Effects of Temperature and Geometry on the Flow Field of the Melt

Ind. Eng. Chem. Res. 2004, 43, 4199-4210

4199

Effects of Temperature and 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

Converging plane jets are used commercially to produce polymeric fibers in the melt blowing process. The behavior of the air flow below the die face is critical for the production process. The die configuration can affect this flow field. Previous work has studied the effects of changing the angle of convergence of the two jets for blunt and sharp die faces, as well as the case of recess or excess (inset or outset) positioning of the sharp die nose for isothermal flow conditions. This study utilized a computational fluid dynamics approach that was validated through experimental data to investigate the effects of nonisothermal conditions on the air flow. The Reynolds stress model was used to simulate the turbulence, and the model parameters were calibrated with the experimental data. The behavior of alternative die designs was also correlated to the die configuration. It was found that, similarly to the isothermal case, the sharper the angle of convergence, the higher the mean air velocity under the die and the higher the intensity of the turbulent velocity fluctuations. The flow field exhibits a self-similar region beyond the point at which the flow from the two jets merges. It was found that the mean temperature and the mean velocity field beyond the merging distance can each be described by a single correlation. In these correlations, the merging distance was used as the length scale, and the mean flow velocity at the merging point was used as the velocity scale. 1. Introduction Dual rectangular jets are commonly used in the industrial melt blowing process for polymer fiber production.1,2 The flow field created by hot air flowing through the two symmetric jets rapidly attenuates molten polymer fibers that are extruded from a row of holes located along the center of the die face (see Figure 1). Over a 1000-fold reduction in fiber diameter can occur in microseconds, and large quantities of hot, pressurized air are needed for the process.3 Therefore, understanding and predicting the behavior of the flow field is important for the design of efficient dies and for the quality of the produced fiber. The flow field that results from a single plane jet has received a great deal of attention because of this field’s theoretical and practical significance.4,5 However, the flow field that results from the combination of more than one jet has not been investigated at equal depth, because of the theoretical difficulties involved. Nasr and Lai6,7 have experimentally investigated the flow field that is created when air flows through two parallel plane jets. The flow field that results from two converging plane jetssthe configuration that is critical to melt blowingshas been studied experimentally by Harpham and Shambaugh8 for blunt and sharp die faces and by Tate and Shambaugh9 for several different jet geometries, including the case of recess or excess (i.e., inset or outset) positioning of the die nose relative to the die face. These studies considered an isothermal air velocity field. However, in addition to the jet geometry, the initial temperature of the air flowing through the dies also has an important effect on the flow field. A later study from the same laboratory (Harpham and Sham* To whom correspondence should be addresses. Tel.: (405)325-6070. Fax: (405)325-5813. E-mail: [email protected].

Figure 1. Cross-sectional view of melt blowing die with blunt nose piece.

baugh10) investigated the temperature field as well as the velocity field for the case of blunt and sharp dies. Recently, with the combination of computational fluid dynamics (CFD) and modern computers, there have been numerical studies of the problem of combined plane jets. The characteristics of parallel jets have been investigated by Anderson and Spall11 and Lai and Nasr.12 These studies quantified the discrepancy between experiments and the results obtained using common turbulence models for the flow field (e.g., the k- model). Using a CFD approach, Krutka et al.13 studied the case of converging plane jets. They used the experimental measurements of Harpham and Shambaugh8 to adjust the default values of the Reynolds stress turbulence model and then simulated the flow field for dies with different convergence angles. In agreement with the experiments, they found that the flow field exhibits the following three zones of develop-

10.1021/ie040043e CCC: $27.50 © 2004 American Chemical Society Published on Web 06/24/2004

