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Ultrasonic Decrosslinking of Peroxide Crosslinked HDPE in Twin Screw Extrusion. I. Process Model and Simulation.

INTRODUCTION

Extensive experimental investigations of ultrasonic decrosslinking of XHDPE were carried out [1-5]. It was found that ultrasonic decrosslinking is a promising technology to obtain melt-processible materials exhibiting excellent mechanical performance. To optimize the scale up of the ultrasonic decrosslinking of XHDPE, a science-based technology is required. Thus, it is important to develop a theoretical model to describe the ultrasonic decrosslinking process of XHDPE.

Ultrasonic devulcanization of rubber vulcanizates is akin to ultrasonic decrosslinking of XHDPE. The theoretical simulations of ultrasonic devulcanization of rubber vulcanizates under static (no flow) condition [6-8] and continuous extrusion condition in the single screw extruder [9-12] were carried out. The simulation of continuous ultrasonic devulcanization of rubber vulcanizates allowed us to calculate distributions of the pressure, velocity, temperature, viscosity, gel fraction, and crosslink density in the ultrasonic treatment zone. Comparison of the simulation with the experimental observation showed a qualitative agreement for ultrasonic devulcanization of rubber under continuous condition [9-12] and quantitative agreement under static condition [8]. However, these earlier simulations described the ultrasonic devulcanization of rubber vulcanizates as a kinetic process whose kinetic parameters were either assumed or determined by the best fitting to the experimental data. Since the dependence of the kinetic parameters on the ultrasonic amplitude is a priori unknown, the potential for the simulation of ultrasonic devulcanization is limited. Thus, it is desirable to develop a theoretical simulation of ultrasonic decrosslinking of XHDPE without using assigned kinetic parameters.

Accordingly, the present study is aimed to build up a process model and develop simulation algorithms for ultrasonic decrosslinking of XHDPE using TSE. This proposed model of decrosslinking of XHDPE in the ultrasonic treatment zone of TSE does not require data on kinetic parameters. The aim of this simulation is to calculate the distribution of pressure, velocity, temperature, viscosity, gel fraction, and crosslink density in the ultrasonic treatment zone in TSE.

THEORETICAL MODELING OF ULTRASONIC DECROSSLINKING OF XHDPE

Statement of the Problem

The ultrasonic decrosslinking of XHDPE using twin-screw extrusion is a complicated process. Figure 1 shows the schematics of the ultrasonic treatment zone in TSE. The polymer flows through the ultrasonic treatment zone consisting of the ultrasonic horn and co-rotating cylindrical elements with a smooth surface. A curved surface of the ultrasonic horn provides a uniform gap between the surfaces of the ultrasonic horn and cylindrical elements. Due to the high aspect ratio of the planar dimensions (x, z) to the thickness (y) of ultrasonic treatment zone in TSE, the flow of polymer can be approximated as a helical flow. The ultrasonic horn provides the longitudinal ultrasonic waves propagating in the direction y being perpendicular to the extrusion direction, z. The imposition of the ultrasonic waves on flowing XHDPE provides energy for its ultrasonic decrosslinking.

Theoretical modeling of ultrasonic decrosslinking of XHDPE using TSE includes four aspects. First, the imposition of ultrasonic waves induces the dynamics of bubble cavitation in XHDPE and decrosslinked XHDPE. Second, the propagation of ultrasound in decrosslinked XHDPE generates an ultrasonic dissipation. Third, due to the ultrasonically induced bubble dynamics the crosslinked network is ruptured. Fourth, in the presence of the viscous flow and ultrasonic dissipation, the flow of decrosslinked XHDPE in the ultrasonic treatment zone is nonisothermal with its shear viscosity being greatly affected by the gel fraction and crosslink density.

Modeling of Bubble Dynamics. The modeling of the dynamics of a spherical bubble oscillation in an infinite viscoelastic material subjected to ultrasound was studied earlier [6]. It used a viscoelastic constitutive equation to describe the viscoelasticity of polymer. Various viscoelastic constitutive equations including Lodge model [13], upper convected Maxwell model [13, 14] and K-BKZ model [15] were employed to predict the bubble dynamics in a viscoelastic material. For purpose of the simplicity, in the present study the viscoelasticity of XHDPE and decrosslinked XHDPE was described by the Lodge model [13]:

[mathematical expression not reproducible] (1)

where [??](t) and P(t) are the stress tensor and its spherical part at time of t, G(t) is the relaxation shear modulus, is the memory function, [??](t) and [[??].sub.*](t,s) are the Finger tensor and relative Finger tensor, respectively, and [??] is the unit tensor.

Let us consider a single spherical gas bubble in a non-deformed polymer. Since the gas bubble is in its equilibrium state, the force balance at the interface between the bubble and polymer is given as

[P.sub.c] = [P.sub.0] 2[sigma]/[R.sub.0] (2)

where [P.sub.c] is the pressure at the wall of the bubble in polymer, [P.sub.0] is the pressure in the bubble in the equilibrium state, a is the surface tension at the interface between the gas bubble and polymer, and [R.sub.0] is the bubble radius in the equilibrium state (atmospheric pressure). It is convenient to express the bubble radius during its oscillation as

[r.sub.0](t)=[R.sub.0]x(t) (3)

where [r.sub.0](t) and x(t) is the bubble radius and relative bubble radius at the time of t, respectively.

The equation of motion governing the dynamics of the bubble in an infinite viscoelastic body described by the Lodge model is given as [6]

[mathematical expression not reproducible] (4)

where [rho] is the polymer density, [P.sub.[infinity]](t) is the ambient pressure in the polymer far from the bubble. Assuming the thermodynamic behavior of the bubble is adiabatic, the [P.sub.c][x(t)] is defined:

[P.sub.c][x(t)] = ([P.sub.0] + 2[sigma]/[R.sub.0])[x.sup.3[gamma]](t) = 2[sigma]/[R.sub.0]x(t) (5)

where [gamma] is the ratio of the specific heats of the gas. Due to the presence of ultrasound, the ambient pressure in polymer is time dependent and given as

[P.sub.[infinity]](t)=P-[P.sub.A] sin [omega]t (6)

where P and [P.sub.A] are the hydrostatic and acoustic pressure, respectively and [omega] is the ultrasonic frequency. The determination of the acoustic pressure will be discussed in the next section. The shear relaxation modulus, G(t) and the storage, G'([omega]), and loss, G"([omega]), moduli of the polymer is described by the generalized Maxwell model:

[mathematical expression not reproducible]

G'([omega])=[G.sub.e] + [n.summation over (i+1)] [G.sub.i] [[omega].sup.2][[lambda].sup.2.sub.i]/1 + [[omega].sup.2][[lambda].sup.2.sub.i] (7b)

