Modeling of the Melt Blowing Performance of Slot Dies - Industrial

Vishnu T. Marla, and Robert L. Shambaugh*. School of Chemical Engineering and Materials Science, University of Oklahoma, Norman, Oklahoma 73019...
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Ind. Eng. Chem. Res. 2004, 43, 2789-2797

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Modeling of the Melt Blowing Performance of Slot Dies Vishnu T. Marla and Robert L. Shambaugh* School of Chemical Engineering and Materials Science, University of Oklahoma, Norman, Oklahoma 73019

The dual-jet slot die is the most common device used for producing melt-blown fibers. In this die, high-velocity air impinges upon a polymer stream. A comprehensive model was applied to predict the performance of these slot dies. This model involves the simultaneous solution of the momentum, energy, and continuity equations that are applied to the fiber threadline. Predicted parameters include fiber attenuation, vibration frequency, vibration amplitude, temperature, and stress. Experimental measurements were taken during melt blowing with an actual slot die. These measurements compared very well with the simulated results. Introduction Melt blowing is a single-step process used to manufacture nonwoven mats or webs from molten polymer. Nonwoven mats or webs formed in this process find diverse applications in absorbent products, upholstery, geotextiles, and many other uses. Key to the melt blowing process is a melt blowing die whereby the molten polymer is impacted by hot gas. A slot, or “Exxon”, die is the most common die used in commercial melt blowing (see Figure 1). Melt blowing is a highly complex process that is difficult to model. Melt blowing has similarities to conventional melt spinning, and the development of a model for melt blowing has its origins in models for melt spinning.1-9 A recent paper by Marla and Shambaugh10 develops a model that can predict the three-dimensional motion of a fiber as it is being formed below a melt blowing die. The model involves the simultaneous solution of the momentum, energy, and continuity equations. The model equations are solved numerically with both Newtonian and viscoelastic constitutive equations. Predicted parameters include fiber attenuation, vibration frequency, vibration amplitude, temperature, and stress. In this paper, this new model will be used to predict the performance of a common slot die. In addition, the model’s predictions will be compared to experimental results. The goal of this effort is to demonstrate the use of a model that will predict, a priori, the performance of common melt blowing dies. Air Flow Field as an Envelope (Boundary) Condition. Critical inputs to the model of Marla and Shambaugh are the air velocity and temperature at any position below the melt blowing die. One method of obtaining these velocity and temperature fields is through experimental measurements. Pitot tubes, laser Doppler velocimeters, thermocouples, and other devices can be used to determine the flow fields below various die geometries. These measurements can then be correlated, and the correlations can be used as inputs to the model. Another way of obtaining the flow and temperature fields is by using CFD (computational fluid dynamics) applied to the gas side of the melt blowing process. If the fields are obtained this way, then some experimental measurements should still be taken to guarantee that the CFD results are properly calibrated. * To whom correspondence should be addressed. Tel: (405) 325-6070. Fax: (405) 325-5813. E-mail: [email protected].

Figure 1. A typical slot, or “Exxon”, die that is used to produce melt blown fibers.

A cross-sectional view of a melt blowing die is shown in Figure 2 (compare to the perspective view in Figure 1). The center section is called the nose piece; since the nose piece in Figure 2 is flat on the die face, this particular die is called a blunt die. When there is no flat present, the die is known is a sharp die (see Figure 3). Harpham and Shambaugh11,12 and Tate and Shambaugh13 have done experimental measurements on dies with the geometries shown in Figures 2 and 3. These investigators used pitot tubes and fine thermocouples to measure the velocity and temperature fields below the melt blowing dies. More recently, Krutka et al.14 made use of computational fluid dynamics (CFD) in predicting the velocity field of melt blowing dies; they validated their results with the experimental data of Harpham and Shambaugh11,12 and Tate and Shambaugh.13 CFD simulations take less time in comparison to data obtained from experiments. The CFD simulations also can describe phenomena that are difficult to measure in the laboratory (such as the flow field at distances very close to the die face).