4200 Ind. Eng. Chem. Res., Vol. 43, No. 15, 2004

ment for the blunt dies: (a) a region in which the flow field from each jet is not strongly affected by the other jet; (b) a merging region, where the two jets are merging into one; and (c) a self-similar region, where the flow field is similar to that resulting from a single jet. For the sharp dies, only the last two regions were observed. Krutka et al.13 (henceforth, this contribution will be referred to as KSP-a) also found that the mean velocity of the air increases as the angle between the die face and the jet slot decreases and that the turbulence fluctuations also become stronger. In more recent work, Krutka et al.14 (henceforth, this contribution will be referred to as KSP-b) simulated converging isothermal jets for inset and outset dies and developed descriptive correlations for the calculation of the mean velocity field as a function of the level of the die nose recess or excess. In our present work, the flow fields generated by two different variations of the die commonly known in industry as the Exxon slot die were analyzed. The two variations were the blunt die and the sharp die. For these variations, the CFD approach described in KSPa13 and KSP-b14 was implemented for the case of nonisothermal air flow. Similarly to the isothermal studies discussed in KSP-a,13 the angle that the jets make with the die face was altered. Both blunt and sharp dies with convergence angles of 45°, 50°, 60°, and 70° were simulated, and their performances were compared. However, unlike the isothermal cases discussed in KSP-a,13 the initial temperature of the air entering the jets was higher than the ambient air temperature. The nonisothermal flow fields from different jet geometries were compared in terms of centerline temperature decay, velocity decay, turbulence intensity, turbulent kinetic energy, UV Reynolds stress, and turbulent dissipation. The information collected from the simulations was used to correlate the angle between the inlet jets and the die face with the flow field characteristics for different jets. Use of CFD to study the changes in the flow field that result from changing the temperature of the inlet air and the jet geometry offers both time and financial savings over experimentation. In addition, CFD can be used to study all sections of the flow field, including the area close to the die face where laboratory equipment often cannot be used to obtain accurate measurements. 2. Numerical Modeling and Simulation Parameters The CFD package used to complete the simulations was Fluent 6.0. Most of the computational conditions and parameters that needed to be specified in order to simulate the flow field were determined in previous studies (KSP-a13 and KSP-b14). For the work herein, the parameters of the computational model for the simulation of the turbulent velocity in nonisothermal cases were calibrated using the experimental results from Harpham and Shambaugh10 (henceforth, this work will be referred to as HS). These parameters are the factors C1 and C2 that appear in the dissipation equation (see sections 2.3 and 3.1 below) and the turbulent Prandtl number. The details concerning the CFD procedures, the determination of the computational domain size, the grid size, and the choice of turbulent models can be found in KSP-a13 and KSP-b.14 However, a brief summary of the CFD procedures for the present paper is given here. 2.1. Problem Configuration. Figure 1 shows the cross-sectional geometry of a standard blunt die, and

Figure 2. Cross-sectional view of melt blowing die with sharp nose piece.

Figure 2 presents the cross-sectional geometry of a typical sharp die. Although the dimensions of these two dies are similar, the blunt die has a flattened nose piece between the two inlet jets. In contrast, the dual jet outlets in the sharp die form a point (a line in 3D) as they meet at the die face. As might be expected, the geometrical differences in these die types result in flow field differences. However, the die fields share important characteristics. The outlets in both types of dies are symmetrically placed relative to a line of symmetry that corresponds to the z axis in Figures 1 and 2. The flow field along this line of symmetry is responsible for the diameter (attenuation), production rate, and strength of polymer fibers because the line of symmetry generally represents the path of the polymer. Under some operating conditions, the fiber will vibrate in the x direction; see Marla and Shambaugh (2003).15 However, even with vibration, the average fiber position corresponds to the line of symmetry. Therefore, most attention is paid to this part of the flow field. The previous CFD work of KSP-a13 and KSP-b14 considered the effect of die geometry on the flow field. In the work described herein, we explored the effect of air inlet temperature. As in the work described in KSPa,13 we investigated the effect that the angle between the jets and the die face has on the flow field - with the additional complication of nonisothermal operation. For both the blunt and sharp dies, we simulated the flow field for angles of 45°, 50°, 60°, and 70°. The different sharp and blunt dies were simulated using a 2D computational domain. For locations near the center (center in terms of the y direction) of the die, this assumption is valid, because industrial slot dies are very long in the y direction (the direction perpendicular to the planes in Figures 1 and 2) relative to the slot width b. In the work of HS,10 the aspect ratio was 114.7, which is much higher than the recommended5 minimum value of 50 in order to assume 2D flow. Actual industrial jets are even wider, with widths on the order of 0.5 m or more. There are some end effects (at the limits of y) in these dies, but because the die slots are so long in the y direction, the effect of the slot ends on the flow field conditions along the polymer path can be neglected. The use of a 2D computational domain requires much less simulation time than its 3D equivalent. 2.2. Grid Generation. Figure 3 presents the basic computational domain and grid used for the different simulations. For the current work, we used the same

Ind. Eng. Chem. Res., Vol. 43, No. 15, 2004 4201 Table 1. Numbers of Cells, Numbers of Iterations, and Approximate Running Times die type

angle (deg)

cells

isothermal iterations

isothermal computational time

nonisothermal iterations

nonisothermal computational time

blunt blunt blunt blunt sharp sharp sharp sharp

45 50 60 70 45 50 60 70

112 909 112 770 112 514 112 386 110 187 109 931 109 675 109 547

30 308 31 560 34 111 31 010 21 137 45 590 44 095 43 760

14:50:00 14:50:00 15:40:00 14:00:00 10:30:00 21:10:00 20:30:00 20:40:00

24 441 25 630 26 191 24 280 58 288 85 550 36 630 34 930

31:50:00 33:30:00 35:20:00 26:40:00 62:10:00 71:10:00 39:40:00 35:30:00

Figure 3. 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-D have cell sides of 0.2, 0.1, and 0.05 mm, respectively.

