Article pubs.acs.org/IECR
Numerical Simulation and Experimental Study of the Polypropylene Polymer Air Drawing Model and Air Jet Flow Field Model in the Spunbonding Nonwoven Process Bo Zhao* College of Textiles, Zhongyuan University of Technology, No. 41 Zhongyuan Road, Henan, Zhengzhou, 450007, People’s Republic of China ABSTRACT: The air drawing model of the polypropylene polymer and the model of the air jet flow field in a spunbonding process are presented and solved by introducing the numerical computation results of the air jet flow field of an aerodynamic device. The air jet flow field model is simulated by means of the finite difference method. The effect of the density and specific heat capacity of polymer melt at a constant pressure changing with the polymer temperature on the fiber diameter was studied. We find that the variation of the density and the specific heat capacity of polymer melt at constant pressure with polymer temperature greatly effects the fiber diameter. Compared with the existing drawing model, the new ones include more processing parameters and are more accurate. The newly developed formulas were incorporated into a spunbonding theoretical model to predict the fiber diameter of nonwoven web. The numerical simulation computation results of the distributions of the zcomponents of the air velocity match quite well with the experimental data. The air drawing model of polymer is solved with the help of the distributions of the air velocity measured by particle image velocimetry (PIV). The model’s predictions of the filament fiber diameters, crystallinities, and birefringences coincide well with the experimental data. It can be concluded that the higher initial air temperature can yield finer filament fiber diameter and the higher initial air velocity can produce finer fiber diameter as well. The experimental results show that the agreement between the results and experimental data is much better, which verifies the reliability of these models. At the same time, also, they reveal great prospects for this work in the field of computer assisted design (CAD) of the spunbonding process.
1. INTRODUCTION The spunbonding process is used commercially as a single-step technology for converting polymer resin into nonwoven web, which dates back to the 1950’s.1−4 In the spunbonding process, a molten stream of polymer is extruded from the screw extruder and rapidly attenuated into filament fiber using an aerodynamic device with the aid of a high-velocity cool air stream.5−7 The fiber diameter is strongly affected by the air jet flow field developed from the spunbonding attenuator as shown in Figure 1. The fiber diameter is affected by many processing parameters. Earlier researchers have studied air drawing models of the polymer spunbonding process, especially Spruiell and coworkers.8,9 However, the reported air drawing model was quite elementary, and all these models mentioned above were not based on using analytical and numerical methods. To achieve the best web performance, therefore, most research on the spunbonding process is focused on the effect of process parameters on the fiber diameter and the structures of spunbonding equipment; these studies can help better understand the technology of the spunbonding process. However, seldom is research conducted to deal with the air drawing model of polymers and the model of the air jet flow field in the spunbonding process which is of significant importance. At the same time, as far as the author knows, some reports exist in the open literature regarding the air drawing model of the spunbonding polymer which are only simple introductions. In fact, the fiber diameter is strongly affected by the air drawing model of polymers and the model of the air jet flow field. Moreover, there is not any literature © 2013 American Chemical Society
regarding the study of the air drawing model of polymers and the model of the air jet flow field, which strongly affect the fiber diameter and web performance of spunbonding nonwoven materials. In this paper, an air drawing model of polymers will be established based on a numerical computational method and verified by numerical results with the experimental data obtained with university equipment. We established the air drawing model of polymers in the spunbonding process which is based on numerical simulation computation results of the air jet flow field, and we adopt the finite difference method to simulate the air jet flow field. The effect of variation in polymer density and polymer specific heat capacity with polymer temperature at constant pressure on fiber diameter is also studied. The fiber diameter can be predicted with the aid of the established air drawing model. We also investigate the effects of process parameters such as the air initial temperature and the air initial velocity on fiber diameter and verify the model reliability of these relationships. The results present great prospects for this research in the field of computer assisted design (CAD) of the spunbonding process, technology, and equipment. Received: Revised: Accepted: Published: 11061
July 23, 2012 March 7, 2013 July 1, 2013 July 1, 2013 dx.doi.org/10.1021/ie3019593 | Ind. Eng. Chem. Res. 2013, 52, 11061−11069
Industrial & Engineering Chemistry Research
Article
Figure 1. Flat narrow slot passage of drafting assembly in spunbonding process. (a) L1= the steady flow segment length; (b) L2= the jet orifice length; (c) L3= the drafting segment length; (d) L4= the outlet orifice length; (e) L5= the drafting segment width; (f) L6= the jet orifice highness; (g) L7= the drafting segment highness (h1/h2).
