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Computational Simulation of the Fiber Movement in the Melt-Blowing Process Xiaomei Wang and Qinfei Ke* College of Textiles, Donghua University, 1882 West Yan’an Road, Shanghai 200051, People’s Republic of China
An air/fiber two-phase flow model is used to simulate the broken fiber movement during meltblowing process. The computation is based on the results from simulations of single-phase air flow field developed by two converging plane jets for a blunt-type die using a computational fluid dynamics approach. The specific parallel straight and tilted straight fiber configurations are used as the initial fiber shape. The simulated fiber trajectories qualitatively coincide with images captured by other researchers. Two melt-blowing processing parameters, the initial air velocity and the air slot angle, are studied in terms of their influence on the fiber motion in the flow field. 1. Introduction The melt-blowing process is commercially used as a single-step technology to produce nonwoven webs, in which a molten polymer strand is rapidly attenuated into very fine diameter fiber by the high-velocity airstream (see Figure 1). The melt-blown web is extremely suitable for filtering material, oil absorbent, hygienic material, etc. To achieve the best web performance and exploit the full potential of melt-blowing technology, many studies have been devoted to theoretical and experimental investigations of melt blowing. Take theoretical research for example, 1-D, 2-D, and even 3-D models for melt blowing have been developed.1-3 These models help better understand the mechanism of melt blowing through prediction for the thermal and mechanical behavior of the fiber stream as it travels from the die to the collector. However, all these models mentioned above were based on the assumption of continuous filament produced in melt blowing. In fact, fiber breakage usually occurs during melt blowing.2,4 Moreover, there is hardly any literature regarding the broken fibers motion, which strongly affects the produced web structure. In this work, we numerically simulated the broken fiber movement in the flow field created by the dual rectangular jets of a die, showing the process that causes the fiber flow to diverge and the fiber to curl and to bundle. In melt-blowing process, the broken fiber motion in the space from die to collector is a two-phase flow problem. The fibers constitute the particle phase, and air is the fluid phase. Due to the quite different properties (e.g., large aspect ratio, elasticity, and flexibility) from the fiber to the general particle, the fiber must be modeled before computational simulation of its motion. Models that can be used for fiber include the “multisphere” chain model, the “node-chain” model, and the “bead-elastic rod” chain model.5-7 For the multi-sphere model, too many spheres are needed because of the large aspect ratio of fiber, which would make the approach very computationally expensive. In the node-chain * To whom correspndence should be addressed. E-mail:
[email protected]. Telephone: ++86-021-62378835.
Figure 1. Melt-blowing process. The dotted lines, a and b, are two typical initial positions of fiber, and the fiber ends closer to the die face are the trailing ends.
Figure 2. Schematic of fiber model.
model, a straight distance of the adjacent nodes, instead of the actual length of fiber section, is used to calculate the air drag force exerted on the fiber. So when the adjoining nodes are very close to each other due to fiber section bend, the straight end-end distance is fairly short. In this case, the skin friction force of the fiber segment acted by the air will be close to zero, which is obviously not true. Therefore, the bead-elastic rod model is an excellent fiber model by comparison. However, in our work, another model called “sphere-spring” was chosen, which can best describe the fiber movement characteristics and has ever been modeled as the individual macromolecules in polymer dynamics research.8 In this model, the springs reflect the elasticity of fiber, and the bending deflection in successive springs describes the fiber flexibility. 2. Fiber Model and Motion Equation The sphere-spring model for a fiber is shown in Figure 2, in which S donates the sphere and C represents the connector. The fiber is considered to be such a configuration that consists of n + 1 spheres joined by n weightless springs. The mass of the ith sphere (mi) is governed by fiber linear density and section lengths between the adjacent
10.1021/ie048878c CCC: $30.25 © 2005 American Chemical Society Published on Web 04/26/2005
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spheres and is expressed as
mi )
Ff i-1 (l + li) 2
(1)
