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Distribution of the Residence Time
The same details as for the calculation of the residence time have to be given for the residence time distribution.
Realization of the theoretical principles
The overlapping of pressure and drag flow in the screw channel create a complicate, multi-dimensional speed profile in the screw channel. From this, difference speeds of different flows result. Thus, a pronounced residence time spectrum of the material in the aggregate occurs.
In order to quantify and compare the mixing effects in an extruder due to the residence time spectrum, the distribution of the residence time of the plasticization units can be referred to the residence time behavior of the ideal mixer. In order to do this, the surface under the curves of the residence time distribution are calculated and compared.
Thus a reference of the cumulated residence time ($F(Θ)$) of the system regarded can be made referring to both reference systems (clot flow and ideal mixer). The result is the following characteristic value $β$:
$$ \beta = 1 - \frac{\Delta A_{\mathrm{Screw/clot}}}{\Delta A_{\mathrm{ideal/clot}}} $$
With the clot flow ($β$=0) all particles have the same residence time:
$$ \Theta = 1 $$
$Θ$ (Theta) describes the dimensionless residence time.
The cumulated residence time distribution $F(Θ)$ of the ideal mixer ($β$ = 1) is defined as:
$$ F(\Theta)_{\mathrm{ideal}} = 1-e^{-\Theta}$$
Thus, for the calculation of the mixing characteristic value $β$ the following surfaces under the residence time profiles the result is:
$$ \beta = 1- \frac{\Delta A_{\mathrm{Screw/clot}}}{0,368} $$
With thermally insensitive materials a wide residence time distribution (great $β$) with a short dimensionless minimum residence time is to be strived for in order to improve the melt homogeneity. With thermally sensitive materials a small residence time distribution (little $β$) with a great dimensionless minimal residence time is to be strived for and the melt homogeneity is to be improved by mixing sections, e. g. the faceted mixing section or the toothed disc mixing section.