Next-Generation Modeling of Melt Blowing - American Chemical Society

ARTICLE pubs.acs.org/IECR

Next-Generation Modeling of Melt Blowing Brent R. Shambaugh, Dimitrios V. Papavassiliou, and Robert L. Shambaugh* School of Chemical, Biological, and Materials Engineering, The University of Oklahoma, 100 East Boyd Street, SEC T335, Norman, Oklahoma 73019, United States ABSTRACT: An advanced model has been developed for predicting the behavior of a fiber as it is formed in the melt blowing process. The model involves the simultaneous solution of the momentum, energy, and continuity equations. Crystallization effects are included. The model equations are solved numerically. As a bottom boundary condition in this solution, a “stop point” is assumed. The stop point is the point where (a) the fiber stress is zero, and (b) the air velocity and fiber velocity are equal. Predicted parameters include fiber diameter, velocity, temperature, stress, and crystallinity. The predicted results show that very little online crystallization takes place under typical melt blowing conditions.

’ INTRODUCTION Melt blowing is used commercially as a one-step process for converting polymer resin directly into a nonwoven mat of fibers. Figure 1 shows a typical melt blowing process in which a molten polymer is pressurized and forced through a fine capillary. The polymer stream is then rapidly attenuated by the drawing force exerted by high velocity air. The fiber collects and forms a mat on an open screen. Because weaving (or knitting) is not used to form the mat, the mat is referred to as a nonwoven. Typical examples of this material are Pall Corporation’s cartridge filters and Kimberly-Clark’s Spunguard surgical gowns. A closely related material, Du Pont’s flash-spun Tyvek, is used for house wrap, sterile packaging, high-strength envelopes, and many other uses. The primary difference between melt blowing and conventional melt spinning (see Figure 2) is that a draw roll, rather than a gas stream, provides the attenuating force in melt spinning. In melt spinning, the gas into which the fiber is spun not only exerts a drag force on the filament but also cools the filament. In contrast, the gas used in melt blowing provides a substantial forwarding force and, because the gas is heated approximately to the polymer temperature, the gas prevents polymer solidification at distances close to the melt blowing die. Lack of a draw roll permits melt blowing at very high speeds: 30,000 m/min is possible with any common melt blowing head, while commercial melt spinning operations run at maximum of 6,000 m/min. Researchers in high-speed melt spinning have achieved, with difficulty, speeds of 12,000 m/min. Special, high-performance mechanical windups are required for this research. In the present work, a comprehensive mathematical model is applied to the process of melt blowing. For all positions along the threadline, this model can predict fiber diameter, fiber velocity, fiber temperature, fiber stress, and fiber crystallinity. As an example of the model’s use, the model is applied to the melt blowing of polypropylene. Predicted fiber diameters are compared to experimental data from previous researchers.

literature on melt spinning is in order. Though melt spinning was developed commercially by Du Pont in the 1930s, only since the 1960s have mathematical models been developed for melt spinning. Ziabicki and Kedzierska,1,2 Ziabicki,3 Kase and Matsuo,4 and Matovich and Pearson5 developed the basic momentum continuity and energy balances that apply to a spinning threadline. However, they applied these equations only to low molecular weight, inelastic fluids. Since most real polymers have material properties that are highly dependent on deformation rate, Fisher and Denn6 extended previous work to include polymers with a power law viscosity and constant shear modulus. Gagon and Denn7 modeled the melt spinning of a viscoelastic fluid with the inclusion of heat transfer, inertial, and air drag effects. Other work has included the effects of polymer crystallization.8,9 The energy of deformation was also added,8 though this effect was found to be small. Because of the complexity of the equations involved in melt spinning, a numerical scheme is generally needed to solve the equations. However, a recent paper by Ziabicki and Jarecki10 produced a closed-form, analytical solution for the melt spinning of polyethylene terephthalate. Their model had to be simplified to permit this analytical solution. For example, a Newtonian constitutive equation was used, and the air drag and heat of crystallization were neglected. However, the solution gave the following very interesting result: multivalued solutions appeared when the polymer was crystallized under stress. Melt blowing (see Figure 1) is particularly difficult to model because the lower end of the spinline is free to move. Thus, establishing a lower boundary condition is tricky. Uyttendaele and Shambaugh 11 solved this problem by assuming that the spinline has a “freeze point” where attenuation ceases. The rheological force at this freeze point is balanced by the gravity and air drag forces acting on the frozen segment of the filament. Uyttendaele and Shambaugh’s model applies to one-dimensional melt blowing wherein the fiber does not vibrate. In later work, Rao and Shambaugh12 and Marla and Shambaugh13 extended the

’ MODELING WORK ON MELT BLOWING Because melt blowing has much in common with conventional melt spinning (compare Figures 1 and 2), a brief review of the

Received: April 20, 2011 Accepted: September 13, 2011 Revised: July 24, 2011 Published: October 07, 2011

r 2011 American Chemical Society

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dx.doi.org/10.1021/ie200836a | Ind. Eng. Chem. Res. 2011, 50, 12233–12245

Industrial & Engineering Chemistry Research

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Figure 1. Melt blowing process.

melt blowing model to two and three dimensions (i.e., the fiber can vibrate). Continuity, Momentum, and Energy Equations for Melt Blowing. The application of the continuity equation to the spinline gives Ff Az vfz ¼ W ¼ constant

ð1Þ

where Az = fiber cross-sectional area, m2 vfz = axial fiber velocity, m/s W = polymer mass flow rate, kg/s Ff = polymer density, kg/m3 The differential form of the momentum equation is " # d dz2 zz v2 π ðτ τxx Þ ¼ jπdz Cf Fa rel dz 4 2 þ Ff Q

dvf z πdz2 Fg dz 4 f

Figure 2. Melt spinning process.

where Rerel is the air Reynolds number. Matsui developed his correlation for melt spinning speeds up to 6000 m/min. Matsui’s correlation was modified for use in melt blowing (Majumdar and Shambaugh15) at higher spinning speeds. For melt blowing, we define Rerel as

where z = axial position, m τzz, τxx = components of the extra stress in the spinning and transverse directions, respectively, Pa Cf = air drag coefficient Fa = air density, kg/m3 vrel = difference in velocity between the filament and forwarding air, m/s dz = fiber diameter, m Q = polymer flow rate, m3 /s g = gravitational acceleration, m/s2 This equation is the same as that in Uyttendaele and Shambaugh.11 The j factor accounts for the fact that, near the spinneret, the gas exerts a positive (downward) force on the filament, but further down the threadline the force exerted by the gas is negative. The j is negative when vaz > vfz and positive when vaz < vfz (vaz is the air velocity). The drag coefficient Cf was correlated by Matsui14 with the relation Cf ¼ βðRerel Þα

