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Ultrasonic Decrosslinking of Peroxide Crosslinked HDPE in Twin Screw Extrusion. II. Simulation and Experiment.

INTRODUCTION

In Part I of this study [1], a process model for the ultrasonic decrosslinking of XHDPE was proposed. The proposed model was developed based on a modification of the model for ultrasonic devulcanization of rubber vulcanizates under the static conditions [2, 3]. Simulation algorithms for the ultrasonic decrosslinking under the static (no flow) condition and during flow in TSE were developed based on the proposed process model.

In the present study (Part II), numerical simulations of ultrasonic decrosslinking of XHDPE under the static condition and using TSE are carried out. The simulation results of the ultrasonic decrosslinking of XHDPE using TSE are compared with the experimental measurements to validate the proposed process model. On the basis of this study, some possible directions for the development a more comprehensive model are proposed.

EXPERIMENTAL

Materials

HDPE used was a rotational molding grade, 35 mesh powder (Paxon 7004, Exxon Mobil, Baytown, TX). Crosslinking agent was dicumyl peroxide in the pellet form (DC-40, Akrochem, Akron, OH). It contained 40 wt% peroxide and 60 wt% calcium carbonate as a carrier. Tetrakis (methylene [3,5-di-tert-butyl-4-hydroxyhydro cinnamate]) methane (Antioxidant 1010, Akrochem, Akron, OH) was used as a stabilizer during the swelling and rheology test. Xylene (mixture of isomers, ACS Reagent Grade, Sigma-Aldrich, WI) was used as a solvent in the swelling test. Details of preparation of XHDPE were reported elsewhere [4, 5],

Decrosslinking of XHDPE via TSE

Figure 1 shows a schematic of an ultrasonic corotating TSE (a) along with compounding (b) and decrosslinking (c) screws used in experiments. The TSE (PRISM US LAB 16, L/D = 24, Thermo Electron Corporation, MA) has screw diameters of 16 mm. It is equipped with a water-cooled horn installed in the barrel. A circular die with a diameter of 3 mm and a length of 11 mm was used. The horn provides the longitudinal ultrasonic waves orthogonal to the flow direction at a frequency of 40 kHz. The horn is attached to a booster and a convenor connected to a power supply of 800 W (2000 bdc, Branson Ultrasonic, CT). The barrel and die temperatures were set up at 200[degrees]C. The temperature of the watercooled horn is maintained at 145[degrees]C. Ultrasonic decrosslinking of XHDPE using the compounding screws was carried out at a screw speed of 20 rpm, ultrasonic amplitudes of 5, 7.5, and 10 pm and flow rates of 3.75, 6.5, and 8.25 g/min, corresponding to the mean residence time in the ultrasonic treatment zone of 22.2, 12.8, and 10.1 s, respectively. Ultrasonic decrosslinking of XHDPE using the decrosslinking screws was carried out at ultrasonic amplitudes of 5, 7.5, 10, and 13 pm, a screw speed of 200 rpm and a flow rate of 6.5 g/min corresponding to a mean residence time of 12.8 s. For the compounding screws four kneading elements with stagger angle of 90[degrees] were placed between the exit of the ultrasonic treatment zone and tip of the pressure transducer at the downstream section of the ultrasonic treatment zone. Since the pressure gradient of the polymer flow in the kneading elements with staggering angle of 90[degrees] is low [6], the pressure measured at the downstream section of the ultrasonic treatment zone is approximated as the pressure at the exit of the ultrasonic treatment zone. However, such an approximation cannot be made for the decrosslinking screwdue to the presence of reverse conveying elements. For this reason simulation of decrosslinking in the decrosslinking screw was not conducted. All other procedures and equipment were same as described in earlier studies [4, 5],

Preparation of Specimens and Their Characterization

The compression molded sheets of XHDPE and decrosslinked XHDPE were prepared by following the same procedure as described in Refs. [4, 5], The sheets were molded at a temperature of 200[degrees]C and a pressure of 50 MPa for 20 min.

Swelling of XHDPE and decrosslinked XHDPE was performed by following ASTM D 2765, test method C. The gel fraction of XHDPE and decrosslinked XHDPE was taken as the ratio between weights of the original sample to the dried swollen sample. The crosslink density of XHDPE and decrosslinked XHDPE was determined by Flory-Rehner equation [7].

A capillary rheometer (Rosand RH7, Malvern Instruments, UK) was used to measure the steady state shear viscosity of decrosslinked XHDPE at a temperature of 200[degrees]C using dies of a diameter of 1 mm and L/D ratio of 8, 16, and 24, respectively. The shear viscosity was measured in the apparent shear rate range of 20-200 [s.sup.-1]. Bagley and Rabinovitsch-Weissenberg corrections were applied to calculate the true shear stress and shear rate. The presence of the wall slip during flow of the decrosslinked XHDPE was checked by using dies of diameters of 1, 1.5, and 2 mm at L/ D= 16. It was found that the differences of the pressure drop measured using these dies were within the experimental error indicating the absence of slippage at the wall.

A stress-controlled Discover Hybrid Rheometer (DHR-2, TA Instruments, New Castle, DE) equipped with 25 mm parallel plates to perform the small amplitude oscillatory shear (SAOS) test of decrosslinked XHDPE at temperatures of 140[degrees]C, 150[degrees]C, and 160[degrees]C. A circular disk of a diameter of 25 mm was cut from the compression molded sheet. A gap of 2 mm was used for the test. By measuring dynamic properties at different gaps (1.5 and 2 mm), it was proven that no slippage occurred during the test. The frequency sweep was in the range from 0.1 to 100 rad/s at a shear stress amplitude of 500 Pa being in the linear range of material behavior. The complex viscosity of the decrosslinked XHDPE obtained at different temperatures is used to calculate the activation energy of viscous flow.

EXPERIMENTAL

Gel Fraction and Crosslink Density

Figure 2 shows the gel fraction (a) and crosslink density (b) of the decrosslinked XHDPE obtained by ultrasonic TSE as a function of the ultrasonic amplitude at flow rates of 3.75, 6.5, and 8.25 g/min. The gel fraction and crosslink density of XHDPE obtained by the TSE without imposition of ultrasound are also indicated. It is seen from Fig. 2 that the rotation of screw causes significant decrease of the gel fraction and crosslink density even without imposition of ultrasound due to high shearing effects.

