Modifying Air Fields To Improve Melt Blowing - Industrial

Article pubs.acs.org/IECR

Modifying Air Fields To Improve Melt Blowing Brent R. Shambaugh, Dimitrios V. Papavassiliou, and Robert L. Shambaugh* School of Chemical, Biological, and Materials Engineering, The University of Oklahoma, 100 East Boyd Street, SEC T335 Norman, Oklahoma 73019, United States ABSTRACT: An advanced model for melt blowing was used to predict the effects of modified air fields on fiber formation. The model, which was developed previously, involves the simultaneous solution of the momentum, energy, and continuity equations. Crystallization effects were included. The model equations were solved numerically. Simulations were done for two classes of modified air fields. The first class of modified air field has a plateau of constant velocity and temperature. For wide plateaus placed near the die face, the effect of the plateau is substantial. Fiber diameters are reduced (by up to two times) versus diameters for when there is no plateau. The second class of modified air field has a quench (or a plateau-quench). Quenching was simulated as a step drop in air temperature. The simulation showed that quenching can enhance online crystallinity, though fiber attenuation is reduced when quenching is used.

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INTRODUCTION In the process known as melt blowing, high-velocity air is used to attenuate polymer streams into fine fibers. These fibers are used commercially as insulation, sorbent material, filters, and numerous other purposes. Normally, both the air field temperature and air field velocity steadily decay as the distance from the die increases. However, in a recent paper (Shambaugh et al.1), it was suggested that melt blowing could be improved if the air field was modified by placing a “plateau” in the air field. This plateau would consist of a certain range of constant air temperature and constant air velocity along the fiber threadline. An example of a 2 cm plateau is shown in Figure 1. In the

In the present paper, we start from the modeling work of Shambaugh et al.1 to further explore the use of air field modification to improve melt blowing. Specific goals include (a) reducing air cost and (b) improving fiber properties through online crystallization.

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AIR AND TEMPERATURE PLATEAUS IN MELT BLOWING Plateau Placement. Melt blowing has been modeled with a set of differential equations that include basic momentum, energy, structure (crystallinity), and constitutive equations. Recent modeling work by Shambaugh et al.1 and others2−5 solves these equations numerically. In Shambaugh et al.’s work, calculations were done along the threadline from the die face to a stop point. The stop point is defined as the point where the fiber stress is zero and the fiber velocity equals the air velocity. The simulation gives profiles (along the threadline) of fiber diameter, fiber velocity, fiber temperature, fiber stress, and fiber crystallinity. (See Shambaugh et al. for the details of the model development and implementation. This model includes the thin fiber assumption wherein the fiber velocity and temperature are uniform in the radial direction.) A plateau can be mathematically described by letting the air velocity and temperature decrease according to the usual correlations until some value z1 is reached.1 Then, the velocity and temperature are arbitrarily kept constant over a plateau range from z1 to z2. At z2 the velocity and temperature are allowed to continue to decrease according to the usual correlations. With the use of a new parameter z′, the plateau is defined as z′ = z

Figure 1. Modification of the air field with a plateau. The air velocity and the air temperature are shown with a 2 cm plateau placed between z = 2 cm and z = 4 cm. The air velocity and air temperature at the die face are, respectively, 110 m/s and 368 °C.

z′ = z1

for z < z1

for z1 ≤ z ≤ z 2

z′ = z1 − (z 2 − z1)

for z > z 2

1

work of Shambaugh et al., the placement of a plateau resulted in smaller fiber diameters for a given air flow rate (it was shown that the same final fiber diameter can be produced with up to 63% less air!). Ergo, air costs can be substantially reduced with the presence of a plateau. © 2012 American Chemical Society

Received: Revised: Accepted: Published: 3472

November 1, 2011 January 7, 2012 January 11, 2012 February 16, 2012 dx.doi.org/10.1021/ie202501u | Ind. Eng.Chem. Res. 2012, 51, 3472−3482

Industrial & Engineering Chemistry Research

Article

This z′ used in place of z in the usual air velocity and air temperature correlations. The usual velocity correlations are (Majumdar and Shambaugh6) vjo/vo = 0.077Z(dAN) + 1.00 vjo/vo = 0.249Z(dAN) + 0.468

for Z(dAN) < 3.093 for Z(dAN) ≥ 3.093

(1) (2)

where dAN = outer diameter of annular die orifice, mm; vo = air velocity along the threadline, m/s; and vjo = va,die = air velocity at the die (z = 0), m/s; Z(dAN) = z/dAN((ρa∞/ρa0)1/2); ρao = air density along the center line downstream from the nozzle, kg/m3; and ρa∞ = air density at ambient conditions, kg/m3. For the temperature profile along the threadline, the analogous correlations are6 θjo/θo = 0.033Z(dAN) + 1.00

for (dAN) < 4.644

(3)

θjo/θo = 0.221Z(dAN) + 0.127

for Z(dAN) ≥ 4.644

(4)