Rerel ¼

ð2Þ

ð3Þ

dz jvrel j νaz

ð4Þ

where vrel = vaz vfz . This definition of vrel implies that, along the spinline, there is only a vertical component of air velocity. This is a reasonable approximation because of the symmetry of the air discharge about the spinneret. Note that vrel is positive when the air is moving faster than the filament. Energy Equation. We include crystallization in the energy equation. To simplify the energy equation, we assume steady state, no conductive resistance in the radial direction, no conduction in the axial direction, and no viscous dissipation. Then the equation of energy is simply Ff Cpf vf z

dTf z 4hz dX ¼ ðTfz Taz Þ þ Ff vfz ΔH dz dz dz

ð5Þ

where hz = convective heat transfer coefficient, J/(s m2 K) Tfz = filament temperature, K Taz = air temperature, K Cpf = heat capacity of the fiber, J/(kg K) ΔH = heat of crystallization of polymer, J/kg X = crystal fraction in polymer 12234

dx.doi.org/10.1021/ie200836a |Ind. Eng. Chem. Res. 2011, 50, 12233–12245


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The left side of eq 5 accounts for the energy required to heat or cool the polymer mass. The first term on the right side accounts for convective heat transfer to or from the fiber surface, and the second term on the right side accounts for the energy associated with polymer crystallization. Forms of this equation are given in Spruiell9 and Jarecki and Ziabicki.16 In line with nearly all other researchers, we have not included the viscous dissipation term that is in the energy equation of Jarecki and Ziabicki. This dissipation term has been found to be marginal in melt spinning.17 Be careful: the dotted X in Jarecki and Ziabicki is dX/dt, not dX/dz. To estimate the heat transfer coefficient, we use the following correlation for the Nusselt number: m

Nu ¼ γðRerel Þ

ð6Þ

This relation was developed by Kase and Matsuo4 for the parallel flow of gas along a cylinder. Structure Equation. Because we are including the effect of crystallization on the process, we need an equation to describe the development of crystallinity along the threadline. An expression that is often used for describing crystallinity development in melt spinning is based on the differential form of the Nakamara equation.9,18,19 The equation is n 1=n dθ 1 ¼ nKst ð1 θÞ ln ð7Þ dt 1θ where θ = X/X∞ = fraction of possible crystallinity X∞ = crystallinity at the termination of crystallization Kst = crystallization rate function n = Avrami index With the use of the above equation, a mass balance on the crystalline fraction of the threadline gives n 1=n vfz dX dθ 1 ¼ vf z ¼ nKst ð1 θÞ ln ð8Þ dz 1θ x∞ dz

A = stress-induced crystallization coefficient (see Jarecki et al.17) Equation 10 applies when the temperature lies between the glass transition temperature and the melting point (Tg < T < Tm). Elsewhere, Kst(z) = 0. The fa in eq 10 is in turn defined by the equation (Jarecki et al.17)

2 3 !2 Copt 4 3 Copt Δna ðzÞ 1 Copt 2 fa ðzÞ ¼ ¼ 1 ΔpðzÞ Δp ðzÞ :::5ΔpðzÞ Δnoa 7 Δnoa 7 Δnoa Δnoa

ð11Þ where Δna = birefringence Δnao = birefringence of perfectly oriented amorphous polymer Copt = stress-optical coefficient, m2/N Constitutive Equations. For a Newtonian fluid, the τzz and τxx components of the stress tensor are11,22 τzz ¼ 2η

τxx ¼ η

ð13Þ

τzz ¼

∑i τzzi

ð14Þ

τxx ¼

∑i τxxi

ð15Þ

¼ 2Gi λi

dvf z dz

ð16Þ

dτxx xx dvf z i þ ð1 X Ki τxx þ λ v Þτ i fz pt i i dz dz ¼ Gi λi and

# ðTðzÞ Tmax Þ2 exp½Afa2 ðzÞ K st ðzÞ ¼ K st ðT, ΔpÞ ¼ Kmax exp 4ln 2 1=2 D2

where Δp = τzz τxx = spinline stress or normal stress difference, Pa Kmax = the maximum crystallization rate, 1/s D1/2 = the half width of the crystallization function, K Tmax = the temperature at which maximum crystallization occurs, K fa = amorphous orientation factor

dvf z dz

dτzz zz dvf z i 2ð1 X Ki τzz þ λ v Þτ i fz pt i i dz dz

"

ð10Þ

ð12Þ

Equations 2, 5, 9, 12, and 13 are the five equations that describe Newtonian, nonisothermal melt blowing. The viscoelastic behavior of a polymer in an elongational flow field can be modeled with the following set of constitutive equations:23

A form of this equation appears in Jarecki and Ziabicki.16 However, in their equation the term X∞ is missing on the left side of the equation. The same equation, also without X∞, is shown in Ziabicki et al.8 as their eq 44. However, Bhuvanesh and Gupta20 do show a form of the Nakamara equation (our eq 7) that includes the missing X∞ . Solving for dX/dz in eq 8 gives n 1=n dX X∞ 1 ¼ nKst ð1 θÞ ln ð9Þ dz 1θ vf z Nested within eqs 8 and 9 are several other equations. The factor Kst(z) is the crystallization rate function that is dependent on the temperature and tensile stress. This factor is often expressed as21

dvf z dz

Ki ¼ exp

dvfz dz

E xx ð2τ þ τzz i Þ Gi i

ð17Þ ð18Þ

where i = 1, 2, 3,... These equations are derived from the application of the Phan-Thien and Tanner rheological model to a simple elongational flow. The Phan-Thien and Tanner model allows the use of a discrete spectrum of relaxation times. The factor Xpt is a viscous shear thinning parameter, and E is a parameter related to stress saturation at high extension rates. When both parameters are set equal to zero, the common (upper convected) Maxwell fluid model is recovered. For our calculations, we assume i = 1 in eqs 1418. 12235



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Equations 2, 5, 9, 16, and 17 describe the mechanical and thermal history of a viscoelastic fluid in the melt blowing process; see Uyttendaele and Shambaugh.11 This Uyttendaele and Shambaugh 1D model works best at relatively low speed melt blowing wherein the fiber does not vibrate to a significant degree. However, even at higher melt blowing speeds, this model gives a good first-approximation of the process behavior.13 The equations discussed so far can be equally applied to either melt blowing or melt spinning. The boundary conditions constitute the big difference in the solutions of the melt blowing versus the melt spinning process. The boundary conditions will be discussed below. Input Equations and Parameters. Polymer Density. The density of isotactic polypropylene is assumed to be a function of both temperature and crystallinity. With a linear mixing rule, the overall polymer density (expressed as the specific volume) is 1 1 1 ¼ ð1 XÞ þ X F Fam Fc

β ¼ 0:78 α ¼ 0:61 Heat Transfer Coefficient. To estimate the heat transfer coefficient in eq 6, we used the following parameters:4,28 γ ¼ 0:42 m ¼ 0:334