This is due to the presence of three kneading zones in the screws. The imposition of ultrasound further decreases the gel fraction and crosslink density of the decrosslinked XHDPE with the effect increasing with the ultrasonic amplitude. The increase of the flow rate from 3.75 to 8.25 g/min slightly reduces the gel fraction of the decrosslinked XHDPE indicating more decrosslinking at the higher flow rate. This seems to be in contradiction with reduction of the residence time with the flow rate that should lead to a reduction of the decrosslinking effect. The possible reason for the observed phenomenon will be discussed in the later section based on the model.

Rheological Behavior

Figure 3 shows the steady state shear viscosity of the decrosslinked XHDPE as a function of the shear rate at a temperature of 200[degrees]C obtained by TSE using the compounding screws and decrosslinking screws without and with ultrasonic treatment at amplitudes of 10 and 13 pm. The gel fraction of the decrosslinked XHDPE sample is also indicated in Fig. 3 and taken from Refs. [4, 5]. The dotted lines represent the fitting of the viscosity data to the power-law equation:

[eta] = K[([??]/[[??].sub.0]).sup.n-1] (1)

where [eta] is the shear viscosity, [??] is the shear rate, [[??].sub.0] = 1 [s.sup.-1], K is the consistency, n is the power-law index. It should be noted that the value of K is the temperature-dependent function as follows:

[mathematical expression not reproducible] (2)

where [T.sub.b] is the material parameter, R is the gas constant, and E is the activation energy of viscous flow.

Figure 4 shows the dependence of the reduced complex viscosity on the reduced frequency of the decrosslinked XHDPE obtained from TSE using the compounding screws without ultrasonic treatment (a), with ultrasonic treatment at an amplitude of 10 [micro]m (b) and using the decrosslinking screws without ultrasonic treatment (c), with ultrasonic treatment at amplitudes of 10 [micro]m, (d) and 13 [micro]m, (e) at a flow rate of 6.5 g/min at a reference temperature of 160[degrees]C. The dashed lines represent the fittings by Eqs. 1 and 2. It should be noted that three of the five decrosslinked XHDPE samples shown in Figs. 3 and 4 are obtained by TSE using the decrosslinking screws. The measured viscosities of the decrosslinked XHDPE by TSE using the compounding and decrosslinking screws provide a reliable quantitative relationship between the material parameters of the powerlaw model of the decrosslinked XHDPE within wide range of variation of the gel fraction. It is seen in Figs. 3 and 4 that there is a strong shear thinning behavior of the decrosslinked XHDPE caused by the presence of the gel. The steady state shear viscosity and complex viscosity of the decrosslinked XHDPE decreases with a decrease of the gel fraction. The material parameters of the power-law model with the Arrhenius temperature dependence of the decrosslinked XHDPE obtained by TSE using compounding and decrosslinking screws without and with ultrasonic treatment at different gel fractions are tabulated in Table 1. As seen from Table 1, the values of A and n from shear and complex viscosity are different with each other indicating a failure of the Cox-Merz rule. This is due to the presence of gel in the decrosslinked XHDPE.

Figure 5 shows the material parameters, including A(a), n (b), and [T.sub.b] (c), of the power-law model for shear viscosity of the decrosslinked XHDPE as function of the gel fraction. The dotted lines represent the fitting of the dependence of material parameters on the gel fraction by the following equations:

A=0.0012[e.sup.7.4035[zeta]], [T.sub.b] = 7733-1455.1[zeta], n = 0.3308-0.1871[zeta] (3)

It is seen from Fig. 5 that the Eq. 3 adequately describe the dependence of A, n and [T.sub.b] on the gel fraction. Thus, Eq. 3 is used for the flow simulation.

SIMULATION RESULTS

Material and Model Parameters

All the material and model parameters used for simulations of ultrasonic decrosslinking of XHDPE under static conditions are tabulated in Tables 2 and 3. Dimensions of ultrasonic treatment zone are given in Table 4. It should be noted that the dynamic properties of XHDPE at the ultrasonic frequency of 40 kHz are not measured and unavailable in literature. The shear storage modulus in the glassy state required in simulation was calculated using the measured storage Young's modulus of XHDPE below its glass transition temperature using the Poisson ratio of 0.46 of the HDPE [8]. Since the ultrasonic frequency is within the rubbery plateau, the value of [G.sup.0.sub.N] of the linear PE is taken as the shear modulus in the rubbery state. The shear loss tangent of XHDPE at 200[degrees]C at the ultrasonic frequency of 40 kHz is also not available. The shear loss tangent of the molten LLDPE at a frequency of 2 x [10.sup.4] rad/s at a temperature of 150[degrees]C, measured by a piezoelectric rotary vibrator in Ref. 9, is used as the shear loss tangent of XHDPE. It was found that the variation of the shear loss tangent from 0.03 to 0.2 did not affect the predicted results. Also, the longitudinal loss tangent of XHDPE at the ultrasonic frequency of 40 kHz is not available in literature. The longitudinal loss tangent of HDPE at a frequency of 5 MHz is 0.03 [10]. The longitudinal loss tangent of a polymer at a frequency within its rubbery plateau is approximately equal to its bulk loss tangent, tan [[delta].sub.K]:

tan [[delta].sub.L] = L"/L' = K" + 4/3 G"/K' + 4/3 G' [approximately equal to] K"/K' = tan [[delta].sub.K] (3)

This is due to the fact that values of the bulk storage, K', and loss, K", moduli are, respectively, by orders of the magnitude higher than those of G' and G". The bulk loss tangent of a polymer increases with the frequency within its rubbery plateau [17, 18]. Since the ultrasonic frequency used in our experiments is within the frequency range of the rubbery plateau of PE and two orders of the magnitude lower than 5 MHz, it is reasonable to anticipate that the longitudinal loss tangent of XHDPE at the ultrasonic frequency used is lower than 0.03. Thus, we assign the value of tan [[delta].sub.L] = 0.007 for XHDPE at the ultrasonic frequency of 40 kHz. Also, the initial radius of bubble in the present simulation is taken 25 [micro]m, based on earlier simulation of devulcanization of SBR vulcanizates [2].

