The closed mathematical description of the general equations mentioned above requires restrictions of the system of differential equations by assumptions and boundary conditions.

For the models mentioned below we made the following prerequisites:

• The screw geometry can be described by a channel model
• The screw channel is completely full
• The flow is steady state and fully developed
• The flow is laminar
• The melt is incompressible
• In- and outflow effects are neglected
• The melt sticks to the wall
• Inertia terms are neglected
• Normal stresses are neglected
• Velocity components normal to the barrel wall are neglected

The analytical description of the melt conveying in co-rotating twin screw extruders is very complicated due to the self cleaning, tightly intermeshing screw profile. Since the profile is only defined piecewise. Because of this difficulty Ansahl [Ans93] replaced the real channel geometry by a rectangular channel with the same cross section.

In this section we will show a comparison of the flow in a real twin screw channel with that of a rectangular channel for a one-dimensional flow of a Newtonian fluid.

Figure: Coordinate system for the analysis of the melt conveying

Using the prerequisites mentioned above, the equilibrium of mass and the coordinate system shown in the figure, the momentum equation is reduced to:

$$-\frac{\partial p}{\partial z} - \eta \cdot \frac{\partial^2 v_z}{\partial y^2} = 0 \tag{1}$$

Solving this equation we get the velocity profile in the screw channel:

$$v_z(x,y) = \frac{1}{2\eta} \cdot \frac{\Delta p}{\Delta z}[y^2 + h(x) \cdot y] + v_{0z}\left[1 + \frac{y}{h(x)}\right] \tag{2}$$

With $v_{0z}$ being the velocity component in screw channel direction and h(x) the channel profile.

The velocity profile is not only dependent on the channel depth but also on the channel width, since the channel height of co-rotating twin screw extruder depends on the coordinate in the channel width direction.

Integrating eqn. (2) over the cross sectional area we get the flow rate:

$$\dot{V} = \int_{-\frac{b_{max}}{2}}^{\frac{b_{max}}{2}} \int_{-h(x)}^{0} v_z(x,y)dxdy \tag{3}$$

This integration can only be done analytically if h(x) = const. For twin screw channels this has to be done numerically.

By introducing a dimensionless throughput:

$$\pi_{\dot{V}} = \frac{\dot{V}}{\frac{1}{2}v_{0z}b_{max}\bar{h}} \tag{4}$$

and a dimensionless pressure gradient

$$\pi_p = \frac{\bar{h}^{1+n} \cdot \Delta p}{6Kv_{0z}^n \cdot Z} \tag{5}$$

a comparison of both channel geometries is made possible.

The figure shows the dimensionless throughput over the dimensionless pressure gradient for a one-dimensional flow of a Newtonian fluid. The profile for the rectangular channel intersects both axis at a value of 1.

Figure: Pressure – throughput behavior of different channel profiles for a one dimensional flow of a Newtonian fluid.

The profile for the twin screw channel intersects the profile for the rectangular channel at the point $p_{pS}$. Below this point, in the area $\pi_p < \pi_{pS}$ higher dimensionless throughputs can be obtained. Above point $\pi_p < \pi_{pS}$ smaller dimensionless throughputs can be obtained in a twin screw channel.

The Finite Element Method (FEM) is an approximation method for the solution of differential equations. When applying this method, a continuum with infinite degrees of freedom is replaced by a set of finite elements with finite degrees of freedom [Chu82].

The description of the flow using the FEM is based as with the analytical solution, on the conservative equations. We used the FE program POLYFLOW [NN93] which was specially designed for the simulation of polymer flows. Flow problems can be either 2 dimensional, 2 ½ dimensional, 3 dimensional, steady state or transient. It is also possible to describe the flow behavior of polymer melts using different viscosity laws.

Analogous to the analytical solution, simplifications are needed to reduce the number of calculations. With the exception of the flow dimension, the same pre-requisites and simplifying assumptions were used to calculate the flow numerically as were used for the calculations shown above. Only in this instance we did not calculate a one dimensional flow but a 2 ½ dimensional flow. The figure shows the screw geometry of a single flighted, a double flighted and a triple flighted screw profile. For both the meshes used, the calculations and the stream lines are shown. We used a power law as a constitutive equation.

Figure: Meshes and streamlines for single, double and triple flighted screw elements.

And as a result of imposing a normal flow rate, we obtain the flow field and the normal pressure gradient from the finite element simulations. With the dimensionless characteristics in eqns. (4) and (5) we get the following figure comparing the flow in the twin screw channel and a rectangular channel with the same cross section (see figure).