G"([omega])=[G.sub.e] + [n.summation over (i+1)] [G.sub.i] [omega][[lambda].sub.i]/1 + [[omega].sup.2][[lambda].sup.2.sub.i] (7c)

where [G.sub.i] and [[lambda].sub.i] are the shear modulus and relaxation time of i-th mode and [G.sub.e] is the equilibrium shear modulus. Typically, the value of [G.sub.i] and [[lambda].sub.i] can be calculated by fitting the master curve of the experimentally measured storage and loss moduli to the generalized Maxwell model. The master curve of the storage and loss moduli is typically obtained by measuring the dynamic moduli as a function of the frequency at various temperatures and shifting them to a reference temperature using the temperature-frequency superposition. However, the crystallization of XHDPE and decrosslinked XHDPE reduces the accessible temperature range of dynamic test leading to an insufficient experimental measurement to construct a master curve. Another possibility to determine the value of [G.sub.i] and [[lambda].sub.i] is to derive them based on a molecular linear viscoelastic model, such as Doi-Edward model [16] and its modification [17]. Indeed, these models provide a realistic prediction of the relaxation modulus function of uncrosslinked polymers over a wide range of time. However, due to the presence of the crosslinks in XHDPE and decrosslinked XHDPE, these models are not suitable in our case. Therefore, without availability of sufficient experimental data and realistic model, the linear viscoelasticity of XHDPE, and decrosslinked XHDPE is approximated by a standard viscoelastic solid:

G(t) = ([G.sub.u] - [G.sub.e])[e.sup.-t/[lambda]] + [G.sub.e] (8)

where [G.sub.u] is the elastic modulus in the glassy state. It is known from literature [17] that the ultrasonic frequency (40 kHz) used in the present study is within the frequency range corresponding to the rubbery plateau of PE. It should be noted that our preliminary results indicated that the presence of modes with relaxation times by an order of the magnitude lower and higher than the reciprocal of the ultrasonic frequency has, respectively, insignificant and significant effects on the numerical simulation of bubble dynamics. Therefore, only the relaxation processes of XHDPE and decrosslinked XHDPE in their rubbery plateau and transition region to the glassy state are of interest. Accordingly, the storage modulus of PE in the rubbery plateau, [G.sup.0.sub.N], is used as the [G.sub.e] value of XHDPE and decrosslinked XHDPE. Consequently, Eq. 8 is rewritten as

G(t) = ([G.sub.u] - [G.sup.0.sub.N])[e.sup.-t/[lambda]] + [G.sup.0.sub.N] (9)

The value of [lambda] can be determined if the shear loss tangent, Tan [[delta].sub.s], at the ultrasonic frequency is known. Substitution of G(t) defined by Eq. 9 into Eq. 4 leads to a system of the ordinary differential equations governing the bubble dynamics as [6]

[partial derivative]x(t)/[partial derivative]t =u(t) (10a)

[rho][R.sup.2.sub.0](x(t)[partial derivative]u(t)/[partial derivative]t + 3/2 [u.sup.2](t))+[G.sup.0.sub.N]/ 2(5 - 4/x(t) - 1/[x.sup.4]I(t)) (10b)

[mathematical expression not reproducible] (10c)

[partial derivative][z.sub.2](t)/[partial derivative] = 4 u(t)/x(t-1/[lambda] + 4 u(t)/x(t))[z.sub.2](t) (10d)

where [z.sub.1](t) and [z.sub.2](t) in Eqs. 10c and 10d are auxiliary functions describing the relaxation process of XHDPE and decrosslinked XHDPE. Equation 10 is solved by using the following initial conditions:

x(0) =[x.sub.0], u(0)=0, [z.sub.1](0)=[z.sub.2](0)=0 (11)

The [x.sub.0] can be determined by solving the following Eq. 6:

[G.sup.0.sub.N]/2(5 - 4/[x.sub.0] - 1/[x.sup.4.sub.0]) = [P.sub.c]([x.sub.0]) - P (12)

Modeling of Ultrasonic Power Consumption. The ultrasonic power consumption is due to the energy loss during the propagation of ultrasonic waves in the polymer. It is dependent on the viscoelasticity of the polymer. In case of the ultrasonic decrosslinking of XHDPE, the propagation of ultrasonic waves in a thin layer of XHDPE, as in the present case, is very different from that in a bulk XHDPE. In the present study, the thickness of the ultrasonic treatment zone is small (2.54 mm) in comparison with the wavelength of the ultrasound in XHDPE and decrosslinked XHDPE (22.5 mm). Since the sound impedance of the metal is by several orders of magnitude higher than that of XHDPE and decrosslinked XHDPE, a strong reflection of ultrasound occurs at the interface. In the previous study on the ultrasonic devulcanization of rubber vulcanizates under the static condition [6], a theoretical model was developed to describe the propagation of ultrasound in a thin layer of the polymer containing a small amount of bubbles. This model provides the ultrasonic power consumed by the ultrasonic horn. Here we briefly introduce the model. Figure 2 shows a schematic of a thin layer of the bubble-filled polymer placed between a metal plate and an ultrasonic horn. The thickness of this thin layer of the bubble-filled polymer is h. For the sake of simplicity, a uniform distribution of bubbles of same sizes is assumed. Due to the presence of the bubbles, the sound velocity in the bubble-filled polymer, c*, is lower than that in polymer without bubbles, c. Under condition of a small volume fraction of bubbles, [[phi].sub.0]

[[phi].sub.0] = 4[pi]/3 [bar.n][R.sup.3.sub.0 [??] 1 (13)

where [bar.n] is the concentration of the bubbles. The sound velocity in the bubble-filled polymer is [6]:

[c.sub.*] = c/[(1 + [[phi].sub.0][x.sup.3.sub.0](P)[rho][c.sup.2]X(P)).sup.1/2] (14)

where X(P) is the compressibility of the bubble in a polymer. The X(P) is calculated by the following equation:

X(P) = 1/[phi][absolute value of d[phi]/dP] =3[[absolute value of [x.sub.0] dP/d[x.sub.0]].sup.-1] (15)

The value of dP/d[x.sub.0] can be obtained by rewriting Eq. 12 and taking the derivative with respect to [x.sub.0]. Based on the linear acoustic theory, the acoustic pressure in a thin layer of the bubble filled polymer at the ultrasonic amplitude of [A.sub.u] as a function of the position along thickness direction is

[mathematical expression not reproducible] (16a)

[Z.sub.m] = [[rho].sub.m][c.sub.m] (16b)

[Z.sub.p] = [rho][c.sub.*]/1 + i tan [[delta].sub.L]/2], (16c)

where tan [[delta].sub.L] is the longitudinal loss tangent of the polymer at the frequency of [omega], [[rho].sub.m], and [c.sub.m] are the density and sound

velocity in the metal used to make the ultrasonic horn, respectively. [Z.sub.m] and [Z.sub.p] are the sound impedance of the metal and polymer, respectively. The value of tan [[delta].sub.L] is defined as [18]

tan[[delta].sub.L] = L"/L' = K"/K' + 4/3 G"/K' + 4/3 G' (17)

where L' and L" are the storage and loss longitudinal moduli at a frequency of [omega], respectively. K' and K" are the storage and loss bulk moduli at a frequency of [omega], respectively. Finally, the ultrasonic power consumption of the ultrasonic horn, J, operating at a frequency of co and an ultrasonic amplitude of [A.sub.u] is given as [6]

[mathematical expression not reproducible] (18a)

q = [bar.[omega]/[c.sub.*](1 + i tan [[delta].sub.L]/2) (18b)

where q is the wave number of the ultrasound in the polymer.