Ultrasonic Decrosslinking of XHDPE Under Static Condition Without Thermal Degradation

The simulation of ultrasonic decrosslinking of XHDPE under static condition without thermal degradation is performed at a temperature of 200[degrees]C. The effect of the temperature rise caused by the ultrasonic dissipation is neglected. This simulation is carried out to find the dependence of the bubble dynamics, gel fraction, and crosslink density on the ultrasonic amplitude, hydrostatic pressure, bubble volume fraction, and treatment time. The initial gel fraction and crosslink density are, respectively, assigned to be 0.63 and 0.0117 kmol/[m.sup.3], based on the measured values for the sample obtained from TSE without an imposition of the ultrasound.

Figure 6 shows the predicted relative bubble radius as a function of dimensionless time at various ultrasonic amplitudes of 5, 7.5, and 10 [micro]m at a hydrostatic pressure of 1 MPa and a bubble volume fraction of 0.67%. The dimensionless time t/[t.sub.p] is defined as the treatment time, t, divided by the ultrasonic period, [t.sub.p]. It is seen from Fig. 6 that the initial relative bubble radius is below 1. The latter is due to a bubble compression caused by the hydrostatic pressure being higher than the atmospheric pressure. The bubble dynamics shows a periodic steady state after two ultrasonic periods. The amplitude of bubble oscillation increases with the ultrasonic amplitude. The driving force of bubble dynamics is the acoustic pressure. The more intensive bubble dynamics at higher amplitudes is due to the fact that the acoustic pressure is proportional to the ultrasonic amplitude.

Figure 7 shows the predicted relative bubble radius as a function of dimensionless time at various hydrostatic pressures of 0.5, 1, 2, and 3 MPa at an ultrasonic amplitude of 10 [micro]m and a bubble volume fraction of 0.67%. It is seen from Fig. 7 that the initial relative bubble radius decreases with the hydrostatic pressure. Also, after the bubble dynamics attains the periodic steady state both the maximum and minimum relative bubble radii decrease with the hydrostatic pressure. To quantify how the intensity of bubble dynamics vary with the hydrostatic pressure, the variation of bubble volume with respect to its initial volume, i.e., [x.sup.3.sub.max] - [x.sup.3.sub.min], is considered. The values of at hydrostatic pressures of 0.5, 1, 2, and 3 MPa are 1.221, 1.1719, 1.0616, and 0.9405, respectively. It is clear that the significant increase of the hydrostatic pressure leads to an insignificant decrease of the intensity of bubble dynamics. This is due to the fact that the intensity of bubble dynamics is affected by both the hydrostatic and acoustic pressures. Indeed, an increase of the hydrostatic pressure suppresses the bubble dynamics. In contrast, the acoustic pressure, which is the driving force for the bubble dynamics, increases with the hydrostatic pressure. The acoustic pressure is proportional to the sound velocity, which in turn depends on the volume fraction of bubbles that decreases with the hydrostatic pressure. Therefore, the sound velocity increases with hydrostatic pressure.

Figure 8 shows the predicted relative bubble radius as a function of the dimensionless treatment time, [t.sub.r]=t/[t.sub.p] at various bubble volume fractions of 0.33%, 0.67%, and 1.33% at an ultrasonic amplitude of 10 [micro]m and a hydrostatic pressure of 1 MPa. It is seen from Fig. 8 that the maximum relative bubble radius decreases with an increase of the bubble volume fraction, while the minimum increases. Obviously, this is caused by a decrease of the acoustic pressure with the bubble volume fraction.

Figure 9 shows the predicted gel fraction (a) and crosslink density (b) of the ultrasonically decrosslinked XHDPE as a function of the treatment time at various ultrasonic amplitudes of 5, 7.5, and 10 [micro]m at a hydrostatic pressure of 1 MPa and a bubble volume fraction of 0.67%. The initial gel fraction and crosslink density are 0.63 and 0.0117 kmol/[m.sup.3], respectively. It is seen from Fig. 9 that the gel fraction and crosslink density of the ultrasonically decrosslinked XHDPE decreases with the treatment time and the ultrasonic amplitude. It is found that the following sigmoidal equations provide an excellent fitting of the predicted gel fraction and crosslink density of the ultrasonically decrosslinked XHDPE.

[zeta] = [[zeta].sub.0]/[(1 + [at.sup.b.sub.r]).sup.c] (4)

v = [v.sub.0]/[(1 + [at.sup.b.sub.r).sup.c] (5)

where a, b, and c are fitting parameters. The asymptotic analysis of Eqs. 4 and 5 shows that the values of the gel fraction and crosslink density of the ultrasonically decrosslinked XHDPE are zero at the infinite treatment time and equal to initial values at zero treatment time. This analysis indicates that the sigmoidal functions correctly describe the asymptotic behavior of the dependence of the gel fraction and crosslink density of the ultrasonically decrosslinked XHDPE on the treatment time. The fitting parameters of Eqs. 4 and 5 for the dependence of the gel fraction and crosslinked density on the residence time, shown in Fig. 9, are tabulated in Table 5. It is seen from Table 5 that the value of a increases with the amplitude, whereas values of b, and c do not depend on the amplitude, bubble volume fraction, and pressure.

Figure 10 shows the predicted gel fraction (a) and crosslink density (b) of the ultrasonically decrosslinked XHDPE as a function of the treatment time at various hydrostatic pressures of 0.5, 1, 2, and 3 MPa at an ultrasonic amplitude of 10 [micro]m and a bubble volume fraction of 0.67%. Dotted lines represent fittings by Eqs. 4 and 5. It is seen from Fig. 10 that the gel fraction and crosslink density of the ultrasonically decrosslinked XHDPE only slightly increases with the hydrostatic pressure. This is caused by the insignificant decrease of the intensity of the bubble dynamics with the hydrostatic pressure, as indicated in Fig. 7. As seen from Table 5, the value of a decreases with the hydrostatic pressure, whereas values of b and c are not affected.

