JP7692338B2 - Method and program for simulating an extruder - Google Patents

Description

Application of Article 30, Paragraph 2 of the Patent Act Polymers 2021, 13, 3892. https://doi. org/10.3390/polym13223892

The present invention relates to a simulation method and a simulation program for an extruder that predicts the pressure, residence time, resin filling rate, resin viscosity, temperature, etc. inside the extruder and after pyrolysis when the extruder screw is rotated at high speed to pyrolyze the resin.

Meltblown nonwoven fabrics, which are used for filtration media, medical cloth, sound absorbing materials, cushioning materials, etc., are manufactured by passing molten resin through a nozzle with small holes of about 0.1 mm. Because the nozzle will break if high pressure is applied to it, low-viscosity grade resins with extremely high melt flow rates (hereinafter referred to as MFR) are used as raw materials for the manufacture of meltblown nonwoven fabrics. However, resins such as low-viscosity polypropylene (hereinafter referred to as PP) with an MFR value of about 1000 g/10 min are special grades that are not widely available on the market, and are several times more expensive than normal grades. Therefore, it would be useful and economical to manufacture low-viscosity resins by reducing the molecular weight of general-purpose resins that are easy to obtain and inexpensive.

One of the known methods for reducing the molecular weight of resins is the use of organic peroxides. In this method, a twin-screw extruder is used to mix the resin (polymer) with an organic peroxide to chemically decompose the resin. However, since organic peroxides are highly reactive, it is difficult to control the decomposition of the resin. Furthermore, since the decomposition of the resin produces decomposition residues of the peroxide, resins reduced in molecular weight by this method are not suitable for use in medical or sanitary products. In addition, organic peroxides are classified as hazardous materials, and therefore require careful handling and storage.

To reduce the molecular weight of resins without using organic peroxides, we have proposed a manufacturing method in which high shear forces are applied to molten resin to reduce the molecular weight (Patent Document 1). This manufacturing method reduces the molecular weight of resins by inducing shear heating through shear stress, and is excellent in that it allows the reduction in molecular weight to be controlled by the screw rotation speed, making it possible to quickly and easily produce resins of different grades (molecular weights).

A simulation method based on the configuration and operating conditions of an extruder has also been proposed for the processing of resin using an extruder. For example, Non-Patent Document 1 describes a simulation method for a twin-screw extruder.

Patent No. 6949342

Fel, E.; Massardier, V.; Melis, F.; Vergnes, B.; Cassagnau, P. Residence Time Distribution in a High Shear Twin Screw Extruder. Int. Polym. Proc. 2014, 29, 71-80.

However, the simulation method of Non-Patent Document 1 is limited to rotation speeds of up to 1000 RPM (revolutions per minute) and does not take into account the reduction in molecular weight of the resin due to the processing in the extruder. For this reason, it is difficult to accurately simulate the case where the viscosity of the resin is suddenly reduced by processing at a high speed rotation of about 2000 to 4000 RPM, as in the manufacturing method described in Patent Document 1.
The present invention aims to provide a simulation method and a simulation program for an extruder that can accurately predict the pressure inside the extruder and after processing, the residence time, the resin filling rate, the viscosity and temperature of the resin, etc., when the viscosity of the resin is reduced by a high-speed rotating screw.

The extruder simulation method of the present invention includes a Nusselt number calculation step for calculating the Nusselt number of the screw for each combination of the screw rotation speed and barrel temperature of the extruder, an estimate value setting step for setting estimated values of the pressure, temperature and viscosity of the resin using the outlet of the extruder as a target part, a flow rate/pressure gradient calculation step for calculating the flow rate and pressure gradient of the resin at the target part using the physical properties of the resin and the configuration and operating conditions of the extruder, a temperature gradient/viscosity gradient calculation step for calculating the temperature gradient and viscosity gradient at the target part using the Nusselt number, the flow rate and the pressure gradient, a pressure/temperature/viscosity calculation step for calculating the pressure, temperature and viscosity of an adjacent part located a predetermined distance away from the target part on the inlet side of the extruder using the pressure gradient, the temperature gradient and the viscosity gradient, and a part determination step for determining whether the adjacent part is the inlet of the extruder. If it is determined in the part determination step that the adjacent part is not the inlet of the extruder, If the adjacent part is determined to be the inlet of the extruder in the part determination step, a comparison step is performed in which the calculated value calculated in the pressure, temperature, and viscosity calculation step is compared with the actual measured value at the inlet. If the calculated value does not match the actual measured value in the comparison step, the process returns to the estimated value setting step, and another estimated value is set for the pressure, temperature, and viscosity of the resin using a root-finding algorithm so that the residual is small. The above steps are repeated until the calculated value matches the actual measured value. If the calculated value matches the actual measured value in the comparison step, the pressure, temperature, and viscosity of the resin set as the estimated value in the estimated value setting step are used as the predicted value.

