Effects of Die Geometry on the Flow Field of the Melt-Blowing Process

Ind. Eng. Chem. Res. 2003, 42, 5541-5553

5541

Effects of Die Geometry on the Flow Field of the Melt-Blowing Process Holly M. Krutka, Robert L. Shambaugh,* and Dimitrios V. Papavassiliou School of Chemical Engineering and Materials Science, The University of Oklahoma, 100 East Boyd Street, SEC T335, Norman, Oklahoma 73019

Sharp dies are often used commercially to produce polymeric fibers in the melt-blowing process. In these sharp dies, the flow field results from two similar converging plane jet nozzles with no space between the nozzles. This study utilizes a computational fluid dynamics approach that is validated through experimental data to investigate the effect of recess or excess (inset or outset) of the die nose on the flow field. The Reynolds Stress Model is used to simulate the turbulence, and the model parameters are calibrated with experimental data. The flow field downstream from the sharp die is found to exhibit (a) a merging region, which includes a maximum in turbulence intensity, and (b) a self-similar region. The behavior of alternative die designs is correlated to the die configuration. The more that the nose piece is recessed, the larger is the mean velocity under the die, but at the same time the turbulence becomes stronger. 1. Introduction Dual rectangular jets are commonly used in industry to fabricate polymer fibers in the melt-blowing process.1,2 Air is propelled through two converging jets to rapidly attenuate a molten strand of polymer below the die face. Figure 1 shows a cross-section of a sharp flush die, which is frequently used in this process.3 The flow field created by the dual rectangular jets strongly affects the size and the strength of the polymer fiber. Therefore, it is important to understand this complicated flow field to improve the conditions for polymer extrusion and to optimize die design. In general, a high mean velocity at the centerline (x ) 0) is desired in order to rapidly attenuate the polymer fibers. In addition, turbulence intensity along this centerline must be minimized, since strong velocity fluctuations in the flow field can cause the polymer fibers to break off and/or stick to the die face. The flow field that results from two parallel plane jets has been studied previously in the experimental investigation of Nasr and Lai4,5 and the numerical investigations of Anderson and Spall6 and Lai and Nasr.7 The flow field that results from two converging plane jetss the configuration that is critical to melt-blowingshas been studied experimentally by Harpham and Shambaugh8,9 for blunt and flush sharp die faces and by Tate and Shambaugh3 for several different jet geometries. Tate and Shambaugh measured the z-component of the mean velocity at different positions in the flow field that exists downstream from a sharp die (both inset and outset). Computational fluid dynamics (CFD) has recently offered a complementary, additional method of studying such flow fields.10 Simulations using CFD can be completed in a fraction of the time required for laboratory experiments, while the cost of running each simulation is much lower than the cost of laboratory experiments. Also, CFD simulations allow for close examination of regions of the flow field that are difficult to test in the laboratory, such as the area very close to * To whom correspondence should be addressed. Tel.: (405) 325-6070. Fax: (405) 325-5813. E-mail: [email protected].

Figure 1. Melt-blowing die with a flush nose piece.

the die face or nose piece. However, as shown by Krutka et al.,10 experimental data are necessary to properly calibrate the CFD simulations. In the work described herein, the flow fields of several types of sharp dies (i.e., inset and outset dies) were simulated using the computational fluid dynamics package Fluent 6.0. These CFD results were validated with the laboratory measurements of Tate and Shambaugh.3 In addition to the mean velocity, the turbulence properties of the flow fieldsincluding the turbulence intensity, the turbulent kinetic energy, and the turbulence dissipation rateswere examined. Furthermore, predictive correlations for the effect of the nose piece location on the flow field were developed. 2. Numerical Modeling and Simulation Parameters Several variations of the sharp die are used in industrial melt-blowing. The inset die is characterized by a nose piece that is recessed above the die face. Figure 2 shows the geometry of a standard inset die. Observe that, unlike the situation with the flush die (Figure 1), in the inset die the slot face width bo is not equal to the distance b (which equals h/2). Since actual commercial dies are “large” in the ydirection (the direction perpendicular to the plane of

10.1021/ie030457s CCC: $25.00 © 2003 American Chemical Society Published on Web 10/04/2003

5542 Ind. Eng. Chem. Res., Vol. 42, No. 22, 2003

Figure 2. Melt-blowing die with an inset nose piece. Figure 5. Turbulence intensity and velocity decay for different grid resolutions (inset a ) -bo/2 die). Table 1. Summary of the Problem Configuration and the Computational Requirements for Each Numerical Experiment

Figure 3. Melt-blowing die with an outset nose piece.

Figure 4. Computational domain and grid refinement regions. In section A, the outermost region, the length of the sides of the quadrilateral cells is 0.4 mm. Sections B, C, and D have, respectively, cell sides of 0.2, 0.1, and 0.05 mm.