Figure 11 shows the predicted gel fraction (a) and crosslink density (b) of the ultrasonically decrosslinked XHDPE as a function of the treatment time at various bubble volume fractions of 0.33%, 0.67%, and 1.33% at an ultrasonic amplitude of 10 [micro]m and a hydrostatic pressure of 1 MPa. Dotted lines represent fittings by Eqs. 4 and 5. It is seen from Fig. 11 that the gel fraction and crosslink density of the ultrasonically decrosslinked XHDPE decrease with the volume fraction of bubbles. As seen from Eq. 32 shown in Part I of this study [1], the crosslink density of the ultrasonically decrosslinked XHDPE decreases with the intensity of bubble dynamics and bubble volume fraction. Although the intensity of bubble dynamics at a higher bubble volume fraction is weaker, as seen from Fig. 8, the effect of the bubble volume fraction prevails over the effect of bubble dynamics. Consequently, a lower gel fraction and crosslink density of the ultrasonically decrosslinked XHDPE is seen in case of a higher bubble volume fraction. It is seen from Table 5 the value of a increases with the bubble volume fraction whereas that of b and c are not affected. On the basis of the simulation of ultrasonic decrosslinking of XHDPE under static condition presented here, it is elucidated how the bubble dynamics, gel fraction, and crosslink density of the decrosslinked XHDPE depends on the ultrasonic amplitude, hydrostatic pressure, and bubble volume fraction.

Ultrasonic Decrosslinking of XHDPE in TSE Without Thermal Degradation

The simulation is carried out based on the proposed simulation algorithm for ultrasonic decrosslinking of XHDPE in TSE without thermal degradation. The processing and thermal boundary conditions are tabulated in Table 6. It should be noted that the initial gel fraction and crosslink density of XHDPE in this simulation are, respectively, 0.63 and 0.0117 kmol/[m.sup.3]. These values correspond to those obtained at a flow rate of 6.5 g/min without ultrasonic treatment.

Figure 12 shows the gapwise temperature at axial distances of z/1 = 0, 0.2, 0.4, 0.6, 0.8, and 1.0 from the entrance of the ultrasonic treatment zone at ultrasonic amplitudes of 5 [micro]m (a), 7.5 [micro]m (b), and 10 [micro]m (c) at a flow rate of 6.5 g/min. It is seen from Fig. 12 that the temperature significantly increases with the distance from the entrance and ultrasonic amplitude. This is caused by both the ultrasonic and viscous flow dissipations. Because of the asymmetric thermal boundary condition, as indicated in Table 6, the gapwise temperature distribution becomes also asymmetric within the ultrasonic treatment zone. Also, the maximum of the gapwise temperature distribution is observed. This maximum increases as the polymer moves along the ultrasonic treatment zone.

Figure 13 shows the gapwise distribution of the velocity in the circumferential direction at axial distances of z/1 = 0, 0.2, 0.4, 0.6, 0.8, and 1.0 from the entrance of ultrasonic treatment zone at ultrasonic amplitudes of 5 [micro]m (a), 7.5 [micro]m (b), and 10 pm (c) at a flow rate of 6.5 g/min. A linear distribution of velocity in the circumferential direction is seen at the entrance of the ultrasonic treatment zone due to the uniformity of the temperature distribution. The distribution of the velocity in the circumferential direction exhibits plateau in the extended entrance and exit regions of the ultrasonic treatment zone. The width of this plateau with the axial distance at the entrance and exit regions increases. The width of this plateau at the entrance region decreases with the ultrasonic amplitude. The emergence of the plateau and the increase of its width are caused by the variation of the gapwise temperature distribution with the axial distance. As seen from Fig. 12, the temperature is lower near the entrance and exit regions leading to a higher viscosity near the wall in these regions. To maintain the momentum balance, a lower velocity gradient emerges near the wall causing a plateau of the velocity distribution. This phenomenon becomes more profound when the difference between the temperature in the core region and near the walls becomes more significant.

Figure 14 shows the gapwise distribution of the velocity gradient at axial distances of z/1 = 0, 0.2, 0.4, 0.6, 0.8, and 1.0 from the entrance of ultrasonic treatment zone at ultrasonic amplitudes of 5 [micro]m (a), 7.5 [micro]m (b), and 10 [micro]m (c) at a flow rate of 6.5 g/min. The gapwise distribution of the velocity gradient with the distance from the entrance of the ultrasonic treatment zone is observed to be asymmetric. The velocity gradient near the horn surface (y/h = 0) approaches to zero due to the significantly lower temperature at this surface than that at the screw surface (y/h = 1.0). With an increase of the distance from the entrance, the peak of the gapwise distribution of the velocity gradient becomes sharper and moves toward the middle point of the gap. This is caused by the continuous temperature rise in the core region as the polymer moves toward the exit of the ultrasonic treatment zone. The decrease of the maximum value of the velocity gradient with the ultrasonic amplitude is caused by the decrease of gel fraction at higher amplitude leading to a decrease of viscosity.

Figure 15 shows a gapwise distribution of the velocity in the extrusion direction at axial distances of z/1 = 0, 0.2, 0.4, 0.6, 0.8, and 1.0 from the entrance of ultrasonic treatment zone at ultrasonic amplitudes of 5 [micro]m (a), 7.5 [micro]m (b), and 10 [micro]m (c) at a flow rate of 6.5 g/min. The distribution of the velocity in the extrusion direction, seen at the entrance of the ultrasonic treatment zone, is typical of a power-law fluid. This is due to the assumed isothermal boundary conditions of the flow at the entrance of the ultrasonic treatment zone. As the polymer moves forward, the distribution of the velocity in the extrusion direction becomes asymmetric due to the imposed asymmetric thermal boundary conditions during the nonisothermal flow. A plateau of the velocity distribution in the extrusion direction is seen in an extended region near the horn surface due to the fact that the temperature of the horn surface is significantly lower than that of the rotating shaft surface. A strong correlation between the velocity in the extrusion direction and the velocity gradient in the circumferential direction is seen by comparing Figs. 15 and 14. Clearly, a higher velocity gradient in the circumferential direction leads to a lower viscosity due to shear thinning. This increases the velocity in the extrusion direction.

