Mathematical modeling of poly(ethylene terephthalate) air drawing in spunbonding nonwoven process

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1 Indian Journal o Fibre & Textile Research Vol. 35, June 2010, pp Mathematical modeling o poly(ethylene terephthalate) air drawing in spunbonding nonwoven process Zhao Bo a College o Textiles, Zhongyuan University o Technology, Henan, Zhengzhou, , P R China Received 20 August 2009; accepted 24 September 2009 The air drawing model o PET polymer in a spunbonding process has been established to predict the ibre diameter and analysed by introducing the numerical computation results o air-jet low ield o the lat narrow slot passage o the aerodynamic device. The inluence o density and speciic heat capacity o polymer melt at constant pressure changing with polymer temperature on the ibre diameter have been studied. The eects o spunbonding processing parameters and drating assembly design parameters on the ibre diameter have been discussed. A lower polymer throughput rate, higher polymer melt temperature, higher primary air temperature, higher air initial speed, medium smaller venturi gap, higher air suction speed, higher quench pressure, smaller highness o the drating segment, larger length o the drating segment, larger length o jet oriice o the drating assembly, and bigger highness o jet oriice o the drating assembly are ound to be beneicial to the air drawing o spunbonding polymer melt as they produce iner ibres. The predicted ibre diameters agree with the experimental data. The indings show great prospects or this research in the ield o computer assisted design o spunbonding process, technology and equipment. Keywords: Air drawing, Computer modeling, Fibre, Nonwoven, Poly(ethylene terephthalate), Spunbonding 1 Introduction Spunbonding process is being used commercially as a single-step technology or converting polymer resin into nonwoven web, which dates back to 1950 s (re.1). In spunbonding process, a molten stream o polymer is extruded rom the screw extruder and rapidly attenuated into ibre using aerodynamic device with the aid o high-velocity cool air stream 1-3. The ibre diameter is aected by many processing parameters. Several researchers 4, 5 have studied the air drawing models o polymer spunbonding process. However, the reported air drawing model was quite elementary. All the models studied so ar were not based on using analytical and numerical methods. In the present work, an air drawing model based on numerical computational method has been established that diers rom those o the others as reported in literature 5-7. The eect o variation in polymer density and polymer speciic heat capacity with polymer temperature at constant pressure on ibre diameter is studied. At the same time, the eect o spunbonding processing parameters and drating assembly design parameters on the ibre diameter has also been investigated. The predicted ibre diameters agree well with the experimental data. It is ound that the model a zhaobohenan@sina.com; zhaobohenan@163.com can be applied to predict the drawing eects successully. The results show great prospects or this research in the ield o computer assisted design (CAD) o spunbonding process, technology and equipment. 2 Theoretical Considerations o Air Drawing Model The air drawing model o polymer consists o a continuity equation, a momentum equation, an energy equation, and a constitutive equation 4-5, 8. The polymer melt conditions, considered as regular distribution and steady state 9-12, and the surrounding air conditions (velocity and temperature) regarded as given unctions o axial position, are obtained by numerical simulation. Nearly all experimental data are well it by a drag orce correlation o the ollowing orms. In the literature 5-7, the density ρ and speciic heat capacity o polymer melt at constant pressure C P are considered to be constant. In act, they vary with polymer temperature T. In this work, we establish an air drawing model that diers rom those o the others, considering the eects o variation in polymer density and polymer speciic heat capacity with polymer temperature at constant pressure. The continuity, momentum, energy, dynamics balance, and constitutive equation along with the boundary conditions used or the study are discussed below.

