Unterschiede

Hier werden die Unterschiede zwischen zwei Versionen angezeigt.

--- en:grundlagenhandbuch:geometriegroessen_des_gleichdrall-doppelschneckenextruders [2026/05/04 15:36] – [Geometry of Tightly Intermeshing, Co-rotating Twin Screw Extruders] neelest
+++ en:grundlagenhandbuch:geometriegroessen_des_gleichdrall-doppelschneckenextruders [2026/06/16 14:37] (aktuell) – [Geometry of Tightly Intermeshing, Co-rotating Twin Screw Extruders] deppe2
@@ Zeile 9: / Zeile 9: @@
   * Modular setup of both screw and barrel (see figure)
-{{ :grundlagenhandbuch:geometriegroessen_des_gleichdrall-doppelschneckenextruders:eingesetzte_schneckenelemente:en_sigma150_dlg_grundlagenhandbuch_geometriegroessen_000.svg?700%nolink |}}
+{{ :en:grundlagenhandbuch:de_sigma150_dlg_grundlagenhandbuch_geometriegroessen_000.svg?nolink&700 |}}
 **Figure:** Modular setup of both screw and barre
@@ Zeile 38: / Zeile 38: @@
 Intermeshing angle:
-$$\Omega = 2 \arccos\left(\frac{a}{D_s}\right)$$
+$$\Omega = 2 \arccos\left(\frac{a}{D_s}\right)\tag{1}$$
 Flight angle:
-$$\phi = \frac{\pi}{i} - \Omega$$
+$$\phi = \frac{\pi}{i} - \Omega\tag{2}$$
 Pitch angle:
-$$\varphi_s = \arctan\left(\frac{t}{\pi D_s}\right)$$
+$$\varphi_s = \arctan\left(\frac{t}{\pi D_s}\right)\tag{3}$$
 Maximum flight width:
-$$e_{max} = \frac{t\phi \cos(\varphi_s)}{2\pi}$$
+$$e_{max} = \frac{t\phi \cos(\varphi_s)}{2\pi}\tag{4}$$
 Maximum channel width:
-$$b_{max} = \frac{t \cos(\varphi_s)}{i} - e_{max}$$
+$$b_{max} = \frac{t \cos(\varphi_s)}{i} - e_{max}\tag{5}$$
 for $0 \leq x \leq \frac{e_{max}}{2}$
-$$h(x) = h_{max} = D_z - a$$
+$$h(x) = h_{max} = D_z - a\tag{6}$$
 for $\frac{e_{max}}{2} \leq x \leq \frac{b_{max}}{2}$
 Channel depth:
-$$h(x) = \frac{D_z}{a}\left[1 + \cos\left(\frac{2\pi\left(|x| - \frac{e_{max}}{2}\right)}{t \cos(\varphi_s)}\right)\right] - \sqrt{a^2 - \left(\frac{D_z}{2}\right)^2 - \sin^2\left(\frac{2\pi\left(|x| - \frac{e_{max}}{2}\right)}{t \cos(\varphi_s)}\right)}$$
+$$h(x) = \frac{D_z}{a}\left[1 + \cos\left(\frac{2\pi\left(|x| - \frac{e_{max}}{2}\right)}{t \cos(\varphi_s)}\right)\right] - \sqrt{a^2 - \left(\frac{D_z}{2}\right)^2 - \sin^2\left(\frac{2\pi\left(|x| - \frac{e_{max}}{2}\right)}{t \cos(\varphi_s)}\right)}\tag{7}$$
 for $\frac{b_{max}}{2} \leq x \leq \frac{(b_{max} + e_{max})}{2}$
-$$h(x) = 0$$
+$$h(x) = 0\tag{8}$$
 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.
