a numerical model of the motion of a curved flexible …
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A NUMERICAL MODEL OF THE MOTION OF A CURVED FLEXIBLE FIBRE IN A SHEAR FLOW
Inna Galperin
A thesis submitted in conformity with the requirements for the degree of Master of Applied Science
Graduate Department of Chernical Engineering and AppIied Chemistry University of Toronto
O Copyright by Inna Galperin 1997
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Inna GaIperin, M. A. Sc., 1997
Department of Chernical Engineering and Applied Chemistry University of Toronto
Abstract
A general method to simulate the motion of initially curled flexible fibres in 2-D
flows is developed. Equations of motion are formulated for a curled, elastic fibre in a fluid
flow using Hamilton's Principle. The non-linear integro-differentid equations contain an
unknown deflection variable that is discretized using a half-range Fourier Sine series
representation. The first (kincmatic) approximation is obtained by setting the inertia terms
to zero. The resulting inertialess equations are solved using a Runge-Kutta method
irnplemented in a C++ code.
The validity of the numerical model was tested against analytic theory and existing
transient results for essentially straight fibres, i.e. fibres with smdl radii of curvature and
large angles. Excellent agreement was achieved. The numericd model was then applied to
simulate fibre motion into a screen slot. For similar fibre parameters, it was found that
significantly more curled fibres entered the slot than their straight counterparts.
Acknowledgments
1 am greatly indebted to Professor D. C. S. Kuhn for his supervision and guidance
throughout the duration of my thesis. 1 would also Iike to thmk everyone associated with
the Pulp and Paper Centre for allowing me the opportunity to hone my research skills and
to participate in the technology tour and the conferences that have contributed very much
to my learning experience.
1 would like to acknowledge Yuri Lawryshyn for his input and advice, and
Abitibi-Consolidated for providing a workplace when 1 needed it. Many thanks to my
colleagues for their help and for a pleasant working environment. 1 would also like to
thank my friends here in Toronto for their good attitude and for the necessary social
distractions that have made my work and stay a valuable personal experience.
Finally, 1 would like to express my gratitude to my parents, sister, and grandfather
for their warmth, kindness and support, both in work, and in life.
Abstract
Acknowledgments
Table of Contents
1 Introduction
2 Literature Review
2.1 Previous Fibre Flow Models
2.2 Harrnonic Analysis
2.3 Fluid Forces
2.4 Application to Pulp Screening
3 Mathematical Model
3.1 Fibre Model
3.2 Hamilton's Principle
3.2.1 Kinetic Energy
3.2.2 Potential Energy
3.2.3 Virtual Work
3.3 Exact Integro-Differential Equations of Motion
4 Orthogonal Expansion of Deflection Function
4.1 Circular Ring
4.2 Circular Arc
5 Kinematic Inertialess Equations
5.1 First Kinematic Approximation
ii
iii
6 Numerical Model
6.1 Runge-Kutta Method
6.2 Computational Algorithm
7 Model Validation and Applications
7.1 Model Validation
7.1. I Static Analysis
7.1.2 Dynamic Curved Fibre Motion
7.1.3 Results and Discussion
7.2 Fibre Motion into Slot
7.2.1 Fibre Passage Values: Hfcrit
7.2.2 Discussion of Results
7.2.3 Application to Pulp Screening
8 Concluding Remarks
8.1 Surnmary of Numerical Model
8.2 Applications
8.3 Suggested Future Work
9 References
A Inertialess Equations for m = 3 Modes of Motion
Chapter 1
Introduction
Many business, commercial, and academic enterprises need high quality paper of
good strength, printability, and optical properties. Thus, fibre quality, in particular fibre
flexibility, is an important concem for the pulp and paper industry. It is generally believed
that longer, more flexible fibres have superior bonding characteristics to their shorter, stiffer
counterparts, and thus contribute to better paper quality. Pulp screening processes attempt to
separate or fractionate short, stiff fibres, and shives from the more desirable flexible ones.
Shives are unseparated fibre bundles that give rise to uneven paper morphology, and can
adversely affect the printability and optical properties of paper. In addition, shives may
accumulate stress concentrations which can contribute to poor paper strength.
The prior work of Lawryshyn and Kuhn [27] examined the Flow of an individual
fibre by modeling it as a flexible cylinder with straight initial shape. Howevcr, due to
various chernical and mechanical processes that fibres tend to undergo during papermaking,
many fibres assume a curled initial shape (see Page et al. [35]) . Thus, the goal of this
motion of an initially curled flexible fibre in a shear flow. Equations of motion are
developed for a curled flexible fibre and are subsequently solved by existing numerical
methods and compared to prior results. Throughout the formulation of the equations, small
deflection theory is assumed and the fluid forces acting on the fibre are estimated based on
the first order approximations of Cox [6 j.
The equations of motion are non-linear and integro-differential in form, containing
an unknown deflection variable. The deflection variable is represented as a function and
discretized by the Rayleigh-Ritz assumed modes method. A C++ implemented Runge-Kutta
routine solves for the time parameters of the unknown deflection function. Ln the test case,
the sme shear flow and fibre parameters are used as those ernployed for the straight fibre
case to allow for cornparison to prior theoretical and computational results. This new curled
fibre analysis is then applied to consider the motion of fibres in a channel flow with a dot,
i.e. a simplified screen geometry.
Chapter 2 presents a literature review encompassing the topics discussed in this thesis.
Chapter 3 presents the analytical work used to formulate the equations of motion. Chapter 4
proposes an orthogonal expansion for the unknown deflection function. The poverning
equations, capable of describing the full motion and deflection of a curled, flexible fibre are
simplified to the inertialess case in Chapter 5. Chapter 6 presents the numerical procedure
and the algorithm used to dynamically solve the mathematical model. Comparison to prior
results and applications of the model are presented in Chapter 7 . Chapter 8 discusses the
Chapter 2
Literature Review
Previous fibre flow models of Gooding [15], Kumar [25], Olson [34], and
Lawryshyn [26], examine the flow of fibres through narrow apertures. Kumar and
Gooding's models are experimental and study the effect of fibre, flow, and slot parameters
on fibre passage through slot geometries. Olson, and Lawryshyn's models are of a
theoreticai nature, and model fibre passage through narrow apertures. Al1 four models are
discussed in the first section.
This thesis presents a mathematical and numerical mode1 for the motion of a curled
flexible fibre in pulp flow, with applications to the study of pulp screening. Fundamentals of
mathematics and geometry found in many textbooks such as Stewart [48] and [49], were
used to arrive at a fibre model. Hamilton's Principle (Meirovitch [30]), used to formulate
the equations of motion, required representations for the energy expressions, a topic found in
standard dynamics texts like Meriarn [32]. The resulting integro-differential equations of
motion contain an unknown deflection variable, as well as a general representation for the
Harmonic analysis, a topic found in rnany texts (e.g. Greenberg [18]), is outlined in the
second section and was used to discretize the deflection function. Cox's [6 ] first
approximation is used for the fluid forces and is presented in the second section. The fourth
section comprises a general discussion on pulp screening.
