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Ind. Eng. Chem. Res. 2008, 47, 8754–8761
Three-Dimensional Fluid Flow in the Processing of Fine Fibers William P. Klinzing* Design and Engineering SerVices, 3M Company, 900 Bush AVenue, St. Paul, Minnesota 55144-1000
Ephraim M. Sparrow Department of Mechanical Engineering, UniVersity of Minnesota, 111 Church Street, SE, Minneapolis, Minnesota 55455-0111
In this paper, a multidimensional numerical simulation has been used to investigate the creation of fine fiber by the melt-blown process. The problem involves highly complex fluid flow and convective heat transfer processes. The fine fiber is created by the use of high-velocity, obliquely impinging air jets whose high shear forces stretch a polymer extrudate in the partially fluid state. High-temperature air is used to maintain the fluidity of the polymeric material as it exits the die. The model which was developed and implemented for the simulation closely reflects the physical situation employed in the actual production of fine fibers. The actual configuration of fine fiber production is a linear array of orifices from which the partially fluid polymer emerges. The array contains inherent symmetries which enable the problem to be tractable. Notwithstanding this, the numerical simulation still required multimillions of control volumes to achieve results of practical relevance. The results of the simulation indicate that, contrary to physical intuition, the shear exerted by the airflow is relatively uniform around the circumference of a given fiber. Although the interfiber spacing is small, it does not promote unfavorable fiber-to-fiber interactions. 1. Introduction Many important manufacturing operations are made up of a large number of individual identical processes which are executed in parallel. A significant case in point is the manufacture of fine fiber. Whereas the published literature has focused on the fluid flow and heat transfer processes that are relevant to the production of individual fine fibers, the actual production of the fibers is carried out using equipment in which thousands of fibers are produced simultaneously in a linear array. The array of fibers is produced by means of a linear array of orifices through each of which a still-amorphous polymer strand emerges. High-velocity air in the form of a pair of slot jets stretches the fiber by means of fluid friction. The linear array is bracketed by a pair of parallel slot jets, one to each side of the array. A review of the open literature revealed only one blownfiber process, aside from that described here, for producing fine fiber in a parallel arrangement. That process, termed the Schwarz method,1 is fundamentally different from that described in the preceding paragraph. In the Schwarz process, orifices deployed in a square arrangement provide egress for the still-amorphous polymer strands. Each such strand is surrounded by a dedicated air jet in the shape of an annulus which need not necessarily be cylindrical. Investigation of the fluid flow and heat transfer characteristics of the Schwarz method has been performed by Schambaugh and co-workers. In a recent paper,2 these investigators have dealt with the air flow corresponding to a square array of cylindrical annular jets in the absence of the fiber strands. In a second paper,3 consideration was given to a single polymeric strand enveloped by a cylindrical annular jet. The geometry of the Schwarz method, a square array, is significantly different from the geometry of the linear process considered here. In addition, the air jets used to stretch the polymer strands also are fundamentally different in the two * To whom correspondence should be addressed. Tel.: (651) 7785187. Fax: (651) 778-6906. E-mail:
[email protected].
methods, one being an annular jet and the other being a linear slot jet. Therefore, the aforementioned work by Schambaugh and co-workers, while clarifying the Schwarz method, is unrelated to the fluid flow and heat transfer that are encountered in the linear array of fibers that is of concern to the present investigation. The authors are pleased to acknowledge the appearance of a relevant journal article4 that was published after the review process for the present paper had been completed. Indeed, the authors are grateful to one of the reviewers for pointing out the existence of that article. The features of that article and their relationship to what is presented here will be assessed later in the text when the geometric aspects of the respective problems are illustrated. To clarify the physical situation to be investigated here, it is useful to make reference to Figure 1. Shown there is the essence of the linear, fine-fiber production process. There is a succession of linearly deployed openings in the flat face of a die. Emerging from each opening is a still-amorphous polymer strand. The pair of slot jets, previously noted, is seen to bracket the emerging strands. The initial temperature of the slot jets is maintained at a value that is more or less equal to that of the merging strand. This elevated temperature ensures the malleability of the polymer in the region adjacent to the die face. In that region, the jet velocities are at their highest values, as is the fluid shear that they impose on the strands. Because of this, the strands undergo considerable stretching in that region. As the jets spread, the velocity and temperature of the air diminish, with the result that the strand stiffens as it solidifies and the imposed shear decreases. The work to be presented here is a comprehensive numerical simulation of the fluid flow and heat transfer which play a central role in the production of melt-blown fine fiber by the linear production process displayed in a highly schematic view in Figure 1. Because of the significant temperature variations and the high velocities that are encountered, the properties of the air must be treated as being variable. As a result, the fluid flow
10.1021/ie800099f CCC: $40.75 2008 American Chemical Society Published on Web 10/28/2008
Ind. Eng. Chem. Res., Vol. 47, No. 22, 2008 8755
Figure 3. (a) Detail of the fiber exit zone. (b) Reproduction of Figure 1b from a recently published article (reproduced with permission, copyright American Chemical Society, 2008).4
Figure 1. Schematic diagram of the essential features of the linear, finefiber production process.
