7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
Direct Lagrangian Tracking of Fibers in a Human Nasal Airway
K. T. Shanley*, G. Ahmadi*, P. K. Hopke**, and Y. S. Chengt
Department of Mechanical and Aeronautical Engineering, Clarkson University, Potsdam, NY 13699, USA
** Department of Chemical and Biomolecular Engineering, Clarkson University, Potsdam, NY 13699, USA
SLovelace Respiratory Research Institute, Albuquerque, NM
ktshanley@ gmail.com, gahmadit@clarkson.edu, hopkepk@Oclarkson.edu, and 5 s c w~i un In i oiy
Keywords: Lagrangian particle tracking, nonspherical particles, fibers, human nose, breathing
Abstract
The human nose has the important task of removing particles from inhaled air before it reaches the more
sensitive regions of the respiratory system where serious damage may occur. Its complex geometry makes analysis
difficult, though many investigators have employed innovative experimental and numerical techniques to help
quantify deposition efficiency as a function of particle size and flow rate. While earlier work mostly assumed that all
particles are spherical, transport and deposition of ellipsoidal fibers in the nasal cavity is addressed in the present
study. Building on the work of Asgharian and Anjilvel (1995) that expanded the equations of motion for an ellipsoid
suspended in a simple shear flow developed by Jeffery (1922) to more general flows, Shanley and Ahmadi (2009)
developed a module for the commercial software FLUENTTM enabling the direct Lagrangian tracking of ellipsoidal
fibers in general low Reynolds number flows.
The present study applied the module described by Shanley and Ahmadi (2009) to track fibers inhaled
through the nose. A 3D volume was constructed from MRI scans of an anonymous, 181.6 cm, 120.1 kg, male,
human subject. The flow field was solved with the steady laminar model in FLUENT. Lagrangian tracking of dilute
concentrations of spherical particles was performed with the FLUENT Discrete Phase Model (DPM) and fibers
using a series of User Defined Functions (UDF's). Deposition efficiency trends were similar for both particle types
with the fibers having slightly lower deposition efficiencies. In general, fiber deposition trends agree with those of
the modified drag approach taken by Inthavong et al. (2008, 2009).
Introduction
A majority of breathed air passes through
the nose before entering the respiratory tract. While
in the nasal cavity air is warmed, humidified, and
filtered. These three treatments prepare the air for
entry into the lungs.
Increases in computational power and
innovative tools have made possible the use of
computational fluid dynamics (CFD) to investigate
the flow field inside the nasal airway. Keyhani et al.
(1995) and Subramanian et al. (1998) were among
the first to use CFD to simulate flow through a
realistic nasal airway. Both reported observing the
majority of the flow by volume passing through the
lower portions of the airway (i.e. middle and inferior
meatuses) and a peak in velocity magnitude in the
valve. These results are in agreement with previously
published in vitro experiments (e.g. Swift and
Proctor, 1977).
Zamankhan et al. (2006) and Shanley et al.
(2008) used CFD to solve the flow field in a novel
nasal anatomy. Both used the Lagrangian particle
tracking algorithm in FLUENTTM to predict
deposition efficiency in the nasal airway. Shanley et
al. (2008) simulated inertial particles and showed
their deposition efficiency to be a function of Stokes
number. Zamankhan et al. (2006) simulated
diffusional particles and showed their deposition
efficiency to be a function of Pecklet and Reynolds
numbers. Both identified 0.1 ptm to 1 pm diameter
particles as being the most difficult to trap.
Shi et al. (2006) simulated the effects of
breathing patterns on nanoparticle deposition. They
showed unsteady effects to influence deposition and
developed a quasisteady velocity profile that yielded
deposition efficiencies similar to in vitro studies
conducted under equivalent steady flow conditions.
Shi et al. (2007) simulated the deposition of inertial
particles with different wall roughness models. Their
results showed the deposition efficiency to vary
significantly with roughness.
Due to the prevalence of fibrous particles in
manufacturing and construction processes their health
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
effects in humans have garnered much attention
recently. The anisotropic geometry of the fiber makes
the dynamics of the problem significantly more
complicated than that of a sphere.
