The geometry of most co-rotating twin screw extruders built today, is based on the tightly intermeshing profile by [Boo78]. Co-rotating twin screw extruders consist of two parallel screws which have the same geometry. Both screws rotate with the same rotational speed and have the same outer diameter. Due to this, each point on one screw makes contact with the other screw.

One can characterise co-rotating twin screw extruders by:

• Axial open screw channels
• Self-cleaning
• A resistance in the intermeshing region
• Modular setup of both screw and barrel (see figure)

Figure: Modular setup of both screw and barre

Conveying elements are generated by twisting the profile cross-section in the opposite rotation direction of the screws. Reconveying elements are twisted in the rotation direction of the screws.

Kneading discs are based on the same profile as conveying elements. They are not twisted but extruded. In conveying kneading blocks, the kneading discs are staggered in the opposite rotational direction to the screws. In reconveying kneading blocks the discs are staggered in the direction of the screws and in neutral kneading blocks the discs are staggered with an angle of $180°/$(number of flights).

Due to the tightly intermeshing, self-cleaning profile of the elements, a thorough description of the profile can be obtained if the following data is known:

• Screw diameter $D_a$
• Barrel diameter $D_z$
• Centerline distance $a$
• Number of flights $i$
• Pitch $t$

All other data (see figure) (channel depth $h$, pitch angle $\varphi_s$, channel width $b$ and flight width $e$) can be calculated from this data

Figure: Side view and cross section of a tightly intermeshing conveying element [Ans93]

The table shows the equations needed to calculate the channel geometry.

Intermeshing angle: $$\Omega = 2 \arccos\left(\frac{a}{D_s}\right)\tag{1}$$

Flight angle: $$\phi = \frac{\pi}{i} - \Omega\tag{2}$$

Pitch angle: $$\varphi_s = \arctan\left(\frac{t}{\pi D_s}\right)\tag{3}$$

Maximum flight width: $$e_{max} = \frac{t\phi \cos(\varphi_s)}{2\pi}\tag{4}$$

Maximum channel width: $$b_{max} = \frac{t \cos(\varphi_s)}{i} - e_{max}\tag{5}$$

for $0 \leq x \leq \frac{e_{max}}{2}$

$$h(x) = h_{max} = D_z - a\tag{6}$$

for $\frac{e_{max}}{2} \leq x \leq \frac{b_{max}}{2}$

Channel depth: $$h(x) = \frac{D_z}{a}\left[1 + \cos\left(\frac{2\pi\left(|x| - \frac{e_{max}}{2}\right)}{t \cos(\varphi_s)}\right)\right] - \sqrt{a^2 - \left(\frac{D_z}{2}\right)^2 - \sin^2\left(\frac{2\pi\left(|x| - \frac{e_{max}}{2}\right)}{t \cos(\varphi_s)}\right)}\tag{7}$$

for $\frac{b_{max}}{2} \leq x \leq \frac{(b_{max} + e_{max})}{2}$

$$h(x) = 0\tag{8}$$

The maximum number of flights of the screw elements in a given machine depends on the ratio of screw diameter $D_S$ to centerline distance $a$, since the flight angle $f$ has to be greater than 0. Usually the ratio $1/2 \cdot \sqrt{2}$ can be found, since two flighted profiles are used in almost all machines. The closer the ratio $a/D_S$ is to 1, the shallower the channel depth is. The greater the number of flights, the smaller the flight angle and the average channel depth.

Figure: Tightly intermeshing screw elements shown at different positions

The figure shows that one, two and three flighted elements are closely intermeshing in different positions.

The axial free cross section is very important geometrical data. The figure shows how this value is determined:

Figure: Free cross section of a tightly intermeshing screw profile

The faces highlighted in the figure can be used to determine the free cross section of the tightly intermeshing screw profile ($s_R = 0$) as follows:

$$A_1 = \frac{1}{8}\phi D_s^2 \tag{9}$$

$$A_2 = \frac{1}{8}\phi(2a - D_s)^2 \tag{10}$$

$$A_3 = \frac{1}{4}\Omega a^2 \tag{11}$$

$$A_4 = \frac{1}{4}aD_s \sin\left(\frac{\Omega}{2}\right) \tag{12}$$

$$A_{Fr} = (A_1 + A_2)i + (A_3 - A_4)2i \tag{13}$$

$$A_{zyl} = \frac{1}{4}(2\pi - \Omega)D_s^2 + \frac{1}{2}aD_s \sin\left(\frac{\Omega}{2}\right)\tag{14}$$

$$A_{fr} = A_{zyl} - 2A_{Fr} \tag{15}$$

From eqn. 9-15 we can see that the free cross section is not dependent on the screw pitch.

