RJS-2008-shayam-perera.dvi RUHUNA JOURNAL OF SCIENCE Vol. III, September 2008, pp. 44–52 http://www.ruh.ac.lk/rjs/ I S S N 1800-279X © 2008 Faculty of Science University of Ruhuna. Comparison of Flow-induced Crystallization Melt Spinning Processes S.S.N. Perera Department of Mathematics, University of Colombo, Colombo 03, Sri Lanka Correspondece: ssnp@maths.cmb.ac.lk Abstract. The melt spinning process for artificial fibers has been studied by many research groups throughout the world during the last four decades. However, comparison of flow-induced crystalliza- tion melt spinning processes has not yet been treated in the literature. In this study, we analyse the dynamics of the flow induced crystallization melt spinning process. Further, we study the sensitivity of the process with respect to fluid shear modulus. Non-Newtonian and Maxwell-Oldroyd models are used to describe the rheology of the polymer the fiber is made of. It has been found that the flow-induced crystallization Maxwell-Oldroyd model has an upper bound for the final velocity. Key words : Fiber spinning, Non-Newtonian, Maxwell-Oldroyd 1. Introduction The fiber spinning process is used to make all types of synthetic textile fibers (nylon, polyester, rayon, etc.). In the melt spinning version of the process, molten polymer is extruded a die called a spinneret to create a thin long fiber. Far away from the spinneret, the fiber is wrapped around a drum, which pulls it away at a pre–determined take–up speed. The take–up speed is much higher than the extrusion speed; in industrial processes the take–up speed is about 50m/s and the extrusion speed is about 10m/s, see (2, 4). The ratio between the take–up speed vL and the extrusion speed v0 is called draw–ratio and denoted by d = vL/v0 > 1 and hence the filament is stretched considerably in length and therefore decreases in diameter. The ambient atmosphere temperature is below the polymer solidi- fication temperature such that the polymer is cooled and solidifies before the take–up, see Figure 1. In industrial processes a whole bundle of hundreds of single filaments is extruded and spun in parallel, however for the analysis we consider a single filament. The dynamics of melt spinning processes has been studied by many research groups throughout the world during the last decades starting with the early works of Kase and Matsuo (3) and Ziabicki (10). Despite great scientific progress in the dynamics of fiber formation processes, especially in flow induced crystallization process, there are still some unsettled issues. For example, comparison of flow-induced crystallization melt spinning processes has not yet been treated in the literature. The sensitivity of the Maxwell-Oldroyd model (both isothermal and non-isothermal) with respect to the characteristic relaxation 44 S.S.N. Perera: Comparison of Flow-induced ... Ruhuna Journal of Science III, pp. 44–52, (2008) 45 Figure 1 Sketch of the melt spinning process. time has been discussed in the literature without considering the crystallization process (7, 8). To investigate the same concept with the crystallization process is quite interesting from both the theoretical and industrial points of view because it is closely related to high quality control of products and its theoretical analysis involves the fundamental understanding of the nonlinear dynamics of the process. In this study, we analyse the behaviour of the flow induced crystallization melt spinning process using non-Newtonian and Maxwell-Oldroyd models. Further, we study the sensitivity of the flow induced crystallization process with respect to the fluid shear modulus. 