress acting on the wall;
Page 27
(P1-P2)nD 2
3.2.6
where
=
TW(rDL)
Tw is the wall shear stress, and for a Newtonian fluid
is given by;
du
Tw =
3.2.7
Adr
wall
which can be approximated by representing
the velocity
gradient at the wall as the ratio of a characteristic
the bulk velocity, by a characteristic
velocity,
tube diameter.
length,
the
Combining the approximation with equations
3.2.6 gives the pressure drop as;
3.2.7 and
Del
3.2.8
P =P 1
P = Const.*PLU/D 2
2
Comparing equation 3.2.8 and 3.2.3 shows that the two
forms agree if the unknown constant
in equation 3.2.8
is taken
to be 32.
3.3 Fully Developed Turbulent Pipe Flow
Laminar flow in a smooth straight tube will remain despite
the presence of slight vibrations and other perturbations
encountered
stable.
in "real world flows" for as long as the flow is
Instabilities occur
in steady flows when the inertial
effects dominate the viscous effects.
Stability is typically
characterized by the ratio of these two effects
the Reynolds number.
pipe flows
and
in the form of
A Reynolds number near 2300 for steady
is the value at which the flow becomes unstable,
therefore subject
to turbulence.
Page 28
in expiration both
Turbulent flows are expected to exist
because of the high Reynolds numbers encountered,
and
significant flow disturbances which are present.
As will be
later,
discussed
by spatial
expected
the conditions for
the
transition are complicated
accelerations and secondary flows,
both of which are
to higher Reynolds numbers.
to delay transition
No theory based on first principles is yet available to
describe turbulent flows, and,
the small
scale of turbulence
renders computer models impractical for a complete solution of
Experimental results have,
all but
the simplest of flows.
though,
lead to a detailed understanding of structure of
turbulence.
The pressure loss
been well documented
in turbulent smooth-walled pipe flows has
(Schlichting 1979),
and can be explained
with the scaling arguments outlined above for
laminar flows.
Referring to Figure 3.3-1, unlike laminar flows,
turbulent
flows are characterized by a relatively blunt velocity profile
scale eddies efficiently
The small
with a thin boundary layer.
transfer momentum radially to
smooth the velocity in the core.
Imposition of a no-slip boundary at the wall,
generates a
region of laminar flow with high shear close to
Estimation of the shear stress
in the laminar
acts on the wall
is then given by;
3.3.1
-w
where S
is the
du
= )4 dr
the wall.
sublayer which
u(S)
0
wall
8
laminar sub-layer thickness,
and u(S)
is the
Page 29
velocity at the boundary layer edge,
3.3.2
8 a P/v*
3.3.3
u(S)
and are given by;
a V*
where v* is the characteristic velocity for turbulent flows;
3.3.4
v*
/
((Tw / g)
Experimental measurements of velocity distributions and wall
(see Schlichting) shear
3.3.5
v* =
stress give v* to be;
.1823
[U
/r
1/8
combining 3.2.4 and 3.3.1-3.3.5 gives the familiar Blasius
friction relation for smooth walled turbulent pipe flows;
'4
3.3.6
Del P = 0.3146(4gU2)(L/D)/Re
where Re is the tube Reynolds number Re = Up/gD.
notation,
equation 3.3.6 can be written more concisely as;
*
3.3.7
Using the p*
Del P = 0.157 P
1.75
(L/D) Re
For the case where surface roughness extends into the flow
past the
laminar sublayer, this relation no
longer holds.
transitional region exists when the mean roughness height,
is between the laminar sublayer
A
Ks,
, yv*/nu=5 to about yv*/nu =70.
For Ks above yv*/nu=70 the flow is considered fully rough,
the increased velocity gradient at the wall generates the
and
Page 30
pressure drop given by the expression;
Del
3.3.8
P= '4P L/D Re2
2Log(r/Ks)
+
1.74
Schroter and Sudlow (1969) have noted that bronchial walls
are coated with a liquid mucus
thin, and
layer which
therfore hydraulically smooth.
is
"microsopically"
Macroscopic
corrugations in the trachea and main bronchi are present due to
cartilidge rings.
The effective Ks generated by these
corrugations is relatively
low due to
their wide spacing.
These corrugations are therefore unlikely to generate fully
rough flow.
Consequently
apply in the lung, rather
it
is more probable that 3.3.7 will
than 3.3.8.
3.4 Steady Entrance Flow
The entrance type phenomena expected
similar
in a bifurcation are
to those for the steady entrance flow problem in a
straight tube, but more complicated.
The entrance phenomena
are a result of the effects of viscosity diffusing
the wall
to adjust the inlet velocity profile
developed final form.
in-ward from
to its fully
The extent of the difference between the
inlet profile and the fully developed profile will therefore
determine the magnitude of entrance effects on the flow.
The entrance analogy in expiratory flows can be seen by
considering a lung as being roughly modelled by the geometry
shown in Figure 3.4-1.
The area changes experienced as
the
Page 31
flow proceeds toward the "trachea" can be approximated as a
series of straight tube sections, L/D = 3.5,
in from the wall as the
flow proceeds in each straight section.
section,
The
joining the tubes.
area ratio contraction, An/An+1=1.2,
viscous effects start to propagate
with a constant
In the contraction
the flow accelerates, and the viscous boundary layer
Once in the straight section,
thins to maintain continuity.
the boundary
layer continues to grow,
but starts from a non-
zero value.
Rivas and Shapiro
(1956)
in an analysis of ASME test
nozzles accounted for the non-zero
initial boundary layer
in the straight tube section downstream of a
thickness
The
bellmouth entrance with an "equivalent length" concept.
equivalent
length, Leq,
is the
required to develop a boundary
that found
length of straight tubing
layer
thickness equivalent
to
in the inlet of the straight section downstream of
the bellmouth.
In general,
on the flow rate,
the equivalent length
and bellmouth geometry.
is dependent
The loss downstream
of the entrance can then be calculated as the difference in a
tube of the actual
length plus Leq,
and a tube of
length Leq.
Applying entrance flow theory with the added correction of
an equivalent
length to expiratory flow is difficult.
The
difficulty arises due to the presence of secondary flows,
nonuniform
required
inlet velocity profiles,
to calculate Leq.
and the flow details
Despite these problems,
flow analogies have been applied with some success
1977)
to
the study of
inspiratory flows,
entrance
(Pedley
and are expected to
Page 32
provide some understanding of expiratory flows.
A sizeable amount of literature has been devoted to
Boussinesq
study of flow in the entrance of a circular pipe.
is the first
the
to be cited as having made a theoretical
investigation of the problem in
1891.
Since that time many
experimental studies have been done, and correspond well
Most current analyses of the problem
theoretical predictions.
are numerical,
to
but the more simple boundary
Boussinesq provides insight
layer analysis of
the flow character for
to
large
Reynolds numbers.
Investigations
(Shapiro et al
1954) have shown that for
the thin boundary layers near the entrance,
the falling
pressure gradient has negligible effect on the boundary layer
Flat plate,
development.
Blasius type, boundary layer growth
therefore occurs, which gives the boundary layer
3.4.1
S(x)=1.72DV(x/DRe)
where D
axis,
is the tube diameter,
x the distance along the tube
In
and Re is the tube Reynolds number.
approximation,
the first
the core velocity can be taken to be constant,
which allows the wall
shear rate to be written as;
Tw(x)
=
-
-
dr
Re
U
du
3.4.2
thickness as;
wall
S(x)
=
JU
xD
which can be integrated to give the loss over a
length of pipe L;
Page 33
A2
Del
3.4.3
P
4
R3
Re
const.
L/D
where the published experiments place the constant as 4.87.
is higher
Note that the typically reported value is 6.87, which
includes the effects of the changing velocity
because it
profile.
Incorporating
the equivalent entrance length and P*
into equation 3.4.3 gives;
Del
3.4.4
P = 4.87 P *
[
(L/D + Leq/D)
-
The validity of this expression for straight
confined to
as the entrance
the thin boundary
For cases
in which
in the boundary
the product of the
10E5,
turbulence is
layer and the growth pattern of the
is altered.
This boundary layer transition to turbulence may arise
expiratory flows.
of 3.5,
low
On the
layer approximation fails
Reynolds number and L/D ratio exceeds about
boundary layer
is
length approaches the tube length, and
Poiseuille flow results.
present
tubes
intermediate values of Reynolds number.
end of the range,
Re
Leq/D]
Assuming an Leq
near
in
1, with an L/D ratio
the transition would occur on the order Re'10,000.
3.5 Steady flow in a convergent 2-D channel.
Another geometry that can be used to model both a single
bifurcation and the entire airway is a convergent channel.
This representation
is not very accurate, but provides a means
to approximate the effect of a continuing balance between a
Page 34
boundary
layer growth and thinning.
The growth of the boundary
layer results from the diffusion of viscosity as the fluid
passes through
while the boundary layer
the network,
results from the increasing velocities imposed by
thinning
the
progessively smaller flow area.
has summarized the exact solution to
(1979)
Schlichting
The equations can
Navier-Stokes equation for the 2-D channel.
be considerably simplified for this case in which the flow is
and the flow profile depends only on the
everywhere radial,
azimuthal angle,
Defining a
The geometry is shown in Figure 3.5-1.
0.
local Reynolds number as;
Uo h
Re =
3.5.1
Nu
where Uo
is the local centerline velocity,
Normalizing the velocity by Uo,
channel height.
distance by h,
and h
is the local
and the radial
allows the N-S equations to be written in terms
of a single non-linear differential equation;
2
62u-
+ 4 U
--
-K 2a
+
Re
3.5.2
802
2o
Re
with the boundary condition;
3.5.4
a
G(
S&( 0
) =
0
....
No slip at
=
0 ......
Centerline symmetry
with the definitions;
3.5.5
LA
=
the walls.
)
3.5.3
U(0)
=
U(0,r)/Uo(r)
Page 35
-1
a = Tan
3.5.6
( '4h/r ) z uh/r
The convergence angle, a,
the lung
which
is difficult to define because of the difficulty in
converting from a three-dimensional
to a two-dimensional
One approach to define an appropriate value of
geometry.
the lung
is appropriate for use in
is
to write the area of the
a for
lung in the exponential
form;
3.5.7
]
A = A exp [0-
L
=
which can be linearized about a typical bifurcation with an L/D
of 3.5 and an area ratio of 1.2,
0.052
giving o as approximately
(representing an included divergence angle of about 60).
The constant K in equation 3.5.2 is the
pressure gradient,
imposed wall
in the form;
3
r
K =
SP
-
3.5.8
g Nu 2 Sr
wall
Equation 3.5.2 can be solved numerically,
for the extremes in Reynolds numbers.
to 3.5.2 is presented
or analytically
The numerical solution
in Figure 3.5-2 for the dimensionless
velocity profile for a range of Reynolds numbers and the
typical pulmonary value of a.
The profile
is seen to progress
from a Poiseuille type profile at low Reynolds numbers to a
profile characterized by a blunt core and thin boundary layer
at higher Reynolds numbers.
The pressure flow characteristics of the geometry also
Page 36
follow the same progression from a Poiseuille form to a
layer form can be seen most
The boundary
boundary layer form.
easily by solving 3.5.2
the high Reynolds number case and
for
The high Reynolds number
evaluating the wall friction.
limit
solution to 3.5.2 is;
-1
3.5.9
u
=
3Tanh
2
(Re a/2)
(1 -
+ Tanh
O/a)
which generates a shear stress at the wall
V(2/3)
corresponding to
the pressure drop;
2
Del P
a cos(
=
3.5.10
Re
0(O)
L
-/3
Del P
a
pois.
Using the value a = 0.052,
numerically evaluating 0(O)
corresponding to the lung, and
to be approximately 1.15,
gives the
asymptotic frictional relationship to be;
3.5.11
Del P Z Del P
.06
Re
-/Re
A relationship which can be applied to
>> 1
lower Reynolds
numbers must be obtained from the direct solution to 3.5.2.
3.6 Flow in a Curved Tube
The generation of secondary flows downstream of a
bifurcation during expiration can also be described by analogy
to curved tube flow.
Figure 3.6-1
shows the secondary flow
Page 37
pattern established by the centrifugal forces generated by the
core flow as the flow proceeds around the bend
in a curved
The resulting pressure gradient across the tube cross-
tube.
section causes fluid
in the core to move outward where the
slowing moving fluid at the wall has
insufficient axial
and is driven
velocity to generate the same pressure gradient,
to preserve continuity.
inward
The four cell structure found in the parent
bifurcation,
shown in Figure 3.1-1,
tube of a
can be generated by
approximating the bifurcation geometry as two curved tubes
This geometry is depicted in Figure
oriented back to back.
and shows that for symmetric flows a balanced four cell
3.6-2,
is expected.
pattern
A basic difference which must be noted between curved tube
in a bifurcation
and flow
is that
in a bifurcation both the
geometry and area change as the flow proceeds along the bend.
the driving factors
Despite these differences the analysis of
which
influence curved tube flow should provide insight as to
the nature of secondary flows
hypothesis
indicated
in expiratory flow.
This
(1986)
which
is supported by the -work of Jan
similarities between the velocity profiles in quasi-
steady oscillatory flow in a bifurcation those predicted with
curved
For
tube flow theory.
curved tube flow the natural
toroidal one,
coordinate system is a
as shown in Figure 3.6-3.
vectors are in the r,
The orthogonal unit
a, and s directions where r is measured
radially outward from the center
of the cross-section,
a is the
Page 38
angle between the radius vector and the plane of symetery,
s is measured along the tube axis.
and
In this coordinate system
the steady non-dimensional Navier-Stokes equation take the form
The dimensional
in Equation 12 of Berger et al.
( denoted with a
parameters
') are nondimensionalized
by;
r
=
r'/a
s = RcO/a
p
= p'/(gU 2
u
=
u'/U
v =
w
= w'/U
)
given
3.6.1
Berger,
Talbot and Yao
v'/U
outlined a simplification of
(1983)
these equations for the case of gradual
curvature, Sr
<< 1, by neglecting all terms of order Sr' or higher.
= r/Rc
The
are necessary to drive the
centrifugal force terms which
secondary motions were retained
in the analysis by rescaling
the axial velocity as;
u = u'/
3.6.2
[Udr
I
and for convenience the axial distance is rescaled as
z = sr
3.6.3
Introducing the Dean number,
Dn,
which
square root of the inertia and centrifugal
is the ratio of
the
force product to
the viscous force;
'4
3.6.4
Dn
= Re Sr
the resulting equation, along with
fully developed flow is given
the continuity condition for
in equation 13 of Berger et al.
This equation was obtained by recognizing
that the velocities
Page 39
no
longer vary with axial position.
Inspection of
this
is linear.
equation indicates that the axial pressure gradient
A stream function is introduced and,
following cross-
differentiation to eliminate the pressure, equations 15 and 16
of Berger et al are obtained.
For
small values of Dn the equations can be solved by
expanding the solution in powers of Dn about the Poiseuille
when expressed
in terms of
pressure difference is given by Dean ( 1927,1928);
[
Del P
3.6.5
Del
Ppois.J
where Del Ppois.
4
2
1 +.0306(K/576)
-. 0110(K/576)
is the the Poiseuille pressure drop
tube with the same flow rate,
and K
+
.
the axial
straight
The result,
tube profile.
straight
in a
is the original
form of the Dean number defined by Dean as;
a
[
Rc
L
KE=
3.6.6
where Wmax
Wmax a
Nu
2
2
is the maximum velocity in a straight pipe of the
same radius resulting from the same
imposed pressure gradient.
The relationship between Dn and K depends on the ratio of the
flux in the straight tube at a specified pressure gradient,
Qs,
and
the flux
in the curved
Qs
3.6.7
Dn
'
pressure gradient, Qc;
E'/ K]
Qc
tube with the same imposed
Page 40
where the ratio Qc/Qs,is (Berger et al);
6
4
2
Qs
= 1 -. 0306(K/576) +.0120(K/576) + O(K/576)
Qc
3.6.8
Dean's series expansion solution is valid up
to Kz576.
Figure 3.6-4 presents the results of deans solution for the
pressure loss in a curved tube normalized to that
in a
Poiseuille flow with the same bulk flow rate verses Dn.
The
graph was generated by calculating the flow ratio based on a
value of K,
and then the finding the Dn corresponding to K and
It
the flow ratio.
is interesting
to note that for Dn S 10,
effects of curvature on the pressure gradient are almost
negligible.
solution,
limit of applicability for Dean's
Even at the high
the effects are small,
in the flow resistance at K=576,
generating only a 2% increase
corresponding to Dn-16.
The nature of the flow and the scaling of Dean's solution
for
the secondary flow can be understood by considering the
The axial velocity establishes a
forces acting on the fluid.
pressure gradient which is balanced by the shear
wall
induced by the secondary flows.
stress at
The resulting pressure
balance can be written as;
2
3.6.9
r
g;U
Rc
Vsec
^
r
which gives the secondary velocity scaling found by Dean;
Vsec
Sr Re
3.6.10
U
the
Page 41
At higher values of Dean number, numerical
been utilized to solve equation 3.6.7.
studies have
The most expansive of
these numerical studies is by Collins and Dennis
(1975) who
were able to extend Dean's solution up to Dn=5000.
numerical
studies suggest that the structure of
tube flow at high Dean numbers
rotational
The
the curved
is that of an inviscid
core surrounded by a thin boundary layer at the
tube wall.
An understanding of the nature and flow characteristics
which
typify the high Dean number curved tube flow can be
gained by investigating the asymptotic
equations.
Pedley
(1980)
formed the asymptotic
scaling equation 3.6.7 with
streamfunction, 0,
thickness,
3.6.11a
limits of the governing
limits by
the asymptotic form for
Dean number, Dn,
and boundary
the
layer
S;
Dn
/K
U
a
3.6.11b
3.6.11c
'
-1K /Dn
S
K
Note; Pedley used a dean number, D, defined
as /K, and normalized velocities by Nu/a rather
These differences in notation account
than U.
for in the somewhat unusual notation presented
in 3.6.14 to preserve the form of Pedley's
analysis.
For
the core flow,
the viscous effects are negligible,
the pressure gradient established by the centrifugal
forces
and
Page 42
terms: equation 3.6.7a therefore
are balanced by the inertial
gives;
= a + a
1
3.6.12
there is a balance between the
In the boundary layer,
inertia,
3.6.13
viscous and the centrifugal force terms gives;
4T
+
a
2a
-
=
2(a
-
a)
+
3T
=
T
Solving the three equations for the three unknowns gives;
3.6.14
T
=
a
which can be inverted
3.6.15a
/K
3.6. 15b
S
3.6.15c
8
The pressure drop
Del
*'a
1/3
to give
the parameters in terms of Dn;
1/cJ
1.5
=
Dn
1/K /Dn
/K
=
=
Dn
= Dn
Dn
is therefore seen to go as;
L du
L
P = Tw - = P D dr wall
D
U L
*L
SP
-
3.6.16
'
=
8 D
- U Dn
D
In contrast to the secondary velocity scaling at
low Dean
numbers, equation 3.6.15 allows the secondary velocity scaling
to be written directly as;
Page 43
U Dn
U
3.6.17
U
Vsec
Dn
giving the ratio of secondary velocity
a constant,
in the core as
to that
unlike the low dean case in which
the ratio
increases with Dean number.
These asymptotic forms have been verified by numerous
experimental and numerical studies and are summarized by
Berger et al
Results of the experimental studies for
(1983).
in Figure 3.6-5.
losses are shown
the frictional
The
experimental data is sden to be correlated well by the Hasson
correlation below curvature dependent branch points which are
identified as transitions to turbulence.
The correlation is
valid down to a Dn of about 20 where Dean's solution asymptote
to Poiseuille flow.
The Hasson correlation
*L
3.6.18
Del P = 32 P
r
-
D
Ito
(1959),
is;
0.556 + 0.0969 Dn
Re
LJ
experimentally documented the frictional
in turbulent flows for a curved
tube.
loss
The transition to
turbulence was found to be retarded by the action of the
secondary flows.
The exact reason for
transition is still
not clear,
but
it
the postponement of
is generally believed
that the action of secondary flows to thin the boundary layer
is similar to
the effect of a favorable pressure gradient.
the range of experiments 1/15 > Sr
critical Reynolds number to be;
> 1/9000,
Ito reports the
For
Page 44
0.32
(Re)crit = 2E4 Sr
3.6.19
The experiments were correlated by a Blasius type
expression with an additional curvature effect;
* L
3.6.20
Del P
=
4 P
r-3/8
-'4
0.029 + 0.034 Sr
Sr
-
D
2
Dn
Re
L
validated over the experimental range;
1.5
300 > Dn Sr
Re Sr
=
2
> .034
1.5
For
larger values of Dn Sr
,
the expression becomes;
1/5
3.6.21
Del P =
'
P
L
- 0.316
D
2
Sr
Re
Dn
While the typical curvature encountered
is below the range of
Ito's experiments,
in the
lung,
it can be used to
estimate a critical reynolds number of about 10,000.
places the Re Sr 2 product at 200 at
that 3.6.20
is the most
'1/7,
transition,
This
indicating
likely form applicable to pulmonary
conditions.
3.7 Kinetic Energy Dissipation
The case of expiratory flow
through a bifurcation which is
embedded within an airway can be considered
fully developed
the sense that the entering velocity profile will be similar
in
to
Page 45
the exiting velocity profile.
fully developed symmetric
This must be the case for a
flow within a
"large"
airway system
because the inlet of a bifurcation will be the outlet of
another bifurcation.
The most striking feature of this situation is that a
total
of eight vortices in the two
within the bifurcation to only four
mechanism for this conversion
most likely be considered
inlet flows are converted
in the outlet.
is not clear,
The
although
it can
to fall between two extreme
alternatives:
a Each of the eight inlet vortices are dissipated
through viscous action, and four more are
generated through the conversion of potential to
kinetic energy.
n The kinetic energy of the inlet vortices are
conserved by the enhancement of the two outer
vortices of each daughter tube by the two inner
vortices feeding them through the secondary flow.
The dissipation in the first case would be similar
nature to fully rough turbulent flows in which
in
large eddies
are generated and then dissipated due to viscous action.
The
generation and dissipation of the secondary flow vorticies
will not occur
randomly as in turbulent flow,
once at each bifurcation.
flow is
but will occur
Another difference from turbulent
the orientation of the secondary velocities.
In
bifurcations, the vorticies are oriented along the axis of
flow, which
is not always the situation in turbulent flows
(Schlichting, 1979).
An estimation of the magnitude of
the pressure loss that
Page 46
would result
if the energy in the inlet vorticies were
dissipated is the magnitude of kinetic energy;
Del P a '4gVsec2
3.7.1
where Vsec
flow.
is the average secondary velocity in the
The magnitude of the secondary flows was given in
1/2 of the bulk flow, U.
section 3.1 to be between 1/5 and
Using
inlet
1/4 as a typical value,
the pressure loss across a single
generation can be approximated as;
Del P
3.7.2
~
(1/32)
P*Re 2
Conservation of the kinetic energy in the second case
would result
in no induced pressure drop.
3.8 Definition of Pressure Drop
Before estimates for the form of the pressure drop
in
bifurcating flow can be made based on the effects discussed
above,
the pressure drop must first be defined.
The
difficulty in defining the pressure drop comes because the
pressure across a
leg of a bifurcation
strong secondary flows and
is not uniform.
The
tube curvature create a pressure
variation within the tube cross section which are a substantial
fraction of those across the bifurcation
(Pedley 1980).
The nature of the flow does not allow many simplifying
assumptions such as inviscid or
low Reynolds number flow to be
Page 47
The pressure loss
made to help in defining pressure losses.
due to
the bulk behavior of the fluid
integral
is desired, which allows
loss of information.
techniques to be used without
The two
integral approaches which are candidates for the
analysis are momentum and energy balances.
are difficult to
implement
in the analysis because the walls
of a bifurcation are inherently curved,
introduce unknown wall
and as a result,
Energy balances are
tensile forces.
"accounting" for changes in
suited to the analysis,
well
Momentum balances
kinetic energy flux by the rate at which work
is done on the
bifurcation by pressure and viscous forces.
The steady-state energy equation for an incompressible
fluid can be written in integral form as;
{-
V2
3.8.1
pu da
pu da +
-
=
I
Vol
I
0
& dVol
Where the subscript I represents
V2
gu dA
-gu
-
dA
0
integration over the region of
inflow and 0 the region of outflow, dA is a differential area,
Vol
is the static pressure,
is the axial velocity component,
u
is the control volume,
is the viscous dissipation function,(
and V is magnitude of the
see Potter and Foss pp 192),
velocity;
3.8.2
where Vsec
3.8.3
V2
=
U 2 + Vsec
2
is the magnitude of the secondary velocity;
Vsec 2 =
p
v2 +
w2
where v and w are the secondary velocity components.
