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en:grundlagenhandbuch:schmelzefoerderung [2026/05/24 17:04] – [Pressure - Throughput Model for Turbine Mixing 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 975: Zeile 967:
 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 983: Zeile 975:
 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. 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 1097: Zeile 1089:
 $$\pi_{\dot{V},intermeshing,dGME} = A_{RE,Z,0} + A_{RE,Z,1} \cdot \pi_{p,zw} + A_{RE,Z,2} \cdot \pi_{p,zw}^2 + A_{RE,Z,3} \cdot \pi_{p,zw}^3 + A_{RE,Z,4} \cdot \pi_{p,zw}^5 \tag{131}$$ $$\pi_{\dot{V},intermeshing,dGME} = A_{RE,Z,0} + A_{RE,Z,1} \cdot \pi_{p,zw} + A_{RE,Z,2} \cdot \pi_{p,zw}^2 + A_{RE,Z,3} \cdot \pi_{p,zw}^3 + A_{RE,Z,4} \cdot \pi_{p,zw}^5 \tag{131}$$
  
-{{ :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 1129: Zeile 1121:
 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 1139: 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 1147: 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 1251: Zeile 1243:
 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 1262: 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 1310: Zeile 1302:
 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.