Modeling of Decrosslinking. Due to the presence of the dynamics of bubble oscillations induced by the ultrasonic treatment, the main chains in XHDPE around the bubble are deformed. For a material point at the distance of [bar.R] from the center of a bubble ([bar.R] [greater than or equal to] [R.sub.0]), its current distance from the center of a pulsating bubble is [7]:

[mathematical expression not reproducible] (19)

The change of the local length caused by the dynamics of the bubble is characterized by using the stretch tensor [??]. Since the expansion and contraction motion of a spherical bubble is equal-biaxial and XHDPE is considered to be incompressible, the explicit form of the stretch tensor [??] is given [7]:

[mathematical expression not reproducible] (20)

Assuming the affine deformation of the main chains between the crosslinks in XHDPE around a pulsating bubble and the end-to-end vector of the polymer chain in its equilibrium state is [??] = {[y.sub.1], [y.sub.2], [y.sub.3]}, the end-to-end vector of the main chain in its deformed state is

[mathematical expression not reproducible] (21)

The breakage of the polymer chains between the crosslinks occurs when the magnitude of its end-to-end vector in its deformed state is larger than its contour length ([mathematical expression not reproducible]). Since the both the end-to-end vector of the polymer chains between crosslinks in its equilibrium state and the contour length of the polymer chains between crosslinks are not uniquely defined, but obey certain distributions, the mathematical expression of the fraction of the unbroken polymer chains at a distance of [bar.R] from the center of an oscillating bubble is [7]

[mathematical expression not reproducible] (22)

where P([??]) and f(L) are probability distribution function of [??] and L, respectively. 8 is the Heaviside function

[mathematical expression not reproducible] (23)

Since XHDPE is prepared by crosslinking of HDPE using peroxide, the crosslinking reaction is random. For a polymer chain consisting of m monomeric units, the explicit form of f(L) is

f(L) = f(m) = [e.sup.-m/N]/N (24)

where N is the average number of monomeric units between crosslinks. The probability distribution of the polymer chains between crosslinks is approximated by the Gaussian distribution. Thus, the explicit form of P([??]) in Eq. 22 is

[mathematical expression not reproducible] (25)

where [L.sub.m] is the length of the monomeric unit of XHDPE. It is interesting to note that the derivation of Lodge model is also based on the assumptions of the Gaussian distribution of the polymer chains and affine deformation. Thus, the assumption of Lodge model is consistent with the assumptions describing the breakage of polymer chains in the network. Substitution of Eqs. 24 and 25 into Eq. 22 and its subsequent integration gives [7]

[mathematical expression not reproducible] (26)

In the equilibrium state [[epsilon].sub.min] = [[epsilon].sub.max] = 1, accordingly, F(1,1) should be equal to 1. However, due to the deficiency of the Gaussian distribution, F(1, 1) is slightly lower than 1 [7]. To correct this deficiency, the fraction of the unbroken polymer chains at a distance of R from the center of a pulsating bubble is [7]

[F.sub.c]([[epsilon].sub.min], [[epsilon].sub.max]) = F([[epsilon].sub.min], [[epsilon].sub.max])/F(1,1) (27)

Accordingly, the fraction of the broken polymer chains at a distance of R from the center of a pulsating bubble is

[PHI]([[epsilon].sub.min], [[epsilon].sub.max]) = 1 - [F.sub.c] ([[epsilon].sub.min], [[epsilon].sub.max]) = 1 - F([[epsilon].sub.min], [[epsilon].sub.max])/F(1,1) (28)

If the crosslinks remain intact and only breakage of main chains between crosslinks occurs, the normalized crosslink density of the gel in XHDPE after one ultrasonic cycle is given as

[mathematical expression not reproducible] (29)

where [v.sub.XHDPE] and [v.sub.DXHDPE] are the crosslink density of XHDPE and decrosslinked XHDPE, respectively. In fact, the upper limit of the integral in Eq. 29 should be cut off at the distance of [R.sub.max]. Since [bar.n][R.sup.3.sub.0] [??] 1, then [R.sub.max] [congruent to] [[bar.n].sup.-1/3]. In case of a weak bubble dynamics induced by ultrasound, i.e. the maximum, [x.sub.max], and minimum, [x.sub.min], relative bubble radius are not far away from the unity, an analytical solution of Eq. 29 is obtained

[v.sub.DXHDPE]/[v.sub.DXHDPE] = 1 - 8[pi]/[3.sup.5/2] [x.sup.3.sub.max] - [x.sup.3.sub.min]/1 + 3N/2 [absolute value of [bar.n][R.sup.3.sub.0]1n([bar.n][R.sup.3.sub.0])] (30)

It should be pointed out that XHDPE experiences not only ultrasonic decrosslinking but also mechanical decrosslinking caused by the rotation of the screws during extrusion of XHDPE [2-5]. Therefore, in this study, a more appropriate definition of the normalized crosslink density of ultrasonically decrosslinked XHDPE is defined as

[v.sub.DXHDPE]/[v.sub.0] = 1 - 8[pi]/[3.sup.5/2] [x.sup.3.sub.max] - [x.sup.3.sub.min]/1 + 3N/2 [absolute value of [bar.n][R.sup.3.sub.0]1n([bar.n][R.sup.3.sub.0])] (31)

where [V.sub.0] is the crosslink density of the material entering the ultrasonic treatment zone. Equation 31 is only applicable for the material entering the ultrasonic treatment zone whose gel fraction is 1. However, it is known that the gel fraction of the material entering the ultrasonic treatment zone in TSE is less than 1 due to the occurrence of mechanical decrosslinking. Therefore, Eq. 31 is modified to take into account this change in the gel fraction:

[v.sub.DXHDPE]/[v.sub.0] = 1 - [[zeta].sub.0] 8[pi]/[3.sup.5/2] [x.sup.3.sub.max] - [x.sup.3.sub.min]/1 + 3N/2] [absolute value of [bar.n][R.sup.3.sub.0]ln([bar.n][R.sup.3.sub.0])] (32)

where [[zeta].sub.0] is the gel fraction of the material entering the ultrasonic treatment zone. Using Eq. 32 the normalized crosslink density of the ultrasonically decrosslinked XHDPE can be calculated during one oscillation period, [t.sub.p] = 2.5 X [10.sup.-5] s, in the present case.

To incorporate the effect of the residence time, the model is modified as described below.

