Leakage-Flow Models for Screw Extruders

Abstract

Many theoretical analyses of extrusion ignore the effect of the flight clearance when predicting the pumping capability of a screw. This might be reasonable for conventional extruder screws with "normal" clearances but leads to errors when more advanced screw designs are considered. We present new leakage-flow models that allow the effect of the flight clearance to be included in the analysis of melt-conveying zones. Rather than directly correcting the drag and pressure flows, we derived regression models to predict locally the shear-thinning flow through the flight clearance. Using a hybrid modeling approach that includes analytical, numerical, and data-based modeling techniques enabled us to construct fast and accurate regressions for calculating flow rate and dissipation rate in the leakage gap. Using the novel regression models in combination with network theory, the new approximations consider the effect of the flight clearance in the predictions of pumping capability, power consumption and temperature development without modifying the equations for the down-channel flow. Unlike other approaches, our method is not limited to any specific screw designs or processing conditions.

Keywords: extrusion; leakage flow; modeling and simulation; polymer processing.

Figure 1

The development of leakage-flow models to predict flow rate and dissipation in the leakage gap of a discretized screw channel.

Figure 2

Schematic work-flow chart of the hybrid modeling approach, including analytical, numerical, and data-based modeling.

Figure 3

The top view of the unwound screw channel.

Figure 4

Flat-plate approximation of the unwound screw channel (a) and the flight clearance (b). In the representation of the flight clearance, the flow channel was rotated clockwise by 90°: $x$ is the direction across and $z$ the direction along the clearance.

Figure 5

The volume flow rate ${\Pi }_V^{\delta }$ as a function of pressure gradient ${\Pi }_{p,x}^{\delta }$: Influence of power-law index for $t/D_b=0.5$ (a) and $t/D_b=1.0$ (b).

Figure 6

The volume flow rate ${\Pi }_V^{\delta }$ as a function of pressure gradient ${\Pi }_{p,x}^{\delta }$: Influence of screw-pitch ratio for $n=0.2$. The scaling of the diagrams was adjusted to better visualize the influence for smaller pressure gradients (a) and larger pressure gradients (b).

Figure 7

Dimensionless dissipation ${\mathit{\Pi }}_{\mathit{Q}}^{\mathit{\delta }}$ as a function of the dimensionless pressure gradient ${\mathit{\Pi }}_{\mathit{p},\mathit{x}}^{\mathit{\delta }}$ (a) and as a function of the dimensionless volume flow rate ${\mathit{\Pi }}_{\mathit{V}}^{\mathit{\delta }}$. (b) The influence of the power-law index for a square-pitched screw with $\mathit{t}/{\mathit{D}}_{\mathit{b}}=\mathbf{1.0}$.

Figure 8

Dimensionless dissipation ${\mathit{\Pi }}_{\mathit{Q}}^{\mathit{\delta }}$ as a function of the dimensionless pressure gradient ${\mathit{\Pi }}_{\mathit{p},\mathit{x}}^{\mathit{\delta }}$ (a) and as a function of the dimensionless volume flow rate ${\mathit{\Pi }}_{\mathit{V}}^{\mathit{\delta }}$. (b) The influence of the screw-pitch ratio for a polymer melt with power-law index $\mathit{n}=\mathbf{0.2}$.

Figure 9

The dimensionless volume flow rate ${\mathit{\Pi }}_{\mathit{V}}^{\mathit{\delta }}$ (a) and dimensionless dissipation ${\mathit{\Pi }}_{\mathit{Q}}^{\mathit{\delta }}$ (b) as functions of the dimensionless pressure gradient ${\mathit{\Pi }}_{\mathit{p},\mathit{x}}^{\mathit{\delta }}$ for a square-pitched screw with $\mathit{t}/{\mathit{D}}_{\mathit{b}}=\mathbf{1.0}$ and a polymer melt with the power-law index $n=0.2$.

Figure 10

The corrected dimensionless volume flow rate $\mathsf{\Delta }{\mathit{\Pi }}_{\mathit{V}}^{\delta }$ (a) and corrected dimensionless dissipation $\mathsf{\Delta }{\mathit{\Pi }}_{\mathit{Q}}^{\mathit{\delta }}$ (b) as functions of the dimensionless volume flow rate ${\mathit{\Pi }}_{\mathit{V}}^{\mathit{\delta }}$ for a square-pitched screw with $\mathit{t}/{\mathit{D}}_{\mathit{b}}=\mathbf{1.0}$ and a polymer melt with power-law index $\mathit{n}=\mathbf{0.2}$.

Figure 11

A comparison of approximated results obtained from $\mathsf{\Delta }{\Pi }_V^{\delta }=f\left(t/D_b, n, {\Pi }_{p,x}^{\delta }\right)$ and numerical solutions for $t/D_b=0.50$ (a) and $t/D_b=1.23$ (b). The points indicate numerical results, and the continuous lines approximated solutions.

Figure 12

A comparison of approximated results obtained from $\mathsf{\Delta }{\Pi }_Q^{\delta }=f\left(t/D_b, n, {\Pi }_V^{\delta }\right)$ and numerical solutions for $t/D_b=0.50$ (a) and $t/D_b=1.23$ (b). The points indicate numerical results, and the continuous lines approximated solutions.

Figure 13

A scatter plot of $\mathsf{\Delta }{\Pi }_V^{\delta }=f\left(t/D_b, n, {\Pi }_{p,x}^{\delta }\right)$: training and test set (a) and validation set (b). The dashed lines indicate an absolute error of 0.07.

Figure 14

A scatter plot of $\mathsf{\Delta }{\Pi }_Q^{\delta }=f\left(t/D_b, n, {\Pi }_V^{\delta }\right)$: training and test set (a) and validation set (b). The dashed lines indicate a relative error of 10%.