The intermeshing region of the screw is bordered off at both the left and right flights of the two screws. The free cross section in the intermeshing region changes periodically due to the rotation of the screws (see figure).

Figure: Free axial cutting plane in the intermeshing region by different screw positions

The average free cross section in the intermeshing region can be calculated according to Booy [Boo78] as follows:

$$\bar{A}_{zw} = mD_s^2 \tag{16} $$

$$m = \frac{1}{2}\left[\left(\frac{a}{D_s}\right)\sin\left(\frac{\Omega}{2}\right) - \left(1 - \frac{\phi}{\pi}\right) - \left[\phi\left(\left(\frac{a}{D_s}\right)^2 - \left(\frac{a}{D_s}\right) + \frac{1}{2}\right) + \Omega\left(\frac{a}{D_s}\right)^2 - \left(\frac{a}{D_s}\right)\sin\left(\frac{\Omega}{2}\right)\right]\right] \tag{16b}$$

Where $m$ is an average intermeshing flow-coefficient. Using equation 16 and the axial length of the intermeshing region we are able to obtain the free volume in the intermeshing region.

$$V_{zw} = \bar{A}_{zw} \cdot L_{zw} = mD_s^2 \frac{\Omega t}{\pi} \tag{17}$$

By cutting the intermeshing region at an angle that equals the pitch angle $j_S$, we get the profile shown in the figure. During the handing over of the material from one screw to the other, the material is deflected about an angle $\gamma$

$$\gamma = \pi - 2 \cdot \arctan\left(\frac{\tan\left(\frac{\Omega}{2}\right)}{\cos(\varphi_s)}\right)\tag{18}$$

Figure: Intersection of the screws by the cut under the helix angle $j_S$

The figure shows the deflection angle depending on the ratio of pitch to screw diameter and centerline distance to screw diameter.

Figure: Deflection angle depending on the ratio $t/D_S$ and $a/D_S$

In order to evaluate the conveying characteristics, one has to investigate whether the co-rotating twin screw extruder is a system in which the axial is open, like a single screw extruder or whether it is a system in which the axial is closed, like a gear pump.

In order to get a closed chamber like in a counter rotating twin screw extruder or a gear pump, the flight of one screw has to block the channel of the other screw.

Co-rotating twin screw extruders are axial open systems. The flows in the different channels are nevertheless separated from each other.

Quite frequently the channel model, known from the single screw theory, is used to describe the melt flow in co-rotating twin screw extruders.

Hwang [Hwa82] proposed to use an unwound channel to calculate the flow using the finite element method. For the calculations a kinematic reversal is used, i.e. the barrel which is not actually moving is regarded as moving and the screw which is actually moving is regarded as being stationary.

Figure: Unwound twin screw channel

The model used for reconveying elements is shown in the figure. For the control volume with the Nodes A, B and C we get the number of $k$ parallel screw channels:

$$k = 2i - 1 + \frac{\phi i}{\pi} \tag{19}$$

Figure: Channel model for reconveying elements (number of flights $i=2$) [Ans93]

The barrel moves with the velocity $v_0$ which is equal to the circumferential velocity of the screw.

$$v_0 = D_s\pi n_0 \tag{20}$$

One can split this velocity into one component in the channel direction

$$v_{0z} = v_0 \cos \varphi_s \tag{21}$$

And into one component orthogonal to the channel direction

$$v_{0x} = v_0 \sin \varphi_s \tag{22}$$

The model mentioned above can also be used for conveying and reconveying kneading blocks but only if the model shown in the figure is used [Ans93].

Figure: Model for conveying and reconveying kneading blocks [Ans93]

The discrete geometry is replaced by a continuous one. This can be done since the profile of conveying elements and kneading blocks are identical.

The pitch angle $\varphi_{S,Kn}$ is calculated using the following equation:

$$\varphi_{S,Kn} = \left(\frac{2b_s}{aD_s}\right) \tag{23}$$

Where $b_S$ is the width of one kneading disc and $\alpha$ is the staggering angle. The figure shows the channel model for conveying kneading blocks, which is performed analogous to screw conveying elements.