Page 48
The integral of the viscous dissipation function over
control
volume represents the total
viscosity,
energy dissipation by
later reference is defined as;
and for
{l dVol
DE
3.8.4
the
Vol
The bulk properties which are measured experimentally are
the flow rate,
R, defined as;
Q, and average static pressure,
1
{
p dA
3.8.5
Q
3.8.6
j
u dA
The average pressure term can be introduced
rewriting the
pressure,
local static pressure as the sum of the average
R, and a pertubation, p'.
the pressure work
Jpu
3.8.7
where Pc
This substitution allows
terms to be written;
dA =
{
[R + p'] u dA = Q
is a correction to
accounts for
into 3.8.1 by
C R
+ Pc
I
the average pressure which
the non-uniform pressure profile,
and
is defined
as;
1
3.8.8
Pc
-
p'u dA
The kinetic energy flux terms
in 3.8.1
in terms of bulk properties by introducing
can also be written
the secondary
Page 49
velocity correction factor,
factor,
fa;
1
3.8.9
F
3.8.10
f
{-
L
+
(u/U)
(v/U)2
dA
3
Eu/U]
a
2
(w/U)
1
where U
and the axial velocity shape
fs,
dA
A
is the average velocity, defined as U
= Q/A.
Introduction of these terms and dividing through by the
flow rate,
written
Q,
allows the steady-state energy equation to be
in the form:
3
3.8.11
R
=
0
D
/Q
+
0
0
CQ
I
+ f
f
A
s
I102
+ Pc
a J
I
I
The terms on the right hand side of 3.8.10 represent the
relative contributions of viscosity,
and pressure uniformity correction to
static pressure respectively,
3.8.12
kinetic energy changes,
the drop
or;
Del R = Del Pv + Del Pk + Del Pc
where the pressure loss terms are defined as:
0
3.8.13
Del
Pk = '4Q2(f +f
a
3.8.14
Del Pv =
D
/ Q
10
)/A2
s
I
in the average
Page 50
0
Del Pc = Pc
3.8.15
I
3.9 Summary of Pressure Loss Correlations
the above
To compare the order of magnitude of each of
effects,
the pressure drop
in an
idealized bifurcation was
The pressure drop was calculated assuming that
calculated.
theoretical pressure
loss form for a particular
applied to each branch in the bifurcation.
flow condition
Results can then
to the experimentally determined pressure losses
be compared
and any similarities in either form or magnitude can serve as
an indicator of flow similarities.
Table 3.9-1,
for
below,
summarizes the expressions developed
the cases of interest,
normalizing factor
presented
is the loss if the flow were everywhere
fully developed Poiseuille flow.
because
it
The
in normalized form.
This normalization was chosen
is the anticipated asymptotic
experimental results,
lower
limit for
the
and indicates the extent to which the
flow resistance is enhanced over Poiseuille flow.
Figure 3.9-1
presents the results for
in a branch of a bifurcation vs tracheal
comparison,
the factor of
the normalized loss
Reynolds number.
For
three band of results found by the
previous investigators presented
in Figure 2.3-1
is also shown.
The magnitude and form of the results are seen to be most
similar to curved tube, 2-D convergent channel,
flow theory.
and entrance
Page 51
Table 3.9-1
Del Pv/Del Ppois
Case
0.556+0.096Re Sr
Curved tube flow
Re > 60
.152
Entrance flow
Re >> 1
-
2-D Channel flow
Re >> 1
-
/D
2
'
'
(L+Leq) -Leq
L
Ln(An-1/An)
Re
/3
L/D
3/4
Turbulent pipe flow
Smooth walls,Re>2000
Turbulent curved
tube flow
2
Re > .034/8r
Kinetic Energy Diss.
4.9E-3 Re
4/5
1/10
4.9E-3 Sr
9.76E-4 Re
Re
Re
Page 52
4.
Experimental Methods
4.1 Subtraction Method
The experimental approach developed to quantify the
pressure drop across a single bifurcation embedded within an
airway system is termed the
"subtraction method".
Development
of this new approach was deemed necessary because of the
restrictive assumptions imposed by
the methods of previous
investigations.
The strong secondary velocities and wall curvature typical
of expiratory flows establish
pressure variations.
(1972)
and Akhavan
"substantial" cross-sectional
Experiments by Jaffrin and Hennessey
(1980),
found that
in oscillatory flows the
cross-sectional pressures can vary from 20% to
pressure difference across a bifurcation.
study exists for expiratory flows,
100% of the
While no similar
large cross-sectional
pressure variations can be expected to make a direct
measurement of pressure
loss across a bifurcation difficult.
Previous investigators have typically avoided
by basing the
this problem
loss in a single bifurcation on the measured
pressure-flow curve for an entire airway network.
Errors in
measuring pressures, upstream in plenums and downstream in
plenums or straight tube sections, can be expected to be small
in comparison to the overall pressure
this method requires that
single bifurcation,
be assumed,
loss.
Unfortunately,
some algebraic form for the
typically that found for
without substantiation.
loss
in a
the entire network
Since the character of the
Page 53
entire network represents a weighted average over the entire
airway,
the overall relationship may not simultaneously reflect
the relationship
In addition,
in each bifurcation.
effects may be erroneously
included,
and
entrance
it will be difficult
at best to distinguish the existence of different flow regimes.
The subtraction technique was also developed with the
intention of extending the model
asymmetric effects.
Basing the
experiments to
include
loss in a single bifurcation
on systemic characteristics becomes extremely difficult with
flow asymmetries.
The subtraction method for symmetric flow conditions is
shown schematically in Figure 4.1-1.
The method requires an
airway system which consists of a set of matched upstream
airways and a single test bifurcation.
The upstream airways
are required to produce a boundary condition for
the flow into
the test bifurcation which can be considered
"typical" of the
inlet flows to a bifurcation embedded in the
lung.
The scheme
to determine the pressure drop of the test bifurcation alone is
as follows;
w The pressure-flow relationship for the identical
upstream airway network (peripheral to the shaded zone in
Figure 4.4-1) is measured over the desired range of flow
rates.
a Two identical upstream airway networks are then attached
to the daughter branches of the test bifurcation.
a The pressure-flow relationship of the new combined airway
system, consisting of the upstream airway networks and
The range of flow
the test bifurcation, is measured.
used to test the
range
the
rates should be about double
upstream airways.
Page 54
that the flow rate through
is equal to half of
systems
each of the upstream airway
the total.
a The symmetry condition ensures
m The pressure drop due to the test bifurcation can then
be calculated by taking the difference in pressure drops
between the combined airway at a specified flow rate,
and the pressure drop for the upstream airways at half
the flow rate.
The subtraction method requires that the pressure-flow
relationship of the upstream airways be fully documented,
and
it assumes that the addition of the test bifurcation does not
alter this relationship.
This assumption must be justified in
light of the experimental measurements of the pressures in a
finite pipe bend,
Ito
(1960),
demonstrating that
the effect of
a pipe bend propagates on the order of a pipe diameter upstream
of the bend.
Figure 4.1-2 presents Ito's results for
the
pressure distribution in a finite pipe bend with straight
upstream and downstream tangents.
The pipe bends used by Ito were severely curved, with a
radius ratio
gradient
Sr=1/3.7.
As a result, a very strong pressure
is established at the bend entrance which generates
secondary flows.
The subtraction technique is not expected to
be significantly effected by this effect for
several
reasons;
" The idealized bifurcations which make-up the airways are
separated by straight sections of about a diameter in
length.
" The blunt velocity profiles typical in expiratory flows
offer a much thinner boundary layer with the low
velocities that are most affected by adverse pressure
gradients.
" The radius ratio in the test bifurcation is more gradual
than used by Ito, 1/7 vs 1/3.7.
Page 55
Finally, it can be seen from Ito's results that while
the variations in pressure can be substantial, the mean
pressure variation is significantly lower that the mean
pressure loss across the bend.
Subtraction of two values obtained
may result
if the overall
in significant error
signal/noise ratio
is low.
in separate experiments
experimental
The signal/noise ratio for the
subtraction experiments is given by the difference in signals
from two experiments divided by the sum of the noise in each
experiment.
High signal/noise ratio therefore require not only
that the experimental equipment be capable of low noise levels,
but also that the experiments be designed to provide
significant differences between signals.
4.2 Downstream Boundary Condition
The downstream boundary condition imposed on the terminal
airway
is an important consideration in determining the signal
differences between experiments.
Selection of the downstream
boundary condition is therefore an important consideration,
while assumed not to have an influence on the pressure-flow
relationship of the airways upstream of
the exit.
Three downstream boundary conditions were evaluated,
long straight tube,
reservoir.
a conical
Exhausting
principle reasons.
and exhausting
into a
into a reservoir was chosen for two
The first
additional unknown frictional
system.
diffuser,
a
The second is that
is that
it
introduces no
dissipation sources to
the
in evaluating the kinetic energy
flux exhausted into the reservoir,
a better understanding of
Page 56
the flow nature might be obtained.
This selection was made despite the introduction of slight
errors,
options.
that some advantages are present with the other
and
Error are
introduced with the assumption that
the
effect of the uniform pressure plenum does not propagate
upstream into the "tracheal" branch,
previous section,
This error
in the
and that a pressure difference between the
imposed hydrostatic head
exist.
as discussed
in the tank and
is proportional
and streamline curvature,
the jet pressure may
to the kinetic energy flux
and is therefore expected to be small
because the curvature in the exiting stream is only significant
at
low Reynolds numbers,
flux
where the corresponding kinetic energy
is small.
The long straight
tube has the advantage that
establishes a Poiseuille flow profile,
allowing a side pressure
tap to measure the average cross-sectional
generating a known kinetic energy flux.
it
pressure,
A conical
and
diffuser
theoretically would allow recovery of the kinetic energy head,
leaving
and
systemic pressure
losses as a result of friction alone
therefor maximizing the relative difference between
experimental
signals.
4.3 Subtraction Method Analysis
Analysis of the subtraction technique demonstrates that
it
is a viable method to measure the area-averaged static pressure
difference at a given flow rate across a single bifurcation.
It remains
to separate that pressure difference into a
Page 57
component attributable to friction and one attributable to the
To make this distinction,
convective acceleration.
based definition of pressure drop and
developed
in 3.7 can be applied to
the energy
components
its two
the airway systems defined
in Figure 4.1-1.
Consider first the airway system consisting of the two
The total
upstream airways and the test bifurcation.
pressure loss of the combined system,
static
R):, at a flow rate
(Del
Q, can be written as the sum of the loss in the test
bifurcation and the loss upstream of the test bifurcation;
4.3.1
(Del
R),
(Del
=
F)j-: +
(Del
is comprised of just the two
the system which
Now consider
upstream airways.
, can be seen to be the same as
(Del
just one of the upstream airways,
the total
pressure drop for
(Del R)U,
at '4 the flow rate.
Rewriting 4.3.1
pressure loss in the test bifurcation,
given by
the difference in
flow rate of
R)s.- 1
to give the
it can be seen to be
the driving pressure required for a
Q for the combined airway system and the driving
pressure required for a flow rate of Q/2 for
just one upstream
airway.
4.3.2
(Del
R)
=
I-E
(Del
R)
-
C
Recalling equation 3.7.12,
0
(Del R)
U
Q/2
the static pressure loss across
the test bifurcation can be written as the sum of the viscous
and kinetic energy
losses plus a correction term for
the non-
Page 58
uniform cross-sectional pressure distribution;
(Del R)x..-
4.3.3
Pv + Del
Del
=
Pk + Del
the viscous pressure
write
with 4.3.2 to
which can be combined
Pc
in the test bifurcation as;
loss
4.3.4
Del
[
Pv
The
3.7.13,
it
-
R)
C
written for
=
K
(Del
R)
loss,
Del Pk -
-
U
0
kinetic energy
Del Pk
4.3.5
where
(Del
Del Pc
Q/2
Del Pk,
can be evaluated with
the case of symmetric flow conditions as;
*
2
P*Re
is understood
*
(fa+fs)
that
the
E
2
[ P Re
(fa+fs)
I denotes evaluation
subscript
of the terms at the inlet of the test bifurcation and E for
the outlet, or exit.
Applying the principle of continuity,
conditions,
equation 4.3.5 can be written;
4
*Ei
4.3.6
where e
Del
local flow
the
and recalling fs and fa are functions of
Pk =
'[ P ReI
JE
L
-
(fa+fs)
L
V4E
E
is defined as the ratio of the parent
(fa+fs)
:IJ
to daughter
diameters.
Equations 4.3.6 and 4.3.4 combine to give the viscous
tube
Page 59
Application of the
pressure loss in the test bifurcation.
expressions requires the difference in driving pressures from
two different experiments,
the
and a functional dependence of
(fa+fs) sum and the pressure correction term on the exiting
The correction factors will be
flow rate or Reynolds number.
explored
in greater detail
in Chapter 5.
4.4 Quasi-steady Test Apparatus
The quasi-steady experimental system developed to measure
static pressure-flow characteristics of an airway
the total
system is shown in Figure 4.4-1.
The method
is considered
quasi-steady because a slowly varying hydrostatic head is used
to generate the flows to cover the desired range of Reynolds
numbers.
The quasi-steady flow
is driven by the hydrostatic head
and resisted by fluid acceleration and friction within the
test airway.
The desired range of flow rates will be shown to
depend on the characteristic pressure P*;
viscosity,
density and airway size.
of the pressure loss at a given flow,
a function of fluid
P* determines the scale
and can be set to
amplify the pressures into an easier range to measure with
conventional
pressure transducers.
The driving pressure head
is generated in a circular
constant diameter fluid filled reservoir.
principles
therefore dictate the flow rate through the airway
will be proportional to
varies.
Continuity
the rate at which
For sufficiently low flow rates,
the reservoir
level
pressure gradients
Page 60
within the reservoir will only be due to hydrostatic forces,
generating
to the fluid
proportional
bottom of
the
a pressure at
system can
the
instantaneous flow rate in
The
in the reservoir.
level
is
the reservoir which
therefore be
determined by the time derivative of the measured hydrostatic
the
head at
base of
the reservoir.
Hydrostatic conditions also exist
exhaust tank for sufficiently
tank
the exhaust
documented phenomena,
107 ),
and
( see for
-
'
2
Q
=
w
gravity,
weir.
the plenum and
The
level
Flow over
in
weir
a
example Sabersky,
is
pp
is given by;
4.4.1
where Q
low flow rates.
is maintained by a
a well
in both
- Cd
3
1.5
h
(2g)
is the width of the weir, g
is the flow rate, w
is
an experimentally determined discharge coefficient
Cd
(typically '0.62)
and h is the fluid height above the weir
level.
The driving pressure across
is given
the pressure difference between
by
plenum and
the airway system
the exhaust tank.
transducer
lines leading
measures the driving pressure.
pressure-flow relationship
of flow
the reservoir
For quasi-steady hydrostatic
conditions, pressure taps with fluid filled
differential
time
at any
for
the airway system
over a
to a
The
range
conditions can therefore be determined by recording
pressure signals over
The variation
time.
in downstream
level
with
flow rate
can be
two
Page 61
made infinitesimally small relative to
the change in the
In this
the weir width.
plenum level by increasing
limit,
the
experiment would be simplified because only one pressure
signal would be required to
driving pressure.
monitor both the flow rate and
size limitations and the desired
Practical
accuracy prohibit this experimental simplification of
eliminating one pressure measurement for these experiments.
4.5 Experimental Scaling
Analysis of the quasi-steady experiments
establish
the experimental scale for which
is necessary to
the quasi-steady
approximation can be supported, and determine the required
fluid properties.
The governing equations for an approximate
analysis are;
one-dimmensional model
Continuity
dD
= Qt
= Qweir + Ae dt
dt
-dH
Qr = Ar
4.5.1
where
-
Qr is the flow in the reservoir, Ar the reservoir area,
H the reservoir height, Qweir
the area of the exhaust
tank over the weir
tank,
level,
is the flow over the weir, Ae is
the level
D is
and Qt
of the exhaust
is the flow rate
"tracheal" tube.
Weir flow
4.5.2
where k
1.5
Qweir = k Dw
is given from 4.4.1
Motion along a streamline
as k=
(Euler)
(2/3) Cd w
(2g)
in the
Page 62
SV
8V
+ Fr
+ V-
4.5.3
where 8
Ss
Ss
st
is the partial
velocity magnitude,
Fr
+ ggZ)
1 &(P
is a frictional
Z is the height
in the gravity field,
Integrating
streamline from the free surface
g,
is the distance
loss per unit mass, and s
along the streamline.
of
is the
differential operator, V
4.5.3 along a hypothetical
in the reservoir to the exit
the "trachea", gives;
t V
4.5.4
2
r2
+
-d
-
+
V
r
Lt
Jr t
1t
P t
J
P
-
atm
gH +-JFr ds
r
where the subscript t refers to the tracheal values,
and r
refers to the values on the free surface of the reservoir.
For
large exhaust tank areas,
assumed
tracheal
to exist,
and using
a hydrostatic
head can be
a gage pressure referenced to the
pressure if the exhaust tank were at the weir level,
the pressure term can be written;
~
1
4.5.5
-
Pt -
P
= gDw
Defining a characteristic length,
L*:
jAt
4.5.6
where A
L*
A
ds
is the cross-sectional flow area,
equation 4.5.4 can be written;
such that V= Q/A,
Page 63
2
L* dQ
+
- At dt
4.5.7
t
2
Q
1
A
2
-(At/Ar)
+ gDw +
Fr ds = gH
L
t
and
introducing continuity to give the relation between tank
height and flow;
dQ
= -Ar
dt
dH
Q = -Ar-
4.5.8
and
dt
and writing the frictional
d2H
dt2
loss in the form which
it
is
typically available;
t*
[ Fr ds = P
Jr
Del P
4.5.9
fr
where Fn(Re)
number
Fn(Re)
represents some function of
the tracheal Reynolds
in the airway.
in a form which
Together these expressions give 4.5.7
convective and
demonstrates the balance of the temporal,
frictional
terms to the driving head;
*
-gL
Ar dH2
- At dt2
Temporal Acc.
+
r
Ag-
2
dH
dt
At
r
Convective Acceleration
which is a one dimensional
with a coupling term Dw,
continuity.
2
J1-[At/Ar)
]
*
4.5.10
+ P Fn[Rel
=
Frictional
Losses
gg(H-Dw)
Driving
Head
non-linear differential equation,
given by
the weir equation and
Assuming that the exiting kinetic energy is
dissipated before flowing over
the weir,
the weir height
is
Page 64
given by;
dDw
4.5.11
-
dt
Ae
Dw
-
Ae
3
Ldt
1.5
(2g)
2 Cd w
-dH
Ar
--
Equations 4.5.10 and 4.5.11
given an approximate frictional
can be solved numerically
relation.
the purposes of
For
ensuring that quasi-steady conditions exist,
only an order
magnitude analysis of the equations is needed,
however.
of
The
analysis needs to demonstrate that the pressure drops
terms are small
associated with the temporal
both the spatial
in comparison to
convective acceleration terms and
the
frictional terms.
From the results of Hardin and Yu (1980),
loss experienced
in a 3 generation airway model,
in Figure 1.3-1,
"idealized bifurcation"
1.6
*
Del P
Giving a low
limit functional
Fn
'
where the tracheal
Re =
2
1.24 Re 2
dependence Fn as approximately;
1.6
or
Re
10 Re
Reynolds number
4 Ar
4.5.14
is;
L
2
4.5.13
+
25.3 Re
P
=
fr
based on the
can be used as an
The relationship
order of magnitude estimate.
4.5.12
the frictional
D
N
h D Nu
is given by;
-dH
dt
ldt
The order of magnitude analysis of 4.5.11
indicates that
Page 65
the exhaust tank
to
can be considered small
level Dw,
long as the tank width
the reservoir head, H, as
r
* Ar d 2 H
= -L
--gL
4
At dt2
4.5.15
it takes for the tank
time
gL
*Nu
-D
Re
T*
time, and can be estimated as the
is a characteristic
where T*
4.5.10 can be written;
dRe
dt
*Nu
D
large
is
To estimate the order of
compared to the tracheal diameter.
magnitude of the temporal terms,
as compared
level
to drop a distance which
flushes the airway system;
4.5.16
T*
'-
L
L
giving the
inertial
*
At 2
L
loss
Ar d2H
--
L
4.5.17
= L
J
-11*
=L
Nu At
*
-1
dH
dt
-
-Re
D
Ar
D
Ar
-
*
L At
Nu Re
in 4.5.15 as;
R
Re
-
*P
Ar
At dt2
4.5.14,
Substituting 4.5.17,
magnitude of each term
and 4.5.13 gives the order of
in 4.5.10;
*
P
At
--
2
2
*
P
Re
I-
[At/Ar)
1.6
*
2
Re
10 P
Re
*
4.5.18
P
LArJ
Temporal
Acceleration
Convective Acceleration
Comparison of terms shows that
terms will be small
if;
Frictional
Losses
Driving
Head
the temporal acceleration
in comparison to convective accelerations
Page 66
<< 1
4.5.19
Ar
losses to
and comparison of the frictional
indicates that
the temporal
losses
the quasi-steady approximation requires the
potentially more restrictive condition that;
0.4
Re
4.5.20
[
At
]
<< 10
LAr J
For
the maximum Reynolds number under
Re=10,000,
investigation,
this condition reduces to the same of 4.5.19,
indicating that the quasi-steady approximation
is valid for
a
large reservoir tank.
test apparatus with a sufficiently
The scale of the experiment and
the range of fluid
properties required to generate measurable pressure drops and
remain quasi-steady can therefore be estimated from equation
4.5.10 by neglecting the temporal
terms.
The second order
differential equation therefore reduces to the approximate
algebraic relation;
H -
2
Dw
'4Re
=
4.5.21
1.6
+ 20 Re
H*
where H* is define as;
Nu
g; 9
g
2
H* =_
4.5.22
For
P*
the maximum Reynolds number of
and frictional
10,000,
the convective
losses in 4.5.18 are on the same order,
and
Page 67
4.5.21
gives the required driving head to be;
H z
4.5.23
At
the lower
10E8 H*
range of Reynolds numbers, near
the
100,
reservoir height can similarly be approximated by;
4.5.24
H Z 10E4 H*
Imposing practical
limits on the experiments, such as a
reservoir height of approximately 100 cm,
pressures of about
diameters about
1% full
scale, or
1 cm,
and minimum measured
and tracheal
1 in, (2.54 cm) the fluid properties required
are for a fluid with
"substantially" higher density than air,
because buoyancy forces have been neglected,
and
kinematic
viscosities in the range;
4.5.25
.05 : Nu
S .5
Other practical considerations such as cost,
and safety
limit the possible choices.
availability
The selection of
water-glycerin mixtures stand-out as an excellent choice.
Figure 4.5-1 presents the kinematic viscosity of waterglycerin as a function of water content.
The desired
kinematic viscosity range falls between 20% and 70% water
content by weight.
4.6 Data Collection
A Digital Equipment corp.
MINC-23 digital
lab computer was
Page 68
used to digitize and record the pressure signals.
equipped with a -5
The MINC
is
to +5 volt 16 bit A-D converter, with 4
reserved bits for configuration information,
giving a
The differential pressure
digitation level of 2.44 mvolts.
was measured with a Valydyne differential pressure transducer
model
DP 103-10 ) driven with a Valydyne carrier-demodulator
(model CD 15) The hydrostatic reservoir pressure head was
measured with a Honneywell Microswitch pressure transducer,
(model 143PC
conditioner.
op-amps,
),
through an analog signal
and passed
The analog signal conditioner
and provides a signal offset and
is based on two 751
Both
amplification.
the signal conditioner and the carrier demodulator allowed the
output of each transducer
to cover
the full
-5
to +5 volt scale
of the A-D converter to minimize digitation error.
Before an experimental
water-glycerin mixture
#200)
drip
run,
the kinematic viscosity of the
is measured with a Cannon-Fenske
type viscometer.
The simple drip viscometer was
calibrated with a Haake ( Model RV12
rate of 346
1/sec to
(Tube
) viscometer at a shear
insure accurate results.
A typical
mixture was used to generate a temperature viscosity
correlation,
shown in Figure 4.6-1.
temperatures experienced,
Cannon-Fenske viscometer
For
the range of fluid
the viscosity measured with the
( Tube #200),
partially submerged in
the exhaust tank for temperature equlibration,
and the
viscosity interpolated based on the exhaust tank temperature
were in agreement
An experiment
to within
1%.
is begun by filling
the reservoir to the
Page 69
level required to generate the maximum desired Reynolds number,
necessary,
to empty
and
outputs are checked and scaled if
The transducer
see 4.5.21.
and the flow gate is removed.
is recorded,
The time for
and the low transducer output
also scaled to utilize the full -5
The reservoir is then refilled,
the tank
is checked,
to +5 voltage range.
stopping at about
10 different
levels to record and re-calibrate the pressure
tank
transducers.
The re-calibration is necessary not only because
the offsets and amplification may have been changed,
but also
to maintain the desired accuracy in tank and differential head
levels as a result of potential changes in transducer
amplification with ambient temperature changes.
Experience demonstrated that slight non-linearities, on the
order of I to 2%,
in the pressure signals
to about 5% in the results.
introduced errors up
The errors appeared primarily as
slight oscillations when pressure differences were normalized
by the square of the calculated flow rate.
The digital signal
processing easily allowed the non-linearities to be corrected
by fitting the transducer calibration curve to higher order
curves.
the small
A second order curve provided enough correction for
non-linearities in the differential height reading,
while a third order curve was necessary for the reservoir
level
transducer.
Figure 4.6-2 presents the typical
differences between the tank
and
levels measured with a manometer,
those predicted with the non-linear transducer
calibrations.
represented by
The errors are less than the digitation errors
2mvolts'
.03cm, and errors
in reading the
Page 70
.05cm.
manometer,
Once the tank
calibrations,
is full after
the experiment
finishing the transducer
is ready to begin.
The data
acquisition is started as the flow gate was removed.