drawing coefficient of filament fiber as a function of the Reynolds number, which will vary with axial position. Cf was given by Majumdar and Matsui15,16 with the following correlation:
2. AIR DRAWING MATHEMATICAL MODEL OF SPUNBONDING POLYMERS Our air drawing model of spunbonding polymers consists of a continuity equation, a momentum equation, an energy equation, a constitutive equation, a crystallization kinetics equation, and a birefringence equation;7−10 the surrounding air conditions (velocity and temperature) are considered as given functions of axial position,11−13 which are obtained by numerical simulation. In the previous literature,5,6,9 the density and specific heat capacity of polymer melt at constant pressure are considered to be constant. In fact, they vary with polymer temperature. In this work, we establish an air drawing model that differs from those of the others, considering the effects of density and polymer specific heat capacity of polymer melt at constant pressure changing with polymer temperature on fiber diameter. In this paper, an improved air drawing model of polymers in the spunbonding process was established. The effects of the processing parameters on the fiber diameter are investigated. The continuity, momentum equation, energy, constitutive, crystallization kinetics, and birefringence equations, along with the boundary conditions used for the study, are also discussed below. Continuity Equation. π Q = D2Vρf (1) 4 Where Q is the polymer mass flow rate, D is the filament fiber diameter, V is the filament fiber mean velocity in the spinning direction, and ρf is the specific density of the polymer, this quantity depends on temperature. As the polymer density varies with polymer temperature, the following correlation14 is also introduced: 1 ρf = (2) 1.145 + 0.000903 × T
Cf = β(Red)−n
Where Red is the Reynolds number based on the diameter of the fiber, β and n5,8 are the fitted constants, and the values of β and n should be 0.37 and 0.61, respectively. The empirical relationship expression of Reynolds number is as follows: Red =
ρa D|(Va − V )| μa
(5)
Where ρa is the density of the air, (Va − V) is the relative velocity between the moving filament fiber and air, and μa is the kinematic viscosity coefficient of the air. As the force on the filament fiber parallel to the air stream can then be found from the following relation: Fdrag =
1 ρ (Va − V )2 Cf πDL 2 a
(6)
Where Fdrag is the air drag force on the filament fiber, ρa is the air density, D is the filament fiber diameter, and L is the filament fiber length. Energy Equation. πDh(T − Ta) ΔHf dX dT =− + dz QCpf Cpf dz
(7)
5
Where T is the polymer temperature, Ta is the air temperature, h is the convective heat transfer coefficient, which will vary with axial position, Cpf5 is the specific heat capacity of the polymer melt, which will vary with temperature at the constant pressure, ΔHf is the heat of fusion of the polymer, and X is the degree of crystallinity. The first right-hand side term of eq 7 describes the decrease in the temperature due to heat loss by the fiber to the cooling medium and the second term represents the opposite effect that crystallization has on temperature due to the release of the latent heat of fusion. The value for the convective heat transfer coefficient can be calculated from the following relationship.
Where T is the polymer temperature. Momentum Equation. dFrheo 1 dV π = jπρa Cf (Va − V )2 D + Q − ρf D2g dz 2 dz 4
(4)
(3)
5
Where Frheo is the rheological force, z denotes the spinning axial direction, ρa is the air density, ρf is the polymer density, V is the filament fiber velocity, and Va is the air velocity. The drag results from the friction between the filament fiber and air when the two are moving with different velocities, because the high air suction in the chamber results in high axial air velocities; therefore, the relative velocity is used in the term instead of the actual velocity of the filament fiber. The term j5 is the sign carrier of the air drag force; j is −1 for Va > V and +1 for Va < V. Here, g is the gravitational acceleration, and Cf5 is the air
⎡
h = 1.47 ×
1/6 ⎡ ⎛ Vc ⎞2 ⎤ ⎢ ⎥ Kc = 1 + 8⎜ ⎟ ⎢⎣ ⎝ Va − V ⎠ ⎥⎦
11062
⎤0.256
(V − V ) 10−4⎢ aπ 2 ⎥ ⎢⎣ ⎥⎦ D 4
Kc (8)
(9)
dx.doi.org/10.1021/ie3019593 | Ind. Eng. Chem. Res. 2013, 52, 11061−11069
Industrial & Engineering Chemistry Research
Article
Where h5,16 is the convective heat transfer coefficient, Kc is the correction factor, and Vc is the side blow air velocity. The correlation of the specific heat capacity of the polymer melt changing with the polymer temperature14 is as follows: Cpf = 0.3669 + 0.00242T
F = Fasp + Fgrav − Fdrag − Finertia
Where F is the force on the filament fiber, Fasp is the tension exerted by the aerodynamic drawing device, Fgrav is the gravitational force, Fdrag is the air drag on the filament fiber, and Finertia is the nertial force.
