Unterschiede

Hier werden die Unterschiede zwischen zwei Versionen angezeigt.

--- en:grundlagenhandbuch:verweilzeitberechnung [2026/05/21 22:32] – [Average Residence Time] neelest
+++ en:grundlagenhandbuch:verweilzeitberechnung [2026/05/24 18:07] (aktuell) – [Analysis of the Residence Time Distribution] neelest
@@ Zeile 73: / Zeile 73: @@
 [[en:grundlagenhandbuch:verweilzeitberechnung#references |[PA90]]].
-$$F(\Theta) = \left\{1 - e^{-c_1\left(\frac{\Theta-\Theta_1}{1-\Theta_1}\right)^{c_2}}\right\} \cdot \left\{1 - e^{-c_3\left(\frac{\Theta-\Theta_1}{1-\Theta_1}\right)^{c_4}}\right\} \tag{1}$$
+$$F(\Theta) = \left\{1 - e^{-c_1\left(\frac{\Theta-\Theta_1}{1-\Theta_1}\right)^{c_2}}\right\} \cdot \left\{1 - e^{-c_3\left(\frac{\Theta-\Theta_1}{1-\Theta_1}\right)^{c_4}}\right\} \tag{11}$$
 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.
@@ Zeile 79: / 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 $F(\Theta)$,
-$$F(\Theta) = \int_{\Theta_1}^{\Theta} f(\Theta) d\Theta \tag{2}$$
+$$F(\Theta) = \int_{\Theta_1}^{\Theta} f(\Theta) d\Theta \tag{12}$$
 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.
@@ Zeile 97: / Zeile 97: @@
   * 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:grundlagenhandbuch:verweilzeitberechnung#references |[PA90]]], [[en:grundlagenhandbuch:verweilzeitberechnung#references |[PA90]]], [[en:grundlagenhandbuch:verweilzeitberechnung#references |[Eng87]]].
-  * the distribution function has to take into account the case of a pure plug flow with the boundary conditions:
+  * The distribution function must account for pure block flow with the boundary conditions
-$$F(\Theta) = \begin{cases} 0, & \text{für: } \Theta < 1 \\ 1, & \text{für: } \Theta \geq 1 \end{cases} \tag{3}$$
+$$F(\Theta) = \begin{cases} 0, & \text{for: } \Theta < 1 \\ 1, & \text{for: } \Theta \geq 1 \end{cases}\tag{13}$$
-$$\bar{\Theta} = \Theta_1 = 1 \tag{4}$$
+,
-  * the distribution function has to take into account the case of the so called ideal mixer with $\Theta_1 = 1$ using the parameters $c_1 = c_2 = 1$.
+  * Furthermore, for pure block flow, the following applies: $\bar{\Theta} = \Theta_1 = 1$,
-  * the variance $\sigma^2$ of the probability function:
+  * 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
-$$\sigma^2 = \int_{\Theta_1}^{\infty} (\Theta - 1)^2 f(\Theta)d\Theta \tag{5}$$
+$$\sigma^2 = \int_{\Theta_1}^{\infty} (\Theta - 1)^2 f(\Theta)d\Theta\tag{14}$$
-has to decrease with the minimum dimensionless residence time and has to take the following values:
+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 $\Theta_1 = 0$,
-$$\sigma^2(\Theta_1 = 1) = 0 \tag{6}$$
+  * for physical reasons relating to the solid content, the variance $\sigma^2$ of the probability density function must always be smaller than that of a melt extruder. The melt extruder was discussed in [[en:grundlagenhandbuch:verweilzeitberechnung#references |[PL85]]].
-$$\sigma^2(\Theta_1 = 0) = 1 \tag{7}$$
-  * the variance of the probability function has to be smaller for plasticating extruders than for melt extruders due to the amount of solid particles present. More information referring to melt extruders can be found in [[en:grundlagenhandbuch:verweilzeitberechnung#references |[PL85]]].
 ===== References =====