Figure: Channel model for conveying kneading blocks [Ans93]

The figure shows the channel model for a neutral kneading block. The conveying component of the circumferential velocity is missing.

Figure: Channel model for a neutral kneading block

The leakage flows $\dot{V}_c$ do not contribute to the axial pressure loss. These leakage flows can be calculated under the assumption of a pure drag flow in the radial clearance.

As shown in the figure, the only difference between the traditional screw profile and the pushing flight profile is the face $A_{sch}$.

Figure: Comparison of pushing flight profile and tightly intermeshing, self wiping profile

In addition to the data needed for the traditional screw profiles, the following data is required to describe the geometry of these elements:

• Screw Diameter $D_a$
• Barrel Diameter $D_z$
• Centerline Distance $a$
• Number of Flights $i$
• Pitch $t$

Two additional values are needed (see figure):

• Chamfer angle $f_s$ and
• Radius $r$

Figure: Definition of the additional geometry sizes of the pushing flight profiles

All other values, except the channel depth, needed to describe the cross section of the screw profile can be calculated using the equations displayed in table.

Due to the face $A_{sch}$, the free cross-section of the pushing flight profile is larger than the one of the traditional profile.

$$A_{fr} = A_{zyl} - 2 \cdot A_{Pr}\tag{24}$$

Where $A_{zyl}$ is the cross section of the barrel and $A_{pr}$ the cross section of the screw profiles. While $A_{zyl}$ can be calculated using the eqn. $A_{zyl} = \frac{1}{4}(2\pi - \Omega)D_s^2 + \frac{1}{2}aD_s \sin\left(\frac{\Omega}{2}\right)$, the cross section of the pushing flight element $A_{pr}$ has to be calculated using eqn. 25

$$A_{Pr} = A_{Pr,the} - 2 \cdot i \cdot A_{Sch}\tag{25}$$

with $A_{Pr,the}$ the cross section of the traditional profile (equation $A_{fr} = (A_1 + A_2)i + (A_3 - A_4)2i$).

Figure: Calculation of the face $A_{sch}$

The face $A_{sch}$ can be calculated using the following equations (see figure):

$$A_{schub} = A_{ABC} - A_{MB/C} + A_{MB/C/} - A_{ABC/} - A_{B/B/B//}\tag{26}$$

with:

$$A_{ABC} = \frac{\Omega}{4} \cdot a^2 \tag{27}$$

$$A_{MB'C} = \left[\Omega + \frac{\pi}{2} - \alpha - \arcsin\left(\frac{R_a}{R_i} \cdot \cos(\alpha)\right)\right] \cdot \frac{R_i^2}{2} \tag{28}$$

$$A_{MB'C'} = \frac{1}{2}\left\{a \cdot \sin(\Omega/2) + R_i \cdot \cos\left[\alpha + \arcsin\left(\frac{R_a}{R_i} \cdot \cos(\alpha)\right)\right]\right\} \cdot \frac{r - R_i \cdot \cos\left(\Omega + \phi + \arcsin\left[\frac{R_a}{R_i} \cos(\alpha)\right]\right)}{\cos(\alpha - \Omega)} + R_a \cdot \cos(\alpha) \tag{29}$$

$$A_{ABC'} = \frac{1}{2} \frac{a \cdot \sin(\Omega/2)}{\cos(\alpha - \Omega)} \cdot a \cdot \cos(-\alpha + \Omega/2) \tag{30}$$

$$A_{B'B'B''} = \frac{1}{2}r^2 \sqrt{1 - \frac{\left[r - R_i \cdot \cos\left\{\Omega + \phi + \arcsin\left(\frac{R_a}{R_i} \cos(\alpha)\right)\right\}\right]^2}{(R_i + r)^2}}$$

$$- \frac{1}{2}R_i^2 \sin\left\{-\Omega - \phi - \arcsin\left[\frac{R_a}{R_i} \cos(\alpha)\right] - \arccos \frac{r - R_i \cdot \cos\left(\Omega + \phi + \arcsin\left[\frac{R_a}{R_i} \cos(\alpha)\right]\right)}{R_i + r}\right\} +$$

$$+ \frac{1}{2}R_i \left[2 + 2 \cos\left(-\Omega - \phi - \arcsin\left[\frac{R_a}{R_i} \cos(\alpha)\right] - \arccos \frac{r - R_i \cdot \cos\left(\Omega + \phi + \arcsin\left[\frac{R_a}{R_i} \cos(\alpha)\right]\right)}{R_i + r}\right)\right]$$