Figure: Pressure – throughput behavior of different channel profiles for a 2 ½ dimensional flow.

The analytical solution shows the intersecting profiles for the twin screw channel and the rectangular channel. A significant difference can be seen for reconveying elements.

As a result of these differences between the flow behavior in the real twin screw profile and in a rectangular channel it appears necessary to make a model that takes into account the real channel geometry.

The Finite Element Method describes multi-dimensional flow fields. The applicability of this method is currently limited to the evaluation of certain screw sections. An overall simulation of one screw configuration is currently not possible.

The analytical description has the advantage of offering both simple and closed solutions. Due to the simplifications used for these models, differences between the real behavior and the model predictions can be observed.

To obtain a closed solution with an accurate level of prediction, we used the method of approximating the results of Finite Element simulations. The numerical results can be used for the calculation of machines which were not simulated, if the machine data is within the variation range of the data basis.

The dimensionless throughput and the dimensionless pressure gradient are related as follows:

$$\pi_V = f(\pi_P) \tag{6}$$

This relationship was investigated for different assumptions.

Potente [Pot83] investigated the 2 ½ dimensional flow in rectangular channels on the basis of numerical simulations. He found an approximation equation describing the pressure – throughput behavior. This approximation is based on a simple linear equation.

$$\pi_V = Y_1 - Y_2 \cdot \pi_P \tag{7}$$

The constants $Y_1$ and $Y_2$ take into account the transverse flow and the influence of the radial clearance.

The pressure – throughput diagram for the 2 ½ dimensional case is shown in Figure [Tad79] together with the prediction (dotted lines). The linear equation enables differences for low power law exponents to be observed.

Figure: Dimensionless Flow Rate vs Dimensionless Pressure Gradient (2 dimensional flow in a rectangular channel) [Tad79]

The variation range of the influencing variables limits the scope of application for approximation equations. The influencing variables for co-rotating twin screw extruders are the number of flights $i$, the centerline ratio $CL = a/(D_S/2)$ , the ratio of pitch and screw diameter $t/D_S$ and finally the power law exponent $n$.

For the Finite Element simulations the influencing variables varied in the following ways:

• Number of flights: $1 \leq i \leq 3$
• Pitch: $0.5 \leq t / D_S \leq 3$ (Conveying elements)
• $0.5 \leq t / D_S \leq 2$ (Reconveying elements)
• Centerline Ratio: $1.65 \leq CL \leq 1.85$
• Power Law Exponent: $0.2 \leq n \leq 1$

We used only those combinations which result into a tightly intermeshing screw profile. In total 14496 data points were achieved.

The resulting characteristic field was split into three sections. For each section an approximation equation was found.

Conveying element, $\pi_P \geq 0$:

$$\pi_V = c_{0P} + c_{1P} \cdot \pi_P + c_{2P} \cdot \pi_P^2 + c_{3P} \cdot \pi_P^3 \tag{8}$$

Conveying element, $\pi_P \leq 0$:

$$\pi_V = c_{0N} + c_{1N} \cdot \pi_P + c_{2N} \cdot \pi_P^3 + c_{3N} \cdot \pi_P^3 \tag{9}$$

Reconveying element, $\pi_P \geq 0$:

$$\pi_V = c_0 + c_1 \cdot \pi_P + c_2 \cdot \pi_P^2 \tag{10}$$

These equations were fitted to the characteristic profiles. The following boundary conditions were used for this fit:

Figure: Position of the boundary conditions for conveying and reconveying elements

The figure shows the position of the boundary conditions used in the positive and the negative range of the dimensionless pressure gradient.

Boundary conditions for conveying elements ($\pi_P \geq 0$):

Boundary Condition 1: $\pi_P(\pi_V = 0) = \pi_{P0}$

Boundary Condition 2: Slope at the point of inflection $\pi_V'W$

Boundary Condition 3: Position of the point of inflection $\pi_{PW}$

Boundary Condition 4: $\pi_V(\pi_P = 0,15) = \pi_{V15}$

Boundary Condition 5: $\pi_V(\pi_P = 0) = \pi_{V0}$

Boundary conditions for conveying elements ($\pi_P \leq 0$):

Boundary Condition 1: $\pi_V(\pi_P = 0) = \pi_{V0}$

Boundary Condition 2: $\pi_P(\pi_V = 1,0) = \pi_{P10F}$

Boundary Condition 3: $\pi_P(\pi_V = 1,5) = \pi_{P15}$

Boundary Condition 4: $\pi_P(\pi_V = 2,0) = \pi_{P20}$

Boundary conditions for reconveying elements:

Boundary Condition 1: $\pi_P(\pi_V = 0) = \pi_{P0}$

Boundary Condition 2: $\pi_P(\pi_V = 1,0) = \pi_{P10R}$

Boundary Condition 3: $\pi'_V(\pi_V = 1,0) = \pi_{V'10}$

Since we were not able to obtain pressure throughput values for all boundary conditions, we had to take the missing values from equations that can be de-rived numerically. These boundary conditions are the slope at the point of inflection $\pi_V'W$, position of the point of inflection $\pi_{PW}$, the value for $\pi_{V0}$, the value for $\pi_P = 0,15$ ($\pi_{V15}$) and the value for $\pi_V = 1,0$ ($\pi_{V'10}$).