10.1021/ie030767a CCC: $27.50 © 2004 American Chemical Society Published on Web 04/24/2004

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sions (2.02 and 0 mm) that straddle this 1.32-mm value (compare Figures 3 and 4). Since the correlations for the blunt and the sharp dies are of similar form, correlations used for our experimental die were calculated by applying the following simple interpolation formula:

Correlation for experimental die ) (1.32/2.02) × correlation for blunt die + (2.02 - 1.32)/2.02 × correlation for sharp die

(1)

We will henceforth refer to the die used in our experiments as the experimental die. For the blunt die, the air velocity and temperature correlations are (Harpham and Shambaugh13) Figure 2. A cross-sectional view of a slot die with a blunt nosepiece. All dimensions are in mm.

vo ) 1.47(Y(h))-0.624 vjo

(

( )(

x v ) exp -0.6749 vo x1/2

2

(2)

( ) ))

1 + 0.027

x x1/2

4

θo ) 1.20(Y(h))-0.615 θjo

(

(4)

( )(

x θ ) exp -0.6749 θo t1/2

2

(3)

1 + 0.027

( ) )) x t1/2

4

(5)

See the Nomenclature Section for the definitions of the symbols; in eq 2, the symbol y is used instead of the symbol z used by Harpham and Shambaugh. Similarly, for the sharp die (Tate and Shambaugh14) the velocity correlations are

Figure 3. A cross-sectional view of a slot die with a sharp nosepiece. All dimensions are in mm.

vo ) 2.88(y(h))-0.532 vjo

(

( )(

x v ) exp -0.6749 vo x1/2

2

(6)

( ) ))

1 + 0.027

x x1/2

4

(7)

The temperature correlations for the sharp die are15

θo ) 1.53(Y(h))-0.511 θjo

(

( )(

x θ ) exp -0.6749 θo t1/2

Figure 4. The blunt die used in our experiments (the “experimental die”). All dimensions are in mm.

Air Property Correlations Combined with CFD Results. Our experiments were run on a blunt die with a 1.32-mm flat on the nosepiece (see Figure 4; additional experimental details are given in a section below). The correlations developed by Harpham and Shambaugh11,12 and Tate and Shambaugh13 were done for flat dimen-

2

1 + 0.027

(8)

( ) )) x

t1/2

4

(9)

Again, see the Nomenclature Section for the definitions of the symbols. Equations 2-7 were used in eq 1 to produce velocity and temperature correlations for our experimental die. The velocity, v, in the above correlations is the downward (y direction) velocity. Lateral (x direction) velocity might also play a role in melt blowing. These velocities are much smaller than downward velocities, and are difficult to measure. However, the aforementioned paper by Krutka et al.14 predicted lateral velocities via CFD. This work by Krutka et al. was based on (and calibrated with) the very same geometries used to develop the correlations listed in eqs 2-9. Hence, one can with good confidence use these CFD predictions of lateral velocity. Figure 5 shows lateral velocity profiles for the conditions shown on the figure. These plots were

Ind. Eng. Chem. Res., Vol. 43, No. 11, 2004 2791

Figure 5. Typical lateral velocity profiles for the blunt die shown in Figure 2. These profiles are from the CFD simulations of Krutka et al. (2003).