2. Melt Spinning Models Considering the basic conservation laws for the mass, momentum and energy of the viscous polymer jet, one can obtain the following set of equations, by averaging over the cross– section of the slender fiber, see (4, 5, 6, 7). ρAv = W0 . (1a) ρAv dv dz = dAτ dz − √ A π Cd ρair v2 + ρAg , (1b) ρCpv dT dz = − 2α √ π √ A (T − T∞) + ρ∆H v dφ dz , (1c) In the mass balance (1a), A denotes the cross–sectional area of the fiber, and v is the velocity of the fiber along the spinline. The density ρ of the polymer is assumed to be constant. In the momentum balance (1b), z denotes the coordinate along the spinline and the axial stress τ is related via the constitutive equations (for the non-Newtonian case equation (1d) and the Maxwell-Oldroyd case equation (1e)) τ = 3η dv dz , (1d) S.S.N. Perera: Comparison of Flow-induced ... 46 Ruhuna Journal of Science III, pp. 44–52, (2008) τ + λ ( v dτ dz − 2τ dv dz ) = 3η dv dz (1e) to the viscosity η and characteristic relaxation time λ. In the energy equation (1c), T and Cp denote the temperature and the heat capacity of the polymer, T∞ is the temperature of the quench air and α denotes the heat transfer coefficient between the fiber and the quench air. According to (4), we assume the following relation for the heat transfer coefficient α = 0.21 R0 κRe 1 3 air [ 1 + 64v2c v2 ] 1 6 depending on the Reynolds–number of the quench air flow Reair = 2vρair ηair √ A π . Here R0 is the radius of the spinneret, ρair, ηair and κ represent respectively the density, viscosity and heat conductivity of the air and vc is the velocity of the quench air. The crystallization process generates an enthalpy by change and this is represented by third term of the equation (1c) and ∆H is the specific heat of fusion of a perfect crystal and φ is the degree of crystallinity. According to (4), the model for the evolution of φ is given by v dφ dz = (φ∞ − φ)Kmax exp [ −4 ln 2 ( T − Tmax D )2 ] . (1f) Here φ∞ is the ultimate crystallinity, Kmax the maximum crystallization rate, Tmax the fluid temperature having the maximum crystallization rate and D denotes the crystallization half width temperature range. The viscosity and characteristic relaxation time are given by η = η0 exp [ Ea RG ( 1 T − 1 T0 )] , (1g) λ = λ0 exp [ Ea RG ( 1 T − 1 T0 )] . (1h) Here η0 > 0 is the zero shear viscosity at the initial temperature T0, Ea denotes the activation energy, RG is equal to the gas constant and λ0 = η0 G (G is the fluid shear modulus). The system (1) is subject to the boundary conditions v = v0 T = T0 φ = 0 at z = 0 (1i) v = vL at z = L (1j) where L denotes the length of the spinline. S.S.N. Perera: Comparison of Flow-induced ... Ruhuna Journal of Science III, pp. 44–52, (2008) 47 3. Dimensionless Form Introducing the dimensionless quantities z∗ = z L , v∗ = v v0 , z∗ = z L , T ∗ = T T0 , A∗ = A A0 , τ∗ = τL η0v0 , φ∗ = φ φ∞ , the system (1) can be formulated in dimensionless form. Dropping the star and considering the non-Newtonian and Maxwell-Oldroyd cases the system can be presented as follows dv dz = τ 3η , (2a) dτ dz = Re ( 1 3η vτ − Fr−1 + C1v 5 2 ) + 1 3η τ2 v , (2b) dT dz = −C2 (T − T∞)√ v + ∆H φ∞ T0Cp dφ dz , (2c) dφ dz = KmaxL v0 ( 1 − φ v ) exp [ −4 ln 2 ( T − Tmax D )2 ] . (2d) Rev dv dz = dτ dz − τ v dv dz + Re ( Fr−1 −C1v 5 2 ) , (3a) 3η dv dz = τ + De ( v dτ dz − 2τ dv dz ) , (3b) dT dz = −C2 (T − T∞)√ v + ∆H φ∞ T0Cp dφ dz , (3c) dφ dz = KmaxL v0 ( 1 − φ v ) exp [ −4 ln 2 ( T − Tmax D )2 ] . (3d) In system (2) and (3), Re = ρLv0η0 is the Reynolds number, Fr −1 = gL v20 is the inverse of the Froude number, C1 = Cd ρair L √ π ρ √ A0 is the scaled drag coefficient and C2 = 2αL √ π ρCpv0 √ A0 denotes the scaled heat transfer coefficient. The Deborah number De is given by De = λ0v0 L exp [ Ea RGT0 ( 1 T − 1 )] . The systems (2) and (3), are subject to the boundary conditions v(0) = 1 T (0) = 1 and φ(0) = 0, v(1) = d, where d is the draw ratio. S.S.N. Perera: Comparison of Flow-induced ... 