grid resolution that was found to produce grid-independent solutions in KSP-b.14 There are four different areas of grid resolution: The jets and the computational domain very close to the die face were assigned the finest resolution because of the importance of these areas during the polymer formation and because of the smaller-scale events that take place in this region. Furthermore, the mean velocity and the turbulence quantities exhibit maxima within this region. In this section of the grid (section D in Figure 3), the quadrilateral cells have a length of 0.05 mm. The next level of resolution within the grid, section C, is composed of quadrilateral cells with a side length of 0.10 mm. The cells in section B have a side length of 0.20 mm, while the most coarse region, section A, contains quadrilateral cells with a side length of 0.40 mm. Significant computational time savings resulted from using this gradation of grid cell sizes. The computational domain, excluding the inlet jet, is 30 mm wide in the x direction and 100 mm long in the z direction. The x axis extends across the die face, perpendicular to the capillary array, while the z axis extends along the line of symmetry, as shown in Figures 1-3. The origin of the coordinate system is in the middle of the die face, which is where the jet and the line of symmetry meet for the sharp die (see Figures 1 and 2). Although the jet slots have different angles with respect to the die face, they all have an assigned (computational) height in the z direction of 5 mm, as suggested by KSPa13 (i.e., the jet slots are located between z ) 0 and z )

-5 mm). The inlet face width of all of the jets is b ) 0.65 mm. For the sharp dies, h is equal to 2b. Table 1 contains the number of cells in the computational domain, the number of iterations until convergence, and the approximate running time for each simulation. For each die, the simulation was completed in two stages. First, the isothermal, incompressible case was completed. Then, with the converged incompressible velocity field prediction as an initial starting point, the compressible model was enabled, and the nonisothermal simulation was completed. 2.3. Turbulence Modeling. The previous work of KSP-a13 examined the applicability of different turbulence models for the accurate simulation of the flow field that results from two converging jets. The predictions of the standard k- model, the realizable k- model, the renormalization theory-based k- model, and the Reynolds stress model (RSM) were compared with experimental results. It was found that the RSM model predictions were closest to the experimentally measured flow fields. For low velocities relative to the speed of sound and nonisothermal conditions, the RSM model equation for the transport of Reynolds stresses is given by16,17

∂ ∂ (FUkuiuj) ) [Fu u u + p(δkjui + δikuj)] + ∂xk ∂xk i j k ∂Uj ∂Ui ∂ ∂ µ (uiuj) - F uiuk + ujuk ∂xk ∂xk ∂xk ∂xk

[

] (

(

Gij + p

)

)

∂ui ∂uj ∂ui ∂uj + - 2µ (1) ∂xj ∂xi ∂xk ∂xk

The summation convention is used in the above equation. In addition to the Reynolds stress transport equations, the dissipation rate is modeled by the dissipation equation in the standard k- model

F

[( ) ]

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

(2)

For the work described herein, the constants C1 and C2 were altered from their commonly used default values. To do this, we used a calibration procedure that compared simulation results with experimental measurements. Comparison to temperature decay and velocity decay measurements from HS10 (see the next section) for nonisothermal jets led to the adoption of the values C1 ) 1.30 and C2 ) 2.05. The energy equation must also be considered for the nonisothermal examples that are being examined in this paper. The following equation is the energy equation solved in the RSM model for the case of no

4202 Ind. Eng. Chem. Res., Vol. 43, No. 15, 2004

heat source due to chemical reactions

∂ ∂ (FE) + [ui(FE + p)] ) ∂t ∂xi

[(

)

]

cpµt ∂T ∂ kc + + uj(τij)eff (3) ∂xi Prt ∂xi For compressible, nonisothermal simulations, the parameters used to describe the thermophysical properties of the fluid in CFD are different from the parameters used to simulate the isothermal cases previously studied in KSP-a13 and KSP-b.14 For the nonisothermal cases, the density in the energy equation was determined using the ideal gas law. In addition, the thermal compressibility, cp, and viscosity were calculated using the kinetic theory. All other parameters were held constant. To expedite the simulations, the flow field for each different die type was first allowed to converge for a noncompressible, isothermal case. After the residuals of all of the equations reached 10-5, the energy equation was enabled, and the flow was made compressible. Then, the simulation was run until all residuals of the viscous model equations reached 10-5 (except for the continuity equation, for which the residuals were required to reach 10-6).

Figure 4. Comparison of simulated centerline velocity to experimental data throughout computational domain for 60° blunt die for simulations with different Prt values. Θjo ) 100 K for both the simulations and experimental data. The curves shown are simulations run with Vjo ) 23.2 m/s.