@@ Zeile 83: / Zeile 83: @@
 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:
-$$A_1 = \frac{1}{8}\phi D_s^2 \tag{1}$$
+$$A_1 = \frac{1}{8}\phi D_s^2 \tag{9}$$
-$$A_2 = \frac{1}{8}\phi(2a - D_s)^2 \tag{2}$$
+$$A_2 = \frac{1}{8}\phi(2a - D_s)^2 \tag{10}$$
-$$A_3 = \frac{1}{4}\Omega a^2 \tag{3}$$
+$$A_3 = \frac{1}{4}\Omega a^2 \tag{11}$$
-$$A_4 = \frac{1}{4}aD_s \sin\left(\frac{\Omega}{2}\right) \tag{4}$$
+$$A_4 = \frac{1}{4}aD_s \sin\left(\frac{\Omega}{2}\right) \tag{12}$$
-$$A_{Fr} = (A_1 + A_2)i + (A_3 - A_4)2i \tag{5}$$
+$$A_{Fr} = (A_1 + A_2)i + (A_3 - A_4)2i \tag{13}$$
-$$A_{zyl} = \frac{1}{4}(2\pi - \Omega)D_s^2 + \frac{1}{2}aD_s \sin\left(\frac{\Omega}{2}\right)\tag{6}$$
+$$A_{zyl} = \frac{1}{4}(2\pi - \Omega)D_s^2 + \frac{1}{2}aD_s \sin\left(\frac{\Omega}{2}\right)\tag{14}$$
-$$A_{fr} = A_{zyl} - 2A_{Fr} \tag{7}$$
+$$A_{fr} = A_{zyl} - 2A_{Fr} \tag{15}$$
-From eqn. 1 - 7 we can see that the free cross section is not dependent on the screw pitch.
+From eqn. 9-15 we can see that the free cross section is not dependent on the screw pitch.
 ==== Intermeshing Region ====
@@ Zeile 109: / Zeile 109: @@
 The average free cross section in the intermeshing region can be calculated according to Booy  [[en:grundlagenhandbuch:geometriegroessen_des_gleichdrall-doppelschneckenextruders#references|[Boo78]]] as follows:
-$$\bar{A}_{zw} = mD_s^2 \tag{8} $$
+$$\bar{A}_{zw} = mD_s^2 \tag{16} $$
-$$m = \frac{1}{2}\left[\left(\frac{a}{D_s}\right)\sin\left(\frac{\Omega}{2}\right) - \left(1 - \frac{\phi}{\pi}\right) - \left[\phi\left(\left(\frac{a}{D_s}\right)^2 - \left(\frac{a}{D_s}\right) + \frac{1}{2}\right) + \Omega\left(\frac{a}{D_s}\right)^2 - \left(\frac{a}{D_s}\right)\sin\left(\frac{\Omega}{2}\right)\right]\right] \tag{9}$$
+$$m = \frac{1}{2}\left[\left(\frac{a}{D_s}\right)\sin\left(\frac{\Omega}{2}\right) - \left(1 - \frac{\phi}{\pi}\right) - \left[\phi\left(\left(\frac{a}{D_s}\right)^2 - \left(\frac{a}{D_s}\right) + \frac{1}{2}\right) + \Omega\left(\frac{a}{D_s}\right)^2 - \left(\frac{a}{D_s}\right)\sin\left(\frac{\Omega}{2}\right)\right]\right] \tag{16b}$$
-Where $m$ is an average intermeshing flow-coefficient. Using equation (8) and the axial length of the intermeshing region we are able to obtain the free volume in the intermeshing region.
+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.
-$$V_{zw} = \bar{A}_{zw} \cdot L_{zw} = mD_s^2 \frac{\Omega t}{\pi} \tag{10}$$
+$$V_{zw} = \bar{A}_{zw} \cdot L_{zw} = mD_s^2 \frac{\Omega t}{\pi} \tag{17}$$
 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 $\gamma$
-$$\gamma = \pi - 2 \cdot \arctan\left(\frac{\tan\left(\frac{\Omega}{2}\right)}{\cos(\varphi_s)}\right)\tag{11}$$
+$$\gamma = \pi - 2 \cdot \arctan\left(\frac{\tan\left(\frac{\Omega}{2}\right)}{\cos(\varphi_s)}\right)\tag{18}$$
 {{ :grundlagenhandbuch:geometriegroessen_des_gleichdrall-doppelschneckenextruders:eingesetzte_schneckenelemente:de_sigma150_dlg_grundlagenhandbuch_geometriegroessen_005.svg?400%nolink |}}
@@ Zeile 151: / Zeile 151: @@
 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:
-$$k = 2i - 1 + \frac{\phi i}{\pi} \tag{12}$$
+$$k = 2i - 1 + \frac{\phi i}{\pi} \tag{19}$$
 {{ :grundlagenhandbuch:geometriegroessen_des_gleichdrall-doppelschneckenextruders:eingesetzte_schneckenelemente:de_sigma150_dlg_grundlagenhandbuch_geometriegroessen_008.svg?400%nolink |}}
@@ Zeile 159: / Zeile 159: @@
 The barrel moves with the velocity $v_0$ which is equal to the circumferential velocity of the screw.
