Mathematical modeling and numerical simulation for nucleated solution flow through slit die in foam extrusion.
Continuous polymeric foam extrusion process involves flow of gas-dissolved polymer through a die. Producing a well-foamed, freely expanded product requires processing conditions that would shift the foaming point to the die exit or beyond it . This is practically achieved by selecting high extrusion pressures and temperatures at which gas is held in solution while transiting the die. The die may be viewed as a device within a processing sequence that stands between gas-dissolved single-phase system and a nucleated phase-separated system disturbing the thermodynamic equilibrium of the gas-dissolved polymer to initiate simultaneous nucleation and growth of nucleated sites into a cellular structure. The importance of this is reflected in the number of studies [2-13] on the flow of gas-dissolved systems across dies.
These studies are largely directed toward single-phase systems, fewer toward both single and biphase (gas-polymer) systems, and still fewer toward biphase alone. The interests of those that focus on single-phase systems [2-4] is in solution rheometry and usually address a design feature that provides sufficient back pressure at the die exit. A primary requirement of solution viscosity studies is to ensure the existence of single-phase system while rheological measurements are carried out.
Blyler and Kwei  have investigated the flow behavior of polyethylene melts containing dissolved gases using a capillary rheometer. Their analysis for single-phase system indicates that the viscosity reduction effected by the gas can be accounted by free volume viscosity theory. The effects of die geometry on nucleation and expansion ratios have been extensively studied by Park and coworkers [6, 7]. It is reported that the pressure drop rate significantly influences the cell density of the resulting cellular product.
On both single and biphase systems, Han et al. [8, 9] have shown that gas charged molten polymer exhibit a nonlinear axial pressure profile across a slit die. The departure from the linear pressure profile along the die signals phase separation, rising volumetric flow rates accompanied by growth of microbubbles. In contrast, single-phase incompressible systems show linear pressure drops corresponding to constant volumetric flow rates. Kraynik  has described phase change in terms of two dimensionless parameters using a phenomenological flow model. Fridman et al.  have constructed a connection between theoretical and experimental dependence of uniphase and biphase flow of gas-containing polymer melts. They note that in the region of the uniphase state, the compressibility of the gas-dissolved melt is the same as that of pure polymer matrices. The intersection of the critical pressure with the solution pressure determines the phase separation point along the die length.
Baldwin et al.  have developed process flow models of nucleated two-phase systems by using a postnucleated die shaping system. They verified the feasibility of shaping a nucleated polymer-gas solution by modularizing the nucleation step from cell growth control through the use of a nucleating nozzle. Their four process design models are applied to the polystyrene-C[O.sub.2] system and classified based on a dimensionless ratio of the characteristic flow rate to the characteristic gas diffusion rate. The characteristic flow rate is a function of position along the die and is not readily ascertained from experiments for any given system. Further, their work is based on experimental data for a single processing temperature. An approximated theoretical boundary condition exit pressure value is used in their simulation. Although the effect of viscosity reduction on simulation results is discussed, it is not built into their modeling work and they recommend going into detailed approaches.
The purpose of the present study is to investigate comprehensively the effects of temperature, viscosity reduction, and gas nonideality for two-phase flow in a three pressure tapping slit die for the PP-C[O.sub.2] system using a single screw foam extrusion set up designed in house. Unlike the work of Baldwin et al.  where a theoretical exit pressure value is assumed, the use of a die with multiple pressure tapping along the die enable experimental values to be used in the simulation and also facilitates comparison with theoretical flow models. The process design framework developed by Baldwin et al.  along with its assumptions is extended here by factoring viscosity reduction and gas nonideality expressed by the Redlich-Kwong equation of state. Using expressions for temperature and concentration dependency on viscosity, surface tension, diffusivity, and solubility, results are simulated and compared over a range of experimental values of processing temperatures.
In foam extrusion simulation work by Shimoda et al. , the bubble growth rates for PP-isobutane system is analyzed in tandem with multiple bubble growth models for fully developed flow in a slit die. Bubble growth models are coupled with a heterogeneous nucleation model to seamlessly describe the foam extrusion process from a single-phase gas-dissolved system to a phase-separated system within the die. Microbubble models are applied to a macro flow model with an attempt to elucidate the effect of influence volumes on bubble radii. While investigation of a complex heterogeneous nucleation mechanism necessitates the use of a multiple microbubble model, the physical connection between multiple microbubble models and the macro model is blurred by applying bulk concentrations rather than the time dependency of concentration. In addition, their bubble growth model uses a momentum conservation governing equation with the slight omission of the power law index. This is likely to have an impact on bubble radii by as much as a factor of the power law index itself.