Inthavong et al. (2008) published the first
simulated deposition of fibers in a nasal airway. They
used two different empirical correlations to modify
the drag coefficient on a sphere. The Haider and
Levenspiel (1989) correlation defined the drag
coefficient as a function of particle Reynolds number
and the ratio of the surface area of an equivalent
volume sphere to the surface area of the particle. The
TranCong et al. (2 1'4) correlation assumed the fiber
to be represented by spherical particles coagulated in
a cylindrical bar configuration. Degree of circularity
(Wadell, 1933) was used to account for the non
spherical shape of the fiber. Differences between the
predictions in fiber deposition efficiency of the two
correlations were observed. The general trends of
both, however, matched well with the experimental
data from Su and Cheng (2005).
The present study investigated the
deposition efficiency of both spherical and fibrous
particles in the human nasal airway. Two nasal
airway models were constructed from MRI data and
meshed for CFD. A steady laminar flow model was
used to simulate inhalation. Volumetric flow rates
were varied from 2 to 20 liter per minute. After the
flow fields were calculated discrete particles were
injected from the nostril surface. A uniformly
distributed sample was introduced and each particle
was tracked individually. The FLUENT DPM was
used to track spherical particles while a series of
UDF's were used to track the motions of fibers. For a
constant flow rate larger particles were trapped more
easily and for a constant size increasing flow rate was
shown to drive entrapment. While this implies
impaction driven deposition, anatomy appeared to
play a significant role as well. Good agreement was
shared between the presented simulations and
previously published experimental and numerical
work.
Flow Volume Models
For the purposes of this study two novel
computational nasal models have been created from
MRI scans obtained from the Lovelace Respiratory
Research Laboratory, Albuquerque, NM. Data
representing the left (Model I) and right (Model II)
passageways of an adult male, mass 120.2 kg, height
181.6 cm, have been reconstructed into separate 3D
computational volumes.
A similar method to that of Subramanian et
al. (1998) and Zamankhan et al. (2006) was
employed to generate both models. The coronal
Nasopharynx
Slice
Figure 1: 3D tetrahdral mash of Model I
with 2D sllce from the odt plane.
crosssections were loaded into the CAD package,
Mechanical Desktop (Autodesk, Inc., San Rafael
CA). Key perimeter coordinates were identified and
used to create adjoining surfaces between the slices.
The solid model was exported in STL To prepare the
volumes for CFD analysis the following method was
performed to fill each volume with sufficiently small
tetrahedral elements.
1. Import the STL file into MagicsRP
(Materialise Inc., Ann Arbor, MI.) and perform a
surface retriangulaition to obtain smaller less
skewed triangular faces on the surface.
2. Perform a "splitbase" refinement procedure to
improve surface mesh quality.
3. Smooth the model to reduce sharp edges.
4. Extrude the nostril and nasopharynx surfaces
then perform a cut to obtain a flat selectablee)
inlet and outlet surface.
5. Repeat steps 2 and 3 until satisfied with mesh
quality.
6. Export new STL file.
7. Import new STL file into TGridTM (Fluent Inc.,
Lebanon, NH)
8. Perform a "surface wrap" procedure to assure
connectivity information is present and impose
water proofing.
9. Merge all duplicate nodes and delete any unused
nodes.
10. Initialize a volume mesh.
11. Refine until satisfied with the size/number of the
tetrahedral elements.
12. Write a FLUENT 5/6 mesh file (*.msh).
An unstructured tetrahedral mesh was
required to fill the volumes due to their complex
geometry. Both models contain roughly 950,000
tetrahedral elements, further refinement proved
unbeneficial. The final volume mesh for Model I is
shown in Figure 1. The grid sizing is difficult to see,
so an enlarged slice taken from the entrance to the
nasopharynx is also shown on Figure 1. In the
enlarged slice localized refinement can be seen near
the walls and in highly complex areas.
Governing Equations
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
Incompressible Flow
The conservation of mass and momentum
equations for steady, laminar, incompressible flows
are defined as follows.
1 2
u Vu = V + vV u + g
P (1)
Vu =0 (2)
In Eqs. (1) and (2) u is the threedimensional
fluid velocity vector (m/s), p is the fluid density taken
to be a constant for air under sea level static
conditions, v is the kinematic viscosity of air also
taken to be a constant, and g is the three dimensional
gravitational acceleration vector.
