Melt spinning of continuous fibers by cold air attenuation: II. Theoretical modeling

Abstract

A modified melt spinning apparatus with a high-speed air nozzle was designed and fabricated to produce continuous polypropylene fibers by cold air drawing only. The experimental studies based on this novel process were described in Part I of this paper series. In this succeeding paper, a one-dimensional non-isothermal Newtonian fluid model is formulated that can be used to analyze the fiber attenuation mechanisms. Unlike melt spinning, the position of the model geometry for the lower boundary is undetermined due to the unknown position of the fiber freezing point. In this model, a method of iteration on the initial model geometry was used to determine the fiber freezing point at which the fiber temperature reaches its freezing point and the velocity gradient equals zero. The results showed good agreement between simulated results and experimental data for the relationship between processing conditions and resulting fiber diameter.

Keywords

Melt spinning continuous fibers cold air attenuation polypropylene non-isothermal flow freezing point

As discussed in Part I of this paper series,¹ ultra fine, continuous fibers can be produced by cold air drawing only, and the experimental results show that the final fiber diameter depends on the polymer viscosity, processing temperature, and polymer and air volume flow rates. It is therefore desirable to formulate a theoretical model to predict the fiber diameter based on processing conditions and material properties. This novel process is modified from melt blowing and melt spinning processes. In both melt spinning and melt blowing, the polymer melt is extruded out of the extrusion orifice to form a fiber, which is immediately attenuated by external forces. In melt spinning, the external force is the drag force applied by a drawing roller. The air surrounding the fiber applies a retarding force on the fiber. In melt blowing, the external force is provided by a high-speed air stream interacting with the fiber surface. Numerical models for melt spinning with air flow as the retarding force have been available for some decades. The models for melt blowing were mostly derived from the same governing equations by applying a forwarding air drag force and different boundary conditions. In this paper, the background on melt spinning modeling is introduced first, and a comparison between melt spinning and melt blowing modeling is then presented.

Earlier studies on melt spinning modeling mainly focused on how the fiber is cooled along the spinline during the attenuation process. Andrews² established a group of heat flow equations and solved them approximately with the aid of empirical data. With this model, he predicted the temperature profile as a function of the distance from the spinneret for both the surface and the centerline of the fiber. The results were quite accurate, although the predicted radial temperature distribution was only correct to within 20%. A similar work in this field was contributed by Kase and Matsuo,^3,4 who developed fundamental equations on the dynamics of melt spinning. Steady-state and transient solutions of these fundamental equations were derived and compared with experimental results. One important contribution of their work is the correction of the heat transfer coefficient for the melt spinning process. They experimentally measured the heat transfer coefficient by hanging a heated wire with diameter of 0.2 mm in an air flow. By recording and converting the cooling of the wire, the Nusselt number (Nu) was fitted with the Reynolds number (Re_D) and was compared with other research results. The data fitting yielded a relation of Nu = 0.764Re_D^0.38. The air drag coefficient is an important parameter to describe the air drag force acting on the fiber during melt spinning and melt blowing. Matsui⁵ developed a relationship between the air drag coefficient C_f and the Reynolds number Re_D based on the turbulent boundary layer theory:

C f = β {Re}_{D}^{- n}

(1)

Re D = \frac{ρ a UD}{μ a}

(2)

where β and n are constant coefficients, ρ_a is the air density, μ_a the air viscosity, U the fiber velocity, and D the fiber diameter. With β = 0.37 and n = 0.61, a satisfactory agreement between theoretical analysis and this empirical relation was observed for spinning speeds in the range of 300–6000 m/min in melt spinning. Majumdar and Shambaugh⁶ designed an instrument to experimentally measure the air drag force under a wide range of conditions. The fiber diameter varied from 13 to 305 µm and the air velocity changed from 36 to 300 m/s. The experimental results were found to fit well with an empirical relation:

C f = 0.78 {Re}_{D}^{- 0.61}

(3)

Based on Kase and Matsuo’s effort, more recent studies on modeling of melt spinning expanded the one-dimensional (1D) model into two-dimensional (2D) models. These 2D models also included the combined effects of flow-induced crystallization, molecular orientation, non-isothermal viscoelasticity, and air drag.