where Ff is the fiber linear density, and li-1 and li are the lengths of the two springs jointing this sphere. For i ) 1 or i ) n + 1, l1 ) ln+1 ) 0. The springs and the spheres are assumed to have the same diameter as that of the fiber. The restoring bending force Fib, exerted on sphere i, depends on the fiber rigidity, and is given by eq 2, which has been explained in detail by Zeng and Yu:6
Fib ) -B∆yi
(2)
where B is considered as a constant related to the flexural rigidity of the fiber for simplicity, although it is strongly dependent on temperature during meltblowing process, and ∆yi is the bending deflection of the section. The fiber immersed in a flow field are mainly subjected to air drag force, gravitational force, buoyant force, added mass force, pressure gradient, Saffman force, turbulence fluctuation force, and thermophoretic force. In the work described herein, only air drag force is considered, while the other forces are neglected for the sake of simplicity. The air drag due to the velocity difference between air and fiber is composed of skin friction drag and form drag, which, for the ith connector, can be calculated respectively by
F if
1 ) CfFAit|ut - vit|(ut - vit) 2
(3)
(4)
where F if and F ip are respectively the skin friction drag and form drag; Cf and Cp are coefficients of skin friction drag and form drag (they can be expressed as eqs 5 and 69,10); F is the air density; Ait and Ain are respectively the surface area and the projected area of the fiber section in the fiber direction; ut and vit are respectively the air and sphere velocities in the tangential direction of fiber axis; un and vin are respectively the air and sphere velocities in the direction normal to fiber axis.
Cf ) 0.78 × (Rel)-0.61 Cp ) 6.96 × (Red)-0.44
(5)
() d d0
0.404
(6)
where Rel and Red are relative air Reynolds numbers based on fiber diameter for skin friction drag and form drag, respectively. The total drag exerted on the fiber section is
Fid ) Fif + Fip
(7)
The air drag exerted on each sphere is determined by the composition of two drags contributed by the two springs connecting this sphere, as written by
1 + Fid) Fid ) (Fi-1 2 d
mi
dvi ) Fib + Fid + Fi-1,i + Fi,i+1 dt
(9)
where vi is the velocity of the ith sphere, Fi-1,i and Fi,i+1 are forces from adjacent spheres (i.e., the recovering elastic force of the springs, F i-1 and F ie), which can be e figured out during the procedure of calculating the fiber motion. 3. Numerical Simulation To simulate the fiber movement using the preceding equation, the fluid phase equations need to be computed for obtaining the air velocity at any position in a flow field in which the fiber moves. The governing equations for the flow phase include mass conservation equation, momentum conservation equation, and energy conservation equation. However, the energy conservation equation does not need to be solved during our simulation for two reasons: there is little influence of temperature on the air velocity development and thermophoretic force resulting from temperature gradient is neglected when doing fiber kinematic analysis. So, the governing equations of the fluid phase are
mass conversation: ∂ ∂F + (Fu ) ) 0 ∂t ∂xi i
(10)
where F is fluid density and ui is velocity in the i direction, and
and
1 F ip ) CpFAin|un - vin|(un - vin) 2
For a fiber model, substituting the forces mentioned above, we obtain the following equation of force balance to describe fiber movement:
(8)
momentum conservation: ∂ ∂ ∂p ∂τij (Fu ) + (Fuiuj) ) + + Ffi ∂t i ∂xj ∂xi ∂xj
(11)
where p is pressure, τij is a stress tensor, and fi is the body force per unit volume. Krutka et al.11,12 simulated the velocity flow field developed by blunt dies and tip dies using the computational fluid dynamics software FLUENT. Chen13 also has ever applied PHOENICS software to implement velocity and temperature field simulation for blunt die type. Both their simulation results agree with experimental data well. In this work, we adopted FLUENT 6.0 software to achieve numerical simulation of the velocity field resulting from dual rectangular jets of a blunt die, as shown in Figure 1. The two air slots of the die were 0.65 mm width. The distance between the outer edges of the slots was 3.32 mm, and the air slot angle (β, shown in Figure 1) was 60°. The coordinate system for simulations is also shown in Figure 1. The origin of the system is at the center of the face of the die. The x direction is along the threadline perpendicular to the die face, and the y direction is transverse to the row of the polymer capillaries. The dimensions of the computational domain were Lx ) 150 mm and Ly ) 30 mm. The Reynolds stress model with the modified constants C1 ) 1.24, C2 ) 2.05 was selected as the turbulence model, as suggested by Krutka et al.11 Figure 3 illustrates part of the simulation results described by 2-D velocity vector field.