The simulation method of the present invention can accurately reflect the screw rotation speed and barrel temperature dependency of heat transfer from the barrel by calculating the Nusselt number of the screw for each combination of the extruder screw rotation speed and barrel temperature. Therefore, by using this Nusselt number to calculate the temperature gradient and viscosity gradient in the target area, it is possible to accurately predict the viscosity and temperature of the resin after processing and inside the device, even when the viscosity of the resin is suddenly reduced by the high speed rotation of the screw.

1 is a flowchart of a simulation method according to an embodiment. FIG. 2 is a schematic diagram of a screw shape of a high shear extruder. FIG. 1 is a schematic diagram showing an overall image of an apparatus used in the examples. FIG. 4 is a schematic diagram showing the configuration of a twin-screw extruder in the apparatus of FIG. 3. FIG. 4 is a schematic diagram showing a screw element of the high shear extruder in the apparatus of FIG. 3. 1 is a graph showing viscosity data for PP1. 1 is a graph showing viscosity data for PP2. 1 is a graph showing the results of thermogravimetric analysis of PP1. 1 is a graph showing the results of thermogravimetric analysis of PP1. FIG. 2 is a schematic diagram showing an outline of the screw configuration of the high shear extruder used in Example 1. FIG. 2 is a schematic diagram showing an outline of the screw configuration of the high shear extruder used in Example 2. FIG. 1 is a schematic diagram showing an outline of the screw configuration of a high-shear extruder used in Example 3. 13 is a graph showing simulated and measured results for Example 2, where 13A shows temperature and 13B shows zero shear viscosity at a reference temperature. 14 is a graph showing simulated and measured results for Example 2, where 14A shows temperature and 14B shows zero shear viscosity at a reference temperature. 1 is a calibration curve (calibration curve) created to specify the MFR value of a resin.

Hereinafter, embodiments of the present invention will be described with reference to the drawings as appropriate.
<Flow of resin inside the extruder>
A one-dimensional model of the resin flow inside the extruder in the case of a non-Newtonian fluid is described below. In the simulation of the present invention, the resin flow inside the extruder was assumed to be a fully developed flow in a steady state. Since the ratio of the inner diameter to the outer diameter of the screw is close to 1, the curvature of the screw groove was assumed to be negligible.

<Screw>
Figure 2 is a schematic diagram showing the screw profile of a single screw extruder. The screw was modeled as a rectangular groove (channel) with a cross section of height H and width W. The barrel was modeled as a moving plate covering the screw groove. In this case, the moving speed V of the plate is expressed as V = πDN (D is the inner diameter of the barrel and N is the rotation speed of the screw).

In the case of a rectangular groove, the length direction of the groove is z. In a shallow and wide groove, the flow velocity ^vz of the resin in the z direction can be treated as a function of the depth direction y of the groove. The equation of motion of the resin is shown below. Note that in each equation described below, each symbol has the same meaning.
where p is the pressure and η is the viscosity of the resin (fluid). The right-hand side is the pressure gradient.

The boundary conditions are as follows:
Φ is the screw angle, V is the barrel speed as seen from the stationary screw system, and ^vz is the z-component of V (see FIG. 2). When the target portion of the screw is completely filled with resin, the pressure gradient α is a finite value. Since the flow velocity ^vz of the resin is a function of the pressure gradient, α can be determined using the following equation for the flow velocity.

Q is the extrusion speed (extrusion amount) of the resin [m ³ /s (seconds)], and W is the width of the groove in the screw (see FIG. 2).

When the target portion of the screw is in an unfilled state where the target portion is partially filled with resin, the pressure gradient α is zero. Therefore, the flow of resin in the unfilled region is expressed by the following simple formula (4) based on formula (1).

The filling rate f in the target area to be subjected to the computational evaluation is defined by the following formula.

The viscosity of a molten resin is generally a function of shear rate as shown in the following formula:

In the following simulation method, a Cross model is used as the viscosity model of the non-Newtonian fluid, however, the viscosity model of the non-Newtonian fluid is not limited to the Cross model and other models may be used.
η ₀ , τ _* , and n are model parameters that vary depending on the type of resin. The zero shear viscosity η ₀ depends on the temperature T.

We assume that η ₀ follows the Arrhenius law.
T _* is a model parameter, and η _r is the zero shear viscosity of the resin at the reference temperature T _r (hereinafter, referred to as the reference zero shear viscosity). Hereinafter, the reference zero shear viscosity will be referred to as the viscosity.

The equation of motion for the resin is shown below.
is the density of the resin in a molten state [kg/m ³ ], and C _p is the specific heat of the resin [J/(kg·K)].
q is the heat flux [W/m ² ], and κ is the thermal conductivity of the resin [W/(m·K)].

The initial conditions are shown below.
T _i is the temperature of the resin at the inlet of the extruder.

The boundary conditions are as follows:
T _b is the barrel temperature.

By partially integrating equation (8) with respect to y and using equation (1), the following equation is obtained.
τ＝ ^η∂vz /∂y is the shear stress. The temperature T in equation (13) is strictly the average temperature defined by the following equation. (Bird, RB; Armstrong, RC; Hassager, O. Fluid Mechanics. Dynamics of Polymeric Liquids, 2nd ed.; John Wiley and Sons Inc: New York, NY, USA, 1987; Volume 1.)