Figure 2), a 2-D approximation of the flow field is adequate to describe the flow field for practical distances below the die. The aspect ratio (i.e., die slot length divided by the die slot width) for such dies is much

die type

a

no. of cells

inset inset inset inset outset outset outset outset outset

-bo/4 ) -0.1625 mm -bo/2 ) -0.325 mm -bo ) -0.65 mm -5bo/4 ) -0.8125 mm bo/4 ) 0.1625 mm bo/2 ) 0.325 mm bo ) 0.65 mm 5bo/4 ) 0.8125 mm 3bo/2 ) 0.975 mm

112 000 112 657 113 952 112 342 111 418 109 789 110 011 109 137 110 581

CPU time on no. of 2.2 GHz iterations Pentium4 (h) 27 892 28 597 28 337 34 020 43 542 30 640 29 691 30 173 29 598

18:10 18:30 21:00 23:45 29:50 21:15 20:50 20:00 21:20

larger than the value of 50 recommended11 for statistically 2-D plane jets that are free of end effects at the center of the nozzle. A plane of symmetry (x ) 0) exists between the two jets. In a 2-D representation of the die, this plane of symmetry becomes a line that coincides with the path of the polymer. Although extensive laboratory tests have been conducted by Tate and Shambaugh3 (henceforth this contribution will be referred to as TS) for a recession of a ) bo/2 ) 0.325 mm, the effect of other magnitudes of this recession has not been investigated. In the present study, inset dies with the following recess values have been simulated: a ) bo/4, bo/2, bo, and 5bo/4, where bo ) 0.65 mm. The data of TS for the case of an inset die with a recession of 0.325 mm have been used to validate the current simulations of an inset die. Both the experimentally tested flow fields of TS and the present simulations neglect the presence of the polymer fiber along the line of symmetry. The outset die is another modification of the sharp die. The outset die consists of a nose piece that extends below the die face; see Figure 3. Similar to their work with the inset die, TS conducted measurements for an outset die with a nose piece that extended 0.325 mm below the die face. In the present work with CFD, outset dies were examined with the following nose piece extensions below the die face: a ) bo/4, bo/2, bo, 5bo/4, and 3bo/2 (where bo ) 0.65 mm). Detailed information about the CFD procedures, the determination of the computational domain size, grid size, and the choice of turbulent models, can be found in Krutka et al.10 (henceforth in the present paper, this previous work will be referred to as KSP). However, a brief summary of the CFD procedures for the present paper is given here.

Ind. Eng. Chem. Res., Vol. 42, No. 22, 2003 5543

Figure 6. Comparison of CFD results with experiments: (a) dimensionless mean velocity close to the die face and (b) dimensionless mean velocity farther from the die face. Part b shows the equation Vo/Vjo ) 3.66*(z/h)-0.558, which is the least-squares fit of the experimental data to a power law equation. A similar fit for the CFD results is the equation Vo/Vjo ) 4.39*(z/h)-0.637 (this equation is not shown in part b).

2.1. Problem Configuration. The computational domain, shown in Figure 4, is based on a 2-D representation of the experimental setup of TS. Gridding of this domain follows the methodology used in KSP: areas near the air discharge slots have finer gridding. The coordinate system for the experiments and the simulations is shown in Figures 1-3. The origin of the system is at the center of the face of the die. The y-direction is along the axes of the nosepiece and the slots (perpendicular to the plane of the figures), the x-direction is transverse to the major axis of the slots, and the z-direction is the primary direction of the flow under the die. The right boundary of the computational domain has been designated as a line of symmetry,12 allowing the computational domain, and therefore the simulation time, to be cut in half. Air enters through the velocity inlet at the top of the jet. The bottom and left boundaries are pressure outlets, while the other boundaries in the computational domain are set as walls. The experimental studies of TS for 60° and 70° sharp dies showed that the flow field downstream from the nozzles becomes self-similar (i.e., the velocities and Reynolds stresses are scaled with the mean centerline

Figure 7. Dimensionless mean centerline velocity: (a) inset dies, (b) outset dies, and (c) maximum mean centerline velocity as a function of the die configuration (R ) 0.99998 for the equation shown on the graph).

velocity in the z-direction, Vo [Vo ) V(0, 0, z)], and with the distance from the centerline at which the mean velocity is half of its maximum, x1/2, defined as the point at which 1/2Vo ) V(x1/2, 0, z), when z/h > 1.8). Therefore, the size of our computational domain was defined so that the two converging jets have enough space to reach the self-similar region; the domain was 100 mm long in the z-direction and 30 mm wide in the x-direction. The angle between the jet inlet and the top wall (the die face) was 60°, and the jet inlet had a height of 5 mm, except

5544 Ind. Eng. Chem. Res., Vol. 42, No. 22, 2003 Table 2. Coefficients in Eq 6 for the Die Configurations Examined in This Work die type

a

c1

m

R2

inset inset inset inset flush outset outset outset outset outset

-5bo/4 ) -0.8125 mm -bo ) -0.65 mm -bo/2 ) -0.325 mm -bo/4 ) -0.1625 mm 0 mm bo/4 ) 0.1625 mm bo/2 ) 0.325 mm bo ) 0.65 mm 5bo/4 ) 0.8125 mm 3bo/2 ) 0.975 mm