The mean residence time in the ultrasonic treatment zone, [t.sub.r], is much longer than the period of oscillation. The value of [t.sub.r] contains k cycles, that is, k = [bar.[omega][t.sub.r]/2[pi]. Accordingly, the values of [v.sub.DXHDPE], [zeta]DXHDPE, [x.sub.max], [x.sub.min], and N are functions of the residence time, that is, [v.sub.DXHDPE](i), [zeta]DXHDPE(i), [x.sub.max](i), [x.sub.min](i), and N(i), where i is the sequence of the cycles in [t.sub.r]. Thus, the normalized crosslink density of the ultrasonically decrosslinked XHDPE after the ultrasonic treatment for time of [t.sub.r] is expressed as

[mathematical expression not reproducible] (33)

The value of N(i) is calculated from the crosslink density as

N(i) = [rho]/[v.sub.DXHDPE](i) x [M.sub.0] (34)

where [M.sub.0] is the molecular weight of ethylene. To calculate the crosslink density of the ultrasonically decrosslinked XHDPE based on Eqs. 33 and 34, the value of [[zeta].sub.DXHDPE](i) needs to be determined. According to the assumption that only breakage of the main chains occurs during the ultrasonic decrosslinking of XHDPE, the gel fraction of the ultrasonically decrosslinked XHDPE at the end of ;th period within the residence time [t.sub.r] is calculated based on the Horikx functions describing the normalized crosslink density vs. the normalized gel fraction [19]:

1 - [v.sub.DXHDPE](i)/[v.sub.0] = (1 - (1 - (1 - [[zeta].sub.DXHDPE(i)).sup.1/2).sup.2]/ (1 - (1 - [[zeta].sub.0]).sup.1/2]).sup.2] (35)

Thus, the gel fraction and crosslink density of the ultrasonically decrosslinked XHDPE is determined by using Eqs. 33-35 under assumption that only breakage of main chains occurs. Based on the experimental observations [1-5], this assumption is not necessarily true. Accordingly, it is convenient to model the breakage of crosslinks based on the assumption that affine deformation of crosslinks occurs during decrosslinking of XHDPE. Indeed, the modeling of the breakage of crosslinks is made by using this approach. However, the model predicts that more than 90% of crosslinks are broken within 10 ultrasonic cycles. The latter significantly over predicts the breakage of crosslinks in comparison with experimental findings [1-5]. The reason of such an over prediction is due to the assumption of the affine deformation. Since the crosslinks in XHDPE are C-C bonds, the crosslinks are easily broken even under the imposition of a small affine deformation. Thus, the criterion for the breakage of crosslinks has to be altered such that crosslinks are not easily broken. Unfortunately, no explicit criterion is available from the classical polymer physics. Thus, the modeling of the breakage of crosslinks is not addressed in the present study.

Modeling of Nonisothermal Flow. In the previous discussion, it is assumed that XHDPE contains small amount of bubbles. Clearly, the flow behavior of the bubble-filled polymer melt is different from the bubble-free polymer melt. The volume fraction of the bubbles is small (typically <2%), the effect of bubbles on the flow behavior of polymer melt is neglected. Also, for the sake of simplicity, the density of XHDPE is assumed to be independent of the temperature.

To carry out the flow simulation, a proper constitutive equation is needed. It is known that Lodge model is unable to predict the shear thinning behavior of a polymer under steady state shear flow [20]. In fact, a strong shear thinning behavior is seen for the decrosslinked XHDPE [1]. Therefore, the flow behavior of the decrosslinked XHDPE is described by a power-law fluid model exhibiting an Arrhenius dependence on the temperature:

[mathematical expression not reproducible] (36)

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, [T.sub.b] is the material parameter related to the activation energy of viscous flow, E and R is the gas constant. Clearly, the values of K, n, [T.sub.b], and E are affected by the gel fraction, crosslink density and molecular structure of the decrosslinked XHDPE. It is assumed that these values are functions of the gel fraction:

K=K([zeta]), n=n([zeta]), [T.sub.b] = [T.sub.b]([zeta]) (37)

The explicit forms of Eq. 37 will be determined by fitting to the measured viscosity.

The circumferential direction of rotating cylindrical elements, gapwise, and extrusion directions in the ultrasonic treatment zone were defined earlier as x, y, and z, respectively (see Fig. 1). Since the flow in the ultrasonic treatment zone is helical [21], the equation of motion governing the helical flow of the decrosslinked XHDPE is given as

d[[??].sub.yx]/dy = d([eta] [dv.sub.x]/dy)/dy = 0 (38)

d[[??].sub.yz]/dy = d([eta] [dv.sub.z]/dy)/dy = dP/dz (39)

where [??] is the deviatoric part of stress tensor, dP/dz is the pressure gradient, [v.sub.x] and [v.sub.z] are the velocity component in the circumferential and extrusion direction, respectively. The shear rate is the second invariant of the rate of deformation tensor given as

[??] = [I.sub.2]([??]) = [([([dv.sub.x]/dy).sup.2] = [([dv.sub.z]/dy).sup.2]).sup.1/2] (40)

Since, in the present study, the value of [dv.sub.x]/dy and [dv.sub.z]/dy is in the order of 10 [s.sup.-1] and 1 [s.sup.-1], respectively, the following condition is satisfied:

[dv.sub.x]/dy [??] [dv.sub.z]/dy (41)

Thus, Eq. 40 is further simplified as

[??] [congruent to] [dv.sub.x]/dy (42)

Substitution of Eqs. 36 and 42 into Eq. 38, leads to the following equation for the value of [dv.sub.x]/dy:

[d.sup.2][v.sub.x]/[dy.sup.2] - 1/n [T.sub.b]/[T.sup.2] [partial derivative]T/[partial derivative]y [dv.sub.x]/dy = 0 (43)

Equation 43 can be solved once the values of [partial derivative]T/[partial derivative]y and T are known.

For the sake of simplicity, a no slip boundary condition is assumed for solving Eq. 39. After its integration one obtains:

[v.sub.z] = 1/2[eta] dP/dz [y.sup.2] - 1/[eta] y dP/dz [[integral].sup.h.sub.0] y/[eta]dy/[[integral].sup.h.sub.0]dy/[eta] (44)

The equation of continuity is

Q = w [[integral].sup.h.sub.0][v.sub.z]dz (45)

where Q is the volumetric flow rate. After substitution of Eq. 44 into Eq. 45, one obtains:

[mathematical expression not reproducible] (46)

From Eq. 46, the value of dP/dz can be calculated as

[mathematical expression not reproducible] (47)