The second and last terms on the right hand side of equation (13) arise from shear heating. The first term on the right hand side of equation (13) represents the heat transfer from the fluid to the barrel, which for a one-dimensional problem can be modeled as follows:
Nu is the Nusselt number, which is a dimensionless quantity of the heat transfer coefficient.

When the operating conditions of the extruder are selected from a wide range of values for the screw speed and barrel temperature, it is not appropriate to set the Nusselt number as a constant under all the operating conditions. Therefore, for each condition of the screw speed and barrel temperature, a uniform temperature distribution is assumed at the inlet of the extruder, and the flow velocity is assumed to be a simple shear flow (equation (4)). The heat conduction equation (8) and the boundary condition equations (11) and (12) are solved by calculation to calculate the temperature distribution and heat flux in the y and z directions. The Nusselt number of the target part is then obtained by the following equation (Bird, RB; Armstrong, RC; Hassager, O. Fluid Mechanics. Dynamics of Polymeric Liquids, 2nd ed.; John Wiley and Sons Inc: New York, NY, USA, 1987; Volume 1.).

Nu _Z is the local Nusselt number at z, H is the height of the screw groove (m), q is the heat flux [W/m ² ], z is the coordinate in the extension direction of the screw path, y is the coordinate in the depth direction of the screw groove, κ is the thermal conductivity of the resin [W/(m·k)], and Tb is the barrel temperature [K].

In the simulation of the present invention, the average Nusselt number shown in the following formula was used.
L _P is the path length of the screw [m], and z is the coordinate in the extension direction of the screw path.

Thailand
The die and through-hole used in the examples described below, which have holes with a circular cross section perpendicular to the z-direction, will be described below. The equation of motion in the die is given below.
r is the radial coordinate of the hole, and z is the axial coordinate.

The boundary conditions are as follows:
R is the radius of the die. As stated above for the screw, the pressure gradient ∂p/∂z ≡ α is derived from the expression for the extrusion rate (throughput, m ³ /sec).

The equation of motion is shown below.
Equation (22) is an equation for the heat flux at the inner wall of the die, and Tω is the temperature of the die. The average Nusselt number for a power-law fluid, expressed by the following equation, was used.
κ is the thermal conductivity of the resin [W/(m·k)], l is the length of the die or through hole, n is the viscosity index when assuming a power law fluid and is a parameter that appears in the Cross model, ρ is the molten density of the resin [kg/m ³ ], C _p is the specific heat of the resin [J/(kg·K)], and Q is the extrusion speed of the resin.

<Viscosity reduction model>
At high rotation speeds and high barrel temperatures, the resin (polymer) is thermally decomposed due to increased shear heating and heat conduction. Various studies have been conducted on the thermal decomposition of resin (Kim, B.; White, JL Simulation of Thermal Degradation, Peroxide Induced Degradation, and Maleation of Polypropylene in a Modular Co-Rotating Twin Screw Extruder. Polym. Eng. Sci. 1997, 37, 576., etc.), but this simulation adopts the simplest model. That is, it is assumed that the resin is composed of a linear polymer and that the thermal decomposition of the polymer follows a random cutting process.

A polymer is modeled as a long chain with node number N, where N represents the number of backbone carbon atoms (Rubinstein, M.; Colby, RH Polymer Physics; Oxford University Press Inc: NY.,2003). If the number of polymers with node number N is represented by _nN , the total number of nodes in the polymer is given by the following equation:

If the scission of the main chain follows a first-order reaction with a reaction rate k and no re-polymerization occurs, the relationship shown in the following formula is established.
t is the time.
The reaction rate was assumed to follow the Arrhenius law.
A is the frequency factor, E is the activation energy, _Rg is the gas constant, and T is the resin temperature [K].

If we ignore the amount of resin volatilization, the number of all nodes in the resin, as shown in the following equation, does not change over time.

Since the number average molecular weight is given by the following equation (26), the time evolution equation (27) for the number average molecular weight can be obtained (Suehiro, T.; O'shima, E. A kinetic study on the random scission of a polymer. Kobunshi Ronbunshu. 1977, 34, 3, 241-248.).

When the number average molecular weight is sufficiently large, equation (27) becomes:

Since it is known empirically that the zero shear viscosity is proportional to the 3.4th power of the number average molecular weight, the following formula (29) holds: In formula (29), the proportionality coefficient between the zero shear viscosity and N is absorbed into the frequency factor A in the viscosity reduction rate model.

When the resin is flowing, the time derivative can be replaced with the material derivative, and in a steady state, the following equation (30) holds.
S is the cross-sectional area of the screw groove or die.