8.4989 7.3297 6.2589 6.0333 5.5037 5.1988 5.4609 5.3653 5.0143 5.3595

-0.6195 -0.6403 -0.6466 -0.6533 -0.6385 -0.6232 -0.6429 -0.6418 -0.6206 -0.6427

0.9996 0.999 0.999 0.998 0.9996 0.9988 0.9967 0.9957 0.9965 0.9948

in the case of inset dies. For these dies, the total height of the jet was the magnitude of the nose piece recess plus the original jet height of 5 mm. The domain size was similar to that used in KSP. 2.2. Grid Generation. Quadrilateral cells were used to construct the grid in the interior space of the domain. The grid resolution was increased around the area of most interest, which is where the flow fields from the two jets converge just below the die face. Both the centerline mean velocity and the turbulence intensity reach a maximum within this region. Four regions of grid refinement were adopted in order to reduce computational time relative to the KSP runs that utilized two grid refinement regions. In the innermost region, the length of the sides of the quadrilateral cells was 0.05 mm. Outside this region, there was a second section where the grid resolution was larger and the length of the sides of the quadrilateral cells was 0.1 mm. The next (third) level of grid resolution was in the area farther from the line of symmetry; this level consisted of quadrilateral cells with sides of 0.2 mm. Finally, in the fourth section of the computational domain (the domain that is farthest from the die), the length of the sides of the cells was 0.4 mm. Figure 4 shows the four different regions of grid refinement and the regions’ dimensions. Figure 5 compares the turbulence intensity and the mean velocity decay results for a die with a recess of a ) 0.325 mm for different grid resolutions. The grid resolution designated as fine is the one used by KSP. The two simulations show excellent agreement. Therefore, the coarse grid was used for our subsequent calculations in order to minimize computational effort. (An even coarser grid was tested but was not used because the simulated results did not match the results shown in Figure 5.) The total number of cells in the coarse computational domains for the different inset and outset dies was approximately 113 000, which reduced the number of grid cells to half of those used in KSP. The exact number of quadrilateral cells, the necessary number of iterations to reach convergence, and the required CPU time for the inset and outset simulations are given in Table 1. 2.3. Turbulence Modeling. The previous work of KSP has shown that the Reynolds Stress Model (RSM) is most appropriate for the accurate simulation of the flow field that results from two converging plane air jets. The RSM is based on the solution of transport equations for the individual Reynolds stresses. The equations from the Reynolds averaging of the momentum equation are multiplied by a velocity fluctuation, and thus these equations are exact. In the 2-D, stationary state, isothermal case with a low Mach number, such as the inset and outset die flow field simulations, the Reynolds

Figure 8. Dimensionless velocity profiles within the self-similar region (z ) 50 mm) of the flow below the die and comparison to experimental measurements for a flush die: (a) inset dies and (b) outset dies.

stress equations take the following form:12,13

∂ ∂ (FUkuiuj) ) [Fu u u + Fj(δkjui + δikuj)] + ∂xk ∂xk i j k ∂Uj ∂Ui ∂ ∂ µ (uiuj) - F uiuk + ujuk + ∂xk ∂xk ∂xk ∂xk ∂ui ∂uj ∂ui ∂uj + - 2µ (1) p ∂xj ∂xi ∂xk ∂xk

[

] (

(

)

)

The summation convention is used in the above equation. With an equation that is the same as the dissipation rate equation for the standard k- model, the dissipation rate is modeled as follows:

[( ) ]

µt ∂ D ∂ 2 µ+ + C1 Gk - C2F ) Dt ∂xi σ ∂xi k k

F

(2)

The suggested values of the constants are C1 ) 1.44, C2 ) 1.92, and σ1 ) 1.0 (as recommended by Fluent12). However, KSP determined that C1 and C2 in the RSM model need to be modified from their default values. The constant C1 was changed from 1.44 to 1.24, and C2 was changed from 1.92 to 2.05. Changing C1 and C2 allowed for the accurate simulation of the centerline mean velocity close to the die as well as the accurate mean velocity decay throughout the computational domain for


Figure 9. Contour plots of the mean z-velocity below the die face: (a) inset die with a ) -bo/2 and (b) outset die with a ) bo/2. These contours can be scaled with the size of the slot width (bo ) 0.65 mm; see Figures 1-3).

the case of both a blunt die and a sharp flush die. These updated values of C1 and C2 were used as the starting point for the current simulations. After comparing CFD results with experimental data for the die with an inset of 0.325 mm and the die with an outset of 0.325 mm, we found that further update of C1 and C2 was not necessary. The residuals of the model equations were required to reach 10-5 to achieve convergence. For the different simulations, approximately 20 000-30 000 iterations were necessary to meet this chosen convergence criterion. Table 1 details both the computational time and the exact number of iterations for each simulation.