2.1 Previous Fibre Flow Models
Kumar' s [25] largely experimental approach studied the ability of aqueous fibre
suspensions to penetrate a single aperture located in a flow channel as well as multiple
apertures in a device simulating a commercial pulp screen cross-section. The flow
conditions used approximated that of commercial screening processes which have a large
channel velocity compared to the aperture velocity. The aperture dimensions were greater
than a fibre's diameter, but Iess than a fibre's length. Kurnar's experimentaI measurements
of fibre passage were related to a ratio of fibre length to aperture width (WW) for both stiff
and flexible fibres.
Gooding's [ 1 51 experimental approach examined the influence of fibre, flow, and
slot variables on fibre passage through an aperture. He found that reduced fibre stiffness and
length increase fibre entry. Gooding also obtained images of fibre trajectories through high
speed ciné-photography. He observed two effects: the "wall effect" whereby a layer of flow
that passes through the dot is depleted of large fibres, and an "entry effect" where short,
findings suggest that shorter or more flexible fibres pass through an aperture more reridily
than longer or stiffer fibres.
Olson 1341 developed a theoretical model to determine the passage ratio of stiff fibres
of 1, 2, and 3 mm lengths through narrow apertures. His model made use of the wail effect
and the turning effect witnessed by Gooding. Olson's mode1 was compatible to prior
experimental results.
Lawryshyn's [26] theoretical model studied the static and dynamic behavior of pulp
fibres. He modeled the flow of initially straight, flexible fibres through small apertures by
developing a set of non-Iinear integro-differential equations of motion. Lawryshyn's
theoretical mode1 was found to agree with static deflection beam theory.
2.2 Harrnonic Analysis
Al1 bodies having mass and elasticity are capable of vibration. There are two classes
of vibration: free vibration and forced vibration. Free vibration t'îkes place under the
absence of external excitation, whereas forced vibration occurs under excitation by external
forces. In free vibration, the body oscillates under the action of forces inherent in the system
itself. A thorough discussion of these topics c m be found in Thomson [52].
motion, wil1 oscillate at one or more of its natural frequencies in free vibration. Oscillations
may repeat themselves regularly or irregularly. Oscillatory motion repeated rit regular time
intervais is known as periodic motion, with the time interval T called the period of
oscillation.
The simplest form of periodic motion is harrnonic motion. In harmonic motion, the
acceleration is proportional and opposite to the displacement and can be characterized solely
by sine and cosine terms. In multi-degree-of-freedom systems, vibrations of several
different frequencies can exist simultaneously resulting in a complex waveform that is
repeated periodically as s hown in Figure 2.2- 1.
Figure 2.2-1. Complex Periodic Waveform.
of sine and cosine terms. Thus a general periodic function x(t) of period r c m be
represented by the Fourier series:
where
The deflection variable in the mathematical model can be approximated by a
deflection function and discretized by the Rayleigh-Ritz method (see Meirovitch [30]). A
Fourier expansion is used for the deflection function; this topic is treated extensively in the
literature (e.g. Greenberg [ 183).
2.3 FIuid Forces
The numerical model of Chapter 6 develops the kinernütic inertialess equations.
These equations require an explicit formulation for the fluid forces acting on the fibre.
Several approxi mate analyticril solutions for the hydrodynarnic forces acting on a long,
slender body are available in the literature. Cox [6] developed expressions for the flriid force
was taken to be negligible. Batchelor [Il calculated the resultant fluid force and couple
required to sustain translational and rotational motion in a straight, rigid slender body. De
Mestre [IO] presented low-Reynolds number results for the drag and torque on a rigid
slender cylinder translating near a plane wall. Recent work by Khayat and Cox [22],
extended [6] and [7] by formulating equations for the fluid force and torque acting on a
slender body with non-negligible fluid inertia. In the lirniting case of 1ow Reynolds nurnber,
the results for both cases were found to agree. This work will use Cox's first order
approximation for the normal and tangential fluid force per unit length acting on the body as
where
p is the viscosity of water,
Un and Ut are the normal and tangential components of the fluid velocity,
V,, and are the normal and tangential cornponents of the fibre velocity,
L is the fibre length,
2.4 Application to Pulp Screening
Previous work on the topic of pulp screening has been both experimental and
niinerical in nature. Gooding [15] and Kurnar [25] studied the passage of aqueous fibre
suspensions through an aperture or slot geometry. Because the aperture width is larger than
the minimum diameter of many shives, Kumar concluded that the screening process does not
rely on physical obstruction but rather on unknown mechanisms and fibre properties such as
fibre flexibility. Gooding, in characterizing a wall and an entry effect, found that shorter
andor more flexible fibres enter the aperture more readily. Thus, both Gooding and
Kumar's experimental approach showed the importance of fibre flexibility in the passage of
fibres through narrow apertures.
By approximating flow conditions similx to pulp screening, Olson [34] examined
the passage of fibres through narrow apertures by developing a theoretical mode1 for the
motion of rigid fibres entering a dot. Lawryshyn (261 studied the motion of an irzitirilly
strnight flexible fibre in a channel flow with a slot, and examined the effect of fibre
flexibility. His mode1 proved the occurrence of pulp screening based on fibre flexibility. To
date, no known mode1 of the motion of iizitinlly ccrrled flexible fibres in channel and slot
geometries is known to exist. In view of the fact that many fibres produced through the
Chapter 3
Mathematical Model
The equations governing the motion of a curved flexible fibre in a 2-D plane are
developed in this chapter. The curved fibre is modeled as a cylinder of a given flexibility
with curved initial shape, and the exact equations of motion for the fibre are developed using
Hamilton's Principle. The proposed mode1 expands upon the specific case of a straight,
flexible fibre considered by Lawryshyn and Kuhn [27] to the more general case of a curved
flexible fibre.
3.1 Fibre Model
A curled fibre is modeled as an arc of a circle of constant radius of curvature r'.
Figure 3.1 - 1 shows the coordinate systems and the magnified geometry of a deflected curved
fibre, where X, Y are absolute coordinates; x, y are relative coordinates; Ir1 = constant; 6 is
' From this point forward, the term "curved" fibre will be used to describe a "curled" fibre.
are considered.
Figure 3.1-1. Geometry and coordinate systems of a typical curved fibre.
The position vector R of any point on the fibre is thus given by
where Ro is the position vector of the origin of the relative coordinate system x-y, r and a are
location parameters, u ( a t ) is the deflection of the fibre at a point defined by the polar
coordinates r and a. The projection of (3.1) ont0 the X and Y axes yields
Here, the variables Xo, Yo, u and 8 depend on time t, and define the absolute motion of a
fibre. Taking the time derivative of equation (3.2) yields
where R is the time derivative of R.