<Calculation method>
FIG. 1 is a flowchart of the simulation method of this embodiment. In this embodiment, when the resin temperature and viscosity data at the inlet of the extruder are given, the pressure p, temperature T, and reference zero shear viscosity η _r at the reference temperature (hereinafter, also referred to as viscosity as appropriate) at all positions in the axial direction of the extruder are calculated. In the following, the axial coordinate in the extruder is indicated by z. Note that the order of each step shown below is an example, and each step may be performed in an order different from that shown in FIG. 1, except when the value obtained and calculated in the previous step is used.

First, the Nusselt number Nu is calculated in the Nusselt number calculation step S10. To calculate Nu, equation (17) is used when the part for which the resin pressure, temperature, and viscosity are to be calculated is the screw, and equation (23) is used when the part is the die or through hole. The simulation method of the present invention calculates the Nusselt number Nu in the screw for each combination of the screw rotation speed and barrel temperature of the extruder. In S10, the average Nusselt number calculated using equation (17) is used as the Nusselt number in the screw, and the Nusselt number calculated using equation (23) is used as the Nusselt number in the die and through hole, which are parts other than the screw.

Next, an estimate value setting step S11 is performed in which estimated values of the pressure, temperature, and viscosity of the resin are set with the extruder outlet as the target site. In S11, the site ^zO to be calculated is set to the extruder outlet _zf ( ^zO = _zf ). Since the pressure p at _zf is atmospheric pressure (ambient pressure), the estimated value ^pO is set to 0 ( ^pO = 0). Then, ^the temperature _Tf and viscosity _ηrf at _zf are set as the estimated values ^T0 and _ηr0 of the temperature T and viscosity _ηr ( ^T0 = _{Tf0 , ηrO = ηrf0 )} ^.

Next, a target part determination step S12 is performed to determine whether the target part is the screw or another part. In the case of an extruder equipped with a screw, a die, and a through hole, in S12, it is determined whether the target part z ^O is the screw, the die, or the through hole.

If it is determined in S12 that the outlet _zf is a die or a through hole, the process proceeds to S13, where the resin flow velocity ^vz and the pressure gradient ∂p/∂z are calculated. These calculations are performed by an iterative method by giving appropriate estimated values to the flow velocity ^vz and the pressure gradient ∂p/∂z. In this embodiment, an analytical solution for the case of a power-law fluid is selected as the estimated value. By solving equations (18) and (20), the resin flow velocity ^vz and the pressure gradient ∂p/∂z are calculated so that the actual resin extrusion velocity (extrusion amount) Q from the extruder matches the calculated resin extrusion velocity _Qth .

In the flow rate and pressure gradient calculation step S14, the flow rate ^vz and pressure gradient ∂p/∂z of the resin in the target part are calculated using the physical properties of the resin, the configuration of the extruder, and the operating conditions. In FIG. 2, the flow rate ^vz and pressure gradient ∂p/∂z are calculated based on the results calculated in S13. Note that when the target part is a die or a through hole, the filling rate f of the resin-filled region is 1.

In temperature gradient/viscosity gradient calculation step S15, the temperature gradient ∂T/∂z and the viscosity gradient ∂ηr/∂z at the target part z = ^zO are calculated using the Nusselt number Nu calculated in S10, and the flow velocity ^vz and pressure gradient _∂p /∂z calculated in S14. When ^zO is a die or a through hole, equations (21) and (30) are used to calculate ∂T/∂z and _∂ηr /∂z.

If it is determined in S12 that the target portion z ^O is a screw, the process proceeds to the filling level determination step S20. In S20, it is determined whether or not the pressure p ^O is greater than 0, that is, whether or not the pressure p ^O is positive or equal to or less than zero.

When p ⁰ >0 (positive), the entire target portion of the screw is filled with resin. Therefore, similarly to the above-mentioned case of the die and through hole, the flow rate v ^z and pressure gradient ∂p/∂z of the resin are calculated by S13 and S14. Then, in the temperature gradient and viscosity gradient calculation step S15, ∂T/∂z and ∂η _r /∂z at z = z ^O are calculated using the Nusselt number Nu calculated in S10 and v ^z and ∂p/∂z of S14. However, when the target portion is a screw, in S13, formulas (1) and (3) are used instead of formulas (18) and (20), and in S15, formulas (13) and (30) are used instead of formulas (21) and (30). In this case, the filling rate f of the resin-filled region is 1.

When p ^O >0 but is 0 or negative (p ^O ≦0), the target portion of the screw is a non-filled region partially filled with resin. Therefore, in S21, the pressure p ^O of the target portion is reset to 0, and the pressure gradient ∂p/∂z is also set to 0. In the non-filled region, the resin flow velocity v ^z is given by equation (4), and the filling rate f is given by equation (5). Based on these, in flow velocity/pressure gradient calculation step S22, the resin flow velocity v ^z , pressure gradient ∂p/∂z, and filling rate f are calculated. Then, in temperature gradient/viscosity gradient calculation step S15, ∂T/∂z and ∂η _r /∂z at z=z ^O are calculated using the Nusselt number Nu calculated in S10, and v ^z and ∂p/∂z calculated in S22. In S15, equations (13) and (30) are used, as in the filled region.