3. Results and Discussion 3.1. Comparison with Experimental Results. Figure 6a presents the dimensionless mean z-velocity, Vo/Vjo, along the line of symmetry, where Vo is the mean z-velocity at x ) 0 and Vjo is the nominal discharge velocity. The simulation and laboratory results close to the die face are presented for the case of an inset die with a recession of bo/2 ) 0.325 mm. Figure 6b compares the experimental and simulated dimensionless centerline velocity decay farther from the die face. Good agreement is observed in the flow field both close to and far from the die face. The exponential fit to the experi-


Figure 10. Turbulence intensity across the centerline: (a) inset dies, (b) outset dies, and (c) maximum turbulence intensity as a function of the die configuration (R ) 0.99894 for the equation shown on the graph).

mentally recorded velocity decay is (TS)

Vo/Vjo ) 3.66*(z/h)-0.558

(3)

while the exponential fit from the present CFD results is

Vo/Vjo ) 4.39*(z/h)-0.637

(4)

The simulated flow fields from all of the different inset

dies were generated using the same volumetric flow rate as in TS, which is 100 L/min. The slot width d was 0.563 mm for all simulations. The air moving through the jets was set at a temperature of 21 °C and 1 atm; these were the conditions used by TS. Even though the air flow in an industrial melt-blowing process is not isothermal, the understanding and modeling of the air flow in isothermal conditions is of significant scientific value, since there are no theoretical results available for the case of two converging plane jets. (Also, as shown by the experimental results of Harpham and Shambaugh,9 the flow patterns of nonisothermal converging jets are a natural extension of the patterns for isothermal converging jets.) The z-component of the air velocity at the jet inlet (see Figure 4) was set at 17.3 m/s (this is the nominal discharge velocity Vjo). To compare the flow fields created by various inset and outset jets, the same model parameters were used. Since the model parameters previously determined by KSP simulate the inset die with accuracy, these parameters were used for the simulations of all die geometries. The velocity, turbulence intensity, Reynolds stresses, turbulent kinetic energy, and turbulence dissipation rate in the flow field were calculated to determine the effect of alternate die geometries on the flow field. 3.2. Mean Velocity. The centerline velocity in the flow field is important, because the path of the polymer fiber during attenuation generally follows the line of symmetry between the two jets. A higher maximum air velocity in the z-direction leads to an increased rate of fiber attenuation (i.e., a finer fiber) for a given air flowratesthis is economically desirable. For the different dies, z ) 0 is set at the die face (i.e., the plane of the air plates as shown in Figures 1-3); therefore, the amount that the dies are inset or outset is measured from the die face. The parameter “a” in Figures 2 and 3 is negative for inset dies and positive for outset dies. For different inset dies Figure 7a shows how the nose piece recession (inset) affects the dimensionless centerline velocity. Increasing the nose piece recession clearly leads to an increase in the maximum centerline velocity; the velocity at the highest recession (a ) -5bo/4) is triple the velocity for a flush die (a ) 0). This result is expected, because increasing the amount of recession of the nose piece leads to a smaller inlet jet width b. Since the same rate of volumetric air flow must pass through a smaller opening, the mean velocity increases. For the outset die, the centerline velocity is related to the extension of the nose piece below the die face. Figure 7b compares this centerline velocity for five different outset dies. The maximum centerline velocity is found to decrease as the nose piece extends beyond the die face. However, as a comparison of part a with part b of Figure 7 shows, the quantitative change in the velocity profiles is less for the case of outset versus inset. Besides the obvious differences at the velocity maxima, all along the velocity profiles there are large differences between the centerline velocities for different inset dies. In contrast, the outset dies have very similar centerline velocity profilessexcept for the (relatively small) profile differences at positions near the velocity maximums. A further generalization, for both parts a and b of Figure 7, is that the effect of moving the nose piece decreases as the nose piece moves in the positive z-direction. Figure 7c shows how die inset or outset correlates with the maximum centerline velocity. To better com-


Figure 11. Contour plots of the turbulence intensity below the die face: (a) inset die with a ) -bo/2 and (b) outset die with a ) bo/2. These contours can be scaled with the size of the slot width (bo ) 0.65 mm; see Figures 1-3).

pare with a “standard” flush die, the maximum velocity along the line of symmetry has been divided by the maximum centerline velocity achieved by the flush die. The abscissa in Figure 7c is the dimensionless recession of the nose piece, a/d. Figure 7c illustrates a significant decrease in the maximum of the mean centerline velocity as the nose piece is moved in the positive z-direction. Also shown in Figure 7c is the following empirical equation that was fit to the data:

(Vo/Vjo)max (Vo/Vjo)max-flush

) -2.5383

(da + 2.3523)

0.79013 + 1.7615

(5)

This equation is an excellent fit (R ) 0.99998). Since the nominal velocity Vjo is constant for all the simulations, the left side of eq 5 could also be written as simply (Vo)max/(Vo)max-flush. Equation 5 is useful for predicting maximum velocities for inset and outset values that were not considered in the simulations (and the experiments of TS).

