Unterschiede

Hier werden die Unterschiede zwischen zwei Versionen angezeigt.

--- en:berechnungen:verweilzeitverteilung [2024/10/30 13:17] – neelest
+++ en:berechnungen:verweilzeitverteilung [2025/09/03 11:50] (aktuell) – [Agglomerate size reduction] neelest
@@ Zeile 1: / Zeile 1: @@
-=====Distribution of the Residence Time=====
+=====Mixed indicators=====
-The same details as for the calculation of the residence time have to be given for the residence time distribution.
+-> [[en:grafische_darstellung_der_ergebnisse:mischverhalten|Graphical representation of the mixed key figures]]
+REX/PSI provides both analytical mixing indicators and regression models based on numerical simulations of the mixing effect of different zone types.
-====Realization of the theoretical principles====
+===== Analytical mixed indicators =====
-The overlapping of pressure and drag flow in the screw channel create a complicate, multi-dimensional speed profile in the screw channel. From this, difference speeds of different flows result. Thus, a pronounced residence time spectrum of the material in the aggregate occurs.
+In the [[en:grafische_darstellung_der_ergebnisse:kurzbericht|short report]] and in the [[en:grafische_darstellung_der_ergebnisse:mischverhalten|Mixture triangle]] three mixing ratios are displayed for the calculated screw.
-In order to quantify and compare the mixing effects in an extruder due to the residence time spectrum, the distribution of the residence time of the plasticization units can be referred to the residence time behavior of the ideal mixer. In order to do this, the surface under the curves of the residence time distribution are calculated and compared.
+==== Transverse mixing ratio ====
-Thus a reference of the cumulated residence time ($F(Θ)$)  of the system regarded can be made referring to both reference systems (clot flow and ideal mixer). The result is the  following characteristic value $β$:
+The transverse mixing ratio $\pi_{LSM}$ is calculated as follows:
+$$\pi_{LSM} = \left[ 1 + \left(c_0 \overline{\gamma} \right)^2 \right]^{-1/2}$$
+$$\text{with}$$
+$$\overline{\gamma} = \frac{2 \left(1.03 + 0.074n \right) L}{\cos\varphi \sin\varphi} \cdot \frac{1}{h_1 \, \pi_\dot{m}}$$
+$$\text{and}$$
+$$c_0 = c_1 \sqrt{\frac{h_1}{h_0}} \left(\frac{T_z}{T_0}\right)^{c_2} \frac{n^{c_3}}{\pi_\dot{m}}$$
-$$ \beta = 1 - \frac{\Delta A_{\mathrm{Screw/clot}}}{\Delta A_{\mathrm{ideal/clot}}} $$
+with c1=0.08367, c2=0.7067 and c3=0.344
-With the clot flow ($β$=0) all particles have the same residence time:
+==== Longitudinal mixing ratio ====
-$$ \Theta = 1 $$
+The longitudinal mixing ratio $\sigma^2$ is calculated as follows
-$Θ$ (Theta) describes the dimensionless residence time.
+$$\sigma^2 = 1 + \exp(c_1 \Theta_{min}^{c_2}) - \exp(c_3 \Theta_{min})$$
-The cumulated residence time distribution $F(Θ)$ of the ideal mixer ($β$ = 1) is defined as:
+with the dimensionless minimum residence time $\Theta_{min} = \frac{t_{min}}{\bar{t}}$ and
-$$ F(\Theta)_{\mathrm{ideal}} = 1-e^{-\Theta}$$
+^ extruder type ^ c1 ^ c2 ^ c3 ^
+| melt extruder | 0.2476 | 0.6014 | 0.8301 |
-Thus, for the calculation of the mixing characteristic value $β$ the following surfaces under the residence time profiles the result is:
+| Conv. plasticising extruder | -4.8828 | 2.1287 | 0.0028505 |
+| grooved barrel extruder | -2.368 | 1.1268 | 0.1186 |
-$$ \beta = 1- \frac{\Delta A_{\mathrm{Screw/clot}}}{0,368} $$
+==== Agglomerate size reduction ====
-With thermally insensitive materials a wide residence time distribution (great $β$)  with a short dimensionless minimum residence time is to be strived for in order to improve the melt homogeneity. With thermally sensitive materials a small residence time distribution (little $β$) with a great dimensionless minimal residence time is to be strived for and the melt homogeneity is to be improved by mixing sections, e. g. the faceted mixing section or the toothed disc mixing section.
+The agglomerate size reduction ratio $Z$ is calculated as follows:
+$$Z=\left[ 1+\left( \frac{t}{c_{31}}\right)^{c_{21}}\left( ln\left( \frac{\tau}{\tau_{min}}\right)\right)^{1/c_{11}}\right]^{-1}$$
+with the current shear stress $\tau$, the minimum shear stress $\tau_{min}$ and the three decay constants $c$. The minimum shear stress and the three decay constants are [[en:materialdaten:zusatzstoffe|Material parameters]]. REX offers default values at the push of a button.