(10)
Constitutive Equation. In this work, the simplest constitutive equation, Newtonian fluid relationship is used because research shows that the temperature dependence on viscosity is the most dominant effect while the type of the constitutive equation is secondary;11,12 therefore, for the sake of computational simplicity, the Newtonian fluid constitutive equation is introduced in our model:
Frheo =
σ=
π 2 dV Dη 4 dz
3. EXPERIMENTAL SECTION 3.1. Numerical Methods for Solving the Air Jet Flow Field Model of Spunbonding. The air jet flow field model is solved by using the finite difference method. The SIMPLE (semi-implicit method for pressure-linked equation) algorithm19 is utilized to solve the problem of velocity pressure couple and the staggered grid is presented to avoid tooth-like distributions of velocity and pressure. The preferred difference scheme for space independent variables is the second-order upwind difference scheme and the TDMA (tri-diagonal matrix algorithm) method19 is used to solve the difference equations. With the help of numerical simulations of the air jet flow field, we can determine the distributions of the z-component of air velocity Va and air temperature Ta along the axial position z. Then we can solve the air drawing model of polymers using a fourth-order Runge−Kutta method. The computational domain is rectangular where the coordinate origin is in the center of the flat narrow slot passage of drafting assembly. Lengths of the z- and y-direction of the computational domain are 875 and 60 mm, respectively. There are 2350 grids in the z-direction and 180 grids in the ydirection. 3.2. Materials Preparation and Process Parameters. The polymer used in the experimental runs is YS-835 polypropylene (PP) pellet with a melt flow index (MI) of 35 g/min. With the aid of numerical simulations of the air jet flow field, the air drawing model of spunbonding polymers can be solved by using a fourth-order Runge−Kutta method. The experiments of spunbonding nonwoven equipment were carried out at our university. The spunbonding process parameters of concern were as following: the spinneret hole diameter,, the initial air velocity, the initial air temperature, the polymer throughput rate, polymer melt temperature, primary air temperature, quench pressure, venturi gap, and blower speed, and the variation ranges of the parameters are 0.45 mm, 45−65 m/s, 11.4−18.2 °C, 0.10−0.56 g/min·hole, 210−330 °C, 4.4−18.8 °C, 100−360 Pa, 10−42 mm, and 2150−2850 rpm, respectively. To condense the discussions and comparison, a group of fundamental parameters was assumed during the computations: polymer throughput rate is 0.16 g/min·hole, polymer melt temperature is 305 °C, primary air temperature is 13.2 °C, venturi gap is 25 mm, quench pressure is 230 Pa, and blower speed is 2450 rpm, etc., when one processing parameters was varied, the fundamental values of the other process parameters were kept unchanged. 3.3. Test Methods and Measurement. The image analysis method was employed to measure the fiber diameter. The images of nonwoven samples were acquired with a Questar three-dimensional video frequency microscope (Questar Corp., New Hope, PA) with an enlargement factor of 600 and a depth of focus of 1 mm and then processed with Image-Pro Plus image analysis software (Media Cybernetics, Inc., Silver Spring, MD) to measure the fiber diameter. The image processing includes enhancement, smoothing, binarization, and filtering. The fibers of the spunbonding nonwoven material are regarded
(11)
Frheo (πD2 /4)
(12)
η = 48.71 exp[2650/(T + 273)]exp[4(X /X∞)2 ]
(13)
Where η is the shear viscosity of the air. Crystallization Kinetics Equation. The variation of the degree of crystallinity along the spinline is given by the following equation. 5
⎤ ⎡ d ⎛ X ⎞ n1k ⎛ X ⎞⎢ ⎛ ⎛ X ⎞⎞⎥ ⎟⎟⎟ ⎟ ln⎜⎜1/⎜1 − ⎜1 − ⎟= ⎜ dz ⎝ X∞ ⎠ V ⎝ X∞ ⎠⎢⎣ ⎝ ⎝ X∞ ⎠⎠⎥⎦
n1− 1/ n1
(14)
X∞5
5
Where is the maximum crystallinity, X/X∞ is the relative crystallinity, the ratio of the absolute crystallinity over the ultimate crystallinity of material, n15 is the Avrami index, and k5 is the crystallization rate. Birefringence Equation. Δnam = Copσ
(15)
⎛ X ⎞ X f Δc 0 Δn = ⎜1 − ⎟Δnam + X∞ ⎠ X∞ c ⎝
(16)
fa = Δnam /Δa 0
(17)
Δa0 13−15
13−15
Where Cop is the stress optical coefficient, is the intrinsic amorphous value, Δ c 0 13−15 is the crystalline birefringence value. Boundary Conditions. V (0) = V0 , T (0) = T0 , 5
F(0) = F0 ,
D(0) = D0 ,
Frheo(L) = 0
(19)
5−7
X(0) = 0, (18)
5
Here L is the chamber length, F0 is the initial rheological force of the polymer melt, V05 is the initial velocity of the polymer melt, D0 is the initial diameter of the polymer melt, and T05 is the initial temperature of the polymer melt. The freezing-point5,19,20 is defined as the boundary condition where the rheological force is considered to be the sum of the cumulative gravitational and air drawing force acting upon the frozen part of the fiber. Beyond the freezing point, the fiber diameter keeps constant until the fibers are laid down on a web screen. Dynamic Balance Equation. 11063
dx.doi.org/10.1021/ie3019593 | Ind. Eng. Chem. Res. 2013, 52, 11061−11069
Industrial & Engineering Chemistry Research
Article
Figure 2. Computational domain and boundary conditions.