Figure 2. Fiber diameter profiles for when a 2 cm plateau is present in the air field. The air velocity at the die face is 110 m/s. The initial fiber diameter (die swell diameter) used in the calculations was 949 μm, the iteration step size was 1 mm, and Mw = 165 000.

where θjo = excess air temperature above ambient at die exit, °C; and θo = excess air temperature above ambient along the center line (the z axis), °C. Experimentally, one can envision that placing a tube or set of louvers below the melt blowing die can in fact approximate a plateau of constant velocity and temperature. Secondary air streams could also be used to maintain constant velocity and temperature for the length of a plateau. Figure 1 is a plot of the air velocity and air temperature change when a 2 cm plateau is placed between z = 2 and z = 4 cm. Note that, beyond z = 4 cm, the air velocity and air temperature decrease according to the established correlations except that the curve has been shifted by 2 cm. The melt blowing of polypropylene was simulated with the presence of a 2 cm plateau. For this simulation, the air velocity was 110 m/s, and the polymer used was polypropylene with molecular weight Mw = 165 000. Figure 2 is a plot of fiber diameter predictions from this simulation. There are eight curves on this figure. Seven of the curves are for the placement of the plateau at various positions along the threadline, and the eighth curve is the base condition (no plateau). There is little change in the diameter profile when the plateau is placed far down the theadline. For example, for the plateau between z = 7.5 cm and z = 9.5 cm, the stop point only moves from 23.4 cm (for no plateau) to 22.8 cm in the presence of the plateau. Concurrently, the fiber diameter only decreases from 53.7 to 50.7 μm. The effect is even less for simulations run with the plateau even farther down the threadline (these results are not shown on this graph). However, as seen in Figure 2, the effect of the plateau is much more pronounced when the plateau is placed closer to the die. For the closest placement of the plateau (between z = 0 cm and z = 2 cm), the stop point moves back to 8.9 cm and the final fiber diameter is 29.3 μm. Figure 3 is a plot of the fiber velocity profiles in the presence of 2 cm plateaus. As was shown for the diameter profiles of Figure 2, the effect of a plateau becomes progressively more pronounced as the plateau is moved closer to the die face. For no plateau, the maximum velocity is 4.8 m/s. For the plateau

Figure 3. Fiber velocity profiles for when a 2 cm plateau is present in the air field. The air velocity is 110 m/s.

placed right at the die face, the maximum velocity is 16.2 m/s at a stop point of 8.9 cm. The fiber temperature profiles in the presence of a 2 cm plateau are plotted in Figure 4. Similar to what happens for the diameter and velocity profiles, the temperature profiles shift only slightly for plateaus placed far from the die. However, for the plateau placed right at the die face, the temperature profile is up to 50 °C higher than the temperature profile with no plateau present. So, in the critical region of fiber attenuation (the first few centimeters below the die), the fiber is hotter, less viscous, and more easily drawn. Figure 5 is a plot of fiber stress in the presence of a 2 cm plateau. As can be seen, when the plateau is placed closer to the die, the maximum stress increases and the position of the maximum moves closer to the die face. For no plateau, the maximum stress is 2.36 × 104 Pa at z = 3.4 cm. In contrast, for the best plateau position at the die face, the maximum stress is four times greater (9.89 × 104 Pa) and the position of this maximum is z = 2.5 cm. With a plateau, fiber stress is higher (see Figure 5) and fiber temperature is higher (see Figure 4). Thus, it is expected that 3473

dx.doi.org/10.1021/ie202501u | Ind. Eng.Chem. Res. 2012, 51, 3472−3482


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Figure 6. Fiber crystallinity profiles for when a 2 cm plateau is present in the air field (where, for example, 1.6e-7 represents 1.6 × 10−7). The air velocity is 110 m/s.

Figure 4. Fiber temperature profiles for when a 2 cm plateau is present in the air field. The air velocity is 110 m/s.

Figure 5. Fiber stress profiles for when a 2 cm plateau is present in the air field (where, for example, 1.2e+5 represents 1.2 × 105). The air velocity is 110 m/s.

Figure 7. Fiber diameter profiles for when a 2 cm plateau is present in the air field. These simulations are for a higher air velocity of 200 m/s.

fiber attenuation is enhanced in the presence of a plateau, and that is indeed what Figure 2 shows. Figure 6 shows fiber crystallinity in the presence of a plateau. Only for the case of no plateau or a plateau placed far from the die are there any changes in fiber crystallinity, and these changes are very small (fractional crystallinities are 1.4 × 10−7 or less). This behavior can be explained by examining Figure 4. With a plateau placed near the die, the fiber temperatures are simply too high to permit any online crystallization. Plateau simulation results were also produced for a higher gas velocity of 200 m/s. Figure 7 shows fiber diameter predictions for the melt blowing of polypropylene with conditions (other than gas velocity) the same as those for Figure 2. For no plateau, the final diameter is 29.4 μm at a stop point of 12.7 cm. For the closest placement of the plateau (between z = 0 cm and z = 2 cm), the stop point moves back to 4.5 cm, and the final fiber diameter is 17.3 μm. For the higher gas velocity, Figure 8 is a plot of the velocity profiles for different plateau placements. Without a plateau, the final fiber velocity is 16.1 m/s at z = 12.7 cm. With the optimum placement of the plateau at the die face, the final fiber velocity triples to 47.8 m/s at z = 6.2 cm.