The commercial package FLUENT was
used to numerically solve Eqs (1) and (2). FLUENT
employs a finite volume algorithm. Spatial
discretization of the conservation equations was
performed with the upwind scheme to a second order
accuracy, pressurevelocity coupling was performed
with the SIMPLE scheme, and derivatives were
computed via the GreenGauss Node Based method
(Fluent User's Manual, 2007).
At the boundaries the walls were taken to be
smooth, rigid, stationary, and viscous. A uniform
profile as deemed suitable by Keyhani et al. (1995)
was applied to the nostril surface. The entrance
surface to the nasopharynx was held at a constant
atmospheric static pressure. Inlet flow velocity was
varied to correspond with volumetric flow rates of 2
to 20 liters per minute per model. This range was
chosen to depict the full range of nasal breathing for
a single passage. Humans typically switch to oral
breathing once circa 35 liters per minute overall is
required.
Lagrangian particle tracking
A direct Lagrangian approach was used to
track individual particles in dilute concentrations. For
spherical particles position and velocity were
obtained by integrating the following equations of
motion.
dv
X = 
dt (3)
dv
= FD +FL +g
dt (4)
Here, x represents the three coordinates of
the particle position vector (xi, x2, x3), and v is the
threedimensional particle velocity vector (vi, v2, v3).
The terms on the righthandside of Eq. (4)
represent the drag, lift, and gravitational
sedimentation per unit mass, respectively
Drag on a small sphere was presented in
Hinds (1989) to be
1 CD Ren
FD (u V)
r 24 (5)
Here is the particle relaxation time, CD is the drag
coefficient, and Rep is the particle Reynolds number;
Saffman (1965) provided an expression for
the shear induced lift on a small sphere to be given
as,
2Kv di
FL =  14 (u v)
Sd(dlkdkl)14 (6)
Here S is the solidtogas density ratio, K is the
Boltzmann constant, and D is the deformation rate
tensor.
Similarly, for fibrous particles the following
coupled equations of motion were solved.
dv +(1mf m)g
mp d = FD +FL +(mf mp
dt (7)
dt
d(IM .(o)= Th
dt (8)
Eq. (7) represents the translational motion
and is similar to that of a sphere with different
definitions of the drag and lift forces to account for
the anisotropic geometry of the fiber.
Hydrodynamic drag on a fiber is dependent
on Reynolds number and orientation as defined
below.
FD = dpK (u v)
Lift due to shear on a nonspherical body is
given as
FL = 2 (K.L.K)(uv)
4j iu/y
(10)
In the limiting case of a sphere (i.e. P = 1) Eq. (10)
reduces to the finding of Saffman (1965).
Eq. (8) represents the rotational motion of
the fiber. Hydrodynamic torque (Th) brought about by
strain rate and vorticity in the fluid drives the
rotational motions.
A detailed formulation of these equations
can be found in the appendix
To simulate a uniform concentration of
particles entering the domain a lattice of equally
spaced particles was released at the nostril surface.
The distance between each particle center of mass
was 50 gm and the minimum distance between a
particle center of mass and a wall was 20 pm. This
resulted in the release of 4,500 particles per trial. The
walls were set to trap without rebound on contact to
simulate the capture ability of the mucosal layer
inside the human nasal airway.
Local fluid velocities were taken to be the
velocity at the center of mass of the particle in the
absence of the particle. The presence of the particle
did not alter the flow field. This is known as oneway
coupling and suitable when studying dilute
concentrations of particles. To obtain this quantity
interpolation between cell centers must be performed.
A Taylor expansion was used within the domain;
however, a user defined interpolation scheme was
used in cells that contact walls (Longest et al. 2007)
Integration of Eqs. (3) and (4) were carried
out with the FLUENT Discrete Phase Model while
integration of Eqs. (7) and (8) was conducted with a
series of UDF's in FLUENT. The classic 4th order
RungeKutta method with an adapting time step was
used to perform numerical integration. This method
was chosen for its high order accuracy and
reasonable efficiency.