as cylinders because their cross sections are nearly round. Twenty fibers are chosen to measure their diameters in each grid, so altogether there are 200 fibers to be measured in 10 grids. The mean value of the diameters of 200 fibers was considered as the fiber diameter of the polypropylene (PP) nonwoven sample. The experimental samples were subject to conditioning at 65% RH and 20 ± 5 °C for 24 h, and then, the testing was carried out. The samples size of the nonwoven fabrics used (produced) on the university’s equipment was 1 m × 1 m, as there are random irregularities of fiber diameter in spunbonding nonwoven material, samples have to be taken randomly on an area of 1 m × 1 m.The length and width of the nonwoven fabrics are divided into ten equal parts, thus dividing the nonwoven fabric into 100 grids. We developed a random number generator program that can generate a group of ten random integers from 1 to 100 automatically each time it runs. One group of random integers is selected randomly, corresponding to the grid number. The corresponding grids of the samples are taken out. As a result, random sampling from the 1 m × 1 m nonwoven fabric is realized and the sample quantity is 10. 3.4. Drafting Assemly Parameters. The flat narrow slot passage of the drafting assembly is shown in Figure 2. The drafting assembly parameters are the steady flow segment length L1 = 40 cm, the jet orifice length L2 = 10 cm, the drafting segment length L3 = 27.5 cm, the outlet orifice length L4 = 10 cm, the drafting segment width L5 = 30 cm, the jet orifice highness L6 = 6 cm, and the drafting segment highness L7 = 3.333./2.333 cm (h1/h2).
⎡ 0 ⎤ ⎢ ⎥ Sϕ = ⎢−∇p+S ⎥ ⎢⎣ S ⎥⎦ T
Where ρa is the air density, u is the velocity vector, φ stands for the conserved property, Γφ is the diffusion coefficient, Sφ is the source term for the variable φ, T is the temperature, μ is the dynamic viscosity, k is the heat conductivity, c is the specific heat of the fluid, p is the pressure, S is the source term, and ST represents the viscosity disssipation rate. And to complete the system of the equations, the ideal gas equation is used to close the equation system: p = ρa RT
(21)
Where R is the gas constant. 4.2. Air Jet Flow Field Theoretical Model of the Spunbonding Process. The air jet flow field of the spunbonding process is considered to be a steady and inviscid flow. The flow field is assumed to be two-dimensional. As the k−ε model is the most widely used turbulence model in engineering computations in the case of high Reynolds numbers, it is taken as the preferred turbulence model. The air flow field of model consists of a continuity equation, a momentum equation, an energy equation, a turbulent kinetic energy equation, and a turbulent dissipation rate equation.16−18 In this paper, we adopted a k−ε standard model for computation. Below are some of the details, values of the constants of standard k−ε model are Cu = 0.09, Cε1 = 1.44, and Cε2 = 1.92.19 The turbulent Prandtl numbers are as follows: σt = 0.9, σk = 1.0, and σε = 1.3,19 where σt, σk, and σε are Prandtl numbers of turbulence, turbulent kinetic energy, and turbulent dissipation rate, respectively. Continuity Equation.
4. NUMERICAL SIMULATION OF THE AIR FLOW FIELD MATHEMATICAL MODEL IN THE SPUNBONDING PROCESS Because the fiber diameter of the spunbonding process is intensely affected by the air jet flow field, in this paper, we established a model of air jet flow field to simulate the flow field and solved numerically via the finite difference method. The distribution of the centerline z-component of air velocity is demonstrated. 4.1. Governing Equations. Since high-velocity compressed air jet flow is forced into the drafting assembly (attenuator) through the attenuator injectors from the air reservoirs device, its Reynolds number (Re = 10 000) is large, so an inviscid perfect gas flow in the abstence of body forces is considered. In addition, steady adiabatic flow is also assumed since the air drawing process occuring in the drafting assembly is very short. Therefore, the generalized governing equations19 are expressed as follows: div(ρa uϕ − Γϕ grad ϕ) = Sϕ
⎡ 0 ⎤ ⎢ ⎥ Γϕ = ⎢ μ ⎥ ⎢⎣ k /c ⎥⎦
⎡1⎤ ϕ = ⎢u⎥ ⎢ ⎥ ⎣T ⎦
∂(ρa Va) ∂z
+
∂(ρa Ua) ∂y
=0
(22)
Where Va is the z-component of the air velocity, Ua is the ycomponent of the air velocity, ρa is the air density, z is the axial position, and y is the transversal position. Momentum Equation in the z-Direction. ∂(ρa VaVa) ∂z
+
∂(ρa VaUa)
∂y ⎡ ⎡ ∂(ρ Va) ⎤⎤ ∂P ∂ = − a + 2 ⎢(μa + μt )⎢ a ⎥⎥ ∂z ∂z ⎢⎣ ⎣ ∂z ⎦⎥⎦
(20)
+
In which 11064
⎡ ∂(Va) ∂(Ua) ⎤⎫ ∂⎧ ⎨(μa + μt )⎢ + ⎥⎬ ∂y ⎩ ∂z ⎦⎭ ⎣ ∂y ⎪
⎪
⎪
⎪
(23)