The present study differentiates from the work of Shimoda et al.  by focusing on microrheological contributions to bubble growth of many independent single bubbles within a much simpler two-phase flow system. By basing the analysis on a quasi-static approximation and using transient interfacial concentrations rather than the computational friendly but less informative bulk (initial) concentrations, bubble growth within a die is simulated and compared with final cell sizes.
In the development of Baldwin et al.'s  flow models, some assumptions are made with a view to obtain approximate estimates of pressure drops and volumetric flow rates. These assumptions are (1) bubbles grow in uniform spheres, (2) there is no coalescence of bubbles, (3) there is no inter bubble interactions, (4) there is no change in polymer density as a result of gas dissolution, (5) concentration gradients around an individual bubble are nonexistent, and (6) gas within bubbles is ideal.
By these assumptions, every spaced out spherical bubble will grow independent of each other but dependent on the properties of the interface with gas-dissolved polymer. Accordingly, it becomes necessary to determine interfacial property changes with localized concentrations. To this effect, Baldwin et al.'s  models are first refined by introducing viscosity and surface tension changes as a result of changes to the equilibrium interface concentration.
Besides bulk concentrations and extrusion speeds, temperature is an important processing variable in a foam extrusion process. Hence, gas transport across a bubble will depend on interfacial property changes with temperature also. Therefore, a second refinement to Baldwin et al.'s  models is introduced by accounting for temperature changes to property variables.
Third, it is felt that assumption (6) needs to be addressed in nucleated solutions where nucleation and expansion is achieved by supercritical fluids. Nonideal gas expansion considerations can become important when the reference position of the gas from its critical state is changed over processing conditions.
The aforementioned features are considered in the following sections.
A numerical friendly form of Redlich-Kwong cubic equation  of state in compressibility factor, Z, is used to model nonideal gas behavior. For a gas with critical temperature [T.sub.c] and critical pressure [P.sub.c], [P.sub.g]VT relationship of C[O.sub.2] is expressed by
[Z.sup.3] - [Z.sup.2] + (A* - B* - [B*.sup.2])Z - A*B* = 0 (1)
A* = [a[P.sub.8]]/[[R.sup.2][T.sup.2]]
B* = [b[P.sub.g]]/[RT]
a = [0.42748[R.sup.2][T.sub.c.sup.2.5]]/[[P.sub.c][T.sup.0.5]]
b = [0.08664R[T.sub.c]]/[P.sub.c].
The gas pressure of mass [m.sub.g] in the vapor phase, [P.sub.g], is given by
[P.sub.g] = Z[[[m.sub.g]RT]/[V.sub.g]] (2)
where R is the universal gas constant.
Introduction of Compressibility Factor Z in Baldwin et al.'s  Flow Design Models
Baldwin et al.  consider four process design flow models--concurrent, post, pre, and partial prediffusion cases that throw light on cell growth control. In the concurrent case, bubble growth is influenced simultaneously by transport of solute as well as gas expansion through progressive pressure drop through the die. The postdiffusion model depicts bubble growth to be primarily driven by the gas pressure inside stable nuclei with no transport. The prediffusion (PD) case characterizes fast diffusive systems where transport is a dominant driving force in bubble growth. An intermediate partial prediffusion (PPD) model is on the basis of a known inlet pressure. The postdiffusion case is also a close representative of single-phase flow and is hence excluded from the analysis here. The four cases make the assumption that the diffused gas is at quasi-static equilibrium. Both mechanical and chemical equilibrium are assumed to exist under quasi-static conditions. For the concurrent diffusion (CD), PD, and PPD flow models of Baldwin et al. that serve as a framework here, the equations and numerical scheme is now modified to include compressibility factors. The mass of gas in the vapor phase, [m.sub.g], for each of the flow models following a chemical equilibrium condition of Henry's law with constant, [K.sub.p], is given by
[m.sub.g,conc] = ([c.sub.[infinity]] - [P.sub.g]/[K.sub.p])[m.sub.p] (3)
[m.sub.g,prediff] = [c.sub.[infinity]][m.sub.p] (4)
[m.sub.g,ppd] = ([c.sub.[infinity]] - [P.sub.g0]/[K.sub.p])[m.sub.p] (5)
[m.sub.g] is mass of polymer, [c.sub.[infinity]] is concentration of gas on solute-free basis.