3. Results and Discussion 3.1. Comparison with Experimental Results. In KSP-a,13 the CFD parameters were established to accurately model the experimentally measured flow fields. However, with the addition of the energy equation and the compressible flow conditions, these parameters had to be reevaluated. The constant default value for the turbulent Prandtl number, Prt (see eq 3), was also reevaluated because the default value resulted in a simulation where the temperature decayed more slowly than experimentally observed. The value of Prt also affects the Reynolds stress production term due to buoyancy (Gij in eq 1). The turbulent Prandtl number represents the dispersion of momentum due to turbulence relative to the dispersion of heat. Therefore, to increase heat dispersion, Prt had to be adjusted to a lower value. A number of different Prt values were tested in the range of 0.20-0.85 (the default value of Prt is 0.85). A comparison of the velocity decays found for simulations with different Prt values is given in Figure 4. In addition, the comparison of the centerline temperature decay for simulations using different Prt values is shown in Figure 5a,b. In Figures 4 and 5, the curves shown are simulations run with a nominal discharge velocity of Vjo ) 23.2 m/s and an air temperature in excess of ambient conditions of Θjo ) 100 Κ. Considering the experimentally obtained functions as the correct solution (for example, the experimentally obtained function for temperature is labeled “HS” in Figure 5), one can determine the mean square root of the error for each simulation and for both the centerline temperature and velocity. As can be seen in Figure 5, Prt values of 0.30, 0.35 and 0.40 give simulated temperature profiles that visually look much closer to the HS function than profiles predicted with the other Prt values. The root-mean-squared error for the temperature decay and the velocity decay at z/h > 10 are given in Table 2. (The errors for velocity and temperature are higher for the other values of Prt that are shown in

Figure 5. Comparison of simulated centerline temperature decay to an experimental correlation for 60° blunt die for (a) 0.20 e Prt e 0.40, (b) 0.50 e Prt e 0.85. The simulations were performed with Θjo ) 100 K for both the simulations and experimental fit. The curves shown are simulations run with Vjo ) 23.2 m/s.

Figures 4 and 5.) Because the error for the temperature decay is 1 order of magnitude less for Prt ) 0.30, whereas the error for the velocity is of the same order of magnitude for these turbulent Prandtl numbers, Prt

Ind. Eng. Chem. Res., Vol. 43, No. 15, 2004 4203 Table 2. Root-Mean-Squared (RMS) Errors for Velocity and Temperature for Prt ) 0.30, 0.35, and 0.40 Prt ) 0.30

error for Vo/Vjo error for Θo/Θjo

Prt ) 0.35

Prt ) 0.40

Vjo ) 23.2 m/s

Vjo ) 35 m/s

average RMS error

Vjo ) 23.2 m/s

Vjo ) 35 m/s

average RMS error

Vjo ) 23.2 m/s

0.067 0.005

0.075 0.006

0.071 0.0056

0.065 0.014

0.033 0.108

0.049 0.061

0.048 0.047

Table 3. Reynolds Stress Model Parameters Available in the FLUENT Software and Modifications Made to Agree with the Experimental Results default KSP-a,13 KSP-b14 present work

Cµ

C1

C2

C1ps

C2ps

C′1ps

C′2ps

σk

σ

Prt

wall Prt

0.09 0.09 0.09

1.44 1.24 1.30

1.92 2.05 2.05

1.8 1.8 1.8

0.6 0.6 0.6

0.5 0.5 0.5

0.3 0.3 0.3

1 1 1

1.3 1.3 1.3

0.85 N/A 0.3

0.85 N/A 0.85

Figure 6. Dimensionless centerline velocity as a function of distance below the die face. The simulated curves and the experimental curves are shown for a 60° blunt die with Prt ) 0.30.

different inlet air temperatures. In all cases, good agreement is exhibited between these simulations and the laboratory data from HS.10 The CFD simulations are able to successfully reproduce this velocity decay. Because the cases being considered are not isothermal, the centerline temperature decay is also important when considering the ability of the CFD simulation to match laboratory results. The experimental correlation shown in Figure 7 is from HS;10 this HS equation matched several different inlet air temperature and velocity cases. Heskestad18 and Gutmark and Wugnanski19 found experimentally that the mean velocity and the Reynolds stresses for a plane jet become self-similar beyond z/b ) 40. In this region, they scaled the velocities and the Reynolds stresses with Vo, and they scaled the distances with x1/2. Figure 8 shows the simulated dimensionless mean velocity of the different inset dies at different z locations well below the die face. Also shown in Figure 8 is the following correlation suggested by HS10 that accurately represents their experimentally determined mean velocity at distances well below the die face

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

Figure 7. Dimensionless centerline temperature as a function of distance below the die face. The simulated curves and the experimental curves are shown for a 60° blunt die with Prt ) 0.30.