dx.doi.org/10.1021/ie3019593 | Ind. Eng. Chem. Res. 2013, 52, 11061−11069
Industrial & Engineering Chemistry Research
Article
Here, μt = Cuρa(ka2/εa). Where, Pa is the air pressure, μa is the kinetic viscosity of air, μt is the turbulent viscosity of air, Cu is the constant of k−ε model, g is the gravitational acceleration, εa is the turbulent dissipation rate of air, and ka is the dissipation rate of turbulent kinetic energy of air. Momentum Equation in the y Direction. ∂(ρa UaVa) ∂z
+
system centerline. So, the plane is chosen as the computation area, the following boundary conditions correlation is introduced. (1) The conditions of upstream sections without the inlet are Va = 0,
∂(ρa UaUa)
=
⎪
⎪
⎪
⎪
k= (24)
∂z =
∂y
Va = 0,
∂(ρa kaUa)
k = 0, (30)
∂T = 0, ∂y
∂k = 0, ∂y
∂ε = 0, ∂y
⎛ ∂U ⎞ ⎛ ∂V μt ⎡ ⎛ ∂Va ⎞2 ∂U ⎞ ⎢2⎜ ⎟ + 2⎜ a ⎟ + ⎜ a + a ⎟ ⎥ ρa ⎢⎣ ⎝ ∂z ⎠ ∂z ⎠ ⎥⎦ ⎝ ∂y ⎠ ⎝ ∂y
∂Va = 0, ∂z ∂T =0 ∂z
2⎤
∂Ua = 0, ∂z
∂k = 0, ∂z
∂ε = 0, ∂z (32)
4.4. Comparison of Theoretical Results with Experimental Data. Experiments are carried out on the air jet flow field of the spunbonding attenuator shown in Figure 1. In order to testify the air flow field model, we measured and predicted the effects of spunbonding process parameters on fiber diameter. Particle image velocimetry (PIV-2100), produced by Denmark Dantec Inc., is utilized to measure the air velocity. In the application of PIV-2100, the air was seeded with oil soot (lampblack) aerosol particles in order for the laser light to be scattered and measured. Experiments are carried out on the flow field of spunbonding. The air velocity is 15 m/s. The initial air temperature is kept unalterable in the measurements. The flat narrow slot passage of drafting assembly is shown in Figure 1. The distributions of the centerline z-component of air velocity along the z-axis are demonstrated in Figure 3. The experiment data are represented by black dots, and the numerical computation results are illustrated by red dots. It can be found that the numerical computation results obtained with the air jet flow field model match well with the experimental data, which prove that the air jet flow field model can be used to predict the flow field of the spunbonding attenuator. We can see from Figure 3 that the z-components of the air velocity change gradually (by degrees) along the z-axis after the air ejects from inlet to outlet of the spunbonding drafting assembly. At z-positions (71.75 cm) of the
(26)
∂(ρa εaUa) ∂y
μ ⎞ ∂(ε ) ⎤ μ ⎞ ∂(ε ) ⎤ ∂ ⎡⎛ ∂ ⎡⎛ ⎢⎜μa + t ⎟ a ⎥ + ⎢⎜μa + t ⎟ a ⎥ σε ⎠ ∂z ⎥⎦ ∂y ⎢⎣⎝ σε ⎠ ∂y ⎥⎦ ∂z ⎢⎣⎝ ε + ρa (Cε1Pk − Cε2εa) a ka
(31)
(5) The outer boundaries conditions (y-direction) are
Where σk is the Prandtl number of turbulent kinetic energy, εa is the turbulent dissipation rate of air, and ka is the turbulent kinetic energy of air. Equation of Dissipation Rate of Turbulent Kinetic Energy.
=
T = TW ,
Ua = 0
∂y
2
∂z
(29)
Ua = 0,
∂Va = 0, ∂y
Here,
+
k3/2 , l
(4) The centerline conditions (z-direction) are
+ (Pk − εa)ρa
∂(ρa εaVa)
ε = Cu3/4
ε=0
(25)
μ ⎞ ∂(ka) ⎤ μ ⎞ ∂(ka) ⎤ ∂ ⎡⎛ ∂ ⎡⎛ ⎢⎜μa + t ⎟ ⎥+ ⎢⎜μa + t ⎟ ⎥ σk ⎠ ∂z ⎥⎦ ∂y ⎢⎣⎝ σk ⎠ ∂y ⎥⎦ ∂z ⎢⎣⎝
Pk =
3 2 (V0 + U0 2), 2
Ta = T0 ,
Here, N = [2L5L6/(L5 + L6)]. Where, V0 is the z-component of the initial air velocity, U0 is the y-component of the initial air velocity, T0 is the initial air temperature, Cu = 0.09, L5 is the drafting segment width, L6 is the jet orifice highness, and N is the inlet characteristic dimension. The conditions of the wall sections are
Where σt is the Prandtl number of turbulence. Turbulent Kinetic Energy Equation. +
Ua = U0 ,
l = 0.07N
∂(ρa TaUa)
μ ⎞ ∂(Ta) ⎤ μ ⎞ ∂(Ta) ⎤ ∂ ⎡⎛ ∂ ⎡⎛ ⎢⎜μa + t ⎟ ⎥+ ⎢⎜μa + t ⎟ ⎥ σt ⎠ ∂z ⎥⎦ ∂y ⎢⎣⎝ σt ⎠ ∂y ⎥⎦ ∂z ⎢⎣⎝
∂(ρa kaVa)
(28)
Va = V0 ,
Where Pa is the pressure of the air, ρa is the density of the air, μa is the kinetic viscosity of the air, and μt is the turbulent viscosity of the air. Energy Equation. ∂z
k = 0,
(2) The conditions of upstream sections with the inlet are
⎡ ∂(Va) ∂(Ua) ⎤⎫ ∂⎧ ⎨(μa + μt )⎢ + + ⎥⎬ ∂z ⎩ ∂z ⎦⎭ ⎣ ∂y
+
∂Ta = 0, ∂y
ε=0
∂y ∂P ∂(Ua) ⎤ ∂⎡ = − a + 2 ⎢(μa + μt ) ⎥ ∂y ∂y ⎣ ∂y ⎦
∂(ρa TaVa)
Ua = 0,
(27)
Where σε is Prandtl number of dissipation rate of turbulent kinetic energy, Cu, Cε1, and Cε2 are the constants of turbulent model, respectively. 4.3. Boundary Conditions of the Air Flow Field Model. Figure 2 shows the computational domain and boundary conditions; the flat narrow slot passage of drafting assembly is shown in Figure 1. As the flow field is symmetrical along the 11065
dx.doi.org/10.1021/ie3019593 | Ind. Eng. Chem. Res. 2013, 52, 11061−11069
Industrial & Engineering Chemistry Research
Article
drawing model of polymer in spunbonding process established in our study.