For positions away from the die face, the CFD data for the velocity decay of the two converging jets can be correlated with the following relation that is similar to the one used for a plane 2-D jet14,15,10 (see also eqs 3 and 4 in the previous section):

(Vo/Vjo) ) c1(z/d)-m

(6)

The length scale used here is the distance d instead of h that was used in eqs 3 and 4, because d (as well as bo) is common for all types of dies, while h changes. Table 2 presents empirical fits of the coefficients c1 and m for the different dies examined here. The exponent has values between 0.6195 and 0.6533; these are different from the exponent of 0.5 suggested for a single two-dimensional turbulent jet.14 For the range of inset dies (from the lowest a until a ) 0), the parameter c1 decreases from about 8.5-5.5. Then, as a is increased further into the range of the outset dies, the parameter c1 stays essentially constant. This behavior parallels what was discussed above for inset versus outset dies: more change occurs for the inset than for the outset die


distances well below the die face:

V ) exp[-0.6749(x/x1/2)2(1 + 0.027(x/x1/2)4)] (7) Vo

Figure 12. Dimensionless Reynolds stress profiles across the centerline: (a) inset dies, (b) outset dies, and (c) maximum Reynolds stress as a function of the die configuration (R ) 0.99894 for the equation shown on the graph).

(e.g., see Figure 7c). Heskestad16 and Gutmark and Wugnanski17 found experimentally that the mean velocity and the Reynolds stresses for a plane jet become selfsimilar beyond z/bo ) 40. In this region, they scaled the velocities and the Reynolds stresses with Vo, and they scaled the distances with x1/2. Figure 8a shows the simulated dimensionless mean velocity of the different inset dies at z ) 50 mm (z/bo ) 76.92, a position well below the die face). Also shown in Figure 8a is the following correlation used by TS that accurately represents their experimentally determined mean velocity at

TS found that eq 7 well-fit their data taken for the flush geometry (they did not take corresponding data for inset or outset dies). Equation 7 was originally developed by Bradbury18 for rectangular jets. Figure 8a shows that the level of nose piece recession plays a very small role in the simulated flow field, since all the inset dies have very similar mean velocities at the 50 mm distance. In addition, V/Vo for the inset dies at z ) 50 mm shows excellent agreement with the experimental V/Vo for the flush die at this distance from the die face. Similarly, Figure 8b compares the simulated mean velocity of the different outset dies at z ) 50 mm to the experimental results represented by eq 7. As with the inset dies, the simulated flow fields for the different outset dies are very close to the flow field obtained experimentally by TS at long distances from the die face. As described earlier, Figures 7 and 8 demonstrate that the flow field resulting from the two converging plane jets exhibits a region of development that depends on the configuration of the die face, followed by a region of self-similarity in which the flow field does not “remember” its origin and behaves like the flow field from a plane jet. The effect of the placement of the nose piece below or above the die face is minimal at a distance far from the die. Figure 9a,b illustrates this transition. Figure 9a,b presents, respectively, the simulated contour plots of the mean z-velocity for inset and outset dies with, respectively, a ) -bo/2 and a ) bo/2. Though the flow fields are quite different near the die face, the similarity in the velocity field away from the die face is evident. At these large distances from the die face, mean velocity decay can be easily determined from eq 6 used in conjunction with Table 2. For “a” values not given in the table, interpolation/extrapolation can be used. 3.3. Fluctuating Velocity Field. In addition to the velocity maximum achieved by different dies, the fluctuating velocity field must also be considered for the optimization of die performance. Figure 10a compares the turbulence intensity, q, along the line of symmetry for the different inset dies. The turbulence intensity is a measure of the relative strength of the velocity fluctuations. The maximum turbulence intensity increases as the amount of recession above the die face increases, and it exhibits two local maxima: one at the die face (z ) 0) and one at a location downstream from the die face. The turbulence in the flow field increases as the air moves through the region where the two jets meet. At the first part of this region of intersection (right at the die face), the air flow is forced to accelerate as the air streams mix and move through a smaller space, which increases the turbulence intensity and results in the first observed intensity maximum. After the fluid exits the constriction at the die face, the turbulence intensity starts to decrease, and the intensity shows a local minimum at a location that corresponds to the location at which the mean velocity exhibits its maximum. After this minimum, the intensity increases again, because the turbulence production, which is proportional to the slope of the mean velocity, increases. The maximum turbulence intensity exhibited by the die with an inset of 5bo/4 () 0.8125 mm) is significantly


Figure 13. Contour plots of the Reynolds stress below the die face: (a) inset die with a ) -bo/2 and (b) outset die with a ) bo/2. These contours can be scaled with the size of the slot width (bo ) 0.65 mm; see Figures 1-3).

