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en:grundlagenhandbuch:schmelzefoerderung [2026/05/24 16:39] – [Pressure-Throughput Model for Threaded Elements] neelesten:grundlagenhandbuch:schmelzefoerderung [2026/09/04 09:53] (aktuell) – [Flow in Simple Geometries] paal
Zeile 1: Zeile 1:
-====== Melt conveying======+====== Melt Conveying======
  
 The basis of the general mathematical description of flow is the equilibrium of mass, momentum and energy. A flow is described, if at any place and at any time the velocity vector and the values pressure and temperature in the flow domain are known. For the calculation of these values, the conservative equations and the constitutive equations are combined  [[en:grundlagenhandbuch:schmelzefoerderung#literatur|[BSL60]]],  [[en:grundlagenhandbuch:schmelzefoerderung#literatur|[BAH77]]],  [[en:grundlagenhandbuch:schmelzefoerderung#literatur|[Böh81]]],  [[en:grundlagenhandbuch:schmelzefoerderung#literatur|[Jis82]]]. The basis of the general mathematical description of flow is the equilibrium of mass, momentum and energy. A flow is described, if at any place and at any time the velocity vector and the values pressure and temperature in the flow domain are known. For the calculation of these values, the conservative equations and the constitutive equations are combined  [[en:grundlagenhandbuch:schmelzefoerderung#literatur|[BSL60]]],  [[en:grundlagenhandbuch:schmelzefoerderung#literatur|[BAH77]]],  [[en:grundlagenhandbuch:schmelzefoerderung#literatur|[Böh81]]],  [[en:grundlagenhandbuch:schmelzefoerderung#literatur|[Jis82]]].
Zeile 633: Zeile 633:
 The analysis of existing models to specify the pressure-throughput behavior of rectangular and twin screw flow channels gives evidence that these models are only suitable to a limited extent to describe mixing elements. This shall be explained on the basis of the results of a three-dimensional finite element simulation of a threaded-mixing element. The analysis of existing models to specify the pressure-throughput behavior of rectangular and twin screw flow channels gives evidence that these models are only suitable to a limited extent to describe mixing elements. This shall be explained on the basis of the results of a three-dimensional finite element simulation of a threaded-mixing element.
  
-{{ :en:grundlagenhandbuch:en_sigma150_dlg_grundlagenhandbuch_schmelzefoerderung_023.png?nolink |}}+{{ :en:grundlagenhandbuch:en_sigma150_dlg_grundlagenhandbuch_schmelzefoerderung_030.svg?nolink&600 |}}
  
 **Figure:** Velocity profiles in a threaded mixing element. **Figure:** Velocity profiles in a threaded mixing element.
Zeile 649: Zeile 649:
 To specify the throughput behavior of the screw elements, first of all, the basic geometry (twin screw channel, rectangular channel and disk elements) is analyzed and modelled. Subsequently, the models are superposed and integrated into an overall model for the description of the throughput behavior of the screw elements. To specify the throughput behavior of the screw elements, first of all, the basic geometry (twin screw channel, rectangular channel and disk elements) is analyzed and modelled. Subsequently, the models are superposed and integrated into an overall model for the description of the throughput behavior of the screw elements.
  
-^ Geometry ^ Parameter ^ Formula ^ +{{ :en:grundlagenhandbuch:en_sigma150_dlg_grundlagenhandbuch_schmelzefoerderung_031.svg?nolink&600 |}}
-| Rectangular channels | Ratio of channel width to channel depth | $\frac{b}{h}$ | +
-| | Ratio of pitch to screw diameter | $\frac{t}{D_s} = \pi \cdot \tan(\varphi_s)$ | +
-| Twin screw channels | Ratio of axial distance to screw diameter | $CL = \frac{2 \cdot a}{D_s}$ | +
-| | Ratio of pitch to screw diameter | $\frac{t}{D_s} = \pi \cdot \tan(\varphi_s)$ | +
-| | Number of flights | $i$ | +
-| Disk elements | Ratio of axial distance to cylinder diameter | $CL = \frac{2 \cdot a}{D_z}$ | +
-| | Ratio of the diameter of the larger disk to the cylinder diameter | $k_1 = \frac{D_a}{D_z}$ | +
-| | Ratio of the diameter of the smaller disk to the diameter of the larger disk | $k_2 = \frac{D_i}{D_a}$ |+
  
 **Table:** Dimensionless parameters for the characterization of the geometry. **Table:** Dimensionless parameters for the characterization of the geometry.
Zeile 678: Zeile 670:
 Due to the closely intermeshing geometry this requirement cannot be transferred to twin screw channels, because channel width and channel height can-not be varied independently. They are rather dependent on the pitch, the axial distance, the screw diameter, and the number of flights (see figure). Equation 67, however, can only be valid, if the channel width is independent of the pitch. Due to the closely intermeshing geometry this requirement cannot be transferred to twin screw channels, because channel width and channel height can-not be varied independently. They are rather dependent on the pitch, the axial distance, the screw diameter, and the number of flights (see figure). Equation 67, however, can only be valid, if the channel width is independent of the pitch.
  
-{{ :grundlagenhandbuch:schmelzefoerderung:regressionsansatz:de_sigma150_dlg_grundlagenhandbuch_schmelzefoerderung_032.png?nolink |}}+{{ :en:grundlagenhandbuch:en_sigma150_dlg_grundlagenhandbuch_schmelzefoerderung_032.svg?nolink&600 |}}
  
 **Figure:** Influence of the helix angle on the geometry of twin screw channels. **Figure:** Influence of the helix angle on the geometry of twin screw channels.
Zeile 710: Zeile 702:
   * **Section III:** In this area a negative pressure gradient and a positive volume throughput can be observed.   * **Section III:** In this area a negative pressure gradient and a positive volume throughput can be observed.
   *    * 
-{{ :en:grundlagenhandbuch:en_sigma150_dlg_grundlagenhandbuch_schmelzefoerderung_024.png?nolink |}}+{{ :en:grundlagenhandbuch:en_sigma150_dlg_grundlagenhandbuch_schmelzefoerderung_033.svg?nolink&600 |}}
  
 **Figure:** Boundary conditions for the modelling of the pressure-throughput behavior. **Figure:** Boundary conditions for the modelling of the pressure-throughput behavior.
Zeile 771: Zeile 763:
 In the figure the calculated dimensionless pressure-throughput characteristics are contrasted with the Polyflow results of varying exponents of the power law. It can be seen, that the model offers a sufficiently good description for a wide area. In the figure the calculated dimensionless pressure-throughput characteristics are contrasted with the Polyflow results of varying exponents of the power law. It can be seen, that the model offers a sufficiently good description for a wide area.
  
-{{ :en:grundlagenhandbuch:en_sigma150_dlg_grundlagenhandbuch_schmelzefoerderung_025.png?nolink |}}+{{ :en:grundlagenhandbuch:en_sigma150_dlg_grundlagenhandbuch_schmelzefoerderung_034.svg?nolink&600 |}}
  
 **Figure:** Comparison of the calculated and the predicted pressure-throughput characteristics for varying exponents of the power law. **Figure:** Comparison of the calculated and the predicted pressure-throughput characteristics for varying exponents of the power law.
Zeile 781: Zeile 773:
 i.e. if the characteristic curves of the conveying channels are known, those of the reconveying channels can be deduced from this basis. i.e. if the characteristic curves of the conveying channels are known, those of the reconveying channels can be deduced from this basis.
  
-The characteristic curve to specify the transportation neutral elements correspond with the specification in zone III, as the drag conveying capacity equals zero and as, therefore, the element is always overrun. The figure displays the pressure-throughput characteristics for the three types of transport and varying exponents of the power law. It can be recognized that the above outlined transformation of the characteristics allow a sufficiently good description of the Poly-flow results.+The characteristic curve to specify the neutral elements correspond with the specification in zone III, as the drag conveying capacity equals zero and as, therefore, the element is always overrun. The figure displays the pressure-throughput characteristics for the three types of transport and varying exponents of the power law. It can be recognized that the above outlined transformation of the characteristics allow a sufficiently good description of the Poly-flow results.
  
-{{ :en:grundlagenhandbuch:en_sigma150_dlg_grundlagenhandbuch_schmelzefoerderung_026.png?nolink |}}+{{ :en:grundlagenhandbuch:en_sigma150_dlg_grundlagenhandbuch_schmelzefoerderung_035.svg?nolink&600 |}}
  
 **Figure:** Comparison of the calculated and the predicted pressure-throughput characteristics of conveying, reconveying and neutral rectangular channels. **Figure:** Comparison of the calculated and the predicted pressure-throughput characteristics of conveying, reconveying and neutral rectangular channels.
Zeile 808: Zeile 800:
 In the figure the dimensionless pressure-throughput characteristics are illustrated for varying exponents of the power law. It can be recognized that the model yields a sufficiently good description for a wide area. In the figure the dimensionless pressure-throughput characteristics are illustrated for varying exponents of the power law. It can be recognized that the model yields a sufficiently good description for a wide area.
  
