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The SPE Extrusion Division

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ICIPC – Plastics and Rubber Institute

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Specific heat capacity describes the amount of energy that it takes to raise the temperature of a polymer. LDPE requires an average of 2.596 J/(g.K) in the molten state, while PS requires 1.95 J/(g.K), [1]. Therefore, it takes almost 33 % more energy to raise the temperature of 1 g of LDPE than 1 g of PS. Additionally, it means that more power is necessary to process LDPE than PS at the same extrusion processing conditions like output in kg/h and melt temperature. It is well known that the specific heat capacity is the derivative of enthalpy with respect to temperature.

Thermal conductivity is the ability of the polymer to conduct heat. In general, the polymers exhibit a low thermal conductivity. LDPE has an average thermal conductivity of 0.25 W/(m.K) in the molten state, while PS is 0.14 W/(m.K), [1]. Because LDPE has a higher thermal conductivity than PS, the heat flux for LDPE from barrel to polymer is higher.

Solid and melt density can be obtained from p-v-T diagram, for example, Tait´s equation of state. The density is found from reciprocal of specific volume. Nevertheless, other authors have shown in the literature that Tait´s equation does not work well for semicrystalline polymers [2]. The research work of Sanchez-Cho on equation of state for polymers seems to be until now the most accurate equation of state [2, 3].

The solid state enthalpy (H

Where,

A: Parameter in J/g

B: Parameter in K

C Parameter in K

D: Parameter in K

T: Temperature in K

The enthalpy for molten state (H

Where,

F: Parameter in J/(g.K)

E: Parameter in J/(g.K

T: Temperature in K

The specific heat capacity for solid state (Cp

Where the parameters A, B, C and D are taken from the enthalpy function,

A: Parameter in J/g

B: Parameter in K

C: Parameter in K

D: Parameter in K

T: Temperature in K

The specific heat capacity for molten state (Cp

Where the parameter E is taken from the enthalpy function,

E: Parameter in J/(g.K)

T: Temperature in K

The heat of fusion for semicrystalline polymers can be obtained graphically from the enthalpy function as shown in Fig. 1 for a HDPE. The heat of fusion is the amount of heat energy that is required to convert a polymer from a solid to a liquid or melt.

The thermal conductivity (k) from the solid state to the molten state is proposed as one function with six parameters:

Where,

A: Parameter in W/(m.K)

B: Parameter in K

C: Parameter in K

D: Dimensionless parameter

E: Dimensionless parameter

F: Parameter in W/(m.K)

T: Temperature in K

This function is suitable for the range from room temperature up to typical melt temperatures in the single screw extrusion process.

The solid state density (ρ

Where,

A: Parameter in cm

B: Parameter in K

C: Parameter in bar

D: Parameter in K

E: Parameter in K

F: Parameter in bar

T: Temperature in K

p: Pressure in bar

The density for molten state (ρ

Where,

A: Parameter in cm

B: Parameter in bar

C: Parameter in (bar.cm3) / (g.K)

T: Temperature in K

p: Pressure in bar

The thermal diffusivity (α) can be calculated from previous functions, such as, specific heat, thermal conductivity and density.

Other Authors have investigated extensions of some of the above mentioned equations or functions, [5, 6, 7]. The functions described in this work have been found more suitable for simulation and design of screws in single screw extrusion, and they were obtained looking for small number of parameters, what it is appropriate for the industrial practice.

There is no fundamental theory behind the equations presented in this paper. This is reasonable because in practice the polymers are normally not pure but mixed with additives and reinforcing agents, blended with other polymers to combine their properties, etc. This will alter the polymer physics and complicate an appropriate fundamental theory.

The previous approximation functions contribute to an easier single screw calculation to simulate the performance of a single screw extruder with regard to output, pressure build-up, melting rate, heat transfer and melt temperature. These thermal functions can be used for the calculations in the screw melting zone, screw metering zone and extrusion die, Fig. 2, specially, for the prediction of pressure build-up and melt temperature profile.

A 10 mg of sample was heated from an initial temperature up to a final temperature under high purity nitrogen, using a heating rate of 20°C/min. An isotherm at final temperature is required during 5 minutes.

A: -971.49 J/g

B: 350.44 K

C: 475.11 K

D: 15.18 K

E: 0.0034 J/(g.K

F: -303.155 J/g

The Fig. 3 shows the measured data and the obtained approximation functions. From the figure it is possible to observe that the approximation functions match the data very well.