The data
acquisition rate was fixed
to give 500 sets of voltage readings
over the time required for
the tank to empty.
The data sets,
along with the calibration values and experimental
were then stored on floppy disks for later
conditions
transmission and
analysis.
4.7 Data Reduction
The data sets obtained in each experiment were reduced to
obtain the non-dimensional pressure-flow relationship of the
test airway.
The digital scheme implemented with a fortran
code first converts the data from a set of voltages to
reservoir level vs time and non-dimensional differential
pressure vs time by multiplying by the appropriate combination
of the stored values of the transducer calibrations,
viscosity,
tracheal diameter,
The data near
kinematic
and data acquisition rate.
the start of the experiment are discarded
the flow becomes quasi-steady.
The time scale for
quasi-steady state to be established
is also given by T*
until
defined in 4.5.16, which can also be interpreted as the time
necessary for the fluid
initially
in the network to be
convected out.
The curve fitting errors outlined above are approximately
constant
throughout the signal range.
Percentage errors are
Page 71
therefore greater at the lower signal
these errors,
levels.
To minimize
the
the experiments are allowed to run until
tank level reaches zero.
Any non-zero asymptotic reading
in
the differential pressure signal is assumed to be an offset,
and
is subtracted from the results to
improve
Offsets greater than one or two digitation levels are
reading.
taken as an indication that the calibration
valid,
is no
longer
and both the experiment and calibration are repeated.
Data below some critical tank level,
or
the low pressure
1% full
scale,
typically 1 cm is used,
are also discarded due to the high
percentage errors expected.
The height vs time data is converted to a Reynolds number
vs time form by evaluating Equation 4.5.14 at each point.
Calculation of the time derivative of
the reservoir height
data proved to be the most sensitive step
in scheme.
in
Digitation errors and experimental noise resulted
significant errors in calculating the derivative with simple
finite difference schemes.
To minimize these errors,
fitting technique was developed to
point.
a curve
calculate the slope at a
A second order curve was fit
to a specified
points before and after each data point.
number of
The curve fit was
then analytically differentiated and then evaluated at the
time corresponding to the data point.
Typically
10 points on
either side of the time being evaluated were used,
corresponding
to 21 points out of the 500 total points
involved in each curve fit.
The numbers of points used
in the averaging represents a
Page 72
compromise between competing factors.
in the averaging decreases,
increases as the number of points
but
Point to point noise
inaccuracies in the curve fit appear when averaged over
too wide a range.
Additionally,
in the curve fit are
included
as the number of points
increased, data at the ends of
is lost.
the curves
The dimensionless pressure vs time,
the Reynolds
and
number vs time data are then assembled parametrically to
generate the dimensionless pressure-flow relationship for the
The data are stored in this form for later use in the
airway.
implementation of the subtraction technique.
4.8 Technique Verification
Verification of both the quasi-steady experimental
approach
the subtraction technique was deemed necessary to validate the
subtraction technique.
For
this purpose,
into a straight
through a bellmouth entry
understood
flow from a reservoir
tube was selected
( Rivas and Shapiro
because it
is both well
and can be
implemented on the same scale as
Two
the airway system.
sets of experiments with different water-glycerin
mixtures,
and therefore Reynolds number ranges,
in the geometry depicted
in Figure 4.8-1.
entrance was placed on the
diameter
1958 ),
tubes,
one with a
were conducted
The same bellmouth
inlet of two different
length of 6.3 inches,
1 inch
and the other
9.8 inches.
Figure 4.8-2 presents the plot of reservoir height verses
time for the short entrance at
two different mixtures with
Page 73
2
kinematic viscosities of 0.125 and 0.475 cm /s.
both cases sufficient time is allowed for
the zero
pressure signal
The asymptotic region of the differential
is amplified in Figure 4.8-3.
level,
digitation
The amplification shows that a
The offset
small offset exists.
to
is about
equal
to
improve the accuracy of the low flow results.
to be small
the bulk of
in comparison to the signal.
Applying the curve fitting
previously described,
derivative,
but can be
The noise over
the signal can be more clearly seen by taking the
difference between the actual signal and
the mean signal.
technique at each point as
but in addition to evaluating the
the mean height value can be approximated by the
curve fit value.
The difference between the curve fit values
and the actual values of the height data
4.8-4.
the
so can be subtracted from the differential
The noise in the signals is difficult to see,
noted
level
limit to be achieved.
asymptotic
signal
Note that in
is shown
in Figure
The noise levels are seen to be primarily a result of
digitation errors,
.03cm, which remain approximately constant
over the range of reservoir
levels.
Figure 4.8-5 presents the calculated results for Reynolds
number vs time for two experiments with the short entrance.
Parametrically combining the dimensionless pressure data and
Reynolds number data gives the dimensionless pressure-flow
relationship.
The combined experimental
relationships are
shown in Figure 4.8-6 for both experiments.
the theoretical prediction for
the geometry
For comparison,
is also shown.
The
Page 74
close agreement,
to within 1-2%,
the quasi-steady technique.
the validity of
demonstrates
The validity of the non-
dimensionalization is demonstrated by the overlapping region of
results which correspond to substantially different flow rates.
To test
the validity of the subtraction technique,
tests were repeated for the longer
the additional
loss in the longer
3.5 inches calculated.
and the
tube due to
the additional
The combined results for
tube- at the two viscosities
Once again,
entrance,
the results fall
is also shown
the
longer
the
in Figure 4.8-6.
within 1-2% of theory,
and
the
overlapping region confirms the non-dimensionalization.
The tubes are both the same diameter, allowing the
additional
pressure drop due to the additional
length to be
calculated from the simple difference
in the pressure flow
relationship for the two geometries.
The results are
summarized
in Figure 4.8-6,
showing the
losses in each
experiment and the difference which represents the loss in the
additional
length of tubing.
The
loss in the additional
length
of tubing alone is shown in Figure 4.8-7, along with a
published empirical correlation,
characteristic pressure, P*.
theory are seen to be within
normalized to the
The error between the results and
10%.
Important characteristics of the results are;
" Experimental results fall within about 10% of the
theoretical prediction over the Reynolds range from 100
to 10,000
" The accuracy of the results is achieved despite a
difference in signals ranging from about 10 to 15%.
This can therefore be seen as a severe test of the
subtraction technique because signal differences
Page 75
expected in the airways are
* The Reynolds number
larger.
and P* properly scale the results.
The small error between theory and experiment confirms the
analysis which
indicated that
the temporal
negligible portion of the total
experimental results.
Further confirmation can
loss.
the temporal
be found by calculating
accelerations are a
terms based on the actual
The pressure drop was calculated to be
no more than .03 cm HaO, which places
its magnitude below the
system resolution level.
4.9 Airway Models
The dimensions of the symmetric airway model built to
typify symmetric airways of the
The model
lung are given in Table 4.9-1.
consists of symmetric bifurcations modeled after
The bifurcations are
"typical" bifurcation in Figure 1.3-1.
coupled with rotary junctions which allow generation to
generation orientations to vary, as well
addition or removal of generations
as allowing the
to the airway network.
Table 4.9-1
Flow
Percentage
Generation
Number
Diameter
cm
Length
cm
Number of
Branches
0
2.44
4.31
1
100
1
1.85
6.47
2
50
2
1.39
4.85
4
25
3
1.08
3.78
8
12.5
4*
1.00
2.00
16
6.25
*
the
Note Generation 4 was not used in all
experiments
Page 76
Construction of the bifurcations which make-up
was achieved with casting techniques and a planar,
the airways
two piece,
The two piece airway model was
three generation airway model.
designed and built by Drs J. Hammersley and D.E. Olson
in clear plexiglas
Each half of the airway was cut
Mich.).
(Univ of
using a numerically controlled milling machine, with a series
of ball-end end mills.
The circular cross-section of the
branches, which are smoothly blended at
junctions,
are formed by the two
the bifurcation
halves and the branching
geometry is formed by the tool path.
The geometric scaling for
Pedley (1977),
nominal
with
the airway is
two exceptions.
that given by
The availability of only
size end mills restricts the values of the tube
The resulting daughter-parent tube diameter
diameters.
ratios vary slightly from one generation to
the value of 0.78 given by Pedley.
0.75 for the first
two generations,
another and from
The diameter ratio's are
and 0.778 for
the third.
The second difference is that the radius of the tube
curvature
is fixed,
and does not
taper from the bifurcation to
the point at which the branching angle is met.
radius of curvature is used
instead,
value of the curvature radius
This difference is significant
A constant
representing
to parent
a typical
tube radius of 7.
and advantageous from an
experimental standpoint because it generates a constant
diameter straight section of tubing, L/D
bifurcation.
described
~
1, between each
The straight section isolates the airways,
in the subtraction technique discussion, and
as
allows
Page 77
the circular collars forming
the rotary junctions
to be cut on
a lathe.
Construction of the individual bifurcations began by
making a wax cast of the planar airway formed by the void
by the two halves of the plexiglas model.
CT
Bridgeport,
),
The wax had a 3%
( Aqua-wax # 261-8105,
shrink ratio when cooled
left
Gesswiein Corp,
allowing the airway to easily be removed when
the plexiglas model
is separated.
A wax bifurcation is cut out
Aluminum collars are placed on the straight section
at the ends of the daughter
and parent tubes.
wax bifurcation are then placed
mold,
and a casting compound
#2850 FT with catalyst #11)
is parted,
the casting
is poured into
Cumming
the mold.
is removed and placed
L/D
The collars and
in a specially built
( Emmerson and
,
two piece
,Stycast
The mold
in boiling water
to remove the wax.
Figure 4.9-1 shows the cross-sectional view of the final
bifurcation.
Smooth transitions between bifurcations are
assured by the concentric collars which are
tube center-line by the
'
indexed to the
diameter straight sections at the
ends of the parent and daughter branches.
The concentricity
also allows the bifurcations to be rotated while maintaining
the smooth
transition.
,
of the airway at the mid-section between bifurcation junctions.
Page 78
5. Estimation of Correction Factors
5.1 Axial
Velocity Profile Factor
Factors were introduced
into the steady one-dimensional
3 to
energy equation in Chapter
The axial
three-dimensionality in the flow.
factor,
fa,
velocity profile
and can be written
in equation 3.7.10,
is defined
the
include the effects of
in the equivalent form;
fa =
5.1.1
P3
uudA
1
IQ
I'
]
LU
U2Q
Inspection of 5.1.1
actual
i
[
J
d
A
shows that fa represents the ratio of
kinetic energy flux to
same flow rate with a blunt
the flux
that would occur
velocity profile.
can therefore be considered an
at the
The value of fa
indication of the "bluntness"
the velocity profile, decreasing to
1 as
the profile becomes
flat.
Evaluation of fa,
and
its Reynolds number dependence,
requires detailed measurements of the velocity profile.
The
laborious nature of velocity mapping techniques and the need to
evaluate fa at every flow rate tested places
practical number of experiments for which
evaluated.
Alternately,
limits on the
fa could be
the dependence of fa on more easily
measured bulk experimental quantities can be
investigated,
correlations and/or theoretical predictions developed
calculate fa.
and
to
The resulting measurements or estimates of
fa
Page 79
if the errors
based on bulk properties can be used,
introduced
are within an acceptable range.
Two methods have been investigated to estimate the value
of fa.
The first method utilized 5.1.1
the exiting momentum flux,
approximate relation between fa and
which is a measurable experimental
to develop an
quantity.
The relation
between fa and the momentum flux can be seen by writing the
axial
velocity, u,
a perturbation,
&.
as the sum of the average velocity, U,
Substitution
into 5.1.1
GI
5.1.2
where aa
gives;
3 dA
-
fa = 3a - 2 +
is the normalized axial momentum flux, defined as the
ratio of the actual momentum flux to the flux that would occur
at the same flow rate with a blunt profile.
The normalized
flux, aa, is given by;
5.1.3
aa
J
=
2 dA
2 dA
u
-
[+j
=
The exiting momentum flux,
M,
-
II
and
can therefore be written in
terms of aa as;
5.1.4
M = aa C gU2]
Application of the momentum equation to the control volume
shown in Figure 5.1-1,
relates the momentum flux to a
experimentally measurable quantity,
momentum flux,
M,
a restraining force.
is balanced by the force created by the
The
Page 80
impinges on a stationary plate which
exiting fluid as it
The momentum correction factor,
the flow 900.
turns
aa, can be
determined along with the flow rate and driving pressure by
measuring the restraining force in addition to
identified.
pressures already
Relating (a to
fa requires that the
it should be small
in comparison to
which are on the order
small
integral quantity in
Inspection of the integral
5.1.2 be known.
the two
term indicates
the other terms
in 5.1.2,
The velocity perturbation,
'1.
in comparison to U, and
G,
The
the area can therefore be expected
U/U integrated over
to be small
is
the average value of U- when
integrated over the cross-section is defined to be zero.
cube of
that
in comparison to
1 because the values are small,
and the values are both positive and negative over the crossThe axial velocity correction factor can therefore be
section.
in terms of the
approximated
number,
momentum flux and Reynolds
two experimentally determined quantities as;
M
5.1.5
fa = 3aa -
2 = 3
2
P* Re
- 2
The second method developed to determine fa based on bulk
properties utilizes published velocity profiles measured
expiratory flows.
allows direct
developed.
The results for the velocity profiles
integration of 5.1.1
correlations of
in
numerically,
and possible
fa with experimental conditions to be
Only a
limited amount of experimental data on the
velocity profiles in expiratory flows is available to evaluate
Page 81
the integrals.
The published velocity profiles available,
Figures 5.1-2a to g, were scaled,
integrals evaluated
and the
with finite difference approximations.
see
The errors introduced
by the scaling process and the integration approximation are
but are estimated to be on the order of
difficult to quantify,
10%.
The results for a range of expiratory flow conditions were
found
to correlate well
Additionally,
with the tracheal
the results provided confirmation of the
approximation in arriving at 5.1.5.
fa and
aa,
Reynolds number.
The resulting values of
and the range of experimental conditions and
Reynolds numbers at which the values are evaluated are
summarized
in Table
5.1-1.
Table 5.1-1
Reference
Jan
1986
Chang and
Menon
1985
Configuration
Quasi-steady, single
bifurcation
Asymmetric human
upper airway cast
Hardin
and Yu
1980
Symmetric idealized
pulmonary model
Reynolds
1982
Asymmetric human
airway cast
Tracheal
Reynolds #
aa
fa
200
1.15
1.47
1060
1.11
1.32
5712
1.07
1.21
6000
1.03
1.09
6811
1.02
1.08
2760
1.06
1.17
2260
1.09
1.22
8000
1.04
1.13
12200
1.04
1.13
Page 82
fa,
The cross-plot of aa and
along with the predicted
relation is shown in Figure 5.1-3.
The small deviations of
the experimentally determined relation between
fa and aa from
the prediction are within the approximate error band of
Included
predictions of the
in the Figure are also theoretical
relation between
fa, and (a for the self-similar velocity
profiles established
in
flow in a long straight
considered
10%.
laminar and fully developed turbulent
tube.
These comparisons can be
to cover the profile extremes expected in
expiratory flows and further validates the assumptions used to
generate 5.1.5.
Figure 5.1-4 presents the cross-plot of fa and
The plot shows fa decreasing with
Reynolds number.
Reynolds number, confirming
the observations
increasing
that velocity
profiles become more blunt as the Reynolds number
A
tracheal
increases.
least squares curve fit gives the approximate relation;
5.1.6
+
6.54
Fa = 1.05
,/Re
The behavior of
fa can be explained in terms of the
observed expiratory flow characteristics.
profile is typified by a blunt core and
similar
The axial velocity
thin boundary
layer,
to both entrance type flows and high Dean number
curved tube flows.
In both cases,
the boundary
layer
is seen
to thin as the flow increases for a fixed geometry.
Estimation of
fa values and a confirmation of the boundary
Page 83
layer analogy can be generated by considering the effect of a
kinetic energy flux.
thinning boundary layer of exiting
flow profile can be approximated as shown
blunt core with velocity Uo,
in Figure 5.1-5,
a
and a linearly decreasing
velocity in the boundary layer.
Integration of the velocity
profile for a given value of the normalized boundary
thickness,
The
layer
S'=&/a, gives the average flow velocity and exiting
kinetic energy flux to be;
5.1.7
5.1.8
U
= Uo
( 1 -
dA
3
= Uo
u
S'
1
2
+ - S'
3
2,
33
1 -
-
8'
+
-8
5
2
which can be combined to give fa as;
2
E 1 - 1.5 8' +0.6 V'
5.1.9
1
fa =
[
1 -
S'
For entrance flows,
+ 0.333
'2
3
the boundary layer thickness can be
approximated by the Blasius flat plate boundary
(Equation 3.4.1)
S'<<1.
layer growth
with free stream velocity Uo, as
long as
Figure 5.1-6 shows the prediction of fa for the
Blasius relation at tube lengths of
1.75, and
3.5 diameters.
The relation is seen to be relatively insensitive to
over the narrow range,
fa values found
length
and provides an excellent prediction of
in expiratory flows.
For high Dean number curved tube flows,
the boundary layer was
Page 84
shown
in 3.5 to be;
5.1.10
where C
'
= C
[
Re Sr
is an unknown constant near
1, and
is based on the average tube velocity.
the Reynolds number
To evaluate C,
high Dean values of fa calculated by Olson
(1971)
the
can be used
to compare the results of 5.1.9 with the boundary layer
Olson's experimental
thickness given by 5.1.10.
are well fit for Dean numbers greater
be approximately equal
to 2.
values of fa
than 200 if C
is taken to
Applying 5.1.10 to the typical
values of fa based on the curved
lung geometry with Sr=1/7,
tube boundary layers are also shown in Figure 5.1-6 to predict
the form and magnitude of the expiratory values of fa.
In a similar
fashion,
the results for
fa based on the
velocity profiles calculated for the 2-D convergent channel
also shown in Figure 5.1-6.
The ability of the curved
are
tube,
entrance and 2-D convergent channel profiles to so closely
predict the results for the expiratory data strongly supports
the observation that expiratory flow can be characterized by a
blunt profile and thin boundary layer.
The consistent
predictions for fa for a wide range of geometries also supports
the use of the correlation in 5.1.6 to predict fa for
idealized bifurcation.
therefore not likely
variation
Errors in using this form for
to be large,
in determining
fa are
and can be estimated from the
in the results to be within about t5%.
The error
the
fa experimentally from
Page 85
measurements of exiting momentum flux can be estimated to be
significantly higher.
2% error
only
Achieving
5% error overall
in the Reynolds number measurement,
and allowing
requires
that the momentum flux be measurable to within accuracies of
about
1%.
This accuracy is highly unlikely due to the difficulties
associated with measuring forces,
and the noises introduced by
turbulence and other fluid dynamic effects in the exhaust
tank.
High accuracies require that the diverter plate be
large to
edges.
insure that no axial momentum "leak"
around the
Small pressure variations within the exhaust tank as a
result of the fluid motions therefore can exert
large forces
when working over the large surface of the diverter plate.
view of these uncertainties in the direct measurement
the correlation with Reynolds number,
5.1.6,
of
In
aa,
will be used to
reduce the experimental results.
5.2 Secondary Velocity Correction factor
The secondary velocity correction factor,
the kinetic energy flux contained
the exiting flows.
in the secondary velocites of
The magnitude of fs is the ratio of kinetic
energy flux contained
to the axial
fs, represents
in the secondary velocities,
normalized
kinetic energy flux that would occur with the same
flow rate and a blunt axial velocity profile.
The value of Fs is defined
written as;
in equation 3.9.7 and can be
Page 86
[
2
fs =
5.2.1
Vsec
LU
where Vsec
2 u
dA
- -A
JU
is the magnitude of the secondary velocity.
Evaluation of
fs
is more difficult
than for
fa because of
the lack of experimental data on secondary flow profiles in
expiratory flow.
The secondary velocity profiles have been
visually studied, but are difficult experiments to quantify
because the secondary velocities are typically a fraction of
the
A number
velocity magnitude.
local
but only two studies have
conducted on inspiratory profiles,
been published
which present
of studies have been
measurements on secondary flows
in expiration.
and
Chang
in a model
Menon
(1985)
measured
velocities
the secondary
of the human upper airways,
and
Jan
the secondary velocity profiles in an single model
in oscillatory flow.
shown
in
The secondary
Figure 5.2-1a to
integrated numerically similarly to
fa.
5.2.1
bifurcation
velocity profiles are
These published
c.
measured
(1986)
results can be
the method described for
An alternate technique is to approximate the
integral
in
as;
Vsec
fs
5.2.2
The
results for
agreement with
normalized
integrals
=
2 u
-
2
dA
-
the numerical
the estimate based
~
(Vsec/U)ave
integrals are
on the
secondary velocity magnitude.
and estimates are
summarized
in good
of
the
results for
the
average value
The
in Table 5.2-1.
Page 87
Table 5.2-1
Vsec/U
Estimated
fs
Reynolds #
Integrated
fs
Chang & Menon
(Station 3)
1060
.04
0.215
.046
Chang & Menon
(Station 2s)
1060
.04
0.214
.046
.028
0.18
.032
Tracheal
Investigator
166
Jan
(ave)
The magnitude of the secondary velocities are most likely
established by
the curved tube type phenomena.
The secondary
velocity scaling for curve tube flow was shown to be linear
This scaling explains the
with U for high Dean numbers.
constant Vsec/U,
as seen in Table 5.2-1, and allow fs to be
estimated as a constant for high flow rates.
fs z 0.04
5.2.3
This approximation is not
where the scaling changes,
reduced relative to U.
scaling
lower flow rates,
valid at
and the secondary velocities are
But,
as seen
in table 5.2-1,
is approximately valid down to a Reynolds number of
about 200 for a typical bifurcation.
At
these lower reynolds
numbers,
the kinetic energy flux becomes small
to other
terms in the energy equation,
error
the
in comparison
which minimizes any
in approximating fs as a constant.
Page 88
5.3 Pressure Correction
The pressure correction term
introduced
in equation 3.7.7
accounts for the effect of non-uniform pressure and axial
in using
velocity profiles
the
a
simple area averaged pressure
The correction becomes necessary
energy equation.
in
in the
subtraction experiments because average pressures and not
velocity weighted pressures are specified with the hydrostatic
heads.
the pressure correction
Evaluation of
due
to
complete
the
is difficult
term
measurements of
lack of experimental
Estimates
static pressure distributions in expiratory flows.
of
the correction term of
errors
the correction terms are likely
as
however,
in
sufficient accuracy
the estimate are therefore only
can be made,
to be small
order
second
and
system
errors.
To
estimate the order
to
can be considered
streamlines.
normal
to
the streamlines
the pressure perturbations
result of
the
in
perpendicular
df
Rs
variations
the streamline,
to
pressure gradient
and
the streamline.
across the
the
in
the flow
pressure gradients
to be written;
is the velocity magnitude, Rs
curvature
curvature
g V2
-
-
dP
Pc,
equation allows
The Euler
5.3.1
where V
be a
of
n
is
is the radius of
in the direction
The magnitude
tube can approximated
and using
by
an average radius
of the pressure
integrating
of curvature,
the
Page 89
giving;
U2
C a g
p'
-
5.3.2
Rs
where C
is the tube
and a
1,
near
is an unknown constant,
radius.
The radius of curvature for a streamline can be written in
the vector equation form ( Hildebrant
) ;
3
Rs =
5.3.3
I
IA X V
is a result of the velocity
where the acceleration vector A
vector V changing with position in a steady flow.
Evaluating
allows the curvature
5.3.3 in the torrodial coordinate system,
radius to be written in terms of the tube radius,
and the
velocity components;
5.3.4
(v/w)
1 +
Rs = r
J
2
U
2
I
For expiratory flows,
velocity components,
V
V2
+
2
+ W2
W2
tangential secondary
the radial and
v & w,
+
are on the same scale,
and
the
square of the axial velocity is typically large compared to
the
square of
the secondary velocities
(v
2
+w
2
),
which
allows
an average radius of curvature to be approximated as;
5.3.5
Rs Z
/2 a [ V/Vsec
]2
combining with 5.3.2 gives;
2
5.3.6
p' I z -A
Experimental
C g Vsec
support for
=
rU2
V2 C
(Vsec/U) 2
this scaling can be demonstrated
Page 90
the tube
by measurements of the pressure variations along
While this
wall.
is not equivalent to establishing the true
pressure variations,
Figure
it should reflect the magnitude.
5.3-1 presents the pressure difference measured
in two
pressure taps at the top and side of the daughter branch of a
bifurcation with planar symmetric upstream airways.
pressure difference wall at these wall positions,
900
was measured instead of the hydrostatic
The
separated by
pressure
otherwise the experiment was conducted as described
difference,
in Chapter 3.
as they should
These two points were chosen,
reflect the maximum pressure variation along the wall given the
expiratory flow profiles previously presented.
The
experimental results are fit reasonable well by the curve;
confirming
.06 P
2
2
*
Del P z
5.3.7
Re
0.12 '4gU
the P* and Reynolds number scaling and allows the
constant C in 5.3.6 to be approximately evaluated as;
C = 2//2
5.3.8
.12 / EVsec/U]
where an additional factor of 2
2
= 3.67
is introduced
to reflect
that
the pressure perturbation along the wall will be about half of
the maximum.
order
The results confirm that the constant C
is of
1.