Figure 8. Fiber velocity profiles for when a 2 cm plateau is present in the air field. The air velocity is 200 m/s.

Figure 9 shows the fiber temperature profiles for the higher gas velocity. For the plateau near the die face, the temperature profile is up to 43 °C higher than the profile for the no plateau case. 3474



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Figure 9. Fiber temperature profiles for when a 2 cm plateau is present in the air field. The air velocity is 200 m/s.

Figure 11. Fiber crystallinity profiles for when a 2 cm plateau is present in the air field. The air velocity is 200 m/s.

This shift in profile has about the same magnitude as the comparable shift in the profile for the low gas velocity case; see Figure 4. Fiber stress profiles for the higher gas velocity are shown in Figure 10. For no plateau, the maximum stress is 9.69 × 104 Pa

110 m/s, Figure 12 summarizes stop point results for plateaus ranging from 0.5 to 5.5 cm in width. In Figure 12, the stop

Figure 12. Stop point as a function of the start (z position) of the plateau. Plateau widths range from 0.5 to 5.5 cm. The air velocity is 110 m/s.

Figure 10. Fiber stress profiles for when a 2 cm plateau is present in the air field (where, for example, 4e+5 represents 4 × 105). The air velocity is 200 m/s.

point is plotted as a function of the start of the plateau. The start of the plateau is the z position of the beginning of the plateau. Thus, a start of plateau of zero indicates that the plateau begins at the die face (at z = 0). Figure 12 shows that wider plateaus have greater effect on the stop point. This is as expected, since a plateau of zero width is essentially the base case with no plateau. In Figure 12, this base case is represented by the horizontal dotted line at an ordinate value of 23.5 cm. Besides the effect of the plateau width, the placement of the plateau along the threadline is quite important. As discussed in the previous section, and as seen in Figure 12, the best placement is close to the die. For the simulations, we were able to place the 0.5, 1, 2, and 4 cm plateaus at the die face. However, the 5.5 cm plateau was unstable (in terms of the computer simulation) when placed at the die. A stable simulation did result from placing the beginning of the 5.5 cm plateau at z ≥ 2 cm. Larger plateau widths (6 cm and greater) were also tried, but these caused instability in the simulation. Physically, this indicates that very low fiber viscosity cannot be

at z = 2.3 cm. For the best plateau position at the die face, the maximum stress is 3.57 × 105 Pa, and the position of this maximum is at a slightly lower value of z = 2.1 cm. However, the position of maximum stress does not monotonically decrease as is the situation for the 110 m/s gas velocity (see Figure 4). Instead, as the plateau is placed closer and closer to the die, the position of maximum stress first increases and then decreases. The stress profile for the plateau placed between z = 3 cm and z = 5 cm has the largest z location for its maximum (this maximum occurs at z = 3.7 cm). For 200 m/s air velocity, Figure 11 shows fiber crystallinity profiles for when a 2 cm plateau is present in the air field. The coincident, zero-value profiles show that no crystallization occurs under the conditions used to produce this figure. Effect of Plateau Width and Position. Figures 2−11 show the effect of a 2 cm plateau on profiles of fiber diameter, velocity, temperature, stress, and crystallinity. Many other plateau widths were also tested. For an air velocity of 3475



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tolerated over too great a threadline range. The most significant perturbation of the stop point in Figure 12 is for the 4 cm plateau located right at the die face. In this, the stop point occurs at z = 7 cm, which is less than a third of 23.5 cm (the stop point when no plateau is present). Figure 12 also shows that locating part of the plateau at z ≥ 7.5 cm has little effect on the stop point. Figure 13 is a plot of the final fiber diameters corresponding to the cases considered in Figure 12. As can be seen, wide

right at the die face, the stop point is only 6 cm, which is about half of 12.7 cm, the stop point when there is no plateau present. Locating part of the plateau at z ≥ 4.5 cm has little effect on the stop point. For the higher gas velocity, Figure 15 is a plot of the final fiber diameters for the range of plateau widths. Again, the most

Figure 15. Final fiber diameter as a function of the start (z position) of the plateau. Plateau widths range from 0.5 to 2 cm. The air velocity is 200 m/s. Figure 13. Final fiber diameter as a function of the start (z position) of the plateau. Plateau widths range from 0.5 to 5.5 cm. The air velocity is 110 m/s.

significant change occurs when the plateau is wide and the plateau is close to the die. For the 2 cm plateau placed at the die face, the fiber diameter is 17 μm. Without a plateau, the diameter is 29.4 μm. Plateaus work because (a) the attenuating fiber is exposed to higher velocity air for a longer period of time and (b) the fiber is hotterand less viscouswhen the attenuating gas force is applied. Plateaus work best at positions near the die because a plateau of given width shifts the air field the most when the plateau is near the die (see Figure 1). As stated earlier, placing a tube or set of louvers below the melt blowing die is one possible method for approximating a plateau of constant velocity and temperature.

plateaus placed close to the die can significantly reduce the fiber diameter. For the 4 cm plateau placed at the die, the final fiber diameter is only 20 μm, a value significantly lower than the 53.8 μm diameter for when no plateau is present. For a higher gas velocity of 200 m/s, Figure 14 is a summary of stop point results for plateau widths ranging from 0.5 to 2 cm.