3.2 Hamilton's Principle
There exist various methods of forrnulating the equations of motion. Newtonian
mechanics uses Newton's second law, F = ma , to derive the equations of motion in terms of
vector quantities for each mass separately. As implied by the equation, forces appear
explicitly in Newtonian rnechanics, see for example Meriam [32]. Analytical mechanics
uses scalar quantities, namely the kinetic and potentiai energies and the virtual work, instead
of vectors to formulate the equations of motion. This approach is especially useful and very
powerful when describing systems of more than one particle, and distributed mass systems
such as a fibre in a fluid. Within analytical mechanics, many representations are possible
such as the Lagrangian formulation, or Hamilton's principle (Meirovitch [30]). The
used, but the Lagrange equations make use of generalized forces, whereas HamiIton's
principle does not. Because of the difficulty of identifying generalized forces in our
analysis, Hamilton's principle is used in this application.
Hamilton's principle is a way of formulating the equations of motion using the scalar
quantities of kinetic and potential energies and the virtual work for a systern moving along a
varied path. A systern of particles travels dong the varied path Ri ( t ) + GRi ( t ) made up of
the true path Ri( t ) and the virtual displacements 6 R i ( t ) , which are possible imaginary
displacements respecting constraints at frozen time. The varied path is chosen such that it
coincides with the true path at the two times t = t , and t = t 2 , implying that
SRi ( t , ) = 6Ri ( t , ) = O . Hamilton's principle rnay thus be represented as [30];
where ST is the variation in kinetic energy,
m/ is the variation in potential energy,
SW is the virtual work,
6Ri ( t ) is the variation from the true path, or the virtual displacement at frozen tirne.
energy, and the virtual work.
3.2.1 Kinetic Energy
The kinetic energy of the fibre system is
where dm is the elementary mass and L is the length of a fibre. Denoting v = volume, p =
constant density, A = constant cross-sectional area, and da = elementary angle of a fibre, we
obtain, for the case of a circula arc
dm = pdv = pArda
Substituting (3.6) into (3.5), the expression for the kinetic energy becomes
whcre a increases from O to any arbitrary a,, I n:
expand the position vector R as shown in Figure 3.1- 1. Using expression (3.3), this yields
When (3.8) is substituted into (3.7), the following expression for the kinetic energy emerges
The kinetic energy of (3.9) is made up of 6 terms. The first term is the kinetic
energy due to pure translation, i.e., rigid body translational motion. The second term is due
to pure rotation, i.e., rigid body rotational motion, and the third term is due to pure vibration.
The subsequent three terms are the so-called mixed terrns. The fourth term is due to
rotation and vibration. The fifth term is due to both vibration and translation, and the sixth
term is a combination of al1 three motions: transhtion, vibration, and rotation.
are functions of a remain inside the integration sign. Thus, simplifying (3.9). and writing
those terms without integrals first, we obtain
xo2 + yn2 ( A ) 7 = p4ra0( ) + pAra0 - + pAr26(% sin 8 + ko cos ~ ) ( m s a, - 1)
2
I a" I a 0
+ p ~ r ' 9 ( c cos8 - x0 sinf3)sina0 + - p ~ r l r i ' d a + - p ~ & ' Iu'da O 2 O
an a,
+p~r'&' 1 irda + p ~ r ( $ cos 8 - x0 sin û ) j Ù sin a& O O
a, a,,
+ p ~ r ( $ sine+ X, c o s 0 ) ~ ~ c o s a d a + ~ ~ d ( ~ c o s e - ~ ~ s i n 8 ) l u c o s a d a
an
-p~ré (Y , sin û + xo cos e)J u sin d a
Now,
where u(-) and ù(.) indicate that T is a functional of these parameters, Le. T depends on
these parameters as they Vary through an angle a. Thus, the variation of kinetic energy has
the form
expressed as
where
Ti,) = pAra,x0 - pAr28sin 0 sin a, + pAr28 cos cos a, - 1)
T, = - p ~ r ' O ( ~ o sin 8 + x0 ccs0) sin a, + p ~ r ' 9 ( ~ o cos 8 - x0 sin û)(cos a, - I )
a0 a11
+ p ~ r ( % cos 0 - x0 sin 0 ) l ti cos ada - p ~ r ( ~ o sin 0 + X, cos 0 ) l li sin culs
a l , ' L I )
- p ~ & ( % cos - X, sin e)J LL sin d a - p ~ & ( Y ; sin 0 + X, cos 0 ) j u cos ada ,
a,
6 q = sin 0 + x0 cos û ) j cos a~rida O
a11 an
cos9 - x0 sin0)lsina6uda + p ~ r j Ù6da . O O
3.2.2 Potential Energy
The potential energy applicable to this study is the potential energy of bending, i.e.,
the elastic strain energy. This is the energy stored in the fibre when it is deflected from its
undeformed shape, and for a fibre represented by an arc of a circle is, see Timoshenko [53],
where the expression has been modified such that cc increases from O to any arbitrary a,, 5 n.
Here, E is Young's modulus, 4 is the moment of inertia of the cross-section with respect to a
alu principal axis paraIlel to the z axis, and u" = -
aoc- '
Through a similar process as that used for the kinetic energy in (3.12)' the variation
of potential energy V i n (3.20) with respect to u"and u, is found to be
EI. "" EI. "" 8V = -J(u"+u)du"da+-J(uM+u)suda O O
3.2.3 Virtual Work
The virtual work is found by considering the forces acting on an initialIy undeflected
fibre, which is subjected to virtual displacements, i.e., possible displacements where time is
frozen. The expression for virtual work for a distributed system takes the following form:
fluid forces, and point forces due to the contact of the fibre with solid boundaries. As shown
in Figure 3.2.3-l., al1 forces acting on the fibre may be expressed in terms of radial and
tangential components.
O Figure 3.2.3-1. Forces acting on an initially undeflected fibre.
Here FN denotes the normal component, and FT denotes the tangential component of the
distributed and the point forces, f,, f,, and F,,, F,, respectively. The normal component is
defined positive in the outward direction, and the tangentiai component is positive in the
counter-clockwise direction, as shown in Fig. 3.2.3- 1.
The transformation from tangentialhormal coordinates to global Cartesian
coordinates is
The virtual work perforrned over the elementary virtual displacements 6R = (a,
6Rv) by the real distributed forces f,,f,, and the real point forces F., F, as defined in Figure
3.2.3- 1 ., is expressed as,
6 W = r j[ f (a) cos@ + 0) - f, (a) sin@ + SR, @)da O
+x [Fn, sin(a, + 9) + F,, cos(a , + O ) ] SR, (a, ) J
w here
Taking partial derivatives from (3.2), and substituting them into (3.26) and (3.27),
gives
(3.28)
SRy = 6% + ( r + u) cos(a + 0)60 + sin@ + 9)Su . (3.29)
Substituting (3.28) and (3.29) into (3.25) yields the following expression for the virtual work
w here
Ur)
Fx = r 1 [f, (a) cos(a + 0) - f , (a) sin(a + e)]da O
3.3 Exact Integro-Differential Equations of Motion
Tn this section, we formulate the equations of motion in the integro-differential form in terms
of the generalized coordinates Xo, Y*, O, u. For Hamilton's principle t o be satisfied over any
period of time, the integrand of (3.4) must be equal to zero, i.e.