6. EFFECTS OF AIR PARAMETERS ON THE FIBER DIAMETER 6.1. Relationship between the Initial Air Temperature and Fiber Diameter. Figure 5 shows the effect of the initial
Figure 3. Distributions of centerline z-component of the air velocity along the z-axis. Figure 5. Effects of the air initial temperature on the fiber diameter.
spunbonding attenuator, the highest air velocity is attained, and at z positions very near the outlet of the spunbonding drafting assembly, the air velocity decreases rapidly along the z-axis. After the air ejects from the outlet of the drafting assembly, the lowest air velocity is attained. Figure 4 is the vector distributon of air velocity in the air jet flow field. We can see from the figure that along the narrow region of the spunbonding attenuator centerline, especially at a 71.75 cm distance from the spunbonding attenuator, the air velocity is the highest, and in the region a very long distance from the spinneret hole, the air velocity is much higher than in other regions, so the polymer melt is drawn rapidly in this region.
air temperature on fiber diameter which changes with the polymer melt temperature. The initial air temperature is 11.4, 14.8, and 18.2 °C, in turn from top to bottom. As can be seen, the higher the initial air temperature, the finer the fiber diameter will be; higher initial air temperature will cause fibers to be attenuated much further. When the initial air temperature is increased to 18.2 °C, the final fiber diameter is 23.6% smaller than that of 11.4 °C. This is primarily due to the fact that the initial air temperature increased first, the air drawing force increased, and the degree of drawing increased which yielded a finer fiber diameter. Second, when initial air temperature increased, the filament fiber cooled more slowly along the spinline, the drawing (tensile) time of polymer extended (lengthened) which result in a finer fiber diameter. Third, when the initial air temperature increased, the viscosity and stress decreased which produced a finer fiber diameter. 6.2. Relationship between the Initial Air Velocity and Fiber Diameter. Figure 6 gives the initial air velocity on fiber
5. NUMERICAL SIMULATION OF THE AIR DRAWING MODEL OF POLYMERS OF THE SPUNBONDING PROCESS With the help of numerical simulations of the air jet flow field, the distributions of the z-component of air velocity can be found. The air drawing model of spunbonding polymer can be solved by using a fourth-order Runge−Kutta method. Experiments are performed on spunbonding at our university. The spinneret hole diameter is 0.45 mm, the initial air velocity is 55 m/s, the initial air temperature is 14.8 °C, the polymer throughput rate is 0.32 and the initial polymer melt temperature is 290 °C. When one processing parameter varies, the other fundamental values are kept fixed and are the fundamental values. The image analysis method is employed to measure the fiber diameter. Diameters of 200 randomly chosen fibers were averaged to obtain the mean fiber diameter. The measured fiber diameter is 22.421 μm, and the computed fiber diameter is 21.934 μm. The measured value coincides with the numerical computation result well, which proves the reliability and accuracy of the air
Figure 6. Effects of the air initial velocity on the fiber diameter.
Figure 4. Vector distribution of air velocity in the flow field. 11066
dx.doi.org/10.1021/ie3019593 | Ind. Eng. Chem. Res. 2013, 52, 11061−11069
Industrial & Engineering Chemistry Research
Article
Figure 7. Relationship between fiber diameter, polymer melt temperature, fiber crystallinity, fiber birefringence, and polymer spinning linear velocity and distance from spinneret with polymer throughput rate.
