The following simplifications are made for the modelling of the turbine mixing elements:

• The gaps in the intermeshing region are equal ($s_{F1} \approx s_{F2}$).
• The flow in the grooves can be described by the flow in a similar rectangular channel

Figure: Real turbine mixing elements and the simplified model

The models describing the pressure throughput relationships in rectangular channels originate from the single screw theory. These equations can not be applied here since the width to height ratio is comparatively small and the pitch of the grooves is larger than the pitch usually used for single screw extruders. Furthermore one can find different conveying directions in turbine mixer elements. There are conveying, reconveying and neutral turbine mixer elements.

For the Finite Element simulations the influencing variables varied as follows:

• $b/h$: $0.05 \leq b/h \leq 40$
• $t/d$: $0.5 \leq t/d \leq 10$ (only conveying and reconveying)
• $n$: $0.2 \leq n \leq 1$
• $\pi_V$: $0 \leq \pi_V \leq 10$

The pressure throughput behavior of the three different flow directions can be described by the following polynomial equation:

$$\pi_V = A_{R,0} \left(\frac{t}{D_a}, \frac{b}{h}, n\right) + A_{R,1} \left(\frac{t}{D_a}, \frac{b}{h}, n\right) \cdot \pi_P + A_{R,2} \left(\frac{t}{D_a}, \frac{b}{h}, n\right) \cdot \pi_P^2 + A_{R,3} \left(\frac{t}{D_a}, \frac{b}{h}, n\right) \cdot \pi_P^3 \tag{1}$$

where the parameter $A_{R,0} = 0$ for neutral elements. In the figures the profiles of the dimensionless pressure gradients are shown.

Figure: Comparison of approximated and numerically determined dimensionless flow rates for conveying rectangular channels

Figure: Comparison of approximated and numerically determined dimensionless flow rates for neutral rectangular channels

Figure: Comparison of approximated and numerically determined dimensionless flow rates for reconveying rectangular channels

The pressure – throughput relationship of the turbine mixing element follows from the superposition of the flow rates in a comparable blister disc and the flow rates in the rectangular channels

$$\dot{V}_{tooth\ mixing\ element} = \dot{V}_{disc} + i \cdot \dot{V}_{rectangle} \tag{2}$$

The following equation describes the pressure throughput relationship for turbine mixing elements results from the above equation:

$$\pi_V = Y_0 + Y_1 \cdot \pi_P + Y_2 \cdot \pi_P^2 + Y_3 \cdot \pi_P^3 \tag{3}$$

with:

$$\pi_V = \frac{\dot{V}}{\frac{1}{2} \cdot n_0 \cdot A_{free} \cdot D_a} \tag{4}$$

$$\pi_P = \frac{\Delta p}{L} \cdot \frac{\overline{s_R}}{K \cdot n_0^n} \tag{5}$$

and

$$Y_0 = A_{R,0} \cdot i \cdot \pi_{geo,V}$$

$$Y_1 = A_{B,1} + A_{R,1} \cdot i \cdot \pi_{geo,V} \cdot \pi_{geo,p} \tag{6}$$

$$Y_2 = A_{B,2} + A_{R,2} \cdot i \cdot \pi_{geo,V} \cdot \pi_{geo,p}^2$$

$$Y_3 = A_{R,3} \cdot i \cdot \pi_{geo,V} \cdot \pi_{geo,p}^3$$

The coefficients $p_{geo,V}$ and $p_{geo,p}$ are combined to couple the flow rates and the pressure gradients. They are defined as follows:

$$\pi_{geo,V} = \frac{\frac{1}{2} \cdot h \cdot b \cdot v_{0z}}{\frac{1}{2} \cdot A_{free} \cdot D_a \cdot n_0} \tag{10}$$

and:

$$\pi_{geo,p} = \frac{h^{1+n}}{6 \cdot (\pi \cdot D_a \cdot \cos(\varphi_z))^n} \tag{11}$$

Depending on the conveying direction different pressure throughput behaviors are resulting (see figures).

Figure: Pressure throughput behavior for conveying turbine mixing elements

Figure: Pressure throughput behavior for neutral turbine mixing elements

Figure: Pressure throughput behavior for reconveying turbine mixing elements

The comparison of the experimentally determined pressure differences and the predicted differences using a model show a significant correlation (see figure).

Figure: Comparison between experimentally determined data and the model predictions