Unterschiede

Hier werden die Unterschiede zwischen zwei Versionen angezeigt.

--- en:berechnungen:leistung_und_schubspannungen [2024/10/30 13:17] – neelest
+++ en:berechnungen:leistung_und_schubspannungen [2025/07/03 13:34] (aktuell) – cschall
@@ Zeile 1: / Zeile 1: @@
 ======Power and wall shear stress======
-The same conditions have to be filled for the calculation of the power and wall shear
+-> [[en:grafische_darstellung_der_ergebnisse:leistung|For a graphical representation of the performance curve]]
-stress as for the calculation of the melting profile.
-**Realization of the theoretical principles**
+Three models can be selected for power calculation in REX/PSI:
+  * Isothermal performance model
+  * Non-isothermal performance model //(recommended model for melt extruders)//
+  * Enthalpy model //(recommended model for plasticising extruders)//
-In the power calculation a distinction is drawn between the incipient fusing and the
+All models are based on the results of the throughput, melting and temperature calculations. The enthalpy model also requires the pressure curve.\\
-melting section. In the incipient fusing section, i. e. from the start of the first heating
+The accuracy of the power calculation therefore depends on the other curves. If experimental values are already known (e.g. flow rate), it is advisable to [[en:eingabe_der_verfahrensparameter:eingabe_der_verfahrensparameter|specify these for the power calculation]].
-section through to the point of melt pool formation, the power can be established via
-an average melt layer thickness, assuming a pure drag flow in the melt film. The
-power conversion in the pure solids section between the hopper and the first heating
-section is neglected, since no pressure has built up till then and the frictional forces of
-the solids at the barrel wall are negligibly small compared to the frictional forces of
-the melt film.
-The calculation for the other sections results in an addition of the wall shear stresses
+For an accurate power calculation, the [[en:eingabe_der_zylinderdaten:eingabe_der_zylinderdaten|cylinder inner wall temperatures]] must be specified in a practical manner.
-at the barrel. Similar to the calculation of the pressure throughput approximation
-equations are used, which are based on numeric examinations. A distinction is drawn
-between the melting section, where high wall shear stresses in the melt film occur
-and the pure melt conveying.
-The overall drive power required (without transmission losses) and the overall torque
+The temperature control power is calculated in the same way for all models and is made up of the melt film and the melt pool. The temperature gradient on the cylinder wall is determined for both areas and the required temperature heating/cooling power is determined from this using an energy balance. The resulting outputs are
-is calculated from the local power requirement.
+[[en:grafische_darstellung_der_ergebnisse:leistung|summarised for the respective heating zones]]. The power calculation only takes into account the heat flow between the inner cylinder wall and the melt, but not heat losses (of any kind) to the outside.
-The heating/cooling power consists of components from the melt film and the melt
+The following models calculate the drive power and the required torque of the extruder. Power losses (e.g. in the gearbox) are not taken into account. \\
-pool. The temperature gradient at the barrel wall is established for both sections and
+The total power of the extruder is made up of the drive power and the **heating** heating zones.
-from this the necessary heating/cooling power is determined via a temperature
-balance. The power at each single calculation point is added up for the heating
+===== Isothermal power model =====
-section in question. The heating/cooling power calculation only allows for the heating
-flow between the band heater and the melt, but not for any heat losses (of any type)
+When calculating the power, a distinction is made between the melting-delay and melting area. In the melting-delay area, i.e. from the start of the first heating zone to the location of the melt pool formation, the power can be determined via an average melt layer thickness assuming a pure drag flow in the melt film. The power conversion in the pure solids area between the hopper and the first heating zone is neglected, as no pressure has yet built up there and the solids friction forces on the cylinder wall are negligibly small compared to the melt film friction forces.
-towards the outside.
+The calculation for the further zones results in a summation of the wall shear stresses on the cylinder. Approximation equations based on numerical analyses are used here. A distinction is made here between the melting area, where high shear stresses occur in the melt film, and pure melt conveying.
+<details><summary>Sources</summary>
+  * Obermann, Christian: Theoretische und experimentelle Untersuchungen zum Durchsatz- und Leistungsverhalten von Glattrohr-Plastifiziereinheiten. Dissertation, University of Paderborn, 2000
+  * Potente, H., Obermann, C.: Screw Drive Power of Single Screw Plasticating Units With Smooth Barrels. International Polymer Processing, Vol. 14, no. 1, 1999, pp. 21-27
+</details>
+===== Non-isothermal power model =====
+The non-isothermal power model is an extension of the isothermal power model. \\
+The model is based on an extensive test plan of non-isothermal FEM flow simulations. The simulation results were analysed and regressed in relation to the drive power and implemented in the theoretical model approach of the isothermal model.