ð19Þ

As functions of temperature, the amorphous and crystalline fractions are 1 ¼ 1:145 þ 9:03 104 T Fam

ð20Þ

1 ¼ 1:059 þ 4:5 104 T Fc

ð21Þ

where F = overall density, g/cm3 Fam = amorphous density, g/cm3 Fc = crystalline density, g/cm3 In eqs 1921, the T is in °C. Equations 1921 were developed by Yamada et al.24 and used by Bhuvanesh and Gupta20 in an analysis of melt spinning. Heat Capacity. Jarecki and Ziabicki16 used an equation for polypropylene that gave Cp as a function of temperature, but not crystallinity. Bhuvanesh and Gupta20 included the effects of both temperature and crystallinity on Cp. We will use the equations of Bhuvanesh and Gupta. These equations are Cp ¼ Cam ð1 XÞ þ Cc X

ð22Þ

Cc ¼ 0:318 þ 2:66 103 T

ð23Þ

Cam ¼ 0:502 þ 8 104 T

ð24Þ

for T = 110 to 200 °C Cam ¼ 0:440 þ 2 103 T

appears that Bhavanesh and Gupta mistakenly used a ΔH value for the energy involved in partial crystallization—about 50%— compared to the theoretical energy of 100% crystallization. We will use the recent value from Mark.25 For the drag coefficient Cf (eq 3), we used the following values from Majumdar and Shambaugh:15

ð25Þ

for T = 0 to110 °C where Cp = overall heat capacity, cal g1 K1 Cc = heat capacity of crystalline fraction, cal g1 K1 Cam= heat capacity of amorphous fraction, cal g1 K1 The units of heat capacity are mistakenly given as cal/g in the nomenclature section of Bhuvanesh and Gupta.20 Heat of Crystallization. Jarecki and Ziabicki16 used a value of 1.65 105 J/kg for ΔH. This value is from Mark.25 Uyttendaele and Shambaugh11 assumed a value of 146.5 J/g for ΔH; this value was from Huda et al.26 Bhuvanesh and Gupta20 used a value of 20.1 cal/g (84.1 J/g) that was from Kamide and Nakamura.27 It

Polymer Viscosity. A number of formulations are available to describe the viscosity of the polymer as a function of temperature, crystallinity, and molecular weight. For example, see Spruiell,9 Bhuvanesh and Gupta,20 and Jarecki and Ziabicki.16 We used the following formulation from Jarecki and Ziabicki: ηðzÞ ¼ ηmelt ðTref ; Mw, ref Þ

Mw Mw, ref

!3:4

0 B ED 1 1 B exp k TðzÞ Tref @

1 C 1 C XðzÞA 1 0:1

ð26Þ for T > Tg where η(z) = polymer viscosity, Pa 3 s ηmelt(Tref; Mw.ref) = viscosity at reference temperature and reference molecular weight, Pa 3 s Mw = weight average molecular weight, g/mol Mw,ref = reference weight average molecular weight, g/mol Tref = reference temperature, K ED/k = activation energy/Boltzmann constant, K When T < Tg, η = ∞. For isotactic polypropylene, we used the following values from Jarecki and Ziabicki:16 ηmelt(Tref; Mw,ref) = 3000 Pa s Mw,ref = 300 000 g/mol Tref = 493 K ED/k = 5.292 103 K Tg = 253 K Relaxation Time. For the modeling of non-Newtonian behavior, relaxation time, τ, is needed. Knowing G, the elastic modulus, allows the calculation of τ from the relation τ = η/G. (For η we can use eq 26.) Based on an equation for G, the following relation for τ is given by Jarecki and Ziabicki:16 τðzÞ ¼

ηðzÞ Tref Mw ¼ τmelt ðTref ; Mw, ref Þ GðzÞ TðzÞ Mw, ref

0

1

XðzÞ B B 3:2 X∞ @

C 1 C XðzÞA 1 0:1

!3:4 exp

ED 1 1 k TðzÞ Tref

ð27Þ

where τ = relaxation time, s G = elastic modulus, Pa 12236


Industrial & Engineering Chemistry Research τmelt(Tref; Mw.ref) = relaxation time at reference temperature and reference molecular weight, s X∞ = maximum degree of crystallinity achievable by the polymer In the Phan-Thien equations (eqs 16 and 17), the symbol λi (not τ) is used for the relaxation times (since there can be more than one). However, since we only consider the case where there is one relaxation time, then λi = λ = τ . Also, the quantity Giλi in these equations is equal to η when i = 1. Then, in eqs 16 and 17, we use eq 27 to calculate λi = τ, and we use eq 26 to calculate Giλi = η . In the version of our eq 27 that is given in Jarecki and Ziabicki16 (their eq 32), they include a critical degree of crystallinity, X*. They then define X* = 0.1. In our equation, we have already made this substitution. Also, although not stated by Jarecki and Ziabicki, eq 27 is applicable for T > Tg. For T < Tg, τ = ∞. Based on the work of Lee et al.29 and Kolb et al.,30 Jarecki and Ziabicki16 assumed the following values for isotactic polypropylene: τmelt(Tref; Mw.ref) = 0.035 s at a reference temperature of 493 K and weight average molecular weight of 300 000 X ∞ ¼ 0:55 Parameters for Structure Equations. The structure equations (eqs 911) have a number of input parameters. In eq 9 the Avrami index, n, was assumed to be equal to one. This is in accordance with Bhuvanesh and Gupta.20 For eq 10, the following values were assumed:16,20,21,31 K max ¼ 0:55 s1 D1=2 ¼ 60 K T max ¼ 338 K For the factor A, the stress-induced crystallization coefficient, Jarecki and Ziabicki16 state, “The factor A is reported only for PET [42], with the values in the range 100 1000.” Their reference [42] is Alfonso et al.32 No further statement is made in the Jarecki and Ziabicki article concerning just what value of A was used in their calculations. For our calculations we let A = 40 103, which is the value used by Bhuvanesh and Gupta20 for the melt spinning of polypropylene (they use the symbol “C” instead of “A”). For the parameters in eq 11, we assumed the following for isotactic polypropylene: Δna o ¼ 60:0 103 Copt ¼ 9:0 1010 m2 =N These values are from Samuels,33 and these values are used by Jarecki and Ziabicki.16 Parameters for Phan-Thien Equations. We used the following Phan-Thien parameters for the melt spinning of polypropylene: X pt ¼ 0:6 E ¼ 0:015 These values were used by Shin et al.34 for the melt spinning of polypropylene. Jarecki and Ziabicki16 used these same values for the melt blowing of polypropylene. Uyttendaele and Shambaugh11 used values of Xpt = 0.1 and E = 0.015 for the melt blowing of polypropylene.

