Flow Patterns in a Screw Extruder (COMSOL) Consider the screw extruder just discussed in Example 6.5. Solve the equations of motion to determine as precisely as possible the flow pattern given approximately in Fig. E6.5.3(b), normal to the flight axis. Note that the pressure cannot be allowed to “f

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Flow Patterns in a Screw Extruder (COMSOL)

Consider the screw extruder just discussed in Example 6.5. Solve the equations of motion to determine as precisely as possible the flow pattern given approximately in Fig. E6.5.3(b), normal to the flight axis. Note that the pressure cannot be allowed to "float" but must be specified to some arbitrary value at any one point in the cross section shown in Fig. E6.6.1, such as the midpoint B.

Based on the comments at the end of Example 6.5, use the following values, which relate to the extrusion of a polyolefin:

Polymer properties: [latex] ho=800 \mathrm{~kg} / \mathrm{m}^{3}, \mu=500 \mathrm{~Pa} \mathrm{~s}\left(\mathrm{~N} \mathrm{~s} / \mathrm{m}^{2} ight)[/latex].
Rotational speed [latex]N=65 \mathrm{rpm}[/latex], so that [latex]\omega=2 \pi imes 65 / 60=6.8 \mathrm{rad} / \mathrm{s}[/latex].
Channel depth [latex]h=5 \mathrm{~mm}=0.005 \mathrm{~m}[/latex].
For a square pitch, the pitch [latex]P[/latex] and screw diameter [latex]D[/latex] are identical, taken to be [latex]100 \mathrm{~mm}=0.1 \mathrm{~m}[/latex]. The distance between flights is [latex]W=P \cos heta=[/latex] [latex]0.1 imes \cos 17.66^{\circ}=0.0953 \mathrm{~m}[/latex]. For simplicity in our example, we shall round up slightly and take [latex]W=0.1 \mathrm{~m}[/latex].
The relative velocity is [latex]V=\omega r=\omega D / 2=6.8 imes 0.1 / 2=0.340 \mathrm{~m} / \mathrm{s}[/latex], with [latex]x[/latex] component [latex]V_{x}=-0.340 imes \sin heta=-0.340 imes \sin 17.66^{\circ}=-0.103 \doteq-0.1 \mathrm{~m} / \mathrm{s}[/latex]
After solving for the flow pattern, make the following plots:

(a) A picture that shows 10 streamlines, one with the true aspect ratio and another with the [latex]y[/latex]-axis "stretched" by a factor of five.

(b) An arrow plot of the [latex]x[/latex] velocities.

(c) A contour plot of the pressure distribution.

(d) A cross-section plot of the [latex]x[/latex] velocity between the midpoints A [latex](0.05,0.005)[/latex] of the upper moving surface and B [latex](0.05,0)[/latex] of the lower stationary surface.

Accepted Answer

See Chapter 14 for more details about COMSOL Multiphysics. All mouse-clicks are left-clicks (the same as Select) unless specifically denoted as right-click (R).

Select the Physics

Select Model Wizard, 2D, Fluid Flow (the little rotating triangle is called a "glyph"), Single-Phase Flow, Laminar Flow, Add. Note COMSOL's symbols for pressure and velocity. Study, Stationary, Done.

Define the Parameters

R-Global Definitions, Parameters. Enter [latex]r h o 1=800\left[\mathrm{~kg} / \mathrm{m}^{3} ight], m u 1=500[/latex] [latex]\left[\mathrm{Pa}^{*} \mathrm{~s} ight], V x=-0.1[\mathrm{~m} / \mathrm{s}][/latex] under the Name/Expression columns. (The names rho and [latex]m u[/latex] are reserved by COMSOL for its own use, and so are unavailable to us.)

Establish the Geometry

R-Geometry, Rectangle. Note Corner is at [latex]x=0, y=0[/latex]. Width [latex]=0.1[/latex], Height [latex]=0.005[/latex]. Build Selected. R-Geometry, Point (this is where we shall specify [latex]p=0[/latex] ). Enter [latex]0.05[/latex] and [latex]0.00[/latex] for [latex]x[/latex] and [latex]y[/latex]. Build Selected.

Define the Fluid Properties

Under Component1, R-Materials, Blank Material. Enter rho1 and mu1 as values. To see the equations being solved, R-click Laminar Flow and select the Equations glyph in the Settings window.

Define the Boundary Conditions

R-Laminar Flow, Wall, Wall 2. Hover over the top wall (turns red), L-click to select. Boundary Condition, Sliding Wall. Enter [latex]V x[/latex] for [latex]U w[/latex]. Set pressure constraint with R-Laminar Flow, Points, Pressure Point Constraint. Hover over bottom center point and select with L-click (note Pressure is zero). Point 1 should appear in the Active Window.

Create the Mesh and Solve the Problem

Mesh 1, Element size, Coarse, Build All. R-Mesh 1, Statistics and note 14,695 elements. To see some of the elements, use the Zoom icon three times, then Zoom Extents to revert.
R-Study, Compute (takes about 4 seconds). Gives velocity magnitude, not of interest here.