The equation of energy conservation is defined as

[mathematical expression not reproducible] (48)

where [C.sub.p] is the heat capacity of the decrosslinked XHDPE, [[??].sub.s] is the rate of energy consumption caused by the breakage of main chains and [[??].sub.ud] is the rate of the ultrasonic energy dissipation during ultrasonic decrosslinking of XHDPE. According to the study [5], the concentration of broken main chains of the ultrasonically decrosslinked XHDPE is equal to the decrease of the crosslink density. The dissociation energy of C-C bond is 345 kJ/mole. Thus, for the ultrasonic decrosslinking of XHDPE in TSE at a flow rate of 8.25 g/min and an ultrasonic amplitude of 10 [micro]m (used in experiments), the values of [[??].sub.s] =0.26 J/s [cm.sup.3] and [[??].sub.ud] = 31.6 J/s [cm.sup.3]. Thus, the value of [[??].sub.s] is neglected during the simulation. Strictly speaking, the value of [[??].sub.ud] is a function of the gapwise position [6]:

[[??].sub.ud] = [omega] [P.sup.2.sub.A](y)/[rho][c.sup.2.sub.*] tan [[delta].sub.L]/2 (49)

To simplify the numerical calculation, the value of [[??].sub.ud] is assumed to be independent of the gapwise position. Thus, Eq. 48 is further simplified as

[rho][C.sub.p][v.sub.z] [partial derivative]T/[partial derivative]z = k p[partial derivative].sup.2]T/[partial derivative][y.sup.2] + [eta][[??].sup.2] + J/h (50)

Simulation Algorithms of Ultrasonic Decrosslinking of XHDPE Introductory Remarks. Based on the above-described models, including the ultrasonically induced bubble dynamics, ultrasonic power consumption, breakage of the network and nonisothermal flow, it is now possible to develop the simulation algorithms describing ultrasonic decrosslinking of XHDPE. In the following sections, three simulation algorithms are introduced to describe ultrasonic decrosslinking of XHDPE under the static condition without thermal degradation, ultrasonic decrosslinking of XHDPE in TSE without and with thermal degradation.

Decrosslinking Under Static Condition Without Thermal Degradation. As the first step in the simulation of the ultrasonic decrosslinking of XHDPE in TSE, a simulation algorithm of the ultrasonic decrosslinking of XHDPE under static condition is considered. The difference between the ultrasonic decrosslinking in TSE and under static condition is the absence of flow under static condition. It should be noted that the propagation of ultrasound in the polymer melt is not significantly affected by flow. This is due to the fact that the propagation of ultrasound in the polymer melt is a dynamic process accomplished by the local segmental motion of polymer, whereas the flow of polymer melt is achieved by the reptation of polymer molecules. Obviously, the flow of polymer melt is a dynamics process at a larger scale compared with the local segmental motion of polymer. In fact, measurements of the sound velocity in HDPE melts at an ultrasonic frequency showed that the shearing flow has an insignificant effect on the sound velocity and longitudinal loss tangent

[22]. Thus, it is reasonable to conclude that the ultrasonic decrosslinking of XHDPE under the static condition exhibits a great deal of similarity with that in TSE and may serves as a good starting point for the simulation of ultrasonic decrosslinking of XHDPE in TSE.

The simulation algorithm of ultrasonic decrosslinking of XHDPE under the static condition is based on the theoretical framework established in the previous sections. Figure 3 shows a computation flow chart of ultrasonic decrosslinking of XHDPE under the static condition. The simulation program starts by reading the physical properties of XHDPE including density, surface tension at the interface between the air and XHDPE, acoustic properties of XHDPE including the sound velocity and longitudinal loss tangent at the ultrasonic frequency of 40 kHz and dynamic properties ([G.sub.u], [G.sup.0.sub.N], and tan [[delta].sub.s]) of XHDPE. Then, the program reads the initial gel fraction and crosslink density of XHDPE. The processing variables of the ultrasonic decrosslinking, including the ultrasonic frequency, ultrasonic amplitude, hydrostatic pressure, and residence time, are also taken as inputs of the program. To carry out the simulation, the value of [R.sub.0] and [bar.n] are required. The earlier experimental studies of the ultrasonic devulcanization of SBR vulcanizates [23] and peroxide crosslinked silicone rubber [24] under the static conditions showed that the imposition of ultrasound leads to the formation of bubbles. Thus, the simulation of ultrasonically induced nucleation of bubbles is carried out for SBR vulcanizates [25]. The simulation was based on the cluster theory describing the nucleation of gas bubbles in a gas-saturated system. However, the simulation has shown a significant underestimation of the size and volume fraction of the bubbles in comparison with the experimental observation. Therefore, for the sake of simplicity, it is assumed that a small amount of bubbles of uniform size exists in XHDPE in the equilibrium state. Accordingly, in the present study the values of [R.sub.0] and [bar.n] are assigned.

Using the above mentioned parameters, the value of [x.sub.0] is determined by solving Eq. 12. The value of [P.sub.A] for XHDPE is calculated by Eq. 16 under the assumption that its variation in the gapwise direction is negligible. The value of [P.sub.A] calculated at z = 0.5 h. Since the imposition of ultrasound leads to the bubble dynamics in XHDPE, the bubbles volume fraction becomes a function of time. Consequently, the value of [P.sub.A] for XHDPE also becomes a function of time affecting the bubble dynamics.

To incorporate the effect of the time dependency of [P.sub.A] into the calculation of bubble dynamics, the each ultrasonic period is further divided into M sub-periods. For the large value of M, the value of [P.sub.A] is assumed to be a constant within a particular sub-period. Equation 10 is used to calculate x(t) within each sub-period. The value of x(t) at the end of each sub-period is used as the starting value, xo, for the next sub-period and then [P.sub.A] in the next sub-period is evaluated. The calculation of x(t) within the time of [t.sub.p] is done by looping this computational scheme for kM times. The values [x.sub.min] and [x.sub.max] within each period are determined and used to calculate the gel fraction and crosslink density of the ultrasonically decrosslinked XHDPE by means of Eqs. 33-35.

It should be noted that values of the density, surface tension, sound velocity, and longitudinal loss tangent of XHDPE are not available. Since these physical and acoustic properties are dependent on the chemical composition of the monomer and insignificantly affected by the chain architecture, it is assumed that these properties for XHDPE and decrosslinked XHDPE are same as those of HDPE. Since the crosslink density of XHDPE and decrosslinked XHDPE is lower in comparison with their physical entanglement density, the effect of the crosslinked network on the value of [G.sup.0.sub.N] is negligible. Thus, the values of [G.sup.0.sub.N] for XHDPE and decrosslinked XHDPE is taken to be equal to that the linear PE. It was shown experimentally that dynamic properties of the decrosslinked XHDPE in the frequency range below their rubbery plateau are significantly different from those of XHDPE. In contrast, dynamic properties of lightly crosslinked polymers in the rubbery plateau region are not strongly affected by the presence of crosslinks. Therefore, the tan Ss of HDPE is taken to be equal of that of XHDPE and decrosslinked XHDPE. For the same reason, the value of [G.sub.u] of the decrosslinked XHDPE is assumed to be equal of that of XHDPE.

