Simulation of Hollow Fiber Spinning - ACS Symposium Series (ACS

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Simulation of Hollow Fiber Spinning Downloaded by UNIV OF TEXAS EL PASO on November 8, 2014 | http://pubs.acs.org Publication Date (Web): October 31, 2011 | doi: 10.1021/bk-2011-1078.ch009

Y. Su1 and G. G. Lipscomb* Chemical and Environmental Engineering Department, University of Toledo, 2801 West Bancroft Street, Toledo, OH 43606–3390 1Current address: Emhart Glass, 123 Great Pond Drive, Windsor, CT 06095 *E-mail: [email protected]

Fiber spinning processes are used to manufacture hollow fibers for the membrane industry. Ideally, the spinning process produces fiber of controlled size and morphology. Theoretical models of the process provide tools for evaluating the dependence of fiber properties on process variables and thereby facilitate process control and optimization. The purposes of this contribution are to: 1) review one- and two-dimensional simulations of solid fiber and hollow fiber spinning and 2) review recent contributions to the theory including predictions of recirculation within the fiber bore and analytical mass transfer calculations using boundary layer theory and the superposition principle.

Introduction Fibers, solid and hollow, are found in myriad applications. The first man-made fibers date back to the late nineteenth century (1–4). Numerous experimental studies have been performed to understand the fabrication processes (5–14). However, these studies tend to be system-dependent, expensive, and time consuming. Synthetic solid fibers are used primarily as textiles. Hollow fibers are used in a variety of membrane separation processes of industrial importance (15–17). The hollow fiber form is preferred to the alternative flat sheet form because of its superior surface area to volume ratio which makes their production of great commercial interest.

© 2011 American Chemical Society In Modern Applications in Membrane Science and Technology; Escobar, I., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 2011.

Downloaded by UNIV OF TEXAS EL PASO on November 8, 2014 | http://pubs.acs.org Publication Date (Web): October 31, 2011 | doi: 10.1021/bk-2011-1078.ch009

A spinning process is used to manufacture most synthetic fibers. Typically, a viscous fiber-forming liquid, or spin dope, is extruded to form continuous filaments through the holes of a spinneret. Upon passing through a quench step, the nascent fiber solidifies to form a fiber with sufficient mechanical strength to be handled mechanically and ultimately wrapped on a spool. Spinning processes may be categorized by the type of spin dope. In melt spinning processes, a neat polymer is melted and extruded at an elevated temperature. In solution spinning process, the polymer is dissolved in one or more solvents before extrusion. Spinning processes can be classified further by the nature of the quench step. A liquid bath is used to quench the filament in wet processes while a gas is used in dry processes. Both types of quench are used in dry-wet spinning processes where the filament passes through a gas filled quench gap before entering a liquid quench bath. After solidification, fibers are collected with a take-up device for storage or additional post-spinning treatments. A typical solution, dry-wet hollow fiber spinning process is illustrated in Figure 1. In hollow fiber spinning processes, a second fluid is fed to the spinneret. This fluid may be a gas or liquid and is co-extruded with the spin dope to create a liquid filament possessing a bore (inner region occupied by the second fluid) and clad (outer region occupied by the spin dope) structure. Upon clad solidification, the bore fluid is removed to leave a hollow fiber. In melt spinning processes, inert gases are used as the bore fluid almost exclusively; when the fibers are potted and the fiber ends opened to provide access to the fiber lumens, the bore gas can be removed. In solution spinning processes, bore gases or liquids commonly are used. If a liquid quench bath is used, the liquid in the bath may displace the bore gas or liquid. Subsequent drying to set the fiber structure and produce a dry fiber displaces the bore fluid with the gas used for drying. Potting the fibers and opening the fiber ends allows removal of this gas as described for melt spinning processes. The relative flow rates of the bore and clad fluids, in addition to the draw ratio (i.e., the ratio of final take-up velocity to exit velocity from the spinneret), determine fiber size. Furthermore, mass and heat transfer between the bore and clad fluids and between the clad and quench fluids can lead to phase separation in the clad and an asymmetric pore structure upon solvent removal. As the synthetic fiber industry grew, manufacturers increased capacity and reduced cost by utilizing multi-hole spinnerets. However, spinning multiple filaments can adversely affect uniformity of fiber size and morphology due to variation in the conditions experienced by each filament. This non-uniformity affects the quality and performance of products made with the filaments, e.g., variation in hollow fiber size and transport properties can be highly detrimental to the performance of hollow fiber membrane gas separation modules (18–20). Therefore, a fundamental understanding of the spinning process is needed to help optimize and improve the manufacturing process.