Assuming that the gas bubbles are uniform, spherical, and noncoalescing, (assumptions (1), (2), and (3)) the volume of gas is related to the bubble radius R by
[V.sub.g] = [[[[rho].sub.c][m.sub.p]]/[[rho].sub.p]]([4/3][pi][R.sup.3]) (6)
[[rho].sub.c] is the nucleation density (number of cells per unit volume of unfoamed polymer) of a polymer with density [[rho].sub.p].
From a known value of the volume of gas relative to mass of polymer, [V.sub.g]/[m.sub.p], the radius of the bubble R is determined from Eq. 7.
R = [[[3[[rho].sub.p]]/[4[pi][[rho].sub.c]]]([V.sub.g]/[m.sub.p])][.sup.1/3]. (7)
Under mechanical equilibrium considerations,
[P.sub.g] = [[2[sigma]]/R] + P (8)
where [sigma] is the surface tension of the polymer solution at the interface.
The gas pressure equations are now
[P.sub.g,conc] = P + 2[sigma][[[3[[rho].sub.p]Z([P.sub.g])RT]/[4[pi][[rho].sub.c][P.sub.g]]]([c.sub.[infinity]] - [P.sub.g]/[K.sub.p])][.sup.-1/3] (9)
[P.sub.g,prediff] = P + 2[sigma][[[3[[rho].sub.p]Z([P.sub.g])RT]/[4[pi][[rho].sub.c][P.sub.g]]][c.sub.[infinity]]][.sup.-1/3] (10)
[P.sub.g0,ppd] = [P.sub.1] + 2[sigma][[[3[[rho].sub.p]Z([P.sub.g0])RT]/[4[pi][[rho].sub.c][P.sub.g0]]]([c.sub.[infinity]] - [P.sub.g0]/[K.sub.p])][.sup.-1/3] (11)
[P.sub.g,ppd] = P + 2[sigma][[[3[[rho].sub.p]Z([P.sub.g])RT]/[4[pi][[rho].sub.c][P.sub.g]]]([c.sub.[infinity]] - [P.sub.g0]/[K.sub.p])][.sup.-1/3]. (12)
The total pressure drop for a steady, fully developed power law fluid through a constant cross-section slit of height 2B and width W across a slit die can be obtained by a finite differencing technique of N (= i) sections of length [DELTA]z using the following equations of localized section suffixed variables
[P.sub.i+1] [approximately equal to] [P.sub.i] + [DELTA][P.sub.i+1] (13)
[DELTA][P.sub.i+1] [approximately equal to] [[[2 + 1/n]/[2W[B.sup.2]]][Q.sub.i]][.sup.n][[m.sub.fi]/B][DELTA]z. (14)
The local volumetric flow rate is given by
[Q.sub.i] = [dot.m]/[[rho].sub.m,i] [approximately equal to] [dot.m.sub.p]/[[rho].sub.m,i] [approximately equal to] [dot.m.sub.p](([V.sub.g]/[m.sub.p])[.sub.i] + [1/[[rho].sub.p]]) (15)
where [dot.m], mass flow rate, is approximated to mass flow rate of polymer, [dot.m.sub.p]. Suffix m denotes properties of the two-phase mixture.
Solution Viscosity Modeling
Solution viscosity modeling is considered important because viscosity dependence on concentration of solute is significant. Based on assumption (5), solution viscosity is modeled using equilibrium wall concentrations as the immediate envelope deterring bubble expansion.
The local power law coefficient for foam rheology is modeled as
[m.sub.fi] = m[1 - [1/[1 + [1/[[rho].sub.p]]([m.sub.p]/[V.sub.g])[.sub.i]]]] x VR[F.sub.i] (16)
where VR[F.sub.i] is the viscosity reduction factor for the section i.
Gendron et al.  describe the effect of temperature on shear viscosity of single-phase systems by a generalized plasticization index (PI[theta]) defined as
P[I.sub.[theta]] = -[[[delta] ln[[eta].sub.s]]/[[delta][theta]]] = 37.5
where [[eta].sub.s] is the single-phase solution viscosity and the reduced molar fraction for a solute-free concentration, c, of C[O.sub.2] in PP is given by
[delta][theta] = 0.478c.
A localized viscosity reduction factor depending on the quasi-static equilibrium concentration is then arrived as
VR[F.sub.i] = |[[eta].sub.s]/[[eta].sub.p]|[.sub.T,c,i] = [e.sup.[[-17.94[P.sub.g]]/[K.sub.p]]]. (17)
The power law consistency index, m, representing temperature dependency is expressed as
m = [a.sub.1] exp(-[a.sub.2]T). (18)
The values of constants [a.sub.1] and [a.sub.2] are determined for polypropylene from slit die rheometry for the range of processing temperatures and extruder speeds.