Figures 16 and 17, respectively, show the gapwise distribution of the gel fraction and crosslink density at axial distances of z/1 = 0, 0.2, 0.4, 0.6, 0.8, and 1.0 from the entrance of the ultrasonic treatment zone at ultrasonic amplitudes of 5 [micro]m (a), 7.5 [micro]m (b), and 10 [micro]m (c) at a flow rate of 6.5 g/min. It is seen from these figures that the gel fraction and crosslink density decrease with the distance from the entrance of the ultrasonic treatment zone and with the ultrasonic amplitude. The asymmetric distributions of the gel fraction and crosslink density are due to the asymmetric distribution of the velocity in the extrusion direction. Also, the shape of the gel fraction and crosslink density distribution curves shows a strong correlation with that of the velocity in the extrusion direction. This is due to the fact that the gel fraction and crosslink density are significantly affected by the residence time which is inversely proportional to the velocity in the extrusion direction.

It should also be noted that the evaluation of the predicted gel fraction and crosslink density of the ultrasonically decrosslinked XHDPE is done by using the Eqs. 4 to 5 and not using Eqs. 33 and 35 shown in Part I of this study [1], Certainly, calculations based on Eqs. 33 to 35 are preferable. However, calculations using Eqs. 33 and 35 are extremely time-consuming since the evaluation of the crosslink density can be done only through iterative technique. Because of a large residence time of the material element near the wall, the computation time using Eq. 33 becomes excessively high (over weeks) and cannot be reduced by means of parallel computing. Obviously, this algorithm is computationally very time-consuming. It should be noted that Eqs. 4 and 5 are able to describe the dependence of the predicted gel fraction and crosslink density on the residence time at a constant hydrostatic pressure with one set of fitting parameters. As discussed earlier, the effect of the hydrostatic pressure on the predicted gel fraction and crosslink density is not significant. Consequently, Eqs. 4 and 5 are used for the simulation with the fitting parameters corresponding to the decrosslinking of XHDPE at a hydrostatic pressure of 1 MPa.

Figure 18 shows the predicted hydrostatic pressure as a function of the axial distance from the entrance of the ultrasonic treatment zone at ultrasonic amplitudes of 5, 7.5, and 10 [micro]m. It is seen from Fig. 18 that both the pressure and pressure gradient significantly decrease with the ultrasonic amplitude. The decrease is due to a reduction of the gel fraction and crosslink density of the ultrasonically decrosslinked XHDPE and an increase of the material temperature with the amplitude, leading to a lower viscosity.

Comparison of Experimental and Simulated Results of Ultrasonic Decrosslinking of XHDPE in TSE Without Thermal Degradation

In this section the results of simulations of the ultrasonic decrosslinking of XHDPE in TSE without thermal degradation are compared with the experimental data. This is of the great interest, since such a comparison is the best way to find the validity of the proposed model. It should be noted that the proposed model is not suitable to correctly predict the gel fraction and crosslink density of the ultrasonically decrosslinked XHDPE, if the breakage of crosslinks and main chains occurs simultaneously. This is due to the assumption that only breakage of the main chains occurs. Figure 19 shows the normalized gel fraction versus normalized crosslink density of the ultrasonically decrosslinked XHDPE by TSE using compounding screws at flow rates of 3.75, 6.5, and 8.25 g/min and various ultrasonic amplitudes. The dotted line represents fitting of these experimental data by the Horikx function in the case of occurrence of the breakage of main chains only [19]. This is given by the following equation:

1 - [v.sub.mech+ultra]/[v.sub.mech] = 1 - [(1 - (1 - [[zeta].sub.mech+ultra]).sup.1/2]).sup.2]/[(1 - (1 - [[zeta].sub.mech]).sup.1/2]).sup.2]

where [[zeta].sub.mech] and [v.sub.mech], [[zeta].sub.mech+ultra], and [v.sub.mech+ultra] are the gel fraction and crosslink density of the decrosslinked XHDPE obtained by the mechanical degradation in TSE without ultrasonic treatment and by combined effect of the mechanical and ultrasonic degradation, respectively. It is seen from Fig. 19 that not all experimental values of the decrosslinked XHDPE coincide with the main chain breakage curve. The data points lying below the main chain breakage curve indicate that there is a contribution of the thermal degradation to the gel fraction and crosslink density. Also, the data lying above the main chain breakage curve indicate that there is a contribution of the crosslink breakage.

It should be noted that the bubble volume fraction is unknown and can only be assigned for the present simulation. The modeling of the ultrasonic power consumption, gel fraction, and crosslink density of the ultrasonically decrosslinked XHDPE indicated that these values are affected by the bubble volume fraction. Clearly, the simulation of the ultrasonic power consumption is more reliable due to less assumption that is made in its modeling. Therefore, different bubble volume fractions are used to simulate ultrasonic decrosslinking of XHDPE to obtain the best fit of the ultrasonic power consumption. Then, the gel fraction and crosslink density are predicted using the same bubble volume fraction.

Figure 20 shows the measured and predicted ultrasonic power consumption during ultrasonic decrosslinking of XHDPE (a), gel fraction (b), and crosslinked density (c) of the ultrasonically decrosslinked XHDPE at a How rate of 6.5 g/min as function of the ultrasonic amplitude at various bubble volume fractions of 0.33%, 0.67%, and 1.33%. It is seen from Fig. 20a that the predicted value of ultrasonic power consumption decreases with the decrease of the bubble volume fraction. This is in agreement with the experimental observation indicating an increase of the power consumption with the ultrasonic amplitude. A very good agreement between the predicted and measured ultrasonic power consumption is observed at amplitudes of 5 and 7.5 [micro]m with a small deviation at an amplitude of 10 [micro]m when a bubble volume fraction of 1.33% is used in the simulation. It is seen from Fig. 20b and c that the predicted gel fraction and crosslink density decrease with the ultrasonic amplitude and the bubble volume fraction. Since a bubble volume fraction of 1.33% provides a best prediction of ultrasonic power consumption, the predicted gel fraction and crosslink density are obtained using a bubble volume fraction of 1.33%. It is seen from Fig. 20b that the gel fraction at amplitudes of 5 and 7.5 pm is underpredicted, but overpredicted at an amplitude of 10 pm. However, as seen from Fig. 20c, there is a very good agreement between the predicted and measured crosslink density. It seems that the proposed model is able to correctly predict the crosslink density, but not the gel fraction. The discrepancy between the predicted and measured gel fraction is due to the deficiency of the proposed model, namely due to the assumption that only breakage of the main chains occurs without the occurrence of the breakage of the crosslinks and thermal degradation. A deficiency of the pre diction at amplitudes of 5 and 7.5 [micro]m can only be fixed by modeling of decrosslinking by including the breakage of crosslinks. The deficiency of the prediction at an amplitude of 10 [micro]m can possibly be eliminated by including of the thermal degradation. Nevertheless, the proposed model is able to predict the ultrasonic power consumption and crosslink density quantitatively and the gel fraction qualitatively using only assumed values of the bubble volume fraction and bubble radius. This is a significant improvement in comparison with the ultrasonic devulcanization model proposed earlier [2, 3, 20-24].