130 In Modern Applications in Membrane Science and Technology; Escobar, I., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 2011.


Figure 1. Schematic drawing of a typical solution dry-wet hollow fiber spinning process.

Theoretical analyses of the spinning process typically rely upon dividing the process into four regions along the axial (draw) direction. Figure 2 illustrates these regions for hollow fiber spinning; the same regions are present in solid fiber spinning. Region 1, the shear flow region, is inside the spinneret where the fluid flow commonly is assumed to be fully developed shear flow. Region 2, the flow rearrangement or transition region, is a short region from the spinneret exit to the point of maximum die swell, where the flow undergoes a transition from shear flow to uniaxial extensional flow. Region 3, the draw zone, is where the fiber is drawn down to the desired final size due to the tension applied by the take-up device or the influence of gravity. Heat and mass transfer occur in the draw zone which may result in phase separation in the clad, affect fiber size, and lead to the transition from Region 3 to Region 4. Region 4, the solidification region, is where the nascent fiber solidifies when the temperature and composition of the fiber change such that the filament temperature is below the glass transition or crystallization temperature. For melt spinning processes, the solidification point is a function of temperature only. However, for solution spinning processes, the glass transition or crystallization temperature is a function of polymer/solvent composition so the point of solidification will depend on both temperature and composition. Dimensional and structural changes after this point typically are assumed to be negligible.



Figure 2. Hollow fiber spinning regions.

Theoretical Models for Solid Fiber Spinning The earliest theoretical analyses of the spinning process are attributed to Andrews (21) and Ziabicki (22–26) and coworkers. Andrews developed an energy balance for filaments in the melt spinning process. The fiber diameter profile along the axial direction was determined empirically from experimental observations. Ziabicki and coworkers developed a theoretical analysis from rigorous conservation of mass, momentum and energy principles. Following the efforts of Andrews and Ziabicki, notable contributions were made by Kase (27, 28), Matovich (29), Lamonte (30), Denn (31) and coworkers to identify the key physical and mathematical assumptions in theoretical models of the spinning process and thereby establish the limits for their use. This early work focused on melt spinning of solid fibers and the effects of fluid rheology, especially the viscoelastic nature of polymer melts. To make the analyses tractable, researchers simplified the problem by reducing the physical dimensions from two to one with the help of the thin filament analysis (TFA). The fundamental assumption of the TFA is that the fiber diameter is small compared to the length of draw zone, which in turn implies radial variations of 132 In Modern Applications in Membrane Science and Technology; Escobar, I., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 2011.


spinline variables are small compared to axial variations. These variables are replaced with radially averaged quantities defined by eq (1), where φ represents a spinline variable, A is the filament cross-sectional area, and R is the filament radius.

The TFA equations are obtained by radially averaging each axisymmetric conservation equation f (momentum, continuity, or energy) as defined in eq (2) or, equivalently, by assuming the axial velocity is a function of axial position only in the draw zone.