Since 61: m/, and SW are functions or functionals of the following variables,
6V = 6 v(u(-), un(-)) ,
6W = 6 w(x,, Y,, 9, u(-)) .
to formulate the equations of motion using (3.35), in terms of the generalized coordinates XO,
Yo, 8, and u, it is first necessary to express the variation of kinetic energy 6T as a function of
Xo, Yo, 8, u(-) , and the vinual potential energy m/ as a functional of u(-) .
Equation (3.13) is thus expanded to the form:
d Since the derivative operation - and the variation operation 6 , the latter
dt
corresponding to frozen (fixed) time, are commutative, (3.37) can be rewritten in the form
To use this expression in Hamilton's principle (3.4), the variations SXtl(t), &Yo(i),
&Hi), are taken in such a way that they vanish at the ends of the time interval [t,, I I ] in (3.4),
over the period [t ,, tJ, yields:
since Tko ( t , )Gx , ( t2 ) = O and T ~ , ~ ( ~ , ) G x , ( ~ , ) = O . Comparing 6T in (3.38) and (3.39), we see
that the first term in (3.38) can be written in the form,
leaving the variation &(t) free of differentiation. Due to the linearity of (3.4) and (3.38),
the above process can be done separately and independently for every mixed term containing
d the double operation - 6( ) , yielding explicii formulae linear in independent2 variations
dt
&(t), mt), and GB(t), of 6r. The expression for 6T c m now be written in the reduced
form of
' Independent variations &(t), 6Y0(t), b&f), & ( K I ) , are at frozen time and fixed a and tliereforc do not depend on any other values or parameters.
an "0
- p ~ r ( ë sin 0 + 0' cos O)/ u cos otda + p ~ r ( - 9 cos 0 + g2 sin 8) u sin cxda . O O
dT. Y, - - pArao% + p ~ r 2 ( 0 sin 0 + 0' cos û)(cos a, - 1)
dt
a, a0
+2p~r6 cos 8 i cos adcc - 2 p ~ & sin 0 / li sin ada O O
an a,
+p~r (O cos 0 - 9' sin û ) l u COS ada - p ~ r ( 8 sin 8 + 8' cos 8) l u sin cula ,
+ p ~ r ' [ ( ~ cos 8 - X, sin 0) - i)(YO sin 9 + xO cos 0 sin a, )l
an ~ I I
+ p ~ r ( $ cos 9 - x0 sin 0) u cos ada - p ~ r ( $ sin 0 + xO cos 0 ) l Ù sin ada
an
+ p ~ r [ ( $ , cos 8 - x0 sin 0) - é(c sin 8 + X, cos û)] j u cos cxda O
d There is still one dependent variation in (3.41), due to the presence of 6u = -(au),
dt
where 6U depends on 61.4. However, with the same choice of 6u(a,t) vanishing at t, and t~
of the time interval, we can integrate the last term of (3.38) by parts with respect to time,
d with the term of (3.19) instead of in (3.39). The operator - is transferred from
dt
&(a,t) ont0 adjacent time dependent factors with a similar change of sign as in (3.39).
This is possible if we choose an appropriate Gu(a,r) such that &(a,r,) = O , and
6u(a,t2) = O for al1 a E [O, aO] , so that the integrated term with respect to tirne vanishes as
in (3.39). This yields the term in (3.4 1) with the independent variation &c(a,r) in the
form:
a11 a11
r [ (g sin 0 + x0 cos 0) + 8(- cos 0 + x0 sin O ) ] 1 cos a6udcc - p~ r ~ 6 u d u a o
The variation of kinetic energy 6T corresponding to the independent virtual
displacements can now be expressed as
where Tl is given by (3.42) and T2 is given by (3.43). Coefficient T3 is found by cornbining
(3.16) and (3.44) as follows:
T, = - p ~ r ~ a , ë - P ~ r 2 [ ( ÿ ~ sin 8 + .&, cos 0)](cos a, - 1) - p ~ r ' [(c cos0 - sin e)] sin a, (3.47)
a, ' 4 1
cos 0 - X, sin 0 ) ] / u cos c<da + p ~ r [ ( ~ sin 8 + X, cos û)]l u sin ocda O O
The variation of potential energy m/ of (3.2 1) can be expressed as
an
6V=- (u" + u)(6uW + Gu)& rJ O
Substituting the virtual work (3.30), the variation in kinetic energy (3.46) and the
variation of potential energy (3.49) into (3.33, collecting like terms with respect to &(t),
&(t), Wt), and the sum of terms containing 6u, 6u': gives the exact integro-differential
equations of motion for a flexible cylinder with curved initiai shape.
equations of motion for the generalized coordinates a,, 6Y,,, 68, and 6u are found to be:
for :
a0 a11
+ 2 p ~ r 6 sin 0 u cos ada + ZPA& cos 8 1 Ù sin cida O O
for 6 Y o :
Fy - p ~ r a o - pAr2 (0 sin 0 + 0' cos 0)(cos a, - 1)
-2pAr-b cos0 Ici cos ada + Z ~ A A sin 0 [ù sin cxda O O
F, - pAr3a0ë - p ~ r ' [ ( C sin 0 + X, cos û)](cos u, - 1 ) - p ~ r ' [ ( C cos e - % sir1 O ) ] sin a, (3.52)
for terms containing 6u:
a )
- p ~ r [ 2 0 ( - 6 cos 0 + x0 sin 0 ) + ( f., sin 0 + & cos û)] 1 cos a8uda O
a,
- p ~ r [ 2 9 ( % sin 0 + X, cos^) - (E cos 0 - & sin O)] 1 sin a 6 u h
Where, Fx, Fy, Fe are taken from (3.31), (3.32), and (3.33).
In equation (3.53), arbitrary variations 611 and dependent values u" and 6u"cannot
be canceled out, since they depend explicitly on a. Moreover, equations (3.50) to (3.53)
contain unknown functions I r , ri, ü, u" under the integral signs. This requires further
analysis and simplification, and is presented in the following chapters.
Chapter 4
Orthogonal Expansion of Deflection Function
In deriving the governing equations of Chapter 3, the deflection relating to the lateral
vibration of the fibre was denoted as u. In order to numerically solve the equations, it is first
necessary to develop an explicit expression for the deflection variable u by representing it as
a function rr = u(a, t ) . A solution for plane flexural vibrations of a circular ring in
Timoshenko [53] will be modified to apply to a circular arc.
4.1 Circular Ring
Timoshenko [53] used harmonic analysis to develop a particular solution for the
radial deflection of a circular ring. For a circular ring with the configuration as shown in
Figure 4.1-I., the radial deflection u(9,t) can be represented by a Fourier series without a
constant term. The constant term corresponds to a constant initial deflection which is absent
in an initially undeflected circular ring. Thus, we have
34
Figure 4.1-1. Circular Ring.
w here
t is time,
a , ( t ) , a2 (t).. . . . bl (t).b2 ( t ) . . . . are functions of tirne,
O is an angle deterrnining the position of a point on the ring,
MW, t ) is the radial deflection of the ring at tirne t and at position 8.