and heat transfer problems must be solved simultaneously. As can be seen from Figure 1, the flow is clearly three-dimensional. In view of the high velocities that are necessary for the proper functioning of the process, the flow is turbulent and compressible. Of particular interest is the determination of the magnitude and the variations, both circumferentially and axially, of the fluid shear that is exerted by the slot jets on the surface of the polymer strand.
Figure 2. Planform view of the physical situation.
A planform view of the physical situation is conveyed in Figure 2. This figure elaborates what has been shown pictorially in Figure 1. In this view, the linearly aligned row of polymer-strand emergent apertures is seen to be deployed along the apex of a die. The slot jets flank the apex. Further clarification is provided by the view defined by section A-A in Figure 2. That sectional view is presented in Figure 3a. That figure provides details of the fiber exit zone. Of particular note is the positioning of the die tip (apex) with respect to the die face. This positioning, with the die tip recessed with respect to the face, is called setback. Alternative positions of the tip with respect to the face of the die have been studied previously.5 That investigation was carried out for a simplified, two-dimensional version of the actual fine-
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fiber production process. The investigated alternatives included the die tip situated forward of the face and the tip aligned with the face. It was found that the setback position yielded the most advantageous shear stress and temperature variations along the length of a surrogate fiber. Another feature revealed in Figure 3a is that the slot jets impinge on the fiber at a location that is upstream of the die face. At this point, it is relevant to compare the situation being studied here with that of the very recently published article4 cited in the Introduction. The geometry of the physical situation dealt with in that article is presented in Figure 3b, which is a precise reproduction of Figure 1b of the article. The contrast between the present geometry and that of the article is clearly evident by comparing parts a and b of Figure 3. The setup that is shown in Figure 3a is currently in use for the production of fine fiber at 3M Company. On the other hand, it is not known to the authors where the geometry of Figure 3b is being used. One major difference that is clearly evident is the position of the die apex from which the fiber emerges. In the present configuration, the apex is situated back from the die face, while in the configuration of the article the apex is flush with the die face. The setback creates a flow emerging from the die face which is very different from that which emerges from the die face in Figure 3b. In particular, the oblique jets converge within a confined space in the present configuration, whereas the convergence of the jets in the configuration of Figure 3b occurs in an open space. As already noted in a previous paper,5 the authors have compared the shear stresses acting on the fiber corresponding to the setback position with those corresponding to the flush configuration. It was found that substantially higher shear stresses were obtained using the setback positioning. The just-noted mismatch in geometry precludes any meaningful comparisons between the present results and those of the published literature. The slot jets perform the dual functions of stretching the fiber and maintaining an elevated temperature during the stretching process. The ambient in which the processing occurs is air at room temperature. The extent of the ambient that relates to the processing of any