extracted from the CFD work of Krutka et al. for a blunt die. Similar plots (not shown on the Figure) were also developed for a sharp die. Lai and Nasr16 used laser Doppler anemometry (LDA) to show that parallel plane jets have lateral velocities similar to those shown in Figure 5. To produce Figure 5, the entire set of data containing the lateral velocity profiles at each y was obtained from the CFD simulations of Krutka et al. These simulations were based on a two-dimensional grid with cell sizes of 122.5 × 122.5 µm and smaller. It would have been computationally intensive to use this entire data set in our model through the use of a search algorithm at every time step (each time step is ∼1 µs.). To simplify computations, correlations were developed for the lateral velocity profiles. These correlations were developed for y positions that were 2 mm apart, that is, y ) 2, 4, 6, etc. (A 2-mm distance is fine enough to allow, via interpolation between the correlations, the determination of velocity at any point in the flow field; 2 mm is also the length of the control volume in the model of Marla and Shambaugh.10) When the lateral velocities for all values of x were plotted at y increments of 2 mm, the velocity profiles were observed to be similar in form. A suitable functional form to fit these data is the asymmetric double sigmoid curve (ADS).17 This curve is an asymmetric peak function of the following form:

f(x) ) a +

b × x - c + d /2 1 + exp e

[

)]

(

[

1-

1 x - c - d /2 1 + exp f

(

]

)

(10)

The curves in Figure 5 were developed by fitting the data at several y planes to the ADS form. The ADS is a six-parameter curve (a, b, c, d, e, f ), and a regression analysis gives the ADS parameters for every y plane. For a large range of y values, Table SI-1 (see Supporting Information) lists the ADS parameters obtained from the regression along with the coefficient of determination for each profile. As can be seen, the R2 value is 0.99 in most cases. A similar data analysis was done with the data from the sharp die, and the ADS parameters

Figure 6. Lateral velocity profiles for the blunt die at y ) 20 mm. CFD simulations are shown for the isothermal case and for the case when Ta,die ) 100 °C. Also shown is the nonisothermal curve produced when the isothermal CFD simulation is modified by a density correction factor.

obtained from this regression are shown in Table SI-2. Again, the R2 value is 0.99 in most cases. One additional factor must be considered. The CFD work of Krutka et al. was done for dual rectangular jets with ambient air flowing at 100 slpm (standard liters per minute measured at conditions of 21°C and 1 atm pressure). However, the melt blowing experiments that we are modeling are performed at elevated air temperatures. Hence, a correction has to be made to account for the nonisothermal nature of the air jets. Now, for a fixed airflow rate (fixed slpm), one would expect that a higher air temperature causes a decrease in the air density and a corresponding increase in the air velocity. This correction factor is given by the average velocity at ambient conditions multiplied by Fa∞/Fa, which is the density at ambient conditions divided by the density at the temperature of interest. Hence, if 100 slpm is fed to our experimental die (see Experimental Details section), the average discharge velocity is 17.3 m/s at ambient conditions, in comparison to an average discharge velocity of 35.37 m/s at 330 °C. Though it is accurate to apply this correction factor for the average discharge velocity at the die face, is this simple correction factor suitable for modifying the lateral velocity (isothermal) correlations in Tables SI-1 and SI-2? Figure 6 shows an isothermal lateral velocity profile as predicted by the parameters in Table SI-1 for the position y ) 20 mm. Also shown in Figure 6 are two lateral velocity profiles for an air feed temperature of 100 °C. The first of these two nonisothermal profiles was produced by multiplying the isothermal velocity by the correction factor Fa∞/Fa (a variable which is determined at the respective temperature along the threadline). The second of these nonisothermal curves is from unpublished CFD work of Krutka.18 Since the two nonisothermal curves match each other quite well, it was assumed that applying the correction factor Fa∞/Fa is a reasonable approach to predicting lateral velocities at nonisothermal conditions. Thus, for both blunt and sharp dies, the correlations (from eq 10 and Tables SI-1 and SI-2) for predicting lateral air velocities were modified with the factor Fa∞/Fa. Then, with the use of eq 1, the actual lateral air velocity at any point below the die was calculated and used in our model.

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the low end for commercial fiber production, the experimental results (e.g., amplitude measurement, etc.) are representative of behavior over a much larger range of operating conditions. The polymer used in the experiments was 88 MFR Fina Dypro isotactic polypropylene. Although this material has a lower MFR than many polymers that are used commercially to produce melt-blown fibers, this 88 MFR material has been used commercially to produce meltblown fibers. The polymer had an Mw of 165 000 and an Mn of 41 500. The zero shear rate viscosity of this polymer can be described by the relation

(

ηf ) 0.001985 exp

Figure 7. The experimental equipment.