In accordance with the Euler-Bernoulli or Thin Bearn Theory [3], shear deformations are not
taken into account, and thus the tangential deflection, v(8, t), is not considered.
Decornposition of rr (8 . t ) . (4.1), is based on the well-known trigonornetric orthogonal
system
with the constant first term of the systern omitted. The constant term i = O, corresponds to a
pure, biased, radial deflection which is not considered in a statically balanced system.
The system (4.2) is orthogonal on [0,2n], i.e. it satisfies the following orthogonality
conditions:
j cosmûsinmûdû = O, Qm O
2 II 2 Ir
jcos' mûdû = jsin2 m6dû = a, 'dm O O
However, these conditions are valid only for the interval of (O, 2n). Our fibre is
modeled as a circular arc increasing from angle O to any arbitrary angle a, I n as shown in
Figure 4.1-2.
Figure 4.1-2. Circular Arc.
4.2 Circular Arc
In order that orthogonaiity conditions be satisfied on an interval of (O,@), an elementary
substitution is necessary. For the circular arc of Figure 4.1-2., with a increasing from O to
1 L any arbitrary a, 5 n , let 8 = -a. Furthemore, for the undeflected fibre u = O at a = O , a
a0
condition that is not satisfied by the cosine terrns in the system. The cosine terms are thus
ornitted, leaving only the sine terrns. This results in both a simpler and more accurate
representation of our deflection function. Making these changes in the system (4.2) yields
the following orthogonal system:
{.in[$ a), .in 2 [ ~ .), . . .}
where
7, ( t ) are functions of time,
a. is half of the extended period.
Taking a finite sum of m first terms in (4.5). the deflection u(a,t) can be represented in the
following rnanner:
or in vector forrn,
where Qi(a) are the admissible functions [s in(ka) , sin(2ka), sin(3ka), . . . , sin(mka)] that
depend only on position, and q(t) are the displacement coefficients
JL k -- - , and m is the highest mode of natural vibration. a0
Equation (4.6) represents the shape up to the i-th mode of natural, radial vibrations of
a circular arc of angle a, with both ends free. This equation will be used to discretize the
deflection function u, and thus the equations of motion obtained for a curled, flexible fibre in
a shear flow. The equations of Chapter 3 contain terms with the infinitesimal displacements
624 and 6u" . These terms, in the format of equation (4.6), can be found in the following
manner:
6u = eT& + T ~ ~ Q
but since oniy lateral vibrations are of concern, 6QT = 0, and
6u = Q'&
also,
where Q(a) and q t ) are colurnn vectors.
Chapter 5
Kinematic Inertialess Equations
With the deflection function being discretized in Chapter 4 by (4.7), and the
infinitesimal virtual displacements being given by (4.9) and (4. IO), the integro-differential
equations of Chapter 3 will now result in m + 3 coupled ordinary differential equations of
the second order. For rn = 3 , this corresponds to a dynamic system in 6 - LI, with six
equations and six unknowns.
5.1 First Kinematic Approximation
The equations of motion of Chapter 3 constitute the exact integro-differential
equations of motion for an initially cuwed flexible fibre of arbitrary mass. The absolute
mass of a pulp fibre in suspension is small when compared to its translational velocity and
acceleration, and to its rotational and vibrational components, and can be neglected (i.e. set
analysis considerably.
The resulting equations of motion become:
for S.Xa :
for f i :
for 68:
for terms containing 6u:
Substituting the expressions for the forces. namely (3.3 I) , (3.32), (3.33), and (3.34),
the resulting equations become:
for :
r j[ fn (a) sin(a + 0) + f ; (a) cos(a + da + c [ F , sin(a + 8) + F , cus(cr, + O ) ] = 0 ; O i
for 68:
for t ems containing au, the equation remains unchanged as:
The expressions of the fluid forces in slow and medium flows are linear with respect
to the relative velocities of the fluid and the fibre, and are given as,
where U , and U, are the normal and tangential components of the fluid velocity, V, and V,
are the normal and tangentiai components of the fibre velocity, and k, and k, are constants
that depend on the geometry of the fibre, see (2.2) and (2.3).
As in [26], the functions k, and kt can be taken as Cox's [6] first order
approximation:
This section develops the inertialess equations which can be represented in matrix
notation. Substituting expressions (5.9) - (5.12) for the fluid forces into equations (5.5) -
(5.8), and simplifying, we have:
for :
a n
rkn 7 cos(a + 0)da - rk, y sin(a + 9 ) d a = O O
for & :
a 1 a 1
rk,, / y, sin(a + 8 ) d a + rk, 1 cm(a + 0)da =
for 6t3
a t EI- a't a t
rk,, j ~ , G u ( a ) d a + -J; (u" + u)(Gu" + Su)da = rk, 1 ~ , ~ c r ( a ) d a + 2 ~,,,&(a, ) . (5.16) O O O J
The above equations can be put into the following matrix form:
The terrns on the right hmd side of equations (5.13) - (5.16) correspond to the
fluid velocities and the contact forces. They will be used to make up the F, vector. The
terms on the left hand side of equations (5.13) - (5.16) correspond to the fibre veIocities and
will be used to make up the sirnplified M, matrix and the K vector for the discretized
coordinates. The expressions developed in Chapter 4 are used to discretize the equations.
Namely,
m m
.(a, f ) = 2, ( t ) ~ , (a) = 7, ( t ) s in( ika)
with
term, can now be calculated using expression (5.18) and (5.19).
After simplification of the equations, the F, matrix for rrz modes of motion is given
as:
(5.20)
a,, a n an au
rk, c o s 8 j ~ ~ cosudn- rk,, s i n û I ~ , sinadu-rk, c o s 8 1 ~ , s inada- rk, s i n û l ~ , cosada O O O O
a n QI! 1 rk,, cos 04 Un sin &a + rk, sin 8 J Un c m aria + rk, cos 9 1 U, cos or?a - rk, sin 0 sin ado O O O
The diagonal stiffness matrix K is of size rn x rn , and after discretization is given as:
Substituting the necessary expressions for Q, and Q , ~ in the above expression for K, yields:
The vector of generalized coordinates q is given as:
Matrix Mv(q) is a (3+ m) x (3+ m) square matrix of coefficients to the fibre
velocities. The directions for the fibre velocities V. and V, are the same as those in Figure
3.2.3-1. for the forces FN and FT, and are derived by inverting (3.23):
convenience as:
Substituting equations (5.25) - (5.28) into (5.13) - (5.16), yields a system of six
equations for six unknowns and can be represented in matrix notation. Much simplification
and manipulation is required to obtain Mv in a closed analytical form. The 3 + rn rows of
the ( 3 + m ) x (3+ m ) Mv matrix are presented according to the following scheme:
Where, Row 1 is made up of:
Row 2 is made up of:
. . . ~ ( i ) l = rk,, (b, cos0 -n i sin O) .
i= l
D A2 = (rk,, - rk , )2 ,
)Il
T ( i ) Z = rk,, ( r i i cos9 + bi sin 8) . i=I
m rn I
C3 = rJk,a, , + 2 r 2 k , C c i C + rk, ç liri , i=l i = l
~ ( 3 + j ) = rk, (b, cos9 - u , sin 9 ) ,
I , is the integral inside the summation sign, and is solved by cornputer.
