The following screw elements are implemented in SIGMA:

• Threaded elements
• Pushing flight elements
• Threaded mixing elements (conveying / reconveying)
• V-Mixing elements
• Kneading blocks
• Eccentrical kneading blocks
• Distance elements
• Blister elements
• Tooth mixing elements
• Transition Elements
• Transition Conveying Elements

To define a new screw element, choose the New element button in the Machine configuration and select the kind of element you would like to define. Then the dialog box for entering the values will open up.

As with barrel elements, already created screw elements can be modified by opening the corresponding dialog through double-clicking the element to be edited, by selecting the icon Edit properties of the selected element, or by making the desired changes directly in the Element Properties list.

For seven of the ten available elements, the dialogs provide the option to define the geometry precisely through additional inputs. As an alternative to specifying the volume correction factor, the actual screw diameter and either the pitch angle or the root angle of the screw profile are entered, with the other angle being calculated automatically.

The input screen for defining and modifying threaded elements (both forward and return elements) contains all the values required to clearly describe tightly meshing profiles.

The specific meaning of the variables to be defined can be seen in the drawing of a pair of elements in the figure.

Please note the following when entering data:

• The centre distance cannot be changed in this screen, as it is predetermined by the general machine data already entered.
• The length of the screw element and the pitch can be entered either in absolute values ([mm]) or in multiples of the grid dimension.
• The volume correction factor is a measure of the deviation of the theoretically close-fitting geometry used in SIGMA from the actual geometry.
• (As an alternative to the volume correction factor, additional inputs can be specified to define the geometry exactly. In this case, the actual screw diameter and either the cutting angle or the base angle of the screw profile must be entered. The other angle is calculated.

The geometry of push edge elements differs from the geometry of conveyor elements only in that it has an additional free surface. In SIGMA, this surface is characterised by two parameters, the angle of inclination $$\phi_s$$ and the radius of curvature r. Otherwise, the same specifications as for conveyor and return elements apply.

Unlike conveyor elements, screw mixing elements have grooves cut into their flanks. In addition to the pitch and number of grooves, the geometry of the groove in the normal section must also be defined. To ensure maximum flexibility, the geometry of the groove has been simplified by using a rectangular cross-section (characterised by groove width b and groove depth h).

The dialog box of v-mixing elements is similar to the one of threaded mixing elements. It differs in two aspects: The Real diameter of the screw is additionally needed and the volume correction factor is fixed. The input mask of V-mixing elements is shown in the figure.

When entering kneading blocks, please note the following:

* In order to describe kneading blocks completely, in addition to length and diameter, the number of kneading discs to be combined into a kneading block must also be specified. * Furthermore, the offset angle is required. The offset angle may only be entered in the range

$$-\frac{180°}{z} \leq \text{Staggering angle} \leq \frac{180°}{z}$$

where i is the number of elements. Entering negative offset angles characterises a kneading block with a damming effect, while positive offset angles characterise a kneading block with a conveying effect. Conveying-neutral kneading blocks are obtained by entering an offset angle of 180°/i.

Unlike conventional kneading blocks, shoulder kneading block have narrower kneading discs, which creates a larger leakage gap. In SIGMA, the width of the kneading discs of the shoulder kneading elements is assumed to be half the width of the conventional kneading blocks. It follows that the input masks for the two elements do not differ.

• In order to fully describe kneading blocks, in addition to length and diameter, the number of kneading discs to be combined into a kneading block must also be specified.
• In addition, the offset angle is required. The offset angle may only be entered in the range $$-\frac{180°}{i} \leq \text{offset angle} \leq \frac{180°}{i}$$, where i is the number of elements. Entering negative offset angles characterises a kneading block with a damming effect, while positive offset angles characterise a kneading block with a conveying effect. Conveying-neutral kneading blocks are obtained by entering an offset angle of $$\frac{180°}{i}$$.

Furthermore, the eccentricity is defined by the outer diameter $$D_a$$.

In practice, eccentric kneading discs are often used in co-rotating twin-screw extruders to increase the melting capacity.

It is known that only one ridge in a profile is in contact with the housing. If the ridge angle is zero, the eccentric profile can be created by shifting. The displacement is called eccentricity e. This generated eccentric profile is also close-fitting. The shapes of the contours are not changed after the displacement. The following image shows the displacement of the profiles.

With these boundary conditions, the channel model in the following image also results for the eccentric kneading block as the basis for the calculation and the volume flow balance.

If the comb angle is zero, the diameter ratio of the three-course profile is at the maximum point.

