Unterschiede
Hier werden die Unterschiede zwischen zwei Versionen angezeigt.
| Beide Seiten der vorigen RevisionVorhergehende ÜberarbeitungNächste Überarbeitung | Vorhergehende Überarbeitung | ||
| en:grundlagenhandbuch:verweilzeitberechnung [2026/04/15 07:57] – [Minimum Residence Time] neelest | en:grundlagenhandbuch:verweilzeitberechnung [2026/05/24 18:07] (aktuell) – [Analysis of the Residence Time Distribution] neelest | ||
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| Zeile 49: | Zeile 49: | ||
| The average residence time is defined as the ratio of the filled volume in one pair of screw elements to the volumetric flow rate: | The average residence time is defined as the ratio of the filled volume in one pair of screw elements to the volumetric flow rate: | ||
| - | $$\bar{t} = \frac{A_{fr}L_{Be}\bar{f}}{\dot{V}} \tag{1}$$ | + | $$\bar{t} = \frac{A_{fr}L_{Be}\bar{f}}{\dot{V}} \tag{8}$$ |
| For a constant density follows: | For a constant density follows: | ||
| - | $$\bar{t} = \frac{m}{\dot{m}} \tag{2}$$ | + | $$\bar{t} = \frac{m}{\dot{m}} \tag{9}$$ |
| The overall average residence time is calculated by totalling the calculated minimum residence times in all sections of constant geometry. | The overall average residence time is calculated by totalling the calculated minimum residence times in all sections of constant geometry. | ||
| - | $$\bar{t} = \sum_i \left(\frac{A_{fr}L_{Be}\bar{f}}{\dot{V}}\right)_i \tag{3}$$ | + | $$\bar{t} = \sum_i \left(\frac{A_{fr}L_{Be}\bar{f}}{\dot{V}}\right)_i \tag{10}$$ |
| ===== Analysis of the Residence Time Distribution ===== | ===== Analysis of the Residence Time Distribution ===== | ||
| - | The residence time distribution of all screw machines can be described using a double Weibull distribution [[en: | + | The residence time distribution of all screw machines can be described using a double Weibull distribution [[en: |
| + | [[en: | ||
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| + | [[en: | ||
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| + | [[en: | ||
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| - | $$F(\Theta) = \left\{1 - e^{-c_1\left(\frac{\Theta-\Theta_1}{1-\Theta_1}\right)^{c_2}}\right\} \cdot \left\{1 - e^{-c_3\left(\frac{\Theta-\Theta_1}{1-\Theta_1}\right)^{c_4}}\right\} \tag{1}$$ | + | $$F(\Theta) = \left\{1 - e^{-c_1\left(\frac{\Theta-\Theta_1}{1-\Theta_1}\right)^{c_2}}\right\} \cdot \left\{1 - e^{-c_3\left(\frac{\Theta-\Theta_1}{1-\Theta_1}\right)^{c_4}}\right\} \tag{11}$$ |
| This function depends on the minimum dimensionless residence time $\Theta_1$ and the parameter $c_1$ to $c_4$, which differ according to the type of machinery. | This function depends on the minimum dimensionless residence time $\Theta_1$ and the parameter $c_1$ to $c_4$, which differ according to the type of machinery. | ||
| Zeile 69: | Zeile 79: | ||
| The double Weibull distribution for one machine depends on the dimensionless residence time: $\Theta = t / \bar{t}$ and the dimensionless minimum residence time. The differentiation of the double Weibull distribution $F(\Theta)$ results in the probability density function $f(\Theta)$. The integration of the probability density function $f(\Theta)$ within the limits $\Theta_1$ and $\Theta_2$ will result in the double Weibull distribution $F(\Theta)$, | The double Weibull distribution for one machine depends on the dimensionless residence time: $\Theta = t / \bar{t}$ and the dimensionless minimum residence time. The differentiation of the double Weibull distribution $F(\Theta)$ results in the probability density function $f(\Theta)$. The integration of the probability density function $f(\Theta)$ within the limits $\Theta_1$ and $\Theta_2$ will result in the double Weibull distribution $F(\Theta)$, | ||