)

5754 Tf + 273

(11)

This relationship was verified with experiments on a cone and plate viscometer for measuring the zero shear viscosity of the polypropylene used in our experiments. These experiments were performed over a wide range of temperatures using a Dynamic Stress Rheometer (model SR-5000, Rheometric Scientific Inc.), and an empirical correlation of the Arrhenius form was developed. Equation 12 was used for calculating the zero shear viscosity in all our simulations. For the non-Newtonian (Phan-Thien) simulations, we assumed G ) 28 kPa, X ) 0.1, and E ) 0.015 (see PhanThien20).

Experimental Details Figure 7 shows the experimental setup. A Brabender extruder of 19.1 mm (0.75 in.) diameter and 381 mm length was used to melt and pressurize the polymer. The barrel had a 20:1 L/D ratio and a 3:1 compression ratio. The polymer exiting from the extruder was then fed to a modified Zenith pump, which pumps controlled quantities of molten polymer through a single-hole melt blowing die. The polymer capillary had an inside diameter of 0.420 mm. The two air slots were set flush with the nosepiece, and the slot widths were 0.65 mm (see Figure 4). Each slot had a length of 74.6 mm (2.94 in.). The die assembly was heated with two 250 W cartridge heaters. A thermal mass flow meter was used to measure the air flow rate, which was maintained at 100 slpm (21 °C and atmospheric pressure). For our die, 100 slpm corresponds to an isothermal die face velocity of 17.3 m/s. Details of the heating equipment for the air lines can be found in Harpham and Shambaugh.12 From the cross-sectional area of the slots and the air temperature, the air velocity at the die exit was calculated and was used as a boundary condition in our simulations. The measurement of the fiber cone diameter was done by taking multiple-image photographs. The camera used was a Nikon N90S equipped with a 105-mm Nikon macro lens. The film used was Kodak Tri-X 400. The illumination for the photographs was provided by a GenRad 1546 digital strobe. Refer to Chhabra and Shambaugh19 for details on the camera settings and the technique for the measurement of the fiber cone diameter. For our measurements, we used a strobe frequency of 5 flashes per second and a camera exposure time of 30 s. This gave 150 multiple-fiber images on each picture, and the cone diameter was determined by measuring the diameter of this multiple image. To facilitate visualization and measurement of the fiber motion and comparison with the model, experimental air velocities were limited to a maximum of ∼35 m/s (at 330 °C). Though this gas velocity is at

Model Results The simulations were done on a Dell Precision Workstation 530 with a dual 2.2 GHz Intel Xeon processor. The simulations were performed for a simulated time of 15 s. Typically, each simulation took about 6-8 h of wall clock time. Unless otherwise mentioned, the parameters plotted on the graphs are averages of the parameter values between t ) 5 and 15 s (simulation time). The correlations discussed above were used to describe the air field. The air field is uniform in the w direction. Of course, an actual die of finite length has end effects where the velocity field does change. The 3D model of Marla and Shambaugh was used for all our simulations; except where noted, refer to their paper for details on equations, boundary and initial conditions, and method of solution. For the fiber, initial sideways velocity, position, or both are needed as initial conditions. For nearly all of our runs, we specified an initial slope of ∆x/∆y ) 10-5 in the x direction and a slope of ∆w/∆y ) 0 in the w direction. If we instead let ∆x/∆y ) 10-5 and ∆w/∆y ) 10-5, the simulation results are essentially the same. This is because of the uniformity of the flow field in the w direction. If both slopes are set equal to 0, then the simulation shows no fiber vibration; this is a metastable condition (see Marla and Shambaugh). Figure 8 compares the predicted fiber attenuation for the sharp, the blunt, and the experimental dies. The sharp die predicts the fastest attenuation rate, with a final fiber diameter of 74 µm, whereas the blunt die predicts the slowest attenuation rate, with a final fiber diameter of 113 µm. This is expected because of the higher centerline velocities of the sharp die. Tate and Shambaugh13 observed that, for a given set of operating conditions, the sharp die produces higher centerline velocities in comparison to the blunt die. Hence, the aerodynamic drag force exerted on the polymer would be higher in a sharp die. For the experimental die the