) 0.30 was chosen for the rest of the simulations. Table 3 summarizes the default Fluent parameters, the parameters used by KSP-a13 and KSP-b14 in their isothermal studies, and the parameters used in our nonisothermal study. Figure 6 compares the simulated centerline velocity of the 60° blunt die, with an inlet temperature above ambient conditions, to experimental data at distances of z/h > 2 below the die face. Figure 7 shows a similar comparison of the simulated centerline temperature decay with experimental measurements. The 60° blunt die was simulated with different inlet air flow rates and

A correlation similar in form to eq 4 was originally developed by Bradbury20 for rectangular jets. HS10 found that eq 4 fit their data recorded for the 60° nonisothermal sharp die geometry. 3.2. Determination of Length Scale. CFD was used to simulate the temperature decay of flow fields using our best-fit Prt value of 0.30. The CFD model with the modified parameters was used subsequently to simulate eight different dies that have not been previously examined experimentally at nonisothermal conditions. These eight dies include blunt and sharp dies with angles of φ ) 45°, 50°, 60°, and 70°; the dies were examined with Vjo ) 23.2 m/s and an inlet air temperature 100 K above ambient conditions (i.e., Θjo ) 100, where ambient temperature is taken as 294 K). The length scale used to compare the results is the distance zmax from the die face at which the mean centerline velocity reaches its maximum for each flow field. The quantity zmax was used as the length scale for comparisons, as both the dimensionless velocity and the dimensionless temperature scale well with this quantity. The physical meaning of zmax is that it indicates the distance at which the two converging jets merge. Beyond zmax, the flow field is self-similar (as found in KSP-a13 and in Harpham and Shambaugh8). Figure 9 shows the dimensionless centerline velocities for the different dies. By using a length scale of zmax, the different velocity

4204 Ind. Eng. Chem. Res., Vol. 43, No. 15, 2004

Figure 8. Simulated dimensionless velocity profiles within the self-similar region (z > zmax) of the flow below the die: (a) z ) 5, 10, and 15 mm; (b) z ) 20, 25, and 30 mm. The experimental measurements from HS10 are shown for comparison. The simulations were run for a sharp 60° die with Prt ) 0.30 and Vjo ) 23.2 m/s.

decays can be fit to the following curve

(Vo/Vjo) (Vo/Vjo)z)zmax

( )

) 1.1978

z

-0.6257

zmax

(5)

Likewise, the temperature decays for all of the different dies also scale well with the length scale zmax, as seen in Figure 10. The curve fit that describes the dimensionless centerline temperature decay for all of the different dies is

Θo/Θjo ) 1.0074(z/zmax)-0.5537

(6)

The quantity zmax is different for each die. Figure 11 shows the relationship between zmax and the die angle φ for several simulated die geometries. The relationship is linear, and the slope (but not the intercept) is the same for both the blunt and sharp dies. The relationship for the simulated blunt dies is

zmax ) 0.0600φ - 0.0500

(7)

3.3. Mean Velocity. The simulated dimensionless velocity decay along the line of symmetry is shown for the different blunt dies in Figure 12a. The maximum dimensionless mean velocity for each die is inversely related to the die angle φ: the higher the φ, the lower the mean velocity for the range of angles φ examined in our study. Such behavior has also been observed experimentally and numerically for isothermal jets (see Harpham and Shambaugh8 and KSP-a13). The increase in dimensionless centerline velocity is attributed to the partial conversion of the x component of the velocity into z velocity as the two individual jets merge. For the same nominal discharge velocity Vjo, the x component of the velocity increases as the angle φ becomes lower, resulting in a higher z velocity for a lower φ than for the dies with a higher φ. Figure 12b shows that the simulated centerline velocity decay for the different sharp dies is similar to that for the blunt dies. For both the blunt and sharp dies, the 45° die has the highest maximum, whereas the 70° die has the lowest maximum. Each of the different dies has a different centerline velocity decay that can be described by the following equation

Vo/Vjo ) a(z/zmax)b

and that for the sharp dies is

zmax ) 0.0629φ - 0.875.

Figure 9. Use of zmax as a length scale: (a) 45° and 50° dies, (b) 60° and 70° dies. For the centerline velocity, the power-law curve fit is (Vo/Vjo)/(Vo/Vjo)z)zmax ) 1.1978(z/zmax)-0.6257 with R2 ) 0.9905. The simulations were run for a die with Prt ) 0.30, Θjo ) 100 K, and Vjo ) 23.2 m/s.