-{{ :en:grundlagenhandbuch:en_sigma150_dlg_grundlagenhandbuch_schmelzefoerderung_027.png?nolink |}}+{{ :en:grundlagenhandbuch:en_sigma150_dlg_grundlagenhandbuch_schmelzefoerderung_036.svg?nolink&600 |}}
  
 **Figure:** Comparison of the calculated and the predicted pressure-throughput characteristics for varying exponents of the Power Law. **Figure:** Comparison of the calculated and the predicted pressure-throughput characteristics for varying exponents of the Power Law.
Zeile 814: Zeile 806:
 The description model of the throughput behavior of twin screw channels can be applied to conveying and reconveying channels. A comparison of the model predictions and the Polyflow results for both conveying and reconveying channels is illustrated in the figure. For all three transportation types a sufficiently good description can be identified. The description model of the throughput behavior of twin screw channels can be applied to conveying and reconveying channels. A comparison of the model predictions and the Polyflow results for both conveying and reconveying channels is illustrated in the figure. For all three transportation types a sufficiently good description can be identified.
  
-{{ :en:grundlagenhandbuch:en_sigma150_dlg_grundlagenhandbuch_schmelzefoerderung_028.png?nolink |}}+{{ :en:grundlagenhandbuch:en_sigma150_dlg_grundlagenhandbuch_schmelzefoerderung_037.svg?nolink&600 |}}
  
 **Figure:** Comparison of the calculated and the predicted pressure-throughput characteristics for conveying, reconveying and transportation neutral twin screw channels. **Figure:** Comparison of the calculated and the predicted pressure-throughput characteristics for conveying, reconveying and transportation neutral twin screw channels.
Zeile 824: Zeile 816:
 $$\pi_\dot{V} = C_{SE,1} \cdot \pi_p + C_{SE,2} \cdot \pi_p^2 + C_{SE,3} \cdot \pi_p^3 \tag{83}$$ $$\pi_\dot{V} = C_{SE,1} \cdot \pi_p + C_{SE,2} \cdot \pi_p^2 + C_{SE,3} \cdot \pi_p^3 \tag{83}$$
  
-{{ :en:grundlagenhandbuch:en_sigma150_dlg_grundlagenhandbuch_schmelzefoerderung_029.png?nolink |}}+{{ :en:grundlagenhandbuch:en_sigma150_dlg_grundlagenhandbuch_schmelzefoerderung_038.svg?nolink&600 |}}
  
  
Zeile 848: Zeile 840:
 For the description of the flow in the channel area a channel model is used, in which the screw and the barrel are uncoiled along the barrel wall. Additionally, a kinematic reversal is carried out, i.e. the rotating screw is represented by stationary, parallel channels above which moves the unwounded barrel (see figure). For the description of the flow in the channel area a channel model is used, in which the screw and the barrel are uncoiled along the barrel wall. Additionally, a kinematic reversal is carried out, i.e. the rotating screw is represented by stationary, parallel channels above which moves the unwounded barrel (see figure).
  
-{{ :en:grundlagenhandbuch:en_sigma150_dlg_grundlagenhandbuch_schmelzefoerderung_030.png?nolink |}}+{{ :en:grundlagenhandbuch:en_sigma150_dlg_grundlagenhandbuch_schmelzefoerderung_039.svg?nolink&600 |}}
  
 **Figure:** Conveyor channel model for double-flighted conveying threaded elements  [[en:grundlagenhandbuch:schmelzefoerderung#literatur|[Ans93]]] **Figure:** Conveyor channel model for double-flighted conveying threaded elements  [[en:grundlagenhandbuch:schmelzefoerderung#literatur|[Ans93]]]
Zeile 956: Zeile 948:
 In the figure the theoretical pressure-throughput characteristics are illustrated for conveying and reconveying threaded elements. In the figure the theoretical pressure-throughput characteristics are illustrated for conveying and reconveying threaded elements.
  
-{{ :en:grundlagenhandbuch:en_sigma150_dlg_grundlagenhandbuch_schmelzefoerderung_031.png?nolink |}}+{{ :en:grundlagenhandbuch:en_sigma150_dlg_grundlagenhandbuch_schmelzefoerderung_040.svg?nolink&600 |}}
  
 **Figure:** Pressure-throughput behavior of conveying and reconveying threaded elements. **Figure:** Pressure-throughput behavior of conveying and reconveying threaded elements.
Zeile 962: Zeile 954:
 In the figure the model predictions are compared with the experimental results. The experiments were performed with a model extruder using silicon oil as melt. In the figure the model predictions are compared with the experimental results. The experiments were performed with a model extruder using silicon oil as melt.
  
-{{ :en:grundlagenhandbuch:en_sigma150_dlg_grundlagenhandbuch_schmelzefoerderung_032.png?nolink |}}+{{ :en:grundlagenhandbuch:en_sigma150_dlg_grundlagenhandbuch_schmelzefoerderung_041.svg?nolink&600 |}}
  
 **Figure:** Comparison of calculated and measured pressure gradients in threaded elements. **Figure:** Comparison of calculated and measured pressure gradients in threaded elements.
Zeile 971: Zeile 963:
 In the figure the variation of the axial velocities in a conveying, closely intermeshing thread mixing element are illustrated on two different levels. Out of the variation of the velocities it can be noticed that the flow in the grooves and in the intermeshing zone of both screws significantly differs from the flow in the channel area. In the intermeshing zone axial velocities can be detected, which are considerably higher than those in the channel area. In the grooves, however, another effect can be observed. As the grooves are of a back-flow design (in this case), in this area a back-flow is to be recognized accordingly. In the modelling, at first, a distinction is made between the intermeshing zone and the channel zone. In the figure the variation of the axial velocities in a conveying, closely intermeshing thread mixing element are illustrated on two different levels. Out of the variation of the velocities it can be noticed that the flow in the grooves and in the intermeshing zone of both screws significantly differs from the flow in the channel area. In the intermeshing zone axial velocities can be detected, which are considerably higher than those in the channel area. In the grooves, however, another effect can be observed. As the grooves are of a back-flow design (in this case), in this area a back-flow is to be recognized accordingly. In the modelling, at first, a distinction is made between the intermeshing zone and the channel zone.
  
-$$\dot{V}_{dGME} = \dot{V}_{channel,dGME} + \dot{V}_{intermeshing\ area,dGME} \tag{1}$$+$$\dot{V}_{dGME} = \dot{V}_{channel,dGME} + \dot{V}_{intermeshing\ area,dGME} \tag{104}$$
  
 The influence of the grooves on the behavior of the elements is then taken account of in the description of the separate areas. The influence of the grooves on the behavior of the elements is then taken account of in the description of the separate areas.
  
-{{ :en:grundlagenhandbuch:en_sigma150_dlg_grundlagenhandbuch_schmelzefoerderung_033.png?nolink |}}+{{ :en:grundlagenhandbuch:en_sigma150_dlg_grundlagenhandbuch_schmelzefoerderung_042.svg?nolink&600 |}}
  
 **Figure:** Calculated velocity fields in a tightly intermeshing threaded mixing element. **Figure:** Calculated velocity fields in a tightly intermeshing threaded mixing element.
Zeile 981: Zeile 973:
 === Channel Model for the Channel Area === === Channel Model for the Channel Area ===
  
-The channel model for tightly intermeshing threaded mixing elements can be compared to that of threaded elements. The flow in the grooves was accounted for by an additional volume throughput in the balance (Eq. 2) of the marked control room ABC illustrated in see figure.+The channel model for tightly intermeshing threaded mixing elements can be compared to that of threaded elements. The flow in the grooves was accounted for by an additional volume throughput in the balance 105 of the marked control room ABC illustrated in see figure.
  
-{{ :grundlagenhandbuch:schmelzefoerderung:regressionsansatz:de_sigma150_dlg_grundlagenhandbuch_schmelzefoerderung_043.png?nolink |}}+{{ :en:grundlagenhandbuch:en_sigma150_dlg_grundlagenhandbuch_schmelzefoerderung_043.svg?nolink&600 |}}
  
 **Figure:** Channel model for a reconveying, intermeshing threaded mixing element. **Figure:** Channel model for a reconveying, intermeshing threaded mixing element.
Zeile 989: Zeile 981:
 A volume flow balance in the control room as depicted in the figure results in the following equation: A volume flow balance in the control room as depicted in the figure results in the following equation:
  
-$$\dot{V}_{channel,dGME} = k \cdot \dot{V}_{z,dGME} + \dot{V}_{x,dGME} + l \cdot \dot{V}_{groove,dGME} \tag{2}$$+$$\dot{V}_{channel,dGME} = k \cdot \dot{V}_{z,dGME} + \dot{V}_{x,dGME} + l \cdot \dot{V}_{groove,dGME} \tag{105}$$
  
 For the description of the flow in channel direction and in the grooves all models developed in chapter Rectangular channels and chapter Twin Screw Channels respectively are used in a general form: For the description of the flow in channel direction and in the grooves all models developed in chapter Rectangular channels and chapter Twin Screw Channels respectively are used in a general form:
  
-$$\pi_{\dot{V},z,dGME} = A_{DSE,0} + A_{DSE,1} \cdot \pi_{p,z} + A_{DSE,2} \cdot \pi_{p,z}^2 + A_{DSE,3} \cdot \pi_{p,z}^3 + A_{DSE,4} \cdot \pi_{p,z}^5 \tag{3}$$+$$\pi_{\dot{V},z,dGME} = A_{DSE,0} + A_{DSE,1} \cdot \pi_{p,z} + A_{DSE,2} \cdot \pi_{p,z}^2 + A_{DSE,3} \cdot \pi_{p,z}^3 + A_{DSE,4} \cdot \pi_{p,z}^5 \tag{106}$$
  