+//The non-isothermal performance model is recommended for melt extruders.//
+<details><summary>Sources</summary>
+  * Bornemann, Markus: Erweiterung der modelltheoretischen Grundlagen zur Durchsatz- und Leistungsberechnung von Einschneckenplastifiziereinheiten. Dissertation, University of Paderborn, 2011
+</details>
+===== Enthalpy power model =====
+The enthalpy model is a fundamentally different approach and only takes into account the conservation of energy (1st law of thermodynamics).
+Accordingly, the following applies:
+$P_{total} = P_{drive} + \dot{Q} = \dot{m}*\Delta h$
+and therefore also:
+$P_{drive} = \Delta h_{temperature} * \dot{m}_{melt} + \Delta p * \dot{V} - \dot{Q}$
+//The enthalpy model is recommended for plating extruders.//
+===== Calculation of the heat flows =====
+In order to calculate the drive power, the energy supply and cooling through the cylinder temperature control must also be known. For this reason, the heat flows along the screw are calculated. The calculation of the heat flow is divided into 2 areas: The heat flow due to forced convection of the melt in the melt pool as well as in fully melt-filled channels ($Q_{SW}$) and the heat flow over the melt film ($Q_{SF}$) in the melting area.
+{{ :berechnungen:leistung_und_schubspannungen:abb_heizleistung_001_en.svg?nolink&600 |}}
+==== Calculation of convection in the melt vortex ====
+The calculation of convection in the melt pool (SW) is based on the assumption of a plane wall with longitudinal flow around it. This results in the following dimensionless key figures:
+$$\text{The Reynolds number } Re=\frac{L_{char} \cdot \rho \cdot v_0}{\eta}$$
+$$\text{The Prandtl number } Pr = \frac{\eta \cdot c_p}{\lambda}$$
+$$\text{The Nusselt number } Nu = 0.6774 \cdot \sqrt{Re} \cdot Pr^{1/3}$$
+with the characteristic length $L_{char}$, which corresponds to the channel height, the density $\rho$, the circumferential speed $v_0$, the viscosity $\eta$, the heat capacity $c_p$ and the thermal conductivity $\lambda$. The heat transfer coefficient $\alpha$ can also be calculated from the Nusselt number:
+$$\alpha = \frac{Nu \cdot \lambda}{L_{char}}$$
+The heat flow results from the heat flow density $\dot q$, the temperature difference between the melt and the cylinder $\Delta T$ and the area of the heat transfer $A = b \cdot (1-y) \cdot Z$ with the channel width $b$, the solid bed width $y$ and the channel length $Z$
+$$\dot q_{SW} = \alpha \cdot \Delta T$$
+$$\dot Q_{SW} =\dot q_{SW} \cdot A$$
+==== Calculation of heat conduction via the melt film ====
+The calculation of the heat flux density by heat conduction in the melt film (SF) is derived from the analytically calculated temperature profile in the melt film. The temperature gradient on the cylinder is used for this:
+$$\dot q_{SF} = \frac{dT}{d \xi} \Bigg \vert_{\xi=1} \cdot \frac{\lambda}{\overline \delta}$$
+$$\dot q_{SF} = (T_Z-T_{Fl}) \left\{ 1 + Br_{SW} \left[ \frac{\left[ (1-A)e^A - 1 \right]}{A^2} \left( \frac{A}{e^A-1} \right)^{1+n} \right] \right\} \cdot \frac{\lambda}{\overline \delta}$$
+$$\text{with}$$
+$$A = \frac{\beta}{n} (T_Z-T_{Fl})$$
+$$\text{and}$$
+$$Br_{SF} = \frac{K \cdot v_{rel}^{1+n} \cdot \bar{\delta}^{1-n}}{\lambda (T_Z - T_{Fl})}$$
+with the thermal conductivity $\lambda$, the mean melt film thickness $\overline \delta$, the cylinder temperature $T_Z$, the melting temperature $T_{Fl}$, the Brinkmann number in the melt film $Br_{SF}$, the flow law exponent $n$, the consistency factor $K$ and the relative velocity between cylinder and solid bed $v_{rel}$.
+The heat flow is also calculated from the heat flow density by multiplying it by the area of the solid bed:
+$$\dot Q_{SF} = \dot q_{SF} \cdot b \cdot y \cdot Z$$
+with the channel width $b$, the solid bed width $y$ and the channel length $Z$.
 ===Further topics===
   * [[en:berechnungen:einfache_berechnung|]]
+  * [[en:berechnungen:prozess_iterieren]]
   * [[en:berechnungen:durchsatz|]]
   * [[en:berechnungen:druckverlauf|]]
@@ Zeile 39: / Zeile 105: @@
   * [[en:berechnungen:temperaturverlauf|]]
   * [[en:berechnungen:leistung_und_schubspannungen|]]
+  * [[en:berechnungen:schergeschwindigkeit]]
   * [[en:berechnungen:verweilzeit|]]
   * [[en:berechnungen:verweilzeitverteilung|]]