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J.B.Rattan
ABHISHEK PUBLICATIONS
CHANDIGARH-17
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© Author
First Edition : 2008
Published by :
Abhishek Publications
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Printed at : Mehra Offset, New Delhi
CONTENTS
1. Technological Analysis ofS pinning 1
2. Applications ofT extile Mechanics 18
3. Liquid Crystal Polymers 50
4. TextileMachines 60
5. Circular Motion 99
6. Transmission ofM otion 161
7. Ultra-Fine Fibers 198
8. Optical Fibers 211
9. Textile Impacts ofI mpulsive Forces 226
10 Solution Spinning 242
--------One
Technological Analysis
of Spinning
In the beginning of the history of spinning, progress in
spinning technique was mainly made by accumulating
empirical facts; that is to say, by repeating a set of procedures
such as setting a spinning, condition and measuring the
resultant properties and structures of the spun fibers. In melt
spinning, we can predict the diameter and temperature and
the tension in a running filament if the spinning conditions
and the rheological properties of a polymer used in the
spinning process are given; the predicted values are, of
course, in good agreement with the experimental results.
Such a prediction, however, can be made only when no
significant crystallisation occurs during the spinning process.
In the theory of Kase and Matsuo, mean values of
temperature and of stress over the whole area of a transverse
section of the filament were used. In the process of high
speed spinning, however, the variables such as temperature,
stress, orientation and crystallinity must be expressed as
functions of the radial distance from the central axis of the
filament as well as the distance from the spinneret. For
example, consideration of the radial distribution of these
variables is inevitable in discllssing inhomogenous structures
such as skin-core structure. In order to understand the
spinning process, it is indispensable for us to know how
2 Modern Textile Technology
structure will be formed during the process as weB as to carry
out the detailed technological analysis of the process.
For the technological analysis of spinning, three fundamental
equations are derived from the conversion of energy, the
conversion of momentum and the conversion of matter,
re~pectively. Here, for simplification, the two differential
operators are defined as follows:
a .a a
. k
V '=I-+J-+ -
ax ay az (1)
a a a a
D
- = - + v - + V -+ V
at ' ax ay 'az
Dt
Y
where t is the time, (x,y,z) the spatial co-ordinates, V(V. .. V,> V)
the velocity of polymer, and i. j and k are the unit vectors in
the x, y, and z directions, respectively.
Fundamental equations
Conversion of energy
Consider an arbitrary small-volume element dv fixed in space
along the spinning path and an enthalpy change in the
element. The fundamental equation for conservation of
energy can then be derived.
Inflow ()f "eat t"roug" conduction
We assume that the thermal conductivity of the polymer Kc is
independent of temperature T. The net inflow of heat in unit
time conducted through the xy-plane into the element, the
centre of which is located at (x,y,z), is expressed as
(3)
Technological Allalysis of Spinning 3
Accordingly, by summing such quantitIes for all the three
directions. heat transferred into the element in unit time is
KcV2Tdv (V2: Laplacian) (4)
Inflow of heat accompanying transfer of matter
For the polymer, the specific heat at constant pressure, the
heat of crystallisation, the crystallinity and the density are
expressed as C . tM/, X and p respectively. We assume that
p
C and tM/ are independent of T and that the enthalpy per unit
'
p
mass (H) is a function of T and X.
Accordingly, by summation of such quantities for all the three
directions,
-V.(p HV)dv (5)
=
Using the equation of continuity dR/dt -V.(pV) (modified
equation 4),
dp
=
-V·(p HV)dv H--pV·VH (6)
dt
Conservation of enthalpy
Here, we direct our attention to the conservation of enthalpy
within dv
(7)
= =
Since dH/dT C and dH/dX -VH, (8)
p
=
VH C VT-VH.VX
p
Substitution of this relation into equation 7 gives
4 Modern Textile Technology
Conservation of matter
The following equation is derived on the basis of the balance
of the matter which flows into the volume element.
Dp
-=-pV·V (to)
Dt
Assuming that the polymer is not a compressible material,
=
!l.V 0 (II)
Conservation of momentum
Based on the balance of the momentum which flows into the
volume element the following equation is obtained
DV
p-= - VP-[V .p] + pg (12)
Dt
where P is the pressure, in a normal sense, of isotropic fluid.
p the excess stress tensor and g the acceleration of gravity.
[V.p] is a vector, and, for example. one of its components is
expressed as
(13)
Under the assumption of non-compressibility. P has no
physical meaning. but will be determined by the boundary
condition. When the spinning direction is chosen as the
z-axis. the quantity. P:.2- P xx, corresponds to the tensile stress.
In addition to the three equations corresponding to
conservation of energy, matter and momentum, we must take
account of constitutive equations (rheological equations).
equations of crystallisation kinetics, the equation concerning
Technological Analysis oJ Spinning 5
molecular orientation (birefringence Dn) and the
thermodynamic equation of state. Before turning, to the
discussion of each of tJ'lese equations, we shall direct our
attention to the number of unknown variables and of
equations. The important equations governing the boundary
conditions are the equation of thermal conduction on the
surface of filament and that of air resistance.
Constitutive equations
Various complicated formulae have been proposed so far as
constitutive equations which relate the excess stress tensor p
to the thermal and deformation history. The simplest example
is Newton's equation of viscosity.
Equation of crystallisation kinetics
The nucleation rate of polymers at a constant temperature is
greatly accelerated by molecular orientation. For the present,
however, there is no general formula expressing the
quantitative relation between the crystallisation rate and
molecular orientation. Furthermore, the crystallisation under
molecular orientation may be different from ordinary
unoriented crystallisation, which is expressed in terms of the
nucleation rate, the growth rate of the nucleus and the mode
of geometrical growth. At any rate, the process of structural
change in oriented crystallisation has never been clarified.
This will be discussed in a subsequent part of this
chapter. If crystallisation kinetics are described in the form of
the Avrami equation, X= 1- exp(-K,At"), with increasing
molecular orientation the rate constant KA increases rapidly
and the Avrami index n decreases to unity or even below. In
reality, the A vrami equation applies only in the early stages of
crystallisation. In addition, it should be noted that secondary
crystallisation becomes prominent in the advanced stages of
crysta1l1sation. Adopting the birefringence bJz (or the tensile
