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[i\1]®@]@[[[Ji) lJ@xrnHD@ lJ@©[}u[Ji)®D®~W J.B.Rattan ABHISHEK PUBLICATIONS CHANDIGARH-17 All rights reserved. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without permission in writing from the publisher. © Author First Edition : 2008 Published by : Abhishek Publications 57-59, Sector 17-C, Chandigarh Ph. : 707562 Fax: 0172-704668 E-mail: [email protected] 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

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