Next, the process proceeds to pressure, temperature, and viscosity calculation step S16. In S16, the pressure p = p O - (∂p / ∂z) δp, temperature T = T ^O - ( _∂T / ∂z) δz, and viscosity η ^r = η r O - (∂η r / ∂z) δz at an adjacent part z ^O - δz that is a predetermined distance δz away from the target part z O on the inlet side of the extruder are calculated using the pressure gradient ∂p / ∂z of _S14 , the _temperature gradient ^∂T / ∂z, and the viscosity gradient ∂η ^r / ∂z _of S15. Here, δz is the size of the discretization on the z axis.

In the part determination step S17, it is determined whether the adjacent part z=z ^o -δz is 0, that is, whether it is the inlet of the extruder.
If it is determined in S17 that the adjacent portion z ^O -δz is not the inlet of the extruder, the process proceeds to S18. In S18, the adjacent portion z ^O -δz is set as a new target portion z ^O (z ^O =z), with pressure p ^O , temperature T ^O and viscosity η _r set to p = p ^O , T = T ^O and η _r = η _r ^O , respectively, and the process proceeds to S12. Then, using the pressure p, temperature T and viscosity η _r calculated in the pressure, temperature and viscosity calculation step S16, the flow rate and pressure gradient calculation step S14, temperature gradient and viscosity gradient calculation step S15, pressure, temperature and viscosity calculation step S16 and portion determination step S17 are performed again.

In this manner, the process returns to S12 via S18 and repeats the steps of calculating the pressure, temperature, and viscosity for each part of the extruder until it is determined in S17 that z = 0, i.e., that the target part is the inlet of the extruder.

If it is determined in the part determination step S17 that z=0, i.e., that the part is the inlet of the extruder, the process proceeds to a comparison step S23, in which the calculated values in the pressure, temperature, and viscosity calculation step S16 are compared with the measured values at the inlet. In S23, it is determined whether the calculated values of temperature T and viscosity _ηr at the inlet match with the measured values of temperature T _i and viscosity η _ri . Here, "match" means that the calculated values and the measured values are sufficiently close, for example, that the difference between the two is equal to or less than a predetermined threshold value. The threshold value may be set appropriately, for example, the relative error between the calculated values and the measured values may be set to about 10 ^-6 .

If the calculated values do not match the actual measured values in the comparison step S23, the process returns to the estimated value setting step S11, where different estimated values are set for the pressure, temperature, and viscosity of the resin, and the above steps are repeated until the calculated values match the actual measured values. The different estimated values are calculated using a root-finding algorithm that reduces the residual. An example of a root-finding algorithm is the Newton method.

If the calculated values match the actual measured values in the comparison step S23, the pressure, temperature, and viscosity of the resin set as the estimated values in the estimated value setting step S11 are used as the simulation results.

In this manner, the calculation of p, T, and _ηr at z ^o -dz is repeated along the z-axis of the extruder to obtain p, T, and _ηr at the inlet of the extruder where z=0. T _f0 and η _rf0 at the outlet of the extruder are repeatedly updated so that T and _ηr obtained as calculated values at z=0, which are set as estimated values in S11, approach T _i and η _ri obtained as actual measured values. After the calculation is repeated, when the calculated values T and _ηr match the actual measured values T _i and η _ri , p, T, and _ηr at the outlet of the extruder and at each position along the z-axis are obtained as predicted values of the simulation results.

In the embodiment, an example has been described in which p, T, and _ηr at the exit of the extruder and at each position along the z axis are obtained as simulation results. However, it is possible to obtain the residence time (cumulative residence time), the filling rate of the resin, in addition to the pressure, the viscosity and temperature of the resin inside the extruder and after processing.
The present invention can also be embodied as a program for causing a computer to execute the above-mentioned simulation method for a single screw extruder.

<Device>
FIG. 3 shows a schematic overview of the apparatus used in the examples. As shown in the figure, an apparatus was used in which two different extruders were connected in series via a single pipe with a diameter of 10 mm and a length of 150 mm. The resin supplied from the resin supply section was melted in the twin-screw extruder and then supplied to the inlet of the high-shear extruder, which is a single-screw extruder. The first extruder was a co-rotating twin-screw extruder (L/D=48.5, D=26 mm, manufactured by Shibaura Machine Co., Ltd.) and was used to melt the resin. The resin temperature at the outlet of the twin-screw extruder was 468 K (195° C.) in all examples.

Fig. 4 is a schematic diagram showing the configuration and temperature conditions of the twin-screw extruder, and the arrows in the figure indicate the direction of resin flow. The twin-screw extruder used had a length of 1260 mm and an inner diameter of the barrel of 26 mm. The extrusion speed (resin supply speed, extrusion amount) and rotation speed N in the examples are shown in Table 1.

