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The minimum residence time $t_1$ is defined as the period between the entrance and the first exit of the material at the screw tip. In a rectangular channel there is a section where no rotational flow occurs, regardless of the type of the flow. In the case of a Newtonian fluid the area is always located at $y/h = 2/3$. Particles moving on this particular streamline are exhibited to the minimum residence time. Based on this observation, the results presented by Bigg and Middleman [1] Lappe [2, 3] proposed the following equation to calculate the minimum residence time:

$$\Theta_1 = \frac{3}{4}\pi_v^{0.23(1-n)} \tag{1}$$

where $\Theta_1$ is the dimensionless minimum residence time, namely the ratio of the minimum residence time to the average residence time, $\pi_v$ is the dimensionless throughput and n the power law index. In the figure one can see the influence of the power law index on the minimum dimensionless residence time.

Figure: Influence of the Power Law index on the minimum dimensionless residence time [3-5]

$\Theta_1$ decreases for constant dimensionless throughputs $\pi_v < 1$ with decreasing power law index n. One can see that the ratio of minimum and average residence time for a Newtonian fluid is always 0.75. This means that one cannot influence the dimensionless residence time by changing machine or processing parameters.

In order to calculate the minimum residence time one has to distinguish the following cases:

• 1. Solids Conveying Section
  • Case 1: f < 1 (partially filled)

$$\Theta_1 = \frac{1}{2} \tag{2}$$

• Case 2: f = 1 (fully filled)

$$\Theta_1 = 1 \tag{3}$$

• 2. Melt Conveying Section
  • Case 1: f < 1

$$\Theta_1 = \frac{1}{2} \tag{4}$$

• Case 2: f = 1

$$\Theta_1 = \frac{3}{4}\pi_v^{0.23(1-n)} \tag{5}$$

$$\Theta_1 = 1 \tag{6}$$

The overall minimum residence time is calculated by totalling the calculated minimum residence times in all sections of constant geometry.

$$t_1 = \sum_i (t_1)_i \tag{7}$$

The average residence time is defined as the ratio of the filled volume in one pair of screw elements to the volumetric flow rate:

$$\bar{t} = \frac{A_{Fr}L_{Be}\bar{f}}{\dot{V}} \tag{1}$$

For a constant density follows:

$$\bar{t} = \frac{m}{\dot{m}} \tag{2}$$

The overall average residence time is calculated by totalling the calculated minimum residence times in all sections of constant geometry.

$$\bar{t} = \sum_i \left(\frac{A_{Fr}L_{Be}\bar{f}}{\dot{V}}\right)_i \tag{3}$$

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].