Ind. Eng. Chem. Res., Vol. 43, No. 11, 2004 2793

Figure 8. The model predictions of the threadline diameter profile for three different die geometries.

Figure 9. The model predictions of the amplitude of fiber vibration. For this and all subsequent figures, the “experimental” die geometry was used.

predicted fiber attenuation curve is intermediate between the other two curves, which is reasonable considering that the experimental die’s geometry is intermediate. The horizontal dotted line in Figure 8 is the measured final diameter of the fibers produced during actual experiments run with the same conditions used in the simulation. This off-line diameter was determined with a Nikon microscope with a calibrated eyepiece. As can be seen on the figure, the predicted final diameter (for the experimental die) is very close to the actual measured diameter. Figure 9 shows the predicted fiber amplitude for the three die geometries. The model predicts higher cone diameters as the die becomes sharper. At 4.5 cm from the die, the fiber amplitude is 0.392 mm for the blunt die, 0.513 for the experimental die, and 0.635 mm for the sharp die. Rao and Shambaugh (1993) observed that higher air velocities result in an increase in the fiber amplitude. Since the centerline velocity at any point below the die becomes higher as the die becomes sharper, our results parallel those of Rao and Shambaugh. Figure 9 also shows the predicted fiber amplitude when the lateral air velocity is not considered (i.e., when vx,air ) 0). Without the lateral air velocities, the predicted fiber amplitudes are more than an order of magnitude less. For example, for the experimental die,

Figure 10. The model predictions of the frequency of fiber vibration.

the fiber amplitude at 4.5 cm from the die is 0.513 mm when lateral velocity is included versus 0.045 mm when lateral velocity is 0. As discussed below, actual experimental measurements of fiber amplitude show that the inclusion of lateral velocity brings the simulation fairly close to the experimental results. The inclusion of lateral velocities affects both the fiber amplitude and frequency (these lateral velocities are only on the order of ∼0.1 m/s). However, our simulations showed that the prediction of other properties, such as diameter, temperature, and stress, are affected very little by the inclusion of lateral velocities. Our results parallel the work of Marla and Shambaugh (2003), who observed that larger predicted fiber amplitudes do not correspond to significant changes in diameter, temperature, and stress. Figure 10 shows the predicted fiber frequency along the threadline for three different dies. As the plots show, a continuous fiber must vibrate at the same frequency at any point along the threadline. The frequency predicted for the experimental die lies between the values for the sharp and blunt die. The highest frequency geometry (20 Hz) is for the sharp die, whereas the lowest frequency (12 Hz) is for the blunt die. Chhabra and Shambaugh19 used laser Doppler velocimetry (LDV) to measure the vibration frequency below a melt blowing die similar to our experimental die. They measured frequencies in the range of 10-40 Hz. Their experimental measurements compare well with our simulations. Effect of Polymer Flow rate. Figure 11 illustrates how polymer flow rate can affect the predicted fiber diameter along the threadline. Lower polymer flow rates result in higher attenuation rates and finer fiber diameters, a commonly observed result in both melt blowing and melt spinning. Figure 12 shows the predicted temperature profiles at the same three polymer flow rates used in Figure 11. The fibers cool more quickly at the lower polymer flow rates. For example, at y ) 10 cm, the fiber temperatures are 184, 225, and 251 °C for polymer flow rates of 0.23, 0.55, and 1.1 g/min, respectively. Apparently, the lower thermal inertia of the finer fiber results in an increase in the cooling rate of the fiber. Figure 13 compares experimental and simulated results for (maximum) fiber vibration amplitude at different polymer flow rates. The trend is the same for both experiment and simulation: the fiber amplitude increases as the polymer flow rate decreases. The model

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Figure 11. The effect of polymer flow rate on fiber attenuation.