| - | $$F(\Theta) = \int_{\Theta_1}^{\Theta} f(\Theta) d\Theta \tag{2}$$ | + | $$F(\Theta) = \int_{\Theta_1}^{\Theta} f(\Theta) d\Theta \tag{12}$$ |
| which is the sum of all tracer particles, which have left the screw machine. The values of this function are limited to values between 0 and 1. | which is the sum of all tracer particles, which have left the screw machine. The values of this function are limited to values between 0 and 1. | ||
| - | The determination of the parameters $c_1$ to $c_4$ has to follow specific boundary conditions [[en: | + | The determination of the parameters $c_1$ to $c_4$ has to follow specific boundary conditions [[en: |
| + | [[en: | ||
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| * the values of the minimum dimensionless residence time $\Theta_1$ can only be within the range between 0 and 1. | * the values of the minimum dimensionless residence time $\Theta_1$ can only be within the range between 0 and 1. | ||
| * the average dimensionless residence time $\bar{\Theta}$ has the value 1. | * the average dimensionless residence time $\bar{\Theta}$ has the value 1. | ||
| - | * characteristic values of the distribution profiles have to fit in with the experimental results. Characteristic values are e.g. the value for $\Theta = 1$ and the position of the maximum of the probability density function [[en: | + | * characteristic values of the distribution profiles have to fit in with the experimental results. Characteristic values are e.g. the value for $\Theta = 1$ and the position of the maximum of the probability density function [[en: |
| - | * the distribution function has to take into account the case of a pure plug flow with the boundary conditions: | + | |
| - | $$F(\Theta) = \begin{cases} 0, & \text{für: } \Theta < 1 \\ 1, & \text{für: } \Theta \geq 1 \end{cases} \tag{3}$$ | + | * The distribution function must account for pure block flow with the boundary conditions |
| - | $$\bar{\Theta} | + | $$F(\Theta) = \begin{cases} 0, & \text{for: } \Theta |
| - | * the distribution function has to take into account the case of the so called ideal mixer with $\Theta_1 = 1$ using the parameters $c_1 = c_2 = 1$. | + | , |
| - | * the variance $\sigma^2$ of the probability function: | + | |
| - | $$\sigma^2 = \int_{\Theta_1}^{\infty} (\Theta - 1)^2 f(\Theta)d\Theta \tag{5}$$ | + | * Furthermore, |
| + | * the distribution function must describe the limiting case of the so-called ideal mixer with $\Theta_1 = 0$ via $c_1 = c_2 = 1$, | ||
| + | * the variance $\sigma^2$ of the probability density function | ||
| - | has to decrease with the minimum dimensionless residence time and has to take the following values: | + | $$\sigma^2 = \int_{\Theta_1}^{\infty} (\Theta - 1)^2 f(\Theta)d\Theta\tag{14}$$ |
| - | $$\sigma^2(\Theta_1 = 1) = 0 \tag{6}$$ | + | must decrease monotonically as a function of the shortest dimensionless residence time $\Theta_1$ and take the value 0 for $\Theta_1 = 1$ or the value 1 for $\Theta_1 |
| - | $$\sigma^2(\Theta_1 = 0) = 1 \tag{7}$$ | + | * for physical reasons relating to the solid content, the variance |
| - | + | ||
| - | * the variance | + | |
| ===== References ===== | ===== References ===== | ||
| - | [1] Middleman S.; Bigg, D.: Mixing in a Screw Extruder. A Model for Residence Time Distribution, Industrial Engineering Chemical Found 1974, 13. | + | [Eng87] Engelhardt, M., et al.: Unveröffentlichte Studien- und Diplomarbeiten an der Universität Paderborn, KTP 1987-1991 |