Compared with melt spinning, melt blowing is more difficult to simulate since its lower boundary is moving and it needs to be determined during the simulation. Extensive modeling studies were conducted by Shambaugh’s group.^7–9 Their first attempt⁸ was to develop a 1D model based on continuity, momentum, and energy equations. These governing equations are essentially the same as those used in melt spinning simulation. The only difference is in the air drag force in the momentum equation, which plays a dominant role on fiber attenuation in melt blowing. For the boundary conditions, the upper boundary is identical to that of melt spinning, where a constant known value of fiber velocity and temperature are imposed. A “freeze point” method was used in their study to define the lower boundary condition. The position of the fiber freeze point was specified at z = 5 cm where the attenuation of fiber becomes negligible. It was claimed that the selection of z = 5 cm gives good fit to the data and a better converged solution. Compared with the experimental data, this model provided an excellent prediction of fiber diameter profile in the region of high fiber attenuation with a slight over prediction on the final fiber diameter. The Phan-Thien–Tanner viscoelastic model was also applied in this model to compare with the Newtonian fluid model, but little difference was found between these two constitutive models. It was pointed out by Shambaugh et al. that the discrepancy in the predicted final fiber diameter from the 1D model was due to fiber vibration at distances far away from the spinneret. To consider this vibration effect, Rao and Shambaugh⁹ developed a 2D melt blowing model. The previous “freeze point” method cannot be applied to the lower boundary condition since this 2D model included fiber vibration. Therefore, a new method called “stop point” was assumed. At this “stop point”, the fiber velocity is equal to the air velocity. The predictions based on these two different methods were virtually identical and thus proved the accuracy of the “stop point” method. By comparing with the results from the 1D model, the new 2D model showed little differences in diameter or temperature predictions. Over prediction of fiber final diameter was observed and attributed to the under-predicted fiber vibration amplitudes. This model can be used to estimate the experimental conditions that cause fiber breakage.

As suggested from the literature survey, models for melt spinning can be applied to the simulation of the melt blowing process with two modifications: reversing the direction of the air drag force from “against” to “with” the polymer flow direction; and using a moving boundary condition for the lower boundary, by either the “stop point” method or the “freeze point” method. Based on the literature, the 1D model is satisfactory to simulate the melt blowing process, whereas the 2D models do not provide better predictions on the fiber diameter and temperature profile despite that they dramatically increase the computation time. For polypropylene, the Newtonian constitutive model can provide similar predictions as that using the Phan-Thien–Tanner model.

Model formulation

Model description

In this study, a 1D non-isothermal Newtonian fluid model was chosen to simulate the modified melt spinning process under cold air attenuation. Figure 1 describes the process and forces acting on the fiber. The air drag force applied on the fiber surface is in the same direction as the gravity force. These two forces are counterbalanced by the rheological force and inertia force. Since cold air was used in this process, forced convection heat transfer occurred on the interface between polymer melt and cold air.

Figure 1.

Schematic of the melt spinning process by cold air attenuation and force balance on a fiber segment.

Governing equations

The continuous fiber obeys mass conservation at any position beneath the spinneret orifice. The density of polymer only changes about 10% between the melt and the solid state. Neglecting the change in polymer density, mass conservation can be simplified to volume conservation so that the polymer volume flow rate remains constant anywhere:

π r 2 V f = Q

(4)

where r is the fiber radius, V_f the fiber velocity, and Q the polymer volume flow rate.

The differential form of the momentum equation can be formulated based on force balance:

\frac{d}{dz} [π r 2 (τ zz - τ rr)] + π (2 r) C f ρ a \frac{(V a - V f) 2}{2} - ρ f Q \frac{d V f}{dz} + π r 2 ρ f g = 0

(5)

where τ_zz and τ_rr are stresses in the z and r directions, respectively, C_f is the air drag coefficient, ρ_f the polymer density, ρ_a the air density, V_a the air velocity, and g the gravitational acceleration.