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LIMITS OF CRYSTALLIZATION

The above discussion shows how air field plateaus can produce finer filaments for a given air flow rate. Or, conversely, plateaus can produce the same size fibers with less air. However, the plateaus produce no significant change in crystallinity. Perhaps there is a flow field modification that can increase online crystallization of fibers. As mentioned above, the simulation involves the simultaneous solution of a set of differential equations. One of these equations is a structure equation that accounts for the change in crystallinity along the threadline. This equation, which is based on the differential form of the Nakamara equation, 7−9 is as follows:1 Figure 14. Stop point as a function of the start (z position) of the plateau. Plateau widths range from 0.5 to 2 cm. The air velocity is 200 m/s.

⎡ ⎛ 1 ⎞⎤(n − 1)/ n X∞ dX ⎟ = nK st(1 − θ)⎢ln⎜ ⎣ ⎝ 1 − θ ⎠⎥⎦ vfz dz

Wider plateaus were also tried, but these simulations were unstable. The reasons for this were as previously stated concerning the simulations for Figure 12. However, the stable range of plateaus is more limited for the higher gas velocity used in the simulations used to produce Figure 14. For a 2 cm plateau placed

(5)

where X = fractional crystallinity; θ = X/X∞ = the fraction of possible crystallinity; X∞ = crystallinity at the termination of crystallization; z = the axial position along the threadline, m; 3476



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vfz = axial fiber velocity, m/s; Kst = crystallization rate function, s−1; and n = the Avrami index. Nested within eq 5 are several other equations. The factor Kst(z) is the crystallization rate function that is dependent on the temperature and tensile stress. This factor is often expressed as10 K st(z) = K st(T , Δp) ⎡ (T (z) − Tmax )2 ⎤ ⎥ exp[Af 2 (z)] = K max exp⎢ − 4 ln 2 a ⎢⎣ ⎥⎦ D1/2 2

The third factor (C2) is a function only of fiber stress. C2 has a minimum value of 1 when there is no stress. Because C2 is exponential, the value of C2 can become quite large for high stress. Table 1 shows values of C2 for some typical values of stress during melt blowing. Stress values from Figures 5 and 10 were used in the calculations. As Table 1 shows, the contribution of stress to the crystallization rate can be phenomenally high.

(6)

Table 1. Values of Factor C2

where Δp = τzz − τxx = spinline stress or normal stress difference, Pa; Kmax = the maximum crystallization rate, s−1; D1/2 = the halfwidth of the crystallization function, K; Tmax = the temperature at which maximum crystallization occurs, K; fa = the amorphous orientation factor; and A = the stress-induced crystallization coefficient (see Jarecki et al.11). Equation 6 applies when the temperature lies between the glass transition temperature and the melting point (Tg < T < Tm) . Elsewhere, Kst(z) = 0. The fa in eq 6 is in turn defined by the equation (Jarecki et al.11) fa (z) =

a

2 ⎤ 1 ⎛ Copt ⎞ ⎜ ⎟ Δp2 (z) − ...⎥Δp(z) ⎥ 7 ⎝ Δna° ⎠ ⎦

−1

Kmax = 0.55 s D1/2 = 60 K Tmax = 338 K n=1

parameter value

references

3, 12−14 3, 12−14 3, 12−14 12

A = 40 × 10 Δna° = 60.0 × 10−3 Copt = 9.0 × 10−10 m2/N

12 3, 15 3, 15

With 2 cm plateau between z = 0 cm and z = 2 cm.

(9)

⎡1 − θ ⎤ f − ln⎢ ⎥ = K stt ⎣ 1 − θo ⎦

(10)

where θf is the final crystal fraction and θo is the initial crystal fraction. The half-period is designated by the time needed to go from θo = 0 to θf = 0.5. On the basis of eq 10, Table 2 shows the half-period for various temperatures. Table 2. Crystallization under Zero Stress Conditions

To understand the model’s limits on fiber crystallization, we can examine Kst(z), the crystallization rate function. We rewrite eq 6 as

K st(z) = K st(T ,Δp) = K maxC1C 2

1 1.65 4210 3.27 × 1047

where t = time. Equation 9 is useful for the discussion at hand since eq 9 expresses crystallinity change as a function of time. In contrast, eq 5 expresses crystallinity change in terms of position along the threadline. Since, as stated above, we assume that n = 1, then eq 9 can be integrated to get the following:

(7)

references

3

C2

0 2.36 × 104 9.69 × 104 3.57 × 105

⎡ ⎛ 1 ⎞⎤(n − 1)/ n dθ ⎟ = nK st(1 − θ)⎢ln⎜ ⎣ ⎝ 1 − θ ⎠⎥⎦ dt

where Δna = birefringence; Δna° = birefringence of perfectly oriented amorphous polymer; and Copt = the stress-optical coefficient, m2/N. The structure equations (eqs 5−7) have a number of input parameters. Values are needed for these parameters. We used the same values as were used previously by Shambaugh et al.1 These values are as follows: parameter value

Δp (Pa)

0 110 200 200 a

For no stress (Δp = 0) conditions, how long does it take for crystallization to take place? As stated above, eq 5 was derived from the differential form of the Nakamara equation.7−9 The Nakamara equation is

Copt ⎡ C Δna(z) ⎢1 − 3 opt Δp(z) = Δna° Δna° ⎢⎣ 7 Δna° −

initial air velocity (m/s)

temperature (°C)

C1

Kst (s−1)

half-period (s)

20 65 125 250

0.2102 1 0.0625 3.569 × 10−12

0.1156 0.55 0.03438 1.963 × 10−12

5.995 1.260 20.16 3.532 × 1011

(8)

As can be seen, at the optimum crystallization temperature of 65 °C it takes only 1.26 s to reach θf = 0.5. At room temperature, T = 20 °C, it takes about 6 s. So, as is well-known, polypropylene crystallizes while sitting on the collection screen. At 125 °C, which is above the optimum temperature by an amount D1/2 = 60 °C, the half-period is 20 s. Since it takes less than a second for a fiber to go from the die to the collection screen, not much crystallization (under low-stress conditions) can take place while the fiber travels to the collection screen. What about enhancement of the crystallization rate with stress (according to eq 6)? Well, at 300 °C, a temperature typical of what the fiber experiences when the stress is maximum (compare Figure 9 with Figure 10), the half-life is huge, and crystallization is thus unlikely. The range of temperatures included in Table 2 is typical of the melt blowing simulations

where

⎡ (T (z) − Tmax )2 ⎤ ⎥ C1 = exp⎢ − 4 ln 2 ⎢⎣ ⎥⎦ D1/2 2

C 2 = exp[Afa 2 (z)] The first factor (Kmax) on the right side of eq 8 is a characteristic of the polymer and cannot be changed. The second factor (C1) is a function only of the temperature. C1 is a distribution function that has a maximum value of 1 when T(z) = Tmax = 338 K. For temperatures relatively near to 338 K (this nearness is defined by D1/2 = 60 K), C1 is large enough to give a significant Kst . 3477



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the goal is to maintain a higher velocity and temperature at a given z). Figures 16−18 are plots of some possible air fields for quenching. Figure 16 shows the air field for when there is an

discussed in this paper. For ordinary melt blowing (see Figures 4 and 9), fiber temperatures go from 310 °C at the die to stop point temperatures of 173 and 231 °C for initial gas velocities of, respectively, 110 and 200 m/s. We saw in Figure 6 that there was a small amount of crystallization that took place near the end of the threadline. High stress did not contribute to this crystallization, since this crystallization took place where the stress was quite low (Figure 5). However, the temperature was relatively cool (Figure 4), and the fiber was relatively slow (and exposure time was relatively long). Thus, there was enough time (as shown in Table 2) to cause this crystallization (albeit a very small level of crystallization).

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QUENCH What we would like to do is to have high stress occur at the same time that the fiber temperature is low enough (near enough to Tmax = 65 C) to allow crystallization to occur along the threadline. Except for the one impractical case mentioned above, neither ordinary melt blowing nor melt blowing with a plateau can cause much crystallization to occur. But, perhaps a rapid temperature quench can cause crystallization. It would be wise to simulate that which we believe could be experimentally duplicated. So, what if, at some point along the threadline, we injected cold air. We assume complete mixing of the cold air and attenuating air at the injection point where z = z1. Then, the air temperature would exhibit a step decrease of temperature; we will call this ΔT. Now, if the cold air were injected in the negative radial direction, then the mixing of the melt blowing air with cold air with little z momentum would slow down the attenuating air stream. However, if the cold air were injected parallel to the threadline and at the same velocity as the attenuating air (at the z1 of injection), then the velocity of the attenuating air stream would not change (though the volumetric flow would increase). If we desired, by increasing or decreasing the injection velocity we could also decrease or increase the attenuating air velocity beyond z = z1. (The assumption of perfect mixing of the cold air and attenuating air at z = z1 is similar to the common assumption of perfect mixing in a CSTR [continuous stirred tank reactor] in discussions of chemical reactors. Though it is impossible to achieve a perfect CSTR, the CSTR assumption serves as a useful starting point for calculations.) So, experimentally we might be able to produce an air field wherein the temperature exhibits a step drop of ΔT, but the velocity does not have a step decrease or increase. We will then assume that, beyond the injection point, the air velocity continues to be described by eqs 1 and 2. How might we modify the temperature correlations (eqs 3 and 4) to handle the step drop in temperature? Well, for our plateau simulations we used the above-described shift in the abscissa value from z to z′. However, for quenching we must deal with a change in the ordinate value. Our approach is to first specify a ΔT and then determine a Δz that produces this ΔT. We know that this Δz will be positive because we are causing the temperature profile (beyond z1) to shift to the left at z1. In other words, we need to put a larger z into the temperature formulas (eqs 3 and 4) in order to get the temperatures for z values beyond z1. Of course, we are also assuming that the decay in air temperature beyond z1 is described by eqs 3 and 4 (with z′, a shifted value of z, in the formula). For quenching, the Δz is positive, while for the plateaus described above, Δz is negative (because with plateaus