representative fiber in the array is bounded by the presence of the neighboring fibers as illustrated in Figure 1. A careful consideration of Figure 1 suggests that the relevant ambient for the production of any one fiber in the array is bounded by symmetry planes that are situated halfway between the fiber in question and its immediate neighbors. The bounding of the ambient that was described in the preceding paragraph limits the solution domain that has to be considered in the analysis of a representative fiber in the array. In fact, the solution relevant to a representative fiber is also the solution for the entire array of fibers. Some of the dimensions of the die system that is pictured in Figures 1-3a are (a) angle of the oblique duct with respect to the horizontal ) 60°, (b) w/d ) 0.625, (c) L /d ) 235, (d) (channel width)/d ) 1.5, (e) (setback)/d ) 1.0, and (f) Reynolds number at the inlet of the oblique channel ) 7350. The condition assigned at the inlet of the channel was a given gauge pressure of 136 000 Pa. In addition, the inlet temperature was 580 K. 2. Governing Equations and Turbulence Model The governing equations appropriate to the problem being considered are the compressible, variable-property form of the RANS equations for momentum conservation, the RANS
equations for energy conservation, and the equation for mass conservation. Those equations are as follows: x-momentum: ∂ 2µ ∂x eff
(
∂ ∂p ∂ ( 2) ∂ Fu + (FuV) + (Fuw) ) - + ∂x ∂y ∂z ∂x ∂u 2 ∂u ∂V ∂ b - µ div V + + µ + ∂x 3 eff ∂y eff ∂y ∂x ∂ ∂u ∂w + µ (1) ∂z eff ∂z ∂x
[ (
)
)]
[ (
y-momentum: ∂ µ ∂x eff
[ (
)]
∂ ∂p ∂ ∂ (FVu) + (FV2) + (FVw) ) - + ∂x ∂y ∂z ∂y ∂V ∂u ∂V 2 ∂ b + - µ div V + + 2µ ∂x ∂y ∂y eff ∂y 3 eff ∂ ∂V ∂w + µ (2) ∂z eff ∂z ∂y
)]
(
)
[ (
)]
∂ ∂ ∂ ∂p (FVw) + (FVw) + (Fw2) ) - + ∂x ∂y ∂z ∂z ∂ ∂w ∂u ∂w ∂V ∂ + + µ + µ + ∂x eff ∂x ∂z ∂y eff ∂y ∂z ∂ ∂w 2 - µ div b 2µ V (3) ∂z eff ∂z 3 eff
z-momentum:
[ (
)]
[ ( (
)]
)
∂(Fu) ∂(FV) ∂(Fw) + + )0 ∂x ∂y ∂z
mass:
(4)
[ ∂x∂ (FuT) + ∂y∂ (FVT) + ∂z∂ (FwT)] ) βT(u ∂p∂x + ∂p ∂p ∂ ∂T ∂T ∂T ∂ ∂ V + w ) + [ (k + k + k +φ ∂y ∂z ∂x ∂x ) ∂y ( ∂y ) ∂z ( ∂z )]
energy: cp
eff
eff
eff
(5) where φ is the dissipation function: 2
2
2
2
[ ( ∂u∂x ) + 2( ∂V∂y ) + 2( ∂w∂z ) + ( ∂V∂x + ∂u∂y ) + ∂u ∂w ∂V ∂u ∂V ∂w 2 + + + + + - µ (6) ( ∂w ∂x ∂z ) ( ∂y ∂z ) ] 3 ( ∂x ∂y ∂z )
φ ) µeff 2
2
2
2
eff
For an ideal gas, such as air that is being considered here βT ) 1 (7) It is relevant to note the three-dimensional nature of these equations and the tight coupling that exists among them. Within these equations are the effective values of the viscosity and thermal conductivity, µeff and keff, respectively. These quantities are defined as µeff ) µ + µt
(8a)
keff ) k + kt
(8b)
where µ and k are molecular quantities, and µt and kt denote turbulence quantities. For the present simulation, µ, k, and cp are regarded as constants. In view of the complexity of the fluid flow being considered, it is necessary to select a turbulence model with great care rather than making an ad hoc selection. There appears to be no prior numerical investigations of fluid flows that closely approximate that considered here, so that little direct guidance can be gained from the literature. In view of this, the authors have carried out an extensive study of turbulence models for a flow that bears some resemblance to the present one.6 On the basis of this search, it was concluded that the wall jet is the nearest flow configuration to the present. In the wall-jet literature, there are experimental data for both the cylindrical wall jet and the plane wall jet. In a separate study, the authors investigated the efficacy
Ind. Eng. Chem. Res., Vol. 47, No. 22, 2008 8757
Figure 4. (a) Entire solution domain. (b) Enlarged view of the simulated fiber.