The air drag coefficient, C_f, is defined by Matsui:⁵

C f = β {Re}_{D}^{- n}

(6)

where Re_D, the Reynolds number, is given by:

Re D = \frac{ρ a V a (2 r)}{μ a}

(7)

where β and n are constant coefficients, and μ_a is the air viscosity. According to Majumdar and Shambaugh,⁶ β = 0.78 and n = 0.61 are appropriate values for melt blowing. These parameters should also be applicable to the process under consideration, since the air dynamics environment is similar.

A simplified steady-state energy equation is used in this study:

ρ f C pf V f \frac{d T f}{dz} = - \frac{4 h c}{2 r} (T f - T a)

(8)

where C_pf is the polymer heat capacity, T_f the fiber temperature, T_a the air temperature, and h_c the convective heat transfer coefficient. Because the forced convective heat transfer is dominant, the heat transfer effects resulted from conduction, radiation, and viscous heating are neglected.

The heat transfer coefficient, h_c, can be determined from the following correlation:

Nu = γ {Re}_{D}^{m}

(9)

where Nu, the Nusselt number, is defined as

Nu = \frac{h c (2 r)}{k a}

(10)

where k_a is the thermal conductivity of air, and Re the Reynolds number defined in Equation (7). The two fitting constants, γ and m, are equal to 0.764 and 0.38, respectively, according to Matsuo and Kase.³ These relations were developed based on experimental data in melt spinning for the parallel flow of gas along a cylinder. It can also be applied to the melt blowing process.⁸

Constitutive equations

For a Newtonian fluid, the stresses τ_zz and τ_rr are related to the polymer deformation rate as follows:

τ zz = 2 η \frac{d V f}{dz}

(11)

τ rr = - η \frac{d V f}{dz}

(12)

where η is the zero shear viscosity of the polymer. As a non-isothermal model, the temperature dependence of viscosity is considered using the Arrhenius law by the following equation:

For PP115

η = 0.002087 \exp (\frac{5530}{T f})

(13)

For PP12

η = 0.002946 \exp (\frac{6347}{T f})

(14)

where T_f is the fiber temperature in Kelvin. The coefficients in the two equations were obtained by fitting the experimental viscosity data given in Part I of this paper series.¹

Boundary conditions

Two boundary conditions can be specified for this model. The upper boundary located just beneath the die orifice can be defined as

V f = V f 0 at z = 0 T f = T f 0 at z = 0

(15)

The lower boundary is located at the position where the fiber solidifies and the fiber attenuation stops. Unlike the traditional melt spinning process whose lower boundary condition is fixed by the take-up speed, the lower boundary presented in this model is a moving boundary and needs to be determined during the calculation. Two conditions based on temperature and velocity are used to define the lower boundary:

\frac{d V f}{dz} = 0 at z = L s T f = T s at z = L s

(16)

where L_s is the stop point position, which is an unknown variable, and T_s is the fiber solidification temperature, which is equal to 135℃ for polypropylene in this study. At z = L_s, the fiber is cooled down to its solidification temperature T_s and the fiber velocity gradient in the z direction vanishes to mark the stop point for fiber attenuation. This lower boundary condition is appropriate to describe the fiber solidification phenomenon, since it determines the solidification point based on two variables, the temperature and the velocity gradient.

The selection of 135℃ as the solidification temperature was based on the results of differential scanning calorimetry (DSC) and rheological measurements. It was found that at 135℃ polypropylene started to solidify when cooled from melt in a parallel-plate rheometer. DSC results also indicated that polypropylene melt started to crystallize at this temperature. Although stress-induced crystallization may affect fiber freezing during melt spinning, this effect was neglected in this study for simplicity.

Computation method

The MATLAB built-in solver “bvp4c” was used to solve the system of three first-order ordinary differential equations. The iteration method described previously was used to simulate one case of the non-isothermal model of fiber attenuation. The material properties and processing conditions are listed in Table 1.

Table 1.