$$\cdot \sqrt{\left[2 - 2 \frac{r - R_i \cdot \cos\left(\Omega + \phi + \arcsin\left[\frac{R_a}{R_i} \cos(\alpha)\right]\right)}{R_i + r} \cos\left(\frac{\Omega}{2} + \frac{\phi}{2} + \frac{1}{2}\arcsin\left[\frac{R_a}{R_i} \cos(\alpha)\right]\right)\right]}$$

$$- \frac{1}{2}r^2 \arccos \frac{r - R_i \cdot \cos(\phi + \Omega + \arcsin\left[\frac{R_a}{R_i} \cos(\alpha)\right])}{R_i + r}$$

$$- \frac{1}{2}R_i^2 \left(-\phi - \Omega + \pi - \arcsin\left[\frac{R_a}{R_i} \cos(\alpha)\right] - \arccos \frac{r - R_i \cdot \cos(\phi + \Omega + \arcsin\left[\frac{R_a}{R_i} \cos(\alpha)\right])}{R_i + r}\right)\tag{31}$$

Since the maximum channel depths of both the pushing flight element and the traditional conveying element are identical, we can calculate the average channel depth using the following equation:

$$\bar{h} = \frac{A_{free}}{b_{max}}\tag{32}$$

Due to the face $A_{sch}$, the cross section in the intermeshing region of the two screws increases (see figure).

Figure: Free cross section in the intermeshing region of two and three flighted elements

This area can be calculated according to Booy [Boo78] using the following equation:

$$A_{zw} = \left(\frac{D_s}{2}\right)^2 \cdot \sin(\Omega) - \frac{A_{Fr}}{i} \cdot \left(1 - \frac{\phi \cdot i}{\pi}\right)\tag{33}$$

Shouldered kneading blocks differ from traditional kneading blocks in the length of their discs. The width of the discs in shouldered kneading blocks is only half as wide.

This causes an additional leakage flow rate to occur. It is necessary to distinguish between the leakage flow rates in the radial clearance and the leakage flow between the kneading discs.

The model for the traditional kneading blocks, is based on the replacement of the discrete geometry (kneading discs) with a continuous geometry. One can find grooves within flights of this assumed continuous geometry. These grooves are characterized by their height $h_{Nut}$, their length $l_{Nut}$ and their width $b_{Nut}$. The width is half the width of the discs of traditional kneading blocks ($b_{Kn}$).

$$b_{groove} = \frac{1}{2} \cdot b_{Kn} \tag{34}$$

$$l_{groove} = \frac{e}{\cos \varphi_s}\tag{35}$$

The height of the groove results from the staggering angle. The staggering angle needs to be larger than the flight angle in order to form a groove. If the staggering angle is smaller than the flight angle then no groove can be calculated.

For staggering $\Phi \leq \alpha \leq 45°$ we use a linear relationship between staggering angle and depth of the groove. After conducting experimental investigations we found that it is necessary to apply a factor of 0.5 to that relationship.

$$h_{groove} = 0.5 \cdot \left(\frac{\bar{h}}{\frac{1}{4} \cdot \pi - \Phi} \cdot \alpha - \frac{\Phi \cdot \bar{h}}{\frac{1}{4} \cdot \pi - \Phi} + s_R\right)\tag{36}$$

Figure: Channel model for conveying shouldered kneading blocks

The channel model for conveying shouldered kneading blocks is shown in the figure. We get the following equilibrium of flow rates within the control volume A-B-C:

$$\dot{V} = k \cdot \dot{V}_{channel} \pm \dot{V}_{gap} \pm l \cdot \dot{V}_{groove}\tag{37}$$

Where $l$ is the number of grooves between the points A and B. The number of the grooves corresponds to the number of kneading discs per rotation, hence follows:

$$l = \frac{b_{Kn}}{2 \cdot \pi} \cdot (2 \cdot \pi - \Omega) \cdot \cos \varphi_s\tag{38}$$

With $b_{Kn}$ being the width of the kneading discs.

The width of the flights $b_{Steg}$ is shortened by the total sum of all groove widths $l$.

$$b_{threads} = (2 \cdot \pi - \Omega) \cdot D_s \cdot \cos \varphi_s - l \cdot b_{groove}\tag{39}$$

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

It is known that in a profile only one comb is scraped off with the housing. If the comb angle is zero, the eccentric profile can be generated by a displacement. The displacement is called eccentricity $e$. This generated eccentric profile is also tightly intermeshing. The shapes of the contours are not changed after the displacement. The following figure shows the displacement of the profiles.