-$$v_0 = D_s\pi n_0 \tag{13}$$
+$$v_0 = D_s\pi n_0 \tag{20}$$
 One can split this velocity into one component in the channel direction
-$$v_{0z} = v_0 \cos \varphi_s \tag{14}$$
+$$v_{0z} = v_0 \cos \varphi_s \tag{21}$$
 And into one component orthogonal to the channel direction
-$$v_{0x} = v_0 \sin \varphi_s \tag{15}$$
+$$v_{0x} = v_0 \sin \varphi_s \tag{22}$$
 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  [[en:grundlagenhandbuch:geometriegroessen_des_gleichdrall-doppelschneckenextruders#references|[Ans93]]].
@@ Zeile 179: / Zeile 179: @@
 The pitch angle $\varphi_{S,Kn}$ is calculated using the following equation:
-$$\varphi_{S,Kn} = \left(\frac{2b_s}{aD_s}\right) \tag{16}$$
+$$\varphi_{S,Kn} = \left(\frac{2b_s}{aD_s}\right) \tag{23}$$
 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.
@@ Zeile 226: / Zeile 226: @@
 Due to the face $A_{sch}$, the free cross-section of the pushing flight profile is larger than the one of the traditional profile.
-$$A_{fr} = A_{zyl} - 2 \cdot A_{Pr}\tag{1}$$
+$$A_{fr} = A_{zyl} - 2 \cdot A_{Pr}\tag{24}$$
-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. (2)
+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
-$$A_{Pr} = A_{Pr,the} - 2 \cdot i \cdot A_{Sch}\tag{2}$$
+$$A_{Pr} = A_{Pr,the} - 2 \cdot i \cdot A_{Sch}\tag{25}$$
 with $A_{Pr,the}$ the cross section of the traditional profile (equation $A_{fr} = (A_1 + A_2)i + (A_3 - A_4)2i$).
@@ Zeile 240: / Zeile 240: @@
 The face $A_{sch}$ can be calculated using the following equations (see figure):
-$$A_{schub} = A_{ABC} - A_{MB/C} + A_{MB/C/} - A_{ABC/} - A_{B/B/B//}\tag{3}$$
+$$A_{schub} = A_{ABC} - A_{MB/C} + A_{MB/C/} - A_{ABC/} - A_{B/B/B//}\tag{26}$$
 with:
-$$A_{ABC} = \frac{\Omega}{4} \cdot a^2\tag{4}$$
+$$A_{ABC} = \frac{\Omega}{4} \cdot a^2 \tag{27}$$
-$$A_{MB/C} = \left[\Omega + \frac{\pi}{2} - \alpha - \arcsin\left(\frac{R_a}{R_i} \cdot \cos(\alpha)\right)\right] \cdot \frac{R_i^2}{2}\tag{5}$$
+$$A_{MB'C} = \left[\Omega + \frac{\pi}{2} - \alpha - \arcsin\left(\frac{R_a}{R_i} \cdot \cos(\alpha)\right)\right] \cdot \frac{R_i^2}{2} \tag{28}$$
-$$A_{MB/C/} = \frac{1}{2}\left\{\frac{a \cdot \sin(\Omega/2)}{\cos(\alpha - \Omega)} + \frac{R_i \cdot \cos\left[\alpha + \arcsin\left(\frac{R_a}{R_i} \cdot \cos(\alpha)\right)\right]}{\cos(\alpha)}\right\} \cdot R_a \cdot \cos(\alpha)\tag{6}$$
+$$A_{MB'C'} = \frac{1}{2}\left\{a \cdot \sin(\Omega/2) + R_i \cdot \cos\left[\alpha + \arcsin\left(\frac{R_a}{R_i} \cdot \cos(\alpha)\right)\right]\right\} \cdot \frac{r - R_i \cdot \cos\left(\Omega + \phi + \arcsin\left[\frac{R_a}{R_i} \cos(\alpha)\right]\right)}{\cos(\alpha - \Omega)} + R_a \cdot \cos(\alpha) \tag{29}$$
-$$A_{ABC/} = \frac{1}{2} \cdot \frac{a \cdot \sin(\Omega/2)}{\cos(\alpha - \Omega)} \cdot a \cdot \cos(-\alpha + \Omega/2)\tag{7}$$
+$$A_{ABC'} = \frac{1}{2} \frac{a \cdot \sin(\Omega/2)}{\cos(\alpha - \Omega)} \cdot a \cdot \cos(-\alpha + \Omega/2) \tag{30}$$