Material properties and processing conditions for non-isothermal case

Density (Kg/m³)	Viscosity (Pa s)	Thermal conductivity (W/K m)	Heat capacity (J/K kg)	Initial temperature (K)	Initial velocity (m/s)
PP115	900	0.002087e^5530/T	0.12	2000	513	0.005
Air	1.2	0.00002	0.027	1000	300	22

T is the polymer temperature.

As discussed previously, the model geometry length has a significant influence on the fiber velocity profile. Figure 2 shows the simulation results of a case study in which an initial value of 1 mm is used for the stop point position for the iteration. It shows that the fiber stops attenuation at a distance of 4 mm from the spinneret orifice and the final fiber diameter is 0.473 mm, which is far larger than the experiment result, 8 µm. In this calculation, the model geometry length L was assigned an initial value of 1 mm and then it was increased gradually with an increment of 1 mm. Using a short model geometry length (L = 1 mm) to start the iteration was responsible for the over prediction of the final fiber diameter. In the simulation, the fiber velocity gradient along the spinline initially increases to some maximum value and then decreases to converge to the lower boundary condition (dV_f/dz = 0). When a short model geometry length is used, the velocity gradient can only reach a very small maximum value and then it starts decreasing towards the value 0. In addition, the short model geometry length also predicts fast cooling on the fiber. The equation below shows the reorganized energy equation derived from Equation (8) with air drag force C_f and Reynolds number Re_D defined in Equations (6) and (7):

\frac{d T f}{dz} = - 0.764 \cdot \frac{π K a}{ρ f C pf Q} \cdot (\frac{2 ρ a V a}{μ a}) 0.38 \cdot r 0.38 \cdot (T f - T a)

(17)

The variables in this equation are r, T_f, and z. The remaining parameters are constant. It is obvious that a large value in r will draw a large temperature gradient, which makes cooling faster. With an assigned short model geometry length, the fiber is quickly cooled down to the solidification temperature, which stops the iterative calculation. Therefore, such an iteration method is not appropriate for this process and a relatively longer model geometry length is suggested.

Figure 2.

Model prediction of fiber diameter and temperature versus distance from the spinneret (processing temperature = 240℃, air velocity = 22 m/s, initial fiber velocity = 0.005 m/s and initial stop point position = 1 mm). The results do not converge correctly when using an initial value of 1 mm for the stop point position in the iteration.

To address this problem, a modified iteration method was developed. Firstly, an initial model geometry length L was set to 5 cm for the previous case. The selection of 5 cm was based on the study of Uyttendaele and Shambaugh,⁸ whose simulation results show that it provides a good fit and permits the solution to converge. Then a non-isothermal model is simulated based on this guessed model geometry length. The simulated fiber temperature profile is then used to identify the solidification point L_s. If this solidification point exactly falls on the starting position of the plateau region for the fiber diameter, it indicates that the initially guessed model geometry length is reasonable. This is because the viscosity becomes infinitely large after solidification and further attenuation is negligible. Therefore, the transition from melt to solid is determined based on the solidification temperature and attenuation plateau. If the transition point falls on the attenuation region, it means that the initially guessed model length value is too small, and the value of the model geometry length needs to be increased gradually until it matches the value of the solidification point. On the other hand, the initial value of model geometry length needs to be decreased if the solidification point falls after the starting position of the fiber diameter plateau region.

With this new iteration method, calculation was done on the previous case-study example and the simulated results are shown in Figure 3. With a solidification temperature of 135℃, the solidification position is found to be L_s = 1 cm, which coincides with the starting position of the fiber diameter plateau region. Based on this method, the final fiber diameter was determined as 7.5 µm, a value very close to the experimental result of 6.8 µm. The favorable match with the experimental data suggests that the modified method can provide reasonable prediction for the process under consideration.

Figure 3.

Non-isothermal model: fiber diameter and temperature versus distance from the spinneret (processing temperature = 240℃, air velocity = 22 m/s, initial fiber velocity = 0.005 m/s and stop point position = 5 cm).