The quality of the approximation depends predominantly on the quality of the approximation of the boundary conditions. The influence of each variable was investigated individually. Due to this we found one equation for each variable. The actual approximation function was found by the recursive insertion of all equations into one another.

Figure: Comparison between approximated and numerical dimensionless pressure throughput behavior.

The figure provides an excellent comparison between the numerical data and the model predictions. The approximated characteristics (lines) were determined for the same flow rates as the numerical (symbols). The positive dimensionless pressure gradients show a good fit. Small differences between the model predictions and the numerical data can be found in the range of negative dimensionless pressure gradients. These differences, especially for power law exponents $n > 0.7$, can be explained by the chosen boundary conditions.

The figure shows the comparison of the model predictions, with the experimental data, which was obtained using polypropylene. In Figure the diagram is the same, although this time polystyrene was used. Both diagrams are plotted for three flighted conveying elements of a RZE 85.

Figure: Comparison between experimental data and model predictions for a Polypropylene

Figure: Comparison between experimental data and model predictions for a Polystyrene

The experimental data can be found close to or on the right hand side of the approximated line. The trend is basically the same. This means, that the model slightly underestimates the pressure build-up behavior of co-rotating twin screw extruders.

However, there are limitations in the production of the tightly intermeshing screw profile. Due to this there are clearances between the barrel and the screw flights. The material can flow in the neighboring channels. A model that takes into account the leakage flows was published in [Ans93].

In eqn. (11) equilibrium of flow rates for the channel model is shown.

$$\dot{V}_{ges} = k\dot{V}_z \mp \dot{V}_x \tag{11}$$

Where $\dot{V}_{ges}$ is the imposed flow rate to the system, $\dot{V}_z$ the flow rate in the channel and $\dot{V}_x$ the leakage flow rate, and k the number of parallel channels. The negative sign represents for conveying elements, the positive sign represents reconveying elements. Equation (11) can be rewritten in dimensionless form if it is divided by the drag flow rate.

$$\pi_{V,channel} = \pi_{V,z} \mp \pi_k\pi_{V,x} \tag{12}$$

with $p_k$ being a constant that takes into account the leakage length and the leakage height $s_R$.

$$\pi_k = \frac{(2\pi - \Omega)D_S \cos \varphi_S s_R \tan \varphi_S}{b_{max} \bar{h}} \tag{13}$$

For the description of the flow rate over the screw tip a one dimensional flow is assumed. This flow can be described by the following linear equation [Ans93]:

$$\pi_{\dot{V} ,x} = C_{0x} - C_{1x}\pi_P \tag{14}$$

the constants $C_{0x}$ and $C_{1x}$ depend on the screw element. For conveying elements eqn (15) can be used [Pot83]:

$$C_{0x} = 1 \qquad C_{1x} = \frac{1}{n^{0.94}} \tag{15}$$

for reconveying elements eqn. (19) is used [NN93]:

$$C_{0x} = -\frac{(0.31 + 0.69n)}{n} e^{1-n} \qquad C_{1x} = \frac{1}{n}e^{1-n} \tag{16}$$

From these relationships we get a description for the pressure throughput relationship within a screw element:

Conveying element: $\pi_P \geq 0$:

$$\pi_{V,ges} = c_{0P} - \pi_k c_{0x} + (c_{1P} + \pi_k c_{1x}) \cdot \pi_P + c_{2P} \cdot \pi_P^2 + c_{3P} \cdot \pi_P^3 \tag{17}$$

Conveying element: $\pi_P \leq 0$:

$$\pi_{V,ges} = c_{0N} - \pi_k c_{0x} + (c_{1N} + \pi_k c_{1x})\pi_P + c_{2N}\pi_P^2 + c_{3N}\pi_P^3 \tag{18}$$

Reconveying element: $\pi_p \geq 0$:

$$\pi_{V,ges} = c_0 - \pi_k c_{0x} + (c_1 + \pi_k c_{1x})\pi_P + c_2\pi_P^2 \tag{19}$$

These equations are solved for the dimensionless pressure gradient. We take into account that there is more than one solution. As there is only one incorrect solution, it is very simple to determine.