-$$A_{ABC} = \frac{\Omega}{4} \cdot a^2 \tag{4}$$
+$$A_{B'B'B''} = \frac{1}{2}r^2 \sqrt{1 - \frac{\left[r - R_i \cdot \cos\left\{\Omega + \phi + \arcsin\left(\frac{R_a}{R_i} \cos(\alpha)\right)\right\}\right]^2}{(R_i + r)^2}}$$
-$$A_{MB'C} = \left[\Omega + \frac{\pi}{2} - \alpha - \arcsin\left(\frac{R_a}{R_i} \cdot \cos(\alpha)\right)\right] \cdot \frac{R_i^2}{2} \tag{5}$$
-$$A_{MB'C'} = \frac{1}{2}\left\{a \cdot \sin(\Omega/2) + R_i \cdot \cos\left[\alpha + \arcsin\left(\frac{R_a}{R_i} \cdot \cos(\alpha)\right)\right]\right\} \cdot \frac{r - R_i \cdot \cos\left(\Omega + \phi + \arcsin\left[\frac{R_a}{R_i} \cos(\alpha)\right]\right)}{\cos(\alpha - \Omega)} + R_a \cdot \cos(\alpha) \tag{6}$$
-$$A_{ABC'} = \frac{1}{2} \frac{a \cdot \sin(\Omega/2)}{\cos(\alpha - \Omega)} \cdot a \cdot \cos(-\alpha + \Omega/2) \tag{7}$$
-$$A_{B'B'B''} = \frac{1}{2}r^2 \sqrt{1 - \frac{\left[r - R_i \cdot \cos\left\{\Omega + \phi + \arcsin\left(\frac{R_a}{R_i} \cos(\alpha)\right)\right\}\right]^2}{(R_i + r)^2}} \tag{8}$$
 $$- \frac{1}{2}R_i^2 \sin\left\{-\Omega - \phi - \arcsin\left[\frac{R_a}{R_i} \cos(\alpha)\right] - \arccos \frac{r - R_i \cdot \cos\left(\Omega + \phi + \arcsin\left[\frac{R_a}{R_i} \cos(\alpha)\right]\right)}{R_i + r}\right\} +$$
@@ Zeile 270: / Zeile 262: @@
 $$- \frac{1}{2}r^2 \arccos \frac{r - R_i \cdot \cos(\phi + \Omega + \arcsin\left[\frac{R_a}{R_i} \cos(\alpha)\right])}{R_i + r}$$
-$$- \frac{1}{2}R_i^2 \left(-\phi - \Omega + \pi - \arcsin\left[\frac{R_a}{R_i} \cos(\alpha)\right] - \arccos \frac{r - R_i \cdot \cos(\phi + \Omega + \arcsin\left[\frac{R_a}{R_i} \cos(\alpha)\right])}{R_i + r}\right)$$
+$$- \frac{1}{2}R_i^2 \left(-\phi - \Omega + \pi - \arcsin\left[\frac{R_a}{R_i} \cos(\alpha)\right] - \arccos \frac{r - R_i \cdot \cos(\phi + \Omega + \arcsin\left[\frac{R_a}{R_i} \cos(\alpha)\right])}{R_i + r}\right)\tag{31}$$
 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:
-$$\bar{h} = \frac{A_{free}}{b_{max}}\tag{9}$$
+$$\bar{h} = \frac{A_{free}}{b_{max}}\tag{32}$$
 ==== Intermeshing Region of the Two Screws ====
@@ Zeile 285: / Zeile 278: @@
 This area can be calculated according to Booy  [[en:grundlagenhandbuch:geometriegroessen_des_gleichdrall-doppelschneckenextruders#references|[Boo78]]] using the following equation:
-$$A_{zw} = \left(\frac{D_s}{2}\right)^2 \cdot \sin(\Omega) - \frac{A_{Fr}}{i} \cdot \left(1 - \frac{\phi \cdot i}{\pi}\right)\tag{10}$$
+$$A_{zw} = \left(\frac{D_s}{2}\right)^2 \cdot \sin(\Omega) - \frac{A_{Fr}}{i} \cdot \left(1 - \frac{\phi \cdot i}{\pi}\right)\tag{33}$$
 ==== Shouldered Kneading Blocks ====
@@ Zeile 295: / Zeile 288: @@
 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 ($b_{Kn}$).
-$$b_{groove} = \frac{1}{2} \cdot b_{Kn} \tag{11}$$
+$$b_{groove} = \frac{1}{2} \cdot b_{Kn} \tag{34}$$
-$$l_{groove} = \frac{e}{\cos \varphi_s}\tag{12}$$
+$$l_{groove} = \frac{e}{\cos \varphi_s}\tag{35}$$
 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.