Simulation results

Effect of processing temperature

Figure 4 shows the simulation results based on the previous case but with a 20℃ decrease in the processing temperature. The iteration indicates that 4.5 cm is a proper value for the model geometry length. The decrease in processing temperature causes an increase in polymer viscosity that reduces fiber attenuation. Hence, the model geometry length shifts from 5 cm to 4.5 cm when the temperature decreases by 20℃. As a result, the final fiber diameter increases to 11 µm. Likewise, the non-isothermal model also predicts that the fiber diameter decreases with the increase of processing temperature.

Figure 4.

Non-isothermal model: fiber diameter and temperature versus distance from the spinneret (processing temperature = 220℃ and stop point position = 4.5 cm).

Effect of air flow speed

The experimental results show that the fiber diameter decreases with the increase of air flow velocity. Once the air flow velocity exceeds a certain critical value, however, there is no further decrease in fiber diameter. This is because the air velocity affects two competing factors: the air drag force and the cooling rate, both increasing with the increase of the air velocity. When the air velocity or air/polymer velocity ratio is relatively small, the increase of air drag force is dominant over that of cooling rate. Once some critical point is exceeded, the increase of cooling rate becomes significant with comparable effect as the increase of air drag force. Hence, the competitive effects resulting from these two factors counterbalance each other with an overall negligible influence on fiber attenuation.

The experimental data is based on very low polymer flow velocities and therefore the air/polymer velocity ratios are very large in values. Three different levels of air velocity (10, 22, and 35 m/s) were chosen for the comparative studies, while the other parameters were kept the same (Table 1). For air velocity of 10 m/s, the predicted final fiber diameter was 9.5 µm, as shown in Figure 5. The case of 22 m/s was simulated in the previous section and the predicted final diameter was about 7.5 µm. When the air speed increases to 35 m/s, the predicted final fiber diameter is 7.0 µm, as shown in Figure 6. The simulation results agree well with the experimental observations, including that only a minor change on the final diameter occurs when the air velocity increases from 22 to 35 m/s. In addition, the polymer solidification point corresponding to a temperature of 135℃ shifts from 1.4 to 1.1 cm and to 0.9 cm when the air velocity increases from 10 to 22 m/s and to 35 m/s. This phenomenon also indicates that the cooling rate increases with the increase of air velocity.

Figure 5.

Non-isothermal model: fiber diameter and temperature versus distance from the spinneret (air velocity = 10 m/s and stop point position = 7 cm).

Figure 6.

Non-isothermal model: fiber diameter and temperature versus distance from the spinneret (air velocity = 35 m/s and stop point position = 5 cm).

Effect of initial fiber velocity

Figure 7 shows the simulation results when the initial fiber velocity increases from 0.005 to 0.01 m/s. The simulation results of the original case with initial fiber velocity 0.005 m/s is provided in Figure 3. By doubling the initial fiber velocity, the final fiber diameter is found to increase from 7.5 to 10.5 µm. The increase of initial fiber velocity causes an increase in the final fiber diameter, mainly due to a higher rate of polymer flow. Meanwhile, the increase of initial fiber velocity also slows down the cooling rate. As indicated in the simulation results, the solidification point shifts from 1.1 to 1.9 cm as the cooling rate is decreased. Although higher fiber temperature improves the fiber attenuation, it does not completely offset the effect of higher polymer flow rate.

Figure 7.

Non-isothermal model: fiber diameter and temperature versus distance from the spinneret (initial fiber velocity = 0.01 m/s and stop point position = 9 cm).