The two-phase viscosity, [[eta].sub.f], is related to the shear rate [dot.V] by the power law relationship
[[eta].sub.s] = [m.sub.fi][dot.V.sup.n-1]. (19)
Temperature and Concentration Dependency on Surface Tension
In their study of polymer melt surface tensions using Axisymmetric Drop Shape Analysis, Kwok et al.  express the temperature dependency on surface tension, [sigma], by the equation
[sigma] = (-0.051[T.sub.cefi] + 30.38)/1000. (20)
This equation was fitted to polypropylene (MW = 318,000, MFI = 2.0) data for the temperatures between 180 and 210[degrees]C.
A correlation developed by Goel and Beckman  for surface tension of swollen PMMA-C[O.sub.2] mixtures assuming negligible surface tension of supercritical C[O.sub.2] is given by
[[sigma].sub.m] = [sigma]([[rho].sub.m]/[[rho].sub.p])(1 - c)[.sup.4] (21)
where c is the gas concentration on solute-free basis.
Consistent with assumption (4), [[rho].sub.m] = [[rho].sub.p] and applying interface equilibrium concentrations, the surface tension of PP-C[O.sub.2], [[sigma].sub.m], is expressed as
[[sigma].sub.m] = [sigma](1 - [[P.sub.g]/[K.sub.p]])[.sup.4]. (22)
Temperature Dependency on Henry's Law Constants
Solubility with increasing temperatures is expressed by Sato et al.  in their equation for Henry's law constants as
ln(1/[K.sub.p]) = 6.255 + 3.706([T.sub.c]/T)[.sup.2]. (23)
Henry's law constants for PP (MW = 220,000) are determined by a pressure decay method for the temperatures ranging between 160 and 200[degrees]C.
Temperature and Concentration Dependency on Diffusivity
The diffusivity of C[O.sub.2] in PP  increases with temperature and is given by an Arrhenius type relationship for the temperature range of 188-224[degrees]C by
D = [D.sub.0] exp(-[E.sub.d]/RT). (24)
The mutual diffusion coefficient D has a weak dependence on concentration of C[O.sub.2] .
By assumption (4), the density of gas-dissolved mixture is independent of concentration, decreasing with increasing temperatures and represented empirically  by
[[rho].sub.m] = [[rho].sub.p] = 1000/(1.142 + 0.00094T). (25)
An iterative numerical scheme to solve the above equations over finite sections of the die is shown in Figure 1. The scheme uses simulation parameters for the PP-C[O.sub.2] system shown in Table 1. Simulation results are compared against the arbitrarily chosen reference cases of CD, PD, and PPD models. Inputs to the simulation are experimental values of mass flow rates, temperature, and gas concentration. Volumetric injection rates are converted to mass flow rates using the density of supercritical C[O.sub.2]. The ratio of mass flow rate of gas to the mass flow rate of polymer is the solute-free concentration. Exit pressure ([P.sub.3]) values are entered as boundary conditions for the simulation.
[FIGURE 1 OMITTED]
The iteration proceeds from the section (i = 1) nearest to the die exit. For CD and PPD models, [P.sub.g] is calculated using function zero solver of MATLAB version 6 with Eqs. 9 and 10 applying a function tolerance of le-30. The solver uses a combination of bisection and inverse quadratic interpolation algorithm. The equations require a calculation of Z, which is a function of [P.sub.g] as given in Eq. 1. Z is calculated from nonlinear zero solver with options using the Levenberg Marquardt algorithm and tolerance for Z set to le-4. In the case of PPD, the inlet gas pressure, [P.sub.g0], is calculated (Eq. 11) with known [P.sub.1] using Z, which is now a function of [P.sub.g0]. Equation 12 is then used to calculate [P.sub.g], Z being a function of [P.sub.g]. Equations 3-5 are used to calculate [m.sub.g], mass of gas in vapor phase for each of the diffusion flow models. The viscosity reduction factor (VRF) for this finite section of the die is calculated next and then [m.sub.f] using Eq. 16. [V.sub.g] is calculated from Eq. 2 of the nonideal gas and R, radius of the bubble, is calculated from Eq. 7. Experimental values of [a.sub.1],[a.sub.2],n,[c.sub.[infinity]],[dot.m.sub.p] and cell density [[rho].sub.c] shown in Table 2 for the temperatures investigated are used as inputs for simulation. [DELTA]P and Q are calculated using Eqs. 14 and 15 respectively. The local power law coefficient [m*.sub.fi] and VR[F.sub.i] is calculated from Eqs. 16 and 17 respectively. Equations 20-25 are used to calculate the values of surface tension, Henry's law constant, and diffusivity at the temperatures and localized concentrations. The exit pressure for the next section is arrived using Eq. 13. The iteration proceeds through the next section advancing incrementally towards the die inlet until the last section (i = 50) is reached.