+===== Numerical mixing ratios =====
+The numerical mixing quality is calculated from the following two parameters for determining the
+dispersive and distributive mixing quality, which are weighted from 100% - good to 0% - bad.
+weighted:
+The numerical mixing quality calculation is currently available for cross-hole mixing, diamond mixing, spiral shear elements and metering,
+spiral shear elements and metreing zones. The calculation is
+only for zones in which a completely melted melt is present, as the basis of the
+CFD simulations form the basis of the key figure. This will be continuously
+expanded in the future.
+====1. Dispersive mixing effect====
+The dispersive mixing quality is based on a regression equation for the so-called Mixing Index
+according to Manas, which was determined by means of numerical investigations using a CCD test plan.
+was determined. The mixing index according to Manas is a quantitative measure for describing the mixing
+mixing quality of numerical investigations, which allows conclusions to be drawn about the dispersive
+mixing behaviour. The index is determined from the deformation gradient and the
+vortex tensor:
+\[
+\lambda = \frac{|\Gamma|}{|\Gamma| + |\omega|}
+\]
+\[
+\nabla \vec{v} =
+\begin{pmatrix}
+\frac{\partial v_x}{\partial x} & \frac{\partial v_x}{\partial y} & \frac{\partial v_x}{\partial z} \\
+\frac{\partial v_y}{\partial x} & \frac{\partial v_y}{\partial y} & \frac{\partial v_y}{\partial z} \\
+\frac{\partial v_z}{\partial x} & \frac{\partial v_z}{\partial y} & \frac{\partial v_z}{\partial z}
+\end{pmatrix}
+\]
+\[
+\Gamma = \frac{\left(\nabla \vec{v} + \nabla \vec{v}^T\right)}{2}
+\]
+\[
+\omega = \frac{\left(\nabla \vec{v} - \nabla \vec{v}^T\right)}{2}
+\]
+\(\lambda\): Manas-number\\
+\(\Gamma\): Deformation gradient tensor\\
+\(\omega\): Vortex tensor\\
+\(\nabla \vec{v}\): Velocity gradient\\
+The mana number characterises the type of flow and is divided as follows
+subdivided:\\
+  * λ = 1 pure elongation
+  * λ = 0.5 pure shear flow
+  * λ = 0 pure rotation
+====2. Distributive mixing effect====
+The distributive mixing quality is based on a regression equation determined using a CCD test plan for the evaluation method of a particle distribution based on the Delaunay triangulation, which was carried out as follows:\\ \\
+Based on the numerically calculated flow area, the particle trajectory of a particle distribution defined at the beginning of the flow area or at the beginning of the geometry can be calculated using the so-called particle tracking method. The particle distribution localised at the end of the flow region in the two-dimensional cross-section was then examined for homogeneity. For this purpose, a triangular mesh was created using Delaunay triangulation. At this point, the correlation that homogeneous area contents of the spanning triangles are accompanied by a homogeneously distributed particle distribution is used as a parameter for the evaluation or as a measure of the mixing quality. The so-called coefficient of variation, which relates the standard deviation of the triangular areas to the mean area, is used as the evaluation parameter. If all particles are evenly distributed, the coefficient of variation is zero (mixing quality 100 %); the coefficient of variation of the initial distribution (all particles in one half of the channel) is defined as 0 %.
+{{ :mischen:rex171_mischen_en_003.png?nolink |}}
+====3. Thermal mixing effect====
+An additional parameter is calculated for the cross-hole mixing section, which evaluates the effectiveness of the radial temperature equalisation. This is necessary because a targeted temperature exchange radial to the channel does not correlate with the results of the distributive mixing effect. For mixing parts that aim for general mixing (circumferential, radial and longitudinal direction), the thermal mixing effect correlates with the distributive mixing effect. For the cross-hole mixing part, which forces a pure exchange between fluid layers at the base of the screw and the barrel wall, a separate consideration is required. By definition, no direct statement about temperature distributions can be obtained from the isothermal CFD simulations. For this reason, the trajectory of each individual particle is analysed as part of particle tracking. The position in the channel is evaluated over the course of the mixing section and the Graetz number is analysed in order to map the influence of the operating point. A high mixing effect is achieved when all particles have spent sufficient time on the cylinder wall and screw base and effectively switch between the two areas.
+{{ :mischen:rex171_mischen_en_001.png?nolink |}}
+{{ :mischen:rex171_mischen_en_002.png?nolink&600 |}}
 ===Further topics===
   * [[en:berechnungen:einfache_berechnung|]]
+  * [[en:berechnungen:prozess_iterieren]]
   * [[en:berechnungen:durchsatz|]]
   * [[en:berechnungen:druckverlauf|]]
@@ Zeile 38: / Zeile 110: @@
   * [[en:berechnungen:temperaturverlauf|]]
   * [[en:berechnungen:leistung_und_schubspannungen|]]
+  * [[en:berechnungen:schergeschwindigkeit]]
   * [[en:berechnungen:verweilzeit|]]
   * [[en:berechnungen:verweilzeitverteilung|]]