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Boundary Conditions. The equations developed above apply equally well for melt blowing or melt spinning. In conventional melt spinning, the numerical calculation is started at the die face and proceeds down the threadline to the fiber collection point. At the die face, the polymer flow rate, polymer temperature, and fiber diameter (either the spinneret diameter or the die swell diameter) are known. However, the initial stress Fo is unknown. But, the speed of the takeup roll (at the end of the threadline) is known; this fixes the final velocity of the filament. So, to solve the equations, an initial value of Fo is guessed, the calculation proceeds down the threadline, and the calculated final fiber velocity is checked against the windup speed. If the speeds do not match, a new value for Fo is selected, and threadline calculations are repeated until the final velocity is close to the windup speed. Similarly, calculations for spunbonding, which uses an air Venturi instead of a windup roll, can be done. However, when running actual experiments, controlling the final fiber speed in the Venturi is not as easy as controlling the windup speed of the mechanical roll in a melt spinning system. An off-line measurement of fiber diameter below the Venturi (which gives fiber velocity from the continuity equation) is often the simplest way of measuring fiber speed. For melt blowing, the boundary conditions are more complicated. As with melt spinning, the initial stress Fo is unknown and must be guessed. To account for the bottom boundary condition, the concept of the “stop point” was developed for 1D, 2D, and 3D melt blowing. The 1D model of Uyttendaele and Shambaugh11 applies to melt blowing wherein the fiber does not vibrate. (The Uyttendaele paper actually used a “freeze point” rather than a stop point, but the concepts are very close.) Thus, the 1D model is like a melt spinning threadline wherein the fiber position (in a lateral sense) is controlled. Of course, some minor vibration does occur in both melt spinning and 1D melt blowing. In later work, Rao and Shambaugh12 extended the melt blowing model to account for fiber motion in a plane, and Marla and Shambaugh13 extended the melt blowing model to fiber motion in three dimensions. In melt blowing, the high velocity air stream (and not a windup roll) drives the attenuation and motion of the fiber (compare Figures 1 and 2). So, near the spinneret, the air is moving faster than the fiber. But, as the fiber speed increases along the threadline, the air speed decreases. At some point, these speeds become equal. We call this the stop point. Now, as we know from freshman physics, you cannot push on a rope. Ergo, when the stop point is reached, the stress in the filament must also be zero. So, our solution procedure is as follows: 1. Guess Fo 2. Iterate down the threadline until the fiber velocity equals the air velocity 3. Check if fiber stress is equal to zero at this velocity crossover. If not, guess a new Fo and repeat the iteration. When the velocity crossover occurs simultaneously with zero stress, we have found the stop point. In a recent 3-part paper (Zachara and Lewandowski,35 Jarecki and Ziabicki,16 and Jarecki and Lewandowski36), 1D melt blowing was considered. In this research, the 1D model was like the 1D Uyttendaele and Shambaugh11 model, except that polymer crystallization was considered. They referred to the experimental work of Bresee and Ko37 wherein a “critical distance of coiling” was discussed. This distance is where the filament velocity exceeds the air velocity and the fiber begins to bend. In a very recent follow up paper by Jarecki et al.,38 further work on this 12237


Industrial & Engineering Chemistry Research critical distance is considered. (Similarly, other researchers39 have described the “initial straight part of the jet” where there is parallel flow of the fiber and the air stream in the zone just below the melt blowing die.) However, as a bottom boundary condition, they16,35,36,38 did not use the stop point concept. Instead, they assumed that stress was zero at the collector screen. For many of their calculations,16,35,36 they assumed that the screen was 20 cm from the die. See Figures 13 and 14 in Jarecki and Lewandowski.36 In their most recent paper,38 the collector position was varied over a wide range. Air Velocity and Temperature Profiles. Because the fiber threadline is exposed to air as the threadline moves between the spinneret and the collection point, profiles of the air temperature and air velocity are needed for all positions along the threadline. These profiles serve as boundary, or envelope, conditions for the solution of the above-described equations. For melt spinning of a single filament, the air is often considered to be stationary and at room temperature. (Multiple filament modeling is more complex because the many filaments heat and “pump” the air.) Then, for the calculation of the Reynolds number in eq 4, the relative velocity is simply the filament velocity. For the case of melt blowing, more complex air-field formulas are needed. In the 1D work of Uyttendaele and Shambaugh,11 a Pitot tube was used to measure air velocity below the melt blowing die. Because of the nature of the 1D model, only measurements along the z axis were needed. Also, to avoid damaging the Pitot tube, measurements were taken with the polymer stream shut off. Thus, this measuring scheme mirrored the assumption that, with the melt blowing of a single filament (or a single row of filaments as in a slot die), the presence of the polymer filament does not significantly perturb the air field. Uyttendaele and Shambaugh also took air temperature measurements with a fine thermocouple. For the 2D model of Rao and Shambaugh,12 velocity and temperature profiles were needed in a plane containing the z axis (see Figure 1). Similar to Uyttendaele and Shambaugh,11 Rao and Shambaugh used correlations from actual experimental measurements of velocity and air fields below an annular die. Specifically, they used the correlations developed by Majumdar40 and Majumdar and Shambaugh.41 For the 3D model of Marla and Shambaugh,13 the Majumdar40 and Majumdar and Shambaugh41 correlations were again used for the annular die geometry considered in the 3D model (this was possible since the correlations give the velocity and temperature fields in 3D). In a subsequent paper on the modeling of the slot or “Exxon” die (a common commercial die), Marla and Shambaugh42 used slot die correlations for the downward air velocity and the air temperature. These correlations were from Harpham and Shambaugh,43,44 and Tate and Shambaugh.45,46 For the smaller—and hard to measure—lateral velocities, Marla and Shambaugh42 developed correlations based on the CFD predictions of Krutka et al.47 As mentioned above, 1D melt blowing was considered in a recent 3-part paper (Zachara and Lewandowski,35 Jarecki and Ziabicki,16 and Jarecki and Lewandowski36). For the air field, they used Fluent with a k ε aerodynamic model for convergent air jets from a slot die. Then, in a manner similar to that used (for lateral velocities only) by Marla and Shambaugh,42 they used the resulting numerical data as the basis for analytical expressions for air velocity and temperature. These analytical expressions were then used as boundary, or envelope, conditions in the solution of the polymer-side equations for melt blowing.