Figure: Comparison of the pressure-throughput characteristics with and without leakage flow for conveying elements

Figure: Comparison of the pressure-throughput characteristics with and without leakage flow for reconveying elements

The figures show the influence of the leakage flow on the dimensionless pressure throughput relationship for conveying and reconveying elements. The figures are based on the geometrical data of the ZSK 30 with a radial clearance of $s_R = 0.075$ mm and a pitch of 20 mm. The maximum influence of the leakage flow can be found at the point of $p_P = 0$ (pure drag flow). When $p_P$ is increased the influence of the leakage flow decreases. Due to the superposition of the polynomial equation for the channel flow and the linear equation for the leakage flow, the characteristic profiles intersect small power law exponents.

Due to their conveying behavior, neutral kneading blocks are a special case. As the assembly of the kneading discs takes place at the staggering angle $\alpha=\pi/i$ there is no conveying direction. For the evaluation of the rheological behavior they must be regarded separately. This procedure is repeated for conveying elements.

The Finite Element simulations are based once again on those of the channel model. It is assumed that the pressure throughput behavior is the same as it would be for separate channels in series. With this assumption 2 ½ dimensional simulations are possible.

The figure shows the unwound channel geometry, together with the mesh used for the simulation

Figure: Finite Element mesh for neutral kneading blocks.

The results of the Finite Element simulations can be described by the dimensionless equation;

$$\pi_{PFN} = f(\pi_{\dot{V}FN}, n) \tag{20}$$

with the dimensionless throughput

$$\pi_{\dot{V}FN} = \frac{\dot{V}}{A_{channel}v_0} \tag{21}$$

and the dimensionless pressure gradient

$$\pi_{PFN} = \frac{A_{channel}^{\frac{1+n}{n}} \cdot \Delta p}{v_0^n K_{0T} \cdot L} \tag{22}$$

The figure clearly displays the results of the Finite Element Simulations (symbols) for different Power Law exponents n.

Figure: Pressure – throughput behavior of neutral kneading blocks

We get a linear relationship for a Power Law exponent n=1. The circulation flow has in this case no influence on the pressure throughput relationship. This case can be described by the equation:

$$\pi_{PFN,N} = \pi_\dot{V} \cdot (a_1 CL^3 + a_2 CL^2 + a_3) \tag{23}$$

Pseudoplastic materials can be described with the equation:

$$\pi_{PFN} = \pi_{PFN,N}^{n^{0,5}}(1 - n)\Delta\pi_P \tag{24}$$

with:

$$ \pi_P = \left(\pi_\dot{V} \frac{a_4}{CL^5} - a_5\right)\pi_\dot{V}^4 \frac{n}{CL^3} + \left(a_6n + a_7n^{\frac{1}{5}}CL^2 + \pi_\dot{V}^{\frac{3}{2}}a_8n^5CL^6\right)\pi_\dot{V}^{\frac{1}{2}}CL^4$$

$$+ \left[\left(\pi_\dot{V}^{2}a_9CL^{\frac{1}{2}} + a_{10}CL^4\right)\pi_\dot{V} + \pi_\dot{V}^n a_{11}n^4 + \pi_\dot{V}^{\frac{n}{5}}a_{12}\right]nCL \tag{25} $$

The constants $a_1$ to $a_{12}$ are shown in the table.

Table: Constants for the calculation of the dimensionless Pressure gradient for neutral kneading blocks

$a_1 = 370$ $a_2 = -873.733$ $a_3 = 2.49\pi^5$ $a_4 = 363$

$a_5 = -113$ $a_6 = 2.6$ $a_7 = -0.3083$ $a_8 = 1/22.5$

$a_9 = 13.3$ $a_{10} = -1.749$ $a_{11} = 2.928$ $a_{12} = 0.6589$

The range of validity of eqn. (25) depends on the centerline ratio.

For $CL \leq 1.75$

$$0 \leq \pi_{VFN} \leq 2 \tag{26}$$

and for $1.75 < CL \leq 1.85$:

$$0 \leq \pi_{VFN} \leq 3 \tag{27}$$

Figure: Comparison between approximated and numerical dimensionless pressure gradients for neutral kneading elements

The figure shows a comparison between the model predictions and the Finite Element results. The figure shows a scatter plot between the numerically determined results and the model predictions. From both figures we can see that the model is able to describe the numerically determined data.