of five candidate turbulence models for predicting the experimental results for these two wall-jet configurations. These models are (a) standard k- (ske) (b) realizable k- (rke) (c) renormalized group k- (RNG) (d) shear stress transport (SST) (e) Reynolds stress model (RSM) The first three models listed above are variants of the k- family. Each one of these models was employed in three different forms: (1) unenhanced; (2) near-wall treatment; (3) near-wall plus pressure gradient treatment. The clear outcome of that investigation was that the k- turbulence model with enhanced wall treatment [Fluent version 6.1.36] provided predictions that were in closest agreement with the available experimental data for both the cylindrical and plane wall jets. On this basis, that turbulence model is adopted here. 2.1. Boundary Conditions. The boundary conditions that are needed to complete the specification of the problem will now be presented. On all of the solid surfaces, the velocities are zero in accordance with the no-slip and impermeability conditions, and there is no heat flow (adiabatic). The fiber is approximated as a nonmoving, rigid cylinder of diameter d, so that it, too, presents zero velocity and adiabatic surfaces to the flow. On symmetry surfaces, the normal derivatives of the independent variables are zero. On those surfaces of the solution domain which extend into the far field, boundary conditions of 1 atm and ambient temperature are applied. At the inlet of the obliquely inclined channels, the pressure is specified. 3. Numerical Simulation In view of the complexity of the problem being considered, a numerical solution is mandatory. The first step in formulating
the numerical simulation is the selection of the space in which the solution is to be carried out. Such a space is commonly denoted as the solution domain. The solution domain selected here is illustrated in Figure 4. Part a of the figure shows the entire solution domain, while part b conveys an enlarged view of the simulated fiber and illustrates the sectioning of the simulated fiber by symmetry planes. Of particular note is the size of the solution domain which extends beneath the die face. In order to achieve a solution that would be independent of the size of the extended domain, numerical experiments were performed. The dimensions of the extended solution domain that satisfied the independence condition can be visualized by reference to Figure 4a. In particular, the vertical and horizontal dimensions of the frontfacing symmetry plane are 2.5L and 2.1L, respectively, where L is exhibited in Figure 2. The position of the simulated fiber corresponds to the left-hand boundary of the aforementioned symmetry plane. The solution domain is outlined in Figure 4a by lines of heavier weight. The faces of the domain are labeled according to their function. Attention is first called to the positioning of the x-, y-, and z-coordinate system in Figures 4a, 4b, and 2. The relationship of these figures to each other can be identified by noting the location of the coordinates in each figure. For example, the shaded rectangle in Figure 2, which represents the solution domain as viewed in the x-y plane, can be related to the three-dimensional view of the symmetry domain in Figure 4a. Also, the location of the fiber can be identified by noting the locations of the coordinates in Figure 4. The view shown in Figure 4a takes advantage of the inherent symmetries of the physical situation. In particular, the physical situation can be assembled from Figure 4a by reflecting it twice: once about the x-z plane and again about the y-z plane.
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Attention will first be turned to the symmetry planes. If reference is first made to Figure 2, it may be noted that the base and the two long edges of the shaded rectangle represent symmetry planes. In Figure 4a, the base is the hidden vertical symmetry plane that bounds the solution domain at the left. Furthermore, the long right-hand edge of the shaded region in Figure 2 corresponds to the forward-facing symmetry plane in Figure 4a, and the long left-hand edge of the shaded region corresponds to the backward-facing symmetry plane in Figure 4a. A further connection between the two figures is that the top of the shaded region in Figure 2 is represented in Figure 4a by the right-facing plane labeled as ambient. By inspection of Figure 4, it can be seen that the simulated fiber lies along the z-axis in Figure 4a. Another important feature shown in Figure 4a is the channel which conveys air and creates a jet at its exit. There are, in reality, two such channels, as shown in Figure 3a. However, it is necessary that only one of the channels be present in the solution domain because of the symmetry of the physical situation. Still another feature to be noted is the top face of the block-like portion of the solution domain depicted in Figure 4a. That face is comprised of the die face, shown hatched at the left, and is extended to the right in order to account for the presence of the ambient. The complicated geometry of the solution domain required a very fine mesh for its proper resolution. All told, a total of approximately 7.8 million control volumes were deployed throughout the domain. Another factor which contributed significantly to the use of such a large number of control volumes is the requirement of the enhanced-wall treatment for extremely fine resolution in the neighborhood of the bounding walls and the surface of the fiber. The smallest control volumes were used adjacent to the fiber proper. With increasing radial distance from the fiber, a graduated, enlarging mesh was used in