Results and Discussion
CFD Results
A series of CFD analyses were performed to
gain understanding of the airflow field patterns inside
the nasal models under steadystate conditions. The
volumetric flow rate was varied from 2 to 20 liters
per minute for each flow volume. The results of the
simulations are presented in this section.
Figure 2 shows the pathlines traveled by
massless tracer particles injected from the nostril
surface. The number of pathlines have been
coarsened for the image; however, the general trend
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
of the data was preserved. A majority of the air
passed through the lower and middle meatuses. Less
air passed through the superior meatus, but very little
air traveled through the olfactory bulb. This
qualitative observation is in agreement with much of
the previously published work on nasal casts (e.g.
Swift and Proctor, 1977; Subramanian et al. 1995;
Keyhani et al. 1996)
Both models showed the presence of a
recirculating region in the vestibule which appeared
for all flow rates. A much smaller recirculating
region in the olfactory area of Model I was observed
for moderate flow rates. This recirculation
disappeared at high flow rates. Model II showed
Model II
Figure 2: Pathllnes traveled by massless
tracer particles Injected fom the nostril
surface at 6, 12, and 18 liter per minute
volumetric flow rates.
greater access to the superior regions of the cavity
Velocity Magnitude (m/s)
O. n
z x
3.2U
4.bU
u n 0
'9 _1
Model I: Left Airway
Model II: Right Airway
Figure 3: Ajrflow velocity contours on coronal crosesctlons of Model I (left) and Model II
(right) under a volumetric flow rate of 20 liter per minute.
including the olfactory bulb though no recirculation
was seen in this case. The olfactory bulb of Model I
was smaller in volume than that of Model II. This not
only prevented air from being drawn in, but caused
the air that did enter to recirculate.
Contours of velocity magnitude are shown
on several crosssectional slices throughout both
nasal airways in Figure 3. Probing of the data
revealed that Model I reached a peak velocity
magnitude of approximately 9.5 m/s and this was
observed in the nasal valve. Similarly Model II also
reached peak velocity magnitude in the nasal valve
with a magnitude of approximately 9.0 m/s. These
specific values were observed for a flow rate of 20
liter per minute, but the maximum velocity
magnitude was always found to occur in the nasal
valve. The valve is the narrowest crosssection of the
airway and in order to preserve mass flow the
velocity must increase here. This observation was
first noted by Swift and Proctor, 1977.
Particle Deposition I i ,. ,. .,. r
Lagrangian particle tracking was performed
to predict particle filtration efficiency in the nasal
passages given the airflow fields described above. A
uniform concentration of particles was introduced at
the nostril surface for diameters ranging from 1 pm to
10 pm and densities from 1 g/cm3 to 3 g/cm3. For the
dilute concentrations studied a oneway coupling
assumption was made. Deposition occurred upon
contact. The deposition efficiency was defined as the
number of particles deposited on walls divided by the
number of particles introduced at the nostril (inlet).
The aerodynamic diameter, dae, of a particle
is a convenient measure of a particle's size. For a
spherical particle the aerodynamic diameter is
defined as follows.
dae= dp /PH/ 2 (11)
However, for the nonspherical shape of a
fiber the aerodynamic diameter was defined by Wang
et al. (2008).
dae = f Pf [Ln(2)0.80685]
2 PH20
(12)
In Eqs (11) and (12) dp is the crosssectional
diameter of a spherical particle,df is the cross
sectional diameter or minor axis of the fiber, pp or pf
is the density of the particle, pH20 is the density of
water, and P is the major axis to minor axis aspect
ratio of the fiber.
Similarly the deposition data for the fibrous
particles was consistently shown to be to the right of
their spherical counterparts. This is due to the fact
that elongated fibers have a natural tendency to align
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
themselves with the direction of flow (Wang et al.
2008). When parallel to a wall a fiber can come
closer to the wall than a sphere with equivalent
aerodynamic diameter without making contact. This
allows for more fibers to pass through than spheres, a
phenomenon which poses great concern to the health
of humans as some fibers are known to cause serious
health effects once they reach the lungs.