The expressions in the Mv matrix make use of the following constants:
1 i ( i k - 1 ) sin(ik + I ) c ~ , - ?[ (ik-1) ( ik + 1) 1
sin 2a,
sin' a, 17
Chapter 6
Numerical Mode1
A numerical mode1 to simulate curved fibre motion implementing the kinematic
inertialess equations of Chapter 5 is developed in this chapter. A C++ code utilizing the
Runge-Kutta method is used for the simulations.
6.1 Runge-Kutta Method
The equations of the analysis, equations (5.7) and (5.8) contain the harmonic
representation of the deflection function u. These equations will be solved numerically for
the normal modes of u and stepped through the flow field. The numerical methods of
current interest are concerned with obtaining approximate sotutions for initial value
problems of the form
These approximate solutions are found at particular, discrete points of x using only the
operations of addition, subtraction, multiplication, division, and functional evaluations.
These points are: .uo, x, , x? , . . ., xn where x, - xn-, = x, - .u, = x, - xo = h ; Iz being an
arbitrary stepsize specified by the user. In general, a smaller h value yields a more accurate
approximate solution. Numerical methods are referred to as being of order n; n being zi
positive integer refening to the exactness of a particular numerical method for polynornials
of degree n or less. Thus, if the true solution to which a numerical approximation is sought
is of order n or less, the tnie solution and the approximate solution are in fact identical.
Generally, the higher the order of a particular solution, the greater the accuracy.
There are a wide variety of numerical methods currently in use to solve initial value
problems. The particular numerical method deployed depends on the problem at hand. Our
deflection function profits from a relatively simple Fourier series representation, well suited
to the classical 5'h order Runge-Kutta method with Cash-Karp constants and variable step
size.
The Runge-Kutta method is based on the Euler method which advances n solution
from x, to x,,, with a stepsize of h. The Euler method can in fact be thought of as a fïrst
order Runge-Kutta method, usually not recommended because of its low accuracy and
sets of N coupled 1" order differential equations of the form:
where the yi are functions which are derivatives of one another. The Runge-Kutta method is
self starting and combines information from several Euler-type steps to propagate a solution
over an interval. This solution is being used to match the Taylor series expansion of the
function sought up to some higher order.
The 5th order Runge-Kutta method is of the form:
k, =hf ( X , . Y , ) 3
k2 = hf (x,, + cz,h, y,, + h I k , ) ,
k, =hf ( x , +a,h,y, +b,,k,+...+b&),
y,,, = y,! + c , k , + c , k Z +c,k, +c&, +c,k, +& +0(h5) .
= y,, + c'ik, + c*lk2 + c'.fk, + c',k, + c'5k, + cB6k6 + 0 ( h 5 )
and the error estimate is
where the values for the constants ai, b, , ci , c'i , and b, are the Cash-Karp parameters, see
Press et al. [37]. The Runge-Kutta method for systern (6.2) implies for our case,
to give:
(5.29), respectively.
The algorithrn used to solve the motion of a flexibIe curved fibre in a known flow
field is defined in Section 6.2. An existing code for straight fibre flow [26] was modified
such that it applied to the curved fibre case.
6.2 Computational Algorithm
The integro-differential equations of motion of Chapter 3 contain an unknown
variable u which represents the fibre deflection. A harrnonic representation in the form of a
half-range Fourier sine series is used to discretize u, and the equations of motion are solved
via a Cc+ irnplemented Runge-Kutta routine for the generalized coordinates
6 X o , SY,, 60, and 6ti .
The numerical code used for the fibre motion simulations is a modified version of the
original 10 000 line code Lriwryshyn 1261 wrote to simulate flexible fibre motion of initial
straight fibres in a slot geometry. The modifications are necessary in order to consider an
initially curved fibre shape. The primary solution steps are:
-
Read an input file containing the fibre properties and initial positions. fluid parameters,
Runge-Kutta parameters including the desired number of steps and accuracies, time
parameters, and output files.
Set the initial Runge-Kutta conditions and fibre and fluid constants.
Cal1 the Runge-Kutta driver routine.
Update Mv, F v , and K
Solve for new fibre positions, i.e. y,,,
Estimate error using equation (6.5).
Keep the error within the desired bounds by comparing it to the vector of
desired accuracies read in (step 2). Depending on whether the error is greater
or less than the desired accuracy, reduce or increase the stepsize respectively,
and retry the step.
Save fibre shape and velocity data of current time.
Repeat steps 4 and 5 until maximum time or specified position.
The program is able to:
Test the numerical mode1 in a simple shear flow
Simulate the motion of a curved rigid or flexible fibre in complex 2-D
flows.
Simulate the motion of an essentially straight rigid or flexible fibre in
comdex 2-D flows.
Chapter 7
Model Validation and Applications
The dynamic model of the motion of curved flexible fibres will be tested for a well-
defined flow field. An essentially straight fibre case will be cornpared to analytic straight
fibre results and to prior dynamic straight fibre results of Lawryshyn and Kuhn [27]. The
model is then used to simulate the motion of a fibre in a channel and slot geometry so that
the effect of curvature on fibre passage c m be investigated.
7.1 Model Validation
To compare dynarnic curved fibre motion results to static andytic results and
dynamic straight fibre motion results, the motion and finai deflection of a straight fibre in the
flow field defined in Figure 7.1 - 1. is considered.
Figure 7.1-1. Straight fibre flow field orientation.
The straight fibre is oriented vertically in the centre of the global X-Y coordinate axis with
L a = - . Thus the fibre is symmetric with the flow field, and does not translate or rotate but
4
is only deflected by the flow field. To achieve a similar test case as that for a straight fibre,
the curved fibre is also oriented vertically and a sufficiently large radius of curvature and
correspondingly small angle are chosen.
The deflection of a stütically constrained straight beam element is, see [26]:
w here
6 is the deflection,
Fm,, is the maximum applied force,
L is the length of the beam element,
EZ is taken to be I x 10''~ ~ m ' where E is Young's modulus of elasticity and 1 is the
moment of inertia about the z axis.