@@ Zeile 303: / Zeile 296: @@
 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.
-$$h_{groove} = 0.5 \cdot \left(\frac{\bar{h}}{\frac{1}{4} \cdot \pi - \Phi} \cdot \alpha - \frac{\Phi \cdot \bar{h}}{\frac{1}{4} \cdot \pi - \Phi} + s_R\right)\tag{13}$$
+$$h_{groove} = 0.5 \cdot \left(\frac{\bar{h}}{\frac{1}{4} \cdot \pi - \Phi} \cdot \alpha - \frac{\Phi \cdot \bar{h}}{\frac{1}{4} \cdot \pi - \Phi} + s_R\right)\tag{36}$$
 {{ :en:grundlagenhandbuch:geometriegroessen_des_gleichdrall-doppelschneckenextruders:en_sigma150_dlg_grundlagenhandbuch_geometriegroessen_016.svg?600%nolink |}}
@@ Zeile 311: / Zeile 304: @@
 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:
-$$\dot{V} = k \cdot \dot{V}_{channel} \pm \dot{V}_{gap} \pm l \cdot \dot{V}_{groove}\tag{14}$$
+$$\dot{V} = k \cdot \dot{V}_{channel} \pm \dot{V}_{gap} \pm l \cdot \dot{V}_{groove}\tag{37}$$
 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:
-$$l = \frac{b_{Kn}}{2 \cdot \pi} \cdot (2 \cdot \pi - \Omega) \cdot \cos \varphi_s\tag{15}$$
+$$l = \frac{b_{Kn}}{2 \cdot \pi} \cdot (2 \cdot \pi - \Omega) \cdot \cos \varphi_s\tag{38}$$
 With $b_{Kn}$ being the width of the kneading discs.
@@ Zeile 321: / Zeile 314: @@
 The width of the flights $b_{Steg}$ is shortened by the total sum of all groove widths $l$.
-$$b_{threads} = (2 \cdot \pi - \Omega) \cdot D_s \cdot \cos \varphi_s - l \cdot b_{groove}\tag{16}$$
+$$b_{threads} = (2 \cdot \pi - \Omega) \cdot D_s \cdot \cos \varphi_s - l \cdot b_{groove}\tag{39}$$
 ==== Eccentric Kneading Blocks ====
@@ Zeile 337: / Zeile 330: @@
 Maximum diameter ratio of three-way profile ($i=3$)
-$$D_{v,3} = D_{v,3,max} = \frac{1}{2 \cdot \cos\left(\frac{\pi}{2 \cdot 3}\right) - 1} = 1.366  \tag{Equation 3-13}$$
+$$D_{v,3} = D_{v,3,max} = \frac{1}{2 \cdot \cos\left(\frac{\pi}{2 \cdot 3}\right) - 1} = 1.366  \tag{40}$$
 Outer diameter of three-course profile
-$$D_{a,3} = \frac{2 \cdot a}{1 + D_{v,3}} \tag{Equation 3-14}$$
+$$D_{a,3} = \frac{2 \cdot a}{1 + D_{v,3}} \tag{41}$$
 Eccentricity $e$
-$$e = \frac{D_{a,3} - D_{i,3}}{2} \tag{Equation 3-15}$$
+$$e = \frac{D_{a,3} - D_{i,3}}{2} \tag{42}$$
 Inside diameter of three-course profile
-$$D_{i,3} = 2 \cdot a - D_{a,3} \tag{Equation 3-16}$$
+$$D_{i,3} = 2 \cdot a - D_{a,3} \tag{43}$$
@@ Zeile 359: / Zeile 352: @@
 This profile shows that the comb angles are zero. Therefore the flank angles of the standard profile are the same:
-Comb angle: $\angle AOB = \angle BOC = \angle COD = \Omega_3 = 60° \tag{Equation 3-17}$
+Comb angle: $$\angle AOB = \angle BOC = \angle COD = \Omega_3 = 60° \tag{44}$$
-Flank angle: $\phi_{k,3} = \phi_{g,3} = 0 \tag{Equation 3-18}$
+Flank angle: $$\phi_{k,3} = \phi_{g,3} = 0 \tag{45}$$
-Outside radius: $R_{a,3} = \frac{D_{a,3}}{2} \tag{Equation 3-19}$
+Outside radius: $$R_{a,3} = \frac{D_{a,3}}{2} \tag{46}$$
-Inside radius: $R_{i,3} = \frac{D_{i,3}}{2} = \left(a - \frac{D_{a,3}}{2}\right) \tag{Equation 3-20}$
+Inside radius: $$R_{i,3} = \frac{D_{i,3}}{2} = \left(a - \frac{D_{a,3}}{2}\right) \tag{47}$$