Figure 16. Air field with an instantaneous 50 °C quench located at z = 2 cm. Initial air velocity = 110 m/s, and initial air temperature = 368 °C.

Figure 17. Plateau−quench with a 2 cm plateau between z = 0 and z = 2 cm. An instantaneous 50 °C quench is located at z = 2 cm. Initial air velocity = 110 m/s, and initial air temperature = 368 °C.

instantaneous, 50 °C quench at z = 2 cm. Figure 17 shows the air field for when both a plateau and a quench occur. The idea behind Figure 17 is that the advantages of both a plateau and a quench are possible. The plateau is placed at the die face for maximum advantage. Figure 18 also has a plateau at the die face. In addition, Figure 18 has a very large quench of 380 °C, and this quench also occurs at the die face. Simulations were run with a whole range of quenches and plateau−quenches. For the usual base conditions as listed on Figure 2, a quench with ΔT = 20 °C was located at quench positions at z = 2, 4, and 8 cm. The results showed no particular advantage from the quench. The final fiber diameters were almost the same as with no quench. There was a slight increase in fiber crystallinity. For example, a 20 °C quench at 8 cm gave a crystallinity of 3.38 × 10−7, which is about twice as high as the base case of 1.45 × 10−7. However, both of these crystallinity levels are extremely small. With the base conditions listed in Figure 2, simulations were also run for a quench with ΔT = 50 °C and quench positions at z = 1, 2, 3, and 4 cm. Likewise, there were no significant changes in fiber diameter predictions 3478



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Figure 18. Plateau−quench with a 1.9 cm plateau between z = 0 and z = 1.9 cm. An instantaneous 380 C quench is located at the die face. After the quench, the air temperature remains constant. The initial air velocity is 200 m/s, and this velocity stays constant over the plateau.

Figure 19. Fiber diameter profiles in the presence of a plateau and a large quench. The ΔT at the die face is 300, 350, 370, 380, and 390 °C, and the air temperature is kept constant over the entire threadline. The initial fiber velocity is 200 m/s, and the Mw = 165 000. The velocity is constant over a plateau from z = 0 to z = 1.9 cm, and the velocity then decreases according to the usual correlations. (For the ΔT of 380 °C, Figure 18 shows the air field used in the simulations.) The result for no plateau or quench is also shown.

versus the case of no quench. The highest fractional crystallinity achieved (for ΔT = 50 °C) was 2.25 × 10−6 for a quench at 1 cm, which is an order of magnitude higher than the base case level of 1.45 × 10−7. But, this crystallinity level is still insignificant. Plateau−quenches were tried with a quench placed right at the end of the plateau. Plateau widths of 0.5, 1, 2, 4, and 5.5 cm were tested, and these plateaus were placed at various positions along the threadline. The plateau widths and positions that were used were almost identical to the widths and positions shown in Figure 12. The operating conditions listed on Figure 12 were also used. For the quench magnitude, ΔT = 50 °C was used. The results for these plateau simulations were close to the results for where there is a plateau with no quench. The highest fractional crystallinity of 3.91 × 10−7 (which is still very small) occurred with a 0.5 cm plateau placed between 5.0 and 5.5 cm and, of course, the 50 °C quench placed at 5.5 cm. This was higher than the base case crystallinity of 5.13 × 10−8. (For the base case crystallinity, there was no quench, but there was a plateau with the same width and position.) Finally, a plateau was combined with a very large quench placed right below the die face. Figure 18 illustrates this type of plateau−quench, and, for this arrangement, Figure 19 shows the fiber diameter profiles for a range of ΔT values from 300 to 390 °C. A base case with no plateau−quench is also shown on this figure. As the ΔT increases, the stop point increases. For ΔT = 300, 350, 370, 380, and 390 °C, the stop points are, respectively, 14.3, 22.3, 27.0, 30.6, and 32.5 cm. The respective final fiber diameters are 20.6, 26.4, 29.2, 31.2, and 32.3 μm. For the base case, the final fiber diameter is 16.3 μm, and the stop point is 7.8 cm. So, the lower ΔT values (and the base case) give smaller fiber diameters (which is desirable). But, as will be shown as follows, there is no increase in crystallinity at these lower ΔT values, though significant fractional crystallinities can be achieved at higher ΔT values. Figure 20 shows the fiber velocity profiles corresponding to simulation conditions used for Figure 19. Lower fiber velocities result from increased quench ΔT. The fiber temperature profiles are shown in Figure 21. The threadline temperature is much lower when there is a plateau− quench. For the base case, the final fiber temperature is 244 °C. For the quench with ΔT = 300 °C, the final temperature is

Figure 20. Fiber velocity profiles for the same conditions listed for Figure 19.