-$$\pi_{\dot{V},x,dGME} = A_{RE,0} + A_{RE,1} \cdot \pi_{p,x} + A_{RE,2} \cdot \pi_{p,x}^2 + A_{RE,3} \cdot \pi_{p,x}^3 + A_{RE,4} \cdot \pi_{p,x}^5$$+$$\pi_{\dot{V},x,dGME} = A_{RE,0} + A_{RE,1} \cdot \pi_{p,x} + A_{RE,2} \cdot \pi_{p,x}^2 + A_{RE,3} \cdot \pi_{p,x}^3 + A_{RE,4} \cdot \pi_{p,x}^5\tag{107}$$
  
-$$\pi_{\dot{V},groove,dGME} = A_{RE,N,0} + A_{RE,N,1} \cdot \pi_{p,N} + A_{RE,N,2} \cdot \pi_{p,N}^2 + A_{RE,N,3} \cdot \pi_{p,N}^3 + A_{RE,4} \cdot \pi_{p,N}^5$$+$$\pi_{\dot{V},groove,dGME} = A_{RE,N,0} + A_{RE,N,1} \cdot \pi_{p,N} + A_{RE,N,2} \cdot \pi_{p,N}^2 + A_{RE,N,3} \cdot \pi_{p,N}^3 + A_{RE,4} \cdot \pi_{p,N}^5\tag{108}$$
  
 with with
  
-$$\dot{V}_{z,dGME} = \pi_{\dot{V},z} \cdot (\bar{h} \cdot b_{max} \cdot v_0) \tag{6}$$+$$\dot{V}_{z,dGME} = \pi_{\dot{V},z} \cdot (\bar{h} \cdot b_{max} \cdot v_0) \tag{109}$$
  
-$$\dot{V}_{x,dGME} = \pi_{\dot{V},x} \cdot (h_N \cdot b_N \cdot v_0)$$+$$\dot{V}_{x,dGME} = \pi_{\dot{V},x} \cdot (h_N \cdot b_N \cdot v_0)\tag{110}$$
  
-$$\dot{V}_{groove,dGME} = \pi_{\dot{V},N} \cdot (h_N \cdot b_N \cdot v_0)$$+$$\dot{V}_{groove,dGME} = \pi_{\dot{V},N} \cdot (h_N \cdot b_N \cdot v_0)\tag{111}$$
  
 and and
  
-$$\pi_{p,z} = \frac{\Delta p}{\Delta z} \cdot \frac{\bar{h}^{1+n}}{K \cdot v_0^n} \tag{9}$$+$$\pi_{p,z} = \frac{\Delta p}{\Delta z} \cdot \frac{\bar{h}^{1+n}}{K \cdot v_0^n} \tag{112}$$
  
-$$\pi_{p,x} = \frac{\Delta p}{x} \cdot \frac{s_R^{1+n}}{K \cdot v_0^n} \tag{10}$$+$$\pi_{p,x} = \frac{\Delta p}{x} \cdot \frac{s_R^{1+n}}{K \cdot v_0^n} \tag{113}$$
  
-$$\pi_{p,N} = \frac{\Delta p}{x} \cdot \frac{h_N}{K \cdot v_0^n} \tag{11}$$+$$\pi_{p,N} = \frac{\Delta p}{x} \cdot \frac{h_N}{K \cdot v_0^n} \tag{114}$$
  
 and and
  
-$$\pi_{\dot{V},z,dGME} = \frac{\dot{V}_z}{\bar{h} \cdot b_{max} \cdot v_0} \tag{12}$$+$$\pi_{\dot{V},z,dGME} = \frac{\dot{V}_z}{\bar{h} \cdot b_{max} \cdot v_0} \tag{115}$$
  
-$$\pi_{\dot{V},x,dGME} = \frac{\dot{V}_x}{s_R \cdot b_{Steg} \cdot v_0} \tag{13}$$+$$\pi_{\dot{V},x,dGME} = \frac{\dot{V}_x}{s_R \cdot b_{Steg} \cdot v_0} \tag{116}$$
  
-$$\pi_{\dot{V},x,dGME} = \frac{\dot{V}_{Nut}}{h_N \cdot b_N \cdot v_0} \tag{14}$$+$$\pi_{\dot{V},x,dGME} = \frac{\dot{V}_{Nut}}{h_N \cdot b_N \cdot v_0} \tag{117}$$
  
-If now the Equations (3) to (8) are inserted into Equation (2), one receives:+Substituting gives:
  
 $$\dot{V}_{channel,dGME} = k \cdot (\bar{h} \cdot b_{max} \cdot v_0) \cdot A_{DSE,0} + (A_{DSE,1} \cdot \pi_{p,z} + A_{DSE,2} \cdot \pi_{p,z}^2$$ $$\dot{V}_{channel,dGME} = k \cdot (\bar{h} \cdot b_{max} \cdot v_0) \cdot A_{DSE,0} + (A_{DSE,1} \cdot \pi_{p,z} + A_{DSE,2} \cdot \pi_{p,z}^2$$
Zeile 1029: Zeile 1021:
 $$+A_{DSE,3} \cdot \pi_{p,z}^3 + A_{DSE,4} \cdot \pi_{p,z}^5) + (s_R \cdot b_{steg} \cdot v_0) \cdot (A_{RE,0}$$ $$+A_{DSE,3} \cdot \pi_{p,z}^3 + A_{DSE,4} \cdot \pi_{p,z}^5) + (s_R \cdot b_{steg} \cdot v_0) \cdot (A_{RE,0}$$
  
-$$+A_{RE,1} \cdot \pi_{p,x} + A_{RE,2} \cdot \pi_{p,x}^2 + A_{RE,3} \cdot \pi_{p,x}^3 + A_{RE,4} \cdot \pi_{p,x}^5) \tag{15}$$+$$+A_{RE,1} \cdot \pi_{p,x} + A_{RE,2} \cdot \pi_{p,x}^2 + A_{RE,3} \cdot \pi_{p,x}^3 + A_{RE,4} \cdot \pi_{p,x}^5)$$
  
 $$+l \cdot (h_N \cdot b_N \cdot v_0) \cdot (A_{RE,N,0} + A_{RE,N,1} \cdot \pi_{p,N} + A_{RE,N,2} \cdot \pi_{p,N}^2$$ $$+l \cdot (h_N \cdot b_N \cdot v_0) \cdot (A_{RE,N,0} + A_{RE,N,1} \cdot \pi_{p,N} + A_{RE,N,2} \cdot \pi_{p,N}^2$$
  
-$$+A_{RE,N,3} \cdot \pi_{p,N}^3 + A_{RE,4} \cdot \pi_{p,N}^5)$$+$$+A_{RE,N,3} \cdot \pi_{p,N}^3 + A_{RE,4} \cdot \pi_{p,N}^5)\tag{118}$$
  
 Is this equation standardized to $k \cdot (\bar{h} \cdot b_{max} \cdot v_0)$, then follows: Is this equation standardized to $k \cdot (\bar{h} \cdot b_{max} \cdot v_0)$, then follows:
Zeile 1041: Zeile 1033:
 $$+A_{DSE,3} \cdot \pi_{p,z}^3 + A_{DSE,4} \cdot \pi_{p,z}^5) + \frac{(s_R \cdot b_{thread} \cdot v_0)}{k \cdot (\bar{h} \cdot b_{max} \cdot v_0)} \cdot (A_{RE,0}$$ $$+A_{DSE,3} \cdot \pi_{p,z}^3 + A_{DSE,4} \cdot \pi_{p,z}^5) + \frac{(s_R \cdot b_{thread} \cdot v_0)}{k \cdot (\bar{h} \cdot b_{max} \cdot v_0)} \cdot (A_{RE,0}$$
  
-$$+A_{RE,1} \cdot \pi_{p,x} + A_{RE,2} \cdot \pi_{p,x}^2 + A_{RE,3} \cdot \pi_{p,x}^3 + A_{RE,4} \cdot \pi_{p,x}^5) \tag{16}$$+$$+A_{RE,1} \cdot \pi_{p,x} + A_{RE,2} \cdot \pi_{p,x}^2 + A_{RE,3} \cdot \pi_{p,x}^3 + A_{RE,4} \cdot \pi_{p,x}^5) $$
  
 $$+ \frac{l \cdot (h_N \cdot b_N \cdot v_0)}{k \cdot (\bar{h} \cdot b_{max} \cdot v_0)} \cdot (A_{RE,N,0} + A_{RE,N,1} \cdot \pi_{p,N} + A_{RE,N,2} \cdot \pi_{p,N}^2$$ $$+ \frac{l \cdot (h_N \cdot b_N \cdot v_0)}{k \cdot (\bar{h} \cdot b_{max} \cdot v_0)} \cdot (A_{RE,N,0} + A_{RE,N,1} \cdot \pi_{p,N} + A_{RE,N,2} \cdot \pi_{p,N}^2$$
  
-$$+A_{RE,N,3} \cdot \pi_{p,N}^3 + A_{RE,4} \cdot \pi_{p,N}^5)$$+$$+A_{RE,N,3} \cdot \pi_{p,N}^3 + A_{RE,4} \cdot \pi_{p,N}^5)\tag{119}$$
  
 To obtain a comprehensive description of the process behavior the pressure gradient in conveyor channel direction is to be linked with the pressure gradients above the radial clearance and with the pressure gradient in the groove. For this purpose the approach proposed by Ansahl  [[en:grundlagenhandbuch:schmelzefoerderung#literatur|[Ans93]]] is reverted to (see chap. Linkage of the Volume Flow Rates). The linkage of the pressure gradients in the conveyor channel and in the groove can be carried out analogously. It only has to be taken into account that the grooves are inserted in a specific angle in the screw flank, so that the length, in which the pressure gradient is effective, in the grooves differs from the length of the screw flight. Consequently, the pressure gradients can be interlinked as follows. To obtain a comprehensive description of the process behavior the pressure gradient in conveyor channel direction is to be linked with the pressure gradients above the radial clearance and with the pressure gradient in the groove. For this purpose the approach proposed by Ansahl  [[en:grundlagenhandbuch:schmelzefoerderung#literatur|[Ans93]]] is reverted to (see chap. Linkage of the Volume Flow Rates). The linkage of the pressure gradients in the conveyor channel and in the groove can be carried out analogously. It only has to be taken into account that the grooves are inserted in a specific angle in the screw flank, so that the length, in which the pressure gradient is effective, in the grooves differs from the length of the screw flight. Consequently, the pressure gradients can be interlinked as follows.
  