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Figure 4. Simulated trajectories of a fiber moving in the flow field with different initial fiber configurations. The melt-blowing conditions were as follows: D ) 0.3 mm, Q0 ) 0.035 g/s, Tf,0 ) 290 °C, Vj0 ) 100m/s, and Ta,0 ) 310 °C. (a) parallel fiber, (b) tilted fiber with a slope of ∆y/∆x ) 0.01.
Figure 3. Part of the velocity vector field. The initial air velocity (Vj0) ) 100m/s.
Based on the calculated air velocity field with CFD software, the force balance equation of fiber movement (eq 9) can be solved to calculate the trajectories of a broken fiber moving in the flow field as partly shown in Figure 3. During melt blowing, filament breaks randomly due to nonuniformity of the polymer properties and fluctuation of processing conditions. As a result, the location where the filament breaks and the length of broken fiber cannot be determined. So, we assumed the broken fiber to be 30 mm length, and specified the initial conditions for simulating fiber motion as the following: (a) the initial fiber configuration is straight, and the position is indicated as a and b in Figure 1; (b) the initial fiber velocity is obtained through solving the melt-blowing theoretical model founded by Uyttendaele and Shambaugh,1 and all the spheres constituting the fiber have the same velocity. Some important parameters included in the Uyttendaele-Shambaugh model are zero shear rate viscosity, specific heat capacity of polymer melt, and polymer density. The correlations between them and the polymer temperature used in this paper were the same as that used by Chen.13 Another information necessarily needed for solving the model is the distribution of the air temperature along the threadline. It, however, cannot be obtained from the simulated flow field with CFD approach due to the neglect of air temperature during this simulation. So, an empirical formula describing the temperature distribution was directly used in the Uyttendaele-Shambaugh model. The empirical formula is as follows: Ta - Tambient/Ta,0 - Tambient ) 0.8637Z(w)-0.4054, where Ta is the air temperature, Ta,0 is the air temperature at the die exit, Tambient is the air temperature at the ambient conditions, and Z(w) is dimensionless position along the threadline.14 The determination of initial rheological force (F0) for solving the Uyttendaele-Shambaugh model was “method B”, i.e., searching “freezing point” method.14 This method requires to check whether the fiber diameters before and beyond some “point” along the filament are equal to each other when F0 is considered to be the sum of the cumulative gravitational and air drag forces acting upon the frozen part of the filament. If the fiber diameters are found to be the same, the “point” is the so-called “freezing point” and F0 used in this iteration is the appropriate initial rheological force. For the boundary condition, we stop calculating if any sphere moves beyond the domain of the calculated flow field.
To facilitate this simulation, additional assumptions were made: first, the fibers do not interact with one another; second, the diameter is constant along the whole fiber length (the fiber diameter is the final filament diameter obtained with solving the Uyttendaele-Shambaugh model using method B); third, every spring length keeps invariable due to the fact that fiber moves freely and the elastic restoring force is on the order of 103 of bent restoring force and air drag force as stated by Zeng and Yu.6 With the initial and boundary conditions and the assumptions described above, the following procedure was used to carry out simulation of the fiber motion in the flow field. The first step, calculate the Fb and Fd for every sphere according to eqs 2 and 8 in combination with the air velocity of a point where the sphere located. The second step, work out the new coordinates of rj at a time t + ∆t and the velocity of the spheres for the next step by using eq 9 and a position-velocity correlation, rj+1 ) rj + 1/2(vj-1 + vj+1)∆t, under the limiting condition of the constant spring length. Repeating the procedure stated above, we can obtain the motion of all spheres and the configuration of a fiber in evolutionary time. 4. Results and Discussion 4.1. Simulation Results with Two Specific Initial Fiber Configurations. We studied fiber motion in the flow field simulated above with two specific initial fiber configurations: the straight fiber is parallel with the threadline and is slantwise to the threadline (position a and position b shown in Figure 1). The simulated results are presented in Figure 4. Figure 4a,b demonstrates how a broken fiber moves in the flow field at several time stages: panel a is for the case of an initial parallel fiber, and panel b is for the tilted fiber case. As it can be seen from these profiles, the fiber is deflected gradually from the threadline with time (i.e., the inclination angle of fiber is increasing). Here, the inclination angle is defined as an acute angle between the threadline and the straight line formed by the trailing end and leading end of the fiber. Figure 4 further shows that the fiber moves vertically far away from the threadline with its progressive buckling or bending configuration (more obvious in Figure 6). This may be explained as below. After it ejects from the two slots and then converges, the high-speed air is not confined by any solid barrier. So the air moves slower and slower in both x- and y-directions. Therefore, the force exerted on the leading end of fiber is smaller than that on the trailing end in both directions, causing the leading end moves slower than the trailing end in