Figure 12. The effect of polymer flow rate on fiber temperature.

Figure 13. The effect of polymer flow rate on vibration amplitude. Both simulation results and experimental results are given.

underpredicts the fiber amplitude, but the amplitudes are of the same order for each polymer flow rate. The best agreement is found for the 0.23 g/min case in which the experimental fiber amplitude at y ) 4.5 cm from the die is ∼4.3 times the fiber amplitude predicted by the model. At the same y value, the experimental and predicted fiber amplitudes for the 0.55 g/min case are 3.0 and 0.513 mm, respectively, whereas for the 1.1 g/min case, the amplitudes are 2.63 and 0.434 mm, respectively.

The effect of polymer flow rate on the predicted frequency of the fiber is shown in columns 2, 5, and 6 in Table 2. The fiber frequency is constant along the threadline (compare Figure 10). The frequency increases as the polymer flow rate decreases; this is analogous to the effect of polymer flow rate on fiber amplitude. The frequency almost doubles to 23 Hz when the polymer flow rate is reduced from 1.1 to 0.23 g/min. A similar trend was observed with the 2D model of Rao and Shambaugh.13 Rao and Shambaugh worked with an annular die, whereas our work is with a slot die. This suggests that a change in an operating variable produces the same trend in predicted results, regardless of the die geometry. Effect of Polymer Temperature. The effect of initial polymer temperature (Tf,die) on the fiber diameter and fiber temperature is shown in Figure 14. A higher initial polymer temperature results in higher attenuation rates due to lower polymer viscosity. A 35 °C increase in the polymer temperature at the die results in a predicted reduction of the fiber diameter from 95 to 74 µm. The final fiber diameters measured experimentally are very close to the predictions of the model (See Table 1). The figure also shows the temperature profiles when the polymer temperature at the die is changed. A higher polymer temperature at the die results in a higher temperature at any point along the threadline. The temperature profile for the Tf,die ) 295 °C case shows an increase in the y < 1 cm region before the fiber temperature begins to decay. This is because the air temperature (Ta,die ) 330 °C) is higher than the polymer temperature for a few millimeters below the die, which results in an increase in the polymer temperature. Also note that the two temperature profiles are almost parallel to each other, and ∆T is ∼30 °C all along the threadline. Figure 15 shows the effect of initial polymer temperature on the maximum fiber amplitude. Both experimental data and the predictions of the model are plotted on the figure. When the initial polymer temperature is increased 35 °C, the measured fiber amplitude increases from 3.0 to 3.75 mm at y ) 4.5 cm. In comparison, the model predicts an increase from 0.513 to 0.784 mm. The effect on frequency of increasing the initial polymer temperature is shown in columns 2 and 7 of Table 2. The fiber frequency is constant along the threadline (compare Figure 10). The fiber frequency decreases by almost 25% when the initial polymer temperature is raised by 35 °C. Rao and Shambaugh9 observed similar behavior with an annular die. Effect of Air Temperature. The diameter and temperature profiles predicted by the model when the initial air temperature (Ta,die) is varied are shown in Figure 16. A higher air temperature results in a higher attenuation rate and, thus, finer diameter fibers. The model predicts final fiber diameters of 95 and 86 µm, respectively, for Ta,die ) 330 and 380 °C. In addition, for the high air temperature case, the temperature at any point along the threadline is greater. This results in a decrease in the polymer viscosity, which in turn increases the attenuation rate. Again, excellent agreement is observed between the experimental measurements and the model predictions of the final fiber diameters. Figure 17 shows the fiber amplitude when Ta,die is changed. The simulated amplitudes show that increasing Ta,die results in an increase in fiber amplitude. The