-$$\pi_{p,x} = \frac{\Delta p}{\Delta z} \cdot \frac{b_{max} + e_{max}}{e_{max} \cdot \tan(\varphi_s)} \cdot \frac{s_R^{1+n}}{K \cdot v_0^n} = \pi_{geo,GE} \cdot \pi_{p,z} \tag{17}$$+$$\pi_{p,x} = \frac{\Delta p}{\Delta z} \cdot \frac{b_{max} + e_{max}}{e_{max} \cdot \tan(\varphi_s)} \cdot \frac{s_R^{1+n}}{K \cdot v_0^n} = \pi_{geo,GE} \cdot \pi_{p,z}\tag{120}$$
  
-$$\pi_{p,x} = \frac{\Delta p}{\Delta z} \cdot \frac{b_{max} + e_{max}}{e_{max} \cdot \tan(\varphi_s)} \cdot \cos\left(\frac{\pi}{2} - \varphi_s - \varphi_N\right) \cdot \frac{h_N^{1+n}}{K \cdot v_0^n} = \pi_{geo,N} \cdot \pi_{p,z} \tag{18}$$+$$\pi_{p,N} = \frac{\Delta p}{\Delta z} \cdot \frac{b_{max} + e_{max}}{e_{max} \cdot \tan(\varphi_s)} \cdot \cos\left(\frac{\pi}{2} - \varphi_s - \varphi_N\right) \cdot \frac{h_N^{1+n}}{K \cdot v_0^n} = \pi_{geo,N} \cdot \pi_{p,z}\tag{121}$$
  
-with+$$\pi_{p,x} = \frac{\Delta p}{\Delta z} \cdot \frac{b_{max} + e_{max}}{e_{max} \cdot \tan(\varphi_s)} \cdot \frac{s_R^{1+n}}{K \cdot v_0^n} = \pi_{geo,x} \cdot \pi_{p,z}\tag{122}$$
  
-$$\pi_{geo,GE} = \frac{b_{max} + e_{max}}{e_{max} \cdot \tan(\varphi_s)} \cdot \left(\frac{s_R}{\bar{h}}\right)^{1+n} \tag{19}$$+$$\pi_{p,N} = \frac{\Delta p}{\Delta z} \cdot \frac{b_{max} + e_{max}}{e_{max} \cdot \tan(\varphi_s)} \cdot \cos\left(\frac{\pi}{2} - \varphi_s - \varphi_N\right) \cdot \left(\frac{h_N}{\bar{h}}\right)^{1+n} = \pi_{geo,N} \cdot \pi_{p,z}\tag{123}$$
  
-$$\pi_{geo,N} = \frac{\Delta p}{\Delta z} \cdot \frac{b_{max} + e_{max}}{e_{max} \cdot \tan(\varphi_s)} \cdot \cos\left(\frac{\pi}{2} - \varphi_s - \varphi_N\right) \cdot \left(\frac{h_N}{\bar{h}}\right)^{1+n} \tag{20}$$+with:
  
-Out of Equation (19), (20) and (16) one receives:+$$\pi_{geo,GE} = \frac{b_{max} + e_{max}}{e_{max} \cdot \tan(\varphi_s)} \cdot \left(\frac{s_R}{\bar{h}}\right)^{1+n}\tag{124}$$
  
-$$\pi_{\dot{V},channel,dGME} = (A_{DSE,0} + A_{DSE,1} \cdot \pi_{p,z} + A_{DSE,2} \cdot \pi_{p,z}^2 + A_{DSE,3} \cdot \pi_{p,z}^3$$+$$\pi_{geo,N} = \frac{\Delta p \cdot b_{max} + e_{max}}{\Delta z \cdot e_{max} \cdot \tan(\varphi_s)} \cdot \cos\left(\frac{\pi}{2} - \varphi_s - \varphi_N\right) \cdot \left(\frac{h_N}{\bar{h}}\right)^{1+n}\tag{125}$$
  
-$$+A_{DSE,4} \cdot \pi_{p,z}^5) + \pi_1 \cdot (A_{RE,0} + A_{RE,1} \cdot (\pi_{p,z} \cdot \pi_{geo})$$+From the equations, we obtain:
  
-$$+A_{RE,2} \cdot (\pi_{p,z} \cdot \pi_{geo})^2 + A_{RE,3} \cdot (\pi_{p,z} \cdot \pi_{geo})^3 + A_{RE,4} \cdot (\pi_{p,z} \cdot \pi_{geo})^5)(21)$$ +$$\pi_{ \dot V,Kanal,dGME} = \left(A_{DSE,0} + A_{DSE,1} \cdot \pi_{p,z} + A_{DSE,2} \cdot \pi_{p,z}^2 + A_{DSE,3} \cdot \pi_{p,z}^3\right.$$ 
- +$$\left.+A_{DSE,4} \cdot \pi_{p,z}^5\right) + \pi_1 \cdot \left(A_{RE,0} + A_{RE,1} \cdot \left(\pi_{p,z} \cdot \pi_{geo}\right)\right.$$ 
-$$+\pi_2 \cdot (A_{RE,N,0} + A_{RE,N,1} \cdot (\pi_{p,z} \cdot \pi_{geo,N}) + A_{RE,N,2} \cdot (\pi_{p,z} \cdot \pi_{geo,N})^2$$ +$$\left.+A_{RE,2} \cdot \left(\pi_{p,z} \cdot \pi_{geo}\right)^2 + A_{RE,3} \cdot \left(\pi_{p,z} \cdot \pi_{geo}\right)^3 + A_{RE,4} \cdot \left(\pi_{p,z} \cdot \pi_{geo}\right)^5\right)$$ 
- +$$+\pi_2 \cdot \left(A_{RE,N,0} + A_{RE,N,1} \cdot \left(\pi_{p,z} \cdot \pi_{geo,N}\right) + A_{RE,N,2} \cdot \left(\pi_{p,z} \cdot \pi_{geo,N}\right)^2\right.$$ 
-$$+A_{RE,N,3} \cdot (\pi_{p,z} \cdot \pi_{geo,N})^3 + A_{RE,N,4} \cdot (\pi_{p,z} \cdot \pi_{geo,N})^5)$$+$$\left.+A_{RE,N,3} \cdot \left(\pi_{p,z} \cdot \pi_{geo,N}\right)^3 + A_{RE,N,4} \cdot \left(\pi_{p,z} \cdot \pi_{geo,N}\right)^5\right)\tag{126}$$
  
 with with
  
-$$\pi_1 = \frac{(s_R \cdot b_{thread} \cdot v_0)}{k \cdot (\bar{h} \cdot b_{max} \cdot v_0)} \tag{22}$$ +$$\pi_1 = \frac{\left(s_R \cdot b_{Steg} \cdot v_0\right)}{k \cdot \left(\bar{h} \cdot b_{max} \cdot v_0\right)}\tag{127}$$
- +
-$$\pi_2 = \frac{l \cdot (h_N \cdot b_N \cdot v_0)}{k \cdot (\bar{h} \cdot b_{max} \cdot v_0)} \tag{23}$$ +
- +
-the equation to calculate the pressure-throughput behavior of the channel area of threaded elements: +
- +
-$$\pi_{\dot{V},GE} = (A_{DSE,0} + \pi_1 \cdot A_{RE,0} + \pi_2 \cdot A_{RE,N,0})$$ +
- +
-$$+(A_{DSE,1} + \pi_1 \cdot \pi_{geo} \cdot A_{RE,1} + \pi_2 \cdot \pi_{geo,N} \cdot A_{RE,N,1}) \cdot \pi_{p,z}$$+
  
-$$+(A_{DSE,2} + \pi_1 \cdot \pi_{geo}^2 \cdot A_{RE,2} + \pi_2 \cdot \pi_{geo,N}^2 \cdot A_{RE,N,2}) \cdot \pi_{p,z}^2 \tag{24}$$+$$\pi_2 = \frac{l \cdot \left(h_N \cdot b_N \cdot v_0\right)}{k \cdot \left(\bar{h} \cdot b_{max} \cdot v_0\right)}\tag{128}$$
  
-$$+(A_{DSE,3} + \pi_1 \cdot \pi_{geo}^3 \cdot A_{RE,3} + \pi_2 \cdot \pi_{geo,N}^3 \cdot A_{RE,N,3}) \cdot \pi_{p,z}^3$$+the equation for calculating the pressure-flow characteristics of the channel section of threaded elements:
  