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Figure 5. Multiple-image photograph of a fiber cone for 1 e z e 8 cm (i.e., 1 e x e 8 cm, in the coordinate system used in this paper). This figure is adapted from Chhabra and Shambaugh.15
the two directions for every time step. So as expected for the case a, the trailing end of fiber bends up, and so the inclination angle increases. While for case b, the fiber leading end bends up. The reason lies in that the displacement difference between the fiber trailing section and the leading section in x-direction is greater than that in y-direction (because x-velocity decays more rapidly than y-velocity), so the leading end of the titled fiber with the leading section uptilting will continuously bend up. For both cases, the fiber moves outward from the threadline, which is attributed to the divergence of airflow in the space from the die to the collector. Figure 5 shows one of the multiflash photographs provided by the literature.2,15,16 From Figure 5, it can be observed that the fibers move away from the threadline (i.e., the trajectories of fibers are divergent). Compared to all the photographs (others are not given here), the trend of simulated fiber movement by us is very close to that of the experimental work. However, the fiber trajectories of computation and experimentation are not in complete agreement. This may be attributed to the following facts. First, the melt-blowing conditions or the die geometry used in our simulation is not exactly the same as that used in the experimental work. Second, every photograph in the literature contains many overlapped pictures due to multiple exposures, so the fibers possessed much more complicated initial configurations, while we only considered two special cases for release into the air stream. 4.2. Effects of Processing Parameters on Fiber Motion. Fiber motion in the flow field depends critically on the flow pattern, which is predominantly determined by the initial air velocity and the die geometry. For the
die geometry, the blunt type is predominant in the production of melt-blown webs, and effects of the air slot angle on the flow field have been investigated by many researchers.11 So, we selected the initial air velocity and the slot angle as the parameters to be studied. And the initial fiber configurations were set according to case b in Figure 1 during all the following simulations of fiber motion. 4.2.1. Effect of Initial Air Velocity. The initial air velocity affects not only the final fiber diameter but also the flow field features, which have significant influence on the web structures. In our study, three different initial air velocities were considered. As we stated earlier, the initial fiber velocity and fiber diameter are obtained by solving the Uyttendaele-Shambaugh model of melt blowing according to the processing conditions used. In the simulation work, we keep other processing conditions and computational parameters constant, only varying the initial air velocity from 100 to 300 m/s. Figure 6 presents the fiber movement in the simulated flow field (the die geometry was as stated previously) at different time stages. The figure illustrates that the vertical distance of the fiber departs from the threadline increases with the increase of the initial velocity, indicating wedge angle of the fiber flow raises, namely, the fiber vibration amplitude increases. This predictions agrees with the experimental work by Moore et al.17 and also coincides with our deliberate observation with the naked eye in a melt blowing run. Still this figure demonstrates that, with increasing velocity, the fiber inclination angle shows an increasing tendency (in Figure 6c, after it moves beyond 80 mm in x-direction, the fiber curls significantly. In this case, we define the fiber inclination angle as 90 o). However, for the case of 300 m/s initial velocity, the fiber becomes curled or folded, and the fiber flow broadens dramatically after a short time movement. This may be explained by the fact that higher air velocity produces finer fiber with relatively small flexural rigidity, which leads to an easy distortion by the action of axial and tangential air velocity. Under the situation of folding, the fiber prefers to entangle with each other, resulting in big fiber bundle size in the web. 4.2.2. Effect of Slot Angle. Simulations of fiber motion in the flow field at various slot angles (other die parameters were kept constant) are shown in Figure 7.