The specific heat capacity for solid state (Cp

The Fig. 4 shows the obtained approximation functions from measured enthalpy data. This figure illustrates that the approximation functions match the data properly.

The thermal conductivity (k) from the solid state to the molten state for a HDPE reported by VDMA [8] has met the following parameters:

A: 0.19886 W/(m.K)

B: -0.00538 1/K

C: 0.00873 1/K

D: -4.76

E: 16.81785

F: 0.25451 W/(m.K)

The Fig. 5 shows the measured data and the obtained approximation function. From the figure it is possible to observe that the approximation function fits the data satisfactorily.

The solid state density (ρ

A: 1.042 cm

B: 3540 K

C: 2166 bar

D: 422.1 K

E: 8.364 K

F: 353.3 bar

The density for molten state (ρ

A: 0.9537 cm

B: 2435 bar

C: 1.8542 (bar.cm3)/(g.K)

The Fig. 6 shows the measured data and the obtained approximation functions for the solid and the molten state. From the figure it is clear that the approximation functions fit the data very well.

The Fig. 7 is demonstrating the use of the above described thermal functions in combination with analytical calculations, [9], for the prediction of pressure build-up in a single screw of D = 20 mm and L/D = 20 for PE processing. The left ordinate axis shows the screw pitch, the right ordinate axis shows the screw channel depth and the abscissa or X-axis is the axial screw length.

Solids bed profiles (SBP) could be obtained for several polymers and single screws using the experimental values of X/W versus L/D from a patented device compared to analytical melting models using the mentioned thermal functions, [10].

There is no fundamental theory behind the equations shown in this paper. This is reasonable because in practice the polymers are normally not pure but mixed with additives and reinforcing agents, blended with other polymers to combine their properties, etc. This will alter the polymer physics and complicate an appropriate fundamental theory.

The developed approximation functions for thermal properties, such as, specific heat capacity, enthalpy, thermal conductivity and density, fitted the measured data very well. The authors gave an adequate mathematical representation of the measured data, suitable for inclusion in future analytical models and papers.

/2/ Zoller Paul, “PVT Reltionship and Equations of State of Polymers,

/3/ Cho, J. and Sanchez, I., “PVT Relationship and Equations of State of Polymers”

/4/ Spencer, R.S. and Gilmore, G.D., “Equation of state for high polymers”, Journal of Applied Physics, 21, pp. 523-526, (1950).

/5/ Sanchez, I. and Cho, J., “A universal equation state for polymer liquids”, Polymer, Vol. 36, No. 15, pp. 2929 – 2939, 2005

/6/ Naranjo, Alberto, “Measurement of the Thermal Diffusivity of Thermoplastics under Processing Conditions”, IKV-Berichte aus der Kunststoffverarbeitung, Mainz, G., Germany, 2004

/7/ Sedlacek, T., et.al. “On relationship between PVT and rheological measurements of polymer melts”, Annual transactions of the Nordic Rheology Society, Vol. 13, 2005.

/8/ VDMA, “Thermodynamik”, Carl Hanser Verlag, Munich, Germany, 1979

/9/ Grünschloss, E., “Polymer Single Screw Extrusion: Modeling“, Encyclopedia of Materials: Science and Technology, Elsevier Science Ltd., 2001

/10/ Noriega, María del Pilar, et. al., “Method and device to visualize in-line and quantify the polymer melting in plasticating screw machines without significantly affecting its thermal regime”, U.S Patent No. 7314363, January 1, 2008.

L/D Dimensionless Extruder length

X/W Dimensionless screw channel width

H

Hm: Enthalpy of the melt

Cp

Cp

k: Thermal conductivity

ρ

ρ

υ

υ

α: Thermal diffusivity

R: Gas constant, 8.3144 J/(mol.K)

p: Pressure

T: Temperature

LDPE: Low density polyethylene

HDPE: High density polyethylene

PP: Polypropylene

PS: Polystyrene

Figure 1: Graphically obtained heat of fusion for HDPE

Figure 2: Zones in a single screw extruder

Figure 3: Enthalpy of PP- SABIC 505P: Measured data and approximations

Figure 4: Specific heat capacity of PP- SABIC 505P: Approximations

Figure 5: Thermal conductivity of HDPE: Measured data and approximation

Figure 6: p-v-T diagram of PP (Borealis): Measured data and approximations

Figure 7: Pressure build-up (ο) of a Ø20 mm screw for PE, channel depth (∆) and screw pitch (x) versus axial screw length

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