A check on the above numerical scheme reproduces simulation results of Baldwin et al.  by setting VR[F.sub.i] and [Z.sub.i] equal to one for their simulation parameters.
Experimental Set Up
High melt strength polypropylene from Borealis (Daploy WB 130D) is used in the foam extrusion experiments. The properties of the polymer are given in Table 3. The blowing agent is supercritical C[O.sub.2] supplied by BOC.
The experimental set up, shown in Fig. 2, consists of a 19.1-mm single screw Haake extruder. The screw is designed with a seal to prevent back travel of the blowing agent. It has a flightless section at the gas injection port, approximately one third distance from feed, to minimize injection pulsations. Three pineapple mixing sections at the end of the screw cater to effective dispersion of the blowing agent. A Haake melt pump is used to transport the molten gas-polymer mixture to the static mixer. It provides a regulated constant volumetric supply of the mixture for accurate downstream pressure and mass flow rate measurements. The oil cooled, five element 1" diameter static mixer (Dynisco) effectively distributes dissolved blowing agent and simultaneously cools the melt before passage through elevated band heaters that raise the melt to die temperatures.
A three-pressure tapping slit die shown in Fig. 3 with dimensions 20 mm width (W), 101.6 mm length (L), and 2 mm height (2B) is used in the experiments. Liquid C[O.sub.2] is injected into the extruder at a constant volumetric flow rate of 0.3 ml/min using syringe pump ISCO 500D. Die pressure readings are recorded at different temperatures for different screw speeds. For a given screw speed, a suitable melt pump speed that holds injection port barrel pressures to within 500 [+ or -] 50 psi is selected and set for recording the different pressure measurements at experimental die temperatures.
In the experimental scheme described above, blowing agent concentration within the melt can be varied only by changing screw speeds while maintaining constant injection rates. Such a constrained scheme on concentration studies was imposed because of the sensitivity limitations to manual throttling of needle valve to hold syringe pump pressures constant during gas injection.
[FIGURE 2 OMITTED]
Achieving Two-Phase Flow in Extrusion Die
An index of having achieved a well-mixed homogeneous two-phase system of polymer and gas is evidenced by the absence of spurting at the die and very low fluctuations (less than 0.5%) in pressure readings. When the inlet die pressure is below the gas saturation pressure, it is safe to assume that the flow is strictly two-phase. To this end, low temperature at the static mixer that causes increase in viscosity, pressure, and solubility is maintained. The band heaters following the static mixer are set at much higher temperatures to bring the temperature of the melt to that of the die. Of the three screw speeds of 50, 70, and 90 rpm, inlet die pressure readings for 50 rpm alone are below the Henry's law saturation pressures calculated at the gas concentration and range of experimental temperatures. This forms the data set (Table 4) for two-phase flow analysis.
Determination of Cell Density
Cell density is recorded as the experimentally determined value of the number of cells per unit volume of unfoamed polymer from microscopically (SEM) examined cross sections of the foamed sheet. The number of cells per cross-sectional area of the liquid nitrogen cooled fractured sections of the sheet are calculated from a count of bubbles over a specified area. The density of foams is measured by liquid displacement method of ASTM D 792-00. The cell density is calculated using Eqs. 26 and 27.
[FIGURE 3 OMITTED]
[V.sub.f] = 1 - [[[rho].sub.f]/[rho]] (26)
[N.sub.c] = [[N[M.sup.2]]/a][.sup.3/2][1/[1 - [V.sub.f]]] (27)
where [V.sub.f] is void fraction, a is cross-sectional area of sample viewed under magnification, M, and N is the number of cells.
RESULTS AND DISCUSSION
Figures 4-6 show the cross-sectional micrographs of foamed sheets at temperatures of 190, 200, and 210[degrees]C. The cell structures show considerable nonuniformity of bubbles across the cross section and cannot be classified as micro-cellular foams defined by cell densities greater than 1 x [10.sup.10] cells/cc and cell diameters of ~10 [micro]m. In the absence of information on foam morphology progression from the die exit as well as nucleation mechanisms along flow through die, it is taken that the measured cell densities is an actual representation of the cell densities within the die. Underlying this is the assumption that instantaneous nucleation takes place upstream of the die and that no additional nuclei are created by either shear effects or the presence of impurities acting as secondary nucleating agents. Cell densities at 190, 200, and 210[degrees]C (Table 2) however, show a trend consistent with the established positive correlation of cell density versus pressure drop rates for microcellular foaming.