When plotted against the impaction
parameter in Figure 5 the data collapses onto four
distinctive curves; one each for spherical and fibrous
particles in both models. All of the data was close,
but distinctive trends were observed. The intermodel
variation was due to the anatomical differences
previously discussed and consistent with the trends
observed on Figure 4. A shift was also observed
between the spherical and fibrous particles in each
model. This was due to the anisotropy of the fiber
geometry and consistent with the trends observed on
Figure 4. It is interesting to note that the fiber data for
both models was more closely aligned than that of
their spherical counterparts.
Also shown on Figure 5 is the deposition
data from INthavong et al. (2007). They proposed the
use of two different modified drag coefficients to
account for the anisotropy of the fiber. Although their
study was performed on a different nasal model
reconstructed from MRI scans of a different subject,
the data matches well with these simulations.
Summary of Results
Computational fluid dynamics and direct
Lagrangian particle tracking were performed on two
nasal airway models representing the left and right
airways of an anonymous adult human male subject.
The CFD revealed much of the air breathed through
the nose will pass through the much larger, inferior
meatuses nearly bypassing the olfactory bulb
altogether. A peak in velocity magnitude was
observed in both models under all flow rates
examined. Deposition efficiencies wer computed
from the final fate destinations of the spherical and
fibrous particles tracked. Fibers deposited less
frequently than spheres in each model while the
larger Model II trapped fewer particles of both kinds
than Model I.
Acknowledgements
The financial support of the National
Institute for Occupational Safety and Health
(NIOSH) through Grant R01 OH003900 and the
Environmental Protection Agency (EPA) under grant
916716 is gratefully acknowledged.
DISCLAIMER:
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
1.00
Model I: Spherical Particles
Model I: Fibrous Particles *
A Model II: Spherical Particles A
0.75 + Model II: Fibrous Particles
Inthavong et al. (2007)
c 0.50 U
o uE.
0.25
0.00
1000 10000 100000
IP (tm*cm3/s)
Figure 5: Depostllon ofiolenoy of spherical and fibrous prlicles a funotlon of Impactlon
parameter for Model I, Model II, and the previously published work of Inthavong t al. (2007).
This work has not been subjected to either
Agency's required peer or policy review and
therefore does not necessarily reflect the views of the
agencies and no official endorsement shall be
inferred.
Appendix: Kinematics of Fiber Motion
Before discussing the motion of an ellipsoid
it is imperative to introduce two independent
coordinate systems as shown on Figure A.1. The
global coordinate system (x, y, z) is fixed in space
while the local coordinate system (x', y', z') is fixed
on the ellipsoid and will translate and rotate with the
ellipsoid. Orientation of the local coordinate system
and thereby the ellipsoid with respect to the global
coordinate system is defined in terms of three angles
(), y, 0) known as Euler's angles. Figure 1 shows the
Euler angles relating the two coordinate systems to
one another. The rotational motion of the ellipsoid
can now be derived from the time rate of change in
the Euler angles.
The equations of translational and rotational
motions of rigid ellipsoids are coupled, and are given
as
dv
m 
P dt
FD +FL +(mf
mp)
(A.1)
d
(IM o)= Th
dt (A.2)
Eq. (A.1) represents the translational motion
and is similar to that of a sphere. The forces summing
on the right hand side of (A. 1) are hydrodynamic
drag (FD), shear induced lift (FL) and gravitational
sedimentation. Eq. (A.2) represents rotational motion.
Rotational motion is brought about by the
hydrodynamic torque exerted on an ellipsoid by the
fluid strain and vorticity.
The hydrodynamic drag is a dynamic
function dependent on the orientation of the ellipsoid.
It can be defined mathematically as,
FD qd K. (u v) (A.3)
Here K is the translational dyadic,
sometimes referred to as the resistance tensor and its
elements are defined in the particle reference frame
as follows.
(x, y, z) = Global Coordinates
(x', y', z') = Local Coordinates
y IvNUUt;
Figure A4: Relatlonship between global and
local refbrence tames.
(A.4a)
(A.4b)
(A.4c)
kz'z, = ky, ,
where,
fkx'xk 0 0
K'= 0 ky 0
0 0 kz,' (A.5)
K' can be transposed into the global frame
(i.e. K) with the basis transformation as follows.