Display the Streamlines

R-Results, 2D Plot Group. R-2D Plot Group 3, Streamline. Settings window, Positioning, Start point controlled, 20 points, yields Fig. E6.6.2.
Stretch the [latex]y[/latex]-axis to improve visibility by R-Definitions (under Component 1), View. L-click the glyph at the left of View 2 to expand its tree, Axis. Settings, View Scale, Manual, [latex]y[/latex]-scale factor [latex]=5[/latex], Update, Zoom Extents. Streamline 1 then gives Fig. E6.6.3.

Display the Arrow Velocity Plot

View 1. R-click Results, 2D Plot Group. R-click 2D Plot Group 4, Arrow Surface 1. Plot. Scale Factor, move slider to right to give Scale Factor [latex]0.034[/latex], Color black.
In the Graphics window, zoom in and out by either: (a) simultaneously holding the middle mouse button down and moving the mouse forwards and backwards, or (b) clicking on Zoom above the Graphics window and holding the [latex]R[/latex] mouse button down to move the graphics around, giving Fig. E6.6.4.

Display the Isobars

R-click Results and L-click 2D Plot Group.
L-click the dropbox (under Definitions for Component 1) for View and select View [latex]1 .[/latex]
R-click the new 2D Plot Group 5 and L-click Contour.
In Settings, Expression field, enter pressure, [latex]p[/latex]. Levels field, change the Entry Method from Number of Levels to Levels. In the Levels field, enter the pressure range as range [latex](-0.6 \mathrm{e} 6,0.1 \mathrm{e} 6,0.6 \mathrm{e} 6)[/latex], Plot; "range" is a function whose first and last parameters denote the start and end; the middle parameter is the increment.
By comparing the isobars with the color bar, or by selecting Level Labels, note that the pressure varies from approximately [latex]-5.9 imes 10^{5} \mathrm{~Pa}[/latex] at the midpoint of the right end to approximately [latex]5.9 imes 10^{5} \mathrm{~Pa}[/latex] at the midpoint of the left end, and zero in the middle.
Change Coloring to Uniform and the Color to black to generate Fig. E6.6.5.

Construct the Velocity Cross-Plot

Section the data set at the midpoint in the [latex]y[/latex]-direction: R-click Data Sets within the Results tree and select Cut Line 2D. In the Settings window, enter [latex](0.05,0)[/latex] and [latex](0.05,0.005)[/latex] for Points 1 and [latex]2 .[/latex] L-click Plot to display the cut line.
R-click Results and L-click 1D Plot Group 6. In the Settings window, L-click the dropdown dialog and change the data set to Cut Line 2D 1.
R-click the 1D Plot Group 6 and select Line Graph [latex]1 .[/latex]
Within Settings, Expression field, enter [latex]u[/latex], the [latex]x[/latex]-velocity component. Expand the Coloring and Style section by L-clicking the triangular glyph at the left.
Change the Color option to Black by L-clicking the dropdown dialog box.
L-click Plot at the top of the Settings window to update the plot. There is a Recovery option at the bottom of the Settings window under the Quality section. Select Recover Within Domains, which does polynomial smoothing, giving Fig. E6.6.6.

Discussion of Results

The COMSOL results are shown in Figs. E6.6.2-E6.6.6. The arrow plot of Fig. E6.6.4 clearly shows that the velocity profile is virtually unchanged over the whole width of the cross-section. Near the top surface the flow is induced by the moving surface to be in the negative [latex]x[/latex]-direction. Toward the fixed bottom surface there is a flow reversal, so that at any value of [latex]x[/latex] the net flow rate [latex]Q=\int_{0}^{h} v_{x} d y[/latex] is zero.

Fig. E6.6.2, which contains 20 streamlines, now indicates that the essential flow reversal occurs very close to each end of the cross section. To make the flow pattern clearer, the streamlines are repeated in Fig. E6.6.3, in which a larger scale is employed in the [latex]y[/latex]-direction-that is, the [latex]x / y[/latex] geometry is no longer to scale.

The pressure distribution is shown in Fig. E6.6.5 as a series of isobars. Note that the isobars are spaced very evenly and that the pressure increase from right to left (at the end midpoints) is from [latex]-5.9 imes 10^{5} \mathrm{~Pa}[/latex] to [latex]5.9 imes 10^{5} \mathrm{~Pa}[/latex], or [latex]1.18 imes 10^{6}[/latex] Pa total.

Recall that the simplified theory gave the pressure gradient as:

[latex]-\frac{\partial p}{\partial x}=\frac{6 \mu r \omega \sin heta}{h^{2}} .[/latex] (E6.5.9)

Thus, the theoretical increase in pressure from right to left should be:

[latex]0.1\left(\frac{6 imes 500 imes 0.05 imes 6.8 imes \sin 16.77^{\circ}}{0.005^{2}} ight)=1.18 imes 10^{6} \mathrm{~Pa},[/latex] (E6.6.1)

in complete agreement with the more accurate two-dimensional solution of the Navier-Stokes equations shown in Fig. E6.6.5.

Finally, Fig. E6.6.6 shows how [latex]v_{x}[/latex] varies from the midpoint A of the top surface to the midpoint B of the bottom surface. As explained in Eqn. E6.5.7, there are two contributions to this velocity profile:

A Couette flow from right to left, driven by the leftward-moving top surface.
A Poiseuille flow from left to right, driven by the excess pressure at the left.

Question 6.6: Flow Patterns in a Screw Extruder (COMSOL) Consider the scre...

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