Residence time
Residence Time Profile
→ For a graphical representation of the dwell time curve
For the calculation of the residence time the same conditions as for the calculation of the melting profile have to be fulfilled. In addition to this the injection time is required with PSI.
Realization of the theoretical principles
Up to Location of the melt vortex formation (OSW), the calculation of the minimum and average residence time does not differ because a block flow is assumed in this area. From the OSW, a distinction is made between the mean and minimum residence time. The mean residence time results in:
$$\bar{t} = \frac{V}{\dot{V}}$$ with the channel volume $V$ and the volume flow $\dot{V}$.
The minimum residence time results in:
$$\frac{t_{min}}{\bar{t}}=\frac{\dot{\pi}_m^{0,23(1-n)}}{2(1,249-0,589n)}$$
with the dimensionless mass throughput $\dot{\pi}_m$ and the Flow law exponents n.
PSI only: As soon as the calculated dwell time is equal to the dosing time, the standstill time and the injection time are added to it. This procedure is continued until the screw peak is reached.
Residence Time Distribution
→ Graphical representation of the residence time distribution
The residence time distribution describes the time required for the polymer to reach the screw tip. The minimum residence time is calculated as described above. In theory, the maximum residence time is infinite. The cumulative residence time therefore approaches 100% asymptotically.
Implementation of the theoretical principles
The cumulative residence time distribution can be approximated by a double Weibull distribution:
$$ F(\Theta) = \left[1-e^{-C_1 \left(\frac{\Theta-\Theta_{min}}{1-\Theta_{min}}\right)^{C_2}}\right] \left[1-e^{-C_3 \left(\frac{\Theta-\Theta_{min}}{1-\Theta_{min}}\right)^{C_4}}\right]$$
Here, $\Theta$ denotes the dimensionless residence time:
$$\Theta = \frac{t}{\bar{t}}$$
where $\bar{t}$ is the mean residence time and $\Theta_{min} = \frac{t_{min}}{\bar{t}}$ is the minimum dimensionless residence time.
The residence time density is obtained as the derivative of the cumulative residence time distribution:
$$ f(\Theta) = \frac{1}{1-\Theta_{\min}} \left( C_1 C_2 \left(C_1 \cdot x\right)^{C_2-1} e^{-\left(C_1 \cdot x\right)^{C_2}} \left(1-e^{-\left(C_3 \cdot x\right)^{C_4}}\right) + C_3 C_4 \left(C_3 \cdot x\right)^{C_4-1} e^{-\left(C_3 \cdot x\right)^{C_4}} \left(1-e^{-\left(C_1 \cdot x\right)^{C_2}}\right) \right) $$
with $x = \frac{\Theta-\Theta_{\min}}{1-\Theta_{\min}}$
Thus, the residence time distribution depends only on the dimensionless minimum residence time and the parameters $C_1$ to $C_4$.
Melt extruder
For melt extruders, the cumulative residence time distribution simplifies to a single Weibull distribution:
$$ C_1 = -\ln\left(0.3679 \cdot \left(1-\Theta_{\min}\right)^{0.3}\right) $$ $$ C_2 = \begin{cases} 1-\dfrac{2}{3}\Theta_{\min}, & \Theta_{\min} \leq 0.71, \\[6pt] e^{-1.2371\left(C_1-1\right)^{0.6573}}, & 0.71 < \Theta_{\min} \leq 0.978, \\[6pt] 0.2632 \cdot e^{-0.55\left(C_1-2.3145\right)^{0.9}}, & \Theta_{\min} > 0.978. \end{cases} $$ $$ C_3 \to \infty$$ $$ C_4 = 0 $$
For smooth-barrel extruders, the following parameters are used:
$$ C_1 = -\ln\left(\frac{\left(1-\Theta_{\min}\right)^{0.47}}{e}\right) $$ $$ C_2 = \left(1-\Theta_{\min}\right)^{0.6} $$ $$ C_3 = 3.33 + \frac{1}{\left[\sin\left(\pi \Theta_{\min}\right)\right]^{0.9}} $$ $$ C_4 = 1.65 \cdot \left[\sin\left(\pi \Theta_{\min}\right)\right]^{0.25} $$
For grooved-bush extruders, the following parameters are used:
$$ C_1 = -\ln\left(0.36 \cdot \left(1-\Theta_{\min}\right)^{0.121}\right) $$ $$ C_2 = \left(2-C_1\right)^{1.1} $$ $$ C_3 = 5.8 - 1.375 \cdot \left(1-\Theta_{\min}\right)^{0.25} $$ $$ C_4 = 2.3 - 2.0 \cdot \sqrt{1-\Theta_{\min}} $$