2 146 INDIAN J. FIBRE TEXT. RES., JUNE 2010 Continuity Equation Q π 4 2 = D V ρ (1) where Q is the polymer mass low rate (kg/s); D, the ibre diameter (mm); V, the ibre mean velocity (m/s) in the spinning direction; and ρ, the speciic density o the polymer (kg/m 3 ), this quantity depends on temperature. The correlation o polymer density change with polymer temperature is explained using the ollowing relationship: ρ = (2) (5 10 T ) where T is the polymer ibre temperature( C). Momentum Equation df dz rheo 1 ( ) 2 dv π 2 = jπρ ac Va V D + ρ Q ρ D g 2 dz 4 (3) where F rheo is the rheological orce (N); z, the spinning axial position(mm); ρ a, the air density (kg/m 3 ); j, the sign carrier o the air drag orce, the j actor accounts or the act that near the spinneret the air low acts as a positive orce on the polymer melt, but on the ar side o the spinneret, the orce is negative, i.e. j is -1 or V a >V and 1 or V a < V, here V is the ibre velocity and V a, the air velocity (m/s); g, the gravitational acceleration (g/s 2 ); and C, the air riction drawing coeicient o ibre as a unction o the Reynolds number, which will vary with axial position. C was given by Matsui with the ollowing empirical ormula: -n β d (4) C =.(Re ) where Re d is the Reynolds number based on the diameter o the ibre; β and n, the itted constants, the assumed values o β and n should be 0.37 and 0.61 respectively as ar as the spunbonding process is concerned. The expression o the Reynolds number is as ollows: ρ D V Re d = a a µ ( V ) a (5) where µ a is the kinematic viscosity o air; and ρ a, the air density. Energy Equation dt dz π Dh T ( T ) a = (6) ρ QC P where T is the polymer ibre temperature; T a, the air temperature( C); h, the heat transer coeicient [W/(m 2.K)] which will vary with axial position; and C p, the speciic heat capacity o polymer [J/(kg.K)]. The right hand side term o Eq.(6 ) describes the decrease in temperature due to heat loss by the ibre to the cooling medium. Implicit in Eq.(6) is no conductive resistance in the radial direction and no viscous dissipation. The value or the heat transer coeicient can be calculated using the ollowing relationship between N u and Re d : Nu N u = γ (Re ) m (7) a d Dh = (8) K where N u (re. 17) is the Nusselt number at a position (based on ibre diameter); K a, the air heat conduction coeicient; and γ and m, the empirical constants. The assumed values o γ and m are 0.42 and respectively. Like the polymer density, the speciic heat capacity at constant pressure o the polymer melt is not considered as to be constant. In act, it also changes with its temperature. Thereore, the correlation used is given below: C p 4 7 = ( T ) (9) Constitutive Equation Polymer melt used in spunbonding process is a kind o non-newtonian luid. However, the simplest constitutive equation (Newtonian luid relationship) is used in this study, because the temperature dependence on viscosity is the most dominant eect while the type o the constitutive equation is 19, 20 secondary in the modeling o the spunbonding process. Thereore, or the sake o computational simplicity, the ollowing Newtonian luid constitutive equation is introduced in our improved model: π dv = η (10) 4 2 Frheo D dz η = [exp{6021/ ( T + 273)}] (11) where η is the shear viscosity o air (Pa.s) Boundary Conditions V (0) = V, F = F, D = D, T = T, F ( L) = 0 0 (0) (0) (0) 0 (0) 0 rheo (12)