Figure 6. Simulated trajectories of a fiber moving in the flow field for different initial air velocities. The melt-blowing conditions were the same as for Figure 4 (Vj0, excepted). (a) Vj0 ) 100m/s, (b) Vj0 ) 200m/s, (c) Vj0 ) 300m/s.
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Figure 7. Simulated trajectories of a fiber moving in the flow field for different slot angles. The operating conditions were the same as for Figure 4. (a) β ) 70°, (b) β ) 60°, (c) β ) 45°.
As can be seen, the influence of air slot angle is as significant as that of initial air velocity. Again, the degree of the fiber divergence reduces as the slot angle decreases. This behavior coincides with the computational results of the velocity field. When the slot angle decreased, both x and y velocities reduce. Therefore, the total drag force exerted on the fiber is less with the smaller slot angle, resulting in the slower fiber movement in the flow field and, hence, the lower degree of fiber divergence. As far as fiber curve is concerned, when β ) 70°, the significantly looped and folded configuration results in bundling in a fiber itself or between fibers during their movement, which is detrimental to the uniformity of the final web. In another extreme case for β ) 45° the fiber does not curl so severely, which is expected to benefit the improvement of web quality. In this case, however, the turbulence intensity near the threadline is higher (i.e., the velocity fluctuations are stronger)11 causing filament near the die head to be instable and then to stick and jam, which is an obstacle to successful melt blowing. Consequently, we can suggest that the 60° die is optimum. 5. Conclusions We numerically simulated the broken fiber movement in the flow field resulting from two rectangular jets of a die during melt blowing using a two-phase air/fiber model. The trend of our predicted results are very close to that of the laboratory investigation by some researchers, although we did not capture specific fiber trajectories to make comparisons. During its movement, the inclination angle of a fiber is increasing gradually, so is the vertical distance from the threadline. We have considered two specific initial fiber configurations during the simulation-parallel fiber and tilted fiber. Simulations for these two cases can well explain the experimental results reported in the previous literature. Still we have studied the influences of initial air velocity and air slot angle on the fiber motion. We have found that a higher initial air velocity produce bigger fiber flow angle, while the fiber is easy to curl and then even bundle itself. Considering other reasons, the 60° die is the optimum to be used in melt blowing.
Acknowledgment This work is supported by the National Natural Science Foundation of People’s Republic of China (Grant 50276010). The permission by Fluent Inc. to use their software with an educational license is gratefully acknowledged. Nomenclature At, An ) surface area of the fiber section and projected area of the section in the fiber direction, respectively (µm2) Cf, Cp ) coefficients of skin friction drag and form drag, respectively D ) polymer capillary (mm) d ) diameter of fiber (µm) d0 ) median diameter of filaments used in the correlation,8 d0 ) 78 µm fi ) body force per unit volume (m/s2) Fb ) restoring bending force (N) Fe ) restoring elastic force (N) Ff ) skin friction drag force exerted on fiber section (N) Fp ) form drag force exerted on fiber section (N) Fd ) total drag force exerted on fiber section (N) g ) gravitational acceleration (m/s2) i ) ith sphere or spring j ) time step l ) fiber section length (mm) m ) sphere mass n ) number of the sphere p ) pressure (Pa) Q0 ) polymer flow rate (g/s) r ) position of sphere Rel ) relative Reynolds number based on fiber diameter and component of velocity parallel to the fiber axis (Rel ) F(ut - vt)d/µ) Red ) relative Reynolds number based on fiber diameter and component of velocity perpendicular to the fiber axis (Red ) F(un - vn)d/µ) Tf,0 ) polymer temperature at the die exit (°C) Ta ) air temperature (°C) Ta,0 ) air temperature at the die exit (°C) Tambient ) air temperature at the ambient conditions (°C) ui ) air velocity in the ith direction ut, un ) air velocities in tangential and normal direction of fiber axis, respectively (m/s) vt, vn ) fiber velocities in tangential and normal direction of fiber axis, respectively (m/s)