To determine the pressure correction term,
a more
sophisticated analysis is needed because each of the factors
the expression giving Del
So
Pc can be both positive and negative.
that while the scale of the term can be found relatively
easily,
in
the actual value may be much
lower,
or possibly even
Page 91
negative.
the vortex cells can be
To evaluate the pressure term,
considered
PL..,
in the center of curvature,
to have a low pressure
increasing outward toward the wall and
the center of the
The average pressure in the cell can therefore be
tube.
written as;
2
Vsec
P = P
5.3.9
+
dA
1
de--
'gU2
A 0
E is a dummy variable of integration, r is the radial
where
and C
distance from the center of the vortex cell,
correction constant evaluated previously.
is the
The pressure
perturbation can therefore be written as;
r
r
5.3.10
Vsec
p-P=
p=
2
Vsec
2dE
-]
U2
dA
1
dE
]
A 0
0
which allows Pc to be written as;
r
u
Pc
5.3.11
Ysec
j
-
L U
',gCU2=
A
U
L
J
2 de dA
E
--
A
0
Numerical evaluation of the right
hand side of equation
5.3.11 was used to give the approximate pressure correction
term.
The
integrals were difficult
to perform because it
required estimating the center of the
available data
vortex cell.
is sparse to evaluate the
integrals,
The
and
is the
same as the data to evaluate secondary velocity correction
factor.
The results give a value of
.06, which as expected
is
Page 92
small,
and on the order of the secondary velocity correction
factor.
5.4 Correction factor Implementation
The frictional pressure drop in the test bifurcation was
written in equation 4.3.4 as the difference between two
two correction factors.
minus the sum of
experimental values,
The correction factors can be combined
term as
the pressure correction
writing
into one expression by
fs=0.04, Pc/'gU 2 =.06,
to give;
5.4.1
4
2
Del Pk + Del Pc
=
fa+0.1)
*KP*Re
With the aid of 5.2.3 and 5.1.6,
'/4E
(fa+0.1)]
the sum of the correction
factors can be combined into one correction factor,
can be written in terms of the
f,
which
local reynolds number;
5.4.2
1.15
+
6.54
f = fa + 0.1 =
,/Re
this allows the correction factors to be
reasonable degree of accuracy,
only one measurement,
implemented with a
an estimated
insensitive to the geometrical
flows,
in terms of
the tracheal Reynolds numbers.
of 5.4.2 as well as the constant were found
apply with about
5%,
details,
to be relatively
and can be expected
the same level of accuracy to real
as well as those
The form
in idealized bifurcations.
to
pulmonary
Page 93
6.
Symmetric Airway Results
6.1 Planar System Results
The planar system results covered the Reynolds number
range from about 50 to
experiments.
10,000 with two sets of the quasi-steady
The fluid mixture used
in the first set of
experiments had a high glycerin content with a kinematic
viscosity of about 0.45 cm 2 /sec,
resulting in tracheal
numbers from about 50 to 2000.
The fluid mixture used
second set of experiments had a
lower glycerin content,
2
yielding a kinematic viscosity of about 0.125 cm /sec,
tracheal
Reynolds numbers from about 300 to 10,000.
Reynolds
in the
and
The
normalized results from the two sets of experiments were
superimposed to cover the entire Reynolds number range,
over-
lapping from Reynolds numbers of about 300 to 2,000.
Typical experimental
tank height verses time results for
the low and high viscosity mixtures for both the four
generation, and
the three generation "upstream" airway network
are shown in Figure 6.1-1.
and scaled,
tracheal
The noise
The curves were then differentiated
as described in chapter 3,
to generate the
Reynolds number verses normalized pressure difference.
is similar
is equivalent to
to that found for
the entrance test,
and
the level of digitation noise associated with
the data acquisition equipment.
Each experiment was conducted at
reduced and analyzed separately.
checked,
least twice,
and the data
The normalized results were
and found generally to vary by no more than about
1 to
Page 94
Larger variations were taken as an indication of
2%.
experimental error,
This
and the experiments were repeated.
level of error can be expected primarily as the result of
digitation noise in calculating the Reynolds number.
The normalized pressure-flow results were
into
The overlap of the
two overlapping experimental ranges.
two ranges,
as with the entrance test,
then synthesized
confirm both the theory
and conclusion that the experiments scale with P* and Reynolds
number.
The static pressure difference for
the single test
bifurcation is given by subtraction of the experimental results
according to 4.3.2.
Figure 6.1-2,
The subtraction
the total upstream loss,
the test bifurcation diameter P*,
Reynolds number
and
normalized
the corresponding
in the trachea of the test bifurcation.
The static pressure
loss in the test bifurcation can be
seen to represent about 30% of
losses.
in
where the results for the 3 generation experiment
were re-scaled to represent
to
is graphically depicted
the magnitude of the system
This is contrasted with the entrance flow
experimental results,
in which t5% accuracy was achieved with
experiments differing by only about
10%.
Errors
in the
experimental measurements of the test bifurcation can
therefore be conservatively estimated to be within
5%.
The frictional component of the pressure loss in the test
bifurcation can be computed by subtracting
the corrected
kinetic energy and pressure correction terms as given
Figure 6.1-3 presents the total
static
loss,
in 5.4.3.
kinetic energy and
Page 95
pressure correction component,
component.
The components are all
kinetic energy,
normalized by
Kinetic Energy
=
'
g
U2
=
1 P* Re 2
Figure 6.1-3 shows, as would be expected,
frictional
losses dominate for
Additionally,
the nominal
given by convention as;
Nominal
6.1.1
and resulting frictional
the
that the
lower Reynolds numbers.
that the kinetic energy losses become more
important and eventually dominate at the higher Reynolds
numbers.
The transition,
where the contribution of frictional
loss and changes in kinetic energy are about equal, occurs near
a Reynolds number of 5000.
To better understand the nature of
the bifurcation,
the
loss in
re-normalized to
if the flow were everywhere
This loss
Poiseuille-like.
form,
loss can be
the frictional
loss that would occur
the frictional
is
the expected
lower asymptotic
and the normalization demonstrates the extent
to which
the flow is affected by the presence of the bifurcations.
The
re-normalized results are shown in Figure 6.1-4.,
The frictional
loss
is seen to approach the expected
Poiseuille-like form for Reynolds numbers of order
loss smoothly increases with Reynolds numbers,
nearly a
10-fold
100.
and approaches
increase over the Poiseuille form at a
Reynolds number of about
10,000.
6.2 Embedded Bifurcation Assumption
Inclusion of the upstream airways was
The
intended to
Page 96
establish
inlet boundary conditions for the test bifurcation
which could be considered "typical" of an embedded
bifurcation.
equivalent
The embedded bifurcation assumption is
to assuming that the
loss in any bifurcation within
an idealized airway of infinite extent
the
local flow rate.
While it
is only a function of
is impossible to
experimentally verify this assumption,
it can be approximately
tested by comparing the pressure-flow results found in
bifurcations at different generations
To
test the assumption,
in the airway.
the planar experiments conducted
in the four generation system described above were repeated
a
three generation system.
created by removing
airways.
in
The three generation system was
the terminal generations from the upstream
The terminal branches were removed rather
than the
"tracheal" bifurcation so that the same test bifurcation could
be used for both experiments,
thus reducing errors introduced
by slight model differences.
Figure 6.1-4 also presents the results for
pressure loss found for the
test bifurcation with two
generations upstream of the test bifurcation
generation system),
the frictional
(the three
and with three generations upstream.
results are seen to be unaffected by the change
The
in generation,
differing by less than the estimated experimental error of
5%.
6.3 Effect of Upstream Airway Orientation
The bifurcations in the
lung are not oriented
in a planar
Page 97
manner, but at a variety of relative angles, generating a
fully three-dimensional branching network.
To determine the
effect of three-dimensionality on the pressure difference
across a bifurcation,
the bifurcations
were systematically rotated 900
from the planar orientation.
Three sets of experiments with
symmetric
in the upstream airway
increasing levels of
"three-dimensionality" were conducted to estimate
the effect of bifurcation orientation on the pressure loss.
Configurations tested are shown in Figure 6.3-1,
and were
selected so as to preserve the symmetry of the airway system.
System symmetry allows the preceding analysis to be used
reducing the experimental results,
in
maintaining equal flows
down each branch of the system.
Normalized results for the frictional pressure loss in the
test bifurcation for the three non-planar configurations tested
are presented in Figure 6.1-4.
The results for
the frictional
loss were obtained by assuming that the velocity profiles are
unaffected by the change in the orientation of the upstream
bifurcations.
This assumption allows the correction factors
previously developed to be used.
Support for making this approximation was obtained from the
experimental results of Chang and Menon
(1985).
The factors
calculated from Chang and Menon's work were measured
planar pulmonary casts,
and were found
in non-
to fall on the same
correlations as the results for correction factors found from
the symmetric planar systems.
Similarity in the results for the frictional
losses in
the
Page 98
test bifurcation despite the variety of upstream bifurcation
orientations indicates that any three-dimensional effects on
the pressure-flow relationship
The results consistently fall within a band of
negligible.
about
in the test bifurcation are
The
and show no observable trends.
5%,
considered negligible as it
5% band can be
is of the same magnitude as the
estimated error in the system results.
6.4 Effect of Additional Parent Tube Length
Typical
length-to-diameter ratios encountered
lung are not fixed, as
from about 2
in the
idealized model
in the upper airways,
peripheral airways
to up
in the human
at 3.5,
but vary
in the
to about 5
(Nikiforov and Schlesinger 1985).
The
effect of the length to diameter ratio on the frictional
loss
in a bifuration can be estimated by studying the loss in a
straight
tube segment downstream of a bifurcation.
A complete assessment of the effects of length
ratios on the pressure-flow relationship
to diameter
is likely to be more
complicated, as changes in kinetic energy fluxes for a given
flow rate can also be expected
to occur.
Additionally,
possible cascading effects of length-to-diameter ratios are
also neglected.
This experiment was conducted by adding a straight tube
section of length 3.5 diameters to a planar
airway.
Once again,
symmetry is preserved,
three generation
allowing the same
previously described analysis to reduce the experimental
results.
A simplifying difference
in applying the subtraction
Page 99
technique is that
tubing does not
the additional
include a change
loss in
the added
length of
in velocity or kinetic energy.
Because the flow area is constant and the correction factors
are assumed
to depend only on Reynolds numbers, no kinetic
energy change is assumed
to occur.
The pressure flow relations for the airway with and
the additional straight section, along with the
without
difference due to the additional
Figure 6.4-1.
The additional
tube section is shown in
pressure loss,
Poiseuille loss at the same flow is shown
along with the normalized
normalized by the
in Figure 6.4-2,
loss in a planar bifurcation.
Comparison of the results for the two systems demonstrates
the striking similarity.
negligible,
found
as they approximately fall within the
that for
lung,
5% band
assumed to represent the
in the orientation experiments,
The similarity in results
experimental error.
the
Differences in the results are
is an indication
the length-to-diameter ratios typically encountered
the pressure-flow relationship can be considered to
be linearly proportional to
A slight increase
tube length.
loss in the straight
in the measured
tube section at high Reynolds numbers may be an exception.
While the magnitude of the deviation
indicated by the results may lead
is not
though, reflect an
the straight tube.
increased error
Figure 6.4-1
the trend
to significant differences
at Reynolds numbers higher than '10,000.
could,
large,
This deviation
in the results for
shows that
represents only about 5% of the signal
at
the signal
the higher Reynolds
in
Page 100
numbers,
increasing the potential errors.
6.5 Comparison of Results
A summary of all
of the experimental results for the
pressure-flow relationships for symmetric flow conditions
in
the idealized test bifurcation are shown in Figure 6.5-1.
The
results can all be represented by a single band which
represents a prediction to within
5%.
The band of results are
seen to be well represented by the curved tube theoretical
with the bifurcation curvature ratios of 1/7, given
prediction,
by the simple relation;
P
Del
[32 P* L/D
6.5.1
0.556 + 0.057 Re
K Re 1
is valid down to a Reynolds number of about 60, below
which
which,
the frictional
Poiseuille
loss.
loss Del P
The factor K is
is given identically by the
introduced
in the Poiseuille
loss term to account for the difference in daughter/parent
tube Reynolds numbers encountered
in a bifurcation,
allowing
in the expression.
the use of
just tracheal Reynolds number
The factor
K is defined by applying the Poiseuille loss to
leg
each
in the bifurcation,
K
6.5.2
where Dp/Dd
which
=
'4
giving;
1 + '4 (DP/Dd)]
is the parent
tube -
is a constant 1.333 for
the
=
1.092
daughter tube diameter ratio,
idealized bifurcation.
Applying this relationship to each branch of a bifurcation
Page
alters the numerical constants
Reynolds number dependence.
because it
is
101
in 6.5.1 slightly because of the
The first
term remains unchanged
but the second term must be raised to
linear,
0.067 to account for the lower velocity in
to apply the results to each branch of a
Therefore,
bifurcation, basing on the local
parent Reynolds number
6.5.3
the daughter branch.
Del P
P
Del P
Pois.
and not the
Reynolds number,
the form to use is;
=el 0.556 + 0.067 Re
LJ
Application of the entrance flow and 2-D convergent channel
flow pressure-flow predictions are also shown in Figure 6.5-1.
Two empirical entrance-flow predictions are included for
comparison,
one with no equivalent length assumed upstream of
the bifurcation junction,
and the other with an equivalent of
one diameter assumed upstream of the bifurcation
prediction with no assumed upstream equivalent
but can be corrected with
eqivalent
junction.
length
The
is high,
the addition of the assumed constant
length.
The entrance theory with Leq=D predicts all of the results
the results
but fails to predict
for
the test bifurcation well,
for
the straight tube downstream of the bifurcation.
straight tube section does not contain a junction,
therefore no boundary
The
and
layer thinning and re-growth occurs.
The
loss should therefore be given by the application of the
entrance theory over the length of the additional
accounting for
the parent tube
length and
tube,
the addition of the
Page
previously
identified one diameter equivalent length.
equivalent
This was
the
while entrance theory can predict
not found to be the case,
results for
102
the straight tube section,
it requires that no
length be used.
The apparently unpredictable nature of the equivalent
length factor
in the entrance theory suggests that
that provided by
correlation to the experimental results is
curved
tube theory.
The similarity in the form of both
entrance theory and curved tube theory,
suggests
the better
strongly
though,
that while neither analogy may strictly apply,
Reynolds number
to the '
power
the
is the characteristic form for
the pressure-flow behavior of a bifurcation.
though,
The 2-D convergent channel predictions,
consistently predict the trend
The L/D dependence is not
in the experimental
linear, but varies
For
for sufficiently high Reynolds Numbers.
numbers this inverse dependence
prediction due to the additional
substantial
results
as
inversely with L/D
lower Reynolds
is not dominate,
Poiseuille-like form assumed by the flow.
length
results.
due to the
The drop in the
is therefore not as
it could be for very high Reynolds numbers,
and
in a prediction close to both the equivalent curved
tube prediction and experimental results.
6.6 Transition to Turbulence
The range over which
the experiments
in the idealized
bifurcation were conducted varied from, 50 to
10,000.
range contains the Reynolds number values which other
This
Page
investigators have reported for
103
the transition from laminar to
turbulent flow.
By analogy to the sudden increase in pipe flow resistance
that accompanies the transition to turbulence,
many
investigators have used changes in the slope of the Log-Log
plots of the dimensionless pressure-flow behavior of expiratory
flows as an indication of transition to turbulence.
al
(1980)
Hardin et
reported such a transition at a parent Reynolds
number of 4500,
Reynolds numbers of
noted that
Reynolds
D.B.
(1982)
noted two transitions at
1800 and 4300, and
Isabey & Chang
(1981)
the transition occurs between Reynolds numbers of
investigator
1,500 and 4000.
Each
were not sudden,
but gradual,
the transitions
noted that
and attributed
the gradual nature
of the transition to the averaging that occurs in measuring
losses over an entire system.
Other expiration investigators have used flow visualization
and hot-wire anemometry to detect
Chang and Menon
(1985)
transitions to turbulence.
used hot wire anemometry to measure
velocity profiles, and while not directly investigating
transitions
to turbulence,
noted that the flow was not fully
laminar at a Reynolds number of only
(1959)
1060.
West & Hugh-Jones
used die traces in water and noted that
preceeding fully turbulent flow,
"wavering",
occurred in the main bronchus
and trachea at a Reynolds number of 1550.
Inspection of the experimental results for frictional
pressure drop
4,
gives no
in the test bifurcation,
presented
in Figure 6..1-
suggestion of a transition to turbulence.
The
Page 104
results increase smoothly from the Poiseuille form to
the fully
developed curved tube form with no apparent discontinuity
Possible
characteristic of a transition to turbulence.
averaging
in a complete airway which may mask a sudden
transition should not occur with the experimental subtraction
technique which determines the results for a single
bifurcation.
To
further investigate the possible transition to
turbulence
in the test bifurcation,
two
sets of experiments
were conducted.
Hot wire anemometry and die-in-water traces
established that
transition to turbulence occurs in the
Reynolds number range from about
1000 to
1500.
Hot-wire measurements were made by placing the probe
in the center of an extended
axial
positioned at the
location corresponding to the exit of the test
bifurcation.
of a
"trachea",
The probe position along the axis of
the trachea
three generation planar airway, at a distance of
diameters from the bifurcation center.
A.C.
1.75
components of the
hot wire traces recorded on a storage oscilloscope are shown in
Figure 6.6-1 for six different Reynolds numbers.
and position in the cross-section were varied,
significantly effect
the results.
Orientation
and not found
to
The width of the probe and
its support prohibited measurements close to the tube wall,
where velocity fluctuations are expected to be
oscilloscope magnification,
lower.
The
Reynolds number and average peak
velocity fluctuations relative to the bulk flow are summarized
below in Table 6.6.1.
Page 105
Table 6.6.1
Y scale
mv/div.
Reynolds
#
Trace
#
G/U
X scale
sec/div.
1
713
20
.2
o.o0e
2
1037
20
.2
0.033
3
1548
20
.2
0.047
4
1585
20
.2
0I.04e
5
1697
20
.2
0.057
6
2286
50
.1
0.071
The traces show that small random fluctuations typical of
turbulence are present at Reynolds numbers as low as 700,
below
which only the high frequency measurement noise is present.
The fluctuations grow to represent about
7% of the mean
velocity at a Reynolds number of about 2300,
magnitude of the fluctuations were found
constant.
to remain roughly
Selecting a single value at which
is difficult,
above which the
turbulence begins
but the fluctuations appear to grow rapidly until
a Reynolds number of about
transition value.
1000,
which will be considered the
This transition value
is consistent with the
other hot wire results of Chang and Menon with reported
turbulence at a Reynolds number of
1060.
Flow visualization experiements were used to
transition from laminar
dye tracer in water.
identify the
to turbulent flows with an India ink
The original
two piece planar airway
formed by two clear plexiglas halves was used in
the
experiments to allow the flow to be easily viewed.
The airway
Page 106
model was placed in a large water filled container with the
trachea connected to a hose leading to a drain.
India ink dye
was injected at various positions within the airway by
maneuvering a catheter
into position,
and
injecting the dye
from a syringe source.
The flow was driven by
tank,
the hydrostatic head
in the supply
and controlled with a variable resistance valve
downstream of the airway.
A simple bucket and
stopwatch
approach was used to measure the flow rates.
The flow is observed to
although clearly
remain laminar,
not poiseuille-like as secondary flows disperse the dye,
a Reynolds number of about
1500 slight
1400.
up to
At a Reynolds number of about
"wavering", as described by West & Hugh-Jones in
the dye traces occurs at the exit of the trachea.
Reynolds number increases,
As the
the wavering grows stronger and
propagates upstream toward the bifurcation
junction.
The
intensity of the dye wavering, or velocity fluctuations
continue to
increase until
the dye breaks up
into a fully
turbulent flow at a Reynolds number of about 2300.
The dye fluctuations are also seen to have propagated
into
the daughter branch at the same tracheal Reynolds number of
2300,
1500.
which corresponds to a
local Reynolds number of about
The strongly turbulent region also propagates upstream
toward the bifurcation as the Reynolds number
Reynolds number of 3400 in
tube,
the trachea,
increases.
or 2300 in the daughter
the strongly turbulent region is seen to enter the
daughter
tube.
At a
Page 107
This pattern continues to progress upstream as the
"Wavering" starts at each branch
Reynolds number increases.
at a local Reynolds number of 1400,
breaking up
into strong
local Reynolds number of about 2300.
Thus
establishing a transitional Reynolds number of about
1500,
turbulence at a
agreement with the results of West & Hugh-Jones,
higher
in
but slightly
than the transitional value found with hot-wire
techniques.
Discrepancies between the transitional Reynolds number
value between the hot-wire technique and flow visualization
techniques are not large, but appear to be consistent.
A
possible explanation for the difference is that the velocity
fluctuations are a small fraction of the bulk velocity at the
start of transition, and build to more significant
increasing Reynolds numbers.
The fluctuations are therefore
not be easily visualized until
in the time before the fluid
These results are
levels with
they are large enough to grow
leaves the bifurcation.
in general agreement with the results of
the previous investigators with the exception of the noted
transition at or near a Reynolds number of 4500 by D.B.
Reynolds and Hardin & Yu.
lies in the manner in which
identified.
An explanation for this discrepancy
the transition at 4500 was
Hardin & Yu and Reynolds assumed that a change in
the dimensionless static pressure-flow relation signified the
transition to turbulence.
The transition identified by these
can be explained
investigators and others
in terms of the pressure -
flow results.
At a
Page 108
Reynolds number of about 5000,
the results show that
the
dominate factor contributing to the static pressure loss
inertial
changes from a viscous losses to an
transition noted by these
The
investigators may therefore not be a
but a regime transition, supporting the
turbulent transition,
results found
losses.
in this investigation.
6.7 Effect of Airway Collapse
The idealized bifurcation proposed by Pedley,
and used as
the basis for this study presumes a circular cross-section for
the airways.
While this geometry
is generally valid for
neutrally and positively inflated airways
Winter el
al
1982),
1966,
its validity for negative transmural
pressure is not clear.
effect
( Hyatt & Flath
Non-circular airways are
likely to
the characteristics of the pressure-flow relationship,
and are expected to be most
important during expiratory flows
where negative transmural pressures are present.
The cross-sectional geometry of the airways
complicated function of tissue properties,
and
lung inflation.
typical
Figure 6.7-1
tension
in the lung parenchyma act
tissue tethering,
presents a cross-section of a
airway proposed by Hughes et al
forces acting to form the airway.
(1974),
showing
the
Internal airway pressure and
to distend
alveolar pressure acts to collapse the airway.
airway wall
is a
the airway,
while
Tension in the
tissue acts to resists distention under
inflation,
and bending stiffness resists airway collapse under negative
transmural pressures.
Studies of human tracheas and upper
bronchial airways
1975, Khosla 1985)
have
demonstrated consistent crescent collapse patterns.
The
Griscom & Wohl
1983,
Jones et al
results of Khosla is typical,
collapse for
showing
the crescent shape
increasingly negative transmural
collapse patterns are not
These
pressures.
expected to be consistent through-out
the airways because of the additional
tissue support
(
Page 109
asymmetric cartilage
in the upper bronchial airways.
Collapse patterns in all but the upper airways have not
been documented well because of the experimental difficulties.
Casting techniques are difficult to
implement because the
pressure distributions which establish the airway collapse
have not been achived with static conditions necessary for
cast setting.
Optical observations are also difficult to
implement in all but the largest airways.
The small airway
dimensions under collapsed conditions, and
the distortions
caused by hemispherical optics typical of commercially
available bronchioscopes limit their usefulness.
Stereo X-ray-tantalum dust techniques have been
successfully implemented by Hughes et al
(1974)
to study
intrapulmonary airways between 4.8 and 9 mm I.D..
views of thw airways were compared,
geometries were constructed.
et
al for
The stereo
and cross-sectional
The results obtained by Hughes
a nominal 4.8 mm airway at
two different
lung
volumes and a variety of transpulmonary pressures are shown
Figure 6.7-2.
In contrast to
the upper airway collapse,
results show nearly concentric collapse,
except at
the
the low
in
Page 110
lung volumes and high
pressures.
(10
cm)
collapsing transpulmonary
The tedious nature of this approach has limited
the available airway testing,
leaving the exact nature of
airway collapse still unclear.
Without more extensive physiological
information the
effect of airway collapse on the cross-section geometry and
the resulting
influence
on the pressure-flow relationship can
To determine the extent
only be approximate.
to which
"reasonable" airway geometries during collapse may effect
pressure flow realtionship,
the
a simple semi-circular cross-
section geometry was chosen because,
" The crescent type collapse
upper airways.
is typical of the trachea and
" It can be easily constructed with casting techniques and
the two piece airway model.
" It is similar
to the previously tested circular geometry.
The subtraction technique was implemented with the semicircular geometry with the two airway systems consisting of
semi-circular airways.
The results for the static pressure
drop across the test bifurcation as a function of flow rate
were obtained using
the same method as before.
reduce the data in non-dimensional form and
energy and
viscous loss component,
To further
into a kinetic
some additional assumptions
were made.
First,
the characteristic
length used to calculate the
Reynolds number and characteristic pressure P*,
be the hydraulic diameter of the cross-section.
was chosen to
Page 111
Area
D = 4
=4
Perimeter
h
6.7.1
C '4u D2]
D
C
D
1 + 'rT
2/u +1
the correction factors developed and used for the
Secondly,
circular
1
cross-section are
also applied
Additionally,
cross-section.
to
semi-circular
the
results are
the frictional
longer L/D
normalized by a Poiseuille-like form,
which has a
ratio
without a corresponding
due to
the decrease
in diameter
change in length.