In conclusion, all our results show that the air drawing model and the air jet flow field model of the polypropylene polymer we established are satisfactory, which verifies the reliability of these models. 6.3. Effects of Spunbonding Nonwoven Processing Parameters on Fiber Performance. For further analysis, we can see the profiles of these properties in figures. The parts of Figure 7 show the relationship between fiber diameter, polymer melt temperature, fiber crystallinity, fiber birefringence, and polymer spinning linear velocity and distance from spinneret
diameter which changing with the polymer melt temperature. The initial air velocities of 45.0, 55.0, and 65.0 m/s are considered. As Figure 3 shows, the higher the initial air velocity, the finer the fiber diameter; higher initial air velocities will cause fibers to be attenuated much further. When the initial air velocity is increased to 65.0 m/s, the final fiber diameter is 36.3% smaller than that of 45.0 m/s. This is mainly attributed to the fact that the initial air velocity increased, the air drawing force increased, and the degree of drawing increased which yielded a finer fiber diameter. 11067
dx.doi.org/10.1021/ie3019593 | Ind. Eng. Chem. Res. 2013, 52, 11061−11069
Industrial & Engineering Chemistry Research
Article
polymer temperature have critically important effects on fiber diameter. The model predictions can be improved remarkably by considering the influences of polymer temperature on the density and specific heat capacity of polymer melt at constant pressure. Therefore, the variation in density and specific heat capacity of polymer melt at constant pressure with the polymer temperature has very important effects on fiber diameter. It has been concluded that the predicted fiber diameters are much closer to the experimental results, which further confirms the effectiveness of the polymer drawing model established in this paper. From the above analyses, it can be seen that the fiber diameter is directly related to the polymer throughput rate, polymer melt initial temperature, air initial temperature, and air initial speed. The finer the geometric mean of fiber diameter, the more uniform the fiber web.
with polymer throughput rate, as presented by the air drawing model, shown as Figure 7. Figure 7a shows the relationship between fiber diameter and distance from spinneret with polymer throughput rate. This figure shows that the fiber diameter draws down slower with increasing polymer throughput rate. Higher polymer throughput rates also lead to higher fiber diameters. This is primarily due to in this situation each polymer throughput rate has its own air speeds, the axial air flow controls the drag over the entire spinline and also affects the heat loss from fiber. Figure 7b shows the polymer melt temperature profiles for different polymer throughput rates. This figure shows that the cooling rate is lower for higher polymer throughput rates, which is also observed in conventional melt spinning. This is attributed to the fact that the increase in stress is at least partly due to the changing air flow conditions with changing polymer throughput rate. Figure 7c shows the relationship between fiber crystallinity and distance from spinneret with polymer throughput rate. This figure indicates that crystallinity also decreases with increasing polymer throughput rate and once again high polymer throughput rate curves tend to approach each other away from the spinneret. Figure 7d shows the relationship between fiber birefringence and distance from spinneret with polymer throughput rate. The predictions of fiber birefringence are shown in Figure 7d where it is clear that the birefringence is decreasing with increasing polymer throughput rate. Note however that, at high polymer throughput rates, the curves tend to approach each other down the spinline. Figure 7e shows the relationship between polymer spinning linear velocity and distance from spinneret with polymer throughput rate. The polymer spinning linear velocity profiles in the fifth subfigure of Figure 7 indicate that the final velocity increases with increasing polymer throughput rate and the velocity tends to stabilize at a higher distance from the spinneret, which is in agreement with the diameter profiles.
8. CONCLUSIONS The air drawing model of polypropylene (PP) polymer and the model of the air jet flow field in the spunbonding process are founded. We numerically simulated the air jet flow field model with the finite difference method. The computation results of the distributions of the z-components of the air velocity along the spinline during the spunbonding process are in good accord with the experimental data. It can be concluded that variation in the density and specific heat capacity of the polymer melt at constant pressure and temperature has a great effect on the fiber diameter. Therefore, the model predictions can be improved dramatically when the influences of the polymer temperature on the density and specific heat capacity of the polymer melt at constant pressure are considered. Compared with the existing drawing model, the new ones include more processing parameters and are more accurate. The newly developed formulas are introduced into the spunbonding air drawing model to predict the fiber diameter of nonwoven webs. The results predicted are in quite good agreement with the data actually measured, which shows that the new air drawing model is effective and excellent. It is found that a higher air initial speed and a higher air initial temperature can all yield finer fibers. The predicted results coincide well with the actually measured data, which reveal that these models are accurate and also show that this area of research has great potential in the field of computer assisted design in spunbonding nonwoven process and technology.
7. COMPARISON OF IMPROVED MODEL WITH PRIMARY MODEL Table 1 shows the fiber diameter changing by considering the polymer density and the specific heat capacity of polymer melt Table 1. Comparison of Model Predictions between Improved Model and Primary Modela improved model
■
primary model
measured diameter (μm)
predicted diameter (μm)
prediction error (%)
measured diameter (μm)
predicted diameter (μm)
prediction error (%)
22.298 21.382 22.209 23.068 23.119
21.604 20.698 22.986 23.801 22.268
3.11 3.20 3.50 3.18 3.68
22.298 21.382 22.209 23.368 23.119
21.019 20.256 21.042 22.076 21.867
5.74 5.27 5.25 5.51 5.53
a
AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected] or
[email protected]. Notes
The authors declare no competing financial interest.