[FIGURE 4 OMITTED]
[FIGURE 5 OMITTED]
The results for pressure profile simulation of different flow models at temperatures 190, 200, and 210[degrees]C are shown in Figures 7-9 respectively. The pressure readings [P.sub.1], [P.sub.2], and [P.sub.3] are indicated as experimental values on the figures and are well below the saturation pressures ([P.sub.sar]) values calculated from Henry's law constants at the three temperatures. This is indicative of operating within a two-phase regime. With increasing temperatures, all three pressure readings [P.sub.1], [P.sub.2], and [P.sub.3] are lowered as would be normally expected from lowered viscosities.
[FIGURE 6 OMITTED]
[FIGURE 7 OMITTED]
The results show nonlinear pressure profiles, which are a typical characteristic of two-phase flow and has been noted previously in the work of others [8, 9]. The PPD model predicts the highest pressure drop. This is followed by the CD and the PD models.
The PPD case represents a scenario where a certain amount of gas (calculated from the pressure at [P.sub.1]) has transited the die and is guided along by the pressure gradient without further diffusion. The PD case assumes that all the available gas has diffused into nuclei prior to slit flow. This is descriptive of a system where rapid diffusion takes place. The intermittent CD case takes into account simultaneous diffusion with flow through the die. The PD case predicts the lowest pressure drop because of the complete diffusion of gas prior to flow. The CD case predicts a higher pressure drop because a progressive diffusion of gas takes place with flow. The PPD case shows the highest pressure drop based on the experimental pressure value at [P.sub.1], which is used to determine the amount of prediffused gas. Similar trends were noted in the work of Baldwin et al.  in their consideration of ideal gas behavior of expanding gas.
[FIGURE 8 OMITTED]
[FIGURE 9 OMITTED]
At 200[degrees]C, the experimental readings in Fig. 8 fall closest to the PD model, while at 210[degrees]C, experimental pressure readings in Fig. 9 compare quite well with the CD flow model. At lower temperatures of 190[degrees]C, the theoretical models over predict experimental measurements. This deviation may be due to a weakening in assumptions (1)-(3) as a result of temperature effects on nucleation mechanisms.
The inclusion of VRFs is found to be an essential ingredient in the analysis of pressure profiles. Without its inclusion, pressure drop profiles of two-phase flow models may intersect saturation pressure values calculated by Henry's law contradicting the physical meaning of two-phase flow models. If instead of transient equilibrium concentrations, bulk or initial concentrations are used, further reductions in pressure values can be anticipated.
The PD case predicts the highest volumetric flow rate at 190, 200, and 210[degrees]C as shown in Figures 10-12. This is followed by the CD and PPD models. The model that depicts a high value of gas mass in the polymer-gas mixture can be expected to show the highest final volumetric flow rates. Again, the model that represents a high gas mass transfer rate during flow can be expected to show the highest volumetric flow rate increases. Thus the PD case, which assumes that all the available gas has diffused into the nuclei, predicts the highest final volumetric flow rates at any temperature. Similarly, the CD case predicts the highest volumetric flow rate increases. The highest volumetric flow rates correspond to those with lowest pressure drops. Volumetric flow rates increase with temperatures and for the CD and PPD models, they also intersect near inlet die sections.
[FIGURE 10 OMITTED]
A trend similar to volumetric flow rates is observed in the case of bubble radii (Figs. 13-15). This is apparent from the dependencies of both volumetric flow rates and bubble radii on the specific volume, [V.sub.g]/[m.sub.p], through Eqs. 7 and 15. The PD, CD, and PPD model predict exit bubble radius of 118, 110, and 80 [micro]m respectively at 210[degrees]C, which is less than a twofold increase from inlet die conditions. Increases in final bubble radii with temperature is also observed in the work of Shimoda et al.  where an initial condition of bubble radius is used to follow bubble growth.
As temperature increases, the diffusivity of the gas within the polymer increases such that at a constant polymer mass flow rate, transfer of dissolved gas to the gaseous phase is faster. This is reflected in a diffusion flow type characterized by a higher pressure drop. The exit bubble radius is pushed toward an increase by the rising temperatures but marginally held back by the higher pressure drop flow type. Theoretically, exit bubble radius increases by a maximum of 45% with a temperature increase of 20%.