The results
for
the frictional
losses,
normalized
Poiseuille form are shown in Figure 6.7-3.
channel.
entrance type losses,
The results
have employed
are based
curved tube type
and the loss in a 2-D
on the hydraulic
a curvature ratio
of
the
Along with the
experimental results are the predictions for
correlation losses,
to
1/11.45,
and
diameter,
and
a L/D ratio of
5.72.
Despite
geometry,
of
the significant
in cross-sectional
curved
tested Reynolds number
results are
tube correlation
Both
range.
lower due
Reynolds numbers below about
to
Reynolds number
tested
of
300,
200.
tubes with
Schlichting
(1979)
the frictional
the
L/D dependence.
At
the results start to deviate
within
10% at the
This deviation at
non-circular
shows that
most of
entrance and convergent
the
Reynolds numbers can be expected based
in straight
over
the weaker
more significantly, but are still
reduces
loss falls within t5%
the results for the frictional
the empirical
channel
change
on
the
lowest
lower
the results for
flows
cross-sections.
the hydraulic diameter properly
results for
a variety of cross-sections
Page
in the turbulent regime,
results
but
112
typically under-predicts the
in the laminar regime by about
10%.
The ability of the analogy to curve tube flow is striking
given that the semi-circular model
of the test bifurcation, but
also the L/D and radius ratio.
While the tests are not conclusive,
support to use the simple curved
in most any state.
theory at the
significant
regions.
it provides substantial'
tube correlation for the lung
The underprediction of the curved tube
low Reynolds number range
in determining the overall
that the errors are relatively low,
the loss
changed not only the shape
is small
is not expected
to be
pressure drop given
and that
the magnitude of
in comparison to the higher Reynolds number
Page 113
Flow Partition Experiments
7.
7.1 Background
The actual
flow conditions in the lung are not symmetric,
as modeled here, but include asymmetric geometries and flows.
These types of asymmetries have been found and documented
throughout the lung
al
( Horsfield
1971, Weibel
1963,
Snyder et
1981) but their effect on the pressure-flow
characteristics of the lung have not been elucidated.
Asymmetric flow partitioning
significant
to have a
influence on the pressure-flow relationship within
the bifurcation.
is that
is expected
The most obvious effect of asymmetric flows
the pressure drop from each daughter branch in a
bifurcation to the parent branch
In addition,
the mixing of
is
likely to be different.
two steams of unequal velocities
in
the parent branch is likely to generate increased frictional
dissipation over the symmetric flow case.
The lack of published information may be a result of the
deficiencies in the systemic approaches alluded to
The subtraction technique,
to
however,
above.
offers a systematic means
study the effects of asymmetric flow partitioning
in a
single bifurcation.
The expected flow asymmetries in the upper airways have
been documented by Horsfield.
In order
flows in each branch of the lung,
to be proportional
therefore acini,
to
to calculate the
Horsfield assumed the flow
the number of terminal
bronchioles,
which are subtended by the branch.
and
The flow
Page 114
ratios vary from unity to over 4.
substantially higher than unity,
one sigma standard variation of
The average flow ratio
at a value of 2.1,
1.1.
is
with a
These flow ratios do
not represent the velocity difference in the two
joining
streams because the daughter
included.
tube area's are not
This effect will be discussed further
The approach used
in the next chapter.
to understand the effect of flow
asymmetries on the pressure-flow relationship
is parametric,
utilizing the symmetric bifurcation previously studied.
parametric approach and the subtraction technique,
in the studies of symmetric bifurcations,
utilized
are likely to
introduce some errors in modeling the flow.
technique, as has been described before,
The
The subtraction
introduces potential
errors such as neglecting cascading effects.
Further errors
introduced by the parametric approach arise because flow
asymmetries are not
likely to occur alone,
but
in conjunction
with other asymmetries.
Olson
which
(1971)
is not
presented a typical pulmonary bifurcation
symmetric, as has been modeled,
Olson's asymmetric bifurcation model
is shown in Figure 7.1-3,
with daughter tube diameters which are not
differences in resulting flow area's will
flow asymmetries
but asymmetric.
equal.
The
interact with the
to determine velocities in the daughter
branches and therefore kinetic energy fluxes.
While
important to understand the limitations imposed by
parametric approach,
the
insights developed by controlled
variation of parameters are expected to
improve the
it is
Page
115
understanding of pulmonary flows.
7.2 Experimental Procedure
in applying the subtraction
The conceptual approach
technique to the asymmetric flow problem is similar
for the symmetric flow case.
The experimental
be changed to establish the asymmetric flow,
the pressure drops from each of
parent branch.
to that
approach must
and to
isolate
the
the daughter branches to
The experimental apparatus developed which
achieves both the pressure isolation and asymmetric flow
partitioning
is shown in Figure 7.2-1.
The asymmetric flow in
the test bifurcation is established
by specifying the flow in one of the daughter branches,
the other
head.
while
is allowed to vary with the changing hydrostatic
Various flow ratios are therefore generated as the flow
rate in the hydrostatically driven branch changes with the
tank
level.
The prescribed flow, which
separate circuit
to isolate the fluid,
centrifugal pump
( Gould model
a flow restricting valve,
(Fischer and Potter, model
is contained
in a
is generated by a
#3642 size #1
),
metered with
and measured with a Rotometer
10A 1027A).
Upstream boundary
conditions for each daughter branch of the
test bifurcation
are maintained as before with airways distal
to both daughter
branches.
The experimental
techniques developed to measure the
pressure flow characteristics of an airway for the symmetric
flow studies are essentially unchanged
in this case.
Page 116
Differentiation of the tank
level to measures flow rate, and
a differential pressure transducer between the reservoir and
exhaust tank measures the driving pressure.
the
loss in the upstream airway, which
symmetric flow conditions,
is an airway with
from the loss in the hydrostatic
branch gives the loss in the hydrostatic
bifurcation.
rate branch
Subtraction of
leg of the test
The pressure difference in the constant flow
is not measured.
Four valves are positioned to allow the system to be
filled and run with the same pump.
The valve positions in the
run and the fill mode are summarized in Table 7.2-1.
Table 7.2-1
Valve Status
Mode
Gate Status
#1
#2
#3
#4
Fill
Open
Closed
Closed
Open
Closed
Run
Closed
Variable
Open
Closed
Open
Near steady flow rates in the "constant" flow rate branch
are maintained despite the change in operating pressures
resulting from the change
in pressure with hydrostatic
Maintaining a constant inlet pump head, and minimizing
level.
the
pump outlet pressure variations gives the near constant flow
rate.
Constant
inlet pressures are maintained by supplying
the pump from the exhaust
with an overflow weir.
tank which has
its level
maintained
An added benefit of using the exhaust
Page 117
tank as a supply to
the pump
is that
it minimizes air
entrainment in the system which would result
if the aerated
fluid in the storage tank were used.
a high pressure
( several
hundred
inches of water
)
Near constant downstream pressures are maintained by using
centrifugal pump and metering the flow with a high resistance
valve
(#2).
Pump flow studies with varying exit pressures
typical of those expected
showed that
in an experiment
(0 to 50 cmH20),
the flow rate varied by less than 1.5%.
This potential error is combined with two other
additional
level
sources of error
to increase the expected error
in the flow partition experiments.
introduced to
Vibrations
the system by the pump and errors
in reading
the constant flow source flow meter are expected
the approximately t5% error found
to add to
in the symmetric flow
experiments.
7.3 Data Reduction
The subtraction technique,
to give the total
as outlined above,
can be used
static pressure difference across one leg
of a bifurcation during asymmetric flow conditions.
analysis previously developed for
The
the symmetric flow
conditions to decompose the static pressure
loss into
frictional and kinetic energy components can not be used,
it is no
longer
applicable.
While the previously developed analysis does not apply
directly,
the energy based methodology used to develop the
as
Page 118
decomposition procedure can be used for the asymmetric flow
conditions as well.
The control
volume selected to analyze
the flow is shown in Figure 7.3-1.
control
The selection of the
volume is such that;
" The flow enters/exits the control
control surface.
volume normal
to the
" The control volume includes one daughter branch and
parent branch of the bifurcation.
the
" The inlet to the control volume from the second daughter
branch is the same area as the branch itself.
Using subscripts 1,2 and 3 to refer
with the variable flow,
fixed flow,
to
the daughter branch
the second daughter branch with the
and parent branch respectively,
continuity equation and energy equation
allows the
(3.7.1) to be written
as;
7.3.1
Continuity ...
7.3.2
Energy Equation
dVol= -Q
P
A U
+ A U
=
2 2
1 1
U
U
3 3
...
*
2
+ 'f P Re
1L1
1
*2
+ Iuf P Re
2L 2
2 2
o[P
P +
+ Q
J
3
*2
P Re
3 3
3J
'f
Vol
Where the correction factors and the pressure correction
into the factor f,
defined
Note that the correction factors were developed for
symmetric flow conditions,
that
but
it was found
the same correction factors applied
in 5.4.2.
the
( see Chapter 5
)
term have been included
to asymmetric as well
as symmetric flows.
Following
the definition of frictional
pressure drops,
integral of the dissipation function can be written as;
the
Page
7.3.4
dVol = Q
J
Del Pv
1
119
Del
+ Q
1
Pv
3
3
Vol
Del Pv,
where the subscripts on the viscous pressure drops,
refer
to
the leg of the bifurcation in which the dissipation
occurs.
in terms of P,
can be written
Pe,
The pressure,
estimated.
leg 2 must be
the pressure at
Before 7.3.3 can be used,
and two pressure differences;
= P-
P
7.3.5
2
p
1
-
P'
LI
-
J
[L P'1
-
P
The pressure loss in daughter branch #1,
branch #2,
Pe-P'e, are due to frictional
-P
[P
+
L 2
21
2
Pt-P'1 , and
losses alone because
the area of the daughter branch remains constant between
bifurcation
junctions.
The pressure loss in the variable flow
daughter branch can therefore be written as Del Pv1 , to be
consistent with 7.3.4.
Pe,
is more difficult to assess,
the base of each
as flow in two curved tubes.
P' 1
but can be approximated by
considering the flow patterns in the
geometry in the vicinity of the
leg,
-
The difference in pressures at
junction.
The flow
junction can be approximated
The pressure differences are a
result of gradients established by the centrifugal forces,
given to a first approximation by the Euler normal
equation 5.3.1.
equation,
Integrating the Euler equation across the
cross-section and weighting the pressure by the differential
area,
gives the average pressure difference between the
Page
120
centerline of the tube and the bifurcation carina as;
a
r
2-/a
2
-r
2
=g -
dr
J
Rc
'4ra2
-
Del P
7.3.6
a
4U2
U2
c;
3r
Rc
0
where a
is the daughter
curvature,
tube radius, Rc
and the ratio of the two for
is .75*(1/7).
is
the radius of
the test bifurcation
The change in the radius of curvature across
the cross section and the pressure variations due to the
secondary flows have been neglected,
and the axial
velocity
has been approximated by the average axial velocity,
pressure difference across the
U.
The
junction between the two
legs
can be approximated with 7.3.6 as;
- U-U=c
P+Re2
r
P' 1
7.3.7
where r
4 a
4
3Tr Rc
P
2
2
2
4
1
2
3n
*
Rc
3
is defined as the flow ratio, and for
geometry of the ideal bifurcation
r = U
7.3.8
/U
=Q
1
Combining
into
a
2
1
2-
3
LE
+
r]2
the symmetric
is given by;
/Q
1
2
the expression for Pe and
the continuity equation
the energy equation gives;
7.3.9
Del Pv + Del Pv = Del F
1
3
-
Del
2
r2-1
*
4 a
Pk +
- P Re2A /A21
3L 3
1JL E1 + r]2
3nRc 3
where the static pressure drop, Del
pressure drop,
Del
Pk, are given by;
R, and the kinetic energy
Page 121
7.3.10
Del
- P
f=P
1
3
3
+
fr
7.3.11
Del Pk =
'
P
f
Re
3
3
A /A
33
3
(1 +
where Del
f
1
2
2
*
2
r
R, r, and the Reynolds number are experimentally
measured quantities.
7.4 Results
The experiments were conducted at seven different fixed
The fixed flows varied from 0 to 902 cc/sec,
flow rates.
which corresponds to a Reynolds number range in the daughter
tube from 0
to 4390.
The normalized results for
in the test
the total static pressure loss
leg of the bifurcation at 6 of, the 7 different
fixed flow rates in the second bifurcation
Figure 7.4-1.
The pressure loss
is normalized to the kinetic
energy flux, '4P*Re2 , in the parent
bifurcation, and
leg are shown in
tube of the
test
is plotted against the flow ratio, F.
experiment with 0 fixed flow is not shown because r
(The
is
infinite)
The normalized pressure loss
about
+.9,
near a flow ratio of
is seen
.8,
and
decreasing values of the fixed daughter
The pressure loss
is seen
decreases away from
0.8.
to have a maximum of
increases with
tube Reynolds number.
to decreases as F both
increases and
Negative pressure losses occur at
Page
low F,
indicating that
cause a pressure
test
122
the presence of the fixed flow rate can
increase along the direction of flow in the
leg of the bifurcation.
This apparent paradox can be explained by the pressure
changes caused by the fluid accelerations.
For conditions
where the fixed flow rate is substantially higher
than the
variable flow in the test
some
leg of the bifurcation,
pressure recovery occurs as the fixed flow slows upon entering
the
larger parent branch.
This negative pressure loss can be
accounted for by subtracting the predicted pressure loss due
to kinetic
energy changes.
Figure 7.4-2 presents the decomposition of the static
pressure loss,
Pk,
Del
R, into the kinetic energy component, Del
and the frictional component, Del Pv,
experimental
run.
for
a typical
The negative pressure drops are shown to be
predicted by the changes in kinetic energy,
and
leave as a
remainder a positive pressure loss which is attributable to
the affects of friction plus the curvature pressure
correction.
Further data reduction requires partitioning
dissipation between the daughter branch and
The viscous pressure loss
the parent branch.
in the daughter branch can be
approximated by the loss that would occur
flows were symmetric.
the viscous
in the leg
This assumption is valid if
if the
the
effects of the flow asymmetry do not propagate upstream.
results of the symmetric experiments give this loss
daughter branch as;
in the
The
Pv = 32 P
11
Del
7.4.1
123
[.556 +
L
Re
.067/Re]
J
-
Page
D
The viscous pressure loss in the trachea was assumed to
be the sum of two components;
loss for
in Figure 7.4-
is shown
the 6 different experiments
to
The loss is seen to be the same for each experiment,
within about
15%,
and is given by;
2
*
Del Pe = 'P Re
7.4.2
1 -F
2
2[
3
L1
+
3.
The excess
and an excess component.
the flow were symmetric,
if
the loss that would occur
This loss represents the excess kinetic energy flux due
in velocity of the two mixing streams.
to the differences
in the straight
simple momentum analysis of a mixing flow
section of the parent
tube is outlined
gives the potential pressure recovery
owing
to
A
and
in Figure 7.4-4,
in the parent branch
the velocity equilibration as;
7.4.3
P
-P
3
where U -4
2
= ; ['U
+
L
31
3
and Uz. refer to
2
2
U
U
-
L
32
the velocity
+ '12U
3
32
that the fluid
in leg
Using continuity,
1 and 2 attain when in the parent branch.
7.4.3 can be written as;
U
2F
P
-P
3
=
3
UL
3
32
U
-
32]
+ U
31
22
=
- 1
2
U
3
32
-
7.4.4
-2
-U
31
r + 1
2
Page
which
is identical
frictional
124
to the excess loss over and above estimated
losses found experimentally.
The similarity of the
losses indicates that the available kinetic
energy in the
mixing streams due to the bulk velocity differences is not
recovered as the flows mix to
The
a uniform velocity.
15% error represented by the results in Figure 7.4-3
is not reflective of the true predictive capability of the
experimental correlations,
as
it
is a cumulative error applied
to only one component of the pressure
loss.
The excess
loss
was computed by subtracting a friction prediction which has an
5%, and a kinetic energy prediction which
accuracy of about
employed a number of assumptions.
the three components, all
The total
loss is a sum of
of which are approximately of the
same order.
For
1/2.1=.476),
the additional
typical flow ratios of about 2.1
approximately equal
asymmetric pressure loss is
to the change in kinetic,
in an accuracy of about
(or
which results
5% in predicting the static pressure
loss.
The form used to describe the additional pressure
difference created by the asymmetric effects has the distinct
advantage of reducing to
when the flow ratio
the symmetric form automatically
is one.
Computational models can
therefore include the asymmetric pressure loss term,
apply to symmetric conditions with no
and
loss of accuracy.
Page
8.
8.1
125
Implications for
Pulmonary Flows
Introduction
A primary objective of
this work
relationship between pressure
and
expiration from a human lung.
As
were designed,
flow
developed
relationship
lung,
a
and
The next step
is
to use
airway
into
conditions
in
the
distribution are
the experimental
and
Intrinsic to
known well
that
the
The entire
the
enough
the
the overall
the experiments properly
lung,
results.
by numerically assembling
and
the
chapters have
complete lung.
a complete system,
that
make-up
these results to predict
performance can be calculated.
the assumption
describe the pressure-
elements which
presented
network can be simulated
bifurcations
to
The previous
pressure-flow performance of a
during
first step, experiments
a
in the structural
these efforts and
the
flow rate which exists
executed
single bifurcation.
described
is to predict
lung
this approach
simulated
geometry and
to permit
is
the flow
flow
accurate numerical
modeling.
To
test
networks
the accuracy of
geometries will
complicated physiological
human
lung.
Their are
be
these assumptions,
tested first,
then,
simple airway
more
models before simulating a complete
a number
of reasons
for working
with
the simplified models first;
* Geometries and'flow distributions of the simplified
networks
are
known,
and
do
not
have
to
be approximated.
* Experimental results for a range of simplified geometries
are available in the literature, and can serve as a check
Page
126
on the numerical prediction over a wide range of flow
rates.
* The quality of the fit between the numerical predictions
and the observed experimental results can serve as an
indicator of the error in predicting the performance of a
complete human airway.
The first
set of simplified airways which will be tested
are idealized models constructed to simulate only a few
These models are smooth
generations of the airway network.
walled,
branch symmetrically,
and have equal
flows in branches
Because the terminal branches can represent
of the same order.
a significant portion of the pressure drop of the entire
airway,
care
is taken to represent
of these airways accurately.
flow correlations
and
the pressure-flow relation
The numerical model uses entrance
(Rivas & Shapiro 1956)
for
these branches,
the experimentally determined curved tube type correlation
elsewhere for the frictional
dissipation.
A number of physiological airway models are also available
in the
literature for comparison of the measured and predicted
pressure-flow relationships.
These models are
less well
characterized
in terms of both geometry and
distribution,
but the pressure-flow character of the airway
is well
documented.
The models are not smooth walled,
asymmetrically, and sometimes branch
than dichotomously.
well
flow
branch
trichotomously rather
Numerical simulation of these models, as
as a real human airway, require that the equations
written previously for symmetric branching be modified for
daughter branches of unequal
For comparison purposes,
size.
Figure 8.1-1 presents the
Page 127
estimated frictional
pressure drop across a single bifurcation
based on the measured systemic performance by several different
investigators and the results from this study.
the figure shows that
frictional
terms.
Inspection of
this study appears to under-predict
One might
that the systemic
therefore expect
predictions based on the results from this study will
low.
the
also be
If this is not the case, and comparison of the systemic
prediction and actual
experimental results are close,
it would
indicate that basing the performance of a single bifurcation of
the systemic performance can introduce significant errors.
8.2 Pressure Drop correlations
The correlations for the pressure drop components
previously presented strictly apply only
branched bifurcations.
branch asymmetrically,
be modified.
Real airways,
to symmetrically
however,
typically
which requires that these expressions
Asymmetric geometric conditions have not been
tested experimentally,
so the applicability of the asymmetric
forms must be assumed to be at
least approximately accurate.
The geometry of the bifurcation is assumed to be that
given in Figure 8.2-1.
It
is further assumed
ratio of a bifurcation branch
the experimental
is constant at
test bifurcations,
Sr= 1/7.
that the radius
the same value as
The viscous
pressure loss correlation which applies to each branch of
bifurcation therefore remains unchanged.
Kinetic Energy Flux
the
Page
128
Application of the energy equation to determine the
pressure drop due to the changing kinetic energy flux
in a
symmetric bifurcation with symmetric flow conditions is
The expression was modified in 5.4 to
in 3.7.
outlined
and the pressure work due to
the energy in the secondary flows,
pressure variations generated by the secondary flows.
of the equation was further modified to account for
asymmetric flow conditions
interest,
include
in 7.3.
The form
the case of
The case of physiological
asymmetric geometry and flows,
simple extension of the equation, and
can be seen to be a
is given by;
3
2
*
Del P
8.2.1
=
P
KE
Re
3
2
-
f
3
3
1
2
[A /A
3
1J
3
c
Where i
2-
i + r
3
is defined as the daughter tube area ratio;
8.2.2
A / A
1
2
Inspection
of 8.2.1
shows that
it reduces to the previously
developed expressions when the ratios are set equal
symmetric geometry with 4=1,
to 1;
and symmetric flow conditions with
r=1.
Mixinq Loss
The experimental
investigation of additional viscous
dissipation resulting from asymmetric flow conditions was
reported
in chapter 7.
indicated that
The results of the experiments
the pressure
loss can be predicted by assuming
129
Page
the excess momentum in the two
streams
is
lost as the
Applying the momentum equation to the
velocities equilibrate.
straight section of the parent branch of the bifurcation
in
8.2-1 gives;
2
*
8.2.3
Re 2
'4P
Del P =
(r
-)2
(
+ 1)2
-_
4
m
Inspection of 8.2.3 shows that it reduces to the proper
form for
the two simplifying cases.
symmetric bifurcation,
obtained.
No mixing
=1,
For
the case of a
and the earlier result
is
loss occurs when the velocities of the
two daughter branch streams are equal;
V
8.2.4
Q
A
Q
1
1
1
V
A
2
2
2
or when;
= Q
8.2.5
Q
1
A
2
A
1
2
8.3 Prediction of Model Experiments
Two sets of model
experiments are available in the
literature which present measurements of the static pressure
drop across a set of symmetric bifurcations.
(1980)
developed a smooth walled symmetric
Reynolds
(1980)
blown glass tubing.
D.B.
experiments in rigid
"Y" connectors,
Hardin and Yu
lung model from
conducted two
sets of
one with L/D ratios of 5,
the other with L/D ratios of 9.
Prediction of Hardin & Yu's Results
The bifurcations used
in the study are very similar
to
Page
those
in this study, and
130
it could therefore be expected that
the results should also be similar.
pressure drops,
though,
The reported frictional
differ by about a factor of 2.
Figure 8.3-1 presents the range of Hardin and Yu's
experimental results, and the numerical prediction for the
normalized total static pressure drop.
to be good,
to within about
1000 to 10,000,
range.
and is best
The agreement
is seen
10% over the Reynolds range from
in the higher Reynolds number
The agreement is good over the entire range, despite
Hardin and Yu's contention that a transition to turbulence
occurs at a Reynolds number of about 4500.
The agreement between the measured and predicted static
pressure drop
is much better than would be expected given the
difference in calculated frictional pressure drop.
The
agreement
is better
than would be expected for
two primary
reasons.
The first
is that Hardin and Yu assumed that
the
kinetic energy correction factors were unity, underpredicting by about
Yu did not
15%.
The second reason is
include entrance effects
in the peripheral airways
which were a significant fraction of the total
system,
that Hardin and
loss in their
in an overprediction of
and the averaging resulted
true dissipation results for a bifurcation.
Prediction of D.B. Reynolds Results
The experimental results and prediction are excellent
agreement for both airway models,
as shown in Figure 8.3-2.
The agreement supports the linear L/D dependence found in
the
Page 131
this study,
Reynolds,
given the much
longer L/D ratio's tested by
both L/D ratios tested.
and the fit for
The close agreement in
the static pressure
the value of
loss could be expected given the close agreement
losses shown
Yu,
in Figure 8.1-1.
in frictional
Unlike the case for Hardin and
frictional resistances in airways with constant area tubes
Reynolds data
drop off rapidly away from the trachea.
therefore more closely reflects the frictional relations
internal
in the
bifurcations.
8.4 Prediction of Physiological Model Experiments
Two sets of experiments were published which present
the
pressure-flow relationship for a more physiologically
accurate airway network than the
was reported by
Isabey and Chang
central airways of the human lung
Meditech,
Watertown, Ma).
Reynolds and Lee
right
(1979)
idealized models.
The first
in a model of the
(1981)
(Zavala Lung Model,
The second was reported by
in a cast of a human lung below the
intermediate bronchus.
Zavala Lunq Model
A drawing of the Zavala lung model
8.4-1.
The details of the Zavala model
by Slutsky et al
(1980),
including trichotomies,
For
presenting
and
geometry were reported
the branching pattern,
the branch
the purposes of simulating the flow
was reconfigured
is shown in Figure
lengths and diameters.
in the cast,
the model
into the "dichotomous" representation shown
in
Page 132
Figure 8.4-2.
Table 8.4-1
presents the branching pattern and
dimensions of the dichotomous model.
