• Entgasungsoberflächenberechnungen
• Entasungseffizienz

Material degradation and degassing

• Thermal (oxidative, mechanical)
• Hydrolytic degradation

Hydrolytic degradation is the most influential degradation type. This results in the necessity of the degassing process, because the polymer melt tends to absorb humidity.

Furthermore the filling degree influences the degassing surface, which has a big impact on the degassing quality. The residence time is also an influencing Parameter.

The material transport out of the polymer is only possible at very high diffusion coefficients via the reduction of the partial pressure by applying a vacuum. The vacuum reduces the partial pressure of the low molecular weight component (water) and allows the water to leak out from the polymer.

Convection and diffusion to the free surface are responsible for mass transfer. The diffusion coefficient of the water in the polymer is very low. At the phase boundary, between air and polymer, the gradient is greatly increased and again enhanced by degassing.

The vacuum degassing increases the diffusion coefficient, which ultimately leads to a faster mass transport of the low molecular weight component.

Figure 1: Material transport between polymer and gas phase during degassing

Alternatives to increase the diffusion coefficient:

• Addition of additives
• Application of a vacuum in the degassing zone
• Expansion and size of the surfaces at the phase boundary

Decisive for a successful degassing are the applied vacuum and the surface of the melt in the screw channel.

The free surface is composed of the melt pool and melt film, which are constantly renewed by the continuous rotation of the screws.

The renewal is defined by the renewal time which indicates the residence time of a fluid element on the free surface before it is renewed by a screw rotation.

The calculations of the free surface and renewal time are based on the mathematical model of Schuler.

Figure 2: The left figure shows the formation of a melt pool in the screw channel and the right figure shows the position of the melt film in the screw channel.

The renewal time is calculated according to Schuler's model by equation (1)

$$t_{Pool} = \frac{h}{D_a \pi n \sin \varphi} \tag{1}$$

This reflects the ratio of the height h of the channel to the velocity v of the moving surface in the Z direction.

The surface (2) is the result of the multiplication of the height and the development of the channel, taking into account that no melt pool is formed in the contact zone.

$$A_{Pool} = 2 \frac{\pi - \alpha_E}{\pi} \cdot \frac{2hL}{\sin \varphi} \tag{2}$$

For the determination of the melt film, the difference of the shell surface of the eight-shaped housing (filled with melt) and comb surfaces of the screw is drawn up, thereby the surface of the film is determined (see equation 3).

$$A_{Film} = 2 \frac{\pi - \alpha_E}{\pi} D_a \pi L \left(1 - \frac{2e}{t \cos \varphi}\right)(1 - \varepsilon) \tag{3}$$

The surface renewal time is calculated by using Equation (4).

$$t_{Film} = \frac{1}{2n} \tag{4}$$

The Schuler model is characterized by the consideration of the free surfaces in the contact zone of both screws and the inclusion of the screw base in the calculation of the film. The constant height of the channel leads to a decisive disadvantage in the calculation. To take the channel height into account, the modification of the Schuler model is necessary.

During the modification, it is assumed that the polymer is completely melted in the last third of the screw channel and there's a bubble-free layer flow. The essential difference is that here the height of the melt is present as a function of the filling degree, as shown in equation 5.

Figure 3 shows that the channel is divided into three sections. The grey area is the product of the pool.

This can be done by the integration of the formula 5 within the limits of -b/2 to -x.

$$h(x) = \frac{D_a}{2}\left[1 + \cos\left(\frac{2\pi\left(x - \frac{e}{2}\right)}{t \cos \varphi}\right)\right] - \sqrt{a^2 - \left(\frac{D_a}{2}\right)^2\left(\sin\left(\frac{2\pi\left(x - \frac{e}{2}\right)}{t \cos \varphi}\right)\right)^2} \tag{5}$$

Since the unique solution of the integral of Equation 6 is complicated, an approximation is performed.

Due to the complicated geometry, the channel is divided into three different areas. The areas between +/- b/2 to e/2 are referred to as the edge cross section and the interval between -e/2 and + e/2 is defined as the web area.