recognition of the very low velocities that are induced in the far field. The mesh consisted of all hexahedral elements. It was created with the aid of the ICEM software package. The actual numerical calculations were performed by means of Fluent 6.3.26 in conjunction with a cluster of five HP j6750 computers utilizing four parallel processers. Approximately 50 000 iterations were required to obtain a converged solution. Convergence was monitored by means of two separate metrics. One of these is the size of the root-mean-square values of the residuals. The second, and more relevant, are the values of the physical quantities of practical importance. In particular, among these quantities, the most important is the maximum z-direction wall shear stress that the moving fluid exerts on the fiber wall. These quantities were carefully monitored, and when their values no longer changed within a preassigned tolerance, convergence was declared. 4. Results and Discussion 4.1. Wall Shear. The quantity of major practical importance is the z-direction (streamwise) wall shear stress that the moving air exerts on the fiber. The magnitude of the wall shear provides the stretching force needed to produce the fine fiber. The results for this quantity are presented in Figure 5, where the shear stress τ is nondimensionalized by the pressure difference ∆p between the air entering the delivery duct and the ambient. The figure consists of two graphs, the larger of which shows the streamwise distribution for the active length of the fiber. The smaller graph, the inset, provides an expanded view of the stress magnitudes in the neighborhood of the maximum stress. In each of these graphs, there are four curves, each of which corresponds to a
Figure 5. Streamwise and circumferential distributions of the z-direction (streamwise direction) shear stress on the fiber wall.
specific angular position around the circumference of the fiber. The angles extend over the range from 0 to 90°. The definition of these angular values can be seen in Figure 4b. Attention will now be turned to the shear stress results that are presented in the inset in Figure 5. The z/d range of that figure corresponds to the critical region where the fiber undergoes the maximum stretch due to jet-imposed shear. It is seen from the inset that the circumferential variation of the wall shear is surprisingly small. At z/d values that are far downstream of those in the critical region, greater circumferential variations are in evidence. However, those circumferential variations do not significantly affect the quality of the fiber because solidification has already taken place. The insensitivity of the wall shear to the circumferential location would not have intuitively been expected from the geometry of the deployment of the linear array of fiber apertures. If it is recalled that the air jets originate as plane sheets and if note is taken of the small interfiber distances, it would not be expected that the jets would invade the interfiber spaces. Rather, conventional wisdom would have suggested that the interfiber spaces would be starved of moving air. Such a view would have predicted very low values of the wall shear at angles in the neighborhood of 0°. The numerical simulations have demonstrated a contrary vision. The results of Figure 5 clearly indicate a high degree of circumferential uniformity in the critical region of high wall shear stresses. This outcome validates the use of the fabrication technique that is being modeled here. The attainment of near-circumferential uniformity may be attributed, at least in part, to the oblique nature of the slot jets. This angled flow serves to drive fluid into the interfiber spaces. Attention will now be turned to the streamwise variation of the wall shear stress. As has already been noted, there is a zone of very high values of the stress. That zone can be associated with the region of impingement of the angled jets on the fiber surface. At streamwise locations that are upstream of the region of high stress, the lower stress values can be attributed to the flow that is induced by the jets proper. The extent of this motion is throttled by its confinement between the curtains formed by the jets, the setback of the point of emergence of the fiber (Figure 3a), and the tight spacing between the fibers. To escape from this confinement, this fluid may move into the interfiber space, and, in this way, provide a compensatory source of shear in the neighborhood of the 0° location. At locations that are downstream of the region of high shear, the jet flow tends to become parallel to the surface of the fiber
Ind. Eng. Chem. Res., Vol. 47, No. 22, 2008 8759
and, simultaneously, spreading of the jets occurs due to entrainment of the air from the surroundings. It is well-known that impinging flows create a stronger shear than a parallel flow. This effect, coupled with the diminished velocity associated with spreading of the jets, causes a steady decrease of the wall shear stress at increasing streamwise distances. 