138 °C less. For the largest quench (ΔT = 390 °C), the final temperature is 257 °C less. Figure 22 shows the fiber stress profiles corresponding to the simulations used for Figure 19. All of the profiles with plateau− quench have a maximum stress that is about five times as high as the maximum stress for the base case. Also, the plateau− quench profiles show significant stress over a wide range of z positions. At 7 cm from the die face, all of the plateau−quench profiles (except for the slightly lower curve for ΔT = 300 °C) have stress values that are higher than the maximum stress for the base curve with no plateau−quench. Now, observe from Figure 21 that, for all the plateau−quench curves (except for ΔT = 300 °C), the fiber temperature at z = 7 cm is around 65 °C, which is Tmax, the optimum temperature for polymer crystallization. Ergo, we have the proper temperature for crystallization and a stress level that will enhance the crystallization rate. We should get online crystallization. 3479



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Figure 23. Fiber crystallinity profiles for the same conditions listed for Figure 19.

Figure 21. Fiber temperature profiles for the same conditions listed for Figure 19.

We relax the bottom boundary conditions on the threadline. Specifically, we assume that we have a collector in place at z = L and the stress is zero at the collector. This zero stress condition at the collector was described by Bresee and Ko.16 A conundrum inherent in the mathematical placement of the collector screen at L is that, while we assume that the fiber velocity reaches a maximum at L, we also realize that the fiber is stopped when it strikes the collector screen (i.e., the velocity is zero). Now, if we run the simulation with only the zero stress boundary condition, we produce fiber velocity profiles that are dependent on collector position (see Figure 24). For the collector placed at 2, 4, 6, 8, 10, and 12 cm, the final fiber velocities are, respectively, 1.48, 8.28, 13.1, 15.2, 15.9, and 16.1 m/s. It is best to avoid collector positions close to the die, since the fiber never reaches a very high velocity. Low final velocities result in large fiber diameters, an undesirable result.

Figure 22. Fiber stress profiles for the same conditions listed for Figure 19 (where, for example, 1.4e+6 represents 1.4 × 106).

Figure 23 verifies what we expect. We do indeed get significant online crystallization. A crystallization level of 0.082 is possible with ΔT = 380 °C. The plateau and quench conditions for Figure 23 (and Figures 20−22) are listed in the caption for Figure 19.

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COMMENTS ON BOUNDARY CONDITIONS

Our simulations are done with the previously mentioned stop point. As stated, the stop point (literally, the point at which calculations are stopped) is where the fiber stress is zero and the fiber velocity and air velocity are equal. In the simulation, the set of differential equations are solved by first guessing an initial stress Fo at the beginning of the threadline. Then, iteration proceeds down the threadline until (and if) the stress becomes zero. The fiber and air velocities are then checked to see if they match. The presence of a collector screen is not needed in this calculation. But, to get the best fiber attenuation with the melt blowing die, the collector should not be allowed to interfere with the attenuation process. Ergo, the collector should be placed at or beyond the stop point (although there may be fiber collection issues and bonding issues that dictate a different placement of the collector).

Figure 24. Fiber velocity profiles when the collector is placed at various positions below the die. For these calculations, the stop point criterion was not used (see text).

For the simulation of melt blowing, other researchers5 have successfully solved their model equations by using the mathematical placement of a collector at various positions below the die (with the assumption of zero stress at the collector). They then plot the final fiber velocity versus the collector position L. 3480



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Q = polymer flow rate, m3/s Tmax = the temperature at which maximum crystallization occurs, K vfz = axial fiber velocity, m/s v0 = vair = air velocity along the threadline, m/s vjo = va,die = air velocity at the die (z = 0), m/s vL = fiber velocity at collector, m/s X = crystal fraction in polymer X∞ = maximum degree of crystallinity achievable by the polymer Z(dAN) = z/dAN((ρa∞/ρao)1/2) z = axial position along threadline, m z′ = shifted axial position, m z1 = z position at start of plateau, or location of quench, m z2 = z position at end of plateau, m Δz = amount that axial position is shifted, m

Figure 25 is a plot that we produced with this technique. We developed this plot by taking the final fiber velocities and

Greek Letters Figure 25. Final fiber velocities (from Figure 24) plotted versus L, the collector position. The air velocity is also shown. Where these two curves cross is equivalent to our stop point. This is the solution technique used by Jarecki et al.5

corresponding L values from Figure 24. Also on Figure 25 is the air velocity. The crossover of the two lines on Figure 25 is what Jarecki et al.5 call the “edge of the air drawing zone” (a position that they state is optimum). In our terminology, this is the stop point. For our (stop point) simulations of melt blowing for the same conditions used to produce Figures 24 and 25, our simulations gave a stop point of 12.7 cm. This is also the crossover point on Figure 25. Therefore, our method of solution gives the same results as the method used by Jarcecki et al.5