48 Ruhuna Journal of Science III, pp. 44–52, (2008) 4. Numerical Results 4.1. Numerics Both systems ((2) and (3)) of ODE are solved using the Matlab routine ode23tb. This routine uses an implicit method with backward differentiation to solve stiff differential equations. It is an implementation of TR-BDF2 (9), an implicit two stage Runge-Kutta formula where the first stage is a trapezoidal rule step and the second stage is a backward differentiation formula of order two. Since both systems are boundary value problems, the shooting method is used to solve them. 4.2. Shooting Method Now, we present the main steps of the shooting method in general. Let y = (v, τ, T, φ). Then one can write the system (2) in the following form dy dz = f (y, u) , with y1(0) = 1, y1(1) = d, y3(0) = 1, y4(0) = 0, (4) where f (y, u) =       τ 3η Re ( 1 3η vτ − Fr −1 + C1v 5 2 ) −C2 (T −T∞)√ v + ∆H φ∞ T0Cp ϑ KmaxL v0 ( 1−φ v ) exp [ −4 ln 2 ( T −Tmax D )2 ]       , and ϑ is given as follows: ϑ = KmaxL v0 ( 1 − φ v ) exp [ −4 ln 2 ( T − Tmax D )2 ] . Let us make an initial guess s for y2(0) and denote by y(z; s), the solution of the initial value problem dy dz = f (y, u) , with y1(0) = 1, y2(0) = s, y3(0) = 1, y4(0) = 0 . (5) Now we introduce a new dependent variable x(z; s) = ∂y ∂s and define the second system as follows ∂x ∂z = ( ∂ f ∂y ) x with x1(0; s) = 0, x2(0; s) = 1, x3(0; s) = 0, x4(0; s) = 0. (6) The solution y(z; s) of the initial value problem (5) coincides with the solution y(z) of the boundary value state system (4) provided that the value s can be found such that ϕ(s) = y1(1; s) − d = 0. S.S.N. Perera: Comparison of Flow-induced ... Ruhuna Journal of Science III, pp. 44–52, (2008) 49 Using the system (6), ϕ′(s) can be computed as follows ϕ′(s) = x1(1; s). Now, using Newton–iteration, a sequence (sn)n∈N is generated by sn+1 = sn − ϕ(sn) ϕ′(sn) for a given initial guess s0. If the initial guess s0 is a sufficiently good approximation to the required root of ϕ(s) = 0, the theory of the Newton–iteration method ensures that the sequence (sn)n∈N converges to the root s. By rearranging the system (3), the function f (y, u) can be obtained for the Maxwell-Oldroyd model. 4.3. Results Figure 2 shows the spinline velocity, temperature profile and crystallinity index of the non- Newtonian and Maxwell-Oldroyd models. Concerning the temperature profile, one sees a jump in the temperature owing to the heat released due to crystallization. Further, the behaviour of the temperature and crystallinity profiles are close in both cases. 0 0.2 0.4 0.6 0.8 1 15 20 25 30 35 40 45 50 Length m v m /s 0 0.2 0.4 0.6 0.8 1 50 100 150 200 250 300 Length m T e m p e ra tu re ° C 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Length m C ry s ta ll in it iy i n d e x 0.572 0.574 0.576 0.578 0.58 156 158 Non−Newtonian Maxwell−Oldroyd 0.5 0.55 0.36 0.38 0.4 Figure 2 Spinline velocity profile(up-left), spinline temperature (up-right) profile, Crystallinity index (down-left). Figure 3 shows the velocity profile of the Maxwell-Oldroyd model depending on the fluid shear modulus. From this one sees the the velocity profile of the melt spinning process S.S.N. Perera: Comparison of Flow-induced ... 