1
K = A K'A (A.6)
In Eq. (A.5) A is the direction cosine matrix
(DCM). The DCM is obtained by multiplying three
matrices representing three consecutive rotations
about an axis. For the purposes of this formulation an
xyx convention was assumed. With this convention
the total rotation of the ellipsoid is represented by a
rotation of f about the xaxis in the initial frame,
followed by a rotation f about an intermediate yaxis
(line of nodes), finishing with a rotation of f about
the x'axis in the final reference frame. The DCM is
then written in Eq. (A.7) for the xyx convention.
Eqs. (A.5) and (A.6) clearly show the
coupling of translational motion with rotational
motion. The DCM, A, depends on fiber orientation.
To determine orientation, the rotational motion must
be analyzed. The hydrodynamic torques are given
by,
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
A snos mympoimRVain (ago +cl 5a )
h 167/ ,
370 (A.8)
T = h' [216/ + j ,7'y)_2 1bxz
3o0 + o) (A.9)
16nu [( (2 +i 4
Tzh = 16 P 2 a1 2 'z')_ 2 1y'z]
ro+/ oa (A. 10)
In Eqs. (A.8) (A. 10) the vorticity and
strain rate quantities are defined in the global frame
Oux
Exx 
ox
.YY _o^y
uz
S1 u +Uux
tx 2 x By
Sl uz B+ ux
= + !
2yz 28 U 8z O
yz z+ C
3 21 Oy Oz
OdzY2
(A.11)
(A.12)
(A.13)
(A.14)
(A.15)
(A.16)
(A.17)
(A.18)
I 8UY oux
2 x y (A.19)
Eqs. (A. 11) (A.19) can be transposed into
the particle frame, as needed, by Eq. (A.5). The
angular velocity components are defined in the
particle frame to be,
0 d, = V+dcos0
x dt dt
dO dO
S= sin 0 cos  sin y
dt dt
The moment of inertia tensor is
(A.21)
(A.22)
(A.23)
2 (Ox
2[y
if7u2
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
ix', 0 0
IM= 0 y, 0
m 0 0 Iz'
0 0 I (A.24)
with components along the and transverse to the fiber
polar axis being,
d2 2 dcods dO dr 4. P 20
dt t n dt dtj +1, & ao)
*.(cos q7sinm)cot+? ] + (A.27)
p2 '( cos + sin )t e cos2 + (in i )sin} 2}
da 2 dord rsin 0+ re 3 in20 d+ u 20
Sq.co+(7 1 .dO #1 (A.28)
L d1J p +1
*[(t,. cos t& sin #)co?+ sin0, + ainm 2 t ain uV t com)1
dv cotW d Cf +1 do dLv1. u 20cosO
dtF cr 1) dt ddt d
d ( 0cos ? Le n coO+'df 2Oa1
dflo 1 dt )
p 20cosO d# dA
After substituting Eqs. (A. 11) (A. 19) into Eqs.
(A.8) (A. 10) and considerable simplification the
angular acceleration components can be expressed as
in Eqs. (A.27) (A.29).
In Eqs. (A.27) (A.29) the dimensionless
parameters 00o and Yo are taken to be,
0 8 2 + P Ln
(A.30)
8Yo & P+ I L.nK
'/L T (A.31)
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
References
P 05
S 60
(A.25)
I z =prfld 2+ 1)
Iy,=Iz,1l)
120 (A.26)
For an ellipsoid of revolution suspended in a
simple shear flow the shear induced lift can be
expressed as follows.
fL u/ (K.L.K).(uv)
(A.32)
with,
(0.0501 0.0329 0.0000
L= 0.0182 0.0173 0.0000
0.0000 0.0000 0.0373 (A.
(A.33)
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f1 1 a b
* 2 2 in a cos )
a ) b )
cos a2 sin a )2
a b ab
cos a 2 s+in a 2
with,
b a 2  +s sin acos a
Ia a2 b 2
l+a2b2 2( sin2 acos2 a
la 2
(A.34)
(A35 ?
The tilt angle was modified slightly to fit
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a = cos1 a2 + a33
^ (A.36)
Here, a32 and a33 are elements of the DCM
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