3 ZHAO BO: MATHEMATICAL MODELING OF PET AIR DRAWING IN SPUNBONDING NONWOVEN PROCESS 147 where L is the chamber length(mm); F (0), the initial rheological orce o the polymer melt; V 0, the initial velocity o the polymer melt (m/s); D 0, the initial diameter o the polymer melt (mm), and T 0, the initial temperature o the polymer melt. The reezing-point is deined as the boundary condition. The method used or determining the initial rheological orce F (0) is the searching reezing-point method, which requires checking whether the ibre diameters beore and beyond some point along the ibre are equal to each other when F (0) is considered to be the sum o the cumulative gravitational and air drawing orce acting upon the rozen part o the ibre. I the ibre diameters are ound to be the same, the point is called as reezing-point and the appropriate initial rheological orce is used in this iteration. With the help o numerical simulations o the air jet low ield, we can determine the distributions o the z- component o air velocity V a and air temperature T a along the axial position z. Then, the air drawing model o the polymer can be established using a ourth-order Runge--Kutta method. Dynamics Balance Equation F = Fasp + Fgrav Fdrag Finertia (13) where F is the orce on the ilament ibre (N); F asp, the tension exerted by the aerodynamic drawing device; F grav, the gravitational orce; F drag, the air drag on the ibre; and F inertia, the inertial orce. 3 Materials and Methods Poly(ethylene terephthalate) (PET) polymer o the speciication 78 MFI and L1482 was used or the study. 3.1 Test Methods The sample size o the nonwoven abrics produced was 1m 2. As there were irregularities in ibre diameter in spunbonded nonwoven, samples have to be taken randomly on an area o 1m 2. The length and width o the nonwoven abrics are divided into ten equal parts, thus dividing the nonwoven abric into 100 grids. A random number generator program was developed that can generate a group o ten random integers rom 1 to 100 automatically each time it runs. One group o random integers is selected randomly, corresponding to the grid number. The corresponding grids o the samples are then taken out. As a result, random sampling rom the 1m 2 nonwoven abric is realized and the sample quantity is taken as 10. The image analysis method was employed to measure the ibre diameter. The images o nonwoven samples were acquired using a Questar threedimensional video requency microscope (Questar Corp., New Hope, PA) with an enlargement actor o 600 and a ocus depth o 1 mm and then processed with Image-Pro Plus image analysis sotware (Media Cybernetics, Inc., Silver Spring, MD) to measure the ibre diameter. The image processing includes enhancement, smoothing, binarization, and iltering. The ibres o the spunbonding nonwoven are regarded as cylinders because their cross-sections are nearly round. Twenty ibres are chosen to measure their diameters in each grid, so altogether there are 200 ibres to be measured in 10 grids. The mean value o the diameters o 200 ibres was considered. The experimental samples were subjected to conditioning at 65% RH and 20 ± 5 C or 24 h and then the testing was carried out. The melt low index (MFI) experiments o PET were perormed at 270 C using a load capacity o kg, an aperture o capillary tube o mm and a length o capillary tube o 8 mm on RL---11B melt low indexer at ambient room temperature conditions. 3.2 Process Parameters The spunbonding process parameters concerned were the polymer throughput rate, the polymer melt temperature, the primary air temperature, the venturi gap, the basis weight, and the air suction speed 1, 3-4. The optimum values o undamental parameters giving the best results were assumed during the computations, which include the polymer throughput rate o 0.30 g/min/hole, the polymer melt initial temperature o 285 C, the air initial temperature o 15.6 C, the air initial speed o 75 m/s, the venturi gap o 24 mm, the air suction speed o 2500 rpm, and the spinneret hole diameter (D 0 ) o 0.45 mm. When one o the processing parameters was varied, the undamental values o other process parameters were kept constant. 3.3 Drating Assembly Parameters The lat narrow slot passage o drating assembly is shown in Fig.1. The drating assembly parameters are the steady low segment length a=50 cm, the jet oriice length b=10 cm, the drating segment length c=18.5 cm, the outlet oriice length d=9 cm, the drating segment width e=30 cm, the jet oriice highness =5 cm, and the drating segment highness h 1 /h 2 =2.333/2.333 cm.