Note that the radial component of the conservation of momentum equation is multiplied by the radial coordinate (r) before applying eq (2) to yield the TFA equations. The TFA addresses only the draw zone region by assuming the transition region is of negligible length relative to the draw zone and the fiber freezes immediately at the solidification point. This obviates the need for determining the fiber radius and velocity in the transition region. However, it requires a priori knowledge of the magnitude of die swell; the point of maximum die swell is used as the origin of the axial coordinate system for analysis of the draw zone which is nearly coincident with the spinneret face if the transition region is short. The fiber size and velocity are specified as boundary conditions at this point. Most solid fiber spinning processes satisfy the thin filament assumption as the filament diameter typically is at least two orders of magnitude smaller than the draw zone length. Predicted spinline diameter, velocity, and temperature from the TFA generally agree well with experimental observations (28, 30). Textile industries produce solid fibers by solution spinning (32) in addition to melt spinning. Theoretical analyses of solution solid fiber spinning process were developed soon after the melt spinning analyses. Modeling of such processes requires simultaneous solution of the conservation of mass equation for the solvent in addition to the momentum, continuity, and energy conservation equations solved for melt spinning processes. Fok and Griskey (33) examined the concentration profile in the fiber for solid dry spinning processes where the filament was not drawn down - a constant axial velocity was specified throughout the draw zone. Ohzawa et al. (34, 35), Simon (36, 37), Chandler et al. (38), and Guo et al. (39) developed models for the concentration changes that occur during solution spinning that included simultaneous momentum and heat transfer. These analyses yielded cross-sectional average concentrations along with other spinline variables through extensions of the thin filament analysis. Consequently, radial gradients cannot be determined and evaluation of the surface concentration requires experimental measurements. Despite the usefulness of one-dimensional spinning models, two-dimensional models are required to capture radial variations in temperature and composition 133 In Modern Applications in Membrane Science and Technology; Escobar, I., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 2011.


that may affect fiber properties, especially for solution spinning of hollow fiber membranes where radial concentration and temperature gradients determine the asymmetric structure of the fiber (11–13). Initial efforts to capture two-dimensional features relied on hybrid models that combined the thin filament reduction of the momentum and continuity equations with two-dimensional analyses of heat and mass transfer. Doufas and coworkers (40) developed a model for non-isothermal melt solid fiber spinning processes with phase transition by including crystallization kinetics. Brazinsky (41), Sano (42), Guo (43) and coworkers extended previous one-dimensional solution spinning models by solving the two-dimensional conservation of mass equation to determine the concentration field as a function of both radial and axial position within the filament. Advances in computational hardware and software have enabled solution of the full two-dimensional conservation equations for solid fiber spinning. Joo et al. (44) report solutions to the conservation equations for melt solid fiber spinning. Their simulations provide a priori predictions of the radial and axial dependence of all spinline variables as well as crystallinity profiles within the filament. Tsai (45) and coworkers simulated the solution spinning of solid optical fibers. Although filament diameter changes due to solvent evaporation were allowed, dimensional changes due to an imposed draw ratio were not included in their work. Two-dimensional numerical approximations to the governing conservation equations also enable simulation of the entire spinline from the shear flow region in the spinneret to the solidification point. In particular, the simulations allow prediction of the die swell that occurs in the transition region (44, 45). This work provides further verification of the validity of the TFA assumptions for many industrially relevant spinning processes. For example, the predicted axial velocity field confirms the shear flow in the spinneret transforms rapidly into uniaxial extensional flow in the draw zone after a short transition region. Furthermore, radial temperature gradients are small in non-isothermal spinning processes thereby confirming the validity of radial averaging of the conservation of energy equation to obtain the one-dimensional thin filament approximation for prediction of the axial temperature profile.

Theoretical Models for Hollow Fiber Spinning The theoretical studies of solid fiber spinning have established the essential physics of the process and led to the development of simulations that yield a detailed picture of the velocity, temperature, and concentration changes that occur. However, the hollow fiber spinning processes that the membrane separation industry relies upon cannot draw directly from these results - the solid fiber results provide a basis for the development of hollow fiber spinning models and a reference point for validating these models but cannot be used directly. Similar to solid fiber spinning models, researchers first studied hollow fiber spinning with one-dimensional models. The first reported analysis by Freeman et al. (46) examined the draw zone only and assumed the bore fluid was a gas which could be characterized by a constant pressure throughout the draw zone; 134 In Modern Applications in Membrane Science and Technology; Escobar, I., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 2011.


the pressure was used as a boundary condition along the inner fiber surface in a thin filament analysis of the clad. Lipscomb (47) extended the thin filament analysis to include the continuity and conservation of momentum and energy equations for the bore region in addition to the clad. Similar to eqs (1) and (2), thin filament equations were obtained by radially averaging the conservation equations. Eqs (3a) and (3b) were used to define radially averaged values for spinline variables in the clad and bore regions, respectively. Eqs (4a) and (4b) were used to radially average the conservations equations for the clad and bore, respectively. Note that in the thin filament approximation the bore and clad radial average values of velocity, temperature, and concentration are equal.