Ind. Eng. Chem. Res., Vol. 43, No. 11, 2004 2795 Table 1. Summary of Results Obtained from Experiment and Simulation for Different Operating Conditions vibration amplitude at y ) 4.5 cm

final fiber diameter die

m (g/min)

Tf,die (°C)

Ta,die (°C)

sharp blunt experimental experimental experimental experimental experimental

0.55 0.55 0.55 1.10 0.23 0.55 0.55

295 295 295 295 295 330 295

330 330 330 330 330 330 380

experiment (µm)

simulation (µm)

95 ( 3 126 ( 5 30 ( 5 73 ( 3 84 ( 3

72.9-74.2 113.1-113.5 95.5-96.2 134.2-135.5 56.2-56.5 74.0-75.2 85.3-86.4

experiment (mm)

simulation (mm)

3.00 2.625 3.375 3.75 3.215

0.635 0.392 0.513 0.434 0.772 0.784 0.593

Table 2. Vibration Frequencies Predicted by the Model die type experimental

sharp

blunt

experimental experimental

experimental

experimental

experimental

m (g/min) 0.55 0.55 0.55 1.1 0.23 0.55 0.55 0.55 Tf,die (°C) 295 295 295 295 295 330 295 295 air flow rate 100 100 100 100 100 100 100 100 (slpm) Ta,die (°C) 330 330 330 330 330 330 380 330 constitutive Newtonian Newtonian Newtonian Newtonian Newtonian Newtonian Newtonian viscoelastic equation vib freq (Hz) 16 20 12 12 23 12 20 13

Figure 14. The effect of polymer temperature (at the die) on fiber diameter profile and fiber temperature profile.

Figure 16. The effect of air temperature (at the die) on fiber diameter profile and fiber temperature profile.

Figure 15. The effect of polymer temperature on vibration amplitude. Both simulation results and experimental results are given.

Figure 17. The effect of air temperature on vibration amplitude. Both simulation results and experimental results are given.

experimental amplitudes also show an increase, though the effect is relatively small. As observed earlier (Figure 15), the experimental values of fiber amplitude are higher than the simulated values, though the trends are the same. As a comparison of Figures 13, 15, and 17 shows, fiber amplitude increases as Ta,die and Tf,die increase, but

amplitude decreases as polymer flow rate increases. Physically, this makes sense. Higher temperature in either the polymer or the air causes the filament to have a lower viscosity, which means that the filament is easier to move around, and the amplitude is thus larger. In contrast, as polymer flow rate increases, the polymer filament becomes heavier and harder to move around, and the amplitude is thus smaller.

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fiber amplitude is 0.578 mm for the Phan-Thien model, whereas the Newtonian model predicts 0.513 mm. Since the viscoelastic Phan-Thien model is far more complicated than the Newtonian model, the simulations for the Phan-Thien model took almost 3 times the wall clock time required for a Newtonian model. Columns 2 and 9 in Table 2 show that there is a decrease in frequency when the Phan-Thien model is used instead of the Newtonian model. The fiber frequency is constant along the threadline (compare Figure 10). The Phan-Thien model gives a frequency of 13 Hz, whereas use of the Newtonian model gives a frequency of 16 Hz. Conclusions Figure 18. Comparison of Newtonian and viscoelastic predictions of fiber diameter profile and fiber temperature profile.

Figure 19. Comparison of Newtonian and viscoelastic predictions of fiber amplitude.