[FIGURE 11 OMITTED]
[FIGURE 12 OMITTED]
The final bubble radii at each of the temperatures 190, 200, and 210[degrees]C from micrographs almost range by a factor of two because of the nonuniformity of cell structure. Gas lost from foamed sheets with large surface to volume ratios and high extrusion temperatures debit bubble growth especially near the skin. A comparison of bubble radii at temperatures of 190, 200, and 210[degrees]C indicates that the average bubble radius ranges from 50 to 120 [micro]m. This lies within the vicinity of cell radii at [P.sub.3] simulated from the models for the three temperatures. The nonuniform cell sizes make it difficult to confirm the maximum of 45% increase in cell sizes predicted by the models at different temperatures, but observed increases in cell density at lower temperatures prompt the inference that the average bubble radii at lower temperatures may be lower.
[FIGURE 13 OMITTED]
[FIGURE 14 OMITTED]
The volumetric flow rates and bubble radii predicted by CD and PPD models at temperatures of 190 and 200[degrees]C intersect each other and this intersection point is shifted towards the die exit at lower temperatures. In the PPD case, the inlet bubble gas pressure [P.sub.g0] is estimated from the solution pressure reading at [P.sub.1]. When this pressure is above the bubble gas pressure value for CD case, a cross over in volumetric flow rates and radii for the two models is observed.
In the above simulation results for theoretical flow models, the compressibility factor, Z, takes values ranging from 0.96 to 1.03. This is a deviation of ~4% from standard ideal gas behavior. In general, the same deviation would apply to bubble gas pressures, and nonideal gas behavior considerations play a role in differentiating nucleated solution flow models for the PP-C[O.sub.2] system.
[FIGURE 15 OMITTED]
[FIGURE 16 OMITTED]
It is observed that at 190[degrees]C, the measured pressure readings are below the PD case and with rising temperatures, they tend to cross the CD model and move toward a PPD model. This shows that temperature affects two-phase flow characteristics significantly.
In developing a criterion for classifying flow models, Baldwin et al.  consider characteristic diffusion rates that are proportional to the diffusion coefficient and inversely proportional to the square of the diffusion distance. However, in the light of present observations, temperature inclusivity is sought by defining a characteristic rate as the reciprocal of reference time , t
t = [[M.sup.2][K.sub.p.sup.2][R.sup.2]]/[[[rho].sub.p.sup.2][R.sup.2][T.sup.2]D]
[characteristic flow rate]/[characteristic ref time rate] = [Q/(LW2B)]/[([[rho].sub.p.sup.2][R.sup.2][T.sup.2]D)/[M.sup.2][K.sub.p.sup.2][R.sup.2]]. (28)
From Eqs. 7 and 28, it is easily seen that the redefined characteristic rate preserves the basic form of Baldwin et al.'s flow reference time rate of direct dependency on [[rho].sub.c.sup.2/3] and also introduces the effects of temperature.
Because the quotient (characteristic rate) values are dependent on the volumetric flow rates that change across the flow length, they are plotted against the die length. The quotient values for a representative temperature of 190[degrees]C are shown in Fig. 16. This quotient decreases as the temperature increases, indicating a propensity to follow a PPD flow model.
By the above definition of a quotient, temperature dependency of diffusion coefficients and Henry's law constant is important to flow classification of nucleated solutions. In the case of PP-C[O.sub.2] system, the diffusivity of the gas increases with temperature while the solubility decreases. The usefulness of defining such a quotient lies in its ability to describe gas-polymer systems that are "reverse soluble" where the solubility increases with increasing temperature.
The above results and discussions hinge on the theoretical assumptions made earlier under modeling work. Assumption (1) is satisfied if the cell size is small and if microcellular foaming takes place. Assumptions (2) and (3) are satisfied if the inter bubble distances are large or when bubble growth is in its initial stages from its nucleated state. At low gas concentrations, nucleation density is low  and the effects of polymer swell may be neglected, satisfying assumptions (4) and (5). In all, the majority of assumptions are met at low gas concentrations.
In their work with limited data at a single temperature, Baldwin et al.  developed generalized design models of nucleated solutions for both slit and filamentary dies and identified a convenient way of classifying such models.
Here, a numerical analysis of two-phase flow in foam extrusion die for PP-C[O.sub.2] system incorporating more exhaustive and realistic features of gas nonideality and VRFs is presented and verified by experiments at three different temperatures. The analysis, however, does not account for end effects and wall slip effects.