Fmu is estimated from Cox's equation for the normal component of the fluid force, see (2.2),
N where Un = U,*, and V. = O, yielding a value of 9.6 x IO-' - for F,,,. The static deflection
rn
is then determined to be 0.28 mm from equation (7.1)
7.1.2 Dynamic Curved Fibre Motion
To achieve an essentially straight fibre case, a sufficiently large radius of curvature
and srnall angle is selected for the curved fibre. The product of these two values is equal to
3 mm, i.e. the analytic straight fibre length. The flow field is oriented sirnilnrly to that of
geometric parameters for the curved fibre test case in the local x--Y axis are shown in Figure
7.1.2-1.
Figure 7.1.2-1. Curved fibre orientation.
To achieve this orientation, the following values are set:
8= - d 2 ,
Ra = 3 mm,
C R a --- - sin - 4 2 2 '
the global X-Y axes as for the straight fibre case. The flow configuration for the curved fibre
with respect to the gIobal X- Y axes is shown in Figure 7.1.2-2.
Figure 7.1.2-2. Flow configuration with respect to the global X-Y axes.
As a test case, it is assumed,
Xo = -R,
7.1.3 ResuIts and Discussion
The motion of the curved fibre with radius of curvature = 85.9 mm, angle = 2' , and
three modes of motion, is shown in Figure 7.1.3-1. As shown, the fibre is initially
essentially straight at t = O rns and reaches its steady state deflection by ? = 3 ms.
Figure 7.1.3-1. Deflection of fibre for t = O to t = 3 ms, for three modes of motion.
different radius of curvature and angle combinations. As the radius of curvature increases
and the angle decreases, the curves converge to analytic straight fibre results, and to the
dynamic straight fibre case of Lawryshyn and Kuhn [27] as shown in Figure 7.1.3-2. Radius
of curvature and angle combinations below 85.9 mm and 2' respectively, were found to
overlap with the straight fibre case, i.e. there is excellent agreement between the present
results and those of Lawryshyn and Kuhn for essentially straight fibre cases.
-0- Rad = 85.9 mm; Angle = 2 deg
+ Straight Fibre Case (Lawysh yn [ 2 6 ] ) - Analytic Straight Fibre Results [26]
time (ms)
Figure 7.1.3-2. Graph of deflection vs time for fibres of 3 »lm lengths and different radii of
curvature.
and different radii of curvature. It is interesting to note that as the radius of curvature
increases, and thus the fibre length as well, it takes longer for the deflection to mach a steady
state value.
time (ms)
Figure 7.1.3-3. Graph of deflection vs tirne for an angle of 0.0078125' and different radii
of curvature.
- - - - -
curvature and different angles. As the angle and the fibre length increase, the curves take a
longer time to reach a steady state deflection.
- 0.03 125 deg
4 7.812%-3 deg
+ 5.20&-3 deg
+ 3.90625~-3 deg
time (m)
Figure 7.1.3-4. Graph of deflection vs time for a radius of curvature of 22 rn and different
angle combinations.
mm, and angle of 0.5" for r i z = 0, 1, 2, 3, and 4 modes of motion. The tïrst and second
modes are shown to overlap, as are the third and fourth modes. Although one mode of
motion is sufficient to accurately represent the curved fibre for the simple flow field of
Figure 7.1.2-2., it is anticipated that three modes of motion will be needed for more cornplex
cases.
O 0.5 1 1.5 2 2.5 3
time (m)
Figure 7.1.3-5. Graph of deflection vs time for rn = 0, 1,2 ,3 ,4 modes of motion.
A fibre's motion into a channel with a slot serves as a simpIified mode1 of fibre
screening, and is an important application of this research. Several simulations involving the
motion of an initially curved fibre in a channel and slot geometry are run and compared with
the results of Lawryshyn [26] . (The flow field used is shown in Figure 7.2-1. with a slot to
channel velocity of 0.67, i.e. U,r /U, = 0.67 .)
Figure 7.2-1. Strearnline and velocity vector plots of flow field. U, /U,, = 0.67. (Plots
taken from Lawryshyn [26]).
The initial fibre position is shown in Figure 7.2.1-1. The midpoint of the fibre is
located 10.25 mm from the slot centerline, and the values Hfand HEL correspond to the fibre
starting height and the height of the exit layer, respectively. For a fibre to enter the slot, it's
initial position must be below the exit Iayer HEL. Each fibre possesses a specific Hf value
above which dot entry is impossible. This value, called Hf.-r determines fibre passage.
Fibres initiated into the channel at or below their respective HfcNt values enter the dot, but
übove the HI;.,, value do not.
Not to Scale
Figure 7.2.1-1. Initial Fibre Position.
A Streamline of Exit Layer
A ~ i b r e A !
The HI,,, value was determined by a process of trial and error. Simulations were run for
various Hf values until the leading edge of the fibre contacted either the downstream slot
wall, the downstream channel wall, or the dot and channel corner. Contact with the slot
wall ensured slot entry whereas contact with the channel wall held a high probability of non-
Hf Channel ; v V ! !
i
1 !< ! 1 0. 25mm
O.5Omm
! \ ! /!
! t Slot
< 1
I 3 I
lower bound for was determined as the maximum Hf value for which the leading edge
of the fibre contacted the slot wall. The values for the fibre parameters used and their Hf;.rit
values are depicted below.
1. nylon: a = 30'. R = 5.730, EI = 329 x IO-" ~m'; H,, = 0.36
2. kraft: a = 30', R = 6.875, E I = 4.4~10-" ~ m ' ; Hf,, = 0.39
Figures 7.2.1-2 and 7.2.1-3 are typical plots of fibre trajectories used to determine the Hfint
vaiues. Figure 7.2.1-2. shows a kraft fibre entering the slot at its Hf,,, value, while Figure
7.2.1-3. depicts a nylon fibre missing the slot at a value higher than its HfCNf value.
Figure 7.2.1-2. kraft: a = 30°, R = 6.875, EI = 4.4 x 10-" Mn' ; Hf ,,, = 0.39
Figure 7.2.1-3. nylon: a = 30', R = 5.730, El = 329 x 10-" ~ m ' ; Hf,,, = 0.45
7.2.2 Discussion of Results
Lawryshyn [26] proved that for straight fibres, short or flexible ones have higher Hf
values than longer or stiffer fibres. The simulations run in this chapter investigate the effect
of fibre curvature on Hf values. In each of the two cases tested, curvature played a
significant role in determining whether or not the fibre entered the slot. Below are shown
H&,, values for both the straight fibre and curved fibre case of two different fibre types.
Straight fibre parameters:
1. nylon: L = 3mm, EI = 329 x IO-" Nm2; H,,, = 0.20
2. kraft: L = 3.6mm, EI = 4.4~10- l2 Nm2; Hf,, = 0.29
1. nylon: a = 30". R = 5.730, EI = 329 x IO-" ~ n z ' ; Hf,, = 0.36
2. kraft: a = 30°, R = 6.875, EI = 4.4 x 1 O-" ~ m ' ; H,,, = 0.39
In both cases, the 30" fibre curvature served to substantiaily increase the fibre's H f c N f value
from that of a sirnilar fibre with no curvature. Thus, curved fibres tend to have higher HJ
values than their stiff counterparts, allowing for significantly more curved fibres to enter the
slot.