The second extruder is a single-screw high-shear extruder with a barrel diameter of 48 mm and a maximum rotation speed of 3600 rpm. In the high-shear extruder, the polymer undergoes thermal shearing of molecular chains due to shear heat. The screw is composed of multiple screw elements, as in a normal twin-screw extruder. In the screw used in the examples, the difference in shape of the screw elements that compose the screw does not have a large effect on the Nusselt number, so for screws with different screw elements, the shape of the screw element on the most inlet side of the screw, i.e., the most upstream side of the resin, was used. The Nusselt number of the screw was calculated using the total length of the multiple screw elements. In addition, if the screw had a through hole, the Nusselt number of the screw was evaluated by replacing it with one without a through hole.

Fig. 5 is a schematic diagram of a part of a high shear extruder, in which the screw elements are provided with through holes having a circular cross section, and the arrows in the figure indicate the direction of resin flow.
The resin is blocked by the inverted screw and flows into the through hole. The role of these elements is to form a region completely filled with resin in front of the inverted screw while preventing the resin from being subjected to excessive shear force, thereby preventing excessive molecular weight reduction. The resin discharged from the die of the high shear extruder was immediately cooled with water and then dried.

<Raw materials>
Two different grades of homopolypropylene (F-704NP and J107G, manufactured by Prime Polymer) were used. The melt flow rates were 7.0 and 30 g/10 min, respectively. The measurements were performed according to ISO 1133-97. In the following, F-704NP is referred to as PP1 and J107G as PP2. To evaluate the parameters of the viscosity model equations (6) and (7), the shear viscosity was measured with a modular compact rheometer (MCR 102 Anton Paar, Graz, Auatria). The shear rate measurement range was 0.01-100/s, and the measurement temperatures were 463, 473 and 483 K (190, 200 and 210 °C). The reference temperature was set at 473 K (200 °C). The model parameters of PP1 and PP2 obtained as a result of the curve fitting are shown in Table 2.
Figures 6 and 7 show the viscosity data and curves for PP1 and PP2.

To evaluate the activation energy of PP, the Kissinger method was used (Chan, JH; Balke, ST The thermal degradation kinetics of polypropylene: Part III. Thermogravimetric analyses. Polym. Degrad.Stab. 1997, 57, 135-149.). To obtain weight loss data due to thermal decomposition of PP, PP1 was simultaneously measured in air using a thermogravimetric analyzer (STA7200, Hitachi High-Tech Science). The heating rates were 2, 4, and 8 K/min. The sample weight was approximately 10 mg for each measurement.
8 is a graph showing the results of thermogravimetric analysis (TGA) of PP1. For each heating rate β (K/min) in the TGA analysis, the temperature at weight fraction X was measured. The following equation was used in the Kissinger's method.

Here, E [J/mol] is the activation energy.
9 shows the data set (1/T, ln(β/T ² )) and the regression curves when X is fixed at 0.4, 0.5, 0.6, or 0.7. The values of E were calculated for X=0.4, 0.5, 0.6, and 0.7, and the average was 80.2 kJ/mol.

Example 1: Nusselt number study
FIG. 10 shows an outline of the screw configuration of the high shear extruder used in this example. The arrows in the figure indicate the direction of resin flow, and the flow of resin is also indicated by arrows in the following figures. A screw element with a groove of 3 mm depth, 15 mm lead, and 45 mm length was used. A die with a hole diameter of 4 mm and a length of 25 mm was used for the high shear extruder, and the extrusion rate was 4.8 kg/hour. The barrel temperatures were 468 K and 573 K (195° C. and 300° C.). The screw rotation speeds were 100, 1000, and 2000 rpm. PP1 was used as the resin, and the temperature of PP1 at the outlet of the die was measured.

<Molecular chain scission of PP using a high shear extruder>
Example 2
FIG. 11 is a schematic diagram showing the configuration of the screw of the high shear extruder. A screw element with a groove having a depth of 3 mm, a lead of 22.5 mm, and a length of 45 mm was used. The rearmost screw element in the flow direction of the resin has the same groove depth, but is different from the previous screw element in that it has a lead of 15 mm and a length of 30 mm. The Nusselt number of the screw in this example was evaluated by replacing the shape of the rearmost screw element with the shape of the screw element on the most upstream side of the resin. However, other calculations such as flow rate calculation were performed taking into account the geometric shape of the rearmost screw element.

The shaded area in Fig. 11 shows a reverse-thread element with through holes (see Fig. 5). This element has four through holes with a diameter of 2 mm and a length of 45 mm. The high-shear extruder has a die with a diameter of 2 mm and a length of 25 mm. The barrel temperature was set at 468 K (195 °C) and the extrusion rate was 4.8 kg/h. The screw rotation speeds were 2000, 2500, 3000 and 3600 rpm. PP1 was used as the resin. The temperature of PP1 was measured at point P in Fig. 11 and at the die exit. The zero-shear viscosity of PP1 after high-shear treatment was measured at a reference temperature of 473 K (200 °C).