In eqs (3) and (4), Ro is the filament outer radius and Ri is the filament inner radius, i.e., location of bore-clad interface. This approach rigorously accounts for continuity of momentum, mass, and energy fluxes across the bore-clad interface. The TFA analysis allowed simulation of hollow fiber spinning with either a bore gas or liquid through use of an appropriate viscosity value. Moreover, the results allowed prediction of the axial bore pressure profile in contrast to assuming a constant value; the assumption of a constant value leads to physically unrealistic variations in bore gas flow rate. The predicted bore gas pressure dropped below atmospheric pressure in most of the draw zone for a typical PET hollow fiber melt spinning process. Other authors have used different approaches to account for the bore fluid. Chung et al. (48) simulated an isothermal dry-wet hollow fiber spinning process using a liquid bore fluid. Continuity of viscous forces across the bore-clad interface was enforced through use of a drag coefficient. The analysis accounted for mass transfer across the bore-clad interface that would lead to phase inversion but did not include solvent loss from the fiber outer surface due to evaporation. De Rovére et al. (49) considered bore gases only. Moreover, they assumed the bore gas pressure was constant and equal to atmospheric pressure. A thin filament analysis for the clad region allowed prediction of radially averaged field variables. 135 In Modern Applications in Membrane Science and Technology; Escobar, I., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 2011.


Balasubramanian (50) further extended TFA hollow fiber spinning models by including the effect of surface tension. Surface tension was found to have little effect on predicted spinline variables for high-speed melt spinning of PET hollow fibers due to the large draw ratio and concomitant viscous forces. Two-dimensional models of hollow fiber spinning also have been reported. Castellari and coworkers (51) developed a model to predict concentration fields in hollow filaments for simulation of membrane production processes that utilize diffusion induced phase separation to produce asymmetric structures. Although the authors consider radial diffusion in a stationary filament, the time dependence of the solutions can be related to axial variations in concentration if the filament velocity in the draw zone is known. Freeman (46), Oh (52), Housiadas (53) and coworkers report two-dimensional simulations of melt hollow fiber spinning. Freemen et al. (46) studied isothermal hollow fiber spinning with a constant bore gas pressure and provide solutions for the velocity field from the spinneret through the transition region and draw zone to the solidification point. The results provide a priori predictions of die swell. Oh et al. (52) examined the dynamics of the non-isothermal spinning process with a hybrid model. The two-dimensional conservation of mass and momentum equations were solved while the thin filament approximation was used for the conservation of energy equation. Additionally, the transition region was not included in the simulation; experimental die swell measurements were used to specify initial filament dimensions and velocity. Housiadas et al. (53) reported a simulation of annular extrusion, not hollow fiber spinning. Therefore, they examined the effects of dimensional changes due to surface tension and gravity and did not include the effect of an imposed fiber draw down. Berghmans et al. (54) report a two-dimensional analysis of solution hollow fiber spinning to produce asymmetric membranes via the TIPS process. Mass and heat transfer were allowed to occur at the fiber outer surface. However, all physical parameters were assumed to be constant, independent of temperature and concentration. Batarseh (55) extended the work of Berghmans by allowing all physical parameters to vary with temperature and concentration. The model was able to predict fiber outer radius changes due to evaporation, though the reported changes were small for the system considered. Both Berghmans and Batarseh neglected fiber draw down due to gravity or an imposed draw ratio to avoid simultaneous solution of the conservation of momentum equation. They assumed the filament moves at a constant axial velocity through the draw zone. Balasubramanian-Rauckhorst et al. (56) coupled the TFA equations with boundary layer analysis (BLA) to predict concentration profiles within the clad. Experimental measurements of the porosity profile across the fiber cross-section exhibited the same qualitative dependence on processing conditions as theoretical predictions of the concentration profile. The thin filament equations are derived for power law fluids in recent work by Aroon et al. (57). The analysis follows that of Lipscomb (47) with the substitution of the power law fluid constitutive equation for the fluid stress. 136 In Modern Applications in Membrane Science and Technology; Escobar, I., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 2011.