Columns 2 and 8 in Table 2 show how frequency is affected by increasing Ta,die from 330 to 380 °C. The fiber frequency is constant along the threadline (compare Figure 10). The temperature rise causes the simulated frequency to increase from 16 to 20 Hz. Non-Newtonian Results. Figure 18 shows the diameter and temperature profiles predicted by using the Phan-Thien viscoelastic equations in the model. As was done in Marla and Shambaugh (2003), we used values for the viscous shear thinning parameter and stress saturation parameter that were suggested by PhanThien (1978). The previous results from using a Newtonian constitutive equation are also plotted. The profiles obtained from the two constitutive equations are very close. For the Phan-Thien equation, a final fiber diameter of 91 µm is predicted, whereas for the Newtonian equation, the model predicts 95 µm. As mentioned earlier, for melt blowing wherein rapid temperature changes occur, the temperature-dependence of the constitutive equation is more important than nonNewtonian effects. Hence, a Newtonian model does a good job of predicting melt blowing performance, and the values used for the Phan-Thien parameters are not critical. Figure 19 compares the amplitude predicted by the Newtonian equations with the amplitude predicted by the Phan-Thien equations. The Phan-Thien equations predict slightly higher amplitude in comparison to the Newtonian equations. For example, at y ) 4.5 cm, the

Table 1 summarizes the results obtained from experiment and from simulation. In almost every case, the model predicts final fiber diameters that are very close to the values measured from experiments. The fiber amplitude is underpredicted in each case, but is of the same order (about 20-25% of the experimental values). A key factor in the simulation was the inclusion of lateral velocities in the air flow field. This factor allowed the prediction of fiber amplitudes that both followed the same trends as the measured amplitudes and had similar magnitudes. The model utilized average values of velocity. If turbulence had been included, the simulated fiber amplitudes would probably be even closer to the experimental values. Underprediction of amplitude is described in Marla and Shambaugh (2003) for the case of annular dies. In their work, simulated amplitudes were only ∼10% of the experimental values. The inclusion of lateral velocities in our present work probably permitted a closer (but still too low) match of simulated amplitudes with experiment. In addition to the prediction of fiber diameter and fiber amplitude, the simulation also allows the prediction of fiber temperature, fiber stress, and vibration frequency. The simulation also shows that the Newtonian (temperature-dependent) viscosity effects dominate over the viscoelastic effects. This is due to the rapid changes in fiber temperature that occur during melt blowing. Acknowledgment This work was generously supported by an NSF GOALI grant (DMII-0245324). The authors also are most grateful for the financial assistance provided by 3M, Procter & Gamble, and Du Pont. Supporting Information Available: Two tables of ADS parameters for the blunt die and the sharp die. This material is available free of charge via the Internet at http://pubs.acs.org. Nomenclature a, b, c, d, e, f ) the six adjustable constants in the ADS curve (see eq 10) h ) gap between edges of plates, mm m ) polymer mass flow rate, g/min t1/2 ) air temperature half width, °C Ta,die ) air temperature at die (y ) 0), °C Tf ) filament temperature, °C Tf,die ) filament temperature at die (y ) 0), °C va,die ) vjo ) air velocity at the die (y ) 0), m/s vx,air ) vx ) lateral air velocity, m/s

Ind. Eng. Chem. Res., Vol. 43, No. 11, 2004 2797 v ) free stream vertical air velocity, m/s vo ) maximum air velocity at a fixed y, m/s x ) horizontal Cartesian coordinate along x-axis (see Figures 2-4) x1/2 ) air velocity half-width, mm y ) vertical Cartesian coordinate y(h) ) y/h Y(h) ) (y/h)(Fa∞/Fao)1/2 w ) horizontal Cartesian coordinate along w axis (see Figures 2-4) Greek Letters ηf ) zero shear fiber viscosity, Pa‚s θjo ) excess air temperature above ambient at die exit, °C θo ) excess air temperature above ambient along the center line (the y axis), °C Fa ) air density, kg/m3 Fao ) air density along the center line downstream from the nozzle, kg/m3 Fa∞ ) air density at ambient conditions, kg/m3 Subscripts a ) air die ) die f ) fiber x ) property along the x axis y ) property along the y axis

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Received for review October 15, 2003 Revised manuscript received February 13, 2004 Accepted February 16, 2004 IE030767A