Example 3
FIG. 12 is a schematic diagram showing the screw configuration of the high shear extruder used in this example. A screw element with a groove depth of 3 mm, a lead of 15 mm, and a length of 45 mm was used. The shaded parts in FIG. 12 are the same as in Example 2. A high shear extruder equipped with a die with a diameter of 3 mm and a length of 25 mm was used. PP2 was used as the resin. The extrusion rate was fixed at 10 kg/h, the screw rotation speed was fixed at 3600 rpm, and the barrel temperature was set to 578 K or 628 K (300 ° C. or 350 ° C.). The resin temperature was measured at the outlet of the die. The zero shear viscosity of PP2 after high shear treatment was measured at a reference temperature of 473 K (200 ° C.).

<Results>
Table 3 shows the predicted and measured values obtained by the simulation of Example 1. The effect of viscosity reduction is not considered here, since we are only focusing on heat transfer. The calculated Nusselt number was slightly increased when the barrel temperature was increased from 473 K to 578 K, with the screw speed fixed at 100 rpm. As a result, the calculated temperature at the outlet of the high shear extruder, _Tf , also changed from the calculated value obtained with Nu = 8.92 at N = 100 rpm and barrel temperature Tb = 473 K. It was found that the optimization of the Nusselt number brought the predicted values by the simulation closer to the measured values.

When the barrel temperature was fixed at 468 K (195° C.) and the screw rotation speed was increased from 100 rpm to 1000 rpm or more, the value of the Nusselt number obtained by calculation increased. This is because the heat transfer coefficient in the case of forced convection depends on the flow rate, and the resin flowing at high speed is cooled to a large extent by the barrel. As a result, the predicted value of T _f predicted by the simulation method of the present invention was smaller than the predicted value T _f obtained using Nu = 8.92. From this, it can be said that the simulation method of the present invention incorporates the influence of the screw rotation speed and barrel temperature, which were not considered in the past, into the predicted value as the Nusselt number. The large difference in T _f between the simulation results and the actual values at N = 1000 and 2000 rpm is thought to be due to the decrease in resin viscosity due to thermal decomposition.

Decomposition of PP using a high shear extruder
13A and 13B are graphs showing predicted and measured values obtained by the simulation results at 3600 rpm in Example 2. As shown in FIG. 13A and FIG. 13B, when the frequency factor A is set to 0 (not considering viscosity reduction), the predicted temperature and reference zero shear viscosity at the outlet of the high shear extruder do not match the measured values. This is because the reference zero shear viscosity _ηr does not decrease and shear heat generation is overestimated. In contrast, the predicted values by the simulation set to A = 5.7 × 10 ⁵ matched well with the measured values, so in the following, A is fixed at 5.7 × 10 ⁵ .

Table 4 shows the predicted values and the measured values obtained by the simulation of Example 2. At all rotation speeds, the predicted values of Tp and ηrf were in excellent agreement with the measured values.

Figures 14A and 14B show the simulation results of temperature and reference zero shear viscosity in Example 2. As the screw rotation speed increases, the temperature increases due to shear heating, and the viscosity decreases due to thermal decomposition of the resin (molecular chain scission). The viscosity significantly decreases in the through hole and in front of the die. This is because at this position, the screw is completely filled with resin, so a large shear force is applied to the resin and the residence time is longer than in an unfilled state. Inside the through hole and die, the shear force applied to the resin is small and the residence time is short, so the thermal decomposition of the resin is suppressed. In the through hole, the temperature did not change significantly due to the adiabatic state, but the die was maintained at 568 K (195 °C), so the temperature of the resin at the die was lower.

After the resin was discharged from the through hole, the temperature of the resin gradually decreased despite the high screw rotation speed. This was because the viscosity decreased in front of the through hole (upstream of the resin flow), which reduced the shear heat generated in the resin, and heat conduction to the barrel became dominant.

In Example 3, the barrel length was increased and the barrel temperature was changed to 573 K (300°C) or higher. The screw rotation speed was fixed at a maximum value of 3600 rpm. A screw with four dam sections with four through holes was used.

Table 5 shows the simulated and measured values of the outlet temperature and the reference zero shear viscosity at the outlet in Example 3. The simulated predicted value of the outlet temperature was slightly lower than the measured value, as predicted from Example 2, but the simulated result of the reference zero shear viscosity was in good agreement with the measured value.

Figure 15 shows a calibration curve (calibration line) created to determine the MFR value of the resin. Using this calibration curve, the MFR values at Tb = 573K and 623K (300°C, 350°C) were 938g/10min and 2411g/10min. In this way, various low molecular weight PPs with MFR values of 1000g/10min or more could be produced using the high shear processing device.

The present invention can be used, for example, to predict the pressure, residence time, resin filling rate, resin viscosity, and temperature in an extruder and after processing when applying shear force to a commonly used polypropylene resin to produce a low-viscosity polypropylene resin that can be used as a raw material for nonwoven fabrics.