higher than the maximum turbulence intensities caused by the lower inset settings. As the nose piece extends below the die face for the outset dies, the difference between the turbulence intensity profiles for different outset dies decreases. Figure 10b shows the turbulence intensity for the centerline of the different outset die flow fields. The average outset die intensity is an order of magnitude less than the average intensity for the inset dies of Figure 10a (note that the ordinate scale in Figure 10a is logarithmic). Similar to the case with the centerline velocities, there is little difference between the turbulence intensity in the flow fields from the dies with outsets of a ) 5bo/4 and a ) 3bo/2. As shown in Figure 7c, movement of the nose piece downward in the positive z-direction decreases the centerline velocity. Figure 10c illustrates how the centerline turbulence intensity also decreases when the nose piece is moved in the positive z-direction. As was

the case for the velocity profile, the turbulence intensity of the flow tends to a common profile as the nose piece is moved away from inset positions to outset positions. The correlation obtained for the maximum turbulence intensity as a function of the position of the die nose piece is

qmax -3.1618 a ) 0.80824 + 2.6842 + 2.1659 (8) qmax-flush d

(

)

As can be seen in Figure 10c, for the simulations at large outset, there are negligible differences in the turbulence intensity maxima along the line of symmetry. Figure 11a,b presents contour plots of the turbulence intensity for the inset and outset dies with, respectively, a ) -bo/2 and a ) bo/2. The fluctuations are strong as the two jets converge (points A) and also at the locations where the mean velocity increases or decreases at a high rate (i.e., high mean velocity slope, points B).


The dimensionless Reynolds stress profile along the line of symmetry is shown in Figure 12a,b for the different die geometries. Figure 12a shows the centerline profile of the Reynolds stresses for the different inset dies. The difference between these profiles is much more pronounced for the dies with insets of 5bo/4 and bo. However, the dies with insets of bo/2 and bo/4 are very similar. This difference is related to the fact that the Reynolds stress changes sign as the slope of the mean velocity changes sign (i.e., negative Reynolds stress is associated with increasing mean velocity, and positive Reynolds stress is associated with decreasing mean velocity). Hence, if we observe the centerline velocity in Figure 7a, we see that, because of the constriction of the jets at high recess levels, the centerline velocity increases dramatically before the air crosses the die face. The centerline Reynolds stress profiles become much more similar as the nose piece moves in the outset position (Figure 12b). Close to the die, the Reynolds stresses profiles have different magnitudes, although they have similar shapes. However, the outset Reynolds stress profiles become identical farther from the die face. Figure 12c is a plot of the centerline Reynolds stress (RS) maximum for both the inset and outset dies. The maximum in the Reynolds stress decreases as the nose piece is moved downward in the z-direction. The location of the maximum Reynolds stress is of interest, because this location is closely associated with area of the flow where turbulence is produced. The relationship between the Reynolds stress and the placement of the die nose piece is found to be

(uw/Vjo2)max

)

(uw/Vjo2)max-flush -3.2662

(da + 1.9212)

0.61019 + 0.95139

(9)

Figure 13a,b presents contour plots of uw (i.e., the Reynolds stress divided by the density of the fluid) for inset and outset dies with, respectively, a ) -bo/2 and a ) bo/2. The maximum Reynolds stress is not located on the centerline; instead, there are two local maxima (these are negative maxima) near the location where the two air jets converge (see points labeled A in Figure 13). Farther downstream, within the self-similar zone of flow development, the Reynolds stress changes sign and exhibits two maxima in the region next to the converged jet (see points labeled B in Figure 13). Figure 13a shows that maximum turbulence is produced right at the location where the two jets are merging (i.e., just before and during the flow of the air stream through the constricting die face); see points A in Figure 13a. As shown in Figure 13b (for the outset die), the locations of maximum turbulence production (see points A in Figure 13b) are slightly below the nose piece. Figure 14a demonstrates the relationship between nose piece recession and turbulent kinetic energy, k, for different inset dies. The k is nondimensionalized by dividing by the square of the nominal velocity. The turbulent kinetic energy increases significantly as the amount of recession above the die face is increased. However, as the nose piece approaches the die face, the effect of changing the recession decreases. The reason for the appearance of two local maxima along the k profile is the same as the reason for the appearance of

Figure 14. Dimensionless turbulence kinetic energy profiles across the centerline: (a) inset dies, (b) outset dies, and (c) maximum turbulence kinetic energy as a function of the die configuration (R ) 0.99647 for the equation shown on the graph).

two local maxima for the turbulence intensity profiles, i.e., the locations of maxima correspond to locations of maximum mean velocity slopes, and the locations of minima correspond to locations of mean velocity maxima or plateaus. Figure 14b shows the centerline turbulent kinetic energy for the different outset dies. As with the inset dies, the effect of increasing z diminishes for larger values of z. However, the range of k for the outset dies is much smaller than for the inset dies. In addition, the


Figure 15. Contour plots of the turbulence kinetic energy below the die face: (a) inset die with a ) -bo/2 and (b) outset die with a ) bo/2. These contours can be scaled with the size of the slot width (bo ) 0.65 mm; see Figures 1-3).

outset dies exhibit a much smaller maximum k than their inset counterparts. The maximum turbulent kinetic energies achieved by the dies are compared in Figure 14c. The maximum k achieved along the centerline decreases as the die nose piece is moved downward in the z-direction. In fact, the difference between the magnitudes of the turbulent kinetic energies for the outset dies is negligible. The curve fit for the relationship between k maximum and nose piece placement is as follows:

(k/Vjo)max (k/Vjo)max-flush

-3.2712

(da + 2.1929)

) 0.79985 + 2.748

(10)

Figure 15a,b presents contour plots of k for the inset and outset dies with, respectively, a ) -bo/2 and a ) bo/2. One can expect high levels of turbulent kinetic energy in locations that are close to areas that exhibit high Reynolds stresses (since the production of turbu-

lent kinetic energy is the product of the Reynolds stress and the mean velocity slope). A local maximum in the turbulent kinetic energy occurs at the merging location of the converging jets (see points A in Figure 15), which is the area between the location of the two maxima in the Reynolds stresses. The maxima in the turbulent energy are represented by points B in Figure 15, which correspond to areas of high Reynolds stress. The rate of the dissipation of the turbulent kinetic energy can be examined for the different dies. Figure 16a shows the centerline turbulence dissipation rate for the different inset dies. The magnitude of the turbulence dissipation rate for the die with an inset of 5bo/4 is significantly higher close to the die. However, the turbulence dissipation rate profiles are very similar farther from the die face. Figure 16b compares the dissipation rate profiles for the outset dies. The magnitude and shape of these profiles are similar throughout the computational domain and are indistinguishable after z/d ) 30.


different dies are extremely similar, except for inset values of a ) -bo and a ) -5bo/2 (which correspond to placement of the nose piece well above the die face). The relationship between turbulence dissipation rate maximum and placement of the nose piece can be correlated as follows:

(d/Vjo3)max (d/Vjo3)max-flush

)

(da + 1.9336)

0.81443 + 0.97901

-3.8472

(11)

4. Conclusions

Figure 16. Dimensionless turbulence dissipation rate profiles across the centerline: (a) inset dies, (b) outset dies, and (c) maximum turbulence dissipation rate as a function of the die configuration (R ) 0.99825 for the equation shown on the graph).

Figure 16c shows the maximum turbulence dissipation rate for the different inset and outset dies. As expected, the dies that exhibit the largest turbulence intensity also exhibit the largest turbulence dissipation rate. As an example of this, compare the turbulence intensity for a die with an inset recess of 5bo/4 (see Figure 14a) with the dissipation rate at this recess value (in Figure 16c). Clearly, as the nose piece is moved in the positive z-direction, the turbulence dissipation rate profiles become more similar. This is exhibited in Figure 16c wherein the magnitude of the maxima for the

The flow field resulting from two rectangular jets that converge in a sharp die has been predicted using CFD. The simulations have been validated with laboratory data. The numerical procedures detailed in KSP have been followed in the present work. The RSM for turbulence simulation in conjunction with the model parameters suggested by KSP have been used. The flow field exhibits two regions of development that depend on the sharp die configuration. In the first region, the two jets are converging, and the die geometry plays a significant role on the flow field development. Different levels of recess for inset dies result in dramatic differences in the turbulence quantities that are observed within this region. The interaction of the two jets is manifested by the generation of high Reynolds stresses in the area just above (z < 0) and just at the die face (z ) 0). This interaction region is also characterized by a maximum in turbulent kinetic energy and turbulence intensity. The effect of having an outset nose piece diminishes as the nose piece extends farther in the z-direction. The second region of flow development is a self-similar region where flow behaves as if resulting from a single plane jet. In this second region, there is only one maximum of the mean velocity, and this maximum is located at the centerline. The numerical experiments allow the correlation of the die configuration to the flow behavior. A number of simulations were conducted using different inset and outset levels, to investigate the effect of the nose piece position on the flow properties. As the inset level is increased (i.e., as a becomes more negative), the mean velocity is much higher in the centerline for the same air flow rate. This higher velocity may result in higher polymer fiber speeds, which is desirable. However, the turbulence intensity is also dramatically higher, which might be a disadvantage in fiber spinning. In fact, a maximum of turbulence intensity occurs right at the die face, where the constriction is at its smallest and where a polymer fiber might start to vibrate and stick to the die tip. For an outset die, the fluctuations tend to a common profile that occurs when a > bo. If it is desirable to reduce the fluctuations along the path of the polymer fiber, particularly at the die tip, then an outset die appears to be a better option. In fact, the original Harding patent2 suggests that a slight outset will help prevent fiber breakage and polymer accumulation on the die. On the other hand, if one desires to increase the mean velocity along this path without increasing the air consumption of the die, then an inset die is more suitable. Acknowledgment This work was supported by an NSF GOALI grant (DMII-0245324). The support of ConocoPhillips and the