-$$+(A_{DSE,4} + \pi_1 \cdot \pi_{geo}^5 \cdot A_{RE,4} + \pi_2 \cdot \pi_{geo,N}^5 \cdot A_{RE,N,4}) \cdot \pi_{p,z}^5$$+$$\pi_{\dot V,GE} = \left(A_{DSE,0} + \pi_1 \cdot A_{RE,0} + \pi_2 \cdot A_{RE,N,0}\right)$$ 
 +$$+\left(A_{DSE,1} + \pi_1 \cdot \pi_{geo} \cdot A_{RE,1} + \pi_2 \cdot \pi_{geo,N} \cdot A_{RE,N,1}\right) \cdot \pi_{p,z}$$ 
 +$$+\left(A_{DSE,2} + \pi_1 \cdot \pi_{geo}^2 \cdot A_{RE,2} + \pi_2 \cdot \pi_{geo,N}^2 \cdot A_{RE,N,2}\right) \cdot \pi_{p,z}^2$$ 
 +$$+\left(A_{DSE,3} + \pi_1 \cdot \pi_{geo}^3 \cdot A_{RE,3} + \pi_2 \cdot \pi_{geo,N}^3 \cdot A_{RE,N,3}\right) \cdot \pi_{p,z}^3$$ 
 +$$+\left(A_{DSE,4} + \pi_1 \cdot \pi_{geo}^5 \cdot A_{RE,4} + \pi_2 \cdot \pi_{geo,N}^5 \cdot A_{RE,N,4}\right) \cdot \pi_{p,z}^5\tag{129}$$
  
 === Intermeshing Zone === === Intermeshing Zone ===
Zeile 1095: Zeile 1083:
 For the description of the throughput behavior in the intermeshing zone it is assumed that the throughput behavior is substantially dominated by a chamber flow. The grooves in the intermeshing zone alone result in an influence of the pressure gradient on the flow. For the description of the throughput behavior in the intermeshing zone it is assumed that the throughput behavior is substantially dominated by a chamber flow. The grooves in the intermeshing zone alone result in an influence of the pressure gradient on the flow.
  
-$$\dot{V}_{intermeshing\,area,dGME} = \overline A_{intermeshing\,area} \cdot t \cdot n_0 + l_{zw} \cdot \pi_{V,intermeshing\,area} \cdot (h_N \cdot b_N \cdot v_{zw}) \tag{25}$$+$$\dot{V}_{intermeshing\,area,dGME} = \overline A_{intermeshing\,area} \cdot t \cdot n_0 + l_{zw} \cdot \pi_{V,intermeshing\,area} \cdot (h_N \cdot b_N \cdot v_{zw}) \tag{130}$$
  
 To specify the groove flow the description of the rectangular channels is applied: To specify the groove flow the description of the rectangular channels is applied:
  
-$$\pi_{\dot{V},intermeshing,dGME} = A_{RE,Z,0} + A_{RE,Z,1} \cdot \pi_{p,zw} + A_{RE,Z,2} \cdot \pi_{p,zw}^2 + A_{RE,Z,3} \cdot \pi_{p,zw}^3 + A_{RE,Z,4} \cdot \pi_{p,zw}^5 \tag{26}$$+$$\pi_{\dot{V},intermeshing,dGME} = A_{RE,Z,0} + A_{RE,Z,1} \cdot \pi_{p,zw} + A_{RE,Z,2} \cdot \pi_{p,zw}^2 + A_{RE,Z,3} \cdot \pi_{p,zw}^3 + A_{RE,Z,4} \cdot \pi_{p,zw}^5 \tag{131}$$
  
-{{ :en:grundlagenhandbuch:en_sigma150_dlg_grundlagenhandbuch_schmelzefoerderung_034.png?nolink |}}+{{ :en:grundlagenhandbuch:en_sigma150_dlg_grundlagenhandbuch_schmelzefoerderung_044.svg?nolink&600 |}}
  
 **Figure:** Axial velocities in the intermeshing zone of a closely intermeshing threaded mixing element. **Figure:** Axial velocities in the intermeshing zone of a closely intermeshing threaded mixing element.
Zeile 1107: Zeile 1095:
 For the linkage of the pressure gradients in the intermeshing zone and in the conveyor channel area it is assumed that the back pressure in the intermeshing zone is negligible and that the same approach can thus be used as above. Consequently, the following equation applies: For the linkage of the pressure gradients in the intermeshing zone and in the conveyor channel area it is assumed that the back pressure in the intermeshing zone is negligible and that the same approach can thus be used as above. Consequently, the following equation applies:
  
-$$\pi_{p,zw} = \frac{\Delta p}{\Delta z} \cdot \frac{b_{max} + e_{max}}{e_{max} \cdot \tan(\varphi_s)} \cdot \cos\left(\frac{\pi}{2} - \varphi_s - \varphi_N\right) \cdot \frac{h_N^{1+n}}{K \cdot v_0^n} = \pi_{geo,zw} \cdot \pi_{p,z} \tag{27}$$+$$\pi_{p,zw} = \frac{\Delta p}{\Delta z} \cdot \frac{b_{max} + e_{max}}{e_{max} \cdot \tan(\varphi_s)} \cdot \cos\left(\frac{\pi}{2} - \varphi_s - \varphi_N\right) \cdot \frac{h_N^{1+n}}{K \cdot v_0^n} = \pi_{geo,zw} \cdot \pi_{p,z} \tag{132}$$
  
 with with
  
-$$\pi_{geo,zw} = \frac{\Delta p}{\Delta z} \cdot \frac{b_{max} + e_{max}}{e_{max} \cdot \tan(\varphi_s)} \cdot \cos\left(\frac{\pi}{2} - \varphi_s - \varphi_N\right) \cdot \left(\frac{h_N}{\bar{h}} \cdot \frac {v_0}{v_{zw}} \right)^{1+n}$$+$$\pi_{geo,zw} = \frac{\Delta p}{\Delta z} \cdot \frac{b_{max} + e_{max}}{e_{max} \cdot \tan(\varphi_s)} \cdot \cos\left(\frac{\pi}{2} - \varphi_s - \varphi_N\right) \cdot \left(\frac{h_N}{\bar{h}} \cdot \frac {v_0}{v_{zw}} \right)^{1+n}\tag{133}$$
  
 === Linkage of the Models === === Linkage of the Models ===
Zeile 1121: Zeile 1109:
 $$+(A_{DSE,1} + \pi_1 \cdot \pi_{geo} \cdot A_{RE,1} + \pi_2 \cdot \pi_{geo,N} \cdot A_{RE,N,1} + \pi_3 \cdot \pi_{geo,zw} \cdot A_{RE,Z,1}) \cdot \pi_{p,z}$$ $$+(A_{DSE,1} + \pi_1 \cdot \pi_{geo} \cdot A_{RE,1} + \pi_2 \cdot \pi_{geo,N} \cdot A_{RE,N,1} + \pi_3 \cdot \pi_{geo,zw} \cdot A_{RE,Z,1}) \cdot \pi_{p,z}$$
  
-$$+(A_{DSE,2} + \pi_1 \cdot \pi_{geo}^2 \cdot A_{RE,2} + \pi_2 \cdot \pi_{geo,N}^2 \cdot A_{RE,N,2} + \pi_3 \cdot \pi_{geo,zw}^2 \cdot A_{RE,Z,2}) \cdot \pi_{p,z}^2 \tag{29}$$+$$+(A_{DSE,2} + \pi_1 \cdot \pi_{geo}^2 \cdot A_{RE,2} + \pi_2 \cdot \pi_{geo,N}^2 \cdot A_{RE,N,2} + \pi_3 \cdot \pi_{geo,zw}^2 \cdot A_{RE,Z,2}) \cdot \pi_{p,z}^2$$
  
 $$+(A_{DSE,3} + \pi_1 \cdot \pi_{geo}^3 \cdot A_{RE,3} + \pi_2 \cdot \pi_{geo,N}^3 \cdot A_{RE,N,3} + \pi_3 \cdot \pi_{geo,zw}^3 \cdot A_{RE,Z,3}) \cdot \pi_{p,z}^3$$ $$+(A_{DSE,3} + \pi_1 \cdot \pi_{geo}^3 \cdot A_{RE,3} + \pi_2 \cdot \pi_{geo,N}^3 \cdot A_{RE,N,3} + \pi_3 \cdot \pi_{geo,zw}^3 \cdot A_{RE,Z,3}) \cdot \pi_{p,z}^3$$
  
-$$+(A_{DSE,4} + \pi_1 \cdot \pi_{geo}^5 \cdot A_{RE,4} + \pi_2 \cdot \pi_{geo,N}^5 \cdot A_{RE,N,4} + \pi_3 \cdot \pi_{geo,zw}^5 \cdot A_{RE,Z,4}) \cdot \pi_{p,z}^5$$+$$+(A_{DSE,4} + \pi_1 \cdot \pi_{geo}^5 \cdot A_{RE,4} + \pi_2 \cdot \pi_{geo,N}^5 \cdot A_{RE,N,4} + \pi_3 \cdot \pi_{geo,zw}^5 \cdot A_{RE,Z,4}) \cdot \pi_{p,z}^5\tag{134}$$
  
 with with
  
-$$\pi_3 = \frac{l_{zw} \cdot (h_N \cdot b_N \cdot v_{zw})}{k \cdot (\bar{h} \cdot b_{max} \cdot v_0)} \tag{30}$$+$$\pi_3 = \frac{l_{zw} \cdot (h_N \cdot b_N \cdot v_{zw})}{k \cdot (\bar{h} \cdot b_{max} \cdot v_0)} \tag{135}$$
  
 The figure displays a comparison of the measured dimensionless pressure gradient and those calculated by means of the model for a conveying and for a reconveying threaded mixing element. Despite the multiplicity of simplifications it can be observed that the model is in a position to describe the experimental results with sufficient accurateness. The figure displays a comparison of the measured dimensionless pressure gradient and those calculated by means of the model for a conveying and for a reconveying threaded mixing element. Despite the multiplicity of simplifications it can be observed that the model is in a position to describe the experimental results with sufficient accurateness.
  