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Testing and Using the Model. Comparison with Previous Work. As discussed above, Rao and Shambaugh12 and Marla

and Shambaugh13 used annular velocity correlations. For the velocity along the threadline, these correlations are (Majumdar and Shambaugh41) vjo =vo ¼ 0:077ZðdAN Þ þ 1:00

for ZðdAN Þ < 3:093

vjo =vo ¼ 0:249ZðdAN Þ þ 0:468

ð28Þ

for ZðdAN Þ g 3:093 ð29Þ

where dAN = outer diameter of annular die orifice, mm vo = air velocity along the threadline, m/s vjo = va,die = air velocity at the die (z = 0), m/s Z(dAN) = z/dAN ((Fa∞/Fao)1/2) Fao = air density along the center line downstream from the nozzle, kg/m3 Fa∞ = air density at ambient conditions, kg/m3 For the temperature profile along the threadline, the analogous correlations are (Majumdar and Shambaugh41) θjo =θo ¼ 0:033ZðdAN Þ þ 1:00

for ZðdAN Þ < 4:644 ð30Þ

θjo =θo ¼ 0:221ZðdAN Þ þ 0:127

for ZðdAN Þ g 4:644 ð31Þ

where θjo = excess air temperature above ambient at die exit, °C θo = excess air temperature above ambient along the center line (the z axis), °C Rao and Shambaugh12 tried air discharge velocities (die face velocities) of 110, 150, 200, and 300 m/s; Marla and Shambaugh13 tried air discharge velocities of 110, 150, and 200 m/s. For our modeling work, we tried velocities of 110, 150, 200, and 300 m/s. Also, as done by previous researchers,1113 we used the following base operating conditions: va,die =110.3 m/s, Ta,die = 368 °C, Q = 0.658 cm3/min, and Tf,die = 310 °C. Also, a 0.762-mm polymer orifice was assumed, and the assumed maximum polymer die swell was 0.949 mm (the numerical calculations were started at a this die swell diameter). As described above, Newtonian, nonisothermal melt blowing involves the simultaneous numerical solution of eqs 2, 5, 9, 12, and 13. With the conditions described in the above paragraph, the model was used to predict the operating behavior of a die run with discharge air velocities of 110, 150, 200, and 300 m/s. Figure 3 shows the simulated diameter profile along the threadline for this range of air velocities. Also included on Figure 3 are the experimental data from Uyttendaele and Shambaugh.11 These data were produced with experimental conditions that are the same as the inputs used in the simulation. As can be seen, there is an excellent match between the simulation and the experimental data. Figure 4 shows the fiber velocity profiles for the different air velocities. At the 110 m/s air velocity, the maximum fiber velocity is less than 5 m/s, while the highest air velocity (300 m/s) produces a fiber speed of 35 m/s, which is about seven times larger. Observe that the right ends of the profiles correspond to the stop points for the respective conditions. Higher final fiber speeds correspond to stop points located closer to the die. 12238



Figure 3. Effect of air velocity on the model predictions of fiber diameter. For comparison, experimental data for a 110 m/s air velocity are shown. The experimental data are from Uyttendaele and Shambaugh.11 The initial fiber diameter (die swell diameter) used in the calculations was 949 μm, the iteration step size was 1 mm, and Mw = 165 000.

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Figure 5. Fiber temperature profiles for different air velocities.

Figure 6. Fiber stress profiles for different air velocities.

Figure 4. Fiber velocity profiles for different air velocities.

Figure 5 shows the fiber temperatures along the threadline. The fiber temperatures at the stop point are cooler when the air velocities are lower. Figure 6 shows the fiber stress profiles along the threadline. The maximum stress is about 2.36 104 Pa for the 110 m/s air velocity. The maximum stress increases an order of magnitude (to about 2.42 105 Pa) for the 300 m/s air velocity. Also, the position of the maximum stress moves closer to the die as the air velocity increases. Figure 7 shows the fiber crystallinity profiles for the range of air velocities. Crystallinities are very small, though there is some rise in crystallinity for the low, 110 m/s air velocity. For researchers in fiber spinning, this result may seem counterintuitive. After all, it is know that high spinning speeds (approximately 6000 m/min) are required to achieve significant crystallization along the threadline during the melt spinning of common (polyethylene terephthalate) polyester; see Ziabicki and Kawai.48 What Figure 7 shows is that, in melt blowing, the region of high fiber stress— which occurs near the die (see Figure 6)—does not overlap the

Figure 7. Fiber crystallinity profiles for different air velocities.

region where the temperature is optimum for crystallization (see Figure 5). As shown in eq 10, for crystallization to occur the fiber temperature must be relatively close to Tmax = 65 C. For the simulations shown in Figures 47, Mw = 165 000 was used. To examine the effect of molecular weight, Mw = 250 000 12239



Figure 8. Effect of air velocity on the model predictions of fiber diameter. These predictions are for Mw = 250 000. The initial fiber diameter (die swell diameter) used in the calculations was 949 μm, and the iteration step size was 1 mm.

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Figure 10. Fiber temperature profiles for different air velocities. These predictions are for Mw = 250 000.

Figure 11. Fiber stress profiles for different air velocities. These predictions are for Mw = 250 000. Figure 9. Fiber velocity profiles for different air velocities. These predictions are for Mw = 250 000.

was also tested. For this higher molecular weight, Figure 8 shows the effect of air velocity on the model predictions of fiber diameter. A higher molecular weight gives a higher polymer viscosity. This higher viscosity makes the fiber harder to attenuate. Thus, as expected, the diameters in Figure 8 are higher than the corresponding diameters in Figure 4. For fiber velocity, the more viscous polymer has slower spinning speeds; see Figure 9. At an air speed of 300 m/s, the final fiber velocity in Figure 9 is 8.45 m/s, while the corresponding maximum velocity in Figure 5 is 33.7 m/s. For the higher molecular weight polymer, Figure 10 shows the temperature profiles. Since the higher molecular weight polymer has slower spinning speeds, the fibers have a chance to cool much more than the fibers with the higher molecular weight. As Figure 10 shows, for air velocities of 110, 200, and 300 m/s, the final fiber temperatures are 32.9, 53.4, and 106.4 C. The corresponding fiber temperatures from Figure 5 are 173.1, 230.9, and 256.8 C. For the higher molecular weight polymer, Figure 11 shows fiber stress profiles for different air velocities. The filaments exhibit

a maximum stress (at 300 m/s) of about 1.94 105 Pa, which is about eighty percent as large as the maximum stress for the low molecular weight polymer (see Figure 6). Figure 12 shows the crystallinity profiles for the high molecular weight polymer. There is a notable difference between Figure 12 and the corresponding figure (Figure 7) for the low molecular weight polymer. In Figure 12, we see that the crystallinity reaches 6.81 104 for the 300 m/s air velocity, 0.043 for the 200 m/s velocity, and 0.37 for the 110 m/s velocity. (As stated earlier, X∞, the maximum possible crystallinity is 0.55.) Because the high molecular weight fibers are hard to attenuate, the spinning speeds are low, and the fibers take a relatively long time to travel to the collector screen. This longer exposure (at temperatures near the optimum crystallization temperature) causes the higher crystallinity levels shown in Figure 12. However, the spinning speed is too slow to be of commercial use, and the final fiber diameter is very large. Also, the stop point (for an air velocity of 110 m/s) is way out at 210 cm, while the stop point (at 110 m/s) for polymer with Mw = 165 000 is only 23.5 cm. Both our work and the recent work by Jarecki et al.38 are based on the solution of a set of differential equations applied to a threadlline in the melt blowing process. For us and for them, 12240