4 148 INDIAN J. FIBRE TEXT. RES., JUNE Numerical Methods or Analysing Air Flow Field Model The air jet low ield model is analysed by using the inite dierence method, considering the conditions (i) the SIMPLE algorithm is utilized to solve the problem o velocity pressure couple, (ii) the staggered grid is presented to avoid tooth-like distributions o velocity and pressure, (iii) the preerred dierence scheme or space independent variables is the second-order upwind dierence scheme, and (iv) the TDMA (Tri-Diagonal Matrix Algorithm) method is used to solve the dierence equations. Fig. 1 Flat narrow slot passage o drating assembly in spunbonding process [a low segment length, b jet oriice length, c drating segment length, d outlet oriice length, e drating segment width, jet oriice highness, and h 1 & h 2 drating segment highness] The computational domain is rectangular where the coordinate origin is in the center o the lat narrow slot passage o drating assembly. Lengths o the z-direction and y-direction o the computational domain are 875 mm and 50mm respectively. There are 1700 grids in the z-direction and 150 grids in the y-direction. 4 Results and Discussion 4.1 Eect o Various Parameters on Fibre Diameter Experimental Results Eects o Polymer Throughput Rate, Polymer Melt Temperature and Air Suction Speed Figures 2 (a)-(c) show the eect o polymer throughput rate, polymer melt temperature and air suction speed on the ibre diameter. As expected, lower polymer throughput rate, higher polymer melt temperature and higher suction air initial speed produce iner ibre and more rapid air attenuation, which are key actors in controlling the inal ibre diameter Eect o Web Basis Weight Figure 2 (d) shows that the eect o web basis weight on the ibre diameter is not signiicant. The ibre attenuation changes slightly. There is no clear trend observed or the inal ibre diameter. Thereore, Fig. 2 Eects o polymer throughput rate, polymer melt temperature, air suction speed, and web basis weight on ibre diameter

5 ZHAO BO: MATHEMATICAL MODELING OF PET AIR DRAWING IN SPUNBONDING NONWOVEN PROCESS 149 it can be concluded that too low web basis weight contributes little to polymer drawing as ar as spunbonding polymer o poly(ethylene terephthalate) is concerned Eects o Quench Pressure, Venturi Gap, Air Initial Temperature, and Suction Speed Figure 3 shows the eect o quench pressure, venturi gap, air initial temperature, and suction speed on the ibre diameter. As expected, higher quench pressure, higher air initial temperature, higher suction speed, and medium smaller venturi gap yield iner ibre. It can be concluded that the variation in quench pressure, venturi gap, air initial temperature, and suction speed have very important inluence on the ibre diameter Eects o Jet Oriice Length, Jet Oriice Highness, Drating Segment Length, and Drating Segment Highness Figure 4 shows the eect o the drating segment highness, the drating segment length, the jet oriice highness and the jet oriice length o the drating assembly on ibre diameter. Smaller highness o the drating segment, larger length o the drating segment, larger length o jet oriice o the drating assembly, and bigger highness o jet oriice o the drating assembly yield iner ibre. 4.2 Experimental Veriication using Improved Model Relationship between Polymer Throughput Rate and Fibre Diameter Figure 5 shows the eect o polymer throughput rate on ibre diameter. With the decrease in polymer throughput rate, the ibre diameter decreases. As expected, reducing the polymer throughput rate can yield a iner ibre diameter. The inal ibre diameter o or a polymer throughput rate o 0.18 g/min/hole is smaller than the inal ibre diameter o or the high polymer throughput rate (0.48 g/min/hole). This may be explained by the act that the lower the polymer throughput rate, the iner is the ibre. Lower polymer throughput rates give iner ibres and more rapid attenuation Relationship between Polymer Melt Initial Temperature and Fibre Diameter Figure 6 clearly indicates that the polymer melt initial temperature has a striking inluence on ibre diameter. As can be seen, increasing the polymer melt initial temperature yields a iner ibre diameter. The high polymer melt initial temperature (297.5 o C) results in a inal ibre diameter o 21.23, while the low polymer melt initial temperature (267.5 o C) results in a inal ibre diameter o As Fig. 3 Eects o quench pressure, venturi gap, air initial temperature, and suction speed on ibre diameter