It is found that temperature affects flow characteristics of two-phase systems significantly. The effect of increasing temperature is to shift experimental data toward applicability of high pressure drop flow models. At 210[degrees]C, experimental pressure readings are close to the CD theoretical flow model while at 200[degrees]C, and the experimental readings fall closest to the PD model. Experimental pressure readings at 190[degrees]C fall below the CD model. This deviation from theoretical models at lower temperatures may signify the influence of concentration on property variables at lower temperatures.
Simulation results of volumetric flow rates and bubble radii along the die at different temperatures show that they not only increase with increasing temperatures but also intersect each other at lower temperatures. Bubbles following different theoretical mechanisms of growth along the die can momentarily have similar radii. Experimental measurements of volumetric flow rates and bubble radii at different temperatures can throw further light into bubble growth mechanisms.
A quotient that includes temperature for classifying flow models is also introduced. At a representative temperature of 190[degrees]C, the PD and PPD flow models can be clearly demarcated along the entire length of die. The CD model takes on intermediary values bridging the two.
The authors would like to thank Mike Allan for help with the experimental set up and Andrew Chryss for useful discussions.
c Mass fraction of solute in polymer (solute free basis) D Diffusion coefficient ([m.sup.2]/s) [K.sub.p] Henry's law constant (Pa) M Molecular weight (Kg/Kg mole) [P.sub.g] Gas pressure (Pa) R Bubble radius (m) R Gas constant (Pa [m.sup.3]/Kg mole K) T Absolute temperature (K) t Reference time (s) n Power law index m Power law consistency index (Pa [s.sup.n])
[eta] Viscosity (Pa s) [rho] Density (Kg/[m.sup.3]) [sigma] Surface tension (N/m)
g Gas phase p Polymer phase [infinity] Far from bubble surface i Section number s Solution c Critical state cent Centigrade scale f Foam m Mixture or two-phase o Upstream condition
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Cooperative Research Center for Polymers, Notting Hill, Victoria, Australia
S.N. Bhattacharya, A.A. Khan, C. Stephen
Rheology and Materials Processing Center, School of Civil and Chemical Engineering, RMIT University, Melbourne, Australia
Correspondence to: S.N Bhattacharya; e-mail: firstname.lastname@example.org
Contract grant sponsor: Co-operative Research Centre for Polymers.
TABLE 1. Simulation parameters for PP-C[O.sub.2] system. Parameter Value [a.sub.1] 1.3696 x [10.sup.6] [a.sub.2] 0.013 n 0.4865 [P.sub.c] 7.38206 x [10.sup.6]Pa [T.sub.c] 304.19 K B 1 x [10.sup.-3] m W 20 x [10.sup.-3] m L 101.16 x [10.sup.-3] m [DELTA]z 2.0232 x [10.sup.-3] m N 50 M 44 Kgmole/kg [C.sub.[infinity]] 0.013 kg gas/kg polymer [E.sub.d] 3000 kcal/kgmole [D.sub.o] 111.768 x [10.sup.-9] [m.sup.2]/s [dot.m.sub.p] 16.44 g/min TABLE 2. Experimental values of cell densities and foam densities at 190, 200, and 210[degrees]C. [N.sub.c,190[degrees]C] 4.32 x [10.sup.11] cells/[m.sup.3] [N.sub.c,200[degrees]C] 3.78 x [10.sup.11] cells/[m.sup.3] [N.sub.c,210[degrees]C] 3.78 x [10.sup.11] cells/[m.sup.3] [[rho].sub.f,190[degrees]C] 713 kg/[m.sup.3] [[rho].sub.f,200[degrees]C] 718 kg/[m.sup.3] [[rho].sub.f,210[degrees]C] 750 kg/[m.sup.3] [rho] 900 kg/[m.sup.3] TABLE 3. Properties of high melt strength polypropylene. MFI (g/min, 230[degrees]C, 2.16 kg) 2.5 Molecular weight (MW) 490,000 Polydispersity index ([M.sub.w]/[M.sub.n]) 9.6 TABLE 4. Experimental die pressure readings at different temperatures. Temperature ([degrees]C) [P.sub.1] (psi) [P.sub.2] (psi) [P.sub.3] (psi) 190 281 181 88 200 271 177 86 210 253 166 80
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|Author:||Stephen, C.; Bhattacharya, S.N.; Khan, A.A.|
|Publication:||Polymer Engineering and Science|
|Date:||Jun 1, 2006|
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