It is interesting to test the performance of the model assuming the same bubble volume fraction of 1.33% at other flow rates. Figure 21 shows the measured and predicted ultrasonic power consumption during ultrasonic decrosslinking of XHDPE (a), gel fraction (b), and crosslink density (c) of the ultrasonically decrosslinked XHDPE at flow rates of 3.75, 6.5, and 8.25 g/min as functions of the ultrasonic amplitude. It is seen from Fig. 21a that there is a fair agreement between the predicted and measured ultrasonic power consumption at all used flow rates. However, the model cannot provide a quantitative agreement between the predicted and measured gel fraction and crosslink density of the decrosslinked XHDPE obtained at flow rates of 3.75 and 8.25 g/min. It is seen from Fig. 21c that the crosslink density of the ultrasonically decrosslinked XHDPE at a flow rate of 3.75 g/min is overpredicted, although the slope of the amplitude dependence is correct. The gel fraction of the ultrasonically decrosslinked XHDPE at a flow rate of 3.75 g/min is also overpredicted at amplitudes 5 and 7.5 [micro]m, as seen from Fig. 21b. It is seen from Fig. 21c that the model correctly predict the crosslink density of the ultrasonically decrosslinked XHDPE obtained at a flow rate of 8.25 g/min and an amplitude of 5 [micro]m, but underpredicts at amplitudes of 7.5 and 10 [micro]m. Also, as seen from Fig. 21b, the measured gel fraction of the ultrasonically decrosslinked XHDPE obtained at a flow rate of 8.25 g/min is lower than the predicted one. The underprediction of the crosslink density and gel fraction may be fixed by using a higher bubble volume fraction. However, the disagreement between slopes of the amplitude dependence of the crosslink density of ultrasonically decrosslinked XHDPE is due to the absence of a suitable model for the ultrasonically induced bubble nucleation. It should be noted that the model for ultrasonic devulcanization of rubbers reported in earlier studies [21-24] overpredicted the effect of flow rate. In both the model for ultrasonic devulcanization of rubbers and decrosslinking of XHDPE, the ultrasonically induced bubble nucleation is not successfully modeled. It seems that a realistic model describing the ultrasonically induced bubble nucleation is the key to correctly predict the effect of flow rate on the ultrasonic decrosslinking of XHDPE [25].

Comparison of Experimental and Simulated Results of Ultrasonic Decrosslinking of XHDPE in TSE Including Thermal Degradation

As seen from Fig. 20, the measured gel fraction of decrosslinked XHDPE at a flow rate of 6.5 g/min and an amplitude of 10 [micro]m is underpredicted. To improve the prediction, the thermal degradation must be included. Since the material parameter B in Eq. 57 shown in Part I of this study [1], is unknown, the value of B is assigned. Therefore, the model is tested by adjusting the value of B to obtain a best fit to the experimental data. Thus, the simulation of ultrasonic decrosslinking of XHDPE in TSE with inclusion of the thermal degradation is carried out for decrosslinking of XHDPE at a flow rate of 6.5 g/min and an amplitude of 10 [micro]m. The material and model parameters used for the simulation with the thermal degradation are the same as in the simulation without the thermal degradation. The value of B is assigned to be 2.3 X [10.sup.8] [m.sup.3]/(kmol s). The predicted gel fraction and crosslink density from the simulation with and without thermal degradation are tabulated in Table 7. It is seen from Table 7 that both simulations with and without the thermal degradation provide excellent predictions of the crosslink density. However, the simulation with the thermal degradation predicts a slightly higher gel fraction than the measured one, whereas the simulation without the thermal degradation underpredicts the gel fraction. Thus, the model for ultrasonic decrosslinking of XHDPE using TSE with inclusion of the thermal degradation provides a quantitative prediction on both the gel fraction and crosslink density of ultrasonically decrosslinked XHDPE at high amplitude showing an importance of inclusion of the thermal degradation in the simulation.

CONCLUSIONS

The numerical simulation of ultrasonic decrosslinking of XHDPE under static condition and using TSE is carried out using a process model developed based on a modified ultrasonic devulcanization model of rubbers. Unlike the earlier model of ultrasonic devulcanization of rubber vulcanizates requiring assigned kinetic parameters, bubble radius, and bubble volume fraction to simulate the ultrasonic devulcanization process, the present model can simulate the ultrasonic decrosslinking of XHDPE by using an assigned bubble radius and volume bubble fraction only. Distribution of the pressure, velocity, temperature, gel fraction, and crosslink density of the decrosslinked XHDPE in the ultrasonic treatment zone is calculated. Comparison between the simulated and experimental results indicates a quantitative agreement for the ultrasonic power consumption during ultrasonic decrosslinking of XHDPE, but a qualitative agreement with the gel fraction and crosslink density. A quantitative agreement on the gel fraction and crosslink density can be achieved by adjusting the bubble volume fraction. Also, incorporation of the thermal degradation into the simulation improves the prediction at high amplitudes. However, the simulation overpredicts the effect of flow rate on the ultrasonic decrosslinking of XHDPE by using the same set of model parameters, possibly due to the absence of a suitable model for the bubble nucleation.

Future studies should be directed toward a theoretical modeling of the ultrasonically induced bubble nucleation to develop a simulation being free of adjustable parameters. In addition, the development of a model for the ultrasonic decrosslinking of XHDPE with the occurrence of crosslinks breakage is highly desirable.