Figure 12. Fiber crystallinity profiles for different air velocities. These predictions are for Mw = 250 000.

these equations are the momentum, energy, structure, and constitutive equations. There are some differences in our equations and boundary conditions. For one thing, our air field equations are somewhat different from their air field equations. A second factor is that our working equations for polymer properties are slightly different from their working equations (the working equations are described earlier in this paper). A third factor is that our starting diameter (at the top of the threadline) is different from theirs. We use a die swell diameter of 0.949 mm as the starting point for our solution, while their starting point is a polymer orifice diameter of 0.35 mm. A fourth factor is that we solve our equations by searching for a stop point (as described above in this article). In contrast, Jarecki et al. set the collector at a particular position and then solve the differential equations for when the stress is zero at the collector. By selecting a whole range of collector positions, they then can determine when the fiber velocity becomes equal to the air velocity. So, when they place the collector at what they call the edge of the air drawing zone (a position that they state is optimum), they should end up with the same solution as we get. In spite of these differences, it is worthwhile to compare our results with their results. Figure 13 shows stop point as function of air velocity. In this figure, our results are compared with results from Jarecki et al.38 Our results are shown for a low Mw = 165 000 and a high Mw = 250 000. Jarcecki et al.’s results are for a low Mw = 150 000 and a high Mw = 250 000. (We assumed that their results for Mw = 150 000 should be quite close to our results for Mw = 165 000.) As can be seen in Figure 13, for both the low and high molecular weights, there is excellent concordance between our results and their results. For the case of final fiber diameter versus air velocity, Figure 14 is a comparison of our results with the results of Jarecki et al. Here again, data from both a high Mw and a low Mw are shown. For the low Mw, our results are nearly identical with theirs. For the high Mw, our results show the same trend but track higher than the results of Jareck et al. In total, Figures 13 and 14 show that our simulations give results that are very similar to the simulation results of Jarecki et al. Optimizing the Air Field. As stated above, what drives the melt blowing process is the impact of the air stream upon the molten filament. If we can optimize this impact, we can make a better melt blowing die. Now, with our model, we can impose unusual air fields on the filament. Let us try imposing a “plateau” on the air velocity and temperature. For this plateau assumption, the air velocity and

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Figure 13. Comparison of our simulation results with the simulation results of Jarecki et al.38 Stop point versus initial air velocity is shown for Mw = 165 000 and 250 000 (for our results) and Mw = 150 000 and 250 000 (Jarecki et al.’s results).

Figure 14. Final fiber diameter versus initial air velocity for Mw = 165 000 and 250 000 (for our results) and Mw = 150 000 and 250 000 (Jarecki et al.’s results).

temperature decrease according to the usual correlations until some value z1 is reached. Then, the velocity and temperature are arbitrarily kept constant (kept at a “plateau”) over a certain range of z from z1 to z2. Then, at z2 the velocity and temperature will continue to decrease according to the usual correlations. For these simulations, we will not at this point address the issue of how to experimentally impose the arbitrary flow field. However, what we are trying to achieve with a plateau is to increase the length of the fiber that is exposed to very hot air. In this way, we expect to decrease the fiber viscosity and thus increase the fiber attenuation. Mathematically, a new parameter z0 can be used to define the plateau as follows: z0 ¼ z z0 ¼ z1

for z < z1 for z1 e z e z2

z0 ¼ z ðz2 z1Þ for z > z2 12241



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Figure 15. Air velocity and air temperature with a 2-cm plateau placed between z = 2 cm and z = 4 cm. The air velocity and air temperature at the die face are, respectively, 110 m/s and 368 °C. The plots are modified versions of eqs 2831 (see discussion in text).

Figure 17. (a) Fiber diameter profiles when a 4-cm plateau is placed in the air field. These predictions are for Mw = 165 000. (b) An expanded section of panel a.

Figure 16. (a) Fiber diameter profiles when a 2-cm plateau is placed in the air field. These predictions are for Mw = 165 000. (b) An expanded section of panel a.

This z0 is used in place of z in the air velocity and air temperature correlations (eqs 2831). Figure 15 shows the air velocity and air temperature change when a 2-cm plateau is placed between z = 2

and z = 4 cm. Note that, beyond z = 4 cm, the air velocity and air temperature decrease according to the established correlations— except that the curve has been shifted by 2 cm. Figure 16 shows the simulation results for a slot die with a 2-cm plateau placed in the following three locations: z1 = 0 cm to z2 = 2 cm; z1 = 1 cm to z2 = 3 cm; and z1 = 2 cm to z2 = 4 cm. Note that the fiber is more quickly attenuated when a plateau is present. Figure 17 shows diameter profiles for a 4-cm plateau placed at several positions below the die. Table 1 lists the final predicted diameters and stop points for the two simulations of Figures 16 and 17. (As previously stated, the stop point is a bottom boundary condition where the stress is zero and the air velocity equals the fiber velocity.) Also included in Table 1 are the final diameters and stop points for Figure 3. The data of Table 1 show that plateaus can improve melt blowing. For example, for a 2-cm plateau placed between z = 1 and z = 3 cm, the final fiber diameter is less than when no plateau is used. Similar behavior is exhibited for the 4-cm-wide plateaus. In other words, with plateaus a lot less air is required to make the same size fibers. Economically, this is very important, since a large part of the operating cost for a melt blowing system is the air cost (manifested in compressing the air, heating the air, and recycling the hot air). For our preliminary tests, the 4-cm plateau placed between 0 and 4 cm produced the best results. For these tests, the 12242



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Table 1. Final Fiber Diameters and Stop Points for Various Runs max air velocity (m/s)

details for Figure 3 final diameter at stop point (micrometers)

stop point (cm)

110.3

53.82

23.5

150

38.93

16.7

200

29.43

12.7

300

20.36

9.1

details for Figure 16 final diameter at stop point plateau range

(micrometers)

stop point (cm)

02 cm

29.32

8.9

13 cm

32.22

10.3

24 cm

39.4

14.5

no plateau

53.82

23.5

details for Figure 17 final diameter at stop point plateau range

(micrometers)

stop point (cm)

04 cm

20.11

7.1

26 cm

31.1

11.7

no plateau

53.82

23.5

final fiber diameter of 20.11 μm was produced with an air velocity of 110 m/s (see Table 1 and Figure 17). To get almost the same diameter without a plateau (see Table 1 and Figure 3), an air velocity of 300 m/s is required. Now, since initial air velocity is proportional to the mass flow of air to the die, this means that, with a proper plateau, the same final fiber diameter can be produced with 63% less air!