 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: $x_A = R_{i,3}$; $y_A = 0$
+A: $x_A = R_{i,3}; y_A = 0\tag{48}$
-B: $x_B = R_{a,3} \cdot \cos 60° = \frac{1}{2}R_{a,3}$; $y_B = R_{a,3} \cdot \sin 60° = \frac{\sqrt{3}}{2}R_{a,3}$
+B: $x_B = R_{a,3} \cdot \cos 60° = \frac{1}{2}R_{a,3}; y_B = R_{a,3} \cdot \sin 60° = \frac{\sqrt{3}}{2}R_{a,3}\tag{49}$
-C: $x_C = R_{i,3} \cdot \cos 120° = -\frac{1}{2}R_{i,3}$; $y_C = R_{i,3} \cdot \sin 120° = \frac{\sqrt{3}}{2}R_{i,3}$
+C: $x_C = R_{i,3} \cdot \cos 120° = -\frac{1}{2}R_{i,3}; y_C = R_{i,3} \cdot \sin 120° = \frac{\sqrt{3}}{2}R_{i,3}\tag{50}$
-D: $x_D = -R_{a,3}$; $y_D = 0$
+D: $x_D = -R_{a,3}; y_D = 0\tag{51}$
 ==== Screw Mixing Elements ====
@@ Zeile 393: / Zeile 386: @@
 From the control volume ABC we get the equilibrium of flow rates:
-$$\dot{V} = k \cdot \dot{V}_{channel} \pm \dot{V}_{gap} \pm l \cdot \dot{V}_{groove}\tag{21}$$
+$$\dot{V} = k \cdot \dot{V}_{channel} \pm \dot{V}_{gap} \pm l \cdot \dot{V}_{groove}\tag{52}$$
 Where $l$ is the number of grooves between the points A and B. $l$ is calculated using the following equation:
-$$l = \frac{l_{Nut}}{2\pi} \cdot (2\pi - \Omega) \cdot \cos \varphi_s\tag{22}$$
+$$l = \frac{l_{Nut}}{2\pi} \cdot (2\pi - \Omega) \cdot \cos \varphi_s\tag{53}$$
 ==== Blister Elements ====
@@ Zeile 422: / Zeile 415: @@
 These sections are characterized by the following dimensionless numbers:
-$$k_1 = \frac{D_a}{D_z}\tag{23}$$
+$$k_1 = \frac{D_a}{D_z}\tag{54}$$
-$$k_2 = \frac{D_i}{D_a}\tag{24}$$
+$$k_2 = \frac{D_i}{D_a}\tag{55}$$
-$$cl = \frac{a}{D_z/2}\tag{25}$$
+$$cl = \frac{a}{D_z/2}\tag{56}$$
-==== Turbine Mixing Elements ====
+==== Thoothed Mixing Elements ====
-Turbine mixing elements consist of rows of mixing discs  [[en:grundlagenhandbuch:geometriegroessen_des_gleichdrall-doppelschneckenextruders#references|[Boo78]]]. The geometry of the teeth in the mixing disc and the number of rows can vary. The teeth are cut into the mixing disc under a certain angle. Depending on the angle, the turbine mixer element can be conveying, reconveying or neutral.
+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.
 {{ :grundlagenhandbuch:geometriegroessen_des_gleichdrall-doppelschneckenextruders:eingesetzte_schneckenelemente:de_sigma150_dlg_grundlagenhandbuch_geometriegroessen_023.svg?600%nolink |}}
@@ Zeile 440: / Zeile 435: @@
   * Outside diameter $D_a$,
   * Inside diameter $D_i$,
-  * Number of teeth $n$
+  * Number of teeth $n$,
-  * Number of rows
+  * Pitch angle $\varphi_N$,
-  * Pitch angle $j_N$
   * Conveying direction (conveying, reconveying or neutral)
-Since the geometry of the grooves is arbitrary, we decided to simplify the grooves using rectangular channels. Therefore, we require the height and the width of the grooves.
+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.
-NOTE: The lead angle $j_N$ has to be entered with a positive value, even for reconveying elements!
 ===== References =====