The residence time distribution of all screw machines can be described using a double Weibull distribution [2, 6 - 15].

$$F(\Theta) = \left\{1 - e^{-c_1\left(\frac{\Theta-\Theta_1}{1-\Theta_1}\right)^{c_2}}\right\} \cdot \left\{1 - e^{-c_3\left(\frac{\Theta-\Theta_1}{1-\Theta_1}\right)^{c_4}}\right\} \tag{1}$$

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.

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 $F(\Theta)$,

$$F(\Theta) = \int_{\Theta_1}^{\Theta} f(\Theta) d\Theta \tag{2}$$

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 [5,7-13, 15].

• 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.
• 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 [14 - 16].
• the distribution function has to take into account the case of a pure plug flow with the boundary conditions:

$$F(\Theta) = \begin{cases} 0, & \text{für: } \Theta < 1 \\ 1, & \text{für: } \Theta \geq 1 \end{cases} \tag{3}$$

$$\bar{\Theta} = \Theta_1 = 1 \tag{4}$$

• the distribution function has to take into account the case of the so called ideal mixer with $\Theta_1 = 1$ using the parameters $c_1 = c_2 = 1$.
• the variance $\sigma^2$ of the probability function:

$$\sigma^2 = \int_{\Theta_1}^{\infty} (\Theta - 1)^2 f(\Theta)d\Theta \tag{5}$$

has to decrease with the minimum dimensionless residence time and has to take the following values:

$$\sigma^2(\Theta_1 = 1) = 0 \tag{6}$$

$$\sigma^2(\Theta_1 = 0) = 1 \tag{7}$$

• 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 [5].