(8)

(9)

where a and b are constants that have been determined

Ind. Eng. Chem. Res., Vol. 43, No. 15, 2004 4205

Figure 10. Dimensionless temperature decay for blunt and sharp dies: (a) 45° and 50° dies, (b) 60° and 70° dies. The general correlation fitting all temperature decays is Θo/Θjo ) 1.0074(z/ zmax)-0.5537 with R2 ) 0.9831. The simulations were run for a die with Prt ) 0.30, Θjo ) 100 K, and Vjo ) 23.2 m/s.

Figure 12. Dimensionless mean centerline velocity: (a) blunt dies, (b) sharp dies. (c) Maximum mean centerline velocity for different blunt and sharp dies. The simulations were run for a die with Prt ) 0.30, Θjo ) 100 K, and Vjo ) 23.2m/s. Figure 11. Relationship between zmax and φ for different simulated blunt and sharp dies. For the blunt dies, the curve fit is zmax ) 0.0600φ - 0.0500 (R2 ) 1), whereas the curve fit for the sharp dies is zmax ) 0.0629φ - 0.875 (R2 ) 0.9974). The simulations were run for a die with Prt ) 0.30, Θjo ) 100 K, and Vjo ) 23.2 m/s.

through simulation results. The constants a and b are given in Table 4 for eight different die geometries. Figure 12c shows the maximum centerline velocity reached by the dies as a function of die angle φ. For both

the blunt and sharp dies, a power-law decay can adequately model the maximum velocity reached as a function of the die angle. For the blunt die, the decay is

( ) Vo Vjo

) -0.8394 + 11.94φ-0.4378

(10)

z)zmax

with a coefficient of determination R ) 0.998 97. The

4206 Ind. Eng. Chem. Res., Vol. 43, No. 15, 2004 Table 4. Centerline Velocity Decay Constantsa die type

angle (deg)

a

b

R2

blunt blunt blunt blunt sharp sharp sharp sharp

45 50 60 70 45 50 60 70

1.6721 1.58 1.415 1.206 2.7679 2.3175 1.9051 1.7065

-0.6145 -0.6265 -0.6443 -0.6282 -0.6692 -0.643 -0.648 -0.6688

0.998 0.9964 0.9963 0.9998 0.9956 0.9989 0.9992 0.9973

a

Decay: Vo/Vjo ) a(z/zmax)b.

sharp die velocity decay is described by

( ) Vo Vjo

) -1.233 + 17.34φ-0.4475

(11)

z)zmax

with a coefficient of determination R ) 0.999 94. However, it should be noted that these correlations do not apply to φ ) 0°, and should be used with caution for small values of φ, as dies with such angles have been studied neither experimentally nor numerically. This caveat also applies to similar correlations that are discussed throughout this paper. 3.4. Temperature. As described earlier, hot air is used to attenuate the polymer in melt blowing. The temperature of this hot air is a basic control variable in the process. Therefore, it is important to investigate the temperature characteristics of the nonisothermal air flow field below the die. As with the velocity, the centerline temperature is of the most interest for the

process, as the centerline approximates the polymer path. Figure 10 compares the dimensionless temperature decay of all of the blunt and sharp dies, and eq 6 describes the centerline temperature decay for all of the different dies that were simulated. Figure 13a is a contour plot of the mean velocity for the 60° blunt die, and Figure 13b is a contour plot of the temperature for the 60° blunt die. (Static temperature is plotted in Figure 13b because static temperature is the temperature that would be measured experimentally if a temperature probe were used in the flow field.) The velocity and temperature contour plots look quite similar, as might be expected from the fundamental relationship between heat- and momentum-transfer processes. Once the jets merge and the flow is developed below the die, the two contour plots have centerline values that decay at about the same rate. Also, at positions of developed flow (after the jets merge), the velocity or temperature profile at any z position is self-similar and can be described by an equation similar to that shown in Figure 8. However, near the die face, there are some differences between the velocity and temperature profiles. In particular, because the zone between the air inlets and very near to the die face is a velocity recirculation zone (as described in KSP-a13), the velocities are relatively low. However, for the case of the temperature contours, this recirculation area is filled with the hot air that has not decreased in temperature (because of lack of interaction with the ambient air). Thus, the temperature in this recirculation zone remains high. For the case of sharp dies (not shown), the temperature and velocity

Figure 13. (a) Mean velocity contour plot close to die face for a 60° degree blunt die; (b) 60° blunt die temperature contour plot. The simulations were run for a die with Prt ) 0.30, Θjo ) 100 K, and Vjo ) 23.2 m/s.