ACKNOWLEDGMENTS

Authors also wish to thank the ExxonMobil Chemical Company for providing HDPE and the Akrochem Corporation for providing peroxide.

REFERENCES

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Keyuan Huang, Avraam I. Isayev

Department of Polymer Engineering, The University of Akron, Akron, Ohio 44325-0301

Correspondence to: A. I. Isayev; e-mail: aisayev@uakron.edu

Contract grant sponsor: NSF Division of Engineering; contract grant number: CMMI-1131342.

DOI 10.1002/pen.24470

Published online in Wiley Online Library (wileyonlinelibrary.com).

Caption: FIG. 1. (a) Schematic of an ultrasonic corotating TSE and (b) its compounding, and (c) decrosslinking screws.

Caption: FIG. 2. (a) Gel fraction and (b) crosslink density of decrosslinked XHDPE in TSE as a function of the ultrasonic amplitude at flow rates of 3.75, 6.5, and 8.25 g/min.

Caption: FIG. 3. Steady state shear viscosity as a function of shear rate of the decrosslinked XHDPE obtained by TSE using compounding and decrosslinking screws without and with ultrasonic treatment at amplitudes of 10 and 13 [micro]m. The dotted lines represent the fittings by Eq. 1a.

Caption: FIG. 4. (a) Dependence of reduced complex viscosity on reduced frequency of decrosslinked XHDPE obtained from TSE using compounding screws without ultrasonic treatment, (b) with ultrasonic treatment at an amplitude of 10 [micro]m, (c) using decrosslinking screw without ultrasonic treatment, (d) with ultrasonic treatment at amplitudes of 10, and (e) 13 [micro]m at a flow rate of 6.5 g/min at a temperature of 160[degrees]C. The dashed lines represent the fittings by Eq. 1.

Caption: FIG. 5. Material parameters (a) A, (b) n, and (c) [T.sub.b] (b) of a power-law viscosity model of decrosslinked XHDPE as function of the gel fraction. Dotted lines represent fittings by Eq. 2.

Caption: FIG. 6. Predicted relative bubble radius as a function of dimensionless treatment treatment time at ultrasonic amplitudes of 5, 7.5, and 10 [micro]m at a hydrostatic pressure of 1 Mpa, a bubble volume fraction of 0.67%.

Caption: FIG. 7. Predicted relative bubble radius as a function of dimensionless treatment time at hydrostatic pressures of 0.5, 1, 2, and 3 MPa at an ultrasonic amplitude of 10 nm and a bubble volume fraction of 0.67%.

Caption: FIG. 8. Predicted relative bubble radius as a function of dimensionless treatment time at various bubble volume fractions of 0.33%, 0.67%, and 1.33% at an ultrasonic amplitude of 10 [micro]m and hydrostatic pressures of 1 MPa.

Caption: FIG. 9. (a) Predicted gel fraction and (b) crosslink density of the ultrasonically decrosslinked XHDPE as a function of the treatment time at various ultrasonic amplitudes of 5, 7.5, and 10 [micro]m at a hydrostatic pressure of 1 MPa and a bubble volume fraction of 0.67%. Dotted lines represent fittings by Eqs. 4 and 5.

Caption: FIG. 10. (a) Predicted gel fraction and (b) crosslink density of the ultrasonically decrosslinked XHDPE as a function of the treatment time at hydrostatic pressures of 0.5. 1, 2, and 3 MPa at an ultrasonic amplitude of 10 [micro]m and a bubble volume fraction of 0.67%. Dotted lines represent fittings by Eqs. 4 and 5.

Caption: FIG. 11. (a) Predicted gel fraction and (b) crosslink density of the ultrasonically decrosslinked XHDPE as a function of treatment time at bubble volume fractions of 0.33%, 0.67%, and 1.33% at an ultrasonic amplitude of 10 [micro]m and a hydrostatic pressure of 1 MPa. Dotted lines represent fittings by Eqs. 4 and 5.

Caption: FIG. 12. Gapwise temperature distribution at axial distances of z/1 = 0, 0.2, 0.4, 0.6, 0.8, and 1.0 from the entrance of ultrasonic treatment zone at ultrasonic amplitudes of (a) 5 [micro]m, (b) 7.5 [micro]m, and (c) 10 [micro]m at a flow rate of 6.5 g/min.

Caption: FIG. 13. Gapwise distribution of the velocity in the circumferential direction at axial distances of z/1 = 0, 0.2, 0.4, 0.6, 0.8, and 1.0 from the entrance of ultrasonic treatment zone at ultrasonic amplitudes of (a) 5 [micro]m, (b) 7.5, and (c) 10 [micro]m at a flow rate of 6.5 g/min.

Caption: FIG. 14. Gapwise distribution of the velocity gradient at axial distances of z/1 = 0, 0.2, 0.4, 0.6, 0.8, and 1.0 from the entrance of ultrasonic treatment zone at ultrasonic amplitudes of (a) 5 [micro]m, (b) 7.5 [micro]m, and (c) 10 [micro]m at a flow rate of 6.5 g/min.

Caption: FIG. 15. Gapwise distribution of the velocity in the extrusion direction at axial distances of z/1 = 0, 0.2, 0.4, 0.6, 0.8, and 1.0 from the entrance of ultrasonic treatment zone at ultrasonic amplitudes of (a) 5 [micro]m, (b) 7.5 [micro]m, and (c) 10 [micro]mat a flow rate of 6.5 g/min.

Caption: FIG. 16. Gapwise distribution of the gel fraction at axial distances of z/ 1 = 0, 0.2, 0.4, 0.6, 0.8, and 1.0 from the entrance of ultrasonic treatment zone at ultrasonic amplitudes of (a) 5 [micro]m, (b) 7.5 [micro]m, and (c) 10 [micro]m at a flow rate of 6.5 g/min.

Caption: FIG. 17. Gapwise distribution of the crosslink density at axial distances of z/l = 0, 0.2, 0.4, 0.6, 0.8, and 1.0 from the entrance of ultrasonic treatment zone at ultrasonic amplitudes of (a) 5 [micro]m, (b) 7.5 [micro]m, and (c) 10 [micro]m at a flow rate of 6.5 g/min.