Ind. Eng. Chem. Res., Vol. 44, No. 11, 2005 3917 Vj0 ) air velocity at slot exit (m/s) x ) horizontal coordinate y ) vertical coordinate Z(w) ) dimensionless position, Z(w) ) x/w(Fa,∞/F)0.5, where w denotes distance between the outer edges of the slots (mm); Fa,∞ is the air density at ambient conditions (kg/ m3); F is air density (kg/m3) Greek Characters β ) air slot angle (deg) ∆y ) bending deflection (mm) ∆t ) time interval (s) Ff ) fiber linear density (kg/m) F ) air density (kg/m3) τij ) stress tensor (Pa) µ ) air viscosity (kg/cm‚s)
Literature Cited (1) Uyttendaele, M. A. J.; Shambaugh, R. L. Melt blowing: General equation development and experimental verification. AIChE J. 1990, 36 (2), 175-186. (2) Rao, R. S.; Shambaugh, R. L. Vibration and stability in the melt blowing process. Ind. Eng. Chem. Res. 1993, 32 (12), 31003111. (3) Marla, V. T.; Shambaugh, R. L. Three-dimensional model of the melt-blowing process. Ind. Eng. Chem. Res. 2003, 42 (26), 6993-7005. (4) Shambaugh, R. L. A macroscopic view of the melt-blowing process for producing microfiber. Ind. Eng. Chem. Res. 1988, 27 (12), 2363-2372. (5) Yamamoto, S.; Matsuoka, T. A method for the dynamic simulation of rigid and flexible fibers in a flow field. J. Chem. Phys. 1993, 98, 644-650. (6) Zeng, Y. C.; Yu, C. W. Numerical simulation of fiber motion in the nozzle of an air-jet spinning machine. Text. Res. J. 2004, 74 (2), 117-122.
(7) Kong, L. X.; Platfoot, R. A. Computational two-phase air/ fiber flow within transfer channels of rotor spinning machines. Text. Res. J. 1997, 67 (4), 269-278. (8) Bird, R. B.; Curtiss, C. F.; Armstrong, R. C.; Hassager O. Dynamics of Polymeric Liquids, Vol. 2: Kinetic Theory, 2nd ed.; John Wiley & Sons: New York, 1987. (9) Ju, Y. D.; Shambaugh R. L. Air drag on fine filaments at oblique and normal angles to the air stream. Polym. Eng. Sci. 1994, 34 (12), 958-964. (10) Majumdar, B.; Shambaugh, R. L. Air drag on filament in the melt blowing process, J. Rheol. 1990, 34 (4), 591-601. (11) Krutka, H. M.; Shambaugh, R. L.; Papavassiliou D. V. Analysis of a melt-blowing die: comparison of CFD ad experiments. Ind. Eng. Chem. Res. 2003, 41 (20), 5125-5138. (12) Krutka, H. M.; Shambaugh, R. L.; Papavassiliou D. V. Effects of die geometry on the flow field of the melt-blowing process. Ind. Eng. Chem. Res. 2003, 42 (22), 5541-5553. (13) Chen, T. Study on the air drawing in melt blowing nonwoven process. Ph.D. Dissertation, Donghua University, Shanghai, China, 2003. (14) Wang, X. M.; Ke, Q. F. Study on the determination of initial rheological force for theoretical model of melt blowing process. Proceedings of The Textile Institute 83rd World Conference; 2004; pp 1083-1088. (15) Chhabra, R.; Shambaugh, R. L. Experimetal measurements of fiber threadline vibration in the melt-blowing process. Ind. Eng. Chem. Res. 1996, 35 (11), 4366-4374. (16) Shambaugh, R. L.; Chhabra, R. Fiber vibration and distributions in melt blowing. Tappi J. 1998, 81 (3), 199-201. (17) Moore, E. M.; Papavassiliou, D. V.; Shambaugh, R. L. Air velocity, air temperature, fiber vibration and fiber diameter measurements on a practical melt blowing die. Int. Nonwoven J. 2004, Fall, 43-53.
Received for review November 21, 2004 Revised manuscript received March 30, 2005 Accepted March 31, 2005 IE048878C