4.2. Velocity. To support the foregoing finding of the prevalence of near-circumferential uniformity, it is useful to display velocity contour diagrams (isoVels). Since the zcomponent of the wall shear is most strongly influenced by the gradient of the z-velocity normal to the wall, the isovels will be specific to that velocity component. This information is displayed in parts a, b, and c of Figure 6, respectively corresponding to dimensionless streamwise locations z/d ) 6.9, 1.04, and 0.054. Figure 6a corresponds to the most downstream among the three stations chosen for display of information. The figure is a portrayal of a 90° portion of the fiber on which is superimposed the isovels of w, the z-velocity component. The important issue with respect to Figure 6 is the distance between an isovel and the surface of the simulated fiber. If this distance were to be perfectly constant, then it would be expected that the wall shear would be independent of the circumferential position. For this purpose, it is not necessary to specify the velocity magnitude of any isovel since it is only its distance from the surface that is relevant. Inspection of Figure 6 enables identification of those cases where near-circumferential uniformity prevails. In this regard, it appears that Figure 6b indicates that near uniformity is achieved for z/d ) 1.04, which corresponds to the critical zone of high shear (see inset of Figure 5). Further study of these figures also suggests that near uniformity also prevails for z/d ) 6.9. On the other hand, a somewhat different conclusion can be inferred from Figure 6c, which corresponds to a location upstream of where the jets impinge on the fiber. At locations upstream of the impingement zone, the flow pattern is still in the process of formation, as was discussed earlier in connection with the results conveyed by Figure 5. Further insights into the velocity field can be obtained from plots of velocity profiles at strategic locations. One such location is in the space between neighboring fibers. A consultation of Figure 1 shows that such profiles are plotted along the x-direction. The velocity to be plotted is w, the z-direction velocity component. These results are presented in Figure 7. The vertical axis represents the magnitude of the w velocity, conveyed in dimensionless form via the Reynolds number. Velocity profiles are shown for three streamwise locations, z/d ) 0.054, 1.038, and 6.9. The first of these locations is upstream of the zone of maximum shear, the second location is the maximum shear zone, and the third is downstream of that zone. The profiles span the space in the x-direction between the surface of the fiber and the symmetry line that is situated halfway between two neighboring fibers. An examination of Figure 7 reveals noteworthy differences in the velocity profiles for the three selected streamwise locations. First, the magnitudes of the velocities in the space that is upstream of the zone of maximum shear are very small and are more or less uniform except adjacent to the wall of the fiber. As set forth in the discussion following Figure 5, that space is cordoned off from interactions with high-velocity fluid. In the zone of jet impingement characterized by z/d ) 1.04, the magnitudes of the velocities are high due to the momentum carried by the strong jet flow. Note that relatively high velocities prevail in the nearest neighborhood of the wall as the jet scrubs the boundary layer off the surface of the fiber. At downstream
Figure 6. Isovels (contours of constant w): (a) z/d ) 6.9, (b) z/d ) 1.04, and (c) z/d ) 0.054.
locations such as z/d ) 6.9, there is clear evidence of boundary layer development, and its thickness more or less fills the space between the surface of the fiber and the interfiber symmetry line.
8760 Ind. Eng. Chem. Res., Vol. 47, No. 22, 2008
Figure 7. Profiles of the dimensionless w velocity component in the interfiber space. Figure 9. Mesh independence study. The plotted shear values correspond to the circumferential location θ ) 0° (see Figure 4b).
There are some differences in detail, but it is believed that these minor deviations do not in any way detract from the basic findings of the study. 5. Concluding Remarks
Figure 8. Profiles of the dimensionless w velocity component in the direction across the thickness of one of the jets.
A second strategic zone chosen for examination by means of velocity profiles is the flow field across the thickness of one of the jets. In terms of the coordinates defined in Figure 1, the profiles to be presented are functions of the values of y. The profiles span the space from the surface of the fiber outward into the jet. Figure 8 conveys these results. Once again the vertical axis conveys the dimensionless w velocity in terms of the Reynolds number, while the abscissa is the dimensionless distance measured outward from the fiber surface. The figure contains velocity profiles at the same three spanwise locations as were adopted for the profiles of Figure 7. The curve for z/d ) 0.054 does not, in fact, span across the jet thickness because the jet impingement actually occurs downstream of this streamwise location. The flow upstream of the jet impingement is relatively quiescent and more or less uniform across the space (see discussion relevant to Figure 5). At the zone of jet impingement, z/d ) 1.04, the velocity is uniform across the thickness of the jet, but decreases sharply at the outboard edge of the jet because no significant entrainment has yet occurred. On the other hand, entrainment is clearly in evidence at the downstream location characterized by z/d ) 6.9. In fact, the shape of the profile with a low velocity adjacent to the wall followed by a local maximum and a slow drop-off is a well-known characteristic of wall jets. 4.3. Mesh Independence Study. Mesh independence was also investigated. For this purpose, the solution domain was discretized into two additional meshes to supplement the original mesh, which consisted of 7 800 000 control volumes. The supplementary meshes respectively included 2 200 000 and 12 500 000 control volumes. The shear stress results from the original and supplementary meshes are brought together and compared in Figure 9. Inspection of the figure shows that the results from the three meshes are generally mutually supportive.