University of Oklahoma Research Council is also gratefully acknowledged. Nomenclature b ) half of h (see Figures 1-3), mm bo ) face width of the die slot (see Figures 1-3), mm c1 ) empirical constant (eq 6) C1 ) parameter for the RSM model (eq 2) C2 ) parameter for the RSM model (eq 2) d ) width of the die slot (see Figures 1-3), mm h ) gap between edges of air plates (see Figures 1-3), mm k ) turbulent kinetic energy (1/2uiui), m2/s2 l ) die slot length, mm Lx, Lz ) size of computational box in the x- and z-directions, mm P ) pressure, Pa q ) turbulence intensity, (ui2)1/2}/Vjo Q ) air flow rate through both slots, m3/s Re ) Reynolds number (Re ) bVjoF/µ) t ) time, s ui ) velocity fluctuation in the ith direction, m/s Ui ) mean velocity in the ith direction, m/s uw ) Reynolds stress divided by the fluid density, m2/s2 Vz ) velocity in the z-direction, m/s Vo ) z-direction velocity along the symmetry line, m/s Vjo ) nominal discharge velocity defined as Vjo ) Q/(2bol), m/s V ) total vector velocity that includes both the z- and x-components of the velocity, m/s |V| ) magnitude of the total vector velocity, m/s x, y, z ) spatial coordinates, mm x1/2 ) jet half width, which is the distance from the centerline at which the mean velocity becomes half of its value at the centerline, mm Greek characters δij ) Kroeneker’s delta ) dissipation rate of turbulent kinetic energy, m2/s3 q ) angle that either air gap slot makes with the face of the die, deg µ ) viscosity, kg/(m‚s) µt ) turbulent viscosity, kg/(m‚s) ν ) kinematic viscosity, m2/s F ) density, kg/m3 σk ) turbulent Prandtl number for the kinetic energy σ ) turbulent Prandtl number for the dissipation

Literature Cited (1) Buntin, R. R.; Keller, J. P.; Harding, J. W. Nonwoven Mats by Melt Blowing. U.S. Patent 3,849,241, November 19, 1974. (2) Harding, J. W.; Keller, J. P.; Buntin, R. R. Melt-blowing Die for Producing Nonwoven Mats. U. S. Patent 3,825,380, July 23, 1974. (3) Tate, B. D.; Shambaugh, R. L. Modified Dual Rectangular Jets for Fiber Production. Ind. Eng. Chem. Res. 1998, 37, 37723779. (4) Nasr, A.; Lai, J. C. S. Comparison of Flow Characteristics in The Near Field of Two Parallel Plane Jets and an Offset Plane Jet. Phys. Fluids 1997, 9(10), 2919-2931. (5) Nasr, A.; Lai, J. C. S. Two Parallel Plane Jets: Mean Flow and Effects of Acoustic Excitation. Exp. Fluids 1997, 22(3), 251260. (6) Anderson, E. A.; Spall, R. E. Experimental and Numerical Investigation of Two-dimensional Parallel Jets. J. Fluid Eng. Trans. ASME 2001, 123(2), 401-406. (7) Lai, J. C. S.; Nasr, A. Two Parallel Plane Jets: Comparison of the Performance of Three Turbulence Models. P. I. Mech. Eng. G-J Aer. 1998, G6, 379-391. (8) Harpham, A. S.; Shambaugh, R. L. Flow Field of Practical Dual Rectangular Jets. Ind. Eng. Chem. Res. 1996, 35, 3776-3781. (9) Harpham, A. S.; Shambaugh, R. L. Velocity and Temperature Fields of Dual Rectangular Jets. Ind. Eng. Chem. Res. 1997, 36, 3937-3943. (10) Krutka, H. M.; Shambaugh, R. L.; Papavassiliou, D. V. Analysis of a Melt-blowing Die: Comparison of CFD and Experiments. Ind. Eng. Chem. Res. 2002, 41(20), 5125-5138. (11) Pope, S. B. Turbulent Flows; Cambridge University Press: Cambridge, United Kingdom, 2000; pp 134-135. (12) Fluent Inc. Fluent 6.1 User’s Guide, Lebanon, NH, 2003. (13) Durbin, P. A.; Petterson Reif, B. A. Statistical Theory and Modeling for Turbulent Flows; John Wiley and Sons Ltd.: West Sussex, England, 2001; p 149. (14) Schlichting, H. Boundary-Layer Theory, 8th ed.; McGrawHill: New York, 1987; p 179. (15) Mohammed, A.; Shambaugh, R. L. Three-Dimensional Flow Field of a Rectangular Array of Practical Air Jets. Ind. Eng. Chem. Res. 1993, 32, 976-980. (16) Heskestad, G. Hot Wire Measurements in a Plane Turbulent Jet. J. Appl. Mech. 1965, 32, 721-734. (17) Gutmark, E.; Wugnanski, I. The Planar Turbulent Jet. J. Fluid Mech. 1976, 73, 465-495. (18) Bradbury, L. J. S. The Structure of a Self-Preserving Turbulent Plane Jet. J. Fluid Mech. 1965, 23(1), 31-64.

Received for review May 29, 2003 Revised manuscript received August 6, 2003 Accepted August 7, 2003 IE030457S

Effects of Die Geometry on the Flow Field of the Melt-Blowing Process

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