Comparisons between experimental results and simulation results

The non-isothermal Newtonian model was carried out to predict the fiber final diameter based on the processing temperature, initial fiber velocity, and air velocity, as listed in Table 2. The values of material properties are listed in Table 1, including the density, viscosity, thermal conductivity, and heat capacity for both air and polypropylene. Most of the simulated results are close to the experimental results except for cases 1 and 4, which show large deviation from the experiment values. Compared with the remaining cases, these two cases are observed with relatively low air velocity and air/polymer velocity ratio. Low air velocity and air/polymer velocity ratio cause slow cooling of the fiber, as observed in the experimental studies. This slow cooling phenomenon was also observed in the simulation. For those two cases, the solidification point with temperature at 135℃ was found to be located outside the air nozzle, which is about 2 cm in length. Once the air exits the nozzle, the velocity starts decreasing dramatically and the air drag force vanishes after a short distance. In such a case, the fiber attenuation stops outside the nozzle and results in a large fiber diameter. However, in the simulation a constant air velocity is used in this model, and the fiber is attenuated under the same air velocity outside the nozzle until the temperature reaches the solidification temperature. Therefore, a smaller diameter was predicted in simulation compared with that in the experimental study. For the cases with higher air velocities and higher air/polymer velocity ratios, the solidification points were found to be located inside the air nozzle, and therefore the assumed constant airflow velocity is a valid assumption in these cases.

Table 2.

Comparisons between experimental and simulation results of final PP115 fiber diameters under various processing conditions

Case	Processing temperature (℃)	Air velocity (m/s)	Initial fiber velocity (m/s)	Air/polymer velocity ratio	Measured fiber diameter (µm)	Predicted fiber diameter (µm)
1	180	24	0.056	429	143	47
2	180	56	0.035	1600	47	43
3	180	80	0.032	2500	38	42
4	200	24	0.069	348	69	30
5	200	24	0.036	667	40	30
6	200	56	0.036	1556	31	25
7	220	24	0.024	1000	27	20
8	220	24	0.016	1500	21	18
9	220	24	0.008	3000	18	16
10	220	56	0.009	6222	13	11
11	240	24	0.038	632	26	18
12	240	56	0.068	824	20	17
13	240	56	0.017	3294	11	13
14	240	24	0.005	4800	7	7.5

One simple approach to address this problem is to take the fiber diameter at the nozzle exit as the final fiber diameter. With this approach, the predicted final fiber diameter is found to be 97 µm for case 1 and 52 µm for case 4, which are much closer than the previous predictions to the experimental values. To achieve a better prediction for the cases with small air velocity and air/polymer velocity ratio, the air velocity value used in the simulation needs to be corrected to account for air velocity decrease outside the air nozzle.

Some minor deviations were also found in cases 5, 6, 7, and 11. Similar to the melt blown process, the variation of fiber diameter is relatively large, as discussed in Part I of this series.¹ High-speed air flow and a large difference between fiber and air temperatures lead to stochastic effects during fiber attenuation, which takes place within a short time period.

Conclusions

Experimental studies on melt-spinning of continuous fibers by cold air drawing show that the final fiber diameter depends on the polymer viscosity, processing temperature, polymer, and air volume flow rates. To study the mechanisms of fiber attenuation in this novel process, a 1D non-isothermal Newtonian fluid model is formulated. In this model, a method of iteration is presented to determine the fiber freezing point at which the fiber temperature reaches its freezing point and the velocity gradient equals zero. It was found that the heat transfer coefficient derived from the melt spinning process can also be applied to this model. The results showed good agreement between model predictions and experimental data for the relationship between processing conditions and the resulting fiber diameter. This 1D model successfully predicts that the fiber diameter decreases with increase of processing temperature. The model is also used to explain the effects of air and polymer volume flow rates. Such effects are complicated as they not only alter the kinetics of fiber attenuation but also the thermal environment. With an increase of air flow or decrease of polymer extrusion speed, a higher drag force is generated, which leads to faster cooling and reduced fiber attenuation. To predict the final fiber diameters for cases involving low air/polymer volume flow ratios where the fiber solidification point is outside the nozzle, the air speed decrease outside the nozzle should be considered in future modeling effort. Alternatively, as estimation, the nozzle exit point could be used as the fiber freeze point in the analysis. Overall, the theoretical model presented in this paper to predict the final fiber diameter based on processing conditions can be a valuable tool to guide further experimental studies and to lead to optimal processing conditions.

Footnotes

Funding

This work was supported by the US Department of Commerce through the National Textile Center (grant number F06-GT01).

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