Recirculation of Bore Fluid during Hollow Fiber Spinning While the TFA includes the bore fluid flow in the conservation of momentum equation, two-dimensional simulations of the shear, transition, and draw zone regions including the bore fluid were not reported until the work of Su et al. (58). The assumed uniaxial extension of the bore fluid had not been verified. Additionally, the TFA requires a prior knowledge of die swell for quantitative prediction of fiber diameter, temperature, and concentration profiles which is obviated by a two-dimensional simulation. Su et al. used FIDAP, a finite element based computational fluid dynamics software package, to simulate the hollow fiber spinning process for a solution domain that extends from within the spinneret to the solidification point. Such a computational domain allows prediction of die swell as well as two-dimensional, axisymmetric components to the velocity field. The reported analysis was limited to isothermal, melt spinning of Newtonian bore and clad fluids but the procedure could be extended to allow analysis of non-isothermal solution spinning of nonNewtonian fluids. The velocity and pressure fields within the bore and clad regions were calculated by creating an algebraic approximation to the governing partial differential continuity and conservation of momentum equations (eqs (5) and (6), respectively) using the finite element method; these algebraic equations then are solved numerically to obtain the velocity and pressure fields. The subscript i in eqs (5) and (6) and subsequent equations is either c to represent the clad or b to represent the bore. The bore and clad fluid equations are linked by continuity of velocity and stress across the bore-clad interface. At steady-state, two kinematic boundary conditions apply to the fiber outer surface and inner surface, eqs (7) and (8), respectively. These equations determine the fiber outer radius Ro and inner radius Ri.

Note that eq 6 assumes the bore and clad fluids are Newtonian and the viscosities are constant. Eqs (7) and (8) follow immediately from the statement that the normal surface velocity must be zero at steady state. One may readily demonstrate that the r and z components of the unit normal vector are given by 1/(1+(∂R/∂z)2)0.5 and – (∂R/∂z)/ (1+(∂R/∂z)2)0.5, respectively, where R is either inner or outer radius. Setting the dot product of the unit normal and velocity 137 In Modern Applications in Membrane Science and Technology; Escobar, I., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 2011.


vectors to zero yields eqs (7) and (8). The computational domain and boundary conditions are illustrated in Figure 3. The spinning conditions and material properties are listed in Tables I and II.

Figure 3. Computational domain and boundary conditions.

Table I. Base spinning conditions Condition

Value

Rco = spinneret clad outer radius

1.9 × 10-3 m

Rci = spinneret clad inner radius

1.55 × 10-3 m

Rbo = spinneret bore outer radius

9.0 × 10-4 m

Qc = clad volumetric flow rate

15.82 cc/min

Qb = bore volumetric flow rate T0

1.00 sccm (1.51 ccm @ T0, 413 K

= spinning temperature

58 Pa

= initial bore gauge pressure vL = take-up velocity

35 m/min

L = draw zone length

0.20 m

Patm = ambient air pressure

101325 Pa


)

Table II. Polyethylene-dodecanol system material properties Property

Value

ρc = clad density

758 kg/m3


= pre-exponential factor for clad shear viscosity

6.07 × 10-3 Pa s

Ec/Rg = ratio of activation energy to the gas constant for clad shear viscosity

3681 K

ηb = bore gas shear viscosity at spinning temperature

0.22 × 10-4 Pa s

MW = bore gas molecular weight

28.0

Simultaneous solution of eqs (5) through (8) provides filament dimensions and both radial and axial velocity components throughout the computational domain. Predicted fluid streamlines within the solution domain are illustrated in Figure 4 and axial velocity contours are illustrated in Figure 5.

Figure 4. Typical streamline contour plot showing flow near the die exit. The arrows indicate the origin of the coordinate system (58).