The figure shows a comparison between the experimental determined data and the model predictions. The correlation is fairly good, especially if one considers the variance of the experimental data.

Figure: Comparison between approximated and experimental dimensionless pressure gradients for neutral kneading elements

Using the channel model the complicated intermeshing region is replaced by a diamond shaped channel. Since the unwinding of the channel was carried out along the barrel surface, the intermeshing region is below the actual channel. A magnification of that region is shown in the figure.

Figure: Schematic diagram of the intermeshing region as a diamond shaped channel

The length $Z_{ei}$ in the figure is calculated using this equation:

$$Z_{ei} = \frac{L_{ei}}{\sin(\varphi_s)} \tag{28}$$

Using this model the flow resistance in the intermeshing region is taken into account by the narrowing of the channel. This flow resistance is remarkable for single flighted elements, due to the large flight widths. The intermeshing region has taken into account the determination of the pressure throughput behavior.

When performing three-dimensional flow simulations of twin screw elements one obtains a flow profile like the one shown in the figure. In this profile two different sections can be distinguished. Whereas in the intermeshing region quite high positive z-velocities can be found, in the channel region relatively low z-velocities are observed.

While there is apparently a backflow due to the pressure gradient in the channel region there is absolutely no backflow in the intermeshing region. This has to be taken into account for a model in the intermeshing region.

Due to the different flow patterns in the channel and in the intermeshing region, the different flows are modelled separately. From equilibrium of flow rates follows:

$$\dot{V}_{tot} = \dot{V}_{intermeshing area} + k \cdot \dot{V}_{channel} - \dot{V}_{gap} \tag{29}$$

The model for the flow rate in the intermeshing region assumes a chamber conveying in this region. This flow rate results from the volume in the intermeshing region and from the rotational speed.

$$\dot{V}_{intermeshing area} = \bar{V}_{intermeshing area} \cdot n_0 \tag{30}$$

Figure: Flow patterns in a conveying element (ZSK30, conveying element 28/28, PP Stamylan P17E19FC (DSM), V=2814mm³/s=7.53kg/h, no=455 1/min)

The calculation of the pressure – throughput relationship is carried out as described above. The only difference is, that the flow rate used for the calculation is not the flow rate imposed to the system but the imposed flow rate reduced by the flow rate in the intermeshing region.

The calculation of the pressure gradients in blister elements is based on these simplifications:

• The gaps in the intermeshing region are equal ($s_{F1} \approx s_{F2}$).
• The geometry can be described as a series of discs (see figure).

Figure: Geometrical model for blister elements

The geometry can be characterized using the following dimensionless numbers:

$$k1 = \frac{D_i}{D_z} \tag{1}$$

$$k2 = \frac{D_i}{D_a} \tag{2}$$

$$cl = \frac{a}{D_z/2} \tag{3}$$

Analogous to the description of the pressure throughput relationship for conveying elements a dimensionless throughput,

$$\pi_V = \frac{\dot{V}}{0.5 \cdot A_{frei} \cdot D_a \cdot n_0} \tag{4}$$

a dimensionless pressure gradient

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

and the Power Law exponent $n$ was used for the calculation.

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

$k1$: $0.7 \leq k1 \leq 1$
$k2$: $0.7 \leq k2 \leq 0.98$
$Cl$: $1.65 \leq Cl \leq 1.85$
$n$: $0.2 \leq n \leq 1$
$\pi_V$: $0.01 \leq \pi_V \leq 5$

Since we used the power law for this model. The determination of the right average shear rate is crucial for the model. The shear rate is characterized by the dimensionless shear rate $\pi_{\dot{\gamma}}$.

$$\pi_\gamma = \left(\frac{\dot{\gamma}}{n_0}\right)^n \tag{6}$$

It depends on

• the geometry ($k1, k2, cl$)
• the dimensionless flow rate $\pi_V$
• the Power Law Exponent $n$

For the description we used a linear equation:

$$\pi_\gamma = A_0(k1, k2, cl, n) + A_1(k1, k2, cl, n) \cdot \pi_\gamma \tag{7}$$

Figure: Comparison of approximated and numerically determined dimensionless shear rates

The pressure throughput behavior can be described using the following polynomial equation:

$$\pi_V = A_{B,1}(k1, k2, cl, n) \cdot \pi_P + A_{B,2}(k1, k2, cl, n) \cdot \pi_P^2 \tag{8}$$

Figure: Comparison of approximated and numerically determined dimensionless flow rates

A comparison between the predicted pressure gradients and the experimental ones shows a significant correlation (see figure).

Figure: Comparison of approximated and experimentally determined pressure gradients

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