Note that the dimensions reported for
model
the dichotomous
are different from those given by Slutsky et al,
part due to
in
Inspection of a
the translation of trichotomies.
male cast of the Zavala lung model also generated some
dimensional differences because the diameters reported by
Slutsky et al were the major axes for elliptical sections,
and not a hydraulic or area averaged diameter.
Prediction of
Isabey & Chang's Results
Despite the differences in geometry,
the agreement between
the experimental results of Isabey and Chang and
prediction is good.
The comparison in Figure 8.4-3 presents
the experimental results and
the numerical prediction based on
two different flow distribution assumptions.
assumption
The first
is that the flow is distributed according
fraction of alveoli subtended by
1971).
the numerical
the branch
to the
(Horsfeild et al,
The second flow distribution is believed to better
represent the actual experimental conditions by generating a
uniform pressure in the peripheral
airways exposed to the
plenum.
Figure 8.4-4 presents the pressure drop components,
expected,
the static pressure drop is dominated by viscous
dissipation at
about 2200,
Mixing
and as
low Reynolds numbers.
At a Reynolds number of
the kinetic energy component
starts to dominate.
losses are an order of magnitude lower
than the other
Page
133
Table 8.4-1
"Dichotomous" Zavala Lung Model
"Internal"
Branches...
Diameter Length
cm
cm
2. 10
1. 70
1. 80
1. 20
1. 30
1. 20
1. 70
0. 90
1. 20
1. 70
0. 70
1. 20
0. 90
0. 90
1. 20
0. 70
1. 00
0. 90
0. 80
0. 70
0. 40
0. 70
0. 50
0. 70
0. 50
0. 70
0. 70
0. 70
0. 70
0. 80
0. 60
0. 70
0. 50
0. 70
0. 70
0. 80
0. 90
1. 00
7.90
5.20
3.90
3.20
1.90
2.20
1.80
1.80
1.20
0.30
1.30
0.70
0.80
0.30
0.30
1.20
0.60
0.70
1.50
0.80
0.90
0.40
1.20
1.10
0.60
0.90
1.10
1.00
1.10
0.70
0.60
0.40
1.50
2.10
2.30
0.30
0.20
0.20
Branch Numbers
Parent Other D. Daug.#1 Daug.#2
0
1
1
3
2
2
4
6
3
7
5
5
4
8
9
10
6
7
8
9
10
11
11
12
12
13
13
14
14
15
15
16
16
32
32
19
18
17
0
3
2
9
6
5
13
17
4
18
12
11
7
19
20
21
8
10
14
15
16
23
22
25
24
27
26
29
28
31
30
33
32
35
34
77
39
65
2
5
4
7
11
8
10
14
15
16
22
24
26
28
30
32
38
37
36
40
42
44
46
48
50
52
54
56
58
60
62
34
66
68
70
72
74
64
3
6
9
13
12
17
18
19
20
21
23
25
27
29
31
33
65
39
77
41
43
45
47
49
51
53
55
57
59
61
63
35
67
69
71
73
75
76
Branch
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
Page 134
Table 8.4-1
(Continued)
"Dichotomous" Zavala Lung Model
"Terminal"
Branches
Diameter
cm
0.40
0.40
0.50
0.30
0.50
0.40
0.40
0.30
0.40
0.40
0.40
0.50
0.40
0.50
0.50
0.50
0.40
0.40
0.40
0.40
0.40
0.40
0.40
0.50
0.40
0.40
0.40
0.40
0.50
0.40
0.30
0.30
0.40
0.50
0.40
0.40
0.40
0.50
0.40
Length
cm
1.40
2.10
1.90
1.20
1.60
1.70
1.50
1.20
1.20
2.10
1.90
1.60
2.00
2.00
2.00
1.80
1.70
2.20
2.70
1.20
1.50
2.30
2.20
2.00
2.00
1.30
1.00
2.30
2.30
1.20
1.10
1.20
1.10
2.90
3.00
1.20
1.30
1.50
2.60
Branch Numbers
Parent Other D.
18
20
20
21
21
22
22
23
23
24
24
25
25
26
26
27
27
28
28
29
29
30
30
31
31
38
17
33
33
34
34
35
35
36
36
37
37
38
19
37
41
40
43
42
45
44
47
46
49
48
51
50
53
52
55
54
57
56
59
58
61
60
63
62
76
38
67
66
69
68
71
70
73
72
75
74
64
36
Branch
Flow Percentage
2.3064
5.36
5.3685
1.81
1.8253
1.81
1.8253
1.3743
1.3743
3.8865
3.8865
3.1772
3.1772
2.4013
2.4014
2.4014
2.4013
2.3490
2.3503
3.8865
3.8865
1.60
1.6067
3.8865
3.8865
1.56
1.5793
3.1772
3.1772
1.5886
1.5886
1.5886
1.5886
2.591
2.591
2.290
2.290
1.56
2.591
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
Page
135
components.
The prediction is seen to fall
high Reynolds numbers.
The reason for this
the roughness of the Zavala model,
is not expected to be
below the experiments at
which,
important in a real
is assumed
to be
as discussed before,
The
lung.
overprediction at low reynolds number is not great, only
about
10%,
and may be attributable to
inaccuracies
in the
geometry modeling.
Generation of Uniform Pressure Boundary Condition
Given that the Zavala
alveoli,
lung model was not supplied by
but by a uniform pressure plenum,
the flow
distribution calculated by Horsfield et al will
correct.
not be
The more correct flow distribution is the one which
generates uniform pressures in the peripheral branches.
Unfortunately,
the coulpled non-linear equations which
characterize the pressure-flow relation are written to
calculate a pressure drop given a flow rate.
The equations
and calculational scheme cannot easily be inverted to
generate the required flow rates
in an airway to produce a
uniform upstream pressure boundary condition.
To determine this more correct flow distribution,
iterative approach was used.
flow distribution,
Starting with the Horsfield et al
the problem was
resistance matrix and calculating
branch
i,
an
linearized by forming a
the change
given a change in flow rate
in pressure in
in branch
j.
Page 136
8.4.1
R
Delta P / Delta Q
i
j
=
ij
After
inverting
the resistance matrix and multiplying by a
uniform pressure vector,
process had
the flow vector
is given.
to be repeated a number of times because the
problem is not
linear, and the solution only represents an
estimate of the correct flow distribution.
iterations the flow vector converged,
After a number of
and remained constant.
graphical presentation of the resistance matrix
Figure 8.4-5,
showing that
would be expected, but
negative values which
to converge
This
that
the matrix
is shown in
is primarily diagonal as
it also contains off-diagonal
is why a full matrix approach
is needed
to a solution.
The non-linear resistances result in the flow distribution
being a function of flow rate.
The flow distribution is shown
in Figure 8.4-6 for a number of different flow rates,
as the
initial assumed flow distribution.
The flow
distribution is seen to vary markedly from the
form,
but
the pressure-flow results
as well
initial assumed
in Figure 8.4-3 show that
the systemic performance reacts by varying only about
This finding will be
Given that
terminal
lung is tested.
the respiratory tree has upwards of 230,000
branches it becomes prohibitive
resistance matrix.
et al
important when the full
10%.
to form and
invert a
The 10% overprediction with the Horsfield
flow distribution can then be taken as an indication of
the error
introduced by
inaccurate flow assumptions.
A
137
Page
Prediction of Reynolds & Lee Lung Cast Results
To model
asymetric
used.
the lung portion used by Reynolds and Lee,
lung model
airways,
Each BPS
(BPS).
branch of order
1,
2 proposed by Horsfield et al(1971)
The model uses
the central
and one
dimensions of
provide the
al Model
2
a statistical
as an
N bifurcates
N-M,
the branches for
each BPS
is excellent,
though,
10,000.
order
5.
are different
to
N-
The
The Horsfeild et
in Figure 8.4-7.
Lee,
along with
The
in Figure 8.4-8.
with
falling within the experimental
100< Re<
one of
a
the pressure drop up to the right
intermediate bronchus are shown
two
for
where
3 and
results of Reynolds and
the numerical prediction for
range from
varies between
is schematically represented
the
airway,
two branches,
best current pulmonary model.
The experimental
between
average representation
asymmetric
into
where M
was
in bronchopulmonary segments
terminating
is modeled
of order
the
the
agreement
numerical prediction
error band
over
The curvature of
the entire
the prediction
tends to be slightly less than the experimental
results.
The flow distribution was
Horsfield
et
distribution
because of
Zavala
is
al.
the
to
attempt
to generate a
large
lung model
likely
No
was made
results,
be
that
to change
given by
the
flow
uniform pressure boundary condition
number of terminal
introduce no
flow relation.
assumed to
the
more
error
than
in
10%
branches.
the
Based on
the
flow distribution
error
in the pressure
Page 138
8.5 Lung Simulations
of
systemic pressure drops
results for
With
models.
to
predict
a wide
variety of
airway
we can move on with confidence
agreement,
this
in
a complete
the pressure flow characteristics of
lung.
human
lung
Two
models will
be
approaches to
lung model
the
in
The symmetric
lung.
accurate representation of a
these
the effects of
lung
lung,
Comparison of the
its use.
based on
indication of
give an
extremes
is the most used pulmonary model
its complexity has limited
pulmonary predictions
numerically because
and the asymmetric Horsefield et al
provides the most
2
represent
the geometry of
(1963)
of Weibel
its simplicity,
due to
model
modeling
investigated
and
they are both widely used,
but
with published
shown excellent agreement
this study have
results
on the experimental
predictions based
The numerical
two
lung
models should
and
asymmetry,
the
sensitivity of pulmonary predictions on the details of the lung
geometry.
Figure
8.5-1 presents
relationships of
cast
energy flux,
the
verses
agreement between
close;
at worst,
difference
tracheal
is
the pressure-flow
The results are
pressure normalized
and dimensional
gV2,
pressure/flow rate,
The
results for
two pulmonary models.
the
two common forms,
in
the
the
two
lung
models
diameters,
th-e
lung
resulting
kinetic
models
give
in different
in
the
trachea.
is suprisingly
not more than 20% apart.
that
the
airway resistance,
Reynolds number
the
to
A major reason for
different
flow rates
for
the
Page 139
same tracheal Reynolds number.
Upon closer inspection of the results from the two
models
lung
the tracheal diameter difference is seen to account
for most of the difference at
diameters are set equal,
the Weibel model
below the asymmetric model
numbers.
low Reynolds numbers.
results then fall
prediction at high Reynolds
The reason for this
is seen in Figure 8.5-2 which
presents the pressure drop components for
asymmetric model.
If the
the Horsfiel
In the higher Reynolds number range the
pressure loss associated with the mixing due to the model
asymmetry represents about 20%'of the total static pressure
loss.
The symmetry of the Weibel model
of loss,
precludes this type
and results in the static pressure prediction
falling below the prediction of the asymmetric model.
Pressure Distributions
The distribution 'of pressure within the
lung
is important
to determine the relative contribution of each generation
to the overall pressure-flow characteristics.
A difficulty
arises in trying to present pressures and pressure gradients
in
the asymmetric
path chosen.
lung model because
it will
depend on the
Unlike the symmetric airway model,
the static
pressure and viscous dissipation pressure drops will not be the
same
in branches which are
distal
the same number of generations
to the trachea.
Additionally,
and alveoli will
the number of generations between the trachea
also vary depending on the path chosen with
Page
the asymmetric lung model.
140
Figure
8.5-3 presents the
percentage of paths through each of the BPSs with a given
number of branches between the trachea and alveoli for the
horsfield
lung model 2.
Shown are the percentage of paths with
a certain number of total generations following the shortest
and
longest possible path for each BPS.
asymmetric
the
lung model
Note that the
will have many more generations into
lung than the symmetric Weible model which consists of
generations of conducting
To present
18
airways.
the data averaged quantities will
average used for the asymmetric model
be used.
The
is a simple arithmetic
mean of the values at a specified number of generations into
the
lung.
No flow percentage weighting
is used.
Figure 8.5-4
presents a comparison of the static pressures verses
generations
into the lung for a range of flow rates.
of the averaging problem,
asymmetric
lung model
static pressure drop
Because
the averaging static pressure in the
is actually an integration of the average
in each generation.
Comparison of the results from the two models shows
The gradient of static pressure is
generally good agreement.
seen to extend more deeply into
demonstrating the increased
resistance.
difference
the
lung for
low flow rates,
importance of frictional
At higher flow rates the inertial
in the larger airways dominates.
difference exists between
pressure
An interesting
the results of the two models;
pressure gradients are seen to remain at significant
further
into the lung for
the asymmetric
lung model.
the
levels
Page
141
Viscous Pressure Drops
The reason for the more significant
static pressure
gradients deep
inside the
the difference
in frictional characteristics.
lung with the asymmetric model
is
Figure 8.5-5
presents the pressure drop associated with viscous energy
dissipation,
including the curved tube type friction and mixing
losses, vs generation into
Surprisingly,
the asymmetric
maximums for the frictional
first
is similar
symmetric
the
lung model
more deeply in the
shows that two local
pressure drops can occur.
in nature to
lung model
lung for the two models.
the one predicted by the
near generation 5.
The second occurs much
lung, near generation 22.
To ensure that this second maximum does
indeed occur,
is not a manifestation of the averaging process,
viscous pressure drop
entire band of results indicates that
and
it
and
the average
1 standard deviation for four different
flow rates are shown in Figure 8.5-6.
occurs,
The
is therefore not
The behavior of the
this second maximum
likely to be a aberration
generated by the averaging process.
Note that the second maximum is most pronounced at
lower
flow rates, presumably that as the flow rates are increased
with
the higher Reynolds numbers near
which behave as Re1 -
the trachea,
(curved tube type friction)
the terms
and Re 2
(mixing loss terms) become dominate.
To better understand
the reason for
the second maximum,
the
results were recast to present the frictional pressure drop vs
Page 142
the second
local
indicate that
The results once again
Figure 8.5-7.
diameter,
maximum exists,
and did not result from a
peculiarity associated with the distribution of average airway
diameter with generation.
diameter and
Figure 8.5-8 presents the average
lung models.
length to diameter ratios for both
in number of generations,
Apart from the difference
models demonstrate the same general
with generation into
both lung
dimensional characteristics
the lung.
The reason for the second maximum must therefore be based
on the interaction fluid dynamic phenomena and the asymmetry of
the lung
geometry.
the asymmetric
lung geometry,
with unequal flows
mixing
losses,
To analyze how a
local maximum can occur
consider a single bifurcation
in each daughter branch.
which are not
important deep
Neglecting
into the
lung,
the viscous pressure drops are simply the curved tube type
result found for symmetric flow conditions.
the viscous pressure
to
loss in
the daughter branch,
8.4.2
=
. 1 +
I
L
r
the parent branch,
of order N-1,
1.5
Dn-1/Dn
15]Re
L
J
2
The ratio,
of
of order N,
is given by;
3.5 rL/D]n
J
0.743
l,
>>
60
[L/DI
n-1
8.4.3
I1
+
L
2
r
1I=
Dn-1/Dn
J
L
0.794
3
EL/D~n
Re 1 60
[L/D]
n-1
The ratio
factor,
is constructed from three terms;
diameter ratio factor and L/D factor,
nominally near unity.
a flow ratio
which are
The product of these ratios will
in
Page 143
determine the factors for
the Horsfield model 2 verses
generations into the lung.
8.5-9,
and
The results are shown
indicate that the
local maximum
in Figure
is primarily due
to the product of the flow ratio factor and diameter ratio
factor reaching a minimum below
minimum is not
likely to occur
because the flow ratio factor
diameter ratio remains above
1 near generation 22.
The
with the symmetric flow model
is identically unity,
an the
1.
Comparison of Different Flow assumptions
For comparison purposes, Figure
8.5-10 presents the
prediction for the viscous flow resistance verses number of
generations into the Weibel
flow assumptions.
et al
(1970)
for
lung model
for three different
The figure presents the results of Pedley
inspiratory flows,
the results of this study
for expiratory flows and for a baseline comparison the
resistance assuming Poiseuille flow.
The results show that
the three flow assumptions produce
significantly different viscous pressure profiles above about
the
10th
generation.
number drops
Below the
1 0 1,*
low enough that the flow
Poiseuille like.
generation,
the Reynolds
is essentially
The difference in profile between the
inspiratory and expiratorty is due primarily to the difference
in L/D dependence.
Both
inspiration and expiration depend on
Reynolds number to the 1.5 power,
but Pedley et al
found a
square root dependence on L/D where this study found a
dependence.
linear
Page 144
8.6 Physiological Data
The final comparison which can be drawn between the
numerical pulmonary predictions and published data is with
experiments conducted on
living humans.
The available data
is sparse presumably due to the difficulties associated with
simultaneously monitoring alveolar and tracheal
pressures
(not mouth)
to determine the pressure-flow character of the
intrathoracic airway.
The data also
tend to be incomplete,
with such critical parameters as lung volume,
tracheal
diameter not reported, and contain great variability.
Most of the data
TLC,
for which
lung volume,
change with
is not available at the
the airway models are based.
the dimensions of the
lung volume to
was found to be valid for
(1972),
the
the
lung volume, 75%
To correct for
lung can be approximated to
1/3 power.
This approximation
lower airways by Hughes et al
but can only be considered a rough approximation for
the trachea and bronchial
airways which are supported with
cartilage rings.
To determine the lung
prediction,
volume correction for
the dimensional
the numerical
airway resistance can be written in
terms of the dimmensionless Moody
type friction factor for the
airway;
8 g f'
8.6.1
Res =
Q
( T
where Dt
Dt2)2
is the tracheal diameter,
and the friction factor is;
Page 145
Del
8.6.2
fI
P
=
P* Re 2
'
The airway resistance at any
lung volume can therefore be
expressed in terms of the calculated resistance at 75% TLC and
the same flow rate by;
- Vol
Res
8.6.3
=
75% TLC
where f''
TLCj
LL
75% L
LV
Res
LV
14/3
75/
Lf''
-
['
is friction factor calculated at the new Reynolds
number due to the change in tracheal diameter for a fixed flow
rate.
Writing the lung volume,
volume, RV,
and vital
LV,
in terms of the
capacity, VC,
the
lung residual
lung volume at 75% TLC
8.6.4
Vol
.75( RV + VC
=
)
is;
75% TLC
lung volume can also be rewritten as;
LV = RV +
8.6.5
Taking
( LV -
RV
)
The actual
the ratio of 8.6.4 to 8.6.5,
and dividing through by
VC gives;
Vol
75% TLC
.75 ( RV/VC +
1
8.6.6
LV
Where %VC
vital
RV/VC + %VC
lung volume in terms of the percentage of
capacity, and
the ratio of residual
lung
volume to vital
capacity is approximately constant at a value of 0.29 for
the
studies below.
Figures 8.6-1
to
8.6-3 presents the numerical prediction
Page
corrected for
lung volume,
and
available; Hyatt and Wilcox,
et al,
1970;
Ferris et al,
146
the four sets of studies
1963;
Blide et al,
1964;
Vincent
The results show excellent
1964).
agreement given the variability in the physiological
data.
The predictions fall within the experimental observations,
but
typically tend
to be near the high end of the data.
this tendency is not very significant,
the comparison with the Weibel model,
small
tracheal
diameter of
1.6 cm.
it is possibly, as with
due to
the relatively
The agreement appears to be
equally good over the entire flow range tested,
L/sec.
The lung volume correction,
approximation,
While
from
.5 to 2
while only a rough
appears to predict the character of the results
well.
A similar approach can be utilized to address the
simulation of pulmonary flows with gasses other
These
tests have proven useful
augment the pressure drops at
than air.
in some clinical settings to
low flow rates,
making
measurements easier and more reliable with conventional
pressure transducers.
The dimensional
groupings which
characterize the flows are the Reynolds number and
characteristic pressure, P*.
increase P*,
flow rate.
Increasing both Nu and g,
while decreasing Reynolds number for a fixed
The effect on the dimensional resistance at a
fixed flow rate can be expressed similarly to equation 8.6-3;
8.6.7
Res
=
Gas X
where the
[
I
Res
air
-
1
ey nl
friction factors are calculated at the Reynolds
Page 147
number given the flow rate,
and density.
airway diameter,
gas viscosity,
Page
IX.
A new experimental
148
Conclusions
technique was developed to measure the
"typical" bifurcation embedded
pressure-flow character of a
within a lung.
The technique was tested on a section of the
entrance region
in a straight tube flow,
Experimental
flow.
errors of
less than 5% demonstrated
the
and can serve as an estimate of
validity of the technique,
the error
a well understood
in measuring the static pressure difference across
a bifurcation.
Experiments utilizing a range of water-glycerin mixtures
allowed the physiologically important Reynolds number range
to be examined;
50
Re
8,000
Analysis of the results indicate that the character of
the flow within the bifurcation is similar
by mechanisms which
-
thin the boundary layer.
Curved tube flow
Entrance flow in a
convergent channel
layer control/growth similarity is drawn
because of the experimental
-
tube
Flow in a two dimensional
The boundary
to flows typified
results
indicating;
Lack of sensitivity to core flow details
-- Turbulence was observed at a Reynolds number of about
1000, but no change in the frictional pressure flow
character accompanied the transition.
--The orientation upstream bifurcations had no effect
on the pressure-flow results in the test
bifurcation.
-
Velocity profiles were characterized by a kinetic
energy correction factor, fa, which was shown to be
Page 149
similar in form and magnitude to flows with boundary
layer growth.
fa =
-
1.05 + 6.54/-/Re
The form and magnitude of the pressure drop due to
viscous dissipation is also similar to the boundary
layer flows;
Del Pv = Del
.556 +
.067
/Re
poisL
The curved tube flow analogy was found to apply best for
the range of "perturbation" experiments performed,
although the
2D convergent channel flow also closely predicted the observed
behavior.
-
Approximately linear
-
Relatively insensitive to shape with;
the use of a hydraulic diameter
the corresponding change in curvature ratio
in L/D
The preference for the curved
channel
tube analogy over the
analogy also results from the similarity
observed secondary flow patterns.
secondary flows,
2D
in the
The existence of the
and the similar scaling with curved tube flows
presents a strong case for
the
influence of curvature.
The
linear dependence in the frictional term was also demonstrated,
both
in
the "perturbation" experiments,
predict the D.B Reynolds
varied from 5 to 9.
(1980)
to be of
experiments
While the curved
preferred slightly over the
and by the ability to
in which L/D's
tube flow analogy may be
2D channel analogy, each is likely
importance, acting together,
and
not as
isolated
effects.
The effects of asymmetric flows on the pressure-flow
results can be approximated as an additional
loss
150
Page
corresponding to
unequal
the loss
in the excess momentum of
the
velocities.
r
*
Del
P
=
- 1
P Re 2
r+1
L
m
2
]2
J
Numerical model predictions of idealized
lung models were
in excellent agreement with published experimental results
-
Geometric pulmonary models
-
Physiological models and casts
Numerical
simulations of the complete human lung also
demonstrated excellent agreement with the available
phyiological data.
The use of an asymmetric
lung model
shown to be important, explaining the observed
pulmonary dependence of gas viscosity at
was
increased
low flow rates.
The
distribution of the viscous pressure losses demonstrated
that
this increase is due primarily to
lower airways.
the MEFV curve
increased resistance in the
This finding helps to explain the utility of
in diagnosing
lower airway pulmonary disorders.
The dimensional scaling shows that this effect occurs at
Reynolds numbers,
not necessarily
low flow rates,
highlights the utility of using gasses other
generate measurable pressures at
low
which
than air
to
low Reynolds numbers flows.
With the ability of these results to predict pressures
resulting from specified flows,
they can be applied
to
the
collapsible tube flow pulmonary models with more confidence.
Additionally,
general
the deeper understanding of expiratory flows
may prove useful
in a wider context,
applications to mass and thermal
with possible
transfer processes.
in
Page
151
References
Akhavan-Alizadeh, R.
'Pressure-Flow Characteristics
Branching Network During Oscillatory Flow'
Masters
M.I.T.,
of a
Thesis,
1980
Bates D.V., Macklem, P.T., Christie, R.V. 'Jespiratory Function
in Disease; An integrated Study of the Lung', @nd Ed.,
Philadelphia, Saunders Pub., 1971
Berger, S.A, L. Talbot, and L.S. Yao 'Flow in Curved Tubes'
ASME Ann. Rev of Fluid Mech 15:461-512, 1983
Boussingeq,
J.
Compt.
Rend,
113,9,49,
1891
Chang, H.K., A.S. Menon 'Air
Flow Dynamics in the Human
Airways' Aerosols in Medicine, Principles, Diagnosis and
Therapy Elsevier Sci. Pub., Chpt. 4, 1985
Collins W.M., S.C.R. Dennis 'Steady Motion of a Viscous Fluid
in a Curved Tube' Q. J. Mech. Appl. Math. 28:133-56, 1975
Dean,
W.R.
Philos.
'The Streamline motion of
Maq.
30:673-93,
fluid
in
a curved
pipe'
1928
Elad, D., R.D. Kamm, A.H. Shapiro 'Model Simulation
Expiration Part 2: Flow Limitation in The Bronchial
submitted to J. Appl Phis, 1987
of
Forced
Tree',
Ferris, B.G., Mead J. and Opie, L.H. ' Partitioning of
Respiratory Flow Resistance in Man', J. Appl. Physiol.,
653-658, 1964
19(4)
Griscom, N.T., M.B. Wohl 'Tracheal Size and Shape: Effects of
Change in Transmural. Pressure' Radi oloqy 149:27-30, 1983
Hardin,
Airways
J.C., Yu, J.C. 'The Pressure/Flow
on Expiration' Biofluid Mechanics
Relation
in
Bronchial
2:39-55, 1980
Hildebrand F.B. Advanced Calculus for Applications 2nd Edition
Prentice Hall Pub.