Figure 5. Typical axial velocity contour plot showing flow near the die exit. A: -50, B: -35, C: -20, D: 2, E: 12, F: 18, G: 56, H: 60, I: 100 (mm/s). The arrows indicate the origin of the coordinate system (58). Figures 4 and 5 clearly illustrate the flow paths for both the bore and clad fluids. For the clad, the shear flow in the spinneret quickly transitions to uniaxial extensional flow confirming the assumption of the thin filament analysis. However, for the bore fluid the shear flow does not transition quickly to an extensional flow. Instead, a large recirculation region exists that extends from the spinneret exit to ~40% of the draw zone length. The presence of the recirculation region is the result of a simultaneous decrease in the bulk velocity, due to an increase in the flow cross-sectional area, and a velocity increase at the bore-clad interface due to continuity of fluid velocity across the interface. When a low viscosity bore fluid is used at a sufficiently low flow rate, the fluid cannot withstand the large shearing stresses generated in the transition region and a recirculation region forms. Simulation results indicate the presence and size of the recirculation region depends on two dimensionless groups: (1) ηb/ηc and (2) Qb/Qc for a given spinneret geometry at low spinning speeds. The dependence of the dimensionless recirculation size (the ratio of the length of the recirculation region Lrec to the draw zone length L) on the two dimensionless groups is illustrated in Figure 6. Recirculation occurs for combinations of ηb/ηc and Qb/Qc below the zero contour line. The dimensionless recirculation size increases as either ratio decreases. Predicted fiber dimensions are compared with TFA simulations and experimental measurements in Figure 7; the experimental measurements are those reported by Balasubramanian-Rauckhorst et al. (56). Only the fiber outer radius was measured experimentally due to experimental limitations on imaging the fiber cross-section. Furthermore, the maximum fiber outer radius obtained from the two-dimensional simulation was used as the initial TFA outer radius, 140 In Modern Applications in Membrane Science and Technology; Escobar, I., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 2011.


while the initial TFA inner radius was calculated from the specified bore and clad flow rates. Note that the results presented in Figure 7, and subsequent figures that illustrate results from the TFA, assume the draw zone begins at the face of the spinneret. This assumption is consistent with neglect of the flow rearrangement region, Region 2 in Figure 2, as described in the Theoretical Models for Solid Fiber Spinning section. Figure 7 indicates the presence of the bore fluid recirculation region increases the cross-sectional area occupied by the bore fluid which in turn leads to thinner fiber walls in the two-dimensional simulation than the TFA simulation. Figure 7 also reveals that the fiber outer radius decreases much more rapidly for the two-dimensional simulation than for the TFA due to elimination of the recirculation region. Therefore, the predicted fiber outer radius from the two-dimensional simulation is in better agreement with experimental observations than that of the TFA. Significant differences between fiber dimensions still exist in the die swell region. These differences arise primarily from neglecting non-Newtonian effects which can lead to enhanced die-swell relative to Newtonian fluids. Use of an appropriate non-Newtonian fluid constitutive equation should improve agreement. The agreement between theory and experiment in Figure 7 is typical of comparisons reported in the literature. One might use these results to evaluate the ability of future theoretical work to provide enhanced predictive capabilities.

Figure 6. Contour plot of Lrec/L as a function of ln(ηb/ηc) and Qb/Qc (58). 141 In Modern Applications in Membrane Science and Technology; Escobar, I., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 2011.


Figure 7. Two-dimensional simulation (dashed lines), TFA (solid lines), and experimental values (triangles) of fiber outer (upper curves) and inner radii (lower curves) along spinline; experimental values are for outer radius only (58).

Concentration Predictions with Boundary Layer Superposition An approximate solution for the axial and radial variation of concentration in solution hollow fiber spinning was obtained by Balasubramanian-Rauckhorst et al. (56) using boundary layer theory. Assuming the bore fluid rapidly saturates with solvent so mass transfer at the inner fiber radius may be neglected, the boundary layer equation describing mass transfer at the outer clad surface is given by eq (9):

where is the cross-sectional average axial velocity, D is the polymer-diluent (i.e., solvent) mutual diffusion coefficient, and Cd is the diluent mass concentration. Eq (9) follows directly from the species conservation of mass equation by substituting

for the axial velocity in the draw zone and – (r/2)

∂ /∂z for the radial velocity; the radial velocity expression follows directly from the continuity equation. Additionally, mass transfer by axial diffusion is neglected relative to mass transfer by radial diffusion in the boundary layer, i.e., ∂2Cd/∂2z

Simulation of Hollow Fiber Spinning - ACS Symposium Series (ACS

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