7.2.3 Application to Pulp Screening
Pulp screening is concerned with the fractionation of pulp fibres based on their
length or flexibility. Fibre fractionation based on flexibility is of great importance, since
flexible fibres are believed to produce better quality paper. A numerical model to simuiate
the motion of pulp fibres is beneficial to investigate the advantages of a proposed screen
geometry. Presently, no numerical models of initially curved flexible fibres exist. This
research presents a numerical model able to simulate the motion of initially curved flexible
fibres and applies it to a channel and slot geornetry which serves as a simplified screen
model. The effect of curvature on fibre passage has proved to be significant when compared
to sirnilar initially straight fibres. A greater proportion of curved fibres enter the dot than
their straight counterparts.
Chapter 8
Concluding Remarks
8.1 Summary of Numerical Mode1
Elastic fibres are believed to produce better quality paper of superior strength,
printability, and optical properties than stiff fibres. Thus, fibre flexibility is of great interest
to the Pulp and Paper industry. Although much experimental work has been done on
straight fibre motion, very little has been attempted on initially curved fibres. Numerical
research, specifically that of Lawryshyn and Kuhn [27], has modeled the motion of an
initially straight flexible fibre; no numerical modeling of curved fibre motion has been
found in the literature. This research expands upon the previous work of Lawryshyn and
Kuhn to mode1 the motion of an initially cuwed flexible fibre in a shear flow.
The curved flexible fibre is modeled as a circula arc. A set of generalized
coordinates define the absolute motion of any point dong the fibre. Becausc of the
complexity of Our force analysis, Hamilton's Principle is used to formulate the equations of
deveioped and used as the entries to Hamilton's Principle. The equations thus derived
represent the movement and deflection of every point on the fibre.
The mathematical model derived with Hamilton's Principle contains an unknown
deflection variable as a generalized coordinate. To solve for the deflection variable, it is
discretized using the Rayleigh-Ritz assurned modes method. This method discretizes the
function, not the flexible cylinder and is shown to yield faster convergence [54]. By the
method of separation of variables, the deflection function can be represented by a half-range
Fourier Sine Series with time dependent coordinates. Following the sarne approximation as
in past numerical work [26], al1 inertia or mass terms in our equations are set to zero and a
first kinematic approximation is obtained for our equations. These kinematic inertiaiess
equations are written in matrix notation and, since a11 initial boundary conditions are known,
are solved using a Runge-Kutta method implemented in a CU code. The output files of the
code plot and calculate the fibre deflection.
The numerical model for curved fibre motion has been vaIidated by comparing it
with proven results for straight fibre motion [26]. Several simulations of fibres of a
specified length having large radii of curvature and small angles have been run and
compared to a straight fibre of similar length. As the radii increases and the length
decreases, the results show very good convergence to the straight fibre case.
This research can be applied to fibre passage through different vessels , as well as
other measurement and processing techniques. The mode1 has been applied to approximate
the flow of a fibre in a channel and slot geometry; an application important to the study of
pulp screening and fractionation. Several simulations have been run for two different fibres
having a radius of curvature of 30' and compared to similar fibres of a straight
configuration. It was found that fibre curvature significantly affected fibre passage values.
The curved fibres had Idfc,,, values in the range of 35% to 80% greater than their straight
fibre counterparts. The higher the Hkn, value, the greater the number of fibres capable of
entering the slot. Thus, curved fibres have been shown to have a higher rate of slot entry
than straight fibres.
8.3 Suggested Future Work
In this research, a mode1 for the motion of a cumed flexible fibre in a shear flow, as
well as general dynamic equations of motion, have been developed and validated. A first
kinematic approximation was obtained and the equations were solved using a C++
implemented Runge-Kutta code. Several simulations were run for different fibre cases to
investigate the effect of curvature on fibre passage. It would be promising to further
investigate this effect by running more detailed simulations, as well as devising a code for
the general dynarnic system of integro-differential equations with fibres having positive
masses.
in the pulp and paper industry have a kinked initial shape as shown in Figure 8.3-1; very
Iittle research on this configuration has been atternpted thus far.
Figure 8.3-1. Possible kinked fibres.
Although theoretical research on the interaction of fibre networks exists, see for
example Soszynski [44], very little work on numerical modeling of curved and kinked fibre
interaction has been found in the literature. These topics remain a possibility for future
work.
Chapter 9
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Appendix
Inertialess Equations for m = 3 Modes of Motion
For m = 3 modes of motion, the inertialess Mv matrix is 6 x 6 and the
following discretized deflection function applies:
3 3
u(a . t ) = 7, ( t ) ~ , (a) = 7, ( t ) sinika
with
sin 3 k a
ri ( t ) =
9
The F, matrix for m=3 modes of motion is given as:
an an a, a11
rkn cos 0 Un cos orda - rk, sin 8 1 Un sin a d a - rk, cos 0 J U, sin nada - rk, sin 0 1 Ur cos adct
%I a n an *O
rk,, c o s O J ~ , , s i n a d a + r k , , s i n û J ~ , , cosada+rkr c o s û l ~ , c o s a d c r - r k , s i n e j ~ ~ s inada O O O O
Substituting expressions for Q, and in the above expression for K, yields:
EI, a" - j x [ j 2 k 4 - ( I + j . ')k2 + i ] ~ , ' s i n jkas inkadc l for 62, r3 0 j=/
EI an 3 5 E [ ( 3 j ) ' k 4 - (3? + j ' )kz + l]~,' sin j k a s i n 3kccda for 6 ~ 3 r3 * ,=/
The six rows of the 6x6 Mv matrix are presented according to the following scheme:
Here, Row 1 is:
D B l = (rk, - rk , ) - j - .
(A. 10)
(A. 1 1)
Row 2 is:
T ( ~ ) I = rk,, (b, cos0 - a , sin 8) .
~ ( 2 ) 1 = rk, (b, cos8 - a, sin O ) ,
T(3)I = rk, (b, cos8 - a, sin O ) .
D A2 = (rk, - rk,)T ,
T (1)1= rk, (a, COS 0 + 6, siil 19) ,
(A. 13)
(A. 14)
(A. 15)
(A. 16)
(A. 17)
(A. 18)
(A. 19)
Row 5 is made up of:
B4 = rk, (a , COS 8 + b, sin 6) ,
rk sin 2 ka, ~ ( 1 ) 4 = 7(ao -
rk, (sinLa, sin 3kao ~ ( 2 ) 4 = - --
2
A5 = rk,(b, cos8 - a, sine),
85 = rk, (a, cos 9 + & sin 9) .
and Row 6 is:
rk sin 4 k a o 7@)5 = $!-(ao -
A6 = rk, (6, cos0 - a, sin 8) .
B6 = rk, (a , cos9 + 6, sin 8) ,
C 6 = 0 1
sin 6 ka, ~ ( 3 ) 6 = +(a, - 6k . 1
The constants (5.46) - (5.52) apply to the above equations as in Chapter 5.