-{{ :en:grundlagenhandbuch:en_sigma150_dlg_grundlagenhandbuch_schmelzefoerderung_035.png?nolink |}}+{{ :en:grundlagenhandbuch:en_sigma150_dlg_grundlagenhandbuch_schmelzefoerderung_045.svg?nolink&600 |}}
  
 **Figure:** Comparison of the experimentally investigated and the calculated dimensionless pressure gradients. **Figure:** Comparison of the experimentally investigated and the calculated dimensionless pressure gradients.
Zeile 1143: Zeile 1131:
 When examining the flow in the intermeshing zone of these elements it directly attracts the attention that the axial velocities strongly deviate along the element pair. In those areas in the figure marked with an "a" a maximum axial velocity of about +40 mm/s can be identified and in those marked with a "b" an axial velocity of about -40 mm/s. Out of this the simplification can be deduced that the sum of the volume flow rates in two sequenced intermeshing zones approximately equals zero. When examining the flow in the intermeshing zone of these elements it directly attracts the attention that the axial velocities strongly deviate along the element pair. In those areas in the figure marked with an "a" a maximum axial velocity of about +40 mm/s can be identified and in those marked with a "b" an axial velocity of about -40 mm/s. Out of this the simplification can be deduced that the sum of the volume flow rates in two sequenced intermeshing zones approximately equals zero.
  
-{{ :en:grundlagenhandbuch:en_sigma150_dlg_grundlagenhandbuch_schmelzefoerderung_036.png?nolink |}}+{{ :en:grundlagenhandbuch:en_sigma150_dlg_grundlagenhandbuch_schmelzefoerderung_046.svg?nolink&600 |}}
  
 **Figure:** Distribution of the axial velocities in non-intermeshing threaded mixing elements. **Figure:** Distribution of the axial velocities in non-intermeshing threaded mixing elements.
Zeile 1151: Zeile 1139:
 The unwinding of the non-intermeshing threaded mixing element into the section results in the conveyor channel model as depicted in the figure. The unwinding of the non-intermeshing threaded mixing element into the section results in the conveyor channel model as depicted in the figure.
  
-{{ :en:grundlagenhandbuch:en_sigma150_dlg_grundlagenhandbuch_schmelzefoerderung_037.png?nolink |}}+{{ :en:grundlagenhandbuch:en_sigma150_dlg_grundlagenhandbuch_schmelzefoerderung_047.svg?nolink&600 |}}
  
 **Figure:** Conveyor channel model for non-intermeshing threaded mixing elements. **Figure:** Conveyor channel model for non-intermeshing threaded mixing elements.
Zeile 1157: Zeile 1145:
 The balance of the volume flow rates in the control room ABCD yields: The balance of the volume flow rates in the control room ABCD yields:
  
-$$\dot{V}_{tGME} = (k \cdot \dot{V}_{z,fe} + \dot{V}_{x,fe} + l \cdot \dot{V}_{groove,fe}) + (k \cdot \dot{V}_{z,rfe} + \dot{V}_{x,rfe} + l \cdot \dot{V}_{groove,rfe}) \tag{1}$$+$$\dot{V}_{tGME} = (k \cdot \dot{V}_{z,fe} + \dot{V}_{x,fe} + l \cdot \dot{V}_{groove,fe}) + (k \cdot \dot{V}_{z,rfe} + \dot{V}_{x,rfe} + l \cdot \dot{V}_{groove,rfe}) \tag{136}$$
  
 with with
  
-$$\dot{V}_{z,fe} = \pi_{\dot V,z,fe} \cdot (\bar{h} \cdot b_{max} \cdot v_0) \tag{2}$$+$$\dot{V}_{z,fe} = \pi_{\dot V,z,fe} \cdot (\bar{h} \cdot b_{max} \cdot v_0) \tag{137}$$
  
-$$\dot{V}_{x,fe} = \pi_{\dot V,x,fe} \cdot (s_R \cdot b_{thread} \cdot v_0) \tag{3}$$+$$\dot{V}_{x,fe} = \pi_{\dot V,x,fe} \cdot (s_R \cdot b_{thread} \cdot v_0) \tag{138}$$
  
-$$\dot{V}_{groove,fe} = \pi_{\dot V,N,fe} \cdot (h_N \cdot b_N \cdot v_0) \tag{4}$$+$$\dot{V}_{groove,fe} = \pi_{\dot V,N,fe} \cdot (h_N \cdot b_N \cdot v_0) \tag{139}$$
  
-$$\dot{V}_{z,rfe} = \pi_{\dot V,z,rfe} \cdot (\bar{h} \cdot b_{max} \cdot v_0) \tag{5}$$+$$\dot{V}_{z,rfe} = \pi_{\dot V,z,rfe} \cdot (\bar{h} \cdot b_{max} \cdot v_0) \tag{140}$$
  
-$$\dot{V}_{x,rfe} = \pi_{\dot V,x,rfe} \cdot (s_R \cdot b_{thread} \cdot v_0) \tag{6}$$+$$\dot{V}_{x,rfe} = \pi_{\dot V,x,rfe} \cdot (s_R \cdot b_{thread} \cdot v_0) \tag{141}$$
  
-$$\dot{V}_{groove,rfe} = \pi_{\dot V,N,rfe} \cdot (h_N \cdot b_N \cdot v_0) \tag{7}$$+$$\dot{V}_{groove,rfe} = \pi_{\dot V,N,rfe} \cdot (h_N \cdot b_N \cdot v_0) \tag{142}$$
  
 The dimensionless volume flow rates are defined as follows: The dimensionless volume flow rates are defined as follows:
  
-$$\pi_{\dot{V},z,fe} = A_{DSE,fe,0} + A_{DSE,fe,1} \cdot \pi_{p,z,fe} + A_{DSE,fe,2} \cdot \pi_{p,z,fe}^2 \tag{8}$$+$$\pi_{\dot{V},z,fe} = A_{DSE,fe,0} + A_{DSE,fe,1} \cdot \pi_{p,z,fe} + A_{DSE,fe,2} \cdot \pi_{p,z,fe}^2$$
  
-$$+A_{DSE,fe,3} \cdot \pi_{p,z,fe}^3 + A_{DSE,fe,4} \cdot \pi_{p,z,fe}^5$$+$$+A_{DSE,fe,3} \cdot \pi_{p,z,fe}^3 + A_{DSE,fe,4} \cdot \pi_{p,z,fe}^5\tag{143}$$
  
-$$\pi_{\dot{V},x,fe} = A_{RE,fe,0} + A_{RE,fe,1} \cdot \pi_{p,x,fe} + A_{RE,fe,2} \cdot \pi_{p,x,fe}^2 \tag{9}$$+$$\pi_{\dot{V},x,fe} = A_{RE,fe,0} + A_{RE,fe,1} \cdot \pi_{p,x,fe} + A_{RE,fe,2} \cdot \pi_{p,x,fe}^2$$
  
-$$+A_{RE,fe,3} \cdot \pi_{p,x,fe}^3 + A_{RE,fe,4} \cdot \pi_{p,x,fe}^5$$+$$+A_{RE,fe,3} \cdot \pi_{p,x,fe}^3 + A_{RE,fe,4} \cdot \pi_{p,x,fe}^5\tag{144}$$
  
-$$\pi_{\dot{V},N,fe} = A_{RE,N,fe,0} + A_{RE,N,fe,1} \cdot \pi_{p,N} + A_{RE,N,fe,2} \cdot \pi_{p,N}^2 \tag{10}$$+$$\pi_{\dot{V},N,fe} = A_{RE,N,fe,0} + A_{RE,N,fe,1} \cdot \pi_{p,N} + A_{RE,N,fe,2} \cdot \pi_{p,N}^2$$
  
-$$+A_{RE,N,fe,3} \cdot \pi_{p,N}^3 + A_{RE,N,fe,4} \cdot \pi_{p,N}^5$$+$$+A_{RE,N,fe,3} \cdot \pi_{p,N}^3 + A_{RE,N,fe,4} \cdot \pi_{p,N}^5\tag{145}$$
  
-$$\pi_{\dot{V},z,rfe} = A_{DSE,rfe,0} + A_{DSE,rfe,1} \cdot \pi_{p,z,rfe} + A_{DSE,rfe,2} \cdot \pi_{p,z,rfe}^2 \tag{11}$$+$$\pi_{\dot{V},z,rfe} = A_{DSE,rfe,0} + A_{DSE,rfe,1} \cdot \pi_{p,z,rfe} + A_{DSE,rfe,2} \cdot \pi_{p,z,rfe}^2$$
  