■
ACKNOWLEDGMENTS The authors would like to thank professors Zhenhua Lu and Yingzheng Liu of Turbo machinery Institute in Shanghai Jiao Tong University, for the assistance in the design and construction of the experimental apparatus. The permission by Phoenics Inc. to use their software with an educational license is gratefully acknowledged.
There are 100 fiber measurements averaged in Table 1.
■
at constant pressure varying with the polymer temperature. As can be seen, the predictions given by the two models are not the same; the two models are very different from each other. It can be concluded that the variations in density and the specific heat capacity of polymer melt at constant pressure with the 11068
NOMENCLATURE Q = polymer mass flow rate, kg/s D = filament fiber diameter, mm dx.doi.org/10.1021/ie3019593 | Ind. Eng. Chem. Res. 2013, 52, 11061−11069
Industrial & Engineering Chemistry Research
Article
V = filament fiber velocity, m/s Va = air velocity, m/s g = gravitational acceleration, g/s2 Cpf = specific heat capacity of melt, J/kg·K h = convective heat transfer coefficient, W/m2·K T = polymer temperature, °C Ta = the air temperature, °C j = sign carrier of the air drag force; j is −1 for Va > V and 1 for Va < V.
(20) Wang, X. M. Computational simulation of the fiber movement in the melt-blowing process. Ind. Eng. Chem. Res. 2005, 44 (11), 3912−3917.
Greek Symbols
ρf = polymer density, kg/m3 ρa = air density, kg/m3 η = the shear viscosity of air, Pa·s
■
REFERENCES
(1) Malkan, S. R. An overview of spunbonding and meltblowing technologies. Tappi 1994 Nonwovens Conference, Orlando, FL, Feb 14−16, 1994; pp 31−37. (2) Lu, F. M. Fundamentals of ason spunbond technologyan innovation to success. Tappi 1998 Nonwovens Conference, St. Petersburg, FL, March 9−11, 1998; pp 125−132. (3) Bhat, G. S.; Malkan, S. R. Extruded continous filament nonwovens: advances in scientific aspects. J. Appl. Polym. Sci. 2002, 83 (3), 572−585. (4) Smorada, R. L. Spunbonded technology: an historical perspective. INDA J. Nonwovens Res. 1991, 3 (4), 26. (5) Malkan, S. R.; Wadsworth, L.; Davey, C. Parametric studies of the recofil spunbonding process. Int. Nonwovens J. 1994, 6 (2), 24−50. (6) Chen, C. H.; White, J. L.; Spruiell, J. E. Dynamics air drag and orientation development in the spunbonding process for nonwoven fabrics. Textile Res. J. 1983, 53 (1), 44−51. (7) Beyreuther, R.; Malbome, H. J. Spunbonded nonwovens linking innovative polymer, technological and textile research. Melliand Textilber. 1993, 74 (4), 287−289. (8) Hajji, N.; Spruiell, J. E.; Lu, F. M.; Malkan, S. Modeling of the reicofil spunbonding process. INDA J. Nonwovens Res. 1992, 4 (2), 16−21. (9) Misra, S.; Spruiell, J. E.; Richeson, G. C. Investigation of the spunbonding process via mathematical modeling. INDA J. Nonwovens Res. 1993, 5 (3), 13−19. (10) Gagon, D. K.; Dean, M. M. Computer simulation of steady polymer melt spinning. Polym. Eng. Sci. 1981, 21 (13), 844−853. (11) Abbott, L. E.; White, J. L. Melt spinning of high density and low density polyethylene fibers, in development of orientation and crystallinity, and mechanical properties of span fiber. Appl. Polym. Symp. 1973, 20, 247−268. (12) Bankar, V. G.; Spruiell, J. E.; White, J. L. Melt spinning dynamics and rheological properties of nylon6. Appl. Polym. Sci. 1977, 21, 2135−2155. (13) Smith, A. C.; Roberts, W. W. Computational modeling of fiber formation in polypropylene spunbonding with crystallization: comparison with experiments. Int. Nonwovens J. 1994, 61 (1), 31−41. (14) Salem, D. R. Structure formation in polymeric fibers; Carl Hanser Verlag: Munich, Germany, 2001. (15) Majumdar, B. Air drag on filament in the melt blowing process. J. Rheol. 1990, 34, 591−601. (16) Matsui, A. Air drag on a continuous filament in melt spinning. Trans. Soc. Rheol. 1976, 465−473. (17) Wei, T. Y. Studies on the processing of PET meltblown nonwovens and its air drawing mathematics model. Master Thesis, Donghua University, Shanghai, China, 2002. (18) Chen, T. Study on the air drawing in melt blowing nonwoven process. Ph. D. Dissertation, Donghua University, Shanghai, China, 2003. (19) Edinburgh, G.; Harlow, G. Computational fluid dynamics; Pearson Education Limited: New York, 1996. 11069
dx.doi.org/10.1021/ie3019593 | Ind. Eng. Chem. Res. 2013, 52, 11061−11069