Figure: Displacement of profiles with eccentricity

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 ($i=3$)

$$D_{v,3} = D_{v,3,max} = \frac{1}{2 \cdot \cos\left(\frac{\pi}{2 \cdot 3}\right) - 1} = 1.366 \tag{Equation 3-13}$$

Outer diameter of three-course profile

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

Eccentricity $e$

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

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.

Illustration: Profile geometry with $D_v=1.366$. Black: standard profile; red: eccentric profile.

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

Comb angle: $\angle AOB = \angle BOC = \angle COD = \Omega_3 = 60° \tag{Equation 3-17}$

Flank angle: $\phi_{k,3} = \phi_{g,3} = 0 \tag{Equation 3-18}$

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

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

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

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

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

D: $x_D = -R_{a,3}$; $y_D = 0$

The geometrical description of the screw mixing elements is based on the SME element by Krupp Werner & Pfleiderer. This element differs from usual conveying elements in that its grooves are cut under a certain angle in the flights. The geometry of the grooves can be chosen arbitrary. In the case of the SME, the groove is V-shaped. For the calculation of the flow within the grooves we simplify the actual geometry by using a rectangular channel characterized by the depth $h_{Nut}$ and the width $b_{Nut}$.

Figure: Geometry of the groove in the screw flight

The width of the screw flight $b_{Steg}$ is reduced by the total sum of widths of all the grooves. The channel model for the screw mixing element is shown in the figure.

Figure: Channel model for a threaded mixing element

From the control volume ABC we get the equilibrium of flow rates:

$$\dot{V} = k \cdot \dot{V}_{channel} \pm \dot{V}_{gap} \pm l \cdot \dot{V}_{groove}\tag{21}$$

Where $l$ is the number of grooves between the points A and B. $l$ is calculated using the following equation:

$$l = \frac{l_{Nut}}{2\pi} \cdot (2\pi - \Omega) \cdot \cos \varphi_s\tag{22}$$

Blister elements consist of discs, which are arranged in rows. To describe the geometry we require the following data:

Figure: Geometry of blister elements

• Centerline distance $a$,
• Outside diameter $D_a$,
• Inside diameter $D_i$,
• Length of the Blister element $L_B$,
• Length of the disc $L_s$ and
• Chamfer angle $j_s$

The calculation of the blister elements is cut into sections (see figure).

Figure: Slicing the blister element

These sections are characterized by the following dimensionless numbers:

$$k_1 = \frac{D_a}{D_z}\tag{23}$$

$$k_2 = \frac{D_i}{D_a}\tag{24}$$

$$cl = \frac{a}{D_z/2}\tag{25}$$

Toothed mixing elements are screw elements used for incorporating, dispersing, and homogenizing the polymer melt. In principle, they consist of alternating rows of teeth and spacer sleeves. These two sections alternate, whereby the number of tooth rows and the number of teeth along the circumference can vary depending on the design. In the region of the tooth rows, the teeth split the melt streams and intermix them through the rotation of the screw elements.

The teeth are arranged within the tooth rows at an angle $\varphi_N$. At an angle of $\varphi_N = 90°$, the toothed discs are conveying-neutral; at $\varphi_N < 0°$, they are reverse-conveying; and at $\varphi_N > 0°$, they are forward-conveying.

Figure: Geometry of the turbine mixing element

For the description of the geometry we need the following data:

• Centerline distance $a$,
• Outside diameter $D_a$,
• Inside diameter $D_i$,
• Number of teeth $n$,
• Pitch angle $\varphi_N$,
• Conveying direction (conveying, reconveying or neutral)

The area between the teeth is approximated by a rectangular channel. For this purpose, the width and height of the rectangular channel must be defined. It should be noted that the helix angle $\varphi_N$ must ALWAYS be entered as a positive value.

[Ans93] Ansahl, J.: Grundlagen für die Auslegung dichtkämmender Gleichdrall-Doppelschneckenextruder, Dissertation Universität Paderborn, 1993

[Boo78] Booy, M.L.: Geometry of Fully Wiped Twin Screw Equipment, Polymer Engineering and Science, 18(1978)12, 973-984

[Hwa82] Hwang, B.K.: Fluid Flow Studies in Twin-Screw Extruders, Dissertation Universität Delaware, 1982