-$$+A_{DSE,rfe,3} \cdot \pi_{p,z,rfe}^3 + A_{DSE,rfe,4} \cdot \pi_{p,z,rfe}^5$$+$$+A_{DSE,rfe,3} \cdot \pi_{p,z,rfe}^3 + A_{DSE,rfe,4} \cdot \pi_{p,z,rfe}^5\tag{146}$$
  
-$$\pi_{\dot{V},x,rfe} = A_{RE,rfe,0} + A_{RE,rfe,1} \cdot \pi_{p,x,rfe} + A_{RE,rfe,2} \cdot \pi_{p,x,rfe}^2 \tag{12}$$+$$\pi_{\dot{V},x,rfe} = A_{RE,rfe,0} + A_{RE,rfe,1} \cdot \pi_{p,x,rfe} + A_{RE,rfe,2} \cdot \pi_{p,x,rfe}^2$$
  
-$$+A_{RE,rfe,3} \cdot \pi_{p,x,rfe}^3 + A_{RE,rfe,4} \cdot \pi_{p,x,rfe}^5$$+$$+A_{RE,rfe,3} \cdot \pi_{p,x,rfe}^3 + A_{RE,rfe,4} \cdot \pi_{p,x,rfe}^5\tag{147}$$
  
-$$\pi_{\dot{V},N,rfe} = A_{RE,N,rfe,0} + A_{RE,N,rfe,1} \cdot \pi_{p,N,rfe} + A_{RE,N,rfe,2} \cdot \pi_{p,N,rfe}^2 \tag{13}$$+$$\pi_{\dot{V},N,rfe} = A_{RE,N,rfe,0} + A_{RE,N,rfe,1} \cdot \pi_{p,N,rfe} + A_{RE,N,rfe,2} \cdot \pi_{p,N,rfe}^2$$
  
-$$+A_{RE,N,rfe,3} \cdot \pi_{p,N,rfe}^3 + A_{RE,N,rfe,4} \cdot \pi_{p,N,rfe}^5$$+$$+A_{RE,N,rfe,3} \cdot \pi_{p,N,rfe}^3 + A_{RE,N,rfe,4} \cdot \pi_{p,N,rfe}^5\tag{148}$$
  
 with the parameters defined in the table. with the parameters defined in the table.
Zeile 1203: Zeile 1191:
 Simplifying assumed that the isobars operate at right angle to the screw axis the following equation applies: Simplifying assumed that the isobars operate at right angle to the screw axis the following equation applies:
  
-$$\frac{\Delta p_{fe}}{\Delta z} = \frac{\Delta p_{rfe}}{\Delta z} \Leftrightarrow \pi_{p,z,fe} = \pi_{p,z,rfe} \tag{14}$$+$$\frac{\Delta p_{fe}}{\Delta z} = \frac{\Delta p_{rfe}}{\Delta z} \Leftrightarrow \pi_{p,z,fe} = \pi_{p,z,rfe} \tag{149}$$
  
 The pressure gradients in channel direction and at right angle to it are again linked in an analogous manner to the approaches introduced so far. The pressure gradients in channel direction and at right angle to it are again linked in an analogous manner to the approaches introduced so far.
  
-$$\Delta p_x = \frac{\Delta p}{\Delta z} \cdot \frac{b_{max} + e_{max}}{\tan(\varphi_s)} \tag{15}$$+$$\Delta p_x = \frac{\Delta p}{\Delta z} \cdot \frac{b_{max} + e_{max}}{\tan(\varphi_s)} \tag{150}$$
  
 ** Dimensionless parameters for the description of the pressure-throughput behavior of non-intermeshing threaded mixing elements: ** ** Dimensionless parameters for the description of the pressure-throughput behavior of non-intermeshing threaded mixing elements: **
Zeile 1221: Zeile 1209:
 Consequently, the dimensionless factors can also be defined for the linkage of the volume flow rates: Consequently, the dimensionless factors can also be defined for the linkage of the volume flow rates:
  
-$$\pi_{p,x,fe} = \frac{\Delta p}{\Delta z} \cdot \frac{b_{max} + e_{max}}{e_{max} \cdot \tan(\varphi_s)} \cdot \frac{s_R^{1+n}}{K \cdot v_0^n} = \pi_{geo,x} \cdot \pi_{p,z} \tag{17}$$+$$\pi_{p,x,fe} = \frac{\Delta p}{\Delta z} \cdot \frac{b_{max} + e_{max}}{e_{max} \cdot \tan(\varphi_s)} \cdot \frac{s_R^{1+n}}{K \cdot v_0^n} = \pi_{geo,x} \cdot \pi_{p,z} \tag{151}$$
  
-$$\pi_{p,N,fe} = \frac{\Delta p}{\Delta z} \cdot \frac{b_{max} + e_{max}}{e_{max} \cdot \tan(\varphi_s)} \cdot \cos\left(\frac{\pi}{2} - \varphi_s - \varphi_N\right) \cdot \frac{h_N^{1+n}}{K \cdot v_0^n} = \pi_{geo,N} \cdot \pi_{p,z} \tag{18}$$+$$\pi_{p,N,fe} = \frac{\Delta p}{\Delta z} \cdot \frac{b_{max} + e_{max}}{e_{max} \cdot \tan(\varphi_s)} \cdot \cos\left(\frac{\pi}{2} - \varphi_s - \varphi_N\right) \cdot \frac{h_N^{1+n}}{K \cdot v_0^n} = \pi_{geo,N} \cdot \pi_{p,z} \tag{152}$$
  
-$$\pi_{p,x,rfe} = \frac{\Delta p}{\Delta z} \cdot \frac{b_{max} + e_{max}}{e_{max} \cdot \tan(\varphi_s)} \cdot \frac{s_R^{1+n}}{K \cdot v_0^n} = \pi_{geo,x} \cdot \pi_{p,z} \tag{19}$$+$$\pi_{p,x,rfe} = \frac{\Delta p}{\Delta z} \cdot \frac{b_{max} + e_{max}}{e_{max} \cdot \tan(\varphi_s)} \cdot \frac{s_R^{1+n}}{K \cdot v_0^n} = \pi_{geo,x} \cdot \pi_{p,z} \tag{153}$$
  
-$$\pi_{p,N,rfe} = \frac{\Delta p}{\Delta z} \cdot \frac{b_{max} + e_{max}}{e_{max} \cdot \tan(\varphi_s)} \cdot \cos\left(\frac{\pi}{2} - \varphi_s - \varphi_N\right) \cdot \frac{h_N^{1+n}}{K \cdot v_0^n} = \pi_{geo,N} \cdot \pi_{p,z}$$+$$\pi_{p,N,rfe} = \frac{\Delta p}{\Delta z} \cdot \frac{b_{max} + e_{max}}{e_{max} \cdot \tan(\varphi_s)} \cdot \cos\left(\frac{\pi}{2} - \varphi_s - \varphi_N\right) \cdot \frac{h_N^{1+n}}{K \cdot v_0^n} = \pi_{geo,N} \cdot \pi_{p,z}\tag{154}$$
  
 with: with:
  
-$$\pi_{geo,x} = \frac{b_{max} + e_{max}}{e_{max} \cdot \tan(\varphi_s)} \cdot \left(\frac{s_R}{\bar{h}}\right)^{1+n} \tag{20}$$+$$\pi_{geo,x} = \frac{b_{max} + e_{max}}{e_{max} \cdot \tan(\varphi_s)} \cdot \left(\frac{s_R}{\bar{h}}\right)^{1+n} \tag{155}$$
  
-$$\pi_{geo,N} = \frac{\Delta p}{\Delta z} \cdot \frac{b_{max} + e_{max}}{e_{max} \cdot \tan(\varphi_s)} \cdot \cos\left(\frac{\pi}{2} - \varphi_s - \varphi_N\right) \cdot \left(\frac{h_N}{\bar{h}}\right)^{1+n} \tag{21}$$+$$\pi_{geo,N} = \frac{\Delta p}{\Delta z} \cdot \frac{b_{max} + e_{max}}{e_{max} \cdot \tan(\varphi_s)} \cdot \cos\left(\frac{\pi}{2} - \varphi_s - \varphi_N\right) \cdot \left(\frac{h_N}{\bar{h}}\right)^{1+n} \tag{156}$$
  
-Are these equations inserted into the balance (Eq. 1), one receives the model to specify the pressure-throughput behavior in the conveyor channel area:+Are these equations inserted into the balance, one receives the model to specify the pressure-throughput behavior in the conveyor channel area:
  
 $$\pi_{\dot{V},tGME} = \frac{\dot{V}_{tdGME}}{k \cdot (\bar{h} \cdot b_{max} \cdot v_0)} = (A_{DSE,fe,0} + A_{DSE,rfe,0} + \pi_1 \cdot (A_{RE,fe,0} + A_{RE,rfe,0}))$$ $$\pi_{\dot{V},tGME} = \frac{\dot{V}_{tdGME}}{k \cdot (\bar{h} \cdot b_{max} \cdot v_0)} = (A_{DSE,fe,0} + A_{DSE,rfe,0} + \pi_1 \cdot (A_{RE,fe,0} + A_{RE,rfe,0}))$$
Zeile 1251: Zeile 1239:
 $$+(A_{DSE,fe,4} + A_{DSE,rfe,4} + \pi_1 \cdot \pi_{geo,x}^5 \cdot (A_{RE,fe,4} + A_{RE,rfe,4})$$ $$+(A_{DSE,fe,4} + A_{DSE,rfe,4} + \pi_1 \cdot \pi_{geo,x}^5 \cdot (A_{RE,fe,4} + A_{RE,rfe,4})$$
  
-$$+\pi_2 \cdot \pi_{geo,N}^5 \cdot (A_{RE,N,fe,4} + A_{RE,N,rfe,4})) \cdot \pi_{p,z}^5 \tag{22}$$+$$+\pi_2 \cdot \pi_{geo,N}^5 \cdot (A_{RE,N,fe,4} + A_{RE,N,rfe,4})) \cdot \pi_{p,z}^5 \tag{157}$$
  
 The comparison of the model predictions with the experimental results for distinct polymers confirms, that the model yields sufficient precision despite its multitude of simplifications. The comparison of the model predictions with the experimental results for distinct polymers confirms, that the model yields sufficient precision despite its multitude of simplifications.
  