Decrosslinking in TSE Without Thermal Degradation. The ultrasonic decrosslinking of XHDPE in TSE is more complicated than that under the static condition, described above. To simplify the ultrasonic decrosslinking of XHDPE in TSE, the process is approximated as a combination of many steps of the ultrasonic decrosslinking of XHDPE under the static condition. Using this approximation, the simulation of the ultrasonic decrosslinking of XHDPE in TSE is developed. As discussed earlier, the breakage of main chains in XHDPE is modeled as a strain-controlled process. Since the flow in TSE imposes a significant strain on the main chains of XHDPE, the effect of this strain on the configuration of main chains in XHDPE is taken into account by means of the stretch tensor [??] for a simple shear flow:

[mathematical expression not reproducible] (51)

where [[gamma].sub.s] is the shear strain. Clearly, [[??].sub.22] diverges as [[gamma].sub.s] increases. In case of the affine deformation of mains chains of XHDPE in simple shear, all the main chains degrade and gel fraction of the decrosslinked XHDPE becomes zero due to a finite extensibility of the main chains. This contradicts to the experimental observation. Thus, the incorporation of the effect of shear flow into the configuration of main chains in XHDPE requires a constitutive equation describing the non-affine deformation of polymer chains in shear flow. Since the power-law model is unable to describe the non-affine deformation of polymer chains, the effect of shear flow on the configuration of main chains in XHDPE cannot be taken into account realistically. To simplify the problem, it is further assumed that the shear flow has no effect on the configuration of main chains in XHDPE.

Figure 4 shows the computational flow chart of the simulation algorithm of the ultrasonic decrosslinking of XHDPE in TSE without the thermal degradation. The finite difference method is employed for solving differential equations. Numerical quadrature using the trapezoidal rule is utilized to evaluate integrals. A two-dimensional mesh grid is created on the y-z plane. Index i is the index of the grid point in the z-direction. The simulation program starts by reading the input parameters including the flow rate, ultrasonic amplitude, screw speed, gel fraction, and crosslink density of the material entering the ultrasonic treatment zone, the thermal boundary conditions, and parameters of the power-law model. The parameters of the power-law model of the material entering the ultrasonic treatment zone are determined from Eq. 37. With the known temperature, T, at z = 0, the values of [dv.sub.x]/dy and [v.sub.x] at z = 0 are obtained by numerically solving of Eq. 43 by means of the tridiagonal solver using a central difference scheme. Accordingly, [eta] at z = 0 is calculated using Eq. 36 at shear rates determined from Eq. 42. The value of dP/dz and [v.sub.z] at z = 0 is, respectively, calculated by Eqs. 47 and 44. Since the temperature T at z = 1 is unknown, its value is assumed to be equal to that at z = 0. Also, as an initial guess, the values of the gel fraction, crosslink density and the parameters of the power-law model of the decrosslinked XHDPE at z = 1 are assumed to be equal to those at z = 0. The values of [dv.sub.x]/dy, [v.sub.x], [eta], dP/dz, [v.sub.z], [t.sub.r], and P at z = 1 are determined based on the initial guess. The values of [x.sub.min] and [x.sub.max] are determined by solving of Eq. 10 numerically with an aid of Eqs. 5 and 16. A backward differentiation, known as the Gear's method, is used for solving of Eq. 10. The value of J is calculated from Eq. 18. The values of the gel fraction and crosslink density of the decrosslinked XHDPE at z = 1 are, respectively, updated based on Eqs. 32 and 33. The parameters of the power-law model of the decrosslinked XHDPE at z = 1 are updated based on these values of the gel fraction and crosslink density. The temperature T at z = 1 is updated by solving Eq. 50 numerically by using a Crank-Nicolson scheme with successive approximations [26], The updated temperature T is used to re-evaluate [dv.sub.x]/dy, [v.sub.x], [eta], dP/dz, [v.sub.z], [t.sub.r], P, [x.sub.min], [x.sub.max], J, gel fraction, crosslink density, and parameters of the power-law model of the decrosslinked XHDPE at z = 1. The iterative cycles is repeated until a prescribed tolerance between two successive cycles is satisfied. Subsequently, calculations of [dv.sub.x]/dy, [v.sub.x], [eta], dP/dz, [v.sub.z], [t.sub.r], P, [x.sub.min], [x.sub.max], J, gel fraction, crosslink density, and parameters of the power-law model of the decrosslinked XHDPE are continued until the end of ultrasonic treatment zone is reached. The mean gel fraction and crosslink density in the gap of ultrasonically decrosslinked XHDPE is calculated as

[mathematical expression not reproducible] (52)

where [[bar.v].sub.z] is the average velocity in the extrusion direction defined as

[[bar.v].sub.z] [equivalent to] [[bar.v].sub.z](y)= 1/l [[integral].sup.l.sub.0] [v.sub.z](y,z)dz (53)

It should be noted that the pressure at the entrance of the ultrasonic treatment zone of TSE, P(z=0), cannot be measured. In fact, only the value of P(z = [z.sub.max]) can be estimated from the readings recorded by the pressure transducer in the downstream of the ultrasonic treatment zone. Thus, the value of P(z=0) is unknown and the simulation algorithm indicated in Fig. 4 cannot be directly applied for its determination. Two methods are available to resolve this issue. The first one is to redesign the simulation algorithm by using a so-called "die filling" approach typically used for injection molding of polymers and earlier used in continuous ultrasonic devulcanization of rubbers [9], This approach would allow us to use the pressure at the end of the ultrasonic treatment zone as the input of the model. However, this approach is computationally expensive. Another approach is to provide an initial guess of the pressure at the entrance, P(z=0), and then calculate the pressure at the end of the ultrasonic treatment zone, P(z=[z.sub.max]). An iterative feedback loop is needed to continuously update the value of P(z=0) until the difference between the measured and calculated P(z=[z.sub.max]) is within the prescribed tolerance. Such feedback loop can be created by using the idea of bisection algorithm for finding the root of function without an analytical solution. Specifically, a pair of initial guesses of P(z=0) is used, namely [P.sub.1](z=0) and [P.sub.2](z=0) satisfying the condition that [P.sub.1](z=0) < [P.sub.2](z=0). Accordingly, values of [P.sub.1](z=[z.sub.max]) and [P.sub.2](z=[z.sub.max]) corresponding to these initial guesses are calculated. In order to do this calculation, the flow model in TSE is required. Such a model is not available. Therefore, the measured value of P(z=[z.sub.max]) is used as an input with its value lying within the interval of [P.sub.1] (z=[z.sub.max]) and [P.sub.2](z=[z.sub.max]). The difference between the measured and calculated P(z=[z.sub.max]) is evaluated with a tolerance of 0.001. If this tolerance is satisfied, the corrected P(z=0) is obtained, otherwise the initial guess of P(z=0) is updated. If the measured f(z=zmax) is higher or lower than the average of [P.sub.1](z=[z.sub.max]) and [P.sub.2](z=[z.sub.max]), [P.sub.1](z=0) and [P.sub.2](z=0) is replaced by their respective averages. The updated pair of the initial guesses of P(z=0) is used to re-evaluate P(z=[z.sub.max]). The successive cycle is repeated until the convergence between the measured and calculated P(z=[z.sub.max]) is achieved. Although this approach is more complicated and requires the estimation of P(z=0), it is computationally more efficient than the die filling approach.