6 150 INDIAN J. FIBRE TEXT. RES., JUNE 2010 Fig. 4 Eects o the jet oriice length, the jet oriice highness, the drating segment length, and the drating segment highness on ibre diameter Fig. 5 Eects o polymer throughput rate on ibre diameter expected, the higher the spinning temperatures, the slower is the ibre cooling rates. Under the same low rate, the higher polymer melt temperatures have a longer time to deormation and give iner ibres Relationship between Air Initial Temperature and Fibre Diameter Figure 7 shows that the air initial temperature has a critical inluence on the ibre diameter and hence is an important key actor in controlling the ibre diameter. The increase in air initial temperature decreases the Fig. 6 Eects o polymer melt initial temperature on ibre diameter values o ibre diameter. The high air initial temperature (24 C) results in a inal ibre diameter o 23.18, whereas the low air initial temperature (10 C) results in a inal ibre diameter o This is primarily due to the acts that (i) the increase in air initial temperature increases the air drawing orce and degree o drawing, which yield a iner ibre diameter; (ii) when air initial temperature increases, the ibre cools down more slowly along the spinline and the drawing (tensile) time o polymer extends

7 ZHAO BO: MATHEMATICAL MODELING OF PET AIR DRAWING IN SPUNBONDING NONWOVEN PROCESS 151 (lengthened), which result in a iner ibre diameter; and (iii) when the air initial temperature increases, the viscosity and stress decrease, which produce a iner ibre diameter Relationship between Air Initial Speed and Fibre Diameter Figure 8 reveals that there is a signiicant change in the ibre diameter with the variation in air initial speed. A larger air initial speed causes the ibres to be attenuated much more. With the increase in air initial speed, the inal ibre diameter decreases gradually. When the air initial speed increases to 90 m/s, the inal ibre diameter is ound to be 29.8% smaller than that obtained when the air initial speed is 55 m/s. This is mainly attributed to the act that the increase in primary cool air velocity increases the air drawing orce and degree o drawing, which yield a iner ibre diameter Combined Eect o Polymer Throughput Rate and Polymer Melt Initial Temperature Table 1 demonstrates the measured and predicted ibre diameters, when the air initial (primary) temperature (18.4 C) and the air initial speed (75.0m/s) are kept constant while the polymer throughput rate and the polymer melt initial temperature are varied; here the predicted ibre diameters it well with the measured ones. Table 1 shows that a lower polymer throughput rate, and a higher polymer melt initial temperature yield iner ibres. Hence, it can be concluded that the polymer throughput rate and the polymer melt initial temperature are key actors in controlling the inal ibre diameter Combined Eect o Air Initial Temperature and Air Initial Speed Table 2 indicates the measured and predicted ibre diameters, when the polymer throughput rate (0.20 g/min/hole), and the polymer melt initial (primary) temperature (295 C) are kept constant, while the air initial temperature and the air initial speed are varied. It is ound that the predicted ibre diameters coincide well with the measured ones. Table 2 also reveals that increasing the air initial temperature and air initial velocity can decrease the ibre diameter. It is observed that the air initial temperature and the air initial speed have very important inluence on the ibre diameter. The results also show that the air drawing model o polymers we established can predict the drawing eects o spunbonding nonwoven processing. As can be seen rom Tables 1 and 2, the measured value coincides with the numerical computational results well, which urther proves the reliability and Fig. 7 Eects o air initial temperature on ibre diameter Fig. 8 Eects o air initial speed on ibre diameter Polymer throughput rate, g/min /hole Table 1 Combined eect o polymer throughput rate and polymer melt initial temperature on ibre diameter Polymer melt initial temperature, C Air initial Temperature, C Air initial speed, m/s Measured diameter, Predicted diameter Prediction error %