The work reported here is a demonstration of the importance of numerical simulation in the design of significant material production processes. In particular, a less-searching evaluation of the blown-fiber process might suggest that the use of plane jets to stretch a cylindrical fiber would result in severe circumferential variations in the fiber-stretching process. The results of the present numerical simulations reveal a different picture. It was found that the circumferential variation of the shear stress exerted by the air-jet fluid on the fiber surface is moderate, especially in the critical region of highest shear. This outcome validates the concept underlying the process. It also leads to the conclusion that other processes which require a large number of indiVidual annular jets to achieve axisymmetric shear distributions do not provide improved fiber-stretching characteristics compared to those afforded by the use of two planar jets. The angular orientation of the paired planar jets is seen to create a focal impingement zone which gives rise to a maximum shear stress. This zone of high shear occurs while the polymeric filaments are still in a fluidized state. It is the confluence of high shear and the fluidic polymeric state which enables the required stretching and attainment of the requisite small diameter. The results exhibited in velocity contour diagrams clearly support the presence of circumferential shear stress uniformity in the high-shear region. Another unexpected but significant result is the fact that the interfiber spacing, although very small, does not promote unfavorable fiber-to-fiber interactions. On the contrary, it appears that the space between adjacent fibers could be diminished without inducing negative consequences. Further work is needed to evaluate the extent of such a decrease in the interfiber separation. The outcomes that have been highlighted in the preceding paragraphs serve as ample testimony to the importance of making use of advanced simulation tools to guide practical design. Acknowledgment The authors gratefully acknowledge the use of 3M computing and experimental facilities for the execution of the research.
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Nomenclature cp ) specific heat d ) diameter of polymer orifice k ) molecular thermal conduction keff ) effective thermal conductivity kt ) turbulent thermal conductivity L ) half-width of die face p ) pressure ∆p ) driving pressure differential T ) temperature W ) half-spacing between adjacent polymer orifices x ) longitudinal coordinate y ) transverse coordinate z ) coordinate perpendicular to the die face u ) x-direction velocity component V ) y-direction velocity component w ) z-direction velocity component Greek Symbols β ) coefficient of thermal expansion µ ) molecular viscosity µeff ) µ + µt µt ) turbulent viscosity F ) density
τ ) wall shear stress φ ) dissipation function (conversion of mechanical energy into thermal energy)
Literature Cited (1) Schwarz, E. C. A. Apparatus and process for melt-blowing a fiber forming thermoplastic polymer and product produced thereby. U.S. Patent 4,380,570, 1983. (2) Krutka, H.; Shambaugh, R.; Papavassiliou, D. Analysis of the temperature field from multiple jets in the Schwarz melt-blowing die using computational fluid dynamics. Ind. Eng. Chem. Res. 2006, 45, 5098–5109. (3) Krutka, H.; Shambaugh, R.; Papavassiliou, D. Effects of the polymer fiber on the flow field from an annular melt-blowing die. Ind. Eng. Chem. Res. 2007, 46, 655–666. (4) Krutka, H.; Shambaugh, R.; Papavassiliou, D. Effects of the polymer fiber on the flow field from a slot melt blowing die. Ind. Eng. Chem. Res. 2008, 47, 935–945. (5) Klinzing, W.; Sparrow, E. Geometric factors in the processing of fine fiber by fluid shear and heat transfer. To be published in Heat Transfer Engineering. (6) Klinzing, W.; Sparrow, E. Evaluation of turbulence models for external flows. To be published in Numerical heat Transfer.
ReceiVed for reView February 14, 2008 ReVised manuscript receiVed July 18, 2008 Accepted July 23, 2008 IE800099F
