Unterschiede

Hier werden die Unterschiede zwischen zwei Versionen angezeigt.

--- en:berechnungen:leistung_und_schubspannungen [2025/02/27 20:55] – neelest
+++ en:berechnungen:leistung_und_schubspannungen [2025/07/03 13:34] (aktuell) – cschall
@@ Zeile 11: / Zeile 11: @@
 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]].
-For an accurate power calculation, the [[en:eingabe_der_zylinderdaten:eingabe_der_zylinderdaten|cylinder wall temperatures]] in particular must be specified in a practical manner.
+For an accurate power calculation, the [[en:eingabe_der_zylinderdaten:eingabe_der_zylinderdaten|cylinder inner wall temperatures]] must be specified in a practical manner.
-The temperature control power is calculated in the same way for all models and is made up of the melt film and the melt vortex. The temperature gradient on the cylinder wall is determined for both areas and the required temperature control power is determined from this using an energy balance. The resulting outputs at each calculation point are
+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
-[[en:grafische_darstellung_der_ergebnisse:leistung|summarised for the respective heating zones]]. The temperature control power calculation only takes into account the heat flow between the heating band and the melt, but not heat losses (of any kind) to the outside.
+[[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 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. \\
@@ Zeile 21: / Zeile 21: @@
 ===== Isothermal power model =====
-When calculating the power, a distinction is made between the melting and melting-on areas. In the melting area, i.e. from the start of the first heating zone to the location of the melt vortex formation, the power can be determined via an average melt layer thickness assuming a pure drag current in the melt film.
+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.
-film can be determined. 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.
-The calculation for the other zones results in a summation of the wall shear stresses on the cylinder. Similar to the pressure throughput calculation, 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.
+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>
@@ Zeile 52: / Zeile 51: @@
 and therefore also:
-$P_{drive} = \delta h_{temperature} * \dot{m}_{melt} + \delta p * \dot{V} - \dot{Q}$
+$P_{drive} = \Delta h_{temperature} * \dot{m}_{melt} + \Delta p * \dot{V} - \dot{Q}$
 //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 63: / Zeile 105: @@
   * [[en:berechnungen:temperaturverlauf|]]
   * [[en:berechnungen:leistung_und_schubspannungen|]]
+  * [[en:berechnungen:schergeschwindigkeit]]
   * [[en:berechnungen:verweilzeit|]]
   * [[en:berechnungen:verweilzeitverteilung|]]