Horsfield, K., Dart, G., Olson, D.E., Filley, G.F. and Cumming,
G. 'Models of the Human Bronchial Tree', J. Appl Physiol.,
31(2)
207-217,
1971
Hughes, J.M., Hoppin
F.J. Mead,J. 'Effect of Lung Inflation on
Bronchial Length and Diameter in Excised Lungs', J Appl.
Physiol.
32(1) 25-35
1972
Hughes, J.M.B., J.A. Hazel, A.G. Wilson, B.J.B. Grant and
N.B.Pride 'Stability of Intrapulmonary Bronchial Dimensions
during Expiratory Flow in Excised Lungs' J. Appl. Physiol.
37:684-694,
1974
Page
152
Hutchinson,J.
'On the Capacity of Lungs and on the Respiratory
Function with View of Establishing a Precise and Easy Method of
Detecting Disease by Spirometer.
Trans. Med.-Chir. Soc. London
29:137-252,
1846.
Hyatt, R.E., D.P. Schilder and D.L. Fry 'Relationship Between
Maximum Expiratory Flow and Degree of Lung Inflation' J. Appl.
Physiol.
13:331-336,
1958
Hyatt, R.E., R.E. Flath 'Influence of Lung Parenchyma on
Pressure-Diameter Behavior of Dog Bronchi'
J. Airol. Phsiol.
21(5):1448-1452, 1966
Hyatt, R.E., Wilcox, R.E. 'The Pressure-Flow Relationships of
the Intrathoracic Airway in Man', J. of Clinical Inv., 42(1)
1963
Hyatt,R.E.
55:1-8,
'Expiratory Flow Limitation',
J.jAppl.
Physiol.,
1983
Isabey, D., Chang, H.K. 'Steady and Unsteady Pressure-Flow
Relationships in Central Airways' J. Appl Physiol.: Resp. Env.
Exerc. Physiol., 51(5) 1338-1348, 1981
Ito H. 'Friction Factors for Turbulent Flow in Curved
Trans of the ASME, J. Basic Eng. 81:123-134, 1959
Ito,
H.
'Pressure Losses in Smooth Pipe Bends'
ASME, J. Basic Eng. 82:131-143, 1960
Jaffrin, M.Y.,
of the Central
Path.
Respir.
Pipes'
Trans of
the
T.V. Hennessey 'Pressure Distribution in a Model
Airways for Sinusoidal Flow'
Bull. Physio.8:375-90, 1972
Jaffrin, M. Y. , Kessic, P. 'Airway resistance: A Fluid
Mechanical Approach' MIT, Dept. Mech. Eng., Fluid Mechanics
Lab. Publ. No. 72-12, 1972
Jan, D.L., 'A Study of Oscillatory Flow
Bifurcation' PhD thesis, M.I.T., 1986
Through
a
Bronchial
Jones, J.G. , R.B. Fraser, J.A. Nadel 'Prediction of Maximum
Expiratory Flow Rate from Area-Transmural Pressure Curve of
Compressed Airway' J. Appl. Physiol. 38:1002-1011, 1975
Khosla S.
'An Exploration into the Qualitative Nature of
Bronchial Collapse' M.I.T. B.S. Thesis, 1985
Lambert,
R.K.,
Wilson, T.A., Hyatt, R.E., and Rodarte, J.R.
for Expiratory Flow' J. Appl. Physiol.
Computational Model
52:44-56, 1982
'A
Page
Nikiforov,
the
A.I.,
Human Upper
R.B.
Schlesinger
Bronchial
Tree'
153
'Morphometric variab ility of
Respir. Physiol.,
59, 289-299,
1985
Olson D.E.
Fluid Mechanics Relevant to Respiration - Flow
Within Curved or Elliptical Tubes and Bifurcating Systems.
Ph.d Thesis, Imperial College, London, 1971
Patera,
A.L.,
Distribution
Biomech.
Eng.
E.M.
Afify
'An Exper imental Study of Velocity
in a Human Lung
105:381--388,
Cast'
Trans.
of the
ASME,
J.
of
1983
Pedley, T.J., Schroter, R.C, Sudlow, M.F.
Pressure Drop in Models of Human Airways'
9:371-376, 1970a
'Energy Losses and
Respir. Physiol.
Pedley, T.J., Schroter, R.C, Sudlow, M.F.
'The Prediction of
Pressure Drop and Variation of Resistance within the Human
Bronchial Airways' Respir. Physiol. 9:387-405, 1970b
Pedley, T.J., Schroter, R.C, Sudlow, M.F. 'Flow and Pressure
Drop in Systems of Repeatedly Branching tubes' J_._Fluid Mech.
46:365-83, 1971
Pedley, T.J. 'Pulmonary Fluid Dynamics'
Ann. Rev. Fluid Mech. 9:229-274. 1977
Trans.
Pedley, T.J.
pp. 160-234.
The Fluid Mechanics of Large
Camb. Univ. Press, 1980
Potter
J.F
M.C.,
Foss Fluid Mechanics
of
Blood
the ASME,
Vessles Chpt
4,
John Wiley & Sons,1975
Reynolds, D.B., and Lee J.S.
'Modeling Study of the PressureFlow relationship of the bronchial tree (Abstract), Federation
Proc., 38:1444, 1979
Reynolds, D.B. 'Modeling the Pressure-flow Relation of
Bifurcating Networks' Biofluid Mechanics, 2:57-75, 1980
Reynolds, D.B. 'Steady Expiratory Flow-Pressure relationship in
a Model of the Human Bronchial Tree' Trans. of the ASME, J. of
Biomech.
Eng.
104:153-158,
1982
Rivas, M.A. , A.H. Shapiro 'On the Theory of Discharge
Coefficients for Rounded-Entrance Flowmeters and Venturies'
Tran. of the ASME, J. Fluid Mech. 78:4,489-497, 1956
Rohrerf. 'Der Str6mungswiderstand in den Menschlichen
Atemwegen' Pflagers Arch. ges Physiol. 162:225-259, 1915
Scherer,
P.W.
'A
Model
Bronchial Bifurcation'
5:223-229
1972
for High Reynolds Number flow
Trans.
of
the ASME,
J.
of
in a Human
Biomechanics
154
Page
Schilder,
D.P. , A. Roberts and D.L. Fry 'Effect of Gas Density
and Viscosity on the Maximal Expiratory Flow-Volume
Relationships' J. Clin. Invest. 42:1705-1713, 1963
Schlichting,
H.
Boundary
Layer
Theory
McGraw-Hill,
1979
Schroter, R.C., M.F. Sudlow 'Flow Patterns in Models of the
Human Bronchial Airways' Respir. Physiol. 7:341-355, 1969
Shapiro, A.H., R.
Laminar Region of
Siegel,
S.J.
Kline
'Friction factor
in the
the Second U.S.
June, 1954
Smooth Tubes' Proceedings of
National Congress of AMolied Mechanics 733-741
Shapiro, A.H. 'Physiological and Medical Aspects of Flow in
Collapsible Tubes' Proc. Sixth Canadian Congress of Applied
Mechanics Vancouver, 1977
Shapiro, A.H. 'Steady Flow in Collapsible Tubes'
ASME J. of Biomech._Enq. 99:126-147, 1977
Trans.
of
the
Slutsky, A.S., Berdine, G.C., D razen, J.M.
'Steady Flow in a
Model of Human Central Airways'
J. Appl Physiol.: Resp. Env.
Exerc. Physiol., 49(3): 417-423
1980
Snyder, B., D.R. Dantzker, M.J.
Symmetric Cascades of Branches'
Jaeger
J.
'Flow Partitioning
Apl. Phys.
51:598-606
in
1981
Staats, B.A., T.A. Wilson, S.J. Lai-Fook, J.R. Rodarte and R.E.
Hyatt 'Viscosity and Density Dependence During Maximal Flow in
Man' J.
Appl. Physiol.: Respirat. Environ. Exercise Physiol.
48:313-319, 1980
Vincent, N.J., Knudson,
J. 'Factors Influencing
Physiol., 29(2)
West,
J.B.,
P.
Bronchial Tree'
Weibel, E.R.
Verlag 1963
R. ,Leith, D.E.,Macklem, P.T. and
Pulmonary Resistance', J. Appl.
236-243
Hugh-Jones
J. Appl.
Mead
1970
'Patterns of
Physiol.
Gas Flow
14:753-759,
in
the
Upper
1959
Morphology of the Human Lung Berlin:
Springer-
Wille, S.O. 'Numerical Simulations of Steady Flow Inside a
Three Dimensional Aortic Bifurcation Model' J. Biomed. Eng.
6:49-55, 1984
Winter, D.C.,
R.L. Pimmel, B.F. Lewis ' Direct In-Vivo
Measurements o f the Elastic Properties of Canine Trachea
Biomech. Eng., Trans A.S.M.E. 104:311-313, 1982
J.
Wood, L.D.H and A.C. Bryan 'Effect of Increased Ambient
Pressure on Flow-Volume Curve of the Lung' J. Appl. Physiol.
27:4-8, 1969
155
0. 5at
R
=
6 a
Figure 1.3-1
Idealized dimensions of a pulmonary
bifurcation, after Pedley (1971)
156
Flow
Limited
Effort
Dependent
10
Vmax
Increasing
Effort
8
LPS
6
4
~
f
2
0
100
80
60
B
/o VITAL
40
20
0
CAPACITY
Figure 2.1-1
A typical maximal expiratory flow volume
(MEFV) curve, and the effect of expiratory effort, after
Bates et al (1971)
157
Previous Investigat ions
Normalized Frictional Loss in Each Leg of a Bifurcation
102
I
I
I
I
I
I
I I i
I
I
I
-- T
I
I
I
Leend
&
A-A Hardin
Yu (1980)
B-B Reynolds (1980)
F
c-c
Reyno.Lds
(1982)
D-D Reynolds & Lee (1979)
M
0
p4
H
10'
I
I
0
04
........
100
102
I
I
I
I
I
I
- -
I
3
10
.
I
I
I I
10,
Reynolds Number
Figure 2.3-1
Comparison of published frictional
pressure losses for the idealized bifurcation.
158
Figure 3.1-1
Typical four cell secondary flow pattern
in the parent branch downstream of a bifurcation.
/
///
/ /4ZzZ/
/,/
zzg
~zz
Z
dX)
-r (D
(ia
P
)
P (a
1
2
Umax
2
)
//
2
Figure 3.2-1
Velocity profile in a fully developed
laminar tube flow, and the resulting force balance acting on
a section of the tube.
Umax
Figure 3.3-1
Velocity profile in fully developed
turbulent pipe flow.
159
L
An
LA
= 3.5
/D
n
n-i
n
A
=1.2
n
n-1n
Stepwise contracted tube pulmonary model,
Figure 3.4-1
with constant L/D ratio's of 3.5, and area ratio's of 1.2.
r
Figure 3.5-1
Representation of laminar flow
Dimensional convergent channel.
in a 2-
160
Convergent Channel Flow
Velocity Profiles
1.6
Re
-
312 -U~h/lv
719
1.4
1509
3407
1.2
5262
-
1.0
ca
8
6
.2
0
0
.5
1.0
1.5
Angular
2.0
2.5
Position
(5
3.0
3.5
4.0
4.5
Deg Channel)
Figure 3.5-2
Computed velocity profiles in a 2-D
convergent channel flow for a range of flow conditions.
5.0
161
Outside
of Bend
Inside
of Bend
Figure 3.6-1
curved tube flow.
Secondary flow pattern in fully developed
ZDG
4C
0C
Figure 3.6-2
Generation of the four cell secondary flow
pattern typical of expiratory flows from modeling a
bifurcation as two curved tubes oriented back to back.
162
S=RO
R
>
Cr
0
Figure 3.6-3
curved
tube flow.
Toroidal
coordinate system
used
to analyze
163
Friction Loss in a Curved Tube
Dean Solution
1.025
Del P/Del P Poiseuille
1.02 F
1.015
F
1.01 I
1.005 F
I
1
1
I
~L-t<IIII
I
10
1111111
100
Dean Number
Figure 3.6-4
Prediction of the pressure loss in a
curved tube, normalized by th e Poiseuille loss at the same
flow rate from the theory of Dean (1927).
1
1.0
/HASSON
CORRELATION (1955)
*
0o
5040
= 0.0969K" 2
0/
+ 0.556
s
100,
200*
,
648 250
0.50
.0,
R/o
z2050
00
p
1/4
2- .47136K
0%
0
2
3'
4
l0o, 0 K
Friction ratios for various curvature ratios. (, White (1929); 9, Adler (1934): A,
t6 (1959) (K. 2D = (2ao/i')(a/R)1 2 ). From Van Dyke (1978).
Experimental results for the pressure loss
Figure 3.6-5
in a curved tube, normalized by the Poiseuille loss that
would occur at the same flow rate, after Berger et al (1983)
165
COMPARISON OF EFFECTS
Del P/ Poiseuille Del P
I VV
Band of Bifurcation
Results
10
1
100
1000
*
C. TUBE
ENTR.
*
TURB. C.T.
2-D Chan.
10000
+
TURB. S.T.
Figure 3.9-1
Pressure loss correlations for curved
tubes, entrance flow, 2-D convergent channel flow, and
turbulent flows in straight and curved tubes
166
Subtraction Technique
Schematic
Plenum P,
Y
D=.75 Dr
L= L.67 Dr
L= 1.75 Dr
Dr =2.47 cm
Static Pressure
Drop
Test Bifurcation
0
ps =(Pu-PO)=(PR-PO)-(P -P)
Schematic representation of the
n4. 1-d
Fibtre
od
met
ion
or symmetric flow conditions.
subrac
10
00
10
I
-0,2
c
.
20
0-90'
R/r a3,7
Ro- 2x10
4
30
0
PIEZOMETER READINGS
-X-----
OUTER SIDE
+---
INNER SIDE
--- e--- TraP
~0
------- BOTTOM
-0,61
h
-
(v t/2g)
04
GRADIENT IN
STRAIGHT PIPE
-1,0
-1,2
-1,4
DOWNSTREAM TANGENT
-
UPSTREAM TANGENT-l,
Figure 4.1-2
bend between two
BEND
Pressure distributions owing to a 90*
long straight tube, after Ito (1960)
pipe
168
Quasi - Steady Test Apparatus
Iese
d (P/ g)
dt
i(/
Water /Glycerin
7/
IGate
/
/
Weir
/S
/.
/
Zj
/'
I
-I
Collection
F ump
test
Tank
Return
Loop
Schematic of the quasi-steady experimental
Figure 4.4-1
apparatus and pressure measurement locations.
169
Kinematic Viscosity of Glycerin
Data from C.R.C.,
20c
Kinematic Viscosity - Cs
100
10
I
0
I
I
I
20
40
60
80
100
% Water Content
Figure 4.5-1
Kinematic viscosity of water-glycerin
mixtures as a percentage of water content, from C.R.C.
170
Viscosity
Temperature
-
Measured with Haake Viscometer
Kinematic viscosity
cm-2/s
0.7
a
0.6
S
U
U
3
CRC Value
U
No
0.5
.
20c
U@
0.4
Measure Viscosity c
U
79% Water
-
21% Glycerin -
0.3
0.21-
0.1
10
I
I
iI
I
Ii
Ii
I
12
14
16
18
20
22
24
Deg C
(
Temperature
)
0
Figure 4.6-1
Measured kinimatic viscosity for a 21% by
water glycerin-water mixture as a function of temperature
using Haake viscometer, and C.R.C. value.
171
Calibration Errors
cm H20
0.06
Error in Fit - cm
0.04
4
0.02
0
--
-0.02
-+
-0.04
-0.06
0
20
40
60
80
100
120
140
Manometer Reading -cm
"Res.
-+/-
Height
+
Differ. Head
Dig. Error
Figure 4.6-2
Typical errors between tank level
calibration manometer readings and predicted levels based on
the curve fits and output voltages.
172
Bellmouth Entrance Geometry
R=D
:-
L
TD
-
10*D ---
Exhaust
Tube
Tank
Bellmouti
Collar
L
Reservoir
Wall
Plenum
Figure 4.8-1
Entrance test experimental bellmouth and
straight tube geometry.
173
Reservoir Tank Level vs. Time
Short Entrance Test
160
I
I
I
I
,
I
,I
I
I
I
,
I
I
I
140
Vise.= 0.125 ern*2/sec
-
120
100
80
Vise.= 0.475 (m**2/see
60
LU
(n)
LU
40
20
0
-20
i
0
20
I
I I
40
I
60
I
I
I I
80
100
SEC0NOS
TIME -
i
l I
120
iI
Ii
140
160
180
Tank level vs time measurements for the
Figure 4.8-2
short entrance test (L/D = 6.3) at kinematic viscosities of
0.475 and 0.125 cm 2 /sec
174
Differential Height Signal
Typical Asymptotic Region Data
.030
-.025
.020
.015
M:
U-
I III II IiII
IIII11
1, 1
L
.010
A
uJ
-E
.005
0
-. 005
i i i ii i
.010
'
45 5
t
'
i
460
'
i'
465
ii'
i
470
III
'
''
'
ii
475
480
OPTR POINT INDEX
'
485
'
490
495
500
Asymptotic region of the differential
Figure 4.8-3
pressure signal, showing a small offset error on the order
the digitation errors.
of
175
Error in Curve Fit
Curve Fit Value - Data Value
.06
II
II
I
I
I
.05
.04
.03
Digitation Error Level
I-t
.02
1:
.01
L.
0
U-
uLJ
.01
.02
uLJ
.03
Digitation Error Level
4
-
-. 05
-. 06
07
10
'
20
'
30
40
.-
50
TIME
60
-
,
70
80
,
90
,
-.
100
110
SECONDS
Figure 4.8-4
Error between the curve fit and the actual
measured tank level, note that the magnitude of the error are
typically around the digitation level, and not systematic.
176
Calculated Reynolds Number vs Time
Short Entrance Experiments
9000
I
I
I
I
I
I
I
I
T-
1
I
I
I
I
I
I
I
I
8000
7000
Vise.= 0.125 cm**2/sec
6000
U)
5000
4000
LU
3000
2000 ~
Visc.= 0.475 crn**2/rec
4
1000
0
10
20
30
40
50
60
70
80
90
100
110
TIME - SECONDS
Figure 4.8-5
Calculated Reynolds numbers vs time for
the short tube entrance test with kinematic viscosities of
0.125 and 0.475 cm 2 /sec.
177
Quasi-Steady Ent rancq
xperiments
Static pressure loss normalized by P*-pv /D , vs Reynolds number
108
L/D-9.8
10 7
IN
CC-
106
CC
LU-
10S
Legend
A-A L/D-9.8
v-0.478 cm2/~/sec
%j
V-.12 UM2 /sec
C-C L/D-6.3 v-0.475 cm2 /sec
D-D L/D-6.3 v-0.125 cm /see
E-E Difference in Results
B-B
I
'
104 1
10 2
L/ID-W.0
1 03
II
II
10,
REYNOLDS NUMBER
Figure 4.8-6
Combined dimensionless pressure vs
Reynolds number for both entrance tests and the subtraction
of the two, and theoretical predictions.
178
Entrance
Result.
Theory & Experimental
Static pressure loss normalized by P*-pv 2/D 2 , vs Reynolds number
For the entrance section of a tube from L/D-6.3 to L/D-9.8
107
-
106
0-
105
Legend
A-A Experimental
Data
B-B Entrance Theory
104
-
-'
'
'
'
'
'
'
i
' ':
10 3
102
:
i
i i i
10'
REYNOLDS NUMBER
Figure 4.8-7
Results for
length of tubing and theory
the
loss
in the additional
179
Model Bifurcation Construction
Aluminum
Collars
Airway
I
Stycast
Alumiuum
Collar
Cross-section of a typical bifurcation
Figure 4.9-1
showing the cast section and aluminum coupling collars.
180
Momentum Analys is
Control Volume
,-Force Plate
F
Restraining
Force
Ai rway
Hydrostatic
Pressure
Hydrostatic
Pressure
wall
Flow Exits Normal
To Control Surface
Figure 5.1-1
Momentum analysis of an exiting
impinging on a flat plat and turning 900.
jet
181
Figure 5.1-2 A)
Jan, Re - 200
.000
Figure 5.1-2 B)
1060
Station 2S, Re
L
0.2
0.7
1.6
1.8
0
dorsal
&
,
&
Figure 5.1-2
Axial velocity profiles used to calculate
the axial correction factor fa. A) Jan, Re=200 ;B) Chang
Meno n, Re= 1060; C) Chang & Menon, Re=5712; D) Chang & Menon
Re=6000; E) Chang & Menon, Re=681; F) Hardin
Yu, Re=2760;
G) D.B. Reynolds, Re= 2260, Re=8000 and Re=12200
182
A
L
4
M
5
p
9
Figure 5.1-2 C)
Station 9, Re -
1.5
1.5
5712
0
-1.0
0
1.0
-1.0
0
Figure 5.1-2 D)
0
r0 R
1.0
1.61
Station 4, Re - 6000
0
0.81
L-M
A-P
01
-1
00
Figure 5.1-2 E)
Station 5, Re - 6f
1.5
__
1
1.-
-1.0
0
rJR
.
r|R
0
0
0
1.0
-1.0
0
r~
/R
rR
..1
1.0
183
o
Figure 5.1-2 F)
Hardin & YU
Re
2760
INPLANE OF BIFURCATION
o NORMAL TO PLANE OF BIFURCATION
VELOCITY U. m/s
2. 5 r
PP
0
= 2760
I
2.
1.5
1.0
0.5
Figure 5.1-2 G)
D.B. Reynolds
Re - 2260
I
I
-
~I I I
L
-16 -14 -12 710 -8
EDGE
I
-6 -4 -2
1 2 4 6 8
DISTANCE FROM CENTER. mm
2.0
Re = 8000
Re = 12200
10 112 14
EDGE
0
R9 1 =12200
o
Re m 8050
X
Re, = 2260
16
Il
1.5 F
87E
.2
0
0
E
I
*
I,
I,
1
p
I
II
it.I
0.5
rI
II
0
1.0
0.5
0
Radial position, r/r,
0.5
1.0
184
Fa vs Ba
Experiments and Estimation
Fa
2
Poiseuille
Flow
1.8
1.6
Fa =3Ba
-2
1.4
1.2
Turbulent Flow
1
I
-I
I
1
1.3
1.2
1.1
1.4
Ba
9
Experimental Data
a
as calculated from
Cross plot of fa and
Figure 5.1-3
the numerical integrals of published velocity profiles, and
approximate relation.
185
Fa vs. Reynolds Number
Fa
2
1.8 P
Curve Fit
1.6
1.05
+
6.54/Sqrt(Re)
U
1.4
mu
U
1.2 F
U
U
a I
1
100
I
~ A .. I LiL I I
---I--
1000
Experimental Data
Figure 5.1-4
vs
tracheal
Results of the numerical
Reynolds number,
and
curve
fit.
integrals of F a
186
Boundary Layer Appoxima t i on
2a
Uo
\\
Figure 5.1-5
77777777
Approximate flow profile used to estimate
the mean flow rate and
thickness.
fa factor
based
on boundary
layer
187
Fa Comparison
Experiment and Theory
Fa
2
Flow Type
1.5
Entrxance, L/D = 3.5
2-D Channel, alpha = .052
Curved Tube, Curvature Ratio = 1/7
Entrance, L/D = 1.75
1.6
1.4
Data
1.2 -Experimental
100
1000
10000
10000
Reynolds Number
Figure 5.1-6
Prediction of fa based on the boundary
layer approximation for an entrance prediction, at L/D=1.75
and 3.5, curved tube flow, 6r=1/7, and 2-D convergent channel
flow, x=.05.
188
L
Figure 5.2-1 A)
Figure 5.2-1 B)
dorsal
L
p
A
0.1
0.5
H+
R
P-
dorsirl
R
A
0.1 0.5
Figure 5.2-1 0)
I
'
~
/
-
,..
I.-
\ IL'
/
Tx
,
\
II
/
\\
.\
/~x
~
-
-
-
I
A
I
I
~.
I
/ ~
,.
.~
\
-
Secondary velocity profiles used to
Figure 5.2-1
numerically evaluate the secondary velocity correction
factor, fs; A) Chang & Menon Re=1060, @station 3, B)Chang
Menon Re=106O @station 2s, C)Jan Re=200.
&
-
\ tI I /
-
\
/
iiI
II
189
Diametric
Pressure
Variat
ion
106
2
*
AP
.06
P Re
105
10
103i
10 2
-
-
103
10,
Reynolds number
Normalized pressure difference measured
Figure 5.3-1
between the top and side of bifurcation leg during expiration
vs local Reynolds number.
190
Reservoir Level vs Time
160
140
Generations 4
120
100
u-O
LU-
Generations 4
Nil = .125
-
U
-
1
Nu = .475
-
80
2
-
-
X:
-
Nu = .125
Nu = .475-
60
CD
LU-
40
20
-20
0
.
0
20
40
60
80 100 120 140 160 180 200 220 240 260 280 300 320 340
TIME - SECONOS
Experimental results for the reservoir
Figure 6.1-1
level vs time for the experiments, Gen's 4-2, Nu=.475; Gen's
4-2, Nu=.125; Gen's 4-1, Nu=.475; Gen's 4-1, Nu=.125.