** Maximum diameter ratio of three-way profile (i=3)** $$D_{v,3} = D_{a,3 \max} = \frac{1}{2 \cdot \cos\left(\frac{\pi}{2 \cdot i}\right) - 1} = 1.366 \tag{Equation 3-13}$$

Outer diameter of three-course profile

$$D_{a,3} = \frac{2 \cdot a}{1 + \frac{1}{D_{v,3}}} \tag{Equation 3-14}$$

Eccentricity $e$

$$e = \frac{D_{a,2} - D_{a,3}}{2} \tag{Equation 3-15}$$

Inside diameter of three-course profile

$$D_{i,3} = 2 \cdot a - D_{a,3} \tag{Equation 3-16}$$

The graph below shows the standard profile and the eccentric profile. The black profile is centered and symmetrical over the rotation center O point. It is also symmetrical over 3 axes: AD, BE and CF. On the other hand, the red profile is symmetrical only over x-axis.

This profile shows that the comb angles are zero. Therefore the flank angles of the standard profile are the same:

$$\phi_{k,3} = \phi_{g,3} = 0 \tag{Equation 3-17}$$

Comb angle:

$$\angle AOB = \angle BOC = \angle COD = \Omega_3 = 60° \tag{Equation 3-18}$$ Flank angle:

$$R_{a,3} = \frac{D_{a,3}}{2} \tag{Equation 3-19}$$ Outside radius:

$$R_{i,3} = \frac{D_{i,3}}{2} = \left(a - \frac{D_{a,3}}{2}\right) \tag{Equation 3-20}$$ Inside radius:

The coordinates of the points in the three-course standard profile can be calculated by these geometry sizes. Because the profile is symmetrical, only half the contour is selected as the calculation object.

A: $$x_A = R_{i,3}; \quad y_A = 0$$

B: $$x_B = R_{i,3} \cdot \cos 60° = \frac{1}{2}R_{i,3}; \quad y_B = R_{i,3} \cdot \sin 60° = \frac{\sqrt{3}}{2}R_{i,3}$$

C: $$x_C = R_{i,3} \cdot \cos 120° = -\frac{1}{2}R_{i,3}; \quad y_C = R_{i,3} \cdot \sin 120° = \frac{\sqrt{3}}{2}R_{i,3}$$

D: $$x_D = -R_{i,3}; \quad y_D = 0$$

Creating a new distance element is almost the same as creating the elements mentioned beforehand. All data which have to be entered are shown in the figure.

All parameters that are needed for a complete definition of blister elements, is shown in the figure. At this the Width of the blister disc Ls can not be larger than the half of the element length.

Blister disc consist of discs placed shifted in a row. For a complete description of the geometry the following geometry sizes are needed:

• Centerline Distance a,
• Outer Diameter Da,
• Inner Diameter Di,
• Length of the complete element LEle,
• Length of the disc Ls and
• Chamfer Angle φs

For the calculation the blister element is divided into discs.

For the calculation, the blister element is divided into slices. The following dimensionless parameters are used in the calculation to characterise these slices:

$$k_1 = \frac{D_a}{D_z} \tag{3.39}$$

$$k_2 = \frac{D_i}{D_a} \tag{3.40}$$

$$cl = \frac{a}{D_z/2} \tag{3.41}$$

The entering mask for tooth mixing elements is shown in the figure. SIGMA makes the simplifying assumption that the gap between the toothed discs is negligible.

While defining Transition kneading elements a few aspects are to be considered:

• In order to define kneading blocks completely, the Number of discs of one kneading block is to determine in addition to Length and Diameter.
• Furthermore, the Staggering angle has to be entered for the discs. The staggering angle is only permitted in the range of

$$-\frac{180°}{i} \leq \text{Versatzwinkel} \leq \frac{180°}{i}$$

with i as Number of flights of the element. 2-3 describes a transition element with double-flighted to triple-flighted construction, and 3-2 vice versa. A negative angle identifies a reconveying kneading block, in contrast positive angle characterizes a conveying kneading block. Neutral kneading blocks are indicated by a Staggering angle of $$\frac{180°}{i}$$. The transition angle $$\alpha'$$ describes the staggering of the last disc.

The input mask for defining and changing threaded elements (both conveying and re-conveying elements) contains all values that are necessary for the unambiguous description of tightly intermeshing profiles.

What is meant in detail by the variables to be defined can be seen from the drawing of an element pair in the illustration.

Please note the following when entering the values:

• The centerline distance cannot be changed in this mask, as it is predetermined by the general machine data already entered.
• The input of the length of the screw element and the pitch can be done either in absolute values ([mm]) or in multiples of the pitch.
• The following transitions can be defined for the number of gears:
  • 1-2: Transition from single pitch to double pitch
  • 2-1: Transition from double pitch to single-pitch
  • 2-3: Transition from double pitch to triple pitch
  • 3-2: Transition from triple pitch to double pitch
• The volume correction factor is a measure of the deviation of the theoretically tightly intermeshing geometry used in SIGMA from the real geometry.
• (As an alternative to the volume correction factor, additional inputs for the exact definition of the geometry can be specified). In this case the real screw diameter and either the flight angle or the base angle of the screw profile. The other angle is calculated.

As soon as all necessary entries have been made, a direct preview is generated from the static preview.