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REFERENCES

(1) Shambaugh, B. R.; Papavassiliou, D. V.; Shambaugh, R. L. NextGeneration Modeling of Melt Blowing. Ind. Eng. Chem. Res. 2011, 50, 12233−12245. (2) Zachara, A.; Lewandowski, Z. Mathematical Modelling of Pneumatic Melt Spinning of Isotactic Polypropylene. Part I. Modelling of the Air Jet Dynamics. Fibres Text. East. Eur. 2008, 16 (No. 4 (69)), 17−23. (3) Jarecki, L. ; Ziabicki, A. Mathematical Modelling of the Pneumatic Melt Spinning of Isotactic Polypropylene. Part II. Dynamic Model of Melt Blowing. Fibres Text. East. Eur. (January/December/A) 2008, 16 (No. 5 (70)), 17−24. (4) Jarecki, L.; Lewandowski, Z.; Mathematical Modelling of Pneumatic Melt Spinning of Isotactic Polypropylene. Part III. Computations of the Process Dynamics. Fibres Text. East. Eur. (January/March) 2009, 17 (No. 1 (72)), 75−80. (5) Jarecki, L.; Ziabicki, A.; Lewandowski, Z.; Blim, A. Dynamics of Air Drawing in the Melt Blowing of Nonwovens from Isotactic Polypropylene by Computer Modeling. J. App. Poly. Sci. 2011, 119, 53−65. (6) Majumdar, B.; Shambaugh, R. L. Velocity and Temperature Fields in Annular Jets. Ind. Eng. Chem. Res. 1991, 30, 1300−1306. (7) Nakamura, K.; Katayama, K.; Amano, T. Some Aspects of Nonisothermal Crystallization of Polymers. II. Consideration of the Isokinetic Condition. J. Appl. Polym. Sci. 1973, 17, 1031−1041. (8) Patel, R. M.; Spruiell, J. E. Crystallization Kinetics during Polymer ProcessingAnalysis of Available Approaches for Process Modeling. Polym. Eng. Sci. 1991, 31 (10), 730−738. (9) Spruiell, J. E. Structure and Property Development During the Melt Spinning of Synthetic Fibers, pp 195−220 In Structure Development During Polymer Processing; Cunha, A. M., Fakirov, S., Eds.; Kluwer Academic: Dordrecht, The Netherlands, 2000. (10) Ziabicki, A. Fundamentals of Fibre Formation; John Wiley and Sons: London, 1976. (11) Jarecki, L.; Ziabicki, A.; Blim, A. Dynamics of Hot-Tube Spinning from Crystallizing Polymer Melts. Comput. Theor. Polym. Sci. 2000, 10 (1−2), 63−72. (12) Bhuvanesh, Y. C.; Gupta, V. B. Computer Simulation of Melt Spinning of Polypropylene Fibers Using a Steady-State Model. J. Appl. Polym. Sci. 1995, 58 (3), 663−674.

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CONCLUSIONS (1) Introducing a plateau into the air field results in substantially higher fiber attenuation for a given air flow rate. (2) Introducing a plateau−quench into the air field can cause online crystallization. However, the required quench ΔT is quite large, and fiber attenuation is less effective in the presence of a plateau−quench.

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ρao = air density along the center line downstream from the nozzle, kg/m3 ρa∞ = air density at ambient conditions, kg/m3 θ = X/X∞ = fraction of possible crystallinity θjo = excess air temperature above ambient at die exit, °C θo = excess air temperature above ambient along the center line (the z axis), °C

AUTHOR INFORMATION

Corresponding Author

*Tel.: (405) 325-6070. Fax: (405) 325-5813. E-mail: shambaugh@ ou.edu.

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NOMENCLATURE A = stress-induced crystallization coefficient Copt = stress-optical coefficient, m2/N dAN = outer diameter of annular die orifice, mm D1/2 = half-width of the crystallization function, K fa = amorphous orientation factor Kmax = the maximum crystallization rate, s−1 Kst = crystallization rate function, s−1 L = collector position, m Mw = weight average molecular weight, g/mol n = Avrami index Δna = birefringence Δnao = birefringence of perfectly oriented amorphous polymer Δp = τzz − τxx = spinline stress or normal stress difference, Pa 3481



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(13) Ziabicki, A., Fundamentals of Fibre Formation; John Wiley and Sons: London, 1976. (14) Magill, J. H. A New Technique for Following Rapid Rates of Crystallization II Isotactic Polypropylene. Polymer 1962, 3, 35−42. (15) Samuels, R. J. Structured Polymer Properties; John Wiley and Sons: New York, 1974; p58. (16) Bresee, R. R.; Ko, W. C. Fiber Formation During Melt Blowing. Int. Nonwovens J. (Summer) 2003, 21

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Modifying Air Fields To Improve Melt Blowing - Industrial

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