191
Planar Bifurcation Results
Static pressure loss/P*tr vs tracheal Reynolds number
108
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
Ul1~
7',"o~oz
/
107
/01 A106
0-.
a:
-LJ
C3
x",
10
IAZ11-A-A 4 Gen.
10'
Leg~end2
Network, v-0.476 cm2 /sec
B-B 4 Gen. Network, v-0.125 cm2 /sec
C-C 3 Gen. Network, v-0.497 cm 2 /seo
D-D 3 Gen. Network, v-0.125 cm /seo
E-E Drop Across Test Bifurcation
103 11
10
i
i
i
i
.
. ,, .
..
102
.
. . . . I
103
101
TRRCHERL REYNOLDS NUMBER
Note: P* and
bifurcation.
Reynolds
number
refer
to
tracheal
values
in the teE
Figure 6.1-2
Subtraction of the three generation
"upstream" airways from the four generation airway to give
the normalized static
pressure loss in
the test
bifurcation.
192
Pressure Drop Components
I I I I
I
-
101
.
I I I I I
I
I
I
I
V v
16P
1/2 p V 2
LiPs
100
APv
A Pk
'
'
10~1
10,
I
........
'
I
1n
2
103
10,
Tracheal Reynolds Number
.
Figure 6.1-3
Decomposition of the total static
pressure loss across a bifurcation into a kinetic energy and
frictional component normalized to '4P*Re2
11,11
193
Symmetric Bifur cat ion Results
Frictional pressure loss/Poiseuille loss vs Reynolds Number
10
cE
Legend
A-A
3 Planar Upstream Airways
B-B
2
Planar Upstream Airways
C-C 3 Upstream Airways, Config. A
D-D 3 Upstream Airways, Config. B
E-E 3 Upstream Airways, Config. C
cr
Uij
-0
100
'
101
'
'
i i
103
TRRCHEAL REYN0LOS NUMBER
102
i
,
101
Figure 6.1-4
Frictional pressure loss across a single
bifurcation normalized to the Poiseuille
loss for the
symmetric flow experiments.
194
4
3
Bifurcation-Bifurcation Oreintation
Bifurcation
Number
Series A
_
_
_
Planar
1
VI
2
VV
_
Series B
_
Series C
_
__
90 Deg. Planar
_
90 Deg. Planar
_V
90 Deg.
_V
3
4
Figure 6.3-1
flow experiments.
Configurations for
the nonplanar symmetric
195
Results for Additional Parent Length
102
I
10'
I
I
I
I
I
I
I
I
I
I
I
I
:
t
Airway with
Additional Parent Tube Length
Airway Alone
04
100
10-1
Loss in Adiitional
Parent T Tbe Length
10-2 1
10 2
I
I
11111,
i
a
10
a
I
A
7
a
a
a
10,
Reynolds Number
Subtraction of the three generation
Figure 6.4-1
"upstream" a i rways from the three generation airway with the
additional tracheal length of L/D =3.5.
196
Pressure Loss in Additional
Parent Tube Length
10
Del P/Del P pois.
Experimental Results
+1-
- I I I r..I'IHlIt ILLE
1
1
10
I
i)
5% Band
I I. I I
I
100
1000
10000
Reynolds Number
Prediction Form
Curved Tube .......
Entrance
Figure 6.4-2
Frictional pressure loss in a straight
tube, L/D =3.5, downstream of a bifurcation,
normalized
the Poiseuille loss in the equivalent
length of tubing.
to
197
Symmetric Bifurcation Results
10
Del P/ Del P pois.
Experimental Results
+/-
5% Band
I II
1
10
1000
100
10000
Reynolds Number
Prediction Form
-
Curved Tube ....... Entrance
Figure 6.5-1
and
2-D
results.
Compar isnn
convergent channel
flow
-
2D Channel
of curved tube,
to
entrance theory
the band of symmetric flow
198
#1
#3
#2
-.
#4
#5
Figure 6.6-1
Photographs of the oscilloscope traces
from the hot wire probe, not transition to turbulence appears
to occur near Re=1000.
199
Pexb
m
PeI(local)
PpI
-- Pe I(b r)-
1
Palv
I
.Pib
BRONCHUS
PARENCHYMA
"Schematic representation of bronchial lumen,
bronchial wall, peribronchial space, limiting membrane
(1m), and surrounding alveolar tissue.
Intrabronchial
(Pib), extrabronchial (Pexb), alveolar (Palv), and pleural
(Ppl) pressures are indicated as well as elastic recoil
pressure of bronchus [Pel(br)] and alveoli locally
[Pel(local)].
Bronchial narrowing during expiratory flow
(indicated by arrow) is illustrated by lighter shading of
bronchial wall."
Figure 6.7-1
Schematic representation of the bronchial
support structure and pressure forces, after Hughes (1974)
200
4
-
3
2-
*
7
6
X
ONI
1
0
S
4
3
2
DISTANCE (mm)
Bronchial area in cross-section (from stereoscopic measurements) at constant lung volume under static [Pib +10 cmH!O (outer
ring)) and dynamic [Pib -10 cmH1O (inner ring), conditions. Note
concentric narrowing. Alveolar pressure +10 cniH-O in both insances. Numbers indicate measurement sites.
z
6
5
C
E
-
LU
42
2i
U
I
0
1
2
-
-
I
n
3
4
5
6
7
DISTANCE (mm)
Bronchial cross-section (from stereoscopic measurements)
approximately
at different lung volumes under static conditions. Noteflattening
in the
some
circular shape at Pib +10 and 0 cmHsO with
cmHtO).
-10
(Pib
lung
collapsed
and
compressed
Figure 6.7-2
cross-sections
inflations.
Stereoscopic measurements of bronchial
at various
transmural
pressures,
and
two
lung
201
Semi-Circular Airway Results
10
Del P/ Del P pois.
Experimental Results
5% Band
......
......
.....
..........
..........
...
.......
..
..
..
.....
.......
.....
........
....
..........
. . . . ............
...
.................
. . .....
. . . .......
.
I
1
100
I
I
_______
I
I
I
I
I
1000
llI
10000
Reynolds Number
Prediction Form
Curved Tube ....... Entrance
Figure 6.7-3
circular
Normalized
test bifurcation
--
frictional
2D Channel
loss
in
the semi-
202
0. 66D
2.5D
45 Deg
I
t
D
25 Deg
0 .875D
5D
Figure 7.1-3
in the
Most
middle airways,
typical
after
asymmetric bifurcation model
Olson
(1971)
203
Quasi-Steady
Test
Apparatus
(Modified for the Flow Partition Experiments)
dt
dC
/
ater/Glycerin
/
Q=AR
d (P/ 'g)
iAesvOi'q
' //
A<
Gate
/o
Weir
#2
#1
Me te r
Col lection
Tank
9LAPS
#3
'ump
Return
LoUop
#4
Experimental apparatus for the flow
Figure 7.2-1
Note separate fluid circuits.
partition experiments.
204
V2,
P 2, A2
Control Volume
P3 3-P2
V3, P3, A3
V1,
P1, Al
Pt,
Control volume selected to analyze the
Figure 7.3-1
bifurcation with asymmetric flow
a
pressure loss in
conditions.
205
Static
Pressure
Drop
Aoross the Vaiable Flow Branoh with Asymmetrio Flow
I
-
.9
I
.8
Id
.7
.6
.5
f-.
p.
.4
-I
Il
.3
.2
.
Legend
A-A Re2 - 4690
B-B Re2 - 3980
C-C Re2 - 3014
D-D Re2 - 2280
E-E Re2 - 1506
F-F Re2 - 662
T
0
I
10
~1
-1
I
I
I
I
I
I
I
I
100
100
I
I
I
I
I
I
a
I
I
I
I
I
I
I
I
I
101
Gamma - F - Q1/Q2
Figure 7.4-1
Results for the static pressure loss
across the variable flow leg of a bifurcation, for six
different fixed flow rates in the other leg vs flow ratio,
F.
206
Pressure Drop Components
(For Fixed Flow Giving Re2-1506)
2.5
Dissipative Pressure
Loss Component
2.0
1.5
1.0
01%
*e
.5
*e
I-.
Statio
0
Pressure
-. 5
Kinetia Energy
Component
-1.0
-2.0
-
-1.5
10 3
10'
Trachel Reynolds Number
Figure 7.4-2 Decomposition of the static pressure loss
across a bifurcation into a kinetic energy component vs
tracheal Reynolds number. (fixed daughter Reynolds # of 1506)
207
Flow Partition Resul t s
-
A
__-
-
1.1
Excess frictional pressure loss/(1/2pV2 ) vs Flow Ratio
1.0
.9
Momentum Excegs Logg
.8
2
.05
r+
.7
Excess AP
1/2pV 2
.6
.5
.4
.3
--
:-op
.2
-
Laend
A-A Re -4390
01
D-D Re -2280
E-E Re 2 -1506
0
-
B-B Re -3721
C-C Re.-3014
F-F Re 2 -662
-. 1
10-I
I IJI,I
t I I I II
I
100
Flow Ratio
Figure 7.4-3 Results for
the mixing of two streams vs
r
101
r - Re 1 / Re 2
the excess pressure loss due to
(flow ratio)
208
Momentum Analys is
(For Equal Daughter Tube Diameters)
U
2
Control Volume
U
32
U
3
U
31
P 3p,
3
3
U
-
P
3
P
L
32
31
U
U
32
J
,
r
-
r
+ 1
2
2
14U
3
+ U
31
U 3
2
32
31
=
3
U 3+
L31
2
-U
U
2
=
[
-
1U
=
3
2
2
2
32.
Momentum analysis of the maximum pressure
Figure 7.4-4
recovery due to the unification of two unequal flows.
209
Comparison of Investigators Results
Viscous Pressure Loss in an
Idealized Bifurcation
Delt.a P
100 Delta P/Poiseuille
10
1
10000
1000
100
Parent Reynolds Number
~~--
Exp's
Hardin & Yu
Reynolds 1980
Reynolds 1982*
Reynolds & Lee 1979
Figure 8.1-1
Comparison of the results for the friction
pressure loss in an idealized bifurcation for this and
previous investigations.
210
D1
Q1
D3
~
Q3
~~~
D2
Figure 8.2-1
bifurcation
used
Geometry of a typical
for
analysis.
asymmetric
211
Pressure-Flow Results
Experiment vs. Prediction
(Idealized 3 Gen. Airway Network)
10
Systemic Delta P/(KE)
Numerical Model
Prediction
Experimental
Range
1
1000
I-IIIII
10000
"Tracheal" Reynolds Number
Figure 8.3-1
Comparison of numerical prediction, and
experimental results of Hardin & YU, for the static pressure
drop across an idealized three generation pulmonary model.
Pressure-Flow Results
Pressure-Flow Results
Experiment vs Prediction
("Y" Connectors, L/D=5)
Experiment vs Prediction
("Y" Connectors, L/D=9)
Delta P/(KE)
Delta P/(KE)
10 ,-
10 r
Numeric al
Prediction
r)
Numerical
Prediction
Experimental Range
1
-I _
I
Experimental Rainge
I
I
I
1000
I
10000
"Tracheal" Reynolds Number
1
1000
10000
Tracheal Reynolds Number
Figure 8.3-2
Comparison of numerical prediction, and
experimental results of D.B Reynolds, for the static pressure
drop across three generations of rigid "Y" connectors.
213
Zavala Lung Cast
TRACHEOBRONCHIAL TREE
B. lit Division
-
Mainstem
Right
Bronchus
{Apical
Right
Upper
Lobe
B
Posterior B2
eg
b
Segment
Antero
pial
131+B2 Poterior
b
c
Segment
B. it Division
Left Mainstan
93
Segen
Bronchus
Left
Upper
Lobe
Upper Division
Lateral
Segment
a
84
{
Segment gS
6
B.chus
b
A.
assal g.
Right
Intermediate
Left
Main
Bronchus
b
D. Left Ineaer
b
BronchuLos
(aBronchus
C
Lower
Lobe
Superior
Segment
Infior
b
Segment
Be Supero
0. Rih nferior
b
Basal Segment
De iAntir
of-Med
t
bb
c
Figure 8.4-1
BasalSeel.
h
M
am Pk.
WMM"4 MA
bE
courtesy
s Iu
Segment
-.
Subsuper8or
Posterior 810
Saul Seg.
.4
Subsuperior
Basal Sag. Be
Basal ".
Dmsion
ao
Medial B7 ab
b
Anterior
Right
n
Mainstemr
gC.
Superior
Sasna
Sell
i.
Right
b
/
Right
Middle
Lobe
b
m7"
b
.2223
ceBe$&[
BBasal Se.
Bepo
Latera
Basal S
B1
strr
seg.
Depiction of the Zavala lung model,
of Medi-tech, Water town Ma.
Left
214
Zavala Lung Model
"Dichotomous'
Branching Pattern
9
50
48
24
62
51
5
63
61
44
12
5
22
45
20
4
3
4041
55
2
2
3
3115
3
60
64
6
23
6
46
54
54
8
39 74
26
52
2
53
0
42
68
14
75
19
3
34
35
69
33
66
70
76
28
77
67
36
72
56
73
2
58
71
Figure 8.4-2
approximation for
47
37
16
43
17 38
Dichotomous branching
1 ung model.
the Zavala
pattern
7
215
Zavala Lung Cast, Results
Experiment vs Prediction
P/KE
100 System Delta
Model Prediction
Horsfield Flow Distribution
Uniform Pressure Flow Dist
10
-
1
100
Experi me nta
Results
1
1000
Tracheal Reynolds Number
10000
Comparison of numerical prediction, and
Figure 8.4-3
experimental results of Isabey and Chang, for the static
pressure drop across the Zavala lung model.
216
Zavala Lung Cast Results
Pressure Drop Components
100
Pressure Drop/(KE)
Uniform Pressure Boundary Condition
Static Pressure
/Drop
10
1
Kmnetic Ener y
Comp onenT
"Viscous" Loss
Component
0.1
Mixing Loss,
Compon ent'
0.01
100
1000
10000
Tracheal R1eynolds Number
Pressure drop components which comprise
Figure 8.4-4
the calculated static pressure drop across the Zavala lung
model for uniform pressure upstream boundary conditions.
217
Zavala Lung Cast,
Resistance Mat.rix
Perspective View
Rij
i=17 -'
17 < BPS # < 33
j= 17
i=33 _-'\j= 3 3
Horizontal View Along Diagonal
Resistance Matrix
Rij = DPi/DQj
Figure 8.4-5
Zavala
lung model.
Calculated
Resistance matrix
for
the
218
Flow Distribution Results
Zavala Lung Cast
Flow Percentage
765-
F2
El
4-
I
32-
1
-
0
II
39
jil
I I I
42
45
I
I
I
48
I
I
I I
51
I
54
I
I
I
57
I~I
I
I
60
III
I hi1ii
I1
63
--r- I
66
11
69
72
75
Terminal Branch Number
LID
H&C
C
Re=100
LIM Re=4k
E
Re=10k
Figure 8.4-6
dA comparison of the flow distributions for
the Zavala lung model.
219
Asynietiric
Lung Model
Horsfield et al Model 2
Central Airway Branching
BPS #31
BPS #24
BPS #30
BPS #25
BPS #20
III's V/22
BPS#2
BPS #23
BPS #26w
BPS #18
BPS #17
BPS #21
BPS #19
BPS #32
BPS
#3:3
BPS #28
BPS #29
Bronchopulmonary Sepmne n t B ranr rhinar
Se ff ria e n t Brq-riql-iiin
Orde r N
Order N-I
Figure 8.4-7
according to the
Ord er N-M
(3 < M < 5)
Asymmetric branching pattern for the
Hc rsfeild et al (1971) lung Model 2.
lung
220
Pressure--Flow Results
Experiment vs. Prediction
(Human Lung Cast Below R.I.B.)
Delta P/KE
100
Numerical Model
Prediction
10
Experimental
Range
I
I
I
I
L..
L
I II
L..
L.I
I
I
1000
100
I
I
II
II
II
10000
Reynolds Number at R.I.B.
and
Comparison of numerical prediction,
static
the
for
Lee,
experimental results of Reynolds and
pressure drop across a pulmonary cast below the right
intermediate bronchus.
Figure 8.4-8
221
Pulmonary Pressure-Flow Prediction
Resistance and Friction Factors
100 DP/KE
Res. cm H20/(L/S)
Horsfield et al Model
Weibel Model
10
1.2
1
0.8
- 0.6
0.4
- 0.2
1
100
-LI
0
1000
10000
Tracheal Reynolds Number
Figure 8.5-1
Comparison of the overall pressure-flow
characteristics for the Weibel and Horsfield lung models.
222
Pulmonary Pressure Drop Compone
Horsfield Asymmetric Pulmonary Model
Delta P/KE
100
Static Pressure
Drop
10
"Viscous" LosS
Component
Kinetic Energy
1
Mixing Loss
-
0.1
100
Component,
1000
10000
Tracheal Reynolds Number
Pressure dr-op components which comprise
Figure 8.5-2
pressure drop across the Horsfield lung
the calculated static
Horsfield & Cunining Model 2
Distribution of Number of Generations
% of B.P.S. Paths
/
z
z
I
1-
I
0.8-
I
/
1.2
0.6
0.4-
K
L
0.2-
0-
1 I~I 1 1 I 1 I
-
1
I
223
I
I
I
I
s
-
I
I
I
I
I
I
I
I
I
I
1
1
15 16 17 18 1920212223242526272829303132
Number of Generations into the Lung
"Short Path"
"Long Path"
Percentage of "long" and "short" paths
Figure 8.5-3
terminating at a given number of bifurcations into the lung.
Weibel Lung Model Results
Horsfield et al Lung Model Results
Static Pressure vs. Generation
Average Static Pressure vs. Generation
Static Pressure/KE
20
Avearge Static Pressure/KE
20
Flow Rate
e .125 L/S
Flow Rate
0.125 L/S
.25 L/S
0 .5 L/S
-1
L/S
'4k .25 L/S
0 .5 L/S
I L/S
I
L/S
10
/
-+
10
S
2 L/S
.....
5
0
0
0
5
10
15
Generations into the Lung
20
0
5
10
15
20
25
Generations into the Lung
Static pressure vs generation into the
Figure 8.5-4
and Horsfield lung models for
Weibel
the
both
lung for
rates.
several flow
30
35
Weibel Lung Model Results
Horsfield & Cumming Results
Viscous Pressure Drop vs. Generation
Viscous Pressure Drops vs Generation
Viscous P Drop/KE
2
1.6
Average Viscous P Drop/KE
1.4
1.51
1.2'
Flow Rat e
Flow Rat
,.125 L/S
'a .25 L/S
.5 L/S
41 L/S
* 2 L/S
1
0.8
A\
/
N)
- .125 L/S
'
0.6.
.25 L/S
S.5 L/S
2L/S
2
0.4'
0.5
0.2
A
0
0
5
10
15
Generations into the Lung
20
0
5
10
15
20
25
Generations into the Lung
Normal ized viscous pressure drop vs
Figure 8.5-5
generation into the lung for both the Weibel and Horsfield
lung models for several f low rates.
30
35
226
Horsfield & (irminig
Model
Viscous PrCssUIre Loss Distribution
Flow Rate = 125cc/See
Flow Rate
Viscous Pressure Drop/(KE)
1.4
1
1.6
= 250cc/Sec
Viscous Pressure Drop/(KE)
1.2
1.4
I Std
Average +/-
A
1
Average +/-
Std
1.2
0.8
1
0.8
...
..-..
-,
0.6
0.6
1AV
0.4
0.2
02
/.
0
4
7
10
13
18
19
22
25
Ed
1
0
4
7
10
'
.'...'..'..
13-
16
19
2U
2
26
Generations into the Lung
Generations into the Lung
Flow Rate = 500cc/Sec
Flow Rate = I L/Sec
Viscous Pressure Drop/(KE)
1.2
31
Viscous Pressure Drop/(KE)
1
+/- 1 Std
0.6
0.8
0.4
0.4
0.2
0.2
0
10 t413 16 19 E 26 B
Generations into the Lung
7
I Std
Average +/0.0
0
31
i
4
i
7
10
13
18
1
22
i
26
-
Average
0.8
28
31
Generations into the Lung
The variation of normalized viscous
Figure 8.5-6
pressure drop vs generation into the lung for Horsfield
models for several flow rates.
lung
227
Horsfield & Cumming Model 2 Results
Viscous Pressure Drops vs Diameter
1.6
Average Viscous Pressure Drop/KE
1.4
F low Rate
.125 L/S
.25 L/S
*
1.2
1
.5 L/S
1 L/S
2 L/S
0.8
0.6
0.4
0.2
0
0.01
IL
0.1
I
iI I
1
Average Airway Diameter cm
Figure 8.5-7
Normalized viscous pressure drop vs
average airway diameter for the Horsfield lung model at
several flow rates.
228
Diameter vs. Generation
2
Diameter (im)
1.6
H&C Model 2 Data
I
(Average +-I-
1
Std.)
Weibel Model
0.6
30
25
20
15
10
Genertationu into the Lung
0
Comparison of Lung Models
L/D Ratio vs Generation
Length/Diameter
6
H&C Model 2 Da ta
(Average
I/- I Std.)
-t
3
Weibel Model
-
0
0
5
10
15
20
25
30
35
Genertations into the Lung
Average and variation of the diameter and
Figure 8.5-8
L/D ratio vs generations into the lung for both the Weibel
and HOrsfiels lung models.
Viscous Pressure Drop Factors
vs. Generations into the Lung
(Low Reynolds # Form)
Viscous Pressure Drop Factors
vs. Generations into the Lung
(Large Reynolds # Form)
Factors
Factors
4
Diameter Ratio Factors
3
Diameter Ratio Factors
N,
N,
In
L/D Ratio
Factor
2
L/D cra~
1
Flow Ratio Fa 2tor
0
5
10
15
A
I
I
III
0
Flow Ratio Factor
20
25
30
Generations into the Lung
Estimates based on averaged data from
Horsfield et al
35
0
5
10
15
20
25
30
Generations into the Lung
Estimates based on averaged data from
Horsfield et al
High and low Reynolds numbers Ratio
Figure 8.5-9
factors vs generation for the horsfield et al lung model.
35
230
Weibel Lung Model Comparison
Expiration, Inspir-ation and Poiseuille Flows
Flow Rate = 0.125 L/S
Flow Rate = 0.25 L/S
Visc. Res (cm HZO/(L/S))
O.tAlo
0.01.
Vise. Res (cm H20/(L/S))
-
0.0 3
0.04
0.02
0.03
0.01
0.01
0.02
0.01
0.01
0.006
0
0
6
to
15
20
0
6
to
is
20
Generations into the Lung
Generations into the Lung
Flow Rate = 1.0 L/S
Flow Rate = 0.5 L/S
Visc. Res (cm H2O/(L/S))
Visc. Res (cm H20/(L/S))
0.1
0.06
.
0.08
0.05
0.02
0.04
0.03
0.04
0.02
0.02
0 .01
00
0
6
10
16
20
0
* Expiration
o
5
10
15
20
Generations into the Lung
Generations into the Lung
Inspiration
* Poiseuille
Comparison of the effect of flow types;
Figure 8.5-10
inspiration, expiration and Poiseuille flows.
Airway Resistance Comparison
Flow Rate = 0.5 L/sec
3
Res. cm H20/(L/S)
Airway Resistance Comparison
Flow Rate = 500 cc/sec
Res. em H20/(L/S)
1.2
Experiment
vs.
X
2.5
1
Experiment
vs.
Numerical Prediction
x
0.8 F
Numerical Prediction
4..
1.5
0.6
0.4
~
-
0. .5
I0
0
20
30
40
50
60
70
80
90
100
20
30
40
% Vital Capacity
Pred
-
Sub. RV
-
Sub. KH
-
50
60
70
80
90
% Vital Capacity
Sub. JT
Sub. FH
Experimental Data of Blide et al (1964)
-
Prod.
-"
Sub.
-
#3
Sub. #1
Sub. #4
-
Sub. #2
Experimental Data of Vincent et al, 1970
Comparison of the numerical prediction for
Figure 8.6-1
pulmonary resistance based on the horsfield lung model and
the experiements of Blide et al, and Vincent et al.
100
232
Airway ResisLance Comparison
Flow Rate = 1 L/sec
3.5
Res. cm H20/(L/S)
Experiment
vs.
3
Numerical Prediction
2.5
2
+.4
------ ----
0.5
0
20
30
40
~--Pred.
.x Sub. #4
50
60
-
1
70
% Vita
CaIpaeity
",-, Sub. #1
Sub. #2
0
Sub. #5
A
80
90
100
Sub. #3
Sub. #6
Figure 8.6-2
Comparison of the numerical prediction for
pulmonary resistance based on the horsfield lung model and
the experiements of Hyatt and Wilcox.
233
Airway Resistance Comparison
@ 50% Vital Capicity
1.6
Res, cm H20/(L/S)
1.4
1.2
1
0.8
0.6
Experiment
vs.
0.4
Numerical Prediction
0.2
0
0
0.5
1
I
I
1.5
2
2.5
Flow Rate L/see
Prediction
-I- Ferris et al
Figure 8.6-3
Comparison of the numerical prediction for
pulmonary resistance based on the horsfield lung model and
the experiements of Ferris et al.