Ind. Eng. Chem. Res., Vol. 43, No. 15, 2004 4207

contour plots are even more similar: both the temperature and velocity contour plots have their highest values just below the die face, where the jets meet. 3.5. Fluctuating Velocity Field. Although the mean temperature and velocity are very important to consider when choosing the die geometry that will provide the most beneficial practical uses, the turbulence throughout the flow field is also of interest. Strong velocity fluctuations can lead to the polymer breaking off and sticking to the die face; this causes fiber clumping, “shot”, and other undesirable characteristics in the fiber web. Figure 14a presents the turbulence intensity throughout the computational domain for the different blunt dies. The maximum turbulence intensity is reached for the die with φ ) 45°, and the lowest maximum turbulence intensity is reached for the die with φ ) 70°. This same trend also occurs with the sharp dies; see Figure 14b. For both blunt and sharp dies, the maximum turbulence intensity reached is inversely related to the angle. Figure 14c shows the maximum turbulence intensity reached as a function of the angle φ. Both the blunt and sharp dies exhibit a relationship between the maximum turbulence intensity and the die angle that can be described by a power-law expression. For the blunt dies, this expression is

qmax ) -0.5499 + 4.344φ-0.3794

(12)

with R ) 0.9952. For the sharp dies, the expression is

qmax ) -0.5963 + 4.256φ-0.4056

(13)

with R ) 0.9754. The Reynolds stresses (Fu′w′) can also be considered throughout the computational domain. The dimensionless centerline Reynolds stresses for the blunt and sharp dies are shown in parts a and b, respectively, of Figure 15. Although the different geometries have different general shapes for the centerline Reynolds stresses, the maximum Reynolds stress that is achieved is inversely related to the die angle φ. For both types of dies, the maximum Reynolds stress is achieved by the 45° die, and the lowest maximum is achieved by the 70° die. The maximum Reynolds stress for the different dies is compared in Figure 15c as a function of the die angle. For both the blunt and sharp dies, a power-law correlation can predict the maximum Reynolds stress from the die angle. For the blunt dies, the curve fit is

u′w′/Vjo2 ) -0.000 898 2 + 2.225φ-1.652

(14)

with R ) 0.999 13, whereas the curve fit for the sharp dies is

u′w′/Vjo2 ) -0.000 637 5 + 12.01φ-1.960

(15)

with R ) 0.999 98. The turbulent kinetic energy shows characteristics similar to those of the turbulence intensity. Parts a and b, respectively, of Figure 16 show the centerline dimensionless turbulent kinetic energy for the blunt and sharp dies. As seen in the previously discussed turbulence properties, the maximum turbulent kinetic energy is inversely related to the die angle φ. The turbulent kinetic energy created by the blunt dies is significantly higher than the turbulent kinetic energy that results

Figure 14. Turbulence intensity along the centerline: (a) blunt dies, (b) sharp dies. (c) Maximum turbulence intensity as a function of the die configuration. The curve fit for the blunt dies is qmax ) -0.5499 + 4.344φ-0.3794 (R2 ) 0.9952), whereas the curve fit for the sharp dies is qmax ) -0.5963 + 4.526φ-0.4056 (R2 ) 0.9724). The simulations were run for a die with Prt ) 0.30, Θjo ) 100 K, and Vjo ) 23.2 m/s.

from the sharp dies. Figure 16c shows a correlation between the die angle and the maximum turbulent kinetic energy achieved by the different dies. For the blunt dies, the correlation is

(k/Vjo2)max ) 0.4531 + 1808φ-2.289

(16)

4208 Ind. Eng. Chem. Res., Vol. 43, No. 15, 2004

Figure 15. Dimensionless Reynolds stress profiles along the centerline: (a) blunt dies, (b) sharp dies. (c) Maximum Reynolds stress as a function of the die configuration. The simulations were run for a die with Prt ) 0.30, Θjo ) 100 K, and Vjo ) 23.2 m/s.

with R ) 0.999 96, whereas the curve fit for the sharp dies is

(k/Vjo2)max ) 0.023 08 + 4138φ-3.233

(17)

with R ) 0.997 48. Finally, the dimensionless centerline turbulence dissipation rate of the flow field was examined. This quantity is important because it explains how quickly the turbulence decreases in the area of interest. Parts a and b of Figure 17 compare the dimensionless center-

Figure 16. Dimensionless turbulent kinetic energy profiles along the centerline: (a) blunt dies, (b) sharp dies. (c) Maximum turbulence kinetic energy as a function of the die geometry. The simulations were run for a die with Prt ) 0.30, Θjo ) 100 K, and Vjo ) 23.2 m/s.

line turbulence dissipation for the blunt and sharp dies, respectively. The turbulence dissipation is related inversely to the die angle. This trend can be explained by considering the other turbulence characteristics. For example, the 45° die exhibits higher turbulent kinetic energy than other dies for z < zmax, but the 45° die also exhibits comparable levels of turbulent kinetic energy for z > zmax. Thus, the rate of dissipation for the 45° die