-{{ :en:grundlagenhandbuch:en_sigma150_dlg_grundlagenhandbuch_schmelzefoerderung_038.png?nolink |}}+{{ :en:grundlagenhandbuch:en_sigma150_dlg_grundlagenhandbuch_schmelzefoerderung_048.svg?nolink&600 |}}
  
 **Figure:** Comparison of the calculated and the measured dimensionless pressure gradients for a non-intermeshing threaded mixing element. **Figure:** Comparison of the calculated and the measured dimensionless pressure gradients for a non-intermeshing threaded mixing element.
Zeile 1266: Zeile 1254:
   * The flow in the grooves can be described sufficiently accurate with a flow in the rectangular channels.   * The flow in the grooves can be described sufficiently accurate with a flow in the rectangular channels.
  
-{{ :grundlagenhandbuch:schmelzefoerderung:regressionsansatz:de_sigma150_dlg_grundlagenhandbuch_schmelzefoerderung_049.png?nolink |}}+{{ :en:grundlagenhandbuch:en_sigma150_dlg_grundlagenhandbuch_schmelzefoerderung_049.svg?nolink&600 |}}
  
 **Figure:** Real turbine mixing elements geometry and the geometrical substitution model. **Figure:** Real turbine mixing elements geometry and the geometrical substitution model.
Zeile 1272: Zeile 1260:
 The description of the process behavior is based upon the superimposition of the volume flow rates in the grooves and in the disk area. The description of the process behavior is based upon the superimposition of the volume flow rates in the grooves and in the disk area.
  
-$$\dot{V}_{ZE} = \dot{V}_{disc} + l_N \cdot \dot{V}_{groove} \tag{1}$$+$$\dot{V}_{ZE} = \dot{V}_{disc} + l_N \cdot \dot{V}_{groove} \tag{158}$$
  
 To specify the volume flow rate the approach is applied as introduced in chapter Pressure-Throughput Model for Threaded Elements: To specify the volume flow rate the approach is applied as introduced in chapter Pressure-Throughput Model for Threaded Elements:
  
-$$\dot{V}_{disc} = A_{free,SE} \cdot D_a \cdot n_0 \cdot (A_{SE,1} \cdot \pi_{p,SE} + A_{SE,2} \cdot \pi_{p,SE}^2 + A_{SE,3} \cdot \pi_{p,SE}^5) \tag{2}$$+$$\dot{V}_{disc} = A_{free,SE} \cdot D_a \cdot n_0 \cdot (A_{SE,1} \cdot \pi_{p,SE} + A_{SE,2} \cdot \pi_{p,SE}^2 + A_{SE,3} \cdot \pi_{p,SE}^5) \tag{59}$$
  
 The modeling of the throughput behavior in the grooves is performed by virtue of the approaches presented in chapter Pressure-Throughput Model for Threaded Elements: The modeling of the throughput behavior in the grooves is performed by virtue of the approaches presented in chapter Pressure-Throughput Model for Threaded Elements:
  
-$$\dot{V}_{groove} = h_N \cdot b_N \cdot v_0 \cdot (A_{RE,0} + A_{RE,1} \cdot \pi_{p,N} + A_{RE,2} \cdot \pi_{p,N}^2 + A_{RE,3} \cdot \pi_{p,N}^3 + A_{RE,4} \cdot \pi_{p,N}^5) \tag{3}$$+$$\dot{V}_{groove} = h_N \cdot b_N \cdot v_0 \cdot (A_{RE,0} + A_{RE,1} \cdot \pi_{p,N} + A_{RE,2} \cdot \pi_{p,N}^2 + A_{RE,3} \cdot \pi_{p,N}^3 + A_{RE,4} \cdot \pi_{p,N}^5) \tag{160}$$
  
 Out of this results the following equation to describe the pressure-throughput behavior of toothed mixing elements: Out of this results the following equation to describe the pressure-throughput behavior of toothed mixing elements:
  
-$$\pi_{\dot V,ZE} = A_{ZE,0} + A_{ZE,1} \cdot \pi_p + A_{ZE,2} \cdot \pi_p^2 + A_{ZE,3} \cdot \pi_p^3 + A_{ZE,4} \cdot \pi_p^5 \tag{4}$$+$$\pi_{\dot V,ZE} = A_{ZE,0} + A_{ZE,1} \cdot \pi_p + A_{ZE,2} \cdot \pi_p^2 + A_{ZE,3} \cdot \pi_p^3 + A_{ZE,4} \cdot \pi_p^5 \tag{161}$$
  
 with: with:
  
-$$\pi_{\dot V,ZE} = \frac{\dot{V}_{ZE}}{A_{fr} \cdot D_a \cdot n_0} \tag{5}$$+$$\pi_{\dot V,ZE} = \frac{\dot{V}_{ZE}}{A_{fr} \cdot D_a \cdot n_0} \tag{162}$$
  
-$$\pi_p = \frac{\Delta p}{L} \cdot \frac{D_a}{K \cdot n_0^n} \tag{6}$$+$$\pi_p = \frac{\Delta p}{L} \cdot \frac{D_a}{K \cdot n_0^n} \tag{163}$$
  
 and and
  
-$$A_{ZE,0} = A_{RE,0} \cdot i_N \cdot \pi_{geo,V} \tag{7}$$+$$A_{ZE,0} = A_{RE,0} \cdot i_N \cdot \pi_{geo,V} \tag{164}$$
  
-$$A_{ZE,1} = A_{SE,1} + A_{RE,1} \cdot i_N \cdot \pi_{geo,V} \cdot \pi_{geo,p} \tag{8}$$+$$A_{ZE,1} = A_{SE,1} + A_{RE,1} \cdot i_N \cdot \pi_{geo,V} \cdot \pi_{geo,p} \tag{165}$$
  
-$$A_{ZE,2} = A_{RE,2} \cdot i_N \cdot \pi_{geo,V} \cdot \pi_{geo,p}^2 \tag{9}$$+$$A_{ZE,2} = A_{RE,2} \cdot i_N \cdot \pi_{geo,V} \cdot \pi_{geo,p}^2 \tag{166}$$
  
-$$A_{ZE,3} = A_{SE,2} + A_{RE,3} \cdot i_N \cdot \pi_{geo,V} \cdot \pi_{geo,p}^3 \tag{10}$$+$$A_{ZE,3} = A_{SE,2} + A_{RE,3} \cdot i_N \cdot \pi_{geo,V} \cdot \pi_{geo,p}^3 \tag{167}$$
  
-$$A_{ZE,4} = A_{SE,3} + A_{RE,4} \cdot i_N \cdot \pi_{geo,V} \cdot \pi_{geo,p}^5 \tag{11}$$+$$A_{ZE,4} = A_{SE,3} + A_{RE,4} \cdot i_N \cdot \pi_{geo,V} \cdot \pi_{geo,p}^5 \tag{168}$$
  
 The two factors $\pi_{geo,V}$ and $\pi_{geo,p}$ are both necessitated for the linkage of the two volume flow rates. They are defined as: The two factors $\pi_{geo,V}$ and $\pi_{geo,p}$ are both necessitated for the linkage of the two volume flow rates. They are defined as:
  
-$$\pi_{geo,V} = \frac{h_N \cdot b_N \cdot v_0}{A_{free} \cdot D_a \cdot n_0} \tag{12}$$+$$\pi_{geo,V} = \frac{h_N \cdot b_N \cdot v_0}{A_{free} \cdot D_a \cdot n_0} \tag{169}$$
  
 and: and:
  
-$$\pi_{geo,p} = \sin(\varphi_{groove}) \cdot \pi^n \cdot \frac{D_a^{1+n}}{h^{1+n}} \tag{13}$$+$$\pi_{geo,p} = \sin(\varphi_{groove}) \cdot \pi^n \cdot \frac{D_a^{1+n}}{h^{1+n}} \tag{170}$$
  
 The figure shows a comparison of the model predictions with the experimental results. It can be recognized that the model is in a position to describe the experimental results with satisfactory precision. The figure shows a comparison of the model predictions with the experimental results. It can be recognized that the model is in a position to describe the experimental results with satisfactory precision.
  
-{{ :en:grundlagenhandbuch:en_sigma150_dlg_grundlagenhandbuch_schmelzefoerderung_039.png?nolink |}}+{{ :en:grundlagenhandbuch:en_sigma150_dlg_grundlagenhandbuch_schmelzefoerderung_050.svg?nolink&600 |}}
  
 **Figure:** Comparison of experimental and approximated dimensionless pressure gradients of conveying turbine mixing elements. **Figure:** Comparison of experimental and approximated dimensionless pressure gradients of conveying turbine mixing elements.