Claims (4)

A Nusselt number calculation step of calculating the Nusselt number in the screw for each combination of the screw rotation speed and barrel temperature of the extruder;
an estimated value setting step of setting estimated values of pressure, temperature and viscosity of a resin with an outlet of the extruder as a target site;
a flow velocity and pressure gradient calculation step of calculating a flow velocity and a pressure gradient of the resin in the target portion using physical properties of the resin, a configuration of the extruder, and operating conditions of the extruder;
a temperature gradient/viscosity gradient calculation step of calculating a temperature gradient and a viscosity gradient in the target portion using the Nusselt number, the flow velocity, and the pressure gradient;
a pressure/temperature/viscosity calculation step of calculating the pressure, temperature, and viscosity of an adjacent portion separated by a predetermined distance from the target portion on the inlet side of the extruder by using the pressure gradient, the temperature gradient, and the viscosity gradient;
and a part determination step of determining whether the adjacent part is an inlet of the extruder,
When it is determined in the part determination step that the adjacent part is not the inlet of the extruder, the adjacent part is treated as a new target part, and the flow rate and pressure gradient calculation step, the temperature gradient and viscosity gradient calculation step, the pressure, temperature and viscosity calculation step, and the part determination step are performed using the values calculated in the pressure, temperature and viscosity calculation step;
When the adjacent part is determined to be an inlet of the extruder in the part determination step, a comparison step is performed in which the calculated values calculated in the pressure, temperature, and viscosity calculation step are compared with actual measured values at the inlet,
If the calculated values do not match the measured values in the comparison step, the process returns to the estimated value setting step, and another estimated value is set for the pressure, temperature, and viscosity of the resin using a root-finding algorithm so as to reduce the residual. The above steps are repeated until the calculated values match the measured values.
if the calculated values and the actual measured values match in the comparing step, the pressure, temperature, and viscosity of the resin set as the estimated values in the estimated value setting step are set as predicted values;
How to simulate an extruder. The method further includes a target part determination step of determining whether the target part is a screw or another part, and a filling degree determination step of determining whether the pressure is positive or below zero when the target part is determined to be the screw in the target part determination step.
The method for simulating an extruder according to claim 1 . The Nusselt number calculation step
The Nusselt number Nu in the screw is calculated using the following formula (17),
(In formula (17), L _p is the path length of the screw, z is the coordinate in the extension direction of the screw path, and Nu _Z is expressed by the following formula (16).)
(In formula (16), H is the height of the screw groove, q is the heat flux, z is the coordinate in the extension direction of the screw path, y is the coordinate in the depth direction of the screw groove, κ is the thermal conductivity of the resin, and Tb is the barrel temperature. T(z) is expressed by the following formula (14).
(In formula (14), H is the height of the screw groove, y is the coordinate in the depth direction of the screw groove, ^vz is the flow velocity of the resin in the extension direction of the screw path, and T is the temperature of the resin.)
The Nusselt number Nu at the portion other than the screw is calculated using the following formula (23).
(In equation (23), κ is the thermal conductivity of the resin, l is the length of the die or through hole, n is the viscosity index when a power law fluid is assumed, ρ is the melt density of the resin [kg/m ³ ], _Cp is the specific heat of the resin [J/(kg·K)], and Q is the extrusion speed of the resin (m ³ /sec).)
The method for simulating an extruder according to claim 2. A Nusselt number calculation step of calculating the Nusselt number of the screw for each combination of the screw rotation speed and barrel temperature of the extruder;
an estimated value setting step of setting the pressure, temperature and viscosity of the resin with the outlet of the extruder as a target site;
a flow velocity and pressure gradient calculation step of calculating a flow velocity and a pressure gradient of the resin in the target portion using physical properties of the resin, a configuration of the extruder, and operating conditions of the extruder;
a temperature gradient/viscosity gradient calculation step of calculating a temperature gradient and a viscosity gradient in the target portion using the Nusselt number, the flow velocity, and the pressure gradient;
a pressure/temperature/viscosity calculation step of calculating the pressure, temperature, and viscosity of an adjacent portion separated by a predetermined distance from the target portion on the inlet side of the extruder by using the pressure gradient, the temperature gradient, and the viscosity gradient;
and a part determination step of determining whether the adjacent part is an inlet of the extruder,
When it is determined in the part determination step that the adjacent part is not the inlet of the extruder, the adjacent part is treated as a new target part, and the flow rate and pressure gradient calculation step, the temperature gradient and viscosity gradient calculation step, the pressure, temperature and viscosity calculation step, and the part determination step are performed using the values calculated in the pressure, temperature and viscosity calculation step;
When the adjacent part is determined to be an inlet of the extruder in the part determination step, a comparison step is performed in which the calculated values calculated in the pressure, temperature, and viscosity calculation step are compared with actual measured values at the inlet,
If the calculated values do not match the measured values in the comparison step, the process returns to the estimated value setting step, and another estimated value is set for the pressure, temperature, and viscosity of the resin using a root-finding algorithm so as to reduce the residual. The above steps are repeated until the calculated values match the measured values.
if the calculated values and the actual measured values match in the comparing step, the pressure, temperature, and viscosity of the resin set as the estimated values in the estimated value setting step are set as predicted values;
A simulation program that causes a computer to execute a simulation method for an extruder.