Ind. Eng. Chem. Res., Vol. 43, No. 15, 2004 4209

distance as a length scale results in correlations that apply for all types of die configurations (for the third region of the flow). The mean centerline temperature can also be predicted with a correlation that uses zmax as a length scale. The effect of decreasing the angle between the jet and the die face is to increase both the dimensionless mean velocity and the turbulence fluctuations. This higher mean velocity might result in higher polymer fiber speeds, which is desirable, but higher turbulence intensity might be a disadvantage in fiber spinning. If it is desirable to reduce the fluctuations along the path of the polymer fiber, particularly at the die tip, then a blunt die appears to be a better option. On the other hand, if one desires to increase the mean velocity along this path without increasing the air consumption of the die, then a sharp die is more suitable. The numerical simulations allow the correlation of the die configuration to the flow behavior. The location of the merging point as a function of φ is provided, as are the maximum values of the mean velocity and of the turbulence characteristics as a function of φ. The procedure for calculating the mean temperature and the mean velocity along the centerline can be summarized as follows: (a) calculate zmax from Figure 11 (i.e., use eqs 7 or 8), (b) calculate the maximum velocity from eqs 9 or 10, (c) use eq 5 to calculate the mean velocity decay, and (d) use eq 6 to calculate the mean temperature decay. Acknowledgment

Figure 17. Dimensionless turbulence dissipation rate profiles along the centerline: (a) blunt dies, (b) sharp dies. The simulations were run for a die with Prt ) 0.30, Θjo ) 100 K, and Vjo ) 23.2 m/s.

at z/zmaz ≈ 1 is higher than the rates for the dies of higher angles. 4. Conclusions The flow fields resulting from two nonisothermal, rectangular jets that converge in a blunt and a sharp die configuration have been predicted using CFD. The RSM was used for turbulence simulations in conjunction with the procedures suggested by KSP-a13 and KSP-b14 to calibrate the values of the model parameters with laboratory measurements. The flow field for the nonisothermal blunt die exhibits three regions of development; this behavior is similar to the regions of development observed for the isothermal flow field. In the first region, the flow field that results from each jet is discernible. In the second region, the two jets are merging, and a maximum mean velocity is observed. The merging distance, at which the velocity reaches the maximum, can be used as a length scale for the process. The third flow region is a self-similar region in which the flow field is independent of its origin. The flow field resulting from the sharp dies exhibits only the latter two regions of development. Different angles of convergence of the jets result in dramatic differences in the turbulence quantities that are observed within the first and second regions, but using the merging

This work was supported by an NSF GOALI grant (DMII-0245324). The support of 3M, Procter & Gamble, and ConocoPhillips is gratefully acknowledged, as well as the permission by Fluent Inc. to use the FLUENT 1.0 software with an educational license. Two of the coauthors (H.M.K. and D.V.P.) also acknowledge the support of the University of Oklahoma Research Council. Nomenclature b ) nozzle width (see Figures 1 and 2), mm bo ) face width of the die slot, mm cp ) heat capacity, J/K C1ps ) coefficient for the slow pressure-strain term of the Reynolds stress model C2ps ) coefficient for the rapid pressure-strain term of the Reynolds stress model C′1ps, C′2ps ) coefficients for the modeling of the wall reflection effects on the pressure-strain term of the Reynolds stress model C1 ) parameter for the RSM model (eq 2) C2 ) parameter for the RSM model (eq 2) Cµ ) coefficient for the modeling of turbulent viscosity E ) energy, J Gij ) production of i,j Reynolds stresses due to buoyancy (eq 3) h ) gap between edges of air plates (see Figures 1 and 2), mm k ) turbulent kinetic energy, (1/2uiui), m2/s2 kc ) thermal conductivity, J/(m‚s‚K) l ) length of air slots in the y direction, m P ) pressure, Pa Prt ) turbulent Prandtl number, Prt ) (eddy viscosity)/ (eddy thermal diffusivity) q ) turbulence intensity, 1/3(ui2)1/2/Vjo Q ) air flow rate through both slots, m3/s

4210 Ind. Eng. Chem. Res., Vol. 43, No. 15, 2004 Re ) Reynolds number (Re ) bVjoF/µ) t ) time, s T ) temperature, K 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 ) distance from centerline where mean velocity is onehalf the magnitude of the centerline velocity, mm zmax ) merging distance below the die face at which the dimensionless mean velocity reaches a maximum Greek Characters δij ) Kroeneker’s delta  ) dissipation rate of turbulent kinetic energy, m2/s3 Θ ) difference between the centerline air temperature and the ambient temperature, K Θjo ) difference between the jet inlet air temperature and the ambient temperature, K φ ) 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 kinetic energy dissipation (τij)eff ) i,jth component of the effective deviatoric stress tensor, Pa

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Received for review January 30, 2004 Revised manuscript received May 7, 2004 Accepted May 11, 2004 IE040043E