Decrosslinking in TSE With Thermal Degradation. The thermal degradation of PE may occur at elevated temperatures by possible radical mechanisms consisting of three steps including initiation, propagation, and termination [27], According to this mechanism, the initiation of radicals during the thermal degradation of XHDPE is mainly caused by the breakage of C-C bonds during the decrosslinking. Thus, the concentration of radicals in XHDPE during the thermal degradation is expected to correlate with the decrease of the gel fraction and crosslink density. The P scission reaction during the thermal degradation is mainly responsible for the reduction of the molecular weight of PE [27]. In case of the decrosslinking of XHDPE, the [beta] scission reaction causes a decrease of the gel fraction. This occurs because of the breakage of main chains generates the dangling chains with radicals at their ends. These radicals cause the degradation of the dangling chains by the [beta] scission reaction. The termination of radicals may lead to an increase of the gel fraction when the recombination of two macroradicals occurs with one macroradical in the crosslinked network and another macroradical in the sol. However, the [beta] scission reaction has a prevailing effect on the gel fraction of the decrosslinked XHDPE than the termination of radicals. Assuming the intermolecular hydrogen transfer reaction is negligible, the thermal degradation of XHDPE during decrosslinking mainly occurs by means of a radical degradation reaction of dangling chains.

In the previous section, the decrosslinking of XHDPE is assumed to occur by the breakage of main chains only. The initial and current number of main chains in XHDPE and decrosslinked XHDPE, respectively, is [[zeta].sub.0][v.sub.0] and [zeta]v. Therefore, in this case the concentration of macroradicals of XHDPE, [R], during decrosslinking is

[R]=2([[zeta].sub.0][v.sub.0] - [[zeta]v) (54)

where [[zeta].sub.0][v.sub.0] - [zeta]v is the number of broken main chains in XHDPE with coefficient of 2 indicating that two macroradicals are created during the breakage of one main chain. Since the [beta] scission reaction is the first order reaction, the following kinetic function is used to describe the decrease of the gel fraction during the thermal degradation:

d[zeta]/dt=k[R]=2k([[zeta].sub.0][v.sub.0] - [zeta]v) (55)

where k is a rate constant exhibiting an Arrhenius temperature dependence:

[mathematical expression not reproducible] (56)

where B is a pre-exponential factor and [E.sub.[beta]] is the activation energy of the [beta] scission reaction of XHDPE. Thus, the gel fraction of the decrosslinked XHDPE subjected to both the ultrasonic decrosslinking and thermal degradation, [[zeta].sub.TD], is

[mathematical expression not reproducible] (57)

In case of the thermal degradation of XHDPE accompanied by radical degradation of dangling chains only, the number of the effective elastic chains in the degraded network does not change, but the gel fraction is reduced. Consequently, the crosslink density of the decrosslinked XHDPE with inclusion of the thermal degradation, [v.sub.TD], is

[v.sub.TD] = [zeta]v/[[zeta].sub.TD] (58)

It should be noted that, for the sake of simplicity, this approach neglects the possibility of the occurrence of the intermolecular hydrogen transfer.

The simulation of ultrasonic decrosslinking of XHDPE in TSE is a coupled effect of the ultrasonic decrosslinking of XHDPE and the thermal degradation. The gel fraction and crosslinked density of the ultrasonically decrosslinked XHDPE are calculated by Eqs. 33-35. In addition, the Horikx function is unable to predict the gel fraction-crosslink density relationship of the decrosslinked XHDPE affected by the thermally degradation. Thus, it is assumed here that the ultrasonic decrosslinking and thermal degradation of XHDPE are two independent processes. Accordingly, simulation of the ultrasonic decrosslinking of XHDPE without the thermal degradation is carried out first. Then, the gel fraction and crosslink density of the decrosslinked XHDPE with the thermal degradation are calculated by Eqs. 57 and 58.

All the simulation algorithms are implemented by using MATLAB. Equation 10 is solved by using an ordinary differential equation solver called ODE115 in MATLAB. The numerical algorithm to solve Eqs. 43 and 50 is coded by authors. The integral is evaluated using the trapezoidal rule. A finite difference mesh of 100 x 100 has been adopted for the simulation of the ultrasonic decrosslinking of XHDPE in TSE. A PC equipped with a Quadcore CPU running at a clock speed of 2.2 GHz and 8 GB memory (XPS 15 L502X, Dell Inc., TX) is used to carry out simulation. The tolerances used to evaluate the bubble dynamics, temperature, pressure gradient and pressure at the end of ultrasonic treatment zone are [10.sup.-6], [10.sup.-6], [10.sup.-3], and [10.sup.-3], respectively. The relative error of predictions is less than 10 The typical computation time of the simulation of the ultrasonic decrosslinking under the static condition and in TSE is, respectively, 1 and 60 min.

Example of Simulation

As an example, the simulation of ultrasonic decrosslinking of XHDPE in TSE without the thermal degradation is carried out and presented below. The material parameters, dimensions of ultrasonic treatment zone, and processing and thermal boundary conditions used in the simulation are given in Tables 1-4. The initial gel fraction and crosslink density of XHDPE used in the simulation are, respectively, 0.63 and 0.0117 kmol/[m.sup.3].

Figure 5 shows the gapwise distribution of the gel fraction (a) and crosslink density (b) at axial distances of z/l = 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 and a flow rate of 6.5 g/min. It is seen from Fig. 5 that the gel fraction and crosslink density decrease with the distance from the entrance of the ultrasonic treatment zone. A higher calculated gel fraction and crosslink density near the middle of gap than those near the wall is seen. The calculated gel fraction and crosslink density are strongly correlated with the residence time distribution of the material point in the gap. The asymmetric distributions of the gel fraction and crosslink density are due to the asymmetric distribution of the velocity in the extrusion direction caused by asymmetric thermal boundary conditions. A detailed discussion of simulation results in comparison with experiment is presented in Part II of this study.

CONCLUSIONS

Theoretical modeling of the ultrasonic decrosslinking of XHDPE under the static condition and in TSE is presented. 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 modeling simulates the ultrasonic decrosslinking of XHDPE using only the assigned bubble radius and bubble volume fraction. A simulation algorithm describing the ultrasonic decrosslinking of XHDPE in TSE is developed. Using this algorithm distribution of the pressure, velocity, temperature, gel fraction, and crosslink density of the decrosslinked XHDPE in the ultrasonic treatment zone can be simulated. The thermal degradation of XHDPE during the ultrasonic decrosslinking is simplified by considering the [beta] scission of dangling chains. The results of these simulations along with comparison to experiment will be presented in Part II of this study.