’ CONCLUSIONS 1. A comprehensive model has been developed for the melt blowing process. This model, which includes crystallization, involves the simultaneous solution of continuity, momentum, and energy balances applied to the moving threadline. The stop point concept was used in the solution, and predicted results showed that, for practical melt blowing conditions, very little online crystallization takes place during melt blowing. 2. Predictions from the model suggest that, by placing a plateau in the air field, substantial reductions in air consumption are possible for the melt blowing process. ’ AUTHOR INFORMATION Corresponding Author

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

’ NOMENCLATURE A = stress-induced crystallization coefficient (see Jarecki et al.17) Az = fiber cross-sectional area, m2 Cam = heat capacity of amorphous fraction, cal g1 K1 Cc = heat capacity of crystalline fraction, cal g1 K1 Cf = air drag coefficient

Copt = stress-optical coefficient, m2/N Cp = overall heat capacity, cal g1 K1 Cpf = heat capacity of the fiber, J kg1 K1 dz = fiber diameter, m dAN = outer diameter of annular air orifice, mm D1/2 = the half width of the crystallization function, K E = Phan-Thien and Tanner model parameter related to stress saturation at high extension rates ED/k = activation energy divided by Boltzmann constant, K fa = amorphous orientation factor g = gravitational acceleration, m/s2 G = elastic modulus, Pa hz = convective heat transfer coefficient, J/(m2 K s) ΔH = heat of crystallization of polymer, J/kg Ki = Phan-Thien and Tanner model parameter Kmax = the maximum crystallization rate, s1 Kst = crystallization rate function, s1 m = constant in Nusselt number correlation; see eq 6 Mw = weight average molecular weight, g/mol Mw,ref = reference weight average molecular weight, g/mol n = Avrami index Δna = birefringence Δnao = birefringence of perfectly oriented amorphous polymer Nu = Nusselt number Δp = τzz τxx = spinline stress or normal stress difference, Pa Q = polymer flow rate, m3 /s Rerel = Reynolds number based on relative velocity; see eq 4 Tfz = filament temperature, K Taz = air temperature, K Tmax = the temperature at which maximum crystallization occurs, K Tref = reference temperature, K vaz = axial air velocity, m/s vfz = axial fiber velocity, m/s vrel = vaz vfz = difference in velocity between the filament and forwarding air, m/s W = polymer mass flow rate, kg/s X = crystal fraction in polymer X∞ = maximum degree of crystallinity achievable by the polymer Xpt = Phan-Thien and Tanner model parameter related to viscous shear thinning z = axial position, m z0 = shifted axial position, m z1 = z position at start of plateau, m z2 = z position at end of plateau, m Z(dAN) = z/dAN ((Fa∞/Fao))1/2 Greek Letters

α = constant in Matsui relation; see eq 3 β = constant in Matsui relation; see eq 3 γ = leading constant in Nusselt number correlation; see eq 6 η(z) = polymer viscosity, Pa 3 s ηmelt(Tref; Mw.ref) = viscosity at reference temperature and reference molecular weight, Pa 3 s θ = X/X∞ = fraction of possible crystallinity θjo = excess air temperature above ambient at die exit, °C θo = excess air temperature above ambient along the center line (the z axis), °C λi = stress relaxation time, s νaz = kinematic air viscosity, m2 /s F = overall polymer density, g/cm3 Fa = air density, kg/m3 12243


Industrial & Engineering Chemistry Research Fao = air density along the center line downstream from the nozzle, kg/m3 Fa∞ = air density at ambient conditions, kg/m3 Fam = amorphous density, g/cm3 Fc = crystalline density, g/cm3 Ff = polymer density, g/cm3 τ = relaxation time, s τzz, τxx = components of the extra stress in the spinning and transverse directions, respectively, Pa τmelt(Tref; Mw,ref) = relaxation time at reference temperature and reference molecular weight, s θjo = excess air temperature above ambient at die exit, °C θo = excess air temperature above ambient along the center line (the z axis), °C

’ REFERENCES (1) Ziabicki, A.; Kedzierska, K. Mechanical Aspects of Fibre Spinning Process in Molten Polymers: 1. Stream Diameter and Velocity Distribution Along the Spinning Way. Kolloid-Z 1960, 171, 51. (2) Ziabicki, A.; Kedzierska, K. Mechanical Aspects of Fibre Spinning Process in Molten Polymers: 2. Stream Broadening after the Exit from the Channel of Spinneret. Kolloid-Z 1961, 171, 111. (3) Ziabicki, A. Mechanical Aspects of Fibre Spinning Process in Molten Polymers: 3. Tensile Force and Stress. Kolloid-Z 1961, 175, 14. (4) Kase, S.; Matsuo, T. Studies on Melt Spinning. I. Fundamental Equations on the Dynamics of Melt Spinning. J. Polym. Sci. 1965, 3 (7), 2541–2554. (5) Matovich, M. A.; Pearson, J. R. A. Spinning a Molten Threadline: Steady State Isothermal Viscous Flows. Ind. Eng. Chem. Fundam. 1969, 8 (3), 512. (6) Fisher, R. J.; Denn, M. M. A Theory of Isothermal Melt Spinning and Draw Resonance. AIChE J. 1976, 22 (6), 236. (7) Gagon, D. K.; Denn, M. M. Computer Simulation of Steady Polymer Melt Spinning. Polym. Eng. Sci. 1981, 21, 844. (8) Ziabicki, A.; Jarecki, L.; Wasiak, A. Dynamic Model of Melt Spinning. Comput. Theor. Polym. Sci. 1998, 8 (l/2), 143–157. (9) Spruiell, J. E. Structure and Property Development During the Melt Spinning of Synthetic Fibers. In Structure Development During Polymer Processing; Cunha, A. M., Fakirov, S., Eds. Kluwer Academic Publishers: Netherlands, 2000; pp 195220. (10) Ziabicki, A.; Jarecki, L. Structure-controlled Bifurcation in Mathematical Modelling of Fibre Spinning. Arch. Mech. 2006, 58 (45), 459–475. (11) Uyttendaele, M. A. J.; Shambaugh, R. L. Melt Blowing: General Equation Development and Experimental Verification. AIChE J. 1990, 36 (2), 175–186. (12) Rao, R.; Shambaugh, R. L. Vibration and Stability in the Melt Blowing Process. Ind. Eng. Chem. Res. 1993, 32 (12), 3100–3111. (13) Marla, V. T.; Shambaugh, R. L. Three-dimensional Model of the Melt Blowing Process. Ind. Eng. Chem. Res. 2003, 42 (26), 6993–7005. (14) Matsui, M. Air Drag on a Continuous Filament in Melt Spinning. Trans. Soc. Rheol. 1976, 20 (3), 465. (15) Majumdar, B.; Shambaugh, R. L. Air Drag on Filaments in the Melt Blowing Process. J. Rheol. 1990, 34, 591. (16) Jarecki, L.; Ziabicki, A. Mathematical Modelling of the Pneumatic Melt Spinning of Isotactic Polypropylene. Part II. Dynamic Model of Melt Blowing. Fibres Textiles East. Eur. 2008, 16 (5 (70)), 17–24. (17) Jarecki, L.; Ziabicki, A.; Blim, A. Dynamics of hot-tube spinning from crystallizing polymer melts. Comput. Theor. Polym. Sci. 2000, 10 (12), 63–72. (18) Nakamura, K.; Katayama, K.; Amano, T. Some Aspects of Nonisothermal Crystallization of Polymers. II. Consideration of the Isokinetic Condition. J. Appl. Polym. Sci. 1973, 17, 1031–1041. (19) Patel, R. M.; Spruiell, J. E. Crystallization Kinetics during Polymer Processing -Analysis of Available Approaches for Process Modeling. Polym. Eng. Sci. 1991, 31 (10), 730–738.