Caption: FIG. 18. Hydrostatic pressure as a function of the distance from the entrance of the ultrasonic treatment zone at ultrasonic amplitudes of 5, 7.5, and 10 [micro]m.

Caption: FIG. 19. Normalized gel fraction versus normalized crosslink density of the ultrasonically decrosslinked XHDPE by TSE using compounding screws at flow rates of 3.75, 6.5, and 8.25 g/min and various ultrasonic amplitudes. The dotted line represents the fitting by Eq. 6.

Caption: FIG. 20. Measured and predicted ultrasonic power consumption during ultrasonic decrosslinking of (a) XHDPE, (b) gel fraction, and (c) crosslinked density of the ultrasonically decrosslinked XHDPE as a function of the ultrasonic amplitude at bubble volume fractions of 0.33%, 0.67%, and 1.33% at a flow rate of 6.5 g/min.

Caption: FIG. 21. Measured and predicted ultrasonic power consumption during ultrasonic decrosslinking of (a) XHDPE, (b) gel fraction, and (c) crosslinked density of the ultrasonically decrosslinked XHDPE as a function of the ultrasonic amplitude at flow rates of 3.75, 6.5, and 8.25 g/min at a bubble volume fractions of 1.33%.
TABLE 1. Material parameters of a power-law model and gel fraction
of decrosslinked XHDPE.

                              Shear viscosity     Complex viscosity

Type of screw    Amplitude    A (Pa s)     n      A (Pa s)     n
                 ([micro]m)
Compounding          0         0.1184    0.2132   0.04146    0.1238
                     10        0.0607    0.2241   0.0247     0.1439
Decrosslinking       0         0.0557    0.2459   0.02385    0.1681
                     10        0.0129    0.2743   0.00589    0.2156
                     13        0.0037    0.2991   0.001432   0.2981

Type of screw    [T.sub.b]    E (kJ/mol)     Gel
                    (K)                    fraction
Compounding         6886        57.25        0.63
                    6982        58.05        0.53
Decrosslinking      6900        57.37        0.47
                    7254        60.31        0.35
                    7552        62.79        0.15

TABLE 2. Material parameters used in simulation of ultrasonic
decrosslinking of XHDPE.

Material   Description                                  Value
XHDPE      Density (kg/[m.sup.3])                        730
           Surface tension (N/m)                        0.028
           Storage Young's                              2,500
           Modulus at [omega]=[infinity] (MPa)
           Poission ratio at [omega]=[infinity]         0.46
           Shear modulus at [omega]=[infinity] (MPa)     856
           Shear modulus at [omega]=0 (MPa)              2.3
           Shear loss tangent at 40 kHz                  0.1
           Bulk sound velocity (m/s)                     900
           Longitudinal loss tangent at 40 kHz          0.007
           Initial bubble radius ([micro]m)              25
           Heat capacity (J/kg/K)                       2,876
           Thermal conductivity (W/m/K)                 0.22
Titanium   Density (kg/[m.sup.3])                       4,506
           Bulk sound velocity (m/s)                    6,100
Air        Ratio of the specific heats of the gas        1.4

Material   Reference source
XHDPE             11
                  12
            Measured value

                  8
            Measured value
                  8
                  9
                  15
            Assigned value
            Assigned value
                  13
                  13
Titanium          14
                  14
Air               2

TABLE 3. Model parameters used in simulation of ultrasonic
decrosslinking of XHDPE.

Description                            Value        Reference source
Initial bubble radius ([micro]m)         25          Assigned value
Activation energy of [beta]             122                16
  scission reaction (kJ/mol)
Ultrasonic frequency (Hz)          4 x [10.sup.4]       Measured

TABLE 4. Dimensions of ultrasonic treatment zone.

Height (h) (mm)   2.54
Length (l) (mm)    28
Width (w) (mm)     22

TABLE 5. Fitting parameters of Eqs. 4 and 5.

                                 Hydrostatic   Bubble volume
Object of fitting   Amplitude     pressure     fraction (%)
                    ([micro]m)      (MPa)
Gel fraction            5             1            0.67
                       7.5
                        10
                                     0.5
                                      2
                                      3
                                      1            0.33
                                                   1.33
Crosslink density       5             1            0.67
                       7.5
                        10
                                     0.5
                                      2
                                      3
                                      1            0.33
                                                   1.33

Object of fitting   a, [s.sup.-b]     b        c

Gel fraction           0.0182       0.9977   0.3230
                       0.0273       0.9970   0.3238
                       0.0364       0.9963   0.3243
                       0.0379       0.9963   0.3244
                       0.0330       0.9965   0.3241
                       0.0293       0.9968   0.3239
                       0.0219       0.9975   0.3233
                       0.0570       0.9954   0.3252
Crosslink density      0.0220       1.0062   0.6952
                       0.0332       1.0072   0.6932
                       0.0444       1.0079   0.6919
                       0.0464       1.0079   0.6917
                       0.0402       1.0076   0.6923
                       0.0356       1.0074   0.6929
                       0.0265       1.0067   0.6943
                       0.0702       1.0086   0.6902

TABLE 6. Processing and thermal boundary conditions used in
Simulation of ultrasonic decrosslinking of XHDPE in TSE
without thermal degradation.

Screw speed (rpm)                                          200
Flow rate (g/min)                                          6.5
Ultrasonic amplitude ([micro]m)                       5, 7.5, and 10
Temperature of horn ([degrees]C)                           145
Temperature of screws ([degrees]C)                         200
Temperature of polymer melt at the                         200
 entrance of ultrasonic treatment zone ([degrees]C)

TABLE 7. Measured and predicted gel fraction and crosslink density of
decrosslinked XHDPE with and without inclusion of thermal degradation at
a flow rate of 6.5 g/min and an amplitude of 10 [micro]m.

                                                      Crosslink
                                           Gel      density (kmol/
                                         fraction     [m.sup.3])
Measured                                  0.5327        0.0086
Predicted without thermal degradation     0.5626        0.0089
Predicted with thermal degradation        0.5427        0.0092
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Author:Huang, Keyuan; Isayev, Avraam I.
Publication:Polymer Engineering and Science
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Date:Oct 1, 2017
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