ACKNOWLEDGMENT

Authors wish to thank the Exxon Mobil Chemical Company for providing HDPE and the Akrochem Corporation for providing peroxide.
NOMENCLATURE

c                       Sound velocity in the polymer without bubble
[C.sub.*]               Sound velocity in the bubble filled polymer
[c.sub.m]               Sound velocity in the metal used to make the
                        ultrasonic horn
f(L)                    Probability distribution function of L
h                       Thickness of thin layer bubble filled polymer
k                       rate constant of [i scission reaction of XHDPE
m                       number of monomeric units between crosslinks
n                       power-law index
[bar.n]                 concentration of the bubbles
q                       wave number of the ultrasound in the polymer
[??]                    end-to-end vector of the main chain between
                        crosslinks in its equilibrium state
[r.sub.0](t)            time-dependent bubble radius
t                       time
[t.sub.p]               duration of one oscillation period at the
                        ultrasonic frequency
[t.sub.r]               mean residence time in the ultrasonic
                        treatment zone
tan [[delta].sub.L]     longitudinal loss tangent of the polymer at
                        the ultrasonic frequency
tan [[delta].sub.s]     shear loss tangent of polymer at the
                        ultrasonic frequency
v                       velocity vector
x(t)                    time-dependent relative bubble radius
[x.sub.max]             maximum relative bubble radius
[x.sub.min]             minimum relative bubble radius
x                       circumferential direction coordinate in planar
                        representation of ultrasonic treatment zone
y                       thickness direction coordinate of ultrasonic
                        treatment zone
z                       extrusion direction coordinate in ultrasonic
                        treatment zone
[A.sub.u]               ultrasonic amplitude
B                       pre-exponential factor of rate constant of
                        [1 scission reaction of XHDPE
[C.sub.p]               heat capacity of the decrosslinked XHDPE
E                       activation energy of viscous flow
[E.sub.[beta]]          activation energy of [i scission reaction of
G(t)                    XHDPE relaxation shear modulus
[partial derivativeG    memory function
(t-s)/[partial
derivative]s
[G.sup.[theta].sub.N]   storage modulus of polyethylene in the rubbery
                        plateau
G'([omega])             storage modulus
G"([omega])             loss modulus
[G.sub.e]               equilibrium shear modulus
[G.sub.i]               shear modulus of ith mode in generalized
                        Maxwell model
[G.sub.u]               elastic modulus in the glassy state
[??](t)                 Finger tensor
[[??].sub.*](t,s)       relative Finger tensor
[??]                    unit tensor
[I.sub.2]([??])         second invariant of the rate of deformation
                        tensor
J                       power consumption of the ultrasonic horn
K                       consistency
K'                      storage bulk modulus of polymer at the
                        ultrasonic frequency
K"                      loss bulk modulus of polymer at the ultrasonic
                        frequency
L                       contour length of the main chain between
                        crosslinks
[L.sub.m]               length of the monomeric unit of XHDPE
L'                      storage longitudinal modulus of polymer at the
                        ultrasonic frequency
L"                      loss longitudinal modulus of polymer at the
                        ultrasonic frequency
[M.sub.0]               molecular weight of ethylene
N                       average number of monomeric units between
                        crosslinks
Q                       volumetric flow rate
Qs                      rate of energy consumption caused by the
                        breakage of main chains
[[??].sub.ud]           rate of ultrasonic energy dissipation during
                        ultrasonic state decrosslinking of XHDPE
P                       hydrostatic pressure
[P.sub.A]               acoustic pressure
[P.sub.c]               pressure at the wall of the bubble in polymer
[P.sub.0]               pressure in the bubble in the equilibrium
                        state
P(t)                    spherical part of time dependent stress tensor
[P.sub.[infinity]](t)   ambient pressure in the polymer far from the
                        bubble
P([??])                 probability distribution function of r
R                       gas constant
[bar.R]                 distance between a material point and center
                        of a bubble
[R.sub.0]               bubble radius in the equilibrium state
[absolute value of R]   concentration of macroradicals of XHDPE
                        caused by main chain degradation
T                       temperature
[T.sub.b]               material parameter related to the activation
                        energy of viscous flow
[??]                    stretch tensor
[Z.sub.m]               sound impedance of the metal
[Z.sub.p]               sound impedance of the polymer
[gamma]                 ratio of the specific heats of the gas
[[gamma].sub.s]         shear strain
[??]                    shear rate
[[zeta].sub.0]          gel fraction of material entering the
                        ultrasonic treatment zone
[[zeta].sub.DXHDPE]     gel fraction of decrosslinked XHDPE
[[zeta].sub.TD]         gel fraction of the decrosslinked XHDPE
                        subjected to both ultrasonic decrosslinking
[eta]                   and thermal degradation shear viscosity
[theta]                 Heaviside function
[[lambda].sub.i]        relaxation time of i-th mode in generalized
                        Maxwell model
[v.sub.0]               crosslink density of material entering the
                        ultrasonic treatment zone
[v.sub.XHDPE]           crosslink density of XHDPE
[v.sub.DXHDPE]          crosslink density of decrosslinked XHDPE
[v.sub.TD]              crosslink density of the decrosslinked XHDPE
                        with inclusion of the thermal degradation
[rho]                   polymer density
[[rho].sub.m]           density of metal used to make the ultrasonic
                        horn
[sigma]                 surface tension at the interface between the
                        gas bubble and polymer
[??](t)                 time-dependent stress tensor
[??]                    deviatoric part of stress tensor
[[phi].sub.0]           volume fraction of bubbles in a polymer
X(P)                    compressibility of the bubble in a polymer
[omega]                 ultrasonic frequency or frequency in
                        oscillatory flow


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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.24487

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

Caption: FIG. 1. Schematics of the ultrasonic treatment zone in TSE.

Caption: FIG. 2. Schematic of a thin layer of the polymer containing bubbles placed between a metal plate and an ultrasonic horn.

Caption: FIG. 3. Computation flow chart of ultrasonic decrosslinking of XHDPE under the static condition.

Caption: FIG. 4. Computation flow chart of the simulation algorithm of ultrasonic decrosslinking of XHDPE.

Caption: FIG. 5. Gapwise distribution of the gel fraction (a) and crosslink density (b) 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 5 (am and a flow rate of 6.5 g/min.
TABLE 1. 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 modulus at [omega]=[infinity], MPa   2,500
           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

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

Description                                 Value
Initial bubble radius, [micro]m               25
Bubble fraction, %                           1.33
Ultrasonic frequency, Hz                4 X [10.sup.4]

TABLE 3. Dimensions of ultrasonic treatment zone.

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

TABLE 4. 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
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
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Author:Huang, Keyuan; Isayev, Avraam I.
Publication:Polymer Engineering and Science
Article Type:Report
Date:Oct 1, 2017
Words:9958
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