50 Ruhuna Journal of Science III, pp. 44–52, (2008) 0 0.2 0.4 0.6 0.8 1 15 20 25 30 35 40 45 50 Length m v m /s 0.175 0.18 0.185 43.8 44 44.2 G=1.1⋅ 105 G=9⋅ 104 G=8⋅ 104 G=7⋅ 104 Figure 3 Velocity profile depending on fluid shear modulus (pa). has no significant variation with respect to the fluid shear modulus. But in the simulation process we experienced difficulties when the fluid shear modulus decreased. We noticed that it is needed to use higher initial guesses for the stress variable for the lower value of the fluid shear modulus. Figure 4 visualizes the final velocity vs the initial guess for stress in different fluid shear modulus. One sees from this that for a particular fluid shear modulus value, the final velocity approaches a fixed value. For example, if we consider G = 4 · 104 pa, then the final velocity approaches 38 m/s. In other words, in this case (i.e. G = 4 · 104 pa) if we set the final velocity as 50 m/s then theoretically ODE system cannot be solved. The fluid shear modulus is related to the characteristic relaxation time; lower G yields higher λ. We can expect this behaviour since the Maxwell-Oldroyd model (without the crystallization process) has an upper bound for the final take-up velocity which depends on the characteristic relaxation time (see (7)). This means that the flow induced crystallization Maxwell-Oldroyd model also has an upper bound for the final take-up velocity. 5. Conclusions We compared the velocity, temperature and crystallization index profiles of the flow induced crystallization melt spinning process using non-Newtonian and Maxwell-Oldroyd models. The quantitative behaviour of the Maxwell-Oldroyd case is similar to the non-Newtonian case. But the qualitative behavior of the Maxwell-Oldroyd model is totally different for S.S.N. Perera: Comparison of Flow-induced ... Ruhuna Journal of Science III, pp. 44–52, (2008) 51 0 500 1000 1500 2000 20 25 30 35 40 45 50 55 60 65 Initial guess for stress F in a l v e lo c it y m /s G=4⋅ 104 G=5⋅ 104 G=6⋅ 104 G=8⋅ 104 Figure 4 Final velocity depending on the initial guesses for the stress. the lower values of the fluid shear modulus. Using numerical simulation, we have seen that the flow induced crystallization Maxwell-Oldroyd model cannot be solved with any arbitrary final velocity; i.e., The flow induced crystallization Maxwell-Oldroyd model has an upper bound for the final take-up velocity which depends on the material properties of the polymer. Theoretically, setting an arbitrary value for the final velocity may yield the spinning process unstable. Instability leads to irregular fibers or induces breakage of the individual filaments of the spinline. Clearly, this investigation is important from an industrial point of view. References [1]Bird RB, Amstrong RC, Hassager O (1987) Dynamics of Polymeric Liquids. 2nd edition, Volume 1: Fluid Mechanics, John Wiley & Sons. [2]Brünig H, Roland H, Blechschmidt D (1997) High Filament Velocities in the Underpressure Spunbonding Nonwoven Process. IFJ:129-134, December 1997. [3]Kase S, Matsuo T (1965) Studies on Melt Spinning, Fundamental Equations on the Dynamics of Melt Spinning. J. Polym. Sci. Part A 3: 2541-2554. [4]Langtangen HP (1997) Derivation of a Mathematical Model for Fiber Spinning. Department of Mathematics, Mechanics Division, University of Oslo, December 1997. S.S.N. Perera: Comparison of Flow-induced ... 52 Ruhuna Journal of Science III, pp. 44–52, (2008) [5]Lee JS, Jung HW, Hyun JC, Seriven LE (2005) Simple Indicator of Draw Resonance Instability in Melt Spinning Processes. AIChE J., Vol. 51, No. 10:2869-2874. [6]Lee JS, Shin DM, Jung HW, Hyun JC (2005) Transient Solution of the Dynamics in Low-Speed Fiber Spinning Process Accompanied by Flow-induced Crystallization. J. Non-Newtonian Fluid Mech. 130:110-116. [7]Perera SSN (2008) Phase-Space Analysis of Melt Spinning Processes, Nihon Reoroji Gakkaishi, Vol: 36 No. 4:161-166. [8]Perera SSN (2009) Viscoelastic Effect in the Non-Isothermal Melt Spinning Processes, Applied Mathematical Sciences, Vol: 3 No: 4: 177-186. [9]Shampine LF, Reichelt MW (1997) The MATLAB ODE Suite. SIAM J. Sci. Comput., Vol: 18:1-22. [10]Ziabicki A (1976) Fundamentals of Fiber Formation. Wiley-Interscience, New York.