8 152 INDIAN J. FIBRE TEXT. RES., JUNE 2010 Polymer throughput rate, g/min /hole Table 2 Combined eect o air initial temperature and air initial speed on ibre diameter Polymer melt initial temperature, C Air initial Temperature, C Air initial speed, m/s Measured diameter, Predicted diameter, Prediction error % Table 3 Comparison o model predictions between improved model and primary model Improved Model Primary Model Measured diameter Predicted diameter Prediction error Measured diameter Predicted diameter Prediction error % % accuracy o the air drawing model o polymer in spunbonding process established in our study. 4.3 Comparison o Improved Model with Primary Model Model predictions consider the inluence o the polymer temperature on the density and speciic heat capacity o polymer melt at constant pressure. The spunbonding process parameters here are the polymer throughput rate ( g/min/hole), the polymer melt initial (primary) temperature ( C), the primary air temperature ( C), the primary air velocity ( m/s), the air suction speed ( rpm), the web basis weight ( g/m 2 ), the quench pressure ( Pa), and the venturi gap (10-30mm). Table 3 shows the ibre diameter changing by considering the polymer density and the speciic heat capacity o polymer melt at constant pressure varying with the polymer temperature. As can be seen, the predictions given by the two models are not the same, two models have much dierence rom each other. It can be concluded that the variations in density and the speciic heat capacity o polymer melt at constant pressure with the polymer temperature have critical important eects on ibre diameter. The model predictions can be improved remarkably by considering the inluence o polymer temperature on the density and speciic heat capacity o polymer melt at constant pressure. Thereore, the variation in density and speciic heat capacity o polymer melt at constant pressure with the polymer temperature has very important eects on ibre diameter. It has been concluded that the predicted ibre diameters are much closer to the experimental results, which urther conirms the eectiveness o the polymer drawing model established in this paper. From the above analyses, it can be seen that the ibre diameter is directly related to the polymer throughput rate, polymer melt initial temperature, air initial temperature, and air initial speed. The iner the geometric mean o ibre diameter, the more uniorm is the ibre web. 5 Conclusions The air drawing model o the poly(ethylene terephthalate) polymer in a spunbonding process is established and demonstrated by the experimental results. The eects o the processing parameters and drating assembly design parameters on the ibre diameter are discussed. The eects o the most process parameters on the ibre diameter are very signiicant. The experimental results indicate that lower polymer throughput rate, higher polymer melt temperature, higher primary air temperature, higher air initial speed, medium smaller venturi gap, higher air suction speed, higher quench pressure, smaller highness and larger length o the drating segment, larger length o jet oriice o the drating assembly, and bigger highness o jet oriice o the drating assembly can all yield iner ibres. It can be concluded that the variations in density and speciic heat capacity o polymer melt at constant pressure with polymer temperature have much eects on ibre diameter. As compared to existing drawing model, the new model includes more processing parameters and is more accurate. The newly developed ormulas were

9 ZHAO BO: MATHEMATICAL MODELING OF PET AIR DRAWING IN SPUNBONDING NONWOVEN PROCESS 153 incorporated into a spunbonding theoretical model to predict the ibre diameter o nonwoven web. The predicted results coincide with the actually measured data quite well, which shows that the new air drawing model is eective and excellent. At the same time, the study results also show great prospects or this research in the ield o computer assisted design o spunbonding technology. Acknowledgement The author is thankul to the Phoenics Inc. or providing the sotware with an educational license. Reerences 1 Malkan S R, Wadsworth L & Davey C, Int Nonwovens J, 6 (1994) Chen C H, White J L & Spruiell J E, Text Res J, 53 (1983) Beyreuther R & Malcome H J, Melliand Textilber, 74 (1993) Hajji N, Spruiell J E, Lu F M & Malkan S R, INDA J Nonwovens Res, 4 (1992) Misra S, Spruiell J E & Richeson G C, INDA J Nonwovens Res, 5 (1993) Bhuvanesh Y C & Gupta V B,J Appl Polym Sci, 58 (1995) Patel R M & Spruiell J E, Polym Eng Sci, 31 (1991) Gagon D K, Denn M M & Morton M D, Polym Eng Sci, 21 (1981) Smith A C & Roberts W W, Int Nonwovens J, 6 (1994) Ju Y D, Polym Eng Sci, 34 (1994) Jeon B S, Text Res J, 71 (2001) Miller C, AIChE J, 50 (2004) Ziabicki A & Kawai H, High-Speed Fiber Spinning Science and Engineering Aspects (John Wiley & Sons, Inc.), Shenoy A V & Nadkarni V M, Text Res J, 54 (1984) Taehwan oh, Polym Eng Sci, 46 (2006) Dutta A & Nadkarni V M, Text Res J, 54 (1984) Mastui M, Trans Soc Rheol, 20 (1976) Majumdar B & Shambaugh R L, J Rheol, 34 (1990) Abbott L E & White J E, Appl Polym Symp, 20 (1973) Bankar V G, Spruiell J E & White J L, J Appl Polym Sci, 21 (1977) 2135.