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(20) Bhuvanesh, Y. C.; Gupta, V. B. Computer Simulation of Melt Spinning of Polypropylene Fibers Using a Steady-State Model. J. Appl. Polym. Sci. 1995, 58 (3), 663–674. (21) Ziabicki, A. Fundamentals of Fibre Formation; J. Wiley: London, 1976. (22) Middleman, S. Fundamentals of Polymer Processing; McGraw-Hill: New York, 1977. (23) Phan-Thien, N. A Nonlinear Network Viscoelastic Model. J. Rheol. 1978, 22 (3), 259. (24) Yamada, N.; Sano, Y.; Nanbu, T. The Analysis of Quenching Process in Polypropylene Melt Spinning. Sen'i Gakkaishi 1966, 22, 197–205. (25) Mark, J. E. Physical Properties of Polymers Handbook; AIP Press: New York, 1996; pp 424, 670. (26) Huda, M. Nurul; Dragaum, H.; Bauer, S.; Mushik, H.; Skalicky, P. A Study of the Crystallinity Index of Polypropylene Fibres. Colloid Polym. Sci. 1985, 263 (9), 730–737. (27) Kamide, K.; Nakamura, K. Heat of Fusion of Isotactic Polypropylene Crystals in Monoclinic Form. Sen'i Gakkaishi 1968, 24 (10), 486–491. (28) Bansal, V.; Shambaugh, R. L. On-line Determination of Diameter and Temperature During Melt Blowing of Polypropylene. Ind. Eng. Chem. Res. 1998, 37 (5), 1799–1806. (29) Lee, J. S.; Shin, D. M.; Jung, H. W.; Hyun, J. C. Transient Solutions of the Dynamics in Fiber Spinning Process Accompanied by Flow-induced Crystallization. J. Non-Newtonian Fluid Mech. 2005, 130, 110. (30) Kolb, R.; Seifert, S.; Stribeck, N.; Zachmann, H. G. Simultaneous Measurements of Small and Wide-angle X-ray Scattering during Low Speed Spinning of Poly(propylene) using Synchrotron Radiation. Polymer 2000, 41, 1497–1505. (31) Magill, J. H. A New Technique for Following Rapid Rates of Crystallization II Isotactic Polypropylene. Polymer 1962, 3, 35–42. (32) Alfonso, G. C.; Verdona, M. P.; Wasiak, A. Crystallization Kinetics of Oriented Poly(ethylene terephthalate) from the Glassy State. Polymer 1978, 19 (6), 711–716. (33) Samuels, R. J. Structured Polymer Properties; J. Wiley: New York, 1974, p 58. (34) Shin, D. M.; Lee, J. S.; Jung, H. W.; Hyun, J. C. Analysis of the Effect of Flow-induced Crystallization on the Stability of Low-speed Spinning Using the Linear Stability Method. Korea-Australia Rheology J. 2005, 17 (2), 63. (35) Zachara, A.; Lewandowski, Z. Mathematical Modelling of Pneumatic Melt Spinning of Isotactic Polypropylene. Part I. Modelling of the Air Jet Dynamics. Fibres Textiles East. Eur. 2008, 16 (4 (69)), 17–23. (36) Jarecki, L.; Lewandowski, Z. Mathematical Modelling of Pneumatic Melt Spinning of Isotactic Polypropylene. Part III. Computations of the Process Dynamics. Fibres Textiles East. Eur. 2009, 17 (1 (72)), 75–80. (37) Bresee, R. R.; Ko, W. C. Fiber Formation during Melt Blowing. Int. Nonwovens J. 2003, No. summer, 21. (38) Jarecki, L.; Ziabicki, A.; Lewandowski, Z.; Blim, A. Dynamics of Air Drawing in the Melt Blowing of Nonwovens from Isotactic Polypropylene by Computer Modeling. J. App. Polym. Sci. 2011, 119, 53–65. (39) Yarin, A. L.; S. Sinha-Ray, S.; Pourdeyhimi, B. Meltblowing: Multiple polymer jets and fiber-size distribution and lay-down patterns. Polymer 2011, 52, 2929–2938. (40) Majumdar, B. Flow Fields for Annular and Multi-Hole Nozzles and Air Drag in the Melt Blowing Process. M.S. Thesis, University of Oklahoma, Norman, OK, 1990. (41) Majumdar, B.; Shambaugh, R. L. Velocity and Temperature Fields in Annular Jets. Ind. Eng. Chem. Res. 1991, 30 (6), 1300–1306. (42) Marla, V. T.; Shambaugh, R. L. Modeling of the Melt Blowing Performance of Slot Dies. Ind. Eng. Chem. Res. 2004, 43 (11), 2789–2797. (43) Harpham, A. S.; Shambaugh, R. L. Flow Field of Practical Dual Rectangular Jets. Ind. Eng. Chem. Res. 1996, 35 (10), 3776–3781. (44) Harpham, A. S.; Shambaugh, R. L. Velocity and Temperature Fields of Dual Rectangular Jets. Ind. Eng. Chem. Res. 1997, 36 (9), 3937–3943. 12244



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(45) Tate, B. D.; Shambaugh, R. L. Modified Dual Rectangular Jets for Fiber Production. Ind. Eng. Chem. Res. 1998, 37 (9), 3772–3779. (46) Tate, B. D.; Shambaugh, R. L. Temperature Fields below Melt Blowing Dies of Various Geometries. Ind. Eng. Chem. Res. 2004, 43 (17), 5405–5410. (47) 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. (48) Ziabicki, A., Kawai, H., Eds. High-speed Fiber Spinning: Science and Engineering Aspects; Wiley: New York, 1985.

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