The solution of the approximation is shown in equation 6.

$$\frac{1}{2}D_a\left[x - \frac{1}{2}\frac{t\cos(\varphi)\sin\left(\frac{2\pi\left(x - \frac{e}{2}\right)}{t\cos(\varphi)}\right)}{\pi}\right] + \frac{1}{4}\frac{D_a^2}{\sqrt{a^2}}\cos\left(\frac{\pi(2x - e)}{t\cos(\varphi)}\right) $$ $$+ \sqrt{a^2 - D_a^2\sin\left(\frac{\pi(2x - e)}{t\cos(\varphi)}\right)^2} + 2\sqrt{\frac{D_a^2}{D_a^2}} - D_a^2\text{Elliptical}\left[\frac{1}{2}\sin\left(\frac{\pi(2x - e)}{t\cos(\varphi)}\right), \sqrt{\frac{D_a^2}{a^2}}\cdot 2\sqrt{\frac{D_a^2}{D_a^2}}\right] - 4a^2\text{Elliptical}\left[\frac{1}{2}\sin\left(\frac{\pi(2x - e)}{t\cos(\varphi)}\right), \sqrt{\frac{D_a^2}{a^2}}\cdot 2\sqrt{\frac{D_a^2}{D_a^2}}\right]\cos(\varphi) \tag{6}$$

Because of the non-usable elliptical function by Excel, a Taylor series development is performed up to the fourth order, which resolves the most important areas sufficiently. The Taylor series development is shown in equation 7.

$$H(x) = Dx - ax + \frac{1}{3}\left(-\frac{D\pi^2}{t^2\cos(\varphi)^2} + \frac{1}{2}\frac{aD^2\pi^2}{t^2\cos(\varphi)^2a^2}\right)\left(x - \frac{1}{2}e\right)^3$$

$$+ \frac{1}{5}\left(\frac{1}{3}\frac{D\pi^4}{t^4\cos(\varphi)^4} - a\left(\frac{2}{3}\frac{D^2\pi^4}{t^4\cos(\varphi)^4a^2} - \frac{1}{8}\frac{D^4\pi^4}{t^4\cos(\varphi)^4a^4}\right)\right)\left(x - \frac{1}{2}e\right)^5 \tag{7}$$

On the basis of the course of the analytical and approximated solution it becomes clear that in the required interval [e/2; B/2], the solution is sufficiently precise, see Figure 4. The irregularity appears only outside the boundary.

Figure 4: Comparison of the analytical and approximated solution

It is clear from Figure 6 that the theoretical calculation is based on the degree of filling. The filling level includes the current operating parameters of the extruder. This basic knowledge makes it possible to determine the channel height on the basis of the approximation equation with the necessary X-coordinate of the melt in equation 7.

Figure 5: Methodology for determining the renewal of surfaces on the basis of theoretical foundations

Figure 6 shows the schematic sequence of the surface calculation for implementation in SIGMA.

Figure 6: Implementation of the surface renewal time and free surface calculation in SIGMA

The degassing efficiency is evaluated with the degassing reference parameter of Schuler (Equation 1. In this case taken place the determination for wetting and non-wetting polymer.

$$\frac{c_{Start} - c_{End}}{c_{Start} - c_{Gleichgewicht}} = \frac{\left(\frac{A_{POOL}}{t_{POOL}} + \frac{A_{FILM}}{t_{FILM}} + \frac{A_{GRUND}}{t_{GRUND}}\right)}{\dot{m}} \tag{Equation 1}$$

After transposing of the equation can be the end concentration of low molecular component determined. Thereby is it possible to take a statement about the degassing efficiency.

For wetting polymer is the Equation 2and for non-wetting polymer is the Equation 3 used.

$$EK_{Benetzend} = \frac{\left(\frac{A_{POOL}}{t_{POOL}} + \frac{A_{FILM}}{t_{FILM}} + \frac{A_{GRUND}}{t_{GRUND}}\right)}{\dot{m}} \tag{Equation 2}$$

$$EK_{Nicht-Benetzend} = \frac{\left(\frac{A_{POOL}}{t_{POOL}} + \frac{A_{FILM}}{t_{FILM}}\right)}{\dot{m}} \tag{Equation 3}$$

The difference between both equation is that the surface area und renewal time at screw root is neglected, because in non-wetting case are not melt at screw root available.