| - | [2] Potente, H.; Lappe, H.: Analysis of the residence time distribution in conventioned plasticising extruders, Plastics and Rubber Processing and Application 1986; 6: 135-140. | + | [HKP89] Hensen, F.; Knappe, W.; Potente, H.: Handbuch der Kunststoff-Extrusionstechnik, |
| - | [3] Lappe, H.: Untersuchung | + | [Kes91] Kessler, H.: Modell |
| - | [4] Potente, H.: An Analysis of Residence Time Distribution in Plasticating Extruders, Advances in Polymer Technology 1984, 4: 147-154. | + | [Koc87] Koch, M.: Berechnung und Auslegung von Nutbuchsenextrudern, Dissertation Universität Paderborn, 1987 |
| - | [5] Potente, H.; Lappe, H.: Verweilzeit- und Längsmischgradgleichungen für Schmelzeextruder, Kunststoffe | + | [Lap85] Lappe, H.: Untersuchung zum Verweilzeitverhalten von Schmelze- und konventionellen Plastifizierextrudern, 1985 |
| - | [6] Hensen, F.; Knappe, W.; Potente, H.: Handbuch der Kunststoff-Extrusionstechnik, Band 1, München, Wien, Hanser Publishers, 1989. | + | [MB74] Middleman, S.: Bigg, D.: Mixing in a Screw Extruder. A Model for Residence Time Distribution, Industrial Engineering Chemical Found, 13(1974)1 |
| - | [7] Potente, H.; Fornefeld, A.; Koch, M.; Schultheis, S. M.: Verfahrenstechnische Auslegung | + | [PA90] Potente, H.; Ansahl, J.: Optimierung |
| - | [8] Schultheis, S.M.: Approximationsgleichungen zur Auslegung | + | [PA90] Potente, H.; Ansahl, J.: Verweilzeitcharakteristik |
| - | [9] Koch, M.: Berechnung und Auslegung von Nutbuchsenextrudern, Dissertation UNIPaderborn, 1987. | + | [PA90] Potente, H.; Ansahl, J.: Residence Time Characteristics of Tightly Intermeshing Co-Rotating Twin Screw Extruders, Kunststoffe German Plastics, 80(1990)8, 29-32 |
| - | [10] Potente, H.; Mitarbeiter: | + | [PFK+86] Potente, H.; Fornefeld, A.; Koch, M.; Schultheis, S.M.: Verfahrenstechnische Auslegung von Plastifizier- und Schmelzeaggregaten - Kunststofftechnisches Seminar, |
| - | [11] Schulte, H.: Grundlagen zur verfahrenstechnischen Auslegung von Spritzgießplastifiziereinheiten, UNI-Paderborn, Dissertation, 1990. | + | [PL85] Potente, H.; Lappe, H.: Verweilzeit- und Längsmischgradgleichungen für Schmelzeextruder, Kunststoffe, 75(1985)11, 855-858 |
| - | [12] Kessler, H.: Modell zum stationären und instationären Mischen | + | [PL86] Potente, H.: Lappe, H.: Analysis of the residence time distribution |
| - | [13] Potente, H.; Ansahl, J.: Optimierung von Schneckenpaaren für die Aufbereitung und Verarbeitung von vorwiegend Polyolefinen auf gleichsinnig drehenden Zweischneckenmaschinen, DFG Forschungsvorhaben Po 171/ 16-1, 1990. | + | [Pot84] Potente, H.: An Analysis of Residence Time Distribution in Plasticating Extruders, Advances in Polymer Technology, 4(1984)2, 147-154 |
| - | [14] Potente, H.; Ansahl, J.: Verweilzeitcharakteristik von dichtkämmenden Gleichdrall-Doppelschneckenextrudern, Kunststoffe, 80(1990)8, 926 - 932 | + | [Pot91] Potente, H.: Rechnergestützte Extruderauslegung, Kunststofftechnisches Seminar, Paderborn, 1991 |
| - | [15] Potente, H.; Ansahl, J.: Residence Time Characteristics of Tightly Intermeshing Co-Rotating Twin Screw Extruders, Kunststoffe German Plastics, 80(1990)8, 29-32 | + | [Sch87] Schultheis, S.M.: Approximationsgleichungen zur Auslegung von gegenläufigen Doppelschneckenextrudern, Dissertation Universität Paderborn, 1987 |
| - | [16] Engelhardt M; et al.: Unveröffentliche Studien- und Diplomarbeiten an der Universität-GH Paderborn, | + | [Sch90] Schulte, H.: Grundlagen zur verfahrenstechnischen Auslegung von Spritzgießplastifiziereinheiten, |